j. nig. soc. phys. sci. 3 (2021) 38–41 journal of the nigerian society of physical sciences velocity distribution of 43ca+ ion cloud in the low temperature limit in a quadrupole penning trap dyavappa b. m.∗ department of physics, government first grade college for women, kolar, india abstract penning trap has electric field created by dc voltage applied between ring and end cap electrodes and magnetic field is applied along symmetry axis, as the electric field confines ions in the axial direction through an electric potential minimum and the magnetic field confines the ions in the radial direction. the trapping potential created by the dc voltage applied between the end cap and ring electrodes in the low temperature limit is cancelled by coulomb interaction of ions and the total energy is mainly kinetic energy of ions. the velocity distribution of 43ca+ ions along axial direction, in radial plane and total velocity distribution due to resulting motion of both axial and radial motion of ions in low temperature limit in a quadrupole penning trap are presented here. these results reveal the properties of 43ca+ ion cloud and are useful to study confining techniques for different types of ions in low temperature limit and a qubit can be encoded in the hyperfine ground states of 43ca+ isotope for ion trap quantum computation. doi:10.46481/jnsps.2021.132 keywords: quadrupole penning trap, velocity distribution function, 43ca+ cloud. article history : received: 08 august 2020 received in revised form: 24 january 2021 accepted for publication: 29 january 2021 published: 27 february 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: b. j. falaye 1. introduction 43ca+ ion has only one valence electron and has simple energy level structure, has long-lived d-states, therefore it can be used for quantum computation [1, 2, 3], for building an ion clock [4] and also for laser cooling. the large fine-structure splitting of ∆νfs = 6.7t hz between the p1/2 and p3/2 states in 43ca+ion as shown in figure 1 allows a large detuning of the raman light fields from the p− levels and thus high fidelity gate operations, as spontaneous emission processes are largely suppressed, which is requirement for ion trap quantum computation. and encoding the qubit in the hyperfine ground states ∗corresponding author tel. no: +91 9483113600 email address: dyavappabm@gmail.com (dyavappa b. m. ) ensures that decay from spontaneous emission is completely avoided and thus, very long coherence times may be achieved potentially and the qubits will ideally depend only in second order on the external magnetic field [5]. a thorough knowledge of different properties of ions including velocity distribution along axial direction, in radial plane and the total velocity distribution is necessary. the quadrupole penning trap is made with two end-cap electrodes and a ring electrode. the equation of ring electrode is r 2 r20 − z2 z20 = +1 and equations of two similar end-cap electrodes is r 2 r20 − z2 z20 = −1, where r0 is the inner radius of the ring electrode in the radial plane and z0 is half of the vertical distance between the two end-cap electrodes such that r0 = √ 2 z0. the trap potential created by the dc voltage applied between 38 dyavappa / j. nig. soc. phys. sci. 3 (2021) 38–41 39 the end cap and ring electrodes is given by [6, 7]. v (r, z) = v0 r20 + 2z 2 0 ( 2z2 − r2 ) (1) the vector potential of the magnetic field b is [4, 5] a = 1 2 (b × r) = 1 2 b(−yx̂ + xŷ) (2) the lorenz force in an electromagnetic field with an electric field e and magnetic field b on 43ca+ion of mass m, charge q, moving with a velocity v is given by [8] f = q [e + (v × b)] (3) f = q [( 2v0 d2 xx̂ + 2v0 d2 yŷ − 4v0 d2 z ẑ ) + b (ẏx̂ − ẋŷ) ] (4) the axial, pure cyclotron, reduced cyclotron and magnetron frequencies of 43ca+ion are given respectively by [9, 10, 11] fz = 1 2π √ 4qv0 md2 , fc = qb 2πm f ′c = fc + √ f 2c − 2 f 2z 2 , fm = fc − √ f 2c − 2 f 2z 2 (5) the velocities of 43ca+ion in the axial direction, radial plane and in space are vz = √ kbt m , vr = √ 2kbt m , v = √ 3kbt m (6) figure 1: energy level scheme of 43ca+ isotope, a qubit can be encoded in the hyperfine ground states for ion trap quantum computation [5] 2. theory we assume that the 43ca+ion cloud is in thermal equilibrium through electrostatic coulomb interaction between ions. the rotation of 43ca+ion cloud in magnetic field is equivalent to neutralization by opposite charge of ions and the distribution of magnetically confined ions in thermal equilibrium without rotation can be treated as ions confined and neutralized by a figure 2: (i): penning trap with 43ca+ions confined in trap space, b: magnetic field, v0: storage voltage, (ii) magnified view of motion of a 43ca+ions confined in trap space in a penning trap showing reduced cyclotron, magnetron and axial motions cylinder of opposite charge. the probability of velocity distribution in thermal-equilibrium is [12] dp ( vr,φ,z ) = 2π (πkt )3/2 ( m 2 )1/2 v × exp [ −m ( v2 −ωφvφ/2 2kbt )] drdφdzdvr dvφdvz (7) where ωφ is the rotational frequency of 43ca+ion cloud as a whole determined by the temperature of the ion cloud. the energy of single 43ca+ion is [12] e = m 2 [ v2r + ( vφ r − qbr 2mc )2 + v2z ] +q [ vt (r, z) + vq (r, z) ] (8) the momenta in radial plane, azimuthal and axial directions are [12] pr = mvr = m dr dt , pφ = mr 2 dφ dt + qb 2c r2 = mrvφ − ωc 2 mr2, pz = m dz dt = mvz (9) 2.1. velocity distribution of 43ca+ ion cloud at the low temperature limit in a penning trap 43ca+ ions are confined in the low temperature limit when the electrostatic potential energy [12] qv q (r, z) � kbt (10) qv t (r, z) + qv q (r, z) + m 2 ωφ ( ωc −ωφ ) r2 = 0 (11) the energy along the axial direction in the low temperature limit is [12] ez = 1 2 mv2z = 1 2 kbt (12) the energy in the radial plane in the low temperature limit is [12] er = 1 2 mv2r + mv2φr2  + ( ω2c 8 ) r2 + ωc 2 vφ (13) 39 dyavappa / j. nig. soc. phys. sci. 3 (2021) 38–41 40 if we neglect coulomb interaction potential vq (r, z) then the probability of velocity distribution in the low temperature limit is [12] dp ( evr,vφ,vz ) = a′′′ex p [ −m ( v2z + v 2 r 2kbt )] dvr d ( vφ r ) dvz (14) the probability of velocity distribution in z-direction in the low temperature limit is [12] dp (vz) = ( 2 πmkbt )1/2 ex p [ − ( mv2z 2kbt )] vzdvz (15) the probability density of velocity distribution in z-direction in the low temperature limit is [12] ρz (vz) = ( 2 πmkbt )1/2 [ ex p ( − 1 2 mv2z kbt )] vz (16) the probability density of velocity distribution in axial direction increases sharply up to √ mv2z /2kbt = 0.854 at ρz = 3.019× 1025, decreases abruptly and remains almost a constant in low temperature limit as shown in figure 3. figure 3: the axial probability density of velocity of 43ca+ion cloud in the low temperature limit the probability of velocity distribution in the radial plane in the low temperature limit is [12] dp ( vr,φ ) = ( 1 kbt ) ex p ( − 1 2 mv2r kbt ) mvr dvr (17) the probability density of velocity distribution in radial plane in the low temperature limit is [12] ρr (vr ) = ( 1 kbt ) ex p ( − 1 2 mv2r kbt ) (18) the probability density of velocity distribution in radial plane increases sharply up to √ mv2r /2kbt = 1.4072 at ρr = 4.89 × 1022, decreases abruptly and remains almost a constant in low temperature limit as shown in figure 4 the total probability of velocity distribution is [12] dp (v) = ( 2m π ) 1 2 ( 1 kbt ) 32 ex p ( − 1 2 mv2 kbt ) vdv (19) figure 4: the radial probability density of velocity of 43ca+ion cloud in the low temperature limit figure 5: the total probability density of velocity of 43ca+ion cloud in the low temperature limit the total probability density of velocity distribution is [12] ρ (v) = ( 2m π ) 1 2 ( 1 kbt ) 3 2 ex p ( − 1 2 mv2 kbt ) v (20) the total probability density of velocity distribution in trapping region increases sharply up to √ mv2/2kbt = 1.4086 at ρ = 3.399 × 1032, decreases abruptly and remains almost a constant in low temperature limit as shown in figure 5. the axial, radial and total probability densities of velocity distribution increase sharply up to √ mv2z /2kbt = 0.854, √ mv2r /2kbt = 1.4072 and √ mv2/2kbt = 1.4086 respectively, at ρz = 3.019 × 10 25, ρr = 4.89×10 22and ρ = 3.399×1032 respectively, beyond which decrease abruptly, remain almost constant for the low temperature limit as shown in the figure 6. the radial probability density of velocity distribution is less than the axial probability density of velocity distribution, which in turn less than the total probability density of velocity distribution. 3. conclusion the probability density of velocity distribution of 43ca+ion cloud along axial direction and in radial plane together results 40 dyavappa / j. nig. soc. phys. sci. 3 (2021) 38–41 41 table 1: the values of axial, radial and total probability densities of velocity of 43ca+ion cloud in the low temperature limit mv2 2kb t √ mv2 2kb t t (k) kbt ( 10−23 j ) ρz (vz) ( 1025 ) ρr (vr ) ( 1022 ) ρ (v) ( 1032 ) 0 0 ∞ ∞ 0 0 0 2 1.4142 4 5.52 2.69719 0.66645 3.3882 4 2 2 2.76 0.992243 4.9034 0.9171 6 2.4495 1.3 1.79 0.365019 2.7814 0.19158 8 2.8284 1 1.38 0.134288 1.3272 0.03359 10 3.1623 0.8 1.1 0.04940 0.61254 0.005714 12 3.4641 0.6667 0.92 0.018174 0.26943 0.0009229 14 3.7417 0.57 0.7866 0.006686 0.11593 0.000149 16 4 0.5 0.69 0.0024595 0.048617 0.00002253 18 4.2426 0.44 0.6 0.0009048 0.020568 0.0000035288 20 4.4721 0.4 0.55 0.00033286 0.0082545 0.000000657 figure 6: the axial, radial and total probability density of velocity of 43ca+ion cloud in the low temperature limit total probability density of velocity distribution under low temperature limit. the radial probability density of velocity distribution is less than the axial probability density of velocity distribution, which in turn less than the total probability density of velocity distribution. these results reveal the velocity properties of the 43ca+ion cloud and are useful to design and carry out experiments on stored 43ca+ ions, with velocity related parameters under low temperature limit and also for ion trap quantum computation in quadrupole penning trap. references [1] a. steane, “the ion trap quantum information processor”, applied physics b: lasers and optics 64 (1997) 623. [2] c. d. bruzewicz et al., “dual-species, multi-qubit logic primitives for ca+/sr+ trapped-ion crystals”, npj. quantum information 5 (2019) 102. [3] kylie foy, “qubits made from strontium and calcium ions can be precisely controlled by technology that already exists, massachusetts institute of technology”, (2020). https://phys.org/news/ 2020-01-qubits-strontium-calcium-ions-precisely.html. [4] c. champeneois, m. bhoussin, c. lisowski, a. knoop, g. hagel, a. vedel & f. vedel, “evaluation of the ultimate performances of a 43ca+ single-ion frequency standard”, physics letters a 331 (2004) 298. [5] r. blatt, h. haffner, c. roos, c. becher & f. schmidt-kaler, “ion trap quantum computing with ca+ ions”, quantum information processing 3 (2004) 1. [6] f. g. major, v. n. gheorghe & g. werth, charged particle traps, physics and techniques of charged particle confinement, springer publishers (2005). [7] p. k. ghosh, ion traps, clarendon press, oxford, (1995) 72. [8] b. m. dyavappa, “spectroscopy of non-neutral plasmas in ion traps” ph.d thesis, bangalore university, (2017). [9] d. datar, b. m. dyavappa, b. l. mahesh, k. t. satyajith & s. ananthamurthy, “energy distribution of electrons under axial motion in a quadrupole penning trap”. can. j. phys 94 (2016) 1245. [10] b. m. dyavappa, “velocity distribution of electrons along the symmetry axis of quadrupole penning trap”, discovery 56 (2020) 138. [11] b. m. dyavappa, d. d. prakash & s. ananthamurthy, “dependence of the confinement time of an electron plasma on the magnetic field in a quadrupole penning trap”, epj techniques and instrumentation 4 (2017) 4. [12] g. z. li & g. werth, “energy distribution of ions in penning trap”, international journal of mass spectrometry and ion processes 121 (1992) 65. 41 j. nig. soc. phys. sci. 2 (2020) 180–185 journal of the nigerian society of physical sciences original research mass resolution of ca, k isotopes and co, n2 and c2h4 isobars in isotopes separator on-line trap mass spectrometry b m dyavappa∗ department of physics, government first grade college for women, kolar, karnataka, india abstract in isotopes separator on-line trap, ions are trapped, cooled, accumulated, bunched and isotopes or isobars are separated, cyclotron frequencies are determined, which are followed by time of flight mass resolution. the mass resolution of isotopes in penning trap mass spectrometry is achieved by the direct excitation of axial motion of ions, driven by rf field at the pure cyclotron frequencies of ions. the design and working of isotopes separator on-line trap which is used for high-accuracy mass spectrometry in the mass resolution of calcium isotopes (40ca+, 42ca+, 44ca+), potassium isotopes (39k+, 41k+) and [28(co)]+, [(28n2)]+, [28(c2h4)]+ isobars found in mixtures is achieved from time of flight mass spectrometry are presented here. keywords: mass spectrometry, mass resolution, isotope online penning trap, resonance spectrum, rf excitation. article history : received: 09 may 2020 received in revised form: 17 july 2020 accepted for publication: 20 july 2020 published: 01 august 2020 c©2020 journal of the nigerian society of physical sciences. all rights reserved. communicated by: o. j. oluwadare 1. introduction the stable six isotopes of calcium are 40ca, 42ca, 43ca, 44ca, 46ca and 48ca out of which 43ca, 46ca, 48ca are rare isotopes found in trace amounts and hence cannot be identified in the isotope-ratio mass spectrum obtained from stimulations. the mass resolving powers of 40ca+, 42ca+, 44ca+ are determined from [1, 2, 3] (m.r.p.)ca+ = mca+ δmca+ (1) where mca+ →mass of ca +and δmca+ →full width at half maximum in mass spectrum. the mass spectrum of ions [28(co)]+, [(28n2)]+ and [28(c2h4)]+ of isobars is obtained by time of ∗corresponding author tel. no: +919483113600 email address: dyavappabm@gmail.com (b m dyavappa ) flight mass spectrometry simulations. the isobars are identified from identifying the values of time of flight in mass spectrum and calculating the mass from [4, 5] m = 2qv [ t f d ]2 , (2) where m → mass of ion, q → charge state of ion, t f → time of flight of ion, d → distance travelled by the ion before reaching the detector. this is compared with the estimated values of masses of the isobars and the corresponding isobar is identified from approximately equating to the estimated value of its mass. the mass determination of 4119 k + isotope in penning trap mass spectrometry is achieved by the excitation of axial motions of same charge state of 3919 k +and 4119 k +ions by driving radio frequency (rf) field at the pure cyclotron frequencies fc (3919 k + ) , fc(4119 k+ ) of 3919 k +and 4119 k + ions respectively as the 180 dyavappa / j. nig. soc. phys. sci. 2 (2020) 180–185 181 magnetic field is known. the mass of most abundant isotope m(3919 k+ ) = 38.963707 amu is well known and hence the mass of less abundant isotope m(4119 k+ ) of 41 19 k + can be determined from the following equation [6]: m(4119 k+ ) = fc (3919 k + ) fc(4119 k+ ) m(3919 k+ ). (3) 2. theory the isotope-ratio mass spectrum of ca isotopes is drawn by using data from the mass spectrometry data base. the accuracy of mass measurements in penning trap is determined by the resolving power of masses of isotopes. the resolving power of masses of isotopes of ions is defined as the ratio of the centre frequency of the resonance line to the full width at half maximum of the resonance line. therefore the mass resolving power (m.r.p.) of masses of isotopes of ions is given by [1, 2, 3] (m.r.p.)ion = m δm = f0 ∆ f1/2 , (4) where m → mass of ion and δm → full width at half maximum in mass spectrum, f0 → centre frequency of the resonance line, ∆ f1/2 → full width at half maximum of the resonance line. the mass spectrum of ions [28(co)]+, [(28n2)]+ and [28(c2h4)]+ of isobars is obtained by time of flight mass spectrometry simulations. in time of flight mass spectrometry ions are accelerated by an electric field of known strength e with potential v and all those ions which have the same charge state q, will have same kinetic energy with velocity v due to the acceleration. the specific charge (q/m) of ions is given by [7] q m = 1 2 [ e2 b2v ] = 1 2 [ v2 v ] , (5) where q → charge state of ion, m → mass of ion, b → magnetic field, e → electric field, v → electric potential, v → velocity of ion. the velocity of the ion accelerated by electric field depends upon its specific charge q/m the time taken by the particle to reach the detector is called time-of-flight t f and it can be measured. the specific charge of ions is determined through the measurement of time to reach the detector in time-of-flight mass spectrometry. heavier ions reach the detector slower than the lighter ones as the mass of moving ion is m ∝ t2f . the time is measured from the instant the ion leaves the cooler ion trap to the instant that reaches the detector, it is used to find specific charge of it and the ion is determined from the known parameters. the time-of-flight is given by [4] t f = d √ 2v √ m q ⇒ m q = 2v [ t f d ]2 ⇒ q m = 1 2v [ d t f ]2 . (6) ∴ m = 2qv [ t f d ]2 , (7) where d → distance traveled by the ion, v → electric potential,m is mass and q is charge state of ion. when the ions are excited by continuously sweeping the rf field, the motional frequencies of ions respond to the external rf at a given step, consequently some of the ions gain enough energy to escape the trap. this change in the motion of ions due to rf field drive causes increase in kinetic energy in the radial plane and can be detected by a time-of-flight technique. the mass resolution of isotopes of ions is related to specific charge and cyclotron angular frequency as [6] m δm ∝ q m b ( trf √ n ) = ωc ( trf √ n ) = 2π fc ( trf √ n ) ⇒ fc∝ 1 m , (8) where q→ charge state of ion, m→ mass of ion, b→ magnetic field, trf→ time of rf drive, n→ number of cycles of rf field, ωc→ cyclotron angular frequency. the resolving power of masses of isotopes of ions is proportional to the time of excitation of rf field, which results in the motional resonances of the isotopes of ions that can be observed in motional resonances spectrum. the lifetime of unstable isotopes limit the time of excitation as they are in very short duration of time. the temporal stability of the magnetic field due to shielding current in pair of coils of wire limit the radial confinement of the stable and long-lived isotopes. the exchange of the ions of isotopes in the trap is required for comparison of cyclotron frequencies of two different ions and measured at different times during which the magnetic field strength changes. superconducting magnets require temperature and pressure stabilization to reduce temporal variation of the magnetic field strength. the mass determination in isotopes separator on-line trap is on the basis of the fact that the two ions of isotopes whose charge state is same but their masses are different. the ratio of their cyclotron frequencies is equal to the inverse of ratio of their masses kept in the same magnetic field [6]. therefore if the charge state of two ions of isotopes is q1=q2 kept in the same magnetic field b then fc1 fc2 = m2 m1 , (9) where fc1 and fc2 are cyclotron frequencies of isotopes of ions of an element with masses m1 and m2 respectively. 3. experimental procedure 3.1. design the isotopes separator on-line trap consists of three ion traps connected end to end together in an order of rf paul trap, cooler penning trap and precision penning trap as shown in figure 1 [8, 9]. the rf quadrupole ion trap consists of 4 rods structure to which a rf field is applied for alternate rods, and 181 dyavappa / j. nig. soc. phys. sci. 2 (2020) 180–185 182 this is used for beam preparation and hence it is also called beam buncher. the cooler penning trap is a large cylindrical penning trap which is placed in the homogeneous magnetic field of superconducting magnets, which is used to cool the ions. the precision penning trap is a quadrupole penning trap in which ions are detected through time of flight. figure 1. schematic diagram of isotopes separator on-line trap [8, 9] 3.2. working a quadrupole penning trap is designed with three-electrode infinite hyperboloid revolution of structure, which consists of two end-cap electrodes and a ring electrode, a homogeneous magnetic field is superposed on electrostatic quadrupole field. the magnetic field b confines ion beam of isotopes of charge state q and different masses in the radial direction, while the electric field quadrupole potential vdc , confines ions in the axial direction, as it prevents the ions from escaping along the magnetic field lines. the motion of trapped ions in a penning trap is not a simply pure cyclotron motion with frequency fc but a combination of three harmonic eigen motions, viz. an axial oscillatory motion with frequency fz, two circular motions called modified cyclotron motion with frequency f ′ c and magnetron motion with frequency fm which are related to each other as [1] fc = f ′ c + fm. (10) the precise value of pure cyclotron frequency in an isotope separator on-line trap is [6] fc = ( q m ) b 2π ⇒ fc ∝ 1 m ( ∵b, q → constants ) (11) the motion of ions of isotopes can be driven by oscillating electric field which changes the amplitudes of the oscillatory motion of ions and azimuthal electric quadrupole field causes the excitation of ion oscillatory motion directly at the side band frequency fc. the mass determination of ion of unknown isotope in isotopes separator on-line trap mass spectrometry is achieved by the direct excitation of axial oscillatory motions of same charge state ions of isotopes at their pure cyclotron frequencies from the relation m2 = m1 fc1 fc2 as the magnetic field is known [6]. 3.3. cooling and bunching in rf quadrupole ion trap a rf field is applied to the 4 rods structure which creates an oscillating quadrupole electric field that confines the ions of isotopes or isobars along the symmetry axis of trap. the rods are segmented and an appropriate shape dc potential is applied to the segments to drag the ions close to the end of the 4-rods structure where the ions are trapped. the first step is stopping and preparation of the high energy of ≈ 30 − 60kev ion beam of isotopes or isobars. the ions of isotopes or isobars are decelerated electro statically by applying repelling potential and then injected into the central region of 4-rods structure being filled with buffer gas. the rf quadrupole ion trap cools the ion beam of isotopes through buffer gas cooling by collisions. the ions of high energy of ≈ 30 − 60kev lose kinetic energy up to a few kev due to the collision with the buffer gas, and then finally accumulated as a small ion cloud of isotopes or isobars in the trapping region. thus cooled ions of isotopes or isobars are accumulated in beam buncher and enter into cooler penning trap later, where contaminants are removed. the cold ion cloud bunch of selected isotopes or isobars of an element can be ejected out off the trapping region, transported and then injected into the cooler trap through a potential adaption in a pulsed drift tube. 3.4. the cooler trap the cooler trap is a large cylindrical penning trap placed in the homogeneous magnetic field of ≈5t superconducting magnets. the ions transported from the rf quadrupole ion trap are captured in the cylindrical penning trap and cooled through mass selection technique. the cooler cylindrical penning trap is optimized for high quality mass selection to resolve isotopes or isobars. isotopes are different species of the same element with same atomic number (same number of protons and electrons) but differ in mass number (the number of nucleons) and hence specific charge (charge to mass ratio) will be different for different isotopes with same charge state, therefore the isotopes travel with different velocities and take different time durations to reach the detector. when ions of isotopes which have same charge state but different masses are trapped in constant magnetic field, then heavier ions of the same charge state reach at lower speeds as [6] fc = q m b ∝ 1 m ∝ v ∝ d t ( ∵b, q are constants ) (12) ∴ m ∝ t (∵d is constant) (13) isobars are different elements with same mass numbers (same number of nucleons) but differ in atomic number (the number 182 dyavappa / j. nig. soc. phys. sci. 2 (2020) 180–185 183 of protons and electrons) and hence specific charge (charge to mass ratio) will be different for different isobars, therefore the isobars travel with different velocities and take different time durations to reach the detector. for ions of isobars which have same mass but different charge state in constant magnetic field, the velocity of ions with higher charge state will also increase [6]. fc = q m b ∝ q ∝ v ∝ d t (∵b, m are constants)(14) ∴ q ∝ 1 t (∵d is constant) (15) the cooled and clean bunches of ions are transferred into the precision penning trap, which are used for highly accurate mass measurements. 3.5. precision penning trap mass spectrometer an azimuthal rf field of frequency frf drives the motion of ions, the amplitude of the cyclotron motion of the stored ions increases due to resonance of driving frequency of rf field with cyclotron frequency ( f rf = fc) in quadrupole penning trap. the rf generator switched to sweep mode is used to feed rf energy into the trap through the antenna. the rf power is kept very low of the order of a few mv to weakly probe the motion of the trapped ion cloud. if the rf power is kept high, then it will resonantly drives the trapped ion cloud in the trap and causes to escape from the trap. when the ions are excited by continuously sweeping the rf field, the motional frequencies of ions respond to the external rf at a particular step, consequently some of the ions gain enough energy to escape from the trap, then the signal height is reduced and appears as a dip in the motional resonance spectrum, which is directly proportional to the number of ions lost from the trap. the cooled ions are ejected from the trap due to the excitation of the motion of ions by rf field, and drift through the inhomogeneous fringe magnetic field b to reach the detector of ions. the magnetic moments of the orbits of ions also increase due to magnetic field. an axial force arising from the inhomogeneous magnetic field increases the axial momenta of the ions by orbital magnetic moments. the time-of-flight of ions is determined as a function of the frequency of the rf field, as ions in resonance with the rf field reach the detector faster than those ions that are not in resonance. the mass can be extracted in conjunction with a reference mass measurement after the determination of the frequency of stored ion from the time-of-flight detection technique . 4. results and discussion 4.1. mass spectrum of calcium isotopes the isotope ratio mass spectrum of calcium isotopes shows 3 peaks as shown in figure 2. the tallest peak corresponds to table 1. mass resolving powers of calcium isotopes ions of calcium isotopes mass resolving power 40 20ca + 794.0675 42 20ca + 789.60086 44 20ca + 900.089 40ca+ as specific charge of it is lesser than that of both of 42ca+ and 44ca+, the second short peak next to it corresponds to 42ca+ as its specific charge is greater than that of 40ca+ and the third short peak next to it corresponds to 44ca+ as its specific charge is greater than that of 42ca+. the mass resolving powers of 40ca+, 42ca+, 44ca+ are [10] (m.r.p.)40 20ca + = mca+ δmca+ = 39.961765 39.981125 − 39.9307996 = 794.06751 ≈ 794.0675 (16) (m.r.p.)42 20ca + = mca+ δmca+ = 41.9586 41.9802842 − 41.9271452 = 41.9586 0.053139 ≈ 789.60086 (17) (m.r.p.)44 20ca + = mca+ δmca+ = 43.9555 43.9763821 − 43.9275475 = 43.9555 0.0488346 ≈ 900.089 (18) therefore the mass resolving powers of 40ca+, 42ca+, 44ca+ as calculated from equation (1) are 794.0675, 789.60086 and 900.089 respectively as presented in table 1. figure 2. isotope ratio mass spectrum of ions of 40ca+, 42ca+, 44ca+ isotopes drawn by using isotope-ratio mass spectrometry data base 4.2. mass spectrum of co, n2 and c2h4 isobars the ions [28(co)]+, [(28n2)]+ and [28(c2h4)]+ of isobars can be produced by collisions of isobars with electrons. the 183 dyavappa / j. nig. soc. phys. sci. 2 (2020) 180–185 184 table 2. masses of isobars calculated from mass spectrum of ions ions of isobars mass (amu)[ 28 (co) ]+ 27.99489[( 28 n2 )]+ 28.00699[ 28 (c2 h4) ]+ 28.031297 mass spectrum of ions [28(co)]+, [(28n2)]+ and [28(c2h4)]+ isobars obtained by time of flight mass spectrometry simulations is shown in figure 3. the mass spectrum consists of three peaks, one corresponds to each of ions [28(co)]+, [(28n2)]+ and [28(c2h4)]+ of isobars [11]. the isobars are identified from identifying the values of time of flight of isobar in mass spectrum and the mass of corresponding isobar is calculated from the value of time of flight. this accurate value of mass is compared with the estimated values of masses of the isobars and the corresponding isobar is identified from approximately equating it to the estimated value of its mass. the masses of ions [28(co)]+, [(28n2)]+ and [28(c2h4)]+ of isobars are calculated from time of flight using equation (19) as shown below. from figure 3 the time of flight that corresponds to first peak is t f = 762.27882 ns, then m = 2qv [ t f d ]2 (19) m = 2 × 1.6 × 10−19 × 25 [ 762.27882 × 10−9 10 × 10−3 ]2 (20) ⇒m = 46.48552 × 10−31kg = 27.99489 amu = mco (21) from figure 3 the time of flight that corresponds to second peak is t f = 762.43137ns, then m = 2 × 1.6 × 10−19 × 25 [ 762.43137 × 10−9 10 × 10−3 ]2 (22) ⇒m = 46.504128×10−31 kg = 28.00699 amu = mn2 (23) from figure 3 the time of flight that corresponds to third peak is t f = 762.77432 ns, then m = 2 × 1.6 × 10−19 × 25 [ 762.77432 × 10−9 10 × 10−3 ]2 (24) ⇒m = 46.54597×10−31kg = 28.031297 amu = mc2 h4 (25) the ions of isobars [28(co)]+, [(28n2)]+ and [28(c2h4)]+ are identified from mass spectrum as shown in figure 3. the mass of the isobar that corresponds to the first, second and third peaks were calculated using equation (19) as 27.99489 amu, 28.00699 amu and 28.031297 amu with corresponding time of flight of 762.27882 ns, 762.43137 ns and 762.77432 ns respectively as presented in table 2. figure 3. mass spectrum of ions [28(co)]+, [(28n2)]+ and [28(c2h4)]+ of isobars drawn by using mass spectrometry data base 4.3. mass spectrum of k isotopes the mass determination of 4119 k + isotope in penning trap mass spectrometry is achieved by the excitation of axial motions of same charge state of 3919 k +and 4119 k +ions by driving rf field at the pure cyclotron frequencies fc (3919 k + ) and fc(4119 k+ ) respectively as the magnetic field is known. as the mass of most abundant isotope m(3919 k+ ) = 38.963707amu is well known, and hence the mass of less abundant isotope m(4119 k+ ) can be determined. the pure cyclotron frequencies of fc (3919 k + ) and fc(4119 k+ ) from figure 4 are 98.426156 khz and 93.624975 khz respectively. the mass of ion 4119 k + of potassium isotope is given by [6] fc (3919 k + ) fc(4119 k+ ) = m(4119 k+ ) m(3919 k+ ) (26) ⇒ m(4119 k+ ) = fc (3919 k + ) fc(4119 k+ ) m(3919 k+ ) = 98.426156 × 103 93.624975 × 103 ×38.963707 × 1.66 × 10−27 (27) ∴ m(4119 k+ ) = 67.996595 × 10 −27kg = 40.9618amu (28) the mass of 4119 k + calculated using equation (27) is 40.9618amu as presented in table 3. 5. conclusion the mass resolving powers of 40ca+, 42ca+, 44ca+ are 794.0675, 789.60086 and 900.089 respectively. the ions of isobars [28(co)]+, [(28n2)]+ and [28(c2h4)]+ are identified in mass spectrum which correspond to time of flights of 762.27882 ns , 762.43137 ns and 762.77432 ns respectively. the pure cyclotron frequencies 184 dyavappa / j. nig. soc. phys. sci. 2 (2020) 180–185 185 table 3. masses of potassium isotopes calculated from mass spectrum of potassium ions ions of potassium isotopes interchange cyclotron frequency values mass (amu) 39 19 k + 98.426156 khz 38.963707 41 19 k + 93.624975 khz 40.9618 figure 4. time of flight detection of cyclotron resonance of isotopes of potassium ions 3919 k +and 4119 k + at magnetic field 0.25t by rf excitation from 90-105 khz drawn by using mass spectrometry data base of ions 3919 k + and 4119 k +from motional resonance spectrum are 98.426156 khz and 93.624975 khz respectively and hence the mass of ion of less abundant potassium isotope 4119 k + is determined to be 40.9618 amu. acknowledgments we thank the referees for the positive enlightening comments and suggestions, which have greatly helped us in making improvements to this paper. references [1] k. blaum, yu. n. novikov & g. werth, “penning traps as a versatile tool for precise experiments in fundamental physics”, contemporary physics, 51 (2010) 149. [2] a. pelander, p. decker, c. baessmann & i. ojanperä, “evaluation of a high resolving power time-of-flight mass spectrometer for drug analysis in terms of resolving power and acquisition rate”, journal of the american society for mass spectrometry, 22 (2011) 379-385. [3] w. a. m. wilfried & r. a. c. correa, interpretation of ms-ms mass spectra of drugs and pesticides, wiley series on mass spectrometry, wiley. [4] time-of-flight mass spectrometry, wikipedia, https://en.wikipedia.org/wiki/time-of-flight_mass_spectrometry [5] mass analyzer time of flight, https://phys.libretexts.org [6] f. wenander, “charge breeding of radioactive ions with ebis and ebit”, jinst 5 (2010) c10004. [7] s. k. singh, electricity and magnetism, http://cnx.org/content/col10909/1.13/, http://creativecommons.org/licenses/by/3.0/, 122 (2009) [8] f. herfurth, j. dilling, a. kellerbauer, g. bollen, s. henry, h. j. kluge, e. lamour, d. lunney, r. b. moore, c. scheidenberger, s. schwarz, g. sikler & j. szerypo, “a linear radiofrequency ion trap for accumulation, bunching, and emittance improvement of radioactive ion beams”, arxiv:nucl-ex/0011021 (2000) [9] m. mukherjee, d. beck, k. blaum, g. bollen, j. dilling, s. george, f. herfurth, a. herlert, a. kellerbauer, h. j. kluge, s. schwarz, l. schweikhard & c. yazidjian, “isoltrap: an on-line penning trap for mass spectrometry on short-lived nuclides”, eur. phys. j. a 35 (2008) 31. [10] s. f. boulyga, “calcium isotope analysis by mass spectrometry”, mass spectrometry reviews, 29 (2010) 685. [11] y. ishida, m. wada, y. matsuo, i. tanihata, a. casares, & h. wollnik, “a time-of-flight mass spectrometer to resolve isobars”, nucl. instr. and meth. in phys. res. b 219-220 (2004) 468. [12] a. finlay, integration of a multi reflection time of flight isobar separator into the titan experiment at triumf, m.sc. thesis, the university of british columbia (2017). 185 j. nig. soc. phys. sci. 3 (2021) 140–143 journal of the nigerian society of physical sciences synthetic characterization of cellulose from moringa oleifera seeds and potential application in water purification a. f. afolabia,∗, s. s. oluyamoa, i. a. fuwapea acondensed matter and statistical physics research unit, department of physics, the federal university of technology, p.m.b. 704, akure, nigeria abstract the use of moringa oleifera seeds for purifying water has been attempted locally in various forms without putting scientific potency of the material into consideration. the cellulose sample isolated from moringa oleifera seed was characterized using x-ray diffraction (xrd), scanning electron microscopy (sem) and fourier transform infrared spectroscopy (ftir). the value of crystallinity index (cir ) from the xrd pattern is 63.1%. the high degree of crystallinity obtained is attributed to the high percentage of crystallinity index, cir (i.e. 63.1%). the morphology revealed aggregates of conical and needle-like structure. the ftir revealed o–h stretching, c–h stretching vibration and c=o bond stretching functional groups. these characteristics are indicative of the potential of the material in water purification. doi:10.46481/jnsps.2021.206 keywords: cellulose, crystallinity, moringa oleifera, morphology, water purification article history : received: 22 april 2021 received in revised form: 27 may 2021 accepted for publication: 03 june 2021 published: 29 august 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: b. j. falaye 1. introduction water is a source of life and human existence depends to a large extend on its availability. water is obtained from different sources such as rain, dam, river, stream, borehole, well and lake e.t.c. the importance of water cannot be over-emphasized in our daily living. it has a broad impact on health, food, energy, economy and also necessary for human survival. the level of purity of water utilized in daily life is very significant since it has a definite effect on human health. likewise, water is an essential component of all living systems. the quality of drinking water has become a major concern since contaminants and toxic compounds are mostly accumulated in the body system thereby causing serious hazard to human health. hence, there is a need ∗corresponding author tel. no: +2347030614850 email address: agafolabi@gmail.com (a. f. afolabi ) for water purification to afford access to potable water for human consumption. imported chemicals for purifying water are expensive and have unfavorable effects on human health [1,2]. therefore, there is a dire need to use locally sourced and environmentally friendly organic material for water purification. moringa oleifera is non-toxic and has an added favourable opportunity over the chemical purification of water due to the medicinal and therapeutics properties such as cholesterol lowering, anti-inflammatory, antiulcer, antioxidant, antidiabetic, antispasmodic, antibacterial, antihypertensive, antiepileptic, antidiabetic, antifungal activity, antitumor and antimicrobial properties [3-6]. moringa has been discovered to be used in different health care products including body and hair conditioners and moisturizers. moringa oleifera is a plant material composed of lignin, hemicellulose and cellulose [7]. cellulose has a degree of polymerization of about 10,000 insoluble polysaccharide which con140 afolabi et al. / j. nig. soc. phys. sci. 3 (2021) 140–143 141 sist of linear chains of glucopyranose units linked by a β-1,4 glycosidic bond. the common formula is (c6h10o5)n [8]. furthermore, cellulose is a prominent structural composition of the cell wall of different plants. cellulose also exists in a broad diversity of class of living, such as algae, fungi, bacteria, and even in some sea animals such as tunicates. a lot has been accomplished in the use of moringa oleifera for water purification, creating general interest in the researcher of moringa oleifera utilization. despite the attractiveness of this research, several challenges remain unresolved in the effective use of moringa oleifera. some of the challenges are: moringa oleifera seeds contain soluble organics that increase the residual organic carbon of the treated water which may serve as food for pathogens (microorganism that can cause harmful effect to the body system) and the large consumption of particles of moringa oleifera seeds precipitate into the body could also pose health challenges to the body. in this research, all the soluble organics and particles in moringa oleifera seeds which could pose health challenges to the body when consumed were eliminated during isolation of cellulose from moringa oleifera seeds which give pure and refined sample. the aim of this research is to identify the intrinsic potential of the cellulose of moringa oleifera seed for the purification of water. 2. materials and methods 2.1. materials the locally sourced organic material used in this research is moringa oleifera seed. it was removed from the shell, dried, grinded with a grinder and sieved to obtain fine particles. purification process was adopted to isolate cellulose and remove components that are not cellulose which include lignin, hemicelluloses, fats and inorganic contaminants. acetic acid, sodium chlorite (naclo2) and sodium hydroxide (naoh) were obtained from pascal scientific ltd and used as analytical chemical reagents. 2.2. methods a liquor ratio of 15:1(v/w) cooking condition was employed, the moringa oleifera seed particles was pulped with 20% of naoh at a temperature of 90◦c for 1 hour 30 minutes. after digestion process, the cooked pulp was filtered, screened and cleaned by rinsing properly with water without alkali. the pulped was left in the oven at 105◦c until the water was completely dried. 200 ml of hot water, 6g of naclo2 and 1.5 ml of acetic acid were mixed with 10g of bone dried sample of pulp in a titration flask. at 70◦c, the mixture was placed in the water bath and heated for 30 minutes. another 6 g of naclo2 and 1.5 ml of acetic acid were mixed and included, submitted to heat for next 30 minutes before putting the water bath power off. the sample remained in the water bath for 24 hours. after digestion, it was filtered and cleaned by rinsing properly with water until the chlorine and the acid were washed away. the sample acquired was left in the oven at 105◦c until the water was completely dried to obtain the cellulose. 3. characterization the crystallinity index of the isolated cellulose from moringa olienfera seeds was obtained using a philips pw diffractometer with cu-kα monochromator at voltage of 15kv, scanned at wavelength λ=1.54å with 2θ angle range from 5◦ to 90◦. the scanning electron microscope which was used to determine surface morphology was achieved using 15 kv accelerated voltage of jeol/eo jsm-6390 and has a resolution up to 100µm. the variation in functional groups was determined by fourier transform infrared (ftir) spectrophotometer induced by various treatments within a wavelength range of 700–4000cm−1. 3.1. theoretical background the interplanar spacing (d-spacing) was obtained as in equation (1) [9,10] d = nλ 2sinθ (1) where, the interplanar spacing of the crystal is d, order of reflection is n, wavelength of the incident x-ray is λ and angle of incidence is θ. the crystallinity index was calculated as following equation (2) [11,12] cir = i200 − iam i200 × 100 (2) where, highest peak intensity of the crystalline fractions is i200 and low intensity peak of the amorphous region is iam. the crystallite size (l) was calculated using scherrer’s equation [13] l = k ×λ b × s inθ (3) where, constant value given as 0.91 isk, wavelength of the incident x-rays is λ, bragg’s angle (◦) is θ, and intensity of the full width at half maximum (fwhm) proportional to a high intensity peak of the diffraction plane is b. 4. results and discussions 4.1. x-ray diffraction (xrd) xrd pattern of isolated cellulose from moringa oleifera seeds revealed crystalline characteristics peaks at 2θ = 14.39◦, 15.33◦, 22.47◦ and 34.50◦, indicating the crystal structure of cellulose i with allomorph cellulose iβ (monoclinic) [14]. the crystalline peaks indicate that the crystal structure is attributed to planes (110), (110), (200) and (004) respectively. it shows that the occurrence of intra and inter-molecular hydrogen bonding in the cellulose through hydroxyl group can ignite the arrangement of crystal order in the cellulose [15]. from the isolated cellulose, the peaks 14.39◦ and 15.33◦ were observed around 15◦ and the peak 15.33◦ was broad due to the amorphous nature of the material used [16,17]. the isolated cellulose shows prominent peak at 22.47◦ which exhibited higher crystallinity because of the efficient elimination of the amorphous parts. the value of crystallinity index(cir ) is 63.1%, 141 afolabi et al. / j. nig. soc. phys. sci. 3 (2021) 140–143 142 figure 1. x-ray diffractogram of isolated cellulose from moringa oleifera seeds figure 2. scanning electron micrograph of isolated cellulose from moringa oleifera seeds crystallite size(l) is 1.95nm, d-spacing is 3.9å and fwhm is 0.07331. since the proportion of crystallinity index is high, then the degree of crystallinity is justified to be high. the high proportion of crystallinity index is ascribed to removal of some of the amorphous constituents and rearrangement of the crystalline regions into a more ordered structure [9]. 4.2. scanning electron micrograph (sem) the morphological features of the cellulose isolated from moringa oleifera seeds are shown in figure 2. the surface morphology showed that the particles have conical and needle-like feature. the isolated cellulose has an average length and diameter of .2µm and 88.9µm respectively. it was disjointed from one another, indicating the total elimination of hemicelluloses and lignin. this is similar to previous researches on the cellulose from oil palm empty fruits bunch extraction and characterization and cellulose nanocrystals from corncob extraction and characterization for application as reinforcing agent nanocomposites [12,18]. 4.3. fourier transform infrared (ftir) spectroscopy figure 3 shows the fourier transform infrared spectra of the isolated cellulose. some important functional groups ocfigure 3. fourier transform infrared (ftir) spectra of isolated cellulose from moringa oleifera seeds cupied by the cellulose which however revealed basic potentials of the material for water purification are highlighted. the spectra showed wide band centered at 3311 cm−1 appointed to o–h stretching. this functional group commonly present in the cellulose. there is also a feature in this region from the nh stretching of amide group which reect the cationic tendency of the cellulose. this is similar to the result of characterization and use of moringa oleifera seeds as a biosorbent for removing metal ions from aqueous effluents [19]. at 2887cm−1, there exist spectra of characteristics of c– h stretching vibration. in the region between 1687cm−1 and 1308cm−1 there are bands appointed to c=o bond stretching. the carbonyl group appears in the structures and there is a band at 1610cm−1 accompanied with the amide group. the presence of hydroxyl, carbonyl and amine groups are responsible for the coagulative capacity in water purification [20,21]. 5. conclusion the xrd determines the high percentage of crystallinity index of the cellulose and the degree of crystallinity was found to be high. the xrd pattern also revealed that the crystal structure is cellulose i with allomorph cellulose iβ (monoclinic). the characterization of isolated cellulose using fourier transform infrared spectroscopy (ftir) revealed o–h stretching, c–h stretching vibration and c=o bond stretching functional groups. the presence of hydroxyl, carbonyl and amine groups are responsible for the coagulative capacity in water purification. acknowledgments the authors gratefully appreciate dr. ige, o.o. and dr. alo, f.i. of the department of material science and engineering, obafemi awolowo university ile-ife, osun state, nigeria for their effort in the analysis of the samples. dr. adekoya mathew, mr. olasoji, m.o. and the department of materials and metallurgical engineering of the federal university of technology, 142 afolabi et al. / j. nig. soc. phys. sci. 3 (2021) 140–143 143 akure, nigeria are also appreciated for their support during the period of the research. references [1] u. a. abdulwahab, s. s. sumaila, w. m. manja, b. opoku & j. ibrahim “assessment on the potential of moringa oleifera seed extract in the clarification of turbid surface water”, international journal of scientific and research 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[21] m. a. idris, m. s. jami, a. m. hammed & p. jamal “moringa oleifera seed extract: a review on its environmental applications”, international journal of applied environmental sciences 11 (2016) 1469. 143 j. nig. soc. phys. sci. 3 (2021) 234–238 journal of the nigerian society of physical sciences review a review of evidence of aerosol transmission of sars-cov-2 particles s. s. aladodoa,b,∗, c. o. akoshileb, j. o. otua acentre for atmospheric research anyigba, kogi state buniversity of ilorin ilorin, nigeria abstract severe acute respiratory syndrome coronavirus 2 (sars-cov-2) causes coronavirus disease (covid-19) through multiple transmission routes and understanding the mode of transmission is very important for its containment and prevention. consequently, inadequate attention has been given to the spread of respiratory droplets in indoor conditions under microclimatologic turbulent wind promoted by aerosol from talking (loud), coughing, sneezing, toilet flushing of an isolation room, and resuspension of the settled virus from the surfaces. to this end, this study is presenting the early review of the process and evidence of aerosol transmission of sars-cov-2 particles. there are significant results of many studies including those under peer review that support aerosol and airborne transmission which government agencies should consider for reducing the transmission rate. doi:10.46481/jnsps.2021.162 keywords: aerosol transmission, sars-cov-2, pandemic, air recirculation, respiratory droplet article history : received: 27 january 2021 received in revised form: 17 june 2021 accepted for publication: 19 july 2021 published: 29 august 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: o. j. abimbola 1. introduction the second wave of the covid-19 disease is already at the peak in some counties in europe while gradually peaking up in some others such as nigeria where some daily confirmed cases are above a thousand. the rate of spread of coronavirus disease (covid-19) caused by severe acute respiratory syndrome coronavirus 2 (sars-cov-2) through the transportation of respiratory droplet, (aerosol) over a wide geographic area gives the reason to be declared a global public health emergency or pandemic by the world health organization (who) [1]. the hazards of the pandemic greatly increase morbidity ∗corresponding author tel. no: +(234)706924xxx email address: aladodoshehu@gmail.com (s. s. aladodo) and mortality globally and cause significant economic, social, and political disruption [2]. as of december 20th, 2020, the total confirmed cases of covid-19 have exceeded 77 million nearly doubled the figure recorded as of october 2020 in 215 countries of the world with over 1.7 million mortality [3]. nigeria has been one of the countries affected by the pandemic with over seventy-eight thousand (78,000) cases recorded as of late december and over one thousand two hundred (1,200) death cutting across the country. the first confirmed case in nigeria was announced on 27 february 2020, when an italian citizen in lagos tested positive for the virus [4]. the multiple routes of transmission of covid-19 disease such as respiratory droplet, contact, fomite, or air make it difficult to independently investigate each route of transmission. evidence have shown 234 aladodo et al. / j. nig. soc. phys. sci. 3 (2021) 234–238 235 that in addition to the mentioned modes of spread, transmission of sars-cov-2 through aerosol is imminent especially in closed indoor settings with poor ventilation and high probability of air recirculation [3, 4, 5, 6, 7]. this study aims to review the mechanism of aerosol suspension, transportation, deposition, and the evidence of aerosol transmission of covid-19. based on the established evidence of aerosol transmission, infection control methods were proposed to mitigate the aerosol transmission of the respiratory particles (sars-cov-2). 2. aerosol sizes, suspension, and deposition the atmospheric aerosol is a suspension of fine solid and/or liquid matter which are either natural or emitted directly into the atmosphere by anthropogenic and biogenic sources or formed indirectly in the atmosphere by gas-to-particle conversion processes. it encompasses a wide range of particle types having different sizes, shapes, compositions, and optical properties [8]. generally, its typical diameters range over four orders of magnitude, from a few nanometres to a few tens of micrometers otherwise referred to as aerodynamic diameter. based on the size, aerosol can be classified into monodisperse (synthesized in the laboratory for experiment) and polydisperse which occur naturally with different sizes usually represented with a relative term particle size distribution. it has two distinct categories, fine and coarse particles. the fine particles are generally referred to as aerosols with diameters less than 2.5 µm, while coarse particles are aerosols above 2.5 µm. these categories can be regrouped into modes. in the domain of coarse particles, there is usually only one mode which is called coarse mode and in the domain of fine particles are usually three modes: nucleation (< 20 nm in diameters), aitken (≈ 20 nm to ≈ 100 nm in diameters), and accumulation modes (≈ 0.1 µm to ≈ 2 µm in diameters) [8]. the size and composition determine the ability of particles to serve as nuclei upon which other droplets in the atmosphere react and settled with. these aerosol processes involve different stages of formation, nucleation, condensation, and coagulation. during nucleation gas molecules or ultra-fine particles (nanoparticlee.g. sars-cov-2) aggregate and can form a cluster that can condense into small liquid particles. this can be of the same molecules nucleating together (homogenous) or the nucleation happens on the surface of foreign particles (heterogeneous e.g. sar-cov-2 and mouth aerosol). when a lot of nucleated particles are formed and super saturation becomes low, condensation takes place instead of nucleation (figure 1). at this point, there is no further formation of new particles. instead, already existing particles start to grow. while coagulation occurs when two aerosol particles come in contact, collide and stick together. a collision can happen due to the brownian motion of the particles in air, gravitational, phoretic, electrical, or other forces which all depend on the aerosol diameters. in the process, externally mixed particles become internally mixed and some small particles are lost due to the formation of larger particles, but the volume of particles is preserved [9]. suspension of aerosol in the atmosphere greatly depends on aerosol type and modes of emission. the coarse mode aerosols are mechanically disintegrated parts of soils and their formation and emission greatly rely on the wind through the dragging of the particles on and off the surface (saltation) at a wind velocity referred to as erosion threshold velocity. for the anthropogenic aerosols that are directly emitted into the atmosphere due to human activities such as coughing, talking, sneezing as in the case of sars-cov-2 with the mouth aerosol. once suspended in the atmosphere, many factors determine the residence time in the air such as physical properties, size, type, altitude range. the suspension period (residence times) of aerosols vary significantly, from a few seconds for very large particles that soon after emission fall back on the ground, to years for sulphate aerosols stable at high altitudes in the stratosphere going through many phases during the cause of staying [10, 11]. during the residence time, aerosol can be transported from emission sources to sink areas. some of the aerosols e.g. desert dust can be transported over very long distances horizontally, commonly over several thousands of kilometers or more which have been proven by several studies with sahara and asian dust over the atlantic ocean, mediterranean sea, and other parts of the globe [13, 14, 15]. some can be transported vertically in layer plumes into the stratosphere such as volcanic ash particles while others reside very close to the sources of emission such as sulphur dioxide (s o2) [16]. the removal mechanism of the aerosol is what is refers as deposition, its mechanism is complex and depends on the locations and properties of the aerosol particles. it can be divided into dry and wet deposition. dry deposition can either be deposition at the surface or gravitational sedimentation while wet deposition is in-cloud and below-cloud scavenging [17]. the dry deposition at the surface is an aerosol deposition process in which particles are removed from the atmosphere by the interaction with the surface, or more precisely with the atmospheric surface layer and a thin layer of air next to the surface (quasi-laminar sublayer). the dry deposition flux directly depends on the aerosol concentration: fdd = υddη (1) where η is the aerosol concentration and υdd is the deposition velocity in ms1. the deposition velocity depends on size, shape, the density of particles, properties of the surface, and the turbulence in the surface layer. the smallest particles in aitken mode (e.g. sars-cov-2) are subjected to brownian motion, collide with the surface and get deposited while the bigger size and coarse mode particles can be deposited through interception and impaction respectively [18]. coarse mode particles can also be subjected to gravitational settling due to the gravitational force that makes the particles fall and because of their large sizes, and it makes their atmospheric lifetime very short. wet removal mechanisms are processes that act on aerosols via atmospheric hydrometeors (cloud droplets, rain, snow, fog) and deposit them to the surface. both in-cloud and below-cloud mechanisms can be efficient in aerosol removal and are reversible, because all hydrometeors that scavenged aerosols can also evaporate, releasing aerosols back into the air [17]. 235 aladodo et al. / j. nig. soc. phys. sci. 3 (2021) 234–238 236 figure 1. possible path of evolution of a particle, from nucleation to coagulation [12]. 3. viral aerosol transmission it has been established by many studies that body fluids can be aerosolized through daily activities such as coughing, and medical procedures. deposited particles can be aerosolized through re-suspension due to floor cleaning, and biological specimens can be aerosolized through improper laboratory procedures. in all, the infectious virus particles can be easily suspended in the atmosphere and as a result infect others [7 and some ref therein]. the suspended infectious aerosol particles (sars-cov-2) can now be transported as either homogenous nucleated, heterogeneous nucleated, condensed, or coagulated aerosols and later deposited on any surface by either dry or wet deposition or directly inhale with the fine aerosol in air. experimental research has demonstrated the variety of respiratory viruses which includes the middle east respiratory syndrome coronavirus (mers-cov), severe acute respiratory syndrome coronavirus (sars-cov), norovirus, and influenza virus could be transmitted by aerosols under certain conditions such as climatic and environmental [19, 20, 21, 22]. 4. evidence of aerosol transmission of sars-cov-2 according to jones and brosseau [23] there were three criteria set for aerosol transmission of the virus to be imminent (1) the mouth aerosol containing virus must be from an infectious person, (2) the virus must be able to exist and infective in aerosol for its residence time, and (3) the target tissues must be accessible to an aerosol with required viral load. several studies have shown in the case of first criteria that sars-cov-2 particles are discharged into the atmosphere through respiratory droplets of infected person [3, 4], and it has been frequently detected in the throat, conjunctival, and anal swabs. when a person coughs, talks or breathes, they throw in any direction between 900 to 300,000 liquid particles from their mouth which ranges in size, and a cough can send them traveling at speeds up to 60 mph which keeps them suspended in the air for some time [24]. since breathing and speaking occur more frequently than coughs and sneezes, they have a critical role in viral transmission, particularly from asymptomatic cases and bigger size droplet (> 50 µm) has a higher probability of containing the virus than smaller size droplet before dehydration [25]. one minute of loud-speaking could produce thousands of oral droplets per second, of these at least 1000 virus-containing droplet nuclei could remain airborne for more than 8 min hence there are high chances of likelihood to be inhaled by others and thus trigger new infections [26]. the viability of the sars-cov-2 virus in aerosol according to the second criteria has been investigated experimentally by several authors. van doremalen et al. [27] demonstrated that sars-cov-2 can survive for more than three hours in the air under control conditions of temperature 21 oc 23 oc and relative humidity 65 %. also, a uk variant of sars-cov-2 could remain viable in aerosols for at least 90 min under experimental conditions (artificial saliva and tissue culture media). another investigation reveals that sars-cov-2 in respire-ablesized aerosols could persist and maintain infectivity for up to 16 h [28]. these are just a few in many studies pointing out the persistence and viability of coronavirus in aerosols. according to song et al. [7] epidemiological studies are difficult to interpret concerning the role of transmission unless other routes can be ruled out. different recent studies have investigated the role of air transmission of sars-cov-2 ruling out other routes such as read [29], shen et al. [30], and lu et al. [31] and airborne route appeared to be a major contributor for the super spreading of the virus in an air recirculation environment. some occurrences of covid-19 disease across the globe where aerosol transmission might have played a role are tabulated in table 1. 5. control and elimination of aerosol transmission the main measure to curtain the aerosol transmission of infectious disease is ventilation. this promotes the air dilution around a source dispersing the aerosol, and the removal of respiratory viruses through brownian motion in an open atmosphere [36, 37, 38]. the use of face masks which is recommended by the nigeria centre for disease control (ncdc) probably blocks some aerosol from leaving the mouth and reduces the chances of inhaling the aerosol contaminated with the virus in the air [4, 7, 24]. 236 aladodo et al. / j. nig. soc. phys. sci. 3 (2021) 234–238 237 table 1. some of the cases indicating possible aerosol transmission of sar-cov-2 virus. place/country date findings author(s) lab (usa) 13/04/2020 sar-cov-2 particles remain infectious in aerosol for up to sixteen hours. fears et al. [28] hospital (china) 08/03/2020 deposited samples in icu and air samples in makeshift hospital patient toilet were positive for sar-cov-2 virus. liu et al. [32] environment hospital (china) 03/04/2020 four surfaces out of one hundred and seven surfaces in the hospital environment sampled tested positive. ding et al. [33] outdoor air (italy) 15/04/2020 sars-cov-2 rna was detected on outdoor particulate matter (pm) suggesting that, in stable atmospheric condition and high concentrations of pm, sars-cov-2 could create clusters with outdoor pm. setti et al. [34] cases public transport (usa) 11/02/2020 two died and at least 103 people have infection among 1,111 crew and 2,460 passengers in the grand princess cruise ship. moriarty et al. [35] restaurant (china) 26/01/2020 aerosol transmission of an outbreak in a restaurant in guangzhou, china which is explainable only to air-conditioned ventilation while the distance between the occupant is greater than one meter. lu et al. 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[38] a. f. stein, r. r. draxler, g. d. rolph, b. j. b. stunder, m. d. cohen & f. ngan, “noaa’s hysplit atmospheric transport and dispersion modeling system”, american meteorological society 96 (2015) 2059. 238 j. nig. soc. phys. sci. 5 (2023) 1222 journal of the nigerian society of physical sciences study of the passivation of defects in the perovskite cell: application to sahelian climate conditions essodossomondom anatea,∗, n’detigma kataa,b, hodo-abalo samaha, a. seidou maigab afaculté des sciences et techniques, université de kara, kara, togo blaboratoire electronique, informatique, télécommunication et energies renouvelables, université gaston berger, saint-louis, sénégal abstract this article is devoted to the study of the performance of the photovoltaic cell based on perovskite (mapbi3) in real conditions of sub-saharan africa. a model of this cell has been made taking into account the integration of defects at the interfaces. after a study of the sensitivity of these defects, a passivation layer was introduced at the interface to improve the performance of the cell. the influence of temperature and irradiance on the performance of perovskite cells was studied on the one hand with defects at the interfaces and on the other hand with the integration of a passivation layer of defects. the results show a decrease of the performance ratio for the non-passivated cell due to the defects present at the interfaces of the said cell. the models developed under scaps-1d were validated by applying it to a real module found in the literature under the same conditions. the performance calculation shows a satisfactory qualitative and quantitative agreement. the results relative to the performance ratios obtained for the simulated models show that perovskite is on the right track for a potential future candidacy to the most suitable technologies for sub-saharan africa. doi:10.46481/jnsps.2023.1250 keywords: modeling, perovskite solar cells, interface defects, performance ratio. article history : received: 30 december 2022 received in revised form: 23 march 2023 accepted for publication: 30 march 2023 published: 14 may 2023 c© 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: k. sakthipandi 1. introduction africa has an almost unlimited potential for solar energy, estimated at about 10 tw [1]. however, africa is among the regions in the world with the lowest coverage of electrical energy. photovoltaics being the ideal alternative, solar fields are covering more and more surfaces using first generation modules which are relatively expensive and which moreover see their yields decrease under the influence of the temperature. studies have shown the production capacity of thin-film technol∗corresponding author tel. no: +22893089314 email address: edso.anate@gmail.com (essodossomondom anate) ogy compared to first generation technology for sub-saharan climate conditions [2,3]. thus, this study is situated in the context of predicting the photovoltaic production under real subsaharan conditions of third generation thin film modules based on perovskite. the challenge is to evaluate the influence of climatic factors, in particular temperature, on the photovoltaic performance of these solar photovoltaic modules. the interest of this technology is its very high efficiency of 25.5% in 2020 [4–6]. in addition, perovskite is a low-cost material with exceptional structural and optoelectronic properties. tolerant in volume defects [6–8], it is not so when it comes to defects at the interfaces which in addition to lowering the yield, 1 anate et al. / j. nig. soc. phys. sci. 5 (2023) 1222 2 makes it less stable in the long term. of neutral, acceptor or donor nature, once at the interfaces, they significantly degrade the performance of the cell [7]. the most effective technique to minimize the influence of defects is passivation which uses materials with specific properties. organic compounds with the carbonyl group are widely used to passivate perovskite cells [9] by exploiting the coordinated bonds based on lewis’s acid-base chemistry [10–12]. wang et al. showed the passivation ability of theophylline (c7h8n4o2) on perovskite cells [9]. in this work, we examine the performance precisely the performance ratio of perovskite cells under sub-saharan african conditions by simulation. three perovskite modules, one of which is real [13] and two others modeled under scaps, are simulated under the same conditions. the purpose of the study of the latter two is to evaluate the impact of defects when the modules are operated in real conditions at high temperatures. 2. methodology we first model the cell under the scaps-1d software version 3.3.1.0 following the fto/etl/perovskite/htl/au structure by integrating at the etl/perovskite and perovskite/htl interfaces the neutral, acceptor and donor defects of the order of 108 1011 cm−2 following street et al. [8,14]. these three types of defects come either from vacancies (vx) such as vma, vpb, vi , or interstitials (xi) like mai, pbi, ii, or substitutions (xy) such as mapb, pbma, mai , pbi , ima, ipb of the constituents (x or y= ma, pb, i) [7] of the perovskite or of foreign elements. they can carry different electric charges behaving as electron acceptor, donor or neutral. then a passivation layer based on theophylline is introduced following the structure fto/etl/theophylline/perovskite/htl/au. the cell with defects at interfaces and the one with passivation layer were used to model tow modules. the data (voc, isc and pmax) from these models in addition to those from the real module which is from literature [13] are used in the hybrid levenbergmarquardt-analytical extraction program proposed by kata et al. to extract the parameters (iph, i0, n, rs and rsh) needed for the simulation under ltspice [15]. finally, the sub-saharan temperature and sunshine conditions are entered into the ltspice module through the translation relations [15]. as an output, a current-voltage characteristic under real conditions is obtained for each measurement. table 1 provides the information on module modeling considerations with reference to the real module. figure 1 shows the method for evaluating the performance ratio of the module. solar irradiance and temperature data are taken from a measurement site in ouagadougou (burkina faso) due to the unavailability of that site in kara (togo) at the moment. these data such as temperature and irradiance are measured at the same time as module i-v characteristic, module voltage and module current by using multimeters simultaneously whereas a pyranometer was used to measure the irradiance as describe by kata et al. [15]. two days are chosen to represent the dry season for one and the rainy season for the other. the figure 1: diagram of the performance ratio evaluation figure 2: effects of acceptor defects density at etl/absorber interface performance evaluation is based on the calculation of the performance ratio (pr) of the three modules. the real efficiency (real) of the ltspice module is determined for each day’s measurement using equation 1. the performance ratio allows comparison of this performance between the three modules under the same conditions. this ratio refers to the ratio of the real efficiency under real conditions (real) to the efficiency under standard test conditions (stc) (stc) in equation 2. this standard test conditions include the temperature of 25◦c, am1.5 and an irradiance of 1000 wm−2. real = pmreal s ∗ greal , (1) 2 anate et al. / j. nig. soc. phys. sci. 5 (2023) 1222 3 table 1: characteristics of the simulated modules pv module base on : passivated cell cell with defects real cell number of cells 55 in series 55 in series 55 in series efficiency (%) 27.87% 24.73% 17.9% voc (v) 71.61 63.86 58.7 isc (a) 0.325 0.323 0.323 vmax (v) 64.9 58.3 48.42 imax (a) 0.318 0.314 0.298 pmax (w) 20.6 18.25 14.43 figure 3: effects of neutral defect density at abs / htl interface figure 4: i-v characteristic of the simulated module pr = real s t c , (2) where pmreal is the maximum power under real conditions, s the surface of the module and greal is the real condition solar irradiance. figure 5: energy band diagrams without (a, b) and with (c, d) passivation 3. results and discussion we studied the sensitivity of the defects at both interfaces varying from 1 to 1015 cm−2. the drop in performance is felt from 1011 cm−2 at the etl/perovskite interface for the three types of defects. figure 2 illustrates this effect for the acceptor type with a drastic drop in voc of 0.66 v, a jsc of 4.53 macm−2, a ff of 78.83% and a yield of 2.35% at 1015 cm−2. at the perovskite/htl interface, the drop already starts at 106 cm−2 for all three types as well. at the perovskite/htl interface (figure 3), we note a decrease in voc of 0.142 v and jsc of 1.915 macm−2 for neutral defect densities ranging from 106 to 109 cm−2 which results in the loss of 2.34% of the ff and especially the yield which falls from 27.17% to 21.66% or a loss of 5.51%. this overall decrease can be explained by a low mobility of holes and therefore a low collection at the interface due to the introduction of larger neutral defects. similar observations are made at other interfaces. by applying the theophylline defect passivation layer whose electronic properties are derived from the work of ejuh et al. [16], the results are presented in graphical form. figure 4 shows the i-v characteristics of the ideal, passivated and nonpassivated cells. a significant improvement in the voc of the passivated cell compared to the non-passivated one is observed. this increase provides information about the recombination reduction. we also observed the behavior of the band diagrams of the two modeled cells. the presence of defects (charged point de3 anate et al. / j. nig. soc. phys. sci. 5 (2023) 1222 4 figure 6: comparison of the electrical characteristics of the cell without and with passivation figure 7: performance ratio for a clear day (dry season) fects) at the interfaces manifests itself on the band diagram as unfavorable band edge curvature (possibly due to unintentional doping effect)[7,17]. passivation reduced this unfavorable band curvature by eliminating the defects and their effect at the interface [17] to promote better performance such as improved open figure 8: performance ratio for an overcast day (rainy season) circuit voltage by widening the potential barrier at the junction. figures 5.a and 5.b show the band diagrams at equilibrium and at voc for a cell with defects, respectively, followed by figures 5.c and 5.d at equilibrium and at voc for a passivated cell, respectively. 4 anate et al. / j. nig. soc. phys. sci. 5 (2023) 1222 5 here we present the performance of the modeled cells before and after passivation. the effect of the three types of defects at the perovskite / htl interface is shown by a decrease in the graphs of figure 6 from 106 cm−2. the effect of passivation on the other hand is illustrated by straight lines giving as information the annihilation of the effect of these defects. we present here on figures 7 and 8 the production capacity in prediction of the performance under real conditions in subsaharan africa. the choice of two days for our study reflects the rainy season for the overcast day and the dry season for the clear day. the performance ratio of the three modules reaches 80% over most of the day for both days (figures 7 and 8). for the clear-sky day, this ratio exceeds 90% between 9 a.m. and 3 p.m., a period during which there is often high irradiation. the strong dependence of the power of the module on the irradiation means that one could recover 90% of the power pmax s t c on more than half of the day, this one could even reach 94% and 96% respectively for the modules with not passivated and passivated cells then 98% for the real module. during the same period, high module temperatures exceeding 47◦c are recorded. however, the highest prs are obtained during this period, moreover, there is no decrease in the ratio for the highest module temperatures of the day which are about 68 ◦c. this means that the perovskite solar modules show only a small decrease in efficiency with high temperature. the real module and the module with passivated cells have very close ratios for both day conditions. however, the non-passivated cell shows much lower ratios over all the measurements. this low ratio is explained by the presence of defects at the interfaces of the cells of this module. the evolution of the performance ratios of the real and passivated modules shows the relevance of the simulated models. moreover, by observing the ratios of the two days we can say that the program responds well to the variations of the solar irradiation. referring to figure 8, it can be seen that perovskite modules have good ratios for overcast skies, which is an asset for some regions of sub-saharan africa that have longer rainy seasons with irregular irradiation. this interesting performance of perovskite allows us to consider perovskite as a potential future candidate of the most suitable technologies for the subregion. 4. conclusion this study was devoted to the evaluation of the performance of perovskite cells under the conditions of sub-saharan africa. different defects at the interfaces, namely neutral, acceptor and donor types, have been studied on the perovskite cell modeled under scaps-1d. the simulation of the model showed a strong sensitivity of the defects at the etl/perovskite and perovskite/htl interfaces of the cell. the results also show the performance improvement by introducing a passivation layer. the simulation of the perovskite solar cell in real conditions of sub-saharan africa is done. the hybrid levenberg-marquardtanalytic parameter extraction method proposed by kata et al. was used to extract the modeling data under ltspice. the performance ratio of perovskite cells can reach 98% at module temperatures around 68 ◦c in sub-saharan african conditions. these results are quite interesting and encouraging for a possible use in the subregion. references [1] bad, “énergies renouvelables: pourquoi l’afrique est la future grande puissance mondiale | banque africaine de développement bâtir aujourd’hui, une meilleure afrique demain,” nov. 22, 2021. 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[17] b. chen, p. n. rudd, s. yang, y. yuan & j. huang, “imperfections and their passivation in halide perovskite solar cells”, chemical society reviews 48 (2019) 3842. 5 j. nig. soc. phys. sci. 5 (2023) 1094 journal of the nigerian society of physical sciences effects of hybrid exchange correlation functional (vwdf3) on the structural, elastic, and electronic properties of transition metal dichalogenides s. a. yamusaa, a. shaarib, i. isahc, u. b. ibrahimd, s. i. kunyac, s. abdulkarimd, y. s. itase,∗, i. m. alsalamhf a department of physics, federal college of education zaria, p.m.b 1041, zaria, kaduna state, nigeria b department of physics, faculty of science, universiti teknologi malaysia c department of science laboratory technology, jigawa state polytechnic, dutse, jigwa state nigeria d faculty of science, physics department kano university of science and technology, wudil, kano, nigeria e department of physics, bauchi state university, gadau, p.m.b. 65 bauchi, nigeria f physics department, faculty of science, university of hail, saudi arabia abstract in this research, the effects of van der waals forces on the structural, elastic, electronic, and optical properties of bulk transition metals dichalcogenides (tmds) were studied using a novel exchange-correlation functional, vdw-df3. this new functional tries to correct the hidden van der waals problems which are not reported by the previous exchange functionals. molybdenum dichalcogenide, mox2 (x = s, se, te) was chosen as a representative transition metal dichalcogenide to compare the performance of the newly designed functional with the other two popular exchange-correlation functional; pbe and rvv10. from the results so far obtained, the analysis of the structural properties generally revealed better performance by vdw-df3 via the provision of information on lattice parameters very closer to the experimental value. for example, the lattice constant obtained by vdw-df3 was 3.161 å which is very close to 3.163 å and 3.160 å experimental and theoretical values respectively. calculations of the electronic properties revealed good performance by vdw-df3 functional. furthermore, new electronic features were revealed for mox2 (x = s, se, te). in terms of optical properties, pbe functional demonstrates lower absorption than vdw-df3, as such it can be reported that vdw-df3 improves photon absorption by tmds. however, our results also revealed that vdw-df3 performed well for mos2 than for mose2 and mote2 because of the lower density observed for the s atom in mos2. doi:10.46481/jnsps.2022.1094 keywords: van der waals, pbe, hexagonal, vdw-df3, dichalcogenides. article history : received: 29 september 2022 received in revised form: 21 november 2022 accepted for publication: 02 december 2022 published: 14 january 2023 c© 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: taoreed owolabi ∗corresponding author tel. no: +2348069316888 email address: yitas@basug.edu.ng ( y. s. itas ) 1. introduction the study of transition metal dichalcogenides has grown in popularity as a condensed matter physics research area and as a potential resource for a range of applications that could 1 s. a. yamusa et al. / j. nig. soc. phys. sci. 5 (2023) 1094 2 affect scientific and high-tech advancement. dft explanations on how electrons in the many-body systems interact with each other depending on the so-called approximations of the exchange-correlation (xc) functional [1]. moreover, much of the successes in dft came from the fact that these functions often produce accurate results. however, there are some situations where failures are reported by many of these functional [2]. therefore, there is a need to understand the accurate xc functional to adequately describe the behavior of the many-body electronic system. one good example of failure by xc is the inability to fully describe a long-range electron interaction also called dispersion forces. van der waals problem by local density approximation (lda) and perdew-burke-ernzerh (pbe) exchange functional remains a challenge that needs urgent improvement [3] the problem of lack of dispersion forces otherwise referred to as van der waals (vdw) forces, is one of the most disturbing problems in dft. therefore it becomes one of the most traded topics in condensed matter physics and material science. it can be understood by the fact that over 800 dispersion-based dft studies were reported in 2011 compared to fewer than 80 in the whole of the 1990s [4]. recent studies have revealed that lda exhibits overestimations of the lattice constants for a and c of 0.4 % and 5.2 %, and gga underestimates c by 4 % while overestimating a by 0.58 %, respectively. this problem persisted in tmds, especially in terms of their mechanical, electronic, and optical characteristics [5]. in this research, a full demonstration of the effects of van der waals forces on the mechanical, electronic, and optical properties of tmds was carried out with mox2 (x = s, se, and te). transition metal dichalcogenides are types of materials that can be metallic or semiconducting [6]. the semiconducting tmds represent the layered materials [7], they can also be classified as direct or indirect energy band gap materials. for example, mos2, mose2, ws2, and wse2 are direct band gap semiconductors. in this study mos2, mose2 and mote2 were considered. molybdenum dichalcogenides mox2 (x = s, se, and te) in the most stable phase 2h-mx2 belong to the space group p63/mmc (194), it has a hexagonal crystal structure with wyckoff positions of 2c and 4f with z coordination value of 0.3779. literature studies revealed that the study of the effect of van der waals forces on this material using the new exchange functional vdw-df3 has not been conducted. although mos2 has been reported as good material for hydrogen storage [8], there are need to explore the potentials of other di-chalcogenides such as mose2 and mote2. based on the obtained results, vdw-df3 may demonstrate promising results on the electronic and optical properties of tmds. table 1 demonstrates the results on the effect of the pre-existing exchange functional with the experimental values. reports from table 1, revealed that there are still more problems to solve regarding the effect of van der waals, for example, the obtained value for the lattice parameter with pbe is 3.680 å, that of rvv10 is 2.186 å which is far away from the experimental value of 3.163 å obtained. therefore there is a need to use better exchange functional such as vdw-df3 for better improvement in the next generations’ sustainability science and technology to pave way for optoelectronic applications. 2. research method the three materials were first optimized using three selected exchange-correlation functional (pbe, rvv10, and vdw-df3) by setting the brillouin zone sample 12 × 12 × 3 monkhorst pack k-mesh and 800 ev plane wave cut-off energy. the optimization was performed till the total energy and force converged to 10−3 ev, this is true for all three materials (mos2, mose2, and mote2). the calculations were performed via ab initio density functional theory (dft) using plane-wave basis as implemented in the quantum-espresso package, this includes the optimization, and determination of equilibrium lattice parameters, and electronic band gaps. an auxiliary package to the quantum espresso thermo_pw [1] was also used in the calculation of the elastic constants and optical properties. to make an accurate comparison, kohn-sham equations were applied by implementing the dft ab initio quantum computing framework within the perdiew-burke-emzahope (pbe) exchange functional [9], rvv10 and our novel vdw-df3 functional. calculations were performed using a non-spin polarized dft to save computational costs. to ensure accurate results in this study, tmds were appropriately relaxed to appropriate geometries. for all three systems, the length and the height were chosen as 12.03 å each. the chiral/translation vectors were constructed such that the maximum force, stress, and displacements were set at 0.05 ev/å each. 3. results and discussion 3.1. structural and elastic properties the equilibrium lattice parameters for the three systems were determined by fitting energy volume in the standard equation of the state. this can also be obtained by polynomial fit to the energy-volume data [10]. the lattice parameter can be determined from equilibrium volume as: a0 = ( v k )1/3 (1) where k is the ratio c/a for the materials mox2 (x = s, se, and te). the crytallogrphic structure of mos2, mose2 and mote2 are presented in figure 1. the lattice parameters were calculated such that the three systems can be viewed as having a hexagonal p6_3/mmc symmetry with a lattice constant of 3.66 å. the mo-s and s-s bond lengths are 2.415 å and 3.131 å respectively which agrees with the available literature [11]. in the case of mose2, the bond lengths of mo-se and se-se atoms were 2.424 å and 3.113 å, respectively. to obtain significant results from our calculations, 2 s. a. yamusa et al. / j. nig. soc. phys. sci. 5 (2023) 1094 3 figure 1: crystal structure of 2h-mx2: (a) unit cell, (b) top view lattice parameters of mote2 were also studied, these were ensured to be 2.423 å and 3.116 å respectively. to further understand the vdw-df3 effects, we calculated the formation energy based on the lattice parameters earlier reported [12, 13]. the results are presented in table 2. the result predicted the output of the effects of van der waals under pbe, rvv10, and the novel vdw-df3 exchange functional. table 1 presented the result of the calculated equilibrium lattice parameter of molybdenum chalcogenide mox2 for the three exchange-correlation functional (pbe, rvv10, and vdw-df3) compared to the available experimental and theoretical results. table 1: the calculated lattice parameters with the three functional are compared with the available experimental and theoretical results pbe (å) rvv10 (å) vdwdf3 (å) theor. results [15] expt. ref. a(å) 3.680 2.186 3.161 3.163 3.160 [14] mos2 c(å) 13.37 13.394 12.296 12.442 12.290 [16] a(å) 2.314 3.523 3.293 3.295 3.288 [17] mose2 c(å) 13.001 13.036 12.918 13.088 12.920 [18] a(å) 3.874 3.489 3.551 3.617 3.520 [19] mote2 c(å) 13.906 13.965 13.817 14.261 13.970 [11] it can be seen that the vdw-df3 successfully describe accurately the lattice parameter of the three systems with only 1 % error. this shows that the functional performed excellently in the determination of the lattice parameters of a bulk mox2. molybdenum chalcogenide mox2 (x = s, se, te) is a hexagonal crystal with 2h-mox2 as the most stable phase. for this type of crystal, there are only five independent elastic constants. the five elastic constants were used to check the stability of the optimized structure using the born stability criteria [20] and to determine the mechanical properties of the materials for the three correlation functional. the calculated properties were compared with the available literature both experimentally and theoretically [21, 22, 23], which is presented in table 2. the born stability criteria were checked using equations (2) (4 [24]. c11 > |c12| (2) 2c213 < c33 (c11 + c12) (3) c44, c66 > 0, (4) where c66 = (c11 − c12) /2 and ci j are the five independent elastic constants for the hexagonal materials. as presented in table 2, vdw-df3 revealed high mechanical stability for all systems. therefore it can be reported that vdw-df3 xc functional significantly improves correction to van der waals problem in tmds. 3.2. electronic properties the electronic band structures of the molybdenum chalcogenides mox2 (x = s, se, te) were calculated along the high symmetry point of the brillouin zone by following the kpaths γ − m − k − γ for all the three systems. results of the three xc functional (pbe, rvv10 and vdw-df3) were obtained as presented in figure 2. the valence band maximum (vbm) and conduction band minimum (cbm) located at γ and between γ − k, respectively, were used to determine the band gaps as shown in table 3. to further explain the efficiency of vdw-df3 xc functional, the electronic band structure and density of states were calculated for mos2, mose2 and mote2 systems, the results presented in figure 2 show that mos2 demonstrated band gap of 0.79 ev, mose 2 revealed 0.88 ev and mote2 was found to be 0.67 ev. this results showed significant improvement in narrowing the band gap of tmds by vdw-df3 xc functional which brought them to new applications for optoelectronics [25]. therefore vdw-df3 xc functional make significant contribution towards turning tmds from wide gap to narrow gap semiconductors. in terms the partial density of states (pdos), calculations were performed to determine contributions by different s, p, d, f orbitals, the results are illustrated in figure 2 (b, d and f). to further elaborate on the nature of the band gap of the three systems, the total density of state (tdos) and partial density of state for the mos2, mose2, and mote2, are illustrated in figure 2. in terms of mos2 (figure 2(b)), the lower valance bands at -6.95 to 0.28 ev are composed mainly of mo4d states and s-3p states, zero states were seen from 0.28 to 0.58 ev. the conduction bands are mainly due to mo-4d and s-3p states located at 0.68 to 3.92 ev, there are lower contributions above 4.002 ev up to the conduction bands. for mose2 (figure 2(d)), the valance bands are composed of mo-4d and se-4p states located at -5.99 to 0.24 ev, the width 0.47 ev to 0.0.36 ev is the fermi level of zero states and above 0.35 ev is mainly composed of mo-4d and se-4p states for conduction bands. for mote2, valance band contribution starts from the energy range of -5.69 to 0 ev mainly by mo-4d te-5p states, fermi level was show from 0 ev to 0.2 ev, and the conduction bands contribution is the energy range above 0.74 ev mainly by mo-4d and te-5p. it can beseen that vdw-df3 was able to 3 s. a. yamusa et al. / j. nig. soc. phys. sci. 5 (2023) 1094 4 table 2: elastic constants in gpa, bulk modulus b in gpa, young modulus e in gpa, and shear modulus g in gpa for three functional for mox2, (x = s, se, and te) material funct. c11 c12 c13 c33 c44 b g e b/g σ pbe 214.39 51.92 13.43 55.27 17.73 57.860 97.784 40.13 1.442 0.218 mos2 rvv10 218.01 53.89 18.86 69.27 4.48 65.303 73.342 27.933 2.338 0.313 vdw-df3 94.21 76.71 25.85 305.07 5.00 79.171 50.102 17.964 4.407 0.395 pbe 78.34 64.68 46.85 403.90 2.36 83.32 83.698 47.07 1.046 5.001 mose2 rvv10 88.32 68.71 53.27 387.93 3.53 88.38 89.207 50.233 1.118 4.994 vdw-df3 87.94 69.88 47.15 459.24 4.31 91.39 91.845 58.691 1.495 4.361 pbe 116.84 30.71 36.83 111.78 25.16 61.577 85.590 33.741 1.825 0.268 mote2 rvv10 119.70 33.14 20.33 69.05 26.98 48.538 82.769 34.039 1.426 0.216 vdw-df3 122.44 31.08 13.33 54.48 27.91 42.193 82.021 34.873 1.210 0.176 figure 2: electronic band structure and density of state of mos2 (a) and (b), mose2 (c) and (d), and (e) and (f) for mote2 using three the fuctionals table 3: the energy gap of mos2, mose2, and mote2 with the three exchangecorrelation functionals system pbe (ev) rvv10 (ev) vdw-df3 (ev) ref. mos2 0.84 0.85 0.79 this work mose2 0.84 0.75 0.88 this work mote2 0.73 0.67 0.67 this work figure out all orbitals contributions as against pbe and rvv10 xc functionals. 3.3. optical properties the optical properties of mos2, mose2, and mote2 bulk crystals with polarization along x-direction (in-plane) are calculated using independent particle approximation by solving time-dependent density-functional theory (tddft) and linear response technique [3], using the sternheimer approach within thermo_ pw code [1], a proprietary branch of the quantum espresso project [4]. the calculated real and imaginary parts of frequency-dependent microscopic dielectric function in the energy range of 0 to 21 ev were plotted. the imaginary parts of the dielectric function of mos2 (figure 4), mose2 (figure 5), and mote2 (figure 6) were obtained from interband transition for the parallel and perpendicular direction of the electric field as computed from equation (3) [5], however, the real part of the frequency-dependent dielectric function was obtained from kramers-kroning relation as shown in equation (6) [2]: �2(ω) = 2πe2 ω�0 σκ,ν,c ∥∥∥∥λ̄. 〈ψck|u.r|ψνk〉∥∥∥∥2 δ (eck − eνk − e) . (5) where λ̄ is the polarization vector of light and the integral is over the brillouin zone, u, ω, e, ψck, ψ ν k are the polarization vector of the incident electric field, frequency of light, the electronic charge, and conduction and valance band wave function at k, respectively. �1(ω) = 1 + 2 π p ∫ ∞ 0 ω′�2(ω′) ω′2 −ω2 dω′, (6) where p denotes the integral’s principle value. the computed real and imaginary dielectric functions of mos2, mose2, and 4 s. a. yamusa et al. / j. nig. soc. phys. sci. 5 (2023) 1094 5 figure 3: real and imaginary dielectric functions for mos2 with respect to the (a) pbe (b) rvv10 and (c) vdw-df3 functionals figure 4: real and imaginary dielectric functions for mose2 with respect to the (d) pbe (e) rvv10 and (f) vdw-df3 functionals mote2 within the three functionals are plotted in figures 3, 4 and 5, respectively. the result shows that the interband transition due to mo-4d and s-3p, se-4p, and te-5p states move to lower energies from mos2, mose2, and mote2, respectively. both mos2, mose2 and mote2 materials show anisotropy [8] in the energy range from 0 to 7.5 ev and isotropy at higher 5 s. a. yamusa et al. / j. nig. soc. phys. sci. 5 (2023) 1094 6 figure 5: real and imaginary dielectric functions for mote2 with respect to the (a) pbe (b) rvv10 and (c) vdw-df3 functionals energy. to further confirm the band gap in the three systems, a bound state can be seen at 2.0 ev, 1.1 ev and 1.2 ev for pbe, rvv10 and vdw-df3 functional respectively, the results obtained by vdw-df3 was found to be in good agreement with previous theoretical results [15]. similar results were also obtained for mose2, mote2, these were presented in figures 4 and 5, respectively. to describe optical absorption, the imaginary dielectrics for all systems were studied. favourable results were obtained by vdw-df3 functional, for example higher optical absorptions were observed for mos2 (figure 3c) at 22.5 cm−1 which corresponds to 2.80 ev, this is the absorption in the visible range, other functionals only demonstrated absorption in the infra-red range, which significantly underestimates the absorption characteristcs of tmds. 4. conclusion to conclude this work, results so far obtained brought out new hidden properties of tmds which failed to be reported by pbe and rvv10 functionals. calculation of the elastic properties revealed that tmds are more stable with vdw-dft3, higher moduli of elasticity such as young’s moduli, shear moduli and bulk moduli were significantly improved with well agreement with theoretical results. from the results of electronic properties, it revealed that bulk tmds can be turned from being a wide gap semi conductors to being a narrow gap semiconductors, it also shows that d’orbital majorly contributed to narrowing the band gap in all the sysyems. higher optical absorption are also repoted by vdw-df3, this brought the systems under study as potential candidates for optoelectronics [25]. acknowledgement the authors acknowledge dr. abdu barde college of vocational and technical study, department of science and laboratory technology, dutse, jigwa state, and the university of technology malaysia for financial support, facilities, and services of high-performance computing on this research work. references [1] i. g. kaplan, modern state of the conventional dft method studies and the limits following from the quantum state of the system and its total spin. density functional theory-recent advances, new perspectives and applications, intechopen, (2022). 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[25] e. e. etim, m. e. khan, o. e. godwin, & g. o. ogofotha, “quantum chemical studies on c4h4n2 isomeric molecular species”, journal of the nigerian society of physical sciences 3 (2021) 429. 7 j. nig. soc. phys. sci. 1 (2019) 51–56 journal of the nigerian society of physical sciences original research modeling of self potential (sp) anomalies over a polarized rod with finite depth extents t. s. fagbemiguna,∗, m. o. olorunfemib, s. a. wahabc adepartment of geophysics, federal university, oye-ekiti, nigeria bdepartment of geology, obafemi awolowo university, ile ife, nigeria cdepartment of applied geophysics, federal university of technology, akure, nigeria abstract modeling is a powerful tool used by geoscientists to provide pre-knowledge about the expectations of any geophysical field measurements. this study generates self potential (sp) anomalies over a typical dyke-like structure to observe the influence of depth of burial and dip on sp anomalies. a computer program was developed from the potential distribution equation of an inclined polarized rod with limited depth extent using visual basic (vb) programming language to produce synthetic data for potential distribution. the potential distribution data were used to generate theoretical sp anomaly curves for a polarized rod for varying depth of burial and dip. twenty sp anomaly curves were generated with different dip values and depth of burial and from these curves superimposed curves were also generated. the anomalies were analyzed for the effect of depth of burial and attitude or dip. the sp anomaly curves generated show that increase in depth of burial causes reduction in the peak negative amplitude of sp anomaly curves. for inclined polarized rod at relatively shallow depth (< 2.0 m), the peak negative amplitude remains virtually the same with a positive shoulder over the down dip side of the target. also as the dip angle decreases from 90◦ for fixed depth of burial, the anomaly curves become asymmetrical. at θ◦, the distance between the peak negative and peak positive amplitude of the anomaly curve is equal to the linear extent of the rod. therefore, this study shows that depth of burial inversely influences the amplitude of self potential (sp) anomalies while dip angle affects the form or symmetry of anomaly curves. keywords: modeling, self potential, polarized rod, geologic dip, depth of burial article history : received: 15 april 2019 received in revised form: 20 may 2019 accepted for publication: 21 may 2019 published: 31 may, 2019 c©2019 journal of the nigerian society of physical sciences. all rights reserved. communicated by: o. j. abimbola 1. introduction the pre-knowledge of the expectation on the field is very essential for any geophysical investigations and in turn, it serves as quality control on interpretation mindset of the geophysicist involved. in lieu of this, modeling or simulation of geophysicalrelated problems has become a vital tool in geophysics and this technique has allowed geoscientists to come up with synthetic ∗corresponding author tel. no: +2348066358839 email address: tsfagbemigun@gmail.com (t. s. fagbemigun ) solutions against any related challenges on the field [1]. numerical simulation of geometric source model involving the self potential (sp) method can yield valuable information related to the field data [2, 3, 4]. this aspect of geophysics enables the generation of theoretical anomalies over geologic structures which can be encountered on the field such as dykes, channels, sandlens and faults. hence, such theoretical anomalies can aid our understanding of field anomalies and the interpretation of same [5, 6]. sp is one of the geophysical methods that makes use of natural electrical sources to study earths subsurface or for surface 51 t. s. fagbemigun et al. / j. nig. soc. phys. sci. 1 (2019) 51–56 52 exploration. the sp method has found several applications in various areas of science and engineering such as mineral exploration [7], well-logging [8, 9], engineering [10, 11, 12], agricultural sciences [13, 14], environmental studies [15, 16, 17]. this study aims to generate sp anomalies over a typical dyke-like feature for different depth extent and attitude (dip) as a means of investigating the influence of depth of burial and attitude on sp anomalies. 2. methodology equation 1 shows the basic equation for potential distribution [18]: v = ±q [ 1 r1 − 1 r2 ] , (1) here v is the potential distribution around the object, r1 is the distance from the point p to the top of the rod, r2 is the distance from the point p to the bottom of the rod and ±q is the charge at either end of the rod (figure 1). however, from figure 1 figure 1: diagrammatic representation of geometric model of self potential for a polarized rod [18]. r1 = ( x2 + z21 ) 1 2 , (2) r2 = ( z22 + (x − l cos α) 2 ) 1 2 . (3) where α is the geologic dip or attitude; l is the length or depth extent; z1 is the depth of burial; z2 is the depth to the bottom of the rod from the ground level; and x is the distance from the centre (o) to observation point. also from figure 1: z2 = z1 + l sin α, (4) therefore, substituting equations 2 and 3 into equation 1, we get: v = ±q  1( x2 + z21 ) 1 2 − 1( z22 + (x − l cos α) 2 ) 1 2  . (5) also, substituting equation 4 into equation 5, we get: v = ±q  1( x2 + z21 ) 1 2 − 1( (z1 + l sin α) 2 + (x − l cos α)2 ) 1 2  . (6) the parameters in equation 6 were considered in generating charge of the rod, depth of burial, geologic dip, length of the rod, depth to the bottom of the rod and distance in order for the theoretical potential distribution data generated to be concise and accurate. visual basic (vb) version 6.0 program was used for this study. potential distribution response was generated at 2 m interval and the polarized rod was inclined at different attitudes and depths of burial. sp profiles were generated for the potential distribution values in milliv olt(mv ) by plotting it against distance (in metre) from the observation point. interpretation was solely based on visualization of amplitude pattern of the sp profiles (qualitative interpretation). 3. results and discussion twenty (20) sp anomaly curves were generated with different dips/attitudes (0◦, 30◦, 60◦ and 90◦) and depths of burial (0.5 m, 2 m, 5 m, 10 m and 20 m). we also generated six (6) superimposed sp anomaly curves for same depth of burial but different dips and six (6) superimposed sp anomaly curves for same geologic dip but different depths of burial. the anomalies were analyzed for the effect of depth of burial and attitude on the amplitude and symmetry of sp anomaly curves. figures 2 and 3 (0◦and 30◦ dip) are replica of each other, except for the difference in the values of points at which they inflected. it is observed that, as the geologic dip increases, the geometry of the curves turned to be cone-like. at points of inflection, values of sp decrease with the increase in depth of burial. for a polarized rod with 0◦, the inflexion points of the anomaly curves are located at the middle of the rod (figure 2). the result of the study by [19] validates this, as their investigation over dykelike structure gave birth to gradual increase to sp values over the target. 52 t. s. fagbemigun et al. / j. nig. soc. phys. sci. 1 (2019) 51–56 53 figure 2: sp profiles for 0◦ geologic dip. at relatively shallow depth (< 2.0 m), the peak negative amplitude remains virtually the same with the shoulder over the down dip side of the target (figures 10 11). the peak positive amplitude shoulder is therefore, maximum on the down dip side. figures 4 and 5 display the sp anomaly curves for 60◦ and 90◦ dips. these curves are characterized with negative peak amplitudes. the negative peak at lower depth of burial is sharp compared to the deeper depth of burial. the maximum and minimum negative sp values are −3.2 mv and −380 mv respectively. in all the anomalies generated except for dip angle of 0◦, the top of the target is located beneath a peak negative amplitude sp (figures 3 5). the sp anomalies for a vertically dipping rod is symmetrical about the top of the rod while the anomalies become asymmetrical as the dip angle is decreased from 90◦ (figures 2 5). the study by [3] corroborates with the results of this study in terms of symmetrical nature and amplitude of the sp anomaly curves. figure 3: sp profiles for 30◦ geologic dip. the superposition of sp anomaly curves of varying depth of burial with the same geologic dip show that, well pronounced negative amplitude is that of 0.5 m with sp value of −380.03 mv . the average maximum sp value of negative amplitude is −380.58 mv. therefore, with same dip angle but varying depth of burial, the sp anomaly decreases in the peak negative amplitude as the depth of burial increased (figures 6 9). with same depth of burial but varying dip angle, the anomaly curves become asymmetrical with decreasing dip angle from 90◦. figures 7 9 show the same signature as discussed by [2] of sp investigation, which revealed weiss anomaly of the maden copper mine, indicated that the source of signature could be spherical. it was established that vertically oriented pipe is associated with or characterized by relatively high-amplitude over the target [20] and this is not far fetch from this study. however, at deeper depth (> 2.0 m), the peak negative amplitude of the sp anomalies decreases in amplitude as dip angle is decreased from 90◦ (figures 12 and 13). the displacement 53 t. s. fagbemigun et al. / j. nig. soc. phys. sci. 1 (2019) 51–56 54 figure 4: sp profiles for 60◦ geologic dip. between the peak negative and peak positive of the sp anomalies generated at relatively shallow depth of burial (< 5.0 m) is equal to the linear extent (l) of the polarized rod or target. 4. conclusion the interpretation of sp anomaly curves shows that the inflection points of the anomaly curves are located at the middle of the rod for a polarized rod with 0◦ dip. the displacement between the peak negative and positive amplitude of the sp anomalies is equal to the linear extent (l) of the polarized rod or target of shallow depth of burial or zero dip. however, the sp anomaly decreases in the peak negative amplitude for fixed dip angle but increasing depth of burial. the anomaly curves become asymmetrical with decreasing dip angle from 90◦ for fixed depth of burial and the anomalies for a vertically dipping rod is symmetrical about the top of target. in conclusion, depth of burial and geologic dip has great influence on the amplitude figure 5: sp profiles for 90◦ geologic dip. figure 6: superposition of sp anomaly curves for 0◦ at different depth of burial. 54 t. s. fagbemigun et al. / j. nig. soc. phys. sci. 1 (2019) 51–56 55 figure 7: superposition of sp anomaly curves for 30◦ at different depth of burial. figure 8: superposition of sp anomaly curves for 60◦ at different depth of burial. figure 9: superposition of sp anomaly curves for 90 at different depth of burial. and the symmetrical nature of self potential anomaly curve over an inclined object. although numerical modeling is limited befigure 10: superposition of sp anomaly curves for 0.5 m depth of burial at different dip. figure 11: superposition of sp anomaly curves for 2 m depth of burial at different dip. figure 12: superposition of sp anomaly curves for 10 m depth of burial at different dip. cause of the interaction required on the field it is assumed the earth material is isotropic and homogenous. therefore, the lim55 t. s. fagbemigun et al. / j. nig. soc. phys. sci. 1 (2019) 51–56 56 figure 13: superposition of sp anomaly curves for 20 m depth of burial at different dip. itations of the numerical modeling are unable to account for the influence of the earths subsurface physical properties and environmental effects. nonetheless, they still serve as frontier in exploration in geosciences as this study would serve as quality control on interpretation mindset of geophysicists. acknowledgments we appreciate mr. o. a. sanuade, mr. j. o. amosun and dr. a. b. eluwole for their advice and contributions towards the success of this work. we also thank the referees for the positive enlightening comments and suggestions, which have greatly helped us in making improvements to this paper references [1] a. bokulich & n. oreskes, models in the geosciences. handbook of model-based 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[15] w. c. robert, geotechnical applications of the self-potential method: report 3: development of self-potential interpretation techniques for seepage detection. us army corps of engineers, washington, dc, 1989. [16] b. åogaåa, m. j. mendecki, w. m. zuberek & m. robak, “application of self potential method in the area contaminated with oil derivatives”, acta geodynamica et geomaterialia 9 (2012) 179. [17] p. soupios & m. karaoulis, application of self-potantial (sp) method for monitoring contaminants movement. 8th congress of the balkan geophysical society, chania, greece, 2015. [18] w. m. telford & l. p. geldart, applied geophysics. new york: cambridge university press, 1990. [19] b. j. dallas & k. james, “regional self potential anomalies at kilauea volcano”, us geological professional paper 1350 (2016) 947 . [20] j. b. rittgers, a. revil, m. karaoulis, m. a. mooney, l. d. slater & e. a atekwana, “self-potential signals generated by the corrosion of buried metallic objects with application to contaminant plumes”, geophysics 78 (2013) 65 . 56 j. nig. soc. phys. sci. 2 (2020) 250–256 journal of the nigerian society of physical sciences alpha decay half-lives of 171−189hg isotopes using modified gamow-like model and temperature dependent proximity potential w. a. yahya∗ department of physics and materials science, kwara state university, malete, kwara state, nigeria abstract the alpha decay half-lives for 171−189hg isotopes have been computed using the gamow-like model (glm), modified gamow-like model (mglm1), temperature-independent coulomb and proximity potential model (cppm), and temperature-dependent coulomb and proximity potential model (cppmt). new variable parameter sets were numerically calculated for the 171−189hg using the modified gamow-like model (termed mglm2). the results of the computed standard deviation indicates that the modified gamow-like model (mglm2) and the temperaturedependent coulomb and proximity potential model give the least deviation from available experimental values, and therefore suggests that the two models (mglm2 and cppmt) are the most suitable for the evaluation of α-decay half-lives for the hg isotopes. doi:10.46481/jnsps.2020.139 keywords: alpha decay, half-life, gamow-like model, proximity potential, radioactive decay article history : received: 26 september 2020 received in revised form: 04 november 2020 accepted for publication: 05 november 2020 published: 15 november 2020 c©2020 journal of the nigerian society of physical sciences. all rights reserved. communicated by: b. j. falaye 1. introduction alpha decay, first discovered in 1899 by rutherford, is one of the crucial decay modes for heavy nuclei [1]. it is one of the important decay modes to describe nuclear structure [2] and it can be successfully described by quantum theory [3]. alpha decay (α-decay) is a crucial decay mode that provides information about the nuclear structure and stability of heavy and superheavy nuclei [4]. it is important in the identification of new heavy and super heavy nuclei (shn) [5], and in the study of nuclear structure and nuclear force [6]. investigations of the α-decay half-lives have been carried out both theoretically and ∗corresponding author tel. no: +2348036289110 email address: wasiu.yahya@gmail.com (w. a. yahya ) experimentally via various approaches. some of the theoretical models that have been employed to study the α-decay halflives are the fission-like model [7], the generalised liquid drop model [8, 9, 10], the effective liquid drop model [11], the modified generalized liquid drop model [6, 12, 13], the coulomb and proximity potential model [14, 15, 16], the gamow-like model [1, 4], and the preformed cluster model [17, 18]. royer [19] developed an analytical formula to calculate the α-decay half-lives by applying a fitting procedure on a set of 373 nuclei. the universal decay law has also been developed by qi et al. [20, 21]. they introduced the new universal decay law (udl) to study α and cluster decay modes. the authors made use of the α-like r-matrix theory and the microscopic mecha250 w. a. yahya / j. nig. soc. phys. sci. 2 (2020) 250–256 251 nism of the charged-particle emission. a generalization of the viola-seaborg formula was given by ren et al. [22] for cluster radioactivity half-lives. modified versions of the ren formulas, called new ren a and new ren b, have also been presented [23]. the new ren a included nuclear isospin asymmetry while new ren b included both nuclear isospin asymmetry and angular momentum. gharaei and zanganeh [24] have studied the half-lives of cluster decay for some isotopes using temperature dependent proximity potential (with the prox. 2010 and prox. zheng potentials). they found that the width and height of the coulomb barrier in the temperature-dependent proximity potential are less than its temperature-independent version. recently, the modified coulomb and proximity potential model was employed to study the α-decay of 186−218po [25]. zdeb et al. [1] proposed a phenomenological model which is based on the gamow theory for the calculation of α-decay half-lives. in the gamow-like model, the square well potential is chosen as the nuclear potential, while the potential of a uniformly charged sphere is taken as the coulomb potential. in ref. [4], the authors presented the α decay half-lives of nuclei with z > 51 (up to z = 120) using a modified form of the gamow-like model. most of the isotopes of mercury (hg) are radioactive with half-lives less than a day. seven of the isotopes are stable. the hg radioisotope with the longest half-life is 194hg, with a halflife of 444 years. some of the applications of hg isotopes are: in medicine, in nuclear gyroscopes and magnetometres. the radioisotope 197hg, for example, is useful for diagnostics of kidney and cerebral diseases [26]. the α-decay half-lives of some astatine isotopes have been studied in ref. [27] using a modified coulomb and proximity potential model with one adjustable parameter. the α-decay half-lives of some mercury isotopes have also been studied in ref. [28] using about 25 different versions of the nuclear potential. in this study, the gamow-like model, its modified version (the modified gamow-like model), the temperature-independent coulomb and proximity potential model (cppm) and the temperaturedependent coulomb and proximity potential model (cppmt) have been employed to calculate the α-decay half-lives of 171−189hg isotopes. the results are compared with the available experimental data. the article is organised as follows. the modified gamow-like model, the temperature-independent coulomb and proximity potential model (cppm) and the temperaturedependent coulomb and proximity potential model (cppmt) are described in section 2 for the calculation of α-decay halflives. the results are presented and discussed in section 3 while the conclusion is given in section 4. 2. theory 2.1. modified gamow-like model in the modified gamow-like model, the interaction potential between the alpha particle and daughter nucleus is given by [4]: v (r) = { −v0, 0 ≤ r ≤ r vh (r) + v`(r), r ≥ r (1) where v0 is the depth of the square well, the hulthen type of screened electrostatic coulomb potential vh = az1z2e2 ear − 1 (2) the centrifugal potential v`(r) = ( ` + 12 )2 ~2 2µr2 (3) z1 and z2 are the proton numbers of the α particle and daughter nucleus, respectively, a is the screening parameter, and ` is the orbital angular momentum that the α particle takes away. the radius of the spherical square well is computed by adding the radii of both the α particle (a1) and the daughter nucleus (a2): r = r0 ( a 1 3 1 + a 1 3 2 ) (4) where r0 is a constant, an adjustable parameter. the α decay half-life is calculated using [1, 4]: t 1 2 = ln 2 λ 10h. (5) here, h is the decay hindrance factor due to the effect of an odd-neutron and/or an odd-proton. the value of h is zero for even-even nuclei. in ref. [4], the values of a, r0, and h were determined to be: a = 7.8 × 10−4, r0 = 1.14 fm, h = 0.3455. (6) for odd-odd nuclei, hnp = 2h. the decay constant λ is obtained via: λ = νp (7) where the penetration probability p is given by p = exp [ − 2 ~ ∫ b r √ 2µ (v (r) − ek) dr ] . (8) here, µ = ma1 a2/(a1 + a2) is the reduced mass of the daughter nucleus and the α particle, m is the nucleon mass, ek = qα(a − 4)/a is the kinetic energy of the emitted α particle. the classical turning point b is obtained through the condition v (b) = ek. in this model, the frequency of assault on the potential barrier is evaluated using ν = ( g + 32 ) ~ 1.2πµr20 , (9) 251 w. a. yahya / j. nig. soc. phys. sci. 2 (2020) 250–256 252 where the parent nucleus radius r0 is obtained via r0 = 1.28a 1/3 − 0.76 + 0.8a−1/3. (10) the main quantum number g is obtained using g =  22 n > 126 20 82 < n ≤ 126 18 n ≤ 82 . (11) 2.2. the coulomb and proximity potential model (cppm) the total interaction potential between the emitted α particle and the daughter nucleus in the cppm contains (for both the touching configuration and for separated fragments) the nuclear, the coulomb and the centrifugal terms [29]: vt (r) = vprox(z) + vc (r) + ~`(` + 1) 2µr2 (12) where µ is the reduced mass of the interaction system and ` is the angular momentum. the coulomb potential vc (r) is defined as: vc (r) = z1z2e 2  1 r for r ≥ rc 1 2rc [ 3 − ( r rc )2] for r ≤ rc . (13) here, z1 and z2 are, respectively, the charge numbers of the daughter and emitted nuclei. the radial distance rc = 1.24 (r1 + r2). the term vprox(z) represents the proximity potential and z = r − c1 − c2 is the distance between the near surfaces of the fragments, where r is the distance between the fragment centres [16]. the presence of the proximity potential causes a reduction in the height of the potential barrier. the proximity potential has been used in the preformed cluster model by malik et al. [30]. the proximity potential vprox can be obtained by calculating the strength of the nuclear interactions between the daughter and emitted α particle: vprox(z) = 4πbγr̄φ ( z b ) mev (14) where the term bγr̄ depends on the geometry and shape of the two nuclei, and the mean curvature radius r̄ is given as r̄ = c1c2 c1 + c2 . (15) the süsmann central radii of fragments c1 and c2 are computed using ci = ri 1 − ( b ri )2 + · · ·  (16) where the diffuseness of nuclear surface b ≈ 1 fm and ri are given by ri = 1.28a 1/3 i − 0.76 + 0.8a −1/3 i fm (i = 1, 2). (17) the universal function φ ( � = zb ) is given in the form [24]: φ(�) = { − 1 2 (� − 2.54) 2 − 0.0852 (� − 2.54)3 � ≤ 1.2511 −3.437 exp (−�/0.75) � ≥ 1.2511 (18) the nuclear surface energy coefficient γ is defined as γ = 1.460734 [ 1 − 4 ( n − z n + z )2] mev/fm2 (19) where n and z denote the neutron and proton numbers of the parent nucleus, respectively. the prox. 2010 potential has been used in this work. according to the wkb approximation [24, 31, 32], the penetration probability p of the emitted α nucleus through the potential barrier is calculated using: p = exp [ − 2 ~ ∫ rout rin √ 2µ [v (r) − q] dr ] (20) where the classical turning points rin and rout are determined from v (rin) = v (rout) = q (21) and the reduced mass µ is calculated using µ = ma1 a2/a, where m is the nucleon mass, a1 and a2 denote the mass numbers of the emitted and daughter nuclei, respectively, and a is the parent nucleus mass number. the α-decay half-life is then computed using t1/2 = ln 2 νp (22) where the assault frequency ν has been taken to be 1020 s−1. 2.2.1. temperature-dependent proximity potential the thermal effects are studied by using the temperature dependent forms of the parameters r, γ and b. they are given by [24]: ri(t ) = ri(t = 0) [ 1 + 0.0005t 2 ] fm (i = 1, 2) (23) γ(t ) = γ(t = 0) [ 1 − t − tb tb ]3/2 (24) b(t ) = b(t = 0) [ 1 + 0.009t 2 ] (25) where tb is the temperature associated with near coulomb barrier energies and b(t = 0) = 1. in this work, we have adopted an alternative form of the temperature dependent surface energy coefficient in the form γ(t ) = γ(0) (1 − 0.07t )2 [33]. the temperature t (in mev) can be obtained from e∗ = ekin + qin = 1 9 at 2 − t (26) where e∗ is the excitation energy of the parent nucleus and a is its mass number, and qin denotes the entrance channel q-value of the system. the kinetic energy of the emitted α particle ekin is obtained from ekin = ( ad/ap ) q. (27) 252 w. a. yahya / j. nig. soc. phys. sci. 2 (2020) 250–256 253 3. results and discussions the α-decay half-lives of the mercury isotopes (z = 80) within the mass range 171 ≤ a ≤ 189 have been calculated using the gamow-like model (glm), modified gamow-like model (mglm1) using the variable parameters of ref. [4], the temperatureindependent coulomb and proximity potential model (cppm), and the temperature-dependent coulomb and proximity potential model (cppmt). the proximity 2010 potential has been employed in the cppm and cppmt calculations. we have also calculated the α-decay half-lives of the isotopes using the modified gamow-like model (mglm2) with new parameter values. the new values for the parameters a, r0, and h in the modified gamow-like model (mglm2) were obtained through least squares fitting procedure. the values of the three adjustable parameters obtained for the hg isotopes are r0 = 1.2457fm, h = 0.1891, a = −2.159 × 10−3 (28) the database have been taken from the nubase2016 [34, 35, 36]. the reaction qα value has been computed via [29]: qα = ∆mp − ∆md − ∆mα + k ( zεp − z ε d ) (29) where ∆mp, ∆mα, and ∆md denote the mass excesses of the parent nucleus, the alpha particle, and the daughter nucleus, respectively. the term k ( zεp − z ε d ) denotes the screening effect of atomic electrons [37]; k = 8.7 ev , ε = 2.517 for z ≥ 60, and k = 13.6 ev , ε = 2.408 for z < 60 [38]. the angular momentum ` are obtained from the selection rule given by [23, 39, 40]: ` =  δ j for even δ j and πd = πp δ j + 1 for odd δ j and πd = πp δ j for odd δ j and πd , πp δ j + 1 for even δ j and πd , πp . (30) here δ j = ∣∣∣ jp − jd∣∣∣, where jd , πd , jp, πp are the spin and parity values of the daughter and parent nuclei, respectively. the alpha-decay half-lives computed for the 19 hg isotopes( 171−189hg ) are shown in table 1. the first four columns show, respectively, the mass number (a), the calculated qα values, the calculated temperature and the experimental α-decay half-lives (expt.) ( log [ t1/2(s) ]) . the last five columns of the table show the computed α-decay half-lives using the glm, mglm1, mglm2, cppm, and cppmt, respectively. all the models give reasonable values of the half-lives when compared to the available experimental results. the root mean square standard deviation σ has been evaluated using the formula: σ = √√√ 1 n n∑ i=1 [( log10 t theory 1/2,i − log10 t expt 1/2,i )2] (31) where t theory1/2,i are the half-lives obtained using the five models and t expt1/2,i are the experimental half-lives. the standard deviation have been calculated in order to compare the agreement between the experimental half-lives and theoretically calculated half-lives using the various models. the computed standard deviations (σ) using the different models are shown in table 2. the second column of the table shows the calculated standard deviations for the hg isotopes. from the table, the mglm2 has the least standard deviation with a value of 0.5013 followed by the glm with a standard deviation value of 0.5885, less than the standard deviation of cppm. the mglm1, cppm, and cppmt have respective standard deviation values of 0.6130, 0.6866, and 0.5949. one observes that the cppmt has a lower standard deviation value compared with the cppm. this shows the importance of using temperature-dependent form of the proximity potential model. all the models give standard deviation values less than 0.7. however, among the five models, the glm and mglm2 seem to be the most suitable for the determination of the α-decay half-lives of the hg isotopes. it should be noted these hg isotopes were studied in ref. [28] using about various proximity potential models. however, the authors considered experimental values of 171−177hg only. this makes it difficult to compare the standard deviations. 90 95 100 105 110 -10 -5 0 5 10 15 neutron number (n) lo g [ t 1 /2 (s )] 171-189 hg expt glm mglm1 mglm2 cppm cppmt figure 1. comparison of the calculated α-decay half-lives of hg isotopes between the various models and experiment. 90 95 100 105 110 4.5 5 5.5 6 6.5 7 7.5 8 neutron number (n) q ( m e v ) 171-189 hg figure 2. plot of the calculated qα values against neutron number (n) for the hg isotopes. the calculated half-lives log [ t1/2(s) ] for the hg isotopes using the five models have been plotted against the neutron 253 w. a. yahya / j. nig. soc. phys. sci. 2 (2020) 250–256 254 table 1. calculated α-decay half-lives, log [ t1/2(s) ] , of hg (z = 80) using glm, mglm1, mglm2, cppm, cppmt. log [ t1/2(s) ] a qα(mev ) t (mev) expt. glm mglm1 mglm2 cppm cppmt 171 7.7011 0.9218 -4.1549 -3.4908 -3.3518 -3.7442 -3.9665 -3.7652 172 7.5571 0.9107 -3.6364 -3.3033 -3.5380 -3.7354 -3.8178 -3.6243 173 7.4111 0.8994 -3.0969 -2.6643 -2.5327 -2.8705 -3.1276 -2.9279 174 7.2661 0.8882 -2.6990 -2.4462 -2.6902 -2.8297 -2.9492 -2.7573 175 7.1051 0.8761 -1.9957 -1.7283 -1.8488 -2.1095 -2.4407 -2.2496 176 6.9301 0.8630 -1.6467 -1.3748 -1.6321 -1.6953 -1.8640 -1.6738 177 6.7686 0.8508 -0.8246 -0.6132 -0.5042 -0.6952 -1.0459 -0.8497 178 6.6105 0.8387 -0.5237 -0.2744 -0.5467 -0.5267 -0.7492 -0.5606 179 6.3935 0.8229 0.1461 0.7478 0.5927 0.5200 0.0658 0.2534 180 6.2916 0.8142 0.7321 0.9145 0.6241 0.7394 0.4550 0.6418 181 6.3175 0.8135 1.1249 1.0050 1.0923 1.0300 0.6013 0.7944 182 6.0288 0.7930 1.8947 1.9638 1.6562 1.8607 1.5193 1.7046 183 6.0717 0.7935 1.9049 1.9747 1.8009 1.8322 1.3184 1.5035 184 5.6951 0.7672 3.4442 3.4213 3.0873 3.4225 2.9941 3.1775 185 5.8061 0.7723 2.9129 3.1016 2.9081 3.0392 2.4621 2.6457 186 5.2376 0.7327 5.7140 5.6765 5.2962 5.8501 5.2691 5.4498 187 5.2628 0.7324 7.9777 5.7360 5.7394 6.1107 5.3940 5.5811 188 4.7405 0.6945 8.7218 8.5148 8.0682 8.9273 8.1268 8.3040 189 4.6699 0.6876 9.1818 9.1566 9.3338 10.0638 9.1191 9.3085 table 2. calculated root means square standard deviation (σ) using the different models. model σ ( hg ) glm 0.5885 mglm1 0.6130 mglm2 0.5013 cppm 0.6866 cppmt 0.5949 90 95 100 105 110 0.65 0.7 0.75 0.8 0.85 0.9 0.95 neutron number (n) t ( m e v ) 171-189 hg figure 3. plot of the calculated temperature (t in mev) against neutron number (n) for the hg isotopes. 90 95 100 105 110 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 neutron number (n) ∆ t 1 /2 glm mglm1 mglm2 cppm cppmt figure 4. plot of the ∆t against neutron number (n) for the hg isotopes using the different models. number in figure 1. from the figure, the half-lives can be seen to increase with increase in neutron number for the hg isotopes. aside n = 107 (corresponding to a = 187), the models give very good descriptions of the half-lives. it should be noted that near double magicity of the parent nucleus 202hg (z = 80, n = 122) was suggested in ref. [28]. figure 2 shows plots of the qα values against neutron number for the hg isotopes while the calculated temperature of the hg plotted against the neutron number in figure 3. the temperature values decrease with increase in neutron number which is equivalent to increase in the half-lives. that is, the calculated temperature is inversely proportional to the α-decay half-lives. 254 w. a. yahya / j. nig. soc. phys. sci. 2 (2020) 250–256 255 the difference between the theoretical and experimental αdecay half-lives have been computed using ∆t1/2 = log10 [ t theor1/2 /t expt 1/2 ] . (32) this factor ( ∆t1/2 ) has been plotted for all the models used in this work in figure 4. it can be observed that most of the points are near zero and within ±0.5. the figure shows that the mglm2 model gives the lowest ∣∣∣∆t1/2∣∣∣ values while the cppm model gives the highest ∣∣∣∆t1/2∣∣∣ values. this agrees with the results shown in table 2. 4. conclusion in this work, the α-decay half-lives of hg isotopes in the mass range 171 ≤ a ≤ 189 have been studied using the gamowlike model (glm), modified gamow-like model (mglm1), the temperature-independent coulomb and proximity potential model (cppm), and the temperature-dependent coulomb and proximity potential model (cppmt). the prox. 2010 proximity potential has been employed for the cppm and cppmt calculations. for the set of isotopes, new parameter values were obtained through a least squares scheme using the modified gamow-like model (termed mglm2). it is shown that the modified gamow-like model (mglm2) with new local parameter values and the glm are the most suitable methods (among the methods considered here) for calculating the α-decay halflives for the hg isotopes. in general, all the models give αdecay half-lives that are in good agreement with the experimental data with the maximum standard deviation values less than 0.7. references [1] a. zdeb, m. warda & k. pomorski, “half-lives for α and cluster radioactivity within a gamow-like model”, phys. rev. c 87 (2013) 024308. 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[40] v. denisov, o. davidovskaya & i. sedykh, “improved parametrization of the unified model for α decay and α capture”, phys. rev. c 92 (2015) 014602. 256 j. nig. soc. phys. sci. 1 (2019) 57–61 journal of the nigerian society of physical sciences original research numerical simulation of sandwiched perovskite-based solar cell using solar cell capacitance simulator (scaps-1d) i. t. belloa,∗, y. a. odedunmoyeb, o. adedokuna,∗∗, h. a. shittua, a. o. awodugbaa adepartment of pure and applied physics, ladoke akintola university of technology, ogbomoso, nigeria. bdepartment of mathematical and physical sciences, osun state university, osogbo, nigeria. abstract due to the superb characteristics of its light-harvesting, the perovskite sensitizer abx3 (a = ch3nh3, b = pb, sn, and x = cl, br, i) has recently attracted great attention. perovskite is composed of inexpensive and earth abundant materials. it is processable at low temperature preferably via the printing techniques. in addition, the charges in the bulk material after light absorption that enhances low loss in energy charge generation and collection were generated freely. in this research work, solar cell capacitance simulator (scaps-1d) was used to harnessing the real device hybrid perovskite (psc) solar cell with material parameters obtained from literatures and experiment used in the definition panel and the arrangement of an hybrid (fto/zno/czts/pscs/czts/htm) model in the scaps-1d simulator. from the simulated results obtained the band gap diagram and other curves were constructed. the efficiency greater than twenty percent (> 20%) was achieved, which shows that having a combination of two different absorber were achievable and calling for great attention from the researchers. keywords: sandwiched, perovskite, efficiency, band gap, harnessing article history : received: 09 april 2019 received in revised form: 22 may 2019 accepted for publication: 27 may 2019 published: 19 june 2019 c©2019 journal of the nigerian society of physical sciences. all rights reserved. communicated by: w. a. yahya 1. introduction the perovskite solar cells (pscs) originally came out as the result of unrelenting efforts on dssc researches. the perovskite solar cells have become a rapidly growing area of the photovoltaic world and of huge desire to the scientific community with its improvement. perovskite solar cells have attracted salient attention of the academic community since the first reported article in 2012 [1]. graphene was introduced into perovskite solar cell and an efficiency of 15.6% was obtained [2]. in 2015, 20.1% efficiency was recorded when the poly∗corresponding author tel. no: +2348062814778 ∗∗corresponding author tel. no: +2347031195750, +2348065976299 email addresses: itbello13@pgschool.lautech.edu.ng (i. t. bello ), oadedokun@lautech.edu.ng (o. adedokun) triarylamine (ptaa) was used as a new htm with another perovskite material, formamidinium iodide (hc(nh2)2pbi3) [3]. there is also a vast potency for better engineering work and effective solar cells which are anticipated to reach excess power conversion efficiency (pce) of over 20 per cent. perovskite solar cells have increased in pce at an unbelievably great rate in comparison with other solar cells. currently, the significant negative aspect of perovskite based solar cells was not known. although the life-times of the cells are not yet proved since there is no evidence to suggest that their life-time is any higher or less than that of pure organic devices. the use of lead in perovskite compound is not ideal since there is potential for a lead alternative to be used in perovskite compound instead, lead can be used in a much smaller amount than that of what is currently present in either lead or cadmium based 57 i. t. bello et al. / j. nig. soc. phys. sci. 1 (2019) 57–61 58 batteries. finally, the optical density of the perovskite materials is yet to be fully discussed, although its optical density was still lower than other active materials but higher than that of silicon. as a result, the light-harvesting perovskite devices require thicker layers which may cause some limitations in the fabrication of a solution processed devices whereby achieving high uniformity with such thick layers will be difficult. improvement of the precursor materials for solution based perovskite deposition and associated coating and processing techniques will be a key development for any solution processed devices will ultimately yield lower production costs. although at present the best perovskite solar cells are vacuum deposited. while vacuum based processes are relatively easy to scale up, the capital equipment cost of doing so can rapidly become astronomical. to achieve a truly low cost-per-watt devices, perovskite solar cells will require to have the much heralded trio of high efficiency, long life-times and low manufacturing costs. perovskite based devices have so far demonstrated enormous potential for achieving this but have not yet been achieved for other thin film technologies [4]. there are many simulation software models used to simulate solar cells devices numerically, such as large-scale atomic/molecular massively parallel simulator (lammps), silvaco atlas, solar cell capacitance simulator (scaps) etc. in this research work, scaps software will be used to simulate a perovskite based solar cell. scaps (solar cell capacitance simulator) is a one-dimensional simulation program with seven semiconductor input layers developed by a group of photovoltaic researcher at the department of electronics and information system, university of gent, belgium [5]. 2. device structure the cell model used in the simulation is n-fto/n-zno/pczts/p-pscs/p-czts/htm. this cell structure consists of fluorine doped tin oxide (sn2o:f), as the window layer, namely, a conductive n-type zno, perovskite (ch3nh3pb3-xclx) and a cu2znsns4 (czts) which are p-type semiconductors. figure 1. shows the solar cells layers structure. the cell illuminated through the transparent conductive oxide (tco), which serves as a window layer, passes across the electron transport layer (ntype zno) which serves as a buffer layer and enters the absorber layer to the hole transport material. 3. methodology and simulations numerical simulation technique of solar cells devices has over the years proved to be a viable tool for studying and understanding the properties of solar cell devices such as the optical, electrical and mechanical properties of complex solar cell devices [5]. it also helps to reduce processing cost and time spent on solar cell device fabrication by providing useful information on how to vary the production parameters to improve the device performance [7][8]. scaps-1d simulator based its simulations on the solutions of the three basics semiconductor equations; poissons equation, continuity equation of electron and continuity equation of hole. scaps-1d software solves these three figure 1: model of sandwiched simulation structure [6]. coupled partial differential equations numerically for the electrostatic potentials electron and hole concentration as a function of positions x. poisson’s equation is given as ∂ ∂x ( � ∂ψ ∂x ) = −q �0 [ p − n + n+d − n − a + ρde f (n, p) q ] (1) where ψ is electrostatic potential, � is dielectric constant and q is an electronic charge. the first two terms in the right are free charge carriers per volume, third and fourth are ionized donor and acceptor-like dopants i.e, localized states and ρde f is defect charge density. thus, the conservation of free electrons and free holes in the device is expressed as continuity equations ∂n ∂t = − ∂jn ∂x + g − un(n, p) (2a) ∂p ∂t = − ∂jp ∂x + g − u p(n, p) (2b) where p,n – free carrier concentrations, nd,a charged dopants, ρde f (n, p)– defect distributions, jn, j−p the electron and hole current densities, un,p− the net recombination rates, gthe generation rate. scaps-1d was used in this work to harnessing the real device hybrid perovskite (psc) solar cell with material parameters defined in table 1.0 which were used in the definition panel of the scaps-1d simulator. the absorption coefficients of the materials used were determined by the simulator based on the input parameters (table 1.0) and the arrangement of the model as allowed by the scaps-1d simulator. from table 1.0, shown above, absorber layers were varied while the other parameters are kept constant. various efficiencies were generated based on the thickness variation of the absorber. all simulations in this work were performed under ambient temperature (300k). the electrical parameters (voc , js c , f f ) and efficiency generated by scaps-1d would then be used to determine the optimum thickness of the absorber layer. from, this, the j-v, c-v, c-f and q.e of the best solar cells from the simulation will be determined and the effect of sandwich in the solar cell. 58 i. t. bello et al. / j. nig. soc. phys. sci. 1 (2019) 57–61 59 table 1: materials parameter used in simulation [7][8][9]. parameters fto zno psc czts thickness (µm) 0.5 0.05 varied varied band gap (ev) 3.5 3.35 1.55 1.55 electron affinity (ev) 4.0 4.21 3.9 4.5 dielectric constant 9 9 6.5 10 conduction band-dos nc(cm−3) 2.2.1018 2.2.1018 2.2.1018 2.2.1018 valence band-dos nv(cm−3) 1.8.1019 1.8.1019 1.8.1019 1.8.1019 electron thermal velocity (cm/s) 1.0.107 1.0.107 3.0.107 1.0.107 hole thermal velocity (cm/s) 1.0.107 1.0.107 3.0.107 1.0.107 electron mobility cm−3 · v−1 · s−1 2.0 25 1.6 100 hole mobility cm−3 · v−1 · s−1 1.0 100 0.2 20 donor density nd(cm−3) 2.0.1019 1.0.1018 0 0 acceptor density na(cm−3) 0 0 6.1018 8.22.1018 figure 2: (a) the band diagram of perovskite. (b) the band diagram of sandwiched perovskite 4. results and discussion 4.1. the bandgap diagram figure 2, show the band diagram of the sandwiched perovskite device. the band gap line up model of the simulated device of an hybrid fto/zno/czts/pscs/czts/htm was constructed from the data obtained from the scaps under the ambient temperature (300k). the band diagram of perovskite depends on the compositional variation of the component entails in the processing and synthesis of the absorber materials such as organic, metal and anion composition of the material. the band gap of the absorbing material is a crucial parameter for photovoltaic actions, as the absorber layer is the key material in any solar cell devices [10]. thus the band alignment is the type ii broken band gap with a band gap of approximately 1.55ev which is concurrent with the theoretical condition as reported by [11]. however, it was shown from figures 2a and 2b above, that the band alignment of perovskite solar cells shows single junction in the band gap while that of sandwiched perovskite band gap shows three junctions which confirmed the presence of a sandwiching materials embedded within the absorber layer. 4.2. j-v curve characteristic of simulated device j-v curves are the parameters used to determine the electrical output power of any solar cells. the j-v curve characteristic was obtained with the simulation of data in the table 1.0 was shown in figure 3 with open circuit voltage (voc) = 0.80v, short circuit current(jsc) = 25.12ma/cm2, fill factor (ff) = 49.99% and percentage conversion efficiency (pce) = 20.09% as the cell output parameters under the standard simulated sunlight of am1.5g and working conditions of ambient temperature and frequency of 106hz. 4.3. effect of variation in the sandwiched absorber layer thickness solar cell absorber layer plays an important roles in the fabrication and harnessing optimum values of the solar cell efficiencies. solar cell device performance depends solely on the electrical characteristic and variation of the absorber thickness. thereby this research simulation work tries to fix out the effect 59 i. t. bello et al. / j. nig. soc. phys. sci. 1 (2019) 57–61 60 figure 3: sandwich i-v curve characteristic of the combination of czts absorber layer and organometallic (perovskite) layer, embedded in one solar cell. it was observed that the short circuit current and percentage conversion efficiency of the sandwiched absorber solar cell were 25.12ma/cm2 and 20.09% which are higher than the simulated perovskite device output in this work. from figure 4, it was observed that figure 4: pscs and sandwiched pce against thickness at 200 nm the efficiency of perovskite is around 15.2% while that of sandwiched perovskite was greater than 16.15% efficiency. also, at 250 nm the efficiencies of 16.52% and 17.58% were of observed for perovskite and sandwiched perovskite respectively. at 300 nm, an efficiency of 17.51% was observed for perovskite solar cell while 18.57% was observed for sandwiched perovskite. however, 18.24% and 19.48% efficiencies were observed at 350 nm for both perovskite and sandwiched perovskite solar cells respectively. lastly, at 400 nm the efficiency of perovskite was observed to be 18.79% and that of sandwiched perovskite was around 20.09%. thus, the apprefigure 5: pscs and sandwiched qe against wavelength ciable increment in the efficiencies values of sandwiched perovskite has shown the positive effects of sandwiching absorber layer the solar cells. 4.4. quantum efficiency of the solar cell the quantum efficiency is the ratio of the number of carriers collected by the solar cell to the number of photons of a given energy incident on the solar cell. however, quantum efficiency is the fraction of the excited carriers that combine radiatively to the total recombination. figure 5 is the q.e plot against the wavelength which showed that more than 90% of the wavelength between 300 nm and 890 nm radiatively recombine and less than 10% of such wavelength recombined through other processes (auger and srh). the results implied that, at the 400nm thickness, sandwiched layer absorbs almost all the incident photons to create the electron-hole pairs and the photogenerated carriers are almost separated and transported to the hole transport materials and electron transport material by the built-in field with minimum recombination. therefore it can be considered that higher thickness is the optimal length of photovoltaic action. the quantum efficiency may be given either as a function of wavelength or as energy. figure 5, showed that sandwiched layer can absorb incident photons up to 800nm, which implied that sandwiched absorber layer can perform better than perovskite layer which can only absorb photons around 750nm because of the higher the wavelength the lower photon energy. 5. conclusion in conclusion, perovskite and sandwiched perovskite-based solar cell has been successfully simulated using one-dimensional solar cell capacitance simulator (scaps-1d). the output results of the simulation were recorded and plotted across the thickness variation of the absorber layers which varies from 60 i. t. bello et al. / j. nig. soc. phys. sci. 1 (2019) 57–61 61 200nm to 400nm. it was found out that the higher the absorber thickness the higher the efficiencies and other electrical parameters output in the solar cell. the efficiencies of 18.79% and 20.09% were recorded for the perovskite and sandwiched perovskite-based solar cells respectively. acknowledgments the authors would like to appreciate prof. marc. burgelman and his co-researchers at the university of gents, belgium for making scaps-1d available for use. one of the authors also thanks the tetfund for grant used in this work and others. the authors also thank the referees for the positive enlightening comments and suggestions, which have greatly helped them in making improvements to this paper. references [1] m. m. lee, j. teuscher, t. miyasaka, t. n. murakami & h. j. snaith “efficient hybrid solar cells based on meso-superstructured organometal halide perovskites”, science 338 (2012) 643. [2] j. t. w. wang, j. m. ball, e. m., barea, a. abate, j. a. alexanderwebber, j. huang, m. saliba, i. mora-sero, j. bisquert & h. j. snaith “low-temperature processed electron collection layers of graphene/tio2 nanocomposites in thin film pscs”, nano letter 14 (2014) 724. [3] w. s. yang, j. h. noh, n. j. jeon, y. c. kim, s. ryu, j. seo & s. i. seok, “high-performance photovoltaic perovskite layers fabricated through intramolecular exchange”, science 348 (2015) 1234. [4] http://www.ossila.com/pages/perovskites-and-perovskite-solar-cells-anintroduction (accessed december, 2017). [5] a. niemegeers, m. burgelman, k. decock, j. verschraegen & s. degrave, “scaps manual”, university of gent, 2014. [6] i. t. bello, m. k. awodele, o. adedokun, o. akinrinola & a. o. awodugba, “modelling and simulation of czts-perovskite sandwiched tandem solar cell”, turkish journal of physics 42 (2018) 321. [7] m. takashi & m. masashi, “device modelling of perovskite solar cells based on structural similarity with thin film inorganic semiconductor solar cells”, journal of applied physics 116 (2014) 054505 [8] m. takashi & m. masashi, “theoretical analysis on effect of band offsets in perovskite solar cells”, solar energy materials & solar cells 133 (2015) 8. [9] m. gloeckler, a. l. fahrenbruch & j. r. sites, “numerical modeling of cigs and cdte solar cells: setting the baseline”, in proc. 3rd world conf. photovoltaic energy conversion, 2003 pp 491. [10] s.j. fonash “solar cell device physics”, 2nd edition elsevier, usa; 2010. [11] w. shockley & h. j. queisser “detailed balance limit of efficiency of pn junction solar cells”, journal of applied physics 32 (1961) 510. 61 j. nig. soc. phys. sci. 2 (2020) 218–227 journal of the nigerian society of physical sciences original research numerical algorithms for direct solution of fourth order ordinary differential equations j. o. kuboye, o. r. elusakin∗, o. f. quadri department of mathematics, federal university oye-ekiti, oye-ekiti, nigeria abstract this paper examines the derivation of hybrid numerical algorithms with step length(k) of five for solving fourth order initial value problems of ordinary differential equations directly. in developing the methods, interpolation and collocation techniques are considered. approximated power series is used as interpolating polynomial and its fourth derivative as the collocating equation. these equations are solved using gaussianelimination approach in finding the unknown variables a j, j=0,...,10 which are substituted into basis function to give continuous implicit scheme. the discrete schemes and its derivatives that form the block are obtained by evaluating continuous implicit scheme at non-interpolating points. the developed methods are of order seven and the results generated when the methods were applied to fourth order initial value problems compared favourably with existing methods. doi:10.46481/jnsps.2020.100 keywords: interpolation, collocation, block methods, fourth order, ordinary differential equations article history : received: 07 may 2020 received in revised form: 10 august 2020 accepted for publication: 15 august 2020 published: 01 november 2020 c©2020 journal of the nigerian society of physical sciences. all rights reserved. communicated by: f. y. eguda 1. introduction the general fourth order initial value problem of ordinary differential equations of the form yiv = f (x, y(x), y′(x), y′′(x), y′′′(x)), y(x0) = y1, y ′(x0) = y2, y ′′(x0) = y3 (1) is considered in this article. in the past, solving fourth order ordinary differential equations (odes) requires reducing ∗corresponding author tel. no: +23480xxxx572 email address: opeyemielusakin21@gmail.com (o. r. elusakin ) the differentials to systems of first order odes and approximate numerical method for the first order would be used to solve the system. this approach is been attached with lots of setbacks which include: computational burden, lots of human effort, complexity in developing computer code which affects the accuracy of the method in terms of error. this was extensively discussed by researchers like awoyemi [1], fatunla [2] and lambert [3]. due to several disadvantages found in reduction method, the direct method of solving odes of higher order was developed by lots of scholars which include akeremale et al. [4], abolarin et al. [5], kuboye et. al [6], omar & kuboye [7], adeyefa [8], abdullahi et al. [9], adeniyi & mohammed [10], olabode [11], adesanya et al.[12], omar & suleiman [13] 218 kuboye et al. / j. nig. soc. phys. sci. 2 (2020) 218–227 219 and familua & omole [14]. specifically, numerical methods for solving equation (1) were proposed by omar and kuboye[15], areo and omole[16] and mohammed[17]. these current methods solved directly equation (1) but its accuracy in terms of error can still be improved. therefore, this paper examines the derivation and implementation of the efficient numerical algorithm for solving fourth order ordinary differential equations directly and it focuses on improving the accuracy of the existing methods. 2. methodology this section considers derivation of block methods for direct solution of fourth order odes. 2.1. derivation of first block method(fbm) power series approximate solution of the form y(x) = k+5∑ j=0 a j x j (2) is used as interpolating polynomial where k=5.the fourth derivative of equation(2) gives: yiv(x) = k+5∑ j=4 j( j − 1)( j − 2)( j − 3)a j x j−4 (3) equation (2) is interpolated at x = xn+i, i = 0(1)2 and 5 2 and equation (3) is collocated at x = xn+i, i = 0(1)5 and 5 2 . interpolation and collocation equations are combined together to give a non-linear system of equations of the form: k+5∑ j=0 a j x j n+i = yn+i k+5∑ j=4 j( j − 1)( j − 2)( j − 3)a j x j−4 n+i = fn+i (4) the unknown variables a′j s in (4) are gotten with the use of gaussian elimination approach which are substituted into equation (2) and this yields a continuous implicit scheme of the form k−3∑ j=0 α j(t)yn+ j + α 5 2 yn+ 52 = h 4 k∑ j=0 β j(t) fn+ j + h 4λ 5 2 fn+ 52 (5) where t = x−xn+k−1h  α0(t) α1(t) α2(t) α 5 2 (t)  =  −9 5 −27 10 −13 10 −1 5 8 34 3 5 2 3 −18 45 2 17 2 1 64 5 208 15 24 5 8 15   t0 t1 t2 t3  (6) 219 kuboye et al. / j. nig. soc. phys. sci. 2 (2020) 218–227 220  β0(t) β1(t) β2(t) β 5 2 (t) β3(t) β4(t) β5(t)  = t  t0 t1 t2 t3 t5 t6 t7 t8 t9 t10  (7) where t =  297000 290304000 404694 290304000 75279 290304000 −102185 290304000 72576 290304000 36288 290304000 −2880 290304000 −7200 290304000 −2080 290304000 −192 290304000 3441528 11612160 5154498 11612160 2433933 11612160 323717 11612160 32256 11612160 15232 11612160 −1728 11612160 −3072 11612160 −800 11612160 −64 11612160 6797304 5806080 10732194 5806080 5745021 5806080 1207573 5806080 −145152 5806080 −6048 5806080 10944 5806080 12384 5806080 2720 5806080 192 5806080 560520 2268000 690174 2268000 419859 2268000 267195 2268000 −129024 2268000 −46592 2268000 11520 2268000 9600 2268000 1920 2268000 128 2268000 1716984 5806080 3760866 5806080 3494877 5806080 153384 5806080 −290304 5806080 −72576 5806080 27072 5806080 16704 5806080 3040 5806080 192 5806080 187272 11612160 129150 11612160 −353709 11612160 −650629 11612160 −169344 11612160 −3136 11612160 20160 11612160 7392 11612160 1120 11612160 64 11612160 428760 290304000 508266 290304000 −76719 290304000 −322779 290304000 290304 290304000 266112 290304000 118080 290304000 2880 290304000 3680 290304000 192 290304000  the coefficient of first and higher derivatives of (5) give α′0(t) α′1(t) α′2(t) α′5 2 (t)  =  27 10 26 10 −6 10 34 3 30 3 6 3 45 2 34 2 −6 2 208 15 144 15 124 15  1 h (8) 220 kuboye et al. / j. nig. soc. phys. sci. 2 (2020) 218–227 221  β′0 β′1 β′2 β′5 2 β′3(t) β′4(t) β′5(t)  = s  t0 t1 t2 t4 t5 t6 t7 t8 t9  (9) where s =  −134898 96768000 −50186 96768000 102185 96768000 −120960 96768000 −72576 96768000 6720 96768000 19200 96768000 6240 96768000 640 96768000 5154498 11612160 4867866 11612160 971151 11612160 161280 11612160 91392 11612160 −12096 11612160 −24576 11612160 −7200 11612160 −640 11612160 3577398 1935360 3830014 1935360 1207573 1935360 −241920 1935360 −120960 1935360 25536 1935360 33024 1935360 8160 1935360 640 1935360 690174 2268000 839718 2268000 801585 2268000 −645120 2268000 −2795520 2268000 80640 2268000 76800 2268000 17280 2268000 1280 2268000 1253622 1935360 2329918 1935360 1533845 1935360 −483840 1935360 −145152 1935360 63168 1935360 44544 1935360 9120 1935360 640 1935360 −129150 11612160 707418 11612160 1951887 11612160 846720 11612160 18816 11612160 −141120 11612160 −59136 11612160 −10080 11612160 −640 11612160 169422 96768000 −51146 96768000 −322775 96768000 4838405 96768000 532224 96768000 275520 96768000 76800 96768000 11040 96768000 640 96768000   α′′0 (t) α′′1 (t) α′′2 (t) α′′5 2 (t)  =  13 5h2 −6 5h2 10 h2 4 h2 17 h2 −6 h2 48 5h2 16 5h2   t0 t1  (10) 221 kuboye et al. / j. nig. soc. phys. sci. 2 (2020) 218–227 222  β′′0 (t) β′′1 (t) β′′2 (t) β′′5 2 (t) β′′3 (t) β′′4 (t) β′′5 (t)  = u  t0 t1 t3 t4 t5 t6 t7 t8  (11) where u =  −25093 48384000 102185 48384000 −241920 48384000 −181440 48384000 20160 48384000 67200 48384000 24960 48384000 2880 48384000 811311 1935360 323717 1935360 107520 1935360 76160 1935360 −12096 1935360 −28672 1935360 −9600 1935360 −960 1935360 1915007 967680 1207573 967680 −483840 967680 −302400 967680 76608 967680 11558 967680 32640 967680 2880 967680 139953 378000 267195 378000 −430080 378000 −232960 378000 80640 378000 89600 378000 23040 378000 1920 378000 1164959 967680 1533845 967680 −967680 967680 −362880 967680 18950 967680 155904 967680 36480 967680 2880 967680 117903 1935360 650629 1935360 564480 1935360 15680 1935360 −141120 1935360 −68992 1935360 −13440 1935360 −960 1935360 −25573 48384000 −322775 48384000 967680 48384000 1330560 48384000 826560 48384000 268800 48384000 44160 48384000 2880 48384000   α′′′0 (t) α′′′1 (t) α′′′2 (t) α′′′5 2 (t)  =  −6 5 4 −6 16 5  t0 (12)  β′′′0 (t) β′′′1 (t) β′′′2 (t) β′′′5 2 (t) β′′′3 (t) β′′′4 (t) β′′′5 (t)  = v  t0 t2 t3 t4 t5 t6 t7  (13) 222 kuboye et al. / j. nig. soc. phys. sci. 2 (2020) 218–227 223 where v =  20437 9676800 −14515 9676800 −145152 9676800 20160 9676800 80640 9676800 3494 9676800 4608 9676800 323717 967680 322560 967680 304640 967680 −60480 967680 −172032 967680 −67200 967680 −7680 967680 1207573 967680 −1451520 967680 −1209600 967680 383040 967680 69350 967680 228480 967680 23040 967680 −53439 75600 258048 75600 186368 75600 −80640 75600 −107520 75600 −32256 75600 −3072 75600 1533845 967680 −2903040 967680 −1451520 967680 947520 967680 935424 967680 255360 967680 23040 967680 650629 1935360 1935360 1693440 62720 1935360 −705600 1935360 −413952 1935360 −94080 1935360 −7680 1935360 −64555 9676800 580608 9676800 1064448 9676800 826560 9676800 322560 9676800 61824 9676800 4608 9676800  discrete schemes and its derivatives are derived by evaluating (5) as well as its derivatives at grid points and non-grid points which are used to form the block yn+1 yn+2 yn+ 52 yn+3 yn+4 yn+5  =  1 1 1 1 1 1  [ yn ] +  1 2 5 2 3 4 5  [ hy′n ] +  1 2 2 25 8 9 2 8 25 2  [ h2y′′n ] +  1 6 4 3 125 48 9 2 32 3 125 6  [ h3y′′′n ] + h4  0 0 0 0 0 579232268000 0 0 0 0 0 1967670875 0 0 0 0 0 1075412518579456 0 0 0 0 0 5850956000 0 0 0 0 0 18534470875 0 0 0 0 0 478759072   fn−4 fn−3 fn− 52 fn−2 fn−1 fn  + h4  413 10368 −8977 90720 9292 70875 −2279 36288 29 3780 −6593 9072000 428 567 −122 81 140288 70875 −2672 2835 4 35 764 70875 33640625 18579456 −30109375 9289728 89375 20736 −19109375 9289728 171875 688128 −437875 18579456 16137 4480 3267 560 6948 875 −243 64 207 448 −4887 112000 28928 2835 −7936 567 1441792 70875 −27136 2835 32 27 −7936 70875 1609375 72576 −484375 18144 23500 567 −671875 36288 15625 6048 −2375 10368   fn+1 fn+2 fn+ 52 fn+3 fn+4 fn+5  (14) 2.2. derivation of second block method(sbm) equation(2) is interpolated at x = xn+i, i = 0(1)2 and 9 4 and equation(3) is collocated at x = xn+i, i = 0(1)5 and 9 4 . the same steps used in deriving the first block method are also employed and this produces the block method 223 kuboye et al. / j. nig. soc. phys. sci. 2 (2020) 218–227 224  yn+1 yn+2 yn+ 94 yn+3 yn+4 yn+5  = [ yn ] +  1 2 9 4 3 4 5  [ hy′n ] +  1 2 2 81 32 9 2 8 25 2  [ h2y′′n ] +  1 6 4 3 243 128 9 2 32 3 125 6  [ h3y′′′n ] + h4  0 0 0 0 0 690432721600 0 0 0 0 0 1169642525 0 0 0 0 0 11919084392936012800 0 0 0 0 0 57935600 0 0 0 0 0 220168505 0 0 0 0 0 568625108864   fn−4 fn−3 fn−94 fn−2 fn−1 fn  + h4  76921 1814400 −1139 6480 594688 3274425 −6749 181440 7421 1270080 −11717 19958400 11248 14175 −7558 2835 8978432 3274425 −1576 2835 344 3969 −1352 155925 3705501897 2936012800 −169057287 41943040 229379121 55193600 −247763043 293601280 540947889 4110417920 −425211849 32296140800 84159 22400 −2349 224 148224 13475 −5031 2240 1377 3920 −8667 246400 150272 14175 −10496 405 8388608 297675 −15872 2835 17888 19845 −256 2835 1668125 72576 −115625 2268 7520000 130977 −378125 36288 509375 254016 −147625 798336   fn+1 fn+2 fn+ 94 fn+3 fn+4 fn+5  (15) 3. analysis of the method properties of the methods are examined in this section. 3.1. order of block methods in finding the order of the block methods,the method proposed by lambert[3] is employed whereby taylor series expansion are used in expanding the y and f-functions and by further comparing the coefficients of h,this gives the block methods to be of order [7, 7, 7, 7, 7, 7]t . 3.2. zero-stability a linear multi-step method is said to be zero-stable if the roots rs, s=1,2,..., n(grid and non grid points) of the first characteristics polynomial defined by p(r) = det(ra ′ − b ′ ) satisfy |rs| < 1 and the root |r| = 1 having multiplicity not exceeding one.(lambert [3]) where a ′ =  1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1  224 kuboye et al. / j. nig. soc. phys. sci. 2 (2020) 218–227 225 , b ′ =  0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1  therefore r=0,0,0,0,0,1. hence the zero-stability of first block method is confirmed which is also applied to the second block method. 3.3. convergence according to awoyemi[1], equation(5) converges if it is zero-stable and consistent.this implies that the developed methods converged. 4. test problems the following fourth order initial value problems[i.v.p] are solved in order to examine the accuracy of the methods problem 1: yiv = x, y(0) = 0, y ′ (0) = 1, y ′′ (0) = y ′′′ = 0, h = 0.1 e xactsolution : y(x) = x 5 120 + x source: mohammed[17] problem 2: yiv − y = 0, y(0) = 1, y ′ (0) = 0, y ′′ (0) = −2, y ′′′ (0) = 0, h = 1320 e xactsolution : y(x) = −14 e x − 1 4 e −x + 32 cos(x) source: areo and omole[16] problem 3: yiv = (y ′ )2 − yy ′′′ − 4x2 + ex(1 − 4x + x2), y(0) = 1, y ′ (0) = 1, y ′′ (0) = 3, y ′′′ (0) = 1, h = 0.132 e xactsolution : y(x) = x2 + ex source: familua and omole [14] the following acronyms are used in the tables below es exact solution cs computed solution fbm – first block method 225 kuboye et al. / j. nig. soc. phys. sci. 2 (2020) 218–227 226 sbm – second block method eim[17] error in mohammed[17] eiao[16]error in areo and omole[16] eiok [15]error in omar and kuboye[15] eifbm error in first block method eisbm error in second block method eifo [14] error in familua and omole [14] table 1. es and cs of fbm for problem 1 x es cs 0.1 0.100000083333333340 0.100000083333333340 0.2 0.200002666666666690 0.200002666666666690 0.3 0.300020250000000040 0.300020250000000040 0.4 0.400085333333333350 0.400085333333333400 0.5 0.500260416666666650 0.500260416666666760 0.6 0.600647999999999960 0.600648000000000070 0.7 0.701400583333333330 0.701400583333333550 0.8 0.802730666666666700 0.802730666666666700 0.9 0.904920750000000050 0.904920750000000160 1.0 1.008333333333333300 1.008333333333333500 table 2. es and cs of sbm for problem 1 x es cs 0.1 0.100000083333333340 0.100000083333333340 0.2 0.200002666666666690 0.200002666666666690 0.3 0.300020250000000040 0.300020250000000100 0.4 0.400085333333333350 0.400085333333333400 0.5 0.500260416666666650 0.500260416666666650 0.6 0.600647999999999960 0.600648000000000070 0.7 0.701400583333333330 0.701400583333333220 0.8 0.802730666666666700 0.802730666666666810 0.9 0.904920750000000050 0.904920750000000160 1.0 1.008333333333333300 1.008333333333333300 5. discussion of results in tables 1 and 2, exact and computed solutions of fbm and sbm for solving problem 1 are shown. table 3 reveals the efficiency of these block methods (eisbm and eisbm) as compared favourably with eim[17] and eiok[15]. furthermore, exact and computed solutions of the newly developed block table 3. comparison of eifbm and eisbm with eim[17] and eiok[15] for solving problem 1 x eifbm eisbm eim(2010) eiok(2016) 0.1 0.0000000e+00 0.0000000e+00 7.000024e-10 1.002087e-12 0.2 0.0000000e+00 0.0000000e+00 8.9999912-10 0.000000e+00 0.3 0.0000000e+00 5.5511151e-17 2.599993e-09 0.000000e+00 0.4 5.5511151e-17 5.5511151e-17 5.100033e-09 0.000000e+00 0.5 1.1102230e-16 0.0000000e+00 7.799979e-09 1.002087e-12 0.6 1.1102230e-16 1.1102230e-16 1.180009e-08 2.755907e-12 0.7 2.2204460e-16 1.1102230e-16 1.180009e-08 3.507306e-12 0.8 0.0000000e+00 1.1102230e-16 1.410006e-08 3.507306e-12 0.9 1.1102230e-16 1.1102230e-16 1.880000e-08 4.175549e-12 1.0 2.2204460e-16 0.0000000e+00 1.008335e-08 4.759970e-12 table 4. es and cs of fbm for problem 2 x es cs 0.0031250 1.000009765628973500 1.000009765628973700 0.0062500 1.000039062563578400 1.000039062563578400 0.0093750 1.000087890946866900 1.000087890946867100 0.0125000 1.000156251017263000 1.000156251017263500 0.0156250 1.000244143108567400 1.000244143108567400 0.0187500 1.000351567649961900 1.000351567649962400 0.0218750 1.000478525166021100 1.000478525166021300 0.0250000 1.000625016276719800 1.000625016276719600 0.0281250 1.000791041697446400 1.000791041697446800 0.0312500 1.000976602239017000 1.000976602239017000 table 5. es and cs of sbm for problem 2 x es cs 0.0031250 1.000009765628973500 1.000009765628973900 0.0062500 1.000039062563578400 1.000039062563578400 0.0093750 1.000087890946866900 1.000087890946866900 0.0125000 1.000156251017263000 1.000156251017263500 0.0156250 1.000244143108567400 1.000244143108567100 0.0187500 1.000351567649961900 1.000351567649962100 0.0218750 1.000478525166021100 1.000478525166021100 0.0250000 1.000625016276719800 1.000625016276719600 0.0281250 1.000791041697446400 1.000791041697445900 0.0312500 1.000976602239017000 1.000976602239017200 table 6. comparison of eifbm and eisbm with eiao[16] for solving problem 2 x eifbm eisbm eiao (2015) 0.0031250 2.2204460e-016 4.4408921e-016 4.440892e-16 0.0062500 0.0000000e+000 0.0000000e+000 2.176037e-14 0.0093750 2.2204460e-016 0.0000000e+000 .771916e-13 0.0125000 4.4408921e-016 4.4408921e-016 7.666090e-13 0.0156250 0.0000000e+000 2.2204460e-016 2.367773e-12 0.0187500 4.4408921e-016 2.2204460e-016 5.932477e-12 0.0218750 2.2204460e-016 0.0000000e+000 1.287681e-11 0.0250000 2.2204460e-016 2.2204460e-016 2.517841e-11 0.0281250 4.4408921e-016 4.4408921e-016 4.546752e-11 0.0312500 0.0000000e+000 2.2204460e-016 7.712331e-11 ’ table 7. es and cs of fbm for problem 3 x es cs 0.103125 1.119264744787591900 1.119264744969084200 0.206250 1.271599493198048500 1.271599504741302500 0.306250 1.452110907065013100 1.452111029006491400 0.406250 1.666216862500122800 1.666217515460942200 0.506250 1.915347109920916500 1.915349507140536000 0.603125 2.191581593606204900 2.191588302867649500 0.703125 2.514440293333696500 2.514456732090109900 0.803125 2.877516387746607200 2.877551937602963200 0.903125 3.282936158805099100 3.283006004031709900 1.003125 3.733049511495175400 3.733176679391747100 226 kuboye et al. / j. nig. soc. phys. sci. 2 (2020) 218–227 227 table 8. es and cs of sbm for problem 3 x es cs 0.103125 1.119264744787591900 1.119264744966372600 0.206250 1.271599493198048500 1.271599504536039800 0.306250 1.452110907065013100 1.452111026692671300 0.406250 1.666216862500122800 1.666217502634746300 0.506250 1.915347109920916500 1.915349459022614800 0.603125 2.191581593606204900 2.191588166231517800 0.703125 2.514440293333696500 2.514456393591375100 0.803125 2.877516387746607200 2.880551715825508700 0.903125 3.282936158805099100 3.283004543615038400 1.003125 3.733049511495175400 3.733174005515217600 table 9. comparison of eifbm and eisbm with eifo [14] for solving problem 3 x eifbm eisbm eifo[14] 0.103125 1.8149238e-010 1.7878077e-010 9.02145880e-10 0.206250 1.1543254e-008 1.1337991e-008 1.216821428e-09 0.306250 1.2194148e-007 1.1962766e-007 1.21681228e-09 0.406250 6.5296082e-007 6.4013462e-007 1.713796095e-09 0.506250 2.3972196e-006 2.3491017e-006 1.481970916e-08 0.603125 6.7092614e-006 6.5726253e-006 3.058338503e-08 0.703125 1.6438756e-005 1.6100258e-005 4.941858156e-08 0.803125 3.5549856e-005 3.5007632e-005 7.128679089e-08 0.903125 6.9845227e-005 6.8384810e-005 1.058773080e-07 1.003125 1.2716790e-004 1.2449402e-004 1.445520074e-07 methods for the solution of problems 2 and 3 are demonstrated in tables 4, 5, 7 and 8. these methods outperform method proposed by areo and omole [16] in terms of accuracy. in addition, the performance of these methods in solving problem 3 is not encouraging as the accuracy is lower when the comparison is made with eifo [14]. however, the capability of these methods in solving the nonlinear equation is established in table 9. finally, it is evident in tables 3, 6 and 9 that sbm is better than fbm in solving fourth order odes. 6. conclusion in this paper, new numerical algorithms for solving fourth order initial value problems of odes via multistep collocation approach were developed. the use of approximated power series as a basis function and its fourth derivatives as collocating equation were considered. the derived methods are efficient in the solution of fourth order odes as depicted in tables 3, 6 and 9. the accuracy of these numerical models is found better compared with some of the existing methods in terms of error. hence, fbm and sbm are viable numerical methods for solving fourth order initial value problems. acknowledgments we thank the referees and editor for the creative comments in making improvements to this paper. references [1] d. a. awoyemi, “class of continuous methods for general second order initial value problems in ordinary differential equations”, international journal of computer mathematics 72 (2009) 29. [2] s. o. fatunla, numerical methods for initial value problems in ordinary differential equations, academic press inc. harcourt brace jovanovich publishers, new york. (1988) [3] j. d. lambert, computational method in ordinary differential equation, john wiley and sons, inc. (1973) [4] o. c. akeremale, j. o. kuboye, s. h. yeak, e. a. abununyi & s. olaiju, ” hybrid-block numerical method for solving second order ordinary differential equations”, international journal of computational analysis 3 (2020) 25. [5] o. e. abolarin, e. o. adeyefa, j. o. kuboye & b. g. ogunware, “a novel multi derivative hybrid method for numerical treatment of higher order ordinary differential equations”, al dar research journal of sustainability 4 (2020) 43. [6] j. o. kuboye, z. omar, o. e. abolarin & r. abdelrahim, “generalized hybrid block method for solving second order ordinary differential equations directly”, research and reports on mathematics 2 (2018) 2. 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[15] j. o. kuboye & z. omar, “new zero-stable block method for direct solution of fourth order ordinary differential equation”, indian journal of science and technology 8 (2015) 1. [16] e. a. areo & e. o. omole, “half-step symmetric continuous hybrids block method for the numerical solutions of fourth order ordinary differential equations”, archives of applied science research 7 (2015) 39. [17] u. mohammed, “a six step block method for solution of fourth order ordinary differential equations”, the pacific journal of science and technology 11 (2010) 259. 227 j. nig. soc. phys. sci. 3 (2021) 116–120 journal of the nigerian society of physical sciences characterizations of galena as potential photosensitizer in a natural dye-sensitized solar cell akinsola samson ibukuna,b,∗, alabi aderemi babatundea, adedayo kayode seunc, nicola coppeded a department of physics, university of ilorin, ilorin, nigeria. b crown-hill university, eiyenkorin, kwara state, nigeria. c department of physics, university of maiduguri, nigeria. d institute of materials for electronics and magnetism, parma, italy. abstract dye is one of the principal parts for high power conversion efficiency in a dye-sensitized solar cell. conspicuous developments have taken place via the work of several researchers in engineering of novel dye structures so as to enhance the performance of the system. the properties of a natural mineral dye were studied in this work. the structure of the dye was determined and discovered to have contains constituents which could enhance better absorption of solar radiation for use in a dye-sensitized solar cell (dssc). the lead sulphide and iron content of the mineral dye studied as revealed by the x-ray diffraction analysis done suggest this. the x-ray fluorescence (xrf) done revealed that the concentration of lead and iron (fe) is high as compared to other elements present in the material, probably as a result of the fact that it is a geological sample (of the earth) and which may even suggest its colour and hence makes it absorbs solar radiation of visible region at its wavelength (around 380 nm – 800 nm). the functional groups present in the dye as obtained from the fourier transform infrared spectroscopy are the amine, carbonyl and the hydroxyl groups, all which confirms the suitability of the dye material in photosensitizing a semiconductor in a dssc. the absorption spectra of the dye within the visible region of electromagnetic radiation shows that the material has high, increased and stable absorption of visible light which is suggesting a more durable natural dye for a dssc than the easily degraded natural dyes of plants source. doi:10.46481/jnsps.2021.184 keywords: galena, mineral, dyes, photosensitizers. article history : received: 26 march 2021 received in revised form: 07 may 2021 accepted for publication: 08 may 2021 published: 29 may 2021 ©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: b. j. falaye ∗corresponding author tel. no: +2348166602544 email addresses: siakinsola711@gmail.com (akinsola samson ibukun ), remi050970@gmail.com (alabi aderemi babatunde), kphysicsq@gmail.com (adedayo kayode seun), nicola.coppede@gmail.com (nicola coppede) 1. introduction as at the end of 2017, roughly 1.8% of the globe electrical energy came from solar photovoltaics (pv ), which has a vital prospect to have a key role in all major future energy matters with an installed capacity of about 5 terawatts by 2050 [1]. dye-sensitized solar cell (dssc) has its genesis from the 116 dawodu / j. nig. soc. phys. sci. 3 (2021) 116–120 117 suggestion of o’regan and gratzel and was classified as the third generation of photovoltaic devices for the conversion of visible light into electrical energy [2]. since the advent of dyesensitized solar cells (dsscs) in 1991, extensive researches are seriously ongoing on it as an alternative to silicon-based solar cells, and even the thin film solar cells; owing to their simple structure, transparency, flexibility and low production cost. regardless of these advantages, the low efficiency of dssc when compared to the long-ranged silicon-based cells is a limitation to their commercial implementation [3]. currently, dssc has the potential of converting photons from sunlight to electrical energy at an efficiency of 13%, according to [4]. a concerted and intensive effort is being put towards the optimization of various components of dssc with the aim of fabricating more efficient and stable cells. dyesensitized solar cells which are liquid-based consist of a fluorine doped tin oxide frontcontact (fto) on glass, nanoparticle photoanode covered in a monolayer of sensitizing dye, a hole conducting electrolyte, and finally graphite or platinum coated fto counter electrode (back contact). in dye-sensitized solar cells, the dye is one of the key components for high conversion efficiencies of power. in recent time, obvious progress has been achieved in the engineering of novel dye structures in order to enhance the performance of the system. for a while, ruthenium based organic complexes have been the most stable and effective dyes used for dsscs. as a result, that these dyes are characterized by its toxicity, relatively expensive, and difficult method of synthesis, increasing activities for using natural dyes have been reported [5]. in particular, the amphiphilic homologues of the pioneering rutheniumbased n-3 dye have beendeveloped. these dyes show several merits when put side by side with the n-3 dye such as: a higher ground state pka of the binding moiety which increases electrostatic binding onto the titanium dioxide surface at lower ph values, the decreased charge on the dye reducing the electrostatic repulsion between adsorbed dye units and hence increasing the dye loading, the oxidation potential of these dyes is shifted cathodically compared to that of the n-3 sensitizer, which increases the reversibility of the ruthenium iii/ii couple, and finally lead to enhanced stability. [4] stated that the sensitizers which are currently used in production of solar cells are transition metal coordination complexes like ruthenium (ii) carboxylated polypyridyl complexes, because of their high charge-transfer absorption within the entire visible range of electromagnetic radiation and highly efficient metal-ligand charge transfer transition (mlct). however, natural dyes are better desired than these synthetic dyes because of being more economical, easily attainable, abundant in supply and environmentally friendly. also, they invariably have large absorption coefficient due to allowed π to π * transitions. these pigments are derived from various figure 1: absorption spectra of galena dye plant parts such as flower petals, leaves, roots and fruits pulp/bark. therelatively quick degradation of even the natural dyes obtained from plants as compared with the metal coordination complexes calls for considering of an alternative natural dye with cost effectiveness and good stability. natural dye can be categorized into biological and mineral dyes. the biological are the ones obtained from plants while mineral dyes are from natural minerals ofthe earth. in this study, dye obtained from natural mineral; galena was characterized and the suitability in absorbing solar radiation for excitation of electrons in generating electricity via a dssc is considered. 2. materials and method rock-like mineral; galena, was obtained from a community market around the location of study, ilorin, nigeria (lat. 8.49280 n, long. 4.59620 e). the natural substance was grinded with an electric industrial grinder into a powder. the dye was separately extracted from the powder using an organic solvent (isopropyl alcohol). the structural property of the dye was studied by carrying out x-ray diffraction (xrd). the quantitative analyses of the dyes were done using the x-ray fluorescence (xrf) technique, to obtain the elemental composition of the dyes. the functional groups present 4 in the dyes were determined using the fourier transform infrared (ftir)spectroscopy. the absorption spectra of the dye was studied within the visible region of the electromagnetic radiation and it was done using the uv-visible spectrophotometer. 3. results and discussion 3.1. optical properties absorption of electromagnetic radiation is the process by which certain energy is being taken up with photon by matter. the absorption spectra of galena dye is given in figure 1. electromagnetic spectrum comprises of radio wave, infrared, visible light, region (about 380 nm – 800nm), since 117 dawodu / j. nig. soc. phys. sci. 3 (2021) 116–120 118 figure 2: absorption spectra of ruthenium-based dye, n-719 (product no. 703214). source: [6] the dye is being studied as a potential photosensitizer in a dye-sensitized solar cell (dssc) which absorbs solar radiation within the visible region of the electromagnetic radiation. it was observed (from figure 1) that the dye has absorption of solar radiation within the visible region. considering figure 2, the absorption of solar radiation, based on the absorbance value of a typical ruthenium-based dye (a synthetic dye) is just a little higher than that of the mineral dye; which shows a promising substitute to the relatively expensive synthetic dye. it is indeed a potential photosensitizer in a dssc, as substitute to dyes of plants sources.in addition, galena is a natural semiconducting material with an energy gap of about 0.4ev. indeed, it’s a strong absorber of solar radiation. the dye extract exhibited a strong absorption broad band in the visible region with a peak at around 408 nm (absorbance value of 0.2424 a.u. inferably, very little composition of the dye for absorbing the electromagnetic radiation was present. further work can still be considered on solvents or process of making the galena powder well dissolved for a uniform analysis by the uv-vis spectroscopy. galena is fundamentally a lead ore i.e.; lead sulphide and lead is metal. this intense absorption in the visible region has been reported for anthocyanin and is the reason for the efficient harvesting of photons in natural dssc. anthocyanins are group of naturally occurring phenolic compounds responsible for the colour of many flowers and fruit.ruthenium-based dye exhibit ligand-centered charge transfer (lcct) transitions (π -π*) as well as metal-to-ligand charge transfer (mlct) transitions (4d π*) that can be observed in the absorption spectra of n-719 dye (figure 2). the absorption bands at lower energiesrepresent the mlct transitions (λ1 and λ2) whereas the more energetically demanding transitions (λ3 and λ4) correspond to lcct transitions. promotion of an electron from π – bonding orbital to an antibonding π orbital* is denoted by π π* transition. section of molecules which can undergo such detectable electron transitions can be referred to asch table 1: elemental composition of galena elements concentration ca < 411.684 sc < 78.741 ti 263.183 ± 50.390 ppm v < 203.917 cr < 141.133 mn < 38.203 fe 965.357 ± 44.279 ppm ni 318.054 ±36.217ppm cu 345.412 ± 21.042ppm zn 141.191 ± 11.137ppm ga 490.431 ± 40.761 ppm pb 3690.413 ± 462.395 ppm se 188.198 ± 32.007 ppm br < 412.070 rb < 26.176 sr < 30.008 y < 1220.770 -romophores since such transitions absorb electromagnetic radiation (light), which may hypothetically be perceived as colour somewhere in the electromagnetic spectrum. the absorption spectra of galena dye given in figure 1 shows absorption bands (408 nm and around 573 nm) at more energetically demanding transitions which is close to lccttransitions within the visible region, hence favoring a good absorption of solar radiation for the operation of a solar cell. 3.2. quantitative analysis the elemental composition of the galena dye was summarized in table 1. from the analysis it was observed that the elements with the prominent concentrations in the dye material are lead (pb) and iron (fe) with 3690.413 ppm and 965.357 ppm respectively. the high concentration of pb and fe in the sample could be as a result of the fact that it has its source from the earth (being a natural mineral). also, the iron concentration in the dye material could be responsible for its lustrous black colour (see figure 3) which could eventually favours it high absorption of electromagnetic radiation in the visible region. although iron (fe), copper (cu), silver (ag) etc. are naturally parts of the common impurities of galena ore. the results discussed under the optical properties and as seen in table 2 justify this fact and also as revealed by the result given by the xrd pattern. colours in the visible region of the electromagnetic spectrum are red, orange, yellow, green, blue, indigo and violet. these colours absorb at different wavelengths of light (table 2), which in turn carry different magnitude of energy.the ma118 dawodu / j. nig. soc. phys. sci. 3 (2021) 116–120 119 figure 3: image of galena (source:[7]) table 2: corresponding wavelength of colour in the visible region elements concentration red 622-780 orange 597-622 yellow 577-597 green 492-577 blue 455-492 violet 390-455 terial being considered has a colour close to the ones within the wavelength range of 390 nm – 577 nm, as it is in table 2. 3.3. fourier transform infrared spectroscopic (ftir) analysis of the dye an ftir spectrum of the galena dye is shown in figure 4. the functional groupspresent in an organic dye responsible for the absorption of solar radiation are actually the amine, hydroxyl and the carbonyl groups. in addition to the high absorption coefficient in the visible region of the electromagnetic spectrum, the presence of hydroxyl and carbonyl anchoring groups in the dye as revealed by the stretching vibrations at 2883.1 cm−1, 2933.4 cm−1, 2970.7 cm−1 and 1654.9 cm−1 respectively will enable their adsorption unto the surface of semiconductor to be used in a dssc. the presence of the aminegroup in the dye is revealed by the vibration at 3332.2 cm−1. the absorption bands for bending vibrations are typically found in the fingerprint region (1400 – 600 cm−1). these vibrations correspond to the likely metalbonded compounds present in the region which characterfigure 4: ft-ir spectra of galena dye figure 5: xrd pattern of galena sample obtained for an organic dye ization carried out using the x-ray diffraction (xrd) technique revealed. 3.4. 3.4 x-ray diffraction characterization the dye material was subjected to x-ray diffraction. the xrd pattern obtained for the dye was given in figure 5. the 2θ peak values considered are as follows: 26.00, 30.10, 43.10 and 52.50 corresponding to diffraction from planes (1 1 1), (2 0 0), (2 2 0) and (3 1 1) respectively for the galena. the xrd patterns confirms the presence of lead sulphide, in the dye material, as thismatches with the jcpds card no. [05-0592]. it is confirmed to be of face-centeredcubic crystal. the multiple peaks obtained from the x-ray diffraction point to the fact that it is also polycrystalline. the crystal plane (2 0 0) is the prominently seen in the xrd pattern. this is in agreement with the report of [8]. the prominent peak in the xrd pattern corresponds to the galena, (pbs) mineral in the galena ore sample as other mineralogical content of the ore could be sphalerite (zns), pyrite (fes2), chalcopyrite (cufes2), etc. the confirmed pbs, a semiconducting material, in the galena ore actually makes it a potential absorber / good photosensitizer in a natural dssc. 119 dawodu / j. nig. soc. phys. sci. 3 (2021) 116–120 120 4. conclusion this research focuses on the properties of a mineral dye which make it suitable as a potential photosensitizer in a dyesensitized solar cell (dssc). the dye, though from a material being used for different purposes, among which is cosmetics in some part of africa for decades, is discovered to possess, through the characterizations carried out, tendencies of being a good absorber of solar radiation in the visible region of electromagnetic radiation. this is expected to yield an improved power conversion efficiency of the cell. references [1] a. le donne, t. vanira & b. simona, "new earth-abundant thin filmsolar cells based on chalcogenides", frontiers in chemistry. 7 (2019) 1. [2] s. a. monzir, b. a. mahmoud, a. naji, m. a. amal, a. t. sofyan, m.e taher & s. e. hatem, “dye-sensitized solar cells using fifteen natural dyes as sensitizers of nanocrystalline tio2", science technology and development34 (2015) 135. doi: 10.3923/std.2015.135.139 [3] h. y. jun, w. b. chung, h. k. kyung & w. c. hyung, “characteristics ofthe dye-sensitized solar cells using tio2 nanotubes treated with ticl4 ", materials 7 (2014) 3522. [4] g. william, k. hyeonggon, s. tajbik, y. sunil, c. tulio, n. fred & u. jamal, “fabrication, optimization and characterization of natural dye sensitized solar cell", scientific reports 7 (2017) 41470. [5] m. a. ahmed hemdan, s. h. m moataz, m. k. y. ghad, m. a. ahmed, s. h. & s. g. k. ahmed, “dye-sensitized solar cells (dsscs) based on extracted natural dyes", journal of nanomaterials. article id 1867271.https://doi.org/10.1155/2019/1867271 [6] d. hans & h. yanek, ruthenium-based dyes for dyesensitized solar cells, dyesol ltd, australia, 2017. [7] http://www.rocksandminerals.com/lead.htm (accessed on 27th september, 2019). [8] m. p. c. kalika, d. kuldeep, d. joyeeta, h. nilakhi, d. purbasha, d. ronita, p. sanchayita, s. trinayan & b. k. sarma, “x-ray diffraction line profileanalysis of chemically synthesized lead sulphide nanocrystals", . materials letters 87 (2012) 84. 120 j. nig. soc. phys. sci. 1 (2019) 82–87 journal of the nigerian society of physical sciences original research the solution of a mathematical model for dengue fever transmission using differential transformation method felix yakubu eguda∗, andrawus james, sunday babuba department of mathematics, federal university, dutse, jigawa state. abstract differential transformation method (dtm) is a very effective tool for solving linear and non-linear ordinary differential equations. this paper uses dtm to solve the mathematical model for the dynamics of dengue fever in a population. the graphical profiles for human population are obtained using maple software. the solution profiles give the long term behavior of dengue fever model which shows that treatment plays a vital role in reducing the disease burden in a population. keywords: dengue fever, mathematical model, differential transformation method, ordinary differential equations article history : received: 15 june 2019 received in revised form: 13 july 2019 accepted for publication: 16 july 2019 published: 03 september 2019 c©2019 journal of the nigerian society of physical sciences. all rights reserved. communicated by: t. latunde 1. introduction dengue fever is an infectious vector borne disease spreading in tropical and subtropical countries with more than 50 million dengue fever cases per year. it is transmitted to humans by the bite of infected aedes mosquitoes. the major vector, aedes aegypti, thrives in tropical regions, mainly in urban areas, closely linked to human populations providing artificial water-holding containers as breeding sites. a second potential vector, aedes albopictus, resides in temperate regions (north america and europe), where it may give rise to occasional dengue outbreaks. [1, 2]. as part of awareness campaigns, different kinds of precautions have been suggested towards preventing mosquito’s bite. some of the precautions that can be taken are: to keep home, environment and surrounding clean, to remove all stagnant water and containers, to cover all containers properly to prevent ∗corresponding author tel. no: +2348160559365 email address: felyak_e@yahoo.co.uk (felix yakubu eguda ) dengue mosquito breeding there, to use mosquito repellents to avoid mosquito bite, to use mosquitoes net around bed while sleeping etc. [1]. different studies have shown the importance of mathematical approaches in understanding dengue disease transmission and evaluating the effectiveness and/or costeffectiveness of control strategies [1]. in recent years, the field of public health has benefited tremendously from the use of mathematical models to study the spread of infectious disease. most epidemiological models are represented using systems of non-linear ordinary differential equations [3]. differential transformation method is a semi-analytical method of solving both linear and nonlinear system of ordinary differential equations (ode) to obtain approximate series solutions. this method which was derived from the taylor’s series expansion has been used to solve problems in mathematics and physics [4], fractional differentialalgebraic equations [5], fourth-order parabolic partial differential equations [6], fractional-order integrodifferential equations [7], differential equation [8] and problems in epidemic models [9]. 82 eguda et al. / j. nig. soc. phys. sci. 1 (2019) 82–87 83 2. materials and methods 2.1. model formulation two populations consisting of human and vector population will be considered in this work. the model sub-divides these populations into a number of mutually-exclusive compartments, as given below. the total population of human and vectors is divided into the following mutually exclusive epidemiological classes, namely, susceptible humans ( s h (t) ), humans with dengue in latent stage ( e1 (t)), humans with dengue ( i1 (t) ), humans treated of dengue ( r1 (t) ), susceptible vectors ( s v (t) ), vectors with latent dengue ( ev (t) ), vectors with dengue ( iv (t) ). 2.2. derivation of model equations let nh (t) and nv (t) denote the total number of humans and vectors at time t, respectively. hence, we have that, nh (t) = s h (t) + e1 (t) + i1 (t) + r1 (t) and nv (t) = s v (t) + ev (t) + iv (t) susceptible humans are recruited at a rate h while the susceptible vectors are recruited at a rate v. susceptible humans contract dengue at a rate λdv = βv h (ηv ev + iv) nh , where ηv < 1 , this accounts for the relative infectiousness of vectors with latent dengue iv compared to vectors in the iv class. susceptible vectors acquire dengue infection from infected humans at a rate λdh = βhv (ηa e1 + ηb i1) nh , where ηa < ηb this accounts for the relative infectiousness of humans with latent dengue e1 compared to humans in the i1 class [10]. the model equations for dengue disease transmission incorporating treatment as a control measure are given below: s h = λh −µh s h −λdv s h e1 = λdv s h − (γ1 + µh ) e1 i1 = γ1 e1 − (τ1 + µh + δd1) i1 r1 = τ1 i1 −µh r1 s v = λv −µv s v −λdh s v ev = λdh s v − (γv + µv ) ev iv = γv ev − (µv + δhv ) iv 2.3. differential transformation method (dtm) an arbitrary function f (t) can be expanded in taylor series about a point t = 0 as f (t) = ∞∑ k=0 tk k! [ dk f dtk ] t=0 (1) the differential transformation of f (t) is defined as f (t) = 1 k! [ dk f dtk ] t=0 (2) then the inverse differential transform is f (t) = ∞∑ k=0 tk f (t) (3) if y (t) and g (t) are two uncorrelated functions with t where y (k) and g (k) are the transformed functions corresponding to y (t) and g (t) then, the fundamental mathematical operations performed by differential transform can be proved easily and are listed as follows table 1: the fundamental mathematical operations by differential transformation method (dtm). source: [11, 12] transformed function original function y (t) = f (t) ± g (t) y (k) = f(k) ± g(k) y (t) = a f (t) y (k) = af(k) y (t) = d f (t)dt y (k) = (k + 1)f(k + 1) y (t) = d 2 f (t) dt2 y (k) = (k + 1)(k + 2)f(k + 2) y (t) = d m f (t) dtm y (k) = (k + 1)(k + 2)...(k + m)f(k + m) y (t) = 1 y (k) = δ(k) y (t) = t y (k) = δ(k − 1) y (t) = tm y (k) = δ(k − m) = { 1, k = m 0, k , m y (t) = f (t) g (t) y (k) = ∑k m=0 g(m) f (k − m) y (t) = e(λt) y (k) = λ k k! y (t) = (1 + t)m y (k) = (m(m−1)...(m−k+1))k! 2.4. analytical solution of the model equations using differential transformation method (dtm) in this section, the differential transformation method (dtm) is employed to solve the system of non-linear differential equations which describe our model for dengue fever. let the model equation be a function q (t), q (t) can be expanded in taylor series about a point t = 0 as q (t) = ∞∑ k=0 tk k! [ dkq dtk ] t=0 , (4) where, q (t) = {sh (t) , e1 (t) , i1 (t) , r1 (t) , s v (t) , ev (t) , iv (t)}(5) the differential transformation of q (t) is defined as q (t) = 1 k! [ dkq dtk ] t=0 (6) 83 eguda et al. / j. nig. soc. phys. sci. 1 (2019) 82–87 84 table 2: values for parameters used for analytical solutions parameter description values unit reference λh, λv recruitment rate into the population of susceptible humans and vectors respectively. 500,10000000 year−1 [14] µh,µv natural death for humans, vectors respectively. 0.02041,0.5 year−1 [13] βv h effective contact rate for 0.5 dengue from vectors to humans 0.5 year−1 [14] βhv effective contact rate for 0.4 dengue from humans to vectors 0.4 year−1 [14] τ1 dengue treatment rate for i1 (0,1) ind−1year−1 [14] γ1 progression rate to active dengue 0.3254 year−1 [14] γv progression rate to active dengue (vectors) 0.03 year−1 [14] δd1 disease induced death dengue 0.365 year −1 [13] δhv disease induced death dengue (vectors) 0 year−1 [14] ηv , ηa, ηb modification parameters for ev, e1, i1 0.4,1.2,0.5 year−1 [13] then the inverse differential transform is q (t) = ∞∑ k=0 tk q (t) . (7) using the fundamental operations of differential transformation method in table 1, we obtain the following recurrence relation of equation (1) as s h (k + 1) = 1 k + 1 [ λh −µh s h (k) − βv hηv nh k∑ m=0 s h (m) ev (k − m) − βv h nh k∑ m=0 s h (m) iv (k − m)  (8) e1 (k + 1) = 1 k + 1 βv hηvnh k∑ m=0 s h (m) ev (k − m) − βv h nh k∑ m=0 s h (m) iv (k − m) − (γ1 + µh ) e1 (k) ] (9) i1 (k + 1) = 1 k + 1 [ γ1 e1 (k) − (τ1 + µh + δd1) i1 ] (10) r1 (k + 1) = 1 k + 1 [ τ1 i1 (k) −µh r1 (k) ] (11) s v (k + 1) = 1 k + 1 λv − βhvηanh k∑ m=0 s v (m) e1 (k − m) − βhvηb nh k∑ m=0 s v (m) i1 (k − m)  (12) ev (k + 1) = 1 k + 1 βhvηanh k∑ m=0 s v (m) e1 (k − m) − βhvηb nh k∑ m=0 s v (m) i1 (k − m) − (γv + µv) ev (k) ] (13) iv (k + 1) = 1 k + 1 [ γv ev (k) − (µv + δhv ) iv (k) ] (14) with the initial conditions s h (0) = 3503, e1 (0) = 490, i1 (0) = 390, r1 (0) = 87, s v (0) = 390, ev (0) = 100, iv (0) = 190 (15) the parameter values are nh = 4470, nv = 610, λh = 500, λv = 1, 000, 000, µh = 0.02041,µv = 0.5,βv h = 0.5,βhv = 0.4, τ1 = 0.75,γ1 = 0.3254,γv = 0.03,δd1 = 0.365, δhv = 0,ηv = 0.4,ηa = 1.2,ηb = 0.5 (16) we consider k = 0, 1, 2, 3. cases a1 to a3 are the variation of different values of τ1 case a1: high dengue treatment rate, τ1 = 0.75 s h (1) = −237.6147983, s h (2) = 12320.67062, s h (3) = −146425.8636, s h (4) = 1347694.18, e1 (1) = −522.0979067, e1 (2) = 6637.369770, e1 (3) = −82219.53607, e1 (4) = 775491.4995, i1 (1) = −283.36390, i1 (2) = 75.92177345, i1 (3) = 691.1992607, i1 (4) = −6884.757898, r1 (1) = 290.72433, r1 (2) = −109.2283043, r1 (3) = 19.72355993, r1 (4) = 129.4992219, s v (1) = −14008.26174, s v (2) = 260942.4174, 84 eguda et al. / j. nig. soc. phys. sci. 1 (2019) 82–87 85 s v (3) = −3239718.545, s v (4) = 30208553.32, ev (1) = −3515.845638, ev (2) = 61669.12070, ev (3) = −717387.0983, ev (4) = 6204920.468, iv (1) = −4742.00, iv (2) = 86488.76230, iv (3) = −1051663.250, iv (4) = 9591046.752. (17) then the closed form of the solution where k = 0, 1, 2, 3 can be written as s h (t) = ∞∑ k=0 s h (k) t k = 3503 − 237.6147983t +12320.67062t2 − 146425.8636t3 +1347694.180t4 e1 (t) = ∞∑ k=0 e1 (k) t k = 490 − 522.0979067t +6637.369770t2 − 82219.53607t3 +775491.4995t4 i1 (t) = ∞∑ k=0 i1 (k) t k = 390 − 283.36390t +75.92177345t2 + 691.1992607t3 −6884.757898t4 r1 (t) = ∞∑ k=0 r1 (k) t k = 87 + 290.72433t −109.2283043t2 + 19.72355993t3 +129.4992219t4 s v (t) = ∞∑ k=0 s v (k) t k = 390 − 14008.26174t +260942.4174t2 − 3239718.545t3 +30208553.32t4 ev (t) = ∞∑ k=0 ev (k) t k = 100 − 3515.845638t +61669.12070t2 − 717387.0983t3 +6204920.468t4 iv (t) = ∞∑ k=0 iv (k) t k = 130 − 4742.00t +86488.76230t2 − 1051663.250t3 +9591046.752t4 (18) case a2: moderate dengue treatment rate, τ1 = 0.5 s h (1) = −237.6147983, s h (2) = 12320.67062, s h (3) = −146421.4193, s h (4) = 1347571.999, e1 (1) = −522.0979067, e1 (2) = 6637.369770, e1 (3) = −82223.98033, e1 (4) = 775613.8858, i1 (1) = −185.86390, i1 (2) = −2.662451550, i1 (3) = 720.7191613, i1 (4) = −6848.453788, r1 (1) = 193.22433, r1 (2) = −48.43782929, r1 (3) = −0.1142032264, r1 (4) = 90.09047788 s v (1) = −14008.26174, s v (2) = 260933.9106, s v (3) = −3239404.789, s v (4) = 30202664.15, ev (1) = −3515.845638, ev (2) = 61660.61400, ev (3) = −71707.62417, ev (4) = 619913.9432, iv (1) = −4742.00, iv (2) = 86488.76230, iv (3) = −1051663.335, iv (4) = 9591049.860. (19) then the closed form of the solution where k = 0, 1, 2, 3 can be written as s h (t) = ∞∑ k=0 s h (k) t k = 3503 − 237.6147983t +12320.67062t2 − 146421.4193t3 +1347571.999t4 e1 (t) = ∞∑ k=0 e1 (k) t k = 490 − 522.0979067t +6637.369770t2 − 82223.98033t3 +775613.8858t4 i1 (t) = ∞∑ k=0 i1 (k) t k = 390 − 185.86390t −2.662451550t2 + 720.7191613t3 −6848.453788t4 r1 (t) = ∞∑ k=0 r1 (k) t k = 87 + 193.22433t −48.43782929t2 − 0.1142032264t3 +90.09047788t4 s v (t) = ∞∑ k=0 s v (k) t k = 390 − 14008.26174t +260933.9106t2 − 3239404.789t3 +30202664.15t4 ev (t) = ∞∑ k=0 ev (k) t k = 100 − 3515.845638t +61660.61400t2 − 717076.2417t3 +6199139.432t4 iv (t) = ∞∑ k=0 iv (k) t k = 130 − 4742.00t +86488.76230t2 − 1051663.335t3 +9591049.860t4 (20) case a3: low dengue treatment rate, τ1 = 0.25 s h (1) = −237.6147983, s h (2) = 12320.67062, s h (3) = −1464169750, s h (4) = 1347571.999, e1 (1) = −522.0979067, e1 (2) = 6637.369770, e1 (3) = −82223.98033, e1 (4) = 775613.8858, i1 (1) = −88.36390, i1 (2) = −56.87167655, i1 (3) = 720.7191613, i1 (4) = −6848.453788, r1 (1) = 95.72433, r1 (2) = −12.02235429, 85 eguda et al. / j. nig. soc. phys. sci. 1 (2019) 82–87 86 r1 (3) = −0.1142032264, r1 (4) = 90.09047788 s v (1) = −14008.26174, s v (2) = 260925.4039, s v (3) = −3239404.789, s v (4) = 30202664.15, ev (1) = −3515.845638, ev (2) = 61652.10730, ev (3) = −71707.62417, ev (4) = 619913.9432, iv (1) = −4742.00, iv (2) = 86488.76230, iv (3) = −1051663.335, iv (4) = 9591049.860. (21) then the closed form of the solution where k = 0, 1, 2, 3 can be written as s h (t) = ∞∑ k=0 s h (k) t k = 3503 − 237.6147983t +12320.67062t2 − 146416.9750t3 +1347450.373t4, e1 (t) = ∞∑ k=0 e1 (k) t k = 490 − 522.0979067t +6637.369770t2 − 82228.42463t3 +775735.7158t4, i1 (t) = ∞∑ k=0 i1 (k) t k = 390 − 88.36390t −56.87167655t2 + 731.9789850t3 −6805.559035t4, r1 (t) = ∞∑ k=0 r1 (k) t k = 87 + 95.72433t −12.02235429t2 − 4.657514297t3 +45.77245152t4, s v (t) = ∞∑ k=0 s v (k) t k = 390 − 14008.26174t +260925.4039t2 − 323909.2451t3 +30196827.38t4, ev (t) = ∞∑ k=0 ev (k) t k = 100 − 3515.845638t +61652.10730t2 − 717666.8033t3 +6193410.4t4, iv (t) = ∞∑ k=0 iv (k) t k = 130 − 4742.00t +86488.76230t2 − 1051663.420t3 +9591052.958t4. (22) 2.5. numerical simulation and graphical representation of the solutions of the model equations the numerical simulation which illustrates the analytical solution of the model is demonstrated using maple software. this is achieved by using some set of parameter values given in the table 2. the following initial conditions for the human populations s h (0) = 3503, e1 (0) = 490, i1 (0) = 390, r1(0) = 87, s v (0) = 390, ev (0) = 100, iv (0) = 190 are considered. figure 1: solution of susceptible population using dtm figure 2: solution of exposed population using dtm figure 3: solution of human population with dengue using dtm 3. discussion of results the figures 1 to 6 give the numerical profiles of the solutions (17), (19) and (21) using dtm. figure 1 shows increase in the population of susceptible individuals while figure 2 indicates a decreasing population of the exposed owing to the progression out of exposed class to class of human with dengue. figure 3 implies a decrease in the population of human infected with dengue which later increases. figure 4 implies the treated 86 eguda et al. / j. nig. soc. phys. sci. 1 (2019) 82–87 87 figure 4: solution of treated human population using dtm figure 5: the effect of different treatment rates on human population with dengue using dtm figure 6: the effect of different treatment rates on treated human population using dtm human population increases sharply to a point and then decreases. figure 5 indicates that increasing the treatment rate of individuals in the infected population leads to a reduction in the number of infected individuals which is due to progression into the treated population while figure 6 shows that increasing the treatment rate of individuals in the infected population leads to a corresponding increase in the treated population. 4. conclusion we formulated a compartmental model to investigate the dynamics of dengue fever in a population with treatment as a control measure. differential transform method (dtm) was employed to obtain the series solution of the model. numerical simulations were carried out to determine the long term behavior of dengue fever model which shows that treatment plays a vital role in reducing the disease burden in a population. the results of the simulations were displayed graphically. acknowledgments we thank the referees for the positive enlightening comments and suggestions, which have greatly helped us in making improvements to this paper. references [1] g. r. phaijoo, & d. b. gurung, “mathematical model of dengue fever with and without awareness in host population”, international journal of advanced engineering research and applications 1 (2015) 2454. 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[14] s. m. garba, a. b. gumel & m.r.abubakar, “backward bifurcations in dengue transmission dynamics”, mathematical biosciences 215 (2008) 11. 87 j. nig. soc. phys. sci. 5 (2023) 1137 journal of the nigerian society of physical sciences robust m-estimators and machine learning algorithms for improving the predictive accuracy of seaweed contaminated big data o. j. ibidojaa,b, f. p. shanb, mukhtarc, j. sulaimand, m. k. m. alib,∗ a department of mathematics, federal university gusau, gusau, nigeria b school of mathematical sciences, universiti sains malaysia 11800 usm, penang, malaysia ci-cefory (local food innovation), universitas sultan ageng tirtayasa indonesia dschool of science and technology, universiti malaysia sabah, kota kinabalu, sabah, malaysia abstract a common problem in regression analysis using ordinary least squares (ols) is the effect of outliers or contaminated data on the estimates of the parameters. a robust method that is not sensitive to outliers and can handle contaminated data is needed. in this study, the objective is to determine the significant parameters that determine the moisture content of the seaweed after drying and develop a hybrid model to reduce the outliers. the data were collected with sensors from the v-groove hybrid solar drier (v-ghsd) at semporna, south-eastern coast of sabah, malaysia. after the second order interaction, we have 435 drying parameters, each parameter has 1914 observations. first, we used four machine learning algorithms, such as random forest, support vector machine, bagging and boosting to determine the significant parameters by selecting 15, 25, 35 and 45 parameters. second, we developed the hybrid model using robust methods such as m. bi-square, m. hampel and m. huber. the results show that there is a significant improvement in the reduction of the number of outliers and better prediction using hybrid model for the contaminated seaweed big data. for the highest variable importance of 45 significant drying parameters of seaweed, the hybrid model bagging m bi-square performs better because it has the lowest percentage of outliers of 4.08 %. doi:10.46481/jnsps.2023.1137 keywords: robust method, hybrid model, machine learning, outliers, big data. article history : received: 22 october 2022 received in revised form: 08 january 2023 accepted for publication: 08 january 2023 published: 04 february 2023 c© 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: tolulope latunde 1. introduction the purpose of regression analysis is to study the relationship between two or more independent variables and a dependent variable. consider a multiple regression model: y = xβ + ε, (1) ∗corresponding author tel. no: +60 14-9543405 email address: majidkhanmajaharali@usm.my (m. k. m. ali ) where y is an n × 1 vector of response variables, x is known as the design matrix of order n × p, β is a p × 1 vector of unknown parameters and ε is an n × 1 vector of identically and independent distributed errors. the ordinary least squares (ols) is popularly used to estimate the unknown parameters in a regression model. according to [1, 2], the ordinary least squares (ols) estimator of β is 1 o. j. ibidoja et al. / j. nig. soc. phys. sci. 5 (2023) 1137 2 obtained as: β̂ = ( x ′ x )−1 x ′ y. (2) observations that deviate from the distribution’s general shape or pattern are called outliers [3]. the relationship between the observed and the dependent variable can be estimated by ols regression, by minimizing the sum of squares [4]. ols also has limitations when the assumptions are violated [5]. estimates from ols are not precise due to the high variances and covariances [6]. the presence of outliers in the data makes the ls estimator unstable, inefficient, and unreliable [7]. agricultural data has outliers because of factors that cannot be regulated, and these outliers will increase the standard errors [4, 8]. the presence of outliers affects the performance of ols, and a robust regression is used [9]. when modelling data using regression analysis, various assumptions are tested but these assumptions are violated. this model needs to be tested on the error structure for the necessary assumptions before prediction [10]. the researcher can transform the variables to fulfil the assumptions, but this cannot eradicate the outliers in the data that affect the forecast and estimate of the parameters [11]. data with outliers is common in the field of agriculture [11, 12]. to overcome this problem, robust estimators have been introduced. m-estimation is the most common method of robust regression, it was introduced by [13], it is a generality to the method of maximum likelihood estimation. before we used the robust methods to reduce the outliers, four machine learning algorithms such as random forest, support vector machine, boosting and bagging are used to select the significant parameters that determine the moisture content of the seaweed. the major contributions of this study are: i. to determine the significant parameters for the moisture content removal of seaweed during drying and reduce the number of outliers. ii. to propose a hybrid model that combines robust mestimators and machine learning models to improve the prediction accuracy. 2. flowchart of the study figure 1 shows the flowchart of the various stages in the study. 2.1. stage i this involves the inclusion of all possible models. n! (n − r)!r! + number of single factor, (3) where n is the number of single factors, r is the number of orders. equation (3) can be used to compute the total number of all possible models. figure 1: flowchart of the procedure for the hybrid model 2.2. stage ii test for the assumptions of linear regression. the residual vs fitted plot, normal q-q plot are kolmogorov-smirnov test are used to verify the assumptions. next, each machine learning model is used to select 15, 25, 35 and 45 highest important variables for optimization and easy comparison, to determine the moisture content removal of the seaweed after drying. we selected the number of variables because features selection can only provide the rank of important variables and does not tell us the number of significant factors [14]. similarly, there is no rule to decide the number of parameters to be included in a prediction model [15]. furthermore, the algorithms cannot tell us the number of significant variables except the ranks [16]. 2.3. stage iii after the selection of the significant parameters, the prediction is done and the validation metrics such as mape, sse, mse and r-square are computed. the outliers are also computed, and the robust method is introduced to build the hybrid model. 3. materials and methods 3.1. data description the data were collected from 8th april 2017 to 12th april 2017, between the hours of 8:00 am to 5:00 pm during the drying of seaweed by using v-groove hybrid solar drier (vghsd) at semporna, south-eastern coast of sabah, malaysia. there are 435 parameters after the inclusion of the second order interaction in this study. 3.2. machine learning algorithms machine learning can learn from data and use the algorithms to understand and forecast the future [17]. machine learning algorithms can be used to determine the rank of significant explanatory variables that contribute significantly to the response variable. these high-ranking variables selected using variable importance can reduce the training time, complexity 2 o. j. ibidoja et al. / j. nig. soc. phys. sci. 5 (2023) 1137 3 of the model and improve accuracy [18]. four machine learning algorithms such as random forest, support vector machine, bagging and boosting are used in this study, to determine the significant parameters that determine the moisture content removal of the seaweed. 3.2.1. random forest a random forest (rf) is a mixture of classification and regression trees (carts). it uses the highest number of votes (classification) or the mean forecasts (regression) of all the trees [19]. it uses the idea of bagging, and it is an ensemble learning method [20], [21]. if l is a learning set ,with a group of n pairs of features, with the output (x1, y1) , (x2, y2) , (x3, y3) . . . (xn, yn ) , if xi ∈ x and yi ∈ y . a class of p-features xi ( f or i = 1, 2, . . . , n) is a n × p matrix x,where the rows i = 1, 2, ..., n relates as xi, with columns j = 1, 2, 3, ..., p as x j. algorithm: for b = 1 to n 1. create a bootstrapped sample d∗b from the training set d. 2. grow the tree by using the m from the bootstrapped sample d∗b. for a specific mode i. select m variables randomly. ii. identify the top split variables and values. iii. divide a node using the top divided variables and values. replicate the steps 1–3 till the stopping conditions are satisfied. 3.2.2. suppor vector machine (svm) support vector machine can be used for regression and classification problems [22]. svm has the capacity to reveal nonlinear connections with kernel function [20, 23]. the svm was developed by cortes & vapnik [24]. a good tutorial and explanations were given by [25, 26]. in support vector regression, the � loss function is usually minimized. beyond this particular bound, a straightforward linear loss function is applied, and any loss less than � is set to zero: l� = f (x) = { 0, if |yi − f (xi) | < � yi − f (xi) − �|, otherwise (4) for instance, suppose f (x) is a linear function f (x) = β0 + xtiβ, then the loss function is given as n∑ i=1 max ( yi − x t iβ−β0 − �, 0 ) (5) the � is the tuning parameter and can be written as the constrained optimization problem: minimize 1 2 ‖β‖2 (6) subject to{ yi − xtiβ−β0 ≤ ε − ( yi − xtiβ−β0 ) ≤ ε . (7) if there are observations who do not lie within the ε band around that regression line,then there is no solution to the problem. the slack variables ζi and ζ∗i are used ,this allows the observations to fall outside the ε band around that regression line. minimize 1 2 ‖β‖2 + k n∑ i=1 ( ζi + ζ ∗ i ) (8) subject to yi − xtiβ−β0 ≤ ε + ζi − ( yi − xtiβ−β0 ) ≤ ε + ζ∗i ζi,ζ ∗ i ≥ 0 (9) 3.2.3. boosting boosting is used to improve the accuracy of algorithms [27]. boosting starts with an algorithm or method to discover the rough rules of thumb. it is called the “base” or “weak” learning algorithm many times. the base learning algorithm creates a new weak prediction rule each time it is called, and after many rounds, the boosting algorithm must merge these weak rules into a singular forecast rule that, ideally, will be significantly more precise than any of the weak rules [28]. suppose we have this model matrix x = [ x1, x2, . . . , xp ] �rn×p, outcomes variable vector y ∈ rn×1. the regression coefficients vector is given as β ∈ rp, the value of predicted for the outcome variable is denoted by xβ, and the residuals are denoted by ε = y − xβ. for regression purposes, least squares boosting (lsb(ε)) gives an accurate description of the data and regularization [27]. the algorithm for lsb(ε) is as follows: algorithm: lsb (ε) choose the rate of learning ε > 0 and iterations number n. define at β̂0 = 0, r̂0 = y, k = 0. 1. do this for 0 ≤ k ≤ n 2. establish the covariates ũ jk and jk as below: ûn = argmin u∈r  n∑ i=1 ( r̂ki − xinu )22 for n = 1, 2, 3, . . . , p, jk ∈ argmin 1≤n≤p n∑ i=1 ( r̂ki − xinũn )2 3. revise the present errors and regression coefficients as: r̂k+1 ← r̂k − ε̃u jk β̂k+1jk ← β̂ k jk + ε̃u jk and β̂ k+1 j ← β̂ k j , j , jk 3.2.4. bagging breiman [29] introduced bagging (bootstrap aggregating) to decrease the variance of classification and regression tree models. it is used to improve the present method and leads to an improvement in the accuracy. bagging is used as an intensive methods to enhance erratic estimation. for a high dimensional 3 o. j. ibidoja et al. / j. nig. soc. phys. sci. 5 (2023) 1137 4 data problems, bagging can be used to find a good model. suppose we have a feature ϕ (x,l) to predict y from x, if there is a training sequence {lk} consisting of n objects , from l distribution, the aim here is to use the {lk} to build a more accurate predictor than ϕ (x,l) as a specific training set predictor ϕ (x,l) [29]. if y is not discreet and we put ϕ (x,lk) with the mean of ϕ (x,lk) over k. we get continually many samples via the bootstrap { l(a) } , an from l, and form { ϕ ( x,l(a) )} . if y is continuous, then ϕa as ϕa (x) = averageϕa ( x,l(a) ) . the { l(a) } will form replicate datasets with m cases are randomly chosen from l and by applying replacement. each (ym, xm) can appear many times in a any specific l(a). the technique to construct ϕ is an important factor to know if bagging improves precision or reliability. theoretically bagging is described as follows: i. build a bootstrap sample l∗i =( y∗i , x ∗ i ) (i = 1, 2, 3, . . . , m) centred on an empirical distribution of these pairs li = (yi, xi) (i = 1, 2, 3, . . . , m). ii. use the plug-in principle to ascertain the bootstrapped predictor θ̂∗m (x); which is, θ̂ ∗ m (x) = gm ( l1, l2, l3, . . . , lm ) (x). iii. θ̂m;b (x) = e∗ [ θ̂∗m (x) ] means the bagged predictor. the bagging algorithm is as follows: input: data d = {(x1, y1) , (x2, y2) , (x3, y3) , . . . , (xm, ym)} ; learning algorithm base l; base learner’s numbers j. process: for j = 1, 2, . . . , j: bs j = bootstrap(d); %create the bootstrap sample from d θ j = l ( bs j ) % train the base learner θ j from the bootstrap sample end output: 1j ∑j j=1 θ j (x) % for regression studies 3.3. robust estimation method outliers are common with contaminated data and how to determine the observations is a challenge. a robust method can deal with the influence of outliers. contaminated data can be analyzed using robust estimation [6], [30, 31, 32]. a robust method is used to solve the problems of traditional methods because of these outliers. to know the best method for the robust estimation methods, m estimation methods m huber, m hampel and m bi-square are compared. the m-estimation method attempts to minimise that the function ρ (•) operates on the residual. m-estimators define: β̂m = argmin β n∑ i=1 ρ (ei (β)). (10) the ρ is ρ−type m-estimation. assume σ is known and the residuals approximate β be ei = yi −βt xi. the β in m-estimate minimizes the objective function: n∑ i=1 ρ { ei (β) σ } . (11) figure 2: (a) residuals vs fitted (b) residuals vs normal q-q the σ robustly estimate and the scale σ̃m in m-estimator has solution: 1 n n∑ i=1 ρ ( ei σ ) = 1 n n∑ i=1 ρ ( yi −βt xi σ ) = k, (12) where the β has the p×1 parameter vector, and then the function ψ yields:∑ i ψ (ei) ∂ei ∂βi , for j = 1, 2, . . . , p. (13) the function ψ (e) = ∂ρ(e) ∂(e) derivatives the influence function. 4 o. j. ibidoja et al. / j. nig. soc. phys. sci. 5 (2023) 1137 5 table 1: robust regression m-estimation description methods objective function weight function bi-square  k2 6 { 1 − [ 1 − ( e k )2]3} for |e| ≤ k k2 6 for |e| > k  [ 1 − ( e k )2]2 for |e| ≤ k 0 f or |e| > k huber { 1 2 e 2 for |e| ≤ k k |e| − 12 k 2 for |e| > k { 1 f or |e| ≤ k k |e| for |e| < k hampel  e2 2 , 0 < |e| < a a |e| − e 2 2 , b < |e| ≤ c −a 2(c−b) (c − e) 2 + a2 (b + c − a) , b < |e| ≤ c  1 f or 0 < |e| < a a |e| for b < |e| ≤ c a c |e|−1 c−b for b < |e| ≤ c table 2: kolmogorov-smirnov test for normality test statistic value p-value remarks 0.1641 2.2e-16 the residuals do not come from a normal distribution. then the weight function defines: w (e) = ψ (e) e , (14) where function ψ (e) states:∑ i w (ei) ei ∂ei ∂βi = 0, for j = 1, 2, . . . , p (15) and the object becomes to obtain the following iterated reweighted least square problem: min ∑ i w ( e(k−1)i ) e2i , (16) where k indicates the iterate number. table 1 shows the summary of the m estimators and their respective weight function. 4. results and discussion from the plot in figure 2a, the residuals vs fitted plot shows that there is no pattern since the residuals did not spread out. there is evidence of non-linearity and heterogeneity. figure 2b shows the normal q-q plot, the residuals are not normally distributed, this also supports the result of kolmogorov-smirnov test in table 2. the possible outliers are the observations 272 and 355. the observation 272 determine more the moisture content removal of the seaweed than the model predict. though, it is an extreme case, but still affect the moisture content removal. the observation 355 has a negative residual and determine less the moisture content removal of the seaweed than the model predicts. the normality assumption is checked with the kolmogorovsmirnov test for a two-taied test. from the results in table 2, the p-value =2.2e-16, which is less than 0.05, it means we have enough evidence to say that the residuals do not come from a normal distribution. this also explains why we have this type of qq plot in figure 2. the results in table 3 are the evaluation of each machine learning algorithm for 15, 25, 35 and 45 high ranking variables that determine the moisture content removal of the seaweed. based on the mean absolute percentage error (mape), mean squared error (mse), r2 and sum of squared error (sse), random forest outperforms support vector machine, bagging and boosting for the 15, 25, 35 and 45 significant parameters. this also confirms the results of [33], where random forest absolutely performed better than the other methods. random forest when 45 significant parameters that determine the moisture content of the seaweed were selected gave mape of 2.125891, mse of 7.330011, r2 of 0.9732063 and sse of 14029.64 gave the best performance. all the validation measures such as mape, mse, r-square and sse imply that significantly better results are obtained by random forest to the determine the moisture content removal of the seaweed. table 4 is the summary of the original model without using robust method and the hybrid models ,which combines machine learning models and robust estimation techniques. it also shows the number and percentage of outliers using 2-sigma limit.the percentage for the outliers is the number of observations outside the 2-sigma limit. it shows the percentage of outliers outside the 2-sigma limit for the original model without using robust method and the hybrid model. this sigma limit can improve the outputs quality and eliminate the source of deficiencies [34]. based on the results in table 4 for the original model, for 5 o. j. ibidoja et al. / j. nig. soc. phys. sci. 5 (2023) 1137 6 table 3: evaluation metrics for the 15, 25, 35 and 45 high-ranking important variables machine learning model high-ranking important variables selected high-ranking important variables selected mape mse r2 sse random forest 15 2.458969 9.910512 0.9637737 18968.72 25 2.337353 9.010273 0.9670644 17245.66 35 2.174667 7.790909 0.9715216 14911.80 45 2.125891 7.330011 0.9732063 14029.64 support vector machine 15 8.614626 45.25618 0.8347612 86620.32 25 7.980399 35.80985 0.8691446 68540.05 35 7.568951 34.00095 0.8757802 65077.81 45 7.351331 32.38644 0.8816661 61987.65 bagging 15 12.25897 74.29053 0.7284423 142192.10 25 9.778194 47.33173 0.8269861 90592.93 35 8.413645 36.41955 0.8668739 69707.02 45 8.151903 33.65611 0.8769752 64417.80 boosting 15 8.168942 142.4542 0.5310293 272657.30 25 8.697362 136.3236 0.5543729 260923.30 35 8.183671 140.1463 0.5368431 268240.10 45 8.203304 134.0864 0.5569358 256641.30 table 4: percentage of outliers outside 2 sigma limits for hybrid models machine learning model robust regression method 15 highest important variables 25 highest important variables 35 highest important variables 45 highest important variables µ± 2σ(%) µ± 2σ (%) µ± 2σ (%) µ± 2σ (%) random forest original 118(6.17) 113(5.90) 112(45.85) 118(6.17) m bi-square 118(6.17) 117(6.11) 75(3.92) 99(5.17) m hampel 72(3.76) 88(4.60) 92(4.81) 93(4.86) m huber 83(4.34) 90(4.70) 88(4.60) 102(5.33) support vector machine original 108(5.64) 98(5.12) 86(4.49) 87(4.55) m bi-square 64(3.34) 18(0.94) 84(4.39) 89(4.65) m hampel 66(3.45) 62(3.24) 85(4.44) 86(4.49) m huber 81(4.23) 83(4.34) 96(5.02) 99(5.17) bagging original 98(5.12) 96(5.02) 97(5.07) 84(4.39) m bi-square 126(6.58) 97(5.07) 95(4.96) 78(4.08) m hampel 101(5.28) 97(5.07) 90(4.70) 85(4.44) m huber 113(5.90) 99(5.17) 97(5.07) 89(4.65) boosting original 193(10.10) 168(8.78) 194(10.12) 194(10.12) m bi-square 77(4.02) 77(4.02) 133(6.95) 79(4.12) m hampel 76(3.97) 76(3.97) 72(3.76) 80(4.18) m huber 83(4.34) 81(4.23) 67(3.50) 85(4.44) 15 highest important variables, the maximum is boosting with 193 (10.1%) outliers, while the minimum is bagging with 98 (5.12%). for the 25 highest important variables, the maximum is boosting 168 (8.78%) , while the minimum is bagging with 96 (5.02%). for the 35 highest important variables, the maximum is boosting 194 (10.12%), while the minimum is 6 o. j. ibidoja et al. / j. nig. soc. phys. sci. 5 (2023) 1137 7 support vector machine with 86 (4.49%). for the 45 highest important variables, the maximum is boosting 194 (10.12%) , while the minimum is bagging with 84 (4.39%). based on this results, bagging with 45 variables importance gave the best performance because it has the lowest number of outliers of 84. based on the results in table 4 for the hybrid model, for the 15 highest important variables, bagging m bi-square has the highest number of outliers of 126 with 6.58% ,while support vector machine m bi-square has the lowest number of outliers with of 64 with 3.34%. for the 25 highest important variable random forest m bi-square has the highest number of outliers of 117 with 6.11% ,while support vector machine m bi-square has the lowest number of outliers with of 18 with 0.94%. for the 35 highest important variable boosting m bi-square has the highest number of outliers of 133 with 6.95% ,while boosting m huber has the lowest number of outliers with of 67 with 3.50%. for the 45 highest important variable random forest m huber has the highest number of outliers of 102 with 5.33% ,while bagging m bi-square has the lowest number of outliers with of 78 with 4.08%. based on this result, bagging m bi-square gave the best performance because it had the lowest number of outliers of 78 and used the highest number of high ranking variables. 5. conclusion the aim of this study is to develop a hybrid model, to forecast seaweed drying parameters that determine the moisture content removal that would enhance the quality of the seaweed. four predictive models such as random forest, support vector machine, bagging and boosting were built with m huber, m hampel and m bi-square to develop a hybrid model that can improve the predictive accuracy of the seaweed contaminated data. in summary, the best model to determine the moisture content removal of the seaweed big data is the bagging m bisquare, it gave the best performance because it had the lowest number of outliers of 78 and used the highest number of high ranking variables. for future study, a hybrid model with imbalanced data or missing values can be investigated. acknowledgement the authors are grateful to the ministry of higher education malaysia for fundamental research grant scheme with project code rgs/1/2022/stg06/usm/02/13 for their assistance. we are also grateful to the editor, associate editor, and anonymous reviewers for their insightful comments and suggestions to improve the quality and clarity of the paper. references [1] d. n. gujarati & d. n. porter, basic econometrics, 4th ed. new york, usa: the mcgraw-hill companies, (2004). 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[34] c. njeru & a. amayo, evaluation of quality control in clinical chemistry using sigma metrics, (2022). 8 j. nig. soc. phys. sci. 3 (2021) 12–16 journal of the nigerian society of physical sciences simple motion pursuit differential game j. adamua,∗, b. m. abdulhamidb, d. t. gbandec, a. s. halliruc adepartment of mathematics, federal university gashua, yobe, nigeria. bdepartment of mathematical sciences, abubakar tafawa balewa university, bauchi, nigeria. cdepartment of mathematical sciences, bayero university, kano, nigeria. abstract we study a simple motion pursuit differential game of many pursuers and one evader in a hilbert space l2. the control functions of the pursuers and evader are subject to integral and geometric constraints respectively. duration of the game is denoted by positive number θ. pursuit is said to be completed if there exist strategies u j of the pursuers p j such that for any admissible control v(·) of the evader e the inequality ‖y(τ)−x j(τ)‖ ≤ l j is satisfied for some j ∈ {1, 2, . . . } and some time τ. in this paper, sufficient condition for completion of pursuit were obtained. consequently strategies of the pursuers that ensure completion of pursuit are constructed. doi:10.46481/jnsps.2021.148 keywords: differential game, pursuer, evader, geometric constraint, integral constraint, hilbert space article history : received: 01 january 2021 received in revised form: 28 january 2021 accepted for publication: 29 january 2021 published: 27 february 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: b. j. falaye 1. introduction in view of the extensive literature on differential games of several player’s with the control functions subjected to either geometric, integral or both constraints. the work in the papers [1-26] and some reference their in attract the attention of many researchers . in many studies of differential game, motion of each player are explicitly stated and considered to be a system of differential equations of the same order. in the papers [2,5,7-9,13,16], motion of each of the player is considered to obey first order differential equation. in other studies such as refs. [4,6,12,15,19], ∗corresponding author tel. no: +2347033885836. email addresses: jamiluadamu88@gmail.com (j. adamu ), mbabdulhamid67@atbu.edu.ng (b. m. abdulhamid), princedavison4@gmail.com (d. t. gbande), aminuhalliru1@gmail.com (a. s. halliru) players’ motions are described by second order differential equations. whereas in refs. [1,23,25] motion of the players are described by first and second order differential equations. in ref, [24], rikhsiev studied simple motion differential game of optimal pursuit with one evader and many pursuers on a closed convex subset of the hilbert space l2. a sufficient condition for optimality of pursuit time is obtained, when the initial position of the evader belong to the interior of the convex hull of the initial position of the pursuers. simple motion differential game of many players with geometric constraints on the control functions of the players is studied in [13]. by using lyapunov function method for an auxiliary problem, they obtained sufficient conditions to find the pursuit time in rn. vagin and petrov in ref. [22] studied a pursuit differential game problem with finite number pursuers and one evader in the hilbert space rn. motions of each player is described by nth order differential equation. control functions of the players are subject to geometric constraints. they 12 adamu et al. / j. nig. soc. phys. sci. 3 (2021) 12–16 13 obtained sufficient condition for completion of pursuit. the work in ref. [26] leong and ibragimov studied simple motion pursuit differential game with m pursuers and one evader on a closed convex subset of the hilbert space l2 . control functions of the players are subjected to integral constraints. the total resource of the pursuers is assumed to be greater than that of the evader. strategy of pursuers were constructed sufficient to complete the pursuit from any initial position. in ref. [23] pursuit differential game problem for the socalled boy(evader) and crocodile (pursuer) in the space rn is studied. boy’s motion is described by first order differential equations and that of the crocodile by second order differential equation. control functions of the pursuer and evader are subject to integral and geometric constraints respectively. they obtained sufficient conditions of completion of pursuit. in this piece of research, we study pursuit differential game problem in a hilbert space l2, where motions of the pursuers and evader described by first and second order differential equations respectively. control functions of the pursuers are subject to integral constrains. whereas, geometric constraint is imposed on the control function of the evader. 2. statement of the problem consider the space l2 = % = (%1,%2, . . . ) : ∞∑ k=1 %2k < ∞  , with inner product 〈·, ·〉 : l2 × l2 → r and norm || · || : l2 → [0, +∞), defined as follows: 〈x, y〉 = ∞∑ k=1 xkyk, ||%|| =  ∞∑ k=1 %2k 1/2 , where x, y,% ∈ l2, respectively. we consider a differential game described by the following equations:{ p j : ẋ j = u j(t), x j(0) = x j0, j ∈ j, e : ÿ = v(t), ẏ(0) = y1, y(0) = y0, (1) where x j, x j0, ui, y, y0, y1, v ∈ l2, u j = (u j1, u j2, . . . ) is a control parameter of the pursuer p j and v = (v1, v2, . . . ) is that of the evader e. here and below j = {1, 2, . . . }, ||x j0 − y0|| > l j, where l j ≥ 0 are given numbers. in the space l2, we define a ball (respectively, sphere) of radius r and center at x0 by b(x0, r) = {x ∈ l2 : ||x − x0|| ≤ r} ( respectively, by s (x0, r) = {x ∈ l2 : ||x − x0|| = r}). definition 2.1. a function u j(t) = (u j1(t), u j2(t), . . . ) with borel measurable coordinates such that∫ θ 0 ||u j(t)|| 2dt ≤ ρ2j, (2) where ρ j is given positive number, is called an admissible control of the jth pursuer. definition 2.2. a function v(t) = (v1(t), v2(t), . . . ) with borel measurable coordinates such that ||v(t)|| ≤ σ, t ≥ 0, (3) is called admissible control of the evader. if the pursuers p j and evader e chose their admissible controls u j(·) and v(·) respectively, the solutions to the dynamic equations (1) are given by: x j(t) = x j0 + ∫ t 0 u j(s)d s, (4) y(t) = y0 + ty1 + ∫ t 0 ∫ r 0 v(s)d sdr. (5) it is not difficult to see that∫ t 0 ∫ r 0 v(s)d sdr = ∫ t 0 (t − s)v(s)d s (6) therefore equation (5) becomes y(t) = y0 + ty1 + ∫ t 0 (t − s)v(s)d s, (7) one can readily see that x j(·), y(·) ∈ c(0,θ; l2) , where c(, 0,θ; l2) is the space of functions h(t) = (h1(t), h2(t), · · · , hk(t), · · · ) ∈ l2, t ≥ 0, such that the following conditions hold: (1) h j(t), 0 ≤ t ≤ θ, j = 1, 2, · · · , are absolutely continuous functions; (2) h(t), 0 ≤ t ≤ θ, is a continuous function in the norm of l2. with this instead of differential game described by (1) we can consider an equivalent differential game with the same control functions described by:{ p : ẋ j = u(t), x j(0) = x j0 e : ẏ = (θ− t)v(t), y(0) = y1θ + y0 = y0. (8) indeed, if the evader uses an admissible control v(t) = (v1(t), v2(t), . . . ), then according to (1), we have y(θ) = y0+y1θ+ ∫ θ 0 ∫ r 0 v(s)d sdr = y0+y1θ+ ∫ θ 0 (θ−t)v(t)dt,(9) and the same result can be obtained by (8) y(θ) = y0 + ∫ θ 0 (θ−t)v(t)dt = y0 +y1θ+ ∫ θ 0 (θ−t)v(t)dt,(10) also, the same argument can be made for the pursuer p j, therefore in the distance ‖y(θ)−x j(θ)‖ we can take either the solution of (1) or the solution of (8). the attainability domain of the pursuer pi from the initial position xi0 up to the time θ is the ball b(x j0,ρi √ θ). indeed, by cauchy-schwartz inequality we have∣∣∣∣∣∣x j(θ) − x j0∣∣∣∣∣∣ 13 adamu et al. / j. nig. soc. phys. sci. 3 (2021) 12–16 14 = ∣∣∣∣∣∣ ∣∣∣∣∣∣x j0 + ∫ θ 0 u j(s)d s − x j0 ∣∣∣∣∣∣ ∣∣∣∣∣∣ = ∣∣∣∣∣∣ ∣∣∣∣∣∣ ∫ θ 0 u j(s)d s ∣∣∣∣∣∣ ∣∣∣∣∣∣ ≤ ∫ θ 0 ||u j(s)||d s ≤ (∫ θ 0 12d s ) 1 2 (∫ θ 0 ||u j(s)d s|| 2 ) 1 2 ≤ ρ j √ θ on the other hand, let x̄ ∈ b(x j0,ρ j √ θ). if the pursuer p j uses the control u j(s) = x̄−x j0 θ , 0 ≤ s ≤ θ, then we have x j(θ) = x j0+ ∫ θ 0 u j(s)d s = x j0+ ∫ θ 0 x̄ − x j0 θ d s = x j0+x̄−x j0 = x̄. in a similar fashion we can show that the attainability domain of the evader e from the initial position y0 up to the time θ is the ball b(y0,σ θ2 2 ). definition 2.3. a strategy of the jth pursuer is a function u j(t, x j, y, v), u j : [0,∞) × l2 × l2 × l2 → l2, such that the system{ ẋ j = u j(t, x, y, v(t)), x j(0) = x j0, ÿ = v(t), ẏ(0) = y1, y(0) = y0 has a unique solution (x j(·), y(·)), and that x j(·), y(·) ∈ c(0,θ; l2), for an arbitrary admissible control v = v(t), 0 ≤ t ≤ θ, of the evader e. a strategy u j is said to be admissible if each control formed by this strategy is admissible. definition 2.4. the system described by (1) in which the controls u j(·) and v(·) satisfy the inequalities (2) and (3) respectively is called game g. definition 2.5. pursuit is said to be completed in l-catch sense in the game g if there exist strategies u j of the pursuer p j such that for any admissible control v(·) of the evader e the inequality ‖y(τ) − x j(τ)‖ ≤ l j is satisfied for some j ∈ {1, 2, . . . } and some time τ ∈ [0,θ]. research problem: in the game g, find sufficient condition for completion of pursuit. we define the half space φ j = { α ∈ l2 : 2〈y0 − x j0,α〉 ≤ θ ( ρ2j −σ 2 θ 3 3 ) + ||y0|| 2 − ∣∣∣∣∣∣x j0∣∣∣∣∣∣2} . 3. main result in this section, we present the main result of the paper. theorem 3.1. if y(θ) ∈ φ j, then pursuit can be completed in the game g. proof to prove this theorem, we first introduce dummy pursuer with state variable z and motion described by the following equation. ż(t) = w(t), z(0) = x j0, where the control function w(t) is such that (∫ θ 0 ||w(t)||2 dt ) 1 2 ≤ ρ̄ = ρ j + l j √ θ . clearly, ρ̄ > ρ j and (ρ̄−ρ j) √ θ = l j for all j ∈ j. we construct the strategy of the dummy pursuer as follows: w(t) = { y0−x j0 θ + (θ− t)v(t), 0 ≤ t ≤ θ, 0, t > θ. (11) to show that the strategy (11) is admissible, we use the fact that y(θ) ∈ φ j. this means 2〈y0 − x j0, y(θ)〉 ≤ θ ( ρ2j −σ 2 θ 3 3 ) + ||y0|| 2 − ||x j0|| 2 (12) in accordance with the inequality (12) and using the state equation of the evader (10), we have: 2 〈 y0 − x j0, ∫ θ 0 (θ− t)v(t)dt 〉 (13) = 2〈y0 − x j0, y(θ) − y0〉 = 2〈y0 − x j0, y(θ)〉− 2〈y0 − x j0, y0〉 = 2〈y0 − x j0, y(θ)〉− 2||y0|| 2 + 2〈x j0, y0〉 ≤ θ ( ρ2j −σ 2 θ 3 3 ) + ||y0|| 2 − ||x j0|| 2 − 2||y0|| 2 + 2〈x j0, y0〉 = θ ( ρ2j −σ 2 θ 3 3 ) − ||y0|| 2 − ||x j0|| 2 + 2〈x j0, y0〉 = θ ( ρ2j −σ 2 θ 3 3 ) − ( ||y0|| 2 + ||x j0|| 2 − 2〈x j0, y0〉 ) = θ ( ρ2j −σ 2 θ 3 3 ) − ||y0 − x j0|| 2. (14) using inequality (13), we have∫ θ 0 ||w(t)||2dt = ∫ θ 0 ∣∣∣∣∣ ∣∣∣∣∣ y0 − x j0θ + (θ− t)v(t) ∣∣∣∣∣ ∣∣∣∣∣2 dt = ∫ θ 0 (∣∣∣∣∣ ∣∣∣∣∣ y0 − x j0θ ∣∣∣∣∣ ∣∣∣∣∣2 + 2 〈 y0 − x j0θ , (θ− t)v(t) 〉 + ||(θ− t)v(t)||2 ) dt = ∫ θ 0 ||y0 − x j0||2 θ2 d s + 2 ∫ θ 0 〈 y0 − x j0 θ , (θ− t)v(t) 〉 dt + ∫ θ 0 (θ− t)2||v(t)||2dt ≤ ||y0 − x j0||2 θ + 2 θ 〈 y0 − x j0, ∫ θ 0 (θ− t)v(t)dt 〉 + σ2 ∫ θ 0 (θ− t)2dt ≤ ||y0 − x j0||2 θ + 1 θ ( θ ( ρ2j −σ 2 θ 3 3 ) − ||y0 − x j0|| 2 ) + σ2 θ3 3 = ρ2j < ρ̄ therefore the strategy (11) is admissible. 14 adamu et al. / j. nig. soc. phys. sci. 3 (2021) 12–16 15 suppose that the dummy pursuer z uses the strategy (11). one can easily see that w(θ) = y(θ). indeed, z(θ) = x j0 + ∫ θ 0 ( y0 − x j0 θ + (θ− t)v(t) ) d s = x j0 + ∫ θ 0 ( y0 − x j0 θ ) dt + ∫ θ 0 (θ− t)v(t)dt = x j0 + y0 − x j0 + ∫ θ 0 (θ− t)v(t)dt = y(θ). using the strategy of the dummy pursuer, we define the strategies of the real pursuers p j, j ∈ j as follows: u j(t) = ρ j ρ̄ w(t). (15) the admissibility of this strategies follows from the fact that the control w(·) is admissible. therefore it is left to show that∣∣∣∣∣∣y(θ) − x j(θ)∣∣∣∣∣∣ ≤ l j. indeed, using cauchy-schwartz inequality we have∣∣∣∣∣∣y(θ) − x j(θ)∣∣∣∣∣∣ = ∣∣∣∣∣∣z(θ) − x j(θ)∣∣∣∣∣∣ = ∣∣∣∣∣∣ ∣∣∣∣∣∣x j0 + ∫ θ 0 w(t)dt − x j0 − ∫ θ 0 u j(t)dt ∣∣∣∣∣∣ ∣∣∣∣∣∣ = ∣∣∣∣∣∣ ∣∣∣∣∣∣ ∫ θ 0 w(t)dt − ∫ θ 0 ρ j ρ̄ w(t)dt ∣∣∣∣∣∣ ∣∣∣∣∣∣ ≤ ∫ θ 0 ∣∣∣∣∣∣ ∣∣∣∣∣∣ ( 1 − ρ j ρ̄ ) w(t) ∣∣∣∣∣∣ ∣∣∣∣∣∣ dt = ( ρ̄−ρ j ρ̄ ) ∫ θ 0 ||w(t)||dt ≤ ( ρ̄−ρ j ρ̄ )  (∫ θ 0 12dt ) 1 2 (∫ θ 0 ||w(t)||2 dt ) 1 2  ≤ ( ρ̄−ρ j ρ̄ ) ρ̄ √ θ = (ρ̄−ρ j) √ θ = l j. this complete the prove of the theorem. illustrative example let ρ j = 5, σ = 8, θ = 1 in the game g. we consider the following initial positions x j0 = (0, 0, . . . , 3, 0, . . . ), y0 = (0, 0, . . . ) of the players, where the number 3 is jth coordinate of the point x j. observe that ρ j √ θ = 5, σθ 2 2 = 4, ∣∣∣∣∣∣x j0 − y0∣∣∣∣∣∣ =( 02 + 02 + · · · + 32 + 02 + . . . ) 1 2 = 3 > 0. we show that y(θ) ∈ φ j. it is suffices to show that the inclusion b(0,σ θ2 2 ) ⊂ ⋃ j∈j b(x j0,ρ j √ θ), holds, where 0 is the origin. indeed, let z = (z1, z2, . . . ) be arbitrary nonnegative point of the ball b(0, 4) : ∑ ∞ j=1 z 2 j ≤ 16. then ∣∣∣∣∣∣z − x j0∣∣∣∣∣∣ = (z21 + · · · + z2j−1 + (3 − z j)2 + z2j+1 + . . . ) 12 =  ∞∑ j=1 z2j + 9 − 6z j  1 2 ≤ ( 16 + 9 − 6z j ) 1 2 = ( 25 − 6z j ) 1 2 ≤ 5. this means that hypothesis of our theorem is satisfied, therefore pursuit can be completed in the game g. 4. conclusion we have studied a simple motion pursuit differential game problem in which countable number of pursuers chase one evader in the hilbert space l2. control function of the pursuers and evader are subject to integral and geometric constraints respectively. pursuers’ motions are described by 1st order differential equations and that of the evader by 2nd order differential equation. in this piece of research the strategies of the pursuers are constructed and sufficient condition for completion of pursuit were obtained. for further research, value of the game and optimality of the pursuit times can also be investigated. acknowledgements the authors would like to thank the reviewers for giving useful comments and suggestions for improvement of this paper. references [1] j. adamu, k. muangehoo, a. j. badakaya, j. rilwan, “on pursuit-evasion differential game problem in a hilbert space”, aims mathematics, 6 (2020) 7467. 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okpara university of agriculture, umudike, nigeria bdepartment of chemistry, federal university of technology owerri, nigeria abstract sulfathiazole (sftz) is an antibacterial drug that contains organosulfur compound. it is used as a short-acting sulfa drug. the metal complexes of sulfa-drug have gained considerable importance due to their pronounced biological activity. the sulfa-drugs have received great attentions because of their therapeutic applications against bacterial infections. mn(ii) complex of sulfathiazole was synthesized by reaction of sulfathiazole with mncl2 ·4h2o. the mn (ii) complex was characterized based on uv, ir, 1h nmr spectroscopy and x-ray powder diffraction. the electronic spectrum of the ligand showed intra charge transfer which were assigned to the chromophores present in the ligand, while that of the complex suggested intra ligand charge transfer (ilct) and ligand to metal charge transfer (lmct). in the ir spectrum of sulfathiazole the n −h stretch of s o2 n h appeared at 3255.23cm−1. in the ir spectrum of the metal complex this band was absent. this suggested the deprotonation of the n − h of s o2 n h during complexation reaction. this showed that sulfathiazole acted as a monodentade ligand. 1h nmr spectrum of [mn(sftz)] complex showed the involvement of nitrogen atom of s o2 n h. the crystal structure of [mn(sftz)] complex belongs to monoclinic system, space group p1, with cell parameters of a = 4.519 å, b = 8.704 å, c = 12.608 å, v = 493.5 å 3 , β = 95.69◦. molecular docking suggested that the ligand/complex binded effectively with the e.coli and s.aureus because their global binding energies were negative. the binding interactions of ligand/complex with e. coli and s. aureus were predicted. molecular docking predicted the feasibility of the biochemical reactions before experimental investigation. it was concluded that sulfathiazole behaved as a monodentate ligand towards mn (ii) ion. the binding energy and interaction of [mn(sftz)] with e.coli and s. aureus have also shown that inhibition of the bacterial species are feasible. the mechanism of action of [mn(sftz)] with e. coli and s. aureus is now well understood. keywords: sulfathiazole, spectra bacteria, complex, docking article history : received: 15 june 2019 received in revised form: 01 september 2019 accepted for publication: 02 september 2019 published: 13 october 2019 c©2019 journal of the nigerian society of physical sciences. all rights reserved. communicated by: w. a. yahya 1. introduction sulfathiazole (figure 1) was the first therapeutic agents used systematically for the cure and prevention of bacterial infections. furthermore, sulfadrugs and their metal complexes, pos∗corresponding author tel. no: +2348065297631 email address: ifeanyiotuokere@gmail.com (i. e. otuokere ) sess many applications as diuretic, antiglaucoma or antiepileptic drugs, among others. sulpha drugs show important biological activity e.g mechanism of action is based on the competitive antagonism of paba (p-aminobenzoic acid) and the sulfanilamide [1, 2]. it has been reported that the activity of the metal complex is much better than the ligand alone [3, 4]. studies on their metal chelates have much physiological and pharmacological relevance because the metal chelates of sulfadrugs have 95 otuokere et al. / j. nig. soc. phys. sci. 1 (2019) 95–102 96 been found to be more bacteriostatic than the drugs themselves [5, 6]. the role of metal ions in living systems has been well established in recent years. the use of transition metal complexes as medicinal compounds has become more and more prominent. these complexes offer a great diversity in their action; they do not only have anti-cancer properties but have also been used as anti-inflammatory, anti-infective and anti-diabetic compounds [7]. metal ions play pivotal roles in many biological processes, and the study of the roles of these metal ions in biological systems falls into the rapidly developing interdisciplinary field known as bioinorganic chemistry. when compared to other branches of natural sciences, bioinorganic chemistry seems to be a young discipline. however, there is a copious amount of information on the effects of metals on biological systems. for instance, the toxicities of metal ions such as mercury, lead and chromium on the environment have been well publicized [8, 9]. metal complexes containing the sulphonamide group has found importance because of their applications as biological, biochemical, analytical, antimicrobial, anticancer, antibacterial, antifungal and antitumor activity [10, 11, 12]. they also find application as antibiotics, anti-inflammatory agents and in the industry as anticorrosion agents [13-17]. molecular complexes of sulfonamides have been reported [18]. syntheses, characterization, thermal and antimicrobial studies of binuclear metal complexes of sulfa-guanidine schiff bases have been reported [19]. the metal complexes of sulfaguanidine were assessed to be more potent than the free ligand [19]. it is in view of this pharmacological importance of sulphonamide that we have reported the synthesis, characterization and molecular docking studies of mn-sulphathiazole. figure 1: structure of sulfathiazole 2. material and methods all chemicals and reagent used in this experimental work were of analytical grade. pure sulfathiazole, and mncl2 ·4h2o salt were all imported from sigma–aldrich laboratories. the solvents are ethanol, methanol, acetone, chloroform, sodium hydroxide, benzene and dimethyl sulfoxide. synthesis of [mn(sftz)]: the complex was prepared following a reported procedure [21]. mn (ii) salt solution was prepared by dissolving 3.96 g(0.02 mol) mncl2 ·4h2o in 25 ml of distilled water. the solution of the metal salt was added slowly with stirring in a separate 20 ml of distilled water containing 5.1 g of sulfathiazole (0.02 mol) at room temperature maintaining the ph between 6.0 6.5 by adding dilute solution of koh. the synthesis was carried out with stirring at room temperature. after 1 hour, the complex separated out. the complexes were washed well with distilled water, recrystallized, filtered and finally dried in vacuum and weighed and melting point recorded. melting points of the complex was determined using mpa160 melting point apparatus. atomic absorption spectroscopy was carried out on duck-2010 spectrometer (duck instrumental company) [20]. infrared spectrum was collected on perkin elmer paragon 1000 ft-ir spectrophotometer (spectrum bx) equipped with cesium iodide window (4000 − 350cm−1) in k br pellets. the uv-visible spectrum was obtained on a perkin elmer (lambda 25) spectrometer (200−800 nm) using distilled water as solvent. the 1 h nuclear magnetic resonance (nmr) spectra were obtained using varian 400 mhz unity inova, using dmso as solvent. in the crystallographic studies, appropriate amounts of the crystal was collected and deposited on bruker d8 diffractometer operating in transmission mode usin germanium monochromated cukα1 radiation, λ = 1.5406 å, linear position-sensitive detector covering 12◦ in 2θ, 2θ mode range 3.5◦ 70◦, step size 0.017◦ and 17 h data collection time. fox software was used for structure determination and refinement. molecular docking: the three-dimensional structure of escherichia coli and staphylococcus aureus and were obtained from the protein data bank, pdb 1e91 and 1stn respectively. the protein structures were subjected to a refinement protocol using molegro molecular viewer. molecular docking was performed using patchdock server: an automatic server for molecular docking [22]. refinement was done in firedock server: an automatic server for fast interaction refinement in molecular docking and processed with molegro molecular viewer [23-26]. 3. results and discussion crystallographic data and structure refinement parameters for [mn(sftz)] is given in table 1, whereas the powdered xray diffraction is shown in figure 2 figure 2: powdered x-ray diffraction of [mn(sftz)]. the crystal structure of [mn(sftz)] complex belongs to monoclinic system, space group p1, with cell parameters of a = 4.519 å, b = 8.704 å, c = 12.608 å, v = 493.5 å 3 , β = 95.69◦. elemental and physical properties of sulfathiazole and its metal complex are shown in table 2 the elemental analysis of sulfathiazole and its mn(i i) complex showed that the experimental values are in agreement with the calculated values the colour of the new product suggested the formation of complex because transition metal complexes 96 otuokere et al. / j. nig. soc. phys. sci. 1 (2019) 95–102 97 table 1: crystal data and structure refinement for sulfathiazole and its (mn(ii) complex parameters [mn(sftz)] temperature (k) 298 wavelength (å) 0.71073 crystal system monoclinic space group p 1 a(å) 4.519 b(å) 8.704 c(å) 12.608 α(◦) 90 β(◦) 95.69 γ(◦) 90 volume (å 3 ) 493.51 (1.0v) table 2: elemental and physical properties of sftz and [mn(sftz)] ligand/complex % mn colour melting point yield (%) found ◦c (calculated) sftz — white 202 – 202.5 — [mn(s ft z)] 17.50 pink 141 142 86 (17.77) are coloured. the change in melting point also indicated the formation of new complex. the infrared spectra data of sulfathiazole and its mn(i i) complex are presented in table 3 figure 3: ir spectrum of sulfathiazole [21]. figure 4: ir spectrum of [mn(sftz)]. a comparison of ir spectrum of sftz and that of the complex was made (figures 3 and 4). the infrared spectrum of sftz showed a broad band at 3354.00 and 3321.00 cm−1 [21]. this band was assigned n − h stretch of the primary amine due to asymmetric and symmetric stretching vibrations of the two n − h bonds. in the ir spectra of the mn(i i) complex, this vibration frequency remained unchanged. this suggested that n h2 was not involved in complexation. vibration frequency 1323.00 cm−1 and 1140.00 cm−1 were assigned to be vas(o=s=o) and vs(o=s=o) in sftz. in the complex, these frequencies showed up at 1319.79 and 1138.39 cm−1 in [mn(sftz)]. it is evident that sulfonyl group was not involved in coordination to mn. in sftz spectrum c − n stretching vibration was observed at 1497.00 cm−1. in the spectrum of the complex, these functional group was observed at 1494.05 cm−1 [mn(sftz)]. this observation suggest that coordination did not occurred through c − n in [mn(sft)]. the n − h stretch of s o2 n h appeared at 3255.23 cm−1 in the free ligand. in the ir spectrum of the metal complex this band was absent. this suggested the deprotonation of the n−h of s o2 n h during complexation reaction. the uv spectral data of sulfathiazole and its mn(i i) complex are presented in table 4, while the spectra are present in figures 5 and 6. figure 5: uv-vis spectrum of sulfathiazole. the uv-vis spectrum of sftz showed a band centered at 269 nm. it was assigned π − π∗ due to intra-ligand charge 97 otuokere et al. / j. nig. soc. phys. sci. 1 (2019) 95–102 98 table 3: infrared spectral data of sulfathiazole and its mn(i i) complex ligand/complex vas(o=s=o) vs(o=s=o) v (n h) v (cn) (n − h) (cm−1) (cm−1) (cm−1) (cm−1) (cm−1) (primary amine) (s o2 n h) sftz 1323.00 1140.00 3354.00, 3321.00 1497.00 3255.23 [mn(s ft z)] 1319.79 1136.39 3350.32, 3320.10 1494.05 absent table 4: the uv spectral data of sulfathiazole and its complex. ligand/metal complex λmax(nm) assignment sft 269 π−π∗(ilct) [mn(s ft z)] 270 π−π∗(ilct) 230 lmct figure 6: uv-vis spectrum of [mn(sftz)]. transfer (ilct).the uv-vis spectrum of [mn(sftz)] showed a band centered at 270 nm which has been assigned ilct due to π−π∗. the chromophores that may exhibit this transition are s=o and c=n. a sharp peak centered at 230 nm suggested ligand to metal charge transfer (lmct). the 1 h − n mr spectral data of sulfathiazole and its mn(i i) complex are presented in table 5. the spectra are shown in figures 7 and 8. figure 7: 1 hn mr spectrum of sulfathiazole [21]. in the 1 hn mr spectrum of sftz, the aromatic protons appeared at 6.51 and 7.43 ppm while the thiazole protons are observed between 6.71 and 7.18 ppm [21]. nh2 protons were observed at 5.80 ppm. in the spectrum of the metal complex, these chemical shifts remained relatively unchanged. in the hnmr spectrum of sftz, the hydrogen that appeared as a singlet at 12.4 ppm is no longer observed in the spectra of the metal complex. this is attributed to the loss of hydrogen atom of figure 8: 1 hn mr spectrum of [mn(sftz)]. the (o2s − n − h) group of sftz when coordination occurred through the nitrogen to the metal centre. based on the uv, ir, 1 hn mr spectra and x-ray powder diffraction, the structure (figure 9) has been proposed for [mn(sftz)]. figure 9: proposed structure of [mn(sftz)]. the solutions tables of the molecular docking are shown in tables 6 9. the crystal structure of the e. coli rna degradosome component enolase and s. aureus nuclease are shown in figures 10 and 11 respectively. the crystal structure of e. coli contains four protein chains (a, b, c and d) and 506 water molecules. the crystal structure of s. aureus nuclease is made up of one protein chain (a) and 83 water molecules. the molecular docking and molecular interactions of sulfathiazole with e. coli are presented in figures 12a and 12b. the molecular docking and molecular interactions of [mn(sftz)] with e. coli are presented in figures 13a and 13b. the molecular docking and molecular interactions of sulfathiazole with s. aureus 98 otuokere et al. / j. nig. soc. phys. sci. 1 (2019) 95–102 99 table 5: 1 h − n mr spectral data of sulfathiazole and its complex. ligand/ thiazole protons o2s − n − h n h aromatic complex (δ ppm) (δ ppm) (δ ppm) (δ ppm) sft 6.71 7.18 12.4 5.80 6.51-7.43 [mn(s ft z)] 7.02 -7.45 absent 5.80 6.51-7.43 nuclease are presented in figures 14a and 14b. . the molecular docking and molecular interactions of [mn(sftz)] with s. aureus nuclease are presented in figures 15a and 15b. figure 10: crystal structure of e. coli rna degradosome component enolase. figure 11: crystal structure of s. aureus nuclease. figure 12: (a) crystal structure of e. coli rna degradosome component enolase docked with sulfathiazole. (b) molecular interactions of sulfathiazole with e. coli rna degradosome component enolase. the best ranking in table 6 is solution 2 with global energy -46.74 kcal/mol. this suggested that sulfathiazole has the figure 13: (a) crystal structure of e. coli rna degradosome component enolase docked with [mn(sftz)]. (b) molecular interactions of [mn(sftz)] with e. coli rna degradosome component enolase. figure 14: (a) crystal structure of s. aureus nuclease docked with sulfathiazole. (b) molecular interactions of sulfathiazole with s. aureus nuclease. figure 15: (a) crystal structure of s. aureus nuclease docked with [mn(sftz)]. (b) molecular interactions of [mn(sftz)] with s. aureus nuclease. ability to inhibit e. coli. the attractive vander waals and atomic contact energy (ace) showed negative values. these suggested that sulfathiazole docked effectively with e. coli. the molecular interactions (figure 12b) show that e. coli formed hydrogen bonding with sulfathiazole using ala 247(c) and glu 250(c). steric interaction between e. coli and sulfathiazole were observed with gly 156(c), glu 157(c), asn 161(c), ala 260(c), asn 162(c), val 163(c), and asp 164(c). the best global energy in table 7 is -40.08 kcal/mol (so99 otuokere et al. / j. nig. soc. phys. sci. 1 (2019) 95–102 100 table 6: solution table of sftz docked with e. coli(vdw = vanderwaals; ace = atomic contact energy). rank solution global attractive repulsive ace number energy vdw vdw (kcal/mol) (kcal/mol) (kcal/mol) (kcal/mol) 1 2 -46.74 -15.12 1.68 -16.11 2 5 -39.59 -14.83 1.49 -12.46 3 9 -34.11 -12.29 1.08 -10.99 4 10 -25.97 -13.63 0.99 -4.88 5 6 -21.88 -13.34 3.23 -3.65 6 1 -21.72 -12.16 0.20 -2.46 7 4 -17.18 -13.71 3.19 0.16 8 7 -11.34 -8.06 2.10 -2.04 9 3 -5.73 -12.05 18.51 -2.15 10 8 3.31 -12.71 29.59 -1.56 table 7: solution table of [mn(sftz)] docked with e. coli(vdw = vanderwaals; ace = atomic contact energy). rank solution global attractive repulsive ace number energy vdw vdw (kcal/mol) (kcal/mol) (kcal/mol) (kcal/mol) 1 9 -40.08 -11.96 3.61 -15.77 2 4 -38.68 -14.27 2.68 -12.57 3 1 -35.93 -13.63 2.29 -11.48 4 6 -33.85 -11.60 1.10 -11.54 5 8 -29.70 -11.47 1.62 -9.17 6 10 -29.18 -9.28 0.71 -10.35 7 3 -20.15 -10.26 1.42 -5.76 8 5 -19.49 -10.86 3.63 -6.83 9 7 -18.03 -12.44 4.59 -2.38 10 2 -10.31 -13.92 8.90 1.35 table 8: solution table of sftz docked with s. aureus (vdw = vanderwaals; ace = atomic contact energy). rank solution global attractive repulsive ace number energy vdw vdw (kcal/mol) (kcal/mol) (kcal/mol) (kcal/mol) 1 1 -25.35 -11.53 3.66 -7.77 2 9 -24.36 -13.36 1.42 -5.00 3 2 -21.31 -9.73 2.95 -6.32 4 6 -19.93 -8.76 1.77 -7.08 5 3 -19.27 -6.79 0.61 -7.54 6 8 -18.63 -11.33 1.99 -3.87 7 7 -16.25 -9.00 2.49 -5.05 8 10 -10.07 -5.26 2.43 -3.60 9 5 -9.98 -8.36 5.39 -4.11 10 4 -9.65 -5.51 2.56 -4.53 100 otuokere et al. / j. nig. soc. phys. sci. 1 (2019) 95–102 101 table 9: solution table of [mn(sftz)] docked with s. aureus (vdw = vanderwaals; ace = atomic contact energy). rank solution global attractive repulsive ace number energy vdw vdw (kcal/mol) (kcal/mol) (kcal/mol) (kcal/mol) 1 4 -23.56 -9.86 4.80 -9.09 2 8 -23.48 -10.23 1.83 -7.94 3 1 -22.65 -11.19 4.93 -8.21 4 5 -16.39 -9.69 1.59 -3.65 5 9 -13.64 -6.72 4.14 -6.44 6 2 -13.17 -6.45 5.68 -5.97 7 6 -11.25 -7.38 2.10 -3.20 8 7 -7.27 -4.41 2.60 -4.43 9 10 -5.71 -4.97 2.29 -1.96 10 3 0.59 -11.04 38.87 -9.47 lution 9). this suggested that [mn(sftz)] has the ability to inhibit e. coli. the attractive vander waals and atomic contact energy (ace) were also predicted. their negative value predicted effective binding. the molecular interactions (figure 13b) showed that e. coli formed hydrogen bonding with [mn(sftz)] through gly 166(b) and hoh 208(b). steric interactions between [mn(sftz)] and e. coli occured with his 158(b), ala 247(b), ser 249(b), gln 166(b) and asp 316(b). the best ranking in table 8 is solution 1 with global energy -25.35 kcal/mol. this suggested that sulfathiazole has the ability to inhibit s. aureus. the attractive vander waals and atomic contact energy (ace) showed negative values. these suggested that sulfathiazole docked effectively with s. aureus. the molecular interactions (figure 14b) showed that s. aureus formed hydrogen bonding with sulfathiazole using hoh 225(a) and hoh 291(a). steric interaction between s. aureus and sulfathiazole were observed with gln 80, lys 116, tyr 115 and pro 117. the best ranking in table 9 is solution 4 with global energy 23.56 kcal/mol. this suggested that [mn(sftz)] has the ability to inhibit s. aureus. the attractive vander waals and atomic contact energy (ace) showed negative values. these suggested that [mn(sftz)] docked effectively with s. aureus. the molecular interactions (figure 15b) show that s. aureus formed hydrogen bonding with [mn(sftz)] using hoh 295(a), hoh 242(a) and glu 52. steric interaction between s. aureus and [mn(sftz)] were observed with pro 42, lys 110, tyr 41, glu 43 and glu 52. 4. conclusion complex of manganese ion with sulfathiazole was successfully synthesized. the colour, ir, uv 1 h nmr spectra and xray powder diffraction suggested that new products were formed. this also shows that sulfathiazole can be used to remove toxic metals from the environment or from the biological system. this is because they can be complexed with sulfathiazole. molecular docking study predicted the binding energies and interactions between the compounds and bacterial strains. it helped us to understand the mechanism of action of the proposed complex. a thorough investigation should be carried out to find out whether the synthesized drugs can be safely used as a metal based anti-bacterial drug for the treatment of bacterial infections. we also recommend toxicology test for the complexes. acknowledgments we thank the referees for the positive enlightening comments and suggestions, which have greatly helped us in making improvements to this paper. references [1] g. m. h. golzar, “synthesis and characterisation of cobalt complex of sulfathiazole with acetic acid”, j. saudi chemical society 17 (2013) 253. 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[26] m. zacharias “accounting for conformational changes during proteinprotein docking”, current opin structural biology 20 (2010) 180. 102 j. nig. soc. phys. sci. 3 (2021) 96–104 journal of the nigerian society of physical sciences stability and sensitivity analysis of dengue-malaria co-infection model in endemic stage solomon akyenyi ayubaa,∗, imam akeyedea, adeyemi sunday olagunjua,b adepartment of mathematics federal university of lafia, nigeria bdepartment of mathematical science bingham university, karu, nigeria abstract in this study, a deterministic co-infection model of dengue virus and malaria fever is proposed. the disease free equilibrium point (dfep) and the basic reproduction number is derived using the next generation matrix method. local and global stability of dfep are analyzed. the results show that the dfep is locally stable if r0dm < 1 but may not be asymptotically stable. from the analysis of secondary data sourced from kenyan region, the value of r0dm computed is 19.70 greater than unity; this implies that dengue virus and malaria fever are endemic in the region. to identify the dominant parameter for the spread and control of the diseases and their co-infection, sensitivity analysis is investigated. from the numerical simulation using maple 17, increase in the rate of recovery for co-infected individual contributes greatly in reducing dengue and malaria infections in the region. decreasing either dengue or malaria contact rate also play a significant role in controlling the co-infection of dengue and malaria in the population. therefore, the center for disease control and policy makers are expected to set out preventive measures in reducing the spread of both diseases and increase the approach of recovery for the co-infected individuals. doi:10.46481/jnsps.2021.196 keywords: co-infection, dengue, malaria, stability analysis, sensitivity analysis, simulation. article history : received: 9 april 2021 received in revised form: 30 april 2021 accepted for publication: 5 may 2021 published: 29 may 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: b. j. falaye 1. introduction the spread of mosquitoes borne diseases has gained concern globally in recent decades because of their recurring outbreaks. millions of people die every year as a result of these infectious diseases and their control has increasingly become a complex issue [1]. dengue virus and malaria fever are common mosquitoes-borne diseases that have become a public health threat in the last few decades with high morbidity and mortality for many patients in various part of the world [2]. the world ∗corresponding author tel. no: +234(0)8130254488 email address: ayubasolomona@gmail.com (solomon akyenyi ayuba) malaria report [3], estimated 229 million cases of malaria in 2019 compared to 228 millions cases in 2018, with 409 000 deaths. 94% of the cases and deaths are reported from subsaharan africa. dengue is currently common in tropical and subtropical regions. the virus have four distinct stereotypes and are transmitted to human through bite of infected aedes mosquitoes (aegyptic & albopictus) [4]. dengue cases reported increased over 8 fold in the last two decades from 505430 cases in the year 2000 to 2.4 million in 2010 and to 4.2 million in 2019 [5]. while dengue is causing devastating impacts on the tropical and subtropical communities, malaria fever is endemic in some of these dengue affected regions there by drastically increases public health burden among the people in tropical 96 ayuba et al. / j. nig. soc. phys. sci. 3 (2021) 96–104 97 communities living at risk of contracting both diseases concurrently. the two pathogens share similar geographical areas, and clinical distinction between them is difficult due to their overlapping symptoms. the work in [6], the researchers proposed a mathematical model to study the transmission dynamics of zika and malaria in malaria-endemic area. in ref. [7] developed a novel mathematical model describing the co-infection dynamics of malaria and typhoid fever. [8] formulated a deterministic co-infection model between malaria and hiv in human population. ref. [9] developed and analyzed the stability of disease free equilibrium point (dfep) of a co-infection model between dengue virus and chikungunya in closed population. [10] developed a mathematical model for dengue-zika co-infection and carried out their synergistic relationship in the presence of prevention and treatment. ref. [11] proposed a co-infection of altered vector infectivity and antibodydependent enhancement of dengue-zika interplay. ref. [12] formulated and analyzed a co-infection model of dengue fever and leptospirosis diseases. in [13], a deterministic model for dengue, malaria and typhoid triple co-infection was developed but limited only to the stability (local and global) analysis. the authors in ref. [14], developed a seir co-infection model of dengue and malaria but only established the local and global stability. in this study, we propose a sir-si deterministic model of dengue virus and malaria co-infection and determine the stability analysis, sensitivity analysis and carryout numerical simulation for the co-infection model. the remainder of this paper is arranged as follows: in section 2, model descriptions, flow diagram (depicting the co-infection interactions) and the model formulation are presented. section 3 is devoted to results and analysis; invariant region, disease free equilibrium point, basic reproduction number, stability analysis, parameters estimation, sensitivity analysis and numerical simulation. discussion of findings is presented in section 4. finally, conclusions are drawn in section 5 and some possible directions for future studies are presented. 2. model formulation 1 the data used in this study are secondarily sourced from [7, 15]. in accordance with previous studies on mathematical model of dengue virus [10, 16, 17, 15] and malaria model [19, 20, 21, 22], we formulate a sir-si deterministic model of dengue and malaria co-infection. in this model, the total human population nh is partitioned into seven classes; susceptible human s h, infected human with dengue virus ihd , infected human with malaria ihm, infected human with both dengue virus and malaria idm, recovery of infected human from dengue virus ,malaria fever and co-infected individuals are rhd , rhm, rdm respectively. the vectors population are subdivided into; susceptible dengue vector s vd , dengue carrier vector ivd , susceptible malaria vector s vm and malaria carrier vector ivm. the recruitment rates for human, dengue and malaria vectors respectively, are λh, λd and λm. the recovery rate from dengue and malaria 1stability and sensitivity analysis of dengue-malaria co-infection model are σ,α, transmission rate of dengue and malaria vectors to human per unit time are ηd, ηm, probability of dengue and malaria vectors to be infected are denoted by ηvd, ηvm respectively. recovered human from malaria become susceptible at γ and acquired immunity ρ rate. the co-infected individuals recover at the rate ψ; but those individuals either recover only from dengue and join rhd with probability of qψ, or recover only from malaria and join rhm with probability of ψl(1 − q), or recover from both diseases and join rdm with the probability of ψ(1 − l)(1 − q). the human natural death rate denote µh while dengue and malaria vectors death rate are µd ,µm respectively. τ, δ are dengue and malaria induced death rates while φ, θ are dengue and malaria related death rates. the following assumptions are made to formulate the co-infection model: the total population is not constant, the susceptible rates are recruited through birth or immigration and the number increases from malaria recovered and co-infectious recovered individuals by losing their temporal immunity. recovered individuals from dengue virus is permanent. figure 2 shown the flow diagram for the interactions between dengue and malaria co-infection model in human population. the time dependent dynamical figure 1. flow diagram depicting dengue virus and malaria co-infection dynamics system associated with the parameters interaction is shown as follows. s ′h = λh + γrhm + πrdm − (ηd ivd +ηm ivm ) nh s h −µhs h i′hd = ηd ivd nh s h − ηm ivm nh ihd − (σ + τ + µh + φ)ihd i′hm = ηm ivm nh s h − ηd ivd nh ihm − (α + ρ + δ + µh + θ)ihm i′dm = ηm ivm nh ihd + ηd ivd nh ihm − (ψ + µh + θ + φ)idm r′hd = σihd + qψidm −µhrhd r′hm = αihm + ψl(1 − q)idm − (γ + µh)rhm r′dm = ψ(1 − l)(1 − q)idm − (π + µh)rdm s ′vd = λd − ηvd (ihd +idm ) nh s vd −µd s vd i′vd = ηvd ihd nh s vd + ηvd idm nh s vd −µd ivd s ′vm = λm − ηvm (ihm +idm ) nh s vm −µms vm i′vm = ηvm ihm nh s vm + ηvm idm nh s vm −µm ivm (1) 97 ayuba et al. / j. nig. soc. phys. sci. 3 (2021) 96–104 98 table 1. parameters description of dengue and malaria co-infection model parameters description λh recruitment rate of human population λd recruitment rate of dengue vectors λm recruitment rate of malaria vectors ρ rate of human acquired immunity from malaria α rate of human recovery from malaria σ rate of human recovery from dengue ψ rate of human recovery from both dengue and malaria γ rate of immunity warning for rhm to become susceptible ηd transmission rate of dengue vectors to human per unit time ηm transmission rate of malaria vectors to human per unit time ηvd probability for dengue vectors to be infected ηvm probability for malaria parasite vectors to be infected qψ proportion of co-infected human recovery from dengue only ψl(1 − q) proportion of co-infected human recovery from malaria only π rate at which rdm become susceptible τ disease induced death rate for human infected with dengue δ disease induced death rate for human infected with malaria φ dengue related death rate θ malaria related death rate µh natural death rate of humans µd natural death rate of dengue vectors µd natural death rate of malaria vectors 3. results and analysis 3.1. invariant regions in this section, we obtain the bounded region of solution for the dengue-malaria model. the total human population is given by nh = s h + ihd + ihm + idm + rhd + rhm + rdm, then n′h = s ′ h + i ′ hd + i ′ hm + r ′ hd + i ′ dm + r ′ hm + r ′ dm (2) =⇒ n′ = λh −µh nh (3) solving equation (3) as t →∞ yields dh = {(s h, ihd, ihm, idm, rhd, rhm, rdm) ∈< 7; 0 ≤ n ≤ λh µh } for the dengue vector population, if there is no spread of infection, then n′d = λd −µd nd (4) dd = {(s vd, ivd ) ∈< 2; nd ≤ λd µd } similarly, for malaria vector population, we obtain n′m = λm −µm nm (5) dm = {(s vm, ivm) ∈< 2; nm ≤ λm µm } therefore, the feasible solution of dengue-malaria model is given by d = {(dh × dd × dm)< 11 + } thus, the solution of dengue-malaria model is bounded in d. theorem 3.1. if at t = 0 and {s h(0), ihd (0), ihm(0), idm(0), rhd (0), rhm(0), rdm(0) , s vd (0), ivd (0), s vm(0), ivm(0)} ≥ 0, then the solution of dengue-malaria model are nonnegative at t > 0. 3.2. existence of disease free equilibrium point 2 to investigate the condition of existence of the disease free equilibrium point and also the asymptotic behaviour of the dengue-malaria co-infection model in this section, we will investigate whether the diseases die out or become endemic. this can only be addressed through the asymptotic behaviour of the diseases. this behaviour depends largely on the equilibrium point, that is time-independent solutions of the system. since these solutions are independent of time, we set the left hand side of system (1) to zero. s ′h = i ′ hd = i ′ hm = i ′ dm = r ′ hd = r ′ hm = r′dm = 0 and s ′ vd = i ′ vd = s vm = i ′ vm = 0. 2stability and sensitivity analysis of dengue-malaria co-infection model 98 ayuba et al. / j. nig. soc. phys. sci. 3 (2021) 96–104 99 thus, the equilibrium point is given by e0dm = [s h(0), ihd (0), ihm(0), idm(0), rhd (0), rhm(0), rdm(0), s vd (0), ivd (0), s vm(0), ivm(0)] = [ λh µh , 0, 0, 0, 0, 0, 0, λd µd , 0, λm µm , 0 ] (6) 3.3. basic reproduction number r0dm the linear stability of the equilibrium point e0dm is established using next generation matrix method on system (1) to obtain the threshold behavior r0dm. hence, we introduce two matrices; matrix a for rates of new infection and b is the transfer rate of in or out of a compartment. taking the partial derivative of the right hand side of (1) at dfep with respect to ihd, ihm, idm, ivd , ivm, we obtain a =  0 0 0 ηdλh µh nh 0 0 0 0 0 ηmλh µh nh 0 0 0 0 0 ηvdλd µd nh 0 ηvdλd µd nh 0 0 0 ηvmλm µm nh ηvmλm µm nh 0 0  b =  −κ1 0 0 0 0 0 −dt 0 0 0 0 0 −κ2 0 0 0 0 0 −µd 0 0 0 0 0 −µm  ∴ b−1 =  − 1 κ1 0 0 0 0 0 − 1dt 0 0 0 0 0 − 1 κ2 0 0 0 0 0 − 1 µd 0 0 0 0 0 − 1 µm  where κ1 = (σ + τ + µh + φ), κ2 = (ψ + µh + θ + φ),dt = (α + ρ + δ + µh + θ) and β = ψ(1− l)(1−q) from equation (1). the basic reproduction number r0dm of dengue-malaria co-infection model is the number of secondary infections of dengue or malaria in the population due to a single dengue or malaria infective individual. the reproduction number is the spectral radius of ab−1 defined as r0dm := p(ab−1), and is given by r0dm = max  √ ηdηvdλdλh µhµ 2 dκ1 n 2 h , √ ηmηvmλmλh µ2mµh dt n 2 h  (7) 3.3.1. local stability of disease free equilibrium point 3 the jacobian matrix j0dm of dengue-malaria model (1) at e0dm is obtained as seen in matrix (8). −µh 0 0 0 0 γ π 0 −ηdλh µh nh 0 −ηmλh µh nh 0 −κ1 0 0 0 0 0 0 ηdλh µh nh 0 0 0 0 −dt 0 0 0 0 0 0 0 ηmλh µh nh 0 0 0 −κ2 0 0 0 0 0 0 0 0 σ 0 qψ −µh 0 0 0 0 0 0 0 0 ρ ψl(1 − q) 0 (−γ−µh ) 0 0 0 0 0 0 0 0 β 0 0 (−π−µh ) 0 0 0 0 0 −ηvdλd µd nh 0 −ηvdλd µd nh 0 0 0 −µd 0 0 0 0 ηvdλd µd nh 0 ηvdλd µd nh 0 0 0 0 −µd 0 0 0 0 −ηvmλm µm nh ηvmλm µm nh 0 0 0 0 0 −µm 0 0 0 ηvmλm µm nh ηvmλm µm nh 0 0 0 0 0 0 −µm  (8) theorem 3.2. the disease free equilibrium e0dm is locally asymptotically stable if r0dm < 1 and unstable if r0dm > 1. 3stability and sensitivity analysis of dengue-malaria co-infection model 99 ayuba et al. / j. nig. soc. phys. sci. 3 (2021) 96–104 100 proof 3.1. . the local stability of e0dm is establish by the jacobian matrix (8) at e0dm. the characteristic polynomial of j0dm is determine by det(j0dm − ti) = (−µh − t) ×(−µh − t) × (−γ−µh − t) ×(−π−µh − t)(−µd − t) ×(−µm − t) × det(ĵ0dm − ti) = 0 where ĵ0dm is given by ĵ0dm =  −κ1 0 0 ηd λh µh nh 0 0 −dt 0 0 ηmλh µh nh 0 0 −κ2 0 0 ηv dλd µd nh 0 ηvdλd µd nh −µd 0 0 ηv mλm µm nh ηv mλm µm nh 0 −µm  using the properties of determinant, we obtain det(ĵ0dm−it) = det  −dt − t 0 ηmλh µh nh 0 0 0 −κ2 − t 0 0 0 ηvmλm µh nh ηvmλm µh nh −µm − t 0 0 0 ηvdλd µh nh 0 −µd − t ηvdλd µh nh 0 0 0 −ηdλh µh nh − t −κ1 − t  det  −dt − t 0 ηmλh µh nh 0 −κ2 − t 0 ηvmλm µm nh ηvmλm µm nh −µm − t  × det −µd − t −ηvdλdµd nhηdλh µh nh −κ1 − t  = 0 (9) the five eigenvalues of j0dm are (−µh − t) × (−µh − t) × (−γ − µh − t)×(−µd − t)×(−µm − t) = 0 and the other five eigenvalues are obtained from the solution of matrix equation (9) by det  −dt − t 0 ηmλh µh nh 0 −κ2 − t 0 ηvmλm µm nh ηvmλm µm nh −µm − t  = 0 det −µd − t −ηvdλdµd nhηdλh µh nh −κ1 − t  = 0 the above determinant becomes t3 − (dt + κ2 + µm)t2 − ( κ2(dt + (dt + κ2) + (1 − r20m)dt µm ) t +(κ2 − r20m)dt µm = 0 (10) t2 + (µd + κ1)t + (1 − r 2 0d )µdκ1 = 0 (11) the above eigenvalues of equation (10) and (11) are also negative. therefore, the disease free equilibrium point are locally asymptotically stable iff r0d < 1 and r0m < 1. 3.3.2. global stability of disease free equilibrium point 4 the global asymptotic stability of the dfep is investigated using carlos castillo-chavez conditions as described in [23]. from the co-infection model (1), we define the time dependent derivatives by x′ = f(x, z) (12) z′ = g(x, z), g(x, 0) = 0 (13) where x = (s h, rhd, rhm, rdm, s vd, s vm) and z = (ihd, ihm, idm, ivd, ivm) denote uninfected and infected populations respectively. to guarantee the global asymptotic stability, the following conditions must be satisfied. (a) x′ = f(x, 0); x∗ is globally stable (b) g(x, z) = dzg(x∗, 0)z − ĝ(x, z), ĝ(x, z) ≥ 0 ∀ x, z ∈ ω theorem 3.3. the equilibrium point e0dm = (x∗, 0) of system (1) is globally asymptotically stable if r0dm ≤ 1 and the conditions (a), (b) are satisfied. proof: f(x, z) and g(x, z) is given by f(x, z) =  λh + γrhm + πrdm − ηd ivd +ηm ivm nh s h −µhs h σrhd + qψidm −µhrhd ρrhm + (1 − qψ)ihm − (γ + µh)rhm βidm − (π + µh)rdm λd − ηvd (ihd +idm ) nh s vd −µd s vd λm − ηvm (ihm +idm ) nh s vm −µms vm  g(x, z) =  ηd ivd nh s h − ηm ivm nh ihd − (σ + τ + µh + φ)ihd ηm ivm nh s h − ηd ivd nh ihm − (α + ρ + δ + µh + θ)ihm ηm ivm nh ihd + ηd ivd nh ihm − (ψ + µh + θ + φ)idm ηvd ihd nh s vd + ηvd idm nh s vd −µd ivd ηvm ihm nh s vm + ηvm idm nh s vm −µm ivm  for x′ = f(x, 0), system (1) is reduced to x′ =  s ′h = λh + πrdm + γrhm −µhs h s vd′ = λd −µd s vd s vm = λm −µms vm with x∗ = ( λh µh , λd µd , λm µm ) (14) given g(x, z) = dzg(x∗, 0)z − ĝ(x, z), ĝ(x, z) ≥ 0 g(x∗, 0) =  −κ1 0 0 ηdλh µh nh 0 0 −dt 0 0 ηmλh µh nh 0 0 −κ2 0 0 ηvdλd µd nh 0 ηvdλd µd nh −µd 0 0 ηvmλm µm nh ηvmλm µm nh 0 −µm  (15) 4stability and sensitivity analysis of dengue-malaria co-infection model 100 ayuba et al. / j. nig. soc. phys. sci. 3 (2021) 96–104 101 dzg(x ∗, 0)z =  −κ1 0 0 ηdλh µh nh 0 0 −dt 0 0 ηmλh µh nh 0 0 −κ2 0 0 ηvdλd µd nh 0 ηvdλd µd nh −µd 0 0 ηvmλm µm nh ηvmλm µm nh 0 −µm  ×  ihd ihm idm ivd ivm  =  −κ1 ihd + ηdλh µh nh ivd −dt ihm + ηmλh µh nh ivm −κ2 idm ηvdλd µd nh ihd + ηvdλd µd nh idm −µd ivd ηvmλm µm nh ihm + ηvmλm µm nh idm −µd ivm  ĝ(x, z) =  ĝ1(x, z) ĝ2(x, z) ĝ3(x, z) ĝ4(x, z) ĝ5(x, z)  =  ηd ivd nh ( λh µh − s h) + ηm ivm ihd nh ηm ivm nh ( λh µh − s h) + ηd ivd ihm nh − ( ηm ivm ihd nh + ηd ivd ihm nh ) ηvd nh ( λd µd − s vd )(ihd + idm) ηvm nh ( λm µm − s vm)(ihm + idm)  (16) since ĝ3(x, z) < 0 in equation (16) and condition (b) requires ĝ(x, z) ≥ 0. hence, condition (b) is not met as ĝ(x, z) < 0 for all x, z ∈ ω. thus, it implies that the defp may not be globally asymptotically stable if r0dm < 1. therefore, the endemic equilibrium exist with dfep if rodm < 1. whence, we can deduced that the dengue-malaria model exhibits backward bifurcation when the basic reproduction number r0dm = 1. 3.4. parameters estimation and sensitivity analysis 3.4.1. parameters estimation and initial value 5 the parameters in table 2 are obtained (or estimated) in line with the work of [7, 15], from kenyan region where malaria and dengue virus are said to be endemic. conservatively, the following initial values are estimated. the total human population is estimated to be 52, 000, 000 and the susceptible human are assumed to be 25, 000, 000 which is about half of the population at the onset of the diseases. for vectors population, 10, 000, 000 is assumed to be susceptible malaria mosquitoes with 2, 000, 000 malaria carrier mosquitoes. dengue susceptible mosquitoes are estimated to 5, 000, 000 and 100, 000 for dengue carrier mosquitoes. therefore, the initial infected human with malaria is estimated to be 10, 000 and infected human with dengue estimate is 5000. 3.5. sensitivity analysis of the model in order to identify the dominant parameter for the spread and control of dengue and malaria infections in the population, we performed the sensitivity analysis. as described in carlos 5stability and sensitivity analysis of dengue-malaria co-infection model table 2. parameters values of dengue-malaria co-infection model parameter value/day source λh 467 [7] µh 0.00004 calculated λd 221056.75 estimated ηd 0.000451 estimated ηvd 0.13502 estimated σ 0.035 estimated π 0.003 estimated τ 0.0245 estimated φ 0.00023 estimated µd 0.00005 calculated ηm 0.000408 [7] ηvm 0.15096 [7] γ 0.06 [0,1] [7] α 0.038 [7] ρ 0.37 [7, 15] δ 0.0019 [7] θ 0.00025 estimated µm 0.00005 calculated castillo-chavez [23], the sensitivity index of r0dm with a parameter say β is expressed as υ r0dm β = ∂rodm ∂β × β r0dm (17) since rodm is defined by r0dm = { √ ηdηvdλdλh µhµ 2 dκ1 n 2 h , √ ηmηvmλmλh µ2mµh dt n 2 h } therefore, we evaluate the sensitivity index of r0d and rom separately as follows: 6 υ r0d ηd = ∂r0d ∂ηd × ηd r0d = 1 2 > 0 υ r0d ηvd = ∂r0d ∂ηvd × ηvd r0d = 1 2 > 0 υ r0d σ = ∂r0d ∂σ × σ r0d = − σ 2κ1 < 0 υ r0d τ = ∂r0d ∂τ × τ r0d = − τ 2κ1 < 0 υ r0d φ = ∂r0d ∂φ × φ r0d = − φ 2κ1 < 0 υ r0d µd = ∂r0d ∂µd × µd r0d = −1 < 0 υ r0d µh = ∂r0d ∂µh × µh r0d = − σ + τ + 2µh + φ 2κ1 < 0 6stability and sensitivity analysis of dengue-malaria co-infection model 101 ayuba et al. / j. nig. soc. phys. sci. 3 (2021) 96–104 102 υ r0m ηm = ∂r0m ∂ηm × ηm r0m = 1 2 > 0 υ r0m ηvm = ∂r0m ∂ηvm × ηvm r0m = 1 2 > 0 υ r0m α = ∂r0m ∂α × α r0m = − α 2dt < 0 υ r0m ρ = ∂r0m ∂ρ × ρ r0m = − ρ 2dt < 0 υ r0m δ = ∂r0m ∂δ × δ r0m = − δ 2dt < 0 υ r0m θ = ∂r0m ∂θ × θ r0m = − θ 2dt < 0 υ r0m µm = ∂r0m ∂µm × µm r0m = − α + ρ + δ + 2µm + θ 2dt < 0 the parameters with positive sensitivity indices are ηd,ηvd,ηm,ηvm and the negative indices includes σ,τ,φ,µd,α,ρ,δ,θ,µm. the positive sign parameters have great influence in the spread of the diseases and their co-infection in the region. whereas, the parameters with negative sign have potential influence on the control of the spread of dengue, malaria and their co-infection. hence, the center for disease control is expected to make policies and control measures in this regard to combat dengue, malaria and their co-infection in an endemic region. 3.6. numerical simulations 3.6.1. effect of malaria recovery rate (α) on infectious (ihm) population as seen in figure 2, it is shown that α plays a significant influence in decreasing malaria infection. when the value of α increases from 0.038 to 1, the infectious population due to malaria decreased, where the contact rate ηm is kept constant. figure 2. effect of malaria recovery rate on infectious population 3.6.2. effect of dengue recovery rate (σ) on infectious (ihd ) population in figure 3, as the value of σ varies from 0.035 to 0.99, the number of dengue infection decreases when the contact rate ηd is kept constant. hence, this can be use by policy makers to combat the disease. figure 3. effect of dengue recovery rate on infectious population 3.6.3. effect of dengue contact rate (ηd ) on co-infectious (idm) population in figure 4, the contact rate of dengue ηd varies from 0.000451 to 0.040451, the number of co-infectious population increases as the recovery rate is kept constant. thus, the center for disease control and policy makers are expected to apply vector control measures and mechanism to reduce the expansion of co-infection in the region. figure 4. effect of dengue contact rate on co-infectious population 3.6.4. effect of dengue-malaria recovery rate (ψ) on co-infectious (idm) population the recovery rate described in dengue-malaria model is either the individual recovery from dengue only, recovery from malaria only or both dengue and malaria infections. as shown in figure 5, increasing ψ play a significant role in reducing both dengue and malaria infections in the region. 102 ayuba et al. / j. nig. soc. phys. sci. 3 (2021) 96–104 103 table 3. parameters value and sensitivity indices parameter sensitivity indice sensitivity index <0d basic reproduction number of dengue µh -ve -0.001787 ηd +ve +0.5 ηvd +ve +0.5 σ -ve -0.001046 τ -ve -0.000732 φ -ve -0.000007 µd -ve -1 <0m basic reproduction number of malaria ηm +ve +0.5 ηvm +ve +0.5 α -ve -0.007794 ρ -ve 0.075885 δ -ve -0.00049 θ -ve -0.00005 µm -ve -1 figure 5. effect of recovery rate on co-infection population 4. discussion in this paper, we develop a deterministic mathematical model that studies the dynamics of dengue virus and malaria fever in an endemic stage. base on the qualitative and numerical analysis of the data sourced from [7, 15] with conservative estimates, the results depict some interesting insights into the underlying relationship between dengue virus and malaria fever and provide information that are useful to combat the diseases. the qualitatively analysis of the model shows that there is a bounded invariant region where the model is mathematical and epidemiological well posed. the basic reproduction number of the model was derived using the next generation matrix method. stability and sensitivity analysis of the disease free equilibrium point (dfep) were established. the result shows that the dfep is locally stable if r0dm < 1 but may not be asymptotically stable. therefore, the endemic equilibrium exist when rodm < 1 with dfep and this implies that the model undergoes backward bifurcation. we demonstrated numerically using maple 17 , the effects of basic parameters for the spread and control of dengue and malaria co-infection. from the results, we conclude that an increase in dengue and malaria recovery rates plays a great role in reducing dengue and malaria infections respectively, in the region. similarly, the recovery rate for co-infectious individuals also contributes greatly to reducing the co-infection in the population if its value increases as seen in figure 5. another findings obtained is that, increasing dengue vectors contact rate has a great influence on spreading the co-infection in the population. we computed the r0dm = 19.70 > 1, indicating that dengue virus and malaria fever are endemic in the area. thus, we recommend that center for disease control set out preventive measures in reducing the spread of both diseases and increase the measures on recovery co-infected individuals. 5. conclusion and recommendation as demonstrated in this study, the co-infection between dengue virus and malaria fever may have devastating impacts in the tropical/subtropical communities. the model helps in identifying distinct features and underlying relationships between dengue and malaria co-infection. this will be of help to policy makers to devise strategies for controlling the diseases. for future studies, we recommend a formulation with optimal control parameters to determine the strategies for mitigating the spread and control of dengue and malaria co-infection. data and materials. the data used for this co-infection model are from previous articles published. acknowledgment the authors are grateful for the data gathered from the previously published articles. 103 ayuba et al. / j. nig. soc. phys. sci. 3 (2021) 96–104 104 references [1] p. yongzhen, l. shgaoying, l. shuping, & l changguo, “a delay seiqr epidemic model with impulse vaccination and the quarantine measure”, computers and mathematics with 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[23] c. c. chavez, z. feng, & w. huang, “on the computation of r and its role on global stability”, //www.researchgate.net/profile/carlos castillochavez2/publication/228915276, biometric unit technical report m1553 (2001) 1. 104 j. nig. soc. phys. sci. 5 (2023) 1081 journal of the nigerian society of physical sciences investigation of point refractivity gradient and geoclimatic factor at 70 m altitude in yenagoa, nigeria y. b. lawala,∗, e. t. omotosob adepartment of physics, university of africa, toru-orua, bayelsa state bdepartment of physics, federal university of technology, akure, ondo state abstract the quality of services provided via inter-terrestrial radio communication links such as gsm networks, wide area network (wan), radio and tv broadcasts is largely influenced by some meteorological parameters such as temperature, pressure and humidity. proper knowledge of these parameters, specifically at microwave antenna heights (about 70m) is important in order to maintain an effective line-of-sight (los) link even during the worst weather conditions. the geoclimatic factor is an important quantity that must be considered in the design of terrestrial links for effective wireless communication. this work utilized satellite data from the european center for medium-range weather forecasts (ecmwf) to compute the point refractivity gradient and geoclimatic factor for yenagoa and its environs. the research was necessitated by the paucity of research on this subject matter for yenegoa. the results of the research show that point refractivity gradient and geoclimatic factor in the study area vary with season. the average point refractivity gradient and geoclimatic factor at 70 m above the ground level are:136.433 n-unit/km and 6.638633e-05 respectively. this implies that radiowaves propagating in this region at the said altitude is most likely to be super refractive in both rain and clear air atmospheric conditions. rain or worst condition refers to the period when atmospheric components such as hydrometeor, lithometeor, aerosol have significant effects on propagated radio signals. clear-air conditions means when maximum possible signal is received such that the most threatening atmospheric components (rain drops) have negligible effects on propagated signal. the results will be useful for radio engineers in the design and configuration of inter-terrestrial microwave links in yenagoa and its environs for optimum quality of service. doi:10.46481/jnsps.2023.1081 keywords: gsm, wan, inter-terrestrial, microwave, geoclimatic factor, refractivity gradient article history : received: 22 september 2022 received in revised form: 27 december 2022 accepted for publication: 07 january 2023 published: 27 january 2023 c© 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: w. a. yahya 1. introduction the condition of the atmosphere is very important for proper planning of terrestrial radio links, navigation and remote sensing installations like radar [1]. the quality of a radio signal and the extent to which it travels within ∗corresponding author tel. no: +2348032412347 email address: lawalyusuf.b@gmail.com (y. b. lawal ) a particular medium are determined by some atmospheric weather parameters which dictate the medium’s refractive index. temperature, pressure, and humidity are the major atmospheric parameters that determine the refractive index in the troposphere [2]. temporal and spatial variations in these parameters have a great effect on the propagation conditions. radio point refractivity gradient and geoclimatic factors are major quantities whose values must be taken into consideration when designing inter-terrestrial radio links. these quantities 1 y. b. lawal & e. t. omotoso / j. nig. soc. phys. sci. 5 (2023) 1081 2 are solely a function of water vapor pressure, temperature, and pressure which are the said atmospheric parameters. precise estimation of refractivity gradient and geoclimatic factor are highly essential in order to determine the multipath fade depth of a communication link [3]. fading is a phenomenon whereby there is a gradual drop in the signal strength of a radio wave as it propagates through the atmosphere. the signal is gradually reduced in the atmosphere because of the various obstacles such as hydrometeors, litheometeors, etc. it encounters. the gradual drop in radio signal results in a fractional power of the transmitted signal reaching a target receiving antenna. multipath fading is a major impairment problem in wireless communication systems such as wireless sensors, mobile telephony, radar systems, tv/radio broadcast etc [3, 4]. the effects of the clear-air fading mechanism due to extreme refractive layers in the atmosphere include, but are not limited to beam spreading, antenna decoupling, surface multipath, and atmospheric multipath [5]. to achieve a seamless inter-terrestrial communication link within a locality, there is a need to carry out accurate estimation of refractivity gradient and geoclimatic factor values. a few research works have been carried out to determine refractivity gradient and geoclimatic factors in some areas in nigeria as presented in table 1. unfortunately, none has attended to the computation of these local parameters in yenagoa except olla and oluwafemi (2018) [6] which predicted them using spatial interpolation. the result is adjudged to be grossly unreliable because only six spatial data points were used generate k-map of nigeria. geoclimatic factor is an important parameter in the planning and design of inter-terrestrial radio links. accurate estimation of geoclimatic factor within a blanket of altitude aids in the identification of worst month condition. thus, leading to proper adjustment of propagation parameters to minimize fade depth margin. the spatial and temporal variability of geoclimatic factor has made it a parameter of interest for local radio engineers. geoclimatic factor is a function of refractivity gradient. consequently, refractivity gradient is a variable that depends on refractivity, a scale-up term for refractive index. according to itu-r, equation (1) expresses the relationship between refractivity n and refractive index n [10]. n = (n − 1) × 106 (1) the value of refractivity gradient determines the degree of curvature of radio signal propagating in the atmosphere for a particular range of altitude. the relationship between refractivity and altitude in the first kilometer above the ground level is linear [11, 12, 13]. for instance, if refractivity n1 is measured at an altitude h1 (km) and another value n2 is obtained at a higher altitude h2 (km) which are less than or equal to 1 km, then the refractivity gradient dn/dh is given by equation (2) [13]. this equation is always negative due to a decrease in the value of refractivity n as altitude increases. dn dh = n2 − n1 h2 − h1 (2) the degree of curvature of radio signals is generally classified into four categories based on the value of refractivity gradient for a given height range [14, 15]. a radio signal is said to be normally refracted if the refractivity gradient is −40 n/km, otherwise, it is abnormally refracted according to equations (3a)(3d) [16]. sub-refraction; ∂n ∂h >−40 n/km (3a) standard refraction; ∂n ∂h = −40 n/km (3b) super refraction;−157n/km < ∂n ∂h <−40 n/km (3c) ducting; ∂n ∂h <−157n/km (3d) figure 1: common classification of atmospheric refraction conditions [17] a sub-refracted radio signal is one with a very large refractivity gradient such that the radio path bends towards the earth but with a curvature less than that of standard refraction as illustrated in figure 1. super refraction is a condition in which the rays bend more rapidly towards the earth when compared with normal refraction. ducting is an extreme case of super refraction in which the degree of curvature of radio signals exceeds that of the earth’s surface curvature. during this condition radio signals especially from radar, may hit the earth surface and suffer multiple reflections instead of hitting the intended target. although ducting may be of great advantage for long-distance non-line-of-sight transmission, such transmitters must be equipped with sufficient transmitting power. 2 y. b. lawal & e. t. omotoso / j. nig. soc. phys. sci. 5 (2023) 1081 3 table 1: reports of previous studies on determination of geoclimatic factors previous studies methodology k-factor for nearest station k-factor for yenagoa remarks etokebe et al., (2016) [7] nimet data was employed to determine g and k at 65m height for calabar. 6.537e05 (calabar) nil emmanuel et al., (2018) [8] ecmwf data was employed to determine g and k at 100m height for 17 stations excluding yenagoa 2.39e04 (port harcourt) nil neither g nor k was computed for yenagoa olla and oluwafemi (2018)[6] nimet data was employed to determine g and k at 65m height for 6 stations excluding yenagoa. the results were used to generate k-map for nigeria 2.791e04 2.791e04 six (6) data points are insufficient to generate data map of nigeria by interpolation and extrapolation techniques oluwafemi and olla (2021)[9] nimet data was employed to determine g and k at 65m height for 6 stations excluding yenagoa. 2.39e04 (port harcourt) nil a propagating radio wave, as shown in figure 1, may miss its target if the actual refractivity gradient within the site is not taken into consideration. 2. methodology 2.1. research location this research work evaluates the necessary point refractivity gradient and geoclimatic factor for the propagation of tropospheric radio waves in yenagoa and its environment. the study area is yenagoa, the capital of bayelsa state, a coastal city in the south-south geo political zone of nigeria. yenagoa is bounded between latitude 4.90 – 4.92◦ n and longitude 6.07 – 6.27◦ e. about 65% of the entire state is covered by water from the atlantic ocean, while the area covered by land is about 15 metres above mean sea level [18, 19]. it is characterized by two climatic seasons: dry and wet seasons. generally, rainfall is experienced in all months of the year in the niger delta region of nigeria. december, january, february and march which are the months with least amount of rainfall make up the dry season [20, 21, 22]. the land mass witnesses a frequent high volume of annual rainfall due to its proximity to the atlantic ocean. this accounts for the reason why attenuation due to rain remains a major threat to radio signals, especially during the rainy season. this research focuses on the determination of radio refractivity gradient and geoclimatic factor which are localized radio propagation parameters. 2.2. data acquisition and computational analysis ten years (2009-2018) monthly meteorological data of yenagoa containing air temperature, relative humidity, pressure and dew point temperature at ground level and 70 m above were retrieved from the archive of the european center for mediumrange weather forecast (ecmwf) [23]. the ecmwf erainterim satellite has a grid and temporal resolutions of 0.75◦ by 0.75◦ lat/long and 24 hours respectively [24]. the values of relative humidity at the two levels were converted to water vapor pressure, e, by using equation (4) while the refractivity was calculated using equation (5) [25]. the data analysis was carried out using microsoft excel software while sorting of the refractivity gradients was accomplished using the necessary empirical equations for determining normal and abnormal refractions (sub-refraction, super-refraction, and ducting). the necessary empirical equations and conditions for the classification of radio refractions in the troposphere are indicated in equations (3a)-(3d). the surface and point refractivities n at 70 m above the ground surface were computed using equation (4) [26]. n = 77.6p t + 3.73 × 105 e t 2 (4) e = h × 6.1121 exp ( 17.502t t+240.97 ) 100 , (5) where p = atmospheric pressure (hpa), t = temperature in degree celsius (◦c), e = water vapour pressure (hpa) defined by equation (5) [27], h = relative humidity (100 %) and t = absolute temperature (k). the refractivity gradients were calculated using equation (2), where n1 is the refractivity at the ground surface (h1), n2 is the refractivity at 70 m height (h2). h1 is the ground surface height (i.e 0 m) and h2 is 70 m. the itu-r recommended formula for computing geoclimatic factor k is given in equation (6) [10, 28, 29] k = 10−4.6−0.0027dn1, (6) where dn1 is the point refractivity gradient, a simple notation representing dndh , the subject of equation (2). the monthly, seasonal, and annual variations of k were studied based on available data. statistical analysis was also carried out to deduce the prevailing type of refraction in the study area and give appropriate recommendations. 3 y. b. lawal & e. t. omotoso / j. nig. soc. phys. sci. 5 (2023) 1081 4 3. results and discussion 3.1. monthly variation of point refractivity gradient the monthly point refractivity gradients for all the months between 2009 and 2018 inclusive were computed and presented in figure 2. according to the classifications of atmospheric refractions in equations (3a)-(3d), it was observed that the prevailing propagation conditions are super-refraction and ducting. similar conditions were reported by [12] for the same station. the results, as presented in figure 2, indicate that refractivity is generally high during the rainy months while low values are predominant in the dry months. the trend of the refractivity gradients shows that the monthly variation has the shape of a stretched letter “m” with double peaks annually. the first peak was observed between may and june which signifies the intense period of the rainy season in the coastal region as reported by [7, 30, 31]. the dip in august could be attributed to the famous august-break which results in a low refractivity gradient due to a decrease in rainfall [31, 32]. the second peak occurred in september which signifies the resumption of frequent rainfall after the august break. figure 2: annual variation of point refractivity gradient 4. seasonal variation of refractivity gradient and kfactor the seasonal variation of refractivity gradient and geoclimatic factor were studied based on the computed monthly values. figure 2 depicts that each month exhibits a unique trend over the years of study. for instance, there is a consistent gradual decrease in the refractivity gradient from january to march of every year. the unique nature of each month informed the decision to average the refractivity gradient values of the corresponding months for all the years. table 2 and figure 3 present the monthly mean of the refractivity gradients from 2009 to 2018. the figure revealed the variation of the refractivity gradient over the two seasons. refractivity gradient rises gradually at the onset of the rainy season specifically from march and becomes fairly steady at a mean value of -68.27 n-units/km between may and july. this is the maximum average refractivity gradient for the entire study period. there is a slight fall in august due to rainfall seizure table 2: monthly average point refractivity gradient and k-factor for yenagoa (2009-2018) months refractivity gradient (n-unit/km) k-factor (k) january -161.604 6.86011e-05 february -227.353 0.000103241 march -214.234 9.51548e-05 april -139.13 5.96557e-05 may -70.0804 3.88344e-05 june -67.9087 3.83137e-05 july -68.2645 3.83985e-05 august -87.6277 4.33106e-05 september -75.8705 4.02578e-05 october -84.3443 4.24355e-05 november -230.899 0.000105542 december -209.879 9.26135e-05 annual average -136.433 6.38633e-05 figure 3: monthly average point refractivity gradient for yenagoa (20092018) associated with this month. the slight fall observed during the august-break is due to the movement of inter-tropical discontinuity (itd) between the northern and southern parts of nigeria. generally, the itd reaches its maximum northward position in august translating to a low amount of precipitation in the south [32]. hence, the refractivity gradient dropped to 87.63 n-units/km. the surge between september and october is due to the resumption of frequent rainfall which signifies the end of rainy seasons in the coastal region. the sharp decline between october and november is occasioned by the cessation of the rainy season. it was observed that the refractivity gradient reduces significantly from -84.34 n-units/km in october to -230.90 n-units/km in november. the month of january is commonly characterized by intense harmattan which causes lower humidity and temperature compared with previous months [33]. this accounts for the slight increase to -161.6 n-units/km in january, followed by a continuous reduction in humidity due to severe solar radiation between february and june [34]. the overall average refractivity gradient during the rainy and dry seasons stands at -75.68 n-units/km and -197.18 n-units/km, respectively. the geoclimatic factor which is an exponential function of 4 y. b. lawal & e. t. omotoso / j. nig. soc. phys. sci. 5 (2023) 1081 5 figure 4: monthly average geoclimatic factor for bayelsa (2009-2018) the point refractivity gradient exhibit a similar trend but in the inverse order. high values prevail during the dry season while low values were dominant during the rainy season as depicted in figure 4. the months of november, february, and march recorded maximum monthly means of 1.06e-04, 1.03e-04, and 9.53e-05, respectively. on the contrary, low values of 3.83e05, 3.84e-05, and 3.88e-05 were recorded in june, july, and may, respectively. the mean values during the rainy and dry seasons are 4.03e-05 and 8.75e-05. the overall annual average point refractivity gradient and geoclimatic factor are 136.43 n-units/km and 6.39e-05, respectively. these results are closely in agreement with the values obtained for calabar by etokebe et. al., (2016) [7]. calabar is also a coastal region in southern nigeria and shares the same climatic features as yenagoa [35]. the results also align closely with the work of [9] which obtained an average geoclimatic factor of 2.39e4 for port harcourt, another coastal city located about 94 km away from yenagoa. 5. conclusion the monthly point refractivity gradient and geoclimatic factor for yenagoa between 2009 and 2018 have been computed using meteorological data retrieved from the archive of ecmwf. the values obtained are similar to results obtained for other coastal stations in nigeria as evidenced in previous studies. although, refractivity gradient is generally low during the dry months, the lowest value was observed in november as shown in figure 4. this implies that extremely poor propagation condition (ducting) is likely to occur in november due to the extreme values of point refractivity gradient and geoclimatic factor. the derived parameters are recommended for link budget calculations in the design of terrestrial microwave links. acknowledgments the authors wish to express their profound gratitude to the ecmwf for providing satellite data used for this research. we also appreciate our colleagues whose comments and suggestions have helped to improve the paper. references [1] m. m. tanko, m. s. liman, w. l. lumbi, u. s. aliyu, & m. u. sarki, “field strength variability mapping of nigeria”, journal of the nigerian society of physical sciences (jnsps) 4 (2022) 746. 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[35] o. j. olaniran & g. n. sumner, “a study of climatic variability in nigeria based on the onset, retreat, and length of the rainy season”, international journal of climatology 9 (1989) 253. 6 j. nig. soc. phys. sci. 3 (2021) 224–233 journal of the nigerian society of physical sciences bayesian multilevel models for count data olumide sunday adesinaa,∗ adepartment of mathematical sciences, redeemer’s university, nigeria abstract the traditional poisson regression model for fitting count data is considered inadequate to fit overor under-dispersed count data and new models have been developed to make up for such inadequacies inherent in the model. in this study, bayesian multi-level model was proposed using the no-u-turn sampler (nuts) sampler to sample from the posterior distribution. a simulation was carried out for both over-and under-dispersed data from discrete weibull distribution. pareto k diagnostics was implemented, and the result showed that under-dispersed and over-dispersed simulated data has all its k value to be less than 0.5, which indicate that all the observations are good. also all waic were the same as loo-ic except for poisson in the over-dispersed simulated data. real-life data set from national health insurance scheme (nhis) was used for further analysis. seven multi-level models were fitted and the geometric model outperformed other model. doi:10.46481/jnsps.2021.168 keywords: count data, health, insurance, dispersion, multilevel models article history : received: 17 february 2021 received in revised form: 09 july 2021 accepted for publication: 30 july 2021 published: 29 august 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: t. latunde 1. introduction count data contains non-negative integers and zero obtained within a fixed period. various studies have been carried out on count data and modelling, [1] applied it to medical data [2] applied it to model microbiome data and many others. frequentist and bayesian estimation have equally been used to model count data, and the widely used is the bayesian estimation technique. there are three major types of count data, the over-dispersed, under-dispersed and over-dispersed. more about types of count data can be found in the study by [3,4]. there are models dedicated to modelling under-dispersion due to their suitability while a model such as negative binomial is suitable for fitting over-dispersion. models such as dirichlet process prior, negative weibull can be used to fit both under-dispersion and ∗corresponding author tel. no: email address: olumidestats@gmail.com (olumide sunday adesina ) over-dispersion. models such as zero inflated and hurdle models can effectively handle over and under-dispersed count data with many zeros. the zero-truncated regression models are specifically designed to fit count data with no zero count. the categorized regression model is designed to fit count data that its response variable is categorized. some of the improved techniques relative to poisson regression model can be found in [5], [6], [3], amongst others. [7] carried out a on hidden markov model in multiple testing on dependent count data, [8] showed that the exponentiatedexponential geometric distribution can be applied to fit underdispersed or over-dispersed count data, in the same manner [4] demonstrated that dirichlet process mixture prior of generalized linear mixed models (dpmglmm) can fit either overdispersed or under-dispersed count data well. [9] sufficiently showed that multi-level zero-inflated poisson (zip) regression model can adequately fit both over-and underdispersed count data that have zero counts. the authors adopted 224 adesina / j. nig. soc. phys. sci. 3 (2021) 224–233 225 em algorithm along with the penalized likelihood and restricted maximum likelihood (reml). [10] adopted multilevel zeroinflated poisson (zip) regression and zero-inflated negative binomial (zinb) and applied the models to fit count data relating to decay, missing and filled teeth of children aged 12 years old. a related and recent study was carried out by [11], the authors proposed multilevel zero-inflated generalized poisson (zigp) that is suitable in fitting both overand under-dispersed count data and compared with multilevel models of zero-inflated poisson, zero-inflated negative binomial. the result showed that the multilevel zigp produced more accurate parameter estimates, particularly for under-dispersed data. in this study, bayesian multilevel modelling was proposed and implemented for some basic distributions used in fitting count data (poisson, negative binomial and geometric), zero inflated and hurdle models, and identify a most suitable model for fitting under-and over-dispersed respectively. the remaining part of this paper is sectionalized as follows; multi-level modelling is described in section 2, parameter estimation and model selection can be found in 3. section 4 is the results of simulation study and that of real-life data. lastly, summary and conclusion in section 5. 2. model description 2.1. multilevel modelling the multilevel modelling technique follows a similar process involved when fitting the generalized linear model. in glm, a link function links the response y variable to the predictor(s), same with multilevel modelling. let ( f ) be an inverse link function that links the response variable y to the predictor, then υ is the linear combination of the predictors transformed by the inverse link function ( f ), and d be a parametric model (distribution), the model can be simply be written as yi ∼ d ( f (υi) ,θ) (1) and linear predictor: υ = xβ + k� (2) the data is made up of the response variable y, x and k, while β, � and θ are the model parameters to be estimated. while β is the fixed effect coefficient at population level, while � is the coefficient at group-level. the bayesian estimation technique by monte carlo markov chain (mcmc) procedure considers � as a parameter relative to maximum likelihood that considers � as error term [12]. 2.2. zero-inflated distribution if y follows zero-inflated poisson (zip) distributions, given by p(y = y) = { ω + (1 −ω) exp(−λ), y = 0 (1 −ω) exp(−λ)λy/y!, y > 0 } (3) where is ω in the range0 < ω < 1 , in order to accommodate more zeros than those allowed under the poisson assumption(ω = 0), and the case of ω < 0imply zero inflated. [9] estimated multi-level parameters of zip regression in the generalized linear mixed models (glmms) context. the authors generalized the zip model so that the model will be able to withstand more complex correlation structure. zero-inflated negative binomial for counts is formed from zip, the mean and variance defined as: e(y ) = (1 −ω)λ = µ, (4) the use of regression models based on zip was established by [13-15 ], [5]. following [5] we have: log(λ) = xβ and log (ω/1 −ω) = zυ (5) where x and z are matrices of covariates, β and υare vector of parameters. assuming two linear predictors are related in some ways, [16] provided a simplest form of (3) which is refers to the zi p(τ)model as follows: log(λ) = xβ, log (ω/1 −ω) = τxβ (6) where τ is a scalar parameter, which implies thatω = (1 + λ−r )−1. following equation (3) in multi-level case, [9] identified the extension of zip model to include random components wi and ui within logistic and poisson linear predictors to take care of dependence of observations in given clusters. the random effects wi and uiare specific to the ith cluster. in a three-level hierarchical situation of yi jk, the kth observation of the jth individual within the ith clusters is measured through random effects associated with the linear predictors as follows: log [ φi jk (1−φi jk ) ] = ξi jk = ati jkα + wi + si j log(λi jk) = γi jk = xti jkβ + ui + vi j (7) the covariates ati jkand x t i jk are not always the same α and β are the corresponding vectors of regression coefficients. si jand vi jare variations at subject level. 2.3. hurdle models if the distribution of y follows zero-truncated poisson distribution it follows that: π0y > 0 p(y = y) = (1 −π0)e−λλy (1 − e−λ)y! y = 0 (8) reparametizing the zero-inflated poisson model in equation (3) withπ0 = ω + (1 − ω)e−λ, [16], gave poisson hurdle regression model is given as; log (λ) = xβ, log[− log(1 −π+) = τxβ (9) where π+ = 1 − π0 is the probability of clearing the “hurdle” and generating a non-zero count. 225 adesina / j. nig. soc. phys. sci. 3 (2021) 224–233 226 2.4. prior distributions prior distribution is specified at population and group-level. at population-level parameters have an improper prior [17]. at group level it is assumed that parameters ε comes from a multivariate normal distribution having zero mean and unknown covariance matrixς. epsilon ∼ n (0,σ) (10) covariances are between group-level parameters are generally of different groupings factors and assumed to be zero. by implication, kand ε can be divided to form several matrices ki and parameter vectors εi , where i indexes grouping factors, thus, the model can be simplified to �i ∼ n (0,σi) (11) sometimes, it can be assumed that group-level parameters for different levels of the same grouping factors are not dependent. if the other level is indexed by j, (11) leads to: �i j ∼ n ( 0, m j ) (12) the covariance matrices m j will become the model parameters. no-u-turn sampler (nuts) by (2014) [18] is used for m j as instead of the inverse-wishart prior distribution used in most studies and packages. inverse-wishart distribution is used because it has good conjugacy characteristics for gibbs-sampler. the choice of inverse-wishart prior distribution was criticized in the studies by [19], and [20]. the parameters of m j is selected in terms of correlation matrix ωj and a vector of standard deviations σ j through, m j = d ( σ j ) ωjd ( σ j ) (13) and d ( σ j ) imply the diagonal matrix with diagonal elements σ j. then, prior would be specified for d ( σ j ) ωjd ( σ j ) . in the case of ωj, lkj-correlation prior by [21] is used, with ζ > 0. that is, ω j ∼ lk j (ζ) sectionparameter estimation and model selection sampling from the posterior require appropriate sampling procedure, two basic sampling procedures are discussed here. first is the hamiltonian monte-carlo (hmc) sampler, also known as hybrid monte-carlo [22-23]. [18] extended hmc to no-uturn sampler (nuts), because hmc has some drawbacks as discussed by [17]. the nuts sampler allows setting parameters, and eliminates the need for hand-tuning, [18] stated that setting the parameters automatically makes it least efficient as compared to a well-tuned hamiltonian monte-carlo. software package by r core team (2020) was used to fit the model with brms package by [17] along with stan processor without which the analysis cannot be run, it can be assessed on http://cran.rproject.org/bin/windows/rtools/. the watanabe-akaike information criteria proposed (waic) by [24] and leave-one-out cross validation loo-cv by [25,26] were used for model selection in this study. the waic was used for estimating the out-of-sample expectation and considered an improvement upon the dic, with waic, correction for effective number of parameters to adjust over-fitting is added. according to [27], waic can be computed in two possible ways, first is calculated using simulation θs, s = 1, .......s and given as pw aic1 = 2 n∑ i=1 log  1s s∑ s=1 log p(yi|θ s)  − 1s s∑ s=1 log p(yi|θ s) (14) for the second waic computation approach, the variance of individual terms in the log predictive density is added up over the ndata points and express as follows: pw aic2 = n∑ i=1 varpost ( log p(yi|θ) ) (15) the advantages of waic over aic and dic was adequately discussed by [27] in the case of leave-one-out cross-validation (loo-cv) in bayesian analysis, the data are repeatedly subdivided into a training set ytrain and a holdout set yholdout with the objective of fitting ytrain yielding a posterior distribution ptrain (θ) = ptrain (θ|ytrain). the bayesian loo-cv estimate of out-of-sample predictive fit is l ppdloo−cv = n∑ i=1 log ppost(−i)(yi) (16) and estimated as n∑ i=1 log  1s s∑ s=1 log p(yi|θ is)  (17) lower waics and loos suggest better model fit. 2.5. pareto-k-diagnostics the shape parameter k of the generalized pareto distribution can be used to assess the reliability and approximate convergence rate of the pareto smoothed importance sampling (psis). it follows that if, k < 0.5(that is, ‘good’) then the central limit theorem holds. similarly, if0.5 ≤ k < 1, (that is, ‘ok’) then the variance of the raw importance ratios is infinite, but the mean exists. in the same manner, if k > 0.7(that is, ‘bad’), unreasonable convergence rates is observed and unreliable monte carlo error estimates, and finally, if k ≥ 1 (that is, ‘very bad’), then neither the variance nor the mean of the raw importance ratios exists. 3. result 3.1. simulation study simulation of over and under-dispersed count data was carried out and the response count variable was obtained from discrete weibull distribution. on simulating count data from discrete weibull (dw) distribution, [28] identified that the parameter β of dw should contain the range0 ≤ β ≤ 1; irrespective the value of parameterq. for under-dispersed count data, β should be specified such that β ≥ 2, irrespective of the value of q. analysis for simulation study was carried using software package by 226 adesina / j. nig. soc. phys. sci. 3 (2021) 224–233 227 figure 1. marginal plot of relationship between encounter and type [29], and r package “dwreg” by [30]. random numbers consisting of 500 observations were generated and two predictors were uniformly generated in interval (0, 1) and (0, 2) for overand under-dispersion respectively. for over dispersed, beta=0.8 and for under-dispersed beta=2.1. the parameter of discrete weibull follows that θ0 = 0.25, θ1 = 0.35, θ2 = 0.5 and the corresponding equation for logit link is as follows log (q (1 − q)) = 0.25 + 0.35x1 + 0.5x2 (18) all pareto k estimates are good (k < 0.5) if p waic estimates greater than 0.4. we recommend trying loo instead. table 1 shows results for under-dispersed simulated count data. the best two performed model are geometric and hurdle poisson distribution indicated with ** and * respectively with lowest waic and loo, following implementation using brms, a package for bayesian multilevel modelling in r. from table 1, it shows that waic=loo for all the models. table 2 shows results for over-dispersed simulated count data. the best two performed model are negative binomial and zero inflated negative binomial distribution indicated with ** and * respectively with lowest waic and loo, following implementation using brms, a package for bayesian multilevel modelling in r. all the model shows that waic=loo expect for poisson model. 3.2. 4.2 application to health insurance data 3.2.1. data description the data set was obtained from national health insurance scheme (nhis), and it contains excess zero count. sample of 116 users of nhis users was obtained from september 2016 to july 2017. response variable is number of encounter (encounter), out of 116 observed, eighty two (82) persons made claims. encounter is the time a user of national health insurance scheme (nhis) visits the health facility, and possibly makes claims. the predictors include type of encounter (type), which is either primary or (secondary= 0, primary= 1) figure 2. marginal plot of relationship between encounter and age figure 3. marginal plot of relationship between encounter and sex figure 4. marginal plot of relationship between encounter and drugsadm primary are users who registered primarily to use the health facility, while secondary are users who were referred from another health facility to that of state hospital ota, because of availability of specialists. another predictor sex (male=1 and female=0), biological age of patients (age). number of drugs administered (drugsadm) that is, both oral and injection. drugs-out-of-stock (drugsos) is another predictor. the 227 adesina / j. nig. soc. phys. sci. 3 (2021) 224–233 228 table 1. simulated under-dispersed count data from discrete weibull model model elpd waic p waic waic elpd loo p loo looic waic=looic poisson est. se -628.1 10.2 1.9 0.1 1256.2 20.3 -628.1 10.2 1.9 0.1 1256.2 20.3 yes negbin est. se -629.3 10.1 1.9 0.1 1258.5 20.2 -629.3 10.1 1.9 0.1 1258.5 20.2 yes geometric est. se -717.3 12.9 0.9 0.1 1434.6** 25.8 -717.3 12.9 0.9 0.1 1434.6** 25.8 yes hurdle poisson est. se -620.5 13.1 3.3 0.2 1241.0* 26.1 -620.5 13.1 3.3 0.2 1241.0* 26.1 yes hurdle negbin est. se 621.2 13.1 3.3 0.2 1242.5 26.2 621.2 13.1 3.3 0.2 1242.5 26.2 yes zero inflated poisson est. se -629.1 10.1 2.0 0.1 1258.3 20.2 -629.1 10.1 2.0 0.1 1258.3 20.2 yes zero inflated negbin est. se 630.2 10.1 1.9 0.1 1260.5 20.2 630.2 10.1 1.9 0.1 1260.5 20.2 yes table 2. simulated over-dispersed count data from discrete weibull model model elpd waic p waic waic elpd loo p loo looic waic=loo poisson est. se -1984.6 108.6 21.6 3.1 3969.2 217.2 -1984.7 108.6 21.7 3.1 3969.4 217.2 no negbin est. se -1217.4 27.9 3.9 0.5 2434.8** 55.8 -1217.4 27.9 3.9 0.5 2434.8** 55.8 yes geometric est. se -1228.7 27.9 4.2 0.6 2457.4 55.8 -1228.7 27.9 4.2 0.6 2457.4 55.8 yes hurdle poisson est. se -1682.3 87.6 19.0 2.9 3364.6 175.1 -1682.3 87.6 19.0 2.9 3364.6 175.1 yes hurdle negbin est. se -1229.7 27.7 4.5 0.4 2459.4 55.5 -1229.7 27.7 4.5 0.4 2459.4 55.5 yes zero inflated poisson est. se -1680.7 87.6 18.7 2.9 3361.3 175.2 -1680.7 87.6 18.7 2.9 3361.4 175.2 yes zero inflated negbin est. se -1218.2 27.9 4.0 0.5 2436.4* 55.9 -1218.2 27.9 4.1 0.5 2436.4* 55.9 yes 228 adesina / j. nig. soc. phys. sci. 3 (2021) 224–233 229 figure 5. marginal plot of relationship between encounter and drugos figure 6. marginal plot of relationship between encounter and type along with age figure 7. marginal plot of relationship between encounter and type along sex idea of national health insurance scheme (nhis) is that treatments and drugs administered attract 10 percent of the total cost. it becomes disadvantage to patients having cases of drugsos. out of 116 users, 97 are primary, while 19 are secondary users, 70 out of 97 (72.16%) primary users did not make claims within the period observed. 4 out of 19 secondary users did not make claims (21.05%). out of 116 nhis users of the health facility, 64 are females and 52 are males. 41 females had zero claims (64.06%), while 33 males had zero claims (63.46%). 11 females and 8 males are secondary users, while 53 females and 44 males are primary users. the data can be obtained https://data.mendeley.com/drafts/6hcf5mf7fy. mean of the response variable (encounter) is 0.7068, while variance is 1.7916 which potentially suggests over-dispersion. further test on the data shows that the dispersion parameter is δ=1.49799, indicating that the data set is over-dispersed with δ>1.the dispersion test was performed using aer package in r by [31]. from table 3 shows results for the real-life over-dispersed count data. the best two performed model are geometric and negative binomial distribution indicated with ** and * respectively with lowest waic and loo, following implementation using brms, a package for bayesian multilevel modelling in r, in all the models waic loo as shown in table 6. as earlier stated; for any observation for k > 0.7indicate unreasonable convergence rates is observed and unreliable monte carlo error estimates. table 4 shows that hurdle negative binomial (4) has the highest of such observations, poisson, negative binomial, and hurdle poisson model has 3 of such observations while geometric, zero inflated poisson and zero inflated negbin has two (2) each. each parameter is presented in table 5 with the posterior mean as the ‘estimate’ and the ‘est.error’ as the standard deviation of the posterior distribution, the table also contain a two-sided 95% credible intervals (l-95% ci and u-95% ci) established on quantiles. from table 5, the ‘intercept’, ‘type’ and ‘type:sex”interaction has a negative posterior mean. for “type”, the model predicts longer periods for encounter for secondary users than primary users; ‘sex’ (0.19) accounts for more encounter than ‘age’ (0.01). “drugsos (0.41) tells us that there significant cases of drugs-out-stock which suggest ineffectiveness of nigeria health insurance scheme. drawing samples from (nuts) follows that for each parameter, efficient sample is a real measure of effective sample size, while rhat is the would-be scale reduction factor on split chains (at convergence, rhat = 1). figure 1 shows that encounter has positive relation with type; the figure suggests that the effect of encounter on type is higher for secondary user of the facility since it is higher on zero end than 1. figure 2 shows that encounter has positive relation with age; the figure suggests that as age increases encounter increases figure 3 shows that encounter has positive relation with sex; the figure suggests that male account for more encounter than female. figure 4 shows that encounter has positive relation with type; number of drugsadm increases with number of encounter figure 5 shows that encounter has positive relation with type; number of drugsos increases significantly with number of encounter figure 6 shows that as primary users of the facility increases, the number of encounter increases across ages 229 adesina / j. nig. soc. phys. sci. 3 (2021) 224–233 230 figure 8. density and trace plots for all covariates figure 7 shows that as primary users of the facility increases, the number of encounter increases across sexes figure 8 shows that the density plots for the tail area of the distribution, which corresponds to l-95% ci and u-95% ci in table 5 and trace plot, the trace plot shows that the chains are stable. 4. summary and conclusion in this study bayesian multi-level model was proposed and implemented. the no-u-turn sampler (nuts) sampler was used to sample from the posterior distribution, and implementations were done using the ‘package brms’ in r. simulation study was carried out for both over-and under-dispersed and response variables were generated through discrete weibull distribution while the predictors generated from uniform distribution. results from under-dispersed revealed that geometric distribution is the most appropriate model to fit count data using multi-level approach. while for over-dispersed simulated data, negative binomial shows to outperform the poisson, geometric, hurdlepoisson, hurdle-negbin, zero-inflated-poisson, zero-inflated-negbin. pareto k diagnostics shows that for under-dispersed and overdispersed simulated data all k are less than 0.5, which makes all the observations to be good, also all waic were the same as loo-ic except for poisson in the over-dispersed simulated data. real-life data set of count of encounter of patients from national health insurance scheme was used for further analysis. the model that performs best was the geometric distribution followed by negative binomial model. contrary to the simulated data not all waic were the same as loo-ic, except for poisson model in the over-dispersed simulated data. the need to carry out loo-ic was informed by having observa230 adesina / j. nig. soc. phys. sci. 3 (2021) 224–233 231 table 3. real life data model elpd waic p waic waic elpd loo p loo looic waic=loo poisson est. se -127.4 15.3 15.9 6.3 254.8 30.6 -129.7 15.9 18.2 7.0 259.3 31.8 no negbin est. se -122.3 12.8 10.6 3.5 244.6* 25.6 -124.0 13.5 12.3 4.3 248.0* 27.0 no geometric est se. -120.8 12.3 7.4 2.3 241.6** 24.6 -121.5 12.5 8.1 2.7 242.9** 25.0 no hurdle poisson est. se -132.6 13.3 11.4 4.2 265.1 26.6 -135.0 14.2 13.8 5.4 269.9 28.5 no hurdle negbin est. se -131.6 12.7 5.0 1.3 263.3 25.5 -133.1 13.3 6.4 2.3 266.2 26.5 no zero inflated poisson est. se -126.7 14.8 12.8 5.1 253.3 29.6 -127.8 15.1 13.9 5.4 255.6 30.1 no zero inflated negbin est. se -122.7 12.9 9.4 3.3 245.3 25.9 -123.4 13.2 10.2 3.7 246.8 26.4 no table 4. pareto k diagnostics model pareto k diag. remark count pct min neff obs. k>0.7 poisson (-inf, 0.5] (0.5, 0.7] (0.7, 1] (1, inf) good ok bad very bad 112 1 2 1 96.6% 0.9 1.7% 0.9% 1367 236 17 4 3 negbin (-inf, 0.5] (0.5, 0.7] (0.7, 1] (1, inf) good ok bad very bad 112 1 3 0 96.6% 0.9% 2.6% 0.0% 1068 446 19 3 geometric (-inf, 0.5] (0.5, 0.7] (0.7, 1] (1, inf) good ok bad very bad 114 0 2 0 98.3% 0.0% 1.7% 0.0% 1036 88 2 hurdle poisson (-inf, 0.5] (0.5, 0.7] (0.7, 1] (1, inf) good ok bad very bad 108 5 1 2 93.1% 4.3% 0.9% 1.7% 859 413 16 7 3 hurdle negbin (-inf, 0.5] (0.5, 0.7] (0.7, 1] (1, inf) good ok bad very bad 109 3 3 1 94.0% 2.6% 2.6% 0.9% 2909 594 135 9 4 zero inflated poisson (-inf, 0.5] (0.5, 0.7] (0.7, 1] (1, inf) good ok bad very bad 111 3 1 1 95.7% 2.6% 0.9% 0.9% 1885 165 26 10 2 zero inflated negbin (-inf, 0.5] (0.5, 0.7] (0.7, 1] (1, inf) good ok bad very bad 113 1 2 0 97.4% 0.9% 1.7% 0.0% 1509 213 50 2 231 adesina / j. nig. soc. phys. sci. 3 (2021) 224–233 232 table 5. population-level effects estimate est. error l-95% ci u-95% ci eff.sample intercept -0.36 0.63 -1.63 0.83 2290 type -1.33 0.65 -2.57 -0.06 1972 age 0.01 0.01 -0.02 0.03 2024 sex 0.19 0.48 -0.73 1.13 2839 drugsadm 0.06 0.03 -0.00 0.13 3444 drugsos 0.41 0.17 0.10 0.74 2609 type:age 0.01 0.02 -0.02 0.05 1818 type:sex -0.18 0.56 -1.26 0.90 2897 tions for all the cases. figures 1 to figure 7 contains marginal plots to identify the relationship between the response variable (encounter) and covariates; type, sex, drugsadm, age, and drugsos. as identified by [8] that geometric family have the ability to model count data effectively, the same has been demonstrated in this study using bayesian multi-level regression approach. future direction can consider fitting multi-level regression model to fit count data using distribution such as the weibull-exponential distribution and exponentiated generalized weibull proposed by [32] and [33] respectively. acknowledgements my appreciation goes to the medical director of the state hospital, ota, ogun state where the data set was obtained. i also appreciate the inputs of reviewers and editors on the manuscript. references [1] m. s. workie & a. m. lakew, “bayesian 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[33] p. e. oguntunde, o. a. odetunmibi & a. o. adejumo, “ on the exponentiated generalized weibull distribution: a generalization of the weibull distribution”, indian journal of science and technology 8 (2015) 1. 233 j. nig. soc. phys. sci. 3 (2022) 26–37 journal of the nigerian society of physical sciences efficient hybrid block method for the numerical solution of second-order partial differential problems via the method of lines olumide o. olaiyaa,, razaq a. azeezb, mark i. modebeia adepartment of mathematics programme, national mathematical centre, abuja, nigeria bdepartment of mathematics, university of abuja, abuja, nigeria abstract this study is therefore aimed at developing classes of efficient numerical integration schemes, for direct solution of second-order partial differential equations (pdes) with the aid of the method of lines. the power series polynomials were used as basis functions for trial solutions in the derivation of the proposed schemes via collocation and interpolation techniques at some appropriately chosen grid and off-grid points the derived schemes are consistent, zero-stable and convergent. the proposed methods perform better in terms of accuracy than some existing methods in the literature. doi:10.46481/jnsps.2021.141 keywords: initial value problem, boundary value problem; block method, linear multistep method, hybrid method, method od lines article history : received: 02 october 2020 received in revised form: 10 december 2020 accepted for publication: 23 january 2021 published: 27 february 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: t. latunde 1. introduction in this work, we consider the second-order pde of the form a ∂y2 ∂x2 + b ∂y2 ∂t2 +c ∂y ∂t = 0, x∈ [a, b] , t > 0 with any of the following initial-boundary conditions y(x,α1) = ξ1, y(x,α2) = ξ2, y(β1t) = ζ1, y(β2, t) = ζ2, y(x,α1) = ξ1, y(β2, t) = ζ2 email address: olaiyaolumide.o@gmail.com (olumide o. olaiya ) , where αi, βi, ζi, ξi, i = 1, 2, a, b, a, b, c are all real constants and a , 0. y ∈ c2 [a, b] × [c, d] , x ∈ (a, b) , t ∈ (c, d). second-order pdes can either be of laplacian, poisson forms which could either be heat or waveform of equations. they find their applications in numerous areas of human endeavours, especially in mathematical sciences and engineering. the developed differential equations require solutions, either in closed form (analytic form) or in numerical forms. in most cases, closed-form of solutions is rare to come by as there exist limited methods for solving such models in the form of differential equations. this brings to light the use of numerical methods/techniques to solve the modelled problem. numerical techniques are numerous and the types know no bounds. they include; the euler method, runge-kutta methods, linear multistep method, shooting method, finite difference method, finite element methods, e.g galerkin method, 26 olaiya et al. / j. nig. soc. phys. sci. 3 (2022) 26–37 27 spectral element method. other methods are; spectral method base on fourier transformation; method of lines reduces the pde to a large system of the ordinary differential equation (ode) boundary element method (bem) based on transforming a pde to an integral equation on the boundary of the domains and it is popular in computational fluid dynamics, the list is endless. authors who have worked extensively on numerical methods for approximation of solution of a differential equation include but no limited to brugnano and trigiante [1], onumanyi et al. [2, 3], jator [4, 5], fatunla [6], yusuph and onumanyi [7], siraj-ul-islam, et al.[8], adewale et al. [9]. we adopt the method of lines approach which is commonly used for solving time-dependent partial differential equations (pdes), whereby the spatial derivatives are replaced by finite difference approximations see ngwane and jator [10]. the method of lines (mols) allows the conversion of pdes into odes by complete or partial discretisation of the independent variables resulting in algebraic equations. if partial discretization is carried out and with only one remaining independent variable, then this results in the system of odes which is an approximation of the original pde. thus, one of the underscored features of the mol is the use of existing, and generally well established, numerical methods for odes, for more literature on this approach see brugnano and trigiante [1], ramos and vigo-aguiar [11], cash [12] and jator and li [13]. 2. derivation of the method second order ordinary differential equation of the form is considered y′′ = f (x, y) (1) subject certain conditions, where a, b are real numbers, f is a continuous function on (a, b) and y ∈ c2[a, b]. a 2-step block methods for the problem of the form (1) is considered. the grid points given by xn, xn+1 = xn + h, xn+2 = xn + 2h, are considered for solving the problem in (1) on the interval [xn, xn+2]. we assume a trial solution y(x) of (1) by a polynomial p(x) given by y(x) ' p(x) = m−1∑ i=0 ai x i (2) which on differentiating yields y′′(x) ' p′′(x) = m−1∑ i=2 i(i − 1)ai x i−2 (3) with the ai ∈ r real unknown parameters to be determined. and m = r + s; r is the number of interpolation points and s is the number of collocation points. 2.1. specification of the method in this work the interval of integration considered is [xn, xn+2], we thus consider two different categories of off-set points viz-aviz the points x i 3 , for i = 1, 2, 4, 5 and x i 4 , for i = 1, 2, 3, 5, 6, 7. 2.1.1. case 1 here, we consider the specification where the off-set points are x i 3 , for i = 1, 2, 4, 5. interpolating (2) at the points x i 3 , for i = 1, 2 implies r = 2 and collocating (3) at points x i 3 , for i = 0(1)6 implies s = 7 so that (2) and (3) becomes y(x) ' p(x) = 8∑ i=0 ai x i = a0 + a1 x + a2 x 2 + · · · + a8 x 8 (4) which on differentiating twice yields y′′(x) ' p′′(x) = 8∑ i=2 i(i − 1)ai x i−2 = 2a2 + 6a3 x + 12a4 x 2 + · · · + 56a8 x 6 (5) 27 olaiya et al. / j. nig. soc. phys. sci. 3 (2022) 26–37 28 from the imposed collocation condition, the following system of algebraic equation is obtained a =  1 xn+ 13 x2 n+ 13 x3 n+ 13 x4 n+ 13 x5 n+ 13 x6 n+ 13 x7 n+ 13 x8 n 13 1 xn+ 23 x2 n+ 23 x3 n+ 23 x4 n+ 23 x5 n+ 23 x6 n+ 23 x7 n+ 23 x8 n 23 0 0 2 6xn 12x2n 20x 3 n 30x 4 n 42x 5 n 56x 6 n 0 0 2 6xn+ 13 12x2 n+ 13 20x3 n+ 13 30x4 n+ 13 42x5 n+ 13 56x6 n+ 13 0 0 2 6xn+ 23 12x2 n+ 23 20x3 n+ 23 30x4 n+ 23 42x5 n+ 23 56x6 n+ 23 0 0 2 6xn+1 12x2n+1 20x 3 n+1 30x 4 n+1 42x 5 n+1 56x 6 n+1 0 0 2 6xn+ 43 12x2 n+ 43 20x3 n+ 43 30x4 n+ 43 42x5 n+ 43 56x6 n+ 43 0 0 2 6xn+ 53 12x2 n+ 53 20x3 n+ 53 30x4 n+ 53 42x5 n+ 53 56x6 n+ 53 0 0 2 6xn+2 12x2n+2 20x 3 n+2 30x 4 n+2 42x 5 n+2 56x 6 n+2  , x=  a0 a1 a2 a3 a4 a5 a6 a7 a8  ;b=  yn+ 13 yn+ 23 h2 fn h2 fn+ 13 h2 fn+ 23 h2 fn+1 h2 fn+ 43 h2 fn+ 53 h2 fn+2  e= (1,x, x2, x3, x4, x5, x6, x7, x8)t where yn ≈ y (xn) , fn ≈ f (xn, yn, y′n). hence, we state the following theorem without proof. theorem 2.1. [14] let (4) and (5) be satisfied, then the 2-step continuous linear hybrid multistep method is equivalent to the equation y(x) =bt (a−1k ) t e (6) where b, a and e are as defined above. applying the above theorem, the following continuous hybrid method is derived y(x) = 2∑ i=1 α i 3 yn+ i3 +h 2 2∑ j=0 β i 3 fn+ i3 (7) where α and β are function of t given as α 1 3 = −3t+2, α 2 3 = 3t−1 β0=h2 [ 21 32 t 6− 459 4480 t− 27 160 t 7+ 12 t 2+ 814480 t 8+ 863108894− 49 40 t 3− 441 320 t 5+ 203120 t 4 ] β 1 3 =h2 [ −243 2240 t 8+ 2728 t 7− 279 80 t 6+ 26140 t 5− 261 40 t 4+3t3−2762360480 t+ 8999 90720 ] β 2 3 =h2 [ 243 896 t 8− 513 224 t 7+ 1233160 t 6− 4149 320 t 5+ 35132 t 4− 15 4 t 3+ 18689120960 t− 769 181440 ] β1=h2 [ −81 224 t 8+ 8128 t 7− 363 40 t 6+ 27920 t 5− 127 12 t 4+ 103 t 3− 139 864 t+ 1987 136080 ] β 4 3 =h2 [ 243 896 t 8− 459 224 t 7+ 963160 t 6− 2763 320 t 5+ 9916 t 4− 15 8 t 3+ 10921120960 t− 1609 181440 ] β 5 3 =h2 [ −243 2240 t 8+ 2735 t 7− 171 80 t 6+ 11740 t 5− 81 40 t 4+ 35 t 3− 347 12096 t+ 263 90720 ] β2=h2 [ 81 4480 t 8− 27 224 t 7+ 51160 t 6− 27 64 t 5+ 137480 t 4− 1 12 t 3+ 479120960 t− 221 544320 ] (8) where t= x−xnh , evaluating (7) at non-interpolating points i.e at the points x=xn+ i3 for i= 0, 3, 4, 5, 6 which is equivalent to t = i 3 yn= 2yn+ 13 −yn+ 23 +h 2 ( 863 108864 fn+ 8999 90720 fn+ 13 − 769 181440 fn+ 23 + 1987 136080 fn+1 − 1609 181440 fn+ 43 + 263 90720 fn+ 53 − 221 544320 fn+2 ) yn+1= −yn+ 13 +2yn+ 23 +h 2 ( −221 544320 fn+ 977 90720 fn+ 13 + 16451 181440 fn+ 23 + 1357 136080 fn+1 + 71181440 fn+ 43 − 31 90720 fn+ 53 + 31 544320 fn+2 ) yn+ 43 = −2yn+ 13 +3yn+ 23 +h 2 ( −137 181440 fn+ 209 10080 fn+ 13 + 433 2240 fn+ 23 + 4927 45360 fn+1 + 25720160 fn+ 43 − 1 672 fn+ 53 + 31 181440 fn+2 ) yn+ 53 = −3yn+ 13 +4yn+ 23 +h 2 ( −19 181440 fn+ 17 560 fn+ 13 + 2987 10080 fn+ 23 + 4927 22680 fn+1 + 3893360 fn+ 43 + 41 5040 fn+ 53 − 11 90720 fn+2 ) yn+2= −4yn+ 13 +5yn+ 23 +h 2 ( −95 54432 fn+ 389 9072 fn+ 13 + 7085 18144 fn+ 23 + 4633 13608 fn+1 − 3893 18144 fn+ 43 + 1061 9072 fn+ 53 + 409 54432 fn+2 ) (9) 28 olaiya et al. / j. nig. soc. phys. sci. 3 (2022) 26–37 29 differentiating α 1 3 , α 2 3 and all the β i 3 , i = 0(1)6 and evaluating the derivative of (8) at the points x=xn+ i3 ,i = 0(1)6, equivalent to t= i3 . hy′n+3yn+ 13 −3yn+ 23 = h 2 [ −459 4480 fn− 27623 60480 fn+ 13 + 18689 120960 fn+23− 139 864 fn+1+ 10921 120960 fn+ 43 − 347 12096 fn+ 53 + 479 120960 fn+2 ] hy ′ n+ 13 +3yn+ 13 −3yn+ 23 = h 2 [ 199 72576 fn− 1973 20160 fn+ 13 − 18 128 fn+23 + 4157 90720 fn+1− 851 40320 fn+ 43 + 416720 fn+ 53 + 289 120960 fn+2 ] y ′ n+ 23 +3yn+ 13 −3yn+ 23 = h 2 [ −731 362880 fn− 13 320 fn+ 13 + 6347 40320 fn+23− 3971 90720 fn+1+ 257 13440 fn+ 43 − 109 20160 fn+ 53 + 253 262880 fn+2 ] hy ′ n+1+3yn+ 13 −3yn+ 23 = h 2 [ −1 1920 fn+ 1537 60480 fn+ 13 + 39587 120960 fn+23 + 4927 30240 fn+1− 2201 120960 fn+ 43 + 2096720 fn+ 53 − 43 120960 fn+2 ] hy ′ n+ 43 +3yn+ 13 −3yn+ 23 = h 2 [ −571 362880 fn+ 691 20160 fn+ 13 + 1299 4480 fn+23 + 33533 90720 fn+1+ 1223 8064 fn+ 43 + 796720 fn+ 53 + 59 51840 fn+2 ] hy ′ n+ 53 +3yn+ 13 −3yn+ 23 = h 2 [ −29 362880 fn+ 51 2240 fn+ 13 + 13313 40320 fn+23 + 5081 18144 fn+1− 5519 13440 fn+ 43 + 73576 fn+ 53 + 1313 362880 fn+2 ] (10) the schemes in (9) and (10) form the requited method for solving (1) numerically. 2.1.2. case 2 here, we consider the specification where the off-set points are also x i 3 , for i= 1, 2, 4, 5. interpolating (2) at the points xi+n, for i= 0, 1 implies r= 2 and collocating (3) at points x i 3 , for i = 0(1)6 implies s= 7 so that (2) and (3) becomes y(x)'p(x) = 8∑ i=0 ai x i =a0+a1 x+a2 x 2 +· · ·+a8 x 8 (11) which on differentiating yields y′′(x)'p′′(x) = 8∑ i=2 i(i−1)ai x i−2 = 2a2+6a3 x+12a4 x 2 +· · ·+56a8 x 6 (12) from the imposed collocation condition, the following system of algebraic equation is obtained a =  1 xn x2n x 3 n x 4 n x 5 n x 6 n x 7 n x 8 n 1 xn+1 x2n+1 x 3 n+1 x 4 n+1 x 5 n+1 x 6 n+1 x 7 n+1 x 8 n+1 0 0 2 6xn 12x2n 20x 3 n 30x 4 n 42x 5 n 56x 6 n 0 0 2 6xn+ 13 12x2 n+ 13 20x3 n+ 13 30x4 n+ 13 42x5 n+ 13 56x6 n+ 13 0 0 2 6xn+ 23 12x2 n+ 23 20x3 n+ 23 30x4 n+ 23 42x5 n+ 23 56x6 n+ 23 0 0 2 6xn+1 12x2n+1 20x 3 n+1 30x 4 n+1 42x 5 n+1 56x 6 n+1 0 0 2 6xn+ 43 12x2 n+ 43 20x3 n+ 43 30x4 n+ 43 42x5 n+ 43 56x6 n+ 43 0 0 2 6xn+ 53 12x2 n+ 53 20x3 n+ 53 30x4 n+ 53 42x5 n+ 53 56x6 n+ 53 0 0 2 6xn+2 12x2n+2 20x 3 n+2 30x 4 n+2 42x 5 n+2 56x 6 n+2  , x=  a0 a1 a2 a3 a4 a5 a6 a7 a8  ;b=  yn yn+1 h2 fn h2 fn+ 13 h2 fn+ 23 h2 fn+1 h2 fn+ 43 h2 fn+ 53 h2 fn+2  e= (1,x, x2, x3, x4, x5, x6, x7, x8)t applying the above theorem, the following continuous hybrid method is derived y(x) = 1∑ i=0 αiyn+i+h 2 2∑ i=0 β i 3 fn+ i3 (13) 29 olaiya et al. / j. nig. soc. phys. sci. 3 (2022) 26–37 30 where α and β are function of t given as α0= −t α1= 1+t β0= 105h2 t−112h2 t3 +56h2 t4 +378h2 t5−252h2 t6−324h2 t7 +243h2 t8 13440 β 1 3 = − 3(−85h2 t−56h2 t3 +42h2 t4 +168h2 t5−168h2 t6−72h2 t7 +81h2 t8) 2240 β 2 3 = 3(347h2 t−560h2 t3 +840h2 t4 +546h2 t5−1092h2 t6−180h2 t7 +405h2 t8) 4480 β1= 563h2 t+1680h2 t2−3430h2 t4 +3528h2 t6−1215h2 t8 3360 β 4 3 = 3(−41h2 t+560h2 t3 +840h2 t4−546h2 t5−1092h2 t6 +180h2 t7 +405h2 t8) 4480 β 5 3 = − 3(−5h2 t+56h2 t3 +42h2 t4−168h2 t5−168h2 t6 +72h2 t7 +81h2 t8) 2240 β2= −11h2 t+112h2 t3 +56h2 t4−378h2 t5−252h2 t6 +324h2 t7 +243h2 t8 13440  (14) where t= x−xn−1h . evaluating (13) at the points x=xn+ i3 for i= 1, 2, 4, 5, 6 which is equivalent to t= −2/3,−1/3, 1/3, 2/3, 1 the following main methods are obtained yn+ 13 = 2yn 3 + yn+1 3 −h 2 ( 2803 fn 544320− 1265 f n+ 13 18144 − 1657 f n+ 23 60480 − 1777 fn+1 136080 + 1049 f n+ 43 181440 − 11 f n+ 53 6048 + 137 fn+2 544320 ) yn+ 23 = yn 3 + 2yn+1 3 −h 2 ( 1291 fn 544320− 1217 f n+ 13 30240 − 10711 f n+ 23 181440 − 1567 fn+1 136080 + 163 f n+ 43 60480 − 67 f n+ 53 90720 + 53 fn+2 544320 ) yn+ 43 = − yn 3 + 4yn+1 3 +h 2 ( 661 fn 272160 + 1789 f n+ 13 45360 + 2147 f n+ 23 30240 + 6817 fn+1 68040 + 841 f n+ 43 90720 − f n+ 53 15120− 11 fn+2 272160 ) yn+ 53 = − 2yn 3 + 5yn+1 3 +h 2 ( 535 fn 108864 + 475 f n+ 13 6048 + 5167 f n+ 23 36288 + 5725 fn+1 27216 + 1321 f n+ 43 12096 + 193 f n+ 53 18144 − 53 fn+2 108864 ) yn+2= −yn+2yn+1+h2 ( 47 fn 6720 + 27 224 fn+ 13 + 459 f n+ 23 2240 + 563 fn+1 1680 + 459 f n+ 43 2240 + 27 224 fn+ 53 + 47 fn+2 6720 ) (15) differentiating (14) and evaluating the derivative of (13) at the points x=xn+ 3i for i= 0(1)6 which is equivalent to t= −1,−2/3,−1/3, 0, 1/3, 2/3, 1 the following additional methods are obtained y′n= − y h n + yn h +1 −h ( 253 fn 2688 − 165 448 fn+ 13 + 267 f n+ 23 4480 − 5 32 fn+1+ 363 f n+ 43 4480 − 57 f n+ 53 2240 + 47 fn+2 13440 ) , y′n+ 13 = − yn h + yn+1 h +h ( 4019h fn 362880 − 571h f n+ 13 60480 − 679h f n+ 23 3456 + 4577h fn+1 90720 − 3673h f n+ 43 120960 + 113h f n+ 53 12096 − 457h fn+2 362880 ) , y′n+ 23 = − yn h + yn+1 h +h ( 2293h fn 362880 + 223h f n+ 13 1728 + 7561h f n+ 23 120960 − 3551h fn+1 90720 + 1193h f n+ 43 120960 − 131h f n+ 53 60480 + 17h fn+2 72576 ) , y′n+1= − yn h + yn+1 h +h ( h fn 128 + 51 448 h fn+ 13 + 1041h f n+ 23 4480 + 563h fn+1 3360 − 123h f n+ 43 4480 + 3 448 h fn+ 53 − 11h fn+2 13440 ) , y′n+ 43 = − yn h + yn+1 h +h ( 2453h fn 362880 + 7421h f n+ 13 60480 + 23593h f n+ 23 120960 + 33953h fn+1 90720 + 3445h f n+ 43 24192 − 103h f n+ 53 12096 + 7h fn+2 10368 ) , y′n+ 53 = − yn h + yn+1 h +h ( 599h fn 72576 + 1345h f n+ 13 12096 + 28459h f n+ 23 120960 + 5165h fn+1 18144 + 48551h f n+ 43 120960 + 1123h f n+ 53 8640 − 1481h fn+2 362880 ) , y′n+2= − yn h + yn+1 h +h ( 47h fn 13440 + 327h f n+ 13 2240 + 111 896 h fn+ 23 + 1651h fn+1 3360 + 93 640 h fn+ 43 + 219 448 h fn+ 53 + 453h fn+2 4480 ) (16) here, it is noteworthy that (9) and (10) are combined to form a block for case 1 while (15) and (16) form another block for case 11. for each case, (1) is solved which is a system of second-order ordinary differential equations resulting from the semidiscretization of a second-order pde. 3. analysis of the method 3.1. order and local truncation error (lte) the lmms (8), and (13) are said to be of order p if c0=c1=c2= · · · =c p+µ−1 = 0, c p+µ,0. 30 olaiya et al. / j. nig. soc. phys. sci. 3 (2022) 26–37 31 here c p+µ is the error constant and c p+µh p+µy( p+µ)(xn) is the principal local truncation error (lte) at the point xn. the c′s are given by c0=α0+α1+α2+· · ·+αk c1= (α0+α1+α2+· · ·+αk) − (β0+β1+· · ·+βk) cq= 1 q! (α1+2 qα2+· · ·+k qαk)− 1 (q−3)! (β1+2 q−1β2+· · ·+k q−3βk),q= 2, 3, . . . the lte associated with any of (8) and (13) is given by the difference operator l[y(x) :h] = 2∑ i=1 α i 3 y(xn+ i 3 )−h2 2∑ j=0 β i 3 y′′(xn+ i 3 ) (17) where y∈c2[a, b] is an arbitrary function. expanding (17) in taylor’s series about the point xn, the expression is obtained as: l[y(xn) :h] =c0y(xn)+c1hy ′(xn)+c2h 2y′′(xn)+· · ·+cρ+2h ρ+2yρ+2(xn) (18) expanding each scheme in (9) and (10), the following principal truncation errors are obtained: c0p+2=− 19y(9)[x]h9 119042784 +o[h]10, c1p+2= 31y(9)[x]h9 1190427840 +o[h]10, c2p+2= y(9)[x]h9 4723920 +o[h]10 c 4 3 p+2= 31y(9)[x]h9 595213920 +o[h]10, c 5 3 p+2= 31y(9)[x]h9 595213920 +o[h]10, c′0 p+2= 8881y(9)[x]h9 5952139200 +o[h]10 c′ 1 3 p+2= − 409y(9)[x]h9 1700611200 +o[h]10, c′ 2 3 p+2= 1201y(9)[x]h9 5952139200 +o[h]10, c′1 p+2= − 463y(9)[x]h9 11904278400 +o[h]10 c′ 4 3 p+2= 1201y(9)[x]h9 5952139200 +o[h]10, c′ 5 3 p+2= − 409y(9)[x]h9 1700611200 +o[h]10, c′2 p+2= 8881y(9)[x]h9 5952139200 +o[h]10 the above blocked method (9) and (10) is of uniform order p= 7 expanding each scheme in (15) and (16), the following principal truncation errors are obtained: c 1 3 p+2= 349y(9)(x)h9 3571283520 +o(h)10, c 2 3 p+2= 2y(9)(x)h9 55801305 +o(h)10, c 4 3 p+2= − 2y(9)(x)h9 55801305 +o(h)10 c 5 3 p+2= − 349y(9)(x)h9 3571283520 +o(h)10, c2p+2= y(9)(x)h9 32659200 +o(h)10, c′0p+2= y(9)(x)h9 765450 +o(h)10 c′ 1 3 p+2= − 1691y(9)(x)h9 3968092800 +o(h)10, c′ 2 3 p+2= y(9)(x)h9 62001450 +o(h)10, c′1p+2− 11y(9)(x)h9 48988800 +o(h)10 c′ 4 3 p+2 y(9)(x)h9 62001450 +o(h)10, c′ 5 3 p+2− 1691y(9)(x)h9 3968092800 +o(h)10, c′2p+2 y(9)(x)h9 765450 ++o(h)10 the above blocked method (15) and (16) is of uniform order p= 7 the lmm (8) (same for (13)) is said to be consistent if it has order p≥1 and the first and second characteristic polynomials which are defined respectively, as ρ(r) = k∑ j=0 α jz j (19) and σ(r) = k∑ j=0 β jz j (20) 31 olaiya et al. / j. nig. soc. phys. sci. 3 (2022) 26–37 32 where r is the principal root, satisfy the following conditions: k∑ j=0 α j= 0 (21) ρ(1) =ρ′(1) = 0 (22) and ρ′′(1) = 2!σ(1) (23) henrici [15], lambert[16]. consider the main method in (9) given as yn+2= −4yn+ 13 +5yn+ 23 +h 2 ( −95 54432 fn+ 389 9072 fn+ 13 + 7085 18144 fn+ 23 + 4633 13608 fn+1 − 3893 18144 fn+ 43 + 1061 9072 fn+ 53 + 409 54432 fn+2 ) (24) the condition (21) is satisfied. the first characteristic equation for (9) is given as: ρ(r) =r2+4r 1 3 −5r 2 3 (25) ρ′(r) = 2r+ 4 3r2/3 − 10 3r1/3 (26) here ρ(r) = 0, ρ′(r) = 0. therefore, (22) is satisfied. the second characteristic polynomial for (9) is given as σ(r) = −95 54432 + 389 9072 r 1 3 + 7085 18144 r 2 3 + 4633 13608 r+ 3893 18144 r 4 3 + 1061 9072 r 5 3 + 409 54432 r2 (27) σ(1) = 10 9 (28) ρ′′(r) = 2− 8 9r5/3 + 10 9r4/3 . ρ′′(r) = 20 9 (29) hence condition (23) is satisfied. conclusively, the hybrid method is consistent. consider the main method in (15) given as yn+2= −yn+2yn+1+h2 ( 47 fn 6720 + 27 224 fn+ 13 + 459 f n+ 23 2240 + 563 fn+1 1680 + 459 f n+ 43 2240 + 27 224 fn+ 53 + 47 fn+2 6720 ) (30) the condition (21) is satisfied. the first characteristic equation for (15) is given as: ρ(r) =r2−2r+1 (31) ρ′(r) = 2r−2 (32) here ρ(1) = 0, ρ′(1) = 0. therefore, (22) is satisfied. the second characteristic polynomial for (15) is given as σ(r) = 47 6720 + 27 224 r 1 3 + 459 2240 r 2 3 + 563 1680 r+ 459 2240 r 4 3 + 27 224 r 5 3 + 47 6720 r2 (33) σ(1) = 1 (34) ρ′′(r) = 2. ρ′′(1) = 2 (35) hence condition (23) is satisfied. conclusively, the hybrid method is consistent. 32 olaiya et al. / j. nig. soc. phys. sci. 3 (2022) 26–37 33 3.2. zero stability to establish hat the methods are zero stable, each of the method in block form are solved simultaneously to obtain all the yi and y′i’s for appropriate index i, (see modebei et al.[17]). for the method (9) and its additional methods in (10), they are taken in block form and solved simultaneously to obtain y i 3 for i = 1(1)6 to obtain the following block method. for block method (9)-(10) yn+ 13 = yn+ hy′n 3 +h 2 ( 28549 fn 1088640 + 275 f n+ 13 5184 − 5717 f n+ 23 120960 + 10621 fn+1 272160 − 7703 f n+ 43 362880 + 403 f n+ 53 60480 − 199 fn+2 217728 ) yn+ 23 = yn+ 2hy′n 3 +h 2 ( 1027 fn 17010 + 194 945 fn+ 13 − 8 81 fn+ 23 + 788 fn+1 8505 − 97 f n+ 43 1890 + 46 f n+ 53 2835 − 19 fn+2 8505 ) yn+1= yn+hy′n+h 2 ( 253 fn 2688 + 165 448 fn+ 13 − 267 f n+ 23 4480 + 5 fn+1 32 − 363 f n+ 43 4480 + 57 f n+ 53 2240 − 47 fn+2 13440 ) yn+ 43 = yn+ 4hy′n 3 +h 2 ( 1088 fn 8505 + 1504 f n+ 13 2835 − 8 945 fn+ 23 + 2624 fn+1 8505 − 8 81 fn+ 43 + 32 945 fn+ 53 − 8 fn+2 1701 ) yn+ 53 = yn+ 5hy′n 3 +h 2 ( 35225 fn 217728 + 8375 f n+ 13 12096 + 3125 f n+ 23 72576 + 25625 fn+1 54432 − 625 f n+ 43 24192 + 275 f n+ 53 5184 − 1375 fn+2 217728 ) yn+2= yn+2hy′n+h 2 ( 41 fn 210 + 6 7 fn+ 13 + 3 35 fn+ 23 + 68 fn+1 105 + 3 70 fn+ 43 + 6 35 fn+ 53 ) y′n+ 13 = y ′ n+h ( 19087 fn 181440 + 2713 f n+ 13 7560 − 15487 f n+ 23 60480 + 586 fn+1 2835 − 6737 f n+ 43 60480 + 263 f n+ 53 7560 − 863 fn+2 181440 ) y′n+ 23 = y ′ n+h ( 1139 fn 11340 + 94 189 fn+ 13 + 11 f n+ 23 3780 + 332 fn+1 2835 − 269 f n+ 43 3780 + 22 945 fn+ 53 − 37 fn+2 11340 ) y′n+1= y ′ n+h ( 137 fn 1344 + 27 56 fn+ 13 + 387 f n+ 23 2240 + 34 fn+1 105 − 243 f n+ 43 2240 + 9 280 fn+ 53 − 29 fn+2 6720 ) y′n+ 43 = y ′ n+h ( 286 fn 2835 + 464 945 fn+ 13 + 128 945 fn+ 23 + 1504 fn+1 2835 + 58 945 fn+ 43 + 16 945 fn+ 53 − 8 fn+2 2835 ) y′n+ 53 = y ′ n+h ( 3715 fn 36288 + 725 f n+ 13 1512 + 2125 f n+ 23 12096 + 250 fn+1 567 + 3875 f n+ 43 12096 + 235 f n+ 53 1512 − 275 fn+2 36288 ) y′n+2= y ′ n+h ( 41 fn 420 + 18 35 fn+ 13 + 9 140 fn+ 23 + 68 fn+1 105 + 9 140 fn+ 43 + 18 35 fn+ 53 + 41 fn+2 420 ) (36) for block method (15)-(16) similar operation is carried out; a numerical method is zero-stable if the solutions remain bounded as h→0, which means that the method does not provide solutions that grow unbounded as the number of steps increases, modebei et al.[17]. to show the zero-stability of the block method (36), we take h→0 the method may be rewritten in matrix form as a0yn=a1yn−1 (37) yn= (y 0 n , y 1 n ) t y 0n = (yn+ 13 , yn+ 23 , yn+1, yn+ 43 , yn+ 53 , yn+2) t y 1n = (y ′ n+ 13 , y′n+ 23 , y ′ n+1, y ′ n+ 43 , y′n+ 53 , y ′ n+2) t for method (36) a0=i12×12 identity matrix and a1=i12×12 matrix given by a1=  a11 0 0 a22  , a11=  1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0  , a22=  0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0  33 olaiya et al. / j. nig. soc. phys. sci. 3 (2022) 26–37 34 the characteristic polynomial of the matrix a11 is given as |a11−λi|, that is λ5(λ−1)= 0 with root λ j= 0 for j= 1, . . . , 5 and λ6= 1. the characteristic polynomial of the matrix a22 is given as |a22−λi|, that is λ5(λ−1) = 0 with root λ j= 0 for j= 1, . . . , 5 and λ6= 1. for method (15)-(16) a0=i12×12 identity matrix and a1=i12×12 matrix given by a1=  a11 0 0 a22  , a11=a22=  1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0  the characteristic polynomial of the matrix aii is given as |aii−λi|, i= 1, 2, that is λ5(λ−1) = 0 with root λ j= 0 for j= 1, . . . , 5 and λ6= 1. definition 3.1. the two step hybrid block method (9)-(10) (or (15)-(16)) is said to be zero stabile if the number of root of the first characteristic equation |ρ(r)| < 1 and if |ρ(r)| = 1, then the multiplicity of ρ(r) must not exceed 2. hence, the are zero stable. 3.3. convergence of the methods definition 3.2. convergence: an lmm is said to be convergent if and only if it is consistent and zero-stable. by the above definition, the derived hybrid methods are convergent. 4. numerical examples in this section, the performance of the developed two-step hybrid block scheme is examined. the exact and approximate solution are tabulated. the tables below show the numerical results of the newly developed scheme with the exact solution for solving the problem and the result of the developed scheme are more accurate than existing methods. for simplicity, method in (9)-(10) would be termed hybrid 2-step block method 1 (h2bm1), and method in (15)-(16) would be termed hybrid 2-step block method 2 (h2bm2) example 1. consider the pde (ngwane and jator [10]). κ ∂y2 ∂x2 − ∂y ∂t = 0 y(0,t) =y(1,t) = 0, y(x, 0) = sinπx+sinωπx, κ>1 (38) the analytic solution is y(x, t) =e−π 2κtsinπx+e−ω 2π2κtsinπx following ngwane and jator [10], (38) becomes dy2m dx2 = 1 κ y(x , tm+1)−y(x , tm−1)] (∆t) (39) ym(0,tm) =ym(1,tm) = 0, ym(x, 0) = sinπx+sinωπx, κ>1 table 1: exact and numerical solution for example 1 x exact h2bm1 error 0.0 1.65341e-9 1.65341e-9 0 0.1 6.16242e-10 6.16242e-10 2.78e-12 0.2 2.29678e-10 2.29678e-10 4.45e-12 0.3 8.56029e-11 8.56029e-11 4.20e-12 0.4 3.19048e-11 3.19048e-11 5.00e-12 0.5 1.18911e-11 1.18911e-11 6.45e-12 0.6 4.43194e-12 4.43194e-12 7.31e-12 0.7 1.65181e-12 1.65181e-12 3.28e-12 0.8 6.15646e-13 6.15646e-13 4.11e-13 0.9 2.29456e-13 2.29456e-13 5.99e-13 table 2: exact and numerical solution for example 1 9 exact h2bm2 error 0.0 1.65341e-9 1.65341e-9 0 0.1 6.16242e-10 6.16242e-10 3.24e-12 0.2 2.29678e-10 2.29678e-10 7.21e-12 0.3 8.56029e-11 8.56029e-11 2.22e-12 0.4 3.19048e-11 3.19048e-11 8.45e-12 0.5 1.18911e-11 1.18911e-11 1.15e-12 0.6 4.43194e-12 4.43194e-12 1.18e-12 0.7 1.65181e-12 1.65181e-12 5.08e-12 0.8 6.15646e-13 6.15646e-13 2.48e-13 0.9 2.29456e-13 2.29456e-13 4.79e-13 where tm=m∆t, m= 0, 1, . . . ,; m ym(x)≈y(x, tm),y(x) = [y0(x), y1(x), . . . ,ym−1(x)] t , hence (39) becomes the system d 2 ym (x) dx2 = f (x , ym) which is in the form of (1), where f (x, tm) =ay+g and a is an m−1 square matrix, g is a vector of constants. bhsda is l -stable block hybrid second derivative algorithm in ngwane and jator [10]. tables 1 and 2 shows the comparison of exact solution and the mothers h2bm1 and h2bm2 respectively. for example 1. table 3 shows the comparison of maximum errors obtained for example 1 using the derived methods and the method in ngwane and jator[10]. this shows the superiorly of the derived methods over existing methods. figure 1 show the surface plots for the exact solution and numerical solutions for example 1. example 2. consider the pde ([14]): ∂y2 ∂x2 + ∂y2 ∂t2 = −32π 2sin(4πx), x∈[0, 1] y(±1,t) =y(x,±1) = 0, t>0 (40) table 3: comparison of maximum errors obtained in different methods for example 1 at t= 1. κ h2bm1 h2bm2 bhsda 1 1.076 × 10−11 1.022 × 10−12 2.64 × 10−6 2 1.024 × 10−12 1.085 × 10−12 1.32 × 10−6 3 1.045 × 10−12 1.045 × 10−12 1.32 × 10−6 5 1.035 × 10−12 1.055 × 10−12 1.32 × 10−6 10 1.019 × 10−12 1.041 × 10−12 1.32 × 10−6 34 olaiya et al. / j. nig. soc. phys. sci. 3 (2022) 26–37 35 figure 1: surface plots for the exact and numerical solution for example 2 table 4: exact and numerical solution using h2bm1 for example 2. κ h2bm1 h2bm2 bhsda x exact h2bm1 error 0.0 6.634320126e-16 6.634320126e-16 0. 0.2 0.5590169122545 0.5590169122520 2.51e-12 0.4 -0.9045084378512 -0.9045084378511 1.00e-12 0.6 0.9045084354472 0.9045084354462 1.00e-12 0.8 -0.5590169311454 -0.5590169311421 3.37e-12 1.0 -4.658833273e-16 -4.658833273e-16 0. for n= 10, x∈[−1, 1] table 5: exact and numerical solution using h2bm2 for example 2. x exact h2bm2 error 0.0 6.634320126e-16 6.634320126e-16 0. 0.2 0.5590169122545 0.5590169122589 4.49e-12 0.4 -0.9045084378512 -0.9045084378577 6.52e-12 0.6 0.9045084354472 0.9045084354428 5.53e-12 0.8 -0.5590169311454 -0.5590169311433 2.12e-12 1.0 -4.658833273e-16 -4.658833273e-16 0. for n= 10, x∈[−1, 1] the analytic solution is y(x, t) = sin4πxsin4πt. following ngwane and jator [10], (38) becomes dy2m dx2 = − y (x , tm+1)−2y (x , tm) +y(x , tm−1)] (2∆t) −32π2sin(4πx) (41) ym(±1,tm) =ym(x,±1) = 0, where tm=m∆t, m= 0, 1, . . . ,m; ym(x)≈y(x, tm),y(x) = [y0(x), y1(x), . . . ,ym−1(x)] t , hence (41) becomes the system d 2 ym (x) dx2 = f (x , ym) which is in the form of (1), where f (x, tm) =ay+g and a is an m−1 square matrix, g is a vector of constants. bvm and bum are boundary value methods and the block unification methods in biala [14]. tables 4-5 shows the comparison of the exact solution and the methods h2bm1 and h2bm2 respectively for example 2. table 6-7 shows the maximum error and cpu time obtained for different methods. table 9 shows the maximum error and cpu time obtained in biala [14]. table 6: comparison of maximum errors obtained in different methods for example 2 at t= 1 n h2bm1 l∞ error cpu time h2bm2 l∞ error cpu time 16 2.257e-7 0.112 5.547e-7 0.121 32 2.787e-7 0.898 1.712e-7 0.871 64 8.234e-7 2.785 2.337e-7 2.662 128 3.114e-7 11.211 4.785e-7 12.009 256 2.779e-7 31.812 1.112e-7 30.101 table 7: comparison of maximum errors obtained in different methods for example 2 at t= 1 n bvm l∞ error cpu time bum l∞ error cpu time 16 9.662e-0 0.483 1.251e-1 0.531 32 2.582e-2 1.235 2.578e-2 1.031 64 6.433e-3 5.358 6.459e-3 5.516 128 1.607e-3 43.641 1.607e-3 46.923 256 2.000e-0 512.843 4.016e-4 532.657 this show that the derived methods performs accurately, superiorly and affluently in terms of the computer time, and errors obtained in examples 2. figure 2 shows the surface plots for the exact and numerical solution for examples 2. example 3. we consider the pde (jator [15]). ∂y2 ∂t2 + ∂y2 ∂x2 = sin(y), x∈[−3, 3] y(x, 0) = 4arctan(e x√ 1−c2 ), yt(x, 0) = − 4ce x√ 1−c2 √ 1−c2 (1+e 2 x√ 1−c2 ) , 0 < t<1 (42) the analytic solution is y(x, t) = 4arctan(sech(x)t), c is velocity of the wave. the problem is solved for c= 0.5, ∆t= 0.125 following ngwane and jator [10], (38) becomes dy2m dx2 = − y (x , tm+1)−2y (x , tm) +y(x , tm−1)] (2∆t) +sin(ym)(43) ym (x, 0) = 4arctan ( e x√ 1−c2 ) , ymt (x, 0) = − 4ce x√ 1−c2 √ 1−c2 1+e2 x√1−c2 , 0 < t<1 35 olaiya et al. / j. nig. soc. phys. sci. 3 (2022) 26–37 36 figure 2: surface plots for the exact and numerical solution for example 3 figure 3: surface plots for the exact and numerical solution for example 3 table 8: exact and numerical solution using different methods for example 3 x exact h2bm1 h2bm2 sbvm 0.125 0.12195641127 0.12195641122 0.12195641178 0.121956 0.25 0.11346954831 0.11346954852 0.11346954877 0.113469 0.375 0.10557254177 0.10557254144 0.10557254129 0.105573 0.5 0.09822454418 0.09822454411 0.09822454412 0.0982256 0.625 0.09138842135 0.09138842122 0.09138842129 0.0913892 0.75 0.08502738529 0.08502738541 0.08502738557 0.0850284 0.875 0.07910885264 0.07910885215 0.07910885213 0.0791101 1.00 0.07360212510 0.07360212548 0.07360212544 0.0736035 table 9: approximate and numerical solution for example 3 x h2bm1 error h2bm2 error sbvm error 0.125 7.2e-10 5.1e-10 1.30e-7 0.25 2.1e-11 4.6e-10 2.94e-7 0.375 3.3e-11 4.8e-10 4.51e-7 0.5 7.2e-11 8.0e-10 6.49e-7 0.625 1.4e-10 5.5e-10 8.41e-7 0.75 1.2e-10 3.7e-10 1.03e-6 0.875 4.9e-10 3.3e-10 1.25e-6 1.00 3.8e-10 7.7e-10 1.44e-6 where tm=m∆t, m= 0, 1, . . . ,m; ym(x)≈y(x, tm),y(x) = [ y0(x), y1(x), . . . ,ym−1(x)] t , hence (43) becomes the system d 2 ym (x) dx2 = f (x , ym) which is in the form of (1). where f (x, tm) =ay+g and a is an m−1 square matrix, g is a vector of constants. sbvm is symmetric boundary value method in jator [15]. table 8 shows the exact and numerical solution using the different methods for example 3 while table 9 shows error obtained for example 3. figure 3 shows the surface plots for the exact and numerical solutions of h2bm1 and h2bm2 for example 3 5. conclusion the development of some numerical schemes has been proposed in this work. this was developed via the interpolation and collocation techniques using power series function as trial solutions. the methods were effectively illustrated some partial differential equations (pde) and the results obtained were accurate. the analysis of the new methods showed that all satisfy the properties of numerical methods for the solution of differential 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[17] m. i. modebei, r. b. adeniyi, s. n. jator & h. c. ramos “a block hybrid integrator for numerically solving fourth-order initial value problems”, applied mathematics and computation 346 (2019) 680. 37 j. nig. soc. phys. sci. 3 (2021) 148–153 journal of the nigerian society of physical sciences synthetic characterization and structural properties of nanocellulose from moringa oleifera seeds a. f. afolabi∗, s. s. oluyamo, i. a. fuwape condensed matter and statistical physics research unit, department of physics, the federal university of technology, p.m.b. 704, akure, nigeria. abstract in this research, nanocellulose is isolated from moringa oleifera seed using acid hydrolysis and the structural properties were determined. x-ray diffraction (xrd) and fourier transform infrared (ftir) spectroscopy were used for the characterization of the isolated nanocellulose. the most noticeable peak is observed at 22.53◦ and the value of the crystallinity index (cir ) from the xrd pattern is 63.1%. the calculated values of hydrogen bond intensity (hbi), lateral order index (loi) and total crystalline index (tci) are 0.93, 1.17and 0.94 respectively exhibited high degree of crystallinity and well arranged cellulose crystal structure. the isolated nanocellulose has an average length and diameter of 14.3 nm and 36.33 nm respectively. furthermore, the ftir peaks revealed the presence of c-h bending, c-o stretching and o-h stretching functional groups. doi:10.46481/jnsps.2021.202 keywords: crystallinity index, crystal structure, hydroxyl group, moringa oleifera, nanocellulose article history : received: 20 april 2021 received in revised form: 14 june 2021 accepted for publication: 27 june 2021 published: 29 august 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: a. h. labulo 1. introduction moringa oleifera is a well known plant material with numerous potential uses which belong to the family of moringaceae [1,2]. moringa oleifera is a plant material composed of organic nutrients, lignin, hemicellulose and cellulose. one of the prominent structural compositions of different green plants cell wall is cellulose. moreover, nanocellulose can be prepared from cellulose [3]. the fact has been established that cellulose with appearance of nanostructures (nanocellulose) is among the paramount organic materials of recent times [4]. nanocellulose exhibits unique characteristics due to the nanoscale size. the properties of the nanocellulose can be tailored to increase their performance for specific applications [5,6]. chemical method ∗corresponding author tel. no: email address: agafolabi@gmail.com (a. f. afolabi ) of treating nanocellulose is based on the source, the resulting material can change in crystal arrangement (crystal structure), degree of crystallinity, morphology and surface chemistry [7]. nanocellulose has been a research key in nanomaterial because it is a sustainable biomaterial which has low toxicity. nanocellulose is isolated using various distinct approaches such as oxidative, acid hydrolysis, oxidative, enzymatic and mechanical treatments of cellulose. the most common approach for isolating nanocellulose from wood and other plant materials is acid hydrolysis [8,9]. many researchers have investigated the isolation of nanocellulose from agricultural residues such as banana [10], sisal [11], tomato peels [12], calotropis procera fibers, onion waste, citrus waste, coconut [13], sesame husk [14], cotton, rice husk [15], oil palm [16], groundnut shells [17], macrophyte typha domingensis, potato peel, jute, spruce bark, agave angustifolia fibers, 148 afolabi et al. / j. nig. soc. phys. sci. 3 (2021) 148–153 149 figure 1. schematic diagram of experimental procedure of nanocellulose mango seed, sugarcane bagasse, corncob, bamboo, straws, soy hulls, olive stones, miscanthus giganteus, kapok and flax fibers. the potential and industrial application of the isolated nanocellulose is based on the structural and other properties of the nanocellulose. the aim of this research is to synthesis, characterize and determine the structural properties of nanocellulose from moringa oleifera seed. 2. materials and methods 2.1. materials the locally sourced organic material (moringa oleifera seeds) was removed from the shells, dried and grinded with a mixer grinder (bajaj gx 10 dlx, mumbai, india). it was sieved to obtain fine particles using a pascal engineering wiley mechanical sieve shaker, england. analytical chemical reagents used are naoh, naclo2, acetic acid and h2so4. the chemical reagents were obtained from the pascal scientific ltd. the schematic representation of experimental procedure is shown in figure 1. 2.2. methods a liquor ratio of 15:1(v/w) cooking condition was employed, the moringa oleifera seed particles was pulped with 20% of naoh at a temperature of 90◦ for 1 hour 30 minutes. after digestion process, the cooked pulp was filtered, screened and cleaned by rinsing properly with water without alkali. the pulped was left in the oven at 105◦c until the water was completely dried. mixture of 200 ml hot water, 6 g of naclo2, 1.5 ml of acetic acid and 10g of bone dried sample of pulp in a titration flask were placed in the water bath at 70◦ and heated for 30 minutes. another 6 g of naclo2 and 1.5 ml of acetic acid were added to the mixture and switched off the water bath after submitted to heat for the next 30 minutes. the sample was left in the water bath for 24 hours. after digestion, it was washed, filtered and cleaned by rinsing properly with water until the chlorine and the acid were washed away. the sample acquired was left in the oven at 105◦until the water was completely dried to obtain the cellulose. 2.3. preparation of nanocellulose the nanocellulose of the sample was prepared by acid hydrolysis in accordance with the method developed by bondeson [18] with little change. the cellulose sample was treated with 60 % sulfuric acid (h2so4). the hydrolysis was conducted by using a hot plate to heat the suspension in a round bottom flask with reflux condenser and intermittently stirred with a magnetic stirrer at an average temperature of 45◦ for 60 minutes. the hydrolyzed cellulose sample was distinctly washed and drained to remove excess h2so4 until the sample was neutral and dried. the reflux condenser was used to cool the acid so that the acid will not escape. 3. characterization the crystallinity index of the isolated nanocellulose from moringa oleifera seeds was acquired by making use of a philips pw diffractometer with cu-kα monochromator at the voltage of 15kv, scanned at wavelength λ=1.54å with 2θ angle range from 5◦ to 90◦. the surface morphology was determined by scanning electron microscope using15 kv accelerated voltage of jeol/eo jsm-6390 and has a resolution up to 100µm. fourier transform infrared (ftir) spectrophotometer was used to determine variation in functional groups induced by various treatments within a wavelength range of 700–4000cm−1. 3.1. theoretical background the interplanar spacing (d-spacing) was obtained as [19,20] d = nλ 2 sin θ (1) where n is the order of reflection, d is the interplanar spacing of the crystal, θ is the angle of incidence and λ is the wavelength of the incident x-ray. the crystallinity index was determined using equation (2) [21,22] cir = i200 − iam i200 × 100 (2) where, low intensity peak of the amorphous region is iam and highest peak intensity of the crystalline fractions is i200. the crystallite size (l) was calculated using scherrer’s equation [23] l = k ×λ b × sin θ (3) where, constant value given as 0.91 is k, bragg’s angle (◦) is θ, wavelength of the incident x-rays is λ and intensity of the full width at half maximum (fwhm) proportional to a high intensity peak of the diffraction plane is b. the surface chains (x) is the proportion of crystallite interior chains [24] was calculated as x = (l − 2h)2 l2 , (4) where l is the crystallite size and h = 0.57 nm is the layer thickness of the surface chain. 149 afolabi et al. / j. nig. soc. phys. sci. 3 (2021) 148–153 150 figure 2. x-ray diffractogram of isolated nanocellulose from moringa oleifera seeds. 4. results and discussion 4.1. x-ray diffraction (xrd) of isolated cellulose and nanocellulose the xrd pattern of the isolated cellulose in figure 2 revealed crystalline characteristics peaks at 2θ = 14.39◦, 15.33◦, 22.47◦ and 34.50◦ while nanocellulose has distinct peaks at 2θ = 14.95◦, 15.01◦, 22.53◦ and 34.67◦ in agreement with isolation and characterization of cellulose nanocrystals from agave angustifolia fibre [25]. the crystalline peaks indicate that the crystal structure is attributed to planes (110), (110), (200) and (004) respectively. furthermore, there is a noticeable crystal peak observed at 50.12◦ similar to the peaks in the xrd results of cellulose and α-cellulose from date palm biomass waste [26]. the peaks at 21.58◦, 24.88◦ and 32.24◦ in the pattern of the cellulose were not noticed in the pattern of nanocellulose in figure 2. this is due to the fact that the bond of the cellulose was broken after the sulfuric acid hydrolysis. the most prominent peaks of the isolated cellulose and nanocellulose are 22.47◦ and 22.53◦. the crystallinity index of isolated cellulose from moringa oleifera seeds (62.6%) is lower than the crystallinity index of the nanocellulose (65.4%), this contributed to high degree of crystallinity of the nanocellulose [27,28,29,30]. additionally, the high crystallinity of nanocellulose depends on the three hydroxyl groups in fundamental chemical structure of cellulose which have potential to instigate large intra and intermolecular hydrogen bonding included in the cellulose chains, granting the crystalline packing of cellulose chains into greatly compact system (crystal structure) [31]. the diffraction peaks of the nanocellulose were narrowed, longer and became sharper due to the efficient elimination of the amorphous parts. this shows that the nanocellulose is highly crystalline [32]. table 1 showed the values of dspacing (d), full width at half maximum (fwhm), crystallinity index (cr i), crystallite size (l), and surface chains (x) also known as the crystalline proportion of the crystallites of the isolated nanocellulose. figure 3. scanning electron micrograph of cellulose from moringa oleifera seeds figure 4. scanning electron micrograph of nanaocellulose from moringa oleifera seeds. 4.2. scanning electron microscopy (sem) analysis of isolated cellulose and nanocellulose figure 3 shows the surface morphological features of the isolated cellulose. the surface of the isolated cellulose from moringa oleifera seeds was rough due to amorphous nature of the materials [28]. the isolated cellulose from moringa oleifera seeds has an average length of 46.20 µm and diameter of 88.90 µm. the particles were dissociated from one another, indicating the elimination of hemicelluloses and lignin. this is similar to the report of nazir et al. [22]. the surface morphology of the isolated nanocellulose from moringa oleifera seeds in figure 4 is predominantly rod-like with conical feature. in addition, the nanocellulose is clean, smooth and disjointed from one another owing to the removal of impurities and non-cellulosic components from the materials. furthermore, non-agglomerated structure of the nanocellulose is expressed as highly porous with noticeable diameters, thus able to provide large surface areas [26]. the isolated nanocellulose has an average length and diameter of 14.30 nm and 36.33 nm respectively. 150 afolabi et al. / j. nig. soc. phys. sci. 3 (2021) 148–153 151 table 1. structural analysis of the isolated nanocellulose from the xrd patterns sample 2θ(◦) d(å) l(nm) fwhm x cr i(%) isolated cellulose 22.43 3.95 1.95 0.07 0.17 62.60 isolated nanocellulose 22.53 3.90 2.13 0.06 0.22 65.40 4.3. fourier transform infrared (ftir) of isolated cellulose and nanocellulose the fourier transform infrared (ftir) spectra of the isolated cellulose and nanocellulose are shown in figure 5. the prospect of the ftir was to ascertain the functional groups of the cellulose and nanocellulose isolated from the moringa oleifera seeds. absorption bands in all spectra of the isolated cellulose were observed at 3335.43 cm−1, 2913.17 cm−1, 2345.16 cm−1, 1577.27 cm−1, 1426.66 cm−1, 1156.49 cm−1, 1015.50 cm−1 and 661.67 cm−1. the spectra of the isolated cellulose showed wide band centered at 3335.43 cm−1 appointed to oh stretching vibration of hydroxyl groups and absorbed water having strong intermolecular hydrogen bonding with alcohol compound class [33,34]. the wide absorption band of the isolated nanocellulose observed at 3340 cm−1 in figure 5 is strong corresponded to the vibration of the o–h group having a compound class of alcohol. this is in agreement with preparation and characterization of novel microstructure cellulosic sawdust material [35]. this functional group commonly present in the cellulose. the characteristics spectra of c–h vibration occur at 2910 cm−1. the absence of peak between 1740 cm−1 and 1726 cm−1 signifies that there is no ester linkage of lignin or ester group of the hemicellulose due to the sulfuric acid hydrolysis [32]. furthermore, disappearance of the hemicellulose and lignin in the ftir spectrum verifies that the cellulose is crystalline. the peaks in the region between 1025 cm−1 – 1321 cm−1 are associated to the c-o stretching [26]. additionally, the band at 664 cm−1 is a characteristic associated with the c-h bending [36]. total crystalline index (tci), hydrogen bond intensity (hbi), lateral order index (loi) and lateral order index (loi) of the nanocellulose from moringa oliefera seeds were obtained from the spectra of the ftir spectroscopy. the calculated values of tci and loi are 0.93 and 1.17 respectively. the values signify more ordered cellulose structure and structure high degree of crystallinity. this result is similar to previous research on green solvent for water hyacinth biomass deconstruction [37]. the other indicator of high degree of intermolecular regularity and ordered nature of cellulose is hbi value. the hbi value of the isolated nanocellulose is 0.94 which indicates high degree of intermolecular regularity. this is in agreement with the result on native cellulose: structure, characterization and thermal properties [23]. this result show that there were additional chains of cellulose in a highly coordinated form which leads to higher hydrogen bond intensity among neighbouring chains of cellulose and produce a more packing structure of cellulose and higher crystallinity. figure 5. fourier transform infrared (ftir) spectra of isolated cellulose from moringa oleifera seeds figure 6. fourier transform infrared (ftir) spectra of isolated nanocellulose from moringa oleifera seeds 5. conclusion the structural properties of the isolated nanocellulose were successfully examined in this research. the isolated cellulose and nanocellulose from moringa oleifera seeds revealed the 151 afolabi et al. / j. nig. soc. phys. sci. 3 (2021) 148–153 152 most prominent peaks at 2θ = 22.47◦ and 22.53◦ respectively. the crystallinity index values were 62.60% and 65.40%. in addition, the nanocellulose is predominantly rod-like with conical feature. the isolated cellulose has an average length of 46.20 µm and diameter of 88.90 µm while the average length and diameter of the obtained nanocellulose are 14.3 nm and 36.33 nm respectively. the ftir revealed the presence of c-o stretching, o-h stretching and c-h bending functional groups. the tci, loi and hbi values of the nanocellulose from moringa oleifera seeds were 0.93, 1.17 and 0.94. these results indicate more ordered cellulose structure and high degree of crystallinity. acknowledgements the authors gratefully appreciate dr. ige, o. o. and dr. alo, f.i. of the department of material science and engineering, obafemi awolowo university ile-ife, osun state, nigeria for 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[37] j. k. singh, r. k. sharma, p. sharma, a. kumar, m. l. khan “imidazolium based ionic liquids: a promising green solvent for water hyacinth biomass deconstruction”. frontiers in chemistry 6 (2018) 548, doi: 10.3389/fchem.2018.00548. 153 j. nig. soc. phys. sci. 5 (2023) 1160 journal of the nigerian society of physical sciences physico-chemical and trace metal analysis in groundwater of nagapattinam region in nagapattinam district of tamilnadu state c. gopia,∗, edward anand ea, a. charlesa, c. manivannanb, s. ponsadai lakshmia, a. josea, m. muthiyanc adepartment of science and humanities, e.g.s. pillay engineering college, nagapattinam, tamil nadu, india. bdepartment of chemistry, puducherry technological university, puducherry, india. c research scholar, department of chemistry, e.g.s. pillay arts & science college, nagapattinam, india. abstract the aim of the present work is to find the quality of water in and around the nagapattinam region and geochemical study of water and its chemical composition with qualitative and quantitatively assessed from the period of post monsoon ( january) in the year 2020. therefore, ten underground water sample were taken from different areas in nagapattinam region and analysed for the following qualities such as color, odour, temperature, electrical conductivity, total dissolved solid, hydrogen ion concentration, calcium, magnesium, chloride, potassium, sodium, nitrate, and sulphate and trace metals like manganese, lead, chromium, copper, iron, arsenic, cadmium and zinc. the physico chemical parameters indicate the quality of ground water varies from bore well to bore well. higher values of any parameter in a borehole indicate that the water is not fit for drinking. therefore, the public is advised that the groundwater source in the study area should be monitored before it is used for domestic and drinking water purposes and that the government should adopt some treatment technology in the current study regions to minimize the hardness and salinity for provide safe water to the public. doi:10.46481/jnsps.2023.1160 keywords: nagapattinam, ground water, monsoon, physico-chemical parameters article history : received: 31 october 2022 received in revised form: 05 january 2023 accepted for publication: 10 january 2023 published: 12 march 2023 c© 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: dr. k. sakthipandi 1. introduction groundwater plays an important role in the ecological functions of various ecosystems. due to human activities, the groundwater system provides proper circulation for water recirculation; water is being abused and contaminated in ordinary ∗corresponding author tel. no: +91 9994648947 email address: gopi@egspec.org (c. gopi ) conceivable. this pollution causes water quality to decline. half of the groundwater used in metropolitan areas in developing countries comes from wells and boreholes, and many in india rely on these resources. contamination of heavy metals with its surrounding could be a major world concern, as a result of toxicity and threat to human life and ecosystem [1]. heavy metals are superimposed with water system from artificial and natural sources [2]. water quality is more important than quantity in any water. groundwater’s physical and chemi1 c. gopi et al. / j. nig. soc. phys. sci. 5 (2023) 1160 2 cal qualities support its use as a source of water for municipal, agricultural, and domestic purposes [3]. nagapattinam district is a part of the south indian state of tamil nadu, forming parts of the cauvery river basins and vennar sub-basins. geographically, the district lies between latitudes 10◦46′16′′ and longitudes 79◦50′50′′78 as shown in figure 1. from june to september starts south west monsoon and north east monsoon begins from october till january. low rain fall observed in southwest period [4] from march to may summer start and end. it is one of the state’s fastest growing cities, rapid urbanization and industry. groundwater supplies have been put under a lot of strain as a result of urbanization. in the major part of the state, depth of the water level in pre-monsoon may 2019 is 2-5 m bgl and post monsoon january 2020 is greater than 2-5 m bgl shown in figure 2. 2. materials and methods in the post-monsoon period of 2020, water samples were taken from ten sample points in nagapattinam and the surrounding area (january) in the depth of 40 ft. apha [5] was used to collect and analyze the samples. for drinking water, all of the settlements in the research region rely on groundwater. during the collecting and handling of the samples, all precautions were taken. polyethylene containers were used to collect groundwater samples. the ph, electrical conductivity was determined on the spot using digital equipments after sampling. water samples were analyzed for chemical parameters such total dissolved solid, electrical conductivity, hydrogen ion concentration, total hardness, calcium, magnesium, sodium, potassium, chloride, bicarbonate, nitrate, phosphate, sulfate and trace metals like iron, manganese, chromium, copper, lead, zinc and cadmium. figure 1: location map of study area 3. results and discussion table 1 lists the locations of ground water sampling stations in the research region. the analytical results of physical and chemical parameters of ground water were compared to the world health organizations 1985 standard guideline values for drinking and public health objectives table 2. figure 2: peizometric surface data for pre monsoon and post monsoon seasons table 1: details of ground water sampling stations in the study area s/n sampling stations sample id source of water 1 kadambanoor s1 bore well 2 sengamangalam s2 bore well 3 palaiyur s3 bore well 4 boothangudi s4 bore well 5 nagoore s5 bore well 6 nagapattinam s6 bore well 7 sikkal s7 bore well 8 pappakovil s8 bore well 9 north poigainallur s9 bore well 10 south poigainallur s10 bore well table 2: who and bis standard of drinking water parameters who (48) (2011) bis 2012 (49) study area samples permissible limit permissible limit min-max (mg/l) ph 6.5-8.5 6.5-8.5 6.5-8.7 ec 500-1500 590-1100 tds 500-1500 500-2000 538-956 th 100-500 200-600 246-722 calcium 75-200 75-200 110-532 magnesium 30-150 30 66-103 magnesium 30-150 30 66-103 sodium 50-200 86-180 potassium 10-dec 0.98-3.08 chloride 250-600 250-1000 154-386 bi carbonate 200-500 241-624 sulphate 250-400 200-400 53-89 phosphate 1.1-3.8 nitrate 45 14-19 3.1. ph and electrical conductivity ph is a measurement of a solution’s acidity or felicity. it is defined as the co (h+) hydrogen ion activity coefficient, which 2 c. gopi et al. / j. nig. soc. phys. sci. 5 (2023) 1160 3 table 3: physico-chemical data of drinking water quality parameters of groundwater samples sample ph ec ca2+ mg2+ na+ k+ cl− hco−3 so 2− 4 no − 3 po 3− 4 tds th sar s-1 6.8 600 350 96 180 1.1 289 256 82 14 1.1 742 615 53 s-2 7.8 700 270 82 155 1.37 352 289 78 15 3.2 853 262 56.9 s-3 7.3 850 327 103 114 1.27 262 456 73 15 2.8 924 551 61.6 s-4 8.7 590 220 92 126 0.98 275 241 89 16 3.1 950 246 59.4 s-5 6.5 1100 233 71 86 1.63 386 250 84 19 3.8 920 316 48.4 s-6 7.3 863 310 72 165 2.08 352 624 85 17 3.4 781 444 45.9 s-7 6.6 846 352 102 131 1.91 240 375 53 18 3.2 732 492 40.7 s-8 8.6 978 110 66 132 1.91 258 425 67 16 2.2 538 258 33.6 s-9 6.7 1005 402 75 172 3.08 154 562 72 19 2.1 676 535 35.6 s-10 7.5 630 532 98 124 1.64 356 342 83 19 2.6 956 722 50.3 mini. 6.5 590 110 66 86 0.98 154 241 53 14 1.1 538 246 33.6 maxi. 8.7 1100 532 103 180 3.08 386 624 89 19 3.8 956 722 61.6 table 4: correlation coefficient values between the water quality parameters of groundwater samples in the study area during monsoon 2020 correlation ca++ mg++ na+ k+ cl− hco−3 so 2− 4 po 3− 4 no − 3 ph ec tds ca++ 1 mg++ 0.106 1 na+ -0.374 -0.07 1 k+ 0.0131 -0.29 0.497 1 cl− -0.032 -0.11 -0.35 -0.68 1 hco−3 -0.087 -0.31 0.367 0.573 -0.37 1 so2−4 -0.445 -0.17 0.002 -0.24 0.556 -0.24 1 po3−4 -0.251 -0.17 -0.6 -0.53 0.481 0.001 0.007 1 no−3 0.228 -0.23 -0.33 0.41 -0.01 0.231 -0.11 0.386 1 ph -0.03 -0.26 -0.03 -0.14 0.121 -0.11 0.45 -0.06 -0.4 1 ec 0.219 -0.63 -0.31 0.293 -0.17 0.434 -0.35 0.303 0.487 -0.32 1 tds 0.204 -0.05 -0.48 -0.62 0.537 -0.42 0.57 0.467 0.078 0.044 -0.39 1 can only be calculated theoretically and cannot be measured empirically. the ph scale is a relative scale. it is relative to a set of standard solution which ph is established by international agreement. the ph level varied from 6.5 to 8.7. the maximum value found at s-4 and minimum value is mainly basic in nature [6]. the variation of electrical conductivity is 590 to 1100 µs/cm .in the minimum value found at s-4 and maximum value found at s-5 is within the desirable limit 3.2. total dissolved solids total dissolved solids (tds) indicate the salinity behavior of ground water sample. tds values varied from 538 to 956. if tds is more than 500 mg/l it is not suitable for drinking [7, 8, 9]. in the present study tds values are higher than the prescribed limit given by who. the tds concentration “found to be in above permissible limit may be due to the leaching of various pollutants into the ground water which can decrease the portability and this may results gastrointestinal irritation in human and also have laxative effect. high level of total dissolved solids may aesthetically be unsatisfactory for bathing and washing purposes” [10]. the tds variation indicates a low concentration at s-8 and high concentration at s-10. tds indicates that there is a low concentration of soluble salt in groundwater that is safe to drink. [11, 12, 13]. 3.3. total hardness the total hardness values are observed in the range of 246 to 722 mg/l post monsoon of 2020. total hardness values are within the maximum permissible limit of world health organization in all the sample station except s-1 and s-10. this may be due to presence of bicarbonates, chloride and sulphates of ca and mg present in the water. the total hardness values are observed in the range of 246 to 722 mg/l post monsoon of 2020. total hardness values are within the maximum permissible limit of world health organization in all the sample station except s-1 and s-10. this may be due to presence of bicarbonates, chloride and sulphates of ca and mg present in the water. 3.4. chloride chloride ion is one the anion present in water and waste water as inorganic compound but chlorine in drinking water does not create harmful even at higher concentration it is harmless. if the concentration exceeds the maximum permissible limit it produces cathartic effect in the samples chlorine ranges from 154 mg/l to 386 mg/l. lower the concentration at s-9 and higher concentration at s-5. as a result, it has a high concentration in groundwater, where temperatures are high and rainfall is low. the porosity and permeability of the soil also play a role in raising the concentration of chlorides [10]. 3 c. gopi et al. / j. nig. soc. phys. sci. 5 (2023) 1160 4 3.5. nitrate concentration of nitrogen in groundwater in the range of 14 to 46 mg/l. it shows the site s-3, s-9 and s-10 are found higher concentration and other sites having lower concentration. but who limit for drinking water standard is 45 mg/l. nitrate concentration higher than this limit unfit for drinking .the present amount of concentration is mainly due to agricultural activities. the usage of larger quantity of nitrogen containing fertilizer in the land which may cause leaching from the root of the plants, soil and accumulate in water. 3.6. sulphate in adults, water containing 1000 mg/l magnesium sulphate serves as a purgative. (bhagavathi perumal and thamarai [14, 15]. sulphate occurring in water due to the municipal and industrial activity nearby discharge. also human activity is one of the major anthropogenic attribute to runoff and rainfall. concentration of sulphate varies from 53 mg/l to 89 mg/l. low concentration observed at s-7 and high concentration s-4. the maximum permissible limit of sulphate in water is 400 mg/l. 3.7. phosphate due to the activities of agriculture and anthropogenic increase the phosphate content in water [16]. the phosphate concentration observed in the groundwater samples from the study area varied from below detection level of 1.1 mg/l. phosphate found moderately low at many locations. 3.8. calcium and magnesium the desirable quantity of calcium is 75mg/l. the ca ionic concentration found low as 110 mg/l in sample station s-8 (532 mg/l at s-10) was observed high concentration but the permissible limit of calcium for water 200 mg/l. except s-8 all other samples show above permissible limit. due to low dissolution of magnesium the concentration is less in ground water than calcium [17]. the magnesium concentration is ranges from 66 to 103 mg/l where higher assessment found at s-3 and lower value found at sample station s-8. the acceptable limit of magnesium in water is 150mg/l. 3.9. sodium and potassium there is no guideline proposed for potassium ion but the concentration of sodium is ranges from 86 to 180 mg/l. s-9 and s-6 are found in maximum and minimum concentration respectively also, the concentration of sodium in ground water influences more in agricultural activity. 3.10. iron iron is the essential element for the organism. it occurs naturally in the environment as its ore like hematite. it acts as the central metal atom in the hemoglobin and transport the oxygen in the blood through organs. the deficiency of iron create anemia. the prescribed limit of iron content in drinking water is 0.30 mg/l by who. in the present study area, the maximum value is 0.26 and minimum value is 0.01.the iron content of the entire sample found below detection limit (bdl). 3.11. manganese manganese is the most abundant metals recover from earth crust in the form of oxides and hydroxides. it behaves as trace element and toxic metal due to the industrial activity, soil erosion, volcanic eruption, and human activity which increase the contaminant in ground water which change the odor and taste of the water also deposit within the pipes may break or form black precipitate. the allowable limit of manganese in ground water is 0.4 mg/l but in the present study area the maximum value is 0.08 and minimum value is 0. the manganese content of the entire sample found below detection limit. all the samples found below permissible limit. 3.12. chromium chromium is one of the most abundant heavy metal in nature it occur in the combined state but it exist as trivalent as well as hexavalent in nature as trace. it acts as removal of glucose from blood. but hexavalent chromium causes allergic reaction on human. tannin and paint industry discharge most of the chromium in ground. who has prescribed 0.05mg/l as prescribed limit. present study all the samples found below detection limit (bdl). 3.13. lead lead is a toxic heavy metal which is present in the natural environment but due to the human and industrial activity the concentration of lead increases day by day. it passes to environment through the vehicular exhaust and may causes serious health problem to child hood below six years. it also causes blood pressure, kidney damage [18]. in this study all the samples are found below the detection level. 3.14. copper copper is one of the common heavy metal found in environment. it enters into groundwater through agricultural wastes, pesticides; industrial waste and it create corrosion on pipes. it is the essential element for human health but high concentration copper in drinking water give liver and kidney damage. the acceptable limit of copper in ground water is 2 mg/l as prescribed by who. in the present study area the maximum value is 0.04 and minimum value is 0.01. the copper content of the entire sample found below detection limit (bdl). 3.15. zinc zinc is an essential trace element. it enters into water on location ore are found. lack of zinc in drinking water results slow growth and diarrhea in children, wounds not heal fastly, suppress the immune system with treating the cold and infection in ear, also preventing low respiratory infections. it may be found in excess due to industrial activity, galvanic industry, and battery production industry. from this is observed to avoid larger quantities of nitrogenous and phosphate fertilizer in agricultural lands. this creates the awareness towards excess use of pesticide [19]. the adverse effect of zinc toxicity is stomach aches; vomiting, fever and diarrhea. all of the samples in this investigation were found to be under the who’s permitted level of 3 mg/l. 4 c. gopi et al. / j. nig. soc. phys. sci. 5 (2023) 1160 5 3.16. cadmium cadmium is the commonly found metal in the world as ores of carbonate, sulphide and zinc. it naturally occurs in environment from the fertilizer, polluted ground water and sewage sludge, mining and effluents from industry. anemia, bronchitis are the adverse effect shown when cadmium concentration higher than the permissible limit. who has prescribed 0.003mg/l as the permissible limit. in the current study sample are found below detection level (bdl). 4. statistical studies correlation studies correlation coefficient is the mutual relationship between the two factors. the direct correlation exists when increase in the value of one parameter is associated with other parameter. the positive correlation observed only when increase in one parameter causes the increase in the other parameter vice versa” [20]. the correlated coefficients between varieties of water quality parameters are measured using the table 4. the correlation co-efficient ‘r’ was calculated using the equation r = n (σxy) − (σx) (σy)√[ nσx2 − (σx)2 ][ nσy2 − (σy)2 ] (1) the value of the correlation coefficient (r) ranges between +1 and -1. if the value ranges from +0.8 to 1.0 and -0.8 to -1.0 has characteristic the parameter is strongly, the value +0.5 to 0.8 and -0.5 to -0.8 has the characteristic the parameter is moderately and the value ranges from +0.00 to 0.5 and -0.00 to -0.5.(13) as the characteristic the parameter is weakly. the strong positive correlation of tds (0.537), (0.57) with chloride and sulphite. weak correlation of tds (0.24), (0.467), (0.0078) and (0.044) with calcium, phosphate, nitrate and ph. the correlation coefficient of ec with calcium, potassium, bicarbonate, phosphate, nitrate. ph is weakly correlated with sulphate (0.45). the correlation coefficient of nitrate is positively correlated with calcium, potassium (0.41) and phosphate. phosphate is weakly correlated with chloride (0.481).bicarbonate and sulphates are positively correlated with sodium. potassium (0.131) and magnesium (0.106) are strongly correlated with calcium. from the result most of the ion positively correlated with no3-. this may be due to increase in agriculture activity, animal, human and poor drainage waste. 5. conclusion groundwater in and around nagapattinam, nagapattinam district, is firm, fresh, and alkaline in character, according to physicochemical investigations. the parameters like, magnesium, sodium, phosphate, sulphate, potassium, electrical conductivity, nitrate, total dissolved solids (tds), and chloride results within the allowable limit. water chemistry signifies that higher ec and tds shown in nearby costal region prescribe saline water traces. almost most of the parameters showed higher values like calcium, ph, total hardness. higher values of total hardness and ph indicate saline water intrusion in the particular area. s1, s3, s9 and s10 location requires some treatment for minimization of those parameters. it may be due to increase in prominent people habits and the pollutants may leach inside the ground water. the majority of parameters were reported less than the allowable limit. the low concentration of ions in the sample does not give any adverse effect for utilize the water for house hold and drinking purposes. except s1, s3, s9 and s10 all other sample in the present region suitable for drinking purposes. trace metal contamination in the present study area showed that s1 to s10 below the permissible limit. statistical application carried out by using the correlation analysis indicates that ec, dissolved solids, calcium, magnesium, sodium, potassium, chlorine, nitrate and phosphate are the dominant ions in the study area due to the leaching of fertilizer impact [21]. the physicochemical parameters found in the entire study indicate that the quality of ground water differs from bore well to bore well. any parameter with a higher value in a borehole indicates that the water is unfit to drink [22].therefore, the public is advised that the groundwater source in the study area should be monitored before it is used for domestic and drinking water purposes and that the government should adopt some treatment technology in the current study region to minimize the hardness and salinity for provide safe water to the public references [1] g. selvarajan, s .punitha, “estimation of physico-chemical parameters of ground water in kilvelur taluk, nagapattinam district, tamilnadu, india”, int. res j environmental sci. 7 (2018) 37. 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[20] s. siddha, paulami sahu, “assessment of groundwater potential of gandhinagar region, gujarat”, j. geological society 1 (2018) 91. https://doi.org/10.1007/s12594-018-0824-y [21] k. o. sodeinde, “waste glass : an excellent adsorbent for crystal violet dye ,pb2+,and cd2+ heavy metals ion decontamination from waste water”, j.nigerian society of physical sciences 3 (2021) 261,https://doi.org/10.46481/jnsps.2021.261 [22] s. p. lakshmi, “pollution status of ground water resources through hydrochemical characteristics a case study from southern india”, j. nigerian society of physical sciences 4 (2022) 751, https://doi.org/10.46481/jnsps.2022.751 6 j. nig. soc. phys. sci. 5 (2023) 1389 journal of the nigerian society of physical sciences understanding the transmission dynamics and control of hiv infection: a mathematical model approach saheed ajaoa,∗, isaac olopadeb, titilayo akinwumia, sunday adewalec, adelani adesanyaa adepartment of mathematics and computer science, elizade university, ilara−mokin, ondo state. nigeria bdepartment of mathematics and statistics, federal university wukari, wukari, taraba state. nigeria. cdepartment of pure and applied mathematics, ladoke akintola university of technology, ogbomoso, oyo state. nigeria. abstract new challenges like the outbreak of new diseases, government policies, war and insurgency etc. present distortion, delay and denial of persons’ access to art, thereby fuelling the spread and increasing the burden of hiv/aids. a mathematical model is presented to study the transmission dynamics and control of hiv infection. the qualitative and quantitative analyses of the model are carried out. it is shown that the disease-free equilibrium of the model is globally asymptotically stable whenever the basic reproduction number is less than unity. it is also shown that a unique endemic equilibrium exists whenever the basic reproduction number exceeds unity and that the model exhibits a forward bifurcation. furthermore, the lyapunov function is used to show that the endemic equilibrium is globally asymptotically stable for a special case of the model whenever the associated basic reproduction number is greater than unity. the model is calibrated to the data on hiv/aids prevalence in nigeria from 1990 to 2019 and it represents reality. the numerical simulations on the global stability of disease-free equilibrium and endemic equilibrium justify the analytic results. the fraction of the detected individuals who are receiving treatment and stay in the treatment class plays a significant role as it influences the population of the latently-infected individuals and aids class as the treatment prevents the individuals from progressing into the aids class. doi:10.46481/jnsps.2023.1389 keywords: hiv/aids, basic reproduction number, global stability, bifurcation, lyapunov function. article history : received: 09 february 2023 received in revised form: 22 march 2023 accepted for publication: 26 march 2023 published: 19 april 2023 c© 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: p. thakur 1. introduction efficient and effective testing is a gateway to hiv treatment and it is an important element of efforts to stop the aids epidemic [1]. a positive diagnosis allows an hiv-infected person to receive antiretroviral therapy (art) [2]. the art sup∗corresponding author tel. no: +2348060548485 email address: saheed.ajao@elizadeuniversity.edu.ng (saheed ajao ) presses the replication of the virus and this prevents transmission to one’s sexual partner. thus, early access to antiretroviral therapy (art) and support for continued treatment is important not only to improve the health status of hiv-infected individuals but also to prevent the transmission of hiv [3]. in 2021, 28.7 million people were receiving art globally, and the global art coverage was 75%. at the end of 2021, only 52% of children aged 0 to 14 years had received art and world health organization recommends that more efforts should be put in place to scale up treatment, most especially for 1 ajao et al. / j. nig. soc. phys. sci. 5 (2023) 1389 2 children and adolescents [3]. in the who african region, where two-thirds of the disease burden exist, access to treatment is an issue for people living with hiv due to many factors. in south africa, discrimination and stigmatization still remain major obstacles to hiv response efforts which also affect children. according to a un report, african children are neglected when it comes to hiv treatment [4]. unaids reported that despite the global scientific advances in providing better treatment for children and adults, children with hiv in indonesia have difficulties accessing antiretroviral therapy. the deeply rooted societal and gender inequalities create barriers to women, children, and adolescents accessing quality prevention and care services and thereby making the situation worse [5]. the report of [6] explained how government policy creates a barrier to hiv treatment. several factors have been identified as barriers to hiv treatment (see [7-10]). the modelling of the transmission of infectious diseases is now influencing the theory and practice of disease management and control. mathematical modelling now plays a significant role in policy decision-making regarding the epidemiology of diseases in many countries [12]. several models have been developed to study the dynamics of hiv transmission (see [1316]). apentang et al [17] studied the impact of the implementation of hiv prevention policies therapy and control strategies among hiv/aids new cases in malaysia. the study revealed that the use of condoms and uncontaminated needle-syringes are important intervention control strategies. yang et al. [18] studied the global dynamics of an hiv model, which incorporates senior male clients and their results showed that diagnosis, treatment and education have a positive impact on controlling hiv transmission, while senior male clients increase the number of new cases of hiv and prolong the time of the outbreak. dubey et al. [19] modelled the role of acquired immune response and antiretroviral therapy in the dynamics of hiv infection. ghosh et al. [20] worked on an hiv/aids model of an si-type with the inclusion of media and self-imposed psychological fear and their results revealed that awareness is more effective in eradicating hiv infection. the existing models in the literature failed to consider the partitioning of detected individuals who are receiving treatment and those who do not access treatment. hence, we proposed a mathematical model to study the transmission dynamics of hiv in nigeria. we assume that a fraction of individuals that are detected moves to the treatment class while the remaining fraction moves to the aids class. the paper is structured as follows: section 2 contains the method used in the study, section 3 has the numerical simulations and discussion of results, while the conclusion follows in section 4. 2. method 2.1. model formulation here, we give the description of how the hiv model is designed and formulated. the overall population of humans at the time (t) is denoted by n(t) and is partitioned into six distinct classes namely: the susceptible population s (t), the hivlatently infected l, the hiv-infected undetected class hu , the hiv-infected detected class hd, the treatment class hw , and the aids class a. thus n(t) = s (t) + l(t) + hu (t) + hd(t) + hw (t) + a(t) (1) the susceptible individuals are assumed to be recruited into the population at the rate π and get infected after effective contact with hiv-infected people in the latent, undetected, detected, treatment and aids classes at the rate λ, which is given by λ = β(l + η1 hu + η2 hd + η3 hw + η4 a) n (2) where β is the contact rate, η1,η2,η4 ≥ 1 and η3 ≤ 1 are the modification parameters which compare the level of transmissibility of the disease in hu, hd, a, and hw classes with respect to people in l class. it is assumed that a fraction � of newly infected individuals progresses to the latently-infected class and the remaining fraction with compromised immunity moves to the hiv-undetected class. a fraction ω of people in the latent class who are detected progresses to the detected class while the other fraction 1 − ω proceeds to the undetected class. the population of the hiv-detected class increases as a result of the detection of the undetected individuals at the rate γ and diminishes as a result of fraction α of detected individuals who are receiving treatment that progresses to treatment class and the remaining fraction 1 −α that progresses to aids class. we assume that those that are receiving treatment move to the latent class at the rate φ. the aids class reduces as a result of death due to the disease at the rate δ. each population size reduces as a result of natural death which occurs in all classes. the flow chart of the model showing the interaction among the classes is depicted in figure 1. thus, with the assumptions above, we present a deterministic model of hiv infection as follows: ds dt = π−λs −µs dl dt = �λs + φhw − (κ + µ)l dhu dt = (1 − �)λs + (1 −ω)κl − (γ + µ)hu (3) dhd dt = ωκl + γhu − (τ + µ)hd dhw dt = ατhd − (φ + µ)hw da dt = (1 −α)τhd − (µ + δ)a where λ = β(l + η1 hu + η2 hd + η3 hw + η4 a) n (4) n = s + l + hu + hd + hw + a (5) for convenience, we re-write the above equation (3) as thus: ds dt = π−λs −µs 2 ajao et al. / j. nig. soc. phys. sci. 5 (2023) 1389 3 figure 1. schematic diagram of the model dl dt = �λs + φhw − t1 l dhu dt = (1 − �)λs + w l − t2 hu (6) dhd dt = xl + γhu − t3 hd dhw dt = y hd − t4 hw da dt = zhd − t5 a where t1 = κ + µ t2 = γ + µ t3 = τ + µ t4 =φ + µ t5 = µ + δ w =(1 −ω)κ x =ωκ y =ατ z = (1 −α)τ 2.2. model analysis 2.2.1. basic properties this section explores the basic dynamical features of the model (3). we claim the following: lemma 2.1. the closed set d = { (s, l, hu, hd, hw, a) ∈ r 6 + : n ≤ π µ } is positively invariant with non-negative initial values in r6+. proof. summing up all the compartments of (3) with δ = 0 we have dn dt = π−µn it follows that dn dt ≤ π−µn then n(t) ≤ n(0)e−µt + π µ (1 − e−µt) if n(0) ≤ π µ , then n(t) ≤ π µ . hence, all solutions of the model having their starting values in d stay there for t > 0. this means that d is positively invariant and in this region, the model is considered to be epidemiologically meaningful and mathematically well-posed. hence, we can study the dynamics of the basic model (3) in d. 2.2.2. stability of the disease-free equilibrium the disease-free equilibrium of the model (6) denoted by e1 is given by e1 = (s 0, l0, hu0, hd0, hw0, a0) = ( π µ , 0, 0, 0, 0, 0 ) (7) by adopting the approach of the next generation matrix method as given by [25], the matrices f (new infection terms) and v (transition terms) are as given below: f =  �β �βη1 �βη2 �βη3 �βη4 (1 − �)β (1 − �)βη1 (1 − �)βη2 (1 − �)βη3 (1 − �)βη4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  (8) and v =  t1 0 0 −φ 0 −w t2 0 0 0 −x −γ t3 0 0 0 0 −y t4 0 0 0 −z 0 t5  (9) then r0 = ρ(fv−1), is given by r0 = β  (1 − �)[t1t3t4t5η1 − xy t5φη1 + t1t4t5γη2 +t1t5yγη3 + t1t4zγη4 + y t5γφ] +�(xt2 + γw)[t4t5η2 + y t5η3 + zt4η4] +�t3t4t5(t2 + wη1)  t5(t1t2t3t4 − xy t2φ− wyγφ) (10) where ρ is the spectral radius of the dominant eigenvalue of the matrix fv−1. hence, using the theorem 2 of [25], the following result is established: lemma 2.2. the disease-free equilibrium of model (6) is locally asymptotically stable whenever the basic reproduction number r0 < 1 and otherwise if r0 > 1 the threshold r0 represents the basic reproduction number of the disease, which is the average number of secondary infections emanating from a single infection source in a population consisting of only the susceptible people [26]. the implication of lemma 2.2 is that a small introduction of infected individuals into the community/population will not produce a substantial outbreak of the disease when the basic reproduction number is less than unity and therefore the disease vanishes. in the next theorem, we show that the disease can be eradicated irrespective of the initial sizes of the sub-populations when r0 < 1 through the exploration of the global stability of the disease-free equilibrium. theorem 2.3. the disease-free equilibrium (7) of the hiv model is globally asymptotically stable whenever r0 < 1 3 ajao et al. / j. nig. soc. phys. sci. 5 (2023) 1389 4 table 1. description of the parameters used in the model (3) parameter description value source π recruitment rate 618893 [21] β contact rate 0.0785 estimated µ natural death rate 0.022 [22] α fraction of detected individuals that are treated 0.29 assumed φ progression rate from treatment class to latent class 0.201 assumed γ detection rate of undetected individuals 0.392 estimated � fraction of newly infected people with uncompromised immunity 0.9 assumed κ progression rate of people in the latent class 0.01 [15] η1,η2,η3,η4 modification parameters 1.17,1.2,0.04,0.18 assumed δ death due to the disease 0.33 [23] τ treatment rate 0.89 [24] ω fraction of latent individuals that are detected 0.1 assumed proof. using the lyapunov function defined by f = e1 l + e2 hu + e3 hd + e4 hw + e5 a (11) by differentiating (11), we have f′ = e1 l ′ + e2 h ′ u + e3 h ′ d + e4 h ′ w + e5 a ′ (12) where e1 = (1 − �)β(xη1 −γ) − n(t2 x + γw) e2 = �β(γ− xη1) −γt1 n e3 = �βs (wη1 + t2) + t1[(1 − �)η1βs + t2 n] e4 = ( φ[(1 − �)βs (xη1 −γ) − n(t2 x + γw)] −βs η1[�(t2 x + γw) + γ(1 − �)] ) t4 e5 =  �βs [(t2 x + γw)(t4η2 + yη3) + t4t3(wη1 + t2)] +(1 − �)βs [t1t4(η2γ + t3η1) − yφ(xη1 −γ) +yη3γ] + n[yφ(t2 x + γw) − t1t2t3t4]  zt4 then by substituting (6) into (12), it becomes f′ = e1(�λs − t1 l + φhw ) + e2((1 − �)λs + w l (13) − t2 hu ) + e3(xl + γhu − t3 hd) + e4(y hd − t4 hw ) + e5(zhd − t5 a) simplifying (13) further leads to f′ =  −nt5(yφt2 x + yφγw − t1t2t3t4) −�βs (t2 x + γw)[zt4η4 + t5t4η2 + t5yη3] −(1 − �)βs [t5t1t4η4γ + t5t3η1 −y t5φ(xη1 −γ) + t5yη3γ + zt4γt1η4] −�βs t5t4t3(wη1 + t2)  a zt4 f′ = t5 n[yφ(t2 x + γw) − t1t2t3t4] zt4 ( s n r0 − 1 ) a f′ ≤ ( t5 n[yφ(t2 x + γw) − t1t2t3t4] zt4 ) (r0 − 1)a since s ≤ n in d, therefore,f′ ≤ 0 if r0 ≤ 1 with f′ = 0 if and only if a = 0, l = 0, hu = 0, hd = 0, hw = 0. also, the largest invariant set in (s, l, hu, hd, hw, a) ∈ d : f′ = 0 is the singleton e1. by lasalle invariance principle [27], every solution having its starting values in d, approaches e1 as t → ∞, and therefore, the disease-free equilibrium is globally asymptotically stable whenever r0 < 1. the theorem implies that the disease can be eliminated irrespective of the initial sizes of the subpopulations of the model whenever the basic reproduction number does not exceed unity. 2.2.3. existence of endemic equilibrium the existence of endemic equilibrium is being investigated here and the condition for the persistence of the disease is being explored. let e2∗ = (s ∗∗, l∗∗, h∗∗u , h ∗∗ d , h ∗∗ w , a ∗∗) represents the endemic equilibrium state. also, let the force of infection at endemic equilibrium be represented by λ∗∗ = β(l∗∗ + η1 h∗∗u + η2 h ∗∗ d + η3 h ∗∗ w + η4 a ∗∗) n∗∗ (14) then solving the model equations in terms of the λ (force of infection) at the steady state, we will get the following: s ∗∗ = π λ∗∗ + µ (15) l∗∗ = [�t2t3t4 + φyγ(1 − �)]λ∗∗s ∗∗ t2(t1t3t4 −φy x) −φyγw = q1λ ∗∗s ∗∗ (16) 4 ajao et al. / j. nig. soc. phys. sci. 5 (2023) 1389 5 h∗∗u = [(1 − �)(t1t3t4 −φy x) + �wt3t4]λ∗∗s ∗∗ t2(t1t3t4 −φy x) −φyγw = q2λ ∗∗s ∗∗ (17) h∗∗d = t4[t1γ(1 − �) + �(xt2 + wγ)]λ∗∗s ∗∗ t2(t1t3t4 −φy x) −φyγw = q3λ ∗∗s ∗∗ (18) h∗∗w = y [t1γ(1 − �) + �(xt2 + wγ)]λ∗∗s ∗∗ t2(t1t3t4 −φy x) −φyγw = q4λ ∗∗s ∗∗ (19) a∗∗ = zt4[t1γ(1 − �) + �(xt2 + wγ)]λ∗∗s ∗∗ t5[t2(t1t3t4 −φy x) −φyγw] = q5λ ∗∗s ∗∗ (20) by the substitution of (16), (17), (18), (19), and (20) into (14), we have λ∗∗ = β(q1 + η1 q2 + η2 q3 + η3 q4 + η4 q5)λ∗∗s ∗∗ s ∗∗ + q2λ∗∗s ∗∗ + q3λ∗∗s ∗∗ + q4λ∗∗s ∗∗ + q5λ∗∗s ∗∗ (21) λ∗∗ = β(q1 + η1 q2 + η2 q3 + η3 q4 + η4 q5)λ∗∗s ∗∗ s ∗∗(1 + vλ∗∗) (22) where v = q1 + q2 + q3 + q4 + q5 q1 = [�t2t3t4 + φyγ(1 − �)] t2(t1t3t4 −φy x) −φyγw q2 = [(1 − �)(t1t3t4 −φy x) + �wt3t4] t2(t1t3t4 −φy x) −φyγw q3 = t4[t1γ(1 − �) + �(xt2 + wγ)] t2(t1t3t4 −φy x) −φyγw q4 = y [t1γ(1 − �) + �(xt2 + wγ)] t2(t1t3t4 −φy x) −φyγw q5 = zt4[t1γ(1 − �) + �(xt2 + wγ)] t5[t2(t1t3t4 −φy x) −φyγw] hence, λ∗∗ = r0 − 1 v > 0 when r0 > 1 ∴ λ∗∗ has a positive unique solution when r0 > 1. hence, the following result is obtained: lemma 2.4. there exists a unique endemic equilibrium of the hiv model equation (6) whenever the basic reproduction number r0 > 1. the above result indicates the existence of forward bifurcation which is verified in the next analysis. 2.2.4. bifurcation analysis the bifurcation is a phenomenon that describes the changes in the behaviour of a dynamical system as a result of changes in the parameter values or initial conditions of the model. this helps to determine if the disease can be cleared off when the basic reproduction number is less than unity. we will adopt the center manifold theory [28] as described by [29][see appendix b] to establish the kind of bifurcation that the model exhibits. the center manifold theory is used to determine the stability of equilibrium and plays a vital role in bifurcation theory because of the changes in behaviour of the system that take place on the center manifold. if β is chosen as the bifurcation parameter for model (6), then at r0 = 1, we have that β = β∗ = t5(t1t2t3t4 − xy t2φ− wyγφ) (1 − �)[t1t3t4t5η1 − xy t5φη1 + t1t4t5γη2 +t1t5yγη3 + t1t4zγη4 + y t5γφ] +�(xt2 + γw)[t4t5η2 + y t5η3 + zt4η4] +�t3t4t5(t2 + wη1)  (23) if the variables of (6) are changed as follows: s = x1, l = x2, hu = x3, hd = x4, hw = x5, a = x6 and we use the vector notation x = (x1, x2, x3, x4, x5, x6)t , then (6) can be re-written in the form d xdt = f(x) where f = ( f1, f2, f3, f4, f5, f6) t such that (6) becomes d x1 dt = π−λx1 −µx1 = f1 d x2 dt = �λx1 + φx5 − t1 x2 = f2 d x3 dt = (1 − �)λx1 + w x2 − t2 x3 = f3 (24) d x4 dt = x x2 + γx3 − t3 x4 = f4 d x5 dt = y x4 − t4 x5 = f5 d x6 dt = z x4 − t5 x6 = f6 (25) the jacobian (24) at disease-free equilibrium e1 is given by j(e1) =  −µ −β∗ −β∗η1 −β ∗η2 −β ∗η3 −β ∗η4 0 �β∗ − t1 �β∗η1 �β∗η2 �β∗η3 + φ �β∗η4 0 (1 − �)β∗ + w (1 − �)β∗η1 − t2 (1 − �)β∗η2 (1 − �)β∗η3 (1 − �)β∗η4 0 x γ −t3 0 0 0 0 0 y −t4 0 0 0 0 z 0 −t5  (26) the matrix (26) has a simple zero eigenvalue at β = β∗ and hence center manifold theory [28] as described by [29] can be used to analyse the dynamics of the system. the jacobian matrix (26) has a right eigenvector denoted by w = (w1, w2, w3, w4, w5, w6)t and a left eigenvector v = (v1, v2, v3, v4, v5, v6)t corresponding to the zero eigenvalue. then w1 = β∗ ( t3t4t5[w2 + η1w3] + (xw2 + γw3) ×[t4t5η2 + t5yη3 + t4zη4] ) t3t4t5µ , w2 = w2 > 0, w3 = w3 > 0 w4 = xw2 + γw3 t3 , w5 = y [xw2 + γw3] t3t4 , w6 = z[xw2 + γw3] t3t5 and v1 = 0, v2 = v2 > 0, v3 = v3 > 0, 5 ajao et al. / j. nig. soc. phys. sci. 5 (2023) 1389 6 figure 2. fitting of hiv model (3) to the data of prevalence cases of hiv/aids infection in nigeria between 1990 and 2019 [30]. 0 10 20 30 0 1 2 3 x 10 5 time(years) d e te c te d cl a ss 0 10 20 30 0 5 10 15 x 10 4 time(years) t re a tm e n t c la ss 0 10 20 30 0 5000 10000 15000 time(years) a id s c la ss 0 20 40 6 7 8 9 10 x 10 7 time(years) s u ce sp ti b le c la ss 0 20 40 0 5 10 15 x 10 4 time(years) l a te n t c la ss 0 20 40 0 1000 2000 3000 4000 time(years) u n d e te ct e d cl a ss figure 3. plot of the different classes of the model when r0 < 1 (r0 = 0.4) with β = 0.0085 and other values used are in table 1 v4 = ( (t4t5η2 + y t5η3 + t4zη4)[�β∗v2 + (1 − �)β∗v3] +φy t5v2 ) t3t4t5 , v5 = β∗η3(�v2 + (1 − �)v3) + φv2 t4 , v6 = β∗η4(�v2 + (1 − �)v3) t5 computation of a and b by finding the associated non-zero partial derivatives of f(x) at disease-free equilibrium, the associated bifurcation coefficients a and b as given by the center manifold the0 20 40 6 7 8 9 10 x 10 7 time(years) s u c e sp ti b le c la ss 0 20 40 0 2 4 6 8 x 10 5 time(years) l a te n t c la ss 0 10 20 30 0 1 2 3 x 10 4 time(years) u n d e te c te d c la ss 0 10 20 30 0 1 2 3 x 10 5 time(years) d e te c te d c la ss 0 10 20 30 0 5 10 15 x 10 4 time(years) t re a tm en t cl a ss 0 10 20 30 0 5000 10000 15000 time(years) a id s cl a ss figure 4. plot of the different classes of the model when r0 > 1 (r0 = 3.3) using the values of the parameters in table 1 0 100 200 300 400 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x 10 7 time (years) l + h u + h d + h w + a figure 5. plot of the total number of the infected classes (l+ hu + hd + hw + a) at different initial conditions when r0 = 0.4, with β = 0.0085 and other values of the parameters used are given in table 1 ory [28][see appendix b], are defined by a = n∑ k,i, j=1 vkwiw j ∂2 fk ∂xi∂x j (0, 0) b = n∑ k,i=1 vkwi ∂2 fk ∂xi∂β∗ (0, 0) then, we obtain a = −2 β∗ µ π  (v2� + v3(1 − �) ) × (w2 + w3 + w4 + w5 + w6) × (η1w3 + η2w4 + η3w5 + η4w6 + w2)  (27) and b = (v2� + v3(1 − �))[w2 + w3η1 + w4η2 + w5η3 + w6η4] (28) 6 ajao et al. / j. nig. soc. phys. sci. 5 (2023) 1389 7 0 100 200 300 400 1 1.5 2 2.5 3 3.5 4 x 10 7 time (years) l + h u + h d + h w + a figure 6. plot of the total number of the infected classes (l+ hu + hd + hw + a) at different initial conditions when r0 = 3.3, with � = 1, ω = 0 and other values of the parameters used are given in table 1 0 5 10 15 20 25 30 0 1 2 3 4 5 6 x 10 5 time (years) l a te n t c la ss α=0.1 α=0.25 α=0.4 α=0.55 α=0.7 figure 7. graph of the population of the latent class against time when the fraction of the detected population receiving treatment is varied from (28), b is positive as usual and a is negative from (27). since a < 0 (negative), then the hiv infection model exhibits a forward bifurcation. this implies that the epidemiological condition r0 < 1 is necessary and sufficient for the elimination of hiv infection. 2.2.5. global stability of the endemic equilibrium we consider the global stability of endemic equilibrium of the hiv infection model for a situation when � = 1 and the fraction of the latently infected people that are detected equals zero (ω = 0) and also we let β∗ = βn . then, with these assumptions the model‘s basic reproduction number when � = 1 and ω = 0 is given by ror = β∗ π ( yγκ t5η3 + zγκ t4η4 + γκ t4t5η2 +κ t3t4t5η1 + t4t3t2t5 ) t5µ (t4t3t2t1 − yγκφ) 0 5 10 15 20 25 30 0 2 4 6 8 10 12 x 10 4 time (years) a id s c la ss α=0.1 α=0.25 α=0.4 α=0.55 α=0.7 figure 8. graph of the population of the aids class against time when the fraction of the detected population receiving treatment is varied also, the model with � = 1 and ω = 0 possesses a unique endemic equilibrium point represented by e∗2r , which is given by e2r ∗ |�=1,ω=0 = (s ∗∗, l∗∗, h∗∗u , h ∗∗ d , h ∗∗ w , a ∗∗) and that s ∗∗ > 0, l∗∗ > 0, h∗∗u > 0, h ∗∗ d > 0, h ∗∗ w > 0 and a ∗∗ > 0 when ror > 1 theorem 2.5. the endemic equilibrium of the reduced model having � = 1 and ω = 0 is globally asymptotically stable whenever ror > 1. proof. using the goh-volterra type of lyapunov function, we have m = s − s ∗∗ − s ∗∗ln s s ∗∗ + l − l∗∗ − l∗∗ln l l∗∗ + c ( hu − h ∗∗ u − h ∗∗ u ln hu h∗∗u ) + d ( hd − h ∗∗ d − h ∗∗ d ln hd h∗∗d ) + e ( hw − h ∗∗ w − h ∗∗ w ln hw h∗∗w ) + f ( a − a∗∗ − a∗∗ln a a∗∗ ) (29) where c = β∗s ∗∗[t3t4t5η1 + γ(t4t5η2 + y t5η3 + zt4η4)] + γyφt5 t2t3t4t5 (30) d = β∗s ∗∗[t4t5η2 + y t5η3 + zt4η4] + yφt5 t3t4t5 (31) e = β∗η3s ∗∗ + φ t4 (32) f = β∗η4s ∗∗ t5 (33) 7 ajao et al. / j. nig. soc. phys. sci. 5 (2023) 1389 8 taking the derivative of (29), we have ṁ = ṡ − s ∗∗ ṡ s + l̇ − l∗∗ l̇ l +( β∗s ∗∗[t3t4t5η1 + γ(t4t5η2 + y t5η3 + zt4η4)] +γyφt5 ) t2t3t4t5 × ( ḣu − h ∗∗ u ḣu hu ) + β∗s ∗∗[t4t5η2 + y t5η3 + zt4η4] + yφt5 t3t4t5 × ( ḣd − hd ∗∗ ḣd hd ) + β∗η3s ∗∗ + φ t4 ( ḣw − h ∗∗ w ḣw hw ) + β∗η4s ∗∗ t5 ( ȧ − a∗∗ ȧ a ) ṁ = 2β∗s ∗∗(l∗∗ + η1 h ∗∗ u + η2 h ∗∗ d + η3 h ∗∗ w + a ∗∗) + 2µs ∗∗ −µs − β∗s ∗∗ 2 s (l∗∗ + η1 h ∗∗ u + η2 h ∗∗ d + η3 h ∗∗ w + a ∗∗) − µs ∗∗ s − β∗s l∗∗ l (l + η1 hu + η2 hd + η3 hw + a) + φh ∗∗ w − hw l∗∗ l − lh∗∗u l∗∗hu ( β∗s ∗∗[η1 h∗∗u + η2 h ∗∗ d +η3 h∗∗w + a ∗∗] + φh∗∗w ) + ( β∗s ∗∗[η1 h ∗∗ u + η2 h ∗∗ d + η3 h ∗∗ w + a ∗∗] + φh∗∗w ) − hu h∗∗d h∗∗u hd ( β∗s ∗∗[η2 h ∗∗ d + η3 h ∗∗ w + a ∗∗] + φh∗∗w ) +( β∗s ∗∗[η2 h ∗∗ d + η3 h ∗∗ w + a ∗∗] + φh∗∗w ) − (β∗η3s ∗∗ + φ)hd h∗∗ 2 w h∗∗d hw β∗η3s ∗∗h∗∗w + φh∗∗w − β∗η4s ∗∗hd a∗∗ 2 f h∗∗d + β∗η4s ∗∗a∗∗ then ṁ =β∗s ∗∗l∗∗ ( 2 − s ∗∗ s − s s ∗∗ ) + µs ∗∗ ( 2 − s s ∗∗ − s ∗∗ s ) + β∗s ∗∗η1 h ∗∗ u ( 3 − s ∗∗ s − hu l∗∗s h∗∗u ls ∗∗ − lh∗∗u l∗∗hu ) + β∗s ∗∗η2 h ∗∗ d ( 4 − s ∗∗ s − hd l∗∗s h∗∗d ls ∗∗ − lh∗∗u l∗∗hu − hu h∗∗d h∗∗u hd ) +β∗s ∗∗η3 h ∗∗ w  5 − s ∗∗ s − hw l∗∗s h∗∗w ls ∗∗ − lh∗∗u l∗∗hu − hu h∗∗d h∗∗u hd − hd h∗∗w h∗∗d hw  +β∗s ∗∗η4 a ∗∗  5 − s ∗∗ s − al∗∗s a∗∗ls ∗∗ − lh∗∗u l∗∗hu − hu h∗∗d h∗∗u hd − hd a∗∗ h∗∗d a  +φh∗∗w ( 4 − hw l∗∗ h∗∗w l − lh∗∗u l∗∗hu − hu h∗∗d h∗∗u hd − hd h∗∗w h∗∗d hw ) the arithmetic mean surpasses the geometric mean, then we have the following inequalities 2 − s s ∗∗ − s ∗∗ s ≤ 0, 2 − s ∗∗ s − s s ∗∗ ≤ 0, 3 − s ∗∗ s − hu l∗∗s h∗∗u ls ∗∗ − lh∗∗u l∗∗hu ≤ 0, 4 − s ∗∗ s − hd l∗∗s h∗∗d ls ∗∗ − lh∗∗u l∗∗hu − hu h∗∗d h∗∗u hd ≤ 0, 5 − s ∗∗ s − hw l∗∗s h∗∗w ls ∗∗ − lh∗∗u l∗∗hu − hu h∗∗d h∗∗u hd − hd h∗∗w h∗∗d hw ≤ 0, 5 − s ∗∗ s − al∗∗s a∗∗ls ∗∗ − lh∗∗u l∗∗hu − hu h∗∗d h∗∗u hd − hd a∗∗ h∗∗d a ≤ 0, 4 − hw l∗∗ h∗∗w l − lh∗∗u l∗∗hu − hu h∗∗d h∗∗u hd − hd h∗∗w h∗∗d hw ≤ 0 therefore l ≤ 0 when ror > 1. hence, by the lasalle invariance principle [27], every solution of the model tends to e2r∗ as t →∞ for r0r > 1. the epidemiological implication of this is that hiv infection will persist in the community irrespective of the initial sizes of the subpopulations of the model whenever r0r > 1. 3. numerical simulations and discussion of results we fit the hiv model to data on hiv/aids prevalence in nigeria from 1990 to 2019 as presented in table 2 (see appendix. a) sourced from [30]. we use the likelihood function to estimate the values of the contact rate (β) and the detection rate of the undetected class (γ). the estimated values of β and γ are 0.0785 and 0.392 respectively. the total population of nigeria in 1990 stood at 95214256 based on [31] and the initial conditions used for the simulations are as follow: s (0) = 94999422, l(0) = 0, hu (0) = 0, hd(0) = 214834, hw (0) = 0 and a(0) = 0. the time interval of 0 to 29 corresponds to the time interval between 1990 to 2019 and the values of the parameters used for the simulations are given in table 1. the results of the numerical simulations of the model are presented in figures 2 8. in figure 2, we fit the hiv model to data in table 2 and also obtain the estimated values for the contact rate (β) and the detection rate (γ). the model fits well with the real data and thus the model represents reality. figure 3 illustrates the behaviour of each compartment when the basic reproduction number is greater than unity. the susceptible class keeps decreasing while the other infected compartments l, hu, hd, hw and a are increasing which indicates the persistence of the hiv infection. figure 4 shows the trajectories of the model when the basic reproduction number is less than unity. the population of the susceptible declines and the infected classes l, hu, hd, hw and a are reducing after some period, which means that the disease can be controlled if r0 < 1. figures 5 and 6 illustrate the verification of the global stability properties of the disease-free equilibrium and endemic 8 ajao et al. / j. nig. soc. phys. sci. 5 (2023) 1389 9 equilibrium. the long-term dynamics of the model as depicted by figure 5 shows that the trajectories converge and the disease vanishes irrespective of the initial sizes of the subpopulations when the basic reproduction number is less than unity and figure 6 shows that the disease persists when the basic reproduction number is greater than unity. in figure 7, the plot shows the population of latently infected individuals when the fraction of the hiv-detected individuals that are receiving treatment is varied. the population of the latently infected increases as this fraction increases. this is due to the fact that hiv infection has no cure but can be managed. the graph in figure 8 shows that the population of the aids class reduces as the fraction of detected individuals receiving treatment increases. 4. conclusion we propose a mathematical model to study the transmission dynamics of hiv and conduct qualitative and quantitative analyses of the model. the model’s disease-free equilibrium is locally asymptotically stable whenever the basic reproduction number is less than unity. also, there exists a unique endemic equilibrium for the model whenever the basic reproduction number is greater than unity and it is shown that the model exhibits forward bifurcation which implies that the necessary condition r0 < 1 is sufficient for the elimination of the disease. using the lyapunov function, we further showed that the disease-free equilibrium and endemic equilibrium are globally asymptotically stable whenever the basic reproduction number is less than unity and greater than unity respectively. the proposed model fits with the data on hiv/aids prevalence in nigeria from 1990 to 2019 as it represents the reality. the simulation shows that the disease can be controlled when the basic reproduction number is less than unity and persists if otherwise. the simulations that illustrate the global stability of the model justify the analytic results. the effect of increasing the fraction of the detected individuals that are receiving treatment is examined and it increases the population of the latent class and reduces the population of the aids class, since the disease has no cure, the treatment is meant to improve the health of a patient by reducing the viral load to an undetected level and prevent 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[13] d. dimitrov, d. wood, a. ulrich, d. a. swan, b. adamson, j. r. lama, j. sanchez & a. duerr, “projected effectiveness of hiv detection during early infection and rapid art initiation among msm and transgender women in peru: a modeling study”, infectious disease modelling 4 (2019) 73. https://doi.org/10.1016/j.idm.2019.04.001 [14] p. ngina, r. w. mbogo, & l. s. luboobi, “hiv drug resistance: insights from mathematical modelling”, applied mathematical modelling 75(2019) 141. https://doi.org/10.1016/j.apm.2019.04.040 [15] l. cai, x. li, m. ghosh, b. guo, “stability analysis of an hiv/aids epidemic model with treatment”, journal of computational and applied mathematics 229 (2009) 313. https://doi.org/10.1016/j.cam.20 08.10.067 [16] a. s. hassan & n. hussaini, “analysis of an hiv hcv simultaneous infection model with time delay”, j. nig. soc. phys. sci. 3 (2021) 1. ht tps://doi.org/10.46481/jnsps.2021.109 [17] o. o. apenteng, p. p. osei, b. oduro, m. p. kwabla & n. a. ismail, “the impact of implementing hiv prevention policies therapy and control strategy among hiv and aids incidence cases in malaysia”, infectious disease modelling 5 (2020) 755. https://doi.org/10.1016/ j.idm.2020.09.009 [18] w. yang, z. shu, j. lam & c. sun, “global dynamics of an hiv model incorporating senior male clients”, applied mathematics and computation 311 (2017) 203. https://doi.org/10.1016/j.amc.2017.05.030 [19] p. dubey, u. s. dubey & b. dubey, “modeling the role of acquired immune response and antiretroviral therapy in the dynamics of hiv infection”, mathematics and computers in simulation 144 (2018) 120. https://doi.org/10.1016/j.matcom.2017.07.006. [20] i. ghosh, p. k. tiwari, s. samanta, i. m. elmojtaba, n. al-salti & j. chattopadhyay, “a simple si-type model for hiv/aids with media and self-imposed psychological fear”, mathematical biosciences, 306 (2018) 9 ajao et al. / j. nig. soc. phys. sci. 5 (2023) 1389 10 160. https://doi.org/10.1016/j.mbs.2018.09.014 [21] the world bank: fertility rate ,total(birth rate per woman). available at https://data.worldbank.org/indicator/sp.dyn.tfrt.in?e nd=1990\&start=1960. accessed on 20/12/2022. [22] the world bank: data: life expectancy at birth,total(years)-nigeria. available at https://data.worldbank.org/indicator/sp.dy n.le00.in?end=2019\&locations=ng\&start=1960. accessed on 20/12/2022. [23] r. aggarwal, “dynamics of hiv-tb co-infection with detection as optimal intervention strategy”, international journal of non-linear mechanics 120 (2020) 103388. https://doi.org/10.1016/j.ijnonlinme c.2019.103388. [24] world health organization, “hiv/aids-protection and early detection, promise of health”. available at https://www.afro.who.int/news/ hivaid-protection-and-early-detection-promise-health. accessed on 20/12/2022 [25] p. van den driessche & j. watmough, “reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission”, mathematical biosciences 180 (2002) 29. [26] s. a. ayuba, i. akeyede & a. s.olagunju, “stability and sensitivity analysis of dengue-malaria co-infection model in endemic stage” journal of the nigerian society of physical sciences 3 (2021) 96. [27] j.p. lasalle, the stability of dynamical systems, regional conference series in applied mathematics, siam, philadelphia, 1976. [28] j. carr, applications of centre manifold theory applied mathematical sciences. springer-verlag, new york, 35 (1982). [29] c. castillo-chavez & b. song, “dynamical models of tuberculosis and their applications”, math biosci eng 1 (2004) 361. [30] our world in data: prevalence, new cases and deaths from hiv/aids, nigeria,1990 to 2019. available at https://ourworldindata.org /grapher/deaths-and-new-cases-of-hiv?country=~nga. accessed on 16/12/2022 [31] our world in data: population, 10000 bce to 2021. available at https: //ourworldindata.org/grapher/population?country=~nga. accessed on 16/12/2022 appendix a: hiv/aids prevalence in nigeria from 1990-2019 as given by [30] table 2. data of hiv/aids prevalence in nigeria from 1990-2019 [30] year 1990 1991 1992 1993 1994 cases 214934 307403 417550 541812 674511 year 1995 1996 1997 1998 1999 cases 808728 940842 1065300 1177623 1271363 year 2000 2001 2002 2003 2004 case 1347177 1406166 1449357 1479819 1500481 year 2005 2006 2007 2008 2009 cases 1515892 1527636 1542107 1558937 1581336 year 2010 2011 2012 2013 2014 cases 1609292 1638694 1670713 1707410 1752498 year 2015 2016 2017 208 2019 cases 1797982 1841027 1882445 1922997 1963044 appendix b: theorem (castillo-chavez and song [29]). consider the following general system of ordinary differential equations with a parameter φ. d x dt = f (x,φ), f : rn × r → rn and f ∈ c2(rn × r) where 0 is an equilibrium point of the system(that is, f (0,φ) = 0 for all φ) and 1. a = dx f (0, 0) is the linearization matrix of the system around the equilibrium 0 with φ evaluated at 0; 2. zero is a simple eigenvalue of a and all other eigenvalues of a have negative real parts; 3. matrix a has a right eigenvector w and a left eigenvector v corresponding to the zero eigenvalue. let fk be the kth component of f and a = n∑ k,i, j=1 vkwiw j ∂2 fk ∂xi∂x j (0, 0) b = n∑ k,i=1 vkwi ∂2 fk ∂xi∂φ (0, 0) then the local dynamics of the system around the equilibrium point 0 is totally determined by the signs of a and b. i. a > 0, b > 0. when φ < 0 with |φ| � 1, 0 is locally asymptotically stable, and there exists a positive unstable equilibrium; when 0 < φ � 1, 0 is unstable and there exists a negative and locally asymptotically stable equilibrium; ii. a < 0, b < 0. when φ < 0 with |φ| � 1, 0 is unstable; when 0 < φ � 1, 0 is locally asymptotically stable, and there exists a positive unstable equilibrium; iii. a > 0, b < 0. when φ < 0 with |φ| � 1, 0 is unstable, and there exists a locally asymptotically stable negative equilibrium; when 0 < φ � 1, 0 is stable, and a positive unstable equilibrium appears; iv. a < 0, b > 0. when φ changes from negative to positive, 0 changes its stability from stable to unstable. correspondingly a negative unstable equilibrium becomes positive and locally asymptotically stable. 10 j. nig. soc. phys. sci. 5 (2023) 1264 journal of the nigerian society of physical sciences solar energy storage by fuel cell technology at abomey-calavi (benin) odilon joseph towanou, hagninou elagnon venance donnou, gabin koto n’gobi∗, augustin enonsi leodé, basile kounouhéwa laboratoire de physique du rayonnement (lpr), physics department, university of abomey-calavi (uac), abomey-calavi, benin republic abstract west africa has a great amount of sunshine power, varying between 5 kwh.m−2.day−1 and 7 kwh.m−2.day−1. this power constitutes high energy source in the region. however, several locations in that area have no access to energy because of the lack of suitable technology and projects exploiting the source. the fundamental problem related to sun power or to renewable energies in general is the lack of efficient technology for energy storage. batteries are generally used for this storage, but once charged, the excess of the energy from the solar photovoltaic panels (pv) is lost. therefore, it is very important to find a system to recover the excess in order to optimize its use. in this context, hydrogen is considered a very promising candidate to fulfill this function and could become a highly developed energy vector in the future. the very numerous works undertaken over the past decade for the production of electricity by hydrogen fuel cells bear witness to this. the objective of this study is to test a more reliable solar energy storage system by using fuel cell technology. to achieve this, three steps have been necessary: (i) make an electrolyser using materials, (ii) produce hydrogen using a system of pv panels and (iii) convert the hydrogen produced into electricity through a fuel cell. the results obtained indicate a production of 0.020m3 of hydrogen after 150 min with a yield of 85.86%. the production of electricity by a 2 v fuel cell gives an efficiency of 0.0042%. even if this value is low, a part of the lost energy has been recovered. in view of these results, the improvement of the device for converting chemical energy into electricity deserves to be deeply explored in west africa. doi:10.46481/jnsps.2023.1264 keywords: hydrogen in fuel cell, electrolysis, energy storage, benin article history : received: 30 november 2022 received in revised form: 12 march 2023 accepted for publication: 14 march 2023 published: 22 april 2023 c© 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: b. j. falaye 1. introduction in recent years, issues related to the energy transition and the carbon-free energy production have aroused several interests and reflections [1]. indeed, fossil fuels are more than ever, largely responsible for the pollution of the atmosphere and the main cause of global warming. the massive introduction of ∗corresponding author tel. no: +22997228700 email address: kotgabin36@yahoo.fr (gabin koto n’gobi) electricity based on renewable energies such as solar energy in the production of power has therefore become a priority for the states. with the quick industrial advancement and decreasing costs, they will play an important role in future energy frameworks [2]. however, the development of these sources of energy, with an intermittent regime, requires the use of reliable storage means in order to avoid the problem of destabilization of the distribution network and to make this production suitable to consumer’s demand [1]. this has therefore led to the emergence of storage as a crucial element in the management 1 towanou et al. / j. nig. soc. phys. sci. 5 (2023) 1264 2 of energy from renewable sources, allowing energy to be evacuated into the grid during peak hours when it is more valuable. the use of energy storage techniques is then becoming increasingly essential to ensure the accessibility of electrical energy in remote regions [3]. lead-acid batteries, which are among the most widely used solar components, cannot withstand high cycle rates, nor store a large amount of energy in a small volume [3]. this is why other types of storage technologies are being developed and implemented. in this context, the hydrogen synthesized from this renewable electricity is considered to be a fairly important storage vector [4]. the combination of solar pv and fuel cell power could offer a feasible solution to the challenge of continuous power supply, especially in geographic areas where renewable resources are abundantly available [4, 5], as in west africa. the depletion of fossil fuel skocks consequently places hydrogen as one of the major energy carriers of the future and the electrolysis of water at low temperature indeed offers prospects for the future with high potential [6, 7]. discovered by sir william grove in 1839, the concern of the fuel cell is not a recent technology. it has been the subject of numerous works since centuries. today, several researchers and industrial companies are still working constantly, giving much more interest to hydrogen production in order to improve energy storage’s performance. authors, such as faias et al. [8], achkari & fadar [3], ibrahim et al. [9], ofualagba et al. [10], zhang et al. [11], staffell et al. [12] , okolie et al. [13], yue et al. [14] found that fuel cell technology is too expensive to compete with cheap methods of generating and storing electricity, but its future development and its advantages need its integration into the list of the main suitable renewable energy sources. other authors have demonstrated the interest of fuel cell application in micro grid systems, based on some attractive characteristics such as being a clean, non-polluting and highly flexible energy resource [5]. pellow et al. [15] estimate that energy storage with a regenerative hydrogen fuel cell represents an attractive technology to reach efficient energy storage. it can lead to the development of more sustainable, efficient and robust hybrid renewable energy systems. according to singla et al. [4] and belmonte et al. [15], benchrifa et al. [1], derbal et al. [17], the production of hydrogen by thermochemical cycles is more promising than the conventional methods of reforming and gasification of fossil resources with the advantage of having lower impact on the environment. rabih [6] has contributed to a better understanding of the electrochemical phenomena, responsible of the storage and release of electricity as well as the conversion of chemical energy into electrical energy. as for akinsola et al. [18], they constructed a fuel cell using three different materials with different electrodes ((bitter leaf and copper electrodes (bcu), bitter leaf and carbon electrodes (bc) and water leaf and carbon electrodes (wc)). the authors then noticed that the cells made from bitter leaf with a carbon electrode have the highest open circuit voltage, short circuit current and generated power and increase with time. it is clear that the storage of energy from the production of hydrogen by the electrolysis of water and its conversion into electricity via the fuel cell is a technology which deserves special attention for its use in the optimal evacuation of electrical energy produced from renewable energy sources. unfortunately, in benin, as in many west african countries the technology is still at an embryonic stage and little work has focused on this domain of research in order to ensure its mastery such as the studies of fopah-lele et al. [19], jumare [20]. the objective of this study is to produce hydrogen from the electrolysis of water in order to supply a fuel cell for the production of electricity. the renewable energy source used is a solar field installed to supply the physics department in the university of abomey-calavi. the electrolyser was made using a well-defined method with suitable materials. the quantity of hydrogen produced according to the chosen experimental protocol is measured using a gas recorder and is stored in an air chamber. from a proton exchange fuel cell (pem) powered by the air chamber, electricity is produced. finally, the production yields of hydrogen and electricity are evaluated. 2. materials and methods 2.1. materials 2.1.1. the electrolyser the production of hydrogen from the electrolysis of water requires the construction of an electrolyser. the material used for this purpose is presented in figure 1. it is composed of plexiglass, vices, nuts, seals, stainless steel plates, a connection pipe, end fittings, a valve and an air chamber. 2.1.2. the gas logger the gas recorder used during the experiment is presented in figure 2. its essential characteristics are as follows: pmax = 0.5bar; qmax = 6m3.h−1; qmin = 0.04m3.h−1; qt = 0.6m3.h−1. 2.1.3. the fuel cell figure 3 shows the fuel cell (phywe, pem fuel cell, order number : 06747.00, germany) that was used to generate electricity in this study. it is a proton exchange membrane (pem) battery with a voltage equal to 2 v. 2.2. methods the experimental protocol for the realization of the electrolyser, production and storage of hydrogen as well as the production of electricity is presented in sections 2.2.1, 2.2.2 and 2.2.3. during the experiment, we can enumerate two phases which are not synchronized. first, hydrogen is produced and stored. then this gas stored at the end of this first phase is used to produce electricity. the yield calculation method for the various operations mentioned above is set out in section 2.2.4. 2.2.1. realization of the electrolyser for the realization of the electrolyser, we first took the measurements of various samples of materials. the cutting of the stainless-steel plates and the plexiglass according to the dimensions (1cm by 1cm by 2mn thick for the stainless-steel plates and 13cm by 13cm by 1cm thick for the plexiglass) was carried out. the stainless-steel plates and the two pieces of plexiglas 2 towanou et al. / j. nig. soc. phys. sci. 5 (2023) 1264 3 figure 1: material used to build the electrolyser (a) plexiglas, screws and nuts, (b) gaskets, (c) stainless steel plates, (d) connection pipe, (e) end caps, (f) valve and g ) inner tube figure 2: gas logger figure 3: fuel cell : (a) view from the side and (b) view from the top have been perforated. the stainless-steel plates were then arranged one after the other, leaving between them approximately 2 mm of space occupied by rubber seals serving as insulation. everything is held together by the two pieces of plexiglas. we thus obtain the electrolyser illustrated in figure 4. figure 4: electrolyser : (a) view from the top, (b) view from the side 2.2.2. production of hydrogen by the electrolysis of water a very specific protocol was followed for the production of hydrogen: • measure a mass m= 4g of sodium bicarbonate in a tank; • add 1.5 l of water to the same tank, shake to homogenize; • carry out the assembly by placing a voltmeter in parallel with the terminals of the electrolyser to measure the electrical voltage; • measure current intensity over time; • close the circuit and start the stopwatch; • during the experiment, check and note the value of the voltage u and the intensity i; • measure the amount of gas produced using the gas logger; • estimate production time. the electrolyser is powered by a mini solar field made up of 8 power solar panels of 180 wc each installed on the roof of the physics department. the current supplied by this source varies during the day. figure 5 shows the assembly carried out as well as the the power source of the electrolyser. under the effect of gravity, the mixture contained in the tank (1) reaches the level of the electrolyser (5) thanks to the connection pipe (3). this mixture undergoes the action of electric current to form a gaseous mixture consisting essentially of hydrogen and oxygen. this gaseous mixture reaches the bottom of the bottle containing water (7) where part of the oxygen dissolves, but the hydrogen which cannot dissolve rises to the surface creating bubbles. it thus continues on its way through the gas meter (8) to be finally stored in the air chamber (2). 2.2.3. production of electricity by the fuel cell the production of electricity by the fuel cell takes place in several stages. it is: • perform the purge (supply the cell with hydrogen for a few seconds to remove impurities from the cell); • close the lower fuel cell valves; • connect the hydrogen tank to the pipe on the anode side; 3 towanou et al. / j. nig. soc. phys. sci. 5 (2023) 1264 4 figure 5: : hydrogen production and storage system: (1) electrolytic solution tank, (2) air chamber, (3) connection pipe, (4) multimeter, (5) electrolyzer, (6) current clamp, (7) bottle containing water, (8) gas logger • the pipe on the cathode side is supplied with oxygen from the air the fuel cell is therefore supplied at the anode by the hydrogen produced and at the cathode by the oxygen in the air. the latter mounted in series with a resistor allowed us to know the variation of the voltage and the intensity produced by the battery as a function of time. figure 6 presents an overview of the test bench made for the production of electricity. 2.2.4. estimation of the efficiency of the electrolyser and the fuel cell during electrolysis of water, the amount of hydrogen released responds to faraday’s first law which states that the amount of substance released during electrolysis at an electrode is proportional to time and electric current. the amount of electricity (q) carried by a current (i) for a duration (∆t) is given by equation 1: q = i × ∆t (1) the experiment lasted for ∆t =150 minutes for a average current evaluated at i = 1.64 a and an average voltage of u = 37 v. the hydrogen production yield is given by [21]: r1 = v (h2ex perimental) v (h2theorical) (2) v (h2ex perimental) is the volume of hydrogen produced by the electrolyser. the theoretical hydrogen volume v (h2)theorical is evaluated as follows: v (h2theorical) = nqvm zf (3) vm is the molar volume (vm = 24l.mol−1 ), f the faraday constant, f = 96,500c.mol−1 and z is the number of electrons necessary to produce a gas molecule. for hydrogen (2h+ + 2e− → h2), z = 2,n is the number of positive plates of the electrolyser. in the case of this study n is equal to 7. the efficiency of the fuel cell is given by the ratio between the energy supplied (e f ) by the cell and that received (er ) during electrolysis: r2 = u i∆t u p ipt (4) up is the average voltage of the fuel cell evaluated to 0.5 v and ip the average current (0.1a), t is the duration of the electricity production experience by the fuel cell (480s). figure 6: : electricity production by the fuel cell 3. results and discussion 3.1. results 3.1.1. evolution of hydrogen production the volume of hydrogen produced noted during the experiment enabled us to collect data on the evolution of this production over time. these data made it possible to obtain the figure 7. at the end of the 150 min of water electrolysis, a volume of 0.02 m3 of cumulative hydrogen was produced. this accumulation can be adjusted by simple linear regression. three production accumulation phases can be reported (0-50min; 50100min; 100-150min). during the first 50 minutes, the cumulative volume of hydrogen has a lower slope evaluated at 8.8×10−5m3.min−1. the hydrogen production at the end of this period is estimated at 0.0044 m3 according to the graph in figure 7. the following 50 min show a less marked linearity in the hydrogen production with a higher slope of the cumulation estimated at 1.12.10−4m3.min−1. indeed, between 50 and 80 min, there is an increase in the accumulation of hydrogen evaluated at 39.73%. but between 80 and 90 min there is a sudden jump in the total from 0.0073 m3 to 0.0096 m3 evaluated at 23.95%. from 90 min to 100 min, there is a very slight increase in the cumulative gas production (0.096 to 0.01 m3). during the last phase (100 min to 150 min), the production of hydrogen doubled. it went from 0.01 m3 to 0.02 m3 with a slope of 2.10−4m3.min−1. we therefore observe an increase in the evolution slope of the cumulative hydrogen production from 8.8.10−5m3.min−1 to 2×10−4m3.min−1 during the three phases. the instantaneous quantity of hydrogen produced and stored is not constant but therefore increases over time. these results are confirmed by the work of rabih [6]. 4 towanou et al. / j. nig. soc. phys. sci. 5 (2023) 1264 5 figure 7: : evolution of the volume of hydrogen produced as a function of time 3.1.2. variation of the intensity of the current at the level of the electrolyser during the hydrogen production phase, the intensity passing through the electrolyser varied as a function of time. figure 8 illustrates this variation over the duration of the experiment. there is a fluctuation of the intensity of the current over time. it reaches its peak after 60 min of experimentation around 2.4 a. the lowest value of the intensity of the current after the start of the operation is 1.2 a observed after 140 min of hydrogen production. this variation in the intensity of the current at the level of the electrolyser would be due to the intermittence of the sunshine which does not prevent the evolution of the production of hydrogen over time. during the experiment, it was also noticed that a decrease in the intensity of the current lowers the production and that an increase leads to an increase in the production. this observation is true because the volume of hydrogen produced at each moment depends on the quantity of electricity. moreover, in the work of yue et al. [14], the authors state that a higher current defines a higher hydrogen production rate. 3.1.3. production of electricity through the fuel cell during the experiment, the supplied current and the voltage at the terminals of the fuel cell were measured. the data collected made it possible to obtain the graphs in figure 9. the voltage and the current intensity measured from the fuel cell show a variation in the form of a bell over time. these two electrical quantities evolve in an increasing way for a period of about 120 s where they reach their peak evaluated at 1.019 v and 0.25 a. after 120 s, the voltage and the intensity decrease until they cancel out after 480 s. as a result, a strong correlation is observed between these two electrical quantities. the voltage across the terminals of the fuel cell is therefore a linear function of the intensity of the current. these results are consistent with the work of saı̈sset et al. [22] and soldi et al. [21]. 3.1.4. efficiency of water electrolysis and power generation the values of the efficiency of the water electrolysis operation, of the electricity production by the fuel cell and of the figure 8: : current intensity as a function of time figure 9: : electrical quantities measured during the production of electricity by the fuel cell as a function of time, a) voltage produced by the fuel cell and b) intensity produced by the fuel cell whole system are presented in table 1. table 1: hydrogen and electricity production efficiency hydrogen production electricity production fuel cell (electrolyser) (fuel cell) electrolyser yield (%) 85.86 0.0042 0.36 the efficiency values of water electrolysis were estimated at 85.86%; that of the fuel cell at 0.0042% and the one of the electrolyser-fuel cell at 0.36%. these values are quite low, in particular those of the cell and consequently of the electrolyserfuel cell system. this could be due to the different losses recorded during the process of transforming chemical energy into electrical energy. 3.2. discussion the electrolysis of water efficiency values and the production of electricity by the fuel cell are compared with the results obtained in other similar studies encountered in the literature. the different yield values observed are summarized by authors in table 2. the values of the production of hydrogen efficiency by the electrolysis of water proposed in the studies of soldi et al. [21] and laurencelle [24] are between 63 and 85%. these values are 5 towanou et al. / j. nig. soc. phys. sci. 5 (2023) 1264 6 table 2: comparison of hydrogen and electricity production yields authors efficiency of fuel cell overall (electrolyser) (%) efficiency (%) performance (%) yilanci et al.. [7] 0.88-9.7 soldi et al. [21] 68.05-85.02 4.36-4.99 tsakiris [23] 59 23 ogawa et al. [28] 40-55 giddey et al. [30] 35-45 gautam and ikram [26] 50-60 ceran [27] 25-45 pellow et al. [15] 47 30 töpler and lehmann [32] 40 hodges et al. [31] 55 35-39 quite close to that obtained in the present study which is 85.8% and thus confirm our results. in view of these results, the experimental protocol adopted for the realization of the electrolyser can be validated. several other authors such as tsakiris [23], laurencelle [24], cheung et al. [25], gautam and ikram [26], pellow et al. [15], ogawa et al. [28], giddey et al. [30], labbé [29], hoogers [31], töpler and lehmann [32], hodges et al. [33], stolten et al. [34], srinivasan [35] studied the power generation efficiency of the proton exchange membrane fuel cell. the values proposed by these authors vary from 35% to 60% and are much higher than the yield obtained in this study, evaluated at 0.0042%. it is therefore noted that the greatest losses observed are concentrated during the conversion of chemical energy into electrical energy via the fuel cell. they could be due to the fact that the electrolyser cells do not produce hydrogen at the storage pressure. according to the work of yilanci et al. [7], soldi et al. [21], tsakiris [23], laurencelle [24], ceran [27], pellow et al. [15], hodges et al. [33], stolten et al. [34] the overall efficiency of electricity production from the proton exchange fuel cell is between 0.88% and 39%. this yield, even though low, is much higher than the yield of the present study estimated at 0.36% and is due to the very low yield observed at the level of the battery. however, it should be noted that according to achkari and fadar [3], even if the idea of storing energy in hydrogen is not desirable by the authors due to the low yield of the technology, they are convinced that the fuel cell is still likely to play a role in the future due to the large storage potential. research efforts will undoubtedly lead to its large-scale use in the years to come. the system proposed in this study deserves to be improved in order to increase the efficiency of the conversion of chemical energy into electrical energy. this involves, for instance, reviewing the heat exchange between the electrolyser cells and their environment, monitoring the increase in the operating temperature of the electrolyser and of the battery, which is generally a source of malfunction in system and gas storage pressure. 4. conclusion in the present study, electricity was produced by a fuel cell fueled by hydrogen obtained by the electrolysis of water. an experimental protocol was followed both for the realization of the electrolyser and for the production of hydrogen and electricity. the yields of these different operations were evaluated and compared with those proposed in the literature. the main results are as follows: • after 150 min of experimentation, a quantity of 0.02 m3 of hydrogen gas has been produced and the evolution curve of this production follows a simple linear regression; • the voltage variation curve as a function of the intensity of the current measured at the terminals of the fuel cell is a linear function; • the gas and electricity production yields are evaluated at 85.86% and 0.0042% respectively. the overall efficiency of the electrolyser-fuel cell system is estimated at 0.36%. these values except those of the electrolyser are quite low and much lower than those encountered in the literature. in short, a part of the energy lost by renewable energy systems via batteries can be recovered. the low yield obtained shows that it is necessary to improve the whole system, possibly size and design a fuel cell or optimize the storage of the hydrogen produced. significant research and development efforts remain to be provided in order to improve the performance of our system and to identify applications that are well suited to their use. hydrogen, as a storage element and as a fuel, offers a concrete solution to the intermittency of renewable energy sources, energy losses in batteries and the depletion of fossil resources while respecting the environment. acknowledgments the authors thank the bachelor school in renewable energy of the faculty of science and technology (fast) at the university of abomey-calavi (uac) through the ”laboratoire de physique du rayonnement (lpr)” and the abomey-calavi university’s mastercard foundation program for funding our participation to the 3rd german-west african conference on sustainable and renewable energy systems (susres) at the university of kara (togo). references [1] r. benchrifa, a. bennouna & d. zejli, “rôle de l’hydrogène dans le stockage de l’électricité a base des énergies renouvelables”, revue des energies renouvelables 7 (2007) 103. 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[35] s. srinivasan, fuel cells: from fundamentals to applications, new york springer, 2006. 7 j. nig. soc. phys. sci. 5 (2023) 1137 journal of the nigerian society of physical sciences electrohydrodynamics convection in dielectric rotating oldroydian nanofluid in porous medium pushap lata sharmaa, mohini kapaltaa, ashok kumara, deepak bainsa, sumit guptab, pankaj thakurc, adepartment of mathematics & statistics, himachal pradesh university, summer hill, shimla, india brajiv gandhi govt college chaura maidan, shimla, india cfaculty of science and technology, icfai university, baddi, solan, india abstract an electrically conducting nanofluid saturated with a uniform porous medium has been tested to determine how rotation affects thermal convection. utilizing the oldroydian model, which incorporates the specific effects of the electric field, brownian motion, thermophoresis and rheological factors for the distribution of nanoparticles that are top-heavy and bottom-heavy, one may use linear stability theory to ensure stability. analysis and graphical representation of the effects of the ac electric field rayleigh number, taylor number, lewis number, modified diffusivity ratio, concentration rayleigh number and medium porosity are provided for both bottom-heavy and top-heavy distribution. doi:10.46481/jnsps.2023.1231 keywords: convection, dielectric, electric field, nanofluid, oldroydian, porous medium, rotation article history : received: 24 november 2022 received in revised form: 18 january 2023 accepted for publication: 28 january 2023 published: 02 march 2023 c© 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: b. j. falaye 1. introduction nanofluids are fluids that include particles with a diameter of less than 100 nm and can stay suspended in them. choi [1] was the first to coin the term. nanofluid is a suspension of normal nano-size particles such as metals (al, cu), oxides ceramics (al2o3, cuo), metal carbides (s ic), nitrides and carbon nanotubes in an aqueous or non-aqueous dispersion media. because of their unique chemical and physical features, nanofluids are now considered the next-generation heat transfer fluid. nanofluids can be used for a variety of things, including nanocomposites, electrical cooling, bio-medicine and email address: pankaj_thakur15@yahoo.co.in (pankaj thakur ) nanostructure production transportation because of their capabilities. many researchers have investigated the properties of nanofluids as well as potential application scenarios. nanofluids are perfect for heat transfer applications since they have better thermal conductivity. because of their superior thermal conductivity, nanofluids are perfect for transferring heat. the effects of particle size, ph and zeta potential on the thermal conductivity of nanofluids have been the subject of several investigations. in nanofluids with a low concentration of metal oxides, the addition of 4 to 5 percent metal oxides by volume results in an increase in thermal conductivity of between 10 and 20 percent. several models have been used to estimate the thermal conductivity of nanofluids. 1 sharma et al. / j. nig. soc. phys. sci. 5 (2023) 1137 2 the first, which relates to the thermal conductivity of particles and fluids was created by maxwell to ascertain the thermal conductivity of colloidal suspension which is analyzed by nield and kuznetsov [2]. sheu [3] has studied the thermal instability in a porous medium layer saturated with a viscoelastic nanofluid and demonstrated that oscillatory instability is possible in both bottom-heavy and top-heavy nanoparticle distributions. the results indicate a conflict between the thermophoresis, brownian diffusion and viscoelasticity processes, leading to oscillatory rather than stationary modes of convection. sharma et al. [4] have investigated the onset of thermal convection in an oldroydian nanofluid layer saturating a porous medium using a darcy-brinkman model revolving vertically with a uniform angular velocity. yu and choi [5] changed the maxwell model by considering the influence of nanolayers on the electromagnetic field. another appealing strategy for heat transfer development in industrial systems is the use of porous media in nanofluids. porous media are normally saturated, hard-open cells that are often filled with fluid to allow fluid to pass through the voids. porous media improve heat conductivity by increasing the contact area between liquid, solid and nanofluid, hence boosting the efficiency of conventional thermal systems. due to its use in many practical applications such as chemical reactors, heat exchangers and fluid filters, the increase in thermal conductivity of nanofluid with the use of porous media has attracted many researchers for the use of materials with high porosity in the current era for many technological problems. natural convection in an ac/dc electric field was studied by jones [6] and chen et al. [7] for electrically accelerated heat movement in fluids and its applications. convective heat transfer through polarized dielectric liquids was studied by stiles et al. [8]. they found that the convection pattern identified by the electric field is similar to that of the well-known bénard cell convection. in their study of the effects of electro-thermal convection on dielectric rotating fluid, shivakumara et al. [9] discovered that an ac electric field accelerates convection initiation and boosts heat transfer. the dielectric nanofluid may be used for instrument transformers, regulating and converter transformers and other electrical devices. sharma et al. [10] have researched of heat convection in a dielectric rheological nanofluid layer employed an ac electric field. a maxwellian model was used to explain the rheology of the nanofluid and it was shown that for bottom-heavy nanoparticle distributions, the electric field and the stress relaxation parameter destabilize both stationary and oscillatory modes. sharma et al. [11] investigation into thermosolutal convection of an elastic-viscous nanofluid in a porous medium with rotation and magnetic field led them to the conclusion that stationary convection is stabilized by the magnetic field and the taylor number, while stationary convection is destabilized by the solutal rayleigh number, nanoparticle rayleigh number, thermo-nanofluid lewis number and modified diffusivity ratio the rivlin-ericksen fluid issue in a darcy-brinkman porous medium was studied by sharma et al. [12]. thermosolutal convection in a porous medium and its relationship to the rotation was examined by sharma et al. [13]. kumar et al. [14] researched thermosolutal convection in jeffrey nanofluid with porous medium and sharma et al. [15] studied electrohydrodynamics convection in dielectric oldroydian nanofluid layer in porous media. for free-free, rigid-free and rigid-rigid boundaries, poonam et al. [16] looked into the electrohydrodynamic convection in thermal instability of jeffrey nanofluid in porous media. shivakumara et al. [17] investigated on the problem darcy-brinkman model of electrical convection in a porous dielectric fluid. chand et al. [18] examined the convection of an electric field saturated by nanofluid in a porous medium. ramanuja et al. [19] researched mhd swcnt-blood nanofluid flow through porous medium in the presence of viscous dissipation and radiation effects. akinpelu et al. [20] studied hydromagnetic double exothermic chemical reactive flow with convective cooling through a porous medium using bimolecular kinetics. this concise survey of the literature shows that research has been done on the current issue, the initiation of thermal convection. 2. mathematical formulation of problem an infinitely extending electrically conducting horizontal layer of an incompressible rotating non-newtonian oldroydian nanofluid of thickness d is taken under gravity g (0, 0, −g) . the rotating vertical angular velocity is ω(0, 0, ω) . the temperatures t and volumetric fractions of nanoparticles φ are taken to be t0 and φ0 at z = 0 and t1 and φ1 at z = d (t0 > t1 and φ1 > φ0). this dielectric nanofluid layer is dominated by a uniform vertical electric field. figure 1: physical configuration 3. governing equations the conservation equations for mass and momentum using boussinesq approximation are ∇.q d = 0, (1) ρ f ε ( 1 + λ ∂ ∂t ) [ ∂ ∂t + 1 ε q d.∇ ] q d =  [−∇p + ( φρp + (1 −φ) ρ f {1 −β (t − t1)} ) g + fe + 2ρ ε (ω× q d)] ( 1 + λ ∂ ∂t ) − µ k1 ( 1 + λ0 ∂ ∂t ) q d  , (2) where q d, p,ε,λ,λ0,φ,ρ f ,ρp,β,µ and k1 are the darcy velocity, pressure, porosity, relaxation time, retardation time, 2 sharma et al. / j. nig. soc. phys. sci. 5 (2023) 1137 3 nanoparticles volume fraction, the density of the base fluid, the density of nanoparticles, coefficient of volume expansion, coefficient of viscosity and medium permeability respectively. fe is the electrical force given by fe = ρe e − 1 2 e2∇k + 1 2 ( ρ ∂k ∂t e2 ) , where ρe is the density of charge, k is the dielectric constant, e is the electric field. the term ρe e is due to the free charge known as coulomb force. here, the term ρe e is neglected as compared to the term −12 e 2∇k for most dielectric fluids. the modified pressure term is p = p − 1 2 ( ρ ∂k ∂t e2 ) , (3) where p is the hydrodynamical pressure. maxwell equations are ∇. (k e) = 0,∇× e = 0. (4) from equation (4), e can be shown as e = −∇ϕ, (5) where ϕ is a measure of electric potential’s root mean square. it is also assumed that k = k0 [ 1 −γ (t − t1) ] . (6) γ > 0, where, 0 < γ∆t << 1. thus, the modified equations of motion for rotating oldroydian nanofluid saturating a porous medium in the presence of an electric field become ρ f ε ( 1 + λ ∂ ∂t ) [ ∂ ∂t + 1 ε q d.∇ ] q d =  ( 1 + λ ∂ ∂t ) [−∇p + ( φρp + (1 −φ) ρ f {1 −β (t − t1)} ) g − 1 2 (e.e)∇k + 2ρ ε (ω× q d)] − µ k1 ( 1 + λ0 ∂ ∂t ) q d  . (7) the nanoparticles’ equation of continuity is[ ∂ ∂t + 1 ε (q d.∇) ] φ = db∇ 2φ + ( dt t1 ) ∇ 2t. (8) a porous medium’s saturating nanofluid’s heat-energy equation is (ρc)m ∂t ∂t + (ρc) f q d.∇t = km∇ 2t + ε(ρc)p [ db∇φ.∇t + ( dt t1 ) ∇t.∇t ] . (9) the boundary conditions appropriate to the problem are w = 0, ∂ϕ ∂z = 0, t = t0, φ = φ0 at z = 0 w = 0, ∂ϕ ∂z = 0, t = t1, φ = φ1 at z = d } . (10) using the non-dimensional variables  (x∗, y∗, z∗) = (x,y,z)d , t ∗ = t αm σ d2 , q ∗ d = q d d αm , p∗ = pk1 µαm ,φ∗ = φ−φ0 φ1−φ0 ,ϕ∗ = ϕ γe0 ∆t d , t∗ = t−t1t0 −t1 , e ∗ = e γe0 ∆t d , k∗ = kk0 , where σ = (ρc)m(ρc) f and αm = km (ρc) f . the non-dimensional forms of equations (1) and (5) (9) are (asterisk is removed for convenience): ∇.q d = 0, (11) 1 va ( 1 + λ1 ∂ ∂t ) [ 1 σ ∂ ∂t + 1 ε q d.∇ ] q d = ( 1 + λ1 ∂ ∂t ) [−∇p − rnφêz − rmêz + rat êz + ret êz −re ∂ϕ ∂z + √ ta(vêx − uêy)] − ( 1 + λ2 ∂ ∂t ) q d  , (12) 1 σ ∂φ ∂t + 1 ε q d.∇φ = 1 le ∇ 2φ + na le ∇ 2t, (13) ∂t ∂t + q d.∇t = ∇2t + nb le ∇φ.∇t + na nb le ∇t.∇t, (14) e = −∇ϕ, (15) k = [ 1 −γt (t0 − t1) ] , (16) where the non-dimensional parameters are λ1 = λαm σd2 is the deborah number, λ2 = λ0αm σd2 is the strain-retardation time parameter, pr = µ ρ f αm is the prandtl number, dr = k1 d2 is the darcy number, va = εpr dr is the vadasz number, le = αm db is the lewis number, ra = ρ f gβdk1 ( t 0−t1 ) µαm is the thermal darcyrayleigh number, rm = [φ0ρp +(1−φ0 )ρ f ]gdk1 µαm is the basic density rayleigh number, rn = (ρp−ρ f )(φ1−φ0 )gdk1 µαm is the concentration rayleigh number, na = dt ( t 0−t1 ) db t 1 (φ1−φ0 ) is the modified diffusivity ratio, nb = ε(ρc)p (φ1−φ0 ) (ρc) f is the modified particle-density increment, √ ta = 2ωρk1 εµ is taylor number and re = kγ2 e20 (t0−t1 ) 2 k1 d2 µαm is the ac electric rayleigh number. in terms of non-dimensional form, boundary conditions (10) transform to w = 0, ∂ϕ ∂z = 0, t = 1, φ = 0 at z = 0 w = 0, ∂ϕ ∂z = 0, t = 0, φ = 1 at z = 1 } . (17) 4. basic state solution the basic state is stated as{ q d = (u, v, w) = (0, 0, 0) , p = pb (z) , t = tb (z) , φ = φb (z) , k = kb (z) , e = eb (z) , ϕ = ϕb (z) . (18) when there is no motion, equations (13) and (14) require the temperature and the volumetric fraction of nanoparticles to satisfy the equations d2φb dz2 + na d2tb dz2 = 0, (19) d2tb dz2 + nb le dφb dz dtb dz + na nb le dtb dz dtb dz = 0. (20) using the boundary conditions (17), equation (19) can be integrated to give φb (z) = −natb + (1 − na) z + na. (21) 3 sharma et al. / j. nig. soc. phys. sci. 5 (2023) 1137 4 substituting φb from equation (21) into equation (20), we get d2tb dz2 + (1 − n a)nb le dtb dz = 0. (22) equation (22) along with the boundary condition (14) gives the solution as tb = e −(1− n a )nb z/le ( 1 − e−(1−na )nb (1−z)/le 1 − e−(1−na )nb/le ) . (23) the terms of second and higher order in the expansion of an exponential function in equation (23) are neglected as they are small and so one gets the best approximate initial stationary state solutions as{ tb = 1 − z, φb = z, kb = 1 + γ∆t z, eb = e0 γ∆t (1+γ∆t z) , ϕb = − e0 (γ∆t )2 log (1 + γ∆t ) , (24) where, e0 = −γ∆tϕ log(1+γ∆t ) at z = 0. 5. the formulae for perturbations by adding minute disturbances to the state variables, we can gently perturb the initial condition indicated by equation (24) so that  q d = (0, 0, 0) + q ′ d (u ′, v′, w′) , t = tb + t ′, φ = φb + φ ′, p = pb + p′, k = kb + k′, e = eb + e′,ϕ = ϕb + ϕ′. (25) using these perturbations given by equation (25) and neglecting the terms of higher powers and products of perturbations (i.e., applying linear stability theory) in equations (11) (16), the resulting linearized non-dimensional perturbed equations are: (( 1 + λ2 ∂ ∂t ) + 1 σva ( 1 + λ1 ∂ ∂t ) ∂ ∂t )2 ∇ 2 + ta ( 1 + λ1 ∂ ∂t )2 ∂2 ∂z2  w′ =  [( 1 + λ2 ∂ ∂t ) + 1 σva ( 1 + λ1 ∂ ∂t ) ∂ ∂t ] ( 1 + λ1 ∂ ∂t )[ −rn∇2hφ ′ + (ra + re)∇2h t ′ − re∇2h ∂ϕ′ ∂z ]  , (26) 1 σ ∂φ′ ∂t + w′ ε = 1 le ∇ 2φ′ + na le ∇ 2t′, (27) (28) ∂t′ ∂z −∇ 2ϕ′ = 0. (29) the boundary conditions (17) for the infinitesimal perturbations become w′ = 0, ∂ 2 w′ ∂z2 = 0, ∂ϕ ′ ∂z = 0, t ′ = 1, φ′ = 0 at z = 0 w′ = 0, ∂ 2 w′ ∂z2 = 0, ∂ϕ ′ ∂z = 0, t ′ = 0, φ′ = 1 at z = 1  . (30) 6. the normal mode analysis for the system of equations (26) (29), the analysis can be made in terms of two-dimensional periodic waves of assigned wave numbers. thus, we assign the quantities describing the dependence on x, y, t of the form exp ( ikx x + iky x + st ) , where kx and ky are the wave numbers in x-direction and ydirection, respectively and a2 = k2x + k 2 x is the resultant wave number, s is the growth rate, which is a complex constant. the above consideration allows us to suppose (w′, t′,φ′,ϕ′) = (w, θ, φ, ψ ) exp ( ikx x + ikyy + st ) . (31) using expression (31), the equations (26) (29), reduces to ( (1 + λ2 s) + 1 σva (1 + λ1 s) s )2 ( d2 − a2 ) + ta(1 + λ1 s) 2 d2  w =  ( (1 + λ2 s) + 1 σva (1 + λ1 s) s ) (1 + λ1 s)( a2rnφ − a2 (ra + re) θ + a2re dψ )  , (32) s σ φ + w ε = 1 le ( d2 − a2 ) φ + na le ( d2 − a2 ) θ, (33) sθ − w = ( d2 − a2 ) θ + nb le (dθ − dφ) − 2na nb le dθ, (34) dθ − ( d2 − a2 ) ψ = 0, (35) where a = √ k2x + k2y. the equations (30) for a free-free boundary are: w = d2w = θ = φ = dψ = 0 at z = 0 and z = 1. (36) therefore{ w = a1sinπz, θ = a2sinπz, φ = a3sinπz, ψ = a4cosπz, (37) where a1, a2, a3 and a4 are the constants. substituting (37) in equations (32) (35) and using the boundary conditions (36), we get a b c d 1 −j − s 0 0 1 ε na j le j le + s σ 0 0 −π 0 −j   a1 a2 a3 a4  =  0 0 0 0  , (38) where a = m2 j + π2ta(1 + λ1 s)2, b = −a2 (1 + λ1 s) m(ra + re), c = a2(1 + λ1 s) mrn, d = −a2π(1 + λ1 s) mre, j = ( π2 + a2 ) , and m = ( (1 + λ2 s) + 1 σva (1 + λ1 s) s ) . where, the thermal darcy-rayleigh number, ra =  − a2 π2 +a2 re − σle σ(π2 +a2)+sle [ π2 +a2 +s ε + (π2 +a2)na le ] rn + π2 +a2 +s a2 [ π 2 +a2 1+λ1 s ( (1 + λ2 s) + 1 σva (1 + λ1 s) s ) + σvaπ2 σva (1+λ2 s)+(1+λ1 s)s (1 + λ1 s) ta]  . (39) 4 sharma et al. / j. nig. soc. phys. sci. 5 (2023) 1137 5 7. stationary convection for the validity of the principle of exchange of stabilities (i.e., steady case), we have s = 0 ( s = r + iω = 0 ⇒ r = ω = 0 ) at the marginal stability. putting s = 0 in equation (39), we get the thermal darcyrayleigh number at which marginally stable steady mode exists, as ra =  (π2 +a2)2 a2 − a2 π2 +a2 re − ( le ε + na ) rn + π2(π2 +a2) a2 ta  , (40) which expresses the stationary thermal darcy-rayleigh number ra as a function of the dimensionless wave number a, electric rayleigh number re, taylor number ta, nanofluid lewis number le, modified diffusivity ratio na, concentration rayleigh number rn and medium porosity ε. it is clear from the equation (40) that ra is independent of stress relaxation time λ1, strain retardation time λ1 for stationary modes since these vanish with the vanishing of s (growth rate). the minimum value of ra is obtained by putting ∂ra ∂a2 = 0 and which on simplification implies that{ (ac) 8 + 2π2(ac) 6 − ( π2re + π2ta ) (ac) 4 −2π2 (1 + ta) (ac) 2 −π2 (1 + ta) } = 0. (41) therefore, the critical wave number ac shows a substantial increase when the electric rayleigh number re increases and is independent of nanoparticles. to obtain the critical wave number we substitute the electric field i.e., re = 0 and we get ac 2 = π2 √ 1 + ta, (42) which is the critical wave number for stationary rayleigh number in the absence of ac electric rayleigh number re. 8. results and discussion to study re, ta, le, na, rn,ε on the stationary convection, we examine the behavior ∂ra ∂re , ∂ra ∂ta , ∂ra ∂le , ∂ra ∂na , ∂ra ∂rn and ∂ra ∂ε analytically. from equation (40) we obtain ∂ra ∂re = − a2( π2 + a2 ), (43) which is always negative for all wave numbers. thus, ac electric field has destabilizing effect for both bottom-heavy and topheavy distribution. equation (40) gives ∂ra ∂ta = π2 ( π2 + a2 ) a2 , (44) which is always positive for all wave numbers. thus, the taylor number has stabilizing effect for both bottom-heavy and topheavy distribution in the system. equation (40) further yields ∂ra ∂le = − rn ε , ∂ra ∂na = −rn. (45) it is clear from equation (45), the nanofluid lewis number le and the modified diffusivity ratio na enhance the stationary convection if rn < 0 and postpone the stationary convection if rn > 0. equation (40) also depicts that ∂ra ∂rn = − ( le ε + na ) , (46) which is always negative for ( le ε + na ) > 0. the nanoparticle rayleigh number postpones the stationary convection for both bottom-heavy and top-heavy configurations. ∂ra ∂ε = lern ε2 . (47) if rn < 0, thus the medium porosity delays the stationary convection for bottom-heavy configuration and if rn > 0 the medium porosity advances the stationary convection. 9. numerical discussion the variation of thermal darcy-rayleigh number with respect to wave number has been plotted using equation (40) for stationary case, whereas the experimental values and the fixed permissible values of the dimensionless parameters are re = 100, ta = 100, le = 200, na = −5 and na = 5, rn = −0.1 and rn = 0.1 and ε = 0.6 . the stationary thermal rayleigh number does not depend upon stress relaxation time and strain retardation time, since it vanishes with the vanishing of s (growth rate). thus, the oldroydian nanofluid acts like a newtonian nanofluid. figures 2 and 3 show the variation of ra for stationary convection with respect to the non-dimensional wave number for three different values of re = 100, 300, 500 for bottom-heavy and top-heavy distribution and fixed permissible values. the graph indicates that the value of ra drops as re rises, indicating that re has a destabilizing influence on both bottom-heavy and top-heavy configurations. figure 2: variations of ra for distinct values of the re for bottom-heavy distribution figures 4 and 5 show that ra increases with an increase in ta which implies that ta has a stabilizing effect on stationary 5 sharma et al. / j. nig. soc. phys. sci. 5 (2023) 1137 6 figure 3: variations of ra for distinct values of re for top-heavy distribution figure 4: variations of ra for distinct values of ta for bottom-heavy arrangement figure 5: variations of ra for distinct values of ta for top-heavy arrangement convection for both bottom-heavy and top-heavy pattern of the system. from figures 6 and 7, it is found from the graphs that with an increase in the values of le, ra increases for bottom-heavy distribution, whereas ra decreases for top-heavy configuration with increase in the values of the lewis number. hence, ra stabilizes the bottom-heavy arrangement and destabilizes the top-heavy arrangement. figures 8 and 9 show that ra increases slightly with increase in na for bottom-heavy arrangement and ra decrease slightly figure 6: variations of stationary ra for different values of le for bottom-heavy distribution figure 7: variations of ra for different values of le for top-heavy distribution with increase in the modified diffusivity ratio for top-heavy arrangement. hence na stabilize the system for bottom-heavy arrangement and destabilize the system for top-heavy arrangement. figure 8: variations of ra for different values of na for bottom-heavy arrangement figures 10 and 11 show the variation of ra for stationary convection with respect to the non-dimensional wave number for different values of rn. it is depicted from the graphs that for the cases of bottom-heavy and top-heavy configuration, ra decreases with the increase in rn which causes the destabilizing 6 sharma et al. / j. nig. soc. phys. sci. 5 (2023) 1137 7 figure 9: variations of ra for different values of na for top-heavy arrangement effect on the system. figure 10: variations of ra for different values of rn for bottom-heavy arrangement figure 11: variations of ra for different values of rn for top-heavy arrangement the effect of the medium porosity ε on ra is displayed in figures 12 and 13. it is found that with an increase in the ε, ra decreases and increases, respectively for bottom-heavy and topheavy configurations. thus, a porous medium destabilizes the system for the bottom-heavy pattern and stabilizes the system for the top-heavy pattern. figure 12: variations of ra for three different values of ε bottom-heavy distribution figure 13: variations of ra for three different values of ε top-heavy distribution 10. conclusions the effect of rotation on thermal convection in an electrically conducting nanofluid saturated by porous medium has been studied using linear stability theory by employing an oldroydian model which incorporates the effects of the electric field, brownian motion, thermophoresis and rheological parameters for bottom-heavy and top-heavy distribution of nanoparticles. the conclusions of the present study are given below: 1. ac electric field has destabilizing for both bottom-heavy and top-heavy distribution of nanoparticles. 2. the taylor number ta has stabilizing for both bottomheavy and top-heavy distribution of nanoparticles. 3. the effect of lewis number (non-dimensional parameter accounting for brownian motion parameter db) tends to stabilize the stationary convection for bottom-heavy distribution and destabilizes for top-heavy configuration. 4. modified diffusivity ratio has stabilized the system for bottom-heavy and destabilized the system for top-heavy configuration. 5. the concentration rayleigh number postpones the stationary convection for both bottom-heavy and top-heavy distribution. 7 sharma et al. / j. nig. soc. phys. sci. 5 (2023) 1137 8 6. medium porosity has destabilizing effect for bottomheavy distribution and stabilizing effect for top-heavy distribution on stationary convection. acknowledgments the third author gratefully acknowledges the financial assistance of csir-hrdg for jrf. references [1] s. u. s. choi, “enhancing thermal conductivity of fluids with nanoparticles”, siginer, d. a., wang, h. p. 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[7] x. chen, j. cheng & x. yin, “advances and applications of electro hydrodynamics”, chinese science bulletin 48 (2003) 1055. [8] p. j. stiles, f. lin & p. j. blennerhassett, convective heat transfer through polarized dielectric liquids, physics of fluids, 5 (1993) 3273. [9] i. s. shivakumara, j. lee, k. vajravelu & m. akkanagamma, “electrothermal convection in a rotating dielectric fluid layer: effect of velocity and temperature boundary conditions”, international journal of heat and mass transfer 55 (2012) 2984. [10] v. sharma, a. chowdhary & u. gupta, “electrothermal convection in dielectric maxwellian nanofluid layer”, journal of applied fluid mechanics11, (2018) 765. [11] p. l. sharma, deepak & a. kumar, “effects of rotation and magnetic field on thermosolutal convection in elastico-viscous walters’ (model b’) nanofluid with porous medium”, stochastic modeling & applications 26 (2022) 21. [12] p. l. sharma, a. kumar & g. c. rana, “on the principle of exchange of stabilities in a darcy brinkman porous medium for a rivlin-ericksen fluid permeated with suspended particles using positive operator method”, stochastic modeling & applications, 26 (2022) 47. [13] p. l. sharma, deepak, a. kumar & p. thakur, “effects of rotation on thermosolutal convection in jeffrey nanofluid with porous medium”, structural integrity and life (2022). [14] a. kumar, p. l. sharma, deepak & p. thakur, “thermosolutal convection in jeffrey nanofluid with porous medium”, structural integrity and life (2022). [15] p. l. sharma, m. kapalta, deepak, a. kumar, v. sharma & p. thakur, “electrohydrodynamics convection in dielectric oldroydian nanofluid layer in a porous medium”, structural integrity and life (2022). [16] p. k. gautam, g. c. rana & h. saxena, “stationary convection in the electrohydrodynamics thermal instability of jeffrey nanofluid layer saturating a porous medium: free-free, rigid-free and rigid-rigid”, journal of porous media 23 (2020) 1043. [17] i. s. shivakumara, n. rudraiah, j. lee & k. hemalatha, “the onset of darcy-brinkman electroconvection in a dielectric fluid saturated porous layer”, transport in porous media 90 (2011) 509. [18] r. chand, g. c. rana & d. yadav, “electrothermo convection in a porous medium saturated by nanofluid”, journal of applied fluid mechanics 9 (2016) 1081. [19] m. ramanuja, j. kavitha, a.sudhakar & n. radhika, “study of mhd swcnt-blood nanofluid flow in presence of viscous dissipation and radiation effects through porous medium”, journal of the nigerian society of physical sciences 5 (2023) 1054. [20] f.o. akinpelu, r.a. oderinu & a.d. ohaegbue, “analysis of hydromagnetic double exothermic chemical reactive flow with convective cooling through a porous medium under bimolecular kinetics”. journal of the nigerian society of physical sciences 4 (2022) 130. 8 j. nig. soc. phys. sci. 5 (2023) 1512 journal of the nigerian society of physical sciences simulation of the movement of groundwater in an aquifer suha ibrahim salih al-ali, nihad jalal kadhem al-awsi computer science and mathematics department, college of computer science and mathematics, tikrit university, tikrit, iraq abstract this study investigates the impact of extracting fresh water from areas where salt water and fresh water meet in tropical regions. traditionally, fresh water is expected to be found above salt water in the ocean or underground. to carry out the investigation, green’s function method is used, and a numerical chart is presented that includes an equation derived from green’s ii matching. the study computes the shape of the interface during water withdrawal and flows through the basins and sources of the line. in addition, this study obtains an analytical solution to the linear problem for the non-withdrawal scenario. finally, the study identifies the maximum rate of water withdrawal before the initial breakthrough of salt water for different density ratios. doi:10.46481/jnsps.2023.1512 keywords: free surface, tropical island, porous medium, freshwater lens, island tub, green’s function. article history : received: 21 april 2023 received in revised form: 19 may 2023 accepted for publication: 22 may 2023 published: 22 june 2023 c© 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: b. j. falaye 1. introduction the provision of drinking water in tropical islands often relies on a freshwater lens trapped in the soil beneath the surface. however, this supply is threatened by various factors, such as the interface between freshwater and saltwater layers, surface channels, and pumping practices. to ensure continued access to freshwater, it is crucial to conserve the freshwater lens and use efficient extraction practices to prevent saltwater intrusion and depletion of the freshwater supply [1]. figure (1) provides a visual representation of this process. according to the united nations, around 2.2 billion people lack access to safe drinking water, and this number is expected to increase due to population growth and climate change [2]. small island developing states email address: suhaibrahim3@tu.edu.iq (suha ibrahim salih al-ali ) (sids) are particularly vulnerable to water scarcity because of their limited resources, geographic isolation, and susceptibility to natural disasters [3]. climate change exacerbates the water scarcity crisis in sids by increasing the frequency and intensity of extreme weather events, causing saltwater intrusion, and reducing precipitation [4]. for example, the pacific island nation of tuvalu faces severe water shortages due to droughts and over-extraction from the freshwater lens, which has led to soil and groundwater salinization [5]. some island communities are responding to these difficulties by implementing creative solutions such as rainwater collection, desalination, and water conservation techniques. these solutions, however, necessitate money and technical skill, both of which may be limited in sids [6, 7]. furthermore, policies that promote water security and sustainability in these vulnerable areas are required [8, 9]. the interface between low-density top liquids and higherdensity liquids in a two-layer water object changes shape when 1 al-ali & al-awsi / j. nig. soc. phys. sci. 5 (2023) 1512 2 figure 1. a diagram indicate the focus of research-pulling freshwater over the underground seawater on an island liquid is preferentially removed from one layer, according to research [10]. one of the primary goals of this research is to validate the deformed interface shape. the goal of this study is to look into the variables that could cause the interface to advance towards the extraction port, resulting in liquid extraction from both levels and mixing. the research focuses on a single island with defined top left and right bounds that contains a freshwater lens atop a saltwater layer [11]. to address this issue, academics employ the darcy act as a foundation, which contains boundary conditions on both sides and the island’s bottom. this study is in the field of hydrology, and it focuses on saltwater intrusion in coastal aquifers, which can cause a variety of environmental and economic problems [12]. the authors assume the presence of a freshwater lens above a saltwater layer, which is a common condition in realworld scenarios like aruba. the findings of the study can be used to develop policies and management methods to reduce saltwater intrusion in coastal aquifers [13]. section 3 of the study employs an analytical technique [14] to address the issue raised in section 2. the fourier series and its properties are used as the principal tool for problem analysis [15, 16]. it was explained in detail how the series’ resulting factors are implemented in matlab to generate a solution. the fourier series is a mathematical approach that uses sine and cosine functions to describe periodic functions as a sum of the sin and cos functions. it is frequently employed in a variety of domains, including signal processing, communication systems, and control theory [17]. the fourier series’ handling qualities are critical to solving the problem stated in section 2, as they allow the authors to analyse the system’s periodic nature. 2. formulation of free surface case problem on the island, there is a porous medium with dimensions of -l < x < l , which contains fluid of varying densities, specifically two layers of freshwater and saltwater. the properties of these fluids can be described using the following equations. ωi = pi + ρigy, y = ξ(x), (1) where i = 1, 2 and y = ξ (x). equation (1) represents the sum of total pressures, which remains constant even when gravity changes. this is called the compression of heads in the two layers. the variables, ρi andpi represent the density and pressure of each layer of fluid, respectively. the interface between the two fluids is denoted by y = ξ (x), and the surface of the less dense fluid is assumed to be in contact with air. since the air is assumed to be in a stable state, the pressure at the horizontal surface of the less dense fluid must be equal to the atmospheric pressure, which is denoted by p1 = 0. additionally, we assume that there is no flow through the fluid surface interface and no flow in the lower layer, i.e., ωi = ρ1gy. (2) when dealing with a non-persistent pressure situation, it is physically impossible to obtain a fixed solution. thus, the pressure along the interface of two fluid layers must match, specifically at the point where y = ξ (x). this means that the pressures of both fluid layers, denoted by p1 and p2, must be equal i.e., p1 = p2, (3) at this interface point. this condition ensures that the fluids are in hydrostatic equilibrium and that there is no net force acting on the interface. assuming no fluid flow across the surface interface and in the lower layer (i.e., ω2 = 0) and with y = ξ (x), we can obtain: p2 = −ρ2gy. (4) by matching the pressures at the fluid interface, we can conclude the following: ω1 = (ρ1 −ρ2) gy, (5) and by dividing (5) by ρ1g to be ω1 = ρ1gzω1 we will have the following equation: ω1 = (1 −ψ). (6) in order to simulate the process of extracting liquids from a freshwater lens on a tropical island, a tub was positioned at coordinates (xs, ys). to achieve this, we applied the following formula to ω1: ω1 → m 4π ln[(x−xs) 2 + (y + ys) 2]. (7) we then applied the equation ω1 = p1gzω1, taking into account that µ = zp1 g as (x, y) approaches (xs, ys). ω1 → µ 4π ln[(x−xs) 2 + (y + ys) 2]. (8) assuming that the tub pressure force and density ratio between the two layers are represented by the balanced ratios of l and ψ, and the island’s aspect ratio is represented by the balanced ratio of µ, we can rewrite the given equations as follows: using the expression ∇ω1 = ρ1 gz z ∇ω1, where q = −κ∇ω1 represents 2 al-ali & al-awsi / j. nig. soc. phys. sci. 5 (2023) 1512 3 the speed, we can use darcy’s law to find the relation between q, z, and ∇ω1: q z t = −κρ1g∇ω1. (9) since t = z ρ1 g , we can simplify equation (9) as follows: q = −∇ω1. (10) equation (10) shows that q is directly proportional to −∇ω1. the parameters l, ψ, and µ have balanced ratios that represent the tub pressure force, the density ratio between the two layers, and the island’s aspect ratio, respectively. 3. freshwater lens on an island with no-withdrawal to solve the linear problem, laplace’s equation is used with the conditions specified in equation ∇ 2ω1 = ∂2ω1 ∂x2 + ∂2ω1 ∂y2 = 0 −l < x < l, ζ (x) < y < 1, (11) which is subject to a range of conditions:ω1 = 1, y = 1 −l < x < lω1 = y, x = l 0 < y < 1. we assume ω1 = ω and use the formula given in equation ω (x, y) = y+ n∑ k=0 s k sinh [( k l ) π (y − 1) ] sin [( k l ) πk ] .(12) and equation ω = (1 −ψ) y = 1 y = ζ (x)ωy − ζ′ (x) ωx = 0, (13) defines additional conditions that must be met, including ω = (1 −ψ)y = 1 and ωy = 0 at y = 0, where we take a small value of ζ assume a large value of ψ. ωy = 1+ n∑ k=0 s k ( k l ) π cosh [( k l ) π (y − 1) ] sin [( k l ) πx ] .(14) at y = 0, equation (14) becomes: −1 = n∑ k=0 s k ( k l ) π cosh ( − k l ) π sin [( k l ) πx ] . (15) to apply orthogonal, we multiply both sides of (15) by sin ( j l ) πx, integrate over the range−l to l, and get: ∫ l −l sin ( j l ) πx d x = ∫ l −l n∑ k=0 −s k ( k l ) π cosh ( − k l ) π sin2 ( j l ) πx d x, (16) assume that ak = s k ( k l ) π cosh ( − k l ) π, i.e. s k = 2[(−1)k − 1] kπ . from which we can conclude that: ak = 2l[(−1)k − 1] (kπ)2 cosh( kl )π . (17) therefore, the fourier series approximation is: ω1 (x, y) = y + n∑ k=0 s k [( k l ) π (y − 1) ] sin [( k l ) πk ] . (18) using (13) and evaluating it at y = 0, we get an approximation of ζ = ω1−ψ : ζ = ω 1 −ψ  n∑ k=0 s k shin ( k l ) π sin [( k l ) πx ] . (19) the linear island problem involves the interface between freshwater and saltwater, and the depth of this interface is determined by the density rate. the ghyben-herzberg relation, which is based on hydrological principles, provides a formula for determining the depth of the interface in a system that is in a constant state of balance. the depth of the interface, according to this relationship, is proportional to the ratio of the freshwater head to the total head, where the total head is the sum of the freshwater head and the depth of the interface below sea level [18]. therefore, if the freshwater head increases, the depth of the interface will also increase, and if the freshwater head decreases, the depth of the interface will decrease as well. the resulting approximation of ζ is given in figure (2) shows the resulting approximation of ζ for an island with width l=100 and a density ratio of 1.01. the measurements were conducted with varying intensity ratios, and it was discovered that the best measurement was achieved at a density value of ψ = 1.01 with n = 300 points. to analyse the interface between freshwater and saltwafigure 2. a diagram determine w-plane and z-plane for free surface condition ter in the linear island problem, researchers use the fourier series approximation, a mathematical technique that breaks down 3 al-ali & al-awsi / j. nig. soc. phys. sci. 5 (2023) 1512 4 complex functions into simpler functions that can be more easily studied. according to the fourier series approximation, the depth of the interface is inversely proportional to the density rate, which means that as the density rate increases, the interface depth decreases, and vice versa. the ghyben-herzberg relation, which states that ζ = ω1−ψ , also supports this relationship between the interface depth and the density rate. in a system in constant balance, when freshwater is present above the interface, the depth of the interface above sea level is equal to the depth of the interface below sea level. both the fourier series approximation and the ghyben-herzberg relation are crucial tools for understanding the interface between freshwater and saltwater in the linear island problem [19, 30]. these principles have practical applications in fields such as hydrology and groundwater management, where researchers can use them to better understand groundwater systems and make informed decisions about resource management and conservation [20]. 3.1. integration of the free surface problem equation to solve a problem numerically, we need to find an appropriate solution that satisfies the governing equation and boundary conditions. in the case of laplace’s equation, we use green’s function f to find a single solution that meets the necessary boundary requirements for the area of concern [21]. green’s function f can be derived using the free surface condition, which is a solution to the equation: ∇ 2 f ( x, y, x0, y0 ) = δ (x − x0, y − y0) , (20) where the function is subject to the boundary conditions: f (±l, y, x0, y0) = 0, 0 < y < 1. (21) f (x, 1, x0, y0) = 0, −l < y < l. (22) to solve this equation using integral equations or finite element methods, we only need to find a solution for on the boundary. this is achieved by using green’s second identity and applying the boundary clauses of equations (21) and (22), which ensures that the integration line of remains on the boundary [22]. the appropriate green’s function f for this problem needs to satisfy the conditions of equations (20), (21), and (22) [23]. one way to find this function is by using the techniques of portfolio conversions of angles in composite variables and applying the solution on the w-plane to the physical (real) plane. we can consider logarithms to be individual at w = w0, where w0 = u0 − iv0 and w̄0 = u0 − iv0. we need a function that evaluates to zero on the real axis, and so we test: f = re [ln (w − w0) − ln(w − w̄0)] . (23) there are different methods to find the appropriate green’s function f for a particular problem, depending on the boundary conditions and the governing equation. some examples of these methods include the method of images, separation of variables, and the method of integral equations. f (z) = a ∫ z 0 (τ− x1) 2l1 (τ− x2) 2l2 . . . (τ− xn) 2ln dτ+c.(24) to convert from the w-plane to the z-plane for each li as shown in figure (3), the conversion should be applied to the inner corners of the polygon. functions of form (24) were used to transform the real axis into a polygonal path. beginning with the assumption that w = 1 corresponds to z = i + l, the appropriate values were offset in (23) to obtain the desired outcome. figure 3. diagram determine w-plane and z-plane for free surface condition z = a ∫ w 1 (w − 1) −1 2 (w + 1) −1 2 dw + i + l = a ∫ w 1 (w − 1) −1 2 dw + i + l. (25) and by using the definitions cosh2θ− sinh2θ = 1 we obtain z = a ∫ w 1 (sinh2θ) −1 2 sinh θ dθ + i + l. (26) to clarify the given expression, let us rewrite it as follows: let w = cosh [ z−(i+l) a ] . since cosh(d) cannot be zero unless d is complex, we can use cosh y = cos iy and set cos (−ila ) = 0, which yields a = −il2π . using this, we can rewrite w as: w = cosh [ z − (i + l) −i2lπ ] = cosh [ π (z − i) 2l − π 2 ] = cos i [ π (z − i) 2l − π 2 ] = cos i ( z − π 2 ) = sin z, (27) and when z = x + iy we will obtain w = sin πx 2l + cosh π(y − 1) 2l +i sinh π(y − 1) 2l cos πx 2l . (28) using the schwarz-christoffel transformation, we can obtain the following function: [missing function]. f = ln ∣∣∣∣∣ w − w0(w − w̄0 ∣∣∣∣∣ , (29) and now let us say that f = sin πx 2l cosh π(y − 1) 2l , (30) g = sinh π(y − 1) 2l cos πx 2l , (31) 4 al-ali & al-awsi / j. nig. soc. phys. sci. 5 (2023) 1512 5 we will obtain a w − w0 w − w̄0 = ( f − f0) + i(g − g0) ( f − f0) + i(g + g0) , (32) and when you take the real part of f, we find that (25) simplified to the follows: f = 1 2 ln ( f − f0) 2 + (g − g)2 ( f − f0) 2 + (g + g)2 . (33) 3.2. derivation of the free surface problem equation we will now derive the complementary equation for the undefined interface, which needs to satisfy the following equation, as mentioned earlier: ∇ 2ω1 = ∂2ω1 ∂x2 + ∂2ω1 ∂y2 = 0 −l < x < l, ζ (x) < y < 1.(34) these equations are subject to the following boundary conditions: f (x) ω1 = 1, y = 1,−l < x < lω1 = y, x = ±l, 0 < y < 1. (35) we also impose the condition that the pressure must match along the interface between saltwater and freshwater where y = ζ (x). furthermore, we assume that there is no flow through the interface, leading to the following conditions: ∂ω1 ∂n ,−l < x < l ω1 = y, x = ±l, 0 < y < 1 (36) when considering fluid dynamics, it is important to keep in mind the effect of negative pressure which results from the pulling of liquids. to address this issue, green’s second identity is commonly used as a tool in solving related equations [24, 225]. by utilizing the condition that the laplacian of a function ∇2ω1 is equal to zero (∇2ω1 = 0), we can simplify the equation and arrive at the expression below: ω1 → µ 2π ln √ (x − xs) 2 + (y − ys) 2. (37) here, f is a function that satisfies laplace’s equation, i.e., ∇2 f = 0, except at a specific point (x0, y0). along the free surface, ∂ω/∂n = 0 and similarly, along the other three boundaries (top, right side, and left side), f = 0. as a result, the second term in equation (1) falls away, leading to:∫ γ ω ∂f ∂n −ω ∂ω ∂n = 0. (38) however, we must also consider what happens along the boundaries s1, se1 se2 , wheres1 is the limit along the boundary and the top, se1 is the ring around the source (i.e., where the liquid passes through), and se2 is the ring around a single point on the surface. therefore, we obtain:∫ s 1 ω ∂f ∂n d s = ∫ se1 f ∂ω ∂n d s = ∫ se2 ω ∂f ∂n d s = 0. (39) assuming that a single point is enclosed within the ring se1 , which causes all the liquid to pass through it, we can consider its specific behaviour. ω → 1 4π ln [ ( fs − f0) 2 + i(gs − g0) 2 ( fs − f0) 2 + i(gs + g0) 2 ] . (40) we start by considering a function of a point (x0, y0). now, let’s consider ∂ω ∂n as an integral of se1 . this integral represents the speed of the liquid out of the tub. we can express this as follows:∫ s 1 −f ∂ω ∂n d s = −f ∫ se1 ∂ω ∂n d s = f ∫ se2 µ 2π . 1 r d s = fµ 2π ∫ 0 2π r. 1 r dθ = fµ, (41) where r = [(x−xs)2 + (y−ys)2] 1 2 is the distance from the source point (xs, ys) to the point (x, y). in other words, the flow from the tub has force µ, so the contribution from the integration of se1 is µf(xs, ys). therefore, we can write: µ 4π ln [ ( f0 − fs) 2 + (g0 − gs) 2 ( f0 − fs) 2 + (g0 + gs) 2 ] + ∫ s1 ω ∂f ∂n d s = ∫ se2 f ∂ω ∂n d s = 0, (42) where ( fs, gs) is the location of the source point, and ( f0, g0) is the location of the point at which we are evaluating the function. we can think of the integral along the se1 line as the integration of a function with a constantly changing value along this line where (x, y) → (x0, y0). ∫ ∂f ∂n d s is ”the flow” of ”the source” due to the effect of f. since the loop around the tub point (x0, y0) is a semicircle, the flow volume due to the source the tub f is q = π. thus, the second integration in (42) could be ωπ(x0, y0). therefore, we can write: µ 4π ln [ ( f0 − fs) 2 + (g0 − gs) 2 ( f0 − fs) 2 + (g0 + gs) 2 ] + ∫ s1 ω ∂f ∂n d s −ωπ(x0, y0) = 0. (43) note that there is a single point of integration on s1 such as (x, y) → (x0, y0), which is the surface point. to deal with this single point, we extract from the method of collecting and subtracting ω0 under integration, giving us the following formula: µ 4π ln [ ( f0 − fs) 2 + (g0 − gs) 2 ( f0 − fs) 2 + (g0 + gs) 2 ] −ωπ(x0, y0) + ∫ s1 (ω−ω0) ∂f ∂n d s + ω0 ∫ se1 ∂f ∂n d s = 0. (44) 4. outcomes and discussion the study examined the impact of density ratios on the formation of the interface and depth of the freshwater lens in free 5 al-ali & al-awsi / j. nig. soc. phys. sci. 5 (2023) 1512 6 surface models. it used both analytical and numerical methods to solve the problem of individual free surfaces. the fourier series and its orthogonal properties were employed to calculate the interface without the need for pumping or pulling on the island. the study also explored the impact of the presence or absence of a source/tub on the island on the maximum pulling rates at different density ratios. with no statistically significant difference between the analytical and numerical solutions, the results demonstrated that the lowest intensity situation resulted in the greatest pulling rate. the findings were consistent with previous research [26, 27]. finally, the study assessed the efficiency of the numerical chart and found that the execution of the program with a low n-value produced results that were just as accurate as those with a high n-value. the research provides valuable insights into the impact of density ratios on free surface models and offers practical solutions for calculating interfaces and freshwater lens depth. the study’s methodology and results can be used in various applications, including environmental management and coastal engineering. 5. conclusion the study aimed to examine the impact of withdrawing water from the freshwater layer of an island that has fixed boundaries at its bottom, left, and right. the unknown interface between two layers of fluids of different densities on an island with consistent boundaries was calculated by integrating relevant parameters such as the island’s pulling rate and density ratio. the study used an analytical approach based on the fourier series to calculate a solution to the linear problem, and it was found to provide a good solution to the non-linear problem formulated. the height of the calculated interface through the analytical approach was consistent with the height predicted by the ghyben-herzberg relation [29, 30, 31], which describes the equilibrium interface between freshwater and saltwater in a coastal aquifer system. interestingly, the study also found that there is a maximum pulling rate for different density ratios after which fixed solutions cease to exist. these findings are consistent with the results of previous studies conducted on similar systems [32]. the study highlights the importance of understanding the complex hydrological processes that occur in coastal aquifer systems and the potential impact of human activities on these systems. references [1] w. li, x. chen, l. xie, z. liu & x. xiong, “ reply to comments on: li et al.(2019)” bioelectrochemical systems for groundwater remediation: “the development trend and research front revealed by bibliometric analysis”, water 11 (2020) 1532. https://doi.org/10.3390/w12061603. 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[32] w. b. ghyben, nota in verband met de voorgenomen putboring nabij amsterdam, tijdschrift van let koninklijk inst. van ing., 1888. 6 j. nig. soc. phys. sci. 4 (2022) 1021 journal of the nigerian society of physical sciences system of non-linear volterra integral equations in a direct-sum of hilbert spaces jabar s. hassana,∗, haider a. majeedb, ghassan ezzulddin arifb a department of mathematics, college of science, salahaddin university erbil / iraq bdepartment of mathematics, tikrit university, college of education for pure sciences, tikrit/ iraq abstract we use the contraction mapping theorem to present the existence and uniqueness of solutions in a short time to a system of non-linear volterra integral equations in a certain type of direct-sum h[a, b] of a hilbert space v [a, b]. we extend the local existence and uniqueness of solutions to the global existence and uniqueness of solutions to the proposed problem. because the kernel function is a transcendental function in h[a, b] on the interval [a, b], the results are novel and very important in numerical approximation. doi:10.46481/jnsps.2022.1021 keywords: system of non-linear integral equations, reproducing kernel hilbert spaces, fixed point theorem article history : received: 20 august 2022 received in revised form: 12 september 2022 accepted for publication: 13 september 2022 published: 01 october 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: b. j. falaye 1. introduction non-linear volterra integral equations have many applications in several fields, such as physics, chemistry, biology, and engineering. for example, particle transport problems in astrophysics theory, electrostatics, potential theory, mathematical problems of radiative steady state, heat transfer problems, and many other mathematical modeling are described by volttera integral equations [1-9]. in this paper, we introduce the following system of non-linear volterra integral equations for t ∈ [a, b]: f (t) = α(t) + ∫ t a f(t, s, f (s), g(s))d s, (1) ∗corresponding author tel. no: email address: jabar.hassan@su.edu.krd (jabar s. hassan ) g(t) = β(t) + ∫ t a g(t, s, f (s), g(s))d s, (2) where α,β ∈ h[a, b] is a direct sum of reproducing kernel hilbert space v [a, b] consisting of those absolutely continuous functions whose derivative is square-integrable on [a, b]. f and g are given functions that satisfy fixed regularity conditions. f and g are unknown functions that need to be determined. recently, reproducing kernel hilbert space methods have been widely studied by many researchers to solve linear and non-linear problems such as partial and ordinary differential equations, as well as integral, fractional, and integral differential equations [4,7,9]. obviously, considering the existence and uniqueness of solutions to such kinds of problems is very important in pure and applied mathematics. in view of the fact that most real phenomena and non-linear problems in the world can not be solved analytically, researchers use numerical methods to obtain their approximate and numerical solutions in an 1 hassan et al. / j. nig. soc. phys. sci. 4 (2022) 1021 2 appropriate space. in this work, we examine the local and global existence and uniqueness of solutions to a system of non-linear volterra integral equations in the reproducing kernel hilbert space h[a, b]. this space is a very favorable space in numerical approximation since its reproducing kernel function is a transcendental function in [a, b]. 2. preliminary notation this section is assigned to present basic notation, definitions, and theorems which will be used later. definition 1. let s , ∅. a hilbert space h of continuous real-valued functions f : s → r is called reproducing kernel hilbert space if there exists a function k : s × s → r in h such that 〈 f (·), k(·, s)〉h = f (s), and k(·, s) ∈ h for all f ∈ h and all s ∈ s . such a function k = k(·, ·) is said to be a reproducing kernel function of h [4,6]. definition 2. let v [a, b] be the space of all absolutely continuous functions f : [a, b] → r such that f ′ ∈ l2[a, b] [4,6]. theorem 1. the function space v [a, b] equipped with the inner product [4] 〈 f1, f2〉v [a,b] = f1(a) f2(a) + ∫ b a f ′1 (t) f ′ 2 (t)dt, and associated with the norm || · || = √ 〈·, ·〉v [a,b], is a reproducing kernel hilbert space and the reproducing kernel function k = k(·, ·) is defined by: k(t,τ) = 1 2 sinh(b − a) ( cosh(τ + t − b + a) + cosh(| τ− t | −b − a) ) . definition 3. the function space h[a, b] = v [a, b] ⊕ v [a, b], consists of those functions ~h : [a, b] → r2 where ~h = (h1, h2) such that h1 and h2 belong to v [a, b]. definition 4. the inner product of the space h[a, b] is defined by : 〈~f ,~g〉h[a,b] = 〈 f1, g1〉v [a,b] + 〈 f2, g2〉v [a,b], where f = ( f1, f2) and g = (g1, g2). such a space is called direct sum of the reproducing kernel hilbert space v [a, b]. 3. existence and uniqueness in this section, we discuss the banach fixed point theorem to show the local and global existence and uniqueness of solutions to (1)-(2). to do this first, we need to introduce some basic tools. let ~h ∈ h[a, b] and let a = {(t, s) : a ≤ s ≤ t ≤ b}. define maps t~h : [a, b] → r and l~h : [a, b] → r by: t~h(t) = ∫ t a f ( t, s,~h(s) ) d s, l~h(t) = ∫ t a g ( t, s,~h(s) ) d s, such that the following conditions are hold for (k = 0, 1): c1) ∂ k ∂tk f and ∂k ∂tk g are uniformly bounded functions on a× r2. c2) for some positive constants m and n such that i) ∣∣∣∣ ∂k∂tk f(x, s, ~f1(s)) − ∂k∂tk f(y, s, ~f2(s))∣∣∣∣ 6 m (|x − y| + ‖~f1 − ~f2‖2 ) ; ii) ∣∣∣∣ ∂k∂tk g(x, s, ~g1(s)) − ∂k∂tk g(y, s, ~g2(s))∣∣∣∣ 6 n (|x − y| + ‖~g1 − ~g2‖2 ) . theorem 2. let ~h ∈ h[a, b]. then t~h ∈ h[a, b]. we first assert that t~h is absolutely continuous in [a, b]. by condition (c1) for (k=0); f is uniformly bounded on a× r2 and condition (c2) part (i) there are positive constants m and m1. let i j = {[a j, b j]}nj=1 be a finite collection of non-over lapping intervals in [a, b], and let ε > 0 such that: n∑ j=1 ∣∣∣∣b j − a j∣∣∣∣ < ε( m1(b − a) + m ). since, n∑ j=1 ∣∣∣∣t~h(b j) − t~h(a j)∣∣∣∣ = n∑ j=1 ∣∣∣∣ ∫ b j a f ( b j, s,~h(s) ) d s − ∫ a j a f ( a j, s,~h(s) ) d s ∣∣∣∣ = n∑ j=1 ∣∣∣∣ ∫ a j a f ( b j, s,~h(s) ) d s + ∫ b j a j f ( b j, s,~h(s) ) d s − ∫ a j a f ( a j, s,~h(s) ) d s ∣∣∣∣ 2 hassan et al. / j. nig. soc. phys. sci. 4 (2022) 1021 3 6 n∑ j=1 ∫ a j a j ∣∣∣∣f(b j, s,~h(s)) − f(a j, s,~h(s))∣∣∣∣d s + ∫ b j a j ∣∣∣∣f(b j, s,~h(s))d s∣∣∣∣ 6 n∑ j=1 ∫ a j a m1 ∣∣∣∣b j − a j∣∣∣∣d s + ∫ b j a j md s = n∑ j=1 ( m1(a j − a)|b j − a j| + m|b j − a j| ) 6 ( m1(b − a) + m ) n∑ j=1 ∣∣∣∣b j − a j∣∣∣∣ < ε. hence, t~h is absolutely continuous on in [a, b]. next, we want to show ∂ ∂t t ~h(·) ∈ l2[a, b]. leibniz rule implies for almost every t ∈ [a, b] that ∂ ∂t t~h(t) =f ( t, t,~h(t) ) + ∫ t a ∂ ∂t f ( t, s,~h(s) ) d s. then, ∫ b a ∣∣∣∣ ∂ ∂t t~h(t) ∣∣∣∣2dt = ∫ b a ∣∣∣∣∣f(t, t,~h(t)) + ∫ t a ∂ ∂t f ( t, s,~h(s) ) d s ∣∣∣∣∣2dt ≤ 2 ∫ b a ∣∣∣∣f(t, t,~h(t))∣∣∣∣2dt + 2 ∫ b a ∣∣∣∣∣ ∫ t a ∂ ∂t f ( t, s,~h(s) ) d s ∣∣∣∣∣2dt. it follows from condition (c1) for (k=0,1 ) there are positive constants n, d and the cauchy-schwartz inequality that∫ b a ∣∣∣∣ ∂ ∂t t~h(t) ∣∣∣∣2dt ≤ 2 ∫ b a n2dt + 2 ∫ b a ( ∫ t a ( ∂ ∂t f ( t, s,~h(s) ))2 d s ∫ t a 12d s ) dt, implies∫ b a ∣∣∣∣ ∂ ∂t t~h(t) ∣∣∣∣2dt ≤ 2 ∫ b a n2dt + 2 ∫ b a ( ∫ t a ( ∂ ∂t f ( t, s,~h(s) ))2 d s ∫ t a 12d s ) dt 6 2n2(b − a) + 2(b − a) ∫ b a ∫ t a ( ∂ ∂t f(t, s,~h(t) )2 d sdt 6 2n2(b − a) + 2(b − a) ∫ b a ∫ b a d2d sdt = 2n2(b − a) + 2d2(b − a)3 < ∞. therefore, t~h belongs to h[a, b] by definitions (2) and (3). similar arguments one can use to show that l~h belongs to h[a, b]. theorem 3. let ~f ∈ h[a, b]. then l ~f ∈ h[a, b]. the proof is analogous to the proof of theorem 2. set α,β ∈ h[a, b]. define operators γ : h[a, b] → h[a, b] and λ : h[a, b] → h[a, b] such that: γ~h(t) = α(t) + t~h(t); λ~h(t) = β(t) + l~h(t); for all ~h ∈ h[a, b]. we divide the interval [a, b]into n equally sub-intervals a ≤ t0 < t1 < ... < tn ≤ b; where 4t = t j − t j−1 j = 1, 2, ..., n and 4t = b−an . the inner product in h[t j, t j + 4t] is defined by: 〈~f ,~g〉h[t j,t j +4t] = 〈 f1, g1〉v [t j,t j +4t] + 〈 f2, g2〉v [t j,t j +4t], for all ~f ,~g ∈ h[t j, t j + 4t]. as a result, we see that the operators γ : h[t j, t j + 4t] → h[t j, t j + 4t] and λ : h[t j, t j + 4t] → h[t j, t j + 4t] become γ~h(µ) = α(µ) + ∫ µ t j f ( µ, s,~h(s) ) d s; λ~h(µ) = β(µ) + ∫ µ t j g ( µ, s,~h(s) ) d s; for all µ ∈ h[t j, t j + 4t]. lemma 1. let ~h ∈ h[t j, t j + 4t] and 4t < 1 [7]. then∥∥∥~h∥∥∥ 2 ≤ √ 2 ∥∥∥~h∥∥∥ h[t j,t j +4t] . remark 1. assume that α(t j) = β(t j) for all j = 0, 1, ..., n − 1. theorem 4. let ~h1,~h2 ∈ h[t j, t j + 4t]. then∥∥∥∥γ~h1 − γ~h2∥∥∥∥ h[t j,t j +4t] 6 δ(4t) ∥∥∥∥~h1 −~h2∥∥∥∥ h[t j,t j +4t] , where δ(4t) ≤ c √ 4t, for some positive constant c. 3 hassan et al. / j. nig. soc. phys. sci. 4 (2022) 1021 4 since∥∥∥∥γ~h1 − γ~h2∥∥∥∥2h[t j,t j +4t] = (γ~h1(t j) − γ~h2(t j))2 + ∫ t j +4t t j ( ∂ ∂t γ~h1(t) − ∂ ∂t γ~h2(t) )2 dt = ( α(t j) −β(t j) )2 + ∫ t j +4t t j ( f ( t, t, ~h1(t) ) − f ( t, t, ~h2(t) ) + ∫ t ti [ ∂ ∂t f ( t, s, ~h1(s) ) − ∂ ∂t f ( t, s, ~h2(s) )] d s )2 dt implies,∥∥∥∥γ~h1 − γ~h2∥∥∥∥2h[t j,t j +4t] 6 2 ∫ t j +4t t j ( f ( t, t, ~h1(t) ) − f ( t, t, ~h2(t) ))2 dt + 2 ∫ t j +4t t j ( ∫ t ti [ ∂ ∂t f ( t, s, ~h1(s) ) − ∂ ∂t f ( t, s, ~h2(s) )] d s )2 dt by (c2) we get constants m1, m2 such that∥∥∥∥γ~h1 − γ~h2∥∥∥∥2 h[t j,t j +4t] ≤ 2 ∫ t j +4t t j m21 ∥∥∥∥~h1(t) −~h2(t)∥∥∥∥2 2 dt + 2 ∫ t j +4t ti ( ∫ t ti m2 ∥∥∥∥~h1(s) −~h2(s)∥∥∥∥ 2 d s )2 dt. then, ∣∣∣∣∣∣∣∣γ~h1 − γ~h2∣∣∣∣∣∣∣∣2 h[t j,t j +4t] 6 2 ∫ t j +4t ti m21 m 2 3 ∥∥∥~h1 −~h2∥∥∥22dt + 2 ∫ t j +4t ti ( ∫ t ti m2 m3 ∥∥∥~h1 −~h2∥∥∥2d s)2dt 6 2m21 m 2 3 ∥∥∥~h1 −~h2∥∥∥22 (4t) + 2 3 m22 m 2 3 ∥∥∥~h1 −~h2∥∥∥22 (4t)3 = 4t ( 2m21 m 2 3 + 2 3 m22 m 2 3 (4t) 2 )∥∥∥∥~h1 −~h2∥∥∥∥2 2 . by using lemma 1 that ∣∣∣∣∣∣∣∣γ~h1 − γ~h2∣∣∣∣∣∣∣∣2 h[t j,t j +4t] 6 δ2(4t) ∥∥∥∥~h1 −~h2∥∥∥∥2h[t j,t j +4t], therefore, ∥∥∥∥γ~h1 − γ~h2∥∥∥∥ h[t j,t j +4t] 6 δ(4t) ∥∥∥∥~h1 −~h2∥∥∥∥ h[t j,t j +4t] , where δ(4t) < c √ 4t, and c = √ 2m23 ( m21 + 1 3 m 2 2 4 2 t ) < √ 2m23 ( m21 + 1 3 m 2 2 ) if 4t < 1. theorem 5. let f, g ∈ h[t j, t j + 4t]. then∥∥∥∥λ f − λg∥∥∥∥ h[t j,t j +4t] 6 σ(4t) ∥∥∥∥ f − g∥∥∥∥ h[t j,t j +4t] , where σ(4t) ≤ c √ 4t, for some positive constant c. the proof is similar to the proof of theorem 4. theorem 6. let f and g satisfy conditions (c1) and (c2). then there exists a unique solution ~h = ( f, g) ∈ h[a, b] to (1) and (2). for all ~h = ( f, g) in the space h[a, b]. it is clear that ~h 7→ γ~h and ~h 7→ λ~h are maps from h[t j, t j +4t] into h[t j, t j +4t]. from theorems 4 and 5; since 4t is an arbitrary positive constant and if we pick 4t small enough such that 4t < 1c2 then we conclude that δ(4t) < 1 and σ(4t) < 1. therefore, by theorems 4 and 5 the operators γ and λ are contraction mapping on h[t j, t j +4t], respectively. it is clear ( h[t j, t j +4t],‖·‖h[t j,t j +4t] ) is a complete matrix space. hence, the banach contraction mapping theorem guarantees that the operators γ and λ have a unique fixed point ~h = ( f, g) in h[t j, t j + 4t]. let ε(4t) = min{δ(4t),σ(4t)}. the existence and uniqueness of solutions in the entire interval [a, b] for (1) and (2) can be achieved by iterating the local existence result. this is accomplished by taking [a,ε(4t)], [ε(4t), 2ε(4t)], ...[nε(4t), b]. 4. conclusion we studied the local and global existence and uniqueness of solutions to a system of non-linear volterra integral equations (1)-(2) in the reproducing kernel hilbert spaces v [a, b] and h[a, b]. the results are very significant in numerical methods since the reproducing kernel function of the space v [a, b] is a smooth function on the compact interval [a, b] and it can be used to solve a wide variety of linear and nonlinear problems. references [1] k. a. dawodu, “extension of admm algorithm in solving optimal control model governed by partial differential equation”, journal of the nigerian society of physical sciences 3 (2021) 105. https://doi.org/10.46481/jnsps.2021.159. [2] v. o. atabo & s. o. adee, “a new special 15-step block method for solving general fourth order ordinary differential equations”, journal of the nigerian society of physical sciences 3 (2021) 308. https://doi.org/10.46481/jnsps.2021.337. [3] m. i. berenguer, d. gamez, a. i. garralda-guillem, m. r. galan & m. s. perez, “biorthogonal systems for solving volterra integral equation systems of the second kind”, journal of computational and applied mathematics 235 (2011) 1875. [4] m. cui & y. lin, “nonlinear numerical analysis in reproducing kernel space”, nova science publishers, inc., 2009 [5] l. dai & r. n jazar, nonlinear approaches in engineering applications springer, 2012. [6] j. s. hassan & d. grow, “new reproducing kernel hilbert spaces on semiinfinite domains with existence and uniqueness results for the nonhomogeneous telegraph equation. mathematical methods in the applied sciences 43 (2020) 9615. 4 hassan et al. / j. nig. soc. phys. sci. 4 (2022) 1021 5 [7] j. s. hassan & d. grow, “stability and approximation of solutions in new reproducing kernel hilbert spaces on a semi-infinite domain”, mathematical methods in the applied sciences 44 (2021) 12442 [8] r. k saeed & j. s hassan, “solving singular integral equations by using collocation method”, mathematical sciences letters 3 (2014) 185. [9] l. h. yang, j. h. shen & y. wang, “the reproducing kernel method for solving the system of the linear volterra integral equations with variable coefficients”, journal of computational and applied mathematics 236 (2012) 2398. 5 j. nig. soc. phys. sci. 5 (2023) 1048 journal of the nigerian society of physical sciences optimization of potassium carbonate-based des as catalyst in the production of biodiesel via transesterification abdulwasiu abdurrahmana,∗, saidu muhammad waziria, olusegun ayoola ajayia, fadimatu nyako dabaib adepartment of chemical engineering, ahmadu bello university, zaria, nigeria bdepartment of chemical engineering, university of abuja, nigeria abstract increasing energy demand necessitates the production of sustainable fuels, which can be in the form of bio-fuels. one of such bio-fuels is biodiesel, which is typically produced via transesterification. the development of homogeneous catalyst that is relatively easy to synthesize, cheap, reusable, and environmentally friendly, is a major issue in transesterification reaction. the use of deep eutectic solvent (des) as catalyst, is believed to be a significant step in the direction of attaining a sustainable bio-economy. in this study, deep eutectic solvent was synthesized from different mole ratios of k2co3/glycerol. the synthesized des was used as catalyst in the transesterification reaction to produce biodiesel from jatropha curcas oil. box-behnken design (bbd) was used to determine the factors that significantly affect the biodiesel yield. optimum fatty acid methyl ester (fame) yield of 98.2845% was achieved at optimum conditions of 1:32.58 mole ratio of k2co3/glycerol, 8.96% w/w concentration of des, and 69.58 minutes. gc-ms analysis revealed that the produced biodiesel contained 98.87% ester content. the properties of the biodiesel produced were characterized and found to agree with those of astm d6751-12 standard. thus, suggesting the synthesized des is a promising catalyst in the transesterification reaction to produce biodiesel from jatropha curcas oil. doi:10.46481/jnsps.2023.1048 keywords: deep eutectic solvent, fatty acid methyl ester, jatropha curcas oil, potassium carbonate, transesterification. article history : received: 08 september 2022 received in revised form: 30 october 2022 accepted for publication: 09 november 2022 published: 21 january 2023 c© 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: k. sakthipandi 1. introduction due to the continuous rise in the world’s population, the demand for global energy keeps growing. increasing energy demand necessitates the production of sustainable energy, which can be in the form of bio-fuels [1-2]. one of such bio-fuels is biodiesel. the direct usage and mixing of raw oils, thermal cracking, micro-emulsions, and transesterification are the ∗corresponding author tel. no: +243 7036088332 email address: acl645035@gmail.com (abdulwasiu abdurrahman) basic ways of producing biodiesel [3]. in particular, transesterification is the most prevalent process for making biodiesel because the resulting biodiesel has higher cetane number, lower emissions, higher combustion efficiency and renewability [4]. acids, alkalis, enzymes, and ionic liquids are used to catalyze the reaction[5]. acid and alkali catalysts are more commonly utilized in the manufacture of biodiesel because of their low cost compared to enzyme catalysts. however, the acid-catalyzed transesterification process necessitates a large mole ratio of methanol to oil, it takes a long time to complete the reaction 1 abdurrahman et al. / j. nig. soc. phys. sci. 5 (2023) 1048 2 compared to alkali catalysts, and the acidic catalysts are caustic and unfriendly to the environment [4, 6-7]. when considering biodiesel production, it is also important to consider the physical state of the catalyst to be employed. while homogeneous catalysts make it difficult to separate catalyst from liquid mixtures, heterogeneous catalysts require harsh operating conditions (such as longer reaction time and high temperature) to produce biodiesel. as already mentioned, ionic liquids (ils) have the potential to be used as catalysts in biodiesel production. the ability to recycle ils at the conclusion of the reaction, and the ease with which products may be separated, are two advantages of using a typical heterogeneous catalyst. therefore, as a result of combining the benefits of homogeneous and heterogeneous catalysts, ils have gained appeal as a catalyst in biodiesel synthesis [8-11]. however, large-scale commercial applications of ionic liquids remain a challenge due to complicated synthesis techniques and high cost of the raw materials needed for the synthesis[12]. a cheaper alternative to ils are the deep eutectic solvents (dess) [13]. due to their potential as an ecologically friendly solvent with favorable features over ionic liquids, such as simplicity of production in high purity at reduced cost, low toxicity, biodegradability, and non-reactivity with water, dess are currently in use in both research and industry [14]. dess have gained much attention in the biodiesel industry, where they may be used as an extracting solvent, catalyst, or co-solvent [15-16]. dess are excellent solvents for the separation of glycerol (by-product) from biodiesel [17]. abott et al. [18] demonstrated that a glycerol-based deep eutectic solvent is effective in separating glycerol and biodiesel from the final reaction mixture generated by ethanolysis in the presence of koh from rape seed and soybean oils. dess were found to be successful in removing glycerol, mono– and diacyl glycerols, and also as an alkali catalyst, for crude biodiesel made from palm oil by hayyanet al. [19]and shahbaz et al. [20-21]. zhao & baker [22] analyzed the feasibility of producing biodiesel by mixing traditional ils with dess. huang et al. [23] discovered a simple and energy-efficient method to initiate commercial cao for biodiesel synthesis with no pre-treatment by adding a novel des that can detach the inert layers of caco3 and ca(oh)2on the commercial cao surface during the reaction to obtain good fame yield, while the reaction was still running. hayyan et al. [24] used a phosphonium-based deep eutectic solvent (p-des) combined with alkali treatment to esterify poor quality crude palm oil. using ideal circumstances, the oil’s free fatty acid content was reduced from 9.5 to 1%. gu et al. [25] developed a choline-based deep eutectic solvent with glycerol as a co-solvent to catalyze transesterification of rapeseed oil for biodiesel synthesis. under optimal circumstances, a 98 % fame yield was attained. granados et al. [26] reported excellent yields of fatty acid alkyl esters of 90.3 and 92.4 by using potassium carbonate at concentrations of 2 and 3mol percent. excess methanol was used to move the reversible reaction’s equilibrium to the product side, while a co-solvent (such as tetrahydro-furan, thf) was added to overcome the mass transport limit in a heterogeneous system [27]. however, these organic solvents (such as methanol and thf) are often volatile, flammable, poisonous, and environmentally hazardous [28]. in addition, the little amount of soap created reduces yield and increases the creation of emulsions in the product, making separation of biodiesel from glycerine more difficult. in an alkali catalyzed chemical transesterification reaction, petracic, [7] investigated the usefulness of a des (choline-chloride: ethyleneglycol with a molar ratio of 1: 2.5) for the extraction of glycerol from biodiesel. it was determined that the dess had little to no effect on the extraction efficiency, hence a further process adjustment to lower the amount of total glycerol and glycerides was advised. in the area of catalysis, alhassan et al. [29] successfully employed chcl:koh, chcl:p-toluenesulfonic acid monohydrate, chcl:glycerol and chcl:fecl3 as catalyst and co-solvent for hydrothermal liquefaction of de-oiled jatropha curcas cake, and later applied chcl:p-toluenesulfonic acid as heterogeneous and homogeneous catalysts to produce biodiesel from pongamia pinnata seed oil [30]. also, chcl:p-toluenesulphonic acid was used as catalyst in co-liquefaction of jatropha curcas seed [31], while, yong et al. [32] utilized chcl:oxalic acid to convert biomass furfural to fumaric acid and maleic acid in the presence of h2o2. although the aforementioned dess performed well as catalyst in the biodiesel production, a side reaction between hydroxyl groups of the salt, and the acids from some types of dess composed of chcl and carboxylic acids was observed [5]. as a result of side esterification reactions observed in choline chloride based-dess, petračić et al. [29] prepared eutectic mixtures des (k2co3 : c2h6o2 = 1 : 10) which was used for feedstock deacidification. a total acid value of the waste cooking oil was reduced from 2.362 mg koh/g to 0.574 mg koh/g. while sander et al. [33] employed potassium carbonate-based solvent (potassium carbonate:ethylene glycol) to lower the total acid number of crude biodiesels using coffee feedstock. the time and mass ratio of des to oil were optimized, and these were shown to be favourable variables for the prospective industrial scale-up of the process. the industrial applications of dess, which are comprised of an organic salt and a hydrous metal salt, are limited [5, 34], hence, there is need to further explore the uses of these classes of dess. since glycerol is a key by-product of the production of biodiesel and the jatropha plant is vastly available and recognized as a significant source of biodiesel [1], des, produced from glycerol and potassium carbonate, was employed as a catalyst in the transesterification reaction to produce biodiesel. in particular, the aim of this research is to investigate the impact of mole ratio, time and concentration of potassium carbonate des on the yield of biodiesel synthesized from jatropha curcas oil. in previous studies, koh was used as the primary catalyst for transesterification, and potassium carbonate based-des as a secondary catalyst for purification, deacidification, separation, and extraction, while in this study, the des was produced from glycerol and potassium carbonate, and it was employed as a catalyst in the transesterification reaction. 2 abdurrahman et al. / j. nig. soc. phys. sci. 5 (2023) 1048 3 2. experimental procedure 2.1. materials jatropha curcas oil, with a free fatty acid (ffa) content of 6.68% was obtained from national research institute for chemical technology narict, zaria, nigeria, while glycerol, methanol, and k2co3 were obtained from romtech scientific supplies company limited, zaria. the chemicals had 98% purity and were employed for the preparation of dess without additional purification or drying. 2.2. synthesis and characterization of des 2.2.1. synthesis of des different molar ratios of potassium carbonate to glycerol (as shown in table 1) were used to produce des samples. in order to combine the salt with the hydrogen bond donor, a magnetic stirrer hot plate was utilized. each des mixture was shaken for 2 hours at 400 rpm at 353 k until a homogeneous transparent colorless liquid was obtained. des samples were produced at atmospheric pressure with moisture content tightly controlled. 2.3. determination of viscosity viscosity and density of des play significant roles in processes involving mass transport. the viscosity of the oil was measured using a brookfield rotary digital viscometer ndj-8s at 40oc. a spindle was attached to the viscometer and set at 60 rpm. 200 ml of the oil was poured into a beaker and the spindle was lowered into the beaker and allowed to attain the same temperature with the sample. the reading at 25% shear rate was taken. 2.3.1. fourier transform infrared (ftir) spectroscopy analysis ftir was utilized to investigate the interactions between the des’s constituents, and determine if des was formed through hydrogen bonding, by observing the stretch or shift in each functional group. the ftir spectroscopy experiments were carried out using microlab pc software of fourier-transform infrared spectrometer (model 630, agilent technology).all samples were scanned over a wave number range of 400-4000 cm−1. the spectra of the samples were recorded in 16 scans at 4 cm−1 resolution and plotted in the transmittance mode. prior to each measurement, the quality of the background signal was evaluated and a background spectrum was recorded using the same settings as for the sample measurement if necessary (residual peaks after cleaning > 0.2 % transmittance). the spectra were submitted to an automatic baseline correction performed with microlab pc software. 2.4. reduction of free fatty acid (esterification) the jatropha oil employed in this study has a significant amount of free fatty acid (ffa) (6.68 %), which is not suitable for the production of biodiesel via transesterification. as a result, it became essential to lower it via esterification. crude jatropha oil was put into a conical flask and heated to 60◦c. a combination of concentrated h2so4 (1% w/w) and methanol (30% v/v) was heated separately at (60◦c) before being added to the heated oil in the flask. the mixture was agitated for an hour and then allowed to settle for another two hours, and then ffa value of the oil was determined. 2.5. experimental design in this study, the reaction temperature was kept constant at 60◦c and the agitation rate was kept constant at 300 rpm, as indicated in the esterification experiment [35]. response surface methodology (rsm) and box–behnken design (bbd) were used to investigate the primary reaction parameters (such as k2co3/glycerol ratio, catalyst (des) concentration, and reaction duration) and optimize the reaction conditions for fatty acid methyl ester yield (fame) production. in the regression and graphical data analysis, the design expert 6.06 program was employed. the model’s statistical analysis was carried out in order to evaluate the analysis of variance (anova). 2.6. transesterification 40g of the esterified jatropha oil was transesterified in conformity with the design layout matrix, shown in table 3. the mole ratio of k2co3/glycerol was in the range of 1:20 to 1:40, time was varied from 30 to 120 minutes and concentration of des varied from 8 to 10% w/w. the mixture was stirred at 300 rpm with a magnetic stirrer hot plate at a temperature of 60◦c. the reaction mixture stabilized into a biphasic system at the end of the reaction. due to variances in viscosity and density between the two products, two layers developed in the separating funnel. the topmost layer was biodiesel (fame), whereas the lower layer was crude glycerol. the separation was allowed to run overnight in order to allow the separation of the fame layer and the free glycerol and other contaminants that can degrade the final quality of biodiesel. 3. result and discussion 3.1. characterization of des different molar ratios of glycerol to potassium carbonate were used to prepare the dess. table 1 shows these ratios along with their abbreviations and observations, during the preparation process. during the synthesis stage, dess samples were formed in a white viscous gel within the first 30 min. after 60 min of mixing, a liquid phase started to appear with some precipitation. therefore, the period of mixing was extended to 120 min in order to get a homogenous liquid phase des. des1 to des8 were not successful, as the two components did not form des, as the products were in either turbid white liquid or a mixture of colourless liquid and solid, throughout the process and after cooling to room temperature. adding more glycerol achieved the necessary balance between the two des constituents and guaranteed complete miscibility. thus, des9, des 10 and des11 remained in colourless liquid phase at room temperature, and the unsuccessful dess were not considered for further investigation in this study. the physical properties of the synthesized des (in particular, des 9) are shown in table 2. des 9 was considered, 3 abdurrahman et al. / j. nig. soc. phys. sci. 5 (2023) 1048 4 table 1. mole ratio and abbreviations of des synthesized mole ratio abbreviation appearance 1:3.5 des 1 turbid white liquid 1:4 des 2 turbid white liquid 1:5 des 3 turbid white liquid 1:6 des 4 turbid white liquid 1:7 des 5 colorless liquid with solids 1:8 des 6 colorless liquid with solids 1:9 des 7 colorless liquid with solids 1.10 des 8 colorless liquid with solids 1.20 des 9 colorless liquid 1.30 des 10 colorless liquid 1.40 des 11 colorless liquid table 2. properties of des synthesized property des synthesized viscosity @ 40◦c 0.428 pa.s density 1.322g/ml ph 10.53 figure 1. ft-ir result of (a) k2co3, (b) glycerol, and (c) des since it was successfully synthesized at a lower mole ratio than des 10 and des 11. the density and viscosity conform to that reported by naser et al. [36]. the ph is important in applications related to catalytic reactions. a ph of 10.53 was obtained, which indicates the basicity of the mixture. this implies that when the des is used as catalyst, the reaction will follow a base-catalyzed transesterification mechanism. figure 1 shows the ft-ir of k2co3, glycerol, and synthesized des. in figure 1 (a, b and c), the region between 3000 and 2800 cm−1shows the existence of c–h stretching bands of the alkanes ch3 and ch2 for the des. the peak at 3022.9cm−1 indicate the absence of o-h in k2co3 in figure 1a, while the presence of o–h stretching bands between 3200 and 3500 cm−1 in figure 1 (b and c) is attributed to hydroxyl group. figure 1 reveals that a shift in the o-h stretching vibration of glycerol indicate that the change in vibrational state occurred because a portion of the cloud of electrons of the oxygen atom was transferred to the hydrogen bond, reducing the force constant. thus, the shift of the o-h stretching vibration (3209.1cm−1) indicates the existence of a hydrogen bond between the glycerol and k2co3 when the des was formed. this is in agreement with the observation reported in the literature[37–41]. thus the ft-ir spectra reveal the intermolecular attraction between the salt and the hydrogen bond donor (glycerol). 3.2. production of biodiesel using des as catalyst prior to the production of biodiesel, the ffa of the jatropha curcas oil was reduced from its initial value (of 6.68%) via esterification. the ffa of the jatropha curcas oil were reduced after the first 3 hours to 2.427 %, after 4 hours to 1.112 %, and then to 0.409 %, which is within the range of standard oil for the production of biodiesel. box–behnken design (bbd) was used to optimize the reaction conditions for the production of fatty acid methyl ester yield (fame), based on the primary reaction parameters (such as k2co3/glycerol mole ratio, catalyst (des) concentration, and reaction duration), as shown in table 3. the esterified oil was transesterified with methanol at a molar ratio of 1:6, utilizing k2co3/glycerol des as a catalyst; with the reaction temperature set at 60◦c, and the system agitated at 300 rpm. fame yields in the range of 88.97–98.15% were obtained at des component ratios of 1:20, 1:30, and 1:40, reaction times ranging from 30 to 120 minutes, and des concentrations ranging from 8 to 10% w/w, as indicated in table 3. 3.2.1. modified quadratic model for transesterification process response surface methodology (rsm), based on bbd, was used to investigate the primary reaction variables. to match the experimental data, a quadratic polynomial equation in terms of real components was established using response surface methods, as illustrated in equation (1). % biodiesel yield = +98.00+2.95a+0.64b+0.44c−4.21a2 − 0.90b2 − 1.31c20.50ab + 0.20ac − 0.38bc (1) where: a mole ratio of k2co3/glycerol, bconc. of des c reaction time. as already mentioned, for the regression and graphical data analysis, the design expert 6.06 program was employed. the model’s statistical analysis was carried out in order to evaluate the analysis of variance (anova). based on the analysis of variance (anova) results (table 4), a second-order polynomial model (equation 1) appears to illustrate the relation between the yield and the important factors. the regression model’s significance is shown by a very high f value (204.34) and a modest p-value (0.0001). a, b, c, a2, b2, c2 are significant model terms. a reasonable determination coefficient (r2 = 0.9962) indicates that the independent variables (k2co3/glycerol 4 abdurrahman et al. / j. nig. soc. phys. sci. 5 (2023) 1048 5 table 3. design layout for the transesterification reaction serial no. (in order of lowest to highest biodiesel yield) run no. mole ratio of k2co3/glycerol conc. of des (%w/w) time (min) actual yield of biodiesel (%w/w) 1 7 20.00 9.00 30.00 88.97 2 1 20.00 8.00 75.00 88.98 3 10 20.00 9.00 120.00 89.65 4 17 20.00 10.00 75.00 91.32 5 14 30.00 8.00 30.00 94.48 6 13 40.00 9.00 30.00 94.90 7 6 40.00 8.00 75.00 95.45 8 2 40.00 10.00 75.00 95.80 9 12 30.00 8.00 120.00 95.90 10 5 30.00 10.00 120.00 96.35 11 11 40.00 9.00 120.00 96.39 12 8 30.00 10.00 30.00 96.45 13 3 30.00 9.00 75.00 97.67 14 15 30.00 9.00 75.00 97.93 15 16 30.00 9.00 75.00 98.10 16 4 30.00 9.00 75.00 98.15 17 9 30.00 9.00 75.00 98.15 figure 2. comparison between the actual (experimental) fame yield and predicted yield molar ratio, catalyst concentration, and reaction duration) can account for 99.62 % of the sample variation in biodiesel generation. to confirm the model validity, the model prediction was compared with experimental data as shown in figure 2. it was found that the model was successful in capturing the correlation between the process parameters to the response with a correlation coefficient.the high adjusted determination coefficient (adj.r2 = 0.9913) verifies the model’s importance, and the comparatively low variation coefficient (cv = 0.32 %) suggests the good accuracy of the experimental data. a precision greater than 4 establishes the model’s adequacy by assessing the signal-to-noise ratio. the three-dimensional graphs of a second-order prediction model for the fame yield response are shown in figures 3 (a, b and c). as shown in figures 3 (a) and (b), the fame production improved significantly when the mole ratio of k2co3/glycerol was adjusted to its midpoint. this is consistent with the results shown in table 4 (the mole ratio of k2co3/glycerol has the highest calculated f-value and the lowest p-value). this is due to the fact that the quantity of salt supplied to glycerol has a substantial impact on the creation of hydrogen bonding, which can lead to improved des activity as a catalyst in the transesterification reaction. when the surplus mole ratio is utilized, it means that the amount of salt utilized was greater than the matching hydrogen bond donor, resulting in the precipitation of the additional salt that was unable to form hydrogen bonds with the hydrogen bond donor. fame yield increases as the concentration of the catalyst (des) increases, as seen in figure 3 (a) and (c). a low catalyst dose does not generate enough methoxide to achieve a high fame yield. while due to probable side reactions such as saponification, an excessive catalyst dose does not result in a high yield. as a result, the optimal concentration zone is depicted in figures 3 (a), (b) and (c). it is important to note that reaction time is a significant operating parameter because of its direct impact on the cost and quality of biodiesel. to obtain a complete reaction, sufficient but not excessive response time must be supplied. the optimal transesterification reaction time was determined to be between 30 and 120 minutes, as indicated in (b) and (c). in particular, after 75 minutes of response time, there is no discernible influence on yield. 3.3. optimization solution one of the main goals of the optimization process is to maximize fame yield. table 6 depicts several optimization solutions. as previously stated, the amount of time and catalyst concentration have a direct impact on the cost and quality of 5 abdurrahman et al. / j. nig. soc. phys. sci. 5 (2023) 1048 6 table 4. anova for selected factorial model source sum of squares df mean squares f value prob > f model 167.03 9 18.57 204.13 < 0.0001 significant a 69.74 1 69.74 766.80 < 0.0001 b 3.27 1 3.27 35.96 0.0005 c 1.53 1 1.53 16.79 0.0046 a2 74.78 1 74.78 822.27 < 0.0001 b2 3.40 1 3.40 37.34 0.0005 c2 7.21 1 7.21 79.22 <0.0001 ab 0.99 1 0.99 10.89 0.0131 ac 0.16 1 0.16 1.80 0.2212 bc 0.58 1 0.58 6.39 0.0393 residual 0.64 7 0.91 lack of fit 0.47 3 0.16 3.70 0.1195 not significant table 5. predicted and adjusted r-squared std. dev. 0.30 r-squared 0.9962 mean variation 94.98 adj r-squared 0.9913 coefficient (c.v.) 0.32 pred r-squared 0.9538 press 7.75 adeq precision 39.784 biodiesel. the mole ratio of k2co3/glycerol also has a significant impact on biodiesel production, therefore a mole ratio of 1:32.58, a concentration of 8.96 percent w/w of des, and a duration of 69.58 minutes are identified as the optimalreaction conditions. based on numerical optimization, as shown in table 6, the optimum fame yield of 98.2845 % is predicted to be attained at a 1:32.58 mole ratio of k2co3/glycerol, 8.96 % w/w concentration of des, and 69.58 minutes. experiments were conducted at the indicated optimal conditions, producing fame yields of 98.20 %, 98.20 %, 98.22 %, with an average value of 98.21 %, as shown in table 7. thus, the experimental and predicted value(s) are in good agreement. the relative error between the anticipated and real data is 0.0789 %, indicating that bbd and rsm successfully achieved the optimization of des-catalyzed biodiesel synthesis from jatrophacurcas oil. 3.4. effect of the effluent des as catalysts the ability to reuse a catalyst is considered vital to lowering biodiesel production costs. thus the catalytic performance of the des, utilized as a catalyst, is an essential metric to consider. the performance of reused des in the transesterification process is reported in table 8. the performance of the des as a catalyst changed significantly after it was reused. deactivation of the hydrogen bond in the des, inability to separate the des from the reaction effluent, and the existence of residual reaction mixture in des might have contributed to a decrease in the des catalytic strength. table 8 shows the fame yield(s) obtained from the initial (synthesized)des (98.22%), from the des obtained from the first run (71.98%), and the des obtained from the second run (53.24%).therefore, following the application of des in two cycles, the yield of fame decreased. (a) (b) (c) figure 3. plots of response surface of fame yield against reaction parameters: (a) k2co3-glycerol mole ratio and conc. of des interaction: (b) k2co3glycerol mole ratio and time: and (c) time and conc. of des interaction 3.5. properties of the biodiesel produced the produced biodiesel was subjected to analysis, to verify its properties. the properties were then compared to the expected standards (astmd6751), as shown in table 9. the biodiesel produced in this study has a viscosity of 4.27mm2/s, 6 abdurrahman et al. / j. nig. soc. phys. sci. 5 (2023) 1048 7 table 6. optimized parameter for the transesterification mole ratio of potasium carbonate/glycerol conc. of des(%w/w) time (min) yield of biodiesel (%w/w) desirability 34.96 8.63 84.67 98.1643 1.000 32.13 8.84 74.77 98.2353 1.000 33.79 9.20 103.67 98.3144 1.000 35.31 9.41 65.35 98.1606 1.000 34.22 8.69 103.56 98.1709 1.000 33.69 8.77 99.53 98.2959 1.000 34.69 8.84 80.65 98.3464 1.000 32.58 8.96 69.58 98.2845 1.000 35.66 8.81 95.63 98.2006 1.000 34.34 8.76 85.51 98.3365 1.000 table 7. optimized conditions and validation for transesterification process predicted optimal conditions and yield a(ratio) b (wt.%) c(min) yield (%w/w) 32.58 8.96 69.58 98.2845 actual experiments (validations) yield 1 (%w/w) yield 2 (%w/w) yield 3 (%w/w) average yield (%w/w) 98.20 98.20 98.22 98.207 predicted yield (%w/w) actual yield (%w/w) deviation 98.2845 98.207 ± 0.1% table 8. comparison of fame yield from the synthesized des and the reused des at optimized conditions s/no catalyst fame yield (%w/w) 1 des 98.22 2 des from run 1 71.98 3 des from run 2 53.24 which is within the astm biodiesel standard range [21]. this is relevant, considering that the atomization of the fuel being injected into an engine combustion chamber is affected by the viscosity of the fuel [42]. another crucial aspect for optimum engine performance is fuel density; the higher the density, the more difficult it is to pump the gasoline. the produced biodiesel has a density of 0.882g/cm3, which is also within the standard range [20]. another essential attribute of fuels is the cetane number, which is a measurement of a diesel’s combustion quality during compression ignition. engine performance, cold starting, warm up, and engine combustion roughness are all affected by the ignition quality, which is determined by the cetane number. the volatility of the fuel is related to the cetane rating, with higher ratings for more volatile fuels. if a high cetane fuel ignites too quickly, it may result in incomplete combustion and smoke; by not giving enough time for the fuel to combine with air for full combustion [42]. the synthesized biodiesel has a value of51.18 cetane number, which is within the acceptable range for use in diesel engines. the acid value, pour point, and cloud point of the jatropha oil biodiesel were all within astm d6751 specifications. despite the fact that astm does not specify a limit for biodiesel saponification value or iodine value, the attributes of the biodiesel generated are very similar to those reported in the literature [15]. gc-ms analysis reveals that the biodiesel produced contains 98.87% ester content and 1.13 % non-ester composition, as shown in table 10. this further confirms the quality of the biodiesel produced. 4. conclusion in the transesterification process to produce biodiesel from jatropha curcas oil, des (made from glycerol and k2co3) was utilized as a catalyst, in this study. the work shows that the des is a promising catalyst in the transesterification reaction, with a biodiesel yield of 98.22%, with ester content of 98.87 %. using response surface methodology (rsm) and box–behnken design (bbd) to investigate the primary reaction parameters, the mole ratio of k2co3/glycerol of 1:32.58, concentration of des of 8.96% w/w and time of 69.58 minutes served as the optimum conditions for the transesterification reaction. also, the study shows that after the third run of reusing the catalyst, a 53.24 % yield of biodiesel was obtained, which shows there is a certain (albeit low) level of reusability of catalyst. nevertheless, the study shows that the synthesized des is a promising catalyst in the transesterification reaction to produce biodiesel. acknowledgement the authors would like to express their gratitude to the administrations of ahmadu bello university, zaria, and university of abuja for their collaborative efforts and permission to publish this work. 7 abdurrahman et al. / j. nig. soc. phys. sci. 5 (2023) 1048 8 table 9. physical properties of jatropha oil biodiesel property produced biodiesel astmd6751 standard density at 400c (g/cm3) 0.882 0.86–0.90 viscosity at 400c (mm2/s) 4.27 1.6 –6.0 acid value (mg koh/g) 0.74 ≤ 0.8 cetane number 51.18 ≥ 47 pour point 0c -2 −15 to 16 cloud point 0c 10 −3.0 to 12 iodine value (mg i/100g oil) 104.133 —– saponification value (mg koh/g oil) 192.8 —– table 10. gc-ms of the produced biodiesel peak no name of the compound molecular formula retention time (min) peak area (%) 1 dodecanoic acid, methyl ester c13h26o2 24.408 0.03 2 methyl tetradecanoate c15h30o2 29.271 0.09 4 hexadecanoic acid, methyl ester c17h32o2 33.791 16.38 5 heptadecanoic acid, methyl ester c18h36o2 35.717 0.22 6 8,11-octadecadienoic acid, methyl c19h34o2 37.332 30.43 7 9-octadecenoic acid, methyl ester c19h36o2 37.479 22.44 8 methyl stearate c19h38o2 37.797 8.96 9 9,12-octadecadienoic acid, ethyl ester c20h36o2 38.392 0.08 10 oleic acid∗ c18h34o2 39.908 0.30 11 glycidyl palmitate c19h36o3 40.857 4.94 12 9-octadecenoic acid (z)-, 2,3dihydroxypropyl ester c21h40o4 43.125 2.72 13 glycidyl oleate c21h38o3 43.987 9.99 14 adipic acid, butyl 3-heptyl ester c22h42o4 44.154 0.85 15 6-octadecenoic acid, (z)∗ c18h34o2 44.571 0.50 16 docosanoic acid, methyl ester c23h46o2 44.721 0.21 17 tetracosanoic acid, methyl ester c25h50o2 47.856 1.52 22 2,2-dimethyl-3-(3,7,16,20tetramethyl-heneicosa-3,7,11,15,19pentaenyl)-oxirane∗ c29h48o 49.525 0.34 total composition 100 total non-ester content (*) 1.13 total ester content 98.87 * indicates non-ester compounds references [1] p. maheshwari, m. b. haider, m. yusuf, j. j. klemeš, a. bokhari, m. beg, a. al-othman, r. kumar, a. k. jaiswal, “a review on latest trends in cleaner biodiesel production: role of feedstock, production methods, and catalysts”, journal of cleaner production 355 (2022) 131588. https://doi.org/10.1016/j.jclepro.2022.131588. 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university, oye-ekiti, nigeria abstract linear programming has served as a great tool in dealing with transportation problems to make a positive difference in economic and social activity. in this work, data of the national union of road transport workers (nurtw) is analysed to minimizing the cost of maintenance and repair of buses taking the route from sango park to different routes. data collected from the park are represented using tables and solved using the maple computer software application. the transportation problem is solved such that the transportation cost is minimized which leads to the profit being maximized. this is achieved by estimating the values of some identified parameters in the problem. this work will be beneficial to every other motor parks controllers to decide on some decision making that may bring to the union profit. this work will help the nurtw in sango to spend less on the vehicles and save more as income. keywords: linear programming, transportation problem, parameters, sensitivity article history : received: 26 september 2019 received in revised form: 26 october 2019 accepted for publication: 29 october 2019 published: 12 november 2019 c©2019 journal of the nigerian society of physical sciences. all rights reserved. communicated by: b. j. falaye 1. introduction one major problem encountered by companies or organizations is the transportation problem. the transportation problem was given shape as a problem in 1871 by a french economist and mathematician called gaspard monge, the transportation problem was first studied in 1920 as a problem by aleskey nikolayevich, [1]. here, we study the sango nurtw which runs the transport service both in ogun state and outside the state. since the transportation problem is to find a way of minimizing cost (or time spent) or to maximize profit, the goal of every transportation is to meet the request of the destination. ∗corresponding author tel. no: +2348032801624 email address: tolulope.latunde@fuoye.edu.ng (tolulope latunde) in this study, we only have two destinations, one destination is seen from the other, they are respectively the place where the vehicles will stop eventually when it leaves the park and the park itself. the optimization of transportation problem can apply to so many areas like the genetic algorithm, networking, modelling and many more. the transportation method in use is a type of linear programming problem that utilises the simplex technique. it is applied to the problems related to the study of the efficient transportation routes i.e., how efficiently the product from different sources of production is transported to the different destination such as the total transportation cost is minimum. here, we revise the solution of the formulated transportation problem until the optimum solution is obtained, thus we varied the values of parameters in the model to study the behaviours of each parameter of the model. 116 latunde et al. / j. nig. soc. phys. sci. 1 (2019) 116–121 117 since usually the objective of the transportation problem is to either minimize the distance to be covered, total transportation cost or to maximize profit. the transportation problem can be defined mathematically as; suppose xi j ≥ 0 is the number of passengers traveling from ith origin to the jth destination the model of the problem given the cost of transportation c will be, minimize z = m∑ i=1 n∑ j=1 ci j xi j (1) s ub ject to n∑ j=1 xi j ≤ ai f or i = 1,2, ...,m i.e supply m∑ i=1 xi j ≤ b j f or j = 1,2, ...,n i.e demand. ∀xi j ≥ 0 if ∑n i=1 ai = ∑n j=1 b j, this means that the number of passengers available is equal to the number of available spaces in the vehicles to transport them. if not, it will be considered unbalanced transportation in this work. 2. literature review in recent times, books have been written, papers published and presentation made diversely on the transportation problem, we generally that in a layman language, the transportation problem is one of the optimization problems. a new way of solving the transportation problem from the park to different destinations was introduced in [2], the method used was readers friendly, easily accessible and can be quickly understood. the last table can be used to decipher the roads that serve as the best to be plied as well it explains the roads that need more vehicles to be added to increase the income rate. the cost coefficient of the objective function was dealt with thoroughly seeing into the multi-objective problems of transportation in [3]. a minimization work from the origin to the destination was carried out. the fuzzy programming technique was utilized to solve the problem that was converted from constraints to a deterministic solvable problem. asase did a great work on the transportation problem also, he used the guinness ghana limited in kumasi as a case study , he explained the transportation problem, created a table and solved using the vogel, north-west corner, least cost method as well as for test for optimality, he used the stepping stone method and the modified distribution method (modi) [4]. a k-objective transportation problem was formulated in [5] by fuzzy numbers and used alpha-cut to obtain a transportation problem in the fuzzy sense expressed in linear programming form. an additive fuzzy programming model for the multiobjective transportation problem was introduced. the method aggregates the membership functions of the objectives to construct the relevant decision function. weights and priorities for non-equivalent objectives are also incorporated in the method. their model gave a non-dominated solution which is nearer to the best-compromised solution. in the paper optimal solution of a transportation problem, a method was developed in [6] to get the initial basic feasible solution (or near to the optimal solution) of transportation problem. also, the algorithm provided in this paper gives the idea for the optimality in comparison with modi method as the flow of steps by step procedure. the paper helps in explaining vividly the algorithm used, the test of optimality and numerous numerical examples. however, some researchers have worked in the areas of sensitivity analysis and its application to transportation. parameter estimation and sensitivity analysis of an optimal control model for capital asset management where the parameters of the formulated model of asset management are classified according to their degree of sensitivity was worked on in [7]. also, the sensitivity of parameters in an optimal control model of the electric power generating system to determine the behaviour of the model’s parameters in [8]. pandian and kavitha proposed a new bound technique for cost sensitivity ranges of solid transportation problems [9]. in this work, the maple software is used to derive the expected income and also determine the best routes that needed to be invested in the most by analysing the sensitivity of the parameters involved in the model. 3. statement of problem given that some passengers are to travel to different destinations from an nurtw transport company, there is a need for the nurtw to optimize the cost of transportation, determine the best ways of allocating vehicles by either increasing or decreasing the number of vehicles on a particular route. the total cost of transporting the passengers from the park i to the destination j is not changing from every other transportation problem. this is given by letting i = park and then j = destinations, prior ai and b j above xi j is the number of passengers to be transported from sango park i to the different destinations: n∑ j=1 ci j xi j = c11 x11 + c12 x12 + ... + c1n x1n (2) the total cost tc of transporting the passengers from sango park to the different destinations. tc = m∑ i=1 n∑ j=1 ci j xi j = c11 x11 + c12 x12 + ... + c1n x1n c21 x21 + c22 x22 + ... + c2n x2n . . . cm1 xm1 + cm2 xm2 + ... + cmn xmn 117 latunde et al. / j. nig. soc. phys. sci. 1 (2019) 116–121 118 the nurtw agrees generally in sango to offer services of transporting her passengers from sango both intra-state and interstate as listed below. intra-state sango-atan sango-ewekoro sango-ifo sango-owode sango-idi-iroko sango-abeokuta sango-ijebu ode sango-ijebu remo sango-iperu inter-state sango-lagos (ikeja) sango-ondo (akure) sango-oyo (ibadan) sango-osun (oshogbo) sango-ekiti (ado-ekiti) 4. data collation and analysis all data were gotten from sango park in march 2019 at sango, ogun state. data were gotten by conversation and interviewing one of the sango nurtw official staff. the destination: this is the destination the vehicles go to from the park. it is the place the passengers paid for a service to be delivered to them. the trip: this is the to and fro movement a vehicle completes weekly. we should note that daily means all the vehicles can travel every day, turn means it depends on the total number of vehicles available and it is determined by how early the vehicles arrive as they will leave respectively as they have come. fuel: this is the average amount of fuel consumption (in litres) per trip. b/rt: (buses per route)these are the number of buses the nurtw for sango agreed to travel in a day. hrs/trip: this is the total number of hours it takes a vehicle to travel to the destination and to travel back to the park if it leaves immediately. t. fare: the t.fare here means transportation fare. in essence, it is the cost a passenger is charged for transportation service, better explained as the transportation fare per person in naira n. table 2 is an average data from different distributors, hence it might be a little bit expensive or cheaper depending on the bargaining power. in table 3, public drivers and sango nurtw mode of servicing their vehicles differ. the road differs, some are bad, some are fair and others are good, on that basis, some drivers will have to service their vehicles fortnightly, some every week and most service their vehicles monthly. this implies that a vehicle will get serviced at the rate of n8700 per month implying that averagely a vehicle will be serviced for n256.7 daily. we carefully noticed that the analysis above in table 4 is not true for all route, as stated earlier above, some route may cost more as some roads are bad e.g the sango to owode-iroko, this road attracts twice the amount most times. fuel : this is gotten by the litres of fuel used in a trip, as at the time of this thesis, a litre is n145, therefore, the fuel consumption cost is the total number of litres multiply by 145 in (naira). parking charges : this is the amount a vehicle driver will pay at the garage of the destination, this is because nurtw, sango does not have a parking space of her own. b/trp : this short form for bus per trip. it is the number of buses per trip that can be on the road at a time. hrs/trip : this is the short form for hours per trip. the time taken for a bus to reach the destination and fro. cvs/day : this is the short form for cost of vehicle service per day. vrm/day : this is the short form for vehicle repair and maintenance per day. income/trip : this is gotten by multiplication of the transport fare twice, twice because a trip is to and fro then multiplied by the number of people in a bus. the total number of passengers in a bus is 14 i.e (t fare × 2 × 14), expense/day : this is gotten from adding the fuel consumption cost + the parking charges + servicing vehicles cost per day + vehicles maintenance per day. income/day : this is gotten from subtracting vehicles expense per day from income for a trip per day. parking charges : this is the multiplication of the amount of parking charges per bus by the number of buses per on the road at a time. fuel : this is the multiplication of fuel used in a trip by the number of buses on the road at a time (b/trp). cvs : this is the cost vehicle service multiplied by b/rt vrm : this is the cost of vehicles repair and maintenance multiplied by b/rt, income/trip : this is gotten by multiplying the income per trip of the bus by the total number of buses that can be on the road at a time. total expense : this is the sum of by parking charges, amount of fuel used, cvs and vrm. total income : this is the difference between income/trip and the total expense. 5. model description the formulation of the problem is based on data gotten from the park and the immediate table 5. nurtw (sango park) allows the maximum of 140 buses in total to travel the road to all road to these respective destinations every day. meanwhile, these buses can be assigned to leave the park. solving the above linear programming problem, we derive n1,069,345.7. however, our goal in this work is to sensitize each parameter of the model to derive optimal allocation. thus from table 6, we increase all routes in twos respectively from the atan destination to the ekiti destination, we keep repeating this process for all the routes till 10 buses are added to all the routes. 118 latunde et al. / j. nig. soc. phys. sci. 1 (2019) 116–121 119 table 1: data on trips s/n destination trip(weekly) fuel(ltrs) b/rt hrs/trip t. fare n 1 atan daily 15 20 1 150 2 ewekoro daily 15 20 1 200 3 ifo daily 20 15 1.5 300 4 owode daily 15 20 1.5 300 5 idiroko daily 25 10 2 400 6 abeokuta turn 30 10 2.5 500 7 ijebu-ode thrice 70 3 5 1500 8 ijebu-remo thrice 75 3 6 1700 9 iperu thrice 75 3 6 1800 10 lagos daily 20 15 2 300 11 ondo five times 35 7 3 700 12 oyo turn 70 2 4 1150 13 osun turn 85 2 5.5 2300 14 ekiti turn 95 2 7 2550 table 2: data on vehicle repair and maintenance (vrm) s/n item cost durability 1 break disk 10500 2 years 2 break lining 3000 1 months 3 break pad 1800 3 weeks 4 car battery 15000 2 years 5 front bearing 7000 5 months 6 fuel pump 5500 2 years 7 release bearing 3000 2 years 8 shock absorber 11000 2 years 9 shock filling 1800 6 months 10 tyre 24000 4 months total 82,600 table 3: cost of vehicle service (cvs) s/n service item amount monthly cost per day 1 engine oil (5ltrs) 5000 166.7 2 oil filter 1700 56.7 3 oil treatment 650 33.3 total 8700 256.7 we then run every addition by the maple software to ascertain the outcome and to determine the profit that will be generated, it was noticed that some routes yielded more income than the others as the buses plying the routes were increased. table 7 represents the optimal result of the formulated model. bus: this is the representation of each route respectively from the atan destination being x1 to the last destination i.e ekiti represented as x14. total no of buses: this is the new total number of buses assumed to be on the road at once from sango park. former b/rt : (former buses per route) this is the total number of buses plying a particular at once before the increase. new b/rt: (new buses per route) this is the total number of buses plying a particular at once after the increase. income generated: this is the result compiled by the maple software after the buses have been respectively increased in naira n. it is important to note that when none of the bus plying the route was increased, the income generated remains the same as the income generated in table 6 i.e n1,069,345.7. the highest income generated is from x14 which is the ekiti destination producing n1,617,172.7. 5.1. discussion on result table 8 shows our recommendation as to know which the routes needs increment of more vehicle to ply them and the outcome of our result is represented in figure 1 where it can be interpreted on the graph to see that the income generated by the x14 being the ekiti destination yields the highest income. figure 1 below shows the income generated by every bus. the buses are labelled respectively from the atan destination as x1 to ekiti as x14. the bus that generated the highest income is the x14, the bus plying ekiti. the problem was solved in maple software and represented in table 8 and figure 1 whereby the optimal value of n1,617,172.7 is obtained instead of the n1,069,345.7 after selecting the optimal allocation of vehicles to best routes by sensitising the parameters of the model. however, to determine the optimal vehicle allocation, we varied the values of parameters representing each vehicle in the model with respect to its location to understand how each parameter behaves. thus we recommend that ekiti routes should get more buses to ply it to optimize the cost transportation. this will yield extra n 547,827 more as profit. 6. conclusion due to some uncertainties in the cost of vrm, cvs and some other input parameters, non-linear constraint optimization problem shall be considered in future works to tackle this. 119 latunde et al. / j. nig. soc. phys. sci. 1 (2019) 116–121 120 table 4: cost of vehicle repair and maintenance per day s/n item item cost durability durability (day) cost (day) 1 break disk 10500 2 years 720 14.6 2 break lining 3000 1 months 30 100 3 break pad 1800 3 weeks 21 85.7 4 car battery 15000 2 years 720 20.8 5 front bearing 7000 5 months 150 46.7 6 fuel pump 5500 2 years 720 7.6 7 release bearing 3000 2 years 720 4.2 8 shock absorber 11000 2 years 720 15.3 9 shock filling 1800 6 months 180 10 10 tyre 24000 4 months 48 500 total 82,600 804.9 table 5: total income per day s/n fuel(ltrs) fuel n parking charges n b/trp hrs/trip t.fare n cvs/day n vrm/day n income/trip n expense/day n income/day n 1 15 2175 0 20 1 150 256.7 804.9 4200 3236.6 963.4 2 15 2175 0 20 1 200 256.7 804.9 5600 3236.6 2363.4 3 15 2900 0 15 1.5 300 256.7 804.9 8400 3169.6 4438.4 4 15 2175 0 20 1.5 300 256.7 804.9 1609.8 4041.5 4358.5 5 25 3625 0 10 2 400 256.7 1609.8 11200 5491.5 5708.5 6 30 4350 150 10 2.5 500 256.7 804.9 14000 5561.6 8438.4 7 70 10150 350 3 6 1500 256.7 1609.8 42000 12366.6 29633.5 8 75 10875 350 3 6 1700 256.7 1609.8 47600 13091.5 34508.5 9 75 10875 300 3 6 1800 256.7 804.9 50400 12236.6 38163.4 10 20 2900 0 15 2 300 256.7 804.9 8400 3961.6 4438.4 11 35 5075 200 7 3 700 256.7 1609.8 19600 7141.5 12458.5 12 70 10150 300 2 4 1150 256.7 1609.8 32200 12316.5 198835 13 85 12325 350 2 5.5 2300 256.7 1609.8 64400 14541.5 49855.5 14 95 13775 350 2 7 2550 256.7 2414.7 71400 16796.4 54603.6 total 645 93525 2350 13850 3593.8 18507.7 387800 103454.54 269815.5 963.4x1 + 2362.4x2 + 4438.4x3 + 4358x4 + 5708.5x5 + 8438.4x6 + 29633.5x7 + 34508.5x8 + 38163x9 + 4438.4x10 + 12458.5x11 + 19883.5x12 + 49855x13 + 54603.6x14 x1 ≤ 20 x2 ≤ 20 x3 ≤ 15 x4 ≤ 20 x5 ≤ 10 x6 ≤ 10 x7 ≤ 3 x8 ≤ 3 x9 ≤ 3 x10 ≤ 15 x11 ≤ 7 x12 ≤ 2 x13 ≤ 2 x14 ≤ 2 x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 + x12 + x13 + x14 ≤ 132 table 6: the total income of all buses on the road at a time s/n dest. parking charges n b/rt fuel t.fare n cvs n vrm n income/trip n total expense n total income n 1 atan 0 20 43500 3000 5134 16098 84000 64732 19268 2 ewekoro 0 20 43500 4000 5134 16098 112000 64732 47268 3 ifo 0 15 43500 4500 3850.5 12073.5 126000 59424 66576 4 owode 0 20 43500 6000 5134 3219.6 168000 80830 87170 5 idiroko 0 10 36250 4000 2567 16098 112000 54915 57085 6 abeokuta 1500 10 43500 5000 2567 8049 140000 55616 84348 7 ijebu-ode 1050 3 30450 4500 770.1 4829.4 126000 39274.5 86725.5 8 ijebu-remo 1050 3 32625 5100 770.1 4829.4 141000 37099.5 88900.5 9 iperu 900 3 32625 5400 770.1 2414.7 151200 36708 114492 10 lagos 0 15 43500 4500 3850.5 12073.5 126000 59424 66576 11 ondo 1400 7 35525 4900 1796.9 11268.6 137200 49990.5 87209.5 12 oyo 600 2 20300 2300 513.4 3219.6 64800 24633 39767 13 osun 700 2 24550 4600 513.4 3219.6 128800 29083 99717 14 ekiti 70 2 27550 5100 513.4 4828 142800 33592.8 109207.2 total 7900 132 455875 62900 33884.4 118318.9 17579400 690054.3 1069345.7 120 latunde et al. / j. nig. soc. phys. sci. 1 (2019) 116–121 121 table 7: result of sensitivity analysis analysis on the model s/n bus total no of buses former b/rt new b/rt income generated n 1 x1 142 20 30 1,080,770.7 2 x2 142 20 30 1,115,520.7 3 x3 142 15 25 1,115,520.7 4 x4 142 20 30 1,114,721.7 5 x5 142 10 20 1,128,221.7 6 x6 142 10 20 1,155,520.7 7 x7 142 3 13 1,367,471.7 8 x8 142 3 13 1,416,221.7 9 x9 142 3 13 1,452,766.7 10 x10 142 15 25 1,115,520.7 11 x11 142 7 17 1,195,721.7 12 x12 142 2 12 1,269,971.7 13 x13 142 2 12 1,569,691.7 14 x14 142 2 12 1,617172.7 table 8: optimal allocation based on sensitivity of the parameters s/n destination former b/rt recommend b/rt 1 atan 20 20 2 ewekoro 20 20 3 ifo 15 15 4 owode 20 20 5 idiroko 10 10 6 abeokuta 10 10 7 ijebu-ode 3 3 8 ijebu-remo 3 3 9 iperu 3 3 10 lagos 15 15 11 ondo 7 7 12 oyo 2 2 13 osun 2 2 14 ekiti 2 12 figure 1: the graph of income generated against number of buses by routes however, as a result of sensitivity analysis carried on the parameters of the formulated model, we recommend that additional buses should be considered to the best route of sango-ekiti to optimize income and minimize cost. references [1] s. sarbjit, “note on transportation problem with new method for resolution of degeneracy”, universal journal of 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[9] p. pandian & k. kavitha, “sensitivity analysis in solid transportation problems”, applied mathematical sciences,6 (2012) 6787. 121 j. nig. soc. phys. sci. 5 (2023) 1392 journal of the nigerian society of physical sciences pre-functions and extended pre-functions of complex variables a. thirumalaia, k. muthunagaia,∗, ritu agarwalb aschool of advanced sciences, vit university, chennai-600 127, tamil nadu, india bdepartment of mathematics, malaviya national institute of technology, jaipur-302017, india abstract pre-functions are functions that possess a sequence { fn(z,β)} which tends to one of the elementary functions as n tends to infinity and β tends to 0. the main objective of this paper is to broaden the scope of pre-functions from functions of a real variable to functions of a complex variable by introducing pre-functions of a complex variable. we have analyzed the pre-functions of a complex variable for their properties. the pre-laguerre, pre-bessel and pre-legendre polynomials of a complex variable have been obtained as special cases. graphs have been used to visualize complex pre-functions. doi:10.46481/jnsps.2023.1427 keywords: pre-exponential functions, pre-trigonometric functions, pre-hyperbolic functions, extended pre-functions. article history : received: 01 march 2023 received in revised form: 09 may 2023 accepted for publication: 10 may 2023 published: 21 may 2023 c© 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: o. adeyeye 1. introduction exponential and logarithmic functions have wide variety of applications in science, medicine, business and many other fields. exponential functions are used to describe growth or decay of a quantity whose rate of change has a relation to its present value. logarithms have the ability to measure quantities that are vastly different but need an easy way to be talked about and compared to. on the other hand, trigonometry can be used in music, roofing a house, cartography, satellite system in naval, aviation industries and in many other fields. deo and howell [1] introduced and studied trigonometry and trigonometry like functions. khandeparkar et al. [2] have introduced and studied pre-functions of a real variable. motivated by their work, we have defined pre-functions of a com∗corresponding author tel. no: +91 9840084991 email address: muthunagai@vit.ac.in (k. muthunagai ) plex variable. functions which possess a sequence { fn(z,β), z ∈ c,β ≥ 0} are called pre-functions of a complex variable z, if they tend to one of the elementary functions as n → ∞ and β → 0. pre-functions also possess some of the properties possessed by the elementary functions but not properties like periodicity. pre-functions were found to be very simple and useful in the study of differential equations. for the methods and solutions of various differential equations one can refer [310]. 2. the pre-exponential function of a complex variable exponential functions play a significant role in almost every branch of mathematics. in this section, we have determined a set of functions called pre-exponential functions, owning a sequence that generalizes the exponential function ex p(z). 1 thirumalai et al. / j. nig. soc. phys. sci. 5 (2023) 1392 2 for any complex number z, the series form of preexponential function is given by, pexp(z,β) = 1 + z1+β γ(2 + β) + z2+β γ(3 + β) + z3+β γ(4 + β) + .... = 1 + ∞∑ n=1 zn+β γ(n + 1 + β) , β ≥ 0, (1) β being the parameter. figure 1: graphs of pexp(z, 0), pexp(z, 1) and pexp(z, 2) figure 1 represents the behaviour of the pre-exponential functions pexp(z, 0), pexp(z, 1) and pexp(z, 2) in order. note that when β = 0, pexp(z, 0) = exp(z). in general, pexp(z, n) = ∗pexp(z, n − 1) − zn n! = pexp(z, n − 2) − zn−1 (n − 1)! ..., n = 1, 2, 3, .. figure 2: z-plane specifically, pexp(z, 1) = exp(z) − z pexp(z, 2) = exp(z) − z − z2 2! pexp(z, 3) = exp(z) − z − z2 2! − z3 3! = exp(z) − s 3 where s 3 is the partial sum of pexp(z, 3). for n ∈ n, pexp(z, n) = exp(z) − s n (2) where s n = ∑n r=1 zr r! . also, note that pexp(z, n) = 1, ∀ z ∈ c as n →∞. replacing z by −z in (1), we have, pexp(−z,β) = 1 − (−1)β { z1+β γ(2 + β) − z2+β γ(3 + β) + z3+β γ(4 + β) − ... } = 1 + (−1)β ∞∑ n=1 (−1)n zn+β γ(n + 1 + β) . (3) in (3), replacing β by 0 pexp(−z, 0) = 1 − z 1! + z2 2! − z3 3! + ... = exp(−z) (4) in short, pexp(−z, n) = exp(−z) − s n where s n = ∑n r=1(−1) r zr r! . 3. pre-trigonometric functions of a complex variable the pre-trigonometric functions of a complex variable are defined by y1(z,β) = pcos(z,β) = 1 − z2+β γ(3 + β) + z4+β γ(5 + β) − z6+β γ(7 + β) + ... = 1 + ∞∑ n=1 (−1)n z2n+β γ(2n + 1 + β) , z ∈ c, β ≥ 0 (5) and y2(z,β) = psin(z,β) = z1+β γ(2 + β) − z3+β γ(4 + β) + z5+β γ(6 + β) − ... = ∞∑ n=0 (−1)n z2n+1+β γ(2n + 2 + β) , z ∈ c, β ≥ 0. (6) table 1 gives the expressions for the pre-trigonometric functions when β takes the values 0,1,2 and 3. 2 thirumalai et al. / j. nig. soc. phys. sci. 5 (2023) 1392 3 figure 3: w-plane figure 4: pcos(z, 1) = 1 − z 3 3! + z5 5! − z7 7! table 1: expressions for the pre-trigonometric functions s.no (z,β) pcos(z,β) psin(z,β) 1 (z, 0) cos(z) sin(z) 2 (z, 1) sin z − z + 1 1 − cos z 3 (z, 2) −cosz − z 2 2 + 2 z − sin z 4 (z, 3) −sin z + z − z 3 6 + 1 cos z + z2 2 − 1 4. euler’s formula for a complex number z, exp(iz) = cos z + i sin z. (7) using (1), the above expression can be rewritten as pexp(iz,β) = 1 − (i)β { z2+β γ(3 + β) − z4+β γ(5 + β) + z6+β γ(7 + β) + ... } +i(i)β { z1+β γ(2 + β) − z3+β γ(4 + β) + z5+β γ(6 + β) − ... } = 1 − (i)β{1 − pcos(z,β) − i psin(z,β)}, β ≥ 0 (8) figure 5: pcos(z, 0.6) = 1 − z 2.6 γ(3.6) + z4.6 γ(5.6) − z6.6 γ(7.6) figure 6: psin(z, 1) = z 2 2! − z4 4! − z6 6! figure 7: psin(z, 0.6) = z 1.6 γ(2.6) − z3.6 γ(4.6) − z5.6 γ(6.6) figure 8: pcosh(z, 1) = 1 + z 3 3! + z5 5! + z7 7! which is the general form of euler’s formula for preexponential function. clearly eiz = pexp(iz, 0) = pcos(z, 0) + i psin(z, 0) = cos z + i sin z. using −iz in place of iz, we have pexp(−iz,β) = 1 − (i)β{1 − pcos(z,β) + i psin(z,β)}, (9) which results in e−iz = cos z − i sin z, when β = 0. 3 thirumalai et al. / j. nig. soc. phys. sci. 5 (2023) 1392 4 figure 9: pcosh(z, 0.7) = 1 + z 2.7 γ(3.7) + z4.7 γ(5.7) + z6.7 γ(7.7) figure 10: psinh(z, 1) = z 2 2! + z4 4! + z6 6! figure 11: psinh(z, 0.7) = z 1.6 γ(2.6) − z3.6 γ(4.6) − z5.6 γ(6.6) figure 12: m3,0(z, 1) = 1 − z4 4! + z7 7! − z1 0 10! 5. relation between pre-circular and preexponential functions using euler’s formula, the relation between circular and exponential functions, we arrive at the following: pcos(z,β) = (−i)β pexp(iz,β) + (i)β pexp(−iz,β) 2 − (i)β + (−i)β 2 + 1 psin(z,β) = (−i)β pexp(iz,β) − (i)β pexp(−iz,β) 2i − (−i)β − (i)β 2i . ptan(z,β) = i (−i)β − (i)β − (−i)β pexp(iz,β) + (i)β pexp(−iz,β) (−i)β pexp(iz,β) + (i)β pexp(−iz,β) − (i)β − (−i)β + 2 psec(z,β) = 2 (−i)β pexp(iz,β) + (i)β pexp(−iz,β) − (i)β − (−i)β + 2 table 2: expressions for the pre-hyperbolic functions s.no (z,β) pcosh(z,β) psinh(z,β) 1 (z, 0) cosh(z) sinh(z) 2 (z,1) sinh z − z + 1 cosh z − 1 3 (z,2) cosh z − z 2 2 sinh z − z 4 (z, 2n) cosh z − ∑n r=1 z2r (2r)! sinh z − ∑n r=1 z2r−1 (2r−1)! pcosec(z,β) = 2i (−i)β pexp(iz,β) − (i)β pexp(−iz,β) + (i)β − (−i)β pcot(z,β) = −i (−i)β pexp(iz,β) + (i)β pexp(−iz,β) − (i)β − (−i)β + 2 (−i)β − (i)β − (−i)β pexp(iz,β) + (i)β pexp(−iz,β) (10) whenever they exist. for β = 0, these results give the relation between circular functions and exponential functions. 6. pre-hyperbolic functions of a complex variable the pre-hyperbolic sine and cosine functions are defined by h1(z,β) = pcosh(z,β) = 1 + z2+β γ(3 + β) + z4+β γ(5 + β) + z6+β γ(7 + β) + ... = 1 + ∞∑ n=1 z2n+β γ(2n + 1 + β) , z ∈ c, (11) and h2(z,β) = psinh(z,β) = z1+β γ(2 + β) + z3+β γ(4 + β) + z5+β γ(6 + β) + ... = ∞∑ n=0 z2n+1+β γ(2n + 2 + β) , z ∈ c. (12) table 2 gives the expressions for the pre-hyperbolic functions when β takes the values 0, 1, 2 and 2n. 7. relation between pre-hyperbolic and preexponential functions pcosh(z,β) = (−1)β pexp(z,β) + pexp(−z,β) 2 − 1 − (−1)β 2 , psinh(z,β) = (−1)β pexp(z,β) − pexp(−z,β) 2 − (−1)β − 1 2 , ptanh(z,β) = (−1)β pexp(z,β) − pexp(−z,β) − (−1)β + 1 (−1)β pexp(z,β) + pexp(−z,β) − 1 + (−1)β , psech(z,β) = 2 (−1)β pexp(z,β) + pexp(−z,β) − 1 + (−1)β pcosech(z,β) = 2 (−1)β pexp(z,β) − pexp(−z,β) − (−1)β + 1 , pcoth(z,β) = (−1)β pexp(z,β) + pexp(−z,β) − 1 + (−1)β (−1)β pexp(z,β) − pexp(−z,β) − (−1)β + 1 , if exists. assigning β the value 0, we find these relations reducing to the relations between exponential and hyperbolic functions. 4 thirumalai et al. / j. nig. soc. phys. sci. 5 (2023) 1392 5 8. extended pre-functions of a complex variable from generalized pre-trigonometric and pre-hyperbolic functions of a complex variable we have, pexp(−z,β) = 1 − (−1)β { z1+β γ(2 + β) − z2+β γ(3 + β) + z3+β γ(4 + β) − .... } = 1 + (−1)β ∞∑ n=1 (−1)n zn+β γ(n + 1 + β) . trisection of the above series leads us to three infinite absolutely convergent series for z ∈ c and β ≥ 0. they are m3,0(z,β) = 1 + ∞∑ n=1 (−1)n z3n+β γ(3n + 1 + β) m3,1(z,β) = ∞∑ n=0 (−1)n z3n+1+β γ(3n + 2 + β) m3,2(z,β) = ∞∑ n=0 (−1)n z3n+2+β γ(3n + 3 + β) , (13) with the initial conditions m3,0(0,β) = 1, m3,1(0,β) = 0, m3,2(0,β) = 0. one can easily verify m′3,0(z,β) = −m3,2(z,β) m′3,1(z,β) = (−1) β z β γ(1 + β) + m3,0(z,β) − 1 m′3,2(z,β) = m3,1(z,β). rewriting the above system in matrix form, we have  m3,0(z,β) m3,1(z,β) m3,2(z,β)  ′ =  0 0 −1 1 0 0 0 1 0   m3,0(z,β) m3,1(z,β) m3,2(z,β)  +  0 (−1)β z β γ(z+β) − 1 0   m3,0(z,β) m3,1(z,β) m3,2(z,β)  =  1 0 0  (14) we find the infinite series represented by m3,0(0,β) = 1, m3,1(0,β) = 0, m3,2(0,β) = 0 to be the solutions of the system of non-homogeneous equations given by (12). the expressions in (11) define the extended pre-trigonometric functions for n = 3 . specifically when β = 1 , we have m3,0(z, 1) = 1 + ∞∑ n=1 (−1)n z3n+1 γ(3n + 2) = m3,1(z, 0) − z + 1 m3,1(z, 1) = ∞∑ n=0 (−1)n z3n+2 γ(3n + 3) = m3,2(z, 0) m3,2(z, 1) = ∞∑ n=0 (−1)n z3n+3 γ(3n + 4) = −m3,0(z, 0) + 1 (15) figure 13: m3,0(z, 0.5) = 1 − z3.5 γ(4.5) + z6.5 γ(7.5) − z10.5 γ(11.5) figure 14: m3,1(z, 1) = z2 2! − z5 5! + z8 8! figure 15: m3,1(z, 0.5) = z1.5 γ(2.5) − z4.5 γ(5.5) − z7.5 γ(8.5) figure 16: m3,2(z, 1) = z3 3! − z6 6! + z9 9! 9. absolute convergence, analyticity and univalence of prefunctions we know that every absolute convergent series is convergent. but the converse is not true. in this section we have discussed about the absolute convergence of pre-exponential function using ratio test. pexp(z,β) = 1 + ∞∑ n=1 zn+β γ(n + 1 + β) 5 thirumalai et al. / j. nig. soc. phys. sci. 5 (2023) 1392 6 figure 17: m3,2(z, 0.5) = z2.5 γ(3.5) − z5.5 γ(6.5) − z8.5 γ(9.5) figure 18: pexp(z, 0.8) = 1 + z 1.8 γ(2.8) + z2.8 γ(3.8) + z3.8 γ(4.8) consider ∣∣∣∣∣ an+1an ∣∣∣∣∣ = ∣∣∣∣∣ γ(n + 1 + β)γ(n + 2 + β) ∣∣∣∣∣ = ∣∣∣∣∣ (n + β)!(n + 1 + β)! ∣∣∣∣∣ = ∣∣∣∣∣ (n + β)!(n + β + 1)(n + β)! ∣∣∣∣∣ = ∣∣∣∣∣ 1n + β + 1 ∣∣∣∣∣ lim n→∞ ∣∣∣∣∣ an+1an ∣∣∣∣∣ = 0 in similar lines the absolute convergence of the other prefunctions can also be proved. as the pre-exponential, pretrigonometric, pre-hyperbolic and extended pre-functions are all polynomials with infinite number of terms, they are analytic throughout the complex plane, (i.e.) they are entire functions. also they are univalent. as any function that is both analytic and univalent is conformal, so are pre-functions. 10. the transformation w = 1pexp( z,β) in this section, we have obtained the image of |pexp(z,β) − 2| = 1 under the transformation w = 1pexp(z,β) . w = 1pexp(z,β) ⇒ pexp(z,β) = 1 w and pexp(z,β) = 1 + ∑ ∞ n=1 zn+β (n+β)! . |pexp(z,β) − 2| = 1 ⇒ ∣∣∣∣∣1 + ∞∑ n=1 zn+β (n + β)! − 2 ∣∣∣∣∣ = ∣∣∣∣∣ 1w − 2 ∣∣∣∣∣ ⇒ ∣∣∣∣∣ ∞∑ n=1 zn+β (n + β)! − 1 ∣∣∣∣∣ = ∣∣∣∣∣ 1 − 2ww ∣∣∣∣∣ approximating the series to only one term we have∣∣∣∣∣ z22 − 1 ∣∣∣∣∣ = ∣∣∣∣∣ 1 − 2ww ∣∣∣∣∣ [n = 1,β = 1] now ∣∣∣∣∣ 1 − 2ww ∣∣∣∣∣ = 1∣∣∣∣∣1 − 2w ∣∣∣∣∣ = |w| |1 − 2(u + iv)| = |u + iv| (u − 2 3 )2 + v2 − 1 9 = 0 which is a circle in the w-plane. figures 2 and 3 are visualization of the given transformation. 11. visualization of certain pre-functions the extended trigonometric functions m3,0(z), m3,1(z) and m3,2(z) are found to be the linear independent solutions of the differential equation z ′′′ + z = 0. the properties possessed by these functions are similar to that of the classical trigonometric functions but for periodicity. due to lack of periodicity we see the graph to be oscillating with interlacing zeros. the parametric equations y1 = m3,0(z), y2 = m3,1(z), y3 = m3,2(z) will generate a surface y31 − y 3 2 + y 3 3 + 3y1y2y3 = r 3. for β = 1, the graphs of pre-trigonometric and extended pre-trigonometric functions are found to be oscillating and at the same time loosing periodicity. the graphs in figures 4-18 show how specific pre-functions behave for some fixed values of β. 11.1. special cases following are some of the special cases obtained as a result of our study about the pre-functions of a complex variable. 1. from the first identity of (13), we obtain m3,0(z1 + z2, 1) = m3,1(z1 + z2, 0)−(z1 + z2) + 1(16) by replacing z by z1 + z2 in it. 2. trisecting the series (1) for pexp(z,β), three infinite absolutely convergent series namely n3,0(z,β), n3,1(z,β), n3,2(z,β) for z ∈ c, β ≥ 0 have been obtained and these series define extended hyperbolic functions for n = 3. proceeding in similar lines as it has been done for n = 2 and n = 3, n-section of the infinite series pexp(−z,β) and pexp(z,β) give rise to generalized extended trigonometric and hyperbolic functions. 6 thirumalai et al. / j. nig. soc. phys. sci. 5 (2023) 1392 7 3. generating function for pre-laguerre polynomial can be obtained from pexp(z,β), by replacing z using −zyz−1 . 1 (1 − z) pexp ( −zy 1 − z ,β ) = 1 (1 − z) { 1 + (−1)β ∞∑ r=0 (−1)r zr+βyr+β (1 − z)r+βγ(r + 1 + β) } = 1 (1 − z) + ∞∑ r=0 (−1)r+β γ(r + 1 + β) zr+βyr+β (1 − z)r+β+1 = 1 (1 − z) + ∞∑ r=0 (−1)r+β γ(r + 1 + β) zr+βyr+β(1 − z)−(r+β+1) = 1 (1 − z) + ∞∑ r=0 (−1)r+β γ(r + 1 + β) zr+βyr+β ∞∑ t=0 (r + t + β)! (r + β)!t! zt = 1 (1 − z) + ∞∑ r,t=0 (−1)r+β (r + t + β)! γ(r + β + 1)(r + β)!t! yr+β ∞∑ t=0 zr+t+β for a fixed value of r and taking r + t = n, the coefficient of zn is (−1)r+β (n + β)! γ(r + β + 1)(r + β)!(n − r)! yr+β. taking all possible values of r into account, the total coefficient of zn is obtained to be n∑ r=0 (−1)r+β (n + β)! γ(r + β + 1)(r + β)!(n − r)! yr+β = ln(y,β), s = n − r ≥ 0 or r ≤ n. here ln(y,β) represents the laguerre polynomial when β = 0. ln(y, 0) = n∑ r=0 (−1)r n! (r)!(n − r)! yr = ln(y) 1 (1 − z) pexp {( −zy 1 − z ,β ) − 1 } = ∞∑ n=0 zn+βln(y,β) 4. we can also obtain the generating function for prebessel polynomial using pre-hyperbolic sine function. in psinh(z,β), replacing z by zx2 , we have (3n + 1 + β)! −(n + 1 −β)! psinh ( zx 2 ,β ) = (3n + 1 + β)! −(n + 1 −β)! { ∞∑ n=0 z2n+1+βx2n+1+β 22n+1+βγ(2n + 2 + β) } = ∞∑ n=0 (3n + 1 + β)!z2n+1+βx2n+1+β −(n + 1 −β)!22n+1+βγ(2n + 1 + β) = ∞∑ n=0 (3n + 1 + β)!z2n+1+βx2n+1+β −(n + 1 −β)!22n+1+β(2n + 1 + β)! for a fixed n and setting k = 2n + 1, the coefficient of zn is n∑ k=0 (k + n + β)! (n − k + β)!(k + β)! xk+β 2k+β = yn(x,β), k ≤ n. here yn(x,β) represents the bessel polynomial when β = 0. yn(x, 0) = n∑ k=0 (k + n)! (n − k)!(k)! ( x 2 )k = yn(x) (3n + 1 + β)! −(n + 1 −β)! psinh ( zx 2 ,β ) = ∞∑ n=0 zn+βyn(x,β) 5. replacement of z using zx2 yields the generating function for pre-legendre polynomial (3n + α + 1)! (3m + n + 1 + α)!(2n + m + 1 + α)! psin ( zx 2 ,α ) = (3n + α + 1)! (3m + n + 1 + α)!(2n + m + 1 + α)! ∗ { ∞∑ n=0 (−1)n z2n+1+α x2n+1+α 22n+1+αγ(2n + 2 + α) } = ∞∑ n=0 (−1)n(3n + α + 1)!z2n+1+α x2n+1+α (3m + n + 1 + α)!(2n + m + 1 + α)!22n+1+αγ(2n + 1 + α) = ∞∑ n=0 (−1)n(3n + α + 1)!z2n+1+α x2n+1+α (3m + n + 1 + α)!(2n + m + 1 + α)!22n+1+α(2n + 1 + α)! (17) fixing n and taking α = −(n + 2m + 1−β), the coefficient of zn is m∑ m=0 (−1)m+β (2n − 2m + β)!xn−2m+β (m + β)!(n − m + β)!(2n−2m+β)(n − 2m + β)! = pn(x,β), (18) m ≤ n. here pn(x,β) represents the legendre polynomial when β = 0. pn(x, 0) = m∑ m=0 (−1)m (2n − 2m)!xn−2m m!(n − m)!(2n−2m)(n − 2m)! = pn(x) =⇒ (3n + α + 1)! (3m + n + 1 + α)!(2n + m + 1 + α)! psin ( zx 2 ,α ) = ∞∑ n=0 zn−2m+βpn(x,β) (19) 12. conclusion in this paper, we have introduced and investigated the properties of pre-functions and extended pre-functions for a complex variable. by fixing the value of β, we were able to graph these pre-functions and extended pre-functions. for suitable choices of z, we observe that these functions reducing to leguerre, bessel, and legendre polynomials, among other special functions. some more special functions can be derived by assuming simple functions for the variable z. references [1] s. g. deo & g. w. howell,”a highway to trigonometry”, bull.marathawada mathematical society 1 (2000) 26-62. 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[7] r. b. khandeparkar, “pretrigonometric and prehyperbolic functions via laplace transforms”, neural, parallel and scientific computations 18 (2010) 423. [8] s. b. dhaigude & d. chandradeepa, “some results on pre-functions and differential equations”, bull. marathawada mathematical society 12 (2011) 18. [9] s. b. dhaigude & c. d. dhaigude, “prefunctions and integral equations via laplace transforms”, international journal of engineering sciences 2 (2013) 204. [10] m. mahmoudi & m. v. kazemi, “solving singular bvps ordinary differential equations by modified homotopy perturbation method”, journal of mathematics and computer science 7 (2013) 138. 8 j. nig. soc. phys. sci. 5 (2023) 1576 journal of the nigerian society of physical sciences isolation, characterization, antimicrobial and theoretical investigation of some bioactive compounds obtained from the bulbs of calotropisprocera m. e. khana,∗, c. e. elumb, a. o. ijeomahb, p. j. amejia, i. g. osigbemhec, e. e. etimf, j. v. anyamb, a. abeld, c. t. agbere a department of chemistry, federal university lokoja, kogi state, nigeria b department of chemistry, federal university of agriculture makurdi, benue state, nigeria c department of industrial chemistry, federal university lokoja, kogi state, nigeria d department of chemistry college of education hong, adamawa state, nigeria e department of chemistry, benue state university, makurdi, benue state, nigeria f department of chemical sciences, federal university wukari, taraba state, nigeria abstract this study characterizes the bioactive molecules from the bulb of calotropisprocera and investigates the antimicrobial activities of the crude extracts. theoretical studies on the two isolated compounds in the crude extract were also accomplished.the bulbs were air dried, pulverized, and subjected to extraction procedures by maceration using 500 ml each of normal-hexane, ethyl acetate and methanol. the crude extracts were further tested onmicroorganisms and phytochemical screening using standard procedures. in addition, the bioactive compounds in the extract were screened against dna gyrase of two gram negative bacterial species; escherichia coli and salmonella typhiusing molecular docking simulation techniques and further subjected to admet profiling,using the swiss adme online server. the crude ethyl acetate extract has the highest effective activity against escherichia coli (mic 2.5mg / ml and mbc/mfc 5mg / ml), staphylococcus aureus (mic 2.5mg/ml), candida albicans, salmonella typhi and candida stellafoidea (mic 5mg/ml). β-amyrin acetate and taraxasterol are the two phytochemicals in the purified white crystalline fractions and were found to fasten to the active sites of dna gyrase of the gram negative bacterial species via hydrophobic and hydrogen bond interactions, with binding activity value of -9.6 kcal/mol and -9.5 kcal/mol, respectively. also, admet investigations of the compounds revealed their sound oral bioavailability and excellent pharmacokinetic and toxicity profiles. the findings of this study could provide a platform for discovering safe and potent antibiotics against pathogenic microbes ravaging our society. doi:10.46481/jnsps.2023.1576 keywords: phytochemical screening, anti-microbial screening, calotropisprocera, pharmacokinetics article history : received: 26 may 2023 received in revised form: 23 june 2023 accepted for publication: 24 june 2023 published: 23 august 2023 © 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: e. a. emile ∗corresponding author tel. no: +234 7031667488 email address: khandora2000@gmail.com, muluh.khan@fulokoja.edu.ng (m. e. khan ) 1. introduction natural products are the most successful precursors of future drug leads [1-5]. it has been, it is, and will continue to 1 khan et al. / j. nig. soc. phys. sci. 5 (2023) 1576 2 provide distinctive structural variety that presents opportunities for discovering, mainly new low molecular weight lead molecules [6]. biosynthesis and a critical analysis of proteins, fatty acids, nucleic acids and carbon derivatives is known as primary metabolism and the molecules are called primary metabolites [7]. the mechanism by which an organism biosynthesizes bioactive molecules or secondary metabolites is a unique one [8]. these metabolites generally, are not actually needed for growth, development or reproduction of an organism but are produced either for adapting the organism to its surrounding environment or to act possibly for defence mechanism against predators in assisting survival of the organism [9]. biosynthesis of these metabolites is gained from the fundamental processes of photosynthesis, glycolysis and the krebs cycle to procure biosynthetic intermediates which, ultimately, results in the formation of these secondary metabolites [7]. though the number of building blocks is limited, the formation of new secondary metabolites is infinite. utmost important constructing blocks employed in the biosynthesis of these metabolites are those obtained from the intermediates: acetyl coenzyme a (acetyl-coa), shikimic acid, mevalonic acid and 1-deoxyxylulose-5-phosphate. these are involved in countless biosynthetic pathways, involving numerous different mechanisms, actions reactions, inactions and interactions. calotropisprocera, asmall tree is widely spread in tropical and subtropical africa [10]. it is called different names in english and locally, like “sodom apple, usher, dead sea apple, swallow-wort, giant milk weed ”tumfafiya or baabaa ambalee in hausa, ewe-bomubomu in yoruba, otokwuru in igbo and konzar atumbagh in tiv [11]. the plant is hardy, pubescent, evergreen, erect, a compact shrub up to 4.5 m tall, and is covered with cottony tomentum. the stem is usually simple, rarely branched, woody at base and covered with fissures. various parts of this plant have the ability to produce large quantities of latex when cut or broken [12]. calotropisprocera (asclepiadaceae) is recognised by most traditional systems of medicine as ‘madar” in unani medicinal system. earliest hindu writers mentioned the plant and its primeval name occurred in the vedic literature was “arka” alluding to the form of leaves, which was used in the sacrificial cremation described by the sanskrit writers [13]. the latex exuded by calotropisprocera is valued for its high medicinal and pharmaceutical activity due to its high content of bioactive compounds such as;cardiac glycosides, alkaloids, terpenes, resins, lipids, flavonoids, tannins and steroids [14]. this latex possesses different biological activities including: anti-inflammatory, analgesic, antitumor, antiviral, hepatoprotective, antiulcer, anthelmintic, insecticidal, herbicidal, antioxidant and spasmolytic activities [15]. theoretical chemistry concepts such as in silicomolecular docking, pharmacokinetic and toxicity profiling has found immense applications in the discovery of new bioactive molecules for the treatment of various diseases [16-18]. molecular docking entails the prediction of the binding interaction of bioactive compounds (ligands) with the active sites of a target macromolecule (receptor). the ligands with the most stable conformations with the target protein are the most promising drug candidates [19]. molecular docking technique has been widely used in pharmaceutical research in recent years because of its fast and cost effectiveness in screening data base of bioactive ligands [20-25, 18]. likewise, pharmacokinetic investigation which is concerned with the fate of therapeutic ligands in the biological system forms an essential component of modern drug discovery. this deals with the absorption, distribution, metabolism, excretion, and toxicity (i.e., admet) potentials of bioactive ligands. in silico admet profiling of drug candidates helps to minimize attrition rates during the preclinical and clinical stages of drug development [26-28].this study is aimed at isolation, characterization and application of in vitro & in silico techniques to explore the bioactive molecules present in bulbs of calotropisprocera. 2. methods 2.1. sample collection the fresh bulbs of the plant identified as calotropisprocera (figure 1) was collected from millionaires’ quarters, lafia l. g a. of nasarawa state nigeria, in august, 2019 and was authenticated at the college of forestry and fisheries, federal university of agriculture makurdi, benue state nigeria with voucher no fh/0086, deposited at the college herbarium. the bulbs were washed with clean water to removed dirt and air dried at room temperature. the dried fruits were then pulverized into coarse powder with mortar and pestle, then sieved with a sieve of 0.55 mm pores and stored in cellophane bags at room temperature until required for experimental use. figure 1: plant and bulbs of calotropisprocera 2.2. extraction 500g of the plant material was weighed and extracted by the process of maceration, allowing the pulverized powdered material to soak in an appropriate solvent in a closed vessel at room temperature. three chosen solvents were employed, in the following order, normal-hexane, ethyl acetate and methanol. the plant sample was macerated with two litres of each of the solvents for 72 hours with agitation. the above was filtered and the filtrate concentrated under reduced atmospheric pressure using a rotary evaporator at 37oc, to recover some of the 2 khan et al. / j. nig. soc. phys. sci. 5 (2023) 1576 3 solvent. this procedure was repeated for the ethyl acetate and methanol extracts. they were then allowed to dry. the various solvents extracts were coded for quick identification during further analysis. 2.3. screening for phytochemicals crude extracts from the used solvents were phytochemically evaluated for the presence of anthraquinones, saponins, tannins, steroids, terpenes, reducing sugars, flavonoids and alkaloids using standard scrutiny as reported by [29, 30]. phytochemical screening results showed the presence of saponins, tannins, steroids/sterols, tannins, terpenoids, flavonoids, reducing sugars, and cardiac glycosides. 2.4. bioassay antibacterial and antifungal activities of the normalhexane, ethylacetate and methanol crude extracts were investigated using clinical isolates of some pathogenic microbes such as: staphylococcus aureus, staphylococcus faecalis, escherichia coli, vancomycin enterococci, neisseria gonorrhoae, profeusmirabili, pseudomonas aeruginosa, salmonella typhi, candida albicans, candidakrusei and candida stellafoidea. dispersion method was used for screening the extracts. mueller hinton agar was used as the growth medium for microorganisms, sterilized at 121°c for 15 min, poured into sterile petri dishes and were cooled and solidified. the said crude extracts (0.4 g) were allowed to dissolve in 10.0 ml of dimethylsulphoxide (dmso) to acquire a concentration of 40.0 mg/ml. a disinfected medium was then seeded with the standard inoculum (0.1 ml) of test microorganisms and were equitably spread over the surface of the medium with sterile swabs. using a 6mm standard cork-borer, a well was cut at the centre of each inoculated medium. a concentration of 5.0 mg/ml of the already weighed crude extract was then launched into each well on the inoculated medium. the inoculated medium was incubated at 37oc for 24 hours, after which the medium was observed for the zones of inhibition which were measured with a transparent ruler [31]. minimum inhibition concentration (mic) of these extracts were carried out using broth agar dilution method. mueller hinton broth was prepared by dispensing 10.0 ml into test tubes and pasteurised at 37°c for 6 hours, then cooled. mcfarland’s turbidity standard scale number 0.5 was prepared to give a turbid solution. normal saline (10.0 ml) which was dispensed into sterile test tubes and the test microbes were inoculated and incubated at 37oc for 24 hours. thereafter, the tubes were observed for turbidity. the lowest concentration of extracts in the sterile broth that indicated no turbidity was recorded as the minimum inhibition concentration (mic) [32]. the minimum bactericidal and minimum fungicidal concentration (mbc and mfc) were carried out to determine the concentration of extracts that could stop growth of tested microorganisms. mueller hinton agar was prepared, pasteurised at 121oc within 15 minutes, poured into sterile petri dishes and allowed to cool. the contents of the test tubes with the evaluated mic were then sub-cultured onto prepared media, incubated at 37oc for 24 hrs, after which the plates were visualised for any colony growth. mbc/mfc plates with lowest concentration of extract, without a colony growth were considered as the mbc/mfc [32]. 2.5. spectroscopic characterization 1h and 13cnmr spectra of β -amyrin acetate (cp17) and taraxasterol (cp35) were run using cdcl3 as solvent on agilent-nmr 500mhz spectrophotometer at strathclyde institute of pharmacy and biomedical sciences, university of strathclyde glasgow, united kingdom. 2.6. molecular optimization and docking procedures antimicrobial studies on the crude extracts revealed that they possess profound bioactivities against escherichia coli and salmonella typhi. also, phytochemical screening divulged the presence of β -amyrin acetate and taraxasterol. guided by these findings, the two bioactive ligands were subjected to geometry optimization using the semi-empirical (pm3) method of spartan’14 software to obtain their minimum energy geometries. molecular docking simulation was used to screen the compounds against dna gyraseof escherichia coli and salmonella typhi. the 3d structure of the target protease (pdb code: 5ztj) was retrieved from protein data bank at www. rcsb. org/pdb. the water molecules, hetero-atoms, and cocrystallized ligands attached to the retrieved protease were removed using the biovia discovery studio interface. the target protein was further processed via the addition of polar hydrogens and kollman charges with the aid of auto dock vina tool v1.5.7. finally, the docking calculation between the ligands and the target dna gyrase was performed using pyrx software of auto dock vina tool by centring the vina search space at x: -26.4401 å, y: 23.0598 åand z: 22.0813 åwith dimensions of x: 51.6292 å, y: 53.0044 åand z: 51.8136 å[26-27, 33-34]. 2.7. drug-likeness estimation this is the evaluation of oral bioavailability of therapeutic compounds. here, in silico technique forms a pivotal part of modern drug discovery owing to the fact that most drugs are administered via the oral route. the assessment of oral bioavailability of β -amyrin acetate and taraxastero were performed using the lipinski’s rule of five and the veber’s rule. according to the lipinski’s rule, a drug must have its molecular weight (mw) ≤ 500, number of hydrogen bond donors (hbd) ≤ 5, octanol/ water partition coefficient log p ≤ 5 and number of hydrogen bond acceptors (hba) ≤ 10 for it to be orally bioavailable and violation of more than one of these indices could translate to poor drug-likeness potentials of a ligand. veber’s rule, on the other hand stipulates that, for a drug to be orally bioavailable, the number of rotatable bonds (nrb) must be < 10 and topological polar surface area (tpsa) must be < 140 å2. the nrb, tpsa, mw, hbd, hba, and log p value of the bioactive ligands were computed using the swiss adme (http: //www.swissadme.ch/www.swissadme.ch/) online tool [2627, 33-34]. 3 khan et al. / j. nig. soc. phys. sci. 5 (2023) 1576 4 2.8. admet profiling the high failure rates of drug candidates at the late stage of drug development has made prediction of pharmacokinetic and toxicity profiles of therapeutic ligands at the early stage of drug development a necessity. the phytochemicals were profiled for their gastrointestinal (gi) absorption, blood brain barrier (bbb) permeation, p-glycoprotein (p-gp) substrate potentials, and cytochrome-p450 enzymes inhibition using the swiss adme online server at http://www.swissadme.ch/ index.php. furthermore, osiris data warrior v5.5.0 chemoinformatics program was used to perform in silico toxicity assay on the ligands using the following toxicity endpoints; mutagenicity, reproductive effect, and irritating effect [26-27, 3334]. 3. results 3.1. phytochemical and antimicrobial screening phytochemical and antimicrobial screening of the crude extract results are presented in tables 1 and 2, respectively. table 1: phytochemical screening of the bulb extacts of calotropisprocera class of compound normalhexane ethyl acetate methanol saponins + + + tannins + + + flavonoids + + + steroids/sterols + + + alkaloids + + + reducing sugars + + + cardiac glycosides + + + anthraquinone + key = (+) indicate, present, (-) indicate, below detectable limits 3.2. elucidated structures of the bioactive compounds figure 2 presents elucidated structures of the two bioactive ligands present in the crude extract of the bulbs of calotropisprocera. experimental and literature data for 1h and 13cnmr analysis are presented in tables s1-s4 in the supplementary file (below). figure 2: elucidated chemical structures of bioactive compounds in the extract 3.3. molecular docking outcome the molecular docking simulation derived binding affinities and diagrams of interaction of the ligands with the active sites of the dna gyrasetarget are presented in figure 3. figure 3: binding affinity and diagrams of interaction of the investigated bioactive ligands with the active sites of dna gyrase 4. oral bioavailability and admet profiles of the phytochemicals oral bioavailability of a therapeutic ligand is a function of some of its physicochemical properties. tables 3 and 4 present 4 khan et al. / j. nig. soc. phys. sci. 5 (2023) 1576 5 table 2: sensitivity/zone of inhibition (mm) of the crude extracts against microorganisms test organism ea meoh hexane ciprofloxacin tetracycline fluconazole nectricillin resistant staph. aureus s (24) s (21) s (20) r (0) s (29) r (0) vancomycin resistant enterococci r (0) r (0) r (0) s (28) s (30) r (0) staphylococcus aureus s (27) s (24) s (21) r (0) s (32) r (0) escherichia coli s (28) s (23) s (20) s (35) r (0) r (0) neisseria gonorrhea s (26) s (22) s (18) r (0) s (25) r (0) profeus mirabilis r (0) r (0) r (0) s (32) r (0) r(0) pseudomonas aeruginosa r (0) r (0) r (0) s (30) s (27) r (0) salmonella typhi s(24) s (21) s (18) s (41) r (0) r (0) candida albicans s (25) s (22) s (20) r (0) r (0) s (32) candida krusei r (0) r (0) r (0) r (0) r (0) s (30) candida stellafoidea s (23) s (20) s (17) r (0) r (0) s (34) legend =� s= sensitive, r= resistance, ea= ethyl acetate extract, meoh= methanol extract, hexane= n-hexane extract, numeric value in brackets = diameter of zone of inhibition in millimetres; drug concentration: ciprofloxacin = 20 µ g, tetracychine = 20 µg, fluconazole = 20 µg the descriptors of oral bioavailability and admet profiles of the ligands, respectively. table 3: oral bioavailability profiles of the phytochemicals ligand rule β-amyrin acetate taraxasterol lipinski’s yes yes hba 2 1 hbd 0 1 mw (gmol-1) 440.7 426.7 clogp(o/w) 7.0 7.1 veber’s yes yes nrb 2 0 tpsa ( å2) 26.3 20.23 hba; hydrogen bond acceptor, hbd; hydrogen bond donor, mw; molecular weight, clogp; consensus octanol water partition coefficient, nrb; number of rotatable bond, tpsa; topological polar surface area 5. discussion phytochemical screening of the crude extracts of the plant divulged the presence of reducing sugars, saponins, steroids, tannis, alkaloids and flavanoids. anthraquinones were below detection level (table 1). this is corroborated with the work of (35), when they analysed the root bark of terminaliaschimperiana. screening for antimicrobials was carried out against nectricillin resistant staphylococcus aureus, vancomycin resistant enterococci, escherichia coli, profeus mirabilis, salmonella typhi, pseudomonas aeruginosa, candida albicans, neisseria gonorrhea, candida krusei and candida stellafoidea. parameters determined were the zone of inhibition (zi), minimum inhibitory concentration (mic) and minimum bactericidal/fungicidal concentration (mbc/mfc). the antibacterial and antifugal results obtained showed that ethyl acetate extract had the highest diameter of zones of inhibition (28 mm) against (35 mm) for the tested microbes. over all, ethyl acetate has the highest efficacy and efficiency of the applied solvents, followed by methanol and the n. hexane. out of the eleven (11) microorganisms employed, seven (7) had various degrees of activities, with the exception of, vancomycin resistant enterococci, profeus mirabilis, pseudomonas aeruginosa , and 5 khan et al. / j. nig. soc. phys. sci. 5 (2023) 1576 6 table 4: in silico pharmacokinetic and toxicity profiles of the phytochemicals ligand cyp450 substrate gia p-gp+ bbb mutagenic irritant reproductive effect tumorigenic logkp (cm/s) β -ac yes yes no no none none none none -2.6 ta yes yes no no none none none none -2.4 gia; gastrointestinal absorption, bbb; blood brain barrier penetration, p-gp+; p-glycoprotein substrate β-ac; β -amyrin acetate, ta; taraxasterol candida krusei, all of which had zero activities, as seen in table 2. the proton analysis (1h nmr spectra) of β -amyrin acetate (table s2), showed the presence of the trademark amyrinolefinic proton (1h-12) resonating as triplet at δ 5.16 ppm due to deshielding caused by the π bond. the proton nmr signals at δ 0.85, 0.98, 0.94, 0.81, 1.11, 0.85, 0.77, and 0.92 ppm depict eight methyl groups all singlets with an additional methyl group at δ 2.02 being the acetate methyl protons. at δ 4.47ppm, the double doublet is due to the lone proton on c-3 bonded equatorial to an acetate ester. broad singlet at δ 0.85 ppm of c-29 and c-30 methyl protons is a pointer to the characteristics of β – amyrin. furthermore, 13c nmr of the lone proton δ 4.47 (table s1), is assigned h-3 by hsqc correlation with δ 80.35 and hmbc3j correlation to methyls δ 15.71 (c-23), δ 28.30 (c-24) and δ 170.94 ppm (c-1’) from the acetyl moiety. hsqc correlation of δ 21.53 to δ 2.02 (3h) is evident of electron rich δ 170.94 ppm (c-1’) of the acetyl moiety. the acetate is attached to the first hexacyclic component at c-3. δ 5.16 and correlates with δ 121.68 ppm on hsqc and is ascribed h-12. it shows a hmbc2j correlation with the secondary carbon δ 23.75 (c-11) and 3j quaternary carbon δ 42.07ppm (c-14). h-12 bonds to the electron rich alkenyl carbon (c-12) and splitting with δ 1.80 (h-11) thus, resolved as triplets. δ 33.42 (c-29) axial and δ 23.86 ppm (c-30) equatorial correlations to 1h-19a, 1h-19b and 3h-28 indicates their association to ring e of amyrin. moreover, δ 0.85 (c-30 & c-29) 2j correlation to the quaternary δ 31.10 ppm (c-20) indicates the β – functionality of these methyls. the compound / molecule was established as β – amyrin acetate (figure 2a), on the basis of spectral data (1d1h-nmr, 13c nmr, 2d hsqc, hmbc spectroscopy, melting point 242oc, and tlc. more so, comparison with literature data (36-37), confirmed the molecule as β –amyrin acetate. studies show this compound to possess profound anti-inflammatory, anti-malarial and anti-rheumatism activities (38) and (39). spectroscopic signals in the pmr (table s4) and 13c nmr (table s3) spectra of taraxasterol were assigned completely using oneand two-dimensional (2d) nmr methods (hsqc and hmbc). the weak-field region of the pmr of the compound contains signals for three protons. based on cross-peaks in the hsqc, the first two protons at 5.10 ppm belong to c-30 with chemical shift 107.25 ppm. the doublet of doublets at 4.70 ppm corresponds with the proton on c-3 (80.75 ppm) of ring a. cross-peaks with c-2 and a quaternary c atom at 37.74 ppm (c-4) are found in the hmbc spectrum for h-3. cross-peaks with protons at 0.78 ppm (1h, triplet; corresponds with the c atom with chemical shift (cs) 55.42 ppm) at hsqc. the protons with chemical shift 0.91 ppm in the hsqc correspond with signals for c atoms with chemical shifts 28.06 and 16.36 ppm of the gem-dimethyls c-23 and c-24. their protons also correlate with tertiary (55.42 ppm) and quaternary (37.74 ppm) c atoms c-5and c-4, respectively. their characteristic atoms c-6 (18.37 ppm) and c-7 (34.13 ppm) are found using the hsqc spectra. protons h-1 and h-5 and protons of the methyl with chemical shift 0.86 ppm, give correlations in the hmbc spectrum with a quaternary c atom with δ 38.01 ppm. these are c-10 and methyl c-25. in the hsqc spectrum, the last signal correlates with h-9 with δ1.33 ppm which, in turn, correlates with c-10 and yet another quaternary c atom with chemical shift 40.62 ppm (c-8), c-8 couples with methyl protons at 0.99 ppm (on c-26 with δ16.36 ppm). the doublet for methyl protons h-29 at 1.08 ppm in the hsqc spectrum corresponds with the c atom at 25.60 ppm(c-29). corresponding cross-peaks in the hmbc spectrum between methyl protons in ring d at 1.08 ppm and c atoms at 38.30 (c-19) and 154.65 ppm (c-20) in addition to c-29 at 25.60 ppm and a proton at 2.15 ppm (h-19) confirm that the assignments were correct. methyl protons with cs 0.94 ppm, which correspond in the hsqc spectrum with the c atom at 19.58 ppm (c-28) have correlations in the hmbc spectrum with c atoms at 34.41 and 48.70 ppm (tables s3 and s4). these assignments in comparison with literature data (40-41), confirmed the compound to be taraxasterol (figure 2b). the presence of carbon spectrum in 145ppm and 178ppm at the 13c nmr indicate the presence of fatty acid impurities. this compound was reported to have many biological properties, including anti-inflammatory and anti-tumour activities [42]. oral bioavailability (drug-likeness) prediction for therapeutic ligands is a fundamental evaluation in novel drug discovery and development owing to the fact that oral delivery remains the most common path of drug delivery into the systemic circulation [34]. this essential parameter was assessed for the two phytochemicals taking cognisance of lipinski’s rule of five and the veber’s rule. the results, thus, presented in table 3, implies that they obey both rules. thus, the extract could be taken through the oral route. a major cause of high attrition rate in drug discovery and development is poor pharmacokinetic and toxicity profiles of drug candidates. a way of circumventing this challenge is early prediction of admet profiles of drug candidates. table 4 presents the in silico admet profile of the two investigated 6 khan et al. / j. nig. soc. phys. sci. 5 (2023) 1576 7 phytochemicals. all the ligands were found to possess good gastrointestinal absorption. the blood brain barrier (bbb) is a layer of endothelial cell that demarcate the brain from blood [44]. the assessment of bbb permeating potentials of the compounds shown in table 4 portrayed that none of the compounds can penetrate the bbb and as such would not have any influence on the central nervous system. also, p-glycoproteins (p-gp) are intracellular and extracellular membrane transporters of xenobiotic in the body [45]. it reduces cellular concentrations of its substrates leading to their poor pharmacokinetic profiles. all the investigated phytochemicals were found to be none substrates of p-gp (table 4). furthermore, cytochrome 450 (cyp450) monooxygenase is a group of enzymes central to the metabolism and excretion of drugs. the bioactive ligands were screened against five isoforms of the enzyme. the result (table 4) revealed that all the phytochemicals are substrate of cyp450 enzyme indicative of their high probabilities of been bio-transformed and eventually made bio available upon oral administration [46]. another important pharmacokinetic parameter worthy of consideration especially for therapeutic compounds that requires transdermal administration is the skin permeability (logkp). the logkp data of β -amyrin acetateand taraxasterol presented in table 4 revealed that both the compounds have poor skin penetration potentials due to the negative values of their logkp (45). additionally, the toxicity profiles (table 4) of the compounds revealed that none of them is mutagenic, tumorigenic, irritating, or pose any adverse effect on the reproductive system. 6. conclusions crude extracts of the bulbs of calotropisprocerawere investigated for the antimicrobial activities against eight bacteria and three fungi species. the extracts were found to exhibit significant inhibitory activities against the microbes with the crude ethyl acetate extract having the highest effective activity against escherichia coli (mic 2.5mg/ml and mbc/mfc 5mg/ml), staphylococcus aureus (mic 2.5mg/ml), candida albicans, salmonella typhiand candida stellafoidea(mic 5mg/ml). the phytochemical screening of the extracts confirmed the presence of β -amyrin acetate and taraxasterol. theoretical studies on the binding interaction of the two identified phytochemicals with the active sites of dna gyrase of escherichia coli and salmonella typhi revealed that the compounds bind to the target macromolecule via hydrophobic and hydrogen bond interactions with binding affinity of -9.6 kcal/mol and -9.5 kcal/mol for β -amyrin acetate and taraxasterol, respectively. the phytochemicals were found to be better inhibitors of dna gyrase when compared with the standard ciprofloxacin ligand which binds to the target macromolecule with binding affinity of -7.6 kcal/mol. in addition, in silico drug-likeness and admet investigations on the compounds showed that they obey both the lipinski’s rule of five and the veber’s rule in addition to displaying excellent pharmacokinetic and toxicity profiles. acknowledgements the authors acknowledge the technical support of strathclyde institute of pharmacy and biomedical sciences, university of strathclyde glasgow, united kingdom and college of forestry and fisheries, federal university of agriculture makurdi, benue state nigeria. references [1] b. b. mishra, & v. k. tiwari, “natural products: an evolving role in future drug discovery”, eur. j. med. chem. 46 (2011) 4769. 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[46] m. f. khan, m. a. bari, m. k. islam, m. s. islam, m. s. kayser, n. nahar, m. al-faruk, & m. a. rashid, “the natural anti-tubercular agents: in silico study of physicochemical, pharmacokinetic and toxicological properties”, j. app. pharm. sci. 7 (2017) 034. appendix supplementary file: 8 khan et al. / j. nig. soc. phys. sci. 5 (2023) 1576 9 table s1: 13c nmr experimental and literature data of β-amyrin acetate position experimental data 13c (δppm) okoyeet al., 2014 13c (δ ppm) ushieet al., 2018 13c (δ ppm) chaturvedula&rakash, 2013 13c (δ ppm) abdullahiet al., 2017 13c (δ ppm) wau, 2016 13c (δ ppm) ercilet al., 2004 13c (δ ppm) 1 38.91 38.79 38.70 39.30 38.60 38.40 38.14 2 26.70 27.44 28.80 28.30 23.90 23.70 27.51 3 80.35 79.24 79.10 79.60 80.60 81.10 79.12 4 38.91 38.99 38.5 39.10 38.60 37.80 38.88 5 55.40 55.37 55.5 55.10 51.50 55.30 55.27 6 18.24 18.58 18.80 18.80 18.20 18.70 18.44 7 32.85 32.85 33.10 33.20 33.30 32.80 33.03 8 40.05 40.21 38.80 39.80 37.80 39.90 38.82 9 47.59 47.43 49.20 48.00 43.30 47.40 47.81 10 37.08 37.15 36.70 37.40 36.80 36.80 37.00 11 23.75 23.75 22.70 23.90 22.70 23.70 23.48 12 121.68 121.93 116.80 122.20 129.80 122.30 121.82 13 145.21 145.41 142.70 145.70 142.70 145.50 145.28 14 42.07 41.92 41.30 42.10 42.80 41.60 42.18 15 26.36 26.36 27.40 26.00 27.90 27.00 26.12 16 26.80 27.14 27.10 26.30 26.20 26.20 27.37 17 32.70 32.70 49.20 32.60 32.50 32.60 32.04 18 48.04 47.84 33.30 48.10 55.60 47.50 47.22 19 46.85 47.03 48.70 47.30 40.80 46.99 46.93 20 31.10 31.30 29.10 31.30 41.70 31.30 31.34 21 38.20 37.35 35.10 34.30 31.30 34.80 34.83 22 34.56 34.94 37.50 37.40 42.80 37.20 37.26 23 15.71 15.71 15.43 16.10 16.70 16.85 28.22 24 28.30 28.31 29.70 28.20 29.70 28.40 15.47 25 16.01 15.80 15.40 16.10 16.60 16.00 15.72 26 16.92 17.01 17.54 17.20 16.10 17.00 16.97 27 26.21 26.62 21.32 24.40 18.10 26.00 25.56 28 28.60 28.62 28.00 28.20 27.90 28.80 28.44 29 33.42 33.56 33.70 33.90 27.50 33.40 33.44 30 23.86 23.91 25.90 23.80 25.30 23.60 23.48 11 170.94 171.53 170.91 171.00 173.71 171.10 171.01 21 21.53 21.53 29.80 21.53 21.20 21.30 21.80 9 khan et al. / j. nig. soc. phys. sci. 5 (2023) 1576 10 table s2: 1h experimental and literature data nmr of β− amyrin acetate position experimental data 1h (δ ppm) okoyeet al., 2014 1h (δ ppm) ushieet al., 2018 1h (δ ppm) abdullahiet al., 2017 1h (δ ppm) ercilet al., 2004 1h (δ ppm) chaturvedula and prakash, 2013 1h (δ ppm) wau, 2016 1h (δ ppm) 1 1.61 1.49 1.31 1.65 2 1.64 1.55 1.67 1.60 1.64 3 4.47 3.20 3.20 4.47 3.23 3.26 4.50 4 5 0.81 0.71 0.86 0.81 0.87 0.69 0.87 6 1.60 1.53 1.58 1.57 1.54 7 1.52 1.28 1.36 8 9 1.63 1.95 1.65 1.58 1.58 10 11 1.80 1.84 1.96 2.25 1.86 12 5.16 5.16 5.50 4.83 5.12 5.18 5.18 13 14 15 2.00 1.99 2.00 16 0.79 1.60 0.97 17 18 2.28 1.89 2.04 2.23 1.97 19 1.63 1.59 1.93 2.24 1.67 20 2.27 21 1.72 1.66 1.31 1.13 1.38 22 0.88 1.63 1.39 23 0.77 0.77 0.82 0.90 0.83 0.77 0.86 24 0.98 0.98 0.84 0.84 0.84 0.91 0.87 25 0.92 0.92 0.93 0.91 0.94 0.94 0.97 26 0.94 0.94 0.95 0.98 0.79 0.76 0.96 27 1.11 1.11 0.97 1.04 0.97 1.21 1.13 28 0.81 0.81 1.00 0.83 0.99 1.09 0.83 29 0.85 0.85 1.10 0.86 0.85 0.87 30 0.85 0.85 1.20 0.87 0.78 0.87 11 21 2.02 2.01 2.07 1.57 2.06 2.05 10 khan et al. / j. nig. soc. phys. sci. 5 (2023) 1576 11 table s3: the 13c nmr experimental and literature data of taraxasterol position experimental data 13c (δ ppm) khalilovet al., 2003 13c (δ ppm) reynoldet al., 1985 13c (δ ppm) alavi and yekta, 2008 13c (δ ppm) shakurova et al., 2008 13c (δ ppm) mahato and kundu, 1994 13c (δ ppm) 1 38.45 38.40 38.44 38.90 38.80 2 23.60 23.60 23.70 27.70 23.70 3 80.75 80.80 80.96 79.00 79.00 4 37.74 37.70 37.79 38.70 80.94 38.80 5 55.42 55.40 55.40 55.20 55.40 6 18.37 18.10 18.18 18.40 18.30 7 34.13 33.90 33.99 34.00 34.10 8 40.62 40.80 40.91 40.80 40.90 9 50.47 50.30 50.39 50.40 50.50 10 37.00 37.00 37.04 37.00 37.10 11 21.50 21.40 21.46 21.50 21.50 12 25.52 25.50 26.15 26.10 26.20 13 38.80 38.80 39.15 39.20 39.20 14 41.85 41.90 42.03 42.10 42.00 15 26.60 26.60 26.64 26.50 26.70 16 39.00 39.10 38.29 38.40 38.30 17 34.41 34.40 34.53 34.40 34.50 18 48.70 48.60 48.63 48.70 48.70 19 38.30 38.30 39.28 39.40 39.40 20 154.65 154.40 154.64 154.60 154.54 154.70 21 25.62 25.40 25.61 25.50 25.60 22 38.93 39.30 38.85 38.90 38.90 23 28.06 27.80 27.94 28.00 28.78 28.00 24 16.36 16.40 16.51 15.40 16.48 15.40 25 15.51 15.40 16.34 16.90 14.54 16.80 26 16.36 16.20 15.89 16.00 16.11 15.90 27 14.85 14.60 14.72 14.90 14.09 14.80 28 19.58 26.10 19.49 19.40 19.53 19.50 29 25.06 19.40 25.49 25.50 25.94 25.50 30 107.25 107.00 107.12 107.20 107.23 107.10 11 khan et al. / j. nig. soc. phys. sci. 5 (2023) 1576 12 table s4: 1h nmr experimental and literature data of taraxasterol postion experimental data 1h (δ ppm) khalilovet al., 2003 1h (δ ppm) alavi and yekta, 2008 1h (δ ppm) mahoto and kundu, 1994 1h (δ ppm) 1 0.92 0.92 2 1.67 1.67 3 4.70 4.70 3.22 3.22 4 5 0.78 0.78 0.77 6 1.52 1.46 7 1.35 1.35 8 9 1.33 1.33 10 11 1.51 1.48 12 1.65 1.65 1.03 13 1.56 1.56 14 15 1.65 1.65 0.93 16 1.17 1.26 17 18 0.99 0.99 19 2.15 2.15 20 21 2.48 2.48 22 1.41 1.41 23 0.91 0.91 0.77 0.71 24 0.91 0.91 0.85 0.77 25 0.86 0.86 0.86 0.86 26 0.99 0.99 1.02 0.97 27 0.96 0.96 0.93 28 0.94 0.94 0.85 0.98 29 1.08 1.08 1.02 0.96 30 5.10 4.79 4.62 4.66 12 j. nig. soc. phys. sci. 5 (2023) 1453 journal of the nigerian society of physical sciences modeling and analysis of a fractional visceral leishmaniosis with caputo and caputo–fabrizio derivatives dalal khalid almutairia, mohamed a.abdoonb, salih yousuf mohamed salihb, shahinaz a.elsamanib, fathelrhman el gumac, mohammed berirb,c,∗ adepartment of mathematics, college of education (majmaah), majmaah university, p.o.box 66, al-majmaah, 11952, saudi arabia. bdepartment of mathematics, faculty of science, bakht al-ruda university, duwaym, sudan. cdepartment of mathematics, faculty of science and arts in baljurashi, albaha university, albaha, saudi arabia. abstract visceral leishmaniosis is one recent example of a global illness that demands our best efforts at understanding. thus, mathematical modeling may be utilized to learn more about and make better epidemic forecasts. by taking into account the caputo and caputo-fabrizio derivatives, a frictional model of visceral leishmaniosis was mathematically examined based on real data from gedaref state, sudan. the stability analysis for caputo and caputo-fabrizio derivatives is analyzed. the suggested ordinary and fractional differential mathematical models are then simulated numerically. using the adams-bashforth method, numerical simulations are conducted. the results demonstrate that the caputo-fabrizio derivative yields more precise solutions for fractional differential equations. doi:10.46481/jnsps.2023.1453 keywords: leishmaniosis, modelling, caputo, caputo–fabrizio, sudan. article history : received: 16 march 2023 received in revised form: 21 may 2023 accepted for publication: 19 june 2023 published: 26 july 2023 c© 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: b. j. falaye 1. introduction visceral leishmaniasis, or kala azar, is a lethal vector-borne illness. india, bangladesh, and nepal have achieved substantial headway in lowering vl cases. east africa has made less progress, especially with south sudan’s continuous endemicity and vl outbreaks during the past 40 years. lack of infrastructure, clinical staff, idps, and hunger have hampered vl management in, and longer-term hazards to diagnostic kits and medications. pentavalent antimonials have been the backbone ∗corresponding author email address: midriss@bu.edu.sa (mohammed berir ) of vl treatment for decades, and resistance to them, as previously demonstrated in the indian subcontinent, provides another major barrier to vl treatment and management. to prevent monotherapy and minimize treatment duration, first-line 30-day sodium stibogluconate ss is substituted with a 17-day injectable combination regimen of ssg and pm in who recommendations in 2010 and sudan ministry of health guidelines in 2011. since 2012, ambisome has been donated to who for these purposes. in east africa, ssg/pm combo treatment had a 5% recurrence rate. relapse may be due to insufficient cellular immunity following therapy due to hiv, tb, or malnutrition, or inadequate treatment resulting in considerable chronic parasitaemia after initial clinical cure. in places like sudan, where 1 almutairi et al. / j. nig. soc. phys. sci. 5 (2023) 1453 2 active patient follow-up is difficult and not common, vl recurrence rates are passively evaluated by vl re-treatment admissions as a proportion of overall vl admissions. passive monitoring shows an increase in re-treatment rates in recent years [1-7]. in the last 30 years, fractional calculus and nonlinear equations has become more well-known and important. fractional differential equations are used in physics, chemical engineering, mathematical biology, and finance [8-16]. simulating a fractional model simultaneously with a caputo derivative and a cf derivative. in addition, modeling and graphing with the fractional derivative is a highly effective technique for demonstrating leishmaniasis using matlab. this could be done to better comprehend the infection. using fractional derivatives as a research strategy for natural occurrences may result in more precise findings than other methods. as a result of this model’s use of a non-singleton kernel, the cf derivative has significantly improved predictive abilities. 2. preliminaries definition 2.1. riemann-liouville fractional integral (rli) operator of order α > 0 for a function y (τ) is given by [17]: dαy (t) = 1 γ(n −α) ∫ t 0 (t −τ)n−α−1yn(τ)dτ = in−αyn(t), t > 0. (1) definition 2.2. for y ∈ h1 (0, t) , t > 0, t > 0, α ∈ (0, 1] then the cf fractional operator [17] is given by cf 0 d α t y (t) = b(α) 1 −α) d dt ∫ t 0 y(τ)e−α t−τ 1−α dτ, 0 < α < 1. (2) in this expression b (α) satisfies the condition b (0) = b (1) = 1. definition 2.3. caputo derivative of order 0 ≤ n − 1 < α < n with the lower limit zero for a function y (τ) is given by [18]: iαy (t) = 1 γ(α) ∫ t 0 (t −τ)α−1y(τ)dτ, t > 0. (3) 3. anthropologic visceral leishmaniosis model with caputo derivative in this section, we describe the leishmaniasis model, which includes four subpopulations:susceptible, infectious, recovered, and recovered with permanent immunity, for the human population, and two compartments for the reservoir population: susceptible and infected. in addition to that, we have two compartments for sandflies: susceptible and infected. the human population is the only population in the model that has permanent immunity. the positivity, reproduction number, and equilibrium solutions of the model that was established in this work have all been determined to be free of leishmaniasis. additionally, the leishmaniasis cases, along with their respective localities and global stability properties, have also been determined. we obtain the model formulation by using a new variable: sh (t) = s h nh , ih (t) = ih nh , ph (t) = ph nh , rh (t) = rh nh , sr (t) = s r nr , ir (t) = ir nr , sv (t) = s v nv , iv (t) = iv nv , sh (t) = s h nh , m = nv nh and n = nv nr . the system of differential equations is given by: c 0 d α t ih = abmiv nh − ( α1 + δ + ah nh −δih ) ih, c 0 d α t ph = (1 −σ)α1ih − ( α2 + β + ah nh −δih ) ph, c 0 d α t ir = abniv sr − ah nh ir, c 0 d α t iv = acihs v + acphs v + acir s v − av nv iv, c 0 d α t sh = ah nh − [ abmiv + ah nh −δih ] sh, c 0 d α t rh = σα1ih + (α2 + β) ph − [ ah nh −δih ] rh, c 0 d α t s r = ar nr − abniv sr − ah nh sr, c 0 d α t sv = av nv − [ acih + acph + av nv ] sv, (4) with initial conditions: sh (0) = c1, ih (0) = c2, rh (0) = c3, sr (0) = c4, ir (0) = c5, sv (0) = c6, iv (0) = c7 . 4. anthropologic visceral leishmaniosis model with fc derivative in this section, we obtain the fractional model formulation under caputo–fabrizio derivatives: sh (t) = s h nh , ih (t) = ih nh , ph (t) = ph nh , rh (t) = rh nh , sr (t) = s r nr , ir (t) = ir nr , sv (t) = s v nv , iv (t) = iv nv , sh (t) = s h nh , m = nvnh and n = nv nr the system of differential equations is given by: fc 0 d α t ih = abmiv nh − ( α1 + δ + ah nh −δih ) ih, fc 0 d α t ph = (1 −σ)α1ih − ( α2 + β + ah nh −δih ) ph, fc 0 d α t ir = abniv sr − ah nh ir, fc 0 d α t iv = acihs v + acphs v + acir s v − av nv iv, fc 0 d α t sh = ah nh − [ abmiv + ah nh −δih ] sh, fc 0 d α t rh = σα1ih + (α2 + β) ph − [ ah nh −δih ] rh, fc 0 d α t s r = ar nr − abniv sr − ah nh sr, fc 0 d α t sv = av nv − [ acih + acph + av nv ] sv, (5) with initial conditions: sh (0) = c1, ih (0) = c2, rh (0) = c3, sr (0) = c4, ir (0) = c5, sv (0) = c6, iv (0) = c7 . 5. stability analysis in this part we discuss the stability of epidemiological model, the equilibrium points, eigenvalues value and the jacobian matrix for the model (1). 2 almutairi et al. / j. nig. soc. phys. sci. 5 (2023) 1453 3 table 1: description of the variables for model. variable description nh (t) human host population nr(t) reservoir host population nv (t) vector population s h (t) susceptible humans ph (t) recovered and have permanent immunity ih (t) infected humans rh (t) recovery humans rs(t) susceptible reservoir ir(t) infected reservoir s v (t) susceptible sandflies iv (t) infected sandflies table 2: parameters values of the leishmaniasis model. parameter description value source a biting rate of sandflies 0.2856 day−1 [16] b progression rate of vl in sandfly 0.22 day−1 [16] c progression rate of vl in human and reservoir 0.0714 day−1 [16] ah human recruitment rate 10.1009 day−1 estimated ar reservoir recruitment rate 19.7795 day−1 estimated av vector recruitment rate 38858.62 day−1 estimated µh natural mortality rate of humans 4.341e − 6 day−1 [2] µr natural mortality rate of reservoirs 0.0017 day−1 [1] µv natural mortality rate of vectors 0.0668 day−1 [1] α1 treatment rate of vl 0.02 [2] α2 pkdl recovery rate without treatment 0.033 [20] σ recovery rate from vl infection after treatment 0.9 [1] 1 −σ developing pkdl rate after treatment 0.1 [1] δ death rate due to vl 0.011 [16] β pkdl recovery rate after treatment 0.9 [1] 5.1. equilibria the equilibrium points of dynamics (5) are computed solving the nonlinear system. table 3: the equilibrium points of the system. ei equilibria e1 (0, 0, 0, 0, 1, 0, 711.58, 1) e2 (32.5436, 0.1132, 711.5801, 22.7897,−29.7254,−1.9314, 0, 0) e3 (−31.1459,−0.0488, 711.58,−21.8109, 33.9641,−1.7693, 0, 0) e4 (85.0303,−72.8759, 711.5801, 59.5451,−82.2121,−71.0578, 0, 0) e5 (2.8182, 0.0062,−2.8244, 0, 0,−1.8244, 714.4046, 0) e6 (2.8182, 0.0062, 711.5801, 252.9373, 0,−1.8244, 0, 0) e7 (0, 0, 0, 0,−7.2892e11, 7.2892e11, 0, 0) e8 (0, 0, 0, 5.1155e8, 0, 0, 0, 0) 5.2. the jacobian matrix for the model: here, we talk about this epidemiological model stability. the disease-free equilibrium point is given as e1 = (0, 0, 0, 0, 1, 0, 711.58, 1) and the endemic equilibrium points e8 = (0, 0, 0, 5.1155e8, 0, 0, 0, 0). j (e1) =  −0.031 0 0 0.0157 0 0 0 0 0.002 −0.933 0 0 0 0 0 0 0 0 −4.297e−8 15907.35 0 0 0 0 0.02 0.02 0.02 −3.438e−11 0 0 0 0 0.011 0 0 −0.0157 −4.297e−8 0 0 0 0.018 0.0933 0 0 0 −4.297e−8 0 0 0 0 −15907.35 0 0 −4.297e−8 0 −0.02 −0.02 0 0 0 0 0 −3.438e−11  (6) j (e2) =  −0.031 0 0 0 8035456.6 0 0 0 0.002 −0.933 0 0 0 0 0 0 0 0 −4.297e−8 −0.0006 0 0 11435742383.06 0 0.000002 0.00002 0.00002 −3.438e−11 0 0 0 14.225 0 0 0 0 −8035456.6 0 0 0 0.028 0.0933 0 0 0 −0.00000003 0 0 0 0 0 0.0006 0 0 −11435742383.06 0 −0.00002 −0.0002 0 0 0 0 0 −0.000000027  (7) table 4: variable values. eigenvalues stability λ (17.833, 0, 0, 0, 0, 0, 0,−17.833) unstable λ∗ (−3.438e−11,−2.8e−8,−2.8e−8, −4.297e−8,−0.31,−0.933,−035456.6,−1.14e10) stable 5.3. the basic reproduction number the basic reproduction number is a baseline statistic in epidemiology and is represented by r0, which stands for the predicted value of the secondary infections rate per time unit. using the equation’s fractional model (1), we have fours infected classes, rewrite the system of equation 1 for the susceptible and 3 almutairi et al. / j. nig. soc. phys. sci. 5 (2023) 1453 4 infected classes in the general form: d x dt = f (x) − v (x) , (8) where f (x) =  abmiv sh 0 abmiv sr ac(ih + ph + ir )sv)  , and v (x) =  (α1 + δ + µh) ih (α2 + β + µh)ph − (1 −σ)α1ih µr ir µviv  . (9) figure 1: systems of fractional orders model for α=0.99. figure 2: systems of fractional orders model for α=1 (first part) now, the jacobian of f (x) and v (x) of the disease free equilibrium point is: f =  0 0 0 abm 0 0 0 0 0 0 0 abm ac ac ac 0  , and v =  α1 + δ + µh 0 0 0 −(1 −σ)α1 α2 + β + µh 0 0 0 0 µr 0 0 0 0 µv  (10) we have r0 = ρ ( fv−1 ) = √√√√√√√√√ ac[µr abm (α2 + δ + µh + (1 −σ) α1) + abn(α1 + δ + µh)(α2 + δ + µh) ] µv µr (α1 + δ + µh)(α2 + δ + µh). (11) lemma 5.1. the disease-free equilibrium e0 is locally asymptotically stable if r0 < 1 and unstable if r0 > 1. 6. numerical simulation and graphical representations this section is devoted to finding the approximate solutions of the proposed models (4) and (5) under fractional operators of caputo and caputo-fabrizio, respectively. we simulate our model using some highly reliable numerical techniques. the finite difference scheme for the initial value problem yields the following numerical techniques for the underlying operators: c xr+1 = x0 + (∆t)ω γ(ω + 1) r∑ k=0 [ (r − k + 1)ω − (r − k)ω ] f (xk) + o ( ∆t2 ) , cf xr+1 = x0 + (1 −δ)f (xr ) + δ∆t r∑ k=0 f (xk) + o ( ∆t2 ) , (12) table 1 shows a description of the variables in the model. figures 1 and 2 were obtained with the caputo (4) and cf methods (5) using the parameters in table 2. tables 3 & 4 show a summary of equilibrium points and the corresponding eigenvalues of the jacobian matrix. 7. conclusion a fractional model was simulated by using a caputo derivative as well as a cf derivative simultaneously. in addition, modeling and graphing with the aid of the fractional derivative is a very effective approach that can be used to show leishmaniasis with the use of matlab. this may be done in order to better understand the infection. when doing research on natural events, using fractional derivatives as a strategy might lead to more precise findings than other approaches. due to the fact that this model employs a non-singleton kernel, the cf derivative has much enhanced prediction capabilities. this research was carried out in the hope that it will be a useful resource for future applications and explorations of simulation by using a caputo derivative, and to investigate new methods such as those in refs. 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[29] m. elbadri, m. abdoon, m. berir & d. almutairi, “a numerical solution and comparative study of the symmetric rossler attractor with the generalized caputo fractional derivative via two different method”, mathematics 11 (2023) 2997. 5 j. nig. soc. phys. sci. 3 (2021) 82–88 journal of the nigerian society of physical sciences comments on “the solution of a mathematical model for dengue fever transmission using differential transformation method: j. nig. soc. phys. sci. 1 (2019) 82-87” gurpreet singh tutejaa,∗, tapshi lalb a zakir husain delhi college, university of delhi b satyawati college, university of delhi abstract the mathematical model for dengue fever transmission studied by [1], has been re-investigated. the differential transformation method (dtm) is used to compute the semi-analytical solutions of the non-linear differential equations of the compartment (sir) model of dengue fever. this epidemiology problem is well-posed. the effect of treatment as a control measure is studied through the growth equations of exposed and infected humans. the inadvertent errors in the recurrence relations (dtm) of equations for dengue disease transmission including initial conditions have been removed. furthermore, the semi-analytic solutions of the model are obtained and verified with the built-in function asymptoticdsolvevalue of wolfram mathematica. it has been found that results obtained from the dtm are valid only for small-time t (t < 1.5), as t becomes large, the human population (exposed and recovered) and infected vector population become negative. doi:10.46481/jnsps.2021.170 keywords: sir model, differential transformation method (dtm), dengue fever, treatment article history : received: 01 march 2021 received in revised form: 2 april 2021 accepted for publication: 01 april 2021 published: 18 may 2021 ©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: b. j. falaye 1. introduction dengue fever is a mosquito-borne flavivirus that is mostly found in tropical and sub-tropical regions of the world. the disease is spread by aedes mosquito due to day-biting [2]. dengue fever is the fastest-spreading vector-borne viral disease affecting 40 per cent of the world’s population, and now endemic in over 100 countries. over the last two decades, the number of dengue cases registered to who has increased from 505,430 cases in 2000 to over 2.4 million in 2010, and ∗corresponding author tel. no: email addresses: gstuteja@gmail.com (gurpreet singh tuteja ), tapshisingh@ymail.com (tapshi lal) 4.2 million in 2019. between 2000 and 2015, the number of registered deaths increased from 960 to 4032 [3]. a second potential vector, aedes albopictus, resides in temperate regions (north america and europe), where it may give rise to occasional dengue outbreaks [4, 5]. the spread of infectious diseases is studied through various epidemiological models, including observational studies, interventional studies apart from mathematical modelling using the compartment model [6]. the pioneering work using the sir model for contagious diseases is done by [7, 8, 9]. in the compartment model, the population is primarily divided into three distinct mutually exclusive compartments: susceptible s(t ), infected/infectious i (t ) and recovered r (t ) at any 82 tuteja & lal / j. nig. soc. phys. sci. 3 (2021) 82–88 83 time t , based on the epidemiological status of the population [10]. for the first time, the dtm was used for solving electrical circuit problems [11]. the application of dtm in finding solutions for the set of non-linear ordinary differential equations obtained using the sir model for various epidemiological diseases including typhoid [12, 13], malaria [14] and seasonal diseases [15] has been studied. in this sir model for dengue disease, we consider two separate but dependent sets of non-linear ordinary differential equations related to human and vector population [16, 17]. the purpose of this paper is to find semi-analytical solutions to the dengue fever (sir) model including the effect of treatment as a measure of control. the semi-analytic solutions are obtained by using the differential transform method (dtm) and are confirmed by using a built-in function: asymptoticdsolvevalue of wolfram mathematica and are discussed graphically also. the paper is organized in the following sections: in section 2, the sir model for dengue fever with model parameters including treatment is briefly described. the existence, uniqueness and positivity of the solution to the epidemiology problem are discussed in section 3. section 4 presents a theoretical concept and implementation of dtm. in section 5, numerical solutions are obtained with a graphical discussion. finally, concluding remarks in section 6. 2. formulation of the problem two sets of populations consisting of human and vector are considered in figure 1. the population is divided into some mutually exclusive compartments as given below. the total human population is divided into the following mutually exclusive epidemiological classes: susceptible humans sh (t ), humans with dengue in latent stage eh (t ), humans infected with dengue ih (t ), humans treated for dengue rh (t ) (recovered), while the vector population is divided into three classes: susceptible vectors s v (t ), vectors with dengue in latent stage e v (t ), vectors with dengue i v (t ). the class of treated vectors (rv (t )) is not taken into consideration. let nh (t ) and nv (t ) denote the total number of humans and vectors at time t , respectively. hence, we have that, nh (t ) = sh (t ) + eh (t ) + ih (t ) + rh (t ) and nv (t ) = s v (t ) + e v (t ) + i v (t ), where nh (t ) > 0, nv (t ) > 0, sh (t ) > 0, s v (t ) > 0, eh (t ) ≥ 0, ih (t ) ≥ 0, rh (t ) ≥ 0, e v (t ) ≥ 0, i v (t ) ≥ 0. the susceptible humans are recruited at a rate λh , while the susceptible vectors are recruited at a rate λv . the susceptible humans’ contract to dengue at a rate: λdv = βv h (ηv e v + i v ) nh , (1) where ηv < 1, this accounts for the relative infectiousness of vectors with latent dengue e v compared to vectors in the i v class. susceptible vectors acquire dengue infection from infected humans (infected blood is passed to a vector through a bite) at a rate: λd h = βh v (ηa eh +ηb ih ) nh , (2) where ηa < ηb this accounts for the relative infectiousness of humans with latent dengue eh compared to humans in the ih class [1]. the model equations for dengue disease transmission including treatment as a control measure are: d sh d t =λh −µh sh −λdv sh , (3) d eh d t = λdv sh − (γh +µh )eh , (4) d ih d t = γh eh − (τh +µh +δdh )ih , (5) d rh d t = τh ih −µh rh , (6) d s v d t =λv −µv s v −λd h s v , (7) d e v d t = λd h s v − (γv +µv )e v , (8) d i v d t = γv e v − (µv +δd v )i v , (9) where λh ,λv are the recruitment rates and µh , µv are natural death rates of susceptible human and vector population, respectively. βv h and βh v are the effective contact rate for dengue from vectors to humans and humans to vectors, respectively. the treatment rate for infected humans is τh . γh and γv are the progression rate of the human and vector population from the latent class (exposed) to the active dengue class, respectively. the disease induced deaths in human and vector are denoted by δdh and δd v . ηv , ηa , ηb are the modification parameters of e v , eh and ih , respectively. 3. existence, uniqueness and positivity of solution we will use the lipchitz condition to verify the existence and uniqueness of solution [18] for the model equations (3)(9): e1 =λh −µh sh −λdv sh , e2 = λdv sh − (γh +µh )eh , e3 = γh eh − (τh +µh +δdh )ih , e4 = τh ih −µh rh , e5 =λv −µv s v −λd h s v , e6 = λd h s v − (γv +µv )e v , e7 = γv e v − (µv +δd v )i v . let b denote the region, |t −t0| ≤ δ, ||x−x0|| ≤ α, where x = (x1, x2, . . . xn ), x0 = (x10, x20, . . . xn0) also suppose that a(t , x ) satisfies the lipschitz condition: ||a(t , x1) − a(t , x2)|| ≤ k||x1 − x2|| whenever the pairs (t , x1), (t , x2) belong to b where k is a positive constant, then there is a positive constant δ ≥ 0, such that there exists a unique and continuous vector solution x (t ) of 83 tuteja & lal / j. nig. soc. phys. sci. 3 (2021) 82–88 84 figure 1: dengue fever model the system in the interval |t − t0| < δ. the condition is satisfied by the requirement that ∂ai ∂x j ,i , j = 1, 2, 3, ..n, be continuous and bounded in b. considering the model equation (3)(9), we are interested in the region 0 ≤ α ≤ r [13]. let b denote the region 0 ≤ α ≤ r , then equations (3) – (9) will have a unique solution if ∂ai ∂x j ,i , j = 1, 2, 3, ..7 are continuous and bounded in b. for e1:∣∣∣∣ ∂e1∂sh ∣∣∣∣ = |− (µh +λdv )| < ∞, ∣∣∣∣ ∂e1∂eh ∣∣∣∣ = 0 < ∞, ∣∣∣∣∂e1∂ih ∣∣∣∣ = 0 < ∞,∣∣∣∣ ∂e1∂rh ∣∣∣∣ = 0 < ∞, ∣∣∣∣∂e1∂s v ∣∣∣∣ = 0 < ∞, ∣∣∣∣ ∂e1∂e v ∣∣∣∣ = 0 < ∞, ∣∣∣∣∂e1∂i v ∣∣∣∣ = 0 < ∞. for e2:∣∣∣∣ ∂e2∂sh ∣∣∣∣ = |λdv | < ∞, ∣∣∣∣ ∂e2∂eh ∣∣∣∣ = |− (γh +µh )| < ∞, ∣∣∣∣∂e2∂ih ∣∣∣∣ = 0 < ∞,∣∣∣∣ ∂e2∂rh ∣∣∣∣ = 0 < ∞, ∣∣∣∣∂e2∂s v ∣∣∣∣ = 0 < ∞, ∣∣∣∣ ∂e2∂e v ∣∣∣∣ = 0 < ∞, ∣∣∣∣∂e2∂i v ∣∣∣∣ = 0 < ∞. for e3:∣∣∣∣ ∂e3∂sh ∣∣∣∣ = 0 < ∞, ∣∣∣∣ ∂e3∂eh ∣∣∣∣ = |γh| < ∞,∣∣∣∣∂e3∂ih ∣∣∣∣ = |− (τh +µh +δdh )| < ∞,∣∣∣∣ ∂e3∂rh ∣∣∣∣ = 0 < ∞, ∣∣∣∣∂e3∂s v ∣∣∣∣ = 0 < ∞, ∣∣∣∣ ∂e3∂e v ∣∣∣∣ = 0 < ∞, ∣∣∣∣∂e3∂i v ∣∣∣∣ = 0 < ∞. for e4:∣∣∣∣ ∂e4∂sh ∣∣∣∣ = 0 < ∞, ∣∣∣∣ ∂e4∂eh ∣∣∣∣ = 0 < ∞, ∣∣∣∣∂e4∂ih ∣∣∣∣ = |τh| < ∞,∣∣∣∣ ∂e4∂rh ∣∣∣∣ = |−µh| < ∞, ∣∣∣∣∂e4∂s v ∣∣∣∣ = 0 < ∞, ∣∣∣∣ ∂e4∂e v ∣∣∣∣ = 0 < ∞,∣∣∣∣∂e4∂i v ∣∣∣∣ = 0 < ∞. these partial derivatives exist, are continuous and bounded, similarly for e5, e6, e7. hence the model has a unique solution. the positivity of the solution is presented in the following theorems: positivity of the solution: we show that the model equations (3)–(9) are biologically and epidemiologically meaningful and well-posed as the solutions of all the stated variables are non-negative [16]. if sh (0) > 0, eh (0) ≥ 0, ih (0) ≥ 0, rh (0) ≥ 0, s v (0) > 0, e v (0) ≥ 0 and i v (0) ≥ 0, then the solution region sh (t ), eh (t ), ih (t ), rh (t ), s v (t ), e v (t ) and i v (t ) of the system of equations (3)–(9) is always non-negative. we consider each differential equation separately and show that its solution is positive. theorem 1: positivity of susceptible human population: consider the differential equation (3): d sh d t =λh − (µh +λdv )sh (t ) ≥ −(µh +λdv )sh (t ), λh > 0 being recruitment rate of humans, we can write as: d sh sh = −(µh +λdv )d t on integrating, the solution is sh = sh0e− ∫ t 0 (µh +λdv )d t . it is clear from the solution that sh (t ) is positive since sh0 = sh (0) > 0 and the exponential function is always positive. theorem 2: positivity of latent human population: consider the differential equation (4): d eh d t = λd h sh (t ) − (γh +µh )eh (t ) ≥ −(γh +µh )eh (t ), sh (t ) is positive in time t and λd h > 0 being humans’ contract rate to dengue, we can write as: d eh eh = −(γh +µh )d t . 84 tuteja & lal / j. nig. soc. phys. sci. 3 (2021) 82–88 85 on integrating, the solution is eh = eh0e− ∫ t 0 (γh +µh )d t . it is clear from the solution that eh (t ) is positive since eh0 = eh (0) ≥ 0 and the exponential function is always positive. theorem 3: positivity of infected human population: consider the differential equation (5): d ih d t = γh eh (t ) − (τh +µh +δdh )ih (t ) ≥ −(τh +µh +δdh )ih (t ), γh being the progression rate of humans from latent class to active dengue class and eh (t ) ≥ 0, we can write as: d ih ih = −(τh +µh +δdh )d t . on integrating, the solution is ih = ih0e− ∫ t 0 (τh +µh +δdh )d t . so, it is clear from the solution that ih (t ) is positive since ih0 = ih (0) ≥ 0 and exponential function is always positive. theorem 4: positivity of recovered human population: consider the differential equation (6): d rh d t = τh ih (t ) −µh rh (t ) ≥ −µh rh (t ), τh > 0 being the treatment rate for infected humans and ih (t ) is positive in time t , we can write as: d rh rh = −µh d t . on integrating, the solution is rh = rh0e− ∫ t 0 µh d t . it is clear from the solution that rh (t ) is positive since rh0 = rh (0) ≥ 0 and the exponential function is always positive. theorem 5: positivity of susceptible vector population: consider the differential equation (7): d s v d t =λv − (µv +λd h )s v (t ) ≥ −(µv +λd h )s v (t ), λv > 0 being recruitment rate of vectors, we can write as: d s v s v = −(µv +λd h )d t . on integrating, the solution is s v = s v 0e− ∫ t 0 (µv +λd h )d t . it is clear from the solution that s v (t ) is positive since s v 0 = s v (0) > 0 and the exponential function is always positive. theorem 6: positivity of latent vector population: consider the differential equation (8): d e v d t = λd h s v (t ) − (γv +µv )e v (t ) ≥ −(γv +µv )e v (t ), s v (t ) is positive in time t and λd h > 0 being vectors’ contract rate to dengue due to infected humans, we can write as: d e v e v = −(γv +µv )d t . on integrating, the solution is e v = e v 0e− ∫ t 0 (γv +µv )d t . it is clear from the solution that e v (t ) is positive since e v 0 = e v (0) ≥ 0 and the exponential function is always positive. theorem 7: positivity of infected vector population: consider the differential equation (9): d i v d t = γv e v (t ) − (µv +δd v )i v (t ) ≥ −(µv +δd v )i v (t ), γv > 0 being the progression rate of vectors from latent class to active dengue class and e v (t ) ≥ 0, we can write as: d i v i v = −(µv +δd v )d t . on integrating, the solution is i v = i v 0e− ∫ t 0 (µv +δd v )d t . so, it is clear from the solution that i v (t ) is positive since i v 0 = i v (0) ≥ 0 and exponential function being positive always. hence, the stated problem is epidemiologically meaningful, well-posed and has a unique solution. 4. differential transform method (dtm) the differential transformation of the k t h derivative of f (x ) is defined as: f (k ) = 1 k ! [ d k f (x ) d x k ] x0 . (10) we obtain, f (x ) = ∞∑ k=0 f (k )(x − x0)k , (11) is called the inverse differential transformation of f(k). in real applications, the function f(x) can be expressed as a finite series and equation (11) can be expressed as: f (x ) = n∑ k=0 f (k )(x − x0)k . (12) so, we have f (x ) = n∑ k=0 (x − x0)k 1 k ! [ d k f (x ) d x k ] x0 . (13) from equations (10) and (11), the following properties are obtained: 1. if z (x ) = f (x ) ± g (x ), then z (k ) = f (k ) ±g (k ). 2. if z (x ) = αf (x ), then z (k ) = αf (k ). 3. if z (x ) = f ′(x ), then z (k ) = (k + 1)f (k + 1). 4. if z (x ) = f ′′(x ), then z (k ) = (k + 1)(k + 2)f (k + 2). 5. if z (x ) = f (l )(x ), then z (k ) = (k+1)(k+2)...(k+l )f (k+l ). 6. if z (x ) = u(x )v (x ), then z (k ) = ∑k l =0 f (l )g (k − l ). 7. if z (x ) = αx l , then z (k ) = αδ(k − l ), where kronecker delta, δ(k − l ) = { 1,k=l 0,k,l using the fundamental operations of differential transformation method, let sh (k ),eh (k ), ih (k ), rh (k ), s v (k ), e v (k ) and i v (k ) denote the differential transformations of sh (t ), eh (t ), 85 tuteja & lal / j. nig. soc. phys. sci. 3 (2021) 82–88 86 ih (t ), rh (t ), s v (t ),e v (t ) and i v (t ) respectively, the recurrence relation to each equation of the system (3)–(9) is: sh (k + 1) = 1 k + 1 { λhδ(k ) −µh sh (k ) −βv hηv nh k∑ m=0 sh (m)e v (k − m) − βv h nh k∑ m=0 sh (m)i v (k − m) } (14) eh (k + 1) = 1 k + 1 { βv hηv nh k∑ m=0 sh (m)e v (k − m) + βv h nh k∑ m=0 sh (m)i v (k − m) − (γh +µh )eh (k ) } (15) ih (k + 1) = 1 k + 1 {γh eh (k ) − (τh +µh +δdh )ih (k )} (16) rh (k + 1) = 1 k + 1 {τh ih (k ) −µh rh (k )} (17) s v (k + 1) = 1 k + 1 { λv δ(k ) −µv s v (k ) − βh v ηa nh k∑ m=0 s v (m)eh (k − m) − βh v ηb nh k∑ m=0 s v (m)ih (k − m) } (18) e v (k + 1) = 1 k + 1 { βh v ηa nh k∑ m=0 s v (m)eh (k − m) + βh v ηb nh k∑ m=0 s v (m)ih (k − m) − (γv +µv )e v (k ) } (19) i v (k + 1) = 1 k + 1 { γv e v (k ) − (µv +δd v )i v (k ) } (20) the recurrence relations (14), (15), (18) and (19) are the rectified forms of (8), (9), (12) and (13) of the study [1]. the semi-analytical solutions and numerical solutions have significantly changed due to these corrections. 5. numerical and graphical simulation of the model equations with the initial conditions sh (0) = 3503, eh (0) = 490, ih (0) = 390, rh (0) = 87, s v (0) = 390,e v (0) = 100, i v (0) = 130, we compute the semi-analytical solutions for k = 4 using following values of the parameters: nh = 4470, nv = 620, λh = 500, λv = 1, 000, 000, µh = 0.02041, µv = 0.5, βv h = 0.5, βh v = 0.4, γh = 0.3254, γv = 0.03, δdh = 0.365, δd v = 0, ηv = 0.4, ηa = 0.2, ηb = 0.5 [5]. 5.1. low dengue treatment(τh = 0.25) sh (t ) = 4∑ k=0 sh (k )t k = 3503 + 361.8919131767338t + 8.365058543813413t 2 − 686.2825200034454t 3 + 197.4722799192922t 4, eh (t ) = 4∑ k=0 eh (k )t k = 490 − 102.83504317673376t + 5.722527622691168t 2 + 685.5659739627513t 3 − 253.23941572498939t 4, i h (t ) = 4∑ k=0 ih (k )t k = 390 − 88.3639t + 11.342391324645417t 2 − 1.7816527943897462t 3 + 56.05381198239061t 4, rh (t ) = 4∑ k=0 rh (k )t k = 87 + 95.72433t − 12.02235428765t 2 + 1.026991360724097t 3 − 0.11659352306745382t 4, sv (t ) = 4∑ k=0 s v (k )t k = 390 + 999794.7744966443t − 263054.4930350626t 2 + 48072.332846142934t 3 − 6858.820737023293t 4, e v (t ) = 4∑ k=0 e v (k )t k = 100 − 42.774496644295304t + 13117.134652512259t 2 − 6547.277795576331t 3 + 1717.2934391692909t 4, i v (t ) = 4∑ k=0 i v (k )t k = 130 − 62.t + 14.85838255033557t 2 + 128.69494943339998t 3 − 65.19145214599749t 4 these solutions are the same as obtained from the in-built function asymptoticdsolvevalue of wolfram mathematica 13. following is the program: a s y m p t o t i c d sol v ev al ue [ {s′h [t ] −λh +µh sh [t ] + (βv hηv /nh )sh [t ] e v [t ] + (βv h /nh )sh [t ]i v [t ] == 0, e ′h [t ] − (βv hηv /nh )sh [t ] e v [t ] − (βv h /nh )sh [t ]i v [t ] + (γh +µh )eh [t ] == 0, i ′h [t ] −γh eh [t ] + (τh +µh +δdh )ih [t ] == 0, r′h [t ] −τh ih [t ] +µh rh [t ] == 0, s′v [t ] −λv +µv s v [t ] + (βh v ηa /nh )s v [t ] eh [t ] + (βh v ηb /nh )s v [t ]ih [t ] == 0, e ′v [t ] − (βh v ηa /nh )s v [t ]eh [t ] − (βh v ηb /nh ) s v [t ]ih [t ] + (γv +µv )e v [t ] == 0, i ′v [t ] −γv e v [t ] + (µv +δd v ) 86 tuteja & lal / j. nig. soc. phys. sci. 3 (2021) 82–88 87 i v [t ] == 0, sh [0] == sh[0], eh [0] == eh [0], ih [0] == i h[0], rh [0] == r h[0], s v [0] == sv [0], e v [0] == e v [0], i v [0] == i v [0]}, {sh [t ], eh [t ], ih [t ], rh [t ], s v [t ], e v [t ], i v [t ]}, {t , 0, 4}] 5.2. moderate dengue treatment(τh = 0.50) sh (t ) = 4∑ k=0 sh (k )t k = 3503 + 361.8919131767338t + 8.365058543813413t 2 − 686.2380770354108t 3 + 254.42405449756384t 4 eh (t ) = 4∑ k=0 eh (k )t k = 490 − 102.83504317673376t + 5.722527622691168t 2 + 685.5215309947168t 3 − 310.1875748678114t 4, i h (t ) = 4∑ k=0 ih (k )t k = 390 − 185.8639t + 65.5516163246454t 2 − 18.725982040526862t 3 + 59.912219486045935t 4, rh (t ) = 4∑ k=0 rh (k )t k = 87 + 193.22433t − 48.43782928765t 2 + 11.25480808602788t 3 − 2.3981754133248154t 4, sv (t ) = 4∑ k=0 s v (k )t k = 390 + 999794.7744966443t − 263053.6423639217t 2 + 49525.708598154626t 3 − 7943.074169380729t 4, e v (t ) = 4∑ k=0 e v (k )t k = 100 − 42.774496644295304t + 13116.28398137132t 2 − 8000.645040876618t 3 + 2812.4460625275524t 4, i v (t ) = 4∑ k=0 i v (k )t k = 130 − 62.t + 14.85838255033557t 2 + 128.6864427219906t 3 − 76.09064314682345t 4 5.3. high dengue treatment(τh = 0.75) sh (t ) = 4∑ k=0 sh (k )t k = 3503 + 361.8919131767338t + 8.365058543813413t 2 − 686.1936340673763t 3 + 311.3702737048312t 4, eh (t ) = 4∑ k=0 eh (k )t k = 490 − 102.83504317673376t + 5.722527622691168t 2 + 685.4770880266824t 3 − 367.13017863962915t 4, figure 2: susceptible, exposed, infected human (τ = 0.5) i h (t ) = 4∑ k=0 ih (k )t k = 390 − 283.3639t + 144.1358413246454t 2 − 53.93038836999731t 3 + 71.07183667576527t 4, rh (t ) = 4∑ k=0 rh (k )t k = 87 + 290.72433t − 109.22830428765t 2 + 36.777076894665t 3 − 10.299602854229525t 4, sv (t ) = 4∑ k=0 s v (k )t k = 390 + 999794.7744966443t − 263053.6423639217t 2 + 50978.94257164284t 3 − 9299.822488317648t 4, e v (t ) = 4∑ k=0 e v (k )t k = 100 − 42.774496644295304t + 13115.43331023038t 2 − 9453.870507653413t 3 + 4180.0925091263725t 4, i v (t ) = 4∑ k=0 i v (k )t k = 130 − 62.t + 14.85838255033557t 2 + 128.6779360105812t 3 − 86.98877080872326t 4. the graph of susceptible, exposed, infected human population with moderate treatment and exposed, infected vectors is plotted against time t figures 2 and 3. it is found that the susceptible and infected human population grows with time t while the exposed human population becomes negative after t = 2.25, being population graph, it can’t be negative (limitation of dtm). similarly, the graph of the infected vectors becomes negative. the effect of the treatment as a control measure can be studied from figures 4 and 5, where the effect of better treatment on infected population is found positive initially, the recovered population also increases initially and then consequently decreases (after t = 1.70, for high treatment rate) due to fast growth of infected population. 6. conclusions the compartmental model for dengue fever with treatment control measure [1] has been re-investigated and the 87 tuteja & lal / j. nig. soc. phys. sci. 3 (2021) 82–88 88 figure 3: exposed and infected vectors figure 4: infected human with different treatment (τ) figure 5: recovered human with different treatment (τ) inadvertent errors have been removed from the recurrence relations of the model equations due to dtm. the existence, uniqueness and positivity of the solutions have been established. the semi-analytical solutions of the model equations are re-computed using the dtm and built-in function asymptoticdsolvevalue of wolfram mathematica and are found to be the same. it has been found that results obtained from the dtm are valid only for a small interval of time t (t < 1.5), as t becomes large, the exposed, recovered human population and infected vector population becomes negative. for smaller t, the better the treatment is, recovery will be faster figure 5. acknowledgments we thank the anonymous referees for the positive enlightening comments and suggestions, which have greatly helped us in making improvements to this paper. references [1] f. y. eguda, a. 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[18] n. r. derrick & s. l. grossman, “differential equation with applications", addison wesley publishing company inc., philippines 1976. 88 j. nig. soc. phys. sci. 3 (2021) 48–58 journal of the nigerian society of physical sciences numerical simulation of copper indium gallium diselenide solar cells using one dimensional scaps software c. o. lawania, g. j. ibeha, o. o. igea, d. elia,b,∗, j. o. emmanuela,c, a. j. ukwenyaa, p. o. oyedared adepartment of physics, nigerian defence academy, kaduna, nigeria bdepartment of physical sciences, greenfield university, kasarami, kaduna, nigeria cdepartment of basic science and general studies, federal college of forestry mechanization, kaduna, nigeria ddepartment of science laboratory technology, federal polytechnic ede, osun state, nigeria abstract the effect of multivalent defect density, thickness of absorber and buffer layer thickness on the performance of cigs solar cells were investigated systematically. the study was carried out using solar cells capacitance simulator (scaps) code, which is capable of solving the basic semiconductor equations. employing numerical modelling, a solar cell with the structure al |zno : al| in2s 3 |cigs | pt was simulated and in it, a double acceptor defect (-2/-1/0) with a density of 1014 cm−3 was set in the absorber in the first instance. this initial device gave a power conversion efficiency (pce) of 25.85 %, short circuit current density (jsc) of 37.9576 macm−2, photovoltage (voc) of 0.7992 v and fill factor (ff) of 85.22 %. when the density of multivalent defect (-2/-1/0) was varied between 1010 cm−3 and 1017 cm−3 the solar cells performance dropped from 26.81 % to 16.87 %.the champion device was with multivalent defect of 1010 cm−3 which shows an enhancement of 3.71 % from the pristine device. on varying the cigs layer thickness from 0.4 µm to 3.6 µm, an increase in pce was observed from 0.4 µm to 1.2 µm then the pce began to decrease beyond a thickness of 1.2 µm. the best pce was recorded with thickness of 1.2 µm which gave jsc of 37.7506 macm−2, voc of 0.8059 v, ff of 85.2655 %. on varying the in2s 3 (buffer) layer thickness from 0.01 µm to 0.08 µm, we observed that there was no significant change in photovoltaic parameters of the solar cells as buffer layer thickness increased. doi:10.46481/jnsps.2021.133 keywords: scaps, cigs, multivalent defect, buffer layer, absorber article history : received: 30 august 2020 received in revised form: 02 february 2021 accepted for publication: 04 february 2021 published: 29 may 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: t. owolabi & b. j. falaye 1. introduction cigs is a quaternary compound semiconductor which is an alloy of cuins e2 (cis) and cugas e2 (cgs) and comprises four elements namely: copper, indium, gallium and selenium. it is a direct band gap material [1] whose energy band gap could be varied from 1.06 ev – 1.70 ev by changing the indium to ∗corresponding author tel. no: +2348063307256 email address: danladielibako@gmail.com (d. eli) gallium ratio [2]. according to hamanche [3], cigs is the most promising candidate for efficient and low-cost solar cells based on the advantages of the optical and electrical properties of the material [3]. in spite of these attractive features of cigs material, there are factors which could affect the performance of solar cells whose absorbers are made from this material. such factors include defect density in semiconductor layers and interfaces of the solar cell, absorber layer and buffer layer thicknesses, bandgap of semiconductor materials, working temperature of the solar cell among others. 48 lawani et al. / j. nig. soc. phys. sci. 3 (2021) 48–58 49 multivalent defects are defects caused by transition metals which usually occur as impurities or as part of the structure of some semiconductors [4]. transition metals are metals which have valence electrons in two shells instead of only one. they therefore exhibit multiple oxidation states and can transit from one charge state to another by accommodating a variable number of electrons in their d-orbitals. when multivalent elements occur as impurities in semiconductors, it is not always clear how many of their valence electrons would be used for charge exchange. for example, if tin (sn) is doped on divalent zinc (zn) or magnesium (mg) site. we are not sure if it will behave as +iv element or a +ii element. there is really the possibility that a multivalent impurity would transit from one oxidation state to another and their transition would appear as deep level inside the band gap of the material. deep levels which are associated with a change in oxidation state tend to deteriorate the electronic properties of semiconductors. as they form recombination centers and carrier traps [4]. we therefore investigate in this work, the effect of multivalent defect density, thickness of absorber and buffer layer on the performance of cigs solar cells using scaps-1d. this software was used to calculate the short circuit current density (jsc), open circuit voltage (voc), fill factor (ff) and efficiency (η) which are photovoltaic parameters used for the assessment of solar cells’ performance. the spectral response of the solar cells in the face of varying defect density, absorber layer thickness and buffer layer thickness were also studied. 2. materials and methods 2.1. cell structure the solar cells simulated have the structure al|zno : al|in2 s 3|cigs |pt| as shown in figure1. the main parts of the cells are cigs absorber and the in2s 3 buffer layers. the cigs absorber is responsible for trapping light from the sun. it is considered environmental friendly because of the absence of cadmium in its structure. the material has a direct band gap and high absorption coefficient requiring just a few micrometers to absorb the maximum incident photon. the wide band gap of this material has also been found to be variable depending on the composition of the cigs material [5]. the in2s 3 is chosen as the buffer since it is stable, has a wider band gap and is considered nontoxic, when compared with other buffers such as cds. it is also transparent and photoconductive [6]. a transparent conductive oxide (tco) layer (acting as the window) made of zno:al is deposited on top of the buffer layer as it is transparent to most of the solar spectrum because of its wide band gap.although zno:b (boron doped zinc oxide) could be more beneficial for the solar cells because it lowers absorption losses leading to an increase in the quantum efficiency of the solar cells[7], zno:al is used in this research because it is a low cost tco and it is also highly conductive. front and back contacts usually made of metallic elements are introduced in the cells’ structure for the conduction of photogenerated charge carriers in and out of the solar cells. al is used as the front contact in this work, because it is lightweight, non-magnetic and corrosion resistant. it is a good conductor of electricity; more conductive figure 1. model of the simulated solar cell. than copper and less expensive than silver. a back contact of pt is preferred to the commonly used molybdenum since it is nontoxic when compared with molybdenum and gives a cell with higher efficiency [8]. 2.2. numerical modelling the method used for this work is numerical simulation. numerical simulation often gives insight into the interpretation of measurements even as it aids in the assessment of the potential merits of a cell structure [7]. some softwares among which is scaps-1d can be used to analyze the effect of the variation of materials parameters, that is, the presence or absence of particular properties or the varying of all properties in the range of values, for the optimization of solar cells’ efficiencies. this helps the producers of solar cells with the needed insight to effect necessary changes in their production methods in order to improve product performance. scaps1d simulation results have a good agreement with the results of existing experimental works [9] and this is the major motivating factor for its use in this research. scaps computes the steady state band diagram, recombination profile and carrier transport in one dimension, based on poisson’s equation (equation 1) together with the continuity equations (equations 2a and 2b) for holes and electrons. ∇· �e = q( p − n + n) (1) ∂n(x, t) ∂t = 1 q ∂jn ∂x + gn(x, t) − rn(x, t) (2a) ∂p(x, t) ∂t = 1 q ∂jp ∂x + g p(x, t) − rp(x, t) (2b) where e = electric field; � = permittivity of semiconductor; q = electronic charge; n = concentration of electrons; p = concentration of holes; n = net charge due to dopants and other trapped charges; jn and j p = current density of electrons and holes; gn and g p = rates of electron and holes generation in 49 lawani et al. / j. nig. soc. phys. sci. 3 (2021) 48–58 50 figure 2. simulation procedure. figure 3. scaps solar cell definition panel. the semiconductor device; rn and rp are rates of electron and hole recombination in the solar cells. figure 2 shows the steps that were taken in the simulation, starting with the launch of scaps. 2.3. simulated parameters the material parameters were selected from experimental results [10,11,12]. table1 gives a summary of the parameters for each layer in the simulation. defect parameters in the layers and interfaces were sourced from literatures [1, 13, 14, 15] as shown in table 2. the work functions of the front contact (al) and back contact (pt) are 4.26 ev and 5.93 ev, respectively [16]. a working temperature of 300 k, solar spectrum am1.5 and a scanning voltage of 0 v 1.3 v were used for all simulations. the solar cell definition panel shown in figure 3 is the environment where the physical properties of the various layers of the solar cells were inputted. figure 4. energy band diagram in the cigs solar cell. 3. results and discussion 3.1. performance parameters from initial simulation in the initial device set up for this work, a multivalent defect in the form of a double acceptor (-2/-1/0) defect with a gaussian energy distribution, defect energy level (et) = {0.1, 0.4} ev above ev and a concentration of 1.0×1014 cm−3, was introduced into the cigs (absorber) layer. this defect which is mainly caused by cui i i (cui i i is a double acceptor and iii represents a group 3 element such as indium or gallium) defect is common in cigs absorbers [17]. details of other defects set in the simulation are given in table 2; the resulting performance parameters of the opencircuit voltage (voc ), shortcircuit current density (js c ), fill factor (ff) and efficiency determined using jv characteristics are compared with those derived from experimental work [18]. the comparison which is shown in table 3 reveals that there is a good agreement between data from calculations and those from experiment hence validating parameters used in the simulation. the j-v curve and quantum efficiency curve are also obtained and shown below. the parameters of the different layers used in the simulation are also given in table 1. in the quantum efficiency (spectral response) curve shown in figure 6, there is an observed increase of spectral response in the short wavelength between 350 nm(0.35 µm) and 400 nm(0.4 µm). the curve reveals a maximum efficiency of approximately 100 % occurring between 400 nm(0.4 µm) and 1000 nm(1 µm) but this high efficiency begins to fall off after 1000 nm(1 µm). this fall is very likely due to incomplete absorption of the long wavelength photons. this analysis pertaining to the quantum efficiency, agrees very much with those reported in literatures [1,19]. 3.2. effect of multivalent defect concentration in cigs (absorber) layer the density of absorber layer defect has a direct effect on photovoltaic cell performance because as the concentration of 50 lawani et al. / j. nig. soc. phys. sci. 3 (2021) 48–58 51 table 1. materials parameters for cigs |in2s 3|zno : al| solar cell [10,11,12]. layer parameter cigs in2s 3 zno : al thickness (µm) 2 0.04 1.6 band gap, eg(ev ) 1.2 2.5 3.3 electron affinity, χe(ev ) 4.25 4.25 4.6 relative permitivity �e 13.6 13.5 9 nc, effective density of states (1/cm3) 2.2 × 1018 1.8 × 1019 2.2 × 1018 nv, effective density of states (1/cm3) 1.8 × 1019 4.0 × 1013 1.8 × 1019 electron mobility, µn(cm2/vs 100 400 100 hole mobility, µp(cm2/vs 25 210 25 acceptor concentration na(1/cm3) 1.0 × 1016 0 0 donor concentration nd(1/cm3) 0 1.0 × 1018 1.0 × 1018 table 2. defect parameters of buffer, window and interfaces [1, 13, 14, 15]. parameters in2s 3 zno:al cigs/in2s 3 in2s 3/zno:al interface interface defect type acceptor 0.04 1.6 capture cross section for electrons σn(cm2) 1.0 × 10−15 1.0 × 10−12 1.0 × 10−14 1.0 × 10−12 capture cross section for holes σh(cm2) 5.0 × 10−13 1.0 × 10−12 1.0 × 10−14 1.0 × 10−15 energetic distribution gaussian gaussian single single energy level with respect to reference (ev) 0.6 above ev 0.6 above ev 0.6 above ev 0.6 above ev characteristic energy level (ev) 0.1 0.1 0.1 0.1 total density (cm−3) 1.0 × 1014 1.772 × 1016 concentration (cm−2) 1.0 × 1010 1.0 × 1010 figure 5. j-v curve of cigs solar cell with initial parameters. defects increase, the minority charge lifetime reduces. this is evident from τ = 1 σvth nt (3) equation 3 above where τ is the minority charge lifetime, σ is electron/hole capture cross section, vth is thermal velocity of electron/holes and nt is total density of defects but σvth = c and c is the capture constant of electron/holes so τ = 1cnt meaning that the life time τ is inversely proportional to the product of defect density and capture constant of the charge carrier. with table 3. results from initial simulation compared with experimental data. voc js c ff efficiency (v ) (ma/cm2) (%) (%) experimental [18] 0.7410 37.8000 80.60 22.60 simulation 0.7992 37.9576 85.22 25.85 figure 6. quantum efficiency curve with initial parameters. reduction in life time, the diffusion length of electrons and holes reduces. diffusion length is the average distance a charge carrier can travel in a semiconductor material before it recombines. di f f usion length, l = √ dτ (4) 51 lawani et al. / j. nig. soc. phys. sci. 3 (2021) 48–58 52 meaning that if defect density increases, τ reduces and diffusion length reduces also. in equation 4, d is the diffusion coefficient for electrons/holes with a reduction in diffusion length, the probability of collection of photogenerated charge carriers at the terminals reduces; this in turn lowers the photocurrent for the solar cell, and increase the chances of recombination hence increasing the recombination loss in the absorber [20]. defects could be introduced either intentionally or unintentionally into semiconductors during the growth process, during processing of the device or from the working environment [21]. theoretical studies which are confirmed by results of measurement show that most of the existing defects in chalcopyrite solar cells, are multivalent in nature [22, 23]. in this study, the impact of varying multivalent defect concentration is observed by choosing the values of the defect density in the range of 1010 cm−3–1017 cm−3. table 4 gives the performance parameters of the cigs solar cells with various values of multivalent defect density in the absorber. it would be observed that the table 4. dependence of cells performance on multivalent defect density in cigs absorber layer. multivalent voc js c ff η defect (v ) (ma/cm2) (%) (%) density (cm−3) 1010 0.82086 37.96297 86.0388 26.8116 1011 0.82083 37.96297 86.0371 26.8102 1012 0.82059 37.96292 86.0196 26.7967 1013 0.81785 37.96244 85.8998 26.6700 1014 0.79924 37.95760 85.2210 25.8537 1015 0.74993 37.90711 83.6588 23.7824 1016 0.70338 37.29294 80.0979 21.0106 1017 0.67284 33.33359 75.2251 16.8716 solar cells’ performance does not change much when the defect density is below 1014 cm−3. this result tallies with the finding of similar study [24] in this regard. figure 7 shows that all electrical performance parameters start degrading at a defect density of ≈ 1015 cm−3. voc goes down from 0.82086 v to 0.67284 v , representing a decrease of 22.38 %. js c falls from 37.96297 ma/cm2 to 33.33359 ma/cm2 corresponding to a decrease of 13.89 %. these drops may be attributed to recombination within localized energy levels created by defects which cause current leakage [25]. as a result, the conversion efficiency goes down from 26.8116 % to 16.8716 % representing a decrease of 58.92 %. since solar cell efficiency is the amount of energy in the form of sunlight that can be converted into electricity by a solar cell, this 58.92 % decrease in conversion efficiency brought about by an increase in concentration of multivalent defect in the absorber layer poses a disadvantage to the functioning of the solar cell. according to a study [26], an efficient solar cell will have a high short circuit current density jsc, a high open circuit voc and a fill factor as close as possible to 1 (or 100 %). the fill factor is a measure of the ideality of a solar cell. in figure 7, the fill factor is observed to depart more from its ideal value of 100 % with an increase in the density of multivalent defect in the absorber leading to less efficiency and ideality of the solar cells. the multivalent defect density which produces an optimum performance of the solar cells is 1010/cm3 at an open circuit voltage voc of 0.8209 v , short circuit current density jsc of 37.96300 ma/cm2, fill factor ff of 86.0388 % and conversion efficiency η of 26.8116 % (as shown in figure 8). this implies that multivalent defect densities (double acceptors, in this case) in cigs solar cells should be controlled in such a way that they do not exceed this value. figure 9 shows the quantum efficiency (qe) as a function of wavelength for different values of defect density in the cigs layer. when the wavelength is in the range of 300 nm(0.3 µm) – 1200 nm(1.2 µm) the absorption efficiency decreases with increased multivalent defect density in the cigs layer. this is because as defect density increases the recombination (which causes loss of charge carriers) phenomena becomes more pronounced and since quantum efficiency is the ratio of the number of carriers collected by the solar cell to the number of incident photons [19], quantum efficiency drops. 3.3. effect of varying in2s 3 (buffer) layer thickness the influence of the thickness of in2s 3 buffer layer on performance of the photovoltaic cell is shown in figure 10. the thickness of the buffer was varied from 0.01 µm through 0.08 µm. although the variation of all photovoltaic parameters with increasing buffer thickness is not very significant, a reduction in the efficiency of the solar cells with increasing thickness of the buffer was noticed, in line with the findings in literatures [27, 28, 29]. this is caused by absorption of some photons in this layer [27] as a large number of short-wave length photons are absorbed before reaching the absorber layer while photons having wavelengths greater than that associated with the band gap of the buffer cannot generate electron-hole pairs and are therefore lost as heat. whereas a thin buffer layer means majority of photons can pass through the buffer into the absorber without being absorbed, increasing the buffer layer thickness causes a drop in efficiency of the solar cells. this is due to photon loss occurring inside the buffer layer. when a smaller number of photons make it through the buffer, less electron-hole pairs are created and this means less electricity is generated. this agrees with the findings [29]. the observed reduction in js c is caused by less production of electron-hole pair as a smaller number of electronhole pairs can reach the absorber layer with increase in buffer layer thickness. a decreased short circuit current means that less photogenerated carriers are produced and this lowers the efficiency of the solar cells. apart from the very little initial decrease in voc when buffer layer thickness is increased from 0.01 µm to 0.02 µm, the open circuit voltage remains constant showing that the buffer layer thickness has little or no effect on voc. from figure 10, there is a slight increment in fill factor when buffer layer thickness is increased from 0.02 µm to 0.03 µm thereafter, ff begins to drop again. these changes must have resulted from valence band discontinuities at the interfaces that appear as spikes [14]. the best efficiency of the solar cells after variation of the buffer thickness is 25.9813 % and this is achieved 52 lawani et al. / j. nig. soc. phys. sci. 3 (2021) 48–58 53 figure 7. variation in performance of cigs solar cells with multivalent defect density. figure 8. j-v curves of cigs solar cells with various values of multivalent defect density. for a thickness of 0.01 µm at an open circuit voltage voc of 0.8030 v , jsc of 37.9591 ma/cm2 and fill factor of 85.2329 % (as shown in figure 11). figure 12 shows the quantum efficiency (qe) as a function of wavelength for different values of buffer (in2s 3) layer table 5. dependence of solar cells’ performance on buffer layer thickness. thickness voc jsc ff η of buffer (v) (ma/cm2) % % (µm) 0.01 0.8030 37.9591 85.2329 25.9813 0.02 0.7997 37.9586 85.2183 25.8698 0.03 0.7993 37.9582 85.2211 25.8556 0.04 0.7992 37.9576 85.2210 25.8537 0.05 0.7992 37.9569 85.2206 25.8529 0.06 0.7992 37.9560 85.2202 25.8521 0.07 0.7992 37.9549 85.2198 25.8512 0.08 0.7992 37.9536 85.2193 25.8502 53 lawani et al. / j. nig. soc. phys. sci. 3 (2021) 48–58 54 figure 9. quantum efficiency as a function of wavelength for different values of defect density in cigs layer. thickness. when the wavelength is in the range of 300 nm (0.3 µm) – 1200 nm (1.2 µm), we observed that the spectral response curves overlap because the absorption efficiency remains constant for all values of buffer layer thickness. this spectral response in relation to varying buffer layer thickness further proves that buffer layer thickness has little or no effect on cigs solar cells investigated. 3.4. effect of varying cigs (absorber) layer thickness an important parameter which also affects the performance of cigs solar cells is the thickness of the absorber layer. the effect of thickness of the absorber layer on the solar cell’s performance parameters voc , js c , ff, pce is seen in figure 13. when the thickness of cigs absorber layer was varied from 0.4 µm to 3.6µm, the solar cell’s efficiency was seen to increase from 23.96 % to 25.94 % for an increase in thickness, of 0.4 µm – 1.2µm respectively. this increase is attributable to absorption of more photons as absorber layer thickness increases. this translates to the production of a significant number of electronhole pairs which then leads to an improvement in efficiency of the solar cells [12]. this is good for the performance of the solar cells since it means that more of the sun’s energy would be converted to electricity in the solar cells. beyond an absorber thickness of 1.2µm, the efficiency begins to drop due to decreased collection of photo-generated charge carriers which is caused by charge recombination. this tallies with similar finding [24] js c increases with increasing absorber thickness since longer wavelength photons are absorbed in thicker layers of the absorber and they enhance the amount of photo-generated carriers which in turn boosts efficiency and therefore produces solar cells which perform better. fill factor ff remains nearly constant but voc kept decreasing as a result of recombination of charge carriers [20] which increases in thicker layers of the absorber. this is not good for the performance of the solar cell as the efficiency of the solar cell has a direct dependence on open circuit voltage v oc. for optimum performance, the absorber thickness of the cigs solar cells should be kept at 1.2 µm. the photovoltaic cell parameters corresponding to this optimum value are v oc of 0.8059 v , jsc of 37.7506 ma/cm2, ff of 85.2655 % and conversion efficiency of 25.9403 % (as shown in figure 14). table 6. dependence of solar cells’ performance on absorber layer thickness. thickness voc jsc ff η of absorber (v) (ma/cm2) % % (µm) 0.4 0.8126 34.5688 85.2896 23.9577 0.8 0.8097 37.2321 85.2886 25.7114 1.2 0.8059 37.7506 85.2655 25.9403 1.6 0.8024 37.9045 85.2379 25.9258 2.0 0.7992 37.9576 85.2210 25.8537 2.4 0.7961 37.9805 85.2398 25.7724 2.8 0.7933 37.9930 85.2467 25.6934 3.2 0.7909 38.9921 85.2450 25.6128 3.6 0.7887 38.0000 85.2375 25.5452 figure 15 shows the quantum efficiency (qe) as a function of wavelength for different values of absorber (cigs) layer thickness. when the wavelength is in the range of 300 nm (0.3 µm) – 1200 nm (1.2 µm) the absorption efficiency increases with increased absorber (cigs) layer thickness. this is because as absorber (cigs) layer thickness increases the number of absorbed photons increases consequently, a higher number of electron-hole pairs are produced and the quantum efficiency increases [1]. for all values of absorber (cigs) layer thickness, it is observed that the spectral response curves show a decrease of the long wave length collection. this is most likely due to incomplete absorption of the long wavelength photons [30]. 3.5. performance of optimized parameters based on the optimized multivalent defect density, absorber layer thickness and buffer layer thickness, an efficiency of 27 %, current density of 37.75 ma/cm3, voltage of 0.829 v and fill factor of 86.26 % were obtained as depicted in figure 16. compared with the experimental data obtained by jackson et al.[18] were an efficiency of 22.6 % was reported, the optimized cell in this work shows an improvement of 16.30 % in efficiency. aside the alkali post deposition treatment (pdt) done on cigs absorbers which were used in the experimental solar cells referred to in table 8 their efficiencies could be improved upon by carefully controlling the concentration of multivalent defect in their absorbers as this form of defect is prevalent in chalcopyrite materials. table 7. optimized parameters of the device. optimized parameters absorber buffer thickness (µm) 1.2 0.01 multivalent defect density (cm3) 1010 – 54 lawani et al. / j. nig. soc. phys. sci. 3 (2021) 48–58 55 figure 10. variation in performance of cigs solar cell with buffer layer thickness. figure 11. j-v curves of cigs solar cells with various values of buffer layer thickness. 4. conclusion in this work, we undertook numerical simulation to investigate multivalent defects and the influence of absorber layer thickness and buffer layer thickness on al|zno : al|in2s 3|cigs figure 12. quantum efficiency as a function of wavelength for different values of buffer (in2s 3) layer thickness. |pt| structured solar cells using scaps code. the efficiency of the initial device which was found to be 25.85 % with a multivalent defect density of 1014 cm−3 experienced a boost to 27 % when the solar cell was optimized with an absorber layer thickness of 1.2 µm, buffer layer thickness of 0.01 µm and a 55 lawani et al. / j. nig. soc. phys. sci. 3 (2021) 48–58 56 figure 13. variation in performance of cigs solar cells with absorber layer thickness. figure 14. j-v curves of cigs solar cell with various values of absorber layer thickness. multivalent defect density of 1010 cm−3 in the cigs absorber layer. the results obtained revealed that when the density of multivalent defect in the absorber was varied from 1010 cm−3 through 1017 cm−3, the efficiency of the cigs photovoltaic cells dropped from 26.81 % to 16.87 % representing a decrease of figure 15. quantum efficiency as a function of wavelength for different values of absorber (cigs) layer thickness. 58.92 %. this result clearly shows how detrimentally multivalent defects can affect the performance of cigs solar cells. as expected, increasing the absorber layer thickness caused an increase in efficiency until an optimal thickness of 1.2 µm was achieved while increase in buffer layer thickness from 0.01 µm 56 lawani et al. / j. nig. soc. phys. sci. 3 (2021) 48–58 57 table 8. photovoltaic parameters corresponding to optimized parameters of the cigs solar cells compared with those of experimental researches. simulation voc jsc ff η (v) (ma/cm2) % % initial 0.7992 37.9576 85.2200 25.8500 optimized absorber layer thickness (µm) 0.8059 37.7506 85.2655 25.9403 optimized buffer layer thickness (µm) 0.8030 37.9591 85.2329 25.9813 optimized multivalent defect density (cm3) 0.8209 37.9630 86.0388 26.8116 final optimization 0.8290 37.7541 86.2600 27.0000 experimental data 0.7570 34.8000 79.1000 20.8000 [31] experimental data 0.7440 36.7000 80.5000 22.0000 [32] experimental data 0.7410 37.8000 80.6000 22.6000 [18] figure 16. j-v curve of cigs solar cell with optimized parameters. through 0.08 µm caused a slight decrease of 0.51 % in 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[31] zws, “thin film photovoltaics success story continues: zws sets european record of 22 percent for cigs cells”, htt ps : //www.zswbw.de/ f ileadmin/useru pload/pdf s/pressemitteilungen /2016/pi07 − 2016 − zs w − cigs 22percent − en.pd f , (2016). 58 j. nig. soc. phys. sci. 5 (2023) 1087 journal of the nigerian society of physical sciences numerical solution of second order fuzzy ordinary differential equations using two-step block method with third and fourth derivatives kashif hussain, oluwaseun adeyeye∗, nazihah ahmad school of quantitative sciences, universiti utara malaysia, kedah, malaysia abstract fuzzy differential equation models are suitable where uncertainty exists for real-world phenomena. numerical techniques are used to provide an approximate solution to these models in the absence of an exact solution. however, existing studies that have developed numerical techniques for solving second-order fuzzy ordinary differential equations (fodes) possess an absolute error accuracy that could be improved. therefore, this article developed a more accurate higher derivative self-starting block scheme for the numerical solution of second-order fodes with fuzzy initial and boundary conditions imposed. linear block approach using taylor series expansion is adopted for the derivation of the proposed method and the basic properties are established using the definitions of stability and consistency for block methods. according to the numerical results, when compared to the exact solution in terms of absolute error, the new method proposed in this article outperformed existing numerical methods. it is thus concluded that the proposed method is effective for solving second-order fodes directly. doi:10.46481/jnsps.2023.1087 keywords: fuzzy initial value problem, fuzzy boundary value problem, second order, two-step, block method, linear, nonlinear article history : received: 24 september 2022 received in revised form: 20 january 2023 accepted for publication: 12 february 2023 published: 04 april 2023 © 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: b. j. falaye 1. introduction second-order differential equations have many applications, especially in the field of engineering, biology, chemistry, electronics, physics, etc. unfortunately, unpredictable scenarios may be encountered which introduced the concept of uncertainty [1] and the application of fuzzy derivatives in fuzzy differential equations (fdes) to handle these situations [2]. there are three differentiations used to describe the differential or derivative of a fuzzy function. the first is the hukuhara derivative ∗corresponding author tel. no: +60 49286354 email address: adeyeye@uum.edu.my (oluwaseun adeyeye) (h-derivative), which was introduced in [3], the second is the seikkala derivative introduced in [4], and the third is the generalized derivative (g-derivative) introduced in [5]. this study focuses on the h-derivative in order to define the differential equations considered in this article, which follows the definition by the authors whose results were considered for comparison in the numerical examples with the newly developed block method. the second-order fode of the form given in the equation below is considered in this article, ŷ′′(x) = f (x, ŷ(x), ŷ′(x)),∀x ∈ [a, b] (1) 1 hussain et al. / j. nig. soc. phys. sci. 5 (2023) 1087 2 from equation 1, ŷ′′(x) = d 2 ŷ dt2 = f (x, ŷ(x), ŷ ′(x)) is a hderivative and ŷ is a fuzzy function of crisp variable x. since the function is fuzzy, there exist solutions known as lower and upper solutions because the parametric form of the α-level is given as ŷ ′′ (x,α) = f (x, ŷ(x,α), ŷ′(x,α)),∀α ∈ [0, 1], where f = min { f (x, ŷ(x,α), ŷ′(x,α)) } and f = max { f (x, ŷ(x,α), ŷ ′ (x,α)) } . the above types of problems in the parametric form of fuzzy function may be difficult to solve directly, and sometimes it is not possible to obtain exact solutions. as a result, researchers were interested in employing various numerical approaches to obtain an approximate solution for second-order fodes. several types of numerical methods developed by numerous researchers for second-order fodes with initial and boundary conditions include the homotopy analysis method in [6, 7], decomposition method [8], laplace and differential transformation method in [9, 10], least-square method [11], and rungekutta method in [12-14]. the biggest drawback of these approaches is the reduction of the second-order fodes to the system of first-order fodes, which leads to computational burden and also impacts solution accuracy. to bypass the rigor of reduction, block methods were introduced for the direct solution of second-order fodes in [15-17]. however, due to the order of the block methods developed by these studies, it is observed that there is still room to improve the accuracy of their obtained results in terms of absolute error. hence, the motivation of this study is to develop a new block method with the presence of two higher derivative terms with the aim of obtaining better accuracy. in comparison to existing methods, the newly developed method has the advantages of better accuracy, being self-starting, and incurring a low computational burden in the development and implementation of the block method. the following is how this article is structured: the essential definitions for fuzzy set theory are presented in section 2, and the construction of the two-step block method with third and fourth derivatives is presented in section 3 with the use of the linear block approach. section 4 highlights the block method’s properties, section 5 considers linear and nonlinear numerical examples, and section 6 concludes the article. 2. preliminaries this section recalls some definitions which will be adopted in this article. the section discusses basic definitions of triangular fuzzy numbers, trapezoidal fuzzy numbers, fuzzy set support, α-level set, and hukuhara differential. these concepts are required to establish the different parameters of the crisp theory’s uncertain behavior. these concepts play an important role when fuzzy differential equations model real-life situations. definition 1: triangular fuzzy number [18] consider three numbers (µ, v, w) ∈ r3,µ ≤ v ≤ w, then m(x) denotes the triangular fuzzy number given as: m(x,µ, v, w) =  0, x < µ x−µ v−µ , µ ≤ x ≤ v w−x w−v , v < x ≤ w 0, x > w (2) the corresponding α-level set is defined as mα = [ µ + α (v −µ) , w −α(w − v) ] ,α ∈ [0, 1]. (3) definition 2: trapezoidal fuzzy numbers [18] consider four numbers (µ, v, w,δ) ∈ r4,µ ≤ v ≤ w ≤ δ, then the trapezoidal fuzzy number m(x) is given as: m(x,µ, v, w,δ) =  0, x < µ x−µ v−µ , µ ≤ x < v 1, v ≤ x ≤ w w−x w−v , w < x ≤ δ 0, x > δ (4) the corresponding α-level set is defined as mα = [ µ + α (v −µ) ,δ−α(δ− w) ] ,α ∈ [0, 1]. (5) definition 3: fuzzy set support [18] a set â has fuzzy set support with x universal set defined as, s upp(â) = { x ∈ x|mâ(x) > 0 } (6) it contains all elements in x which have membership degree of fuzzy element greater than zero. definition 4: α-level set [18] consider that, m ∈ r f , the α-level set is defined as, mα = {x ∈ r|m(x) > 0} , α ∈ [0, 1]cl(suppm), α = 0 , (7) with its closed, bounded interval [m(x), m(x)]. m(x) and m(x) are lower and upper bound of mα respectively. definition 5: hukuhara differential [3] a function f : (u, v) → r f is called h-differentiable, if for h > 0 sufficiently small, then h-difference f (x) − f (x − h), f (x + h) − f (x) exists and ∃ an element f ′(x) ∈ r f such that, lim h→0 f (x) − f (x − h) h = lim h→0 f (x + h) − f (x) h = f ′(x). (8) then f ′(x) is called the h-derivative of f at x. 2 hussain et al. / j. nig. soc. phys. sci. 5 (2023) 1087 3 3. methodology given that the second-order fode defined in equation 1 be a mapping f : r f → r f and ŷ0 ∈ r f with α-level set ŷ0 ∈ (̂ y(0,α), ŷ(0,α) )α α ,α ∈ [0, 1]. the partition of the has the set of grid points 0 = x0 < x1 < x2 <,...,< xn = x with exact solution as ( ŷ (xn,α) )α α = ( ŷ (xn,α), ŷ (xn,α) )α α and approximation solution also denoted as (̂ y(xn,α) )α α = (̂ y(xn,α), ŷ(xn,α) )α α at which points, h = x−x0n , xn = x0 + nh, 0 ≤ n ≤ n. the two-step linear block method with the presence of third and fourth derivatives in second-order form is stated below as, (̂ yn+η )α α =  1∑ v=0 (ηh)v v! ŷ(v)n + 2∑ d=0  2∑ v=0 ψdvη f (d) n+v   α α ,η = 1, 2 (9) with the first derivative expression for the block method form given as (̂ y′n+η )α α = ̂y′n + 2∑ d=0  2∑ v=0 ωdvη f (d) n+v   α α ,η = 1, 2 (10) expanding equations 9 and 10 produces the expressions in equations 11, 12, 13, and , 14. (̂ yn+1 )α α =  ŷn + ĥy ′ n + [ψ001 fn + ψ011 fn+1 + ψ021 fn+2 +ψ101 f ′ n + ψ111 f ′ n+1 + ψ121 f ′ n+2 + ψ201 f ′′ n +ψ211 f ′′ n+1 + ψ221 f ′′ n+2]  α α (11) (̂ yn+2 )α α =  ŷn + 2ĥy ′ n + [ψ002 fn + ψ012 fn+1 + ψ022 fn+2 +ψ102 f ′ n+1 + ψ112 f ′ n+1 + ψ122 f ′ n+2 + ψ202 f ′′ n +ψ212 f ′′ n+1 + ψ222 f ′′ n+2]  α α (12) (̂ y ′ n+1 )α α =  ŷ′n + [ω001 fn + ω011 fn+1 + ω021 fn+2 + ω101 f ′ n +ω111 f ′ n+1 + ω121 f ′ n+2 + ω201 f ′′ n + ω211 f ′′ n+1 +ω221 f ′′ n+2]  α α (13) (̂ y ′ n+2 )α α =  ŷ′n + [ω002 fn + ω012 fn+1 + ω022 fn+2 + ω102 f ′ n +ω112 f ′ n+1 + ω122 f ′ n+2 + ω202 f ′′ n + ω212 f ′′ n+1 +ω222 f ′′ n+2]  α α (14) by applying taylor series expansions (̂ y(x + h; α) )α α =  n∑ i=0 hi i! f i(x; α) α α (15) which is given in [19] to expand each term in equations 11-14 yields (̂ yn+ j )α α = (̂ y(xn + jh; α) )α α =  n∑ i=0 ( jh)i i! f i(xn; α) α α , j = 0, 1, 2, (16) (̂ yn+ j )α α =  ŷ(xn; α) + jĥy ′(xn; α) + ( jh)2 2! ŷ′′(xn; α) + ( jh)3 3! ŷ′′′(xn; α) + .... + ( jh)n n! ŷn(xn; α)  α α . (17) after that, the unknown coefficients ψdvn and ωdvn are obtained from ψdvn = a−1 b and ωdvn = a−1 d, where a =  1 1 1 0 0 0 0 0 0 0 h 2h 1 1 1 0 0 0 0 h 2 2! 22 h2 2! 0 h 2h 1 1 1 0 h 3 3! 23 h3 3! 0 h2 2! 22 h2 2! 0 h 2h 0 h 4 4! 24 h4 4! 0 h3 3! 23 h3 31 0 h2 2! 22 h2 2! 0 h 5 5! 25 h5 5! 0 h4 4! 24 h4 4! 0 h3 3! 23 h3 3! 0 h 6 6! 26 h6 6! 0 h5 5! 25 h5 5! 0 h4 4! 24 h4 4! 0 h 7 7! 27 h7 7! 0 h6 6! 26 h6 6! 0 h5 5! 25 h5 5! 0 h 8 8! 28 h8 8! 0 h7 7! 27 h7 7! 0 h6 61 26 h6 6!  α α , b = α (ηh)2 2! (ηh)3 3! (ηh)4 4! (ηh)5 5! (ηh)6 6! (ηh)7 7! (ηh)8 8! (ηh)9 9! (ηh)10 10!  α , d =  ηh (ηh)2 2! (ηh)3 3! (ηh)4 4! (ηh)5 5! (ηh)6 6! (ηh)7 7! (ηh)8 8! (ηh)9 9!  α α . therefore,  ψ001 ψ011 ψ021 ψ101 ψ111 ψ121 ψ201 ψ211 ψ221  α α =  19h2 60 h2 5 −h2 60 911h3 20160 −16h3 315 113h3 20160 53h4 20160 h4 80 −11h4 20160  ,  ψ002 ψ012 ψ022 ψ102 ψ112 ψ122 ψ202 ψ212 ψ222  α α =  76h2 105 128h2 105 2h2 35 34h3 315 −32h3 315 −2h3 315 2h4 315 16h4 315 0  ,  ω001 ω011 ω021 ω101 ω111 ω121 ω201 ω211 ω221  α α =  5669h 13440 64h 105 −42h 13440 303h2 4480 −1h2 8 47h2 4480 169h3 40320 8h3 315 −41h3 40320  ,  ω002 ω012 ω022 ω102 ω112 ω122 ω202 ω212 ω222  α α =  41h 105 128h 105 41h 105 2h2 35 0 −2h2 35 1h3 315 16h3 315 1h3 315  . the obtained values of the coefficients are substituted in equations 11-14 which is the required two-step block method 3 hussain et al. / j. nig. soc. phys. sci. 5 (2023) 1087 4 with the presence of third and fourth derivatives as given below.(̂ yn+1 )α α = ŷn + ĥy ′ n + h 2 [ 19 60 fn + 1 5 fn+1 − 1 60 fn+2 ] +h3 [ 911 20160 gn − 16 315 gn+1 + 113 20160 gn+2 ] +h4 [ 53 20160 mn + 1 80 mn+1 − 11 20160 mn+2 ] , (̂ yn+2 )α α = ŷn + 2ĥy ′ n + h 2 [ 76 105 fn + 128 105 fn+1 + 2 35 fn+2 ] +h3 [ 34 315 gn − 32 315 gn+1 − 2 315 gn+2 ] + h4 [ 2 315 mn + 16 315 mn+1 ] , (18) (̂ y′n+1 )α α = ŷ ′ n + h [ 5669 13440 fn + 64 105 fn+1 − 421 13440 fn+2 ] +h2 [ 303 4480 gn − 1 8 gn+1 + 47 4480 gn+2 ] +h3 [ 169 40320 mn + 8 315 mn+1 − 41 40320 mn+2 ] , (̂ y′n+2 )α α = ŷ ′ n + h [ 41 105 fn + 128 105 fn+1 + 41 105 fn+2 ] +h2 [ 2 35 gn − 2 35 gn+2 ] + h3 [ 1 315 mn + 16 315 mn+1 + 1 315 mn+2 ] (19) where g = d f (x,α)d x , m = d2 f (x,α) d x . the block method in equation 18 has corrector form,( a0ŷn+k )α α = ( a1ŷn−k )α α + h ( b1ŷ ′ n−k )α α + h2 ( c0 fn+k + c 1 fn−k )α α +h3 ( d0gn+k + d 1gn−k )α α + h4 ( e0 mn+k + e 1 mn−k )α α where, a0 = ( 1 0 0 1 )α α , a1 = ( 0 1 0 1 )α α , b1 = ( 0 1 0 2 )α α , c0 = ( 1 5 −1 60 128 105 2 35 )α α , c1 = ( 0 1960 0 76105 )α α , d0 = ( −16 315 113 20160 −32 315 −2 315 )α α , d1 = ( 0 91120160 0 34315 )α α , e0 = ( 1 80 −11 20160 16 315 −2 315 )α α , e1 = ( 0 5320160 0 2315 )α α , ŷn+k = (̂ yn+1 ŷn+2 )α α , ŷn−k = (̂ yn−1 ŷn )α α , ŷ′n−k = (̂ y′n−1 ŷ′n )α α , fn+k = ( fn+1 fn+2 )α α , fn−k = ( fn−1 fn )α α , gn+k = ( gn+1 gn+2 )α α gn−k = ( gn−1 gn )α α , mn+k = ( mn+1 mn+2 )α α , mn−k = ( mn−1 mn )α α . 4. properties of the proposed method this section will first mention the required definitions and theorems to investigate the properties of the developed two-step third-fourth derivative scheme, and thereafter apply these theorems and definitions to the method. 4.1. convergence and stability properties theorem 1: a block method is convergent iff it is consistent and zerostable. [22] proof the aim of the proof is to show that zero stability and consistency are necessary conditions for convergence. suppose that the block method defined in equation 9 is convergent, the first condition for zero-stability follows by considering equation 1 with a trivial solution ŷ(x) = 0. applying equation 9 to this problem yields the difference equation̂yn+η − 1∑ v=0 (ηh)v v! ŷ(v)n − 2∑ d=0  2∑ v=0 ψdvη f (d) n+v   α α ,η = 1, 2 (20) since the method is assumed to be convergent, for any x > 0, then lim h→0 nh→0 ŷn+η = 0 (21) for all solutions of equation 20 satisfying ŷs = ςs(h), s = 0, 1, ..., k − 1 where lim h→0 ŷs = 0 (22) let ψ = reiφ be a root of the first characteristic polynomial p(ψ) = 0, r ≥ 0, 0 ≤ φ ≤ 2π. it can be verified then that the numbers ŷn+η = hr n cos(nφ) (23) define a solution to equation 20 satisfying equation 22. if φ = 0, φ , π, then ŷn+η − ŷn − ŷ ′ n sin2φ = h2r2n (24) since the left-hand side of this identity converges to 0 as h → 0, n →∞, nh = x the same must be true of the right-hand side; therefore, lim n→∞ ( x n )∞ r2n = 0 (25) this implies that r ≤ 1. in other words, it is proven that any root of the first characteristic polynomial of (9) lies in the closed unit disc. note that any root of the first characteristic polynomial of equation 9 that lies on the unit circle must be simple. for the other condition, which is consistency, let us first show that c0 = 0. consider equation 1 with trivial solution, ŷ(x) = 1. applying equation 9 to this problem yields the difference equation equation 20. choose ŷs = 1, s = 0, 1, ..., k − 1. given that by hypothesis the method is convergent, it is deduced that lim h→0 ŷs = 1 (26) since in the present case ŷn is independent of the choice of h, equation 26 is equivalent to saying that lim h→∞ ŷn = 1, (27) 4 hussain et al. / j. nig. soc. phys. sci. 5 (2023) 1087 5 and passing to the limit n → ∞ in equation 20, it is deduced that αk + αk−1+, ..., +α0 = 0. (28) recalling the definition of c0, equation 28 is equivalent to c0 = 0 (i.e. p(1) = 0). to show that c1 = 0, consider equation 1 with trivial solution, ŷ(x) = x. applying equation 9 to this problem yields the difference equation in equation 20. for a convergent method every solution of equation 20 satisfying lim h→0 ςs(h) = 0, s = 0, 1, ..., k − 1 (29) where ŷs = ςs(h), s = 0, 1, ..., k − 1, must also satisfy lim h→0 ŷn+η = x. (30) since according to the previous theorem zero-stability is necessary for convergence, we may take it for granted that the first characteristic polynomial p(ψ) of the method does not have multiple roots on the unit circle |ψ| = 1, therefore p ′ (1) = kαk+, ..., +2α2 + α1 , 0. (31) let the sequence (xn) n n = 0 be defined by ŷn = knh, where k = ψdkη + ψd(k−1)η+, ..., +ψd2η + ψd1η + ψd0η kαk+, ..., +2α2 + α1 . (32) this sequence clearly satisfies equation 30 and is the solution of equation 20. furthermore, equation 31 implies that x = ŷ(x) = lim h→0 nh=x ŷn+η = lim h→0 nh=x knh = k x (33) c1 = (kαk+, ..., +2α2 + α1) −(ψdkη + ψd(k−1)η+, ..., +ψd2η + ψd1η + ψd0η) = 0. (34) equivalently, p ′ (1) = σ(1). thus, since the necessary conditions in terms of zero-stability and consistency is satisfied, so the block method is convergent. definition 6: consistency [20] a block method is consistent if it has order ρ ≥ 1. definition 7: zero-stability [20] a block method with matrix difference equation in the following form a0ŷn+k = a 1ŷn−k + b 1ŷ ′′ n−k + b 2ŷ ′′ n−k + · · · + b 1ŷ (m−1)n−k +hm ( c0ŷ mn+k + c 1ŷ mn−k )α α + h(m+1) ( d0ŷ (m+1)n+k + d 1ŷ (m+1)n−k )α α +h(m+2) ( e0ŷ (m+2)n+k + e 1ŷ (m+2)n−k )α α , (35) with ŷ an+k = (̂ yan+1, ŷ a n+2, ..., ŷ a n+k )t and ŷ an−k = (̂ yan1(k−1), ŷ a n−(k−2), ..., ŷ a n )t , is zero-stable if the first characteristic polynomial takes form p(ψ) = det(ψv a 0 − a1), (36) and the root of p(ψ) = 0 satisfy |ψv| ≤ 1, v = 1, ..., , k. definition 8: region of absolute stability [26] to obtain the polynomial for the absolute stability region of the block method. the expressions for the corrector take the form: det  −(w)k + a1 + q  k∑ j=0 b jwk− j  + q2  k∑ j=0 thec jwk− j  +q3  k∑ j=0 d jwk− j  + q4  k∑ j=0 e jwk− j    α α , q = λh. the absolute stability region is then obtained by plotting the polynomial roots using the boundary locus technique. if the obtained roots of the polynomial lie in the unit circle, then the block method is absolutely stable and its region is called the region of absolute stability. note that large absolute stability regions mean that large time-step size can be used during the implementation of the method to solve the differential equation [27-29]. definition 9: a-stable according to [20], a numerical method is said to be a-stable if its region of absolute stability contains the whole of the lefthand half-plane. definition 10: l-stable according to [20] a general linear multistep method is lstable if it is a-stable and, in addition, when applied to the scalar test equation ŷ ′ = λy, λ is a complex constant with reλ < 0, it yields ŷn+1 = r(hλ)̂yn, where, |r(hλ)|→ 0 as re(hλ) →∞. however, a clause is encountered as given in the following definition definition 11 according to [21] an a-stable linear multistep method cannot have an order greater than two. therefore, based on definition 8, the properties of a-stability and l-stability cannot be explored for the block methods developed in this article. this is because the block method developed have order greater than two. hence, the stability property with respect to choosing a stepsize value is limited to just absolute stability alone. although, much attention was not placed on choosing h-values from the stability region because the hvalues were chosen the same as the authors for comparison. these definitions for block methods in crisp form is adopted to the proposed method for fodes to prove the convergence properties for the proposed method in the next subsection. 4.2. convergence and stability analysis of proposed method order and error constant the linear operator associated with equation 9 is defined as: l(̂y(x), h) = ̂yn+η − 1∑ v=0 (ηh)v v! ŷ(v)n + 2∑ d=0  2∑ v=0 ψdvη f (d) n+v   α α , η = 1, 2, (37) 5 hussain et al. / j. nig. soc. phys. sci. 5 (2023) 1087 6 with l(̂y(x), h) = c0̂y(xn) + c1ĥy′(xn) + c2h2̂y′′(xn) + ... +cz+1h z+1̂yz+1(xn) + cz+2h z+2̂yz+2(xn)  α α . the method is said to be of order z if c0 = c1 = · · · = cz = cz+1 = 0, cz+2 , 0, and cz+2 is the error constant. following the approach by [28], the order of the two-step third-fourth derivatives block method with corrector equation 18 is nine with an error constant c11 = (3.8174e − 08, 7.617e − 08) t , and the order of the derivative part ten with an error constant c12 = (6.5076e − 08, −7.6349e − 09) t . the derivative formulae will be used to obtain the first derivative term in equation 1. expressing the corrector scheme 18 as blocks using previous definitions for the block methods. a simple iteration has been implemented to approximate the value of ŷn+1 and ŷn+2. in the code, we iterate the corrector to convergent and the convergence test employed, and the order of the correctors in nine [23] zero-stability applying above definition in fuzzy form for the proposed method gives p(ψ) = det(ψv a 0 − a1)αα, (38) p(ψ) = ∣∣∣∣∣∣ψv ( 1 0 0 1 ) − ( 0 1 0 1 )∣∣∣∣∣∣ α α . the root of p(ψ) = 0 satisfies the condition |ψv| ≤ 1, v = 1, 2. convergence the proposed method is convergent because it is zero stable and consistent. absolute stability region the polynomial of the proposed block method to plot its region of absolute stability is obtained as:  ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ ( w 0 0 w2 ) + ( 0 1 0 1 ) + q ( 0 1 0 2 ) +q2 [( 1w 5 1w2 60 128w 105 2w2 35 ) + ( 0 1960 0 76105 )] +q3 [( −16w 315 113w2 20160 −32w 315 −2w2 315 ) + ( 0 91120160 0 34315 )] +q4 [( 1w 80 −11w2 20160 16w 315 0 ) + ( 0 5320160 0 2315 )] ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣  α α , (39) r(w) =   11q8 396900 − 37q7 88200 + 19q6 52920 + 44q5 4725 + 13q4 3600 − 2q3 35 − 9q2 35 + 1]w 3 + [ 43q 8 793800 + 53q7 52920 + 508327q6 33868800 − 31547q5 793800 − 35639q4 100800 − 44q3 45 − 20597q2 20160 − 2q − 1  w  α α the absolute stability region is thus plotted as shown in figure 1, which implies that large time-stepsizes can be utilised with the method. from figure 1, it is seen that for the absolute stability region, all the roots of polynomial lie on the unit circle. figure 1. absolute stability region of proposed method 5. results this section details the application of the developed block method for the solution of second-order (linear and nonlinear) fodes (fivps and fbvps) and the obtained results are compared with the exact solution and existing methods. comparisons between exact and approximate solutions are shown in tables and graphs. x−axis shows the value of the approximation solution, y−axis show the value of α-level values, ŷ, ŷ are the lower and upper bounds of the exact solution respectively, ŷ, ŷ are the lower and upper bounds of the approximate solution respectively, e = ∣∣∣∣ŷ − ŷ∣∣∣∣ computes the absolute error of the lower bound approximation, e = ∣∣∣∣∣ŷ − ŷ ∣∣∣∣∣ computes the absolute error of the upper bound approximation, h is the step size, tsbm: two-step block method with third and fourth derivatives, ebhdef: extended block hybrid backward differentiation formula [16], bdf: block differentiation formula [15], bbdf: block backward differentiation formula [15], oomb: optimization of one-step block method [17], rk5: runge kutta method order five [14], oham: optimal homotopy asymptotic method [7], fdm: finite difference method [30]. example 1. given the second-order linear fivp ŷ′′(x) = −̂y(x), ŷ(0,α) = 0, ŷ′(0,α) = (0.9 + 0.1α, 1.1 − 0.1α), with exact solution ŷ (x,α) = (0.9 + 0.1α) sin(x), ŷ (x,α) = (1.1 − 0.1α) sin(x), 6 hussain et al. / j. nig. soc. phys. sci. 5 (2023) 1087 7 table 1. lower and upper solution of example 1 α tsbm e ebhdef e bbdf e bdf e h = 0.1 h = 0.01 h = 0.01 h = 0.01 0 0.0000e+00 2.8094e-11 5.4048e-08 3.0991e-08 0.2 0.0000e+00 2.8719e-11 5.5249e-08 3.1647e-08 0.4 0.0000e+00 2.9343e-11 5.6450e-08 3.2335e-08 0.6 0.0000e+00 2.9966e-11 5.7651e-08 3.3023e-08 0.8 0.0000e+00 3.0592e-11 5.8853e-08 3.3711e-08 1 0.0000e+00 3.1216e-11 6.0054e-08 3.4399e-08 α tsbm e ebhdef e bbdf e bdf e h = 0.1 h = 0.01 h = 0.01 h = 0.01 0 1.1102e-16 3.4337e-11 5.4048e-08 3.7838e-08 0.2 1.1102e-16 3.3713e-11 5.5249e-08 3.7151e-08 0.4 1.1102e-16 3.3089e-11 5.6450e-08 3.6463e-08 0.6 0.0000e+00 3.2464e-11 5.7651e-08 3.5775e-08 0.8 0.0000e+00 3.1840e-11 6.1255e-08 3.5087e-08 1 0.0000e+00 3.1216e-11 6.0054e-08 3.4399e-08 figure 2. numerical solution of example 1 with lower/upper solution and at x = 1, ŷ (1,α) = [ y (1,α), y (1,α) ] , 0 ≤ α ≤ 1. the results obtained for example 1 are shown in table 1 and figure 2 displays the complete iterations graph with stepsize h = 0.1 and h = 0.01 partition of the time interval x ∈ [0, 1]. it is observed from table 1 that the approximate solution obtained by new proposed method in comparison to the exact solution in terms of absolute error is very impressive, as it give same results as the exact solution at certain points. the results are graphically shown in figure 2. in the figure the behaviour of the linear fivp solution is seen to monotonically increase as shown in the graph. this follows from the property that a function’s output will not appear more than once during the course of a monotonically rising interval. it is worth noting that y(x) rises in lockstep with x. the exact and approximate solutions are also compared using the graph and it shows the approximate solution completely overlapping the exact solution which indicates high accuracy of the proposed method. table 2. lower and upper solution of example 2 α tsbm e ebhdef e bbdf e bdf e h = 0.1 h = 0.01 h = 0.01 h = 0.01 0 6.661338e-16 9.8449e-14 2.4250e-10 1.5988e-10 0.2 3.663736e-15 3.4927e-13 5.7971e-10 3.9122e-10 0.4 1.532108e-14 9.7144e-13 1.2016e-09 3.7933e-09 0.6 5.129230e-14 2.2859e-12 2.2597e-09 2.6125e-09 0.8 1.498801e-13 6.4525e-12 3.9207e-09 6.6967e-08 1 3.850253e-13 4.7628e-12 6.3971e-09 1.1110e-08 α tsbm e ebhdef e bbdf e bdf e h = 0.1 h = 0.01 h = 0.01 h = 0.01 0 1.345191e-13 4.2267e-11 4.07084e-08 1.26440e-07 0.2 7.431833e-12 2.6623e-11 2.98235e-08 9.1889e-08 0.4 3.907541e-12 1.5982e-11 2.13238e-08 7.34552e-08 0.6 2.774669e-12 9.0485e-11 1.48046e-08 3.52946e-08 0.8 9.001688e-13 8.1274e-11 9.93257e-09 1.51728e-08 1 3.854694e-13 4.7628e-12 6.39707e-09 1.11097e-08 figure 3. numerical solution of example 2 with lower/upper solution example 2. given the second-order non-linear fivp ŷ′′(x) = −(̂y′(x))2, ŷ(0,α) = (α, 2 −α), ŷ′(0,α) = (1 + α, 3 −α), with exact solution ŷ (x,α) = ln((xα + x + 1)eα), ŷ (x,α) = ln((3x − xα + 1)eα−2), and at x = 1, ŷ (1,α) = [ y (1,α), y (1,α) ] , 0 ≤ α ≤ 1. the results obtained for example 2 are shown in table 2 and figure 3 displays the complete iterations graph with stepsize h = 0.1 and h = 0.01 partition of the time interval x ∈ [0, 1]. it is observed from table 2 that the approximate solution obtained by the new proposed method in comparison to the exact solution in terms of absolute error is very impressive. just as the previous example, the results graphically shown in figure 3 are monotonically increasing showing the behaviour of the nonlinear fivp. likewise, the approximate solution completely 7 hussain et al. / j. nig. soc. phys. sci. 5 (2023) 1087 8 table 3. lower and upper solution of example 3 α exact solution tsbm e ebhdef e h = 0.1 h = 0.1 0 -0.100004086851013030 1.94289e-16 4.131e-07 0.2 -0.080004095094799887 8.32667e-17 4.137e-07 0.4 -0.060004103338586523 1.59594e-16 4.141e-07 0.6 -0.040004111582373492 6.93889e-17 4.149e-07 0.8 -0.020004119826159905 2.56739e-16 4.149e-07 1 -0.000004128069946763 1.35559e-16 4.161e-07 α exact solution tsbm e ebhdef e h = 0.1 h = 0.1 0 0.100003908832573600 2.77555e-17 9.094e-03 0.2 0.080003917076360453 1.52655e-16 9.094e-03 0.4 0.060003925320147089 4.85722e-17 1.267e-01 0.6 0.040003933563933947 1.66533e-16 8.459e-02 0.8 0.020003941807720582 6.24500e-17 4.291e-02 1 -0.000004128069946763 1.35559e-16 4.161e-07 overlaps the exact solution which indicates high accuracy of the proposed method. example 3. given the second-order linear fbvp ŷ′′(x) + ŷ(x) + x = 0, ŷ(0,α) = ŷ(1,α) = (0.1α−0.1, 0.1−0.1α), with exact solution ŷ (x,α) = −x + (0.1α− 0.1) cos(x) + (1.13376 + 0.054630α) sin(x) ŷ (x,α) = −x + (0.1 − 0.1α) cos(x) + (1.24303 − 0.054630α) sin(x) and at x = 1, ŷ (1,α) = [ y (1,α), y (1,α) ] , 0 ≤ α ≤ 1. the results obtained for example 3 are shown in table 3 and figure 4 displays the complete iterations graph with stepsize h = 0.1 partition of the time interval x ∈ [0, 1]. from table 3 and the graph in figure 4, impressive monotonocally dereasing results are still observed. the absolute error accuracy is high compared with the existing ebhdef method and the overlapping behaviour of the approximate solution with the exact solution is evident. example 4. given the second-order non-linear fbvp ŷ′′(x) = − [̂y′(x)]2 ŷ(x) , x ∈ [0, 1], ŷ(0,α) = (0.9+0.1α, 1.1−0.1α), ŷ(1,α) = (0.9+0.1α, 2.1−0.1α), with exact solution ŷ (x,α) = √ 1.4 + 0.1α √ 0.1(9 + α)2 14 + α + 2x, ŷ (x,α) = √ 1.6 − 0.1α √ −0.1(−11 + α)2 −16 + α + 2x, figure 4. numerical solution of example 3 with lower/upper solution table 4. lower and upper solution of example 4 α tsbm e ebhdef e fdm e h = 0.1 h = 0.008 h = 0.008 0 0.000000e+00 0 0 0.2 4.4408920e-16 2.57e-06 9.27e-07 0.4 2.4424906e-15 2e-06 8.55e-07 0.6 1.5321077e-14 1.26e-06 5.92e-07 0.8 9.6207486e-12 5.88e-07 2.94e-07 1 0.000000e+00 0 0 α tsbm e ebhdef e fdm e h = 0.1 h = 0.1 h = 0.008 0 0.000000e+00 0 0 0.2 4.4408920e-16 2.05e-06 8.15e-07 0.4 8.8817841e-15 1.63e-06 7.65e-07 0.6 1.7763568e-15 1.03e-06 5.35e-07 0.8 1.3322676e-15 4.87e-07 2.67e-07 1 0.000000e+00 0 0 and at x = 1, ŷ (1,α) = [ y (1,α), y (1,α) ] , 0 ≤ α ≤ 1. the results obtained for example 4 are shown in table 4 and figure 5 displays the complete iterations graph with stepsize h = 0.1 and h = 0.008 partition of the time interval x ∈ [0, 1]. it is observed from table 4 that the approximate solution obtained by the new proposed method in comparison to the exact solution in terms of absolute error is very impressive as it give same results as the interval boundaries. the results are graphically shown in figure 5 and the behaviour of the nonlinear fbvp solution is seen to monotonically increase. the comparison of the exact and approximate solutions on the graph also shows high accuracy as the plots overlap. this indicates the high accuracy of the proposed method. in addition, the time in seconds required to compute the approximate solution of the numerical examples is given in the table below. the program code was written with matlab 2015a on a laptop with 8gb ram and intel core i5-3427u cpu. 8 hussain et al. / j. nig. soc. phys. sci. 5 (2023) 1087 9 figure 5. numerical solution of example 4 with lower/upper solution table 5. time taken to compute approximate solutions α example 1 example 2 example 3 example 4 time/sec time/sec time/sec time/sec 0 0.4767 2.0844 0.9204 0.4457 0.2 1.3682 1.9451 1.5024 0.4097 0.4 1.3742 2.0163 1.4550 0.4072 0.6 1.0563 1.9995 1.3675 0.3741 0.8 1.8364 2.1498 1.4603 0.3982 1 0.8937 2.0415 1.4031 0.3813 α example 1 example 2 example 3 example 4 time/sec time/sec time/sec time/sec 0 0.9037 2.2057 1.3941 0.3928 0.2 0.8487 2.0648 1.3404 0.3877 0.4 0.8703 2.7846 1.4259 0.4361 0.6 0.8650 2.1738 1.4331 0.2778 0.8 0.8650 1.9303 1.4310 0.3267 1 0.8937 2.0415 1.4031 0.3813 6. conclusion the major objective of this research to enhance the accuracy of the solution (in terms of absolute error) by developing a numerical technique for solving second order fodes (fivps and fbvps) directly. as a result, this article developed a twostep block method for second-order fodes with the presence of third and fourth derivatives. the proposed method outperforms other methods discovered in the literature as shown in the tables and graphs of the numerical results obtained. in addition, the method eliminates the requirement for complicated subroutines in conventional methods that require starting values or predictors. the proposed block method has proven to be a viable strategy with increased accuracy for solving both linear and nonlinear fivps and fbvps. the method developed using linear block approach with low computational complexity also satisfied all convergence conditions for the block methods. hence, the proposed method in this article is more suitable for obtaining the approximate solutions of second order fivps and fbvps. acknowledgments we thank the referees for 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general third order initial value problems of ordinary differential equations”, int. j. pure appl. math. 86 (2013) 365. http://dx.doi.org/10.12732/ijpam.v86i2.11 [29] j. o. kuboye & z. omar, “numerical solution of third order ordinary differential equations using a seven-step block method”, int. j. math. anal. 9 (2014) 743. http://dx.doi.org/10.12988/ijma.2015.5125 [30] a. f. jameel, a. saaban, & h. h. zureigat, “numerical solution of second-order fuzzy nonlinear two-point boundary value problems using combination of finite difference and newton’s methods”, neural. comput. appl. 30 (2018) 3167. https://doi.org/10.1007/s00521-017-2893-z 10 j. nig. soc. phys. sci. 2 (2020) 205–217 journal of the nigerian society of physical sciences effect of benzophenone on the physicochemical properties of n-cnts synthesized from 1-ferrocenylmethyl (2-methylimidazole) catalyst ayomide hassan labuloa,∗, elijah temitope adesujia, charles ojiefoh oseghalea, elias emeka elemikeb, adamu usmana, akinola kehinde akinolac, enock olugbenga darec adepartment of chemistry, federal university of lafia, lafia, nasarawa state, nigeria bdepartment of chemistry, federal university of petroleum, nigeria cdepartment of chemistry federal university of agriculture, abeokuta, ogun state, nigeria abstract vertically-aligned nitrogen-doped carbon nanotubes (v-n-cnts) were synthesized via the chemical vapour deposition (cvd) technique. 1ferrocenylmethyl(2-methylimidazole) was employed as the source of the fe catalyst and was dissolved in different ratios of acetonitrile/benzophenone feedstock which served as both the carbon, nitrogen, and oxygen sources. the morphological difference in n-cnts was as a result of increased oxygen concentration in the reaction mix and not due to water vapour formation as observed in the oxygen-free experiment, indicating specifically, the impact of oxygen. raman and x-ray photoelectron spectroscopy (xps) revealed surface defects and grafting of oxygen functional groups on the sidewall of n-cnts. the ftir data showed little or no effect as oxygen concentration increases. xps analysis detected the type of nitrogen species (i.e. pyridinic, pyrrolic, graphitic, or molecular nitrogen forms) incorporated in the n-cnt samples. pyrrolic nitrogen was dominant and increased (from 8.6 to 11.8 at.%) as oxygen concentration increases in the reaction precursor. an increase in n content was observed with the introduction of a lower concentration of oxygen, followed by a gradual decrease at higher oxygen concentration. our result suggested that effective control of the reactant mixtures can manipulate the morphology of n-cnts. doi:10.46481/jnsps.2020.105 keywords: chemical vapour deposition, nitrogen-doped carbon nanotubes, 1-ferrocenylmethyl(2-methylimidazole), x-ray photoelectron spectroscopy article history : received: 11 may 2020 received in revised form: 08 august 2020 accepted for publication: 09 august 2020 published: 15 november 2020 c©2020 journal of the nigerian society of physical sciences. all rights reserved. communicated by: b. j. falaye 1. introduction vertically-aligned carbon nanotubes (v-cnts) have been found to be fascinating for various range of applications, such as catalysis (as catalyst support) [1, 2, 3], electronics [4, 5, 7] ∗corresponding author tel. no: +234 8062295936 email address: labulo@yahoo.com (ayomide hassan labulo ) and biological [8, 9, 10] devices. this is as a result of the controllable diameter and surface area of v-cnts [11] which can be explored in the fabrication of materials of particular interest. the major drawbacks of v-cnts are their low selectivity and reactivity at the surface. these drawbacks can be overcome by surface functionalization and nitrogen-doping which tailor their physicochemical properties [12]. doping of cnts 205 labulo et al. / j. nig. soc. phys. sci. 2 (2020) 205–217 206 with heteroatoms, such as b, p, s and n, into the sp2 carbon framework has been reported [13, 14, 15, 16]. these electronrich atoms help fine-tune the electronic properties of cnts [2]. also, nitrogen incorporation into cnts alters the wall thickness, crystallinity and diameter of cnts [17]. the nitrogen embedded into cnts can take various forms. the most common are graphitic-nitrogen, pyrrolic-nitrogen, pyridinic-nitrogen and molecular n2 stuck in the interior of cnt structures [18, 19, 20]. the nitrogen composition largely depends on the solubility of nitrogen in the catalyst nanoparticle during the reaction at a specified temperature. it has been shown that the type and concentration of nitrogen obtained depend on the nature of the catalyst employed (i.e. ferrocene or ferrocenyl derivatives), synthetic temperature, gas flow rate and type of nitrogencontaining precursors [21, 22, 23]. several methods have been employed in the synthesis of n-cnts; namely arc-discharge [24], laser deposition [25] and chemical vapour deposition (cvd) [26]. of these, the cvd technique has been the method commonly used for large-scale n-cnt synthesis [27]. however, the control of the reaction conditions in cvd technique is somewhat tricky, as catalyst poisoning due to limiting carbon diffusion rate and formation of amorphous carbon on fe substrate surface is common [28, 29]. many researchers have reported the introduction of oxygen [30], water [31, 32] and co2 [33], ethyl benzoate [34] among other reaction gases to improve n-cnts quality and catalyst activity [35]. however, an excess level of oxygen-containing species could lead to n-cnts etching [29, 32, 36]. recently, sakurai et al. [37] reported that the introduction of the oxygencontaining molecule (e.g. h2o) during cvd synthesis enhanced the growth of cnts and prolong catalyst lifetime at temperatures above 750 ◦c. this resulted in the removal of amorphous carbon through water vapour etching to give a graphitic nanostructured carbon network [26]. fatuba et al. [38] also reported that the addition of oxygen-containing aromatic compounds (i.e. growth enhancer), such as methyl benzoate and benzaldehyde into the reaction mixture tailored the size and controlled the n-cnts wall numbers and alignment [38]. the essential role of the growth enhancer compared to previously reported approach (such as h2o), is to control the wall numbers, diameters and to reactivate catalyst particles [39, 40, 41]. in this study, we report for the first time, the use of benzophenone in the reactant mixture to modify n-cnts growth and morphology. benzophenone was also employed to improve the solubility of the ferrocenyl imidazolium catalyst in acetonitrile. we elucidate the effect of oxygen on the type of nitrogen incorporated in n-cnts. the morphology, surface area and stability of n-cnts were studied at varying oxygen concentration levels. 2. experimental 2.1. materials and characterization ferrocene (≥ 97%), ferrocenemethanol (98%), 2-methylimidazole (≥ 98.2%) and sodium borohydride (95%), potassium hydrogen phthalate (≥ 99.5%) were obtained from sigma aldrich ltd. south africa. acetonitrile (hplc grade, 99.9%), toluene (≥99.5%) and ethanol (98%) were purchased from merck chemicals south africa. nitric acid (55%) and sulphuric acid (98%) were purchased from saarchem, south africa. 10% hydrogen in argon (purchased from afrox gases, south africa) was used as a carrier gas for the synthesis of n-cnts. images of the synthesized n-cnts were obtained by using scanning electron microscopy (sem) (joel jem 1010) and transmission electron microscopy (tem) (joel jsm 6100). higher magnification images of n-cnts were obtained from high-resolution transmission electron microscope (hrtem). elemental analysis was conducted on a leco chns elemental analyser. the crystallinity of the n-cnts was determined with a rigaku/dmax rb powder x-ray diffractometer using graphite monochromatized high-density cu kα radiation (λ= 0.15406). the thermal stabilities of n-cnts were determined using a q seriest m thermal analyzer tga/dsc (q600). the fourier transform infrared (ftir) spectra of n-cnts were recorded on a perkinelmer spectrum rx1 ftir spectrometer by embedding the samples into kbr pellets. the adsorption-desorption isotherms and surface area of n-cnts were determined on a micrometrics tristar ii surface area analyser. the graphitic nature of the n-cnts was determined by a raman spectrometer (deltanu advantage 532tm). four accumulated spectra were collected to access the homogeneity of the samples. the synthesized n-cnts were purified in nitric acid under microwave irradiation using a cem discover sp microwave instrument. the surface chemical composition of n-cnts was analysed using x-ray photoelectron spectroscopy (xps). xps analysis was conducted on a quantum 2000 instrument using a monochromated al kα source and charge neutralizer, with a pass energy of 117.4 ev. the peaks were deconvoluted using casaxps programme. the surface charge on the n-cnts in ultra-pure water was determined with a malvern zetasizer (ns500). boehm titration was conducted to quantify the acidic functional groups on the n-cnt surfaces. potassium hydrogen phthalate (khp) was used as the primary standard for the standardization of naoh solutions using phenolphthalein as the indicator [42]. 0.20 g of each n-cnts samples were placed in separate bottles. 25 ml naoh (0.05 m), na2co3 (0.025m) and nahco3 (0.05 m) were added to the bottle, sealed and shaken for 24 h. the solutions were then filtered and titrated against standardized hcl or naoh [43]. different functional groups (i.e. phenolic, carboxylic, lactonic and hydroxyl groups) were calculated based on the amount of acid or base consumed. 206 labulo et al. / j. nig. soc. phys. sci. 2 (2020) 205–217 207 2.2. synthesis of 1-ferrocenylmethyl(2-methylimidazole) (fcmech3) the general procedure described by pan et al. [44] was used to synthesize fcmech3. briefly, ferrocenemethanol (1 mm) and 2-methyl-1h-imidazole (1.1 mm) was refluxed in acetic acid for 9 h at 60 ◦c. the product formation was monitored using preparative tlc plates with a solvent system of ch2cl2/meoh (4:1). the product was neutralized with 50% koh in distilled water to remove the acetic acid and then puried by column chromatography. the final product was washed in na2so4 and finally dried under vacuum to obtain yellow crystals. detailed characterization of fcmech3 has been reported in our previous work [45]. 2.3. synthesis of n-cnts n-cnts were synthesized by pyrolyzing fcmech3 catalysts in acetonitrile at 850 oc using the cvd method. in the experiment, different concentrations of benzophenone were added to the reaction mixture to study the effect of oxygen on the growth of n-cnts. the cvd procedure and set-up described by oosthuizen et al. [46] was followed. briefly, 0.25 g of the catalyst was added to 0.5, 1.0, 1.5 and 2.0 g of benzophenone to produce 1, 2, 3 and 4 wt.% oxygen, respectively. the mixture was dissolved in acetonitrile (as carbon and nitrogen source) to make a 10 g solution . the reactant mixture was injected using a syringe at 0.8 ml min−1 through the quartz tube placed in a muffle furnace. the mixture was swept through the tube by 10% hydrogen in argon carrier gas for 100 ml min−1. after 30 min of reaction, the furnace was allowed to cool to room temperature, and the product was collected from the hot region of the furnace. n-cnts from 1-4 wt.% oxygen is denoted as n-cnts-1%, n-cnts-2%, n-cnts-3% and ncnts-4%, respectively. n-cnts-0% was synthesised by dissolving fcmech3 catalyst in acetonitrile. for comparison, ncnts-fe was synthesized using ferrocene and toluene as catalyst and solvent, respectively. 2.4. purification procedure for n-cnts n-cnts were purified using microwave digestion. briefly, n-cnts (0.8 g) were dispersed in nitric acid (6 m) by ultrasonic agitation for 45 min. after sonication, each sample was purified by a microwave assisted irradiation. this was done by placing 50 ml of the dispersed sample in a thermal resistant teflon (milestone (tfm)) vessel on a sample rotor available for 4 vessels. the microwave was set at 100 w power and ramped from room temperature to 100 oc for 30 min. after digestion, the obtained suspension was filtered on 0.1 µm ptfe membrane. the collected solid samples were washed with deionized water until a neutral ph was obtained. afterwards, the n-cnts were washed with alcohol and dried in an oven at 100 ◦c for 24 h. 3. results and discussion 3.1. tem analysis the morphology of n-cnts was studied by tem. the obtained images are shown in figure 1. the incorporation of nitrogen correlated with the bamboo-like structure typical of ncnts [47] (figure 1a-f). the use of fcmech3 as a catalyst in acetonitrile and benzophenone gave mainly clean n-cnts (figure 1) and in good yield (table 1). this could be attributed to the cleaning effect of oxygen as it reacts with amorphous carbon to form co2. n-cnts and carbon sphere (cs) are obtained in toluene solvent. table 1. summary of the effect of oxygen from benzophenone on the yield of n-cnts synthesized by using 1-ferrocenylmethyl[2-methylimidazole] catalyst in acetonitrile at 850 ◦c samples yield (%) n-cnts-0% 74 n-cnts-1% 68 n-cnts-2% 63 n-cnts-3% 61 n-cnts-4% 58 the n-cnts yields decrease as the concentration of oxygen increases due to the formation of co2 from unreacted carbon and oxygen. the tem images of n-cnts-1% and n-cnts2% (figure 1a and b) showed a curly tubular structure. this could be as a result of fe catalyst left inside the n-cnts with smaller diameters [48]. the bamboo compartment of n-cnts1%, n-cnts-2%, n-cnts-3% and n-cnts-4% decreased as the concentration of oxygen increased (table reftab2). all ncnts obtained are opened at the tips, while some region along the tube gave stacked cup-like cones. this suggests that the bamboo structures were obtained by tip growth mechanism [49]. the cup-like cones appear to be more prominent as the oxygen concentrations increased (figure 1c and e). n-cnts-0% and n-cnts-fe exhibit relatively straight tubes and a wall thickness of ∼15 nm. the wall thickness decreases as the concentration of oxygen from benzophenone increases (table 2). this is due to a reduction in the number of corrugated carbon layers and the closure of tubes which resulted in reduced compartment distances [50]. table 2 shows the effect of oxygen on the inner diameter (id) and outer diameter (od) of the synthesized n-cnts. from the results, it is believed that oxygen plays a vital role in modulating the morphology and diameters of n-cnt [39]. the od decrease as the oxygen content in the reaction mixtures increases. this is due to the effect of oxygen on the catalyst leading to a decrease in fe particle size as a result of catalyst migration, sintering, and precipitation processes [51, 52]. it was suggested that oxygen enhances the catalyst activity by removing amorphous carbon which prevents n-cnts surface 207 labulo et al. / j. nig. soc. phys. sci. 2 (2020) 205–217 208 figure 1. tem images of n-cnts obtained from (a) n-cnts-0%, (b) n-cnts-1%, (c) n-cnts-2%, (d) n-cnts-3%, (e) n-cnts-4% and (f) n-cnts-fe poisoning [41]. n-cnts-1%, n-cnts-2% and n-cnts-3% gave smaller id (14±7 nm to 16±5 nm). however, larger id n-cnts was obtained for n-cnts-4% (i.e. 33±8 nm). this is due to excess oxygen content, leading to etching of the outer walls which largely affects n-cnts quality. figure 2 shows the hrtem images of n-cnts with varying oxygen contents. an increase in the d002 interlayer spacing of the graphitic carbon was observed as the od decreases. the interlayer d-spacing increased from 0.339 nm (n-cnts1%) (figure 2b) to 0.344 nm (n-cnts-3%) (figure 2d). the increase in the d002 spacing is due to the curvature of smaller diameter n-cnts and higher strain caused by the structural defect on the nanotube walls [53]. also, the regular bamboo compartment for n-cnts-4% (figure 2f) was destroyed. this is attributed to supersaturation of molten fe catalyst particles with carbon [54]. it could also be as a result of highly reactive oxygen at the surface or within the molten fe nanoparticles which form feo (i.e. fe + o2 →feo + o), leading to etching of the graphitic carbon. 3.2. sem analysis the morphology of n-cnts was analysed using sem. the obtained images are shown in figure 3 (a-f). figure 3 a-e manifested the effect of oxygen on n-cnts growth and alignment. this was as a result of the reaction of oxygen with very reactive hydrogen radical involved in the hydrocarbon-based growth of nanotubes [20]. this helps to scavenge unreactive hydrogen which inhibits the growth of sp2 like graphitic sheets [30]. for example, the vertical alignment was observed for n-cnts-1%, n-cnts-2% and n-cnts-3% (figure 3 b-d) compared to ncnts-fe (figure 3f) the alignment was depleted at higher oxygen concentration (as observed in n-cnts-4%). this could be attributed to the partial oxidization fe-catalyst which reduced catalyst density, leading to reduced n-cnts nucleation [55]. figure 2. eect of oxygen on n-cnt wall thickness and diameters: hrtem images of (a) n-cnts-0%, (b) n-cnts-1%, (c) n-cnts2%, (d) n-cnts-3%, (e) n-cnts-4% and (f) etched wall of n-cnts4% at moderate oxygen concentration (i.e. n-cnts-2%), the nanotubes walls are free of amorphous carbon (figure 3c) as compared to n-cnts-4% (figure 3e) with more amorphous carbon and lesser tubes (table 1). 3.3. thermal studies the thermal stabilities of n-cnts with different oxygen wt.% loading was studied as shown in figure 4. tga analysis was measured in air at 25-1000 ◦c to give an idea of the oxygen content and the purity of the samples. the first mass loss due to loss of water appears before 100 ◦c. n-cnts-1% shows a significant weight loss at 386 ◦c while n-cnts with 2-4 wt.% oxygen showed a weight loss between 390-530 ◦c. n-cnts-0% is the most thermally stable with the decomposition temperature at 589 ◦c. the oxygen treated n-cnts started to decompose at the on-set point between 334 and 430 ◦c (table 3). all n-cnts showed weight loss after decomposition above 87% with a residual mass between 9.6-0.5%. from dtg curves, the maximum mass loss temperature for 1-4% oxygentreated n-cnts is between 392 and 514 ◦c. further investigation by raman, xps and ftir analysis was done. 3.4. crystallinity of n-cnts figure 5 shows the raman spectra of n-cnts-0%, n-cnts1%, n-cnts-2%, n-cnts-3%, n-cnts-4% and n-cnts-fe. the two prominent peaks observed at ∼1330 and ∼1573 cm−1 are assigned to the dand g-bands, respectively. the intensity ratios of the dand g-bands (id/ig ) shows the defect level of graphitic carbon materials [56, 57, 58]. the id/ig ratio of ncnts-0% and n-cnt-fe is 0.74 and 0.66, respectively (table 4). after introduction of oxygen from benzophenone in the reactant precursor, the id/ig ratio increased to 0.97, 0.93, 0.85 and 0.79 for n-cnts-1%, n-cnts-2%, n-cnts-3% and n-cnts4%, respectively. this is as a result of incorporation of surface 208 labulo et al. / j. nig. soc. phys. sci. 2 (2020) 205–217 209 table 2. effect of oxygen on n-cnts diameter and wall thickness oxygen wt. % ave. od±sd (nm) ave. id±sd (nm) wall thickness (nm) ave. compartment distance (nm) n-cnts (%) n-cnts-0% 48±25 38±31 15 18±11 90 n-cnts-1% 37±31 19±5 11 17±9 85 n-cnts-2% 33±21 14±7 9 15±8 76 n-cnts-3% 34±19 16±5 8 13±9 74 n-cnts-4% 41±15 33±8 7 11±8 65 n-cnts-fe 75±16 48±12 14 20±12 83 table 3. thermal features of n-cnts at different oxygen concentration. toxidation refers to the temperature of primary oxidation. entry catalyst on set point (oc) toxidation (oc ) 1 n-cnts-0% 430 572 2 n-cnts-1% 378 450 3 n-cnts-2% 397 410 4 n-cnts-3% 346 428 5 n-cnts-4% 334 420, 514 6 n-cnts-fe 386 392 oxygen functionalities and n atoms which produces more defects and disorders on the graphitic structure of the n-cnts. the lower id/ig ratio in n-cnt-fe and n-cnts-0% indicates that fewer defects are introduced in the carbon lattices due to less nitrogen atom intrusion into the graphitic carbon network compared to n-cnts-1%, n-cnts-2% and n-cnts-3%, respectively. the width of the g-band peak also indicates the level of doping in n-cnts [59, 60]. table 4 shows that the gband width of n-cnts with varying concentrations of oxygen follows the order of n-cnts-1% > n-cnts-2% > n-cnts3% > n-cnts-4% > n-cnts-0% > n-cnts-fe. this result suggested a possible increase in n-doping at lower oxygen concentration. table 4. ig /id ratios of the n-cnts samples d g id/ig n-cnts-0% 1341 1591 0.74 n-cnts-1% 1342 1601 0.97 n-cnts-2% 1354 1599 0.93 n-cnts-3% 1365 1595 0.85 n-cnts-4% 1369 1590 0.79 n-cnt-fe 1374 1581 0.66 3.5. surface chemistry of n-cnts figure 6 shows the ftir spectra of n-cnts from 0-4% of oxygen and n-cnts-fe. peaks at around 2927 and 2625 cm−1 are assigned to the o–h and ch3 stretching vibrations [61], respectively. the prominent band at 2381 cm−1 is assigned to the characteristic absorbance of co2 groups [62], while peaks at 1763, 1567 and 1030 cm−1 are assigned to stretching vibrations of c=o, c=n and c-o functional groups, respectively [63]. the peaks at 1375 cm−1 are assigned to stretching vibrations of c-nh3 [64]. the presence of c=n and c-n functional group on the purified n-cnts indicates the substitution of graphitic sp2 carbon with nitrogen, leading to the bamboo configuration observed in tem images [65]. for n-cnts-0%, the intensity of the c=o band peak at 1763 cm−1 was weaker than that from 1-4% oxygen, which becomes broader as the concentration of 209 labulo et al. / j. nig. soc. phys. sci. 2 (2020) 205–217 210 figure 3. sem images of (a) n-cnts-0%, (b) n-cnts-1%, (c) n-cnts-2%, (d) n-cnts-3%, (e) n-cnts-4% and (f) n-cnts-fe oxygen increases. the increase in the intensity of the c=n peak at 1567 cm−1 for n-cnts from 1-4% oxygen can be related to the increase in nitrogen-doping level, which correlates with raman analysis results (table 4). the results of boehm titration of n-cnts-0%, n-cnts-1%, n-cnts-2%, n-cnts-3%, ncnts-4% and n-cnts-fe are shown in table 5. according to this method, nahco3, na2co3 and naoh, neutralize carboxyl groups, carboxyl groups and lactones; and carboxyl groups, lac210 labulo et al. / j. nig. soc. phys. sci. 2 (2020) 205–217 211 figure 4. (a) tga curves and (b) dta of purified n-cnts synthesized from 0-4% wt. oxygen figure 5. raman spectra of n-cnts tones and phenols, respectively. therefore, different functional groups can be calculated from the volume of acid and bases used. the acid functional groups on n-cnts-1%, n-cnts2%, n-cnts-3% increases a little as the oxygen concentration increases compared to n-cnts-0% and n-cnts-fe, while the amount of basic functional groups significantly increases. this indicates that the oxygen functionalities on the surface of ncnts synthesized in the presence of oxygen are more basic than n-cnts synthesized in acetonitrile only [66]. from the results, n-cnts-1%, n-cnts-2%, n-cnts-3% contains high concentration of basic group (≥ 1.025 mmol/g) (table 5). additionally, n-cnts-2% has the highest concentration of the phenolic group. zeta potential (ζ ) measurement provides information on the adsorption of ions (h+ and oh−) from aqueous suspension and dispersibility which lead to the formation of net charge on figure 6. ftir spectra of n-cnts the n-cnts [67]. these net charges lead to the formation of the electrical double layer which stabilizes the suspension and prevents particle aggregation. the properties of nanoparticles are largely affected by their colloidal stability. nanoparticles with zeta potential less than -25 mv or above +25 mv are said to have a high degree of stability [67]. table 6 shows the variation in the zeta potential of n-cnts-0%, n-cnts1%, n-cnts-2%, n-cnts-3%, n-cnts-4% and n-cnts-fe nanofluids. our result showed that the zeta potential follows the 211 labulo et al. / j. nig. soc. phys. sci. 2 (2020) 205–217 212 table 5. boehm titration of n-cnts samples acidic groups (mmol/g) basic groups (mmol/g) phenolic carboxylic lactonic n-cnts-0% 0.766 0.813 0.070 0.889 n-cnts-1% 0.085 1.025 0.680 1.025 n-cnts-2% 0.181 1.142 0.0826 1.542 n-cnts-3% 0.062 0.851 0.348 1.416 n-cnts-4% 0.016 0.664 0.529 0.784 n-cnts-fe 0.0860 0.612 0.481 0.741 table 6. zeta potentials of n-cnts in ultrapure water samples zeta potential (mv) n-cnts-0% -37.6 n-cnts-1% -51.4 n-cnts-2% -57.0 n-cnts-3% -54.0 n-cnts-4% -43.2 n-cnts-fe -38.8 order n-cnts-2% > n-cnts-1% > n-cnts-3% > n-cntsfe > n-cnts-0% > n-cnts-4%. the zeta potential increases as the concentration of oxygen increases but drops sharply at higher oxygen concentration. according to this measurement, the oxidized n-cnts are negatively charged in the aqueous phase as a result of oxygen-containing functional group ionization [68]. the effect of oxygen on the porosity of n-cnts-0%, ncnts-1%, n-cnts-2%, n-cnts-3% and n-cnts-4% was characterized by bet analysis. the nitrogen-adsorption isotherms of all n-cnts are of type iv with different hysteresis loops in the high-pressure regions (p/po = 0.7–1), suggesting the presence of mesoporous structure [69]. as shown in table 7, the surface areas of n-cnts follows the order: n-cnts-2% > ncnts-1% > n-cnts-3% > n-cnts-fe > n-cnts-0% > ntable 7. bet surface area and pore volume of n-cnts-0%, n-cnts-1%, ncnts-2%, n-cnts-3%, n-cnts-4% and n-cnts-fe samples surface area (m2g−1) pore volume (cm3 g−1) n-cnts-0% 95 0.37 n-cnts-1% 127 0.35 n-cnts-2% 130 0.57 n-cnts-3% 122 0.53 n-cnts-4% 89 0.39 n-cnts-fe 110 0.46 cnts-4%. this indicates that the surface area of the n-cnts can be modified by the introduction of oxygen into the reactant mixture. 3.6. elemental analysis the elemental composition and the bonding environment of the c, o and n species were determined by xps analysis, and the result is presented in table 8. figure 7 shows the highresolution n 1s energy region of selected n-cnts (n-cnts0%, n-cnts-3%, and n-cnts-fe). the deconvolution of the spectra gave three distinct n 1s peaks centred at 398.50, 400.18 and 401.20 ev assigned to pyridinic, pyrrolic and graphitic nitrogen, respectively [70]. a steady increase in the level of nitrogen-doping was observed during the cvd synthesis, followed by a gradual decrease due to an increase in oxygen con212 labulo et al. / j. nig. soc. phys. sci. 2 (2020) 205–217 213 figure 7. xps n 1s spectra of (a) n-cnts-0%, (b) n-cnts-3% and (c) n-cnts-fe centration (table 8); a result consistent with raman and tga data. compared with n-cnts-0% and n-cnts-fe, the ncnts-3% gave higher pyrrolic-nitrogen species which could be attributed to the presence of active site caused by the lower amount of oxygenated species on the graphitic carbon framework [71]. at a low oxygen concentration in benzophenone, pyridinic-n species was formed (table 8). at a high oxygen content, pyrrolic-n was obtained [72]. this may be due to the change in the elemental ratio (c: n: o) in the precursor mixture. the amount of nitrogen incorporated into n-cnts obtained in our study is higher compared to other studies [34, 73]. this was attributed to the higher amount of nitrogen contained in the ferrocenyl imidazolium catalyst. additionally, the decrease in nitrogen content could be ascribed to the presence of o in benzophenone which we believe inhibits nitrogen incorporation into n-cnts. deconvolution of o 1s spectra of n-cnts-0%, n-cnts-3% and n-cnts-fe peaks gave two bands centred at 531.26 and 533.40 ev assigned to c=o and c-o [74], respectively. the elemental analysis results (table 8) corroborate xps result with increased nitrogen-doping triggered by addition of varying amount of oxygen. table 9 shows the detailed analysis of c 1s peaks of ncnts-0%, n-cnts-3% and n-cnts-fe. the deconvoluted c 1s peaks produced five components at 284.3, 285.8, 287.0, 287.9 and 289.4 ev, assigned to c=c, c-c, hydroxyl, carbonyl and carboxyl functional groups, respectively [75, 76]. from the figure 8. xrd patterns of the n-cnts xps analysis, the carboxyl and carbonyl functional groups increase as the sp2 carbon decreases. for example, the atomic percentage of c=o increased from 3.8 (n-cnts-0%) to 8.8% (n-cnts-3%). the oxidation of c=c is confirmed by an increase in c-c components, which led to the formation of new functional groups on n-cnt surfaces. 3.7. powder xrd pattern studies the xrd profiles of n-cnts (i.e. 0-4 wt.% oxygen) and n-cnts-fe showed the crystalline nature of n-cnts (figure 8). all diffraction patterns showed the formation of (002) crystalline carbon plane (i.e. 26◦), indicative of cnts formation [77]. other peaks at 44.5◦, 49.1◦ and 77.6◦ correspond to (100), (221) and (401) reflections of the graphite structure of n-cnts, respectively. the weak peaks at 37.6◦ and 43.5◦ are assigned to fe3c and fe2o3, respectively, which are stuck inside the core of n-cnts [78, 79]. the xrd diffraction pattern for n-cnts showed a decrease in the intensities of (002) peaks as the alignment increases, particularly, from n-cnts-1% to n-cnts-3%. also, the diffraction peak intensities of the (002) plane for ncnts-1%, n-cnts-2% and n-cnts-3% are weaker than those of n-cnts-0% and n-cnts-fe. this shows that n-cnts-fe and n-cnts-0% have fewer structural defects since n-doping create faults in the graphitic layers. this result agrees with the raman results (table 4). the interlayer d-spacing increases from 0.339 to 0.352 nm as oxygen concentration increases (table 10). the increase in the d002 spacing is due to curvature of smaller diameter n-cnts and higher strain caused by the structural defect on the nanotube walls. this result is consistent with d002 spacing obtained from hrtem analysis. 213 labulo et al. / j. nig. soc. phys. sci. 2 (2020) 205–217 214 table 8. relative atomic concentration and nitrogen species distribution from elemental and xps analysis elemental analysis xps analysis samples c (at.%) o (at.%) n (at.%) c (at.%) o (at.%) n (at.%) pyrrolic (at.%) pyridinic (at.%) graphitic (at.%) nitrogen molecule (at. %) n-cnts-0% 80.36 11.62 8.02 72.72 8.36 9.92 8.60 2.30 0.70 0.30 n-cnts-3% 71.82 13.04 15.14 77.00 9.40 13.36 11.80 1.24 0.80 0.20 n-cnts-fe 79.35 15.35 7.30 84.54 7.72 8.27 7.50 0.27 0.60 0.10 table 9. intensities of c 1s peaks samples 284.3 ev 285.8 ev 287.0 ev 287.9 ev 289.4 ev c=c sp2 (%) c-c sp3 (%) c-o (%) c=o (%) cooh (%) n-cnts-0% 75.5 5.9 13.6 3.8 1.2 n-cnts-3% 75.3 4.7 11.4 8.8 0.8 n-cnts-fe 76.9 4.2 14.8 2.5 1.6 table 10. x-ray structural parameters of n-cnts-0%, n-cnts-1%, n-cnts2%, n-cnts-3%, n-cnts-4% and n-cnts-fe. d002 values are obtained from hrtem and correlated with those from the xrd analysis entry samples d002 values (nm) intensity of c002 peaks fwhm at c002 peaks crystalline size (nm) 1 n-cnt-fe 0.348 488.41 2.446 3.06 2 n-cnt-0% 0.333 181.49 1.461 1.17 3 n-cnt-1% 0.339 129.72 1.548 1.41 4 n-cnt-2% 0.340 256.25 2.400 2.54 5 n-cnt-3% 0.344 247.13 2.616 2.61 6 n-cnt-4% 0.352 185.92 2.637 1.91 4. conclusion this study presented the role of oxygen and nitrogen-doping as a promising method to improve the physicochemical properties of n-cnts. this has critical implications for reproducibility in n-cnt synthesis, particularly on the effect of oxygen in diameter and wall thickness control. it can be concluded that the introduction of an appropriate amount of oxygen promotes n-cnts growth with clean walls and reduced diameters. from xps analysis, pyrrolic-n was predominantly incorporated into the crystalline cnt structure at a high oxygen concentration. nitrogen-doping was further confirmed by tga analysis and raman spectroscopy. lastly, the understanding of the effect of oxygen species on the morphology and surface area of ncnts during synthesis is critical in vast numbers of industrially promising supported metal nanoparticles catalyst design. acknowledgments this research was financially supported by the national research foundation (nrf) south africa. we are grateful to the school of chemistry and physics, university of kwazulu-natal (ukzn) for creating a conducive research laboratory for this work. ayomide is grateful to prof. vincent nyamori, prof. 214 labulo et al. / j. nig. soc. phys. sci. 2 (2020) 205–217 215 bernand omondi and mrs rashidat labulo for proofreading this manuscript. references 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[79] t .fu, r. liu, j. lv & z. li, “influence of acid treatment on n-doped multi-walled carbon nanotube supports for fischer-tropsch performance on cobalt catalyst”, fuel process technol 122 (2014) 49. 217 j. nig. soc. phys. sci. 5 (2023) 1054 journal of the nigerian society of physical sciences study of mhd swcnt-blood nanofluid flow in presence of viscous dissipation and radiation effects through porous medium m. ramanujaa,b,∗, j. kavithac, a. sudhakara, v. nagaradhikab adepartment of mathematics, marri laxman reddy institute of technology and management, dundigal, hyderabad – 500 043, india bdepartment of mathematics, gitam institute of technology and management, bangalore, karnataka – 561203, india cdepartment of mathematics d k, government college for women, spsr nellore-524003, india abstract in this analysis, a computational study is conducted to examine the two-dimensional flow of an incompressible mhd swcnt-blood nanofluid, saturated mass and porous medium .in addition to viscous dissipation, thermal radiation is taken into consideration. we developed the mathematical model and useful boundary intensity approximations to diminish the structure of partial differential equations based on the fluid for blood-based swcnt underflow assumptions. converted the partial differential equations by applying corresponding transformations to arrive at ode’s. the above results are solved numerically by the runge-kutta 4th order technique. noticed that there is desirable conformity when interpolated with the numerical one. the effects exhibited the velocity of swcnt-blood nanofluid enhanced for defined standards of the viscosity parameter. rise in temperature when various parameters like prandtl number, eckert number, and slip parameter are applied on swcnt-blood. the impact of fluid flow on blood-based swcnt is discussed graphically, and our results are tabulated along with illustrations. the design concepts, such as the nusselt quantity and the local skin friction, conform to the analytical approach. velocity reductions with an increase in cnt’s volume fraction, whereas enhancement in the blood temperature, is noted, which is directed to the rise in the heat mass transfer rates. doi:10.46481/jnsps.2023.1054 keywords: swcnt, blood, viscous dissipation, radiation, nusselt number, skin friction article history : received: 11 september 2022 received in revised form: 20 november 2022 accepted for publication: 28 november 2022 published: 14 january 2023 c© 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: s. fadugba 1. introduction the study of non-newtonian fluids like water, mineral oil, and ethylene glycol was reported in many papers for more than a decade, and their applications could be found in industrial sectors such as chemical manufacturing, microelectronics, air ∗corresponding author tel. no: +91 9550754250 email address: mramanuja09@gmail.com (m. ramanuja) conditions, engineering, petroleum industry, paper production, aerodynamic heating, coating, and polymer processing, etc. a range of substances such as blood, mud, polymers, and paint depicts a non-newtonian fluid description. however, no single model in literature deals with multiple non-newtonian fluids. but the properties of these fluids are multiple in themselves because of their low thermal conductivity, which hampers their functions during heat exchangers. as a consequence, there may be a demand to expand its thermal conductivity. in contempo1 ramanuja et al. / j. nig. soc. phys. sci. 5 (2023) 1054 2 rary applications, due to commercial aspects, the flow involving casson nanofluid creates critical interest in present-day researchers. many substances in the actual field, like mud, malt, condensed milk, glues, sugar solution, emulsions, soaps, paints, etc., exhibit newtonian fluid properties. but the actual situation is to assemble a single constitutive equation that follows the defined casson nano -fluid’s properties. it also plays a vital function in nuclear physics within geographical flows. many researchers have identified different effects resting on casson nanofluid. salman et al. [1] developed a combination of viscous dissipation with radiation parameters. the cone angle has a significant result on heat transfer and fluid flow conduct within the porous medium. the consequences of viscous dissipation, holmic dissipation, thermal radiation, and mass exchange outcomes on uneven hydromagnetic boundary thickness float of a stretching plane were developed by anjali devi et al. [2]. abd el aziz [3] is interested in the impact of thermal radiation along with mixed heat and mass transfer on hydromagnetic with the flow over a porous stretching level. mhd float was scientifically clarified with radiation via a stretching sheet surrounded by a porous medium, as specified and examined by anjali et al. [4]. makinde et al. [5] have analyzed the chemical response results from the stretching surface in the existence of interior heat invention. the radiation and chemical response outcomes on the mhd boundary layer glide of a stretching surface were examined by seini et al. [6]. abdul et al. [7] have investigated, via a stretching flat surface, the effect of thermal radiation on mhd fluid flow. the pressure of thermal radiation on the mhd flow via the stretching surface was discussed by chen et al. [8]. further, raju et al. [9] concentrated on resting the movement of casson fluid through a slippery wedge and noticed a decrease in the increasing temperature area estimations of the eckert number. electrically directing casson liquid flow over an item that is neither a perfect level/vertical slanted/cone in the presence of a steady, attractive field is also a significant concern. sabetha et al. [10] determined the thermal radiation results on hydromagnetic free convection drift previously and impetuously began vertical plate. hiteesh [11] studied the absence of transverse magnetic discipline, the border layer regular drift and heat transfer of a viscous incompressible fluid resulting from stretching plate with viscous dissipation impact. the convection heat transfer side by way of a constantly shifting heated vertical plate, including suction or injection analyzed by al-sanea [12]. unsteady free convection and mass change glide over a limitless vertical permeable plate in the commentary of suction/injection are to come upon with learning about by takhar et al. [14]. the impact of suction /injection on unsteady free convection couette float and warmth switch of an active viscous fluid with vertical permeable plate is explained by jha et al. [16]. shamshuddin et al. [17] developed a particular cover of flow that isn’t dissipative, and in-depth graphical illustrations are offered for the fine of the magnetic subject parameter. further, shah et al. employed the dissimilar case of nanoparticles. additionally, entropy optimization with activation electricity and chemical response is also studied. the 2nd regulation of thermodynamics is utilized to discover the entropy technology in velocity. heat and mass switch of williamson nanofluid causes a magnetohydrodynamic perimeter layer to move with the flow across a stretched sheet explored by reddy et al. [19]. the magnetohydrodynamic (mhd) flow of casson nanofluid impact over an extended surface was developed by hayat et al. [20]. dawar et al.[22] investigated the mhd casson-nanofluid, carbon nanotube, and radioactive heat transfer revolving channels. ali et al. [21] studied the blood moves with the casson fluid below the influence of mhd magnetohydrodynamics in axis-symmetric cylindrical tubes. the mhd williamson fluid via a bent sheet and below the influence of non-thermal heat source or sink cnts is analyzed by kumar et al. [23]. c. sus [24] for the first time, nanofluids were proposed by elevating nanometer and sized particles interested in the bottom fluid. the (swcnts) single-walled carbon annotates own a better heat transfer assessment, and surface drag compels (mwcnts) multiwall carbon annotates described by haq et al. [25]. further, liu et al. [26] examined glycol, ethylene, and engine oil with the existence of mwcnts, and they concluded that cnts with ethylene glycol have an advanced thermal conductivity. recently, the approach of nanofluid precedent above a stretching sheet was obtained by needed and lee [27]. the boundary deposit flow of nanoparticles concluded a stretching/shrinking exterior had been examined by nadeem bejan [28]. further, hayat et al.[29] obtained the nonmaterial fluid flow in a circulating method. the nanofluid flow throughout the entropy generation considers the circular heat source, which is studied by nouri et al. [30]. das et al. [31] have analyzed the mhd flow of nanofluid through a porous medium. further, sheremet et al. [32] examined the identical fluid in the crimped cavity. the flow of nanofluid in an enlarged porous sheet is observed by alharbi et al. [33]. zueco[34] exploited a community simulation technique (nsm) to check the consequences of viscous dissipation and radiation on unsteady free convection mhd on a vertical porous plate. hamzeh et al. [45] aims to investigate the properties of heat transfer and magnetohydrodynamics casson nanofluid in the presence of a free convection boundary layer fluid flow on a stretching sheet using cnts in human water/blood as the base fluid. the unsteady separated stagnation-point flow of hybrid nanofluid with viscous dissipation and joule heating is investigated numerically in this examined by amira zainal et al. [46]. zainal et al. [47] have investigated viscous dissipation and mhd hybrid nanofluid flow towards an exponentially stretching/shrinking surface. taza et al. [48] have studied heat and mass transmission more conveniently, such as in hybrid-powered engines, pharmaceutical processes, microelectronics, domestic refrigerators, and engine cooling. aditya et al. [49] have been examined by considering buongiorno’s two-component non-homogeneous model with the inclusion of electrification of nanoparticles and viscous dissipation. adedire et al. [50] examined the concentration profiles in the single and the interconnected multiple-compartment systems with sievepartitions to transport chemical species with second-order chemical re2 ramanuja et al. / j. nig. soc. phys. sci. 5 (2023) 1054 3 action kinetics. ramanuja et al. [51] have examined casson nanofluid flow over a growing or contracting porous medium with different permeability and thermal radiation.[54] have obtained numerical experiments show that the methods compete favourably with existing processes and efficiently solve stiff and oscillatory problems. nanofluids are obtained by reacting oxides, metals, carbides, or carbon nanotubes (cnts) with nanoparticles. generally, in base fluids, nanoparticles are regularly floating, consisting of water, kerosene, ethylene glycol, and engine oil in some areas. cnts used inner nanofluids, which are available in two types in carbon nanotubes (swcnts), and colloidal deferment of nanoparticles in a base fluid is used to create these fluids. this model investigates the stagnation flow on swcnt and blood nanofluids of mhd fluid in the existence of a porous medium, viscous dissipation, and investigation under the impact of injection/suction in addition to thermal radiation parameters are studied and analyzed through this examination. numerical techniques solved the resulting equations. symbolic computational software such as matlab bvp4c solver is used. the effects are presented graphically. 2. formation of the problem the physical description of the problem is illustrated in fullydeveloped steady-mixed convection flow of human blood utilized as base fluid, and swcnt as the nanoparticles over a state, incompressible, laminar flow of swcnt-based nanofluids saturated with human blood is embedded in the medium porous surface that allows the liquid to enter or exit during progressive developments or constrictions. the porous plates are separated by distancea. one part of the cross-segment converse to separation by 2a (t)between the walls, which is to a great extent less than the channel’s width and length. a consistent segment of the flow field is shown in the cartesian coordinate system, which is chosen within such a manner as exposed in fig.1. one and the other channel partitions are perceived to have distinct permeability issues and expansion or convention are systematically at a dependents-time velocity v0which represents the uniform suction v0 > 0 and injection v0 < 0channel which is supposed to be infinite in the distance. because of their magnetic characteristics, these nanoparticles are considered. in the proposed issue, the casson liquid model is exposed to blood and swcnts nanoparticles which are scattered into it for upgraded heat move. the above assumptions portrays leading equations in support of the nanofluid flow by 2-dimensional boundary cover equations are assumed as a continuity equation, momentum equation, and the energy equations as mentioned by vijayalakshmi et al. [41]; bestman [42]; srinivas et al. [43] and radha krinshnama charya et al.[44]. ∂u ∂x + ∂v ∂y = 0 (1) figure 1. schematic of problem ∂u ∂t + u ∂u ∂x + v ∂u ∂y = µn f ρn f ( 1 + 1 β ) ( ∂2u ∂x2 + ∂2u ∂y2 ) − µn f φ ρn f k ( 1 + 1 β ) u − b20σn f µn f u ρn f − 1 ρn f ∂p ∂x (2) ∂v ∂t + u ∂v ∂x + v ∂v ∂y = µn f ρn f ( 1 + 1 β ) ( ∂2v ∂x2 + ∂2v ∂y2 ) − µn f φ ρn f k ( 1 + 1 β ) v − b20σn f µn f u ρn f − 1 ρn f ∂p ∂y (3) ∂t ∂t + u ∂t ∂x + v ∂t ∂y = kn f (ρcp )n f ( ∂2t ∂y2 ) − 1 (ρcp )n f ∂qr ∂y + µn f (ρcp )n f ( 1 + 1 β ) ( ∂u ∂y )2 + q′′′ (ρcp )n f + q0 (ρcp )n f (t − t0) (4) where uand v denote the velocity fundamental quantities, with the directions of x-axis andy-axis, pdenote the dimensional pressure t be the time,φ&k are the permeability and porosity of the permeable medium, φ (η) is the dimensionless concentration of the fluid, temperature t ,kn f denote the thermal conductivity of the nanofluid, β be the blood casson fluid parameter,ρn f be the efficient density of the nanofluid, the efficient dynamic viscosity of the nanofluid be µn f , (ρcp )n f denote the heat electrical condenser of the nanofluid, and vdenote the kinematic viscosity. the non-uniform heat absorbed by generation per unit volume q′′′is defined as: q′′′ = b(x0)m+1k vx0 [ f 1(η)a∗(t1 − t0) + b ∗(t − t0)] here a∗represents the velocity of heat transfer for the spacedependent and b∗ represents an exponentially decaying parameter of space and internal temperature-dependent heat absorption. where a0 = v0a−1 and a1 = v1a−1 are the wall permeability quantities;t0, t1are the temperature of the upper and lower 3 ramanuja et al. / j. nig. soc. phys. sci. 5 (2023) 1054 4 walls;twis the temperature taking place at the wall;t∞is the temperature of free stream fluid flow. the substantial effects of the such as nanofluid ρn f ,µn f , (ρcp)n f , and kn f are involved in the outcomes of the distance distribution on cnts are compensated for using spinning oblique nanotubes with a very large axial ratio and given as, which might be outlined. 2.1. mathematical model for the thermal physcial property of a nanofluid table 1 mathematical model for the thermal physical property of a nanofluid the viscosity, density, heat capacitance and the effective thermal conductivity of the nanofluid are defined as given by h.c. brikman [52] and r. i. hamilton et al. [53] respectively: where, nthe nanoparticle is shape factor , vn f = µn f ρn f ,φthe volume of the utility of nanoparticles,ρ f concentration of the base fluid, ρs-be the density of the nanoparticle, µ f -viscosity of the base fluid, (ρc p) f , (ρc p)sthe capacitance heat of the base fluid along with nanoparticle is a combination with solid nanoparticles, and k f ; ksthermal conductivities of the base fluid and nanoparticle correspondingly. the thermo-physical properties of changed base fluids and nano-particles are revealed in table 1. by introducing the complimentary flow utility, the same represents flow velocity components u and v can be written through conditions of the free flow function in flow. u = ∂ψ ∂y and v = − ∂ψ ∂x (5) partial differential equations that are non-linear and condensed addicted to non-linear ode’s deliberated for that purpose the stream function whereψ = ψ(x, y)routinely satisfy continuity equation, indicates stream function appropriate to mass conservation and f (η) is dimensionless flow function. u = xva−2 fn(η, t), v = −va −1 f(η, t) ; ψ = xvf(η, t)/a (6) here η = ya , fn = ∂f ∂η the governing equations (2) are based on these assumptions are given by vijayalakshmi et al. [41]: ∂u ∂t = µn f ρn f ( 1 + 1 β ) ∂2u ∂y2 − µn f φ ρn f k ( 1 + 1 β ) u − b20σn f µn f u ρn f − 1 ρn f ∂p ∂x (7) usage of the irradiative heat flux is basic in rosseland’s estimate for radiation brewster [39], and the thermal flux is defined as: qr = − 4σ∗ 3k∗ ∂t 4 ∂y (8) here and k∗ be the “absorption specific” coefficient, σ∗be the stefan-boltzmann constant. we had been constrained that the temperature variations contained by using the glide are satisfactorily slighter such that the term t 4strength is stated as a direct function of temperature. this is consummated by increasing t 4 in taylor’s sequence about t∞ and ignoring higher-order expressions, thus assuming a small temperature difference in flow given below: t 4 � 4t 3∞t − 3t 4 ∞ (9) using eqs(9) and eqs(8) becomes ∂qr ∂y = −16σ∗t 3∞ 3k∗ ∂2t ∂y2 (10) under these assumptions, the leading equations are given by vijayalakshmi et al. [41]; bestman [42]; srinivas et al. [43] ∂t ∂t = kn f (ρcp )n f ( ∂2t ∂y2 ) + 1 (ρcp )n f  16σ∗t 313k∗ ∂ 2t ∂y2  + µn f (ρcp )n f ( 1 + 1 β ) ( ∂u ∂y )2 + q′′′ (ρcp )n f + q0 (ρcp )n f (t − t0) (11) the temperature of the nanofluid in the channel can be calculated as follows: tw = t∞ + b ( x a )m1 θ(η) (12) the dimensionless form of temperature from eq. (12) where θdimensionless temperature function, ηsimilarity variable, m1 denote the index power-law of the temperature and b is the constant of the fluid. the pressure gradient of the kind by vijayalakshmi et al. [41]; bestman [42]; srinivas et al. [43] and radhakrinshnama charya et al. [44] is thought to generate the pulsatile flow. a(1 + ceiwt) = − 1 ρn f ∂p ∂x (13) by inserting the dimensionless variables and parameters listed below: x = x h , y = y a , t = ωt ′ , p = p aρn f h u = ωu ′ a ,θ(η) = t − t0 t1 − t0 (14) at this moment, we eliminate pressure commencing from equations(12) using (13) and (14) with reference from vijayalakshmi et al. [41] the following is obtained; (1 + ceiωt) = − ∂p ∂x (15) a2 a1r (1 + 1 β ) ∂2u ∂y2 − 1 a1 ∂p ∂x − 1 a1 ( a5 m 2 + a2 da ) u − ∂u ∂t = 0 (16) eqs(12) using (14) (15) eqs becomes:( a4 a3 + 4 3a3 rd ) 1 r pr θ ′′ + a2 a3 eca∗ r ( ∂u ∂y )2 + b∗qh a3r θ− ( ∂θ ∂t ) = 0 (17) 4 ramanuja et al. / j. nig. soc. phys. sci. 5 (2023) 1054 5 table 1. mathematical model for the thermal physical property of a nanofluid physical quantity mathematical model influential dynamic viscosity of the nanofluid µn f = µ f (1 −φ) −2.5 the influential density of the nanofluids ρn f = φps + (1 −φ)p f the heat capacitance of nanofluid (ρcp)n f = φ (ρcp)s + (1 −φ)(ρcp) f thermal conductivity of sphericalnanoparticles approximated kn f = k f [ 2k f +ks−2φ(k f −ks ) 2k f +ks +φ(k f −ks ) ] the electrical conductivity σn f = σ f [ 1 + 3(σ−1)(σ+2)−(σ−1)ϕ ] using the following dimensionless similarity variables, where darcy parameter, the frequency parameter, eckert number, prandtl number, heat source parameter, radiation parameter. da = k a2 , r = ωh2 v f , ec = a2 (c p) f ω2(t1 − t0) pr = (pc p) f vn f k f , qh = q0a2 (ρc f ) f v f , a1 = φ ρs ρ f + (1 −φ) , a2 = (1 −φ) 2.5 , a3 = φ (ρcp)s (ρcp) f (1 −φ), rd = 4t 31 σ ∗ k f k∗ a4 = [ 2k f + ks − 2φ(k f − ks) 2k f + ks + φ(k f − ks) ] a5 = 1 + 3 ( σs σ f − 1 ) φ( 2 + σs σ f ) + ( − σs σ f + 1 ) ϕ  (18) the corresponding boundary conditions are: f (1) = 1, f (−1) = 1, f ′ (1) = 0, and f ′ (−1) = 0. (19) θ(−1) = 1,θ(1) = 0, i f θ(0) = 1 + δθ′(0) (20) it was once initiated that there is an appropriate settlement between analytical and numerical solutions. dimensionless shear stress at the partitions is described as heat transfer. the pace of the partitions is a prerequisite for nusselt quantity non-dimensional, which is characterized by hatami et al. [40] τ = x(1 −φ)−2.5 r ( f ′′(η) ) η=−1,1 (21) nux = − kn f k f ∂t ∂η /(t1 − t0) = −φ2θ(η)η = −1, 1 c f = 2µn f ρ f (uw (x)) ( ∂u ∂y ) y=0 = −φ2θ(η)η = −1, 1 qw = −kn f ∂t ∂y /y = 0 3. results and discussion we investigated in this study the combined property of thermal radiation, heat generation, and viscous dissipation resting on the swcnt and blood nanofluid flow modal that incorporates nanoparticle volume fraction. within the numerical computation, the properties of the blood and swcnt are utilized (reference table 2). the consistency and accuracy of our accurate solutions and numerical trials of the significant parameters are highlighted through graphs in this section. the governing equation (15) and (16) through the boundary conditions (18) and (19) were worked out by employing the rungekutta strategy through the shooting method (matlab solver, bvp4c package software). to achieve these results, mathematical computations are exposed by making an allowance for a distinct norm of non-dimensional governing parameters. the impacts of governing substantial parameters are explored in detail. specifically; eckert number(ec), magnetic parameter (m), nanoparticle volume parameter(φ), heat generation parameter (qh ), prandtl number (pr), darcy number (da), casson parameter (β), and a∗, b∗ are velocities of heat transfer for the spacedependent; with the following assigned values to the respective parameters: \ m = 2, a = 0.5, b = 0.1, m1 = 0.2, r = 2, da = 0.5, qh = 0.1, a ∗ = 0.2, b∗ = 0.2, nr = 0.5, ec = 0.2 through figure 2, the result ofa∗on the temperature distribution θ(ς) is illustrated and from this, we conclude that a∗ starts declining against the temperature profile which is being enlarged after a certain range. it is observed from figure 3 that the impact of b∗on temperature profile θ(ς) decreases. after these factors, the speed profiles are accelerated. it can be seen from figure 4 the difference of eckert number (ec) through the temperature profile. the incidence of eckert number in casson nanofluids enhances the development of thermal vitality, which results in the advance with temperature distributions and in consequence of thermal deposit thickness. the result of the undeniable viscosity of nanofluids supplies vitality from the waft because of the rise in heat electricity through frictional heating and transforms it into interior electricity. variations of various heat-generating parameter values are dependent on the temperature; the profile is revealed in figure.5. when qh grows positively, heat production takes up residence in the thermal limit layer. the casson nanofluid thermal electricity improves due to a large amount of heat. this process raises the thickness of the thermal boundary layer, implying that 5 ramanuja et al. / j. nig. soc. phys. sci. 5 (2023) 1054 6 figure 2. impact ofa∗onθ(ς) figure 3. impact ofb∗on θ(ς) warmness strength is activated and, as a result, the temperature of the fluid rises. the impression of the thermal slip parameter δon the temperature profile and the velocity profile is depicted through figure 6. we observed that the thermal slip parameter leads to increases in the temperature distribution and the thermal boundary layer thickness; besides, this outmost impact is noticed at the outside of the channel. figure 7 shows the impact of the slip parameter on the rate of casson nanofluid. we noticed that the slip parameter δ at a certain point the velocity of the casson fluid enhanced. figure 8 depicts the effect of da on velocity profiles of casson nanofluid, and it is observed that velocity accelerates with rising values of darcy number. the impact of attractive boundary m on velocity and temperature profiles is shown through figure 9 and figure 10. from this, we noticed that the velocfigure 4. impact ofecon θ(ς) figure 5. impact of qh on θ(ς) figure 6. impact of δ on θ(ς) 6 ramanuja et al. / j. nig. soc. phys. sci. 5 (2023) 1054 7 figure 7. impact ofδ on f ′(ζ) figure 8. impact ofdaon f ′(ζ) ity profiles decline for swcnt with rising values of m resulting thickness of the boundary layer is reduced at a faster rate. physically, the lorentz energy, which opposes movement, occurs due to the used transverse magnetic flow and is responsible for reducing fluid velocity. besides, as the temperature profiles are improved, the temperate limit layer thickness expands. the impact of blood parameters β on the velocity and temperature distributions is shown in figure 11 and figure 12; it is noticed that for the increasing value ofβ, the velocity profile decreases for swcnt. it is because of blood with β will increase the plasticity of blood fluid expands with motive the deceleration in velocity. it’s due to the blood’s malleability, when β decreases, the flexibility of the fluid increases, causing the pace to slow down. in addition, the temperature profile of the flow escalates for increasing values of m. in figure 13, the impact of volume fraction φon the temperature profile for human blood-based nanofluid with swcnt is displayed; it is noticed that for both human blood flow and swcnt, as the φ rises, the temperature of the nano-fluid also increases. it is additionally referred to as those changes in φ infigure 9. impact of m on f ′(ζ) figure 10. impact of m on θ(ς) dicating the adjustments in temperature after which shows the significance of nanofluid. the impact of a on velocity and temperature profiles are portrayed through figure 14 and figure 15; it is noticed that the temperature profile θ(ς)escalates and the velocity profile f ′(ζ) decelerates with the impact ofa. in figure 16, the outcomes of the m1 on the temperature profile are displayed; the temperature of the nanofluid decreases to swcnt with increasing value of m1, which leads to a decline in velocity boundary deposit thickness. 3.1. physical properties of base fluid and nano-particles thermo substantial properties regarding the base fluid and nanoparticles of carrier fluid human blood and swcnt nanoparticles are given below table 2 3.2. skin friction and nusselt number for the cases of unsteady contraction/expansion from table 3 it is seen that the coefficients of space with temperature-dependent a∗and b∗with relatively high-temperature 7 ramanuja et al. / j. nig. soc. phys. sci. 5 (2023) 1054 8 table 2. physical properties of base fluid & nanoparticles (blood and swcnt) physical properties solid nanoparticles swcnt base fluid blood cp(j/kg k) 425 3617 κ(w/m k) 6600 0.52 ρ (kg/m3) 2600 1050 figure 11. impact of β on f ′(ζ) figure 12. impact ofβon θ(ς) source/sink, the skin friction coefficient stable in nature whereas the nuselt number will be enhanced. with the improved values of da, both skin friction and nusullt number decrease. the enhanced values βreduce skin friction on velocity, in addition, to enhancing the heat transfer moderately. the pores of skin friction remain stable, convenient and incomplete gradual reduction inside heat transfer velocity used with increasing values ofnr &qh . in this case of m, friction (c f )and nusselts numfigure 13. impact of φon θ(ς) figure 14. impact of a on f ′(ζ) ber decrease. for the effect of the parameters ec, φ anda, both the values of skin friction, nusselts number declines, whereas form1, both skin friction and nusselt number increases. 3.3. skin friction and nusselt number for steady contraction situation in the above table 4, we can see that the enhanced values of a∗as well as b∗have no impact resting on skin friction, although these values concentrated the heat transfer velocity. the skin friction values for the variations in daremained constant whereas local nusselt number values will be increasing. the enhanced values of β covers the oscillatory nature for pores 8 ramanuja et al. / j. nig. soc. phys. sci. 5 (2023) 1054 9 table 3. the impact of various parameters on skin friction ( c f ) and nusselt number (nux)for the cases of unsteady contraction/expansion a∗ b∗ da β nr qh m ec δ φ a m1 c f nux 0.0 19.471388 3.193296 0.2 19.471388 3.290621 0.4 19.471388 3.387946 0.0 19.471388 3.015712 0.2 19.471388 3.290621 0.4 19.471388 3.646890 0.001 66.175569 3.259443 0.05 24.559828 3.307261 0.1 21.627181 3.298364 0.1 19.471388 3.290621 0.3 8.683747 3.562513 0.5 6.435490 3.624260 0.5 19.471388 3290621 1.0 19.471388 5.542735 1.5 19.471388 7.858197 -0.1 19.471388 3.015712 0.0 19.471388 3.144902 0.1 19.471388 3.290621 1 19.246748 3.292558 10 21.302554 3.274551 20 23.677717 3.252965 0.1 19.471388 3.516654 0.2 19.471388 3.290621 0.3 19.471388 3.064588 -0.2 8.072545 3.021844 0.0 11423853 3.118250 0.2 19.471388 3.290621 0.1 19.471388 3.290621 0.3 35.873231 8.780344 0.5 19.180772 27.79143 0.1 34.402685 4.064004 0.3 27.006243 3.661321 0.5 18.710125 3.254065 0.0 19.471388 3.267488 1.0 19.471388 3.401005 2.0 19.471388 3.579480 and skin friction although there is an insignificant variation for heat transfer rate. influence of mis not affecting skin friction whereas heat transfer velocity is decreased for the same m values. there is an oscillatory characteristic in pores and skin friction and a fast increased within the heat transfer velocity which is meant for rising values ofa. raising values of the slip parameter φescalate the both skin friction and nusselt number. for the influence ofm1, the skin friction values remain constant and the local nusselt number decrease. 4. validation similarly, table 4 is organized to explain nu at the employing the runge-kutta strategy through the shooting method estimated for different values of relevant model variables for both swcnts and mwcnts nano fluids. from table 3 and table 4, it can be observed that the weight of rate of heat transport accelerates for a high magnitude of both φ and pr and declines for a higher value of the eckert number(ec). eckert number (ec) is related to the dissipation term, and the more considerable importance of eckert number (ec) enhances the thermal field. therefore, the opposite result for the higher significance of the eckert number (ec) verses nu is perceived table 5 5. conclusion in this article, the stagnation flow on swcnt and blood nanofluids of mhd fluid in the existence of a porous medium, viscous dissipation, and injection/suction in addition to thermal radiation parameters are studied and analyzed through this examination. the resulting equations were solved by numerical techniques. the tables and graph values for the tempera9 ramanuja et al. / j. nig. soc. phys. sci. 5 (2023) 1054 10 table 4. the impact of various parameters on skin friction ( c f ) and nusselt number (nux)for steady contraction situation a∗ b∗ da β nr qh m ec δ φ a m1 c f nux 0.0 19.997073 2.555168 0.2 19.997073 2.647360 0.4 19.997073 2.739552 0.0 19.997073 2.462266 0.2 19.997073 2.647360 0.4 19.997073 2.880635 0.001 68.577628 2.542118 0.05 25.222523 2.654748 0.1 22.208947 2.651121 0.1 19.997073 2.647360 0.3 9.274754 2.907525 0.5 7.085115 2.970431 0.5 19.997073 2.647360 1.0 19.997073 4.751474 1.5 19.997073 6.972890 -0.1 19.997073 2.462266 0.0 19.997073 2.549819 0.1 19.997073 2.647360 1 19.769175 2.649027 10 21.854765 2.633497 20 24.264288 2.614779 0.1 19.997073 2.860118 0.2 19.997073 2.647360 0.3 19.997073 2.434603 0.2 8.119202 2.475313 0.0 11.560063 2.542268 0.2 19.997073 2.647360 0.1 19.997073 2.647360 0.3 36.522610 6.779180 0.5 91.966641 20.72406 0.1 35.32447 3.026218 0.3 27.732563 2.846351 0.5 19.215453 2.625739 0.0 19.997073 2.765707 1.0 19.997073 2.328076 2.0 19.997073 2.100096 table 5. the numerical values of the nusselt number −φ ′ (1) pr ec φ −φ ′ (1) swcnts mwcnts 20 1.5 0.01 0.067257 0.201779 21 0.107184 0.254538 22 0.127169 0.307473 20 1.6 0.067655 0.202565 1.6 0.068047 0.203369 1.5 0.02 0.246742 0.517264 0.03 0.427643 0.835802 ture field, skin-friction coefficient, velocity profiles, local nusselt number with the effects of parameters magnetic enclosure thermal radiation, thermophoresis, prandtl number, permeability parameter, and eckert number are exposed and obtained. the numerous observations of the current examined about the following conclusions. 1. human blood as well as the base fluid is drastically utilised first time to attain the solution for casson nano-fluids. it’s extremely noticed that rate decreases with increasing cnt quantity portion, and advances in cnt quantity division will increase the blood temperature, which affects in and gives an improvement to the heat transfer velocity. 2. the velocity expressed increases through increasing the velocity fraction parameter about the temperature and concentration profiles are decreased and increased bya. 3. an increase of the swcnt solid φquantity section and eckert quantity yields an addition with nano-fluids tem10 ramanuja et al. / j. nig. soc. phys. sci. 5 (2023) 1054 11 figure 15. impact of a on θ(ς) figure 16. iimpact of m1 on θ(ς) perature, leading to the direction of sudden radiation in the heat transfer rates. 4. increasing values of the slip parameterδreduces the pace subject. the concept of base fluid provides parameter enhancement in the temperature. 5. the viscous dissipation affects the flow within the temperature profile and decreases with the insignificant value of the (pr)prandtl quantity. 6. an eckert number (ec)shows a small effect c f decreased but nux has increased with the enlargement of the (ec) eckert number. the temperature distribution θ(η) is increased as the eckert number (ec) and radiation parameter increase. 7. the thermal radiation, heat generation/absorption, and permeability parameters throughout decreases with advancement in prenatal number, the unsteadiness, the suction, and the magnetic parameter. 8. as an essential position in dissipating heat the temperature of the fluid decreases through enhancement with nanoparticle quantity section for swcnt as a result of elevated thermal conductivity. an expansion in the swcnt’s quantity suction increases the casson-nanofluid temperature, which affects inflation in the heat transfer 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[54] m. kida, s. adam, o. o. aduroja & t. p. pantuvo, “numerical solution of stiff and oscillatory problems using third derivative trigonometrically fitted block method”,.j. nig. soc. phys. sci. 4 (2022) 34. appendix u & v fluid flow velocity µ dynamic viscosity ρ density of the fluid ρs be the density of the nanoparticle, p dimensional pressure β blood casson fluid parameter kn f thermal conductivity µ f viscosity of the base fluid k f ; ks thermal conductivities 12 ramanuja et al. / j. nig. soc. phys. sci. 5 (2023) 1054 13 ρn f efficient density of the nanofluid (ρcp )n f heat electrical condenser of the nanofluid v kinematic viscosity k f ; ks thermal conductivities t fluid temperature k porous permeability r radiation da darcy number λ1 jeffrey parameter θ dimensionless temperature 13 j. nig. soc. phys. sci. 5 (2023) 1366 journal of the nigerian society of physical sciences thermal instability of rotating jeffrey nanofluids in porous media with variable gravity pushap lata sharmaa, deepak bainsb, pankaj thakurc,∗ adepartment of mathematics & statistics, himachal pradesh university, summer hill, shimla, india bdepartment of mathematics & statistics, himachal pradesh university, summer hill, shimla, india cfaculty of science and technology, icfai university, baddi, solan, india abstract it is investigated how changes in gravity affect the thermal instability rotating jeffrey nanofluids in porous media. along with the galerkin method and normal mode approach, the darcy model is used. the distinct variable gravity parameters taken in this paper are: h(z) = z2 − 2z, h(z) = −z2, h(z) = −z and h(z) = z and their effects on the jeffrey parameter, taylor number, moderated diffusivity ratio, porosity of porous media, lewis number and nanoparticle rayleigh number on stationary convection have been scrutinized and graphically shown. our finding demonstrates that varying gravity parameter h(z) = z2 − 2z has more stabilising impact on stationary convection. we have also discovered the necessary condition for overstability in the instance of oscillatory convection for this problem. doi:10.46481/jnsps.2023.1366 keywords: jeffrey nanofluid; variable gravity; porous medium; galerkin method; rotation article history : received: 23 january 2023 received in revised form: 06 april 2023 accepted for publication: 10 april 2023 published: 20 may 2023 c© 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: j. ndam 1. introduction choi [1] devised the term “nanofluid”, which was defined as a liquid having a dispersion of submicronic solid particles (nanoparticles). convective transport in nanofluids was an issue that buongiorno [2] examined. he advanced choi [1] work by including mathematical terms. different uses for nanofluid were introduced by tzeng et al. [3], kim et al. [4], routbort et al. [5] and donzelli et al. [6]. the theoretical and experimental findings in chandrasekhar [7] are based on the newtonian fluid’s capacity to convect steadily ∗corresponding author tel. no: +918570975865 email addresses: pl maths@yahoo.in (pushap lata sharma), deepakbains123@gmail.com (deepak bains) in the absence of a porous medium while subject to rotation and a magnetic field. papamarkos et al. [8] described a method based upon octagonally symmetric design and iir digital filterations. a study of non-newtonian nanofluids like rivlinericksen, maxwellian and modified darcy-maxwell model for s.c. is employed by rana et al. [9], chand [10] and singh et al. [11], respectively. linearised stability theory was used by lapwood [12] to investigate convective flow in a porous material. nield et al. [13] introduced convection in porous media. convection with internal heating in a porous material saturated by a nanofluid was examined by nield et al. [14]. the results reveal that the inclusion of nanofluid particles increases the system’s instability. later, nield et al. [15] provided brief introduction to the book nield et al. [13]. tzou [16, 17] investigated how 1 sharma et al. / j. nig. soc. phys. sci. 5 (2023) 1366 2 natural convection affected nanofluids’ thermal instability. several researchers nield et al. [18, 19, 20], sheu [21] and chand et al. [22, 23, 24] used the buongiorno [2] model to investigate nanofluids’ thermal instability in porous media. ramanuja et al. [25] also used porous medium in their problem. the development of objects in an astrophysical plasma environment is caused by thermal instability, which is studied by kaothekar [26] for partly ionised thermal plasma. this plasma has a relation to astrophysical condensations. chand et al. [27] studied t.i. effect on oldroydian nanofluid by considering realistic boundary conditions. nield et al. [20], sharma [28], yadav et al. [29], chand et al. [30, 31], govender [32] and chand et al. [33] examined the of nanofluid’s thermal instability in rotation. some of them examined rotation’s interactions with suspended particles, many non-newtonian fluids, couple-stress rotation’s interactions with porous media, and rotation’s interactions with itself. they discovered that a system’s thermal instability depends heavily on rotation. yadav et al. [34] employed magneto-convection in rotatory layer of nanofluid and electrothermo-convection in a horizontal layer of rotating nanofluid is examined by chand et al. [35]. additionally, several of them created rotation-based industrial applications, including those found in nuclear reactors, power plants, the petroleum sector, geophysics etc. nanofluid oscillating convection in a porous media was explored by chand et al. [36]. gautam et al. [37] established the concepts of free-free, rigid-free, and rigidrigid boundary conditions for the electrohydrodynamic t.i. of a jeffrey nanofluid layer saturating a porous medium and concluded that the rotation parameter stabilises the system for bottom and top-heavy layouts. a porous-medium-saturated jeffrey nanofluid flow was studied by rana [38] for the effects of rotation. for both bottom and top-heavy arrangements and provided evidence that the rotation parameter stabilises the system. sharma et al. [39] studies the electrohydro dynamics convection in dielectric rotating oldroydian nanofluid in porous medium. the idealisation of uniform gravity used in theoretical research, while appropriate for lab applications, is seldom warranted for large-scale convection events happening in the earth’s atmosphere, ocean, or mantle. gravity must thus be viewed as a changeable quantity that changes with distance from a surface or other reference point. pradhan et al. [40] investigated the thermal instability of a fluid layer in a changeable gravitational field and discovered that boosting the gravitational field vertically accelerates the commencement of convection. a porous media with an internal heat source and an inclined temperature gradient was studied by alex et al. [41] to see how changing gravity affected thermal instability. straughan [42] used both linear theory and nonlinear energy theory to analyse the issue for the case of stiff boundaries in a spatially changing gravitational field and discovered that the nonlinear conclusions were remarkably similar to the linear ones. chand et al. [43] looked into how changing gravity would affect a layer of nanofluid in a porous medium and found that the gravity parameter had a big impact on fluid stability. theoretically and visually, chand [10] investigates thermal instability of maxwell non-newtonian fluid with varying gravity. using a higher order galerkin method, yadav [44] investigated the joint effects of variable gravity fields and throughflow on the beginning of convective motion in a porous medium layer. the results showed that both the throughflow and gravity variation parameters serve to delay the motion’s onset. mahajan et al. [45] analyses the effects of several fundamental temperature and concentration gradients on a layer of reactive fluid in a varied gravity field utilising both linear and non-linear analysis. surya et al. [46] examine the thermal instability of a horizontal layer of liquid heated from below that is contained between thermally conductive porous walls under the influence of a fluctuating gravitational field. as limiting examples of the permeability parameters of the borders, the impact of the gravity variation growing vertically upward for various particular situations of the boundary conditions is derived and graphically depicted. in a layer of porous media, the effects of rotation and varying gravitational strengths on the beginning of heat convection were computed by yadav [47]. the findings demonstrate how the gravity variation parameter and the rotation parameter both delay convection’s arrival. with increased rotation and gravity variation parameters, the measurement of the convection cells diminishes. shekhar et al. [48] investigates numerically how varying gravity affects rotational convection in a porous material that is poorly packed. the linear, parabolic, cubic, and exponential functions are taken into account for variations in gravitational force. while the darcy number increases convection cell size, convection cell size falls when the variable gravity parameter and rotation parameter are increased. additionally, it has been found that the system is more stable for exponential gravity functions than for cubic gravity functions. chand et al. [49] investigated the impact of variable gravity on the thermal instability of rotating nanofluids in porous media and discovered that, in the presence of rotation and also for nanoparticle rayleigh numbers, decreasing the gravity parameter has a stabilising effect while increasing it has a destabilising effect. by taking into account its numerous applications in various fields like geophysics, astrophysics, food processing, oil reservoir modeling, building of thermal insulations and nuclear reactors etc. this brief review of the literature leads one to believe that such a problem was nonexistent; hence, the current problem of thermal instability of rotating jeffrey nanofluids in porous media with variable gravity was chosen. 2. mathematical formulation here, we examine a rotating horizontal jeffrey nanofluid layer heated from below in a porous medium with medium permeability k1 and porosity ε and angular velocity ω(0, 0, ω) bordered by plane z = 0 and z = d, working upward under the influence of variavle gravity. furthermore, it is assumed from nield et al. [18] and chand et al. [49] that there is variable gravity along z -direction i.e. g = (1+δh(z))g, where δh(z) is the variable gravity parameter. when the top boundary layer is at z = d, the temperature t and volumetric fraction ϕ of nanoparticles are assumed to be t1 and ϕ1, respectively, with t0 > t1 and ϕ0 > ϕ1. 2 sharma et al. / j. nig. soc. phys. sci. 5 (2023) 1366 3 figure 1: physical configuration for the sake of simplicity, oberbeck-boussinesq approximation is used and darcy’s law is taken to be true by nield et al. [18] and chand et al. [49]. thus from buongiorno [2], chandrasekhar [7], nield et al. [18] and chand et al. [33, 49] the pertinent governing equations for the study of spinning jeffrey nanofluid in porous medium are ∇.q = 0 (1) 0 = −∇p + ( ϕρp + (1 −ϕ) {ρ0 (1 −α (t − t1))} ) g − µ k1(1 + λ) q + 2ρ0 ε (q × ω) (2) for nanoparticle, the continuity equation is given by (buongiorno [2]) ∂ϕ ∂t + 1 ε q.∇ϕ = db∇2ϕ + dt t1 ∇ 2t (3) for the nanofluid, the equation of thermal energy is given by (buongiorno [2] and chand et al. [49]) (ρc)m ∂t ∂t + (ρc) f q.∇t = km∇ 2t +ε (ρc)p [ db∇ϕ.∇t + dt t1 ∇t.∇t ] (4) where q is the fluid velocity, p is the pressure, ρ0 is nanofluid density at z = 0, ρp is nanoparticles density, ϕ is the volume fraction of the nanoparticles, t is temperature, t1 is the reference temperature, α is thermal expansion coefficient, g is gravitational acceleration and k1 is medium fluid permeability, µ is coefficient of viscosity, ε is the porosity of the porous media, λ = λ1 λ2 the jeffrey parameter (which is the ratio of stressrelaxation-time parameter, λ1 to strain-retardation-time parameter, λ2), the fluid’s heat capacity in porous medium is (ρc)m, (ρc)p stands for heat capacity of nanoparticles, (ρc) f stands for fluid’s heat capacity, km is the fluid’s thermal conductivity, the brownian diffusion coefficient is db and dt is nanoparticles’ the thermophoretic diffusion coefficient (chand et al. [49]). we presumed nanoparticles’ temperature and volumetric fraction as constant. thus, boundary conditions (chandrasekhar [7] and nield et al. [18]) are{ w = 0, t = t0, ϕ = ϕ0 at z = 0 w = 0, t = t1, ϕ = ϕ1 at z = d (5) on introducing non-dimensional variables as (chandrasekhar [7]) (x∗, y∗, z∗) = (x, y, z) d , q∗ = q d κm , t∗ = tκm σd2 p∗ = pk1 µκm , ϕ∗ = ϕ−ϕ0 ϕ1 −ϕ0 , t∗ = t − t1 t0 − t1 where κm = km (ρc) f , σ = (ρc)m(ρc) f are fluid’s thermal diffusivity and thermal capacity ratio, respectively. relaxing the star (∗) for 3 sharma et al. / j. nig. soc. phys. sci. 5 (2023) 1366 4 simplification. the reduced non-dimensional form of equations 1,2,3,4 are: ∇.q = 0 (6) 0 = −∇p − 1 1 + λ q − rm(1 + δh(z))k̂ +rd(1 + δh(z))t k̂ − rn(1 + δh(z))ϕk̂ + √ ta ( q × k̂ ) (7) 1 σ ∂ϕ ∂t + 1 ε q.∇ϕ = 1 ln ∇ 2ϕ + na ln ∇ 2t (8) ∂t ∂t + q.∇t = ∇2t + nb ln ∇ϕ.∇t + na nb ln ∇t.∇t (9) where dimensionless parameters are rm = (ρpϕ0 +ρ0 (1−ϕ0 ))gk1 d µκm is density rayleigh number, rd = ρ0α(t0−t1 )gk1 d µκm is rayleigh darcy number rn = (ρp−ρ0 )(ϕ1−ϕ0 )gk1 d µκm is nanoparticle rayleigh number, ta = ( 2ωρd2 µ )2 is taylor number, ln = κm db is lewis number, na = dt (t0−t1 ) db t1 (ϕ1−ϕ0 ) is nanofluid modified diffusivity ratio, nb = ε(ρc)p (ϕ1−ϕ0 ) (ρc) f is modified nanoparticle-density increment. the reduced non-dimensional boundary conditions are:{ w = 0, t = 1, ϕ = 0 at z = 0 w = 0, t = 0, ϕ = 1 at z = 1 (10) 3. basic states and it’s solutions the time independent basic states for nanofluid are expressed as (nield et al. [18, 19] and chand et al. [49]):{ q(u, v, w) = 0 ⇒ u = v = w = 0, p = pb(z), t = tb(z), ϕ = ϕb(z) (11) the basic variable represented by subscript b. using equation (11) in 6), (7,8), (9), these equations reduce to 0 = − d dz pb(z) − rm(1 + δh(z)) + rd(1 + δh(z))tb(z) −rn(1 + δh(z))ϕb(z) (12) d2 dz2 ϕb(z) + na d2 dz2 tb(z) = 0 (13) d2 dz2 tb(z)+ nb ln d dz ϕb(z) d dz tb(z)+ na nb ln ( d dz tb(z) )2 = 0(14) solving equation 13 with boundary conditions equation 10, we get ϕb(z) = (1 − na)z + (1 − tb)na (15) using (15) in equation (14), we have d2 dz2 tb(z) + (1 − na)nb ln d dz tb(z) + na nb ln ( d dz tb(z) )2 = 0 neglecting the higher order term, we have d2 dz2 (tb(z)) + (1 − na)nb ln d dz (tb(z)) = 0 (16) using boundary conditions (10), the solution of differential equation 16 is tb(z) = e− (1−na )nb ln z [ 1 − e− (1−na )nb ln (1−z) ] 1 − e− (1−na )nb ln (17) according to buongiorno [2] hypothesis, the approximated solutions for equations 15 and 17 are given as tb = 1 − z, and ϕb = z (18) these approximated solutions 18 agrees well with the results obtained by nield et al. [18, 19, 20], sheu [21] and chand et al. [49]. 4. perturbation solutions superimposing infinitesimal perturbation on the basic states in ordered to examine the stability of the system, the basic states equation 11 are written in following form (nield et al. [18, 19, 20] and chand et al. [49]){ q(u, v, w) = 0 + q′(u, v, w), p = pb + p′ t = tb + t′ = (1 − z) + t′, ϕ = ϕb + ϕ′ = z + ϕ′ (19) using (19) in equations (6, 7,8,9), and linearize by ignoring the products of primes and for convenience discarding primes (′) . we obtain the reduced equations (6,7,8,9) as ∇.q = 0 (20) 0 = −∇p − 1 1 + λ q − rn(1 + δh(z))ϕk̂ +rd(1 + δh(z))t k̂ + √ ta ( q × k̂ ) (21) 1 σ ∂ϕ ∂t + 1 ε w = na ln ∇ 2t + 1 ln ∇ 2ϕ (22) ∂t ∂t − w = ∇2t − 2 na nb ln ∂t ∂z + nb ln ( ∂t ∂z − ∂ϕ ∂z ) (23) and boundary conditions are ϕ = 0, t = 0, w = 0 at z = 0 and z = 1 (24) it should be noted that rm is unrelated in equations 21,22 and 23, it is simply the basic static pressure gradient measurement. operating equation 21 with k̂.curl.curl, we get (i.e. making use of result curl.curl = grad.div −∇2) 1 1 + λ ∇ 2w = −rn(1 + δh(z))∇ 2 hϕ +rd(1 + δh(z))∇ 2 h t − √ ta ∂ξ ∂z (25) 4 sharma et al. / j. nig. soc. phys. sci. 5 (2023) 1366 5 now eliminating p from equation (21), i.e. by operating it with i ∂ ∂y and further with − j ∂ ∂x , respectively and further solving, we get ξ = (1 + λ) √ ta ∂w ∂z (26) now, using (26) in equation (25), we have 1 1 + λ [ 1 1 + λ ∇ 2w + rn(1 + δh(z))∇ 2 hϕ −rd(1 + δh(z))∇ 2 h t ] + ta ∂2w ∂z2 = 0 (27) 5. stability analysis by normal mode the disturbances analysing by normal mode analysis as follow (chandrasekhar [7]):[ w, t,ϕ ] = [w(z), θ(z), φ(z)] ex p(ikx x + ikyy + nt) (28) where growth rate is represented as n and the wave number along x and y directions are kx and ky, respectively. using equation 28 in equations 22, 23 and 27, we get 1 1 + λ [ 1 1 + λ ( d2 − a2 ) w + rd(1 + δh(z))a 2 θ −rn(1 + δh(z))a 2 φ ] + ta d 2w = 0 (29) 1 ε .w − na ln (d2 − a2)θ + [ n σ − (d2 − a2) ln ] φ = 0 (30) w+ [ (d2 − a2) + nb ln d − 2 na nb ln d − n ] θ− nb ln dφ = 0(31) where d = ddz and −a 2 = k2x + k 2 y = ∂2 ∂x2 + ∂2 ∂y2 , ∇ 2 = d 2 dz2 − a 2 = d2−a2. the a is the dimensionless resulting wave number. the boundary conditions by considering normal mode are written as chandrasekhar [7] (freefree boundary condition) w = d2w = θ = φ = 0 at z = 0 and z = 1 (32) assume that the solutions for w, θ and φ are of the form (chandrasekhar [7]) w = w0 sin(πz), θ = θ0 sin(πz), φ = φ0 sin(πz) (33) these solutions in (33) satisfy the boundary conditions (32). substituting solution (33) into equations (29,30,31) and integrating each equations individually within limits z = 0 to z = 1, we gain the following matrix equation  j 1+λ + (1 + λ)π 2ta − a2rd(1 + δh(z)) a2rn(1 + δh(z)) 1 −(j + n) 0 1 ε na ln j jln + n σ   w0 θ0 φ0  =  0 0 0  (34) where j = π2 + a2 is the entire wave number. the eigenvalue to the system of linear equation 34 is given as rd = [ j 1 + λ + (1 + λ)π2ta ] (j + n) a2(1 + δh(z)) − [ na ln j + (j+n) ε ] j ln + n σ rn (35) 6. stationary convection for steady state, put n = 0 in equation 35, we obtain rd = (π2 + a2)2 a2(1 + λ)(1 + δh(z)) + (π2 + a2)(1 + λ)π2ta a2(1 + δh(z)) − ( na + ln ε ) rn (36) the rayleigh darcy number for stationary convection reveal by the equation 36 is a function of a, λ, δh(z), ta, na, ln, ε, rn. in non-appearance of jeffrey’s nanofluid (λ = 0), the equation 36 reduces to rd = (π2 + a2)2 a2(1 + δh(z)) + (π2 + a2)π2ta a2(1 + δh(z)) − ( na + ln ε ) rn(37) equation 37 agrees well with the results obtained by chand et al. [49] for stationary convection. in non-appearance of jeffrey’s nanofluid (λ = 0) and rotation (ta = 0), the equation 36 reduces to rd = (π2 + a2)2 a2(1 + δh(z)) − ( na + ln ε ) rn (38) equation 38, agrees well with the results obtained by pradhan et al. [40]. in non-appearance of jeffrey’s nanofluid (λ = 0), rotation (ta = 0) and constant gravity (δh(z) = 0), then the equation 36 reduces to rd = (π2 + a2)2 a2 − ( na + ln ε ) rn (39) equation 39, agrees well with the results obtained by nield et al. [18] and chand et al. [49]. according to nield et al. [18], the critical value of equation 36 is accomplished at a = π, so 5 sharma et al. / j. nig. soc. phys. sci. 5 (2023) 1366 6 table 1: on the onset of stationary convection (rd)c variation of constant variables variable gravity’s impact on stationary convection z in graphs λ ta ln rn ε na δ h(z) = z2 − 2z h(z) = −z2 h(z) = −z h(z) = z 0.3 λ 0.6 0 1 100 500 -1 0.6 5 0.5 stabilising stabilising stabilising destabilising 0.9 100 ta 200 0 1 0.6 500 -1 0.6 5 0.5 stabilising stabilising stabilising destabilising 300 100 ln 500 0 1 0.6 100 -1 0.6 5 0.5 stabilising stabilising stabilising destabilising 1000 -1 rn -0.5 0 1 0.6 100 500 0.6 5 0.5 destabilising destabilising destabilising destabilising -0.1 0.3 ε 0.6 0 1 0.6 100 500 -1 5 0.5 destabilising destabilising destabilising destabilising 0.9 1 na 5 0 1 0.6 100 500 -1 0.6 0.5 stabilising stabilising stabilising destabilising 10 figure 2: variability of (rd)c w.r.t. z for distinct values of h(z) by taking distinct values of λ for stationary convection the critical rayleigh-darcy number is specified as (rd)c = 4π2 (1 + λ)(1 + δh(z)) + 2π2(1 + λ)ta 1 + δh(z) − ( na + ln ε ) rn (40) in non-appearance of rotation (ta = 0), jeffrey’s nanofluid (λ = 0), nanoparticles and at constant gravity (δh(z) = 0), we figure 3: variability of (rd)c w.r.t. z for distinct values of h(z) by taking distinct values of ta obtained the rayleigh darcy number given as (rd)c = 4π 2 this agrees well with the results obtained by lapwood [12] for regular field. 7. oscillatory convection here, possibility for oscillatory convection is considered. for oscillatory convection, put n = ini in equation 35, we have rd = [ j 1 + λ + (1 + λ)π2ta ] (j + ini) a2(1 + δh(z)) 6 sharma et al. / j. nig. soc. phys. sci. 5 (2023) 1366 7 figure 4: variability of (rd)c w.r.t. z for distinct values of h(z) by taking distinct values of ln figure 5: variability of (rd)c w.r.t. z for distinct values of h(z) by taking different values of rn − [ na ln j + (j+ini ) ε ] j ln + ini σ rn (41) by equating the real and imaginary components of equation 41, we get a2 jrd(1 + δh(z)) ln + ( na ln + 1 ε ) jrna 2(1 + δh(z)) = [ j 1 + λ + (1 + λ)π2ta ] j2 ln − [ j 1 + λ + (1 + λ)π2ta ] n2i σ (42) and rd σ + rn ε − j a2(1 + δh(z)) × [ j 1 + λ + (1 + λ)π2ta ] ( 1 σ + 1 ln ) = 0 (43) figure 6: variability of (rd)c w.r.t. z for distinct values of h(z) by taking different values of ε figure 7: variability of (rd)c w.r.t. z for distinct values of h(z) by taking different values of na where j = π2 + a2. the frequency of the oscillatory mode is calculated as follows n2i ln a2σ = j2 a2 − jrd(1 + δh(z))[ j 1+λ + (1 + λ)π 2ta ] − j(1 + δh(z))[ j 1+λ + (1 + λ)π 2ta ] (na + ln ε ) rn (44) in order for ni to be real it is necessary that j(1 + δh(z))[ j 1+λ + (1 + λ)π 2ta ] [rd + (na + ln ε ) rn ] 6 j2 a2 (45) where j = π2 + a2. the equations 43,44,45 becomes as the absence of the jeffrey nanofluid (λ = 0), rotation (ta = 0) and constant gravity (δh(z) = 0) rd σ + rn ε − (π2 + a2)2 a2 ( 1 ln + 1 σ ) = 0 (46) 7 sharma et al. / j. nig. soc. phys. sci. 5 (2023) 1366 8 n2i ln a2σ = (π2 + a2)2 a2 − rd − ( na + ln ε ) rn (47) and rd + ( na + ln ε ) rn 6 (π2 + a2)2 a2 (48) the above result obtained in 46,47,48 are good agreement of results obtained by nield et al. [18] and chand et al. [36, 49]. according to nield et al. [18], the critical value of the wave number is accomplished at a = π , therefore, by setting a = π in equations 46,47,48, we obtain the result for the stability boundary case as rd σ + rn ε = 4π2 ( 1 ln + 1 σ ) (49) n2i ln a2σ = 4π2 − [ rd + ( na + ln ε ) rn ] (50) and [ rd + ( na + ln ε ) rn ] 6 4π2 (51) these results obtained in equations 49,50,51 are same as that of obtained by nield et al. [18] for particular case. 8. results and discussion variable gravity factors’ impacts on density rayleigh number, nanoparticle rayleigh number, lewis number, porosity of porous media, modified diffusivity ratio, and rotation on stationary convection have been graphed, and their stabilising or destabilising effect has been explored below. the variable gravity parameters are as follow: h(z) = z2 − 2z, h(z) = −z2, h(z) = −z and h(z) = z. figure 2 shows the graph for (rd)c with respect to z for distinct values of λ = 0.3, 0.6, 0.9 by fixing other parameters as ln = 500, na = 5,ε = 0.6,δ = 0.5, ta = 100, rn = −1. it is discovered that when the gravity parameter changes, such as when it becomes h(z) = z2 − 2z, h(z) = −z2, h(z) = −z it stabilises, however when it becomes h(z) = z, it destabilises. these match those in straughan [42] for the variable gravity parameter. figure 3 depicts the graph for (rd)c with respect to z for various values of ta = 100, 200, 300 setting other parameters like ln = 500, na = 5,ε = 0.6,δ = 0.5, rn = −1,λ = 0.6. it is found that ta has a stabilising impact when the gravity parameters are h(z) = z2 −2z, h(z) = −z2, h(z) = −z, but a destabilising effect when the gravity parameter is h(z) = z. this is in good accord with the finding reported by chand et al. [49], which states that reducing the gravity parameter has a stabilising impact on stationary convection while raising the gravity parameter has a destabilising effect. figure 4 depicts the curve for (rd)c with respect to z for distinct values of ln = 100, 500, 1000 while holding other parameters constant like na = 5,ε = 0.6,δ = 0.5, ta = 100, rn = −1,λ = 0.6. it is found that ln has a stabilising impact when the gravity parameters are h(z) = z2 −2z, h(z) = −z2, h(z) = −z, but a destabilising effect when the gravity parameter is h(z) = z. this is in excellent accord with the finding from chand et al. [49] that reducing the gravity parameter stabilises stationary convection while raising the gravity parameter destabilises it. figure 5 shows that (rd)c decreases with increase in rn (as = −1,−0.5,−0.1). thus rn has destabilizing effect for all variable gravity parameter on stationary convection. figure 6 shows that (rd)c decreases with increase in ε (as = 0.3, 0.6, 0.9). thus ε has destabilizing effect for all variable gravity parameter on stationary convection. figure 7 depicts the graph for (rd)c with respect to z for various values of na = 1, 5, 10 setting other parameters like ln = 500,ε = 0.6,δ = 0.5, ta = 100, rn = −1,λ = 0.6. it is found that na has a stabilising impact when the gravity parameters are h(z) = z2 −2z, h(z) = −z2, h(z) = −z, but a destabilising effect when the gravity parameter is h(z) = z. 9. conclusion this article investigates the thermal instability of spinning jeffrey nanofluids in porous media with changing gravity. the problem is examined for free-free boundary conditions using galerkin technique and normal mode analysis. equation 40 is the essential rayleigh-darcy number for stationary convection, and it has been studied whether this number stabilises or destabilises stationary convection with regard to changing gravity. equation 45 yields the adequate condition for the oscillatory mode’s frequency, while equation 44 also finds the oscillatory mode’s frequency. in table 1, the varied gravity’s effects in stationary convection are illustrated visually by changing one parameter at a time while keeping the other parameters constant by assigning them certain constant values. the main conclusions from table 1 are: 1. jeffrey parameter (λ), taylor number (ta), modified diffusivity ratio (na) and lewis number (ln) have stabilizing effect on stationary convection when variable gravity parameters varies as h(z) = z2 − 2z, h(z) = −z2, h(z) = −z whereas have destabilizing effect when variable gravity parameter varies as h(z) = z. in other words, stationary convection has a stabilising impact for lowering the variable gravity parameters and destabilising the system for raising the variable gravity parameters. 2. when the variable gravity parameter fluctuates as: h(z) = z2 − 2z, h(z) = −z2, h(z) = −z and h(z) = z, the nanoparticle rayleigh number (rn), porosity of porous medium (ε) destabilise the system. 3. variable gravity perameters h(z) = z2 − 2z, h(z) = −z2, h(z) = −z delay the motion’s onset. 4. nb has no effect on (rd)c. 5. the variable gravity parameter h(z) = z2 − 2z has more stabilizing impact on stationary convection rather than other variable gravity parameters taken in this paper. 8 sharma et al. / j. nig. soc. phys. sci. 5 (2023) 1366 9 6. the sufficient condition for the oscillatory mode’s frequency is obtained and is represented by equation 45. acknowledgments the second author gratefully acknowledges the financial assistance of ugc for nfsc. references [1] s.u. choi & j. a. eastman, “enhancing thermal conductivity of fluids with nanoparticles”, argonne national lab.(anl) (1995). 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[49] r. chand, g. rana & s. kango, “effect of variable gravity on thermal instability of rotating nanofluid in porous medium”, fme transactions 43 (2015) 62. 10 j. nig. soc. phys. sci. 1 (2019) 88–94 journal of the nigerian society of physical sciences original research common fixed point theorems for multivalued generalized f-suzuki-contraction mappings in complete strong b−metric spaces yusuf ibrahim∗ department of mathematics, sa’adatu rimi college of education, kumbotso kano, nigeria abstract this paper introduces a new version of multivalued generalized f-suzuki-contraction mapping and then establish some new common fixed point theorems for these new multivalued generalized f-suzuki-contraction mappings in complete strong b−metric spaces. keywords: common fixed point problem, multivalued generalized f-suzuki-contraction mapping, complete strong b−metric space. article history : received: 04 april 2019 received in revised form: 01 september 2019 accepted for publication: 02 september 2019 published: 28 september 2019 c©2019 journal of the nigerian society of physical sciences. all rights reserved. communicated by: b. j. falaye 1. introduction let x be a nonempty set and s ≥ 1 be a given real number. a mapping d : x × x → r∗ is said to be a b-metric if for all x, y, z ∈ x the following conditions are satisfied: 1. d(x, y) = 0 if and only if x = y; 2. d(x, y) = d(y, x); 3. d(x, z) ≤ s[d(x, y) + d(y, z)]. the pair (x, d) is called a b-metric space with constant s. a strong b−metric is a semimetric space (x, d) if there exists s ≥ 1 for which d satisfies the following triangular inequality. d(x, y) ≤ d(x, z) + sd(z, y), f or each x, y, z ∈ x. (1) in 1922, a mathematician banach [1] proved a very important result regarding a contraction mapping, known as the banach contraction principle, which states that every self-mapping t defined on a complete metric space (x, d) satisfying ∗corresponding author tel. no: +2348062814778 email address: danustazz@gmail.com (yusuf ibrahim ) ∀x, y ∈ x, d(t x, t y) ≤ λd(x, y), where λ ∈ (0, 1) has a unique fixed point and for every x0 ∈ x a sequence {tn x0}∞n=1converges to the fixed point. subsequently, in 1962, edelstein [2] proved the following version of the banach contraction principle. let (x, d) be a compact metric space and let t : x → x be a self-mapping. assume that for all x, y ∈ x with x , y, d(x, t x) < d(x, y) =⇒ d(t x, t y) < d(x, y). then t has a unique fixed point in x. in 2012, wardowski [3] introduced a new type of contractions called f-contraction and proved a new fixed point theorem concerning f-contractions. let (x, d) be a metric space. a mapping t : x → x is said to be an f-contraction if there exists τ > 0 such that ∀x, y ∈ x, d(t x, t y) > 0 =⇒ τ + f(d(t x, t y)) ≤ f(d(x, y)), 88 yusuf ibrahim / j. nig. soc. phys. sci. 1 (2019) 88–94 89 where f : r+ → r is a mapping satisfying the following conditions: f1 f is strictly increasing, i.e. for all x, y ∈ r+ such that x < y, f(x) < f(y); f2 for each sequence {αn}∞n=1 of positive numbers, limn→∞ αn = 0 if and only if lim n→∞ f(αn) = −∞; f3 there exists k ∈ (0, 1) such that lim α→0+ αk f(α) = 0. we denote by ζ, the set of all functions satisfying the conditions (f1) − (f3). wardowski [3] then stated a modified version of the banach contraction principle as follows. let (x, d) be a complete metric space and let t : x → x be an f-contraction. then t has a unique fixed point x∗ ∈ x and for every x ∈ x the sequence {tn x}∞n=1 converges to x ∗. in 2014, hossein, p. and poom, k. [15] defined the f-suzuki contraction as follows and gave another version of theorem. let (x, d) be a metric space. a mapping t : x → x is said to be an f-suzuki-contraction if there exists τ > 0 such that for all x, y ∈ x with t x , t y d(x, t x) < d(x, y) =⇒ τ + f(d(t x, t y)) ≤ f(d(x, y)), where f : r+ → r is a mapping satisfying the following conditions: f1 f is strictly increasing, i.e. for all x, y ∈ r+ such that x < y, f(x) < f(y); f2 for each sequence {αn}∞n=1 of positive numbers, limn→∞ αn = 0 if and only if lim n→∞ f(αn) = −∞; f3 f is continuous on (0,∞) we denote by ζ, the set of all functions satisfying the conditions (f1) − (f3). let t be a self-mapping of a complete metric space x into itself. suppose f ∈ ζ and there exists τ > 0 such that ∀x, y ∈ x, d(t x, t y) > 0 =⇒ τ + f(d(t x, t y)) ≤ f(d(x, y)). then t has a unique fixed point x∗ ∈ x and for every x0 ∈ x the sequence {tn x0}∞n=1 converges to x ∗. following this direction of research (see examples, [4, 5, 6, 7, 8, 9, 10, 16, 17]), in this paper, fixed point results of piri and kumam [11], ahmad et al. [9], suzuki [18] and suzuki [19] are extended by introducing common fixed point problem for multivalued generalized f-suzuki-contraction mappings in strong b-metric spaces. definition 1.1. (hardy and rogers [14]) (1) there exist non-negative constants a, satisfying ∑5 i=1 ai < 1 such that, for each x, y ∈ x, d( f (x), f (y)) < a1d(x, y) + a2d(x, f (x)) + a3d(y, f (y)) + a4d(x, f (y)) + a5d(y, f (x)). (2) there exist monotonically decreasing functions ai(t) : (0,∞) → [0, 1) satisfying ∑5 i=1 ai(t) < 1 such that, for each x, y ∈ x, x , y, d( f (x), f (y)) < a1(d(x, y))d(x, f (x)) + a2(d(x, y))d(y, f (y)) + a3(d(x, y))d(x, f (y)) + a4(d(x, y))d(y, f (x)) + a5(d(x, y))d(x, y). (3) for each x, y ∈ x, x , y, d( f (x), f (y)) < max{d(x, y), d(x, f (x)), d(y, f (y)), d(x, f (y)), d(y, f (x))}. lemma 1.1. [13] from definition 1.1, (1) =⇒ (2) =⇒ (3). denote by c b(x), the collection of all nonempty closed and bounded subsets of x and let h be the hausdorff metric with respect to the metric d; that is, h(a, b) = max{sup a∈a d(a, b), sup b∈b d(b, a)} for all a, b ∈ c b(x), where d(a, b) = inf b∈b d(a, b) is the distance from the point a to the subset b. 2. main results definition 2.1. let 0 be the family of all functions f : r+ → r such that: (f1) f is strictly increasing, i.e. for all x, y ∈ r+ such that x < y, f(x) < f(y); (f2) for each sequence {αn}∞n=1 of positive numbers, limn→∞ αn = 0 if and only if lim n→∞ f(αn) = −∞; (f3) f is continuous on (0,∞). definition 2.2. let ψ be the family of all functions ψ : [0,∞) → [0,∞) such that ψ is continuous and ψ(t) = 0 iff t = 0. definition 2.3. let (x, d) be a strong b−metric space. mappings t, s : x → c b(x) are said to be multivalued generalized f-suzuki-contraction on (x, d) if there exists f ∈ 0 and ψ ∈ ψ such that, ∀x, y ∈ x, x , y, 1 1 + s d(x, t x) < d(x, y) and 1 1 + s d(y, s y) < d(y, s t x) ⇒ ψ(nφ(x, y)) + f(s4 h(t x, s y)) ≤ f(nφ(x, y)) in which nφ(x, y) = φ1(d(x, y))(d(x, y)) + φ2(d(x, y))(d(y, s t x)) + φ3(d(x, y)) ( (d(y, t x)) + d(x, s y) 2s ) + φ4(d(x, y)) ( (d(x, s t x)) + h(s t x, s y) 2s ) + φ5(d(x, y))(h(s t x, s y) + h(s t x, t x)) + φ6(d(x, y))(h(s t x, s y) + d(t x, x)) + φ7(d(x, y))(d(t x, y)) + d(y, s y)) (2) for which φ : r+ → [0, 1), with ∑7 i=1 φi(d(x, y)) < 1, is monotonically decreasing function. comsidering the definition s t x := {s y ⊆ c b(x) : ∀y ∈ t x}, we have the following result. theorem 2.1. let (x, d) be a complete strong b−metric space and let t, s : x → c b(x) be multivalued generalized fsuzuki-contraction mappings. then t and s has a common 89 yusuf ibrahim / j. nig. soc. phys. sci. 1 (2019) 88–94 90 fixed point x∗ ∈ x and for every x ∈ x the sequence {t n x}∞n and {s n x}∞n converge to x ∗. proof let x0 = x ∈ x. let xn+1 ∈ t xn and xn+2 ∈ s xn+1 ∀n ∈ n. if there exists n ∈ n such that d(xn, t xn) = d(xn+1, s xn+1) = 0 then xn+1 = xn = x becomes a fixed point of t and s , respectively, therefore the proof is complete. now, suppose that d(xn, t xn) > 0 and d(xn+1, s xn+1) > 0 ∀n ∈ n then the proof will be divided in to two steps. step one. we show that {xn}∞n=1 is a cauchy sequence. let d(xn, t xn) > 0 and d(xn+1, s xn+1) > 0 ∀n ∈ n. (3) therefore, we have that 1 s + 1 d(xn, t xn) < d(xn, t xn) and 1 s + 1 d(xn+1, s xn+1) < d(xn+1, s xn+1) ∀n ∈ n. (4) by definition 2.3, we get f(h(t xn, s xn+1)) ≤ f(nφ(xn, xn+1)) −ψ(nφ(xn, xn+1)). since that nφ(xn, xn+1) = φ1(d(xn, xn+1))(d(xn, xn+1)) + φ2(d(xn, xn+1))(d(xn+1, xn+2)) + φ3(d(xn, xn+1)) ( d(xn, xn+2) 2s ) + φ4(d(xn, xn+1)) ( (d(xn, xn+2)) 2s ) + φ5(d(xn, xn+1))(d(xn+2, xn+1)) + φ6(d(xn, xn+1))(d(xn, xn+1)) + φ7(d(xn, xn+1))(d(xn+2, xn+1) ≤ φ1(d(xn, xn+1))(d(xn, xn+1)) + φ2(d(xn, xn+1))(d(xn+1, xn+2)) + φ3(d(xn, xn+1)) ( d(xn, xn+1) + sd(xn+1, xn+2) 2s ) + φ4(d(xn, xn+1)) ( d(xn, xn+1) + sd(xn+1, xn+2) 2s ) + φ5(d(xn, xn+1))(d(xn+2, xn+1)) + φ6(d(xn, xn+1))(d(xn, xn+1)) + φ7(d(xn, xn+1))(d(xn+2, xn+1) ≤ φ1(d(xn, xn+1))(d(xn, xn+1)) + φ2(d(xn, xn+1))(d(xn+1, xn+2)) + φ3(d(xn, xn+1)) ( s[d(xn, xn+1) + d(xn+1, xn+2)] 2s ) + φ4(d(xn, xn+1)) ( s[d(xn, xn+1) + d(xn+1, xn+2)] 2s ) + φ5(d(xn, xn+1))(d(xn+2, xn+1)) + φ6(d(xn, xn+1))(d(xn, xn+1)) + φ7(d(xn, xn+1))(d(xn+2, xn+1) ≤ φ1(d(xn, xn+1))(d(xn, xn+1)) + φ2(d(xn, xn+1))(d(xn+1, xn+2)) + φ3(d(xn, xn+1))(d(xn, xn+1)) + φ3(d(xn, xn+1))(d(xn+2, xn+1) + φ4(d(xn, xn+1))(d(xn, xn+1)) + φ4(d(xn, xn+1))(d(xn+2, xn+1) + φ5(d(xn, xn+1))(d(xn+2, xn+1)) + φ6(d(xn, xn+1))(d(xn, xn+1)) + φ7(d(xn, xn+1))(d(xn+2, xn+1) = [φ1(d(xn, xn+1)) + φ3(d(xn, xn+1)) + φ4(d(xn, xn+1)) + φ6(d(xn, xn+1))](d(xn, xn+1)) + [φ2(d(xn, xn+1)) + φ3(d(xn, xn+1)) + φ4(d(xn, xn+1)) + φ5(d(xn, xn+1))(d(xn+2, xn+1)) + φ7(d(xn, xn+1))](d(xn+2, xn+1) = φ′(d(xn, xn+1))(d(xn, xn+1)) + φ ′′(d(xn, xn+1))(d(xn+2, xn+1)) (5) then by (5) and definition 2.3, we get f(d(xn+1, xn+2)) ≤ f(φ′(d(xn, xn+1))(d(xn, xn+1)) + φ ′′(d(xn, xn+1))(d(xn+2, xn+1))) −ψ(φ′(d(xn, xn+1))(d(xn, xn+1)) + φ ′′(d(xn, xn+1))(d(xn+2, xn+1))). (6) on contrary, if d(xn+1, xn+2) > d(xn, xn+1), then φ′(d(xn, xn+1))(d(xn, xn+1)) +φ′′(d(xn, xn+1))(d(xn+2, xn+1)) < d(xn+1, xn+2) and therefore (6) becomes f(d(xn+1, xn+2)) ≤ f(d(xn+1, xn+2)) −ψ(d(xn+1, xn+2)). but, from (3) and the fact that ψ(d(xn+1, xn+2)) > 0, this is a contradiction. thus, we conclude that f(d(xn+1, xn+2)) ≤ f(d(xn, xn+1)) −ψ(d(xn, xn+1)) < f(d(xn, xn+1)). (7) by (7) and definition 2.1(f1), we have that d(xn+1, xn+2) < d(xn, xn+1) < d(xn−1, xn) ∀n ∈ n. (8) therefore {d(xn, xn+1)} is a nonnegative decreasing sequence of real numbers. thus there exists γ ≥ 0 such that lim n→∞ d(xn, xn+1) = γ. from (7) as n →∞, we have that f(γ) ≤ f(γ) −ψ(γ). this implies that ψ(γ) = 0 and thus γ = 0. consequently we arrive at lim n→∞ d(xn, t xn) = lim n→∞ d(xn, xn+1) = 0. (9) now, we claim that {xn}∞n=1 is a cauchy sequence. on contrary, we assume that there exists � > 0 and n, m ∈ n such that, for all n ≥ n� and n� < n < m, d(xn, xm) ≥ � and d(xn−1, xm) < �. (10) it implies that � ≤ d(xn, xm) ≤ d(xn, xn−1) + sd(xn−1, xm) < d(xn, xn−1) + s�. (11) by (11) and (9), we have that � ≤ limsup n→∞ d(xn, xm) < s�. (12) 90 yusuf ibrahim / j. nig. soc. phys. sci. 1 (2019) 88–94 91 by triangle inequality, we have that � ≤ d(xn, xm) ≤ d(xn, xm+1) + sd(xm+1, xm) ≤ d(xn, xm) + 2sd(xm+1, xm). (13) by (9),(10), (12) and (13), we have that � ≤ limsup n→∞ d(xn, xm+1) < s�. (14) similarly, we have that � ≤ d(xn, xm) ≤ d(xn, xn+1) + sd(xn+1, xm) ≤ sd(xn, xm) + (s 2 + 1)d(xn, xn+1). (15) by (9),(10), (12) and (15), we have that � ≤ limsup n→∞ d(xn, xn+1) < s�. (16) observe that d(xn, xm+1) ≤ d(xn, xn+1) + sd(xn+1, xm+1) ≤ d(xn, xn+1) + s[d(xn+1, xm) + sd(xm+1, xm)] ≤ d(xn, xn+1) + s[d(xn, xn+1) + sd(xn, xm) + sd(xm+1, xm)]. (17) by (17), we have that � s ≤ limsup n→∞ d(xn+1, xm+1) < s 2�. (18) by (9)and (10), we select n� > 0 ∈ n such that 1 s + 1 d(xn, t xn) < 1 s + 1 � < � ≤ d(xn, xm) ∀n ≥ n(�) ⇔ 1 s + 1 d(xn, t xn) < 1 s + 1 � < d(xn, xm) ∀n ≥ n(�) and 1 s + 1 d(xn+1, s xn+1) < 1 s + 1 � < � s ≤ d(xn+1, xm+1) ∀n ≥ n� ⇔ 1 s + 1 d(xn+1, s xn+1) < 1 s + 1 � < d(xn+1, xm+1) ∀n ≥ n� it follows that from definition 2.3, we have, for every n ≥ n� f(h(xn+1, xm+1)) ≤ f(nφ(xn, xm)) −ψ(nφ(xn, xm)). (19) since that d(xn, xm) ≤ nφ(xn, xm) = φ1(d(xn, xm))(d(xn, xm)) + φ2(d(xn, xm))(d(xn+2, xm)) + φ3(d(xn, xm)) ( d(xn+1, xm) + d(xn, xm+1) 2s ) + φ4(d(xn, xm)) ( (d(xn+2, xn) + d(xn+2, xm+1)) 2s ) + φ5(d(xn, xm))(d(xn+2, xm+1) + d(xn+2, xn+1)) + φ6(d(xn, xm))(d(xn+2, xm+1) + d(xn, xn+1)) + φ7(d(xn, xm))(d(xm, xn+1 + d(xm, xm+1))) ≤ φ1(d(xn, xm))(d(xn, xm)) + φ2(d(xn, xm))(d(xn+2, xn+1) + sd(xn+1, xm)) + φ3(d(xn, xm)) ( d(xn+1, xm) + d(xn, xm+1) 2s ) + φ4(d(xn, xm)) ( (d(xn+2, xn+1) + sd(xn+1, xn) + d(xn+2, xn+1)) + sd(xn+1, xm+1)) 2s ) + φ5(d(xn, xm))(d(xn+2, xn+1) + sd(xn+1, xm+1) + d(xn+2, xn+1)) + φ6(d(xn, xm))(d(xn+2, xn+1) + sd(xn+1, xm+1) + d(xn, xn+1)) + φ7(d(xn, xm))(d(xm, xn+1) + d(xm, xm+1))). (20) by (12), (14), (16), (18) and (20), we have that limsup n→∞ d(xn, xm) ≤ limsup n→∞ nφ(xn, xm) < φ1(�)(s�) + φ2(�)(s 2�) + φ3(�)(�) + φ4(�)( s2� 2 ) + φ5(�)(s 3�) + φ6(�)(s 3�) + φ7(�)(s�) ≤ max{s�, s2�,�, s� 2 , s3�, s�} = s3� and therefore � ≤ limsup n→∞ nφ(xn, xm) < s 3�. (21) similarly � ≤ limin f n→∞ nφ(xn, xm) < s 3�. (22) by (19), (21) and (22), we have that f(s3�) = f(s4 � s ) ≤ f(s4limsup n→∞ d(xn+1, xm+1)) ≤ f(limsup n→∞ nφ(xn, xm)) −ψ(limsup n→∞ nφ(xn, xm)) ≤ f(s3�) −ψ(�). (23) by (23) and the fact that � > 0, this is a contradiction. hence {xn} is a cauchy sequence in x. by completeness of (x, d), {xn}∞n=1 and {xn+1} ∞ n=1 converge to some point x ∗ ∈ x, that is, lim n→∞ d(xn, x ∗) = 0 and lim n→∞ d(xn+1, x ∗) = 0. (24) there exists increasing sequences {nk}, {n + 1k} ⊂ n such that xnk ∈ t x ∗ and xn+1k ∈ s x ∗ for all k ∈ n. since t x∗ and s x∗ are closed and lim n→∞ d(xnk, x ∗) = 0 and lim n→∞ d(xn+1k, x ∗) = 0, we get x∗ ∈ t x∗ and x∗ ∈ s x∗. step two. we show that x∗ is a common fixed point of t and s . it suffices to show that 1 1 + s d(xn, t xn) < d(xn, x ∗) and 1 1 + s d(xn+1, s xn+1) < d(xn+1, x ∗), f or every n ∈ n, (25) 91 yusuf ibrahim / j. nig. soc. phys. sci. 1 (2019) 88–94 92 implies f(d(t x∗, x∗)) ≤ f(nφ(x ∗, t x∗)) −ψ(nφ(x ∗, t x∗)) and f(d(s x∗, x∗)) ≤ f(nφ(s x ∗, x∗)) −ψ(nφ(s x ∗, x∗)), respectively. on contrary, suppose there exists m ∈ n such that 1 1 + s d(xm, t xm) ≥ d(xm, x ∗) or 1 1 + s d(xm+1, s xm+1) ≥ d(xm+1, x ∗). (26) by (26), we have that (s + 1)d(xm, x ∗) ≤ d(xm, t xm) ≤ d(xm, x ∗) + sd(t xm, x ∗) or (s+1)d(xm+1, x ∗) ≤ d(xm+1, s xm+1) ≤ d(xm+1, x ∗)+sd(s xm+1, x ∗), and therefore d(xm, x ∗) ≤ d(t xm, x ∗) = d(xm+1, x ∗) and d(xm+1, x ∗) ≤ d(s xm+1, x ∗) = d(xm+2, x ∗). (27) by (8), (26) and (27), this is a contradiction. hence, (25) holds, and therefore f(d(xn+1, x ∗)) = f(h(t xn, s x ∗)) ≤ f(nφ(xn, x ∗)) −ψ(nφ(xn, x ∗)), (28) and f(d(xn+2, x ∗)) = f(h(s xn+1, t x ∗)) ≤ f(nφ(xn+1, x ∗)) −ψ(nφ(xn+1, x ∗)). (29) since that d(x∗, t x∗) ≤ nφ(xn, x ∗) = φ1(d(xn, x ∗))(d(xn, x ∗)) + φ2(d(xn, x ∗))(d(xn+2, x ∗)) + φ3(d(xn, x ∗)) ( d(xn+1, x∗) + d(xn, s x∗) 2s ) + φ4(d(xn, x ∗)) ( d(xn, s x∗) + d(s x∗, xn+2) 2s ) + φ5(d(xn, x ∗))(d(s x∗, xn+2) + d(xn+1, s x ∗)) + φ6(d(xn, x ∗))(d(xn, xn+1) + d(xn+2, t x ∗)) + φ7(d(xn, x ∗))(d(t x∗, x∗) + d(x∗, xn+1)) ≤ max{(d(xn, x ∗), d(xn+2, x ∗), d(xn+1, x∗) + d(xn, s x∗) 2s , d(xn, s x∗) + sd(s x∗, xn+2) + d(s x∗, xn+2) 2s , d(s x∗, xn+2) + d(xn+1, s x ∗), d(xn, xn+1) + d(xn+2, t x ∗), d(t x∗, x∗) + d(x∗, xn+1)} (30) and d(x∗, s x∗) ≤ nφ(xn+1, x ∗) = φ1(d(xn+1, x ∗))(d(xn+1, x ∗)) + φ2(d(xn+1, x ∗))(d(x∗, xn+3)) + φ3(xn+1, x ∗)) ( d(xn+2, x∗) + d(xn+1, x∗) 2s ) + φ4(d(xn+1, x ∗)) ( d(xn+1, s x∗) + d(s x∗, xn+3) 2s ) + φ5(d(xn+1, x ∗))(d(xn+3, s x ∗) + d(xn+2, s x ∗)) + φ6(d(xn+1, x ∗))(d(xn+1, xn+2) + d(xn+3, s x ∗)) + φ7(d(xn+1, x ∗))(d(s x∗, x∗) + d(x∗, xn+2)) ≤ max{d(xn+1, x ∗), d(x∗, xn+3), d(xn+2, x∗) + d(xn+1, x∗) 2s , d(xn+1, xn+2) + sd(xn+2, s x∗) + d(s x∗, xn+3) 2s , d(xn+3, s x ∗) + d(xn+2, s x ∗), d(xn+1, xn+2) + d(xn+3, s x ∗), d(s x∗, x∗) + d(x∗, xn+2)}. (31) by (24) and (30), we have that lim n→∞ nφ(xn, x ∗) = d(t x∗, x∗). by (24) and (31), we have that lim n→∞ nφ(xn+1, x ∗) = d(s x∗, x∗). by (28)and (29) and by the continuity of f and ψ, we have that f(d(x∗, t x∗)) ≤ f(nφ(x ∗, t x∗)) −ψ(nφ(x ∗, t x∗)), and f(d(x∗, s x∗)) ≤ f(nφ(x ∗, s x∗)) −ψ(nφ(x ∗, s x∗)). hence, since t x∗ and s x∗ are closed then we have x∗ ∈ t x∗ and x∗ ∈ s x∗, that is, x∗ is a fixed point of t and s . in theorem 2.1, when t = s = u, then we have the following result. corollary 2.1.1. let (x, d) be a complete strong b−metric space and let u : x → c b(x) be a multivalued generalized f-suzukicontraction mapping. then u has a fixed point x∗ ∈ x and for every x ∈ x the sequence {u n x}∞n=1 converges to x ∗. in corollary 2.1.1, when u is a single-valued then we have another new result as follows. corollary 2.1.2. let (x, d) be a complete strong b−metric space and let u : x → x be a single-valued generalized f-suzukicontraction mapping. then u has a fixed point x∗ ∈ x and for every x ∈ x the sequence {u n x}∞n=1 converges to x ∗. in theorem 2.1, when t and s are two single-valued then the 92 yusuf ibrahim / j. nig. soc. phys. sci. 1 (2019) 88–94 93 following result holds. corollary 2.1.3. let (x, d) be a complete strong b−metric space and let t, s : x → x be two single-valued generalized fsuzuki-contraction mappings. then t and s have a common fixed point x∗ ∈ x and for every x ∈ x the sequence {t n x}∞n=1 and {s n x}∞n=1 converge to x ∗. in theorem 2.1, when (x, d) is a complete b−metric space then the following new result holds. corollary 2.1.4. let (x, d) be a complete b−metric space and let t, s : x → x be two single-valued generalized f-suzukicontraction mappings. then t and s have a common fixed point x∗ ∈ x and for every x ∈ x the sequence {t n x}∞n=1 and {s n x}∞n=1 converge to x ∗. in corollary 2.1.4, when t = s = u, then we have the following result. corollary 2.1.5. let (x, d) be a complete b−metric space and let u : x → c b(x) be a multivalued generalized f-suzukicontraction mapping. then u has a fixed point x∗ ∈ x and for every x ∈ x the sequence {u n x}∞n=1 converges to x ∗. corollary 2.1.6. let (x, d) be a complete strong b−metric space and let u : x → c b(x) be a multivalued generalized fsuzuki-contraction mapping such that there exists f ∈ 0 and ψ ∈ ψ, ∀x, y ∈ x, x , y, 1s+1 d(x, u x) < d(x, y) ⇒ ψ(n(x, y)) + f(s4d(u x, uy)) ≤ f(n(x, y)) in which n(x, y) = max{d(x, y), d(y, u 2 x), (d(y, u x)) + d(x, uy) 2s , (d(x, uy)) + d(u 2 x, uy) 2s , d(u 2 x, uy) + d(uy, u x), d(u 2 x, uy) + d(u x, x), d(u x, y)) + d(y, uy)}. (32) then u has a fixed point x∗ ∈ x and for every x ∈ x the sequence {u n x}∞n=1 converges to x ∗. proof from lemma 1.1, since (2) ⇒ (32) then by the corollary 2.1.1 the result follows immediately. corollary 2.1.7. let (x, d) be a complete strong b−metric space and let u : x → x be a single-valued generalized f-suzukicontraction mapping such that there exists f ∈ 0 and ψ ∈ ψ, ∀x, y ∈ x, x , y, 1s+1 d(x, u x) < d(x, y) ⇒ ψ(n(x, y)) + f(s4d(u x, uy)) ≤ f(n(x, y)) in which n(x, y) = max{d(x, y), d(y, u 2 x), (d(y, u x)) + d(x, uy) 2s , (d(x, uy)) + d(u 2 x, uy) 2s , d(u 2 x, uy) + d(uy, u x), d(u 2 x, uy) + d(u x, x), d(u x, y)) + d(y, uy)}. (33) then u has a fixed point x∗ ∈ x and for every x ∈ x the sequence {u n x}∞n=1 converges to x ∗. proof from lemma 1.1, since (2) ⇒ (33) then by the corollary 2.1.2 the result holds. corollary 2.1.8. let (x, d) be a complete strong b−metric space and let t, s : x → x be two single-valued generalized fsuzuki-contraction mappings such that there exists f ∈ 0 and ψ ∈ ψ, ∀x, y ∈ x, x , y, 1s+1 d(x, t x) < d(x, y) and 1 s+1 d(y, s x) < d(y, s t x) ⇒ψ(n(x, y))+f(s4 h(t x, s y)) ≤ f(n(x, y)) in which n(x, y) = max{d(x, y), d(y, s t x), (d(y, t x)) + d(x, s y) 2s , (d(x, s y)) + d(s t x, s y) 2s , d(s t x, s y) + d(s y, t x), d(s t x, s y) + d(t x, x), d(t x, y)) + d(y, s y)}. (34) then t and s have a common fixed point x∗ ∈ x and for every x ∈ x the sequence {t n x}∞n=1 and {s n x}∞n=1 converge to x ∗. proof from lemma 1.1, since (2) ⇒ (34) then by the corollary 2.1.4 the result holds. corollary 2.1.9. let (x, d) be a complete b−metric space and let u : x → c b(x) be a multivalued generalized f-suzukicontraction mapping such that there exists f ∈ 0 and ψ ∈ ψ, ∀x, y ∈ x, x , y, 12s d(x, u x) < d(x, y) ⇒ ψ(n(x, y)) + f(s6d(u x, uy)) ≤ f(n(x, y)) in which n(x, y) = max{d(x, y), d(y, u 2 x), (d(y, u x)) + d(x, uy) 2s , (d(x, uy)) + d(u 2 x, uy) 2s , d(u 2 x, uy) + d(uy, u x), d(u 2 x, uy) + d(u x, x), d(u x, y)) + d(y, uy)}. (35) then u has a fixed point x∗ ∈ x and for every x ∈ x the sequence {u n x}∞n=1 converges to x ∗. proof from lemma 1.1, since (2) ⇒ (35) then by the corollary 2.1.5 the result holds. 3. example let x = [0, 1]. t, s : [0, 1] → c b([0, 1]) be defined by t x = [0, x2 ] and s y = [0, y 2 ] such that s t x = [0, x 8 ] for all x ∈ [0, 1]. let d be the usual metric on x. taking f(t) = t10 and let x < y, then ∀x, y ∈ [0, 1] d(x, y) > 0 and d(y, s t x) = |y− x8 | > |y− y 8 | = 7 8 y > y 4 . now, for s = 1, we have that 1 2 d(x, t x) = 0 < d(x, y) and 12 d(y, s y) = y 4 < d(y, s t x). without lose of generality, let φ1(d(x, y)) = φ2(d(x, y)) = φ3(d(x, y)) = 1 5 ; and φ4(d(x, y)) = φ5(d(x, y)) = φ6(d(x, y)) = φ7(d(x, y)) = 1 102 . therefore, we have that f(h(t x, s y)) = ln (h(t x, s y)) + h(t x, s y) = 1 10 ∣∣∣∣∣ y2 − x4 ∣∣∣∣∣ = 110 ∣∣∣∣∣y − y2 − x4 ∣∣∣∣∣ ≤ 1 10 (∣∣∣∣∣y − x4 ∣∣∣∣∣ + ∣∣∣∣∣x − y2 ∣∣∣∣∣) = 1 10  ∣∣∣y − x4 ∣∣∣ + ∣∣∣x − y2 ∣∣∣ 2  + 110  ∣∣∣y − x4 ∣∣∣ + ∣∣∣x − y2 ∣∣∣ 2  93 yusuf ibrahim / j. nig. soc. phys. sci. 1 (2019) 88–94 94 ≤ 1 10  ∣∣∣y − x4 ∣∣∣ + ∣∣∣x − y2 ∣∣∣ 2  + 110  ∣∣∣y − x4 ∣∣∣ + ∣∣∣x − x8 ∣∣∣ + ∣∣∣ x8 − y2 ∣∣∣ 2  = 1 10  ∣∣∣y − x4 ∣∣∣ + ∣∣∣x − y2 ∣∣∣ 2  + 110  ∣∣∣ x 8 − y 2 ∣∣∣ + ∣∣∣x − x8 ∣∣∣ 2  + 1 10 (∣∣∣∣∣ y2 − x8 ∣∣∣∣∣) ≤ 110  ∣∣∣y − x4 ∣∣∣ + ∣∣∣x − y2 ∣∣∣ 2  + 1 10  ∣∣∣ x 8 − y 2 ∣∣∣ + ∣∣∣x − x8 ∣∣∣ 2  + 110 (∣∣∣∣∣ y2 − x8 ∣∣∣∣∣ + ∣∣∣∣∣ x8 − x4 ∣∣∣∣∣) = 1 5  ∣∣∣y − x4 ∣∣∣ + ∣∣∣x − y2 ∣∣∣ 2  + 15  ∣∣∣ x 8 − y 2 ∣∣∣ + ∣∣∣x − x8 ∣∣∣ 2  + 1 10 (∣∣∣∣∣ y2 − x8 ∣∣∣∣∣ + ∣∣∣∣∣ x8 − x4 ∣∣∣∣∣) + 1102 (|x − y|) + 1102 (∣∣∣∣∣y − x8 ∣∣∣∣∣) + 1 102 (∣∣∣∣∣ x8 − y2 ∣∣∣∣∣ + ∣∣∣∣∣ x4 − x ∣∣∣∣∣) + 1102 (∣∣∣∣∣ x4 − y ∣∣∣∣∣ + ∣∣∣∣∣y − y2 ∣∣∣∣∣) − 1 102 [ (|x − y|) + (∣∣∣∣∣y − x8 ∣∣∣∣∣) + ( ∣∣∣∣∣ x8 − y2 ∣∣∣∣∣ + ∣∣∣∣∣ x4 − x ∣∣∣∣∣)] + (∣∣∣∣∣ x4 − y ∣∣∣∣∣ + ∣∣∣∣∣y − y2 ∣∣∣∣∣) − 110 (∣∣∣∣∣ y2 − x8 ∣∣∣∣∣ + ∣∣∣∣∣ x8 − x4 ∣∣∣∣∣) − 1 10  ∣∣∣y − x4 ∣∣∣ + ∣∣∣x − y2 ∣∣∣ 2  − 110  ∣∣∣ x 8 − y 2 ∣∣∣ + ∣∣∣x − x8 ∣∣∣ 2  . = φ1(d(x, y))(d(x, y)) + φ2(d(x, y))(d(y, s t x)) + φ3(d(x, y)) ( (d(y, t x)) + d(x, s y) 2s ) + φ4(d(x, y))( (d(x, s t x)) + d(s t x, s y) 2s ) + φ5(d(x, y))(d(s t x, s y) + d(s t x, t x)) + φ6(d(x, y))(d(s t x, s y) + d(t x, x)) + φ7(d(x, y))(d(t x, y)) + d(y, s y)) −ψ(nφ(x, y)). 4. conclusion fixed point results of piri and kumam [11], ahmad et al. [9], suzuki [18] and suzuki [19] are extended by introducing common fixed point problem for multivalued generalized f-suzukicontraction mappings in strong b-metric spaces. in specific, corollary 2.1.1 and corollary 2.1.2 generalize and extend the work of ahmad et al. [9] and kumam and hossein [5], respectively. acknowledgments i thank the referees for the positive enlightening comments and suggestions, which have greatly helped me in making improvements to this paper. references [1] b. banach, “sur les operations dons les ensembles abstraits et leur application aux equations intégrales”, fundam. math. 3 (1922) 133. [2] m. edelstein, “on fixed and periodic points under contractive mappings”, 37 (1962) 74. [3] d. wardowski, “fixed point theory of a new type of contractive mappings in complete metric spaces”, fixed point theory appl. 2012 (2012) 94. [4] d. wardowski & n. v. dung, “fixed points of f -weak contractions on complete metric spaces”, demonstr. math. 1 (2014) 146. 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[14] g. e. hardy & t. d. rogers, “a generalization of a fixed point theorem of reich”, canad. math. bull. 16 (1973) 201. [15] p. hossein & p. kumam, “some fixed point theorems concerning fcontraction in complete metric spaces”, fixed point theory and applications 2014 (2014) 210. https://doi.org/10.1186/1687-1812-2014-210 [16] t. rasham, a. shoaib, b. a. s. alamri & m. arshad, “multivalued fixed point results for new generalized f-dominated contractive mappings on dislocated metric space with application”, journal of function spaces volume 2018 (2018) 4808764. https://doi.org/10.1155/2018/4808764. [17] t. rasham, a. shoaib, n. hussain, m. arshad & s. u. khan, “common fixed point results for new ciric-type rational multivalued f-contraction with an application”, j. fixed point theory appl. 2018 (2018) 20 https://doi.org/10.1007/s11784-018-0525-6. [18] t. suzuki, “a generalized banach contraction principle that characterizes metric completeness”, proceedings of the american mathematical society 136 (2008) 1861. [19] t. suzuki, “discussion of several contractions by jachymski’s approach”, fixed point theory and applications 2016 (2016) 91. https://doi.org/10.1186/s13663-016-0581-9 94 j. nig. soc. phys. sci. 3 (2021) 66–73 journal of the nigerian society of physical sciences biostimulation effects and temperature variation in stimulated dielectric substance (diabetic blood comparable to non-diabetic blood) based on the specific absorption rate (sar) in laser therapy sylvester j. gemanama,b, nursakinah suardia, barnabas a. ikyob, samson damilola oluwafemia, terver danielb,∗, samuel t. kungurc aschool of physics, universiti sains malaysia (usm), pulau pinang, penang, 11800, malaysia bdepartment of physics, faculty of science, benue state university, makurdi, 102119, nigeria cdepartment of physics, college of education, katsina-ala, benue state, nigeria abstract human blood exposed to irradiation absorbs electromagnetic energy which consequently effect temperature variation. the evaluation of specific absorption rate (sar) of human blood helps to ascertain the values for optimum laser power, time, and temperature variation for fair therapy to avoid blood-irradiation pollution but to enhance its rheological properties when using lasers. prior knowledge of blood sar evaluating its dielectric properties is significant, but this is under investigation. we investigate the appropriate sar threshold value as affected by temperature variation using fundamental blood dielectric parameters to optimize the effect of low-level laser therapy based on physiological and morphological changes of the stimulated diabetic blood. studies were carried out with agilent 4294a impedance analyser at frequencies (40hz – 30 mhz) and designed cells (cuvettes) comprises of electrodes were used in the preand post-irradiations measurements. at different laser power outputs, blood samples were subjected to various irradiation durations using portable laser diode-pumped solid state of wavelength 532 nm. results showed laser at low energy is capable of moderating morphologically the proportion of abnormal diabetic red blood cells. hence, there is a significant effect using a laser at low energy, as non-medicinal therapy in controlling diabetic health conditions. the positive biostimulation effects on the irradiated diabetic blood occurred within absorbance threshold sar values range of 0.140 ≤ 0.695 w/kg and average temperatures range of 24.2 ≤ 28.0◦c before blood saturation absorbance peak. there is morphological stimulation at a laser power of 50 mw for an exposure time of 10–15 minutes and 60 mw for 5–10 minutes of laser therapy that demonstrates better blood rejuvenated conditions. this occurred within the threshold sar of 0.140 ≤ 0.695 w/kg and average temperatures range of 24.2 ≤ 28.0◦c. therefore, the diabetic blood irradiated using laser output powers of 70 and 80 mw during exposure durations of 5,10, 15 and 20 minutes rather bio-inhibits positive blood stimulation which has resulted to crenation due to excessive irradiation. doi:10.46481/jnsps.2021.182 keywords: dielectric properties, specific absorption rates, diabetic blood, low-level laser therapy, impedance, diabetic blood. article history : received: 24 march 2021 received in revised form: 15 april 2021 accepted for publication: 24 april 2021 published: 29 may 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: w. a. yahya 66 gemanam et al. / j. nig. soc. phys. sci. 3 (2021) 66–73 67 1. introduction laser technology has significantly impacted in medicine and medical research studies due to its quick advancement and acceptance of non-invasive treatment techniques [1]. the applications yield an all-encompassing range of biomedical fields. this includes its attempts in the treatment of cancer, diabetes, and other diseases. the low-level laser-induced therapy has demonstrated to be minimally invasive and an encouraging surgical technique in diabetes treatment, tumour treatment in brain, liver, lung and colorectal, etc. [2, 3, 4]. nd:yag lasers and diode lasers with a wavelength close to infrared light are mostly used in the treatments due to blood and tissues ability of photoresponse. the absorbed light can inhibit stimulation and deteriorates blood if not monitored based on the absorption level of the blood, the laser power, and exposure duration. this will aid to check the excessive temperature upsurge build-up in the blood. the optical and blood thermal properties resulted from characteristics of the laser device determined the distribution of temperature and the level of the induced changes in the blood or tissues [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. the specific absorption rate (sar) quantified the amount of energy absorbed by the blood or tissue. it is expressed in watts per kilogram of the blood weight [5, 6, 9, 10]. the emitted laser radiations during medical practices are absorbed by the blood penetrating through the exposed cells and plasma, thereby producing heat around and within the exposed blood or tissue. there is a non-thermal effect which includes changes within the plasma contents and cellular levels. while the thermal effect is capable of causing harm by varying the blood temperature which results in cells and other blood constituents’ damage if not taken into consideration [4, 8, 12, 15]. the sar serves as an optimisation model of the human blood or tissue from any adverse effect and the immediate surroundings. this is done by determining the amount of energy absorbed or dissipated by the blood, for reliable measurement and evaluation of safe limits for electromagnetic energy absorption by human blood [6, 7, 8, 13]. due to the use of headphones, there is a substantial amount of research in the literature that addressed the sar assessment, although little or no document has been done in lasers. collins and his group work on human head sar determinations through magnetic resonance imaging (mri), the authors estimated a head average sar level range of 3.0 – 3.2 w/kg [7]. it is unlikely for a significant temperature increase in the brain to occur as a result of perfusion but sar limits in any 1 g of head tissue may be exceeded. the sar values are relative to measured temperature changes which had a sharp increase in absorbed heat especially at the place near the ear skull due to constantly used of mobile phones [4, 7, 8, 14, 15, 16, 17]. since there are near to nothing that has been researched in the area of low-level laser for diabetes blood, it is pertinent to evaluate the sar value relative to temperature variation for the ∗corresponding author tel. no: +2348067988338 email addresses: gemanamsly@gmail.com (sylvester j. gemanam), terver.daniel@yahoo.co.uk (terver daniel ) diabetes blood during laser therapy. this is to enhance effective irradiation procedures during laser therapy and minimize what may negatively affect blood glucose levels. diabetes mellitus is a complex and chronic disease and increasing health concern as the incidence increases worldwide [18]. it is responsible for the disruption of the lipid profile, especially increased susceptibility to lipid peroxidation, which results in increased atherosclerosis, a major complication of diabetes mellitus [3]. therefore, comparing the diabetes blood sar with that of the non-diabetes blood in terms of temperature variation, then its morphology and physiological analysis of the smear blood cells will aid with low-level laser biostimulation. this research sought to investigate appropriate sar threshold values with respect to temperature variations for the optimal effect of low-level laser therapy in terms of physiological and morphological changes on the diabetes blood. then do the same with non-diabetes human blood and compare to observe any difference in elucidating the effects of low-level laser biostimulation as a way to enhance diabetes mellitus practical non-medicinal therapy and prevention of some blood transfusion diseases. 2. experimental materials and methods a programmable automatic rlc analyser, model 4194a, adjustable frequency ranging from 40 hz to 30 mhz and oscillation level of 500v was used to measure the capacitance, c p and dissipation factor d f with a delayed time of 0.05 sec. the impedance, z and conductance were calculated to determine the blood samples dielectric response using a designed blood cuvette as sample holder with two opposite pure copper electrodes connected to the impedance analyser [6, 7, 8, 9, 10, 19]. the dielectric permittivity was known to be expressed as a complex number ε∗(ω) = ε′(ω) − iε′′(ω) (1) where ε′(ω) is the real part known as relative permittivity or dielectric constant and ε′′(ω) the imaginary part, represents the loss factor as functions in angular frequency, ω. the dielectric constant ε′(ω) was evaluated using: ε′(ω) = c pd ε0 a (2) where, d denotes the distance between the electrodes and a is the cross-sectional area of the electrode. the imaginary part of the dielectric constant, ε′′, was ascertained using the relations [5, 15, 18, 20]: ε′′(ω) = ε′ tan δ (3) σ = ε′′r ωε0 (4) then, s ar = σ|e|2 2ρ (w/kg) (5) where δ is called the loss angle, tan δ is called dielectric loss tangent and for low-loss dielectrics, δ is very small, σ is the 67 gemanam et al. / j. nig. soc. phys. sci. 3 (2021) 66–73 68 conductivity of the blood (s/m), ε0 is the permittivity of free space (8.854×10−12 f/m), ω = 2π f ( f is the frequency in hz), ρ is the density of the blood (kg/m3) and ε ′′ r is the relative permittivity. |e|2 is the electric field strength (field amplitude) (v2/m2) of the wave (beam) [9, 16, 17]. the wave energy is proportional to the square of the amplitude (e2), where the amplitude is proportional to pressure. but in electromagnetic waves, the amplitude is the maximum field strength of the electric and magnetic fields. also, the intensity of an electromagnetic wave and the energy carried is proportional to e2 and b2. therefore, for a continuous sinusoidal electromagnetic wave, the average intensity iav is expressed as given below [5, 15], iav = cε0e2 2 (6) equation (6) is known as the poynting vector magnitude equation, and the irradiance (iav) is relative to electric field squared ( e2 ) given by e2 = 2iav cε0 (7) where c in the equation is the speed of light 3.0 × 108m/s and other symbols retain their usual meanings as explained above [2, 5, 6, 7, 9, 12]. 2.1. research ethical approval and blood sample collection the research approval was given by the university clinic management and controls as well as human ethics approval from jepem usm, under study protocol code of usm /jepem /16060208 before data were collected. the anticoagulant ethylene diamene tetra acetic acid (edta) treated samples of venous blood were collected from 64 patients (32 diabetic patients with average blood fast sugar level of 9.28 mmol/l and 32 non-diabetic patients without other blood related diseases). the sample patients’ age ranges from 22 to 54 years (mean age of 36 years) were used in the study work. all samples were collected from adults with no serious medical history of illness or under medication for major diseases and non-pregnant women. blood samples were collected from the wellness centre of universiti sains malaysia (usm), pulau pinang malaysia, observing all the necessary safety conditions. this was done through the prior consent of the patients. each of the samples was divided into two which were used as control and radiated samples for this study. the blood morphology and physiology conditions of the two grouped samples were properly examined using optical microscopy connected to the desktop scanner. 2.2. samples stimulations and experimental calculations the obtained blood samples from 64 patients were properly stirred and pipettes into the designed cuvettes connected to impedance analyser for data, before and after irradiation. the irradiation was carried out by using a portable diode-pumped solid state laser of wavelength 532 nm at different output powers of 50, 60, 70, and 80 mw. the power adjustment was done using an optical radiation power meter. the exposure duration was timed out for different intervals of 5, 10, 15, and 20 minutes for each of the laser power adjusted. the above equations (1), (2), (3), (4), (5), (6), and (7) were used for the various calculations to ascertain the sar values for both the diabetic and non-diabetic blood. the data for blood capacitance (c p) and dissipation factor (d f ) was gotten within the frequency range of 40 hz to 30 mhz of the impedance analyser. this was carried out after calibration using a standard 100 ω resistor for the instrument load data measurement and phase compensation, within room temperature of 21◦c. also the temperatures of the blood samples were maintained and taken at room temperature. this is because the acidity of blood in vitro varies to such an extent with temperature that for accurate determination of its ph, the inconvenience of working with temperaturecontrolled apparatus it has been the practice to take measurements at room temperature and then make the appropriate corrections [4, 21]. the data were taken as a function of different frequencies corresponding to dielectric response and energy deposited in the blood by the lll irradiation within a different time duration. 3. results the research evaluated the diabetic blood specific absorption rate from blood-laser therapy (blt) by 532 nm wavelength laser under observed laser parameters and sample properties. it is observed that an increase in sar values has direct effects on temperature changes in the blood. however, there is no rapid temperature rise since the absorbed energies are used in breaking down the ions and molecular chains form by the free radicals as a course of photochemical reactions (catalysis) and biostimulation of the blood constituents [2, 17, 22]. the sar values dropped sharply as the turn-off point is attained at blood saturation peak when there is no more energy absorption rather heat is given off to balance the system, then triggered the temperature to its highest points. the temperature build-up gradually as long as there is continuous irradiation therefore the blood temperature becomes excessively high and rather cause more damages to the diabetic blood. table 1 showed blood properties and laser parameters used in the evaluation of the appropriate sar optimal values for biostimulation of both bloods. 3.1. the effect of blood temperature on diabetic blood sar. the values of experimental results demonstrated the corresponding effects of the average maximum blood temperature and its specific absorption rate from different exposure power and durations. within an average temperature range of 24.2 ≤ 28.0◦c the diabetic blood samples were examined to improve the health conditions of erythrocytes morphological structure positively after exposures as shown in table 2. this occurred within a little temperature variation because the photochemical reactions generally do not result in a significant rise in temperature. photochemical effects involved either a change in the course of biochemical reaction due to the presence of an electromagnetic field or photodecomposition due to high energy photons that rupture molecular bonds [23]. there is an 68 gemanam et al. / j. nig. soc. phys. sci. 3 (2021) 66–73 69 table 1. blood and laser parameters used for sar evaluation of both blood samples material properties/laser parameters laser power (mw) exposure duration (mins) calculated av. blood density(kg/m3) av. initial temp. (◦c) final temp. (◦c) laser spot sectional area(m2) laser intensity (w/m2) 50 5 10 15 20 1020 23.2 24.2 27.4 28.0 29.2 1.26 × 10−5 3968.25 60 5 10 15 20 1020 23.1 27.4 27.7 29.2 30.4 1.26×10−5 4761.91 70 5 10 15 20 1020 23.2 28.0 29.2 30.4 30.6 1.26×10−5 5555.56 80 5 10 15 20 1020 23.2 29.2 30.5 30.6 32.9 1.26×10−5 6349.21 observed significant decreased of the immature cells to a slight increase in the erythrocytes count as a result of the reticulocytosis decreased (that’s the decrease rate of the formed reticulocytes). the reticulocytes decreased served as the useful indicator of glycaemic been under control. the radiated blood shows a higher resistance as stimulated by absorbed laser photon light. the absorbed energy also lowers the blood coagulation system, improves the red blood cells into biconcave form and disc liked shape after stimulation forming a normocytic. within this temperature range and corresponding values of diabetic blood absorption rate, there were noticeable optimal positive biostimulation effects on the irradiated samples that can be considered effective for low-level laser therapy. the resulting absorption rate values occurred within the range 0.140 ≤ 0.695 w/kg excepts for exposure to 50 mw for 20 minutes, where the diabetic blood reaches the optimal saturation peak at 15 minutes of exposure and took a downward trend to 0.326 w/kg. this implies raising the blood temperature directly causes the heat transfer in the blood and the specific absorption rate to the saturation point thereafter occur a reverse effect on sar. the radiated blood within this particular exposure has shown more cell damage and haemoglobin released in plasma and twice extracellular potassium than in non-irradiated blood (tables 2, 3, and 4). 3.2. specific absorption rate (sar) of diabetic and non-diabetic blood the experimental results here compared between the specific absorption rate (sar) of non-diabetic to diabetic blood samples calculated via blood parameters: conductivity and blood density. the sar values for both irradiated bloods (diabetes and non-diabetic) samples were observed to increase with increasing frequency untill attains saturated plateau before taking a descending direction. this increase reflects the biological impacts of blood-based thermal effects on key features. the observed blood samples specific absorption rates are shown in figures 1 and 2 below and some highlighted values in tables 2 and 3 which shows the evaluated sar values of the irradiated diabetic blood with the corresponding low-level laser biostimulation effects comparably higher than that of the nondiabetic blood counterparts. the morphological and physiological changes due to the biostimualtion effects of irradiation were captured through a microscope with a charged coupled device (ccd) camera connected to the computer operated by software as in figures 3 and 4. the higher sar values of diabetic blood might be due to the high absorption capability rate of the high glucose present in the blood of diabetic patients [19, 24]. this produces reactive oxygen species (ros) derived from flavin, reduced nicoti69 gemanam et al. / j. nig. soc. phys. sci. 3 (2021) 66–73 70 table 2. the effect of blood temperature on the sar of diabetic blood duration/power (min) 50mw 60mw 70mw 80mw sar (w/kg) av.temp var(◦c) sar (w/kg) av.temp var(◦c) sar (w/kg) av.temp var(◦c) sar (w/kg) av.temp var(◦c) 5 0.615 24.2 0.489 27.4 0.140 28.0 2.064 29.2 10 0.647 27.4 0.695 27.9 1.666 29.2 2.412 30.5 15 0.674 28.0 0.893 29.2 0.837 30.4 1.506 30.6 20 1.195 29.2 0.326 30.4 2.060 30.6 1.552 32.9 table 3. efficacy (optimization) of normal patient’s blood irradiated and sar at characterized frequency of 40hz duration (min) 50mw 60mw 70mw 80mw sar (w/kg) blood physiology sar (w/kg) blood physiology sar (w/kg) blood physiology sar (w/kg) blood physiology 5 0.173 stimulate 0.178 stimulate 0.791 bloat cells 0.121 bloat cells 10 0.528 stimulate 0.390 stimulate 0.427 bloat cells 0.443 bloat cells 15 1.417 bloat cells 0.754 bloat cells 0.576 all crenate 0.941 lake & haemolysed 20 0.631 crenate 0.540 crenate 0.003 lake & haemolysed -0.130 lake & haemolysed table 4. efficacy /optimization of diabetic patient’s blood irradiated and sar at characterized frequency of 40hz duration (min) 50mw 60mw 70mw 80mw sar (w/kg) blood physiology sar (w/kg) blood physiology sar (w/kg) blood physiology sar (w/kg) blood physiology 5 0.615 stimulate 0.489 stimulate 0.140 stimulate 2.064 bloat cells 10 0.647 stimulate 0.695 stimulate 1.666 bloat cells 2.412 all crenate 15 0.674 stimulate 0.893 bloat cells 0.837 all crenate 1.506 lake & haemolysed 20 1.195 bloat cells 0.326 crenate 2.060 lake & haemolysed 1.552 lake & haemolysed namide adenine dinucleotide phosphate (nadph), and hemeprotein due to photosensitization of the cell chromophores. it is a key enabler for the various activation of signalling pathways, and the role of bio-stimulation in cells since laser light depends on them [2, 18, 20, 21, 23] (figures 1,2, 3 and 4). 70 gemanam et al. / j. nig. soc. phys. sci. 3 (2021) 66–73 71 figure 1. frequency characteristics of sar values of non-diabetic blood irradiated using a laser at an output power of 50 mw under different time duration. figure 2. graph of sar values of diabetic blood frequency characterization irradiated using a laser at an output power of 50 mw under different time duration figure 3. (a) smeared control non-diabetic blood morphology and (b) diabetic blood morphology showing bloats rbcs, immature rbcs and some blast cells in blue circle. magnifications 40x 4. discussions this research work was conducted to elucidate the low-level laser biostimulation effects as a way to enhance diabetes mellitus practical non-medicinal therapy and prevention of some figure 4. (a) smeared diabetic blood irradiated within 10 and 15 minutes. (b) smeared diabetic blood stimulated > 15 minutes durations showing cell crenation due to excessive irradiation at 50 mw power output. magnifications 100x. (c) smeared non-diabetic blood irradiated for 15 minutes showing bloat cells circled in blue and cells crenation in red circles. (d) smeared non-diabetic blood stimulated > 15 minutes durations using laser at 50 mw power output showing cell crenation magnifications 100x blood transfusion diseases. this was observed with respect to the blood specific absorption rate relative to temperature variation. the research findings support the appropriate irradiation conditions for low-level laser therapy for diabetic mellitus optimising the efficacy of laser non-invasive treatment of diabetes mellitus. the result findings in figure 1, the irradiated blood identified as sar (5m) reaches a maximum sar value of 1.35 w/kg at 100 khz frequency, also known as absorption peak after stimulation using a laser output power of 50 mw for a period of 5 minutes irradiation. at this point, the blood absorption rate yields the highest level which further frequency characterization and exposure often result in a sudden downward trend of lower sar values. at high frequencies, the blood polarisation is not in the same phase with the applied energy as well as the blood sar, therefore blood releases more heat than its absorption capacity. it is also noted that thermal effects apt to strongly occur beyond this peak. thermal energy causes high blood temperatures associated with an imbalance between heat production and dissipation. within this region, the blood is known to starts deteriorating at the point of becoming shrinking 71 gemanam et al. / j. nig. soc. phys. sci. 3 (2021) 66–73 72 and crenation. the sar peak result value of 1.65 w/kg for 10 minutes of irradiated blood identified as sar (10m) increased with increasing frequency and reached its plateau at 265 khz frequency mark. excess heat was observed when the blood irradiated for 15 minutes; the sar value was extremely high and reached a saturation point at 640 hz with 2.79 w/kg. this even prevents the blood from being polarized at the high oscillation frequency. the detrimental deteriorating effect was observed for the human blood irradiated for 20 minutes since high exposure reversed the sar trend downward right from the polarization stage, therefore, dropping its value instantly. in table 2, the findings show the stimulating effects after irradiating the blood of non-diabetic patients’ samples, with different laser power and exposure duration to compare any occur effect with the diabetic blood. the morphology and blood physiology of the irradiation power of 50 and 60 mw for 5 and 10 minutes irradiation duration have no negative effects on results with slight moderate stimulations there are non-diabetes blood. the exposure for 15 minutes, effects on the irradiated blood are slightly excessive shown bloat and little crenate cells. the results obtained at all the laser powers for 20 minutes’ blood stimulation period have lower sar values while blood physiology was not satisfactory with bloat, crenate, and haemolysis effects (morphology and physiology conditions of the irradiated blood shown in figure 4). this is because the absorption capacity of the blood is saturated and reflects the radiant heat from its absorption at these points resulting in crenation based on the laser intensity level as also shown in table 3. the results in figure 2 presented the sar values of diabetic blood under frequency characterizations of varying laser power output and exposure durations (morphology in figure 3b). in the graphical representations of sar for diabetic blood been irradiated at the power of 50 mw for 5 minutes indicated as sar (5 m) increased with increasing frequency to saturation and absorption peak at the point of 1.906 w/kg. at a frequency of 1.1 khz, the sar values took a downward trend. here, the blood molecules cannot regulate any higher frequency beyond and its rate of absorption retrogressed immediately. before saturation point, exposed diabetic blood morphology observed to be maintained, although it is not sufficient to completely stimulate it. the radiation here does not help to renew the aging process of the plasma and cell membrane of diabetes mellitus. diabetic blood exposed for 10 and 15mins (indicated as sar (10m) and sar(15m) respectively) gives a better stimulating result when examined its morphological and physiological status. the blood temperatures within these exposure durations were moderate (average temperature range of 24.2 ≤ 28.0 ◦c of the irradiated diabetic blood) and the irradiated blood shows a higher resistance, lower coagulation system, and improved the red blood cells to biconcave and disc liked shape that is normocytic as shown in figure 4a. also showed a slight increase in the erythrocytes count and decrease in the reticulocytes. there was no bloat appearance of the blood cells noticed after the exposures and showed an absorption rates range of 0.140 ≤ 0.695 w/kg. within this average temperature range and corresponding values of the absorption rate there were noticeable optimal positive biostimulation effects on the irradiated samples that can be considered effective for the use of low-level lasers for treatment of diabetes mellitus. the blood reaches the maximum sar at frequencies 4.3 khz and 398 hz, and the corresponding sar values were 3.072 and 1.577 w/kg respectively. exposure of 20 minutes indicated as sar (20m) did not produce good results due to excessive heating for long periods of time as observed in figure 4b. and most diabetic blood cells became crenated due to shrinkage of cells that change the cell membrane. it inhibits the process of stimulation that can increase the resistance of cells that protect the k+ ion efflux but accelerates the blood deterioration [12]. the microscopic examination of the blood morphology and physiological conditions in figures 3 and 4 above reveals its stimulus on the basis of different biostimulation effects from sar, temperature variation, and laser parameters. the examined smeared of diabetic blood irradiated using a 50mw output laser power for exposure of less or equal to 5 minutes indicates that energy from the green laser is insufficient for erythrocyte to maintain its shape and correct defect. the effectiveness of diabetic blood stimulations was achieved with a 50 mw stimulated laser output power for a duration of 10 and 15 minutes. the smeared of exposed blood for more than 15 minutes produce unsatisfactory results. where there are free intracellular ca+2 concentrations that affect the efflux of k+ions [12, 16], it results in the crenation of the blood cells as in figure 4. this is due to excess free radicals that caused changes in the molecular chemical structures as well as confirmation by altering h-bonds of membrane proteins [1, 18, 19]. table 4 summarized the stimulating effects of the irradiated diabetic blood based on the morphological/physiological conditions with different laser output power and of time periods with corresponding evaluated sars at 40 hz frequency. there are positive stimulation effects when lower laser output powers of 50 mw for 10 and 15 minutes then 60 mw for 5 and 10 minutes’ duration were used for the irradiations. at an output power of 70 mw, the positive stimulation effect lasted only 5 minutes. although all other laser output power exposures with different duration results are unsatisfactory bloat cells and full of crenation. 5. conclusion we concluded from this research study that the lllt stimulates the diabetic blood to effective cell rejuvenation showing positive biostimulation effects. the specific absorption rate (sar) of blood served as a special tool to regulate and enhance the efficacy of the low-level laser blood therapy. this occurred at absorbance threshold sar values of 0.140 ≤ 0.695 w/kg within an average temperature range of 24.2 ≤ 28.0◦c before saturation absorbance peak for the diabetic blood and the nondiabetic blood show positive stimulation within the sar range of 0.173 ≤ 0.528 w/kg before turn-off peak absorbance point. the blood-laser therapy demonstrated a robust positive influence on the immune system cells and all blood exchange pro72 gemanam et al. / j. nig. soc. phys. sci. 3 (2021) 66–73 73 cesses. these humongous effects help to improve not only the biostimulation as diabetic blood-laser therapy but indirectly capable in reduction of a certain number of transmission-associated graft-versus-host diseases (ta-gvhd) involved during blood transfusion. whereas anything above the absorbance threshold of the sar range inhibits the effectiveness of the laser therapy and causes more deteriorating effects on the diabetic blood. contrary, at higher laser powers and time exposure, diabetic blood shows crenation due to profuse heating. this formed cell membrane shrinkage with an abnormal notching. the diabetic blood shows better stimulation within the low-level laser output powers of 50 mw for 10 and 15 minutes then 60 mw for 5and 10-minutes duration. therefore, lllt has a positive blood stimulation effect by rejuvenating the diabetes blood cells and regulating the glucose negative effects within the blood as 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[25] d. j. vitello, r. m. ripper, m. r. fettiplace, g. l. weinberg, & j. m. vitello, “blood density is nearly equal to water density: a validation study of the gravimetric method of measuring intraoperative blood loss”, journal of veterinary medicine 2015 (2015) 152730. 73 j. nig. soc. phys. sci. 3 (2021) 131–131 journal of the nigerian society of physical sciences corrigendum to “effect of benzophenone on the physicochemical properties of n-cnts synthesized from 1-ferrocenylmethyl (2-methylimidazole) catalyst” [j. nig. soc. phys. sci. 2 (2020) 205-217] ayomide hassan labuloa,∗, elijah temitope adesujia, charles ojiefoh oseghalea, elias emeka elemikeb, adamu usmana, akinola kehinde akinolac, enock olugbenga darec adepartment of chemistry, federal university of lafia, lafia, nasarawa state, nigeria bdepartment of chemistry, federal university of petroleum, nigeria cdepartment of chemistry federal university of agriculture, abeokuta, ogun state, nigeria doi:10.46481/jnsps.2021.213 article history : received: 28 april 2021 accepted for publication: 15 may 2021 published: 29 may 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: b. j. falaye in the acknowledgment section of this article, the first and second sentences that read “this research was financially supported by the national research foundation (nrf) south africa. we are grateful to the school of chemistry and physics, university of kwazulu-natal (ukzn) for creating a conducive research laboratory for this work” should have read “the authors acknowledge the school of chemistry and physics, university of kwazulu-natal (ukzn) for creating a conducive research laboratory for this work”. in addition, the sentence that reads “ayomide is grateful to prof. vincent nyamori, prof. bernand omondi, and mrs. rashidat labulo for proofreading this manuscript” should have read “ayomide is grateful to mrs. rashidat labulo for proofreading this manuscript”. ∗corresponding author tel. no: +234 8062295936 email address: labulo@yahoo.com (ayomide hassan labulo ) doi of the original article: 10.46481/jnsps.2020.105 131 j. nig. soc. phys. sci. 5 (2023) 1350 journal of the nigerian society of physical sciences numerical simulation of nonlinear and non-isothermal liquid chromatography for studying thermal variations in columns packed with core-shell particles abdulaziz g. ahmada,b,∗, nnamdi f. okechia, david u. uchea,c, abdulwasiu o. salaudeend adepartment of mathematics programme, national mathematical centre abuja, nigeria bdepartment of applied mathematics, federal university of technology babura, nigeria cdepartment of mathematics, university of abuja, nigeria ddepartment of applied mathematics (chemistry unit) programme, national mathematical centre abuja, nigeria abstract a high-resolution flux-limiting semi-discrete finite volume scheme (hr-fvs) is applied in this study to numerically approximate the nonlinear and non-isothermal flow of one-dimensional lumped kinetic model (1d-lkm), for a fixed-bed column loaded with core-shell particles. the developed model comprise a system of convection-dominated partial differential for mass and energy balances in the mobile phases coupled with differential equation and algebraic equation in the stationary phase. the solution of the model equations is obtained by utilizing a hr-fvs, the scheme has second-order accuracy even on the grid coarse and its explicit nature has the potential to resolve the arisen sharp discontinuities in the solution profiles. a second-order total variation diminishing (tvd) runge-kutta technique is used to solve the system of odes in time. several forms of a single-solute mixture are produced to investigate the influences of the fractions of core radius on thermal waves and concentration fronts. moreover, a particular criterion is introduced for analyzing the performance of the underlying process and to identify the optimal parameter values of the fraction of core radius. doi:10.46481/jnsps.2023.1350 keywords: non-isothermal chromatography, non-linear isotherm, one-dimensional lumped kinetic model, high-resolution scheme article history : received: 15 january 2023 received in revised form: 06 march 2023 accepted for publication: 10 march 2023 published: 24 april 2023 © 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: j. ndam 1. introduction in recent years, chemical engineers and researchers are increasingly interested in high-performance liquid chromatography (hplc) to further improve the performance of classical column. an innovative and valuable tool used for the separation and quantification of the multi-component mixture due ∗corresponding author tel. no: +234 80326117615 email address: agarbaahmad@yahoo.com (abdulaziz g. ahmad) to the different affinities of adsorption for the components is known as hplc. this technique is commonly used in the chemical, pharmaceutical and food industries where the traditional operations in the thermal unit, such as distillation and extraction, are unsuitable [1–3]. for both large and preparative scales this process is equally popular, especially for the purification of proteins and other higher valuable products. figure 1 illustrates a typical demonstration of a single-column hplc. here, the sample is injected via inner region, which propagates the 1 ahmad et al. / j. nig. soc. phys. sci. 5 (2023) 1350 2 solute along z-direction of the column by advection and axialdispersion. separation efficiency of hplc can be made better by using small diameter particles to be loaded into the column in order to reduced the resistance of intra-particulate mass transfer due to short diffusion distances [4–6]. these loaded core-shell particles were designed and operated in the columns of hplc to contempt the utilization of technical types of equipment. recently, the use of core-shell particles has created significant interest in both analytical and preparative liquid chromatography. it was applied to segregate peptides and other molecules, such as nucleotides, and proteins [5, 7]. aside from that, several theoretical investigations involving the use of core-shell particles were made by analyzing models of liquid chromatography in one-dimensional (1d) forms [1, 3]. kaczmarski and guichon [8] consider the general rate model to substantiate the benefits of thin-shell coated beads. also, the same general rate model is employed by gu et al. [4] to optimize and examine the influences of core radius fraction in isocratic elution of multicomponent mixture. several authors have recently emphasized the importance of studying the differences of core-shell and fully porous particles by formulating and solving a number of chromatographic models via core-shell particles [9–11]. inspired by the work of brandt, et al., [12]. temperature has a wide impact on all chromatography processes, and there are several ways to analyze its significance [13, 14]. for example, increasing the temperature decreases viscosity while rising solubility and diffusivity. moreover, the peak shape, the efficiency of the column, complete-time analysis and retention time has temperature influences due to the thermodynamics, and adsorption kinetics are temperature-dependent [15, 16]. further, the use of a steady condition of temperature during the process increases reproducibility. even though, effects of thermal property are typically ignored in the columns of liquid chromatographic because the effect of heat adsorption is assumed to be inconsiderable, many more chromatographers have figured out that temperature is critical to the process optimization [13]. on contrarily, a number of researchers have investigated thermal impacts in gas chromatography [17–19]. there are additional contributions in the literature that examine the thermal influences in liquid chromatography columns, which are also accessible [3, 13, 14, 20–24]. a number of mathematical models for simulating the underlying chemical process at various extent of complexities are available in the literature to aid in the understanding of physical phenomena [15, 16, 25, 26]. this modeling approach allows us to comprehend what occurs within the column, and also during the separation process. the most influential and widely employed models that exist in the literature include the equilibrium dispersive model (edm), the model based on linearized driving force, the lumped kinetic model (lkm), and the general rate model (grm) [1–3, 27]. the mass transfer rate in the edm is considered to be infinite, but the local concentration’s rate of change in the lkm is presumed to be finite [1–3]. grm is a model that incorporates intra-particle diffusion which includes mass transfer across the interface among the both phases (i.e., stationary and mobile). it is also referred to as the most comprehensive model [1]. a nonlinear 1d-lkm incorporating core-shell particles is numerically approximated in this article. this work extends the analysis of 1d-lkm non-isothermal model [28]. unlike the past study, this prevailing non-isothermal 1d-lkm helps to examine the effect of core radius fractions on thermal and concentration fronts along the axial gradients of the eluent in the column. the formulated model comprises of a system of convection-dominated partial differential for mass and energy balances in the mobile phases coupled with differential and algebraic equations in the stationary phase. the solution of the model is obtained by utilizing a hr-fvs. the scheme deals with integral form of conservation laws and also it has potential to produce numerical result that can resolve sharp discontinuity and achieve higher order accuracy. afterwards, a runge-kutta approach for second-order tvd is used to simulate the system of odes in term of time [30]. some certain test problems are considered to illustrate the simultaneous elution of thermal and concentration fronts. furthermore, we hope that the key parameters that influence the elution profiles within the column are identified. the article arranged as follows. in section 2, a nonlinear and non-isothermal 1d-lkm incorporated with core-shell particles is developed. in section 3, the numerical simulation of the non-isothermal 1d-lkm is obtained for the boundary conditions discussed by dankwerts condition. section 4 outlines a number of simulated case problems, and ultimately, section 5 contains the conclusions. 2. mathematical model formulation of the process in this section, the following assumptions are employed to configure and develop the model equations: (i) the column is considered to be thermally insulated and homogeneously loaded with ore-shell particles, (ii) the fluid is incompressible with constant rate of volumetric flow, (iii) the stationary and mobile phases have negligible interaction between them, (iv) the axial heat conductivity coefficient are independent of the flow rate, (v) the temperature does not affect physical properties such as density, viscosity, heat capacity, or coefficients of transport (e.g. axial dispersion and heat conductivity), (vi) the linear driving force model is utilized to determine the overall adsorption rate, and (vii) the heat transfer resistance of the solid phase is concentrated at the particle surface. let t symbolize the coordinate time while the coordinate in axial direction alongside the column length is donated as z, the symbol rp represents the unvarying size of cored beads parked in the column and fraction of core radius is ηcore = rc/rp where rc symbolized the radius of the inert core. the solute will propagate via the column’s z-direction with advection and axial-dispersion. the one-dimensional equations for the balance of mass and the heat for a solute mixture elution in the mobile phase via column loaded with spherical core beads are 2 ahmad et al. / j. nig. soc. phys. sci. 5 (2023) 1350 3 expressed as ∂c ∂t + u ∂c ∂z − dz ∂2c ∂z2 + f(1 −ηcore)k(φ ∗ −φ) = 0, (1) ∂t ∂t + u ∂t ∂z − λz ξ f ∂2t ∂z2 + f(1 −ηcore) 3hp rpξ f (t − t s) = 0. (2) from the equations above, the symbol c represents the solute mixture in the mobile phase, the interstitial velocity is donated as u, the symbol dz represents the axial-dispersion coefficient, while the symbol � denotes the external porosity and the phase ratio is expressed as f = 1−� � . also, the symbol φ represents the non-equilibrium average loading of concentration in the particular solid phase, in the mobile phase the temperature is indicated as t , and coefficients of heat conductivity along with the axial coordinates is represented as λz. moreover, ξe = ρscsp, ξ f = ρ lclp, ρ s and ρl indicate the solid and liquid phases’ densities per unit volume, respectively, t s stands for the temperature of the stationary phase, while csp and c l p symbolize the solid and liquid phases’ respective heat capacities. in the solid phase, the corresponding mass and heat balance equations are written as ∂φ ∂t =k(φ∗ −φ), (3) ∂t s ∂t = −∆ha ξe ∂φ ∂t + 3hp rpξe (t − t s) . (4) from the equations above, the symbol k stands for the mass transfer rate coefficient, ∆ha represents the enthalpy of adsorption and the coefficient for heat transfer among the mobile and solid phases is symbolized as hp. for the ι-th number of components in the mixture, the symbol φ∗(c, t s) shows a temperature dependency relationship among the specific phase in solid part of concentration at equilibrium φ and the temperature, is defined as [16] φ∗(c, t s) = aref c exp [ −∆ha rg ( 1 t s − 1 tref )] 1 + brefι cι exp [ −∆ha rg ( 1 t s − 1 tref )] , ι = 1, · · · , nc. (5) here, the coefficient aref is symbolizing the henry’s constant at a reference temperature, the symbol brefι is representing the nonlinearity coefficients, the symbol tref represents the reference temperature, and rg is universal gas constant. in addition, the model includes the following additional dimensionless parameters, which minimize the parameters in terms of number, this facilitates the analysis of the model. pez,m = lu dz , pez,h = ξ f lu λz , x = z l , τ = ut l , κ = lk u , βs = 3lhp urpξe , βl = 3lhp urpξ f , (6) the column length is represented by the symbol l, and the peclet-numbers of both heat and the mass transfer via axial direction are indicated by the symbols pez,h and pez,m , accordingly. on utilizing eq. (6) in eqs. (1)-(4), we get the following equations below after some manipulations ∂c ∂τ = − ∂c ∂x + 1 pez,m ∂2c ∂x2 − f(1 −ηcore)κ(φ ∗ −φ), (7) ∂t ∂τ = − ∂t ∂x + 1 pez,h ∂2t ∂x2 − f(1 −ηcore)βl(t − t s), (8) ∂φ ∂τ =κ(φ∗ −φ), (9) ∂t s ∂τ = −∆ha ξe ∂φ ∂τ + βs(t − t s). (10) appropriate initial conditions that are suitable for the computational solution of the model equations (c.f. eqs. (7)-(10)) in the range 0 ≤ x ≤ 1. the following are the initial conditions for an equilibrated column: c(x,τ = 0) = cinit, t (x,τ = 0) = tinit, φ(x,τ = 0) = φ∗init, t s(x,τ = 0) = tinit. (11) here, the initial temperature is represented by tinit, the symbol cinit is the initial equilibrated concentration of the single component solute, and φ∗init is obtained from eq. (5). the inflow conditions which dankwert’s boundary conditions (bcs) investigate at the column inlet are listed below [9, 31]. the injections in the inner circular region are described as follows: c(x = 0,τ) − 1 pez,m ∂ci(x = 0,τ) ∂x = { cinj, if 0 ≤ τ ≤ τinj, 0, if τ > τinj, (12a) t (x = 0,τ) − 1 pez,h ∂t (x = 0,τ) ∂x = { tinj, if 0 ≤ τ ≤ τinj, tref, if τ > τinj. (12b) in this eq. (12a), cinj represents the concentration of the injected component, the dimensionless time of injection is represented by τinj, and tinj represents the temperature of the injected component. moreover, the thermal conditions at the boundary provided by eq. (12b) allow the temperature of the injection sample to fluctuate. further, the injected temperature donated by tinj could be adjusted from the reference temperature of the bulk phase tref . at the right end of the column, the neuman conditions are considered: ∂c(x = 1,τ) ∂x = 0 , ∂t (x = 1,τ) ∂x = 0. (12c) the chromatographic process mathematical model is now complete. the subsequent task is to employ the suggested fluxlimiting hr-fvs in order to solve the developed model equations. 3. numerical scheme in order to approximate the current model equations numerically, a flux-limiting semi-discrete method hr-fvs is applied, 3 ahmad et al. / j. nig. soc. phys. sci. 5 (2023) 1350 4 table 1: values of the model parameters used in the test problems parameters values column length l = 4.0 cm radius of the column r = 0.2 cm radius of solid particle rp = 0.004 cm interstitial velocity u = 1.5 cm/min porosity � = 0.4 density of heat capacity of solid ξf = 4 kj/l density of heat capacity of liquid ξe = 4 kj/l axial dispersion coefficient dz = 0.01 cm2/min axial conductivity coefficient λz = 0.04 kjcm−1min−1 mass transfer coefficient k = 1 cm/min heat transfer coefficient hp = 1 w(cm2k)−1 reference temperature tref = 300 k initial temperature tinit = tref inlet temperature tinj = tref initial concentration cinit = 0 mol/l inlet concentration cinj = 1 mol/l dimensionless injection time τinj = 1.5 adsorption equilibrium constant aref = 1 figure 1: sketch of solute injection in the thermally insulated chromatographic column incorporating core-shell particles which have been widely discussed in the literature for approximating 1d-models [9, 10]. to simulate the ode system in term of time, the second-order tvd runge-kutta method is used. for the derivation of this numerical scheme, let us descritise the computational domain. let n stand for the number descritisation, xl+ 12 to be the interval of left and right boundaries, ∆x is the cell width and xl symbolizes the cell center. further, let us assign the following xn+ 12 = l, x 12 = 0, xl+ 12 = l∆xl, (13) xl = xl−12 − xl+ 12 2 , ∆xl = xl− 12 − xl+ 12 = l n + 1 . (14) the domain of cartesian grid [0, 1] that is fully covered by the cells ψl = xl− 12 − xl+ 12 for l ≥ 1. furthermore, the average initial data wl(0) in each interval are formulated as wl(0) = 1 ∆x ∫ x l+ 12 x l− 12 w(x, 0)d x, w ∈ {c, t, q, q∗}, l = 1, 2, · · · , n. (15) 0 2 4 6 8 10 12 τ 0 0.05 0.1 0.15 0.2 0.25 c (x ,τ ) [m o l/ l] η core = 0.0 η core = 0.4 η core = 0.8 (a) figure 2: these display the non isothermal single-solute elution profile for distinct values of fraction of core radius ηcore. specifically, ∆ha = −10 kj/mol. table 1 lists all of the other parameter values that were chosen once we discretized the computational domain and the associated initial data for τ = 0 is specified for each mesh interval, the subsequent task is to employ the proposed scheme. integrations of eqs. (7)-(10) over ψl give dcl dτ = − ( cl+ 12 − cl− 12 ) ∆xl + 1 ∆xlpez,m [ ( ∂c ∂x ) l+ 12 − ( ∂c ∂x ) l− 12 ] (16) − fκ(φ∗l −φl), dtl dτ = − ( tl+ 12 − tl− 12 ) ∆xl + 1 ∆xlpez,h [ ( ∂t ∂x ) l+ 12 − ( ∂t ∂x ) l− 12 ] + (17) − fβl(tl − t s,l), dφl dτ =κ(φ∗l −φl), (18) dt s,l dτ = −∆ha ξe κ(φ∗l, j −φ1,l, j) + βs(tl − t s,l). (19) the derivatives appearing in eqs. (16) and (17) are approximated as [ ∂c ∂x ] l± 12 = ± [ cl±1 − cl ∆xl ] , (20)[ ∂t ∂x ] l± 12 = ± [ tl±1 − tl ∆xl ] . (21) 3.1. koren hr-fvs according to the first order method, the values of concentration and with temperature of the cell-interface in eqs. (16) 4 ahmad et al. / j. nig. soc. phys. sci. 5 (2023) 1350 5 0 2 4 6 8 10 τ 0 0.02 0.04 0.06 0.08 0.1 0.12 c (x ,τ ) [m o l/ l] pe z,h =600, η core =0.6 pe z,m =50 pe z,m =100 pe z,m =600 (a) ∆ h a = -10 kj/mol 0 2 4 6 8 10 τ 299.6 299.8 300 300.2 300.4 300.6 t (x ,τ ) [k ] pe z,h =600, η core =0.6 pe z,m =50 pe z,m =100 pe z,m =600 (b) ∆ h a = -10 kj/mol figure 3: influence of pez,m on the model of non isothermal for the single-component elution by utilising two different ηcore values. table 1 lists all of the other parameter values that were considered and (17) are approximated as numerically cl+ 12 = cl, cl− 12 = cl−1, tl+ 12 = tl, tl− 12 = tl−1. (22) for the second-order hr-fvs, the developing flux-limiting formula is utilized in eqs. (16) and (17) to numerically approximate the values of temperature and concentration of the cell interface [9, 30]: cl+ 12 = cl + 1 2 φ1(γl+ 12 )(cl − cl−1), (23) tl+ 12 = tl + 1 2 φ2(λl+ 12 )(tl − tl−1), (24) here, the symbols φ1 and φ2 are the flux-limiting formula for the concentrations and temperature, respectively. also, γl+ 12 , j and λl+ 12 , j are the concentration and temperature gradient ratios, accordingly. γl+ 12 , j = cl+1 − cl + δ cl − cl−1 + δ , λl+ 12 = tl+1 − tl + δ tl − tl−1 + δ . (25) in order to avoid division by zero, we have taken the value of δ = 10−10. moreover, the limiting functions are expressed as φ1(γl+ 12 , j) = max [ 0, min ( 2γl+ 12 , j, min ( 1 3 + 2 3 γl+ 12 , j ))] , (26) φ2(λl+ 12 , j) = max [ 0, min ( 2λl+ 12 , j, min ( 1 3 + 2 3 λl+ 12 , j ))] , (27) where φ1(γl+ 12 , j) and φ2(λl+ 12 , j) are sequentially defined for the concentration and temperature. likewise, cl− 12 and tl− 12 can be evaluated by just replacing the index l by l − 1 in the equations above. finally, the built-in rk-45 in matlab is used to simulate the resulting ode-system in eqs. (16) and (17). the entire scheme described above was programmed and simulated in matlab. 4. evaluating criterion for the process performance in this section, we presents an evaluation for the performance criterion that can be utilized to improve product quality, according to the findings of hováth and fellinger [29] research. for applications to industries, preparative chromatographic methods required to be optimized in terms of their yield, productivity and efficiency. to describe this criterion procedure, we used a two-component mixture with component 2 having a higher reference affinity to the solid phase than component 1. i.e. a1ref < a 2 ref . let ζ 1 represent the non-dimensional time when the fraction of component 1 be more than an appropriate benchmark (i.e, c1 < �c1,inj), where � = 10−6. in a similar way, let ζ2 stand for a non-dimensional time when the concentration of component 2 drops below a particular level (c2 < �c2,inj). the amount of time that has elapsed between the injections of consecutive two values, i.e. as cycle time, is represented by ζcyc and is written as ζcyc = ζ2 − ζ1. (28) the cut time is defined as the time it takes to complete component 1 fractionation pur = ∫ ζcut ζ1 c1(x = 1,ζ)dζ∫ ζcut ζ1 [c1(x = 1,ζ)dζ + c2(x = 1,ζ)dζ] , (29) where ck(x = 1,ζ) = 2 ∫ 1 0 ck(x = 1,η,ζ)ηdη, k = 1, 2. (30) the productivity (pr) is defined by the quantity of a required compound manufactured per cycle’s time. also, the purity of 5 ahmad et al. / j. nig. soc. phys. sci. 5 (2023) 1350 6 0 2 4 6 8 10 12 τ 0 0.1 0.2 0.3 0.4 0.5 c (x ,τ ) [m o l/ l] k = 1 min −1 k = 100 min −1 (a) 0 2 4 6 8 10 12 τ 299.4 299.6 299.8 300 300.2 300.4 300.6 300.8 t (x ,τ ) [k ] k = 1 min −1 k = 100 min −1 (b) 0 2 4 6 8 10 12 τ 0 0.05 0.1 0.15 0.2 0.25 0.3 c (x ,τ ) [m o l/ l] h p =0.5 w(cm 2 k) -1 h p =2.0 w(cm 2 k) -1 (c) 0 2 4 6 8 10 12 τ 299.7 299.8 299.9 300 300.1 300.2 300.3 t (x ,, τ ) [k ] h p =0.5 w(cm 2 k) -1 h p =2.0 w(cm 2 k) -1 (d) figure 4: impacts of both coefficients of mass as well as transfer of heat (k and hp) for unchanged ηcore = 0.8 and ∆ha = −10 kj/mol. table 1 lists all of the other parameter values that were considered the suitable peak area is placed equal to 99%. for component 1, it is obtained as pr = ∫ ζcut ζ1 c1(x = 1,ζ)dζ ζcyc . (31) the proportion of considered component evaluated in segregated order and the complete amount of that injected component over the column inlet is referred as the quantity yield (y). for the component 1 case, it is obtained as y = ∫ ζcut ζ1 c1(x = 1,ζ)dζ∫ ζ2 ζ1 c1(x = 1,ζ)dζ . (32) 5. numerical test cases numerous numerical test cases for simulating the influences of ηcore, pez,m , k and hp, on the elution profiles are presented in this section. these characterize the core radius fraction, the axial peclet number, the mass transfer coefficient and the heat transfer coefficient, respectively. in addition, we have taken the nonlinearity coefficients zero (i.e., bι = 0 l/mol in eq. (5)). moreover, a process performance criterion for the parameters ζcyc, ζcut, pr, and y are simulated. furthermore, the solute is injected through the column inner zone in all of the cases we investigated. in the plots in figures 2-7, the solute component is denoted as c. while, the mobile phase temperature is denoted by the symbol t . the values of all needed parameters are taken from the scopes used in the applications of hplc [32] and are classified in table 1. in figure 2, we display the elution profile plots of both concentration and temperature of a single-component solute, for three distinct values of ηcore (i.e. ηcore = 0, 0.4, 0.8). additionally, the value of adsorption enthalpy of the process is taken as ∆ha = −10 kj/mol. it is obvious that operation of nonisothermal results in notable temperature differences within the column, as shown in given figure 2(b). as shown in figure 2(a), these temperature variations have no visible effect on the concentration profile due to the considered low value of ∆ha. it is also worth noting that as ηcore increases from 0 to 0.8 increases, the elution profiles become sharper, and as a result, their retention times reduce accordingly, i.e. by increasing the value of ηcore, the column efficiency gradually improves. at 6 ahmad et al. / j. nig. soc. phys. sci. 5 (2023) 1350 7 0 2 4 6 8 10 12 τ 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 c (x ,τ ) [m o l/ l] η core = 0.0 ∆ h a = -10 kj/mol ∆ h a = -20 kj/mol ∆ h a = -40 kj/mol (a) 0 2 4 6 8 10 12 τ 299 299.5 300 300.5 301 301.5 302 302.5 303 t (x ,τ ) [k ] η core = 0.0 ∆ h a = -10 kj/mol ∆ h a = -20 kj/mol ∆ h a = -40 kj/mol (b) 0 2 4 6 8 10 12 τ 0 0.05 0.1 0.15 0.2 0.25 c (x ,τ ) [m o l/ l] η core = 0.8 ∆ h a = -10 kj/mol ∆ h a = -20 kj/mol ∆ h a = -40 kj/mol (c) 0 2 4 6 8 10 12 τ 299 299.5 300 300.5 301 301.5 t (x ,τ ) [k ] η core = 0.8 ∆ h a = -10 kj/mol ∆ h a = -20 kj/mol ∆ h a = -40 kj/mol (d) figure 5: effects of ∆ha on the model of non-isothermal for single-component elution including two different values of ξcore. specifically, plots in 2-d for the single solute elutions profiles are presented at ξcore = 0 & ξcore = 0.8. table 1 lists all of the other parameter values that were employed the same time, the column’s absorption capacity gradually decreases due to a reduce in the thickness of the layer including in porous. figure 3 demonstrates the impact of the non-dimension parameter pez,m of concentration and temperature on the profiles of elution by utilizing the value of ηcore = 0.6 and an unchanged value of pez,m = 600. the lesser value of pez,m develops wider (spread) peaks, and thus the column efficiency decreases. in contrast, a higher pez,m value results in narrower peaks, which improves column performance. all plots clearly show analogous impacts of different pez,m . figure 4 illustrates the results of k and hp for an unchanged ηcore = 0.8. it is clear that lower values of these dimension parameters result in broader elution profiles, while for high values of these parameters, the profiles are sharpened. the plots also show that the parameter hp has a minor influence on the given profiles, whereas k has a significant impact on both profiles. figure 5 presents the influence of ∆ha on the eluent profile for two distinct ηcore values of a single-solute. as the operating condition for non-isothermal (∆ha = −10, −20 & −40 kj/mol) produces remarkable temperature differences, as it’s clearly observed from figure 5. also, increasing the magnitude of ∆ha affects the solute profile as well, and causing the concentration profile to become sharper while the peak moves upward. further, variation of the particles size of core-shell decreases the retention time of the solute profile by making them narrower and sharper. in figure 6, for isothermal and non-isothermal conditions, we describe a process performance by simulating the following terms: ζcyc, ζcut, pr, and y (c.f. eq. (28)-(29)) over ηcore for ∆ha = −10 kj/mol. it can be clearly observed that the time cycle as well as the cut time is decreasing from 36 to 12 and 97 to 10, respectively, as varies from entirely particles that are porous i.e. ηcore=0.0 to the particles of core shell i.e. ηcore=0.8. moreover, a rise in the productivity can be seen up to ηcore = 0.74 and it gradually falls later. furthermore, the y continuously increases for an upward increment of the ηcore value. so, the cycle time, cut time, productivity and yield profiles are similar 7 ahmad et al. / j. nig. soc. phys. sci. 5 (2023) 1350 8 0 0.2 0.4 0.6 0.8 1 η core 5 10 15 20 25 30 35 40 ζ c y c isothermal non-isothermal (a) 0 0.2 0.4 0.6 0.8 1 η core 0 20 40 60 80 100 ζ c u t isothermal non-isothermal (b) 0 0.2 0.4 0.6 0.8 1 η core 0 1 2 3 4 5 6 p r ×10 -3 isothermal non-isothermal (c) 0 0.2 0.4 0.6 0.8 1 η core 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 y isothermal non-isothermal (d) figure 6: isothermal and non-isothermal comparison for the process performance assessment. here, the value for ∆ha = −40 kj/mol is taken for the model of non isothermal process operating condition. firstly, the plot (a) presents ζcyc, secondly, the plot (b) shows ζcut, thirdly the plot (c) depicts productivity (pr) and the last plot (d) displays yield (y) represents as a functions of ηcore for c1,inj = 1 = c2,inj. table 1 lists all of the other parameter values that were considered for both isothermal and non-isothermal operations, as presented in figure 5. in figure 7, we discussed a performance process to the assessment for both conditions of isothermal and non-isothermal. however, here we change the magnitude of ∆ha from ∆ha = −10 kj/mol to ∆ha = −40 kj/mol over ηcore. the cycle and cut times are both decreasing, as can be seen from varying 36 to 12 and 97 to 10, respectively, when we get away from particles that are completely porous (i.e. ηcore=0.0) to the particles of core-shell (i.e. ηcore=0.8). similarly, the performance ascends upward and drops thereafter around ηcore = 0.74. moreover, the yield rises upward continuously. however, significant deviations in the cycle time and productivity profiles of nonisothermal and isothermal operations are detected. in the case of non-isothermal operating conditions, the reduction in cycle time decelerates slightly, but yield and productivity improve. 6. conclusion the effects of temperature variations were theoretically investigated on fixed-bed columns filled with core-shell particles. for that purpose, a nonlinear 1d-lkm was developed and solved numerically. it was found that rapid and better separations of complex samples can be obtained in columns filled with core-shell particles. the findings showed that profiles with sharper peaks and shorter residence times are produced by core radius fractions with higher values. thus, the diffusion path within the adsorbents was reduced, which boosted the column’s efficiency. furthermore, the numerical results show that heat and concentration front interactions were investigated, which indicates an increase in the process performance assessment. consequently, hplc can be optimized by using core-shell particles with a sufficient core radius fraction loaded in the column. 8 ahmad et al. / j. nig. soc. phys. sci. 5 (2023) 1350 9 0 0.2 0.4 0.6 0.8 1 η core 5 10 15 20 25 30 35 40 ζ c y c non-isothermal isothermal (a) 0 0.2 0.4 0.6 0.8 1 η core 0 20 40 60 80 100 ζ c u t non-isothermal isothermal (b) 0 0.2 0.4 0.6 0.8 1 η core 0 1 2 3 4 5 6 p r ×10 -3 non-isothermal isothermal (c) 0 0.2 0.4 0.6 0.8 1 η core 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 y non-isothermal isothermal (d) figure 7: isothermal and non-isothermal process comparison for the process performance assessment. specifically, the value for ∆ha = −40 kj/mol is taken for the process of non-isothermal operating condition. in the first plot (a) presents ζcyc, moving to second plot (b) shows ζcut, as well as the third plot (c) depicts productivity (pr) and the last plot (d) displays yield (y) represents as functions of ηcore for c1,inj = 1 = c2,inj. table 1 lists all of the other parameter values that were considered acknowledgements the authors would like to express their sincere thanks for the financial support given by the national mathematical centre abuja, nigeria. references [1] g. guiochon, a. felinger, d. g. shirazi & a. m. katti, fundamentals of preparative and nonlinear chromatography, elsevier academic press, new york (2006). 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[32] t. d. vu & a. seidel-morgenstern, “quantifying temperature and flow rate effects on the performance of a fixed-bed chromatographic reactor”, journal of chromatography a 1218 (2011) 8097. 10 j. nig. soc. phys. sci. 5 (2023) 1392 journal of the nigerian society of physical sciences model fitness and predictive accuracy in linear mixed-effects models with latent clusters waheed b. yahyaa, yusuf bellob,∗, abdulrazaq abdulraheemc adepartment of statistics, , university of ilorin, p.m.b. 1515, ilorin, kwara state, nigeria. bdepartment of statistics, federal university, dutsin-ma, p.m.b. 5001, dutsin-ma, katsina state, nigeria. cdepartment of statistics and mathematical sciences, kwara state university, malete, p.m.b. 1530, ilorin, kwara state, nigeria. abstract in clustered data, observations within a cluster show similarity between themselves because they share common features different from observations in the other clusters. in a given population, different clustering may surface because correlation may occur across more than one dimension. the existing multilevel analysis techniques of the primal linear mixed-effect models are limited to natural clusters which are often not realistic to capture in real-life situations. therefore, this paper proposes dual linear mixed models (dlmms) for modeling unobserved latent clusters when such are present in data sets to yield appreciable gains in model fitness and predictive accuracy. the methodology explored the development and analysis of the dual linear mixed models (dlmms) based on the derived latent clusters from the natural clusters using multivariate cluster analysis. a published data set on political analysis was used to demonstrate the efficiency of the proposed models. the proposed dlmms have yielded minimum values of the models’ assessment criteria (akaike information criterion, bayesian information criterion, and root mean squared error), and hence, outperformed the classical plmms in terms of model fitness and predictive accuracy. doi:10.46481/jnsps.2023.1437 keywords: clustered data, primal and dual clusters, linear mixed-effects models, model fitness, predictive accuracy article history : received: 05 march 2023 received in revised form: 13 may 2022 accepted for publication: 14 may 2022 published: 11 june 2023 c© 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: o. adeyeye 1. introduction the multilevel modeling technique follows a similar process involved when fitting the generalized linear model [1]. in particular, a linear mixed model (lmm) is one of the approaches in modeling normally distributed clustered data [2]. in clustered data, observations within a cluster show similarity between themselves because they share common features different from observations in the other clusters. in a given popu∗corresponding author tel. no: +2348134983581 email address: ybello@fudutsinma.edu.ng (yusuf bello) lation, different clustering may surface because correlation may occur across more than one dimension [3]. they further argued that clustering is in essence a design problem, either a sampling design or an experimental design issue. even if data is collected in an unclustered way, there is still natural clustering in the population. as an illustration from nigeria ′ s crime analysis, the initial dataset comprised 36 states grouped into six geo-political zones and 12 police zonal commands that share spatial and socioethnic similarities. however, the optimal number of clusters provided new structure classifications based on crime rates sim1 yahya et al. / j. nig. soc. phys. sci. 5 (2023) 1392 2 ilarities different from the initial spacial and socio-ethnic similarities [4]. it is posited here that the observations in the newly formed clusters based on multivariate clustering similarities are more correlated than in the natural clusters based on sampling and experimental design similarities. the former would better account for the differences between the clusters and improve model fitness and predictive accuracy. the natural clusters and latent clusters are respectively described as ‘primal clusters’ and ‘dual clusters’. the linear mixed-effects models (lmems or simply lmms) on the primal clusters and dual clusters are respectively described as primal linear mixed models (plmms) and dual linear mixed models (dlmms). this paper proposes efficient dlmms for modeling data with latent clusters with appreciable gains in model fitness and predictive accuracy. 2. the concept of lmems on latent clusters and model assessment criteria the general concept of lmems on latent clusters is to maximize correlation of observations within clusters, model fitness and predictive accuracy. the latent clusters were formed from natural clusters using the multivariate cluster analysis. both the natural and dual clusters contain the same observations, although the cluster structures differ. the argument for comparing models formed from same data set with differing data structures was demonstrated by [14]. agglomerative algorithm is a common approach in cluster analysis for classifying observations that share common properties into groups. the algorithm starts by calculating the distances between all pairs of observations followed by stepwise agglomeration of close observations into groups. euclidean distance is the most commonly used distance measure in numerical data, while ward method is the most frequently used linkage method [5]. lmm is a linear model with an extension of accounting for dependency among clustered observations. in biological and social sciences, a model-based cluster analysis utilizes lmm in the grouping of individuals into one of two or more clusters according to their longitudinal behaviour similarities. [6, 7]. in contrast to those studies that utilize expectation-maximization algorithms in cluster formations, this study conjoins lmems and multivariate cluster analysis to develop efficient techniques for modeling unobserved latent groupings in a data set. the degree of clustering in a data set is measured by the intraclass correlation (icc). the icc is the proportion of total variance in the data that is due to the clusters [8]. the argument in the multilevel analysis on latent clusters is that the increasing clustering in dlmm would simultaneously reduce the indices of the model assessment criteria. many indices are available to measure the performances of competing models [9], however, the models’ assessment criteria in this work are the root mean squared error (rmse), akaike information criterion (aic) and the bayesian information criterion (bic). the rmse indicates the absolute fit of the model to the data. smaller values of rmse indicate better fit results. the lmm improves model fitness and predictive performance, and this is because a multilevel model produces fitted values, ŷ, that are on the average closer to the observed y than those obtained by fitting simply the fixed part of the model. again, even in the multilevel analysis, when the estimated random effects tend to be biased towards zero; it pulls the fitted values in the direction of those of the fixed part of the model that results in bias estimates. furthermore, simpler models such as random intercept models produce larger bias relatively than the complex models such as random intercept and slope models [10]. therefore, by analogy, the more the estimated random effects tend to be larger, the more the ŷ moves closer to y, and hence the lower the rmse. the simplest information criterion widely applicable to nonnested models is the aic [11, 12]. this traditional aic is not appropriate in clustered data, and therefore marginal aic (maic) is the most widely used in model selection in lmms [12]. a related criterion to the maic in their marginal likelihoods is the bic [13]. the presence of random effects in lmms results in smaller aic and bic than in the lms [14]. 3. the linear mixed model consider a vector y of data from j clusters, the lmm as define by [2, 15] is y j = x jβ + z jυ j + � j (1) where y j is the n j vector for cluster j, where j = 1, 2, · · · , j is the cluster index, β is the p-vector of fixed effects, υ j is the qvector of random effects for cluster j, x j and z j are respectively the n j × p and n j × q matrices of covariates for the fixed and random effects of full rank. it is assumed that υ j and � j follow independent and multivariate gaussian distributions such that [16]: [ υ j � j ] ∼ n ([ 0 0 ] , [ t 0 0 σ2 i ]) (2) were t is q×q positive definite covariance matrix of the random effects, υ j is assumed independently for each j, � j associated with different clusters are assumed independent of each other, and that � j is assumed independent of υ j [15]. in a marginal model y j = n(x jβ, v j) v j = z jt z ′ j + σ 2 i (3) with a marginal likelihood given as l(y j/β̂, θ̂) = − 1 2 log|v̂ j| − 1 2 (y j − x jβ̂) ′ v̂ j −1 (y j − x jβ̂) (4) 3.1. cluster effects the cluster effect or the dependency among clustered observations is measured by the icc, and is defined as icc = σ2υ σ2υ + σ 2 e (5) where σ2υ and σ 2 e are random effects variance and random error variance respectively. 2 yahya et al. / j. nig. soc. phys. sci. 5 (2023) 1392 3 to show that the cluster effect is higher in the dual clusters than in the primal clusters, that is, icc(d) > icc(p): let p and d describe plmm and dlmm respectively, σ2e(p) = σ 2 e, σ 2 e(d) = σ 2 e − δ1, σ 2 υ(p) = σ 2 υ, σ 2 υ(d) = σ 2 υ + δ2, where δ2 ≥ δ1, δ2 and δ1 are small increment of random effects variance and decrement random error variance, respectively,. case 1: if δ2 = δ1, such that δ2 −δ1 = 0, then icc(d) − icc(p) = σ2υ + δ2 (σ2υ + δ2) + (σ2e −δ1) − σ2υ σ2υ + σ 2 e = σ2υ + δ2 (σ2υ + σ2e ) + (δ2 −δ1) − σ2υ σ2υ + σ 2 e = σ2υ + δ2 (σ2υ + σ2e ) − σ2υ σ2υ + σ 2 e = δ2 (σ2υ + σ2e ) (6) since (6) results in a positive difference, then icc(d) > icc(p) case 2: if δ2 > δ1, such that δ2 −δ1 = δ3 > 0, then icc(d) − icc(p) = σ2υ + δ2 (σ2υ + δ2) + (σ2e −δ1) − σ2υ σ2υ + σ 2 e = σ2υ + δ2 (σ2υ + σ2e ) + (δ2 −δ1) − σ2υ σ2υ + σ 2 e = σ2υ + δ2 (σ2υ + σ2e + δ3) − σ2υ σ2υ + σ 2 e = σ2υ + δ2 (σ + δ3) − σ2υ σ , where σ = σ2υ + σ 2 e = δ2σ 2 e + δ1σ 2 υ σ(σ + δ3) (7) since (7) results in a positive difference, then icc(d) > icc(p) 3.2. root mean squared error the rmse is the difference between observed data and the predicted values from the model, and it is defined as rms e = √√ ∑j j=1 ∑n j i=1(yi j − ŷi j) 2∑j j=1 n j (8) where j is the number of clusters, n j is the number of observations in the jth cluster, yi j and ŷi j are the ith observed and estimated y in jth cluster, respectively [17]. 3.3. marginal akaike information criterion the commonly used information criterion is the aic [11]. this criterion which is based on kullback-leibler distance is defined as aic = −2log[ f (y/ψ̂(y))] + 2k where f (y/ψ̂(y)) is the maximized likelihood, and k is the number of parameters. this aic is not appropriate in clustered data, and hence the maic is widely used in the clustered data [12]. the maic in the lmm uses the likelihood of the implied marginal model y ∼ n(xβ, v ) with v = in + zt z ′ . the number of estimable parameters then is p + q, with β = (β1, · · · ,βp) and q the number of unknown parameters θ in v . thus, the maic is defined as maic = −2log[ f (y/β̂, θ̂)] + 2( p + q) (9) where f (y/β̂, θ̂) is the maximized marginal likelihood. however, the maic is positively biased, and favours smaller models without random effects [18]. 3.4. bayesian information criterion the is obtained by taking the maic (9) and replacing the constant 2 in the penalty by log(n) to obtain bic = −2log[ f (y/β̂, θ̂)] + log(n)( p + q) (10) this definition ensures that bic bears the same relationship to maic for model (1) as bic bears to aic in regression and so should inherit some of its properties [13]. 3.5. cluster effects on model assessment criteria cluster effects in mixed models are explained by the random effects variance of the models, and including the random effects has an effect on the covariance matrix, v j. as an illustration, consider a random intercept model from a data where five observations are taken on each cluster, so that n j = 5 for all j. therefore, z j is a matrix of dimension 5 × 1 and r j = σ2×i5×5. then v j =  1 1 1 1 1  ×σ2υ × ( 1 1 1 1 1 ) + σ2 ×  1 0 · · · · · · 0 0 1 . . . . . . 1 . . . . . . 1 0 0 · · · · · · 0 1  =  σ2 + σ2υ σ 2 υ σ 2 υ σ 2 υ σ 2 υ σ2υ σ 2 + σ2υ σ 2 υ σ 2 υ σ 2 υ σ2υ σ 2 υ σ 2 + σ2υ σ 2 υ σ 2 υ σ2υ σ 2 υ σ 2 υ σ 2 + σ2υ σ 2 υ σ2υ σ 2 υ σ 2 υ σ 2 υ σ 2 + σ2υ  (11) the elements in the diagonal, σ2υ + σ 2, are correlations between two observations from the same cluster, and the elements at off diagonal, σ2υ, are the covariances between any two units on the same cluster. by relating the two terms, the intraclass correlation between two observations from the same cluster is σ2υ/(σ 2 υ + σ 2) [14]. denoting v j as v j(p) and v j(d) for the plmm and dlmm respectively, therefore, if the icc(d) > icc(p) as in (6) and (7), then v j(d) > v j(p). 3 yahya et al. / j. nig. soc. phys. sci. 5 (2023) 1392 4 from (4), we respectively ascribe the marginal likelihood for the plmm and dlmm as l(y j/β̂, θ̂)(p) and l(y j/β̂, θ̂)(d), such that l(y j/β̂, θ̂)(p) = − 1 2 log|v̂ j(p)|− 1 2 (y j−x jβ̂) ′ v̂−1j(p)(y j−x jβ̂)(12) and l(y j/β̂, θ̂)(d) = − 1 2 log|v̂ j(d)|− 1 2 (y j−x jβ̂) ′ v̂−1j(d)(y j−x jβ̂)(13) similar to the basic concept of fraction that a negative fraction increases with the increase in the denominator, the second term in (13), −12 (y j − x jβ̂) ′ v̂−1j(d)(y j − x jβ̂), is relatively higher than in (13) because of the increase of the inverse of the matrix v̂−1j(d) relative to v̂ −1 j(p). again, in the basic concept of logarithm that a negative logarithm decreases with the increase of a number, the first term in (13), −12 log|v̂ j(d)|, is relatively lower than in (12) because of the increase of the inverse of v̂ j(d) relative to v̂ j(p). although, the first term and second term in (13) decrease and increase respectively, the increase outweights the decrease, such that l(y j/β̂, θ̂)(d) > l(y j/β̂, θ̂)(p) (14) the presence of negative sign in the −2log[ f (y/β̂, θ̂)] for the information criteria in (9) and (10) has changed the direction of the inequality in (14), such that l(y j/β̂, θ̂)(d) < l(y j/β̂, θ̂)(p). the p and q are the same in both the plmm and dlmm, and therefore maic(d) < maic(p) and mbic(d) < mbic(p) (15) the lmm improves model fitness and predictive performances because it incorporates clustering effects when estimating the fixed parameter. this adjustment enhances it to produce fitted values, ŷ, that are on the average closer to the observed y than those obtained by fitting simply the fixed part of the model [10]. in our proposal, dlmm has higher clustering effect than plmm that enhances it to produce fitted values, ŷ, that are on the average closer to the observed y than those produced by the plmm. 4. cluster algorithm: the agglomerative algorithm the agglomerative procedure depends on the definition of the distance between two clusters. for a particular case where metric a = (s −1x1 x1, · · · , s −1 xp xp ) is used for the standardization of the variables, the euclidean distance di j between two cases i and j with variable values xi = (xi1, xi2, · · · , xip), x j = (x j1, x j2, · · · , x jp, ) is defined by di j =  p∑ k=1 (xik − x jk)2 s xk xk  1 2 where s xk xk is the variance of the kth component [19]. ward algorithm computes the distance between groups and joins the ones that do not increase a given measure of heterogeneity “too much”so the resulting groups are as homogeneous as possible. if two objects or groups say, p and q, are united, one computes the distance between this new group (object) p + q and group r using the following distance function d(r, p + q) = nr + np nr + np + nq d(r, p) + nr + nq nr + np + nq d(r, q)− nr nr + np + nq d(p, q) (16) the heterogeneity of group r is measured by the inertia inside the group. this inertia is defined as ir = 1 nr ∑nr i=1 d 2(xi, x̄r) [20]. 5. illustration and analysis a published data set on political analysis was used to demonstrate the efficiency of the proposed models. the dataset dcese provided with the ceser r package came from [21]. it contains information on 299 (i = 1, 2, · · · , 299) observations across 47 countries ( j = 1, 2, · · · , 47). the outcome variable is the effective number of electoral parties (enep). the explanatory variables are the number of presidential candidates (enpc), the proximity of presidential and legislative elections (proximity); the effective number of ethnic groups (eneg), the logarithm of average district magnitudes (logmag), and an interaction term between the logarithm of the district magnitude and the number of ethnic groups (logmag eneg = logmag × eneg). 5.1. comparison between primal and dual linear mixed models we begin with a preliminary comparison of the plmm and dlmm using primal and dual cluster data sets with j = 47 number of groups, and subsequently test the significance of the comparison. the comparison is in terms of the variancecovariance components and their impact on model fitness and predictive accuracy. the summary outputs of the models are presented in table 1. it reveals from the summary in table 1 that while σ2e is higher under plmm, the σ2υ and icc are relatively higher under dlmm. there is a 61 percent decrease of σ2e from plmm to dlmm, and respectively 64 and 38 percent increase in σ2υ and icc from plmm to dlmm. the aic, bic and rmse are lower in dlmm than under plmm by 18, 17 and 38 percent, respectively. hence, the proposed dlmm has increased the homogeneity of the observations within clusters and the heterogeneity of the clusters, which in turn increased the model fitness and predictive accuracy. the plmm and dlmm in table 1 are described as ‘full models’ because they compose of significant and nonsignificant explanatory variables. we shall now determine if we can obtain similar gains in the model assessment criteria when only significant variables are included in the models. the 4 yahya et al. / j. nig. soc. phys. sci. 5 (2023) 1392 5 table 1. summary outputs for the lm, plmm and dlmm clustering estimate effect lm plmm dlmm covariate lm plmm dlmm σ2υ 1.8460 5.0900 intercept 1.2374 3.2081 4.5972 σ2e 1.4790 0.5823 enpc 0.8636 0.5092 0.1995 icc 0.5552 0.8973 proximity -0.0173 -0.1921 0.0190 aic 1168.80 1073.09 881.98 eneg -0.1208 -0.1764 -0.3091 bic 1194.70 1102.69 911.58 logmag -0.1982 -0.0956 0.0413 rmse 1.6689 1.1345 0.7024 logmag eneg 0.3663 0.0652 0.0274 models with only significant variables are described as ‘reduced models’. the enpc is the only significant variable in both the plmm and dlmm. the summary of the reduced models is in table 2. the icc in dlmm has increased by 38 percent from plmm, and this increase is the same as it was in the full model. the aic, bic, and rmse have smaller values under dlmm than under plmm by 18, 18, and 38 percent, respectively. similarly, the percentage decrease is almost the same as it was in the full model. although the magnitudes of the aic and bic have reduced when non-significant explanatory variables are excluded in both the full plmm and dlmm; however, the percentage difference between the plmm and dlmm is almost the same in both the full and reduced models. the plmm and dlmm in table 2 are random intercept models, we recast them to random intercept and slope models to assess the effects of increasing complexity in dlmms. the summary of the random intercept and slope models is in table 3. the icc in dlmm has increased by 20 percent from plmm, which is lower than in the random intercept model. the aic, bic and rmse have lower values in dlmm than under plmm by 17, 17 and 32 percents, respectively. a similar percentage differences are recorded between the plmm and dlmm as were in the random intercept models; however, the difference is smaller in rmse. the comparison reveals a superiority of random intercept and slope dlmm over random intercept dlmm in terms of model fitness. the comparison reveals a superiority of random intercept and slope dlmm over random intercept dlmm in terms of model fitness. this coincides with the work of [14] when random intercept and slope model has smaller value of aic than in the random intercept model. the model predictive accuracy is higher in dlmm than in plmm, and also it is higher in random intercept and slope dlmm than in random intercept dlmm. higher predictive accuracy signifies smaller rmse. the above comparison used single sample outcome, j = 47, and hence it has not satisfied statistical testing procedure. therefore, we obtained fifteen sample combinations of the plmms and the corresponding dlmms and compared their respective outcomes. some sample combinations were replicated to explore possible outcome variability. figure 1. random effects variance figure 2. random error variance 5.2. assessing clustering effects between the plmms and dlmm the icc is a function of σ2υ and σ 2 e , and they are presented in table 4 and figures 1 and 2. it shows that σ2e decreases and σ 2 υ increases significantly from plmm to dlmm. the decrease in the σ2e signifies a homogeneity of observations; that is, the increase of correlations/ dependency of observations within the dual clusters. the increase in the σ2υ signifies heterogeneity of clusters; that is, the between cluster variations. the two variance components have greatly affected the icc, which is significantly higher in the dlmms. this signifies higher grouping structure in the dual clusters, and higher clustering effects in the dlmms. 5 yahya et al. / j. nig. soc. phys. sci. 5 (2023) 1392 6 table 2. significant explanatory variable in random intercept models icc aic bic rmse plmm dlmm plmm dlmm plmm dlmm plmm dlmm 0.5603 0.8972 1066.84 876.36 1081.64 891.16 1.1360 0.7053 table 3. significant explanatory variable in random intercept and slope models icc aic bic rmse plmm dlmm plmm dlmm plmm dlmm plmm dlmm 0.7340 0.9149 1052.11 869.71 1074.31 891.91 0.9570 0.6472 table 4. random effects variance, random error variance and icc from plmms and dlmms σ2υ σ 2 e icc cluster plmm dlmm plmm dlmm plmm dlmm 40 2.3330 3.7956 1.0920 0.6377 0.6811 0.8561 41 1.3060 3.6943 1.4520 0.5564 0.4734 0.8691 42 1.7770 4.6620 1.4990 0.5310 0.5425 0.8978 43 1.9850 4.4058 1.1760 0.5158 0.6280 0.8952 43b 1.9740 5.4642 1.4920 0.6113 0.5695 0.8994 44 1.9340 4.8178 1.4780 0.5924 0.5668 0.8905 44b 1.9570 4.8289 1.4510 0.5736 0.5743 0.8938 44c 1.9490 5.6101 1.4790 0.5677 0.5685 0.9081 45 1.9130 5.1335 1.6680 0.6533 0.5343 0.8871 45b 1.8770 5.3469 1.4900 0.5747 0.5574 0.9030 45c 1.8700 5.1142 1.5880 0.5965 0.5407 0.8956 46 1.7570 4.9810 1.6820 0.6280 0.5110 0.8881 46b 1.8580 5.1988 1.4920 0.5783 0.5546 0.9000 46c 1.8800 5.3696 1.5060 0.6018 0.5552 0.8992 47 1.8460 5.0900 1.4790 0.5823 0.5552 0.8973 mean 1.8811 4.9008 1.4683 0.5867 0.5608 0.8920 figure 3. the plot of icc for the model assessment 5.3. assessing model fitness and predictive accuracy between plmms and dlmms the model assessment criteria are presented in table 5 and figures 4, 5 and 6. the dlmms have smaller aic and bic than plmms, this figure 4. the plot of aic for the model assessment indicates a significant gain in model fitness in the dlmms over plmms. in addition to the relative selection of the best-fitted model carried out using the aic and bic, we supplemented the selection with the assessment of the model ′ s predictive accuracy. the rmse is significantly lower in the dlmms, and this 6 yahya et al. / j. nig. soc. phys. sci. 5 (2023) 1392 7 table 5. model assessment criteria from plmms and dlmms aic bic rmse cluster lm plmm dlmm lm plmm dlmm lm plmm dlmm 40 915.9 806.2 720.0 940.2 834.0 747.8 1.609 0.965 0.735 41 847.5 798.0 659.4 871.4 825.2 686.6 1.568 1.117 0.677 42 1057.1 971.1 772.1 1082.3 999.9 800.9 1.670 1.144 0.671 43 1074.0 963.8 799.4 1099.5 993.0 828.6 1.564 1.011 0.663 43b 1084.2 991.8 827.6 1109.5 1020.7 856.5 1.694 1.139 0.720 44 1143.8 1048.0 860.0 1169.5 1077.5 889.5 1.675 1.137 0.711 44b 1034.4 943.0 777.4 1059.4 971.5 805.9 1.696 1.118 0.693 44c 1138.1 1041.8 848.7 1163.8 1071.0 878.1 1.681 1.136 0.696 45 1052.5 979.8 816.0 1077.5 1008.4 844.5 1.743 1.198 0.738 45b 1154.6 1060.3 866.4 1180.4 1089.8 895.9 1.672 1.141 0.699 45c 1094.9 1011.3 826.9 1120.2 1040.3 855.9 1.715 1.174 0.709 46 1064.9 993.7 818.4 1090.0 1022.4 847.1 1.732 1.205 0.723 46b 1156.4 1060.9 868.5 1182.2 1090.4 898.0 1.678 1.140 0.701 46c 1150.6 1056.9 875.8 1176.3 1086.3 905.3 1.683 1.145 0.714 47 1168.8 1073.1 882.0 1194.7 1102.7 911.6 1.669 1.135 0.702 mean 1075.8 986.6 814.6 1101.1 1015.5 843.5 1.670 1.127 0.704 figure 5. the plot of bic for the model assessment figure 6. the plot of rmse for the model assessment explains the significant gain in model predictive accuracy in the proposed dlmm over the existing plmms. 6. conclusion the paper proposed the development and analysis of dlmm on the dual clusters derived from the primal clusters. the clustering similarity in the dual clusters was based on the commonly occurring phenomenon or the experimental designs, and the similarity in the dual clusters was based on the multivariate clustering algorithms. findings revealed that observations in the dual clusters are more correlated than in the primal clusters. the proposed dlmm is relatively more efficient than the classical plmm based on the results of the models’ assessment criteria (aic, bic, and rmse) in which the dlmm yielded minimum values of the assessment criteria. therefore, the proposed dlmm outperformed the classical plmm in terms of model fitness and predictive accuracy. acknowledgment we acknowledge with thanks for the careful reading and suggestions from the referees of this paper. references [1] o. s. adesina, “bayesian multilevel models for count data”, journal of the nigerian society of physical sciences 3 (2021) 224. doi:10.46481/jnsps.2021.168 [2] n. m. laird & j. h. ware, “random-effects models for longitudinal data”, biometrika (1982) 963. 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[17] i. ercanli, a. gunlu & e. z. bas,kent, “mixed effect models for predicting breast height diameter from stump diameter of oriental beech in göldaǧ”, scientia agricola (2014). [18] s. greven & t. kneib, “on the behavior of marginal and conditional akaike information criteria in linear mixed models”, johns hopkins university, department of biostatistics working papers, paper 179. http://www.bepress.com/jhubiostat/paper179/ deposited (2008). [19] w. härdle & l. simar, applied multivariate statistical analysis, 2nd edition, springer-verlag new york (2003). [20] j. h.ward, “hierarchical grouping methods to optimize an objective function”, journal of the american statistical association 58 (1963) 236. doi.org/10.2307/2282967 [21] r. elgie, c. bucur, b. dolez & a. laurent, “proximity, candidates, and presidential power: how directly elected presidents shape the legislative party system”, political research quarterly 67 (2014) 467. doi.org/10.1177/1065912914530514 8 j. nig. soc. phys. sci. 5 (2023) 1371 journal of the nigerian society of physical sciences corrosion inhibition properties of lawsone derivatives againts mild steel: a theoretical study saprizal hadisaputra∗, lalu rudyat telly savalas chemistry education division, fkip, university of mataram. jalan majapahit 62, mataram, 83125, indonesia abstract theoretical studies have been carried out using dft, ab initio mp2 and monte carlo (mc) simulations of corrosion inhibitors from lawsone derivatives against carbon steel. the research focuses on studying the effect of substituent groups in the lawsone structure on the efficiency of corrosion inhibition in mild steel. quantum chemical parameters of lawstone inhibitors in neutral and protonated conditions have been calculated. fukui’s function analysis predicts that the active side of the inhibitor will be adsorbed on the mild steel surface. mc simulation is used to understand the adsorption patterns of lawsone compounds on metal surfaces. the organic inhibitor l-nh2 has better performance as a corrosion inhibitor for mild steel in neutral or protonated conditions. doi:10.46481/jnsps.2023.1371 keywords: lawsone, substituents, dft, mp2, monte carlo, corrosion inhibitors article history : received: 26 january 2023 received in revised form: 21 february 2023 accepted for publication: 30 march 2023 published: 14 june 2023 © 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: k. sakthipandi 1. introduction metals are damaged by corrosion as a result of corrosive environmental interactions [1, 2]. corrosion causes significant economic losses, and harms the environment [3, 4]. the corrosion inhibitors that are currently in use are inorganic and unfriendly to the environment. therefore, organic inhibitors based on eco-friendly natural ingredients prevent potential corrosion. plant extracts have reportedly been used as corrosion inhibitors that are safe for the environment [4]. the majority of natural materials are highly effective inhibitors. it is because natural product molecules like alkaloids and flavaloids, which are sources of π-electrons, also include heteroatoms n, o, and s ∗corresponding author tel. no: email address: rizal@unram.ac.id (saprizal hadisaputra) [4]. the natural chemicals that have adsorbed on the surface of the metal and shield it from corrosion attack are responsible for the high value of corrosion inhibition efficiency. el-etre et al. investigated lawsone as corrosion inhibitors on metals in various environments, including as acids, in experimental study that was previously published. the figure for inhibitory efficiency that was highest was 95.78% for carbon steel, followed by 93.44% for zn in a nacl media and 88.77% for ni in an hcl medium [5,6]. in a different investigation, dananjaya et al. evaluated the lawstone as a corrosion inhibitor on mild steel in an acidic media and discovered that it was 93.14% effective [5]. theoretical investigations have shown that substituents in the inhibitor structure can raise the value of inhibitory efficiency. the mechanism of corrosion inhibition has frequently been explained by theoretical investigations [7]. the challenges that 1 hadisaputra & savalas / j. nig. soc. phys. sci. 5 (2023) 1371 2 experimental research have identified have been successfully addressed by theoretical studies [8]. molecular structure can be determined using quantum chemistry techniques, which can also be used to explain reactivity and electronic structure. the experimental studies that have already been conducted can be supplemented by quantum chemical computations [9]. the current investigation focuses on the electrical characteristics of the quantum parameters of the lawone derivative. it is thought that substituents alter how far electrons shift inside the inhibitor, which in turn affects how well corrosion is inhibited. 2. methodology 2.1. quantum chemical parameters to understand experimental results and explore reaction pathways, quantum chemical simulations have been used extensively [10]. the structural significance of corrosion inhibitors and adsorption on metal surfaces has been satisfactorily described using dft [11–12,13]. gaussian 09 [14] implements for density functional theory (dft) and ab initio mp2 6-311++g (d,p) for quantum chemical calculations in the gas and solution phases. the reactivity of a molecule can be described by quantum chemical characteristics. charge population and condensed fukui function are two more variables that can be used to determine local selectivity [16]. koopmans’ theorem states that the values of ehomo and elumo are related to ionization potential i =− ehomo and electron affinity a=− elumo, respectively. the ability of an atom or collection of atoms to draw electrons toward itself is known as electronegativity χ= i+a2 . atomic resistance to charge transfer is measured by hardness (η) [18, 19]. the hardness value can be used to calculate by η= i−a2 . the quantity of electrons exchanged: ∆n= χfe−χinh2(ηfe+ηinh) , where χfe and χinh represent, respectively, the absolute electronegativities of iron and organic inhibitors. the absolute hardness of iron and organic inhibitors, respectively, are denoted by the symbols ηfe and ηinh. the number of electrons transported is determined using the theoretical values of fe = 7.0 ev and fe = 0 [20, 21]. the fukui function can be used to calculate by f+= q(n + 1) q (n) and f= q(n) – q(n-1) where the charge an atom has upon accepting electrons is denoted by q(n+1). the charge on an atom in a neutral molecular state is defined by the formula q(n). the charge left on an atom after it loses electrons is known as q(n-1) [22, 23]. 2.2. mc simulation calculations material studio 7.0 was used to do mc simulations [24– 25]. the interaction of lawsone derivatives with 100 water molecules on the surface of mild steel was simulated using mc techniques to find the configuration with the lowest adsorption energy [26]. the fe (110) field can be used to depict the surface of mild steel. because it is the most stable and has a medium atomic density, the fe(110) field is employed [27]. a 20 vacuum layer on the c axis and an 8x8 supercell were included in the simulation box (19.859002 x 19.859002 x 34.187956) used to simulate the fe(110) field [28]. the water dissolving action was simulated by adding 100 geometrically optimized water molecules, each of the organic inhibitors (l-h, l-nh2, l-och3, and l-no2), and the fe(110) surface to a simulated box using the kompass force field [25]. to simulate the actual corrosion environment, a mc simulation was performed. 3. results and discussion the lawone was evaluated as a corrosion inhibitor on mild steel in hydrochloric acid environment in earlier studies by dananjaya et al. the weight loss strategy utilized produced a 93.14% inhibitory efficiency score [5]. henna plant extract, an aromatic hydroxyl chemical, is the source of lawsone. the metal surface will be able to create a more stable structure thanks to the lawsone phenol group’s ability to transfer electrons to it. this can inhibit redox reactions and guard against corrosion attacks on metals [29]. in addition, both donor and electron withdrawing groups have the potential to affect the value of inhibition efficiency. first, technique validation was done in this theoretical study. to ensure the accuracy of the technique and the set of bases employed, method validation is done. this is done so that results from theoretical and experimental investigations can be compared. figure 1 depicts the lawsone’s geometric structure. it can be contrasted with the findings of theoretical research and an experimental x-ray study that salunke-gawali et al. [30] previously reported. table 1 shows that there is a 0.0504-unit difference in the bond distance. the basis sets are suitable for application since the results of the variations in the lawsone’s binding distances are fairly modest. the surfaces of mild steel (fe) can adsorb lawsone compounds and substituents (nh2, och3, and no2) under neutral or protonated circumstances [31]. electron transfer is investigated by demonstrating the characteristics of molecular orbitals [15]. ehomo is typically correlated with an organic inhibitor’s ability to provide metals with electrons [32]. table 3 demonstrates that the organic inhibitor l-nh2 has a larger ehomo value and a tendency to be able to donate electrons to fe metal, as measured by -8.6774 ev. the values for electron transport are l-nh2, l-och3, l-h, and l-no2, in that order. as opposed to organic inhibitors l-h, l-och3, and l-no2, l-nh2 is expected to have a higher level of inhibitory efficiency. in addition to accepting electrons from metal d-orbitals, which results in the creation of back bonds, the value of inhibitory efficiency can also be acquired by donating electrons to the empty d-orbitals of metal ions [33]. as a result, accepting electrons from the empty d-orbitals of metals is made easier by a lower elumo value. table 3 shows l-nh2 had the greatest elumo at 0.1850 ev and l-no2 had the lowest at -0.4789 ev. these findings suggest that the organic inhibitors of l-no2 are more likely to receive electrons from the fe d orbital, predicting a decline in efficiency. the reactivity of atoms and molecules can be characterized by their ionization potential [15]. due to the atoms’ low ionization potential, they are able to donate electrons from organic inhibitors to the metal surface by simply releasing their outer electrons. the high ionization potential value indicates that electrons do not easily escape from the outer shell, which means that there is difficulty in the process of transferring electrons 2 hadisaputra & savalas / j. nig. soc. phys. sci. 5 (2023) 1371 3 figure 1: structure of lawsone (r= -nh2, och3, no2) table 1: comparison of the crystal structure of the lawone compound in the experimental study [30] and the theoretical study of dft/6-311++g (d,p) bond (å) exp [30] theory bond (å) exp [30] theory c1-o1 1.220 1.22029 c5-c6 1.394 1.39450 c2-o2 1.264 1.34507 c5-h5 0.950 1.08516 c4-o3 1.259 1.22792 c6-c7 1.390 1.39891 c1-c9 1.483 1.48758 c6-h6 0.950 1.08654 c1-c2 1.530 1.50585 c7-c8 1.389 1.39335 c2-c3 1.388 1.35348 c7-h7 0.950 1.08638 c3-c4 1.407 1.46702 c8-c9 1.393 1.39914 c3-h3 0.950 1.08799 c8h8 0.950 1.08522 c4-c10 1.491 1.49688 c9-c10 1.397 1.40743 c5-c10 1.392 1.39639 table 2: quantum chemical characteristics (in ev) of organic inhibitors in gaseous media estimated by dft and mp2 using a 6-311++g(d,p) level of theory inhibitors ehomo elumo ∆e i a χ η ∆n l-h dft/b3lyp -7.5272 -3.4436 -4.0836 7.5272 3.4436 5.4854 2.0418 0.3709 ab initio mp2 -9.8483 0.1905 -10.0388 9.8483 -0.1905 4.8289 5.0194 0.2163 l-nh2 dft/b3lyp -6.2537 -3.2066 -3.0471 6.2537 3.2066 4.7302 1.5236 0.7449 ab initio mp2 -8.7414 0.3086 -9.0500 8.7414 -0.3086 4.2164 4.5250 0.3076 pp-och3 dft/b3lyp -6.6268 -3.3718 -3.2550 6.6268 3.3718 4.9993 1.6275 0.6147 ab initio mp2 -9.2486 0.1502 -9.3988 9.2486 -0.1502 4.5492 4.6994 0.2608 pp-no2 dft/b3lyp -7.8570 -4.1963 -3.6607 7.8570 4.1963 6.0266 1.8304 0.2659 ab initio mp2 -10.3033 -0.6196 -9.6837 10.3033 0.6196 5.4615 4.8419 0.1589 from organic inhibitors to the fe surface [34]. table 3 shows the low ionization potential value on l-nh2 is 8.6774 ev while the highest ionization potential value is on l-no2 of 9.9471 ev. the organic inhibitor l-nh2 is more reactive to metal fe so that it can cause a strong interaction between organic inhibitors and metal fe. it can be predicted that the organic inhibitor l-nh2 can increase the value of inhibition efficiency. pan et al. have reported that the experimental ionization potential value for 1,4naphthaquinone using ir ld/vuv pims was found to be 9.52 ev [35]. the results obtained were almost close to the ionization potential value of the organic inhibitor l-h (lawsone) of 9.7926 ev using the ab initio method in aqueous media. therefore, the ab intio mp2 at 6-311++g (d,p) method is valid to use. 3 hadisaputra & savalas / j. nig. soc. phys. sci. 5 (2023) 1371 4 table 3: the quantum chemical characteristics (in ev) of organic inhibitors in aqueous environments determined using dft and mp2 with a 6-311++g(d,p) level of theory inhibitors ehomo elumo ∆e i a χ η ∆n l-h dft/b3lyp -7.4219 -3.5035 -3.9184 7.4219 3.5035 5.4627 1.9592 0.3923 ab initio mp2 -9.7926 0.1162 -9.9088 9.7926 -0.1162 4.8382 4.9544 0.2182 l-nh2 dft/b3lyp -6.1835 -3.3157 -2.8678 6.1835 3.3157 4.7496 1.4339 0.7847 ab initio mp2 -8.6774 0.1850 -8.8625 8.6774 -0.1850 4.2462 4.4312 0.3107 l-och3 dft/b3lyp -6.6273 -3.4733 -3.1541 6.6273 3.4733 5.0503 1.5770 0.6182 ab initio mp2 -9.2483 0.0465 -9.2949 9.2483 -0.0465 4.6009 4.6474 0.2581 l-no2 dft/b3lyp -7.8475 -4.0904 -3.7571 7.8475 4.0904 5.9690 1.8785 0.2744 ab initio mp2 -9.9471 -0.4789 -9.4682 9.9471 0.4789 5.2130 4.7341 0.1887 table 4: the quantum chemical characteristics (in ev) of protonated organic inhibitors determined using dft and mp2 in a 6311++g(d,p) in gaseous media inhibitors ehomo elumo ∆e i a χ η ∆n pronated l-h dft/b3lyp -11.6394 -8.2804 -3.3590 11.6394 8.2804 9.9599 1.6795 -0.8812 ab initio mp2 -13.6748 -4.7081 -8.9667 13.6748 4.7081 9.1915 4.4834 -0.2444 pronated l-nh2 dft/b3lyp -10.7471 -8.1474 -2.5998 10.7471 8.1474 9.4473 1.2999 -0.9413 ab initio mp2 -13.1058 -4.7557 -8.3501 13.1058 4.7557 8.9308 4.1750 -0.2312 pronated l-och3 dft/b3lyp -10.9319 -8.2570 -2.6749 10.9319 8.2570 9.5945 1.3374 -0.9699 ab initio mp2 -13.4060 -4.8469 -8.5591 13.4060 4.8469 9.1264 4.2795 -0.2484 pronated l-no2 dft/b3lyp -12.0519 -8.7101 -3.3418 12.0519 8.7101 10.3810 1.6709 -1.0117 ab initio mp2 -14.0941 -5.1383 -8.9558 14.0941 5.1383 9.6162 4.4779 -0.2921 table 5: quantum chemical characteristics (in ev) of protonated organic inhibitors in aqueous media as computed using dft and mp2 in a 6-311++g(d,p) level of theory inhibitors ehomo elumo ∆e i a χ η ∆n pronated l-h dft/b3lyp -8.1177 -4.6654 -3.4523 8.1177 4.6654 6.3915 1.7262 0.1762 ab initio mp2 -10.2720 -1.0814 -9.1907 10.2720 1.0814 5.6767 4.5953 0.1440 pronated l-nh2 dft/b3lyp -7.0450 -4.6338 -2.4112 7.0450 4.6338 5.8394 1.2056 0.4813 ab initio mp2 -9.4592 -1.2120 -8.2472 9.4592 1.2120 5.3356 4.1236 0.2018 pronated l-och3 dft/b3lyp -7.4235 -4.7650 -2.6586 7.4235 4.7650 6.0943 1.3293 0.3407 ab initio mp2 -9.9305 -1.3124 -8.6181 9.9305 1.3124 5.6215 4.3091 0.1600 pronated l-no2 dft/b3lyp -8.4192 -5.0341 -3.3851 8.4192 5.0341 6.7267 1.6925 0.0807 ab initio mp2 -10.4484 -1.4066 -9.0418 10.4484 1.4066 5.9275 4.5209 0.1186 theoretical values of fe = 7 ev and fe = 0 ev can be used to calculate the amount of electron transfer from organic inhibitors to fe metal surfaces. the inhibition efficiency value obtained from the electron donation of organic inhibitors to fe metal coincides with the electron transfer value [37]. organic inhibitors’ ability to give electrons can do so to mild steel’s surface (fe 110). according to table 2-5, the values for electron transfer are as follows: l-nh2 > l-och3 > l-h > l-no2. table 3 shows that the greatest electron transfer value in lnh2 was measured at 0.3107 ev using the mp2/6-311++g 4 hadisaputra & savalas / j. nig. soc. phys. sci. 5 (2023) 1371 5 figure 2: homo, lumo orbitals, mep and esp of the studied molecules (d,p) technique. these findings provide credence to the idea that organic inhibitors can bind to metal surfaces during electron donor-acceptor interactions. l-nh2 is therefore expected to be the best corrosion inhibitor. the homo, lumo, esp, and mep molecular orbital coefficients can represent the region around a molecule, and the electron probability density can provide information about the size and electrophilicity of the molecule [38]. figure 2 illustrates the visualization of homo, lumo, esp, and mep using the dft/6-311++g (d,p) approach to describe the mechanism of adsorption on metal surfaces. in order to identify the reactive side of a molecule, esp visualization offers a visual way for comprehending regions that have higher electron density than other regions. in esp, the color red denotes the highest negative electrostatic potential, blue denotes the most positive electrostatic potential, and green denotes zero electrostatic potential [15]. red, orange, yellow, green, and blue are in ascending order of electrical potential [31]. on the oxygen atom of the lawstone carbonyl in the organic inhibitor l-nh2, there is a yellow hue. it is possible that oxygen atoms will get up on the surface of the metal fe (110) by adsorptive means. however, the fukui function is a more accurate way to assess the electron density of the area of the molecule that is exposed to electrophilic or nucleophilic attack [39]. fukui function was proposed by parr and yang 1984 [40] as a measure of local reactivity indicating the presence of reactive site regions in molecules such as nucleophilic and electrophilic attack [41]. the preferred site for nucleophilic attack is the atom in a molecule that has the maximum functional value (f+) due to its association with elumo. 5 hadisaputra & savalas / j. nig. soc. phys. sci. 5 (2023) 1371 6 table 6: fukui functional analysis of l-h, l-nh2, l-och3, pp-no2 molecules l-h nn n+ f+ fc1 -0.3892 -0.3764 -0.3521 0.0242 0.0129 c2 -0.3095 -0.2896 -0.2386 0.051 0.0199 c3 0.1960 0.2655 0.3007 0.0415 0.0695 c4 0.2357 0.1893 0.2046 0.0153 -0.0464 c5 0.3801 0.3155 0.3413 0.0258 -0.0645 c6 0.1601 0.2399 0.2834 0.0435 0.0798 c7 -0.7657 -0.6214 -0.5695 0.0518 0.1443 c8 0.3974 0.4442 0.5449 0.1007 0.0467 c9 0.0173 0.0293 0.0270 -0.0023 0.0119 c10 -0.8073 -0.7433 -0.7462 -0.0030 0.0641 o11 -0.3682 -0.2058 -0.1222 0.0836 0.1624 o12 -0.3918 -0.2495 -0.1551 0.0944 0.1422 o13 -0.2239 -0.1801 -0.0594 0.1207 0.0437 l-nh2 nn n+ f+ fc1 -0.3939 -0.3661 -0.3428 0.0232 0.0278 c2 -0.3352 -0.3209 -0.3015 0.0194 0.0144 c3 0.1255 0.2094 0.2422 0.0329 0.0839 c4 0.1674 0.1023 0.1057 0.0033 -0.0651 c5 0.7316 0.7209 0.7438 0.0229 -0.0107 c6 0.2215 0.2871 0.3259 0.0388 0.0656 c7 -1.0085 -0.8464 -0.8297 0.0167 0.1621 c8 0.6826 0.6309 0.6519 0.0210 -0.0517 c9 -0.0923 -0.0581 0.0037 0.0618 0.0342 c10 -0.7777 -0.7120 -0.7230 -0.0110 0.0657 o11 -0.3733 -0.2305 -0.1309 0.0996 0.1428 o12 -0.4217 -0.2731 -0.2040 0.0692 0.1486 o13 -0.2321 -0.1923 -0.0572 0.1351 0.0398 n14 -0.5461 -0.4994 -0.3365 0.1629 0.0467 l-och3 nn n+ f+ fc1 -0.3773 -0.3638 -0.3401 0.0237 0.0135 c2 -0.4298 -0.4051 -0.3838 0.0213 0.0247 c3 0.1725 0.2412 0.2745 0.0334 0.0687 c4 0.3175 0.2375 0.2323 -0.0052 -0.0800 c5 0.5314 0.4988 0.4992 0.0004 -0.0327 c6 0.1279 0.1853 0.2101 0.0247 0.0574 c7 -0.2970 -0.1380 -0.0629 0.0751 0.1590 c8 0.1008 0.1620 0.2823 0.1202 0.0612 c9 -0.1741 -0.1835 -0.1872 -0.0037 -0.0093 c10 -0.5862 -0.5174 -0.5504 -0.0330 0.0688 o11 -0.3681 -0.2174 -0.1288 0.0886 0.1507 o12 -0.3942 -0.2484 -0.1674 0.0810 0.1458 o13 -0.2453 -0.1946 -0.0523 0.1423 0.0507 o14 -0.3666 -0.3370 -0.2289 0.1081 0.0296 c15 -0.2889 -0.2944 -0.3132 -0.0188 -0.0056 l-no2 nn n+ f+ fc1 0.3589 -0.3603 -0.3591 0.0012 -0.7192 c2 -0.3791 -0.3404 -0.2778 0.0626 0.0386 c3 0.3035 0.3356 0.3809 0.0453 0.0321 c4 0.3187 0.2703 0.2703 0.0000 -0.0484 c5 0.4743 0.4862 0.5675 0.0814 0.0118 c6 0.1983 0.2693 0.3215 0.0521 0.0711 c7 -0.6848 -0.6217 -0.5946 0.0271 0.0631 c8 0.8653 0.8424 0.8908 0.0484 -0.0229 6 hadisaputra & savalas / j. nig. soc. phys. sci. 5 (2023) 1371 7 c9 -0.3609 -0.2941 -0.2849 0.0092 0.0667 c10 -0.9140 -0.8509 -0.8439 0.0070 0.0631 o11 -0.3236 -0.1723 -0.0865 0.0858 0.1513 o12 -0.2651 -0.1732 -0.0992 0.0740 0.0919 o13 -0.2064 -0.1374 -0.0504 0.0870 0.0690 n14 -0.5459 -0.5019 -0.4841 0.0178 0.0440 o15 0.0364 0.1259 0.2000 0.0741 0.0895 o16 -0.0538 0.0282 0.0892 0.0611 0.0819 figure 3: adsorption of organic inhibitors (l-h, l-nh2, l-och3, l-no2) on ferrous metal surfaces in the mc fe(110)/inhibitor/100h2o system table 7: adsorption energy of organic inhibitors (l-h, l-nh2, l-och3, l-no2) fe(110)/inhibitor/100h2o system using mc simulation systems adsorption energy of inhibitors kcal.mol−1 adsorption energy of water kcal.mol−1 neutral inhibitor fe(110)/l-h /100h2o -129.5897925 -13.84984664 fe(110)/l-nh2 /100h2o -143.4672645 -14.48869095 fe(110)/l-och3 /100h2o -142.3615059 -13.74108729 fe(110)/ l-no2 /100h2o -113.8560443 -12.27065388 protonated inhibitor fe(110)/l-h /100h2o -134.9297789 -13.26018364 fe(110)/l-nh2 /100h2o -140.5591996 -12.53631498 fe(110)/l-och3 /100h2o -139.2186163 -12.76122573 fe(110)/ l-no2 /100h2o -88.50163792 -12.41283883 7 hadisaputra & savalas / j. nig. soc. phys. sci. 5 (2023) 1371 8 the preferred area for electrophilic attack is the atom in the molecule that has the maximum value of the fukui function (f-) because it is associated with ehomo [42]. the value of findicates the ability of an atom to donate electrons to the empty d orbitals of metal fe (110) [8]. table 7 using the dft/6311++g (d,p) method shows that the maximum f+ value of organic inhibitors l-h on atoms (c8, o12, o13), l-nh2 on atoms (o11, o13, n14), loch3 on atoms (c8, o13, o14), l-no2 on atoms (o11, o13, o15). the atom acts as an electron acceptor because of the back-donation from the fe metal surface [8]. the maximum fvalues of organic inhibitors lh, l-nh2 and l-och3 are found on atoms (c7, o11, o12), l-no2 on atoms (o11, o12, n14), respectively. each organic inhibitor on the o11 and o12 atoms tends to donate electrons to the fe (110) surface so as to form coordinate bonds [43-44]. the most effective way for characterizing the interaction between substrate and adsorbate is the mc simulation approach. the lowest desired adsorption energy arrangement for the adsorbate component on the fe(110) surface can be found using mc simulation [45]. figure 3 shows the most stable adsorption energy configuration of each lawsone derivatives under the conditions of 100 water molecules and the fe (110) surface. understanding the nature of the adsorption process can be aided by measuring the distance between the atoms in organic inhibitors and the surfaces of fe metal. the van der waals force is regarded as being the primary interaction in chemical adsorption (chemissorption) if the value of this distance is less than 3.5. these results prove that the organic inhibitor l-nh2 has a distance between oxygen atoms in o1 of 3.256 å, o2 of 3.132 å and o3 of 3.379 å and to the surface of fe (110). the carbonyl (c=o) and hydroxyl (o-h) atoms can donate electrons to the fe (110) surface to form complex compounds. since more oxygen atoms contribute as electron donors to the fe (110) metal surface, the visualization findings of the esp and fukui functions confirm this conclusion. in a mc simulation, the lowest energy system across all systems is sought after [46]. the adsorption energy is linked to the energy generated while the relaxed adsorbate is adsorbed on the substrate [47, 48]. table 7 shows that the highest negative adsorption value for the organic inhibitor l-nh2 is -142.3615 kcal/mol. this is caused by the interaction between the mesomeric effect and the nh2 substituent, which serves as an electron donor and facilitates the transfer of electrons to the vacant fe (110) orbital. in order to create stable coordination bonds, l-nh2 possesses a lone pair of electrons on the oxygen and nitrogen atoms. in order to create the most stable adsorption layer and shield mild steel from corrosion attack, l-nh2 is therefore more likely to be adsorbed on the surface of fe (110) [49, 50]. 4. conclusion the effect of substituents (nh2, och3, no2) to the lawsone structure was studied as a corrosion inhibitor in mild steel using quantum chemical calculations and mc simulations. it is possible to predict quantum chemical characteristics from 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[50] s. hadisaputra, a. a. purwoko, a. hakim, n. prasetyo & s. hamdiani, “corrosion inhibition properties of phenyl phthalimide derivatives against carbon steel in the acidic medium: dft, mp2, and monte carlo simulation studies”, acs omega 7 (2022) 33054. 9 j. nig. soc. phys. sci. 3 (2021) 132–139 journal of the nigerian society of physical sciences development of predictive model for radon-222 estimation in the atmosphere using stepwise regression and grid search based-random forest regression omodele e. olubia,b, ebenezer o. oniyaa,∗, taoreed o. owolabia a physics and electronics department, adekunle ajasin university, akungba-akoko, 342111, ondo state, nigeria. b achievers university, p.m.b 1030, owo, ondo state, nigeria. abstract this work develops predictive models for estimating radon (222rn) activity concentration in the regression (swr). the developed models employ meteorological parameters which include the temperature, pressure, relative and absolute humidity, wind speed and wind direction as descriptors. experimental data of radon concentration and meteorological parameters from two observatories of the korea polar research institute in antarctica (king sejong and jang bogo) have been employed in this work. the performance of the developed models was assessed using three different performance measuring parameters. on the basis of root mean square error (rmse), the gs-rfr shows better performance over the swr. an improvement of 64.09 % and 15.19 % was obtained on the training and test datasets, respectively at king sejong station. at the jang bogo station, an improvement of 75.04 % and 28.04 % was obtained on the training and test datasets, respectively. the precision and robustness of the developed models would be of significant interest in determining the concentration of radon (222rn) activity concentration in the atmosphere for various physical applications especially in regions where field measuring equipment for radon is not available or measurements have been interrupted. doi:10.46481/jnsps.2021.177 keywords: radon, machine learning, meteorological parameters, atmosphere article history : received: 24 march 2021 received in revised form: 15 april 2021 accepted for publication: 24 april 2021 published: 29 may 2021 ©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: o. j. abimbola 1. introduction the importance of radon (rn-222), the only gaseous member of the u-238 series, has been of interest to scientists since the twentieth century when it was first suspected to be a caus -ative agent for lung cancer among miners. the radioactive gas has been a significant subject of research among health and environmental scientists having been characterized as a ∗corresponding author tel. no: +2348035033421 email address: ebenezer.oniya@aaua.edu. (ebenezer o. oniya ) potential indoor source of air pollution. its subsequent classification as a carcinogen has led to investigation and monitoring of the indoor concentration of the gas in several countries of the world [1-8]. the source of the noble gas is from the decay of ra-226 in bedrock and soil and migrates through soil pores by gas-phase diffusion and advection to the surface and its sink process is by radioactive decay [9, 10]. due to some important characteristics of radon as a tracer of atmospheric processes, there has been a growing interest in recent decades in monitoring environmental radon. being a noble gas, it is not chemically reactive with other elements. 132 olubi et al. / j. nig. soc. phys. sci. 3 (2021) 132–139 133 its relative solubility in water and non-attachment to aerosols makes it highly insusceptible to dry or wet atmospheric removal processes. its half-life of 3.82 days is comparable to the life times of short-lived environmental pollutants (e.g nox , so2, co, o3, ch4) and atmospheric residence times of water and aerosols [10]. the noble gas has become very useful as a tracer of the influence of the terrestrial environment on air mass composition. some areas of application of ground-based radon observation include atmospheric, pollution studies and climatic studies [11-16]. observed anomalous behaviour of radon in soil and groundwater during earthquake events has been employed as a precursor for impending earthquakes [17, 18]. despite the progress that has been made in radon instrumentation, access to data on atmospheric radon concentration is still to a large extent, lacking in the public domain. africa for instance, has only one mention of a radon observatory in the literature; an ansto-developed detector installed at a global atmospheric watch (gaw ) station at cape point, south africa [10]. ground based radon measurement methods have not been applied to study atmospheric processes as have been done in europe. as a matter of fact, the only radon time series characterization to have been reported was published recently for the first time on the continent [19]. in the unavailability of measuring equipment, a theoretical approach to developing predictive models for radon concentration in the atmosphere may be a viable step in generating synthetic data for the noble gas, using machine learning to train available experimental data. theoretical models have been developed by several researchers in the literature to predict radon behaviour and concentration for various conditions and applications [17, 18, 20-22]. several studies in the literature have reported the variation of atmospheric radon and its progenies with changes in meteorological parameters like temperature, pressure, humidity and windspeed [23-25]. [26] used these meteorological variables as independent predictors in the development of a multiple linear regression model for estimation and prediction of the time series of radon and thoron progeny concentrations. random forest (rf) methodology is a machine learning technique developed by leo breiman and is useful for classification and prediction problems [27]. its algorithm operates by sampling small divisions of the data, grows a tree predictor that is randomized on each small division, then aggregates these predictors together. it applies bootstrap aggregation and random feature selection to individual classification or regression trees for prediction [28]. apart from the speed and ease of implementation of random forests, their predictions are remarkably accurate, with the ability to process a very large input data whilst dealing with overfitting. they also perform well with small to medium data [29]. their good predictive abilities have made them highly applicable to regression and classification problems in the atmospheric sciences [30-31]. the grid search (gs) is one of the algorithms for hyperparameter optimisation and tuning of models with an expectation of the most accurate results. with a specified subset of the hyperparameters space of the training algorithm, the algorithm conducts a search with the aim of producing the best combination of parameters yielding the most remarkable results. to apply a grid search, boundaries need to be specified because some parameters within the algorithm’s parameter space may contain unlimited values. the high dimensional space problem with grid search algorithms is easily resolved with parallelization of the of the process since the hyperparameters are usually not dependent on each other [32]. multiple stepwise regression is efficient in the selection of contributing factors used in establishing models that can do statistical prediction. the critical objective it sets to achieve is to discover the most cordial relationship between predictor variables that would accurately forecast the predicted variable. the regression process begins with the input of the mostly contributing predictor variable to the prediction model. additional variables are continuously added as long as they are of any essence to the regression equation. [33, 34]. this present work develops stepwise regression (swr) and grid search-based random forest regression (gs-rfr) models through which radon concentration can be estimated and predicted using much more available meteorological parameters (air temperature (at), atmospheric pressure (ap), absolute humidity (ah), relative humidity (rh), wind direction (wd) and wind speed (ws) as predictors. a comparison is also made between both models in terms of performance. the proposed model will help not only to predict radon concentration, it may also help to generate estimated or synthetic radon data that can approximate measured data, for regions that lack measuring instruments for atmospheric radon but have access to meteorological data. it will also help estimate radon data for sites where measurements have been interrupted. 2. theory 2.1. description of random forest regression a random forest is described, according to [35], to consist of n regression trees that are randomized also referred to as a family. for any individual (i-th) tree, the predicted value at the query point y can be represented as mn (x; θi , dn ), where θ1, . . . , θn are independent random variables that are not dependent on dn . resampling of the training set is first done using θ before individual trees are grown. the finite forest estimate for regression as a result of the combination of the trees is mn,n (x;θ1, ...θn , dn ) = 1 n n∑ i =1 mn (x;θi , dn ) (1) in the case of classification, the random forest classification makes use of the majority vote among the classification trees. the forest estimate for classification is mn,n (x;θ1, ...θn , dn ) = { 1 if 1n ∑n i =1 mn (x;θi ,dn )> 1 2 0 if otherwise (2) 133 olubi et al. / j. nig. soc. phys. sci. 3 (2021) 132–139 134 2.2. description of grid search optimization the implementation of the grid search technique involves upper and lower bound vectors v = v1, v2, . . . , vq and w = w1, w2, . . . , wq respectively, defined for each component of hyperparameters h = h1, h2, . . . , hq where q is the number of hyperparameters. the optimization and parameter search procedure involves taking n equally spaced points within the search space over an interval of the form [vi , wi ] which includes of vi and wi . the algorithm searches through n q possible points and a selection of the optimum values results, following the evaluation of each grid point in space [36]. 2.3. stepwise regression based on the forward and backward selection, stepwise regression is a self-determining process for in the selection of independent variables. multiple linear regression (mlr) has the form y = βo +β1 x 1 +β2 x 2 +β3 x 3 ···+βp x p +ε (3) in equation (3), y is the output variable and x 1, x 2, x 3. . . are predictor variables. βi are regression parameters, βo is an intercept and ε is the random error term. the process is summarised below; 1. if after the performance of simple multiple linear regression of n predictor variables, all the variables show remarkable significance, then the whole model containing all n variables is adopted. 2. if results show otherwise, simple n-variable linear regression is performed with each of the predictor variables and the process selects the variable which gives lowest p-value for t-test. 3. a subsequent n−1 variable regression is performed taking the selected variable in step 2 as common. 4. step 3 is repeated with each significant variable becoming added to the model in a stepwise manner. the test for significance by stepwise regression can be applied at two levels. the first being for addition of variables and the second, for removal of variables [37]. 2.4. performance measuring parameters three performance measuring parameters were used to assess the developed models namely correlation coefficient (cc), root mean square error (rmse), mean absolute error (mae). correlation coefficient is defined as c c = ∑n i =1 ( yi ∗ − y ∗ ) ( yi − y ) √∑n i =1 ( yi ∗ − y ∗ )2 √∑n i =1 ( yi − y )2 (4) where where yi ∗ and yi are the mean values of the predicted and actual outputs. rmse is defined as: r m se = √√√√ 1 n n∑ i =1 (yi − yi ∗)2 (5) n represents the number of samples contained in the dataset mae is defined as: m ae = 1 n n∑ i =1 |yi − yi ∗| (6) 3. methods 3.1. description of sites the data used in this work was published by [38], being data measured in two korea antarctic research program stations namely king sejong (ksg) and jang bogo ( jbs). measurements have been done jointly with the australian nuclear science and technology organisation (ansto). the korea polar research institute (kopri) operates the ksg station (62.217◦ s, 58.783◦ w ). the station functions as a regional world meteorological organisation (wmo) global atmospheric watch (gaw ) station. the jbs is 10 m above sea level with coordinates (74.623◦s, 164.228◦e). a detailed geographical description of the sites is seen in [39]. 3.2. radon and meteorological data at jbs, radon measurements have been made using a 1200 l two-filter dual-flow-loop radon detector custom built by ansto. installed approximately 40 m east of the main station, air is sampled at 55 l min−1 through 50 mm high-density polyethylene pipe from approximately 6 m above ground level. in order to avoid thoron (220rn; t0.5 = 55.6 s) from entering into the pipe and contaminating sampled air, a 400 l delay volume is coupled within the sampling line. at approximately 170 m from the radon detector, meteorological data was collected from a 10 m tower with instrumentation composed of a sonic anemometer, temperature and humidity probe, barometer and a windspeed logger. in post processing, all observations are aggregated to hourly values [39]. a radon detector similar in operation to that in jbs but with a volume of 1500 l was used for radon data collection in ksg with meteorological data collected from a nearby observation system [40]. the dataset used was measured between december and february with 1818 and 1955 datapoints for jbs and ksg respectively. table 1 shows the statistical analysis from jbs and ksg. the mean, standard deviation and range are presented. while the mean and range describe the content of the dataset, the correlation coefficients depict the level of linear relationship between the target and predictor variables. both tables indicate weak correlation between ar n and the descriptors indicating that a purely linear regression model may be insufficient to represent the relationship between the descriptors and target. 3.3. computational methodology 3.3.1. swr model a flow chart of the stepwise process is presented in figure 1. whenever a variable x is added in each step, all the predictor variables in the model are assessed for their significance p. if it has been reduced below a specified threshold. 134 olubi et al. / j. nig. soc. phys. sci. 3 (2021) 132–139 135 table 1: statistical analysis of dataset from jbs correlation coefficient mean standard deviation range ar n (bq/m 3) 1 0.937 0.743 5.213 ws (m/s) -0.32 3.723 3.284 17.600 wd (o ) -0.27 167.766 112.667 359.700 at (o c) 0.22 -3.055 3.814 19.300 rh (%) -0.11 57.524 16.121 72.400 ap (hpa) -0.16 982.704 7.370 36.700 ah (g/m3) 0.073 2.317 0.822 4.05 table 2: statistical analysis of dataset from ksg correlation coefficient mean standard deviation range ar n (bq/m 3) 1 63.344 30.921 314.060 ws (m/s) -0.05 7.043 3.463 18.600 wd (o ) -0.09 235.993 96.123 358.500 at (o c) 0.40 -0.023 1.447 11.600 rh (%) 0.12 84.583 8.763 41.200 ap (hpa) -0.02 987.753 9.369 43.000 ah (g/m3) 0.39 4.111 4.111 4.070 figure 1: the stepwise regression flow chart 3.4. building of gs-rfr model the computation of the gs-rfr model development was achieved using python software. the radon concentration and the descriptors, which include (air temperature (at), atmospheric pressure (ap), absolute humidity (ah), relative humidity (rh), wind direction (wd) and wind speed (ws), after randomization, was partitioned into training and testing sets in the ratio of 8:2 respectively. the rfr model development was done with the training set, while the general predictive ability of the model was assessed using the 20% test set. a helpful purpose for randomization is that it enhances computation efficiency by ensuring unbiased spread of data during both the training or testing phase. the performance algorithm was optimized through an optimum selection of hyperparameters using grid search (gs) with cross validation. table 3 below shows the hyperparameters that were tuned as suggested in the literature [41, 42]. during the hyperparameters tuning, the 5-fold cross validation was used as the fitness function. verification of the rf model with the optimum hyperparameters was carried out on the testing set. * 4. 4. results and discussion 4.1. comparison of performance between the swr and gsrfr for the two datasets, the performance of the two models developed by swr and gs-rfr is depicted in figure 1. the predictive capabilities of the two models were assessed using the performance measuring parameters: correlation coefficient (cc), root mean square error (rmse) and mean absolute error (mae). tables 4 and 5 shows the estimated predictive performance for the two regression methods based on correlation coefficient, root mean square error and mean absolute error. figure 2 compares the performance of the test set on the models developed using the data from ksg and jbs. the figures show better performance by the gs-rfr model over the more traditional swr. considering rmse, an improvement of 64.09 % and 15.19 % was obtained on the training and test datasets, respectively at ksg while at jbs, an improvement of 75.04 % and 28.04 % was obtained on the training and test datasets, respectively (table 6). the optimum hyperparameters for the rfr algorithm for each dataset is summarized in table 7. table 4. predictive performance of the two regression models in terms of correlation coefficient (cc), root mean square error (rmse) and mean absolute error (mae) for ksj dataset. the gs-rfr model presents the smallest rmse on the two datasets employed. it also achieved the highest correlation coefficient on both training and test sets. the plots in figure 3 show the correlation between predicted and measured values of radon concentration. it can be seen that the gs-rfr model 135 olubi et al. / j. nig. soc. phys. sci. 3 (2021) 132–139 136 table 3: hyperparameters description no hyperparameters definition 1 max_depth the maximum depth of decision trees (dt). 2 min_samples_split the minimum number of samples for the split 3 min_samples_leaf the minimum number of samples at the leaf node 4 n_estimators the number of trees in the forest of the model 5 max_features the number of features considered during the selection of the best splitting table 4: predictive performance of the two regression models in terms of correlation coefficient (cc), root mean square error (rmse) and mean absolute error (mae) for ksj dataset method cc rmse (bq/m3) mae (bq/m3) training test training test training test gs-rfr 0.99 0.83 8.57 20.26 5.38 14.07 swr 0.64 0.61 23.86 23.89 0.01 0.10 (a) (b) (c) (d) figure 2: performance comparison between the developed models for training (tr) and test (ts) sets on the basis of rmse on (a) ksg training dataset (b) ksg test dataset (c) jbs training dataset (d) jbs test dataset made a potential success in describing the non-linear relationship between atmospheric radon concentration and influencing meteorological parameters considering strong correlation coefficients it achieved. 5. conclusion in this work, modelling of atmospheric radon as done using the more traditional stepwise regression (swr) and a novel 136 olubi et al. / j. nig. soc. phys. sci. 3 (2021) 132–139 137 table 5: predictive performance of the two regression models in terms of correlation coefficient (cc), root mean square error (rmse) and mean absolute error (mae) for ksj dataset method cc rmse (bq/m3) mae (bq/m3) training test training test training test gs-rfr 0.98 0.82 0.17 0.46 0.11 0.32 swr 0.61 0.66 0.66 0.63 0.06 6553 table 6: improvement of gs-rfr over swr in this study jbs ksg training set test set training set test set 75.04 % 28.04 % 64.09 % 15.19 % table 7: selected optimum hyperparameters after the grid search hyperparameters jbs ksg max_depth 2000 2000 min_samples_split 2 3 min_samples_leaf 1 1 n_estimators 650 50 max_features 3 3 (a) (b) (c) (d) figure 3: performance comparison between the developed models for training (tr) and test (ts) sets on the basis of rmse on (a) ksg training dataset (b) ksg test dataset (c) jbs training dataset (d) jbs test dataset. 137 olubi et al. / j. nig. soc. phys. sci. 3 (2021) 132–139 138 grid search based random forest regression (gs-rfr). datasets from two radon stations in antarctica were used in the building of the models. important factors such as air temperature, atmospheric pressure, absolute humidity, relative humidity, wind direction and wind speed were used as predictors. comparing both models, the results show that the gs-rfr model performed better on both datasets in the training and testing phases. it presents a respective training and test improvement of 64.09 % and 15.19 % on one dataset and 75.04 % and 28.04 % on the other. atmospheric radon data, which is finding more relevance today in the atmospheric sciences, is still scarce and not readily available. the precision and robustness of the developed models would be of significant interest in determining the concentration of radon (222rn) activity concentration in the atmosphere for various physical applications especially in regions where field measuring equipment for radon is not available but have meteorological parameters are. acknowledgment the korean polar research institute is acknowledged for making the employed radon and meteorological data available online. we thank the referees for the positive enlightening comments and suggestions, which have greatly helped us in making 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[42] c. qi, q. chen, a. fourie & q. zhang, "an intelligent modelling framework for mechanical properties of cemented paste backfill", minerals engineering 123 (2018) 16. https://doi.org/10.1016/j.mineng.2018.04.010 139 j. nig. soc. phys. sci. 5 (2023) 1109 journal of the nigerian society of physical sciences effect of pre-test drying temperature on the properties of lateritic soils l. o. afolagboyea,∗, z. o. arijea, a. o. talabia, o. o. owoyemib adepartment of geology, ekiti state university, ado-ekiti, nigeria bdepartment of geology, kwara state university, malete, nigeria abstract the properties of residual soils, according to literature, are sensitive to the pre-test drying method given to the sample prior to testing. similarly, residual soils such as laterites/lateritic soils are formed under various climatic conditions; hence, they show different degrees of sensitivity to the pretest drying method. this work is therefore carried out to elucidate the influence of the pre-test drying temperature or method on the properties of three lateritic soils that developed over three different pre-cambrian basement complex rocks from ado-ekiti, sw, nigeria. the soils were subjected to two pre-test drying temperatures before conducting laboratory tests. the pre-test drying temperatures considered in this study include air-drying, oven-drying at 60 ◦c, and oven-drying at 110 ◦c. pre-test drying at 60◦ and 110 ◦c caused particle aggregation (which reduced the soil surface area) and loss of cohesion. consequently, this reduced the specific gravity, optimum moisture content, clay content, consistency limits, and unconfined compressive strength of the lateritic soils. the maximum dry density and sand content increased as the pre-test drying temperature increased. the pre-test drying temperature did not significantly change the plasticity classification of the soils; however, at higher pre-test temperatures, the soils become less plastic. the free swell index of the lateritic soils increased with increasing pre-test drying temperatures (up to 60 ◦c) before decreasing when the temperature rose to 110 ◦c. this study has revealed the effect that pre-test drying temperatures may have on the properties of lateritic soils, and these may produce soil properties that do not likely indicate the actual field performance of the tested soils. doi:10.46481/jnsps.2023.1109 keywords: lateritic soil, pre-test drying temperature, index properties, oven drying, soil classification. article history : received: 05 october 2022 received in revised form: 21 january 2023 accepted for publication: 27 january 2023 published: 04 february 2023 c© 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: oladiran johnson abimbola 1. introduction lateritic soils/laterites are residual soils that are highly weathered. they are usually low in silica but have sufficient concentrations of iron and aluminum sesquioxide to have been cemented to some degree. lateritic soils are restricted to ∗corresponding author tel. no: +2348037857241 email address: afotayour@hotmail.co.uk; lekan.afolagboye@eksu.edu.ng (l. o. afolagboye ) tropical and subtropical regions of the world and occur as the capping of hills; therefore, they provide excellent borrowing areas for extensive use in various construction activities [1]. the properties and behavior of lateritic soils vary because of differences in degree of weathering (laterization), parent rock, climate, position in the soil profile, and topography [2, 3]. in civil engineering, the determination of soil index and engineering properties is important and integral to any engineering construction and design. to determine these 1 l. o. afolagboye et al. / j. nig. soc. phys. sci. 5 (2023) 1109 2 properties, tests such as consistency limits, grain size distribution, compaction, and strength tests are carried out using or on disturbed samples, and the test procedures completely re-mould the samples. that is, the samples must be carefully prepared to the required standards. for instance, one of the general specifications of sample preparation for most of these standard laboratory tests requires either air-drying or drying in an oven at a temperature usually between 60 ◦c to 110 ◦c. this is required to obtain fully dried soil, with the assumption that water present in the soil pore spaces could be removed by heating without destroying or changing the soil composition/structure. however, previous works have shown that this assumption may not be true for all soils [4, 5]. previous workers such as [6, 7, 8] reported that the method of drying and generally the method of sample preparation may significantly affect the index properties of some soils such as weathered, tropical/subtropical residual soils or soils that contain organic matter and halloysite or allophane. terzagi et. al. [9] reported the atterberg limits and grain size distribution of residual soils from indonesia tested at natural moisture content (nmc) and when air dried and oven dried. when tested at nmc, the soil had liquid limit (wl), plasticity index (ip) and amount of fine of 184%, 38%, and 95%, respectively. at nmc, the soil is classified as high plastic or organic silt. after air-drying, the amount of fines, wl and ip decreased to 19 %, 79 %, and 6 %, respectively. when oven-dried, the amount of fines was 15 % and the soil became “non-plastic”. the soil classified as silty sand (sm) when air-dried and oven dried. hence, the index and engineering tests may yield inconsistent results or significantly different results as they are influenced by the degree of pre-test drying temperature (nmc, oven-dried or air-dried) used prior to testing. these changes are attributed to increased cementation due to oxidation of the iron and aluminum sesquioxides or dehydration of allophane and halloysite [10]. in addition, pre-test drying may alter the structure, physical behavior, and clay content of a residual soil by causing aggregation of fine particles. the resultant larger particles remain bonded together even on wetting or after dispersion by standard dispersion techniques [5, 11]. previous studies have shown that lateritic soils are structured and contain significant concentration of iron and aluminum sesquioxides hence their properties are likely to be affected by the pre-test method of drying and generally the method of sample preparation [6, 7, 8]. in addition, the sensitivity of different lateritic soils to pre-test drying, and sample manipulation is different depending on climatic conditions [12]. the present study is therefore aimed at further understanding the properties/behavior of the residual lateritic soil whose behaviors are termed problematic by investigating the effect of pre-test drying temperature on the geotechnical and index properties of lateritic soils. these problematic behaviors can’t be explained by the accepted principles established for temperate soils [13]. in addition, to provide further insights as to why the plasticity classifications of the soil are sometimes not in agreement with the major component of the soil and in situ observations. three different genetically derived lateritic soils were selected for this study. 2. geology of the study area ado-ekiti is one of the areas of southwestern nigeria underlain by the pre-cambrian basement complex rocks. the rocks of the pre-cambrian basement complex are classified as “migmatite gneiss-quartzite complex,” “schist belts”, and “pan-african granites” [14]. except for the schistose rocks, ado-ekiti contains most of the rock lithologies that comprise the precambrian basement complex of southwestern nigeria [15]. these rocks include charnockite, migmatite gneiss, granite, and granite gneiss. migmatite gneiss is characterized by a fine-grained texture and alternating bands of dark and lightcolored minerals. quartzites are ridge-shaped, non-foliated rocks. with only a trace amount of feldspar, quartz makes up the majority of the quartzite’s mineral composition. the constituent minerals in the granites do not exhibit any preferred orientation. they range from having a fine-grained to porphyritic texture. the granites also contain compact crystals that interlock with one another. charnockite has a dark gray color and a texture that ranges from medium to coarse. charnockite can be found along the edges of granites [16, 17]. 3. methodology the lateritic soils used in this study developed over three different rock types. these rocks are granite, charnockite, and quartzite. the granite is made of quartz (66.3 %), biotite (12.2 %), albite (20.3 %), and opaque minerals [16]. quartzite is mainly made up of quartz (about 95.1 %) and other minerals such as feldspar. the charnockite, on the other hand, is made up of quartz (21.3 %), microcline (16.7 %), plagioclase (36.8 %), biotite (16.5 %) and others [17]. disturbed lateritic soil samples (that developed over quartzite and granite) were collected from active burrow pits where lateritic soil is presently being quarried for different construction purposes in ado-ekiti, southwestern, nigeria while samples of lateritic soil that developed over charnockite were collected from a road cut exposure. the samples were collected from the laterite horizon because lateritic soil from this zone is the most preferred for most construction activities [3], [18]. the collected samples were stored in airtight sealed polythene bags to keep their water contents intact. in all, a total of five soil samples were taken from soils that developed over the three rock types. the lateritic soil samples were tested at their natural moisture content (nmc) and three other states obtained through drying. these include air-dried soil, and soils oven dried at 60 ◦c and 110 ◦c. 3.1. drying process the soil samples that were not tested at their nmc were prepared under the following drying conditions: 2 l. o. afolagboye et al. / j. nig. soc. phys. sci. 5 (2023) 1109 3 i. air drying: the soil samples were spread on a clean wide pan and the spread samples were exposed to normal ambient temperature (25 to 30 ◦c) for three — four weeks. during this period, we regularly turned the soil over to avoid local drying out. it is sufficing to say that it takes at least four weeks to reduce the moisture content of the air-dried samples to a relatively constant value. ii. oven drying at 110 ◦c: this entails drying the soil samples to a constant mass in an oven at a temperature of 110 ± 5 ◦c as stipulated by [19]. the period of heating is 24hrs. iii. oven drying at 60 ◦c: this entails drying the soil samples to a constant mass in an oven at a temperature of 60 ± 5 ◦c. during heating, we constantly measured the weight of the samples and stopped the heating process when the weight of the samples became constant. 3.2. index and engineering tests procedure the index and engineering properties of these genetically different lateritic soil samples at their natural moisture content, air dried, and oven dried (60 ◦c and 110 ◦c oven drying) states were determined according to british standard 1377 [19] with some modifications where necessary. the particle size analysis was carried out using british standard 1377-2. to ensure proper segregation of the soil particles, we soaked the soil samples in calgon solution for a day before wet sieving. we conducted the particle size analysis using sieve analyses and hydrometer test. soil fractions retained on and passed through sieve no. 200 (75 µm), respectively, were used for sieve analyses and hydrometer test. the consistency limits were also determined using british standard 1377-2 on soil fraction passing sieve size 425 µm. the determined consistency limits of the soils include liquid limits and plastic limits. before the consistency limits tests, the sieved soil fractions were mixed with water and left to hydrate for 24 hrs. we determined the wl and plastic limit (wp) of the soils using the casagrande percussion cup and thread rolling methods, respectively. for determining the moisture-density relationship of the soil samples, compaction test was carried out on fractions of the soils that passed through 425 µm sieve following the specification of british standard 1377 – 4. we carried out the compaction test using the standard proctor compaction efforts. the specimens for the moisture — density relationship test is about 10.2 cm and 11.2 cm in diameter and height, respectively. we performed the unconfined compression test (uct) on lateritic soil following the specification of british standard 1377 – 7. the specimens were compacted at their optimum moisture content (omc) using the standard proctor compaction effort. the specimens used for the test measured 5 cm and 10 cm in diameter and height, respectively. the compacted soil sample were loaded under a stress-strain controlled condition. the strain rate was set at 1.25 mm/min. we compressed the soil specimen till failure and monitored the deformation at each point of loading. the peak stress attained during loading correspond to the unconfined compressive strength. free swell index (fsi) was carried out on fractions of lateritic soil that pass-through sieve size 0.425 mm. according to rao et. al. [20], fsi can be considered as an index property of expansive soil, and it reflects the potential for expansion of the soil. holtz and gibbs [21] defined the fsi as the ratio of the difference in volumes of soil in water and kerosene to the volume of soil in kerosene. it is mathematically expressed as: fs i = vd − vk vd × 100, (1) where vd (ml) = final volume of soil in a graduated cylinder containing distilled water, vk (ml) = final volume of soil in a graduated cylinder containing kerosene. therefore, fsi is expressed in (%). 4. results and discussion sg is an indicator for engineering behavior of lateritic soils in that, it’s the weighted average of the specific gravities of the minerals which comprise the soil. however, weathering and age of formation of parent rocks are fact factors to be considered while determining sg. table 1 shows the specific gravity of the studied lateritic soils for all pre-test drying methods. at nmc, the table revealed that the average sg values of quartzite derived lateritic soil, charnockite derived lateritic soil and granite derived lateritic soil are 2.648, 2.686 and 2.670, respectively. the difference between the sg of the three lateritic soils may be due to the variation in texture and mineralogy of the parent rocks. compared with lateritic soil tested at nmc, the sg of the three genetically different lateritic soils were lower and decreased as the drying temperature increases from air to oven drying at 110 ◦c. the difference is, however, not significant. sunil and krishnappa [22] studied “the influence of drying on the properties of lateritic soils and observed that the sg of the air and oven dried lateritic soils did not vary significantly from each other. at nmc and all the pretest drying temperatures, the average sg of charnockite-derived lateritic soil is higher compared to the other lateritic soils. this could be attributed to the mineral constituent of the parent rock which contained more heavy and opaque minerals compared to granite and quartzite [16]. generally, the high sg values of the soils are indicative of a high degree of laterization. 4.1. particle size distribution table 2 shows the results of particle size distribution of the lateritic soils as determined at their nmc, after air drying and oven-drying at 60 ◦c and 110 ◦c, respectively. the table reveals that the grain size distribution of the lateritic soils is affected by the parent rock factor. at nmc and all pre-test drying temperature, the lateritic soils are well-graded. the amount of sand size fractions in the lateritic soils are high. quartzite derived-lateritic soils 3 l. o. afolagboye et al. / j. nig. soc. phys. sci. 5 (2023) 1109 4 table 1: effect of pre-test drying temperature on the specific gravity of the lateritic soil parent rock drying method sg average quartzite nmc 2.642 2.648 2.645 2.652 2.653 2.648 air dried 2.643 2.644 2.643 2.643 2.643 2.643 oven dried at 60 ◦c 2.642 2.642 2.645 2.641 2.641 2.642 oven dried at 110 ◦c 2.634 2.634 2.636 2.636 2.631 2.634 charnockite nmc 2.687 2.688 2.685 2.685 2.686 2.686 air dried 2.688 2.687 2.685 2.686 2.684 2.686 oven dried at 60 ◦c 2.675 2.677 2.675 2.676 2.676 2.676 oven dried at 110 ◦c 2.661 2.661 2.664 2.665 2.663 2.663 granite nmc 2.668 2.674 2.672 2.668 2.666 2.670 air dried 2.667 2.668 2.67 2.662 2.664 2.666 oven dried at 60 ◦c 2.664 2.665 2.661 2.659 2.663 2.662 oven dried at 110 ◦c 2.654 2.655 2.654 2.655 2.653 2.654 table 2: grain size distribution of the lateritic soils at different pre-drying temperature parent rock nmc air dried oven dried at 60 ◦c oven dried at 110 ◦c g (%) s (%) si (%) c (%) g (%) s (%) si (%) c (%) g (%) s (%) si (%) c (%) g (%) s (%) si (%) c (%) quartzite 1.3 55.0 19.8 23.9 1.2 56.5 18.5 23.8 1.2 58.6 19.5 20.7 1.1 60.8 19.6 18.5 0.9 54.2 20.0 24.9 0.7 57.5 16.3 25.5 1.1 58.9 19.3 20.7 1.0 60.7 18.7 19.6 0.7 56.0 18.5 24.8 0.6 57.1 17.5 24.8 0.9 58.9 18.3 21.9 0.8 61.4 18.3 19.5 1.0 54.9 20.4 23.7 0.9 57.2 17.3 24.6 1.5 58.4 19.0 21.1 1.5 61.0 18.4 19.1 1.0 54.0 19.8 25.2 1.3 57.0 14.9 26.8 1.2 58.7 19.1 21.0 1.1 61.0 18.8 19.1 average 1.0 54.8 19.7 24.5 0.9 57.1 16.9 25.1 1.2 58.7 19.0 21.1 1.1 61.0 18.8 19.2 charnockite 0.7 46.7 20.7 31.9 0.6 49.2 20.4 29.8 0.6 51.0 20.7 27.7 0.6 53.3 20.5 25.6 0.9 46.1 20.0 33.0 0.8 48.9 19.7 30.6 0.8 50.8 19.9 28.5 0.8 53.2 19.6 26.4 0.5 46.0 20.4 33.1 0.4 49.0 19.7 30.9 0.4 51.1 20.5 28.0 1.1 52.6 20.3 26.0 0.5 45.0 21.8 32.7 0.5 48.8 22.4 28.3 0.5 51.3 21.0 27.2 1.2 52.4 20.9 25.5 0.6 45.0 21.0 33.4 0.8 48.2 21.6 29.4 0.6 51.0 20.6 27.8 0.9 52.9 20.4 25.8 average 0.6 45.8 20.8 32.8 0.6 48.8 20.8 29.8 0.6 51.0 20.5 27.8 0.9 52.9 20.3 25.9 granite 1.4 48.3 20.8 29.5 0.9 50.4 21.4 27.3 0.9 52.0 18.8 28.3 0.8 54.2 18.7 26.3 0.9 48.4 18.5 32.2 1.0 50.1 19.2 29.7 1.0 51.8 17.9 29.3 0.9 53.7 18.1 27.3 0.7 49.0 18.6 31.7 0.6 50.0 19.1 30.3 1.3 51.4 18.2 29.1 1.2 53.6 18.1 27.1 1.0 48.5 19.5 31.0 0.9 50.4 20.4 28.3 1.4 51.2 19.1 28.3 1.3 53.4 19.0 26.3 1.0 48.6 19.3 31.1 0.9 49.6 19.8 29.7 1.2 51.6 18.5 28.7 1.1 53.8 18.5 26.6 average 1.0 48.6 19.3 31.1 0.9 50.1 20.0 29.1 1.2 51.6 18.5 28.7 1.1 53.7 18.5 26.7 nmc, natural moisture content; g, gravel; s, sand; si, silt; c, clay have more than 50% sand content at all the pre-test drying temperatures. from table 2, it could be observed that the amount of clay and sand fractions are affected by method of drying that is the pre-test drying temperature. the percentage sand and clay fractions of the lateritic soils increased and reduced, respectively, with an increase in pre-test drying temperature. for instance, the average clay size fractions decreased from 24.5 to 19.2 %, 32.8 to 25.9 % and 31.1 to 26.7 % in quartzite, charnockite and granite derived lateritic soils respectively. furthermore, it was also observed that the decrease and increase, respectively, in percentage clay and sand contents was mostly influenced by oven-drying at 110 ◦c than oven-drying at 60 ◦c and air drying when compared with nmc. the average percent increase in sand content for lateritic soil derived from quartzite was 4.2 %, 7.12 %, and 11.31 % at drying temperatures of air-drying, 60 ◦c, and 110 ◦c, respectively. the average percent reduction in clay content for soil derived from charnockite was 9.15 %, 15.24 %, and 21.04 % at pre-test drying temperatures of air-drying, 60 ◦c, and 110 ◦c, respectively. basma et al. [6] made similar observation while studying the influence of drying methods on the properties of clays. the silt fractions of the soils, however, remain virtually constant at all the pre-test drying temperature. the decrease and increase in clay and sand fractions, respectively, of the lateritic soils may be attributed to particle aggregation as a result of drying (that is increase in temperature of pre-test drying). according to previous work, drying promotes loss of adsorbed and inter-particle water [23]. this mechanism leads to aggregation of smaller fine particles, inter-particle attraction and separation of small particles [6]. this eventually produce an increase in capillary stress which allows close contact of particles in addition to development of strong coulombic and van der waal bonds which are not easily reversible [5]. 4 l. o. afolagboye et al. / j. nig. soc. phys. sci. 5 (2023) 1109 5 4.2. consistency limits table 3 shows the results of consistency limits of the lateritic soils as determined at their nmc, by air drying and oven-drying at 60 ◦c and 110 ◦c. the wl and ip of charnockite-derived lateritic soil were constantly higher than lateritic soils derived from granite and quartzite at all the pre-test drying temperatures used in this study. the wl and ip of the lateritic soils reduced with increase in pre-test drying temperature (table 3). the significance of this effect is that nmc samples gave the highest wl and ip values while samples oven-dried at 110 ◦c gave the lowest values. in quartzite-, charnockiteand granite-derived lateritic soils, the averages ip decreased from 41.9 %, 56.7 % and 52.6 % when the samples were tested at their nmc to 37.6 %, 50.5 % and 48.0 % when the samples were tested after oven-dried at 110 ◦c, respectively. increase in pre-test drying temperature, according to sunil and deepa [24], leads to aggregation and clustering of soil particles. the agglomeration of particles reduces the soils available surface area available for water interaction. this in turn will make the soil to absorb less water and consequently reduces the wl and ip. the results of particle size distribution also confirmed this observation. as earlier reported, the amount of clay and sand fractions in the lateritic soils are affected by the pre-test drying temperature. the percent clay and sand contents, respectively, decreased and increased as the pre-test drying temperature increases. similar to the grain size distribution, it was also observed that the decrease in wl and ip for the three tested lateritic soils was mostly influenced by oven-drying at 110 ◦c more than oven-drying at 60 ◦c and air drying when compared with nmc. in charnockite-derived lateritic soil, the results in this research show a reduction in the ip when oven dried at 110 ◦c giving the highest reduction of 11.37 % while oven-dried at 60 ◦c and airdried samples gave 8.57 % and 6.41 % reduction from nmc value. in granite-derived lateritic soil, a reduction of 6.16 % (oven-dried at 110 ◦c), 1.38 % (oven dried at 60 ◦c), and 0.22 % (air dried). the sensitivity of a soil, as revealed in the literature, to pre-test drying depends on the type of clay mineral present and its state of hydration[6, 25]. it has been revealed that soils containing kaolinite are less sensitive to pre-test drying [25]. 4.3. plasticity charts the decrease in consistency limits because of increase in pre-test drying temperature may become a significant factor as this may change the classification of the soil. to examine the effect of pretest drying temperature on the plasticity classification of the lateritic soils, the values of wl and ip in table 3 were used to plot the points on casagrande and polidori [26] plasticity charts (figures 1 and 2). it was observed that polidori’s plasticity chart gives a fair classification of lateritic soils based on soil fractions [27]. on the casagrande’s plasticity chart (figure 1), the soils all plotted in the clay zones i.e., above the a-line. the soils are classified as either ci or ch. it was observed that even figure 1: casagrande plasticity classification of the lateritic soil. a) quartzitederived lateritic soil b) charnockite-derived lateritic soil c) granite-derived lateritic soil. cl: inorganic clays of low plasticity; ci: inorganic clays of intermediate plasticity, ch: inorganic clays of high plasticity; ml: inorganic silts of low compressibility; mi: inorganic clays of intermediate plasticity; mh: inorganic silts of high compressibility 5 l. o. afolagboye et al. / j. nig. soc. phys. sci. 5 (2023) 1109 6 table 3: consistency limits of nmc, air dried, and oven dried lateritic samples. parent rock nmc air dried oven dried at 60 ◦c oven dried at 110 ◦c wl (%) wp (%) ip (%) wl (%) wp (%) ip (%) wl (%) wp (%) ip (%) wl (%) wp (%) ip (%) quartzite 42.60 24.40 18.20 41.30 23.30 18.00 40.40 22.90 17.50 37.80 21.20 16.60 41.80 24.40 17.40 40.30 23.10 17.20 40.20 22.50 17.70 37.80 21.10 16.70 42.10 24.10 18.00 41.20 23.40 17.80 40.30 22.60 17.70 37.80 21.30 16.50 41.00 23.90 17.10 40.10 23.10 17.00 39.90 22.10 18.00 37.20 21.40 15.80 41.90 24.20 17.70 41.20 23.10 18.10 40.40 23.20 17.20 37.60 21.20 16.40 average 41.88 24.20 17.68 40.82 23.20 17.62 40.24 22.66 17.58 37.64 21.24 16.41 charnockite 56.30 22.20 34.10 54.30 22.20 32.10 52.70 21.40 31.30 53.50 21.20 32.30 57.10 22.60 34.50 55.20 22.60 32.60 53.70 21.30 32.40 50.10 19.80 30.30 56.80 22.40 34.40 54.60 22.40 32.20 52.40 21.70 30.70 49.40 19.40 30.00 56.50 22.30 34.20 54.70 22.30 32.40 52.20 21.20 31.00 49.20 20.20 29.00 56.70 22.40 34.30 54.30 23.20 31.10 52.80 21.40 31.40 50.50 20.10 30.40 average 56.68 22.38 34.30 54.62 22.54 32.10 52.76 21.40 31.36 50.54 20.14 30.40 granite 53.10 25.10 28.00 52.20 24.10 28.10 51.30 24.20 27.10 48.50 22.30 26.20 52.40 24.90 27.50 51.40 23.50 27.90 51.00 24.10 26.90 48.20 21.50 26.70 51.80 25.40 26.40 50.60 24.10 26.50 50.60 23.20 27.40 47.50 21.50 26.00 53.10 24.70 28.40 52.00 24.50 27.50 50.50 23.00 27.50 47.70 23.00 24.70 52.60 25.00 27.60 51.30 23.70 27.60 50.90 23.80 27.10 47.90 22.10 25.80 average 52.60 25.02 27.58 51.50 23.98 27.52 50.86 23.66 27.20 47.96 22.08 25.88 nmc, natural moisture content; wl, liquid limit; wp, plastic limit; ip, plasticity index though the silt and sand fractions (combined) of the lateritic soils were more than the clay fractions, the lateritic soils are classified as ch or ci soils. the pre-test drying temperature does not change the classification of the quartzite-derived lateritic soil (figure 1a) and charnockite-derived lateritic soil; except for two samples oven-dried at 110 ◦c. the classification of these samples changed from ch to ci (figure 1b). in samples where the classification does not change, a closer look at relative shift in position of the points on the casagrande’s chart distinctly shows that lateritic soils are becoming less plastic as the pre-test drying temperature increases. in granite derived lateritic soils, the pre-test dying temperature change the classification of the soil oven dried at 110 ◦c (figure 1c). on the polidori’s plasticity chart (figure 2), the soils plot above the c-line (silt zones). the soils are classified as either ml or mh. the pre-test drying temperature does not change the classification of the quartzite-derived lateritic soil (figure 2a) and charnockite-derived lateritic soil; except for two samples oven-dried at 110 ◦c. the classification of these samples changed from mh to ml (figure 2b). in granitederived lateritic soils, the pre-test drying temperatures change the classification of the soil oven dried at 110 ◦c (figure 2c). 4.4. compaction parameters the moisture content-dry density relationships of the lateritic soils were obtained at their nmc, air-dried and ovendried (60 ◦ and 110 ◦c) conditions. table 4 shows the results of optimum moisture content (omc) and maximum dry density (mdd) of the lateritic soils. the mdd of charnockite derived lateritic soil were constantly higher than lateritic soils derived from granite and quartzite at all the pre-test drying temperatures. the mdd and omc increased and reduced, respectively, with increase in pre-test drying temperature (table 4). the soil samples tested at their nmc gave the lowest mdd (highest omc) values while samples oven-dried at 110 ◦c gave the highest mdd (lowest omc) values. in quartzite-, charnockiteand granite-derived lateritic soils, the average mdd increased from 1787 kg/m3 1858 kg/m3 and 1703 kg/m3 when the samples were compacted at their nmc to 1926 kg/m3, 1954 kg/m3 and 1857 kg/m3 when the samples were compacted after ovendried at 110 ◦c, respectively. on the other hand, the average omc of quartzite, charnockite and granite derived lateritic soils, respectively, decreased from 21.88 %, 21.68 % and 24.63 % (nmc) to 17.33 %, 18.2 % and 19.56 % when oven-dried at 110 ◦c. these results agree with the findings of previous researchers [11, 24]. the changes in the compaction parameters of the lateritic soils as a result of pre-test drying temperature may be attributed to the effect of particle aggregation and re6 l. o. afolagboye et al. / j. nig. soc. phys. sci. 5 (2023) 1109 7 table 4: compaction parameters of the lateritic soils parent rock nmc air dried oven dried at 60 ◦c oven dried at 110 ◦c omc (%) mdd (kg/m3) omc (%) mdd (kg/m3) omc (%) mdd (kg/m3) omc (%) mdd (kg/m3) quartzite 22.40 1772 21.50 1797 19.00 1874 16.90 1940 21.60 1794 20.80 1819 19.30 1865 17.40 1924 22.10 1781 20.50 1828 19.50 1859 17.70 1915 21.40 1800 20.90 1815 19.20 1867 17.30 1928 21.90 1788 20.95 1811 19.30 1865 17.30 1923 average 21.88 1787 20.93 1814 19.26 1866 17.33 1926 charnockite 21.80 1856 21.20 1870 20.00 1904 18.40 1948 21.40 1864 20.80 1881 19.70 1912 18.20 1954 21.60 1859 20.40 1892 19.90 1906 18.00 1960 21.90 1853 20.80 1880 19.90 1910 18.20 1950 21.70 1857 20.80 1882 19.80 1903 18.10 1958 average 21.68 1858 20.80 1881 19.86 1907 18.20 1954 granite 24.60 1704 24.10 1719 22.80 1760 20.00 1843 24.40 1710 23.70 1732 22.30 1775 19.40 1862 25.00 1692 23.80 1729 23.00 1754 19.30 1865 24.50 1707 23.90 1723 22.70 1764 19.60 1857 24.60 1702 23.90 1728 22.80 1762 19.50 1858 average 24.63 1703 23.87 1727 22.70 1763 19.56 1857 duction in micropores [28]. the reduction in omc may be attributed to the decrease in specific area of the soils brought about by agglomeration of clay fractions to form silt/sand particles. the increase and decrease, respectively, in mdd and omc for the three tested lateritic soils were also mostly affected by oven-drying at 110 ◦c more than oven-drying at 60 ◦c and air drying when compared with nmc. for instance, in charnockite-derived lateritic soil, mdd increased by 5.17 % (oven dried at 110 ◦c), 2.64 % (oven dried at 60 ◦c) and 1.51 % (air dried) when compared to the value at nmc. 4.5. unconfined compressive strength (ucs) the results of the uct for the three genetically different lateritic soils as influenced by different pre-test drying temperatures are shown in table 5. as the pre-test drying temperature increases, the ucs decreased. for lateritic soil derived from quartzite, ucs (average) decreased from an average value of 229.80 kpa (nmc) to 220.18 kpa (oven-dried at 110 ◦c). in lateritic soil derived from charnockite, ucs (average) decreased from 287.60 kpa to 270.68 kpa for nmc and oven-dried at 110 ◦c conditions, respectively. although the mdd of the soils increased with increase in pretest drying temperature, the lateritic soils seemed to lose their strength as the temperature increases. similar observations have already been reported by various researchers [29], [30]. the reduction in strength may be attributed to the alteration/destruction of soil structure and loss of soil cohesion due to aggregation and clustering of the soil particles. lateritic soils table 5: variation of ucs of the lateritic soils with pre-test drying temperature parent rock nmc air dried ucs (kpa) oven dried at 60 ◦c oven dried at 110 ◦c quartzite 229.40 218.81 223.40 218.80 228.40 222.90 221.40 217.90 230.50 229.42 225.90 228.30 230.90 223.40 220.80 215.70 229.80 223.60 222.90 220.20 average 229.80 223.63 222.88 220.18 charnockite 286.80 275.30 272.77 272.30 285.60 275.60 280.80 275.30 290.10 276.40 276.40 264.90 287.90 278.20 270.50 270.20 287.80 276.40 275.10 270.70 average 287.60 276.38 275.12 270.68 granite 263.80 252.40 253.80 252.40 265.90 250.20 252.90 241.80 261.70 253.40 251.40 249.40 264.90 257.10 250.30 245.30 264.10 253.10 252.20 247.30 average 264.08 253.23 252.10 247.23 are known to partially derive their strength from cohesion, increase in particle aggregation due to increase drying tempera7 l. o. afolagboye et al. / j. nig. soc. phys. sci. 5 (2023) 1109 8 figure 2: polidori plasticity classification of the lateritic soil. a) quartzitederived lateritic soil b) charnockite-derived lateritic soil c) granite-derived lateritic soil. cl: inorganic clays of low plasticity, ch: inorganic clays of high plasticity; ml: inorganic silts of low compressibility; mh: inorganic silts of high compressibility ture will likely cause a loss in cohesion, hence a loss in ucs. similar to other properties, oven drying at 110 ◦c produced the highest percentage reduction in ucs. in granite derived lateritic soil, for instance, the average ucs decreased by 6.27 % (oven dried at 110 ◦c), 4.49 % (oven dried at 60 ◦c) and 4.05 % (airdried) when compared to the value at nmc. in quartzite derived lateritic soil, the average ucs decreased by 4.19 %, 3.01 % and 3.23 %, respectively, when oven dried at 110 ◦c, oven dried at 60 ◦c and air-dried. 4.6. free swell index the free swell is a simple test used for estimating the swelling potential of a soil. the values of fsi for the three lateritic soils are shown in table 6. it is observed that the maximum value of fsi for the lateritic soils was at 60 ◦c oven drying. this was followed by fsi of samples oven-dried at 110 ◦c and air-dried samples except in quartzite derived lateritic soil. table 6: free swell index of the lateritic soils parent rock nmc air dried oven dried at 60 ◦c oven dried at 110 ◦c quartzite 38.89 42.11 47.62 42.11 40.00 42.86 45.45 40.00 38.89 42.11 45.45 40.00 36.84 40.00 46.00 40.90 39.00 41.90 45.90 41.20 average 38.72 41.79 46.09 40.84 charnockite 42.11 45.00 52.38 47.37 42.11 45.00 54.55 50.00 44.44 50.00 52.38 47.37 42.11 42.11 53.70 48.30 42.80 46.00 52.50 49.00 average 42.71 45.62 53.10 48.41 granite 42.86 45.45 50.00 47.83 45.45 47.83 50.00 47.83 42.86 45.45 52.17 50.00 43.48 45.83 51.50 48.20 44.00 46.30 49.80 49.00 average 43.73 46.17 50.69 48.57 the increase in fsi up 60 ◦c oven drying may be due to the clay minerals present in the soil developing a high repulsive force with increasing temperature. consequently, according to basma et. al. [6], “this caused the clay particles to separate more and developed a flocculated structure which led to more water being needed to make up for the deficiency upon wetting and hence more swelling”. the decrease in fsi when the pretest temperature increased from 60 ◦c to 110 ◦c may be due to loss of plasticity and increasing aggregation of the soils. 5. conclusions in this present study, the effect of pre-test drying temperature on the properties of lateritic soils was examined. for this 8 l. o. afolagboye et al. / j. nig. soc. phys. sci. 5 (2023) 1109 9 reason, three genetically unrelated lateritic soils were selected for this study. three pre-test drying temperatures were used and this includes air-drying, and oven-drying at both 60 ◦c and 110 ◦c. different tests were carried out on the three lateritic soils to study the influence of the aforesaid pre-test drying temperatures on particle size, specific gravity, consistency limits, free swell, compaction parameters, and strength properties. the index and engineering properties at their nmc are influenced by the parent rock factors. drying the lateritic soils to 110 ◦c reduced the plasticity index, specific gravity, clay content, liquid limit, omc, and unconfined compressive strength of the lateritic soils. the decrease in properties such as plasticity index, omc and unconfined compressive strength may be attributed to particle aggregation (which reduced the soil surface are) and loss of cohesion. this study also revealed that lateritic soils dried at 110 ◦c may lead to underestimation of the ucs. the increased in pre-test drying temperature slightly reduced the silt content and plastic limit of the soils. the pre-test drying temperature did not significantly change the soils’ plasticity classification, however, at higher pre-test temperature (namely 110 ◦c) the soils are generally less plastic. the mdd and sand content of the soils increased as the pre-test drying temperature increases. in general, the free swell index of the lateritic soils increased with increasing pre-test drying temperature (up to 60 ◦c) before decreasing when the temperature rose to 100 ◦c. the study has shown the effect pre-test drying temperature may have on the properties of lateritic soils. however, the fact that in most engineering and earthworks, materials wined are normally stockpiled. this process of stockpiling normally leads to some sort of air-drying. finally, it can be concluded that air-drying seems more suitable as pre-test heating method because it will reflect the in-situ field condition. references [1] r. k. goswami & c. mahanta, “leaching characteristics of residual lateritic soils stabilised with fly ash and lime for geotechnical applications”, waste management 27 (2007) 466. doi: 10.1016/j.wasman.2006.07.006. 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[30] i. c. attah & r. k. etim, “experimental investigation on the effects of elevated temperature on geotechnical behaviour of tropical residual soils”, sn appl sci. 2 (2020) 370. doi: 10.1007/s42452-020-2149-x. 9 j. nig. soc. phys. sci. 5 (2023) 1116 journal of the nigerian society of physical sciences analysis of the bioactive compounds from carica papaya in the management of psoriasis using computational techniques misbaudeen abdul-hammed∗, ibrahim olaide adedotun, tolulope irapada afolabi, ubeydat temitope ismail, praise toluwalase akande, balqees funmilayo issa computational and biophysical chemistry laboratory, department of pure and applied chemistry, ladoke akintola university of technology, p.m.b. 4000, ogbomoso, nigeria abstract psoriasis is a persistent and mysterious autoimmune skin condition that affects 2-3% of the world’s population. currently, topical therapies, light therapy, and systemic drugs are the three main forms of treatment used to lessen inflammation and skin irritation/itching. however, all these treatments are only used to manage the disease each time it surfaces. therefore, the main target of this work is to search for a safer and more effective remedy for psoriasis from the reservoir of phytochemicals present in carica papaya via in silico studies due to its anti-psoriatic and anti-inflammatory properties. reported phytochemicals isolated from carica papaya were subjected to computational simulations using the pyrx docking tool and were docked against janus kinase 1 (jak1) and tumor necrosis factor α (tnfα) target receptors. the results obtained were visualized using pymol, and biovia 2019. analysis of the results identified both chlorogenic acid and coumaroylquinic-acid with docking scores (-8.6 kcal/mol and -7.9 kcal/mol) respectively as potential inhibitors for the jak1 receptor. the identified compounds also possessed excellent admet, drug-likeness, bioactivity, and activity spectra for substances (pass) prediction properties. their binding mode and the molecular interactions with the targets also affirmed their potency. in comparison with the standards (methotrexate and cyclosporine), chlorogenic acid and coumaroylquinic-acid have better admet properties, binding affinities, drug-likeness, pass properties, bioactivities, oral bioavailability, binding mechanism, and interactions with the active site of the target receptor and are hereby recommended for further analysis towards the development of a new therapeutic agent for psoriasis treatment and management. doi:10.46481/jnsps.2023.1116 keywords: psoriasis, carica papaya, molecular docking, anti-inflammatory, skin disorder article history : received: 12 october 2022 received in revised form: 26 december 2022 accepted for publication: 05 january 2023 published: 27 february 2023 © 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: k. sakthipandi 1. introduction psoriasis is a chronic inflammatory noncontagious autoimmune skin condition that results in a rash with itchy, burning, ∗corresponding author tel. no: +234 8069151819 email address: mabdul-hammed@lautech.edu.ng (misbaudeen abdul-hammed) and scaly patches [1]. this disease is common on the skin of the scalp, knees, elbows, lumbosacral regions, and trunks and may appear anywhere on the body’s skin [2]. psoriasis affects 2 to 3% of the world population of any age, skin color, and sex but is more prevalent in adults than children. the condition often starts to manifest around the age of 20. psoriatic arthritis affects 10 to 15% of the population and about 7 mil1 abdul-hammed et al. / j. nig. soc. phys. sci. 5 (2023) 1116 2 lion americans (2%–3% of the population) suffer from psoriasis. each year, between 150,000 and 260,000 new cases are diagnosed [3]. some conditions such as obesity, high blood pressure, and diabetes tend to increase the risk of developing psoriasis [4], while several conditions are linked to psoriasis which includes cardiovascular disease, severe depression, and lymphoma [5, 6]. chronic interactions between invading, activated immune cells and hyperproliferative keratinocytes cause it to occur, which depend heavily on the immune system. psoriatic lesions have high levels of t cells, especially th1 and th17 [7], while dendritic cells that produce tnf and inos also heavily infiltrate psoriatic skin and polarize t cells to the th1 and th17 subtypes [8]. psoriasis can be in minor patches or complete body coverage depending on the degree of severity and type. the degree of severity of psoriasis depends on environmental exposure and family history [9]. as the rate of occurrence of psoriasis is between 2 to 4% of the world’s population, researchers are on the verge of seeking permanent treatments for the disease. the treatment presently available for psoriasis is only used to manage the disease, which are; topical medications, these are often used to treat mild to moderate psoriasis. they include the use of topical corticosteroids, vitamin d analogs, anthralin, retinoids, and calcineurin inhibitors. the skin thins due to the abuse of corticosteroids. anthralin and vitamin d analogs (calcipotriene and calcitriol) slow the development of skin cells, get rid of scales, and smooth the skin. along with other therapies, these analogs relieve mild to severe psoriasis, but they also irritate the skin. similar to topical retinoids, which may reduce inflammation but irritate skin and heighten sensitivity to sunlight. additionally, oral retinoids increase the risk of birth abnormalities and are not advised for use by women who are pregnant or nursing. tacrolimus and pimecrolimus are two calcineurin inhibitors that similarly lessen inflammation and plaque buildup, but they also come with a higher risk of skin cancer [10]. phototherapy (ultraviolet light) which uses uv can lead to thinning of the skin on exposure. although skin cell turnover is slowed by uv exposure, which also lessens scaling and irritation, also small quantities of sunshine each day may help with psoriasis, prolonged contact with the sun can exacerbate the condition and harm the skin [11]. systemic treatments (retinoids, methotrexate, cyclosporine, acitretin, hydroxyurea, fumarates) are used to treat patients with severe psoriasis, but they come with serious side effects. retinoids may result in hair loss and lip irritation. methotrexate treats psoriasis by reducing the growth of skin cells and reducing inflammation, but it can also make you tired and upset your stomach. methotrexate can harm the liver over time and reduce the synthesis of platelets, red blood cells, and white blood cells. cyclosporine has comparable immunosuppressive effects as methotrexate, but it should only be used temporarily due to the danger of infection, cancer, renal issues, and high blood pressure when taken at large dosages or ongoing treatment [12]. each time the disease manifests, all of these therapies are solely employed to control it [13]. so, to effectively treat psoriasis, new and safer chemical agents are thus urgently needed. the need to manage psoriasis has usually been a lifelong one which used to result in a significant cost to mental wellbeing such as higher rates of depression and negative impact on individuals in a society. social exclusion, discrimination, and stigmatization have always been associated. in the research and development of new drugs, phytochemicals are rapidly emerging as significant alternative medicinal and pharmacological agents. as opposed to synthetic medications, they have fewer or no adverse effects after administration, a unique mode of action, and a wide range of chemical constituents, all of which improve their therapeutic interaction with a variety of biological targets [14]. phytochemicals derived from papayas such as flavonoids, terpenoids, tannins, and phenols have been found to have anti-psoriatic and antiinflammatory effects associated with psoriasis [15]. this study aims at investigating the anti-psoriatic and antiinflammatory potential phytochemicals found in the papaya plant against two psoriasis targeted enzymes; jak1 (pdb id: 6n7b) and tnfα (pdb id: 2az5) through molecular docking coupled with admet studies, pharmacokinetic evaluation, drug likeliness among other analyses at a therapeutic dose as used previously in the study on enzyme inhibitors of sarscov2 main protease [16, 17] and human tyrosinase-related protein [18]. 2. materials and methods 2.1. preparation of ligands one hundred and three phytochemicals extracted from carica papaya with their various classes of phytochemicals which are, 18 phenols, 5 amino acids, 2 carotenoids, 9 fatty acyls, 24 fatty acids, 24 flavonoids, 9 steroids, 4 terpenoids, and 3 glycoside, 2 lactones and 3 organosulfur compounds were used in this investigation study. methotrexate and cyclosporine are used as standard. pubchem database (https://pubchem.ncbi.nlm.nhi.gov) [19] was used to obtain the 2d/3d conformers of these ligands and the standard used. the 2d structure of these 103 ligands was converted to 3d using spartan’14 software and the conformational search was also implemented using spartan’14 as well with molecular mechanics in which the stable conformers were carefully chosen and optimized using density functional theory (dft) with b3lyp function and 631+g(d) as a basis. 2.2. preparation of the target receptor the xray structure of tumor necrosis factor alpha (tnf alpha) (pdb id: 2az5) and human janus kinase jak1 (pdb id: 6n7b) (fig 1) was downloaded from the protein data bank with a resolution of the retrieved structure given as 2.10å and 1.81å respectively in protein data bank (pdb) file format. the protein was prepared by removing the impurities including water molecules present using discovery studio software to escape interference. the binding pocket of the initial inhibitors present in 2az5 and 6n7b was used to determine the binding parameters as preferences. 2 abdul-hammed et al. / j. nig. soc. phys. sci. 5 (2023) 1116 3 figure 1. the crystal structure of (a) tumor necrosis factor alpha (tnf-alpha) (pdb id: 2az5) and (b) human janus kinase jak1 (pdb id: 6n7b) 2.3. determination of receptors’ active sites tumor necrosis factor alpha (tnf-alpha) (pdb id: 2az5) and human janus kinase jak1 (pdb id: 6n7b) binding pockets, ligand interactions, and all amino acids in the active site were established using castp (http://sts.bioe.uic.edu/castp/index.html) and biovia discovery studio [20]. concerning the two receptor active sites complexed with their respective ligands, the obtained data were compared and validated against the previously published experimental data [21-23] 2.4. admet profiling and drug likeness analysis absorption, distribution, metabolism, excretion, and toxicity (admet) of the docked ligands were evaluated using the admet sar2 database (http://1mmd.ecust.edeu.cn/admetar2/) (www.admetexp.org) [24], which is a free web tool used in evaluating admet properties while drug-likeness (lipinski rule of 5) were inspected using molinspiration online tool (http://molinspiration.com/) [25]. 2.5. ligands oral bioavailability assessments oral bioavailability assessments of the ligands were achieved using the swissadme web server (http://www.swissadme.ch/) [26]. 2.6. prediction of activity spectra for substances (pass) the biological activities of the ligands and the standard drugs used in this research study were carried out using a web server [27]. 2.7. molecular docking protocol molecular docking and scoring of optimized ligands and the standard drugs against tumor necrosis factor alpha (tnfalpha) (pdb id: 2az5) and human janus kinase jak1 (pdb id: 6n7b) were obtained using pyrx software. the inhibition constants (ki) in µm of the ligands and the standard method were obtained using their binding affinities (∆g) in kcal/mol as shown in (equation 1), thus showing their potency against the target receptors (2az5 and 6n7b). ki = ex p(∆g/rt ) (1) where r= gas constant (1.987×103 kcal/mol); t=298.15k (absolute temperature); ki= inhibition constant and ∆g = binding energy . 3. results and discussion 3.1. structural and active site analysis of prostate cancer target receptors 3.1.1. tumor necrosis factor alpha (tnf alpha) the xray crystallographic structure of tumor necrosis factor alpha (tnf alpha) (pdb id: 2az5) (fig. 1) contains 148 amino acid residues complexed with an inhibitor (6,7 dimethyl3-[(methyl-{2[methyl-({1-[3(trifluoromethyl)phenyl]-1hindol3yl}methyl)-amino] ethyl}amino)methyl]-4-chrome-4-one). the resolution of the protease as revealed by xray diffraction was 2.10 å, crystal dimension is a = 165.25 å, b = 165.25 å, and c = 63.72 å with angles α (900), β (900), and γ (120) respectively. r values (free, work, and observed) are 0.278, 0.220, and 0.2127 respectively. tnfα plays a crucial role in the exacerbation of inflammation in psoriasis. its main function is to control the immune system’s cells. tnf is an endogenous pyrogen that can cause fever, apoptotic cell death, inflammation, cachexia, and cancer while also inhibiting virus replication and triggering il1 and il6producing cells in response to sepsis. several human disorders, including alzheimer’s disease, cancer, severe depression, psoriasis, and inflammatory bowel disease have been linked to dysregulation of tnf production [28-31]. amino acid residue at the active site is as follows leu57, tyr59, ser60, gln61, tyr119, leu120, gly122, tyr151 [21]. 3.1.2. human janus kinase jak1 the x-ray crystallographic structure of human janus kinase jak1 (pdb id: 6n7b) (fig.1) contains 302 amino acid residues complexed with n[3(5chloro2methoxyphenyl)1methyl1hpyrazol4yl]1hpyrazolo[4,3c]pyridine7carboxamide. the resolution of the protease as revealed by x-ray diffraction was 1.81å, crystal dimension is a = 170.28 å, b = 42.78 å, and c = 44.98 å with angles α (900), β (900), and γ (900) respectively. rvalues (free, work, and observed) are 0.264, 0.220, and 0.222 respectively. through interactions with signal transducers and transcriptional activators, the janus kinase (jak) family, which consists of four receptor associated protein tyrosine kinases (jak1, jak2, jak3, and tyk2), is involved in the interferon and cytokine signaling process [32]. seven jak homology domains make up the jak kinases (120130 kda) [33]. the catalytically active region of the protein that is in charge of its physiological action is known as the c-terminal kinase module (jh1) and it has been demonstrated that the catalytically inactive jh2 domain controls the jh1 domain’s activity [34]. two src homology 2 (sh2) domains (jh3 and jh4) are located at the n-terminus, followed by the ferm domain (jh5–jh7). the atp binding site, which is located in the jh1 domain, has been targeted by several small molecule inhibitors. amino acid residue at the active site is as follows leu881, gly887, glu883, gly884, gly887, lys908, glu957, leu959, gly962, glu966, arg1007, asn1008, leu1010, gly1020, asp1021 [21, 22]. 3 abdul-hammed et al. / j. nig. soc. phys. sci. 5 (2023) 1116 4 table 1. admet profiling of the selected hit compounds and standard drug ligands absorption and distribution metabolism extn. toxicity bbb hia logs caco-2 2c19 1a2 3a4 2c9 2d6 b am aot ei ec hi c l-1 0.96 0.99 -1.85 0.79 + + iii + + l-2 -0.44 0.98 -0.56 0.55 + iii + l-3 0.97 0.97 -2.42 0.93 + iii + + l-4 0.9 0.98 -2.58 0.53 iii + l-5 -0.99 0.99 -1.61 0.93 + iii + + l-6 -0.44 0.96 -1.69 0.5 iv + l-7 -0.76 0.98 -1.35 0.92 + iii + + l-8 -0.73 0.97 -0.22 0.85 + iii + + l-9 -0.24 0.91 -2.48 -0.92 iii l-10 0.98 0.96 -1.75 0.62 iii + l-11 -0.3 0.9 -2.46 -0.92 iii l-12 -0.39 0.99 -3.74 -0.95 + iii + l-13 -0.31 0.77 0.45 -0.84 + iii + l-14 -0.44 0.77 0.28 -0.96 + iii + l-15 0.97 0.84 -3.5 0.71 + + iv + + l-16 0.99 0.84 -0.14 0.83 + iii + + l-17 0.56 0.91 -4.04 0.9 + + iv + + l-18 0.96 0.91 -4.04 0.71 + + iv + + l-19 0.97 0.84 -3.5 0.71 + + iv + + l-20 0.99 0.84 -0.14 0.83 + + iii + + l-21 0.97 0.84 -3.5 0.86 + + iv + + l-22 0.97 0.84 -3.5 0.77 + + iv + + l-23 0.97 0.84 -3.5 0.59 + + iv + + l-24 0.97 0.84 -2.02 0.86 + + iii + + l-25 0.98 0.92 -3.67 0.68 iii + + l-26 0.95 0.89 -2.75 -0.7 iii l-27 0.94 0.89 -2.59 -0.66 iii sd-1 -0.99 0.9 -3.06 -0.86 iii sd-2 0.91 0.93 -1.76 -0.85 + iii bbb= blood brain barrier, hia=human intestinal absorption, as =aqueous solubility. extn. = excretion; b=biodegradation (+/-) biodegradable (+), non-biodegradable (-). am =ames mutagenesis (+/-); aot= acute oral toxicity(+/-) acute toxic (+), non acute-toxic (-); hi = human either-a-go-go inhibition (+/-), c=carcinogenicity (+/-) carcinogenic (+), non-carcinogenic (). l1 = 2,6dimethoxyphenol, l2 = gentisyl alcohol, l3 = cinnamic acid, l4 = sinapinic acid, l5 = salicylic acid, l6 = caffeic acid, l7=phydroxybenzoic acid, l8=pcoumaric acid, l9=coumaroylquinic acid, l10=chlorogenic acid, l11=translinalool oxide, l12 = phydroxyl benzoic, l1 = citric acid , l14 = malic acid, l15= nhexadecanoic acid, l16= butanoic acid, l17=linoleic acid, l18=oleic acid, l19=palmitic acid, l20=nbutyric acidl, l21=noctanoic acid, l22=myristic acid l23=stearic acid, l24=nhexanoic acid, l25=cisvaccenic, l26=dehydrocarpaine i, l27=dehydrocarpaine ii, sd1=methotrexate, sd2=cyclosporine 3.2. admet (pharmacokinetics) analysis of the selected compounds adsorption, distribution, metabolism, excretion, and toxicity (admet) profiling of ligands is a crucial step in the early stages of the drug discovery process for expediting the conversion of hits and lead compounds into approved candidates for therapeutic development. a high-quality drug candidate is highlighted by drugs’ efficacies against therapeutic targets in conjunction with good admet profiling at a therapeutic dose [35, 36]. as part of the drug admet profile, a drug must possess good human intestinal absorption (hia), solubility (log s) which ranges between 1 and 5, should be a non-inhibitor of cytochrome p450 enzymes, and should be non-ames toxic (am), non-carcinogenic(c), non-inhibitor of herg(hi), and no or low level of toxicity [37]. all the 103 compounds isolated from carica papaya understudies were screened using admet sar2 webserver, 27 passed the analysis, the result was shown in table 1 and they were subjected to further analysis. notably, all the selected hit compounds and the standard (std) have excellent chances of being absorbed in the human intestine (hia), some of the selected hit compounds and std2 can penetrate the blood brain barrier (bbb+), although only drugs that are specifically targeted for the central nervous sys4 abdul-hammed et al. / j. nig. soc. phys. sci. 5 (2023) 1116 5 table 2. drug likeness properties of the best hits and two standard drugs (sd) compounds heavy atoms (ha) molecular weight (mw) ro5 violations hydrogen bond donor (hbd) hydrogen bond acceptor (hba) milogp l1 10 138.12 0 2 3 1.37 l2 12 164.16 0 2 3 1.43 l3 24 338.31 0 5 8 0.04 l4 11 146.15 0 0 2 2.01 l5 11 154.16 0 1 3 1.34 l6 10 140.14 0 3 3 0.71 l7 11 148.16 0 1 2 1.91 l8 16 224.21 0 2 5 1.26 l9 10 138.12 0 2 3 1.87 l10 12 170.25 0 1 2 1.94 l11 25 354.31 1 6 9 0.45 l12 16 222.28 0 2 3 3.83 l13 13 192.12 0 4 7 1.98 l14 9 134.09 0 3 5 1.57 l15 18 256.43 1 1 2 7.06 l16 6 88.11 0 1 2 1.00 l17 20 280.45 1 1 2 6.86 l18 20 282.47 1 1 2 7.58 l19 18 256.43 1 1 2 7.06 l20 6 88.11 0 1 2 1.00 l21 10 144.21 0 1 2 3.02 l22 16 228.38 1 1 2 6.05 l23 20 284.48 1 1 2 8.07 l24 8 116.16 0 1 2 2.01 l25 27 396.73 1 0 2 9.36 l26 34 476.70 1 1 6 6.60 l27 34 474.69 1 0 6 6.79 sd1 33 454.45 2 7 13 1.97 sd2 85 1202.63 2 5 23 3.61 l1 = 2,6dimethoxyphenol, l2 = gentisyl alcohol, l3 = cinnamic acid, l4 = sinapinic acid, l5 = salicylic acid, l6 = caffeic acid, l7= phydroxybenzoic acid, l8=pcoumaric acid, l9=coumaroylquinic acid, l10=chlorogenic acid, l11=translinalool oxide, l12 = phydroxyl benzoic, l13 = citric acid , l14 = malic acid, l15= nhexadecanoic acid, l16= butanoic acid, l17=linoleic acid, l18=oleic acid, l19=palmitic acid, l20=nbutyric acidl, l21=noctanoic acid, l22=myristic acid l23=stearic acid, l24=nhexanoic acid, l25=cisvaccenic, l26=dehydrocarpaine i, l27=dehydrocarpaine ii, sd1=methotrexate, sd2=cyclosporine tem must penetrate the blood brain barrier; oral drug may not always require to achieve this [38]. and all the hit compounds and std have excellent aqueous solubility (logs) values, falling within the recommended range of (-1 to -5). this shows that the selected hit compounds and the standard have good absorption and distribution potential. the metabolic activities of the selected hit compounds were assessed using microsomal enzyme (cytochrome p450 inhibitors) which catalysed reactions involved in the metabolic activities of the drug. as observed in table 1, l1, l15 to l24 are non-inhibitors of all the cyp450 inhibitors. moreover, critical observation of the results obtained in the table 1 revealed that all the selected hits are non-carcinogenic, furthermore, the potential of a drug molecule to cause mutation in dna is revealed by ames toxicity value and could be a major reason for excluding a drug molecule along the discovery process, as shown in table 1, all the selected hit compounds are non-ames toxic. similarly, the majority of the hit compounds possess type iii acute oral toxicity (ld50) values (slightly toxic) which could easily be converted to type iv (non-toxic) during hit lead optimization. l6, l15, l17, l18, l19, l21, l22, and l23 possess type iv which makes it nontoxic while sd1 possesses type ii which means it is highly toxic. interaction of drug candidates with human ether a-go-go (herg) is one of the important factors to consider in selecting a good drug candidate. a good drug candidate is expected to be a non-inhibitor of herg, because herg inhibition may lead to blockage of the potassium ion channel of the myocardium, which will affect the heart, causing chronic health challenges, and that may lead to death [39]. as observed in table 1, all selected hits and stds are non-herg in5 abdul-hammed et al. / j. nig. soc. phys. sci. 5 (2023) 1116 6 table 3. the docking scoring, binding affinities, and inhibition constant (ki) of the interaction of passed ligands and the standard drug with human janus kinase jak1 (pdb id: 6n7b) compounds binding affinity (∆g), kcal/mol inhibition constant (ki), µm dehydrocarpaine-ii -10.5±0.0 0.02 chlorogenic-acid -8.6±0.0 0.50 dehydrocarpaine-i -7.9±0.0 1.60 coumaroylquinicacid -7.9±0.0 1.60 cis-vaccenic -6.9±0.0 8.8 sinapinic-acid -6.5±0.0 18.8 caffeic-acid -6.6±0.0 15.9 pcoumaric-acid -6.3±0.0 24.2 phydroxyl-benzoicacid -6.4±0.0 20.4 cinnamic-acid -6.1±0.0 33.9 linoleic acid -5.8±0.0 56.2 oleic-acid -5.7±0.0 66.5 translinalool-oxide -5.6±0.0 85.7 stearic acid -5.6±0.0 78.8 citric-acid -5.5±0.0 101.4 myristic-acid -5.4±0.0 110.4 gentisyl alcohol -5.4±0.0 110.4 palmitic-acid -5.3±0.0 130.7 nhexadecanoic-acid -5.2±0.0 168.3 2,6dimethoxyphenol -5.2±0.0 168.3 octanoic-acid -5.1±0.0 183.1 hexanoic-acid -4.5±0.0 504.0 malic-acid -4.4±0.0 596.6 nbutyric-acid -3.9±0.0 1387.0 butanoic-acid -3.9±0.0 1387.0 methotrexate -8.9±0.0 0.36 cyclosporine -8.0±0.0 1.62 hibitors. summarily, all the selected hit compounds and stds show excellent admet properties and are better drug candidates against the target receptors. 3.3. drug-likeness analysis of the selected ligands as proffer by lipinski 2004, orally active drugs must obey the rule of five (ro5) which are, molecular weight (mw) ≤ 500, octanolwater partition coefficient (log p) ≤ 5, hydrogen bond donor (hbd) ≤ 5, and hydrogen bond acceptor ≤ 10 and no more than one violation is allowed [40]. drug-likeness of the selected phytochemicals with standard drugs was carried out to make a model that can successfully predict whether a molecule is druglike or not [20]. out of 27 ligands isolated from carica papaya that passed admet screening, all of them obeyed the lipinski ro5 with violations of 1 and 0 except the two standard drugs having a violation of 2. these properties were estimated by an online server called molinspiration table 4. the docking scoring, binding affinities, and inhibition constant (ki) of the interaction of passed ligands and the standard drug with tumor necrosis factor alpha (tnf alpha) (pdb id: 2az5) compounds binding affinity (∆g), kcal/mol inhibition constant (ki), µm dehydrocarpaine-ii -7.6±0.0 2.7 dehydrocarpaine-i -7.5±0.0 3.2 chlorogenic-acid -6.2±0.0 28.7 coumaroylquinicacid -5.5±0.0 93.3 cinnamic-acid -5.0±0.0 215.0 sinapinic-acid -4.9±0.0 256.9 pcoumaric-acid -4.9±0.0 256.9 cis-vaccenic -4.9±0.0 256.9 caffeic-acid -4.8±0.0 304.1 phydroxyl benzoicacid -4.8±0.0 304.1 translinalool oxide -4.8±0.0 304.1 linoleic acid -4.5±0.0 504.7 stearic acid -4.4±0.0 597.1 oleic-acid -4.4±0.0 649.0 nhexadecanoicacid -4.4±0.0 649.0 palmitic-acid -4.2±0.0 836.8 noctanoic-acid -4.2±0.0 836.8 myristic-acid -4.2±0.0 836.8 citric-acid -4.1±0.0 990.5 gentisyl alcohol -4.0±0.0 1172.6 2,6dimethoxyphenol -3.8±0.0 1643.2 nhexanoic-acid -3.7±0.0 1945.2 malic-acid -3.3±0.0 3819.8 nbutyricacid -3.2±0.0 4521.9 butanoic-acid -3.2±0.0 4521.9 methotrexate -6.4±0.0 23.3 cyclosporine -4.3±0.0 770.9 (http://www.molinspiration.com/) [41], and are shown in table 2. 3.4. molecular docking analysis molecular docking procedures can be used to recognize the interaction between a small ligand and a target molecule and to determine if they could behave in combination as the binding site of two or more constituent molecules with a given structure. a potential active drug is expected to have inhibitory values from 0.1 and 1.0µm and it should not be greater than 10nm. the inhibition constant was calculated using ki = exp [ ∆g/rt]. where ki = inhibition constant, ∆g = binding energy, r = gas constant (1.937×103kcal/mol); t=298.15k (absolute temperature) [42]. figure 1 shows the structure of tumor necrosis factor alpha (tnf alpha) (pdb id: 2az5) and human janus kinase jak1 (pdb id: 6n7b) that was used as the target proteins for this research. the 27 ligands that passed both admet and druglikeness parameters were docked separately with the 6 abdul-hammed et al. / j. nig. soc. phys. sci. 5 (2023) 1116 7 table 5. oral bioavailability analysis of the selected compounds and the standard drug ligands m.f m.w tpsa #r.b xlog p3 esol logs b.s. frac. csp3 #pain alert s.a c1 c28h46n2o4 474.68 77.32å² 0 5.66 -6.35 0.55 0.86 0 7.34 c2 c16h18o9 354.31 164.75å² 5 -0.42 -1.62 0.11 0.38 1 4.16 c3 c28h48n2o4 476.69 76.99å² 0 5.97 -6.56 0.55 0.89 0 7.45 c4 c16h18o8 338.31 144.52å² 5 -0.07 -1.75 0.56 0.38 0 4.07 sd1 c20h22n8o5 454.44 210.54å² 10 -1.85 -1.19 0.11 0.25 0 3.58 sd2 c62h111n11o12 1202.61 278.80å² 15 2.92 -8.15 0.17 0.79 0 10.00 m. f = molecular formular, m.w = molecular weight, #rb = rotatable bond, b.s = bioavailability score, s.a = synthetic accessibility c1=dehydrocarpaine ii, c2=chlorogenic acid , c3=dehydrocarpaine i , c4=coumaroylquinic acid, sd1=methotrexate , sd2=cyclosporine table 6. bioactivity properties of the selected ligands and standard drug with human janus kinase jak1 (pdb id: 6n7b) bioactivity c1 c2 c3 c4 sd1 sd2 autodock vina docking score (kcal/mol) -10.5 -8.6 -7.9 -7.9 -8.9 -8.0 ki (µm) 0.02 0.50 1.60 1.60 0.36 1.62 milog p 6.60 1.94 6.79 1.87 -1.97 3.6f1 ligand efficiency (le)/kcal/mol/heavy atom) 0.31 0.72 0.23 0.79 0.27 0.09 le scale 0.30 0.58 0.30 0.61 0.31 0.03 fit quality (fq) 1.04 1.25 0.78 1.30 0.88 2.96 ligand efficiency dependent lipophilicity (lelp) 21.37 2.71 29.22 2.37 -7.30 38.36 c1=dehydrocarpaine-ii, c2=chlorogenic-acid, c3=dehydrocarpaine-i , c4=coumaroylquinic-acid , sd1=methotrexate, sd2=cyclosporine table 7. bioactivity properties of the selected ligands and standard drug with tumor necrosis factor alpha (tnf alpha) (pdb id: 2az5) bioactivity c1 c2 sd1 sd2 autodock vina docking score (kcal/mol) -7.6 -7.5 -6.4 -4.3 ki (µm) 2.70 3.20 23.30 770.9 milog p 6.60 6.79 -1.97 3.61 ligand efficiency (le) /kcal/mol/heavy atom) 0.22 0.22 0.19 0.05 lescale 0.30 0.30 0.31 0.03 fit quality (fq) 0.75 0.74 0.63 1.59 ligand efficiency dependent lipophilicity (lelp) 30.338 29.92 10.16 71.36 c1=dehydrocarpaine ii, c2=dehydrocarpaine i, sd1=methotrexate, sd2=cyclosporine receptors, (pdb id: 2az5) and (pdb id: 6n7b), the major cytokines (tnfα) exacerbated in psoriasis, and inflammatory pathways particularly jak1 which are responsible for the initiation, progression, and exacerbating the disease’s development. the docking results of the passed ligands with both good admet and drug-likeness profiles were reported in table 3 and 4. dehydrocarpaine-ii had -10.5kcal/mol, chlorogenic-acid had 8.6kcal/mol, dehydrocarpaine-i and coumaroylquinic-acid had -7.9kcal/mol, cis-vaccenic had 6.9kcal/mol while methotrexate and cyclosporine had -8.9kcal/mol and -8.0kcal/mol binding energy values with the target protein (pdb id: 6n7b). dehydrocarpaine-ii and dehydrocarpaine-i had -7.6kcal/mol and -7.5kcal/mol while methotrexate and cyclosporine had table 8. pass prediction of the passed ligands and standards compounds pa pi activity chlorogenic-acid 0.52 0.02 antipsoriatic 0.6 0.03 antiinflammatory 0.7 0.02 immunosuppressant coumaroylquinic-acid 0.51 0.02 antipsoriatic 0.71 0.02 immunosuppressant 0.65 0.02 antiinflammatory methotrexate 0.23 0.11 antipsoriatic cyclosporine 0.86 0 immunosuppressant 0.42 0.19 antieczematic 0.27 0.09 antipsoriatic 0.28 0.18 antiinflammatory -6.4kcal/mol and -4.3kcal/mol binding energy values with the second target protein (pdb id: 2az5). this show that dehydrocarpaine-ii, chlorogenic-acid, and dehydrocarpaine-i have higher binding affinity than the two standard drugs, methotrexate and cyclosporine. 3.5. oral bioavailability analysis of the selected ligands and standard the compounds with good admet and drug-likeness profiles were docked with the choice target receptor. and the compounds that interact with the amino acid residue in the active site pocket were subjected to oral bioavailability analysis obtained through the swissadme web tool (http://www.swissadme.ch/) [26]. the bioavailability radar of the compounds and the standard is presented in figure 2, showing the pink area of the 7 abdul-hammed et al. / j. nig. soc. phys. sci. 5 (2023) 1116 8 table 9. receptor amino acids forming hydrogen bond and other electrostatic/ hydrophobic interaction with passed ligands compounds binding affinity (∆g), kcal/mol 6n7b receptor amino acids forming hbond ligands electrostatic/hydrophobic interactions involved inhibition constant (ki), µm chlorogenic acid -8.6±0.0 phe282, leu959, asn1008, arg1007, val889, leu1010, asp1021, lys908 0.50 coumaroylquinicacid -7.9±0.0 lys908, asp1021, gly887, asp1003, arg1007, glu925 gly1023 1.60 methotrexate -8.9±0.0 his918, gly887, phe886, asp1021, gly1020 ala906, leu1010, met956, gly1023, arg1007, asn1008, val889 0.36 cyclosporine -8.0±0.0 asp880, glu883, arg879, pro960 his918, asn1008, leu1010, ala906, val889, gly882, asp1021, asp921, phe958, leu959, arg1007, leu881, glu966, lys970, asp1003, 886 1.62 radar for the optimum zone for each of the properties (polar, flex, lipo, size, insolu, and insatu). the recommended ranges for the properties as revealed in table 4 are -0.7 and +5.0 for lipophilicity (xlogp3), 500g/mol for molecular weight (mw), 20-130 å2 for total polar surface area (tpsa), ≤6 for solubility (logs), 0.25-1.0 for fraction of carbon in the sp3 hybridization (insatu), and ≤9 for rotatable bond for an effective drug candidate [38]. the molecular weight (<500), as well as the solubility of water (esol logs) for the selected compounds, were analyzed in the acceptable range with an exception for sd2 (1202.61 g/mol). the partition coefficient (xlog p3), a very crucial parameter ranges for all the compounds from -0.07 to 5.66 with an exception for c3 (5.97). the saturation; a fraction of carbons in the sp3 hybridization range from 0.25 to 0.86 and both sd1 and sd2 has rotatable bonds of more than 9 while c2, c4, sd1, and sd2 failed the polarity with tpsa value of 164.75å², 144.52å², 210.54å², and 278.80å² respectively. c2 and c4 can still be orally bioavailable because they are not too flexible while the two standards are predicted not to be orally bioavailable, because too flexible and too polar [26]. the passed ligands are further subjected to other analyses. 3.6. bioactivity test of the selected ligands and standard drug table 3 reveals the bioactivity properties of the selected ligands and standards showing the ligand efficiency (le) with a recommended range of ≥0.3, fit quality (fq) with a recommended range of ≥0.8, and ligand efficiency dependent lipophilicity (lelp) with a recommended range of -10 to 10 [43], which was calculated using eqn, 2-5. all the selected ligands were reported in table 6 and 7, only c2 and c4 in table 6 has an excellent bioactivity profile with all their values within the recfigure 2. the bioavailability radar for the selected hit compounds and standards (c1) dehydrocarpaine-ii; (c2) chlorogenic-acid; (c3) dehydrocarpainei; (c4) coumaroylquinic-acid; (sd1) methotrexate; and (sd2) cyclosporine ommended range and are subjected to further analysis. ligand efficiency (le) = −(b.e)÷heavy atoms (h.a)(2) l.e scale = 0.873e − 0.026 × h.a − 0.064 (3) fq = le ÷ lescale (4) lelp = logp ÷ le (5) 3.7. prediction of activity spectra for substances (pass) biological activity prediction of the selected compounds and standard a computer-based program for an online web server pass software [27] was used for the prediction of the biological activity of the selected compounds. as shown in table 8 the 8 abdul-hammed et al. / j. nig. soc. phys. sci. 5 (2023) 1116 9 table 10. binding mode and binding interaction for passed ligands ligands binding interaction binding mode chlorogenic acid coumaroylquinic acid methotrexate cyclosporine value of the probability to be active must be greater than the probability to be inactive. this works in hand with the activity spectrum concerning the high probability to be active (pa) to the probability to be inactive (pa > pi). all the ligands in table 8 show excellent biological activity against psoriasis, chlorogenic-acid, and coumaroylquinic-acid displayed antipsoriatic activity, anti-inflammatory, and immunosuppressant activity. they both can be further explored in the development of novel drugs for the management, prevention, and curing of psoriasis. 9 abdul-hammed et al. / j. nig. soc. phys. sci. 5 (2023) 1116 10 3.8. binding mode and molecular interactions of the best hit compound and the standard in the lead optimization stage of drug development, the molecular interactions and binding mode involved in the binding of ligands to the target receptors’ active site are of utmost importance. it aids in improving the potency and efficacy of the selected hit compounds. notably, all analyses performed so far on the phytochemicals from carica papaya, chlorogenic-acid, and coumaroylquinic-acid showed outstanding results owing to their excellent binding affinities and inhibition constant, excellent admet properties, drug-likeness properties, bioactive, orally bioavailable analysis and pass analysis. the binding modes of chlorogenic-acid and coumaroylquinic-acid suggest that these compounds neatly fit at the active site of jak1 where lys908, arg1007, asn1008, leu959, gly887, asp1003, and asp1021 particularly stabilize these compounds through conventional h-bonding. hydrophobic/electrostatic interactions are also reported to participate and for 6n7b chlorogenic-acid, the hydrophobic interactions include val889, leu1010, asp1021, and lys908 while for 6n7b coumaroylquinic-acid we have gly1023. similarly, the standard drugs (methotrexate and cyclosporine) formed a conventional hydrogen bond with his918, gly887, phe886, asp1021, gly1020, and asp880, glu883, arg879, pro960. hydrophobic/electrostatic interactions with ala906, leu1010, met956, gly1023, arg1007, asn1008, val889 and his918, phe958, asn1008, leu959, leu1010, arg1007, ala906, leu881, val889, glu966, gly882, lys970, asp1021, asp1003, asp921, phe886. as expected, arg1007 and some other important amino acid residues are common to chlorogenicacid, coumaroylquinic-acid, and the standard drugs (methotrexate and cyclosporine) showing that they shared similar binding pockets and interactions with the active site of human janus kinase jak1. the molecular interaction and binding mode are displayed in the tables below. 4. conclusion the anti-psoriatic potential of carica papaya was explored via in silico studies. the structure-based screening was employed by using molecular docking simulation, admet profiling, lipinski rule of 5 (ro5), and other analysis for the target fishing of phytochemicals isolated from papaya against 2 possible targets of psoriasis. major cytokines, tumor necrosis factorα (tnf-α) exacerbated in psoriasis and inflammatory pathways particularly janus kinase 1 (jak 1). this computational analysis reflects that papaya can serve as excellent antipsoriatic and anti-inflammatory agents by targeting human antiinflammatory molecular targets (jak 1). the results obtained revealed chlorogenic acid (8.6 kcal/mol) and coumaroylquinic acid (7.9 kcal/mol) as probable inhibitors of janus kinase 1 (jak 1) compare to the two standard methotrexate (8.9 kcal/mol) and cyclosporine (8.0 kcal/mol) due to their excellent binding energies, admet profile, drug-likeness, oral bioavailability properties, pass properties, bioactivity, outstanding binding mode and molecular interactions with the target receptor and can serve as promising chemical scaffolds for the development and improvement of inhibitors to treat 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[43] a. l. hopkins, g. m. keserü, p. d. leeson, d. c. rees & c. h. reynolds, “the role of ligand efficiency metrics in drug discovery”, nature reviews drug discovery 13 (2014) 105. 11 j. nig. soc. phys. sci. 5 (2023) 1221 journal of the nigerian society of physical sciences solving fractional variable-order differential equations of the non-singular derivative using jacobi operational matrix m. basima, n. senua,b,∗, a. ahmadianc, z. b. ibrahimb, s. salahshourd ainstitute for mathematical research, universiti putra malaysia, 43400 upm, serdang, malaysia bdepartment of mathematics and statistics universiti putra malaysia, 43400 upm, serdang, malaysia cdecision lab, mediterranea university of reggio calabria, reggio calabria, italy dfaculty of engineering and natural sciences, bahcesehir university, istanbul, turkey abstract this research derives the shifted jacobi operational matrix (jom) with respect to fractional derivatives, implemented with the spectral tau method for the numerical solution of the atangana-baleanu caputo (abc) derivative. the major aspect of this method is that it considerably simplifies problems by reducing them to ones that can be solved by solving a set of algebraic equations. the main advantage of this method is its high robustness and accuracy gained by a small number of jacobi functions. the suggested approaches are applied in solving non-linear and linear abc problems according to initial conditions, and the efficiency and applicability of the proposed method are proved by several test examples. a lot of focus is placed on contrasting the numerical outcomes discovered by the new algorithm together with those discovered by previously well-known methods. keywords: fractional differential equations; atangana-baleanu caputo; variable order; operational matrix. article history : received: 23 november 2022 received in revised form: 31 january 2023 accepted for publication: 14 february 2023 published: 05 may 2023 c© 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: o. adeyeye 1. introduction over the past three decades, the most focus was placed on fractional calculus study, together with its countless implementations in the fields of engineering and physics. fractional differential equations (fdes) can effectively represent the implementations of fractional calculus employed in a variety of fields, which includes signal processing, optics, statistics and probability, electrochemistry of corrosion, control theory of dynamical systems, electrical networks, as well as chemical physics. there have been a number of important early papers on fractional derivatives and fdes, as may be seen in [1, 2]. these ∗corresponding author tel. no: +23480xxxx572 email address: norazak@upm.edu.my (n. senu ) publications offer a systematic explanation of fractional calculus, including its uniqueness and existence, and are regarded as the introduction with respect to the fdes and fractional derivative theory. numerous other scholars have recently focused on the findings of the initial value problem (ivp) and boundary value problem (bvp) solutions for fdes, which can be further read in [3-5]. it is crucial to determine approximate or exact solutions to fdes. we have trouble locating their analytical solutions for any but a small subset of these equations. many different types of differential equations in a variety of fields in science, engineering physical and natural applications can be solved, which are extremely effective [6-11]. other than that, many authors have been inspired to adopt these approaches for various equations because of their high accuracy and simplicity of usage. the collocation, galerkin, as well as tau methods 1 fulafia mis typewriter doi:10.46481/jnsps.2023.1221 basim et al. / j. nig. soc. phys. sci. 5 (2023) 1221 2 are particular spectral methods that are more suitable and frequently employed. when saadatmandi and dehghan [12] used spectral methods in solving multi-term linear as well as non-linear fdes numerically, they instigated the shifted legendre operational matrix with respect to fractional derivatives. to find approximate solutions for multi-term linear and non-linear fdes, doha et al. [13] developed a novel formula that explicitly expresses any fractional-order derivatives of shifted chebyshev polynomials of any degree with respect to the shifted chebyshev polynomials themselves. they combined this formula with tau and collocation spectral methods. recently, bhrawy et al. [14] treated multi-term linear fdes having variable coefficients employing a quadrature-shifted legendre tau technique. the shifted chebyshev operational matrix was recently presented by doha et al. [15] and used in conjunction with spectral methods to solve multi-term linear and non-linear fdes according to ivps and bvps conditions. additionally, in [16, 17], the authors introduced the spectral tau method for the numerical solution of a few fdes, while in [18], pedas and tamme created the spline collocation methods for solving fdes. in recent years, esmaeili and shamsi [19] instigated a direct solution method for solving a particular family of fractional ivps employing a pseudospectral method. moreover, esmaeili et al. [20] introduced a computational method with regard to the müntz polynomials and collocation method for the fdes solution. furthermore, the algorithms utilized in the current work are associated with those applied by saadatmandi and dehghan [12], doha et al. [13-15], as well as bhrawy et al. [14] to create accurate algorithms for a variety of uses. the classical jacobi polynomials, represented by ju,vi (x)(i ≥ 0, u, v > −1) [21] are crucial to the study and use of spectral methods and have been widely employed in mathematical analysis and real-world applications. the benefit of using general jacobi polynomials is that they may be used to calculate solutions using the jacobi parameters a and b (refer to [22]). therefore, to generalize, it is beneficial to perform a systematic study with regards to jacobi polynomials (u, v > −1) having general indexes. this may then be immediately implemented in other contexts rather than generating approximation findings for each specific indices pair. this is the reason we introduce the jacobi polynomials family having indexes u, v > −1 in this work. to solve numerically linear fractional and variable order problems with initial conditions, this work introduces the shifted jacobi operational matrix (jom) of fractional derivative. this method relies on the jacobi tau method. additionally, we present an appropriate technique for approximating the nonlinear fractional and variable order ivps on the interval [0, l] using the spectral shifted jacobi collocation approach relying on jom in order to determine the solution y(t). at (n − m + 1) points, the non-linear fractional and variable orders collocate. these equations produce (n + 1) non-linear algebraic equations that may be resolved employing newton’s iterative method after being combined with m initial conditions. finally, test problems are used to show how accurate the suggested algorithms are. we point out that saadatmandi and dehghan [12] and doha et al. [15] introduced the two shifted legendre and chebyshev operational matrices, correspondingly, and that several more extremely interesting situations may be produced directly as special cases emerging from the shifted jom [23]. as a result, we were inspired to pursue the shifted jacobi polynomials because it is the most generalized of the orthogonal polynomials. this paper is structured accordingly: first, we commence by going over several fundamental information about jacobi polynomials and fractional calculus theory that is necessary for supporting our findings in section 2. the jom for atanganabaleanu caputo (abc) is obtained in section 3. the spectral tau, jom of abc derivative, as well as collocation methods are all applied in section 4 to solve general linear and nonlinear abc. the variable-order abc-derivative jom is found in section 5. the suggested methods are used in several cases in section 6. in addition, section 7 provides a conclusion. 2. basic concepts and notations this section defines the caputo derivative, cf-derivative, as well as atangana-baleanu caputo (abc)-derivative in which fractional order and variable order are concisely highlighted. 2.1. fractional derivatives caputo fractional-order differential equation is expressed by [1] as: cdαy(t) = 1 γ(1 −α) ∫ t 0 y′(s)d s (t − s)α , (1) where 0 < α < 1. the most popular fractional derivative is the caputo derivative, which is frequently used in engineering and science domains. definition 2.1.1. to 0 < α < 1, y(t) ∈ h1(ı, ),  > ı, the fractional-order of the cf-derivative is defined by [29]: cfdαy(t) = m(α) 1 −α ∫ t 0 y′(s)e −α(t−s) 1−α d s, (2) in which m(α) denotes a normalization function. definition 2.1.2. for 0 < α < 1, y(t) ∈ h1(ı, ),  > ı, the fractional-order of the abc-derivative is represented by [30]: abcdαy(t) = m(α) 1 −α ∫ t 0 y ′ (s)eα [ −α(t − s)α 1 −α ] d s, (3) in which 0 < α < 1, m(α) resembles a normalization function, while eα denotes mittag-leffler function. to broaden the abc-derivative for the case of n < α < n + 1 having y(s)(a) = 0 f or s = 1, 2, ..., n, we have abcdαy(t) = abcdα(dny(t)) = m(α) 1 −α ∫ χ 0 y(n+1)(s)eα [ −α(t − s)α 1 −α ] d s, y(n+1)(s) = d(n+1)y(s) = ddαey(s), (4) in which dαe represents ceil α. 2 basim et al. / j. nig. soc. phys. sci. 5 (2023) 1221 3 definition 2.1.3. the fractional integral of the abc-derivative may be expressed by [30]: ab 0 i α t {y(t)} = 1 −α m(α) y(t)+ α m(α)γ(α) ∫ t 0 y(s)(t−s)α−1d s.(5) definition 2.1.4. the abc-derivative having variable-order α(t), 0 < α(t) < 1 of function y(t) may be expressed by [31]: abcdα(t)y(t) = m(α(t)) 1 −α(t) ∫ t 0 y ′ (s)eα(t) [ −α(t) 1 −α(t) (t − s)α(t) ] d s, (6) where eα(t) is the mittag leffer function. theorem 2.1. suppose β > 0. then, the variable-order abc-derivative may be expressed as [31]: abcdα(t)tβ = m(α(t)) 1 −α(t) γ(β+1)tβeα(t),β+1 [ −α(t) 1 −α(t) tα(t) ] .(7) 2.2. some properties of sjps the following recurrence formula can be used to create the famous jacobi polynomials, which are specified in the interval [−1, 1]: j(u,v)i (t) = (u + v + 2i − 1)[u2 − v2 + t(u + v + 2i)(u + v + 2i − 2)] 2i(u + v + i)(u + v + 2i − 2) j(u,v)i−1 (t) − (u + i − 1)(v + i − 1)(u + v + 2i) i(u + v + 2i)(u + v + 2i − 2) j(u,v)i−2 , for i = 2, 3, ..., where j(u,v)0 (t) = 1 and j (u,v) 1 (t) = u + v + 2 2 t + u − v 2 . we now define the shifted jacobi polynomials by instigating the change of variable t = 2tl − 1 to apply these polynomials with respect to the interval t ∈ [0, l]. let the shifted jacobi polynomials j(u,v)i ( 2t l − 1) be expressed by j(u,v)l,i (t). then, j (u,v) l,i (t) can be generated from: j(u,v)l,i (t) = (u + v + 2i − 1)[u2 − v2 + ( 2tl − 1)(u + v + 2i)(u + v + 2i − 2) 2i(u + v + i)(u + v + 2i − 2) j(u,v)l,i−1(t) − (u + i − 1)(v + i − 1)(u + v + 2i) i(u + v + i)(u + v + 2i − 2) j(u,v)l,i−2(t), i = 2, 3, ..., (8) where j(u,v)l,0 (t) = 1 and j (u,v) l,1 (t) = u+v+2 2 ( 2t l − 1) + u−v 2 . the analytic form with regards to the shifted jacobi polynomials j(u,v)l,i (t) of degree i may be expressed as j(u,v)l,i (t) = i∑ k=0 (−1)i−k γ(i + v + 1)γ(i + k + u + v + 1) γ(k + v + 1)γ(i + u + v + 1)(i − k)!k!lk tk, (9) where j(u,v)l,i (0) = (−1) i γ(i+v+1) γ(v+1)i! , j (u,v) l,i (l) = γ(i+u+1) γ(u+1)i! . of these polynomials, the most commonly utilized are the shifted chebyshev polynomials with respect to the first kind tl,i(t), the shifted legendre polynomials pl,i(t), as well as the shifted chebyshev polynomials with respect to the second kind ul,i(t). moreover, for the non-symmetric shifted jacobi polynomials, two essential special cases with regards to shifted chebyshev polynomials of third and fourth kinds vl,i(t) and wl,i(t) are also taken into account. the following relations connect these orthogonal polynomials with respect to the shifted jacobi polynomials. tl,i(t) = i!γ(0.5) γ(i+0.5) j (−0.5,−0.5) l,i (t), pl,i(t) = j (0,0) l,i (t), ul,i(t) = (i+1)!γ(0.5) γ(i+1.5) j 0.5,0.5 l,i (t), vl,i(t) = (2i)!! (2i−1)!! j (0.5,−0.5) l,i (t), wl,i(t) = (2i)!! (2i−1)!! j (−0.5,0.5) l,i (t). the orthogonality condition with respect to the shifted jacobi polynomials is expressed as ∫ l 0 j(u,v)l, j (t)j (u,v) l,k (t)w (u,v) l (t)dt = hk, (10) where w (u,v)l (t) = t v(l − t)u and hk =  lu+v+1 γ(k+u+1)γ(k+v+1) (2k+u+v+1)k!γ(k+u+v+1) , i = j, 0, i , j. let y(t) refers to a polynomial with degree n. now, we may write these in terms of shifted jacobi polynomials given by y(t) = n∑ j=0 c j j (u,v) l, j (t) = c t , (11) in which the coefficients c j are provided as follows c j = 1 h j ∫ l 0 w (u,v)l (t)y(t)j (u,v) l, j (t)dt, j = 0, 1, .... (12) suppose the shifted jacobi coefficient vector c, as well as the shifted jacobi vector φ(t), is expressed as ct = [c0, c1, ..., cn ], (13) 3 basim et al. / j. nig. soc. phys. sci. 5 (2023) 1221 4 and φ(t) = [j(u,v)l,0 (t), j (u,v) l,1 (t), ..., j (u,v) l,n (t)] t , (14) accordingly. therefore, the first-order derivative with respect to the vector φ(t) may be written as dφ(t) dt = d(1)φ(t), (15) in which d(1) denotes the (n + 1) × (n + 1) operational matrix of derivative written as d(1) = (di j) = c1(i, j), i > j,0 otherwise, in which c1(i, j) = lu + v(i + u + v + 1)(i + u + v + 2) j( j + u + 2)i− j−1γ( j + u + v + 1) (i − j − 1)!γ(2 j + u + v + 1) × 3f2  −i + 1 + j, i + j + u + v + 2, j + u + 1 ; 1 j + u + 2, 2 j + u + v + 2  (the proof can be found in [24], and the general definitions of a generalized hypergeometric series, as well as special 3f2, may be found in [25], accordingly on pp. 41 and 103–104). for instance, for even n, we obtain d(1) =  0 0 0 . . . 0 0 c1(1, 0) 0 0 . . . 0 0 c1(2, 0) c1(2, 1) 0 . . . 0 0 c1(3, 0) c1(3, 1) c1(3, 2) . . . 0 0 ... ... ... . . . ... ... c1(n, 0) c1(n, 1) c1(n, 2) . . . c1(n, n − 1) 0.  (16) 3. generalized sjps operational matrix to fractional calculus this section’s major goal is to expand the jacobi operational matrix (jom) of derivatives for atangana-baleanu caputo (abc) [32, 33]. theorem 3.1. suppose ψ(t) vector be sjps defined in eq.(11) such that α > 0. then abcdαψ(t) ' abcd(α)ψ(t), (17) in which dα denotes the operational matrix (m + 1) × (m + 1) that may be expressed as: let φ(t) vector be sjps defined in eq.(14). here, suppose α > 0, then the ϑ in om eq.(18) is obtained using sjps as follows ϑi, j,k = m(α) (1 −α) j∑ ł=0 (−1)i+ j−k+łγ(i + v + 1)γ(i + k + u + v + 1)γ( j + v + 1)γ( j + ł + u + v + 1) h jγ(k + v + 1)γ(i + u + v + 1)(i − k)!lkγ( j + v + 1)γ( j + u + v + 1)( j − ł)!ł!lł a j,ł. (19) proof. abcdαtv = m(α) 1 −α ∫ t 0 y (n+1)(s)eα [ −α(t − s)α 1 −α ] d s, v > 1, v > dαe = m(α) 1 −α ∫ t 0 γ(v + 1) γ(v − n) s v−n−1 eα [ −α(t − s)α 1 −α ] d s i f 0 < α < 1, bαc = n = 0, ψ(t) is solve f or ∫ t 0 s v−1 eα [ −α(t − s)α 1 −α ] d s abcdαju,vl,i (t) = i∑ k=dαe (−1)i− jγ(i + v + 1)γ(i + k + u + v + 1) γ(k + v + 1)γ(i + u + v + 1)(i − k)!k!lk abcdαtk = i∑ k=dαe m(α) 1 −α ψ(t) (−1)i−kγ(i + v + 1)γ(i + k + u + v + 1) γ(k + v + 1)γ(i + u + v + 1)(i − k)!lkγ(k)4 basim et al. / j. nig. soc. phys. sci. 5 (2023) 1221 5 where ψ(t) = µ∑ j=0 ak, j j u,v l, j(t) ak, j = 1 h j j∑ ł=0 γ( j + v + 1)γ( j + ł + u + v + 1) γ( j + v + 1)γ( j + u + v + 1)( j − ł)!ł!lł ∫ 1 0 ψ(t)w (u,v)l t łdt abcdαju,vl,i (t) = i∑ k=dαe µ∑ j=0 (−1)i−kγ(i + v + 1)γ(i + k + u + v + 1) γ(k + v + 1)γ(i + u + v + 1)(i − k)!k!lk ak, j j u,v l, j(t) = m∑ j=0 ( i∑ k=dαe ϑi, j,k)j u,v l, j(t). corollary 1. in the case of u = v = 0, it is apparent that the jom of derivatives for integer calculus aligns with legendre operational matrix of derivatives with respect to integer calculus as gained by saadatmandi and dehghan (refer [12] eq. (11)). corollary 2. in the case of u = v = −0.5, it is evident that the jom of derivatives for integer calculus aligns with chebyshev’s operational matrix of derivatives with respect to integer calculus as gained by doha et al. (refer [15] eq. (3.2)). 4. applications of the operational matrix of fractional derivative this section solves an fde in order to demonstrate the great significance of an operational matrix based on slps of fractional derivatives. we follow the same steps when using sjps. 4.1. linear fdes consider the linear fdes abcdαy(t) = b1 d βky(t) + ... + bk d β1y(t) + bk+1y(t) + bk+2q(t), for k = 1, 2, .... (20) the initial conditions are y(v)0 = dv, v = 0, ..., n, (21) in which bk denotes real constant coefficients with n < α ≤ n+1, 0 < β1 < β2 < ... < βk < α, in which abcdβ refers to the abcderivative of order β. to solve eq.(20), we present an approximation of the function q(t) ' m∑ i=0 qi j (u,v) l,i (t) = q t ψ(t), (22) abcdαy(t) ' ct abcd(α)ψ(t), (23) where q = [q0, ..., qm]t is a known vector. employing eqs.(22),(23) and (13), the residual rm(t) for eq.(20) can be written as rm(t) ' ( ct abcd(α) − bk+1c t − bk+2 q t ) ψ(t). (24) we now establish m − n linear equations as in a typical tau method by applying 〈rm(t),p j(t)〉 = ∫ 1 0 rm(t)pi(t)dt = 0, i = 0, 1, ..., m − n − 1. (25) moreover, by substituting eq.(13) and eq.(14) with eq.(21), we get y0 = c t ψ(0) = d0, y(1)0 = c t d(1)ψ(0) = d1, ... y(n)0 = c t d(n)ψ(0) = dn. (26) eqs. (25) and (26) generate (m−n) linear equations, which may be solved using arbitrary coefficients with respect to the vector c. 4.2. nonlinear fdes consider the non-linear fdes abcdαy(t) = f(t, y(t), dβ1 y(t), ...dβk y(t)). (27) the initial conditions are y(v)0 = dv, v = 0, ..., n, (28) in which n < α ≤ n + 1, 0 < β1 < β2 < ... < βr < α, as well as abcdα resembles the abc-derivative of order α. we put in mind that f can be non-linear in general. ct abcd(α)ψ(t) ' f(t, ct ψ(t), ct d(β1 )ψ(t), ..., ct d(βk )ψ(t)). (29) furthermore, upon substituting eq.(13) and eq.(14) with eq.(28), we now have y0 = c t ψ(0) = d0, y (v)0 = c t d(v)ψ(0) = dv, v = 1, 2, ..., n. (30) to discover the solution y(t), we collocate eq.(29) by employing first (m−n) points shifted legendre roots of p̄m+1(t). these equations along with eq.(30) establish (m + 1) non-linear equations, which can be resolved by employing newton’s iterative method. 5 basim et al. / j. nig. soc. phys. sci. 5 (2023) 1221 6 5. jom of variable-order abc-derivative this section is devoted to tackling the problem using the sjps om of variable order. λ(t) = [1, t, t2, ..., tn]t . (31) thus, the vector ψ(t) can be expressed as: ψ(t) = a(u,v)λ(t), (32) in which a(u,v) is (n + 1) × (n + 1) denotes a square matrix that specified by: (ai, j)0≤i, j≤n =  (−1)n−i γ(n+β+1)γ(n+i+α+β+1) γ(i+β+1)γ(n+α+β+1)γ(n−i+1)γ(i+1)li , i ≥ j, 0, otherwise. (33) hence, by employing eq.(32), we now have λ(t) = a−1ψ(t). (34) using the om for variable-order fractional differential operator dα(t)ψ(t), and eq.(32), we now have dα(t)ψ(t) = dα(t)(aλ(t)) = adα(t)[1, t, t2, ..., tι]t . (35) here, the atangana-baleanu caputo (abc) derivative with respect to the variable order provided in eq.(4) may be employed. then, we may obtain eq.(35) as given below: dα(t)ψ(t) = [0, γ(2) γ(1 −α(t)) t ∞∑ k=0 ( −α(t)1−α(t) t α(t))k γ(kα(t) + 2) , γ(3) γ(1 −α(t)) t2 ∞∑ k=0 ( −α(t)1−α(t) t α(t))k γ(kα(t) + 3) , ..., γ(ι + 1) γ(1 −α(t)) tι ∞∑ k=0 ( −α(t)1−α(t) t α(t))k γ(kα(t) + ι + 1) ]t , dα(t)ψ(t) = ab(t)λ(t), (36) in which b(t) =  0 0 0 . . . 0 0 γ(2) γ(1−α(t)) ∑ ∞ k=0 ( −α(t)1−α(t) t α(t) )k γ(kα(t)+2) 0 . . . 0 0 0 γ(3) γ(1−α(t)) ∑ ∞ k=0 ( −α(t)1−α(t) t α(t) )k γ(kα(t)+3) . . . 0 ... ... ... ... ... 0 0 0 . . . γ(ι+1) γ(1−α(t)) ∑ ∞ k=0 ( −α(t)1−α(t) t α(t) )k γ(kα(t)+ι+1)  (37) substituting eq.(34) into eq.(36), we get dα(t)ψ(t) = ab(t)a−1ψ(t), (38) in which ab(t)a−1 denotes the om of the variable-order abcderivative dα(t)ψ(t). here, the approximate solution may be given as dα(t)y(t) ' dα(t)(ct ψ(t)) = ct dα(t)ψ(t) = ct ab(t)a−1ψ(t),(39) ct ab(t)a−1ψ(t) = f[t, ct ψ(t), ct ad(1) a−1ψ(t), ..., ct ad(n) a−1ψ(t)], 0 ≤ t ≤ 1. (40) here, we employ the collocation points, tu = 2u+1 2n+2 , u = 0, 1, ..., n, in converting the system of equations given in eq.(40) into an algebraic equations system as follows: ct ab(tu)a −1 ψ(tu) = f[tu, c t ψ(tu), c t ad(1) a−1ψ(tu), ..., ct ad(n) a−1ψ(tu)], ct ψ(0) = y0 (41) ultimately, the arbitrary vector c in eq.(9) may be gained by solving the algebraic equations system provided in eq.(41). 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 t 0 0.05 0.1 y( t) a approximate exact 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 t 0 0.05 y( t) b approximate exact 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 t 0 0.05 y( t) c approximate exact figure 1: comparison between exact and approximate solution for (a)α = 0.95, (b)α = 0.9 and (c)α = 0.85 for example 6.2. example 5.1. suppose the following[36] abcdαy(t) = y2(t) − 2(t + 1)−2, 6 basim et al. / j. nig. soc. phys. sci. 5 (2023) 1221 7 α method t = 0.1 t = 0.3 t = 0.5 t = 0.7 t = 0.9 0.85 lom 2.19158e-3 2.89152e-3 4.26507e-3 4.40040e-4 3.13683e-3 prco 2.36581e-3 4.24947e-3 6.00249e-3 7.71664e-3 9.42102e-3 mtslp 1.08027e-2 5.75464e-3 3.35770e-3 1.33299e-3 5.46216e-4 jom(0,0.5) 3.00081e-3 5.85134e-4 2.97585e-3 2.07844e-3 1.57973e-4 jom(0.5,0) 3.28839e-3 1.50328e-3 5.29734e-3 2.26751e-3 1.67970e-3 0.9 lom 1.26158e-3 2.08766e-3 2.57667e-3 5.91395e-4 1.94142e-3 prco 1.18803e-3 2.39605e-3 3.56382e-3 4.72981e-3 5.90684e-3 mtslp 1.10428e-2 7.41747e-3 5.88865e-3 5.57577e-3 3.34146e-3 jom(0,0.5) 2.51233e-3 1.28073e-4 2.28507e-3 1.93179e-3 7.25019e-4 jom(0.5,0) 2.01129e-3 1.21534e-3 3.35330e-3 1.02220e-3 1.41226e-3 0.95 lom 4.88352e-4 1.16544e-3 1.10247e-3 5.09878e-4 8.67245e-4 prco 4.09639e-4 9.83473e-4 1.55920e-3 2.14596e-3 2.74713e-3 mtslp 9.83430e-3 8.58128e-3 7.92322e-3 7.35794e-3 6.82360e-3 jom(0,0.5) 1.77326e-3 6.45112e-4 1.54473e-3 1.53799e-3 1.01582e-3 jom(0.5,0) 8.75527e-4 7.70414e-4 1.55386e-3 1.52607e-4 9.24438e-4 table 1: the absolute error obtained by employing various values of α for example 6.2. α method t = 0.1 t = 0.3 t = 0.5 t = 0.7 t = 0.9 0.85 lom 8.14287e-4 1.92419e-3 1.00572e-2 4.67247e-2 1.15441e-1 prco 1.02731e-3 7.40800e-3 1.19908e-2 5.84856e-2 1.50751e-1 mtslp 4e-3 2.74688e-3 1.97787e-3 1.91297e-2 7.45086e-2 jom(0,0.5) 8.01348e-4 1.93026e-3 1.00587e-2 4.67324e-2 1.15433e-1 jom(0.5,0) 8.16107e-4 1.91932e-3 1.00499e-2 4.67267e-2 1.15445e-1 0.9 lom 6.70490e-4 1.42673e-3 8.20714e-3 3.66835e-2 8.89397e-2 prco 8.64541e-4 1.70507e-3 9.89488e-3 4.55878e-2 1.14652e-1 mtslp 4e-3 2.06924e-3 2.35421e-3 1.18382e-2 3.19227e-1 jom(0,0.5) 6.62878e-4 1.42923e-3 8.20792e-3 3.66879e-2 8.89366e-2 jom(0.5,0) 6.71528e-4 1.42406e-3 8.20355e-3 3.66839e-2 8.89426e-2 0.95 lom 4.13897e-4 7.86685e-4 5.01340e-3 2.15783e-2 5.13325e-2 prco 5.90418e-4 8.64405e-4 6.50201e-3 2.77896e-2 2.90733e-1 mtslp 4e-3 6.57765e-3 3.02391e-3 1.94353e-3 1.90389e-2 jom(0,0.5) 4.11399e-4 7.86925e-4 5.01366e-3 2.15797e-2 5.13320e-2 jom(0.5,0) 2.94860e-4 3.13253e-4 3.61277e-3 1.25339e-2 2.53848e-2 table 2: the absolute error obtained employing various values of α for example 6.3. for y0 = −2 and the exact solution y(t) = −2 (t+1) in case of α = t 0, figure 3 displays the approximate values of α = 0.85, 0.9, 0.95 and m = 6. a good approximation that is comparable to the exact answer can be obtained via an operation matrix based on sjps. 6. numerical examples the numerical examples of linear and non-linear fractionalorder and variable-order scenarios will acquire some solutions in this section. our computational findings will measure the difference between the exact and approximate solutions using absolute error. the matlab r2020b software is used to code and perform all of the numerical programs, whereas the cpu is for the next. • jom jacobi operational matrix method derived in this study. • cpskom chebyshev polynomials with respect to the second kind operational matrix method derived in this study. • lom legendre operational matrix method [26]. • prco predictor-corrector method provided in [27]. • mtslp mixture two-step lagrange polynomial as well 7 basim et al. / j. nig. soc. phys. sci. 5 (2023) 1221 8 α method t = 0.1 t = 0.3 t = 0.5 t = 0.7 t = 0.9 0.85 lom -1.74986 -1.50024 -1.35680 -1.23342 -1.13618 prco -1.60319 -1.52292 -1.41361 -1.32635 -1.25359 mtslp -1.67179 -1.45348 -1.31389 -1.22725 -1.17437 jom(0,0.5) -1.76123 -1.50623 -1.35787 -1.23557 -1.13710 jom(0.5,0) -1.74359 -1.49912 -1.35666 -1.23182 -1.13669 0.9 lom -1.78420 -1.52494 -1.36023 -1.22657 -1.11789 prco -1.65250 -1.54279 -1.42052 -1.32416 -1.24500 mtslp -1.78773 -1.58943 -1.45396 -1.35011 -1.26552 jom(0,0.5) -1.79249 -1.53108 -1.36213 -1.22826 -1.11925 jom(0.5,0) -1.77931 -1.52306 -1.36002 -1.22533 -1.11773 0.95 lom -1.82307 -1.54990 -1.35819 -1.21279 -1.09254 prco -1.71332 -1.56366 -1.42509 -1.31793 -1.23183 mtslp -1.89701 -1.64139 -1.47105 -1.34753 -1.25202 jom(0,0.5) -1.82560 -1.55437 -1.36040 -1.21334 -1.09386 jom(0.5,0) -1.82039 -1.54740 -1.35792 -1.21227 -1.09156 1 exact -1.81818 -1.53846 -1.33333 -1.17647 -1.05263 table 3: the approximate solutions obtained employing various values of α for example 6.4. α method t = 0.1 t = 0.3 t = 0.5 t = 0.7 t = 0.9 0.85 lom 0.13534 0.20345 0.24998 0.28275 0.31040 prco 0.19782 0.22782 0.27041 0.30270 0.32864 mtslp 0.164103 0.22650 0.25520 0.28130 0.30738 jom(0,0.5) 0.12887 0.20271 0.24847 0.28210 0.30963 jom(0.5,0) 0.13737 0.20354 0.25049 0.28286 0.31040 0.9 lom 0.11432 0.19514 0.24890 0.28804 0.31966 prco 0.16976 0.21614 0.26616 0.30394 0.33403 mtslp 0.10613 0.18237 0.23502 0.27718 0.31138 jom(0,0.5) 0.10899 0.19386 0.24741 0.28717 0.31881 jom(0.5,0) 0.11608 0.19533 0.24944 0.28817 0.31989 0.95 lom 0.08994 0.18737 0.24973 0.29612 0.33192 prco 0.13780 0.20414 0.26292 0.30695 0.34158 mtslp 0.051491 0.15966 0.23010 0.28087 0.32011 jom(0,0.5) 0.08644 0.18564 0.24848 0.29510 0.33113 jom(0.5,0) 0.09129 0.18767 0.25026 0.29628 0.33233 1 exact 0.08375 0.19264 0.26323 0.31422 0.35351 table 4: the approximate solutions obtained using different values of α for example 6.5. as the fundamental theorem with respect to fractional calculus stated in [28]. example 6.1. we now consider the bagley–torvik equation governing the motion of a rigid plate immersed in the newtonian fluid given as follows abcd1.5y(t) + d2y(t) + y(t) = t + 1. here, y0 = t0, y ′ 0 = t 0 and y(t) = t + 1 denotes the exact solution. using slps, the approximate solution for m = 3 is y(t) = [1.5 0.5 0 0]ψ(t) = t + 1, which equals to the exact solution. using sjps(0.5,0), the approximate solution for m = 3 is y(t) = [1.4 0.4 0 0]φ(t) = t + 1, which equals to the exact solution. for sjps(0,0.5), the approximate solution for m = 3 is y(t) = [1.6 0.4 0 0]φ(t) = t + 1, which equals to the exact solution. 8 basim et al. / j. nig. soc. phys. sci. 5 (2023) 1221 9 t error error error error error error lom prco mtslp cpskom jom(0,0.5) jom(0.5,0) 0.1 7.71674e-12 1.09802e-4 1.1e-2 2.09480e-9 8.56178e-11 2.50289e-11 0.2 7.77495e-12 4.53577e-4 2.24064e-2 3.26627e-9 8.53458e-11 3.76301e-11 0.3 8.81307e-12 1.09934e-3 3.46062e-2 3.66969e-9 8.97760e-11 3.88097e-11 0.4 2.77689e-11 2.12757e-3 4.60297e-2 3.46035e-9 2.89525e-10 3.28610e-11 0.5 3.48143e-11 3.61364e-3 5.77915e-2 2.79354e-9 3.63677e-10 2.40773e-11 0.6 1.56708e-11 5.63079e-3 6.74063e-2 1.82454e-9 1.62011e-10 1.67516e-11 0.7 4.39398e-11 8.25300e-3 7.82267e-2 7.08644e-10 4.65697e-10 1.51773e-11 0.8 1.58296e-10 1.15567e-2 8.50985e-2 3.98864e-10 1.66967e-9 2.36475e-11 0.9 3.41676e-10 1.56209e-2 9.75044e-2 1.34269e-9 3.60013e-9 4.64555e-11 table 5: the absolute error for example 6.6 for m = 4 t error error error error error erorr lom prco mtslp cpskom jom(0,0.5) jom(0.5,0) 0.1 1.75338e-4 6.54123e-4 e-2 1.75338e-4 1.75338e-4 1.75338e-4 0.2 3.38732e-3 3.95957e-3 2.70108e-2 3.38732e-3 3.38732e-3 3.38732e-3 0.3 9.96953e-3 9.35838e-3 3.72260e-2 9.96953e-3 9.96953e-3 9.96953e-3 0.4 1.88528e-2 1.27167e-2 4.22493e-2 1.88528e-2 1.88528e-2 1.88528e-2 0.5 2.93188e-2 8.50431e-3 4.70844e-2 2.93188e-2 2.93188e-2 2.93188e-2 0.6 4.06491e-2 8.72007e-3 5.81601e-2 4.06490e-2 4.06490e-2 4.06490e-2 0.7 5.21251e-2 4.36191e-2 8.11371e-2 5.21250e-2 5.21250e-2 5.21250e-2 0.8 6.30285e-2 1.00115e-1 1.20052e-1 6.30284e-2 6.30284e-2 6.30284e-2 0.9 7.26408e-2 1.82050e-1 1.77509e-1 7.26406e-2 7.26406e-2 7.26406e-2 table 6: the absolute error for example 6.7 for m = 4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 t -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 y( t) =0.85 =0.9 =0.95 exact figure 2: comparison between approximate solutions for α = 0.85, α = 0.9 and α = 0.95 with the exact solution for example 6.3. example 6.2. suppose we have the following model[34]: abcdαy(t) = −k(1 − y(t)), 0 < α < 1, and y0 = 0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 t -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1 y( t) =0.85 =0.9 =0.95 exact figure 3: comparison of approximate solutions for α = 0.85, α = 0.9 and α = 0.95 with the exact solution in case α = t0 for example 6.4. the exact solution is as follows: y(t) = −k(1 −α) m(α) − k(1 −α) eα ( kα m(α) − k(1 −α) tα ) + [ 1 − eα ( kα m(α) − k(1 −α) tα )] + m(α)y(0) m(α) − k(1 −α) eα ( kα m(α) − k(1 −α) tα ) . 9 basim et al. / j. nig. soc. phys. sci. 5 (2023) 1221 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 t 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 y( t) =0.85 =0.9 =0.95 exact figure 4: comparison between approximate solutions for α = 0.85, α = 0.9 α = 0.95 with exact solution in case of α = t0 for example 6.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 t 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 y( t) exact lom mtslp prco figure 5: the approximate solution and exact solution for example 6.6 for m = 4 table 1 and figure 1 compare the absolute error for the two methods for k is constant k = 0.1,α = 0.85, 0.9, 0.95, and m = 4 respectively. an enhanced approximate solution comparable to the exact solution can be obtained using an operation matrix based on sjps. example 6.3. suppose we have the following[35] abcdαy(t) = −y(t) + t4 − 0.5t3 − 3 γ(4 −α) t3−α + 24 γ(5 −α) t4−α, for y0 = 0, while the exact solution is expressed by y(t) = t4 − 0.5t3. table 2 and figure 2 compare the absolute error for the two approaches for α = 0.85, 0.9, 0.95, and m = 6. a good approximate solution that is comparable to the exact answer can be obtained using an operation matrix based on sjps. example 6.4. suppose we have the following[37] abcdαy(t) = (1 − y(t))4, 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 t 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 y( t) exact lom mtslp prco figure 6: the approximate solution and exact solution for example 6.7 for m = 4 the exact solution in case of α = t0 is expressed by y(t) = 1+3t−(1+6t+9t2 )1/3 1+3t and y0 = 0. the problem is solved with m = 8, and the numerical results are shown in figure 4. example 6.5. consider the following linear variable-order fdes[38]: abcdα(t)y(t) + ety(t) = et(t2 + t3 + 1) + m(α(t)) 1 −α(t) 2t2 eα(t),3(− α(t) 1 −α(t) tα(t)) + m(α(t)) 1 −α(t) 6t3 eα(t),4(− α(t) 1 −α(t) tα(t)), where α(t) = 0.5t + 0.1, y0 = t0 and the exact solution is provided by y(t) = t2 + t3 + 1. the absolute error for m = 4 is shown in table 3. an enhanced approximate solution that is comparable to the exact answer can be obtained using an operation matrix based on sjps. example 6.6. we now consider the following non-linear variable-order fdes given by[39]: abcdα(t)y(t) + y2(t) = t2 + t4 + 2t2−sin(t) γ(3 − sin(t)) . where α(t) = 0.5t + 0.6, y0 = 0 and the exact solution is expressed by y(t) = t2. the absolute error for m = 4 is shown in figure 5. operation matrix relying on sjps may give an enhanced approximate solution that is comparable with the exact solution. 7. conclusion we came up with a general formulation for the fractional and variable order jacobi operational matrix (jom), which is utilized to approximate atangana-baleanu caputo (abc) derivatives in numerical solutions. the shifting jacobi tau and 10 basim et al. / j. nig. soc. phys. sci. 5 (2023) 1221 11 collocation approaches served as the foundation for our strategy. since the abc fractional derivative enables the inclusion of 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[39] d. baleanu, “approximate solutions for solving nonlinear variable-order fractional riccati differential equations”, inst mathematics & informatics (2019). 11 j. nig. soc. phys. sci. 5 (2023) 1263 journal of the nigerian society of physical sciences generation of electricity from a hydraulic turbine in the djonou river (benin) mahouton justine carine adjassaa, gabin koto n’gobia,∗, hagninou elagnon venance donnoua, clément adéyèmi kouchadéa, basile bruno kounouhewaa alaboratoire de physique du rayonnement (lpr), university of abomey-calavi, abomey-calavi, 01 bp 526 cotonou, bénin abstract the shortage of electricity in rural areas despite the hydraulic potential they possess is becoming a challenge for benin. to date, nearly 140,000 people spread over the 42 lakeside villages of this country live in energy inaccessibility, insecurity and poverty. to overcome this situation, the present study is therefore interested in the production of electrical energy on an experimental basis in low water periods thanks to an archimedean screw turbine which operates at low flow rates and height of fall on the river. djonou located in southern benin a few kilometers from the university of abomey-calavi. the geometrical and hydraulic parameters of the screw were therefore determined and the device was modeled using autocard software. a prototype was then made with local recycled materials and tested on the river. the screw specifications indicate an inside and outside radius of 0.072 m and 0.135 m. the length of the screw was set at 0.46 m for a blade radius estimated at 0.137 m. the number of screw blades is equal to 2 with a flow rate of 0.049 m3/s. the inclination angle of the screw is 25◦. the device on the experimental site produces a voltage of 16 v and provides a current of about 0.12 a which can power a 2 w lamp. this performance of the prototype made on a small scale is a reliable indicator of the optimal use of this technology in the national hydraulic network of benin to supply populations with electrical energy. doi:10.46481/jnsps.2023.1263 keywords: hydroelectricity, flow, experimentation, energy inaccessibility, archimedean screw, low water period article history : received: 30 november 2022 received in revised form: 12 march 2023 accepted for publication: 14 march 2023 published: 21 may 2023 c© 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: b. j. falaye 1. introduction technological advances associated with population growth mean that the need for energy continues to increase. this problem is even more sensitive in rural areas where the use of conventional resources often proves to be very costly. in addition, there are several constraints, such as the geographical location of these areas and the very high cost of connecting electricity production sites to the conventional network, thus making the ∗corresponding author tel. no: +22997228700 email address: kotgabin36@yahoo.fr (gabin koto n’gobi ) search for an alternative energy source essential. recent studies and forecasts by the international energy agency (iea) alert us and inform us that the massive use of fossil energy resources will certainly lead to the total depletion of these reserves. the awareness of international opinion of the need to turn to green energies is therefore becoming more and more palpable in terms of the fight against global warming and the protection of the environment. the iea explains that renewable energy resources are like those derived from natural processes and replenished at a rate faster than they are consumed [1]. they can therefore meet the growing energy demand and are ready to 1 adjassa et al. / j. nig. soc. phys. sci. 5 (2023) 1263 2 provide the world with a reliable energy system [2, 3]. among these sources, hydroelectricity is the leading source of renewable electrical energy with an installed global capacity of approximately 4306 terawatt hours per year, i.e. 70% of global renewable energy production and 15.6% of global electricity production in 2019 [4]. unfortunately, the high cost of hydroelectric installations makes it difficult to access in underdeveloped countries, despite the fact that the latter are full of hydraulic resources. this is the case of benin which has an important hydraulic potential unexploited until now. it is therefore necessary to think of means of energy production by hydroelectricity at a lower cost [5, 6, 7]. the archimedean screw which is a technology that makes it possible to exploit low water flow rates and low head heights on hydroelectric sites such as small rivers or streams [8-16] and in areas remote areas that are difficult to access via the national electricity grid [17] is therefore an asset for benin. its design is environmentally friendly and allows fish and other aquatic life to pass through without danger; there is no requirement for deforestation, displacement of people and does not require construction of large dams, penstocks etc. [6, 13, 14, 18, 19]. in order to improve the performance of this technology, several investigations have been carried out in the literature. in the work of yulistiyanto et al. [20], maulana et al. [13], alkistis et al. [19], maulana et al. [15], abdul et al. [9], khan et al. [21], betancour et al. [22], abdillah et al. [23], imawati et al. [24], eswanto et al. [25], erinofiardi et al. [26] the authors studied the influence of the geometric and hydraulic parameters of the archimedes screw, namely the angle of inclination of the shaft, the number of blades, the variations in flow, the slope of the shaft , the direction of the axis of rotation, the speed of rotation, its angle of slope, the diameter ratio between the inner and outer diameters, the length of the axis and the pitch on the power and efficiency of the screw. according to the work of warjito et al. [6], the authors believe that the archimedes turbine has no fixed design theory. several theories and design-simulation procedures have therefore been developed for this turbine by certain authors such as alonso-martinez et al. [10], saroinsong et al. [11], purece, and corlan [12], dellinger et al. [14], slaboch et al. [18], fiardi [27], siswantara et al. [28], yulianto et al. [29], dragomirescu [30], thombare et al. [31], rosly et al. [32], abdullah et al. [33], muller [34], yulianto et al. [35], rohmer et al. [36], simmons et al. [37], erinofiardi et al. [38], yoosefdoost and lubitz [39], hedia et al. [40] followed by an on-site or laboratory experimentation phase discussed by some authors. ubando et al. [41] for their part have examined different methods of manufacturing archimedean screw turbines such as the 3d printing method still in their early stages of development and additive manufacturing as having a relatively lower environmental impact than conventional manufacturing of turbine blades. gogoi et al. [5] and kumar et al. [6] meanwhile have shown that the electrification scenario in rural areas can be improved specially where there is a continuous flow of a river or a canal with small water flow by the installation of the low-cost archimedes screw turbine. darmono and pranoto [42] investigated the numerical analysis of the effect of the number of threads on the turbine blades by the computational fluid dynamics method and ansys fluent software. velásquez et al. [43] used a gravitational vortex hydraulic turbine (gwvht) to determine the optimal position (h) of the slide to increase the efficiency of the hydroelectric plant using computational fluid dynamics (cfd). this hydroelectric technology is low head and has a vertical channel to extract energy from water vortices. based on these findings and to demonstrate the feasibility of using this technology in benin, this study aims to carry out and experiment with the archimedes screw on a small scale on a river in southern benin near the university of abomey-calavi. in a specific way in a first time, the geometrical and hydraulic parameters are determined. then a design of the system under autocard followed by a realization of the device with local materials of recovery are made. finally, on-site experimentation will make it possible to ensure the operation of the turbine and to identify a few electrical quantities. this test, which constitutes a first experience of an archimedean screw turbine in benin, will be carried out during low water periods when the water levels are low in order to better assess the efficiency of the device. 2. materials and methods 2.1. materials 2.1.1. study site the djonou river is located in the south of benin in the commune of abomey-calavi, arrondissement of godomey and about 1 km from the university of abomey-calavi. it crosses the houédonou bridge to reach lake nokoué, which is the largest lake in benin. it can be located at longitude 2◦18′44.291”e and latitude 6◦24′529”n. this area is characterized by two rainy seasons (april-july; october-november) and two dry seasons (december-march; august-september). the average interannual rainfall in the study area is 1100 mm [44]. this site was chosen to test the prototype that will be produced with the aim of producing electricity to supply a domestic load. figure 1 provides an overview of the study region and its location in africa. 2.1.2. data used various measurements were carried out on the study site during the low water period in order to determine the minimum hydraulic parameters of this river. the data collected are, among other things, the speed of the water flow and the depth of the river. the depth varies according to the bathymetry. to assess the flow in the absence of a measuring device, we had defined distances on the water several times and placed a light polystyrene object on it which will have to cover these predefined distances. the movement of the object between two positions was timed. note that this measurement was carried out several times in order to reduce measurement errors. from these two parameters, we estimated the average speed of water flow. from the section of the archimedes screw the flow rate of the water will be determined. table 1 gives some experimental values of the hydraulic parameters of the djonou river during the low water period. 2 adjassa et al. / j. nig. soc. phys. sci. 5 (2023) 1263 3 figure 1: geographical location of the djonou river in godomey in benin. location in africa table 1: some experimental values of hydraulic parameters land length speed of depth of close (m) flow (m/s) the lake (m) 0.5 0.62 0.80 1 0.65 1.25 1.5 0.71 2.5 2 0.79 3.5 2.5 0.83 4 3 0.96 4.8 4 1.25 5 2.2. methods 2.2.1. geometric and hydraulic parameters to determine the geometrical and hydraulic parameters of the archimedes screw, certain characteristics will be fixed to facilitate the study. in figure 2, the screw parameters are shown. the geometric parameters of an archimedean screw are: • the outer radius ra • the inner radius ri • the pitch of the screw s • the total length l • the threaded length lb • the number of blades n • the inclination of the screw β the hydraulic parameters are: • the inflow q • the geodesic head h in the work of [45], the author asserts that the screw performs well when the angle of inclination varies from 22◦ to 45◦ we therefore fixed the value of this angle at β = 25◦. the number of blades used for the design of the turbine is fixed at n equal to 2 referring to the work of maulana et al. [14] who showed that turbines with two blades have a more inclined pressure distribution so that it has better stability. the length of the screw lb is taken equal to 0.46 m. the geodesic drop height is set at 0.3 m depending on the topography of the site. finally, the outer radius ra is 0,135m. in order to determine the various geometric parameters of the screw, the determination of the radius ratio (ρ), the inclination ratio (λ), the volume ratio (v) and the volume ratio per revolution (λ.v) is paramount [8, 16, 17, 28, 45]: ρ = ri ra . (1) λ = s v tan (β) 2πra . (2) v = vu tan (β) πra2 s v , (3) where ri is the inner radius in m; s v denotes the surface in mm2. vu, the volume of the displaced fluid per revolution (m3), is a function of (λ.v ) [12, 46] and given by : vu = 2π2r3a (λ.v ) tan β . (4) the radius ratio ρ must of course be between 0 and 1 [46]. table 2 is a summary of the different values of these parameters depending on the number of blades. from eq.(1), we can deduce the inner radius ri(m) : ri = ρra. (5) the pitch s (m) which constitutes the distance between the blades is determined using the inclination ratio λ contained in table 2 [16, 36, 46]: s = 2piraλ tan (β) . (6) the determination of the distance between the trough and the screw s sp (m) is given by eq. (7) [36, 46]: s sp = 0.0045 √ 2ra. (7) 3 adjassa et al. / j. nig. soc. phys. sci. 5 (2023) 1263 4 figure 2: representation of the hydraulic and geometric parameters of an archimedean screw micropower [45] figure 3: modeling of the archimedean screw rod figure 4: modeling of the (a) central shaft of archimedean screw and (b) the threads around the shaft archimedean screw the dimensions of the trough r (m) are a function of the outside radius of the auger and the distance between the trough and the auger: r = s sp + ra. (8) the hydraulic parameters of the archimedean screw are deter4 adjassa et al. / j. nig. soc. phys. sci. 5 (2023) 1263 5 figure 5: (a) built-in support of the archimedean screw trough and (b) modeling of the archimedean screw contained in the trough figure 6: archimedean screen: (a) prototype of the archimedean screw made and (b) measurement of the tension during the experiment on the site table 2: archimedean screw ratio parameters for different numbers of blades [47] number of radius the inclination volume (m) volume ratio blades ratio ratio ratio per revolution (n) (ρ) (λ) (v) (λ.v ) 1 0.5358 0.1285 0.2811 0.0361 2 0.5369 0.1863 0.2747 0.0512 3 0.5357 0.2217 0.2697 0.0588 4 0.5353 0.2456 0.2667 0.0655 5 0.5352 0.2630 0.2647 0.0696 mined by eqs. (9-15). the volume flow q(m3/s) is calculated by: q = s vv. (9) with s v the area of the right session in (m2) and v the average speed of the fluid flow (m/s). the flow rate q of water flowing through an archimedean screw can be broken down as follows [28, 36]: q = qe + q f + qs, (10) where qe is the effective q f water flow, the leakage rate between the trough and the blades and qs the leakage rate due to overfilling. when the screw is under filled or at the optimal filling point, the flow rate qs is zero. several flow rates are involved in determining the operating flow rate of the screw qe. the most commonly used leakage flow model is that established for the archimedes screw pump. the leakage rate q f is given by equation (11) [34, 46] : q f = 5s spra √ 2ra. (11) the axial transport speed is given by equation (12) [12, 48]: cax = s n 60 , (12) where n(r pm) is the rotational speed of the archimedean screw which is given by [28, 46]: n = 56 (2ra) 2 3 , (13) with cax as the axial speed in m.tr/s, and p is the pitch of the screw. the average wetted surface s moy (m2) orthogonal to the axis of the screw is defined in order to express the flow rate qe as a function of cax: s moy = vu s n . (14) n is the number of blades or threads of the turbine. the flow rate qe is then given by [12, 48]: qe = nvu n 60 . (15) 5 adjassa et al. / j. nig. soc. phys. sci. 5 (2023) 1263 6 2.2.2. mechanical and electrical power the mechanical power of the screw shaft pm is determined from eq. (16) [12, 16, 33]: pm = ηtρwater gqh, (16) where ηt is the efficiency of the screw. in the case of this study, the efficiency of the screw is taken as 92% [32]. h is the geodesic drop height. the torque of the archimedes screw m (n.m) is given by eq. (17): m = 60pm 2πn . (17) the electrical power pe of an archimedean screw turbine is determined as follows: pe = ηg pm. (18) where ηg is the generator efficiency (95%). 3. results and discussion 3.1. results 3.1.1. the characteristics of the archimedean screw in table 3, the dimensions of the archimedean screw are summarized. table 3: archimedean screw dimension settings description dimension ri inner radius (m) 0.072 ra outer radius (m) 0.135 s pitch (m) 0.0107 β angle of inclination of the screw 25 s sp distance between the auger and the trough (m) 0.0023 s v surface of the trough (m2) 0.0592 r radius of the trough (m) 0.137 n rotational speed (rpm) 134 n number of blades 2 cax axial speed (m.tr/s) 0.023 vu per revolution (m3) 0.0053 v water velocity (m/s) 0.83 lb screw length 0.46 s moy average wet surface (m2) 0.46 q f the leak rate (m3/s) 0.000806 qe effective water flow (m3/s) 0.0236 qs the overfill flow (m3/s) 0.024 q river flow (m3/s) 0.049 h drop height (m) 0.30 pm mechanical power (w) 133 m the torque at the screw (n.m) 9.48 pe theorical electrical power (w) 126 the values observed in this study were compared with those obtained by other authors in the literature who have designed and experimented with the archimedean screw (table 4). the geometric and hydraulic characteristics of archimedean screws collected in the works of brada, 1993, 1999 [49, 50], lashofer et al. [51, 52, 53], lubitz et al. [54], lyons [55], yulistiyanto et al. [20], maulana et al. [12], saroinsong et al. [8], alonso-martinez et al. [7], khan, et al. [17], rohmer et al. [36], erinofiardia et al. [38] and dellinger et al. [48] indicate that the interior and exterior radius of the archimedean screw vary respectively from 0.030 m to 0.525 m and from 0.055 m to 0.265 m. the pitch of the screw is between 0.054 m and 1.22 m with a number of screw blades ranging from 1 to 10. the angle of inclination of the screw is generally chosen between 17◦ and 45◦ the length of the screw and the water flow rate can reach 5.3 m and 1.2 m3/s respectively. drop heights varying from 0 to 2.5 m are encountered with rotation speeds of up to 395 rpm. except for the pitch of the screw, these parameters are similar and close to those obtained in this study. these values therefore confirm the results of the present study. 3.1.2. modeling of the archimedean screw under autocard the archimedean screw is the main element of this design because it is the basis for the production of electrical energy. but it is really essential to know that it is she who produces the mechanical energy thanks to the potential energy of the water which causes the latter in its rotation. figure 3 gives an overview of the design of the archimedean screw rod in autocard. figures 4 and 5 show the modeling of the archimedean screw shaft, the threads around the screw shaft, the incorporated support of the trough and the contained archimedean screw respectively. in the trough. 3.1.3. practical realization the description of the essential elements having participated in the realization of the device is as follows: • nets: we used number 45 polyvinyl chloride (pvc) to make the threads (blade) of the screw, taking into account the geometric parameters mentioned. similarly, we used polyvinyl chloride (pvc) number 16 with its covers to make the central axis of the screw on which we place the threads of the screw. pvc was used not only because it is light and easy to move with a small amount of water on its surface, but also because its maintenance will be very simple compared to other materials such as aluminum; • drive shaft: we used a metal rod for the screw drive shaft. it connects the screw to the generator via other elements in order to transmit the rotation speed to the generator; • bolts: they allowed us to fix certain elements of the turbine to their different locations without friction. the elements fixed by the bolts are among others: the shaft of the turbine, the bicycle chainring and the driving tooth; • speed multiplier: we used a number 12 bicycle cog that we attached to the shaft of the screw to drive the generator with a chain and another motor cog with a radius smaller 6 adjassa et al. / j. nig. soc. phys. sci. 5 (2023) 1263 7 table 4: comparacian de las especificaciones para cada diseao del sistema. authors inner outer screw threaded number screw debit drop rotation radius radius pitch length of blades inclination (m3/s) height speed (m) (m) (m) (m) (◦) (m) (rpm) brada [49, 50] 0.525 0.265 1.05 5.3 3 26-34 0-0.35 1.8-2.2 48-79 lashofer et al. [51, 52, 53] 0.403 0.18-0.22 0.8-1.2 3 3-5 18-32 0.02-0.22 0.5-1.7 20-80 lubitz et al. [54]; lyons [55] 0.038 0.078 0.117-0.2 0.584 3 17-35 0.0004-0.0012 0.14-0.28 0-280 yulistiyanto et al. [20] 0.076 0.142 0.22 2 35 0.00364 maulana et al. [16] 0.077 0.143 0.287 2 0.025 295 saroinsong et al. [12] 0.030 0.055 0.132 0.055 3 30 395 alonso-martinez et al. [11] 0.032 0.56 0.972 3.20 3 22 1.2 < 2 73.2 abdullah et al. [33] 0.07 0.13 0.07 1 1 30-45 khan, et al. [21] 0.426 0.80 1.22 5.2 1-10 0.82 2.2 rohmer et al. [36] 0.21 0.42 0.96 3 30 0.15 2.5 erinofiardia et al. [38] 0.032 0.142 0.054 0.646 1 22 0.0012 0.25;0.38;0.41 106 dellinger et al. [48] 0.052 0.09 0.192 0.40 3 18-30 0.001-0.004 0-0.35 present study 0.072 0.135 0.0107 0.46 2 25 0.049 0.30 134 than that of the bicycle. thanks to this association, the transmission ratio will be 8, which means that when the archimedes screw turns once, the generator turns eight times in order to have a high rotation speed, hence the speed multiplier effect; • driving chain: we used the power chain not only because these links are better suited to our system but also because it is available; • generator: it receives the mechanical work provided by the screw to produce electrical energy continuously. we used a generator with 40 w power, 310 v dc voltages, 1500 rpm rotation speed and 0.129 a current; • trough: the trough is the most important element for the safety of auger users. it is the one that will inhabit the archimedes screw and will also reduce the sound noise produced by the archimedes screw. we incorporated the trough as well as the deflector in an aluminum support in order to facilitate the movement of the turbine. figure 6 shows a display of the multimeter (voltage) when the screw is moving following the flow of water on the djonou river. during the experimental phase, the device was able to power a 2 w electric lamp under a voltage of 16 v by supplying a current of approximately 0.12 a. the power produced is estimated at 1.92 w for a flow rate of 0.049 m3/s. 3.2. discussion by comparing the electrical quantities obtained in this study with the experiments carried out in the literature on similar and small-scale devices, we note in the work of yulistiyanto et al. [20], fiardi et al. [27], maulana et al. [16], saroinsong et al. [12], abdullah et al. [33] that the authors obtained output powers estimated respectively at 16.23 w (61.61%); 0.098w; 116.10 w (0.025 m3/s; 55%); 16.97w (350 rpm); 9.03 w (2.06 10−3 m3/s; 72%). similarly, the experimental performances of the screw turbine for very low head hydroelectric resources are presented in the work of erinofiardia et al. [38]. the screw turbine with an outer diameter of 142 mm and a water flow of 0.0012 m3/s with a head of 0.25 m, can produce a maximum power of 1.4 w with an efficiency 49% at 22◦ bank angle. these different powers recorded on small-scale archimedean screw turbines are corroborated by the results obtained in this study. however, some power values are higher than the values presented for this study. the optimal determination of the height of fall, the length of the screw using the angle of inclination and the investigation of the relationship of the input speed to the angular speed of a wheel on the yield, could therefore avoid the overflow leaks noted on our device (leakage rate at overfilling evaluated at 0.024 m3/s) due to loading and thus improve the performance of the turbine produced. 4. conclusion in this study, an archimedean screw hydraulic turbine was designed, built and tested on the djonou river in benin. the characteristics of the device made from local recycled materials made it possible to measure a few electrical quantities, in particular the voltage and the intensity of the current. the main results of this work can be summarized as follows: • the geometrical parameters of the archimedes screw turbine indicate an internal and external radius evaluated respectively at 0.072 m and 0.135 m. the number of screw blades is equal to 2 with the radius of the trough estimated at 0.137 m. the threaded length is 0.46 m for an inclination angle of 25◦; • the hydraulic parameters give a flow rate of 0.049 m3/s for a fall height of 0.3 m; • the theoretical maximum electrical power of the device is 126 w. during the experimental phase on site, the device produced a voltage of 16 v and provides a current 7 adjassa et al. / j. nig. soc. phys. sci. 5 (2023) 1263 8 intensity evaluated at 0.12 a which made it possible to power a lamp of 2w for a flow of 0.049 m3/s. the experimental power is estimated at 1.92 w. this experimentation, which constitutes a pilot phase which will result in large-scale production, requires improvements in order to increase the performance of the archimedes screw, particularly in terms of its geometric parameters. in the future, we are therefore thinking of modifying the geometry of the screw in order to study its impact on electricity production. acknowledgments the authors of this article sincerely thank the bachelor school in renewable energies of the faculty of science and technology (fast) of the university of abomey-calavi (uac) through the “laboratoire de physique du rayonnement (lpr)” and the mastercard of abomey-calavi university for funding our participation to the 3rd german-west african conference on sustainable and renewable energy systems susres at the university of kara in togo. references [1] s. i. akinsola, a. b. alabi, m. a. soliu, & t. akomolafe, “optimization of method and components of enzymatic fuel cells”, nig. soc. phys. sci. 1 (2019) 143. 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[55] m. w. k. lyons, “lab testing and modeling of archimedes screw turbines”. phd thesis, the university of guelph (2014). 9 j. nig. soc. phys. sci. 3 (2021) 262–266 journal of the nigerian society of physical sciences novel developments of zno/sio2 nanocomposite: a nanotechnological approach towards insect vector control ezra abbaa,∗, zaccheus shehub,c, danbature wilson lamayib, kennedy poloma yoriyoa, rifkatu kambel dogarab, nsor charles ayuka adepartment of zoology, faculty of science, gombe state university, gombe, pmb 127, gombe, nigeria bchemistry department, faculty of science, gombe state university, gombe, pmb 127, gombe, nigeria cdepartment of chemistry, college of natural science, makerere university, p.o. box 7062, kampala abstract recently, there is increasing efforts to develop newer and effective larvicides to control mosquito vectors. this study was carried out to examine the efficacy of zno/sio2 nanocomposite synthesized using gum arabic against culex quinquefasciatus larvae. the elemental composition, morphology, functional groups and surface plasmon resonance of the zno/sio2 nanocomposite was analyzed by energy dispersive x-ray analysis (edx), scanning electron microscope (sem), ftir and uv-visible spectroscopy respectively. in bioassay, larvae were exposed to three different concentrations of synthesized zno/sio2 nanocomposite. the mortality rates at concentrations of 10, 20 and 25 were found to be (70%, 80%, 86%), (56%, 64%, 84%) and (44%, 48%, 76%) for 1st , 2nd , and 3rd instar respectively. this study revealed that the synthesized zno/sio2 nanocomposite exhibit high larvicidal activity; 1st instar (lc50=4.024, lc90= 39.273 mg/l), 2nd instar (lc50=8.767, lc90=51.069 mg/l) and 3rd instar (lc50=13.761.lc90=81.809 mg/l) doi:10.46481/jnsps.2021.198 keywords: culex quinquefasciatus, vector control, nanotechnological, zno/sio2 nanocomposite article history : received: 15 april 2021 received in revised form: 05 july 2021 accepted for publication: 24 july 2021 published: 29 august 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: t. o. owolabi 1. introduction application of chemical insecticides in order to kill mosquito larvae or pupae in the water is known as larviciding. larviciding is generally more effective and target-specific than adulticiding (applying chemicals to kill adult mosquitoes). the most common synthetic chemicals used in controlling mosquitoes’ larvae are methoprene, pyrethroids, diflubenzuron, malathion, dichlorodiphenyltrichloroethane (ddt), organophosphate temephos ∗corresponding author tel. no: email addresses: gemanamsly@gmail.com (ezra abba ), ezra.abba@gmail.com (nsor charles ayuk) and as well as phytochemicals [1-3]. however, synthetic chemicals (insecticides) are known to cause serious environmental problem thereby killing non-target organism and affecting human health. moreover, continuous application of synthetic chemicals (insecticides) results in control failures due to development of resistance by the mosquitoes (vectors) [4]. hence, it has been reported that zno and sio2 nanoparticles provides a lay down of a novel green nanotechnology to control insect pest including mosquitoes’ larvae [5-8]. zno/sio2 nanocomposites have been synthesized using various techniques such as chemical vapor deposition, sputtering, chemical etching and sol gel process and they are used in different applications such as an262 ezra et al. / j. nig. soc. phys. sci. 3 (2021) 262–266 263 timicrobial, photonic crystals, photocatalysts, gas sensors, vacuum fluorescent display and varistors etc., [9-19]. but based on our search, there was no report on the effect of zno/sio2 nanocomposites against mosquito larvae except for the individual zno and sio2 nanoparticles. culex quinquefasciatus is a vector of lymphatic filariasis. the breeding site of culex species includes; gutters and water retention sites having organic matter. filariasis has been reported to be a public health problem in africa as well as other part of the world [20-22]. thus, to prevent mosquito bornediseases and improve the quality of public health, it is necessary to control mosquito larvae. in this study, zno/sio2 nanocomposite was synthesized using gum arabic and was tested against the larvae of culex quinquefasciatus. the formation of zno/sio2 nanocomposite was confirmed using ultraviolet– visible (uv–vis) spectrophotometry, scanning electron microscopy (sem) coupled energy dispersive x-ray (edx) spectroscopy and fourier transforms infrared (ftir) spectroscopy. 2. materials and methods 2.1. collection of gum arabic (acacia senegalensis) a fresh a. senegalensis extrudes were collected from billiri local government of gombe state. the gum extract were neatly collected and allowed to dry properly under the sun. the gum arabic was crushed to powder using pestle and mortar. 2.2. synthesis of zno/sio2 nanocomposite one gram (1 g) of gum arabic was poured into a beaker containing 40 ml of distilled water which was magnetically stirred for 10 minutes at 90◦c. following this, 2 g of zn(no3)6 .6h2o and 2 g of silica gel were added and stirred for 30 minutes. upon addition of zn(no3) 6 .6h2o and silica gel the aqueous solution changed to milk colour. with time the solution became viscous. a cloudy formation at the bottom of the beaker indicated the formation of resin. the resin obtained was placed in a furnace at 450 oc for two hours to obtain zno/sio2 nanocomposite [23]. 2.3. characterization of the green synthesized zno/sio2 nanocomposite zno/sio2 nanocomposite was characterized using ultravioletvisible spectroscopy, fourier transformed infrared spectroscopy (ft-ir) and scanning electron microscopy (sem) coupled with energy dispersed x-ray techniques. 2.4. collection of culex quinquefasciatus larvae culex quinquefasciatus mosquito larvae were collected from different areas of gombe metropolis. using ladle and a collection bottle, the ladle was lowered into the water (breeding site) at an angle of about 45o until one side is just below the surface of the water. while dipping, care was taken not to disturb the larvae which may cause them to swim downward. the larvae were maintained and fed in the laboratory with glucose for the larvicidal bioassay. the collection was done based on previous literature [23]. figure 1. uv-visible spectrum for zno/sio2 nanocomposite. 2.5. larvicidal activity of zno/sio2 nanocomposite the test was carried out according to our previous work [24]. 0.1g of zno/sio2 nanocomposite was weighed and diluted with distilled water in a 1000 ml volumetric flask and shaked to obtain 100 mg/l concentration. the bioassay was done by placing different instars (1st – 3rd ) of the larvae into 200 ml of plastic container with four replicates and a control in each of the instars, each replicate comprised of twenty-five larvae. 100 ml of dechlorinated water was added in each of the replicates. finally, 10 mg/l, 20 mg/l and 25 mg/l of the zno/sio2 nanocomposite concentrations were inoculated into each of the replicates. and percentage mortality was calculated as follows: 2.6. statistical analysis percentage mortality, probit analysis, chi square and correlation analysis were calculated and tabulated using (spss, 2016). 3. result and discussion 3.1. ultraviolet visible analysis absorption spectrum of synthesized zno/sio2 nanocomposite at different wave lengths ranging from 260 to 380 nm revealed the maximum absorption wavelength of 280 nm, (figure 1). elsewhere, maximum absorption wavelength of 300 nm was reported for cao/sio2 nanocomposite [25]. this optical property is in the same range with the one in the current. 3.2. ft-ir analysis figure 2 depict the ft-ir spectrum of zno/sio2 nanocomposite analyzed from 450-4000 cm−1 which exhibited prominent peaks at 3457.75, 1654.65, 1067.44, 701.43, and 455.97 cm−1. the band at 1067.44 cm−1 corresponds to asymmetric stretching vibration of si-o-si bond. the peaks at 701.43 cm−1 corresponds to si-oh bond [9-19,26-28]. the band at 3457.75 cm−1 indicates ho-h stretching mode for silanol group and adsorbed water. and band at 1654.65 cm−1 indicates bending mode of adsorbed water. the zn-o and si-o bond is indicated by the peak at 455.97 cm−1 [919, 25-27]. thus, this information confirmed the formation of zno/sio2 nanocomposite. 263 ezra et al. / j. nig. soc. phys. sci. 3 (2021) 262–266 264 figure 2. ft-ir spectrum of zno/sio2 nanocomposite. table 1. larvicidal activity of synthesized zno/sio2 nanocomposite on culex quinquefasciatus larvae. lc50 ; lethal concentration that kills 50% of larvae, lc90 ; lethal concentration that kills 90% of larvae, r: correlation coefficient; χ2 ; chi square instars conc (mg/l) mortality % mortality lc50 (mg/l) lc90 (mg/l) χ2 r 1st instar 10 17.5 70 4.024 39.273 0.076 0.999 20 20 80 25 21.5 86 2nd instar 10 14 56 8.767 51.069 1.543 0.908 20 16 64 25 19 84 3rd instar 10 11 44 13.761 81.809 2.75 0.826 20 12 48 25 19 76 3.3. sem/edx analysis figure, 3a depict sem image, 3b depict elemental data and 3c depict edx spectrum of zno/sio2 nanocomposite. the sem image showed large and dispersed particles of silica coated zno nanoparticles. the edx spectrum showed different element apart from the expected zn, si and o. the presence of other element is due to gum arabic used in the synthesis because gum arabic has been reported to contained; al, ba, ca, fe, k, mg, mn, p, s and sr [28]. from figure 3b, percentage composition of si, zn and o were 38.02, 37.42 and 7.21 % respectively. 3.4. larvicidal test result: the exposure of culex quinquefasciatus larvae to different concentrations of the synthesized zno/sio2 nanocomposite for 24hrs demonstrates their larvicidal efficacy. table 1 show that larval mortality significantly increased with the increase in concentrations of zno/sio2 nanocomposite. the mortality rates of concentrations; 10, 20 and 25 mg/l for 1st instar were 70%, 80%, 86%, 2nd instar were 56%, 64%, 84%, and 3rd instar were 44%, 48% and 76% respectively. this study revealed that the synthesized zno/sio2 nanocomposite larvicidal activity decreases from 1st instar to 3rd instar. thus, lethal concentrations of the nanocomposite on the larvae of culex quinquefasciatus were found to be (lc50=4.024 mg/l, lc90= 39.273 mgl/1), (lc50=8.767 mg/l, lc90=51.069 mg/l) and (lc50=13.761 mg/l, lc90=81.809 mg/l) for 1st, 2nd , and 3rd instar respectively. in our previous studies, ag-co and cu/ni bimetallic nanoparticles were synthesized through green pathway and it larvicidal activities were tested against culex quinquefasciatus larvae. the lc50 for 1st, 2nd , and 3rd instars were 5.237, 9.310 and 13.626 mg/l respectively [29]. and the lc50 for 1st, 2nd , and 3rd instars were 14.75, 18.25 and 18.50 mg/l respectively [30]. hence, zno/sio2 nanocomposite proved to be more effective against culex quinquefasciatus larvae than ag-co and cu/ni bimetallic nanoparticles as reported by [29-30]. 264 ezra et al. / j. nig. soc. phys. sci. 3 (2021) 262–266 265 figure 3. 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[30] l. d. wilson, z. shehu, a. j. mai, b. magaji, m. m. adam & m. a. bunu, “green synthesis, characterization and larvicidal activity of cu/ni bimetallic nanoparticles using fruit extract of palmyra palm”, int. j. chem. mater. res 8 (2020) 20. 266 j. nig. soc. phys. sci. 1 (2019) 72–81 journal of the nigerian society of physical sciences original research simulation and optimization of lead-based perovskite solar cells with cuprous oxide as a p-type inorganic layer d. elia,d,∗, m. y. onimisia, s. garbab, r. u. ugbea, j. a. owolabia, o. o. igea, g. j. ibeha, a. o. muhammedc adepartment of physics, nigerian defence academy, kaduna, nigeria bdepartment of chemistry, nigerian defence academy, kaduna, nigeria cdepartment of physics, bayero university, kano, nigeria ddepartment of physical sciences, greenfield university, kaduna, nigeria abstract the hole transporting material (htm) is responsible for selectively transporting holes and blocking electrons which also plays a crucial role in the efficiency and stability of perovskite solar cells (pscs). spiro-meotad is the most popular material, which is expensive and can be easily affected by moisture contents. there is need to find an alternative htm with sufficiently high resistance to moisture content. in this paper, the influence of some parameters with cuprous oxide (cu2o) as htm was investigated using solar cell capacitance simulator (scaps). these include the influence of doping concentration and thickness of absorber layer, the effect of thickness of etm and htm as well as electron affinities of etm and htm on the performance of the pscs. from the obtained results, it was found that concentration of dopant in absorber layer, thickness of etm and htm and the electron affinity of htm and etm affect the performance of the solar cell. the cell performance improves greatly with the reduction of etm electron affinity and its thickness. upon optimization of parameters, power conversion efficiency for this device was found to be 20.42 % with current density of 22.26 macm−2, voltage of 1.12 v , and fill factor of 82.20 %. the optimized device demonstrates an enhancement of 58.80 %, 2.25 %, 20.40 % and 30.23 % in pce, jsc, ff and voc over the initial cell. the results show that cu2o in lead-based psc as htm is an efficient system and an alternative to spiro-meotad. keywords: perovskite solar cells, inorganic htm, device simulation, cuprous oxide, defect density article history : received: 26 april 2019 received in revised form: 16 may 2019 accepted for publication: 18 may 2019 published: 30 august 2019 c©2019 journal of the nigerian society of physical sciences. all rights reserved. communicated by: b. j. falaye 1. introduction the ecxiting properties, including tuned band gap, small exciton energy, excellent bipolar carrier transport, long charge diffusion, and amazingly high tolerance to defects [1-7], perovskite halides have demonstrated promising abilities for a numerous of optoelectronic applications, including photovolataics, ∗corresponding author tel. no: +2348063307256 email address: danladielibako@gmail.com (d. eli ) light-emmision, photodetectors, x-rays imaging, lasers, gamma ray detection etc [8-14]. perovskite solar cells (pscs) based on lead have demonstrated remarkable breakthrough in almost a decade since after its invention due to its advantages of low cost, high efficiency and simple fabrication process. its efficiency has grown from 3.9 % in 2009 to over 23 % in late 2018 [16, 17]. despite its remarkable attainment, these power conversion efficiencies are still low as compared to inorganic solar cells such as crystalline silicon (c−s i, 25.7 %), gallium arsenide (gaas, 28.8 %) 72 d. eli et al. / j. nig. soc. phys. sci. 1 (2019) 72–81 73 etc [18]. methyl ammonium lead iodide (ch3nh3pbi3) with a band gap of 1.50 ev that covers absorption within wide range of visible spectrum was reported by various experimental and theoretical studies [18, 19]. generally, psc is made up of hole transporting layer, electron transporting layer and absorber layer. the function demonstrated by each layer in psc should be understood in order to enhance the performance of the device [20]. the most routinely used electron transporting material (etm) is t io2 because of its suitable energy level for electron injection, high electron mobility, good stability and environmental friendliness [3, 4, 7, 18]. it is often a difficult task to make good choice of hole transporting materials which are needed for extracting holes effectively from the perovskite layer while preventing electrons from recombination. the most commonly used hole transport material is spiroometad which is organic in nature [21]. it is made up of basically two additives, 4-tert-butylpyridine (tbp) and bis (trifluoromethane) sulfonamide lithium salt (li-tfsi), which are used to improve the conductivity and hole mobility of spiromeotad. the most commonly used htm demonstrate hygroscopic nature, tendency to crystalize, and vulnerability to both moisture and heat, as such must be replaced with a cost effective and stable htm having high hole mobility with ease of synthesis [18, 22, 23, 24]. robust metal oxide [25, 26], carbon [27, 28], and other inorganic materials [18, 29] have shown outstanding behaviours in stabilizing the device, but in the meantime, the optimization of pce in these devices is still necessary for accelerating the commercialization. inorganic p-type semi-conductor such as cu2o is considered to be an alternative to organic htms [30]. pce has been greatly enhanced and reached up to 11.03 % when cu2o film was prepared via a facile process of cu sputtering and controlled thermal oxidation [30, 31]. except for experimental work, it is also equally important to investigate all aspects of the device theoretically in order to fully understand the device mechanism and optimize the device performance. considering cu2o as htm in lead based pscs, very few works have been demonstrated so far. for example, perovskite (ch3nh3pbi3) solar cells with cu2o as htm was simulated using scaps, while only the effect of thickness of the absorber on the performance of pscs was investigated [32, 33]. a device model that involve the simulation of various htms with cu2o inclusive, was done but with no sufficient investigation on various parameters (only thickness of absorber) was carried out [34]. in addition to the thickness of the absorber, there are also many other important parameters which could affect the performance of pscs. these include doping concentration in the absorber layer, thickness of the etm and electron affinity of etm and htm. for example, proper choice of suitable electron affinity of etm and htm can prevent exiton quenching at the interface, thus can assist in enhancing device performance. as such, a comprehensive study of these parameters needs to be investigated in order to uncover further understanding and thus improve device performance. in this paper, simulation of lead based ch3nh3pbi3 pscs with cu2o as htm and tio2 as etm was done with scaps. the influence of all above mentioned parameters on the performance of pscs were studied systematically. 2. device simulation parameters the structure of our simulated psc is considered with layer configuration of glass substrate/tco (transparent conducting oxide)/tio2 (etm) absorber layer ch3 n h3 pbi3/cu2o (htm)/ metal back contact. the structure and the band diagram is shown in figure 1 (a) and (b). from the band structure, the valence band offset at the ch3nh3 pbi3/cu2o interface is +0.08 ev , which can be considered beneficial for the flow of holes to the back-metal contact in order to avoid their recombination with the electrons in the perovskite layer. the conduction band offset is +0.30 ev at the tio2 /ch3nh3 pbi3 interface, which is also necessary for the flow of photo excited electrons to the front electrode. neutral gaussian distribution defect is selected in the absorber layer and characteristic energy is set to be 0.1 ev [18]. two defect interfaces are inserted for carrier recombination. one defect interface is tio2/ch3nh3pbi3 and the other one is ch3nh3pbi3/cu2o. the nature of the defect is set as gaussian and defect density is set as 1 × 1018 cm−3 [18, 32]. table 1 shows the defect parameters which are used in the simulation. basic parameters for each material used in the simulation are summarized in table 2. thermal velocities of hole and electron are selected as 107 cms−1 [18, 32, 35, 36]. the optical reflectance is considered to be zero at the surface and at each interface [18]. parameters are optimized in the study by using control variable method. the initial total defect density of the absorber layer is assumed to be 2.5 × 1013 cm−3. the current density–voltage curve has been drawn with these initial parameters as shown in figure 2(a). the short-circuit current density (jsc) of 21.77 macm−2, open-circuit voltage (voc) of 0.865 v , fill factor (ff) of 68.27 %, and pce of 12.86 % are obtained. the simulated device performance is consistent with the experimental results of lead-based pscs [30, 31]. this consistency shows that input parameters are valid and close to the real device. in the incident photonto-current efficiency (ipce) of the device shown in figure 2(b) which is featured with a high platform between 300 nm and 850 nm with the maximum of 90 % at 570 nm. optical absorption edge is red shifted to 800 nm which corresponds to a band gap of 1.55 ev in ch3 n h3 pbi3. the ipce covers the whole visible spectrum which is closer to the experimental work [30, 31]. 3. results and discussion 3.1. influence of doping concentration (na) of absorber layer in order to enhance the performance of solar cells, doping is a key process considered. depending upon the type of dopants, doping can either be n-type or p-type. like the other crystalline semiconductors, the shallow point defects in absorber could cause unintentional doping at room temperature. the performance of psc can be enhanced by introducing appropriate 73 d. eli et al. / j. nig. soc. phys. sci. 1 (2019) 72–81 74 table 1: defect parameters of interfaces and absorber [18, 32] parameters ch3 n h3 pbi3 t io2/ch3 n h3 pbi3 interface ch3 n h3 pbi3/cu2o interface defect type neutral neutral neutral capture cross section for electrons (cm2) 2 × 10−15 2 × 10−16 2 × 10−15 capture cross section for holes (cm2) 2 × 10−15 2 × 10−16 2 × 10−15 energetic distribution gaussian single single energetic level with respect to ev(ev ) 0.500 0.650 0.650 characteristic energy (ev ) 0.1 0.1 0.1 total density (cm−3) 1 × 1015 − 1 × 1019 1 × 1018 1 × 1018 table 2: simulation parameters of pscs devices parameters fto etm (t io2) absorber htm (cu2o) thickness (µm) 0.4 0.05 0.45 0.15 band gap energy eg (ev ) 3.5 3.26 [32] 1.55[32] 2.17[32] electron affinity χ(ev ) 4.0 4.2[32] 3.9[18] 3.2[32] relative permittivity �r 9 10 6.5 7.11[32] effective conduction band density nc(cm−3) 2.2 × 1018 2.2 × 1018[32] 2.2 × 1018[32] 2.2 × 1018[32] effective valance band density nv(cm−3) 2.2 × 1018 2.2 × 1018[32] 2.2 × 1018[32] 2.2 × 1018[32] electron mobility µn(cm2v−1 s−1) 20 20[18, 32] 2[32, 33] 80[32, 39] hole mobility µp(cm2v−1 s−1) 10 10[18, 32] 2[32, 33] 80[32, 39] donor concentration nd (cm−3) 1 × 1019 1 × 1017 0 0 acceptor concentration na (cm−3) 0 0 1 × 1013[7, 32] 1 × 1018[18, 32] defect density nt (cm−3) 1 × 1015 1 × 1015[18, 32] 2.5 × 1013[18, 32] 1 × 1015[18, 32] figure 1: (a) the structure of perovskite solar cell in the simulation and (b) energy level diagram of cu2o in the device dopant in absorber layer [18, 37]. the self-doping process can be adopted for nor p-type doping in absorber layer. it has been demonstrated experimentally that n-type or p-type self-doping in ch3 n h3 pbi3 lead towards the manipulation of carrier density, majority carrier type and charge transport by changing the thermal annealing or precursor ratios in the solutions [37, 38]. formation of ch3 n h3 pbi3 involves organic and inorganic precursors named methyl ammonium iodide (mai) and lead iodide (pbi2). the ratio between precursors (pbi2/mai) decides the doping of the absorber. upon thermal annealing, pbi2 rich absorber layer is n-doped and pbi2 deficit absorber layer is pdoped [39]. furthermore, ch3 n h3 pbi3 is unstable in air and humidity. when moist air comes in contact with device then pbi2 is generated and oxidation state of lead is changed. this process is the cause of introducing impurities in absorber layer. the effect of doping concentration on the performance of perovskite solar cell is studied by choosing the values of na in the range of 1014–1017cm−3. table 3 gives the pce of psc with various values of doping concentration. it is worth noting that pce is maximum when the value of na is 1 × 1015 cm−3. jsc also has the same behaviour. the results above demonstrate that charge carriers are transported and collected more efficiently at the same irradiance when na of the absorber is 1 × 1015 cm−3. therefore, proper selection of na is necessary for the improvement of performance of pscs. on the other hand, jsc and voc decrease when values of na increases beyond 1×1015 cm−3. the variation in the cell performance with the doping concentration can be explained in terms of built-in electric field which is enhanced with the increase of doping concentration. the charge carriers are separated and increased by the increase of electric field resulting in the enhanced performance of pscs [18, 40]. the decrease in jsc with increasing doping concentration could be explained from the perspective of auger recombina74 d. eli et al. / j. nig. soc. phys. sci. 1 (2019) 72–81 75 figure 2: (a) j–v curve of psc with initial parameters, (b) ipce spectra of the device with initial parameters tion. auger recombination rate increases with further increase of doping density beyond 1 × 1015 cm−3. it is also clear that total recombination rate also increases when doping density increases beyond 1 × 1015 cm−3. the scattering and recombination increases due to increasing doping density thus suppressing hole transportation [18, 41]. therefore, optimum doping density enhances the voc and jsc which in turn increases the pce. while further increase in doping density is not favourable due to high recombination and scattering. there should be lower carrier concentration in lead perovskite so that carrier mobility can increase within the absorber. the optimum performance with jsc of 22.10 macm−2, voc of 0.85 v , ff of 73.97 % and pce of 13.82 % is obtained under the doping density of 1 × 1015 cm−3. the comparison is shown between j–v curves with different value of na in figure 3(b). with the optimization, pce was enhanced by 7.38 %, and jsc increases 1.52 %, as compared with the device having initial value of na = 1 × 1013cm−3. figure 3(a) shows the simulation results by changing the value of doping concentration from 1×1015 to 1×1017cm−3 with respect to photovoltaic parameters (pce, voc, jsc, and ff). figure 3: (a) variation in performance parameters of psc with doping concentration of absorber, (b) j–v curves of psc with different values of doping concentration. 3.2. influence of electron affinity of etm and htm one of the important factor considered in t io2/ perovskite/ cu2o is band offset which becomes a determining factor as to the carrier recombination at the interface and is the measure of voc. by varying the values of electron affinities of t io2(3.7–4.3 ev ) and cu2o(3.1–3.7 ev ), the band offset can be adjusted. figures 4(a) and 5(a) show variation of pce, voc, jsc and ff with electron affinity of etm and htm respectively. the values of 3.7 ev and 3.3 ev give the best pce for t io2 and cu2o respectively. when the electron affinity of etm is high (greater than 3.7 ev ), then voc and jsc decrease slightly. pce of 20.29 %, jsc of 22.55 macm−2, voc of 1.10 v and ff of 81.72 % were obtained upon optimizing value of electron affinity of etm, as shown in table 4 and pce of 13.11 %, jsc of 21.87 macm−2, voc of 0.87 v and ff of 69.31 % were obtained upon optimizing value of electron affinity of htm, as shown in table 5. it is evident that proper etm and htm selection with suitable electron affinity can reduce the recombination of car75 d. eli et al. / j. nig. soc. phys. sci. 1 (2019) 72–81 76 table 3: dependence of solar cell performance on the doping concentration of absorber layer parameters na(cm−3) jsc(macm−2) voc (v ) ff pce (%) 1014 21.80 0.86 69.02 12.99 1015 22.80 0.85 73.97 13.82 1016 21.96 0.79 76.43 13.21 1017 19.20 0.60 73.45 8.40 riers and performance of pscs can further be optimised [42]. figure 4: (a) variation in performance parameters of psc with electron affinity of etm, (b) j–v curves of psc with different values of electron affinity of etm. 3.3. influence of thickness of etm and htm figure 6(a) is the plot of solar cell parameters; voc , js c , ff and pce versus thickness of the etm; t io2. in both cases voc , js c , ff and pce are gradually decreasing due to fractional absorption of incident light by the t io2 layer, the bulk recombination and surface recombination at the interface [15]. thickness of etms has been varied from 0.001 to 0.160 µm which shows a decrease in photovoltaic parameters with increase in etm thickness, as shown in table 6. similarly, figfigure 5: (a) variation in performance parameters of psc with electron affinity of htm, (b) j–v curves of psc with different values of electron affinity of htm. ure 6(b) shows reverse case as voc , js c , ff and pce increase with increase in htm up to 0.02 µm. above 0.02 µm we noticed a constant value for voc , js c , ff and pce, which means the thickness that gives optimum performance is from 0.04 to 0.16 µm. the slightly increase with increase in thickness up to 0.02 µm suggests the higher conductivity of the t io2 and partial absorption of the light. pce of 15.52 %, jsc of 22.10 macm−2, voc of 1.01 v and ff of 69.81 % are obtained at a thickness of 0.001 µm which is the optimized value of htm thickness and pce of 12.87 %, js c of 21.77 macm−2, voc of 0.87 v and ff of 68.27 % are obtained, as shown in table 7. 76 d. eli et al. / j. nig. soc. phys. sci. 1 (2019) 72–81 77 table 4: dependence of solar cell performance on the electron affinity of etm parameters ea(ev ) jsc(macm−2) voc (v ) ff pce (%) 3.7 22.50 1.10 81.72 20.29 3.8 22.49 1.10 81.20 20.10 3.9 22.25 1.09 77.50 18.78 4.0 22.07 1.05 71.14 16.49 4.1 21.93 0.97 69.21 14.64 4.2 21.77 0.87 68.27 12.87 4.3 21.58 0.77 67.36 11.13 table 5: dependence of solar cell performance on the htm parameters ea(ev ) jsc(macm−2) voc (v ) ff pce (%) 3.1 21.59 0.87 64.10 12.01 3.2 21.77 0.87 68.27 12.87 3.3 21.87 0.87 69.31 13.11 3.4 21.88 0.87 68.48 12.96 3.5 21.74 0.86 61.94 11.64 3.6 21.54 0.83 55.03 9.90 3.7 21.28 0.75 51.43 8.16 table 6: dependence of solar cell performance on the etm parameters t (µm) jsc(macm−2) voc (v ) ff pce (%) 0.0010 22.10 1.01 69.81 15.52 0.0025 22.05 0.97 69.37 14.89 0.0050 22.00 0.95 69.12 14.37 0.0100 21.93 0.91 69.84 13.81 0.0200 21.86 0.89 68.50 13.27 0.0400 21.79 0.87 68.33 12.93 0.0800 21.73 0.86 68.27 12.81 0.1600 21.64 0.86 68.30 12.76 table 7: dependence of solar cell performance on the htm parameters t (µm) jsc(macm−2) voc (v ) ff pce (%) 0.0010 21.23 0.82 55.61 9.66 0.0025 21.24 0.82 55.73 9.76 0.0050 21.29 0.84 56.41 10.12 0.0100 21.51 0.87 62.00 11.58 0.0200 21.76 0.87 68.15 12.84 0.0400 21.77 0.87 68.27 12.87 0.0800 21.77 0.87 68.27 12.87 0.1600 21.77 0.87 68.27 12.87 table 8: dependence of solar cell performance on the absorber parameters t (µm) jsc(macm−2) voc (v ) ff pce (%) 0.2 17.57 0.83 74.16 10.78 0.3 20.32 0.85 71.75 12.32 0.4 21.43 0.86 69.51 12.83 0.5 21.99 0.87 67.05 12.82 0.6 22.20 0.87 64.72 12.52 0.7 22.21 0.88 62.50 12.22 0.8 22.10 0.88 60.48 11.82 0.9 21.93 0.87 58.67 11.41 77 d. eli et al. / j. nig. soc. phys. sci. 1 (2019) 72–81 78 table 9: optimized parameters of the device optimized parameters etm(t io2) absorber (ch3 n h3 pbi3) htm(cu2o) doping density (cm−3) – 1 × 1015 – electron affinity (ev ) 3.7 – 3.3 thickness (µm) 0.0010 0.4000 0.1600 table 10: photovoltaic parameters of cu2o based perovskite solar cells reported in the experimental work in the literature and simulated results using scaps. simulation jsc(macm−2) voc (v ) ff pce (%) initial 21.76 0.86 68.26 12.86 optimized na of absorber 22.09 0.85 73.97 13.82 optimized thickness of absorber 21.43 0.86 69.51 12.83 optimized ea of etm 22.55 1.10 81.72 20.29 optimized ea of htm 21.87 0.87 69.31 13.11 optimized thickness of etm 22.10 1.00 69.81 15.52 optimized thickness of htm 21.77 0.87 68.27 12.87 final optimization 22.26 1.12 82.20 20.42 [30] 17.50 0.95 66.20 11.03 [31] 19.02 0.99 73.63 13.97 figure 6: (a) variation in performance parameters of psc with thicknss of etm, (b) j–v curves of psc with different values of thickness of etm. figure 7: (a) variation in performance parameters of psc with thickness of htm, (b) j–v curves of psc with different values of thickness of htm. 3.4. influence of thickness of absorber layer there is another parameter, thickness of absorber layer, which affects the performance of solar cell. the influence of thickness78 d. eli et al. / j. nig. soc. phys. sci. 1 (2019) 72–81 79 figure 8: (a) variation in performance parameters of psc with thickness of absorber, (b) j–v curves of psc with different values of absorber thickness. figure 9: j–v curves of psc with optimized parameters. of absorber on the solar cell parameters; voc , js c , ff and pce is shown in figure 8(b). pce is lower when thickness of the layer is too small (0.2 µm) due to the poor light absorption. pce of pscs increases with the increase of the thickness of the absorber 0.20 to 0.40 µm before it starts decreasing. for thickness beyond 0.4 µm, the collection of photo generated carriers decreased because of charge recombination. the pce of the device increases when thickness of the absorber layer increases. pce decreases when thickness is larger than 0.40 µm. considering, the effect of thickness of the absorber, the optimized parameters are pce of 12.83 %. js c of 21.43 ma/cm2, voc of 0.86 v , and ff of 69.51 %, as shown in table 8. it is evident from literature that pin hole free structure of methyl ammonium lead iodide perovskite can be obtained by using dimethyl sulfoxide (dmso) and using polyethylene glycol (peg) also gives a better effects on the surface morphology [18, 43, 44]. by using solvent retarding method (sr), optimal thick and uniform perovskite film can be deposited [45]. 3.5. performance of optimized parameters at the end, considering all the factors as doping concentration, electron affinity and thickness, we obtained pce to be 20.42 % with current density of 22.26 macm−2, voltage of 1.12 v , and fill factor of 82.20 %, which shows an improvement of 58.80 %, 2.25 %, 20.40 % and 30.23 % in pce, jsc, ff and voc over the initial cell. the final optimized parameters and optimised j–v curve are shown in table 9 and figure 9 respectively. we compared our simulated results with the experiment work published by other researchers and the related data is summarized in table 10. in the literature, the best efficiency of 11.03 % has been achieved for pscs with cu2o as htm. the voc , ff and js c still need to be increased to achieve 20.42 % efficiency. this could be achieved by further improving the film morphology and crystalline quality of both the absorber and cu2o layer. doping of cu2o by replacing either part of cu or part of o by other element might/can further modify the charge carrier concentration and mobility of htm. 4. conclusion in this work, the lead-based perovskite solar cells with cu2o as htm are studied by one dimensional simulation programme. the results show that optimum doping concentration in the absorber layer gives improved pce. high values of doping concentration leads to decrease of pce due to higher recombination rates. to reduce the recombination rates at the interfaces, proper selection is made for the electron affinity of etm and htm. by choosing the electron affinity of etm as 3.7 ev , pce of pscs increases from 12.86 % to 20.29 %, and by choosing the electron affinity of htm as 3.3 ev , pce of pscs increases from 12.86 % to 13.11 %. with the optimised thickness of 0.001 µm, for etm layer, the pce of the device increases from 12.86 % to 15.52 %. with the optimised htm thickness of 0.16 µm, thus, pce increases up to 12.87 %. the overall pce, ff, js c , and voc , of 20.42 %, 82.20 %, 22.26 macm−2, 79 d. eli et al. / j. nig. soc. phys. sci. 1 (2019) 72–81 80 and 1.12 v respectively were obtained by using all optimised parameters. the results show that cu2o as alternate htm has the potential to be used with ch3 n h3 pbi3 and can replace the spiro-meotad which is costly htm for perovskite solar cell. acknowledgments we thank the referees for 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[45] z. yuan, y. yang, z. wu, s. bai, w. xu, t. song, x. gao, f. gao, and b. sun, “approximately 800 nm thick inhole-free perovskite films via facile solvent retarding process for efficient planar solar cells”, acs applied materials interfaces 8 (2016) 34446. 81 j. nig. soc. phys. sci. 3 (2021) 395–405 journal of the nigerian society of physical sciences covid-19 risk factors, economic factors, and epidemiological factors nexus on economic impact: machine learning and structural equation modelling approaches david opeoluwa oyewolaa, emmanuel gbenga dadab,∗, ndunagu juliana ngozic, terang a. u.a, akinwumi s. a.a adepartment of mathematics and computer science, federal university kashere, gombe, nigeria bdepartment of mathematical sciences, university of maiduguri, maiduguri, nigeria cdepartment of computer sciences, national open university of nigeria, nigeria abstract since the declaration of covid-19 as a global pandemic, it has been transmitted to more than 200 nations of the world. the harmful impact of the pandemic on the economy of nations is far greater than anything suffered in almost a century. the main objective of this paper is to apply structural equation modeling (sem) and machine learning (ml) to determine the relationships among covid-19 risk factors, epidemiology factors and economic factors. structural equation modeling is a statistical technique for calculating and evaluating the relationships of manifest and latent variables. it explores the causal relationship between variables and at the same time taking measurement error into account. bagging (bag), boosting (bst), support vector machine (svm), decision tree (dt) and random forest (rf) machine learning techniques was applied to predict the impact of covid-19 risk factors. data from patients who came into contact with coronavirus disease were collected from kaggle database between 23 january 2020 and 24 june 2020. results indicate that covid-19 risk factors have negative effects on epidemiology factors. it also has negative effects on economic factors. doi:10.46481/jnsps.2021.173 keywords: covid-19, structural equation modelling, latent variables, random forest, boosting. article history : received: 14 march 2021 received in revised form: 27 may 2021 accepted for publication: 11 september 2021 published: 29 november 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: b. j. falaye 1. introduction the negative impact of covid-19 is felt by everyone in one way or another. the pandemic has created a situation whereby some people are more likely to experience severe illness because they have medical conditions that increase their risk. these are commonly called risk factors. examples include age, race, gender, poverty and overcrowding, certain occupations and pregnancy [1]. epidemiologic factors are definable entities that have ∗corresponding author tel. no: the potential to bring about a change in a health condition or other defined outcome [2], while macroeconomic factors are a trend or condition that comes from or applies to a broad aspect of an economy rather than a certain population. common macroeconomic factors include gross domestic product, the rate of employment, phase of business cycle, rate of inflation and money supply [3]. from the time when it was first discovered in wuhan in december 2019, the 2019 novel coronavirus also known as covid19 has quickly transmit to all regions, metropolises, and au395 oyewola et al. / j. nig. soc. phys. sci. 3 (2021) 395–405 396 tonomous provinces in china and has infected many countries in asia, europe, oceanic, north america, south america, and africa [4]. this novel virus has posed a serious challenge to preventing the spread of the deadly virus disease in many countries and regions and has had a great impact on economic, financial, commercial, and social development [5]. during the peak of this pandemic, most cities in different nations of the world have embraced closed management techniques which make businesses to operate on their own with little or no influence from the outside world. the purpose of this is to prevent further transmission of the virus and to lower the likelihood of more patients being infected [6, 7]. however, in the past few months, due to the unproductivity of most business and industrial activities in nations of the world, the great number of poor people have been confined to their house, and this has resulted to several social, economic and financial problems. it has also had an enormous impact on the economic and financial development of nations of the world [7]. covid-19 is hypothetically a single-stranded enclosed viruses with an dna with size of around 26 – 32 kilobyte [8]. world health organization (who) declared coronavirus as global public health emergency on 30 january 2020. this is because of to the sudden eruption of respiratory disorder. the novel coronavirus was classified by who as severe acute respiratory syndrome coronavirus 2 (sars-cov-2) and was termed the coronavirus disease 2019 (covid-19) [9]. respiratory symptoms, fever, dry cough, fatigue, sputum production, shortness of breath, sore throat, headache, myalgia or arthralgia, chills, nausea or vomiting, nasal congestion, diarrhea, hemoptysis and conjunctiva congestion are common symptoms of such infection. in critical cases of covid-19 disease, the symptoms can lead to kidney failure, death and severe acute respiratory syndrome [10]. in the light of the above development, the appearance of the novel covid-19 disease had elicited immense concerns on the science and art of preventing disease among the populace. this is also proved to have declining universal socioeconomic effects in due course. if the pandemic is left unchecked to continue wreaking havoc without any vigorous, reliable and sustainable effort or policy to improve health of the populace, then many economies around the globe will witness more reduced economic activities, and many will get poorer than before [11]. the economic impact of covid-19 on the nations of the world cannot be overemphasized. industrial plants and business facilities have been collapsed in a number of affected nations [11]. also, the delivery of goods and services through a transnational corporations’ worldwide network has been interrupted. for instance, the worsening universal economic impact of the covid-19 endemic, and the feud between saudi arabia and russia have made brent crude prices to sell lower $22 per barrel. this happens to be the least selling price since 2003 [12]. with the impending economic downturn as a consequence of the endemic, the situation can only be salvaged if adequate measures are put in place [12]. for people living in underdeveloped and developing countries with densely-populated houses, limited hygiene, and unavailability of funds to ease avoiding contacts with people, the needy are at higher risk of getting infected. furthermore, the world is at the risk of seeing more people fall below the poverty line as a result of the high cost of medical treatment, increased economic shock, financial crisis, and increased number of deaths. as these viruses transcend borders, the global impacts will keep on spreading. it has been reported that about 94% of businesses across the globe have been negatively affected, and are now experiencing covid19 interruptions [12]. while it is expected that the covid-19 threat will sooner or later disappear just like as the ebola, zika, and sars viruses that have plagued the nations of the world in the last few years. nevertheless, social-economic impact will still linger even after the virus is gone. machine learning (ml) algorithms have been applied to solve problems in different domain by analyzing and interpreting large quantity of data. machine learning has aided in the detection and identification of diseases, as well as drug discovery, medical imaging, smart health records, radiotherapy, robotic surgery and pharmaceutical development. many researchers use machine learning algorithms to solve economic and financial problems. supervised learning, unsupervised learning, semi-supervised learning, and reinforcement learning are all forms of machine learning. how well a machine-learning system works depends on the type of data it uses, how well the learning algorithms function. structural equation modelling (sem) is a sophisticated multivariate analytic approach commonly utilized in social sciences. it may be used for a variety of purposes, ranging from determining basic connections between variables to doing more complicated studies of measurement equivalency for simpler notions. a number of sem analytical methods are combined. these include comparisons of variance between and within groups, which are usually linked to anova. it also incorporates path analysis, which involves solving equations that describe the influence of one or more variables on others in order to evaluate the strength of their connection. as a result, path analysis illustrates the predicted causal links between the variables being investigated. since the beginning of the pandemic, some studies have been conducted to provide better understanding of the covid-19 factors that are making negative impact on the global economy. in this paper ml algorithms and structural equation modelling (sem) was applied to predict the impact of covid-19 pandemic on global economy. ml algorithms have proved over the years that they are very efficient and robust algorithm that successfully cope with huge data. they can therefore be used to prudently predict the impact of covid-19 risk factors on economic factors. this paper analyzes the correlation among covid-19 risk factors, economic factors, and epidemiology factors and their impact on the covid-19 crisis. having an understanding of this will further help in policy formulation that will assist in mitigating against the effect of the pandemic. moreover, it has the potential to positively impact labor productivity and economic growth of any nation. the major contributions of this paper are: 1. a survey of different ml algorithms and sem that have been applied to predicting covid-19 risk factors, eco396 oyewola et al. / j. nig. soc. phys. sci. 3 (2021) 395–405 397 nomic factors, and epidemiology factors was presented. 2. application of machine learning techniques and sem on epidemiological data of covid-19 infection cases in south korea was discussed. 3. performance of the all ml algorithms and sem was evaluated using different performance metrics. the rest of this paper is organized as follows: section 2 is the literature review. section 3 discusses the materials and methods used in this work as well as our employed performance measurements. the results and the discussion are presented in section 3 and section 4 is the conclusion. 2. literature review covid-19 has captured the attention of researchers around the world. in this section, we present reviews of the impacts of covid-19 on business, economies, transportation and so on. the coronavirus outbreaks have spread to 215 countries worldwide with a total of 43,824,534 cases, 1,165,290 deaths and 32,205,492 recovered from these diseases by 27 october 2020 [13]. the pandemic has tremendous consequences in the economy, although the exact magnitude of the effect is still uncertain. many countries around the world had already implemented partial or absolute lockdowns [14]. governments take emergency steps to control the epidemic, such as social distancing, quarantine and care of the reported cases to control the illness on one side. the authors [15] suggested image processing of time series crude oil price through the integration of directed acyclic graph to convolutional neural network (dag). the results indicated that combining dag with cnn increases forecast accuracy by 14.18%. and it was established that covid-19 negatively impacts the nigerian crude oil price, suggesting a decreasing trend in the crude oil prices. the effects of financial markets on the covid-19 pandemic was discussed in a study by [16]. during the period of 22 january 2020 through 17 april 2020, the researcher uses covid-19 confirmed cases, death and market prices of stocks from 64 countries. the outcome was that the stock markets have a negative reaction to growth in covid-19 cases. in other words, stock market decreased as the number of reported cases grew. in addition, they find proactive response on financial markets with the rise in the numbers of suspected cases as opposed to the increase in deaths. the developing countries were severely impacted by the covid-19. the authors [17] mentioned the effect of covid-19 on transportation in lagos, nigeria. this study was based on an email and social media administered to the residents of the lagos state from 18th to 24th may 2020, to assess the effects of covid-19 on transport in lagos. findings have shown that the transportation has been negatively affected by the pandemic. there is also a positive association between covid-19 and transportation with its effect on people’s economic, social and religious practices. the estimated rate of deaths in india from sars-cov-2 for 6 weeks from day 0 to 100 on 14 march 2020 was predicted by ghosal et al [18] using multiple and linear regression. findings indicate that week 6 death counts are not statistically significant while week 5 death count is statistically significant. for sixtyfour days, two months and three days the author [19] compiled data from the nigeria center for disease control. they employed three different linear regression such as quadratic, cubic, and quartic. the result shows that quartic linear regression model with an autocorrelated error of order one performed better. sharif et al [20] examined the linkage within spread of covid-19, oil price volatility shock, the stock market, geopolitical risks as well as us economic policy. in their application, they show the unprecedented effect of covid-19 on geopolitics, economic policy instability and the market volatility of lower rate bands, as well as oil price shocks, through the wavelet dependent granger causality tests. the findings show that the covid-19 has a considerably higher impact on the geopolitical risk than on the us financial uncertainty. the authors in [21] examined the effect of covid-19 on the correlation between crude oil and agriculture. the cross-correlation between the crude oil of brent and agriculture future of london sugar, london wheat, usa cotton and usa orange juice have been studied using a multifaceted cross-relationship analysis. the results demonstrated the strongest link between brent crude oil and london sugar future and three other future agricultural markets. they also investigated the impact of covid19 on cross-correlations between crude oil and agriculture. the findings exhibited the greatest cross correlation between brent crude oil and the london sugar future market among other three future agriculture markets. the results showed that covid19 persistence has become stronger and the correlation between the multifractal relations between the crude oil and the sugar future markets are strongest. overall analysis show that covid19 has a strong impact on the correlation between the crude oil and selected future agriculture market and multifractal properties. in this analysis, the authors in [22] employed four different machine learning techniques such as linear regression (lr), least absolute shrinkage and selection operator (lasso), support vector machine (svm), and exponential smoothing (es) to predict the threatening factors of covid-19. the study considered three types of model such as the number of newly infected cases, the number of deaths, and the number of recoveries in the next 10 days. the results showed that the es performed better than the remaining three machine techniques. six machine learning techniques such as decision tree, support vector machine, naive bayes, logistic regression, random forest, and k-nearest neighbor algorithms were used by [23]. the model predicted that covid-19 patients would recover from the virus for minimum and maximum days, that the patients with high levels of risk would not recover from the covid-19 pandemic, the patients with potential recovery and those likely to recover quickly from covid-19. the results show that decision tree performs better than other algorithms. the spread of sars appears to be influenced by the weather. during the first 16 weeks of the pandemic, [24] conducted a worldwide scale research that included 134,871 virologic climatic demographic data from 209 nations. the relationship between covid 19, population density, and climate was studied using structural equation modeling (sem). the findings of 397 oyewola et al. / j. nig. soc. phys. sci. 3 (2021) 395–405 398 the study revealed that the spread of covid 19 is influenced by both climate and population density. the author [25] investigated the factors that might impact public perceptions of indonesia’s pembatasan sosial berskala besar (psbb). partial least squares and structural equation models were used and data were collected from 856 respondents across indonesia’s provinces. the advantages of the psbb, positive perception, negative perception, threatening perceptions of covid-19 and attitude toward the psbb policy were all utilized to evaluate these policies. more than half of the attitudes toward psbb policy implementation can be explained by the model, which takes into account perceived advantages, negative and positive views, and the danger posed by covid-19. the authors in [26] used structural equation modeling to examine the relationship between socio-demographic variables (gender, age, level of education, place of residence, and employment status) and covid-19 preventive behaviour and the threat appraisal of covid-19, fear of covid-19, trust in covid-19 information sources, and covid-19 conspiracy beliefs. covid19 assessment of threat, confidence in covid-19 information sources, and fear of covid-19 are all significant predictors of covid-19 preventative activity, according to the results. covid-19 conspiracy theories have a negative correlation with threat assessment and trust in covid-19 information sources. covid-19 danger assessment has an important and direct role in explaining covid-19 phobia. a study conducted by [27] used structural equation modeling to predict how work-life balance will be affected by factors such as their own health and emotional well-being as well as their current relationship status as well as their location of employment. findings showed that elements including physical and mental health, activities, relationship status, and place of employment directly affected the work-life balance. there was a notable gender gap among dentists, with far fewer women than men. structural equation modeling was used by franzen et al. [28] to simulate a group of young individuals who were polled shortly after the conclusion of switzerland’s initial lockdown. to find out why and how much they helped in averting the pandemic by following the advice to stay at home as much as possible. they believe that people who believe they are at danger, or who have relatives in the risk category, are more likely to follow safety precautions than people who do not believe they are at risk. coronavirus social separation procedures were well followed during the first shutdown, according to research. young individuals felt the virus posed little personal risk, but society as a whole was at risk. furthermore, the findings show that support for preventative measures is the most significant factor in fostering collaboration in the effort to contain the spread of covid-19. 3. methodology 3.1. description of dataset the dataset used in this research comprises of epidemiological data of covid-19 infection cases in south korea which were obtained from kaggle database [29] and macro-economic data was obtained from yahoo finance [30]. the dataset is made up of data from 23/01/2020 to 24/06/2020 recorded daily, patient id, sex, age, country, province, city, infected by, contact number, symptom onset date, confirmed date, released date and state which consists of released, deceased and isolated. in this study, due to the nature of the dataset, we extracted sex, age, state, confirmed date, released date while the macroeconomic dataset obtained from yahoo finance which consists of south korea exchange rate (kr), jakarta composite index (jk), kospi composite index (ks) as shown in table i. it displays sex (male=1, female=2), age (numeric), state (released=1, deceased=2, isolated=3), dr is obtained from subtracting released date from confirmed date. table i shows that sex, ks, jk and kr is negative moderately skewed while age, state and dr is positively skewed. the data distribution of sex, age, state, jk is platykurtic since the data distribution is less than 3 while dr, ks and kr is leptokurtic since the data distribution is greater than 3. sex and state have a very low variance while age, dr, ks, jk, kr have a very high variance. this is an indication that the data points are widely spaced from each other and this may also result in high degree of error. with this intuitive knowledge at hand, table ii is the variance reduction of dataset. fig. 1 shows a scatter plot of matrices, with bivariate scatter plots below the diagonal, histograms on the diagonal and pearson correlation above the diagonal. a scatter plot shows a relationship sex, age, state, dr, jk, ks and kr. the pearson correlation coefficients of the data shows a positive relationship between sex and age, negative relationship between state and dr, positive relationship between ks and jk, jk and kr and ks and kr. also, the diagonal of fig.1 shows the histogram of all the observed variables. there is more female than male in the sex variable, there is more recorded cases of covid-19 within the age of 20s than the age of 90s. in the state variable, there is more released of patient of covid-19 than death. dr, jk, ks and kr are numeric values. 3.2. structural equation modelling structural equation modeling (sem) is also known as the causality model or covariance structural model. it is a method for defining, estimating and evaluating the causality model [31]. it comprises a range of statistical analytical methods including confirmatory factor analysis, variance, covariances, regression and latent growth curve. it is a very broad and linear method of statistical modeling that tests hypotheses according to theories. this model was first presented by wright [32-33]. sem equation are split into measurement model and structural equation model. measurement model primarily tests the correlation between latent and significant variables while structural equation model tests mainly causality among the latent variables. the key characteristics of scientific research are the estimation, relativizing variables and disclosure of causality [34]. moreover, observable or manifest variables such as sex, age, state, dr, jk, ks and kr can be calculated while latent variables such as covid-19 risk factors, epidemiological factors and macroeconomic factors cannot be directly measured as shown in fig. 2. in such cases, regression equalities should be defined which demonstrate how endogenous and exogenous structures are related and which benefits from a statistical technique, which has 398 oyewola et al. / j. nig. soc. phys. sci. 3 (2021) 395–405 399 table 1. variance reduction of dataset variables number of sample minimum median mean variance skewness kurtosis sex 3782 0.00 2.00 1.54 0.25 -t0.19 1.05 age 3782 0.00 40.00 40.38 408.03 0.31 2.33 state 3782 1.00 1.00 1.66 0.87 0.72 1.55 dr 3782 1.00 1.00 10.93 204.23 1.50 5.20 ks 3782 0.00 1938 1815.04 163848.07 -1.18 5.50 jk 3782 0.00 4091 3767.94 3153274.83 -0.24 1.84 kr 3782 0.00 1110 1086.97 19496.45 -3.29 29.63 table 2. variance reduction of dataset variables number of sample minimum median mean variance skewness kurtosis sex 3782 0.00 2.00 1.54 0.25 -0.19 1.05 age 3782 0.00 4.00 4.04 4.08 0.31 2.33 state 3782 1.00 1.00 1.66 0.87 0.72 1.55 dr 3782 0.10 0.10 1.09 2.04 1.50 5.20 ks 3782 0.00 1.93 1.82 0.16 -1.18 5.50 jk 3782 0.00 4.09 3.76 3.15 -0.24 1.84 kr 3782 0.00 1.11 1.08 0.02 -3.29 29.63 a broad range of applications to combine measurement principles such as sem [35]. the risk factors of covid-19 include age, race/ethnicity, gender, some medical conditions, use of certain drugs, poverty and crowding, certain occupations and pregnancy [36]. due to the sparsity of data of covid-19 risk factors, we considered only the sex (sex) and age (age) of epidemiology of south korea covid-19. epidemiology factors consists of state (state) of covid-19 patients and duration (dr) is obtained from subtracting released date from confirmed date of epidemiology of south korea covid-19. covid-19 has catastrophically affected economy. the survival of economy greatly relies on the crude oil price and other macro-economic factors [37]. in this study, we considered three macro-economic factors such as south korea exchange rate (kr), jakarta composite index (jk) and kospi composite index (ks). fig. 3 is the schematic diagram of structural equation modeling of impact of covid-19 risk factors on epidemiology and economic factors. circles are displayed as latent variables; squares are displayed as manifest, measured or observed variables; arrows displayed the paths from latent variables to observed variables; residuals and variances are indicated as double headed arrows. the latent variables are i, y, z while the observed variables are sex, age, state, dr, kr, jk, kr. the paths from latent variables to observed variable are λ1 to λ6. ϕ1 to ϕ3 double headed curve arrows are path from each latent variables while θ1to θ10 are the double headed curve arrows for both observed and latent variables. the structural equations of impact of covid-19 risk factors on epidemiology and economic factors can be represented as: i = α + sexβ1 + ageβ2 + ξ1 (1) y = ρ + stateβ3 + drβ4 + ξ2 (2) z = γ + krβ4 + jkβ5 + ks β6 + ξ3 (3) i = yη + zπ + ξ4 (4) where i is the covid-19 risk factors, y is the epidemiology factors, z is the economic factors, α,ρ,γ are the intercept, β1, β2, β3, β4, β5, β6 are the predictor observable variable, ξ1,ξ2,ξ3, ξ4 are the residual error,η,π are latent predicted variable. 3.3. machine learning 3.3.1. bagging (bag) bagging which is an acronym for bootstrap aggregating is a parallel ensemble technique. it provides a way to decrease the variance of prediction model throughout the training phase by producing extra data. this is accomplished through arbitrary sampling and substitution from the original data. decisions made by multiple learners can be integrated into a single prediction. in the case of classification, it is clearly a vote to combine these decisions. models of bagging bear the same weight as good models of bagging because an executive can use a collection of expert advice based on their previous right predictions to achieve other outcomes. it is considered right which one gets more votes than other groups. if more votes are expected, they are reliable because more votes are present [38]. bag is used in this paper because of its capacity to minimize the variance of 399 oyewola et al. / j. nig. soc. phys. sci. 3 (2021) 395–405 400 table 3. correlation coefficient of observed variables sex age state dr ks jk kr sex 1.00 0.12 -0.03 0.03 0.02 0.02 0.01 age 0.12 1.00 0.08 -0.01 0.08 0.11 0.08 state -0.03 0.08 1.00 -0.28 0.11 -0.01 0.12 dr 0.03 -0.01 -0.28 1.00 0.20 0.25 -0.07 ks 0.02 0.08 0.11 0.20 1.00 0.76 0.15 jk 0.02 0.11 -0.01 0.25 0.76 1.00 0.23 kr 0.01 0.08 0.12 -0.07 0.15 0.23 1.00 table 4. summary of impact of covid-19 risk factor (i) on epidemiology factor (y ) and economic factor (z). latent variables estimate std.err z-value p-value i =∼sex 1.00 age 21.84 16.62 1.32 0.19 y =∼state 1.00 dr -68.89 199.89 -0.35 0.7 z =∼ ks 1.00 jk 5.60 0.21 26.23 0.0 kr 0.10 0.01 13.98 0.0 covariances i ∼∼ y -0.00 0.00 -0.32 0.75 i ∼∼ z 0.00 0.00 1.31 0.19 y ∼∼ z -0.00 0.01 -0.34 0.73 variance sex 0.24 0.01 34.89 0.00 age 1.46 1.98 0.74 0.46 state 0.86 0.03 34.17 0.00 dr -23.53 73.83 -0.32 0.75 ks 0.07 0.00 17.30 0.00 jk 0.08 0.11 0.73 0.47 kr 0.02 0.00 43.29 0.00 i 0.01 0.00 1.27 0.20 y 0.01 0.02 0.34 0.73 z 0.09 0.01 19.94 0.00 a decision tree classifier. bag enables a trade-off balance between variance and bias by reducing the variance and carefully adjusts the prediction to an estimated result. the mathematical equation that depicts the parameters used in bagging is in equation 5. h ( di, c j ) = m∑ m=1 αm hm(di, c j) (5) where hm is the weak classifiers, diis classified to the classes c j and αm is the constant parameter. 3.3.2. boosting (bst) boosting is a successive ensemble technique that decreases bias error and produces outstanding prediction models. the word ’boosting’ describes a set of methods that transforms a poor learner into a strong learner. stochastic gradient boosting (bst) method is a hybrid of boosting and bagging proposed by friedman [39]. bst is a set of learning algorithm with a combination of boosting and decision tree, which classifies the value of all trees by weighing all trees. the new model is constructed along the path of gradient descent of the loss function of the previous tree. it is important to note that the loss function between classification and actual function is reduced by the training function of the classification function [40]. bst technique was selected for use in this paper because the algorithm continues its iteration until a learner with superior results compared to a random guess is achieved. bst approach therefore helps in increasing the capability of machine learning and improving prediction accuracy. the mathematics equation of the loss function is given in equation 6 and 7: ρ (yk, fk (x)) = k∑ k=0 yklog  efk (x)∑k k=1 e fk (x)  (6) ŷk = − [ ∂ρ(yk, fk(x) ∂fk (x) ] = yk − pk(x) (7) where y is the output variable, x is the input variables, k is the 400 oyewola et al. / j. nig. soc. phys. sci. 3 (2021) 395–405 401 table 5. parameters estimate of structural equation modeling parameter value λ1 0.15 λ2 0.80 λ3 0.08 λ4 -3.54 λ5 0.77 λ6 0.99 λ7 0.23 θ1 0.98 θ2 0.36 θ3 0.96 θ4 -11.52 θ5 0.40 θ6 0.03 θ7 0.95 θ8 1.00 θ9 1.00 θ10 1.00 ϕ1 -0.01 ϕ2 -0.07 ϕ3 0.14 table 6. performance evaluation of machine learning algorithm rmse mse mase bag 0.0541 0.0029 0.7758 bst 0.0512 0.0026 0.7685 svm 0.0584 0.0034 0.8954 rf 0.0003 0.0187 0.2229 dt 0.0033 0.0577 0.8902 number of classes, pk(x) is the probability. 3.3.3. support vector machine (svm) svm procedure categorizes both linear and non-linear data. svm uses a non-linear mapping to transform the training set to a high level. in this new dimension, svm explores the ideal linear hyperplane separation as a decision limit by which the tuples of a class of one class are split from another. there are two class data that can be separated by a hyperplane with the proper, non-linear upper dimensional mapping. in contrast to the other approaches, hyperplanes are highly robust for overfitting [41]. svm is considered in this research due to its ability to handle numerous continuous and categorical variable. svm technique is used in this paper because it has a low bias and a high variance, nevertheless, the trade-off can be modified by tuning the c parameter, which determines the number of infractions of the border permitted in training data, raising the bias while reducing the variance. equations 8, 9 and 10 for svm are stated below: wt .x + b = 1 (8) wt .x + b = −1 (9) the set of inequalities can be combined to form: y[wt .x + b] ≥ 1 (10) the equation can be formulated as a minimization problem given in equations 11, 12 and 13: min w,τ j (w,τ) = 1 2 wt w + c n∑ i=1 τ (11) subject y[wt .x + b] ≥ 1 to lagrangian function, we then have l (w, b,α,β) = j (w,τ)− n∑ i=1 α ( y [ wt .x + b ] − 1 + τ ) − n∑ i=1 βτ(12) the optimal point of the lagrangian function is given as max w,β min w,b,τ l(w, b,α,β) (13) differentiate (12), we obtain ∂l ∂w = 0, w = n∑ i=1 αyx (14) ∂l ∂b = 0, n∑ i=1 αy (15) 401 oyewola et al. / j. nig. soc. phys. sci. 3 (2021) 395–405 402 figure 1. pairwise scatter plots of the dataset figure 2. impacts of covid-19 risk factors with respect to epidemiology and economic factor ∂l ∂τ = 0 (16) the quadratic programming problem will be form by substitutfigure 3. schematic diagram of structural equation modeling of impacts of covid-19 risk factors on epidemiology and economic factors. circles are displayed as latent variables; squares are displayed as manifest, measured or observed variables; arrows are displayed as paths from latent variables to observed variables; residuals and variances are indicated as double headed arrows. 402 oyewola et al. / j. nig. soc. phys. sci. 3 (2021) 395–405 403 ing (14), (15) and (16) to equation (12), we then have min α γ (α) = n∑ i=1 α− 1 2 αiα jyiy j k(xi, x j) (17) where k(xi, x j) is the kernel function, w is the weight, x is the input, b is the bias, α,β are the positive real constants. 3.3.4. random forest (rf) random forest (rf) is a decision-making ensemble classifier that has various types of trees. an arbitrary sequence of features at each node is used to evaluate the division to create an individual decision tree. each tree is based on the individual values of a random variable. we are able to shape an rf using bagging along with the selection of the random attribute, using the cart method, in order to increase the trees. rf uses the random linear combination of the input attributes. the subcluster of features is not chosen randomly, but new attributes are created, which reflect a linear combination of existing features [42]. rf model will assist in the construction of numerous decision trees and their merging to produce a more accurate and reliable prediction. rf is an ensemble learning technique that applies the idea of bagging. it provides a compromise between variance and bias by lowering the variance and judiciously finetunes the prediction to a desired result. 3.3.5. decision tree (dt) decision trees (dt) are an easy model which classify by dividing training data into pieces and mainly holding the result of each part [43]. it is a natural non-parametric supervised learning model, also called classification and regression tree (cart) which produces accurate classifications with easily understood regulations. dt models transparency makes them highly relevant for economic and financial purposes. in addition, continuous and discrete data can be dealt with using dt. our choice of dt model in this work is based on it ability to fit the training data flawlessly fine. 3.4. performance evaluation three measures such as root mean square error (rmse), mean square error (mse) and mean absolute scaled error (mase) are used to calculate the prediction efficiency of impacts of covid-19 risks factors with respect to epidemiology factors and economic factors. 3.4.1. root mean square error (rmse) rmse is defined as: rms e = √√ 1 n n∑ n=1 ( in − în )2 (18) mean square error (mse) mse is defined as: ms e = 1 n n∑ n=1 ( in − în )2 (19) mean absolute scaled error (mase) mase is defined as: mas e = 1 n n∑ n=1 |in − în| 1 n−m ∑n n=m+1 |in − în−m| (20) where in is the real covid-19 risk factors, în is the predicted values and m is the seasonal period of in. 4. result and discussion this section presents the experimental results of structural equation modeling and machine learning techniques such as bagging (bag), stochastic gradient boosting (bst), support vector machine (svm), random forest (rf), decision tree for predicting impacts of covid-19 risk factors with respect to epidemiology and economic factors. we compared the performances of the algorithms under consideration using root mean square error (rmse), mean square error (mse) and mean absolute scaled error (mase) to discern which is more accurate in predicting the impacts of covid-19 risk factors. as stated earlier, we used data from the kaggle database for covid-19 infection cases in south korea. statistical correlation analysis was used to determine the strength of the association between observed variables. table iv is the result of correlation coefficients of observed variables. a correlation coefficient of 0.12 was noted between the sex and age of covid-19 risk factors, this indicates a weak positive linear relationship between them. a correlation coefficient of -0.28 was noted between state and dr of epidemiology factors, this indicates a weak negative linear relationship. there is a strong positive relationship between ks and jk while between jk and kr there is a weak positive relationship between them. however, ks and kr also show a weak positive linear relationship of the three economic factors such as jk, ks and kr. table v displays the estimate of latent and observe variable, covariance and variance of the covid-19 risk factor (i), epidemiology (y ) and economic factor (z). the estimate, standard error, z-value and p-value are also shown in the table. the estimate of latent variable i and the observed variables such as sex and age are estimated as 21.84 with a standard error estimate of 16.62, zvalue is 1.52 but it was observed that there is no statistically significant linear dependence of the mean of i with respect to age. this means that no effect was observed. also, the estimate of latent variable y and the observed variable such as state of covid-19 (state) and duration of covid-19 (dr) are estimated as -68.89 with a standard error estimate of 199.89, z−value is -0.35 but it is insignificant at the 0.05 level. there is statistically significant linear dependence of the mean of latent variable z with respect to jk and kr. the covariance result of latent variable indicates that the covariance between i, y and y, z is approximately -0.00, which indicates that the relationship is negative while the covariance result of i, z is approximately 0.00, which indicate that the relationship is positive. the small variance of sex, age, state, jk, ks, kr, i, y, z it shows that the data points appear to be very similar to average and to one another, while dr with high variance shows that the 403 oyewola et al. / j. nig. soc. phys. sci. 3 (2021) 395–405 404 data points are very widespread from the average and from each other. table vi is the parameter estimate of structural equation modeling. there is a positive relationship between latent and observed variable as shown in λ1, λ2, λ3,λ5, λ6, λ7 except in λ4 which shows a negative relationship between them. the parameter estimates of θ1, θ2,θ3,θ5,θ6,θ7, θ8,θ9,θ10 also shows a positive relationship between them except in θ4 which shows a negative relationship. ϕ1, ϕ2, ϕ3 is the impact of covid19 risk factors (i) with respect to epidemiology factors (y ) and economic factors (z). there is a negative effect of covid-19 risk factors (i) on epidemiology factors (y ), negative effect also obtained on epidemiology factors (y ) and economic factors (z) but positive effect was obtained from covid-19 risk factors (i) and economic factors (z). table vii report rmse, mse and mase of the extracted predicted value of covid-19 risk factors. rf outperform the rest of the algorithms with smaller accuracy result than other methods, which means that the approaches are more effective than others. the findings show that rf perform well. 5. conclusion structural equation modeling (sem) has provided a means to understand direct impact of covid-19 risk factors with respect to epidemiology factors and economic factors. latent variable sem has provided the necessary tools for developing several equations to describe the covid-19 behavioural impact framework. it has the potential to quantify and test the relationships between latent and observed variables. they measure the uniformity and plausibility of the assumed model in relation to the findings observed. furthermore, a researcher can examine both direct and mediate relationships. findings indicate that covid-19 risk factors have negative effects on epidemiology factors. it also has negative effects on economic factors. the result indicates that there is no statistically significant linear dependence of the mean of covid-19 with respect to age. this means that no effect was observed. also, the estimate of latent variable epidemiology and the observed variable such as state of covid-19 and duration of covid-19 is insignificant at the 0.05 level. also, there is a negative effect of covid-19 risk factors on epidemiology factors, negative effect also obtained on epidemiology factors and economic factors but positive effect was obtained from covid-19 risk factors and economic 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[43] s. sivakumar, s. venkataraman, r. selvaraj, “predictive modeling of student dropout indicators in educational data mining using improved decision tree”, indian journal of science and technology, 9 (2016) 1. 405 j. nig. soc. phys. sci. 3 (2021) 165–171 journal of the nigerian society of physical sciences entropic system in the relativistic klein-gordon particle c. a. onatea,∗, m. c. onyeajub aphysics programme, department of physical sciences, landmark university, omu-aran, nigeria bdepartment of physics, theoretical physics group, university of port harcourt, choba, nigeria abstract the solutions of kratzer potential plus hellmann potential was obtained under the klein-gordon equation via the parametric nikiforov-uvarov method. the relativistic energy and its corresponding normalized wave functions were fully calculated. the theoretic quantities in terms of the entropic system under the relativistic klein-gordon equation (a spinless particle) for a kratzer-hellmann’s potential model were studied. the effects of a and b respectively (the parameters in the potential that determine the strength of the potential) on each of the entropy were fully examined. the maximum point of stability of a system under the three entropies was determined at the point of intersection between two formulated expressions plotted against a as one of the parameters in the potential. finally, the popular shannon entropy uncertainty relation known as bialynick-birula, mycielski inequality was deduced by generating numerical results. doi:10.46481/jnsps.2021.209 keywords: eigensolutions, bound states, wave equation, theoretic quantity. article history : received: 25 april 2021 received in revised form: 16 june 2021 accepted for publication: 29 june 2021 published: 29 august 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: o. j. oluwadare 1. introduction the understanding of correlations in quantum systems is based on the analytic tools provided by the entropic measures. these entropic measures are shannon entropy, rényi entropy, and tsallis entropy. the most outstanding of the entropic measures is the shannon entropy introduced by shannon [1]. the shannon entropy has several applications in various scientific disciplines. in the concept of information, for instance, it presents a discrete source without memory as a functional that quantifies the uncertainty of a random variable at each discrete time. it is the expected amount of information in a given event drawn from a distribution that serves as a measure of uncertainty or variability that is associated with random variables. shannon ∗corresponding author tel. no: +234(0)7036631325 email address: oaclems14@physicist.net (c. a. onate ) entropy has examined entropic uncertainty and has been tested for different potential models. it serves as another form of heisenberg uncertainty relation. in physics, shannon entropy has been widely reported under the non-relativistic wave equation over the years for different potential models [2-25]. however, all the reports given in refs. [2-25] dwell under the nonrelativistic wave equation leaving out the relativistic wave equation. this motivates the present work. in the present study, the authors want to examine the entropic system under the relativistic klein-gordon equation using kratzer-hellmann potential. the accuracy of shannon entropy for any calculation can be checked by the uncertainty relation of shannon entropy that relates position space and momentum space with the spatial dimension. this is otherwise called bialynick-birula, mycielski (bbm) inequality given as s (ρ) + s (γ) = d(1 + ln π), (1) 165 onate & onyeaju / j. nig. soc. phys. sci. 3 (2021) 165–171 166 where s (ρ) = − −4π δ ∫ inf 0 ρ(r) log ρ(r) dr, s (γ) = − −4π δ ∫ inf 0 γ( p) log γ( p) d p, (2) ρ(r) and γ(p) are probability densities. d refers to the spatial dimension, and ln π is a constant term. in this work, numerical results will be generated for equation (1) to verify whether the results from the relativistic klein-gordon equation will satisfy the bbm inequality. the kratzer-hellmann potential comprises of kratzer potential and hellmann potential. the physical form of kratzer-hellmann potential is v (r) = de ( r − re r )2 − ( a − be−δr r ) , (3) where de is the dissociation energy, re is the equilibrium bond length, a and b are the strengths of the hellmann potential, r is internuclear separation while δ (screening parameter) characterized the range of the hellmann potential. the two sub-sets of the potential (3) have received attention in the bound states and other areas of sciences. the hellmann potential is known to be suitable for the study of inner-shell ionization problems. the potential was equally studied for alkali hydride molecules by varshni and shukla [26]. recently, the hellmann potential was used in ref. [27] as a tensor interaction for the breaking of energy degenerate doublets in the dirac equation. the kratzer potential, on the other hand, forms a potential pocket that is useful for vibrational and rotational energy eigenvalues [28]. the combination of these potentials is considered necessary because of their applications. 2. parametric nikiforov-uvarov method (pnum) this pnum is a straight forward method that uses transformation of variable. the pnum is short and accurate for solving bound state problems. according to tezcan and sever [29], the reference or standard equation for the pnum is( d2 d s2 + c1 − c2 s(1 − c3 s) d d s + −ξ1 s2 + ξ2 s − ξ3 s2(1 − c3 s)2 ) ψ(s) = 0. (4) according to ref. [29], the eigenvalues and eigenfunction respectively can be obtained using [29, 30] nc2 − (2n + 1) c5 + c7 + 2c3c8 + n (n − 1) c3 + (2n + 1) √ c9 + ( 2 √ c9 + c3 (2n + 1) ) √ c8 = 0, (5) ψn,` (s) = nn,` s c12 (1 − c3 s) −c12− c13 c3 ×p ( c10−1, c11 c3 −c10−1 ) n (1 − 2c3 s) ,(6) the parameters in equations (5) and (6) are deduced as follows c4 = 1 − c1 2 , c5 = c2 − 2c3 2 , c6 = c 2 5 + ξ1, c7 = 2c4c5 − ξ2, , c9 = c3 (c7 + c3c8) + c6, c10 = c1 + 2c4 + 2 √ c8, c8 = c 2 4 + ξ3 c11 = c2 − 2c5 + 2 (√ c9 + c3 √ c8 ) , c12 = c4 + √ c8, c13 = c5 − (√ c9 + c3 √ c8 ) (7) 2.1. the klein-gordon equation (kge) with kratzer-hellmann potential the kge is use to describe spinless particles in the domain of relativistic wave equation [31, 32, 33, 34, 35, 36, 37, 38]. the klein-gordon equation for space-time scalar potential s (r) and the time component of the lorentz four-vector potential v (r) arising from minimal coupling, in the relativistic unit (~ = c = 1), reads[ p̂2 + (m + s (r))2 − (e − v (r))2 ] r(r) = 0, (8) where p̂ is the momentum operator, m is the particle’s mass, e is the relativistic energy and r(r) is the wave function. the kge above has a potential 2v in which the nonrelativistic limit cannot give the solutions of the schrödinger equation. a critical investigation was done by alhaidari et al. [39] who proved that s = ±v . this is the nonrelativistic limit for the potential 2v . thus, in the relativistic limit, the interacting potential becomes v instead of 2v . therefore, to obtain a solution of the kleingordon equation for any arbitrary `−state whose energy equation in the nonrelativistic limit equals the solution of the schrödinger equation, equation (8) becomes [39, 40, 41, 42, 43, 44, 45][ p̂2 − m2 + e2 − v (r)(m + e) − `(` + 1) r2 ] r(r) = 0, (9) the solutions of the klein-gordon equation above and some diatomic molecular potential models have been obtained for different molecules [46, 47, 48, 49, 50], and the results compared with experimental values. to get rid of the inverse squared term in equation (9), we need to adopt a suitable approximation scheme. in this work, we adopt the following approximation that is valid for δ � 1, 1 r2 ≈ δ2( 1 − e−δr )2 . (10) plugging equations (3) and (10) into equation (9) and making a simple transformation of the form y = e−δr , equation (9) turns to be [ d2 dy2 + 1 − y y − y2 d dy + ay2 + by + c y2(1 − y)2 ] rn,`(y) = 0, (11) a = υ + bδβ δ2 (12) b = 2υ − (b + a + 2dere)δβ δ2 , (13) c = e2 − m2 + βϑ− `(` + 1)δ2 δ2 , (14) β = m + e, (15) ϑ = 2dereδ− de + aδ− der 2 eδ 2, (16) υ = m2 − e2 + deβ. (17) 166 onate & onyeaju / j. nig. soc. phys. sci. 3 (2021) 165–171 167 comparing equation (11) with equation (4), equation 7 numerically becomes c1 = c2 = c3 = 1, c4 = 0, c5 = − 1 2 , c6 = 1 4 − a, c7 = −b, c8 = −c, c9 = 1 4 − a − b − c, c10 = 1 + 2 √ −c, c11 = 2 ( 1 + √ −c ) + √ 1 − 4(a + b + c), c12 = √ −c, c13 = − 1 2 − 1 2 √ 1 − 4(a + b + c) − √ −c (18) substituting c1 to c9 in equation (18) into equation (5), we have energy equation for the kratzer-hellmann potential as υ −β(ϑ + de) + `(` + 1)δ2 δ2 =  (ϑ− de − bδ)β− n(n + 1) − 1 2 − 2`(` + 1) − ( n + 12 ) √ 1 − 4(a + b + c) 1 + 2n + √ 1 − 4(a + b + c)  2 . (19) the energy equation obtained for kratzer-hellmann potential in equation (19) above has subset energy equations for kratzer potential, yukawa potential, and coulomb potential. however, the wave function for the kratzer-hellmann potential is obtained by substituting c10to c13in eq. 18 into eq. 6 to have r(s) = n sη(1 − s) 1 2 + λ 2 p(n2η,λ)(1 − 2s), (20) where η = √ υ − 2dereδβ− aδβ δ2 + βder2e + `(` + 1), (21) λ = √ (1 + 2`) + 4βder2e. (22) and n is a normalization factor which can easily be calculated using normalization condition. using∫ ∞ 0 |r(r)|2 dr = 1, (23) the normalization factor can easily be obtained. consider the transformation y = e−δr and another transformation of the form x = 2y − 1, with a relation of the form 1 − x = 1 − ( 1−x 2 ) , when invoked on equation (23), using an appropriate integral, we have the normalization factor as n2n,` = − n!2δηγ(2η + λ + n + 1) γ(2η + 1)γ(λ + n + 2) . (24) 2.2. kratzer-hellmann potential and entropies the three entropies mentioned in the introduction will be calculated here using equation (20). 2.2.1. shannon entropy to obtain shannon entropy, we plug equation (20) into equation (2). for s (ρ), we define a transformation of the form s = 1 − y, and using the appropriate integral given in the appendix, we have s (ρ) = 8πη(n!)γ(2η + 1)γ(λ + n + 3)γ(2η + λ + n + 1) γ(2η + n)γ(λ + n + 2)γ(2η + λ + n + 3) × log [ (0.99)2η(0.01)1+λ γ(2η + n + 1) γ(2η + 1) ] . (25) to obtain s (γ), we define x = −1 + 2y and then, using integral and formula in the appendix, we have s (γ) = − 4πγ(2η + n + 1)γ(2η + λ + n + 1) γ(2η + n)γ(2η + λ + n + 2) × log (0.99)2η (0.01)1+λ × [γ(2η + n + 1)]2 n! [ γ(2η + 1) ]2  . (26) 2.2.2. rényi entropy rényi entropy is a generalization of shannon entropy and is defined as [51] rq(ρ) = 1 1 − q log 4π ∫ ∞ 0 ρ(r)q dr. (27) the q is called tsallis index. following the procedures used to obtain shannon entropy for position space, we have rq(ρ) as rq(ρ) = − 2.5314δq−1 1 − q (28) × [ 2η(n!)γ(2η + 1)γ(λ + n + 3)γ(2η + λ + n + 1) γ(2η + n)γ(λ + n + 2)γ(2η + λ + n + 3) ]q . for the momentum space, we follow step by step as in the shannon entropy for momentum space to have rq(γ) as rq(γ) = − 1.2657δq−1 (1 − q) (29)[ 2γ(2η + n + 1)γ(2η + λ + n + 1) γ(2η + 1)γ(2η + λ + n + 2) ]q . 2.2.3. tsallis entropy the tsallis entropy was introduced by tsallis [52]. the concept acts as a basis for generalizing the statistical mechanics. this tsallis entropy is defined as tq(ρ) = 1 q − 1 ( 1 − 4π ∫ ∞ 0 ρ(r)q dr ) , q , 1. (30) the tsallis entropy reduces to the usual boltzmann-gibbs entropy as the tsallis index q approaches one. with the wave function in equation (20) and following previous procedures, we have tsallis entropy for position space as tq(ρ) = 1 q − 1 + 4πδq−1 q − 1 (31)( 2ηγ(2η + 1)γ(λ + n + 3)γ(2η + λ + n + 1)(n!) γ(2η + n)γ(λ + n + 2)γ(2η + λ + n + 3) )q following the procedures to obtain momentum space of shannon entropy, the tsallis entropy for momentum space is obtained as tq(γ) = 1 q − 1 (32) × [ 1 + 2πδq−1 ( 2γ(2η + n + 1)γ(2η + λ + n + 1) γ(2η + 1)γ(2η + λ + n + 2) )q] . 167 onate & onyeaju / j. nig. soc. phys. sci. 3 (2021) 165–171 168 3. results and discussion figure 1. s (ρ) against b figure 2. s (γ) against b in figure 1, we plotted s (ρ) against one of the potential strength at the first excited state with ` = 1, re = 0.1, a = 1, δ = 0.25, de = 2.5 and m = 2de. there is less concentration of electron density and so, less concentration of the wave function which makes the system unstable as the potential strength increases. in figure 2, s (γ) is plotted against the potential strength (b) at the first excited state with ` = 1, re = 0.1, a = 1, δ = 0.25, de = 2.5 and m = 2de. there is a more concentration of the spreading of electron density which leads to more concentration of the wave function as the potential strength goes up. thus, there is stability of the system as b goes up. in figures 3 and 4, we examined the variation of the product of rényi and tsallis entropies respectively against the potential strength (a) at the first excited state with ` = 1, re = 0.1, b = 2, δ = 0.25, de = 2.5 and m = 2de and in both cases, there is an inverse variation between the product of the entropy and the potential strength. the point of intersection for the entropies (shannon, rényi and tsallis) is determined by plotting s = rt /s t ; t = (rt /tt ) − 0.0a against a potential table 1. shannon entropy relation with n = ` = 1, re = 0.1, b = 2, δ = 0.25, de = 2.5 and m = 2de. a s (ρ) s (γ) s t = s (ρ) + s (γ) 1 -1.561445865 10.47908433 8.917638464 2 -0.884937778 10.33123825 9.446300475 3 -0.568858951 10.18281361 9.613954654 4 -0.396386446 10.05310205 9.656715603 5 -0.254891378 9.898107966 9.643216588 table 2. renyi entropy relation with n = ` = 1, re = 0.1, q = b = 2, δ = 0.25, de = 2.5 and m = 2de. a r(ρ) r(γ) rt = r(ρ) + r(γ) 1 0.4291810608 2.803088803 3.232269864 2 0.3386588921 2.886297376 3.224956268 3 0.2792173453 2.936768150 3.215985495 4 0.2373724018 2.970645793 3.208018195 5 0.1937069142 3.004709576 3.198416490 table 3. tsallis entropy relation with n = ` = 1, re = 0.1, q = b = 2, δ = 0.25, de = 2.5 and m = 2de. a t2(ρ) t2(γ) tt = t2(ρ) + t2(γ) 1 0.077255921 8.181988939 8.259244860 2 0.048103385 8.674956863 8.723060249 3 0.032699075 8.980995892 9.013694967 4 0.023632579 9.189394955 9.213027534 5 0.015737688 9.401349043 9.417086731 table 4. shannon entropy s (ρ) for kratzer, coulomb and yukawa potentials at different excited states. n kratzer coulomb yukawa 0 -0.614945865 -0.36740543 -0.373228924 1 -0.489475768 -0.32198215 -0.330603267 2 -0.323785845 -0.19261467 -0.206643987 3 -0.191326241 -0.11715545 -0.130089665 table 5. shannon entropy s (γ) for kratzer, coulomb and yukawa potentials at different excited states. n kratzer coulomb yukawa 0 5.670453776 4.870762243 4.873986446 1 4.879443281 4.163817555 4.192208735 2 3.991974378 3.519430921 3.530045080 3 3.268931407 2.980665799 3.001043671 168 onate & onyeaju / j. nig. soc. phys. sci. 3 (2021) 165–171 169 figure 3. r(ρ)r(γ) against a figure 4. t (ρ)t (γ) against a strength as shown in figure 5. table 1 presented the numerical results for shannon entropy. the results were numerically verified and confirmed the bialynick-birula, mycielski (bbm) inequality that gives a standard relation s (ρ) + s (γ) ≥ d(1 + log π). for d = 1, d(1 + log π) = 1.497206180. however, the lower bound from table 1 is 8.917638464. this verifies the accuracy of the present work. in tables 2 and 3, we numerically presented the rényi entropy and tsallis entropy respectively for position space and momentum space. in both entropies, the position space and momentum space varies inversely with one another. in tables 4 and 5, results of s (ρ) against s (γ) for the subset potentials were given. the results of these subset potentials are similar to the results of the mother potential. the result for kratzer potential was obtained by putting a = b = 0. the result for coulomb potential was obtained by putting b = de = 0. the result for yukawa potential was obtained by putting a = de = 0. the results in tables 1, 2, 3, 4 and 5 showed that a diffused density distribution γ( p) in momentum space is associated with a localized density distribution ρ(r) in the position space or configuration space. the physical meaning is that a decrease in the position space corresponds to an increase in the momentum space and figure 5. entropies (s = rt /s t + 0.03a; t = rt /tt + 0.05) against the potential strength a vice visa. 4. conclusions we calculated the shannon information, rényi information and tsallis information for position and momentum entropies of a relativistic klein-gordon equation. in the first excited state i.e. the shannon uncertainty yields the minimum value of 8.917638464 which maintains the normal condition for the entropic uncertainty relation with respect to bbm inequality. the s (ρ), s (γ), r(ρ), r(γ), t (ρ) and t (γ) plotted against the potential strengths determined the concentration of the electron and wave density. the results obtained for the relativistic kleingordon equation were found to obey bbm inequality. the results for both the rényi entropy and tsallis entropy followed the pattern of the results of the shannon entropy. appendix ∫ 1 0 yα(1 − y)β2 f1(−n, n + 2(α + β); 2α + 1; y) 2dz = n!γ(α + 1)2γ(β + n + 2) βγ(α + n + 1)γ(α + β + n + 2) ∫ 1 −1 ( 1 − x 2 )η ( 1 + x 2 )ν × [ p(η,ν)n (x) ]2 d x = 2γ(η + n + 1)γ(ν + n + 1) n!ηγ(η + ν + 2n + 1)γ(η + ν + n + 1) ∫ 1 −1 ( 1 − x 2 )s ( 1 + x 2 )ν × [ p(s,ν)n (x) ]2 d x = 2γ(s + n + 1)γ(ν + n + 1) n!sγ(s + ν + 2n + 1)γ(s + ν + n + 1) 169 onate & onyeaju / j. nig. soc. phys. sci. 3 (2021) 165–171 170 p(a,b)n (1 − 2x) = γ)a + n + 1 n!γ(a + 1) 2 f1(−n, n + a + b + 1; 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[52] c. tsallis, “possible generalization of boltzmann-gibbs statistics”, journal of statistical probability 52 (1988) 479. 171 j. nig. soc. phys. sci. 5 (2023) 1103 journal of the nigerian society of physical sciences thermal distribution of magneto-tangent hyperbolic flowing fluid over a porous moving sheet: a lie group analysis a. b. disua, s. o. salawub,∗ adepartment of mathematics, national open university, abuja, nigeria bdepartment of mathematics, bowen university, iwo, nigeria abstract an investigation of magneto-hyperbolic tangent fluid motion through a porous sheet which stretches vertically upward with temperature-reliant thermal conductivity is scrutinized in this study. the current model characterizes thermal radiation and the impact of internal heat source in the heat equation plus velocity and thermal slipperation at the wall. the translation of the transport equations is carried out via the scaling lie group technique and the resultant equations are numerically tackled via shooting scheme jointly with fehlberg integration runge-kutta scheme. the results are publicized through various graphs to showcase the reactions of the fluid terms on the thermal and velocity fields. from the investigations, it is found that rising values of the material weissenberg number, slip and suction terms damped the hydrodynamic boundary film whereas the heat field is prompted directly with thermal conductivity. doi:10.46481/jnsps.2023.1103 keywords: thermal conductivity, magneto-tangent hyperbolic liquid, porous sheet, scaling lie group, thermal radiation article history : received: 03 october 2022 received in revised form: 03 november 2022 accepted for publication: 11 december 2022 published: 24 february 2023 © 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: t. latunde 1. introduction magnetohydrodynamic is the interaction of an electromagnetic field with conducting fluids. rheostat flow kinematics is a realistic application of magnetohydrodynamic where a magnetic field is used in conventional fluids since the heat exchange rate is unavailable for some sheet materials. other uses include; thermal insulation, power storage, and so on. in the light of this, abbas et al. [1] investigated a lateral stretching sheet of mhd power-law fluid with differing thermal conductivity. the authors reported a shrinking boundary layer structure with growth ∗corresponding author tel. no: +234 8032056439 email address: kunlesalawu2@gmail.com (s. o. salawu) in the magnetic field term at all phase of fluid categories considered in the report. more so, salawu et al. [2] analyzed magnetohydrodynamic viscous liquid flow across a nonlinear stretchy plate where the model ordinary derivative equations of the system was solved using the collocation-approximation. sarkar and makinde [3] explored the viscous fluids heat transport and magnetohydrodynamic flow across an exponentially stretching layer accounting for viscous dissipation and radiation effect. nadeem et al. [4] explored the magnetohydrodynamic rheological fluid flow on an angular boundary layer flow. the analysis showed that a boost in the magnetic field strength, the regular and tangential velocity profiles diminish while the skin friction coefficient increase. meanwhile, the studies conducted on the flow along a stretch1 disu & salawu / j. nig. soc. phys. sci. 5 (2023) 1103 2 ing sheet has become a great deal of interest among researchers in respect of significant contributions in manufacturing and engineering activities. sakiadis [5] analyzed the continuously moving flat surface on a laminar boundary layer and obtained a computational solution for the boundary layer equations. crane [6] extended this by reporting on the time-independent linearly stretchy flow and specified the solutions in a closed form. such a study was also extended by wang [7] by incorporating partial slip effect, whereas tshivhi et al. [8] investigated such a concept over a flat stretchable sheet when the flow is initiated spontaneously from rest. okedoye et al. [9] analyzed slip fluid motion confined in a permeable stretchable material while the impact of partial slip due to vertical stretching sheet on stagnation-point flow with thermal transport was assessed by zaimi and ishak [10]. salawu et al. [11] examined the crossdiffusion impact on magnetohydrodynamic fluid flow through a stretched sheet with velocity slip. a case of mhd dissipative fluid flow occasioned by a non-linearly stretched material with heat-mass transfer was numerically evaluated by upreti et al. [12]. it was pointed out from the analysis that the thermal field is enlarged by the enhancement of the magnetic field. such an investigation was also extended to the transport of casson liquid configuration in a three-dimensional sheet with pores and joule heating effect by sreenivasulu et al. [13] while fatunmbi and okoya [14] inspected hydromagnetic micropolar fluid thermal transport characteristics over a stretching material featuring the prescribed thermal flux and plate temperature heating conditions. the studies of non-newtonian fluids have inspired scientists and engineers in recent times owing to its many uses in science and technology including food processing, drug and pharmaceutical productions, chemical engineering works and many more, salawu et al. [15-16]. examples of non-newtonian fluids include gels, paints, blood, printers ink, lubricants with polymer ingredients, cosmetics and toiletries. among various non-newtonian fluid theories, there exists the tangent hyperbolic fluid model commonly utilized in numerous laboratory and chemical engineering processes. this fluid model displays a shear thinning attributes such that there exists a decline in the viscosity as the shear rate rises, hassan et al. [17]. this unique feature of tangent hyperbolic fluid makes it a sought after in bio-engineering operations, for instance, the thinning attributes of blood flow in the body serves as a prevention to the obstruction of arteries and veins such that coagulation effect is minimized, alsharif et al. [18]. in view of such striking characteristics, various researchers have applied this fluid model to analyze various flow problems under different configurations. mamatha et al. [19] examined the motion of hydrodynamic tangent hyperbolic liquid mixed with dust particles in a porous stretching plate with convective heating. a numerical evaluation of such a phenomenon towards a stagnation-point in the occurrence of radiative heat, nonlinear convection, haphazard motion and thermo-migration of nanoparticles was scrutinized by khan et al. [20]. meanwhile, the distribution of such a liquid with nanomaterials mixture in a nonlinear stretchable material was scrutinized by mahanthesh and mackolil [21] for a stagnation fluid. these researchers reported a rise in the viscous drag due to enhancement in the power-law index and magnetic field terms. oyelakin and sibanda [22] inspected the influence of exponentially based viscosity on the motion of hyperbolic tangent fluid. the report showed that a decrease in the viscosity triggered a spike in the velocity while lowering the heat and species intensity. sophus and ackerman [23] found point metamorphosis that mapped a given differential equation and introduced the lie group analysis classical approach. this approach brings together nearly every known technique of exact integration for all the associated ordinary and partial differential equations. many researchers employed this technique to determine the similarities among given differential equations. using this technique, the number of variables that control the partial differential systems can be effectively reduced, salawu and dada [24]. the dilution of values transforms the partial differential system into ordinary systems. using lie group analysis approach, convective dynamics problems have been studied on different flow configurations in various science and engineering branches, zakir and zaman [25]. similarly, ullah and zaman [26] engaged this approach while studying the transport and thermal effects of a tangent hyperbolic flowing liquid through a stretched plate with navier slip effect. further, ullah et al. [27] engaged this approach to extend the work of [26] by incorporating suction/injection coupled with heat generation. the authors examined the partial differential equations representing a natural convective unstable flow movement using the lie symmetry transformation approach. the classical lie group transformation is applied twice sequentially in this study to change the transport model into a set of ordinary derivative equation. the above studies however ignored the impact of variable heat conduction in the temperature field. thermal conductivity describes the characteristic quantity of fluids that allows them to conduct heat. for accurate prediction of thermal propagation processes, the influence of temperature-based thermal conductivity has to be considered. shahzad et al. [28] investigated such an effect on a viscous fluid in the existence of a stretching layer by utilizing the shooting process and the perturbation procedure in analyzing the numerical solution. similarly, alsherif et al. [29] took into account the case of a stretching cylinder, considered a viscous fluid flow alongside variable thermal conductivity. the investigation depicted that growing the curvature of the cylinder causes the fluid temperature to rise rapidly. an examination of temperature based thermal conductivity coupled with thermal radiation impact of a viscous fluid in a porous stretching material was evaluated by hayat et al. [30]. ullah et al. [31] reported on power-law convective mhd liqud flow across a linearly stretchy plate alongside thermal conductivity influence. recently, such a concept has been widely investigated by various researchers, aziz and shams [32] on diverse flow configurations and conditions. in view of the discussion above and the consequential applications of essential fluids parameters in manufacturing and engineering works, the present work aim to determine the motion and thermal transport of hydromagnetic tangent hyperbolic liquid over a permeable vertical stretchy surface using lie group analysis approach. in particular, this study extends that of [26, 2 disu & salawu / j. nig. soc. phys. sci. 5 (2023) 1103 3 figure 1. configuration of the flow model 31] by considering porous media with the inclusion of variable heat conductivity, thermal radiation and a buoyancy effect which were ignored by previous authors. a unique similarity transformation approach is developed using the lie group analysis which is adopted for transforming the nonlinear partial derivative transport model into a more simplified ordinary derivative form. the resultant set of outlining equations is numerically tackled using the shooting algorithm in conjunction with fehlberg integration runge-kutta method. the physical characteristics of dimensionless terms obtained are clarified using graphs with appropriate discussion. 2. problem formulation analysis the underlisted assumptions have been identified as crucial for the formulation of the governing equations for the current investigation. is is assumed that the fluid movement is timeindependent, incompressible tangent hyperbolic fluid. the fluid movement is designed in a two-dimensional porous plate which stretches upwardly in a vertical route as displayed in figure 1. the flow is routed in x axis while y axis runs perpendicular to x axis. a restriction is placed on the flow in the region y > 0. there is slippery in the momentum and energy boundary layers. there is a surface mass flux on the sheet having a velocity of vw(x) as expressed in eq. (7). with the imposition of an external magnetic field normal to x direction but ignoring that of the induced magnetic filed influence and electric field as well. likewise, it is supposed that the radiative heat flux is negligible towards the x axis whereas it is applicable along y direction. furthermore, the assumption of varying thermal conductivity is held valid with the inclusion of heat source. other fluid attributes are are constant apart from the non-uniformity of the density in the momentum body force and the thermal conductivity. boussinesq approximation coupled with boundary layer approximation are applied in this study for the derivation of the main equations. for this study, the tangent hyperbolic fluid tensor is described as [26,31] τ = [µ∞ + (µ0 + µ∞) tanh(γγ) m]γ, (1) in eq. (1), τ describes the tensor stress while µ∞ depicts viscous shear rate at infinity whereas µ0 signifies the zero viscous shear rate and γ describes the material constant of timedependent whereas m connotes the power-law exponent while γ is expressed as: γ = ( 1 2 σiς jγi jγ ji ) 1 2 = ( 1 2 π ) 1 2 . (2) in eq. (2), π = 12 tr((∇v ) t + ∇v )2. the case µ∞ = 0 is accounted for owing to low influence of viscosity at infinity. also taking into account the tangent hyperbolic fluid detailing shear thinning characteristics, with the assumption that γγ < 1, eq. (1) then reduces to: τ = µo[γγ m]γ = µo[(1 + γγ− 1) m]γ ≈ µo[(1 + m(γγ− 1))]γ (3) 2.1. the governing equations combining the above-mentioned assumptions for the development of the transport model, eqs. (4-6) describes the transport equations for the present investigation (see [20,26,31]). ∂u ∂x + ∂v ∂y = 0, (4) u ∂u ∂x + v ∂u ∂y = (1 − m) ν ∂2u ∂y2 + √ 2νmγ ( ∂u ∂y ) ∂2u ∂y2 − σb2 ρ u + gβt (t − t∞) − ν kp u, (5) u ∂t ∂x + v ∂t ∂y = 1 ρc p ∂ ∂y ( k(t ) ∂t ∂y ) − 1 ρc p ∂qr ∂y + qo ρc p (t − t∞) + ν c pkp u2 + σb2 ρc p u2, (6) the respective flow boundary constraints are stated below u = cx + β ∂u ∂y , v = vw(x), t = tw + g ∂t ∂y at y = 0, (7) u −→ 0, t −→ t∞ as y −→∞. (8) the thermal flux radiation qr in eq. (6) is indicated in eq. (9) as (see sumalatha and bandari [33]) qr = − ( 4σ∗ 3k∗ ) ∂t 4 ∂y (9) from the above eqs.(4-9), u and v describe flow rtae modules in respect to x and y axes. the symbols kp,β and σ represent porous medium permeability, velocity slip factor and electrical conductivity whereas the density, volumetric thermal expansion coefficient, magnetic flux density and the thermal slip factor are sequentially denoted by ρ,βt , b and g. also, t signals the fluid temperature, g denotes gravitational acceleration, ν is the kinematic viscosity, vw describes surface mass flux, c defines 3 disu & salawu / j. nig. soc. phys. sci. 5 (2023) 1103 4 stretching rate and qo describes coefficient of heat source/sink, σ∗ connotes stefan-boltzmann constant while the coefficient absorption mean is taken as k∗. by the application of the rosseland approximation and assuming that the heat variation is low in the flow field, so that taylor’s series can utilized to expand t 4 to get t 4 ≈ 4t 3∞ − 3t 4 ∞, the temperature-based thermal conductivity is also specified as (see animasaun [34]): k(t ) = k∞[1 + ζ(t − t∞)], (10) in which k∞ denotes the upstream heat conduction, ζ typifies the thermal conductivity parameter. to transmute the outlining flow equations into dimensionless system, the underlisted quantities adopted: x x = ( a ν ) 1 2 , y y = ( a ν ) 1 2 , u u = 1 (aν) 1 2 , v v = 1 (aν) 1 2 , t = (tw − t∞)θ + t∞. (11) dropping the bar and substituting u = ∂ψ ∂y and v = − ∂ψ ∂x into eqs. (5-6) taking cognizance eqs (9) and (10), the underlisted are obtained( ∂ψ ∂y ∂2ψ ∂x∂y − ∂ψ ∂x ∂2ψ ∂y2 ) = (1 − m) ∂3ψ ∂y3 + √ 2maγ ( ∂2ψ ∂y2 ) ∂3ψ ∂y3 − ( σb2 aρ + ν akp ) ∂ψ ∂y + gbt (tw − t∞) a 3 2 ν 1 2 θ, (12) ( ∂ψ ∂y ∂θ ∂x − ∂ψ ∂x ∂θ ∂y ) = ( k∞ µcp (1 + ζθ) + 16σ∗ 3µcpk∗ t 3∞ ) ∂2θ ∂y2 + k∞ µcp ζ ( ∂θ ∂y )2 + qo aρc p θ+ u2wσb 2 aρc p(tw − t∞) ( ∂ψ ∂y )2 + u2wν akpρc p(tw − t∞) ( ∂ψ ∂y )2 , (13) also, the boundary conditions (7-8) transform to: ∂ψ ∂y = c a x + β √ a ν ∂2ψ ∂y2 , ∂2ψ ∂x2 = vw √ aν ,θ = 1 + g √ a ν ∂θ ∂y at y = 0, ∂ψ ∂y → 0, θ → 0 as y →∞. (14) 3. lie group scaling transformations the lie scaling technique depends on theory formulated to find all symmetry transformations that keep the system of equations unchanged. it helps in reducing the number of independent variables and in consequence transforms the pdes to an odes. using this method to generate similarity variables involves finding the invariant solution which does not alter the structure of the given equation under study. in this section, the simplified format of the lie group transformation approach is employed to derive the new similarity transformations for the transport equations. as such, the outlining flow equations can be changed to ordinary derivative equations. following [27,31,35] the transformation variables are defined υ : x∗ = xeεγ1, y∗ = yeεγ2, ψ∗ = ψeεγ3, θ∗ = θeεγ4, γ∗ = γeεγ5 (15) in eq. (15), ε depicts the parameter of the group whereas the transformation variables are represented by γ1,γ2,γ3,γ4,γ5. also, eq. (15) is called point transformation for the set of coordinates system (x, y,ψ,θ, γ) transforms into (x∗, y∗,ψ∗,θ∗, γ∗). the substitution of the transformation eq. (15) into eq. (12) and (13) results to the form: eε(γ1 +2γ2−2γ3 ) ( ∂ψ∗ ∂y∗ ∂2ψ∗ ∂x∗∂y∗ − ∂ψ∗ ∂x∗ ∂2ψ∗ ∂y∗2 ) = eε(3γ2−γ3 )(1 − m) ∂3ψ∗ ∂y∗3 + eε(5γ2−2γ3−γ5 ) ( √ 2mγ ( ∂2ψ∗ ∂y∗2 ) ∂3ψ∗ ∂y∗3 ) − eε(γ2−γ3 ) ( σb2 aρ + ν akp ) ∂ψ∗ ∂y∗ + gbt (tw − t∞) a 3 2 ν 1 2 θ∗e−εγ4, (16) eε(γ1 +γ2−γ3−γ4 ) ( ∂ψ∗ ∂y∗ ∂θ∗ ∂x∗ − ∂ψ∗ ∂x∗ ∂θ∗ ∂y∗ ) = eε(2γ2−γ4 ) ( k∞ µcp (1 + ζθ∗) + 16σ∗ 3µcpk∗ t 3∞ ) ∂2θ∗ ∂y2 + eε(2γ2−2γ4 ) k∞ µcp ζ ( ∂θ∗ ∂y∗ )2 + qo aρc p θ∗e−εγ4 + eε(γ2−γ3 ) ( u2wσb 2 aρc p(tw − t∞) + u2wν akpρc p(tw − t∞) ) ( ∂ψ∗ ∂y∗ )2 , (17) similarly, the boundary conditions transform to: eε(γ2−γ3 ) ∂ψ∗ ∂y∗ = c a e−εγ1 x∗ + β √ a ν ∂2ψ∗ ∂y∗2 eε(2γ2−γ3 ), ∂2ψ∗ ∂x∗2 eε(γ2−γ3 ) = vw √ aν , e−εγ4θ∗ = 1 + g √ a ν ∂θ∗ ∂y∗ eε(γ2−γ4 ) at e−εγ1 y∗ = 0, eε(γ2−γ3 ) ∂ψ∗ ∂y∗ → 0, e−εγ4θ∗ → 0 as y∗ →∞. (18) the preceding system of equations is invariant under the group transformation if the underlisted relationship exist among the exponents: γ1 + 2γ2 − 2γ3 = 3γ2 −γ3 = 5γ2 − 2γ3 −γ5 = γ2 −γ3 = −γ4 (19) γ1 + γ2 −γ3 −γ4 = 2γ2 −γ4 = 2γ2 − 2γ4 = −γ4 (20) 4 disu & salawu / j. nig. soc. phys. sci. 5 (2023) 1103 5 solving eq. (19) and (20) to obtain the following relations: γ1 = γ3,γ2 = 0,γ4 = γ1,γ5 = −γ1 (21) eq. (21) can then be introduced into eq. (15) to obtain the criterion for the transformation as: υ : x∗ = xeεγ1, y∗ = y,ψ∗ = ψeεγ1,θ∗ = θ, γ∗ = γe−εγ1 (22) applying taylor’s series to expand eq. (22) in the power of ε to the first order to obtain: x∗ − x = xεγ1, y ∗ − y = 0, ψ∗ −ψεγ1, θ∗ − θ = 0, γ∗ − γ = −xεγ1, (23) taking eq. (23), the following characteristic equation were obtained: d x xγ1 = dy 0 = dψ xγ1 = dθ 0 = dγ −xγ1 , (24) the following similarity transformations are derived by solving eq. (24) (see ulla and zaman, 2017): η = y, ψ = x f (η), θ = θ(η), γ = x−1γo (25) the non-dimensional odes obtained with corresponding boundary condition via the similarity transformations (25) into eqs. (16-18) are as follows: (1 − m) f ′′′ + mwe f ′′′( f ′′) + f f ′′ − (m2 + da) f ′ − ( f ′)2 + grθ′ = 0 (26) (1 + ζθ + nr)θ′′ + ζθ′2 + pr(qθ + f θ′) + precm2 f ′2 + precda f ′2 = 0 (27) f ′(0) = λ + α f ′′(0), f (0) = s, θ(0) = 1 + θ′(0), (28) f ′ → 0, θ → 0 as η →∞. (29) in eqs. (26-29), we = √ 2aγ symbolizes the weissenberg number, nr = 16σ ∗ 3k∗k t 3 ∞ defines radiation parameter, m 2 = σb 2 aρ denotes hartmann number, q = q0aρcp typifies the heat source/sink factor and b = √ a ν g is the thermal slip parameters whereas α = √ a ν β represents the velocity slip, da = νakp characterizes the darcy number and ζ implies thermal conductivity parameter. the primes signifies differential with respect to η, λ = ca is the stretching parameter, ec = u 2 w c p (tw−t∞) is eckert number, gr = gbt (tw−t∞) a 3 2 ν 1 2 x symbolizes the grashof number, s = vw√ aν is the mass suction and pr = µcp k∞ represents the prandtl number. the incorporated engineering quantities in the current investigation include the wall friction c fx and the thermal gradient nux which are orderly specified in eq. (30) as: ρ(ax)2c fx = τw, nux = xqw k(tw − t∞) (30) table 1. skin friction coefficient as compared with previous studies when we = m = 0 m akbar [36] fathizadeh et al. [37] present values 0 1.00000 1.00000 1.00000 1 −1.41421 −1.41421 −1.41421 5 −2.44948 −2.44948 −2.44949 10 −3.31662 −3.31662 −3.31663 50 −7.14142 −7.14142 −7.14143 100 10.0499 10.0499 10.0499 500 −22.38300 −22.38300 −22.38300 where τw = (1 − m) ∂u ∂y + mγ √ 2 ( ∂u ∂y )2∣∣∣∣∣∣∣ y=0 , qw = −k∞ ( 1 + 16σ∗ 3k∗k∞ t 3∞ ) ∂t ∂y ∣∣∣∣∣∣ y=0 (31) the dimensionless form of eq. (30) are specified in eq. (31) as: re 1 2 c f = [(1 − m) f ′′(0) + m 2 we( f ′′(0))2], re− 1 2 nux = − (1 + nr) θ ′(0), (32) where rex = ax2 ν signifies the local reynolds number. 4. numerical solution eqs. (26-27) comprises of a set nonlinear coupled differential equations with it’s associated wall conditions. owing to the non-linearity nature of the governing equations, eqs (2627) subject to (28-29) are tackled numerically using shooting techniques alongside runge-kutta fehlberg scheme by utilizing a computer algebra symbolic code of maple software. this algorithm relies on the adopted method. except otherwise, the subsequent default values as been adopted for the study based on related previous analysis as n = 0.4, we = 0.3, � = 0.2, nr = 0.3, m = 0.2, da = 0.3, pr = 3.0, q = 0.3, gr = 2.0, ec = 0.01, λ = 0.7, s = 0.3, α = 0.2, b = 0.5. the numerical code’s accuracy is validated by assessing the computational outcomes of the wall drag coefficient c f x offered in this study as compared with previously published works of akbar [36] and fathizadeh et al. [37] in respect to variations in the hartmann number (m). as recorded in table 1, the comparison showed a perfect harmony with the existing data in the literature under limiting circumstances and thus confirming the accuracy of our numerical code. table 2 depicts the influences of some entrenched parameters on the wall friction and heat gradient. as seen, an enhance or decline in the engineering quantities are observed due to the boundary layer viscosity. when the boundary film viscidness is stimulated the wall friction and nusselt effect are raised, but when thinner boundary film viscidness noticed the diffuse more to the ambient leading to a decrease in the wall effects. 5 disu & salawu / j. nig. soc. phys. sci. 5 (2023) 1103 6 table 2. numerical values for the skin friction (c f ), heat gradient (nux) m � λ da we ec q c f nux 0.2 0.2 0.7 0.3 0.3 0.01 0.3 0.8722973797 -0.5937107035 0.5 0.7677915589 -0.56754883934 1.0 0.4376848220 -0.48222768379 0.4 0.9077648755 -0.53183984435 0.7 0.9077648765 -0.53183984435 1.0 0.4765432364 -0.67408773863 1.5 0.2732784056 -0.80313254770 0.7 0.6779856282 -0.54475633672 1.0 0.6779856282 -0.54475633672 0.5 0.8498608064 -0.59162423311 0.7 0.8300639857 -0.58974089749 0.03 0.8735104313 -0.59064728168 0.07 0.8735104313 -0.59064728168 1.0 1.1540832041 0.0316335380 2.0 1.8442011437 1.9856306670 figure 2. plot of da&λ on velocity f ′(η) 5. discussion of outcomes this aspect displays and discusses the reactions of the dimensionless flow rate and energy profiles due to variations in the physical flow parameters. these physical parameters include the stretchy term (λ), grashof (gr), prandtl (pr), weissenberg (we) and darcy (da) numbers, heat source term (q), power-law exponent term (m), velocity slip term (α), radiation parameter (nr), hartmann number (m), mass suction parameter (s ), thermal conductivity parameter (ζ), and temperature slip term (b). figures 2-6 describe the influences of various physical flow parameter on the velocity field. fig. 2 illustrates the effects of figure 3. behaviour gr&we on velocity f ′(η) (da) darcy term on the dimensionless velocity in the existence of stretching parameter (λ). evidently, there is a decrease in the velocity as (da) increases. the flow behaviour in respect to a spike in darcy number (da) stimulates an opposition to the flow distribution that leads to a shrink boundary layer and thereby decelerates the fluid motion. in a related sense, an enhancement in the magnitude of the stretching term (λ) lowers the momentum boundary layer structure and consequently decelerates the locomotion. the impacts of grashof number (gr) and weissenberg term (we) on the dimensionless flow rate profile are presented in fig. 3. it is evident from the graph displayed that the velocity drop significantly by a rise in (we) owing to an increase in the viscosity whereas there is an acceleration in the 6 disu & salawu / j. nig. soc. phys. sci. 5 (2023) 1103 7 figure 4. effect of s &m on velocity f ′(η) figure 5. reaction q&m on velocity f ′(η) fluid motion as grashof number increases due to enhancement in the buoyancy force. as (gr) is raised, the buoyancy force dominates the viscous force and thus encourages the velocity distribution. fig. 4 portrays the impact of mass suction term (s ) coupled with that of the power-law exponent (m) on the velocity distribution. it is evident that enhancing the magnitude of the power-law exponent (m) raised the viscosity and as a result, there is a significant drop in fluid velocity. also, this plot figure 6. effect of α&m on velocity f ′(η) shows raising the magnitude of s and (m), the hydrodynamic boundary structure thickness declines and the velocity decelerates. the evaluation of the heat generation (q) term and the hartmann number (m) is plotted in fig. 5. evidently, an electromagnetic force is produced from the magnetic field interaction with the tangent hyperbolic electrically conducting liquid that create a drag in the flow movement as noted in the plot. the electro-conducting fluid’s interaction with the transverse magnetic field induces a retarding force on the liquid motion. similarly, a hike in (q) induces higher flow velocity rate owing to a decrease in the viscosity. fig. 6 offers the behaviour of (α) on the liquid motion. in this plot, a declining trend is observed in the velocity field as the slip term (α) rises. figures 7-12 offer the variations of some physical terms on the thermal field. firstly, the temperature profile showing the impact of (λ) in the occurrence of radiative heat (nr) parameter is plotted in fig 7. the graph elucidates that advancement in (λ) causes the temperature to fall whereas growing (nr) enhances the thermal profile. an advancement in the radiative heat flux corresponding to a rise in nr while the rosseland mean absorption coefficient declines and as such, the thermal field is enhanced as found in this figure. the results of the prandtl number (pr) and power-law index (m) on the thermal distribution are depicted in fig. 8. the graph demonstrates that a boost in the (pr) number lowers the thermal field by shrinking the energy boundary viscosity structure whereas the thermal propagation improves as (m) rises. the prandtl number connotes the diffusivity of the momentum ratio to the diffusivity of the heat, and also influence the relative momentum shear stress and thermal boundary layer. thus, a boost in the pr implies a reduction in the energy boundary film and consequently leads to a decline in the heat transfer. the reactions of thermal generation term 7 disu & salawu / j. nig. soc. phys. sci. 5 (2023) 1103 8 figure 7. influence of λ&nr on temperature θ(η) (q) and hartmann term (m) on the energy filed are displayed in fig. 9. the graph portrays the fluid temperature exhibiting identical growing patterns on (q) and (m). typically, both parameters cause a rising trend in the thermal boundary layer. an enhancement in (m) induces a higher electromagnetic force which inspires an obstruction to the liquid motion and thus increase frictional heating effect which boosts the surface temperature. similarly, a hike in q is an indication of extra energy being generated and thus, a rise in the temperature as found in this figure. fig. 10 elucidates the reactions of the thermal slip term (b) and weissenberg term (we) on the fluid heat propagation. the temperature boundary structure shrinks and the temperature falls with growth in b whereas the converse occurs with enhancement in we as noticed in this figure. a rise in (b) draws away the fluid from the heated region thereby lowers the temperature whereas as we rises in magnitude a frictional heat is generated due to rising viscosity. the reactions of mass suction term (s ) and power-law exponent (m) are plotted in fig. 11. this plot reveals that with advancement in s , the temperature distribution subsides whereas as (m) increases, the temperature distribution shoots up. likewise, the plot showing the variation in darcy term (da) and thermal conductivity term ζ in respect to temperature is sketched in fig. 12. it is noticeable that an increment in the (da) and ζ boost the temperature distribution due to extra heat generated by the resistance imposed on the fluid flow as da increases. in figure 13, the impact of ecket number (ec) on the heat propagation with variation in hartmann number (m) is established. as seen, temperature distribution is raised due to an induce magnetic joule heating that inspired the tangent hyperbolic fluid flow particles interaction. also, the magnetic joule heating effect is complemented figure 8. reactions of m&pr on temperature θ(η) figure 9. impact of q&m on temperature θ(η) by the porous joule heating that creates fluid friction and resistant to free flow, thus, particles collision and random motion is encouraged to increase heat transfer. therefore, rising heat distribution magnitude is observed all over the flow region. 6. conclusion a computation solution has been performed on the motion and thermal propagation of hydromagnetic tangent hyperbolic 8 disu & salawu / j. nig. soc. phys. sci. 5 (2023) 1103 9 figure 10. impact of b&we on temperature θ(η) figure 11. effect of s &m on θ(η) liquid passing a vertically stretched surface with varying thermal conductivity. the flow model is in steady 2-dimensional and incompressible stretchable plate enclosed in permeable media with the impact of radiative heat and internal thermal energy source. lie group analysis generates the similarity transformation which transformed the coupled differential equations with boundary conditions from partial to ordinary derivative equations, the solution to the equations are the offered computafigure 12. influence of da&ζ on temperature θ(η) figure 13. effect of ec&m on heat field θ(η) tionally via shooting approach alongside fehlberg runge-kutta method. the solutions are given graphically and deliberated while comparison with published studies show good agreement. the study also reveals that: • the fluid velocity accelerates with enhancement in the heat source term q and grashof number gr magnitude. however, augmenting the darcy term da, suction s term, velocity slip α, stretching term λ, hartmann number m, 9 disu & salawu / j. nig. soc. phys. sci. 5 (2023) 1103 10 power-law exponent m as well as weissenberg value we decelerates the velocity profiles. • a damped in the thermal boundary structure and the corresponding surface temperature distribution falls with a rise in the mass suction term s , prandtl number pr and thermal slip term b. • a rising thermal boundary film viscosity structure is formed with increasing the fluid heat conduction term ζ, radiative heat term nr, darcy number as well as the heat source parameter q. references [1] t. abbas, s. rehman, a. s. muhammad & i. utmankhel, ”analysis of mhd carreau fluid flow over a stretching permeable sheet with variable viscosity and thermal conductivity”, physica a: statistical 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[36] n. s. akbar, s. nadeem, r. u. haq & z. h. khan, ”numerical solutions of magnetohydrodynamic boundary layer flow of tangent hyperbolic fluid towards a stretching sheet. indian journal of physics 87 (2013) 1. [37] m. fathizadeh, m. madani, y. khan, n. faraz & s. tutkun, ”an effective modification of the homotopy perturbation method for mhd viscous flow over a stretching sheet”, j. king. saud. university sci. 25 (2013) 107. 11 j. nig. soc. phys. sci. 5 (2023) 1017 journal of the nigerian society of physical sciences an accuracy-preserving block hybrid algorithm for the integration of second-order physical systems with oscillatory solutions joshua sundaya,∗, joel n. ndama, lydia j. kwarib adepartment of mathematics, university of jos, jos 930003, nigeria bdepartment of mathematics, federal college of education, pankshin 933105, nigeria abstract it is a known fact that in most cases, to integrate an oscillatory problem, higher order a-stable methods are often needed. this is because such problems are characterized by stiffness, chaos and damping, thus making them tedious to solve. however, in this research, an accuracy-preserving relatively lower order block hybrid algorithm (bha) is proposed for solution of second-order physical systems with oscillatory solutions. the sixth order algorithm was derived using interpolation and collocation of power series within a single step interval [tn, tn+1]. in order to circumvent the dahlquist-barrier and also obtain an accuracy-preserving algorithm, four off-step points were incorporated within the single step interval. a number of special cases of oscillatory problems were solved using the proposed method and the results obtained clearly showed that it outperformed other existing methods we compared our results with even though the bha is of lower order relative to such methods. some of the second-order physical systems considered were the kepler, bessel and damped problems. some important properties of the bha were also analyzed and the results of the analysis showed that it is consistent, zero-stable and convergent. doi:10.46481/jnsps.2023.1017 keywords: accuracy-preserving, algorithm, block hybrid method, oscillation, physical systems, second-order article history : received: 28 august 2022 received in revised form: 05 december 2022 accepted for publication: 11 december 2022 published: 14 january 2023 c© 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: t. latunde 1. introduction second-order physical systems with oscillatory solutions find applications in diverse areas of human endeavors like engineering and sciences. such systems are applied in vibration of mass-spring systems, astrophysics, control theory, mechanics, circuit theory, biology among others [1, 2]. in this research, an accuracy-preserving bha shall be derived for the direct in∗corresponding author tel. no: +234 7034884482 email address: sundayjo@unijos.edu.ng (joshua sunday) tegration of second order oscillatory physical systems of the form y′′(t) = f ( t, y(t), y′(t) ) (1) subject to the initial conditions y(t0) = y0, y ′(t0) = y ′ 0 (2) on the interval t ∈ [t0, tn ], where f : < × <m → <m, n is an integer and m the dimension of equation (1). equation (1) is assumed to satisfy the uniqueness theorem stated in theorem 1.1. 1 sunday et al. / j. nig. soc. phys. sci. 5 (2023) 1017 2 theorem 1.1 [3] let, y(n)(t) = f ( t, y(t), y′(t), ..., y(n−1)(t) ) , y(k)(t0) = yk (3) where k = 0, 1, ..., (n − 1),y(0) = y, y(t) and f are scalars. let < be the region defined by the inequalities t0 ≤ t ≤ t0 + a, |sk − yk| ≤ bk, k = 0, 1, ..., n − 1, where yk ≥ 0 for k > 0. suppose the function f (t, s0, s1, ..., sn−1) in (3) is non-negative, continuous and non-decreasing in t, and continuous and nondecreasing in sk for each k = 0, 1, ..., n−1 in the region <. if in addition f (t, y0, y1, ..., yn−1) , 0 in < for t > t0, then the initial value problem (3) has at most one solution in <”. the derivation will be carried out via a continuous scheme based on linear multistep method by incorporating four off-step points. the implementation of the algorithm will be effected in a block-by-block mode, thus making it self-starting (i.e. without the need for predictors). equations of the form (1) can be solved by first transforming them into their equivalent system of first order differential equations and then employing an appropriate method [46]. however, one of the setbacks of such approach is that some of the vital properties and characteristics of the higher order differential equations are lost in the course of the conversion. besides, coding such methods are often cumbersome since in most cases subroutines have to be incorporated to provide the starting values. some methods have also been proposed in literature for the direct solution of special second order differential equations. such equations are termed ‘special’ because they do not depend on y′, in order words they are of the form y′′ = f (x, y). these methods often require fewer function evaluations and less memory space [7-9]. over the years, several authors have proposed different methods for the direct solution of oscillatory problems of the form (1). ref[2] proposed an eleventh order block hybrid method for the direct solution of system of second order differential equations including hamiltonian systems. the method was derived from continuous scheme via hybrid method approach with several off-step points. the method was implemented in a block manner. ref[10] formulated a continuous explicit hybrid method for the solution of second order differential equations. the authors interpolated the basis function at both grid and offgrid points while the differential systems were collocated at selected points. the authors further derived starting values of the same order with the methods by adopting taylor series expansion to circumvent the inherent disadvantage of starting values of lower order. ref[11] developed an order eight implicit block method for the solution of second order differential equations. the authors adopted the hermite polynomial as basis function to construct the method that comprises first and second derivatives. the basic properties of the method were analysed and the method was implemented on some linear and nonlinear second order differential equations. other researchers that also developed direct methods for solving problems of the form (1) are [12-23]. 2. derivation and implementation of the bha 2.1. derivation of the bha the accuracy-preserving bha shall be derived by seeking a continuous approximate solution y (t) to the second order oscillatory problems of the form (1) on the interval [tn, tn+1]. this is expressed compactly in a vector form as, y (t) = [ 1 t t2 ... t7 ]  σ0 σ1 σ2 . . . σ7  (4) where σ0,σ1,σ2, ...,σ7are uniquely determined parameters in <n. equation (4) is interpolated at the points (tn+τ, yn+τ) , τ = r, s and collocated at the points (tn+κ, fn+κ) , κ = 0, p, q, r, s, 1, where τ and κare the interpolation and collocation points respectively. the off-step points p, q, r, s are taken as p = 1/5, q = 2/5, r = 3/5 and s = 4/5. the parameters σ0,σ1,σ2, ...,σ7are determined by imposing the conditions y (tn+τ) = yn+τ, τ = r, s (5) y′′ (tn+κ) = fn+κ, κ = 0, p, q, r, s, 1 (6) to obtain the system of equations  1 tn+r t2n+r t 3 n+r t 4 n+r t 5 n+r t 6 n+r t 7 n+r 1 tn+s t2n+s t 3 n+s t 4 n+s t 5 n+s t 6 n+s t 7 n+s 0 0 2 6tn 12t2n 20t 3 n 30t 4 n 42t 5 n 0 0 2 6tn+p 12t2n+p 20t 3 n+p 30t 4 n+p 42t 5 n+p 0 0 2 6tn+q 12t2n+q 20t 3 n+q 30t 4 n+q 42t 5 n+q 0 0 2 6tn+r 12t2n+r 20t 3 n+r 30t 4 n+r 42t 5 n+r 0 0 2 6tn+s 12t2n+s 20t 3 n+s 30t 4 n+s 42t 5 n+s 0 0 2 6tn+1 12t2n+1 20t 3 n+1 30t 4 n+1 42t 5 n+1   σ0 σ1 σ2 . . . σ7  =  yn+r yn+s fn fn+p fn+q fn+r fn+s fn+1  (7) the system (7) is solved using swp 5.5 for the parameters σ′j s and then substituted into equation (4) to get the continuous form 2 sunday et al. / j. nig. soc. phys. sci. 5 (2023) 1017 3 y (t) = [ 1 t t2 ... t7 ]  1 tn+r t2n+r t 3 n+r t 4 n+r t 5 n+r t 6 n+r t 7 n+r 1 tn+s t2n+s t 3 n+s t 4 n+s t 5 n+s t 6 n+s t 7 n+s 0 0 2 6tn 12t2n 20t 3 n 30t 4 n 42t 5 n 0 0 2 6tn+p 12t2n+p 20t 3 n+p 30t 4 n+p 42t 5 n+p 0 0 2 6tn+q 12t2n+q 20t 3 n+q 30t 4 n+q 42t 5 n+q 0 0 2 6tn+r 12t2n+r 20t 3 n+r 30t 4 n+r 42t 5 n+r 0 0 2 6tn+s 12t2n+s 20t 3 n+s 30t 4 n+s 42t 5 n+s 0 0 2 6tn+1 12t2n+1 20t 3 n+1 30t 4 n+1 42t 5 n+1  −1  yn+r yn+s fn fn+p fn+q fn+r fn+s fn+1  (8) on simplification and evaluation of the continuous scheme (8), we obtain y (t) = ∑ τ=r,s ατ(x)yn+τ + h 2  1∑ j=0 β j(t) fn+ j + βk(t) fn+k  , k = p, q, r, s (9) where αr (x) = −5x + 4 αs(x) = −3 + 5x β0(x) = − 625 1008 x 7 + 12548 x 6 − 425 96 x 5 + 12532 x 4 − 137 72 x 3 + 12 x 2 − 397 6300 x + 1 375 βp(x) = 3125 1008 x 7 − 875 72 x 6 + 177596 x 5 − 1925 144 x 4 + 256 x 3 − 14059 50400 x + 127 3000 βq(x) = − 3125 504 x 7 + 162572 x 6 − 1475 48 x 5 + 2675144 x 4 − 25 6 x 3 − 157 1260 x + 29 375 βr (x) = 3125 504 x 7 − 125 6 x 6 + 122548 x 5 − 325 24 x 4 + 259 x 3 − 5813 25200 x + 161 1500 βs(x) = − 3125 1008 x 7 + 1375144 x 6 − 1025 96 x 5 + 1525288 x 4 − 25 24 x 3 − 1 1575 x + 4 375 β1(x) = 625 1008 x 7 − 125 72 x 6 + 17596 x 5 − 125 144 x 4 + 16 x 3 − 107 50400 x − 1 3000  (10) and x is expressed as x = t − tn h (11) solving (9) for the independent solution at the grid points gives the method yn+ j = 1∑ i=0 ( jh)i i ! y(i)n + h 2  1∑ j=0 η j(x) fn+ j + ηk(x) fn+k  , k = p, q, r, s (12) where η0(x) = − 625 144 x 6 + 1258 x 5 − 2125 96 x 4 + 1258 x 3 − 137 24 x 2 + x ηp(x) = 3125 144 x 6 − 875 12 x 5 + 887596 x 4 − 1925 36 x 3 + 252 x 2 ηq(x) = − 3125 72 x 6 + 162512 x 5 − 7375 48 x 4 + 267536 x 3 − 25 2 x 2 ηr (x) = 3125 72 x 6 − 125x5 + 612548 x 4 − 325 6 x 3 + 253 x 2 ηs(x) = − 3125 144 x 6 + 137524 x 5 − 5125 96 x 4 + 152572 x 3 − 25 8 x 2 η1(x) = 625 144 x 6 − 125 12 x 5 + 87596 x 4 − 125 36 x 3 + 12 x 2  (13) on the evaluation of (12) at x = p, q, r, s, 1, we obtain the discrete accuracy-preserving bha whose coefficients are presented in table 1. therefore, the accuracy-preserving bha is given explicitly as, yn+ 15 = yn + 1 5 hy ′ n + h 2 ( 1231 126000 fn + 863 50400 fn+ 15 − 761 63000 fn+ 25 + 941 12600 fn+ 35 − 341 126000 fn+ 45 + 107 252000 fn+1 ) yn+ 25 = yn + 2 5 hy ′ n + h 2 ( 71 3150 fn + 544 7875 fn+ 15 − 37 1575 fn+ 25 + 136 7875 fn+ 35 − 101 15750 fn+ 45 + 8 7875 fn+1 ) yn+ 35 = yn + 3 5 hy ′ n + h 2 ( 123 3500 fn + 3501 28000 fn+ 15 − 9 3500 fn+ 25 + 87 2800 fn+ 35 − 9 875 fn+ 45 + 9 5600 fn+1 ) yn+ 45 = yn + 4 5 hy ′ n + h 2 ( 376 7875 fn + 1424 7875 fn+ 15 + 176 7875 fn+ 25 + 608 7875 fn+ 35 − 16 1575 fn+ 45 + 16 7875 fn+1 ) yn+1 = yn + hy ′ n + h 2 ( 61 1008 fn + 475 2016 fn+ 15 + 25 504 fn+ 25 + 125 1008 fn+ 35 + 25 1008 fn+ 45 + 11 2016 fn+1 )  (14) y ′ n+ 15 = y ′ n + h ( 19 288 fn + 1427 7200 fn+ 15 − 133 1200 fn+ 25 + 241 3600 fn+ 35 − 173 7200 fn+ 45 + 3 800 fn+1 ) y ′ n+ 25 = y ′ n + h ( 14 225 fn + 43 150 fn+ 15 + 7 255 fn+ 25 + 7 255 fn+ 35 − 1 75 fn+ 45 + 1 450 fn+1 ) y ′ n+ 35 = y ′ n + h ( 51 800 fn + 219 800 fn+ 15 + 57 400 fn+ 25 + 57 400 fn+ 35 − 21 800 fn+ 45 + 3 800 fn+1 ) y ′ n+ 45 = y ′ n + h ( 14 225 fn + 64 255 fn+ 15 + 8 75 fn+ 25 + 64 255 fn+ 35 + 14 255 fn+ 45 + (0) fn+1 ) y ′ n+1 = y ′ n + h ( 19 288 fn + 25 96 fn+ 15 + 25 144 fn+ 25 + 25 144 fn+ 35 + 25 96 fn+ 45 + 19 288 fn+1 )  (15) equations (14) and (15) together form the proposed bha. 3 sunday et al. / j. nig. soc. phys. sci. 5 (2023) 1017 4 table 1. coefficients of the bha yn y′n fn fn+p fn+q fn+r fn+s fn+1 yn+p 1 1 5 1231 126000 863 50400 −761 63000 941 12600 −341 126000 107 252000 yn+q 1 2 5 71 3150 544 7875 −37 1575 136 7875 −101 15750 8 7875 yn+r 1 3 5 123 3500 3501 28000 −9 3500 87 2800 −9 875 9 5600 yn+s 1 4 5 376 7875 1424 7875 176 7875 608 7875 −16 1575 16 7875 yn+1 1 1 61 1008 475 2016 25 504 125 1008 25 1008 11 2016 y′n+p 0 1 19 288 1427 7200 −133 1200 241 3600 −173 7200 3 800 y′n+q 0 1 14 225 43 150 7 255 7 255 −1 75 1 450 y′n+r 0 1 51 800 219 800 57 400 57 400 −21 800 3 800 y′n+s 0 1 14 225 64 255 8 75 64 255 14 55 0 y′n+1 0 1 19 288 25 96 25 144 25 144 25 96 19 288 2.2. implementation of the bha unlike the conventional predictor-corrector methods that require starting values, the proposed bha is self-starting. it is implemented in block mode over a nonoverlapping one-step interval [tn, tn+1], n = 0, 1, 2, ..., n−1. firstly, we set the data inputs namely initial conditions, step size and differential equation to be solved. thus at n = 0, the block method is applied simultaneously on the subinterval [t0, t1] to get [ yκi, y ′ κi ]t , i = 1, 2, ....5. applying the values of the known previous block, we get the values of the next subinterval [t1, t2] and the process goes on in this manner until the final subinterval [tn−1, tn ] is obtained. the code was written in matlab 2021a. 3. properties of the bha the properties of the proposed bha shall be analysed in this section. 3.1. order and error constant of the bha definition 3.1 the linear difference operators associated with the bha in equations (14) and (15) are defined as `p {y(tn); h} = y (tn + ph) − ( αr (t)y (tn + rh) + αs(t)y (tn + sh) + h2 ∑1 j=0 ( β j(t) fn+ j + βk(t) fn+k )) , k = p, q, r, s `q {y(tn); h} = y (tn + qh) − ( αr (t)y (tn + rh) + αs(t)y (tn + sh) + h2 ∑1 j=0 ( β j(t) fn+ j + βk(t) fn+k )) , k = p, q, r, s `r {y(tn); h} = y (tn + rh) − ( αr (t)y (tn + rh) + αs(t)y (tn + sh) + h2 ∑1 j=0 ( β j(t) fn+ j + βk(t) fn+k )) , k = p, q, r, s `s {y(tn); h} = y (tn + sh) − ( αr (t)y (tn + rh) + αs(t)y (tn + sh) + h2 ∑1 j=0 ( β j(t) fn+ j + βk(t) fn+k )) , k = p, q, r, s `1 {y(tn); h} = y (tn + h) − ( αr (t)y (tn + rh) + αs(t)y (tn + sh) + h2 ∑1 j=0 ( β j(t) fn+ j + βk(t) fn+k )) , k = p, q, r, s  (16) ` ′ p {y(tn); h} = hy ′ (tn + ph) − ( α ′ r (t)y (tn + rh) + α ′ s(t)y (tn + sh) + h 2 ∑1 j=0 ( β ′ j(t) fn+ j + β ′ k(t) fn+k )) , k = p, q, r, s ` ′ q {y(tn); h} = hy ′ (tn + qh) − ( α ′ r (t)y (tn + rh) + α ′ s(t)y (tn + sh) + h 2 ∑1 j=0 ( β ′ j(t) fn+ j + β ′ k(t) fn+k )) , k = p, q, r, s ` ′ r {y(tn); h} = hy ′ (tn + rh) − ( α ′ r (t)y (tn + rh) + α ′ s(t)y (tn + sh) + h 2 ∑1 j=0 ( β ′ j(t) fn+ j + β ′ k(t) fn+k )) , k = p, q, r, s ` ′ s {y(tn); h} = hy ′ (tn + sh) − ( α ′ r (t)y (tn + rh) + α ′ s(t)y (tn + sh) + h 2 ∑1 j=0 ( β ′ j(t) fn+ j + β ′ k(t) fn+k )) , k = p, q, r, s ` ′ 1 {y(tn); h} = hy ′ (tn + h) − ( α ′ r (t)y (tn + rh) + α ′ s(t)y (tn + sh) + h 2 ∑1 j=0 ( β ′ j(t) fn+ j + β ′ k(t) fn+k )) , k = p, q, r, s  (17) suppose y(t)is differentiable, the terms ` and `′ are expanded as taylor series about the point tn to get `k {y(tn); h} = ckhp+2y(p+2) (tn) + o ( hp+3 ) , k = p, q, r, s ` ′ k {y(tn); h} = c ′ kh p+2y( p+2) (tn) + o ( hp+3 ) , k = p, q, r, s  (18) where p in equation (18) denotes the order of the bha and ` and `′ the error constants. 4 sunday et al. / j. nig. soc. phys. sci. 5 (2023) 1017 5 table 2. order and error constant of the bha y(tn+ j) order error constant j = 1/5 6 −2.1058 × 10−8 j = 2/5 6 −5.1471 × 10−8 j = 3/5 6 −8.0571 × 10−8 j = 4/5 6 −1.0836 × 10−7 j=1 6 −1.4550 × 10−7 therefore, the proposed bha is of order 6. definition 3.2 [24] a linear multistep method for a second order problem is said to be of order pif it satisfies c0=c1=c2=. . . =cp=cp+1=0, cp+2 , 0, where, c0 = ∑k j=0 α j c1 = ∑k j=0 ( jα j −β j ) . . . cp = ∑k j=0 [ 1 p! j pα j − 1 (p−1)! j p−1β j ] , p = 2, 3, ..., q + 1  (19) the parameter cp+2 , 0 is referred to as the error constant of the method with the local truncation error defined as tn+k = cp+2hp+2y(p+2)(tn) + o ( hp+3 ) ”. the bha is expanded in taylor series about the point tn to obtain  ∑ ∞ j=0 ( 15 h) j j! y j n − yn − 1 5 hy ′ n − 1231 126000 h 2y ′′ n − ∑ ∞ j=0 h j+2 j! y j+2 n { 863 50400 ( 1 5 ) j − 761 63000 ( 2 5 ) j + 94112600 ( 3 5 ) j − 341 126000 ( 4 5 ) j + 107252000 (1) j } ∑ ∞ j=0 ( 25 h) j j! y j n − yn − 2 5 hy ′ n − 71 3150 h 2y ′′ n − ∑ ∞ j=0 h j+2 j! y j+2 n { 544 7875 ( 1 5 ) j − 37 1575 ( 2 5 ) j + 1367875 ( 3 5 ) j − 101 15750 ( 4 5 ) j + 87875 (1) j } ∑ ∞ j=0 ( 35 h) j j! y j n − yn − 3 5 hy ′ n − 123 3500 h 2y ′′ n − ∑ ∞ j=0 h j+2 j! y j+2 n { 3501 28000 ( 1 5 ) j − 9 3500 ( 2 5 ) j + 872800 ( 3 5 ) j − 9 875 ( 4 5 ) j + 95600 (1) j } ∑ ∞ j=0 ( 45 h) j j! y j n − yn − 4 5 hy ′ n − 376 7875 h 2y ′′ n − ∑ ∞ j=0 h j+2 j! y j+2 n { 1424 7875 ( 1 5 ) j + 1767875 ( 2 5 ) j + 6087875 ( 3 5 ) j − 16 1575 ( 4 5 ) j + 167875 (1) j } ∑ ∞ j=0 (h) j j! y j n − yn − hy ′ n − 61 1008 h 2y ′′ n − ∑ ∞ j=0 h j+2 j! y j+2 n { 475 2016 ( 1 5 ) j + 25504 ( 2 5 ) j + 1251008 ( 3 5 ) j + 251008 ( 4 5 ) j + 112016 (1) j }  =  0 0 0 0 0  (20) table 2 displays the order and error constant of the proposed bha. 3.2. consistency of the bha definition 3.3 [24] “if a method has orderp ≥ 1, then it is consistent”. thus, the proposed bha is consistent since it is of order 6. 3.3. zero-stability of the bha definition 3.4 [25] “if no root of the characteristic polynomial has a modulus greater than one and every root with modulus one is simple, then such a method is called zero-stable”. in order words, the bha is zero-stable if its first characteristic polynomial ρ(z)satisfies |zd| ≤ 1, d = 1, 2, 3, ..., n. therefore, 5 sunday et al. / j. nig. soc. phys. sci. 5 (2023) 1017 6 figure 1. region of absolute stability of the bha ρ(z) = ∣∣∣∣∣∣∣∣∣∣∣∣∣∣  z 0 0 0 0 0 z 0 0 0 0 0 z 0 0 0 0 0 z 0 0 0 0 0 z  −  0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1  ∣∣∣∣∣∣∣∣∣∣∣∣∣∣ = z4(z − 1) = 0 (21) on the evaluation of equation (21), we obtain z = 0, 0, 0, 0, 1. therefore, the proposed bha is zero-stable. 3.4. convergence of the bha theorem 3.1 [25] “for a method to be convergent, it is necessary and sufficient that it be consistent and zero-stable.” the bha is convergent since it is consistent and zero-stable. 3.5. region of absolute stability of the bha definition 3.5 [26] “the region of absolute stability of a method is the region in the complex z plane, where z = λh.” in such region, the approximate solution y′′ = −λ2y satisfies yi → 0 as i → 0 for any initial condition. definition 3.6 [26] “if the region of absolute stability of method contains the whole of the left half-plane, that is re (hλ) < 0, then the method is said to be a-stable.” using the boundary locus method, the stability polynomial for the proposed bha is h(w) = −h10 (( 8.13 × 10−10 ) w5 + ( 1.00 × 10−8 ) w4 ) − h8 (( 5.02 × 10−8 ) w5 + ( 5.04 × 10−6 ) w4 ) − h6 (( 1.32 × 10−6 ) w5 + ( 7.18 × 10−4 ) w4 ) − h4 (( 3.71 × 10−2 ) w4 − ( 2.00 × 10−4 ) w5 ) − h2 (( 2.00 × 10−2 ) w5 + ( 6.27 × 10−1 ) w4 ) + w5 − 2w4 (22) for more details on boundary locus method, see [27-28]. the region of absolute stability of the bha is shown in figure 1. note that the absolute stability of the proposed bha in figure 1 is the region that lies inside the closed contour. from the figure, it is clear that the bha is a-stable. 4. numerical examples the newly derived bha shall be applied in integrating some second-order physical systems with oscillatory solutions. the following abbreviations shall be used in the tables 3-5 and figures 3-10. t: point of evaluation h: step-size ns: number of steps taken err: maximum global error errh: maximum global error for the hamiltonian ode 45: matlab inbuilt solver vsshm: variable step-size hybrid method of order seven by [13] bhm: order eleven block hybrid method by [2] ibm: order eight implicit block method by [11] bha: newly proposed sixth order block hybrid algorithm let y(tn) and yn be the exact and approximate solutions respectively, then the maximum global error is computed as err = max |y(tn) − yn| and the maximum global error for the hamiltonian as errh = max |hn − h0|. example 1. the system below describes the perturbed kepler problem y ′′ 1 (t) = − y1 (t) (y21 (t)+y22 (t)) 3/2 − (e2 +2e)y1 (t) (y21 (t)+y22 (t)) 5/2 , y1(0) = 1, y ′ 1(0) = 0 y ′′ 2 (t) = − y2 (t) (y21 (t)+y22 (t)) 3/2 − (e2 +2e)y2 (t) (y21 (t)+y22 (t)) 5/2 , y2(0) = 0, y ′ 2(0) = 1 + e  (23) whose exact solution is given by y1(t) = cos ( et + t ) y2(t) = sin ( et + t ) } (24) where e is the eccentricity taken as e = 10−3. the problem (23) has an angular momentum l = y1(t)y ′ 2(t) − y2(t)y ′ 1(t) (25) as a first integral and also has the hamiltonian h = 1 2 ( y ′ 2 1 (t) + y ′ 2 2 (t) ) − 1( y21(t) + y 2 2(t) )1/2 − ( e2 + 2e ) 3 ( y21(t) + y 2 2(t) )3/2 (26) the author in [2] developed an eleventh order block hybrid method to compute the maximum global errors in the solution as well as the maximum global error in the hamiltonian of example 1 at different step-sizes. the newly derived bha was also used to solve the same problem. the results obtained clearly showed that even though the bha is of the sixth order, it performed better than the eleventh order block hybrid method proposed by [2], see table 3. figure 2 shows the plots for mean anomaly against eccentric anomaly at different eccentricities (e=0.01, 0.1, 0.2, 0.4, 0.6 and 0.9) using the newly derived bha. on the other hand, figure 3 shows the exact plots for mean anomaly against eccentric anomaly at the same eccentricities. the plots generated using the bha (see figure 2) 6 sunday et al. / j. nig. soc. phys. sci. 5 (2023) 1017 7 figure 2. mean anomaly against eccentric anomaly (for example 1) at different values of eccentricity using the proposed bha figure 3. exact plots for mean anomaly against eccentric anomaly (for example 1) at different values of eccentricity figure 4. showing the trajectories in the phase plane flow (for example 1) using the proposed bha at n=160 converges to those of the exact plots in figure 3 at the respective eccentricities. it is important to state that the figures 2 and 3 show the relationship between mean anomaly (that is, the angle between line drawn from the sun to perihelion and to a point moving in the orbit) and eccentric anomaly (that is, the angle that describes the position of a body that is moving along an elliptic orbit). furthermore, figures 4, 5 and 6 respectively depict the trajectories in the phase plane flow using proposed bha, bha developed by [2] and the exact flow n = 160. for more details on kepler equations, see [29-35]. example 2. the nonlinear system below describes the aerodynamic damping of a pendulum θ′′ + g l sin θ + ca ml ( θ′ )2 sgn (θ′) = 0 (27) and the linear system below describes the viscous damping of a pendulum θ′′ + cd m ( θ′ )2 + g l θ = 0 (28) to determine the quadratic damping in equations (27) and (28) respectively, the proposed bha is employed in simulating the figure 5. showing the trajectories in the phase plane flow (for example 1) using bhm at n=160 figure 6. showing the exact trajectories in the phase plane flow (for example 1) at n=160 two equations and the simulation results are compared with that of author [1]. the author transformed (27) and (28) to their equivalent first order systems and then applied matlab solver (ode 45). let g = 981cms−2, ca = cd = 14, l = 5cm, m = 10g, θ(0) = 1.57and θ′(0) = 0. the initial value θ is in radians. also, sgn in equation (27) represents signum function; ca and cd are the coefficients of aerodynamic damping and viscous damping respectively. figure 7 shows a simple pendulum with two forces acting on the mass; that is, the weight mg and the tension t . the net force is found to be f = mg sin θ. from simulation results presented in figure 8, it is obvious that quadratic damping causes a rapid initial amplitude reduction but is less effective as time increases and amplitude decreases. on the other hand, linear damping is less effective initially but is more effective as time increases. the plots confirm that the proposed bha is computationally reliable as they agree with those of ode 45. example 3. consider the nonlinear oscillatory system y′′(t) + [ 13 − 12 −12 13 ] y(t) = ∂u ∂y , y(0) = [ −1 1 ] , y′(0) = [ −5 5 ] (29) where u(y) = y1(t)y2(t) (y1(t) + y2(t)) 3and the exact solution 7 sunday et al. / j. nig. soc. phys. sci. 5 (2023) 1017 8 table 3. juxtaposition of maximum global error for example 1 on [0, 100] h bha-err bhm-err bha-errh bhm-errh 1/2 4.39 × 10−15 1.74 × 10−13 2.62 × 10−17 5.27 × 10−15 1/4 3.46 × 10−14 8.62 × 10−13 5.89 × 10−17 1.23 × 10−14 1/8 5.09 × 10−14 3.49 × 10−12 9.22 × 10−17 2.11 × 10−14 1/16 8.13 × 10−14 6.94 × 10−12 2.16 × 10−16 4.19 × 10−14 1/32 1.67 × 10−13 1.92 × 10−12 6.65 × 10−16 5.06 × 10−14 figure 7. a simple pendulum figure 8. solution curves for example 2 depicting quadratic and linear damping given by y(t) = [ −sin 5t − cos 5t sin 5t + cos 5t ] (30) the system (29) has the hamiltonian h (y(t) + y′(t)) = 12 y ′t y′ + 12 y t ,[ 13 − 12 −12 13 ] y(t) + u(y)  (31) the eleventh order block hybrid method formulated by [2] was employed in computing the maximum global errors in the solution as well as the maximum global error in the hamiltonian for example 3 at different step sizes. the proposed bha was also used to solve the same example. the results obtained clearly showed that even though the bha is of the sixth order, it outfigure 9. solution curves for example 4 showing the xand y displacements performed the block hybrid method of order eleven derived by [2], see table 4. example 4. the following nonlinear system describes the trajectory motion of an object in terms of x and y displacements mx′′(t) + cd x ′(t) (( x′(t) )2 + ( y′(t) )2)(1/2) = 0 (32) my′′(t) + cd y ′(t) (( x′(t) )2 + ( y′(t) )2)(1/2) = −w (33) the system was solved using the following parameters of aerodynamic drag cd = {0.1, 0.3, 0.5} n/m. let x(0) = 0, y(0) = 0,x′(0) = 100ms−1, y′(0) = 10ms−1,m = 10kg,w = mg and g = 9.81ms−2. the proposed bha was directly employed in simulating the system and the results compared with that of the author in [36] who applied ode 45 solver. it is obvious from the plots in figure 9 that as cd is increases, the maximum value of the displacement in the ydirection decreases. the value of the displacement in the x direction, for which the displacement in the y direction is maximum, also decreases. the curves generated via the proposed bha converge to those generated using the ode45 solver, implying that the bha is computationally robust and accurate. example 5. consider the bessel’s nonlinear oscillatory problem t2y′′(t) + ty′(t) + ( t2 − 0.25 ) y(t) = 0, y(1) = √ 2 π sin(1), y′(1) = 2 cos(1) − sin(1) √ 2π (34) whose exact solution is given by y(t) = √ 2 πt sin(t) (35) the newly derived sixth order bha was applied on example 5 in the interval [1, 8]. the results obtained were compared with 8 sunday et al. / j. nig. soc. phys. sci. 5 (2023) 1017 9 table 4. juxtaposition of maximum global error for example 3 on [0, 100] h bha-err bhm-err bha-errh bhm-errh 1/2 1.98 × 100 4.58 × 100 1.04 × 103 2.95 × 103 1/4 8.41 × 10−10 7.54 × 10−8 5.24 × 10−10 3.13 × 10−8 1/8 3.55 × 10−12 2.20 × 10−11 3.33 × 10−13 5.61 × 10−12 1/16 2.23 × 10−14 4.95 × 10−14 6.19 × 10−13 1.62 × 10−12 1/32 5.70 × 10−15 9.15 × 10−14 4.89 × 10−14 3.78 × 10−12 table 5. juxtaposition of maximum global error for example 5 on [1, 8] ns bha-err ibm-err vsshm-err 67 1.1984 × 10−17 7.7716 × 10−16 6.5286 × 10−11 82 6.1189 × 10−18 1.8874 × 10−15 1.3679 × 10−11 112 2.8346 × 10−18 1.1601 × 10−15 1.1897 × 10−12 figure 10. accuracy curves for example 5 those of order eight implicit block method (ibm) by [11] and order seven variable step-size hybrid method (vsshm) developed by [13] using the same number of steps. the results of the new sixth order bha performed better than the other two methods, see table 5. figure 10 also showed that the bha is more accurate than the ibm and vsshm. for more details on oscillatory problems, refer to [37-45]. 5. conclusion an attempt has been made in this research to derive an accuracy-preserving bha for the solution of second-order physical systems with oscillatory solutions. problems considered included kepler, bessel and damped systems. the results generated showed that the bha is accuracy-preserving and computationally reliable. simulation results obtained also showed that the new method was in agreement with the matlab inbuilt solver (ode 45). properties such as order, error constant, zero-stability, stability, consistence and convergence of the new bha were also validated. finally, the proposed bha is recommended for the integration of physical systems with oscillatory solutions. further research will focus on the solution of physical models for higher order partial differential equations. acknowledgements we would like to appreciate the university of jos for providing the facilities for conducting this research. we also appreciate the reviewers and editors for all the valuable comments and suggestions, which helped us to improve the quality of the manuscript. references [1] h. soraya, “pendulum with aerodynamic and viscous damping”, journal of applied information, communication and technology 3 (2016) 43. 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[45] j. sunday, c. chigozie, e. o. omole & j. b. gwong, “a pair of three-step hybrid block methods for the solutions of linear and nonlinear first-order systems”, european journal of mathematics and statistics 3 (2022) 14. 10 j. nig. soc. phys. sci. 3 (2021) 354–359 journal of the nigerian society of physical sciences mathematical models and comparative analysis for rice and soya bean irrigation crop water needs: a case study of bida basin niger state, nigeria a. abdulrahim, m. d shehu, e yisa, z. a. ishaq department of mathematics, school of physical science, federal university of technology, minna, niger state, nigeria. abstract in this manuscript, mathematical models for cropping water need (c.w.n) and the size of land for irrigation (s.l.i) were formulated. the solutions of the models for crop water need for soya beans and rice, and the size of land for irrigation (s.l.i) of the two crops was obtained. we fill the gap by considering the size of the irrigation land which is not considered by the food and agriculture organization (f.a.o). the computational method of solutions is carried out to get effective results. the climatic data of the study area (bida basin) under which our research is based includes: rainfall, humidity, sunshine hours, minimum and maximum temperature, evapotranspiration were secondary data collected from nigeria metrological society (nimet). we compared the results of cropwat 8.0 software developed by the food and agriculture organization (f.a. o) and our computational method so that we can arrive at a new finding and better results. the results for the computational method with the size of land for irrigation shows that there is an increase in crop water need for the crops than the results of cropwat 8.0 software developed by the food and agriculture organization (f.a. o) in which the size for the land is not considered. we therefore, recommended that the integral calculus can be used to estimate the irregular shape of the size of the land if the land shape is not in rectangular form before solutions are given for accuracy and effective results. doi:10.46481/jnsps.2021.149 keywords: bida basin, crop water coefficient, evapotranspiration article history : received: 01 february 2021 received in revised form: 01 march 2021 accepted for publication: 07 june 2021 published:29 november 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: o. j. abimbola 1. introduction bida basin lies in the sedimentary terrain of the middle part of nigeria. it has an area of coverage of about 27,000 km2. the area falls under the middle climatic belt which is mainly tropical with an average rainfall of about 1250m. we are therefore considering the crop water need of rice and soya bean on the aquifer of two lithological groups: unconfined and semi confined aquifer in our selected study area. email address: m.shehu@futminna.edu.ng (z. a. ishaq) cropwat 8.0 software is software developed by food and agricultural organisation (f.a.o), used to evaluate farmer’s irrigations, irrigation practices and to estimate crop performance under both rainfall and irrigated condition. our computational method is used as a tool to solve the models and obtain the solutions to get effective results. the weakness of the cropwat 8.0 software developed by the food and agriculture organization (f.a.o) is that the irrigator farmers do not know the size of the land the results of the software is given as the size of land for irrigation (s.l.i) is not considered by the cropwat 8.0 software 354 abdulrahim et al. / j. nig. soc. phys. sci. 3 (2021) 354–359 355 and we fill the gap by considering the size of land for irrigation. many irrigation farmers faced the challenges of what the crop water need of irrigation crops would be before embarking on irrigation water planning. this research aims are to develop a mathematical model for crop water need problems and determine the exact amount of water need for rice and soya beans crops by considering the size of the land and shape. our objective is to use cropwat 8.0 software developed by the food and agricultural organisation (f.a.o) and our computational method to determine the crop water need of two crops and compared the two results to unravel crop water need problems. irrigated agriculture in south africa has not been profitable over the years. even though, it is the highest user of total consumptive water [1]. its economic returns have not been impressive. the sustainable management of irrigation water resources is therefore, a necessity. a common procedure for estimating crop water use is to first determine the daily reference crop evapotranspiration (et0) and then multiply it by a specific crop coefficient(kc), as given by the food and agriculture organisation [2]. [3] determine the crop water requirements for maize in abshege woreda, gurage zone in ethiopia, they determine crop water requirement which is the major food crop of the area, they used the climatological records of sunshine, maximum and minimum temperature, humidity and wind speed were used as secondary data. penman monteith method was used and crop water requirement was estimated using cropwat 8.0 the results show that a maize variety with a growing period of 140 days to maturity would require 423 mm depth of water, while 101mm of water would be required as supplementary irrigation. [4] came up with a quantitative analysis of hydraulic interaction process in the stream – aquifer systems. it revealed both the theoretical and laboratory tests have demonstrated that, the hydraulic connectedness of the stream aquifer system can reach a critical disconnection state. [5] researched on the crop water requirement for agriculture in a typical river basin of india. the results show the crop water requirements is much below the available rainfall and even available groundwater at various location of the river basin, the crop water requirement is required to be increased by increasing the crop production with multiple crops and by using more agriculture land for crop production. with these, there is a need to develop a mathematical model for solving crop water need with the specific land size. [6] develop estimating aquifer hydraulic properties in bida basin central nigeria, using the empirical method. he determined aquifer properties such as hydraulic conductivity, porosity and effective porosity, and coefficient of uniformity. [7] the understanding of crop water need becomes necessary, as it enables efficient use of water and better irrigation practices like scheduling as the supply of water through rainfall is limited in these areas. [8] carried out a study to determine the crop water need of few selected crops for the commanded area, the outcomes of the study are capable of planners of the water resources for future planning and helps save water in satisfying the crop water need. [9] cropwat 8.0 software helps irrigator planner in allocating the water resources in the future. 2. mathematical model for crop water requirement. 2.0.1. metrological data to calculate this, the respective climatic data was collected from the nigeria meteorological station. the data used for et0 computation was the meteorological data obtained from the station; for instance, minimum and maximum temperatures (c), wind speed in km per day, the relative humidity (maximum and minimum, in %) and the hours of sunshine, and the physical data such as altitude, latitude and longitude. the climatic records obtained were then adjusted into the format accepted by cropwat 8.0. the rainfall data collection was also obtained from the meteorological station. rainfall records from a range of years (10–15) were collected to allow for a calculation of crop water need. 2.0.2. soil data the data utilized on the soil characteristics were acquired through laboratory soil analyses done on the soil samples collected. after the collection of all these data, it was entered into the cropwat 8.0 program and saved. crop and irrigation water needs were then calculated using the model for the majorly observed high value crops including rice and soya bean. the weakness of the crop water need developing by the food and agricultural organisation (f.o.a) is that the results for the amount of water need for crops are given without the size of the land, thus. the size of land for irrigation have to consider. 2.1. mathematical derivation of reference crop water need. considering an energy balance at the earth surface equates all incoming and outgoing energy flux. the following governing equation is considered; rn = h + λe + g (1) where rn=energy flux density net incoming radiation (w/m2) h= flux density of latent heat into the air(w/m2) λe= flux density into the water body (w/m2) g = heat flux density into the water body (w/m2) λ= the latent heat of vaporization of water e = the vapour flux density in kg/m2 s where h = c1 (t s − ta) ra (2) and λe = c2 (es − ed ) ra (3) where, c1, c2= constant t s= temperature at a certain height above the surface (k pa) ea= prevailing vapour pressure at the same height as ta (k pa) ra= aerodynamic diffusion resistance. applying the similarity of transport heat and water vapour, we have a bowen ration yield as; h λe = c1(t s − ta) c2(es − ed ) (4) 355 abdulrahim et al. / j. nig. soc. phys. sci. 3 (2021) 354–359 356 equation (4) becomes; h λe = γ ( t s − ta es − ed ) (5) c1 c2 = γ } (6) γ = c pρa λε (7) making c p the subject of the relation in (7) we have: c p = γλε ρa (8) where, ρa = pa tvkr (9) where, tkv = the virtual temperature r = specific gas constant ρa = mean air density at constant pressure kgm−3 c p = specific heat at constant pressure mj k/g/ 0c equation (7) is referred to as psychometric constant (k pa/oc) in determining the surface temperature, we considered the penman monteith equation which is given as es − ea = ∆(t s − ta) (10) from equation (10) t s − ta = es − ea ∆ (11) substituting equation (11) into equation (5), we have: h λe = γ ∆ ( es − ea es − ed ) (12) replacing es − eaby es − ed − ea + ed h λe = γ ∆ ( 1 − ea − ed es − ed ) (13) considering isothermal evaporation λea given as: λea = c2 es − ed ra (14) setting λ = 1in equation (14) ,then ea = c2 es − ed ra (15) replacing c2 by ερa p in equation (15) we have: ea = ερa p ( es − ed ra ) (16) dividing equation (16) by equation (3) we have: ea e = (ea − ed ) (es − ed ) (17) substituting equation (17) into equation (12) we have: h λe = γ ∆ ( 1 − ea e ) (18) replacing e by et0in equation (18) and make h the subject of relation, we have: h = λγet0 ∆ ( 1 − ea et0 ) (19) substitute equation (19) for h in equation (1) and simplify, we have: et0 = 1 λ (rn − g) ∆ + γea ∆ + γ (20) where, et0=open water evaporation rate (kg/m2) ∆= proportionality constant (k pa/oc) rn= net radiation (j/kg) γ= psychometric constant (k pa/oc) ea= isothermal evaporation rate (kg/m2 s) the term 1 λ (rn−g)∆ ∆+γ in equation (20) is called radiation term the term γea ∆+γ in equation (20) is called aerodynamic term substitute equation (7) and (16) into equation (20). simplifying further to obtain et0 = 1 λ ( (rn − g)∆ + c pρa ra (es − ed ) ) (γ + ∆) (21) where, ε= ration of molecular masses of water vapour and dry air (-) pa= density of moist air (kg/m3) ρa= atmospheric pressure (kpa) cp=specific heat of dry air at constant pressure (j/kg.k) considering vapour diffusion rate then equation (16) is express as: ea = εpa pa . ea − ed ra = εpa pa ea − ed rc = εpa pa e0 − ed rc + ra (22) then, εpa pa ea − ed ra = εpa pa e0 − ed rc + ra (23) simplifying equation (23) we have: ea − ed =  e0 − ed1 + rcra  (24) where, ea= isothermal evapotranspiration rate from canopy eo= internal saturated vapour pressure at (cpa) ea= saturated vapour pressure at the leaf surface ed = vapour pressure in the external air ra= aerodynamic resistance (s/m) rc= canopy diffusion resistance (s/m) substitute equation (24) into equation (20) we have: et0 = 1 λ ( (rn − g)∆ + c pρa ra e0 − ed ) ∆ + γ ( 1 + rcra ) (25) 356 abdulrahim et al. / j. nig. soc. phys. sci. 3 (2021) 354–359 357 let γ ( 1 + rc ra ) = γ∗ (26) then equation (25) becomes: et0 = 1 λ ( (rn − g)∆ + c pρa ra e0 − ed ) ∆ + γ∗ (27) where, eto=evaporation rate from dry surface (kg/m2 s) γ∗ = modified psychometric constant (k pa/oc) substitute equations (8) and (9) into equation (27) we have; et0 = 1 λ ( (rn − g)∆ + γελ pa pa tvk r ra e0 − ed ) ∆ + γ∗ (28) simplifying equation (28) we have: et0 = ( 1 λ (rn − g)∆ + γε ra tvk r e0 − ed ) ∆ + γ∗ (29) ra can be expressed as: ra = ln (zm−d)zom ln (zh−d) zoh k2u2 = ln [ 2−0.08 0.01476 ] ln [ 2−0.08 0.001476 ] (0.41)2 u2 = 208 u2 (30) with the following standard values: d = 0.67h, zom = 0.123h, and zov = 0.120m, k = 0.41, h = 0.12. (31) where z = height at which wind speed is measured (m) d = displacement height (m) zom = roughness length for momentum (m) zov = roughness length for water vapour (m) k = von karman constant equal 0.41 u2= wind speed measure at height (m/s) tkv =the virtual temperature r =specific gas constant ρa = mean air density at constant pressure kgm−3 c p = specific heat at constant pressure m j kg−1 0c−1 from equation (30) we have: c pρa ra = γε ratvkr = (86400)(0.622)γλ (ta + 273)(0.287)(208) u2 = γ 900 ta + 273 u2 (32) with the following standard values: ra = (208)u2, tvk = (ta + 273), r = (0.287) (33) from equation (30) we have: 1 λ = 1 2.45 = 0.408 (34) with standard value of λ = 2.45 from equation (27) γ∗ = γ ( 1 + rc ra ) (35) we let, rc = ri laiactive = 100(0.5)(24)(0.12) = 70sm −1 (36) substituting equation (31) and (37) into equation (27), we have; γ∗ = γ (1 + 0.34u2) (37) substitute equation (31), (33), (37) and (38) into equation (30) we have: et0 = 0.408∆(rn − g) + γ 900 t a+273 u2(e0 − ed ) ∆ + γ(1 + 0.34u2) (38) 2.2. mathematical formulation of crop coefficient considering crop coefficient, we let: kcb tab= is the value for (kcb)mid or (kcb)end ke = kr (kc − kcb) (39) simplifying the equation, we have kc = ke + kcb (40) where, kr = 1 (41) ke = soil evaporation coefficient kcb = basal crop coefficient kc= crop coefficient value of kc following rain or irrigation kr = dimensionless evaporation reduction coefficient dependent on the cumulative depth of water depleted (evaporated) from the top so 2.3. mathematical formulation of irrigated area of land (ai) we consider the area of the irrigated land as a rectangular surface; let l = length of the farm b = breadth of the farm l = length of the spacing on the farmland b = breadth of the spacing on the farmland the spacing area of the farm land is considered to be; lb = ( lb pn ) n (42) where, pn = number of plants on the farmland. n = number of seeds per stand. 357 abdulrahim et al. / j. nig. soc. phys. sci. 3 (2021) 354–359 358 table 1. the comparison of result cropwat 8.0 software and the computational method for rice crops month eto(f.a.o) eto kc (fao) kc etcwn (fao) etcwnmm ai (f.a.o) ai hectares november 4.44 4.92 4.27 3.93 1689 9571 nill 569 december 4.45 4.32 3.3 3.12 1515 6721 nill 569 january 4.42 4.25 4.4 3.67 1521 1826 nill 569 february 4.73 4.21 3.24 3.84 1431 1646 nill 569 march 5.17 4.08 5.2 1.29 7560 627 nill 569 table 2. the comparison of result cropwat 8.0 software and the computational method for soya bean month et0f.a,o et0 kc f.a.o kc= kcb + ke etcwnf.a.o etcwn ai f.a.o ai november 4.44 4.33 0.8 0.23 19.6 12.7 nill 569 december 4.45 4.32 2.47 0.3 114.4 128 nill 569 january 4.42 4.25 3.18 0.82 146.5 347 nill 569 february 4.73 4.08 1.14 1.64 36.7 666 nill 569 also, loss of plants on two adjacent rows is; s = ( l l + b b + i ) n (43) the accurate plant population formula becomes: pn = ( lb lb ) n + s (44) substitute (43) into (44) which further simplified to; pn= ( lb + lb + lb + lbi lb ) n (45) furthermore, lb = lbpn − (lb + lb + lbi) (46) replacing lbwith ai; then ai = lbpn − (lb + lb + lbi) (47) where; ai = irrigated area of land. etcwn = et0 × kc × ai (48) where, etcwn =crop water need et0 =reference crop evapotranspiration kc =crop water coefficient ai =crop irrigated area combining equation (39), (41) and (48), the crop water need equation becomes; etcwn = ( 0.408∆(rn−g)+γ 900 t a+273 u2 (es−ea ) ∆+γ(1+0.34u2 ) ) × (kcb + ke) × (lbpn − (lb + lb + lbi)) (49) 3. result and discussion table 4.1 shows the comparison of the result analysis of the outcome of cropwat 8.0 software developed by f.a.o and the computational method for rice irrigation crop water needs, reference evapotranspiration (eto) and crop coefficient (kc) in bida basin irrigation sites. it is observed from the table that, reference evapotranspiration (eto) results in computational method is decreasing as the dry season is biting harder, this would enable the irrigator farmers to know the quantity of water need for crops provided the size of the land is known. while it alternates in cropwat 8.0 software results developed by f.a.o. the crop coefficient (kc) results in our computational method maintains the same behaviour except in march where we have 1.29 (kc) this is due to the harvesting period which is when crops need little or no water, comparing to cropwat 8.0 software developed by f.a.o which is 5.2 (kc) in march. the crop water needs result in computing method is higher than the cropwat 8.0 software developed by f.a.o, this is own to the fact that the quantity of crop water need in each month of the dry season is known to the irrigator farmers within the available land size of 569 hectares table 4.2 shows the comparison of the result analysis of the outcome of cropwat 8.0 software developed by f.a.o and the computational method for soya bean irrigation crop water needs, reference evapotranspiration (eto) and crop coefficient (kc) in bida basin irrigation sites. it is observed from the table that the crop (soya bean) is grown from november to february which is four months, the crop water need (etcwn) in our computational method is greater than crop water need (etcwn) in the cropwat 8.0 software, this is own to the fact that the size of the land for irrigation is considered in the former method. the reference evapotranspiration (eto) results in our computational method are closer to that of cropwat 8.0 software results developed by f.a.o. the crop coefficient (kc) results in our computational method maintain the same behaviour which shows that the crops need little or no water during their harvesting period. the crop water needs result in our computational method is higher than the cropwat 8.0 software developed by f.a.o. the challenges of crop water need (etcwn) faced by the irrigator farmers can be unraveled if the size of the land for irrigation and plant population is known. 358 abdulrahim et al. / j. nig. soc. phys. sci. 3 (2021) 354–359 359 4. conclusion the penman monteith mathematical model for the crop water need and mathematical model for the size of land for irrigation were solved and the results obtained are compared to cropwat 8.0 software results, the climatic data of the study area (bida basin) under which our research is based which include: rainfall, humidity, minimum and maximum temperature, evapotranspiration were collected from nimet and used for the both cropwat 8.0 software and the computational method so that comparative analysis can be made from the two methods. our computational method results show that crop water need for crops can better be estimated if the size of the land for irrigation is considered. the irrigator farmers can inextricably know of the estimated crop water need for a given size of the land before commencing irrigation exercise. we therefore, recommended that integral calculus can be used to estimate the irregular shape of the size of the land if the land shape is not in rectangular form before solutions are given for accuracy and effective results. references [1] m. n. nkodo, f. c. van zyi, h. keuris, & b. schreiner, “proposed national water resources strategy 2 (nwrs) summary cape town”, department of water affair, south africa, 2012. [2] food and agriculture of the united nation state of the rome world’s, https//www.faoorg/011/1035e00htm, 2010. [3] a.y. solomon, m. girma, & m. mengistu, “determination of crop water requirements for maize in abshege wreda, gurage zone, ethiopia”, journal of earth science and climatic change 9 (2018) 2157. [4] w.z. wenken, w. dai, & y. zhao, a quantitative analysis of hydraulic interaction process in stream aquifer system”. journal of science education and agronomy journal 3 (2016) 426. [5] p. rijwana, “a study on the crop water requirement for agriculture in typical river basin of india”, international journal of water research; 2 (2014) 67. [6] i. n. abdullahi, “estimating aquifer hydraulic properties in bida basin, central nigeria using empirical methods”, journal of earth science research 2 (2013) 1. [7] d.m. onyancha, d.m. kgachene, c.k. kironchi, “f.a.o – crop model – based estimation of the crop water need of major crops in mwala machas kos county”, journal of ecology 2 (2018). 80. [8] h. mehanuddin, g.r. nikitha, k. s. praspithishree, l.b. praveen & h.g. manasa, “study on the crop water requirement of selected crops and irrigation scheduling using crop wat 8.0”, journal of science, engineering and technology 7 (2018) 2347. [9] r. doria, c.a. madramostoo, & b.b. medi, “estimation of future crop water requirement for 2020 and 2050 using crop wat 8.0”, ieee eic climate change conference, 2016. 359 j. nig. soc. phys. sci. 5 (2023) 1143 journal of the nigerian society of physical sciences modelling and simulation for the use of natural waste to purified contaminated heavy metals suha ibrahim salih al-alia,, zaidun naji abudib, mohammed nsaif abbasb adepartment of mathematics, college of computer and mathematical science, tikrit university, tikrit, iraq benvironmental engineering department, college of engineering, mustansiriyah university, baghdad, iraq abstract the possibility of recovering one of the famous heavy metal ions, divalent copper, from contaminated aqueous solutions (which simulates wastewater) was studied in this study. the removal method was adsorption technique using a laboratory batch-mode unit, while the used tea leaves were the adsorption media. the adsorption process was performed under various operating conditions and ranges that simulate the natural environmental conditions to determine the ideal values that achieve the maximum removal of copper ions. the acquired results demonstrated that the maximum remediation efficiency was 85%, which was achieved at treatment time, shaking speed, initial concentration, temperature, acid function, and adsorption dose of 90 min, 250 rpm, 70 ppm, 25oc, 4, 4.5g, respectively. the values of the thermodynamic properties demonstrated that adsorption is spontaneous, exothermic and has negative entropy, while adsorption follows langmuir’s model and the second pseudo-model according to the isotherm and kinetic studies, respectively. to conduct the zero residues level concept, the loaded used tea leaves were prepared to study it effect as a simple type of rodenticide by applying it to sprague dawley rats. the results of the test show that the effectiveness of utilizing the residues as rodenticide and the half lethal dose ld50 of the proposed rodenticide were identical to those mentioned in the literature. based on these results, the current study sheds light on the possibility of converting used tea leaves from harmful solid waste to an environmentally friendly substance using it as an effective adsorbent medium for the treatment of water polluted with heavy metals. doi:10.46481/jnsps.2023.1143 keywords: adsorption; copper; used tea leaves; kinetic; isothermal and thermodynamic article history : received: 10 october 2022 received in revised form: 04 december 2022 accepted for publication: 06 december 2022 published: 23 december 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: b. j. falaye 1. introduction the great growth of the world’s population, industrial and agricultural progress in the past century led to the pollution of all elements of the environment (water, air, soil and space) and the depletion of natural wealth sources [1]. in addition, the failure to follow appropriate methods in treating the sources of pollution, and the lack of proper planning have. the problems email address: suhaibrahim3@tu.edu.iq (suha ibrahim salih al-ali ) of pollutant accumulation and depletion of natural resources can be considered among the most important major environmental issues at the present time for developed and developing countries alike [2]. pollution refers the interference of human activities with the resources and energies of the environment, such that this interference endangers human health or natural resources, or puts them in a situation where they are likely to be directly or indirectly exposed to danger [3]. pollution is defined as the presence of foreign substances in any component of the environment, which renders it unsuitable for use 1 al-ali et al. / j. nig. soc. phys. sci. 5 (2023) 1143 2 or limits its use [4]. pollutants are defined as substances, microbes or energies that harm humans or living organisms in general and cause diseases or lead to destruction [5]. in the past, natural ecosystems were able to assimilation of the pollutants, whether in the soil, water, or air, because of the small quantities and low concentrations of pollutants, and the absence of difficult or non-degradable foreign substances [6]. today, natural ecosystems are unable to assimilate these pollutants, and the influence of these contaminants depends on the degree of their concentration in the environment, their chemical, physical and biological characteristics, and the nature of their interaction with each other and with the surrounding environment [4]. water pollution resulting from industrial and human activities is considered to be one of the main issues in the world, as it is the main cause of 80% of diseases worldwide [7]. the reports issued by who show that the safe drinking water does not reach more than 109 inhabitants worldwide, and most of them are from poor and developing third-world countries [8, 9]. it should be noted that thousands of chemicals are released into the environment in general and into water sources without any treatment on a daily basis [10]. these chemicals (even drugs, medicines or pharmaceutical products in high concentrations) will be affected the human body [11]. among these chemicals, heavy metals are among the most dangerous and harmful, even if found in small amounts, because they accumulate inside living tissues [12]. some heavy metals such as iron, copper and zinc are directly related to the vital activities of living organisms, while others such as lead, cadmium and mercury do not play any role in these activities and are therefore classified as toxic elements [13]. however at high concentrations (exceeding the permissible or recommended limit), all heavy metals are toxic to living organisms, regardless of which they are necessary [14]. there are many ways to recover heavy metals from contaminated water, including membrane separation, electrodialysis, ion exchange, precipitation, liquid extraction and adsorption [15]. adsorption method is an effective and excellent technique in removing heavy metals and other pollutants. there are many kinds of pollutants remediate by adsorption such as dyes [16], radioactive materials, inorganic toxins and others. the ability may be attributed to the high surface area provided by various adsorption media such as activated carbon [17], alumina [18] and silica [19]. however, the high cost of these media and the loss of some of their mass in each regeneration step prompted researchers to use alternative materials of appropriate efficiency, available, cheap, and at the same time, without toxicity [20]. among these materials that meet the above conditions are agricultural wastes [21]. agricultural residues are available in large quantities throughout the year as a result of the human consumption of agricultural crops and can be utilize by different methods, such as producing beneficial chemicals [22], fertilizers [23], and [24], and addition of material to concrete [25] and synthesis of nano-materials [16] because of its unique properties, either naturally or gained after treatment [26]. they are low-cost and not invaluable materials with an appropriate surface area and have proven their efficiency in removing many pollutants from water [27], soil [28] and petroleum [29], and non-toxic substances [30]. in addition, its accumulation (from any source) leads to the pollution of various elements in the environment by raising the vital requirement of water or being a medium for the growth of insects harmful to humans [31]. as a result of the foregoing, researchers studied the ability of this waste to treat polluted water including rice husks [32], banana peels [33], and orange peels [34], pomegranate peels [1], watermelon peels [35], lemon peels [36], egg shells [37], algae [38], water hyacinth [39] among others. tea leaves are one of the most common types of agricultural waste. tea is consumed worldwide in large quantities daily. as a result of this consumption, very large quantities of tea leaves are accumulated, which must be disposed of in economical, safe and environmentally friendly ways [40]. this paper studies the adsorption possibility of copper ions (cu+2) from polluted water by using tea leaves as an available, cheap and non-toxic adsorbent media in a laboratory batch-mode unit. the adsorption procedure was performed using various design parameters to determine the optimum values that gave the highest adsorption efficiency. in addition, we studied a suitable method to dispose the toxic residues using a beneficial route. thus, this study presents the concept of safe, efficient and economical simultaneous disposal of more than one type of environmental pollutants at the same time. 2. material and methods 2.1. adsorbent media preparation the raw media of adsorption i.e., the used tea leaves were collected from the domestic use as a batch. each collected batch was washed with two types of water. the first one was performed with excess tap water to remove any dirt or dust that may be stuck. while the second one was conducted by double distilled water to neutralize the adsorption media. the washed leaves were dried for 48 h in a drying oven at 50oc to avert charred. they were then ground in a domestic mill and sieved to obtain fine powder. powder that passed through a 35 mesh sieve was selected as the adsorption medium. the prepared powder was stored in opaque glass containers in a cool dry place until use. the used tea leaves were collected, washed and dried in continuous batches. after ending all things, the media was first kept in opaque glass containers (i.e., amber glass jar with tight lid). these jars are put in a dark drawer at room temperature to prevent light and humidity. 2.2. stock solution to avoid competition between copper and other elements may be found in wastewater, adsorption experiments of copper ions were conducted using laboratory prepared solutions with predetermined concentrations. stock solution of divalent copper ions (1000 ppm) was prepared by dissolving 3.81 g of bluecolored copper (ii) nitrate trihydrate salt of cu(no3)2.3h2o chemical formula and 241.6 g/mol molar mass. a 99.5% purity salt was purchased from indiamart.a stock solution was prepared using a magnetic stirrer under laboratory conditions, which are atmospheric pressure and temperature of 25−28±1◦c 2 al-ali et al. / j. nig. soc. phys. sci. 5 (2023) 1143 3 2.3. calibration curve by atomic absorption spectroscopy (aas) device, the concentration of cu+2 ions are detecting. the detection was achieved by preparing calibration curve for copper ions at a wavelength of 372.0 nm. the process was carried out by preparing a series of known concentrations of copper ions and accurately determining the absorbance for each concentration. the concentration was plotted against the absorbance so that the calibration curve was ready in its final form. calibration curve experiments were conducted in triplicate for each concentration and the average between three readings was taken to reduce the error. the calibration curve of cu+2 ions is explained via figure 1. figure 1. calibration curve of cu+2 ions using aas 2.4. adsorption experiments adsorption experiments of divalent copper ions were conducted using used tea leaves in a batch type adsorption unit consisting of a water bath shaker under various operating parameters. the acidity function, initial concentration of cu+2 ions, shaking speed, adsorbent dose, treatment time and temperature represented the operating conditions of the adsorption process and their values ranged between 1 − 8, 1 − 100 ppm, 100 − 350 rpm, 0.5−5.0 g, 10−120 min, 25−50oc, respectively. the ph was adjusted using a ph meter with 1 m solutions of hcl and naoh. glass beakers (100 ml capacity glass) of a specific concentration and ph were placed in a water bath shaker, and the device was turned on after setting the temperature and shaking speed. after the end of the specified time for the experiment, the sample was purified using filter paper and tested using an aas device. the remaining concentration of unadsorbed cu was determined from the calibration curve. the percentage of copper ions adsorption and the adsorption capacity are calculated using equations 1 and 2 respectively %r = 1 − ( c f ci ) , (1) q = v m ( ci − c f ) , (2) where %r is the percentage of adsorption, c f and ci are the final and initial concentrations of divalent copper ions (ppm), respectively, v is the volume (l), q is the capacity of adsorption (mg/g), m: the adsorbent dose used (g). 2.5. isothermal study the dispersion of copper ions between the solution and the adsorption media can be explained by using theoretical models after the system reaches the equilibrium state. there are many models that explain this relationship, the most important of which are the langmuir and freundlich models. 2.6. langmuir model this model was introduced by the american chemist langmuir at 1916 and it is assumed that the adsorption occurs on one layer only, the energy is distributed in it evenly, and that the number of adsorption sites is specific and identical. when the layer is saturated, the maximum value of adsorption is achieved, in addition to the actuality that the density of the adsorption layer is determined by the same thickness of the particle. equation 3 symbolizes the general formula of the langmuir model, equation 4 symbolizes the linear mathematical formula of this model, and equation 5 describes the formula for the separation factor rl by which the type of adsorption is determined. qe = qmax klce 1 + klce (3) 1 qe = 1 qmax kl 1 ce 1 qmax (4) where: qe: equilibrium adsorption capacity (mg/g) ; kl: binding sites constant of langmuir equation, (l/mg); qmax: maximum adsorption capacity (mg/g); and ce: equilibrium adsorbed concentration (mg/l) rl = 1 1 + klce (5) the value of rl gives a good indication of the nature and form of adsorption, if the value of the separation factor is higher than 1, the adsorption is not preferred, and linear adsorption if its value is 1. the adsorption is preferred if 0 < rl < 1 value, and adsorption is irreversible if rlvalue is 0. by plotting the graph between 1/qe and 1/ce, a linear relationship is resulted through which the langmuir constants kl and qmax are calculated. 2.7. freundlich model a model presented by the german scientist freundlich in 1909 assumes that adsorption gets multiple and heterogeneous layers and adsorption sites can be unsaturated as well as irregular energy levels. equations 6 and 7 describe the general and linear form of the freundlich model, respectively qe = kf ce 1 n (6) lnqe = lnkf 1 n lnce (7) where: kf : constant of freundlich equation [(mg/g).(l/mg)1/n], n: adsorption intensity (−). 3 al-ali et al. / j. nig. soc. phys. sci. 5 (2023) 1143 4 2.8. kinetic study 2.8.1. pseudo first order model (pfo) the assumption of this model represented by the adsorption time rate is of the first order as a result of obtaining adsorption on only one layer. equations 8 and 9 describe the general and linear form of this model, respectively dqt dt = k1(qe − qt) (8) where: k1: first order rate constant (s−1); qe: equilibrium adsorption capacity (mg/g); and qt amount adsorbed onto adsorbent at any time t (mg/g) ln(qe − qt) = lnqe − k1t (9) the relationship shown in equation 10 can be represented mathematically as a straight line by plotting ln(qe − qt) against t its slope equilibrium constant k1. 2.8.2. pseudo second order model (pso) the most important hypothesis of this model is that the adsorption rate time is directly proportional to the number of active sites on the adsorption surface, and it is the model closest to describing adsorption kinetics when physical or chemical interactions occur between the adsorbent and the adsorbent. equation 10 shows the general form and equation 11 describes the linear form of this model. dqt dt = k2(qe − qt) 2 (10) where: qe: is the equilibrium adsorption capacity (mg/g), qt: is the adsorption capacity at any time (mg/g), t : is the time(s) and (k2): is the second order rate constants (g.mg−1.s−1). 1 qt = 1 k2qe2 + 1 qe t (11) the pso model constants can be found by plotting the relationship between t/qt with t, which is the equation of a straight line with intercept and slope 1/(k2qe2) and 1/qe respectively. 3. results and discussions a batch-mode unit was used to study the influence of some design factors, such as ph, shaking speed, temperature, treatment time of the adsorption unit, initial concentration of divalent copper ions, and adsorbent dose (used tea leaves < 700 µm) on the copper ion removal efficiency. the copper ions concentration in the treated simulated aqueous solution after treatment was detected using an aas device and calibration curve, and then the percentage and the capacity of the adsorption were calculated. 3.1. influence of treatment time treatment time is significant design factors that have been studied in experiments to remediate heavy metals from polluted media, owing to the importance of this factor in the unit accessing the state of equilibrium, in which the time affects the adsorption process, and any subsequent period does not affect the adsorption of pollutants. the effect of treatment time was studied in the current study within a range of 10 − 120 minutes, based on similar literature [18].the results of studying of this operating factor indicated that the maximum recovering efficiency of copper ions was achieved at 90 minutes, and this is obviously from figure 2. at the beginning of the adsorption, the ability of used tea leaves (the adsorption media) to adsorb pollutant ions is high because they are virgin and do not contain any kind of competition for the active sites; therefore, adsorption is the maximum possible. over time, the heavy metal ions begin to accumulate on the adsorbent surface and are linked to functional groups with diverse bonds, which restricts the arrival of new ions to the active sites; thus the adsorption decreases gradually until the system reaches the equilibrium state. after equilibrium is achieved, any increase in the treatment time on the adsorption efficiency is not beneficial, and the adsorption capacity is fixed while other variables remain constant at the optimum values. these results have been indicated by [40] in their conclusions. figure 2. influence of treatment time on removal of cu+2 ions 3.2. influence of initial concentration the effect of varying the concentration of cu+2 ions on the removal capacity is shown in figure 3. it is reported -from the figurethat the highest adsorption efficiency was at the lowest initial concentration. at the same time, the adsorption concentration increased until its maximum value was reached and then remained constant. studying the influence of the initial concentration of cu+2 ion is significant for its effect on the ability of the material to be adsorbed; the higher the concentration of the initial pollutant, the lower the adsorption ability according to the constant value of the surface area of the adsorption media, and consequently, the limited effective sites of the adsorption media. the initial concentration increase indicates an increase in the mass of a heavy element in the same volume unit, and this means an increase in competition between a large number of molecules for the same number of active sites that are filled 4 al-ali et al. / j. nig. soc. phys. sci. 5 (2023) 1143 5 with a specified number of molecules. the higher the initial concentration, the greater the number of unadsorbed molecules in the solution that remain free, and thus, the adsorption percentage removal decreases. the results of the current study were confirmed by similar results similar on investigate the adsorption of cadmium by watermelon rinds conducted by [32]. figure 3. initial concentration influence on cu+2 removal 3.3. influence of ph ph is one of the significant factors affecting the adsorption process of any kind of material in general, and heavy metals in particular, as the positive hydrogen ions (h+) are considered a competition for the heavy metal ions on the adsorption site owing to the similarity of the charge. on the other hand, hydroxide ions (oh−) with a negative charge can precipitate positive heavy metal ions in the form of sparingly soluble hydroxide salts if their concentration is high in the solution. the influence of acidity on the efficiency of cu+2 ions adsorption was studied within a range of 1 to 8, while other factors remained constant at their optimum values. the relationship between adsorption efficiency and ph is illustrated in figure 4. it is noted from the figure that the highest adsorption efficiency for copper ions was achieved at a ph value of 4, which was 85%. at low ph values, the adsorption capacity decreases due to the rivalry of h+ with cu+2 ions on the adsorption sites, which decreases with increasing ph value. as the value of the acid function increases, the hydroxide ions increase and the hydrogen ions decrease, which reduces competition for active sites. this situation continues until the maximum value of efficiency is reached. subsequently, the removal efficiency will increase, but due to precipitation in the form of metal hydroxide and not through adsorption on the active sites. this was observed from the precipitation of a bluish-green solid at the bottom of the flask after the acidity function exceeded the value of 4. although the removal percentage reached its ultimate value, 4 was considered to be the optimal value of the acidity function for the adsorption of copper ions. both [21, 38] indicated results similar to the acidity function set results obtained in this study. 3.4. influence of shaking speed the effect of changing the adsorption removal due to varying the shaking speed of the polluted solution by keeping the figure 4. effect of ph on cu+2 removal remaining operational factors at the optimum values is shown in figure 5. it is noted from the above figure that increasing the shaking speed leads to an increase in the ability of the used tea leaves to steadily remove heavy metal steadily between 100 − 250 rpm. increasing the shaking speed increases the chance of ions arriving at active sites by increasing the mass transfer rate or destroying the film layer growing on the surface of the used tea leaves, which reduces the resistance that occurs to the transfer of copper ions. after reaching the maximum value at 250 rpm, the adsorption efficiency did not change. this is because the adsorbent reaches the equilibrium state or the rate of collisions between the used tea leaves increases. thus, the cu+2 ions are adsorbed and then return to the solution after their liberation from the surface of the adsorbent owing to collisions by a process called desorption. in general, the results showed that the best adsorption efficiency for copper ions was recorded at a shaking speed of 250 rpm. these results that we got are compatible with the conclusion of ghulam et. al. [17]. 3.5. influence of adsorbent dose within a range of 0.5 − 5.0 g, the ability of used tea leaves as an adsorbent media to remove copper ions was investigated, and the effect recorded from practical experiments to study this operating factor was represented in figure 6. the obtained results indicate that there is a clear direct relationship between the amount of adsorbent material and the adsorption capacity between 0.5 − 4.5 g. the reason for this result is that an increase in the amount of adsorbent means the increase in the surface area and, in turn, provides more active sites. thus, the chance of adsorption of a larger number of heavy metal molecules will increase and thus reduce their concentration in the solution, thus increasing the adsorption efficiency. the removal continued to increase from 0.5 g of used tea leaves to 4.5 g, and then the removal efficiency remained unchanged. this stability can be explained by the fact that an rising of the amount of material leads to the formation of layers of the material on top of each other, even if there is a sufficient surface area and additional effective sites will remain vacant because there is an obstacle that prevents the arrival of heavy metal ions to the adsorption sites. therefore, the optimal adsorbent dose for the adsorption of copper is 4.5 g, the rest of the variables remaining constant at the 5 al-ali et al. / j. nig. soc. phys. sci. 5 (2023) 1143 6 optimum values. comparable conclusions have been reported by [35]. figure 5. effect of shaking speed on cu+2 removal figure 6. influence of adsorbent dose on cu+2 removal 3.6. influence of temperature it is noted from figure 7 that the temperature factor owning an adverse influence on the adsorption removal of cu+2 ions using the used tea leaves, and that the highest efficiency was achieved at 25oc, and then the efficiency begins to decrease with increasing temperature until it reaches its lowest value at 50oc. temperature plays a vital role in the results of cu+2 ions adsorption, as its increase leads to raise in the kinetic energy of the cu+2 ions on the surface of used tea leaves, making it able to disengage with it and return to the solution again. meanwhile, when the temperature arises, the material breaks down into smaller particles after adsorption, which makes the copper ions free to move in the solution again. this is what was observed when adsorption was tested at 45 and 50oc. same results are recorded by [15]. the results of isotherm study demonstrated that the adsorption of copper ions utilizing the used tea leaves is subject to the langmuir model clearly with higher correlation coefficient (0.999), while the results are identical with the freundlich model with a correlation coefficient of 0.97. this congruence with the langmuir model can be attributed to the small size of the small adsorption particles (< 700 µm) which provide approximately one layer for adsorption. due to this tiny size, the energy distribution can be homogeneous and the figure 7. influence of temperature on cu+2 removal thickness of the layer is the same as the thickness of the particle. it is also shown that the adsorption is preferred type due to the rlvalue. figures 8 and 9 and table 1 show the values of the model constants used in describing the isothermal study. figure 8. langmuir figure 9. freundlich table 1. the values of the model constants used in describing the isothermal models used in this study langmuir freundlich kl qmax rl n kf 0.1295 1.1537 0.1 4.0733 0.388 6 al-ali et al. / j. nig. soc. phys. sci. 5 (2023) 1143 7 figure 10. pfo model figure 11. pso model figures 10 and 11 explain the obtained results of the kinetic study. it is noted that the adsorption is subject to the pso model with a high r2 that is close to 1, while the other model is far from representing the adsorption results due to the low correlation coefficient. this result can be clarified according the number of active sites that the pso model takes into account, as the optimum dose of material determines the possibility of reaching the adsorption to the maximum efficiency. it is assumed that adsorption obtains a single homogeneous layer and this is the same hypothesis of the pso model. the constants values of the kinetic study models used in this study show in table 2. table 2. the values of the model constants used in describing the isothermal models used in this study pfo model pso model k1 qe k2 qe 0.0496 1.2182 0.01712 1.068 3.7. thermodynamic study in any adsorption process, this study is very important as it detects the thermodynamic functions of the adsorption system through which the spontaneity is combined through the values of the free energy of pressure (m g) in addition to the values of enthalpy (m h) and entropy (m s ) changes that show the state of heat emission and the randomness of the adsorption process. these thermodynamic functions can be calculated by finding the equilibrium constant values via equation 11, and then applying the van’t hoff equation described by kad = qe ce (12) by plotting the relationship between kad and 1/t , it is possible to calculate m h from the slope and m s from the intercept, from which the compressive energy m g of the adsorption system can be determined through equations 13 and 14 below lnkad = − m h r 1 t + 4s r (13) m g =m h − t m s (14) where: kad : thermodynamic equilibrium coefficient of adsorption (dimensionless); 4h: enthalpy variation (j/mol) ; r: universal gas constant (8.3144j/mol.k) and m s : entropy variation (j/mol.k); t : absolute temperature (k). figure 12 and table 3 show the numbers of thermodynamic functions for adsorption of copper ions on the surface of the adsorbed tea leaves that were studied in this study. it is cleared from the figure 12 that the values of the equilibrium constant decrease with increasing temperature and the reason for this is that the adsorption efficiency decreases with increasing temperature (as shown previously). as a result, the concentration of copper ions will increase in the solution as a result of the ions returning to it again, in addition to adsorption capacity decreasing in the substance it is also noted from the enthalpy values figure 12. temperature effect on thermodynamic equilibrium coefficient that adsorption is of the chemical type due to the number of enthalpy, which means that the link between the adsorbent material and the surface of the used tea leaves is the result of chemical adsorption that lead to the generation of a new type of bonds that require high energy to break. this also corresponds to the behavior of temperature (referred to above in paragraph ( influence of temperature). as for the entropy function, it is noted that it is a negative value, which means that the copper ions are more uniform in the solution compared to their presence on the adsorbent media surface. this result is also logical due to the 7 al-ali et al. / j. nig. soc. phys. sci. 5 (2023) 1143 8 table 3. the values of thermodynamic functions of this study temperature, entropy varying enthalpy varying gibbs free energy (oc) m s (j/mol.k) m h (j/mol) m g (kj/mol) 25 -19.11375145 30 -17.29230212 35 -15.47085279 40 -364.2898652 -127726.775 -13.64940347 45 -11.82795414 50 -10.00650482 actuality that rising the temperature leads to lower the adsorption efficiency, which means that the greater number of ions will be in the solution the higher the temperature. the obtained results indicate that the adsorption is spontaneous under the studied operational conditions, due to the negative sign of the values of the free energy of compression, which decreases as the temperature of the adsorption system increases. this proves that the adsorption process is self-occurring and does not need any external energy to complete and that low temperature is better to achieve maximum efficiency. 3.8. zero residues level (zrl) concept application after the process of treating the contaminated aqueous solutions at different design conditions, the used tea leaves loaded with the adsorbed divalent copper ions were collected and classified according to the amount of the adsorbed heavy metal and an attempt was made to dispose of them in a safe, useful, low-cost and eco-friendly method. the use of contaminated leaves residue as a rodenticide was tested by feeding it to white sprague dawley rats of the scientific name rattus norvegicus domestica. use the same procedure used by [35] in preparing and preamble the animal cages, as well as the same feeding method. after the end of the laboratory experiments, the half lethal dose (ld50) was calculated and compared with the dose mentioned in the literature and it was within the known range. table 4 shows the half lethal dose (ld50) for rats feeding normal and contaminated used tea leaves. although copper is one of the moderate toxic heavy metals compared to other elements such as lead, chromium, mercury, antimony and others, however, treating laboratory rats with used tea leaves loaded with this metal showed that it has effects that range in severity with increasing dose, leading to the death of rats [24]. the descent of table 4. ld50 dose of proposed rodenticide ld50 (mg/kg) group type of feeding male female standard control provender of nonogroup rats (ordinary) mortality mortality g1 provender of nonorats +used mortality mortality tea leaves g2 provender of 41.7301 38.0689 37-42 rats + cu+2(atsdr, used tea leaves [41] albumin and protein materials with urine, diarrhea, and gradual turbulent skin infections were the corporeal changes that occurred in rats during the period of treatment with pomegranate peels loaded with bismuth metal [14]. as for the anatomical study of laboratory rat’s samples that died due to feeding the used tea leaves loaded with copper, it was found that they suffered from kidney atrophy and that the degree of this atrophy was increasing with increasing the treated dose [11]. 4. conclusion in this research, the efficiency of natural adsorbent ı.e. used tea leaves was tested to remove divalent copper ions from contaminated simulated solutions in a unit of batch-mode and at various design parameters. this available and low-cost substance has proven highly efficient in treating polluted water with different concentrations ranging from 1 − 100 parts per million, with an adsorption rate of 85%. the study showed that the adsorption efficiency rising with rising ph, adsorbent quantity, shaking speed and treatment time, while it decreases with rising initial concentration and temperature. the obtained results showed that the optimal conditions to achieve the highest removal percentage were ph of 4, adsorbent dose of 4.5g, treatment time of 90 minutes, shaking speed of 350r pm, initial concentration of 70 ppm and temperature of 25oc. the isothermal study showed that the adsorption follows the langmuir model with a correlation coefficient very close to 1, while the adsorption was spontaneous and of the 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[41] agency for toxic substances and disease registry (atsdr), (2022), “toxicological profile for copper”, draft for public comment released, april. 9 j. nig. soc. phys. sci. 3 (2021) 455–458 journal of the nigerian society of physical sciences deposition time induced structural and optical properties of lead tin sulphide thin films j. damisaa,∗, j.o. emeghaa, i.l. ikhioyab adepartment of physics, university of benin, benin city, edo state, nigeria bdepartment of physics and astronomy, university of nigeria, nsukka, enugu state, nigeria abstract lead tin sulphide (pb-sn-s) thin films (tfs) were deposited on fluorine-doped tin oxide (fto) substrates via the electrochemical deposition process using lead (ii) nitrate [pb(no3)2], tin (ii) chloride dehydrate [sncl2.2h2o] and thiacetamide [c2h5ns] precursors as sources of lead (pb), tin (sn) and sulphur (s). the solution of all the compounds was harmonized with a stirrer (magnetic) at 300k. in this study, we reported on the improvements in the properties (structural and optical) of pb-sn-s tfs by varying the deposition time. we observed from x-ray diffractometer (xrd) that the prepared material is polycrystalline in nature. uv-vis measurements were done for the optical characterizations and the band gap values were seen to be increasing from 1.52 to 1.54 ev with deposition time. in addition to this, the absorption coefficient and refractive index were also estimated and discussed. doi:10.46481/jnsps.2021.157 keywords: thin films, xrd, band gap, pb-sn-s, refractive index, absorption coefficient article history : received: 19 january 2021 received in revised form: 17 september 2021 accepted for publication: 18 september 2021 published: 29 november 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: b. j. falaye 1. introduction tin sulphide (sns) is a potential material for the production of thin solar cells because of its high absorption coefficient ( α ≈ 104 cm−1 near the fundamental edge), suitable band gap ( eg = 1.1–2.1 ev) and high hole mobility of 90 cm3 v−1 s −1 [1-3]. sns semiconductors could present p or n-type conductivity owing to the preparation conditions and doping materials, which allow the films to be used as an absorption layers in hetero-junction solar cells fabrication [1, 4, 5]. theoretically, sns thin films solar cells can be optimized such that a conversion efficiency of above 25 % can be reached [6]. sns thin films ∗corresponding author tel. no: +234 (0) 7060515540 email address: john.damisa@uniben.edu (i.l. ikhioya) has been prepared by several methods like thermal evaporation, spray pyrolysis, electron beam evaporation, silar, hot injection, aqueous solution, colloidal route, single solid approach, precipitation and electrochemical deposition (ecd) [5-7]. ecd technique was used to fabricate pb-sn-s thin films in this study. the method presents a simple route of depositing tfs due to its low cost of experimental system, uniformity of films thickness as well as its large area deposition at low temperature [8-10]. recently, investigation into new photovoltaic materials with improved efficiency have assumed a considerable interest and researchers are investigating for understanding and engineering the properties of sns tfs for photovoltaic application [5]. doping with elements like lead has shown to enhance the structural as well as the optical properties of sns tfs. the study presents the preparation of lead tin sulphide (pb-sn-s) thin films using 455 damisa et al. / j. nig. soc. phys. sci. 3 (2021) 455–458 456 figure 1. schematic diagram of pb-sn-s deposition process the electrochemical deposition technique. our aim is to develop a better growth approach of new and non-toxic material for the production of low cost solar cells. improvement in the properties of sns tfs due to lead incorporation in sns-system will enhance the device efficiency of the material. particularly, this communication is concerned with the influenced of deposition duration on the properties (structural and optical) of pb-sn-s tfs via the electrochemical deposition technique which hitherto has not been studied using this route. 2. materials and method analytically graded chemical (sigma-aldrich) was used to deposit the pb-sn-s tfs. these chemicals include; lead (ii) nitrate [pb(no3)2], tin (ii) chloride dehydrate [sncl2.2h2o] and thiacetamide [c2h5ns]. the electrochemical deposition bath system consist of a cationic precursor of 0.05 mol of pb (no3)2, 0.01 mol of sncl2.2h2o and an anionic (precursor) of 0.15 mol of c2h5ns mixed in distilled water. a stirrer (magnetic) was used to harmonize the reaction bath. fluorine doped tin oxide (fto) and carbons were employed as the cathode and anode electrodes respectively. the deposition using the electrochemical deposition method was achieved according to the scheme in figure 1. the deposition were repeated for 20, 25, 30, 35 and 40 seconds and the samples were later coded as j0, j1, j2, j3, and j4 respectively. 2.1. characterization of thin films the pb-sn-s films used for this study have good adherent with the fto substrates. the xrd patterns of pb-sn-s tfs were observed using advanced x-ray diffractometer (bruker d8) operating with a wavelength of 1.5406ȧ and, at a scanning range of 15 to 80o. the uv-visible optical measurements of pb-sn-s thin films were done using a uv-1800 spectrophotometer in the range of 300 to 1000 nm at room temperature. the optical band gaps (eg), absorption coefficient (α) as well as refractive index (n) were estimated from the optical data. 3. results and discussion 3.1. x-ray diffraction (xrd) studies the xrd patterns of pb-sn-s tfs are shown in figure 2. although the deposition time is increased, the xrd spectra showed four similar main peaks at 21.70◦, 23.50◦, 24.94◦ and 33.62◦, which correspond to the diffraction peaks of (200), (201), (211) and (221). the presence of higher intensity peaks in the pb-sn-s films with narrower spectral widths indicated that the films are polycrystalline in nature [11, 12]. from the xrd patterns also, it is obvious that the pb-sn-s structures consist of mixtures of several phases including the orthorhombic sn2s3 (jcpds no 014-0619), hexagonal sns2 (jcpds no 0230677) and cubic structure pbs thin films (jcpds 01-0880). as known, the presence of secondary phases within the pb-sn-s system may have deteriorated the structural crystallization as well as the peaks patterns. figure 2. xrd patterns of the prepared pbsns thin film 3.2. optical studies to ascertain the potentials of the electrochemical prepared pb-sn-s tfs for device fabrications, the absorbance was investigated in the spectra wavelength of 300 to 1000 nm. the absorbance a, was measured using the relation in equation(1) [13]; a = log ( 1 t ) (1) figure 3 shows the relationship between the optical absorbance and wavelength of pb-sn-s fts. in the plot, it could be observed that the absorption was decreasing along the wavelength regions. as well from the figure, the absorbance decreases with deposition time. such decrease in absorbance may be due to structural defects such as surface irregularity and defect density in the pb-sn-s system as a result of increase in deposition time [14], as indicated from the xrd measurement. the low absorbing nature of the films consequently indicates an improvement in the transmission [13]. the absorption coefficient (α) of the material was evaluated by means of the equation (2) [15]; α = 1 t ln ( 1 t ) (2) 456 damisa et al. / j. nig. soc. phys. sci. 3 (2021) 455–458 457 table 1. some optical properties of pb-sn-s tfs samples band gap absorption coefficient refractive index (ev) (cm−1) × 104 (n) jo 1.51 6.10 2.944 j1 1.52 15.0 2.939 j2 1.53 17.5 2.934 j3 1.54 13.2 2.929 j4 1.51 6.80 2.944 equation (2) assumed a negligible reflectance while t and t are the respective transmittance and thickness of pb-sn-s films. the observed variation of the absorption coefficient with deposition time is shown in table 1. it was observed that α increased from 6.1 × 104 to 17.5 × 104 cm−1 with deposition time of 30 seconds (sample j2) and was decreased on further increase of deposition time. also from the table, the magnitude of the absorption coefficient was found to be greater than the 104 cm−1 which makes pb-sn-s thin film a better alternative than gaas and cdte as absorber layers in photovoltaic applications [5,16]. the band gap (direct) was determined using the following relafigure 3. absorbance against wavelength of the prepared pbsns thin films tion in equation(3) [17]; (αhv)2 = k (hv − eg) (3) where hv is the photon energy, α is the absorption coefficient, k is a proportionality constant, and eg is the optical band gap. a plot of the square of absorption coefficient against photon energy gives a curve illustrated in figure 4. since the prepared pb-sn-s films is a direct semiconducting material [2], extrapolating the linear portion of figure 4 to the x-axis at x = 0 gives the band gap energy (eg). the band gap obtained was seen to be between 1.52 – 1.54 ev as indicated in table 1. also from the table, the band gap energy values were observed to be increasing with increase in deposition time. generally, band gap energy in semiconducting materials are mostly influenced by their structural defects, crystallinity, impurities, grain sizes as well as grain boundary disorders [18]. consequently, the observed increase in band gap energy in pb-sn-s samples can be explained in terms of the effect of impurities in their lattice system as indicated from the xrd studies. sebastian et al. [2] have reported a band gap range of 1.60 to 1.90 ev for lead doped tin sulphide (sns:pb) tfs grown by varying lead concentration using nebulized spray pyrolysis (nsp) technique. orimi et al. [19] estimated an optical band gap of 1.63 to 1.80 ev for pb1-xsnxs nano-powder using chemical precipitate technique. our obtained values are relatively lower than these values which could be the direct effect of the electrochemical deposition method employed in the preparation of this film. the refractive index (n) is an essential property of optical materials. it is closely related to the electronic polarization of ions as well as the local field within the optical materials [20]. many optoelectronic devices such as switches, modulators, filters, waveguides, solar cells and detectors are based on refractive index [21]. generally, the n of pb-sn-s tfs is related to the optical band gaps. moreover, the refractive index of the prepared pb-sn-s films was estimated using the proposed relation by herve and vandamme, given in equation (3) [21, 22] n = 1 + ( a eg + b )2 1 2 (4) where eg is the band gap, a (13.6 ev), and b (3.4 ev) are constants. the estimated values of the refractive index are indicated in table 1. the results indicated that the n values of pb-sn-s tfs decreased from 2.944 to 2.929 with enhance deposition time. the range of the refractive indices of the material indicate that pb-sn-s film is a ternary material whose properties falls between the binary constituents of pbs (1.8 – 6.0) [23] and sns (3.5 – 5.5) [4], and compares favorable well with values in literature. 4. conclusion thin films of pb-sn-s have been prepared by the electrochemical deposition method and the structural and optical properties were investigated as function of the deposition time. the xrd measurements indicated a polycrystalline film with no much difference in their crystallinity as deposition time increases. the increased deposition time resulted in the variation of the 457 damisa et al. / j. nig. soc. phys. sci. 3 (2021) 455–458 458 figure 4. square of absorption coefficient and photon energy for pb-sn-s tfs absorption coefficient and band gap from 6.1 × 104 to 17.5 × 104 cm−1 and 1.51 to 1.54 ev respectively. the refractive index was observed to decrease with deposition time except for sample j4, and obeyed the harve and vandamme model. the high absorption coefficient obtained for the material suggests that the prepared pb-sn-s would create good absorbing properties in solar cell fabrications. acknowledgments we thank the reviewers as well as the editorial team for their positive contributions and ideas, which have markedly enhanced the value of this paper. references [1] e. guneri, f. gode, c. ulutas, f. kirmizigul, g. altindemir. & c. gumus, “properties of p-type sns thin films prepared by chemical bath deposition”, chalcogenide letter, 7 (2010) 685-694. 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[23] m. m. abbas, a.a. shehab, a. al-samuraee, & n.a. hassan, “effect of deposition time on the optical characteristics of chemically deposited nanostructure pbs thin films”, energy procedia 6 (2011) 241. 458 j. nig. soc. phys. sci. 5 (2023) 1148 journal of the nigerian society of physical sciences an alleviation of cloud congestion analysis of fluid retrial user on matrix analytic method in iot-based application k. nandhini, v. vidhya∗ division of mathematics, vellore institute of technology, chennai, tamilnadu, india abstract cloud computing (cc) and internet of things (iot) are upgrowing human intervention to enhance the daily lifestyle. currently, the heavy loaded traffic congestion is a very big challenge over iot-based applications. for that purpose, the researchers approached various ways to overcome the congestion mechanism in recent years. even though, they have futile to acheive the best resource storage accessing capacity expectation other than, cloud computing. data sharing is a key impediment of cloud computing as well as internet of things. these are the constituent that give rise to the combination of the iot and cloud computing paradigm as iot cloud. though, preserving the missed data during the execution time is a key factor to indulge the retrial queueing theory (rqt), who is facing issue upon accessing cloud service provider (csp) enter into virtual pool to preserve the data for reuse. the paper imposes markov fluid analysis with matrix analytic method (mam) allows the data as continuous length of data rather than individual data to avoid the congestion. the virtual orbit queue follow constant retrial rate discipline, that is, head of the orbital users makes attempt to occupy the server are assumed to be independent and identically distributed (i.i.d). steady-state expression presented to study the behaviour of congestion. an illustrative analysis is produced to gain deep perception into the system model. doi:10.46481/jnsps.2023.1148 keywords: retrial queue, stationary distribution, congestion, markov fluid queue, matrix analytic method, iot cloud computing article history : received: 26 october 2022 received in revised form: 13 january 2023 accepted for publication: 18 january 2023 published: 04 april 2023 c© 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: s. fadugba 1. introduction the rising prevalence of cloud users and their anticipations has resulted in an incredible growth in the state-of-the-art of cloud computing approaches in recent years. a key primary characteristics of cloud computing and the iot environment is better communication. the cloud is a stand-alone platform that enables users to access cloud data and resources via the internet from any location or device. similarly, iot devices may ∗corresponding author tel. no: +91 9791020444 email address: vidhya.v@vit.ac.in (v. vidhya) access data and resources from any location. the necessary components of iot technology include cloud properties like resource pooling, flexibility, and ubiquitous network connectivity. the on-demand and elastic nature of the exertion service enhance service reliability, which is further enhanced by massive resource pooled storage. all of these strong arguments support the necessity to combine iot with cloud computing standards. researchers have worked on numerous projects in this field. the potential and stigma associated with the bifurcation of the iot and cloud were extensively investigated by [15] and [16]. in order to govern and monitor the complex cloud infrastructure, [25] conducted a survey on the cloud integrated iot 1 nandhini & vidhya / j. nig. soc. phys. sci. 5 (2023) 1148 2 dependent supervisory control and data collection security systems. the premium approach was created by [26] and is based on the amount of time the user data should wait whilst being uploaded to the cloud. cloud storage is a critical aspect of cloud-based technologies. as both the user’s data volume and the network bandwidth are increasing at an exponential rate periodically, the device’s capacity cannot keep up with the user’s requirements. people were seeking a brand-new alternative to save their data, and they uncovered cloud storage, which has more powerful floor space. the practice of using cloud storage becomes increasingly sophisticated, and in the foreseeable years ahead, users’ information will be held there instead. cloud storage merely refers to a computing environment where processing and information information is made available. the cloud storage consists of a cluster of distributed file systems, applications, and cooperating network technology. the firms that provide cloud storage services include icloud, dropbox, baidu cloud, amazon web services s3, azure cloud, google drive, and so forth. all such enterprises are successful in persuading an immense number of users by offering sustainable throughputs and many administrations pertaining to well-known applications. the underlying storage in this instance ensures resource pooling for different kinds of data sources. contributions to [17],[19] as well as several sources of literature where numerous sorts of cloud storage and forthcoming concerns have been emphasised and reviewed. traffic congestion is the main concern among these problems. when significant amounts of heterogeneous data are transferred to sensor nodes concurrently in a cloud environment, network bandwidth congestion ensues, which in turn affects cloud application service or a contravene of the service level agreement (sla). few background studies on network congestion were previously known; see[20]. as in meantime, there is indeed a considerable amount of traffic whenever the utilization of cloud data has increased substantially. now, the usage of data of cloud uses have been raised significantly leads to heavy traffic. the user has access to store data directly in the cloud, and the cloud service provider (csp) administers the data there. the partitioning, positioning, and management of information do occurs without the user presiding over the actual storage of their data. due to the heterogeneous dataset with diverse interface and various geographical paths to store the data, the csp independently accesses the data in the cloud, and there is a significant risk that the user will receive inaccurate transmission. as previously stated, data losses will result from the instances taken. information is repeated as a consequence of it though. accessing data from the virtual pool and waiting for csp for web-based services are crucial components of cloud data repetition. due to massive infrastructure and increasing number of cloud primary user and repeated user, collision among the user have been increased. new collision prevention techniques are developing in response to environmental changes. fluid cloud model is the ongoing model used to freely opt the csp providing accessible data into clusters omitting the individual performance and reduce the blocking of network bandwidth congestion, see [21]. therefore, we proposed the model of markov fluid queue (mfq) approach to simplify the matrix analytic method based on retrial queueing theory to validate the result. there is some hope for effective background traffic production in mathematical representations of traffic congestion. the idea behind abstraction is that it aggregates activity requiring less computational work. [22] established the practise of fluid queue analysis in markovian environments. this research focuses on the distinctive elements of network behaviour, like the buffer. use of important sampling is an additional intriguing method for quick stochastic modelling and simulation. e.g. [23]; nevertheless, given that it focuses on its quick estimation of statistics like probability of packet loss, this is unlikely to suit our goals of effectively creating realistic background traffic. the phenomenon of fluid queues has garnered a lot of attention over the last two decades. whilst also modelling ”packet trains” as instead of individual packets, networks can be simulated at a lower computing cost. such fluid queues have been widely regarded as valuable mathematical tools for modelling, for instance, packet speech, video systems with or without background data, computer networks, including call admission control, traffic shaping and modelling of traffic control protocol (tcp), and production and inventory control. refer to [7],[8] . the research presented in this article effectively employs the traffic model used in the previous two articles, where a traffic flow is defined by a piecewise-constant rate function. according to [24], an effort with a specific intent analogous to our own has been made. in order to demonstrate how to make packets and the fluid model work together, a simulation model that is packet-oriented and is based on sde was incorporated. as a packet travels through a region characterised by the fluid approach, the fluid model was utilised to estimate the overall queueing delay. although packet flow data was used to feed the fluid model’s input for the time step, neither the packets themselves nor the fluid in the fluid network directly interacted with the packet streams that passed the fluid networks at the same time. as indicated earlier, fluid models enabled by traditional queuing systems make some progress. there is, however, no article on a fluid queue system that is driven over retrial queue background. the analytical theory of fluid models can be provided by studies on the fluid model over various queueing models. we can offer the fluid model more diversity and flexibility in the design and control of input and output rate by putting forth several retry strategies. for instance, user’s requests are organized into packets and relayed via a network of routers. requests are blocked unless the router is accessible to send packets to the server for decoding the data. in order to minimize the packet latency, the user request flow must therefore be continuous to neglect the individual effect on system performance over the retrial queueing paradigm. the network bandwidth congestion describes with repeated transmission using mam to obtain system stability. the work [1], [2] the author instigated the homogeneity of the fluid queue by explicitly defined the closed form solution. [3], have studied the buffer content distribution by expressing closed form solution of fluid over two distict queues. the study 2 nandhini & vidhya / j. nig. soc. phys. sci. 5 (2023) 1148 3 [4] approaches a catastrophic queueing model in random environment over fluid queue investigates the stability of the system. mao et al. [5] and [6] have studied the fluid vacation model expressing laplacian of the stationary distribution. kapoor, s., and dharmaraja, s. [9] have obtained the transient analysis of fluid queue by exploiting the explicit solution of eigenvalue decomposition and subsequently computing the eigenvector interms of bisection algorithm. n.j starreveld, r. bekker and m. mandjes [10] approaches an fluid queue with finite buffer along with workload process by martingale method. in [11] encouraged markov modulated arrival process by obtaining steady state probabiliy distribution with phase type service and impatient customers. reader can refer from [12],[13],[14] for further reference of fluid queues this paper makes two significant contributions. first, (i) to present a matrix analytic method (mam) based on data repetition phenomena to acquire the data from cloud data warehouse. (ii) to introduce a markov fluid queue (mfq) approach-based retrial queueing mechanism which is used to access the heterogeneous data from various interfaces. the proposed model (iii) developed modified bessel function of first kind which is used to show linearly independent solution leads to gradual stability and (iv) using the fluid queue technique, we show a methodological benefit that might be used to intuitively get the stationary probability of the system states using the matrix-analytic method. finally, by conducting extensive numerical trials, we significantly broaden the stationary performance analyses’ applicability. following of how the remaining of this work is organized: the more pertinent and related research in the field of fluid flow approach and cloud storage has been discussed. system design is discussed with real-world instance in section 1.1 the conceptual mathematical framework is thoroughly explained in section 2 and section 3. section 4 presents the experimental setup and provides justifications with practical augmentation. the conclusion of the suggested work is provided in section 5 with potential improvement. 1.1. system architecture a real-world instance of an online education cloud system is provided for illustration in fig 1. the major components of the system including the user interface module (ui), a cloud database contains virtual machine manager (vmm), a cloud broker, a cloud information service (cis), a cloud server. here cloud user send and receive the data via user interface module. the data storage and retrieval process of the data in the cloud server are the primary responsibility of the user interface module. the cloud database contains the volume of information of cloud users. vmm is responsible for processing the assigned task, and a cloud broker partition the task into cloudlets but in our case, it is evenly spaced packet trunks where each trunk represents multiple packets, that is, ”fluid packet trunks”, a cloud information service (cis) corresponds to obtain the list of resources for the assigned task to provide service and virtual server to fulfill the request of cloud users in connection for online-learning. each virtual machine receives one of the 16 cpu cores that are available. a 1 tb hadoop distributed file system storage will be deployed in the interim to increase the i/o effectiveness of the online learning system. each vm has a preinstalled linux operating system and ”moodle” online learning platform with a root file system size of 50gb. each vm has a 300 gb additional root filesystem mounted in order to retain any lost data and provide an orbital storage pool in case of transmission errors. thus, a maximum of 8 missing information in vms can be transmitted to the server to wait for service. to guarantee the quality of service, unsuccessful task are transmitted to a vm server, which can inspect and make the corrupt root filesystem accessible again. if the server is accessible, a failing virtual machine will proceed right away to access the service in accordance with its system log file. in the event that this does not occur, the failing vm is sent to the server’s storage pool to wait for a filesystem check and reconfiguration. when a failed vm does not access the server, the cloudstack has a 50% (treated as q1) chance to inhibit any failed information due to non-availability of server. the cloudstack may failed to prevent a impatient re-accessed information with probability 50% (treated as p̄1) and cause an error due to excessive congestion of cloud user. in the beginning, all the vm are bootstrapped to speed up the process. let the arrival rate of the failed vm be stepwise function with parameter λ = 1.2/hr. the failed vm can be reaccessed at an instant if the server is accessible. the service time is exponentially distributed with service rate µ = 2.0/hr if the server is in busy period. on the contrary, the failed information in vm is sent to storage pool if the server is busy and repeat its service with exponential amount of retrial time with rate θ = 0.9/hr. 2. a framework of mathematical structure as a way to examine the behaviour of congestion, we take into account a single standby server markovian queueing mechanism with retrial users. figure 2 represents the state transition diagram. the system consist of single server markov fluid queue model with constant retrial rate policy. the customer arrive according to the piecewise-constant process and the inter-arrival time distribution follows an i.i.d exponential distribution with parameter λ. the service time of a user are i.i.d follows an exponential distribution with parameter µ. if a user encounters a free server upon arrival, this user receives service immediately. if a server are busy during an arbitary user arrival epoch, the user join in virtual buffer and wait until a server becomes available with probability q1 and if the server is not accessible, they again enter the so-called orbit after an arbitary amount of time. a user who arrive to the system from orbit is called a retrial user or exit the sytem with complementary probability p̄1, 0 ≤ p̄1 ≤ 1, p̄1 = 1 − p1. 2.1. the process of background queueing model the behaviour of the process under examination can be derived as the conventional irreducible continuous-time markov chain (ctmc) ζt = {(kt, it)}, t ≥ 0, where kt ∈ kt signify the 3 nandhini & vidhya / j. nig. soc. phys. sci. 5 (2023) 1148 4 architecture.png figure 1. overview of system design of cloud data storage transition diagram 2.png figure 2. transition state diagram ζt = {(kt, it )}, t ≥ 0 amount of data packets accumulated in the orbit, kt ≥ 0; it ∈ it designate the state of the underlying process of qbd process, it ∈ {0, 1}, where, it = 0, the state is in operational mode; it = 1, the state is in non-operational mode. theorem 1. the infinitesimal generator q structured as blocktridiagonal matrices of the markov chain ζt, t ≥ 0 q =  a0 c . a b c . a b c . a b c . . . . . . .  4 nandhini & vidhya / j. nig. soc. phys. sci. 5 (2023) 1148 5 where a0 = ( −λ λ µ −(λq1 + µ) ) c = ( 0 0 0 λq1 ) a = ( 0 θ 0 θp̄1 ) b = ( −(λ + θ) λ µ −(λq1 + µ + θp̄1) ) proof. theorem 2.1 is demonstrated by analysing each transition in the markov chain ζt, t ≥ 0 during the interval of infintesimal length, and by reconfiguring the intensities into block matrices. here the intensities of the transition assumed to be continuous entity, termed as fluid. individual users impact will be negligible since the flow will be continuous in the system during an interval of an infintesimal length, the non-diagonal blocks are zero matrix, thus the generator q has the block-tridiagonal structure. the diagonal entries of the 2×2 square block matrix a0, define the intensities of the transition of the chain ζt, t ≥ 0 that does not affect the number of users in the orbit. the intensities of an event are given by −(λq1 + µ) defines the user enter orbit with probability q1 and ends with customers service. the intensities of the former event are defined by the entries of the event that insist arrival of users in the orbit while the latter event defines the service of the user µ. the super-diagonal 2×2 matrix c, defines the intensities of the event are given by λq1, that is, the probability of joining the socalled orbit who finds server busy. the sub-diagonal 2 × 2 matrix a, defines the intensities of the event are given by θ, enter into a retrial orbit with the transition probability (1, 0) → (0, 1) and leave with impatience with θp̄1 the diagonal entries of the matrix b, defines the intensities of the event are given by −(λ+θ) that the arrival of user encounters an busy server, enter into a retrial orbit with probability θ and probability of joining the so-called orbit who finds server busy λq1. the intensity of the former event is arrival of users in the orbit while the latter event describes zero hence the proof 3. markov fluid analysis 1 this section includes the set of governing equation driven by qbd process over fluid retrial queue with non-persistent users. let x be the buffer occupancy at time ′t′. let b(t) denotes the amount of fluid data packets in the buffer at time ′t′. 1the efficacious action of the buffer is defined as η(b(t), k(t), i(t)) = d(b(t)) dt =  r0, (k(t), i(t)) = (k, 0), k ≥ 0 r, (k(t), i(t)) = (k, 1), k ≥ 0 0, b(t) = 0 we know that, r0 ≥ r ≥ 0. the occupancy of the buffer when the server is in non-operational (idle) period is much greater than the buffer occupancy in the operational (busy) period. clearly, buffer occupancy is linear increasing at the rate of r0, as soon as users begin the driving process and the process is still in progress; buffer occupancy is linear increasing at the rate of r according to the driving process stays in operational period. also, the buffer occupancy level cannot depleted until the buffer is empty. as soon as the buffer occupancy level varies, the effective request-service rate of fluid remains unstable. in order to ensure the stability condition, define the average drift condition should satisfy d < 0 d = r0 ∞∑ k=1 fk0 + r ∞∑ k=1 fk1 < 0 the following set of differential equations that satisfy a quasibirth and death process for a stationary joint distribution fki(t) can easily be solved using the conventional approach. define fki(t) is the stationary joint distribution when kt ∈ kt users in the retrial orbit at time ′t′ and it ∈ it represents the affiction of the server r0 df00(t) dt = −λf00(t) + µf01(t) (1) r0 dfk0(t) dt = −λfk0(t) + µfk1(t) − θfk+1,0(t) (2) r df01(t) dt =λf00(t) − (λq1 + µ)f01(t) − θf1,0(t) + θp̄1 f11(t) (3) r df11(t) dt =λq1f01(t) + λf10(t) + θf2,0(t) − (λp1 + µ + θp̄1)f11(t) + θp̄1f21(t) (4) with boundary conditions f00(0) = a1 f01(0) = a2 fkt,it (0) = (kt, it), (kt, it) ∈ ω{(kt, 0) ∪ (kt, 1)} where 0 < a1, a2 < 1 the boundary conditions are feasible to the solution since there is linear growth of customers in the background driving process and it has some non-negative probability distributed around the boundary region. also the buffer gets empty when the buffer occupancy level decreases at the rate of r0. thus the boundary conditions are satisfied. 3.1. markov decision process in this theoretical evaluation, the steady state analysis of the fluid retrial queue is presented with the detailed study of average buffer occupancy expressed in an closed form solution of stationary distribution and buffer occupancy is determined in the form of modified bessel function of first kind. the set of differential equations are governed by quasi birth-death process evaluated interms of infintesimal generator matrix. the 5 nandhini & vidhya / j. nig. soc. phys. sci. 5 (2023) 1148 6 set of matrix quadratic equations defined to provide an explicit solution for joint steady state analytic distribution. now define f0(t) = {f00(t),f01(t)} fk(t) = {fk,0(t),fk,1(t)}, j = 1, 2, ... therefore, f (t) = (f0(t),f1(t),f2(t), ...) the set of differential equation can be rewritten with the joint probability density sequence in the matrix form as df (t) dt λ = f (t)q (5) where λ =  λ0 . . λ1 . λ1 . λ1 . . . .  here λ0 = ( r0 0 0 r0 ) , λ1 = ( r 0 0 r ) to depict the laplace transform of the system of matrix solutions to provide an analytical interpretation to the stationary analysis of the buffer capacity distribution. f̂ (s) = (f̂0(s),f̂1(s),f̂2(s), . . . ) then the above matrix form is given as f̂ (s)(q− sλ) = (−āλ, 0, 0...), ā = (r0a1, r0a2) the set of differential equations is given by f̂0(s)(a − sλ0) −f̂1(s)a = −āλ (6) f̂0(s)c + f̂1(s)(b − sλ1) + f̂2(s)a = 0 (7) f̂k−1(s)c + f̂k(s)(b − sλ1) + f̂k+1(s)a = 0 (8) theorem 2. a necessary sufficient condition for the driving queueing process {kt = kt, it = it; t ≥ 0} is given by ρ = λq1 (λ+θ) θµp where µp = µ(λ + θ) p̄1 proof. from the framework of the generator matrix, we know that {kt = kt, it = it; t ≥ 0} is a quasi birth-death process. let us define l = a + b + c. l = ( 0 0 0 λq1 ) + ( 0 θ 0 θp̄1 ) + ( −(λ + θ) λ 0 λq1 ) = ( −(λ + θ) (λ + θ) µ −µ ) clearly, l is a finite integrable matrix. its steady-state probability vector defined as π = (π0,π1) satisfies (π0,π1)l = 0 where π0 + π1 = 1 the obtained solution is π0 = µ λ+θ and π1 = λ+θ µ a necessary sufficient condition for the driving queueing process {kt = kt, it = it; t ≥ 0} is πce < πae, where e is a two dimensional column vector whose elements are all equal to one. then using simple expression, we obtained the result ρ = λq1 (λ+θ) θµp where µp = µ(λ + θ) p̄1 theorem 3. the evaluation of rate matrix is determined in the form of matrix quadratic equation is given by r2(s)a + r(s)(b− sλ1) + c = 0 yields minimal non-negative solution such as r(s) = ( χ(s) β(s) 0 r(s) ) proof. we make an assumption for the rate matrix r(s) has the structure is of the 2 × 2 matrix form such that r = ( r11 r12 0 r22 ) from the equation above, we obtain,( r211 r12(r11 + r22) 0 r222 ) ( 0 θ 0 θp̄1 ) + ( r11 r12 0 r22 ) ( −(λ + θ + sr) λ µ −(λq1 + µ + θp̄1) + sr ) + ( 0 0 0 λq1 ) = 0 the set of differential equations are determined as follows: r11(λ + θ + sr) + r12µ = 0 (9) r211θ + r12(r11 + r22)θp̄1 + r11(λq1 + µ + θp̄1 + sr) = 0 (10) r22µ = 0 (11) r222θp̄1 − r22(λq1 + µ + θp̄1 + sr) + λq1 = 0 (12) the solution from equation 7 expressed as r22 = r(s) = (λq1 + µ + θp̄1 + sr) ± √ (λq1 + µ + θp̄1 + sr)2 − 4θp̄1λq1 2θp̄1 (13) let the above equation has positive and negative eigenvalues namely,r(s) and r1(s) .since the root that lies inside the unit disc, we consider r(s) for further reference. therefore it satisfies the following recurrsion relation as follows: (λq1 + µ + sr + θp̄1(1 − r)) = µ 1 − r + θp̄1 + sr 1 − r = λq1 r (14) from the equation above, we get, χ(s) = ρ + sr r(s) − λµ θµp (15) where ρ = λq1 (λ+θ+sr) θµp and µp = µ + (λ + θ + sr)p̄1. using the expression given in equation 8, the solution yields, β(s) = (λ + θ + sr)ρ µr(s) −λµ λ + θ + sr µθµp (16) 6 nandhini & vidhya / j. nig. soc. phys. sci. 5 (2023) 1148 7 hence, the rate matrix is given by r(s) =  ρ r(s) − λµ θµp (λ+θ+sr)ρ µr(s) −λµ λ+θ+sr µθµp 0 (λq1 + µ + θp̄1 + sr) ± √ (λq1 +µ+θp̄1 +sr)2−4θp̄1λq1 2θp̄1  = ( χ(s) β(s) 0 r(s) ) ... rn−1 = ( χ(s)n−1 β(s)(χ(s)n−2 + r(s)n−2 + ∑n−3 i=1 χ(s) ir(s)n−2−i) 0 r(s)n−1 ) in general, rn = ( χ(s)n β(s)(χ(s)n−1 + r(s)n−1 + ∑n−2 i=1 χ(s) ir(s)n−1−i) 0 r(s)n ) hence the result is proved theorem 4. the steady state probability distribution of the buffer occupancy in the driving queueing model gives an analytical solution in an laplace domain determined as f̂k0(s) = f̂00(s)χ(s) n f̂k1(s) = f̂00(s)β(s)(χ(s) n−1 + r(s)n−1 n−2∑ i=0 χ(s)ir(s)n−1−i) + f̂01(s)r(s) n proof. without loss of generality, let us assume fk(s) = fk−1(s)r(s) can be rewritten as fk(s) = f0(s)rk(s) from equation 6,7, we have =f̂k−1(s)c + f̂k(s)(b − sr) + f̂k+1(s)a =f̂k−1(s)c + f̂k−1(s)r(s)(b − sr) + f̂k−1r 2(s)a =f̂k−1(s){c + r(s)(b − sr) + r 2(s)a} = 0 =f̂0(s)c + f̂1(s)(b − sr) + f̂2(s)a =f̂0(s)c + f̂0(s)r(s)(b − sr) + f̂0(s)r 2(s)a =f̂0(s){c + r(s)(b − sr) + r 2(s)a} = 0 from equation 6, we get f̂0(s)(a − sr1) −f̂1(s)a = −ā f̂0(s) = ā sr1 − a0 − r(s)a (17) the proof is provided in the appendix with all sufficient conditions. 3.2. stationary analysis of buffer content distribution theorem 5. the stationary distribution of the buffer occupancy level can be expressed analytically to provide explicit solution to the joint steady probability vector[1] f̂ (s) = p(x ≤ x) = ∞∑ k=0 {f̂k0(s) + f̂k1(s)} (18) buffer occupancy.png figure 3. a stationary buffer occupancy distribution f(x) against x proof. taking laplace transform on the above equation, we get f̂ (s) = f̂0(s)e1 + f̂0(s) ∞∑ k=1 rn(s)ek (19) = f̂0(s)e1 + f̂0(s)e(i − r(s)) −1e2 where (i − r(s))−1 = 1 (i −χ(s))(i − r(s)) ( ir (s) β(s) 0 i −χ(s) ) = ( (iχ(s))−1 β(s)(i −χ(s))(i − r(s)) −1 0 (i − r(s))−1 ) f̂ (s) = f̂00(s) n−1∑ j=0 χ(s) j + f̂00(s) n−1∑ j=0 χ(s) jβ(s) ∞∑ k=0 r(s)n + f̂01(s)r(s) n = ∞∑ k=0 ( ρe− (λq1 +µ+θp̄1 ) r x ji j(βx)β j x ( r 2θp̄1 ) j − λµ θµp )∗ kf00(s) + β(x) ∗ ∞∑ k=0 ( r 2θp̄1 )k kik(βx)β j x e− (λq+µ+θp̄1 ) r x ∗ ∞∑ k=0 ( ρe− (λq1 +µ+θp̄1 ) r x ji j(βx)β j x ( r 2θp̄1 ) j − λµ θµp )∗ kf00(s) + f̂01(s) ∗ ∞∑ k=0 ( r 2θp̄1 )k kik(βx)β j x e− (λq1 +µ+θp̄1 ) r x (20) hence, evaluation of buffer content distribution are expressed in terms of f̂00(s) and f̂01(s). therefore the measure of performance can be given using this explicit analytical solution given in the form of modified bessel function of first kind. 4. analysis of statistical illustration to acquire a holistic view of the model stability and performance, this section describe the findings of certain numerical experiments in the aspect of system stability with intrinsic of markov fluid analysis. we examine the steady-state convergence of several measures to their exact theoretical values 7 nandhini & vidhya / j. nig. soc. phys. sci. 5 (2023) 1148 8 of buffer occupancy distribution.png figure 4. dependency of buffer occupancy distribution against the variation of parameters where analytical expressions are given in the depiction for the purpose of model validation. following the model’s validation, we use empirical correlation to conduct a numerical investigation of a few auxiliary measures for the dependence of orbits, arrivals, and service patterns. the simulation is carried out using matlab and is based on the continuous event procedure. as a desired set approach, the following preceding parameters are chosen for the illustrated example of an online education cloud system shown in section 1.1: λ = 1.2/hr, µ = 2.0/hr, θ = 0.9/hr, q1 = 0.5, p̄1 = 0.5. the values are selected in such a way to guarantee sufficient stability condition equation 20 holds true. the graph displayed in figure 3 displays the difference of buffer capacity distribution against time t with retrial rate θ using the appropriate value of θ considered for the purpose of comparison. it is to be noted that the retrial rate θ value increases with increases of buffer capacity distribution, particularly at the values of buffer capacity t more than 0.4. it is strictly convex montonically increasing function proves stability in relation with the dependent variable of λ, µ and θ. in figure 4 depicts the behaviour of buffer content distribution in three dimensions with variation of the system estimation parameters dependence of λ, µ and θ. in particular, figure 4(a) explains the variation of buffer content distribution along with service rate µ and retrial rate θ has validate the result. figure 4(b) depicts the evolution of stationary distribution in three dimension with variation of the arrival λ and retrial rate θ. this figure explains the variation of buffer content distribution increase along with increment of the arrival rate λ and retrial rate θ. it is concluded that the numerical illustration provided has validate the result. figure 4(c) depicts the behaviour of buffer content distribution in three dimensions with variation of the arrival λ and service µ. this figure explains the evolution of buffer content distribution is strictly increasing along with increment of the arrival rate λ and service rate µ. it is to be concluded that the numerical illustration provided has validate the result. to illustrate our result, we simulate m/m/1 type queueing system. the main motivation of our simulation study is to learn the behaviour of the performance measure which are associated with the buffer content distribution against the variation of some system parameters. 8 nandhini & vidhya / j. nig. soc. phys. sci. 5 (2023) 1148 9 5. conclusion in this research, steady-state evaluation of the markovian queueing model with constant retrial rate with non-persistent users is demonstrated. it is displayed that the service and interretrial times are linearly independent of one 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[29] r. sakthi, v. varadharajan & k. mahaboob, “performance measures of state dependent mmpp/m/1 queue”, international journal of engineering and technology 4 (2018) 942. appendix proof of theorem 4. the proof is built on the analysis of buffer occupancy distribution given by f̂0(s) = a1r0, a2r0 s ( r0 0 0 r0 ) − ( −λ λ µ −(λq + µ) ) − ( χ(s) β(s) 0 r(s) ) ( 0 θ 0 θp̄ ) = (a1r0, a2r0) ( sr0 + λ −(λθχ(s) + θp̄β(s)) −µ sr0 + (λq + µ)θp̄r(s) )−1 f̂00(s) = a1r0(sr0 + λq1 + µ− θp̄1r(s)) + a2r0µ (sr0 + λ)(sr0 + λq1 + µ− θp̄1r(s)) −µ(λ + θχ(s) + θp̄1β(s)) f̂01(s) = a1r0(λ + θχ(s) + θp̄1β(s)) + a2r0(sr0 + λ) (sr0 + λ)(sr0 + λq + µ− θp̄1r(s)) −µ(λ + θχ(s) + θp̄1β(s)) 9 nandhini & vidhya / j. nig. soc. phys. sci. 5 (2023) 1148 10 upon factorization which leads to f̂00(s) = a1r0(sr0 + λq1 + µ− θp̄1r(s)) + a2r0µ (sr0 + λ)(sr0 + λq1 + µ− θp̄1r(s)) −µ(λ + θχ(s) + θp̄1β(s)) = ∞∑ k=0 ω(s)k{a2σ0µd(s) − a1σ0 sσ0 + λ } (21) similarly for f̂01(s) f̂01(s) = a1r0(λ + θχ(s) + θp̄β(s)) + a2r0(sr0 + λ) (sr0 + λ)(sr0 + λq + µ− θp̄1r(s)) −µ(λ + θχ(s) + θp̄1β(s)) = (a1r0(λ + θχ(s) + θp̄1β(s))d(s)) + a2r0 sr0 + λq1 + µ− θp̄1r(s) ∞∑ k=0 ω(s)k (22) where ω(s) = µ(λ + θχ(s) + θp̄1β(s)) (sr0 + λ)(sr0 + λq1 + µ− θp̄1r(s)) d(s) = 1 (sr0 + λ)(sr0 + λq1 + µ− θp̄1r(s)) from inversion of (12),(14),(15) with β = 2 √ λθp̄q σ , we have, β(s) = (λ + θ + sr)ρ µr(s) −λµ λ + θ + sr µθµp = e− (λ+θ) r x ( r µ )k xk k! ∗χ(x)∗i ω(s) = µ(λ + θχ(s) + θp̄1β(s)) (sr0 + λ)(sr0 + λq1 + µ− θp̄1r(s)) = µ(λ + θχ(x) + θp̄1β(x)) ∗ ∞∑ i=0 ∞∑ j=0 θp̄i1 ri+2 e− λ r0 xe− (λq1 +µ) r0 x xi i! iii(βx)βi x ( r 2θp̄1 )ie−( λq1 +µ+θp̄1 r )x χ(s) = ρ + sr r(s) − λµ θµp = 1 θp1 ∞∑ k=0 ( µ p̄1r )k xk−1 (k − 1)! e− (λ+θ) r x ( λµ r ) e− (λ+θ) r x − 1 θp1 ∗ kik(βx)βk x e− (λq1 +µ+θp1 ) r x ( r 2θp̄1 )k d(s) = 1 (sr0 + λ)(sr0 + λq1 + µ− θp̄1r(s)) = ∞∑ i=0 ∞∑ j=0 θp̄i1 ri+1 e− (λq1 +µ) r0 x xi+ j (i + j)!  ∗ ii j(βx)βi x ( r 2θp̄1 )ie− (λq1 +µ+θp̄1 ) r x from equation 17, we arrive at the solution in terms of the modified bessel function of the first kind, which signifies explicitly the exponentially growing function achieved the characteristic of the proposed work by significantly increasing the function of the number of packets/signals in the virtual buffer (retrial) f̂00(s) = ( −a1r0 (sr0 + λ) + a2r0µd(s) ) ∞∑ k=0 ω(s)k = ∞∑ k=0 (ω(s))∗k ∗ ( a2 ∞∑ v=0 ( θp̄1 r0 )ve− (λq1 +µ) r0 x xv v! ∗ viv(βx)βv x ( r 2θp̄1 )ve− (λq1 +µ+θp̄1 ) r x ) + a1r0(λ + θχ(x) + θp̄1β(x)) ∗ d(x) f̂01(s) =(a1r0(λ + θχ(s) + θp̄1β(s))d(s)) + a2r0 sr0 + λq1 + µ− θp̄1r(s) ∞∑ k=0 ω(s)k = ∞∑ k=0 (ω(s))∗k ∗ ( a2r0µ∗ d(x) − a1e − λ r0 x ) (f̂k0(s),f̂k1(s)) = (f̂00(s),f̂01(s))r n(s) = (f̂00(s),f̂01(s)) ( χ(s)n ∑n−1 i=0 χ(s) iβ(s)r(s)n−1−i 0 r(s)n ) = ( f̂00(s)χ(s)n f̂00(s) ∑n−1 i=0 χ(s) iβ(s)r(s)n−1−i +f̂01(s)r(s)n ) which is simplified in the form of f̂k0(s) = f̂00(s)χ(s) n (23) f̂k1(s) = f̂00(s) n−1∑ i=0 χ(s)iβ(s)r(s)n−1−i + f̂01(s)r(s) n (24) thus all steady state probabilities are obtained interms of modified bessel function of first kind to determine the flow of fluid in retrial queue along with impatient customers are observed. 10 j. nig. soc. phys. sci. 5 (2023) 1028 journal of the nigerian society of physical sciences evaluation of anfis predictive ability using computed sediment from gullies and dam stephen olushola oladosua,∗, alfred sunday alademomib,c, james bolarinwa olaleyeb, joseph olalekan olusinab, tosin julius salamib adepartment of geomatics, faculty of environmental sciences, university of benin, p.m.b. 1154, edo state, nigeria bdepartment of surveying and geoinformatics, faculty of engineering, university of lagos, p.m.b. 12003, akoka, lagos state, nigeria ccentre for multidisciplinary research and innovation, suite c59, new bannex plaza, wuze 2, abuja, nigeria abstract the study proposed an adaptive neuro-fuzzy inference systems (anfis) model capable of predicting sediment deposited in a dam and sediment loss-in-transit (slit) using the potential of a formulated mathematical relation. the input parameters consist of five members viz: the rainfall, the slope, the particle size, the velocity, and the computed total volume of sediment exited from two prominent gullies for 2017, 2018, and 2019. the outputs are the total volume of sediment deposited at the adjoining ikpoba dam for 2017, 2018, and 2019, respectively. the ordinary least square (ols) regression model on sediment volume retained all covariates with p<0.05, explaining 93.8% of the variability in the dataset. the multicollinearity effect on the dataset was assessed using the variance inflation factor (vif) which was found not to pose a problem for (vif<5). the model was validated using the (mse), the (mae), and the correlation coefficient (r). the best prediction was obtained as: (rmse = 0.0423; r2 = 0.947). the predicted volume of sediment was 842,895.8547m3 with an error of -0.3295344% and the predicted volume of slit was 57,787.98m3 which is an indication that anfis performs satisfactorily in predicting sediment volume for the gullies and the dam respectively. doi:10.46481/jnsps.2023.1028 keywords: anfis, gully erosion, ikpoba dam, sedimentation article history : received: 23 september 2022 received in revised form: 21 october 2022 accepted for publication: 21 october 2022 published: 21 may 2023 © 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: o. j. abimbola 1. introduction there are different ways in which artificial neural networks (ann) and adaptive neuro-fuzzy inference systems (anfis) find application in solving real-life problems that are subject to interpretation using differential or partial differential equations numerically [1]. water resource is one key area where ∗corresponding author tel. no: email address: olushola.oladosu@uniben.edu (stephen olushola oladosu) soft computing, or machine learning and techniques have become indispensable for the modelling of complex, non-linear, and dynamic processes in a hydrological system domain [2]. however, their applications are found in other areas [34]. the aspect of the application of soft computing methodologies in sediment study and water resources in nigeria is quite scanty in literature [5]. gully erosion, one of the major contributor to channel sediment is more profound in the southern part of nigeria and is known to have contributed sizable amount of sediment as in1 oladosu et al. / j. nig. soc. phys. sci. 5 (2023) 1028 2 take into dams such as the one under investigation in this work. notable studies have been conducted in the region in various capacities on gully initiation, development, and remediation in the past as obtained in [6-11]. in 2013, the nigerian government established the nigeria erosion and watershed management project (newmap) to oversee how best to mitigate soil erosion (particularly gully erosion) and land degradation in specific watersheds by inclusive approach [12-13]. most often, actions to salvage the havoc pose by such environmental problem are delayed due to bureaucracy in governance. approximately 6% of nigeria’s total landmass, relied upon by many other sectors of the economy, has either been badly damaged or degraded by gullying activities [13]. the 2% per annum growth in the population as noted by [13] is an indication that more demand will be on land use; hence, more efforts are required in addressing the issue of land degradation and gullying in the country. gully erosion is a process by which soil’s cohesive forces are drastically weakened at particular thresholds by the action of an active run-off leading to the commencement of mass movement and the eventual creation of deep channels with a substantial amount of sediment deposited downslope [14-15]. the development of gullies causes a substantial amount of soil loss, especially in badland where sediment from gullies finds its way into the river channel and is later transported as bedload into a dam. the negative impact, in the long run, is the speedy rate of siltation and a drastic reduction in the useful life of a reservoir which when occurring will require serious attention. gully erosion has wreaked havoc globally and has been a threat to a free, safe, and life-sustaining environment [16-20]. the motivating variables and drivers of gully initiation and development in distinct catchments are present in previous works [21-22]. on a global scale, as reported by [15], [22-23], gully erosion caused between 10 to 94 percent of soil loss, most of which finds its way downstream into a reservoir. sediment modeling using soft computing methodologies, such as ann, wavelet, ga, anfis, c-anfis, and others, is described in the literature. a researcher’s objective is to ascertain whether combining or complementing any models will produce an improved outcome. according to [24-26], ann in a broad sense, encompasses a wide range of network designs and configurations. the most common ann in literature is the multilayer feedforward neural network (mlffnn), which adopts a form of an interconnected perceptron that makes data and calculations flow ultimately in one direction, starting from the input data to the outputs. the most common attributes of an mlffnn consist of an input layer, a single or multiple hidden layers, and an output layer [27-28]. during anfis internal training of the ann network, the inputs of the first layer multiply an initial random weight coefficient. this development prompts the network to progress to the neurons in subsequent layers. the resulting sum is then forwarded to an activation function, which processes and transforms it to the required output. the network error of the predicted output and target output is calculated and again sent from the last layer to the previous one, thereby updating the weight coefficients. this process is called “error-back-propagation” [2829]. in the history of the anfis soft computing technique, [2425] were pioneers. because of its ability to deal with nonlinear phenomena, it is preferred for simulating and modelling complex hydrological systems [29-32]. applications of anfis in diverse fields are quite numerous in the literature. [33], applied an anfis-based approach for the prediction of sediment transport in clean sewers and affirmed a satisfactory result with (r2 = 0.98 and rmse = 0.002431) in comparison to other existing predictors. [31-33], adopted an anfis-based approach for predicting the bed load for four moderately-sized rivers. while other equations failed to produce an accurate result, the anfis results showed an accurate prediction for measured bed-load data based on a regression method used for comparison. this research is aimed at proposing an anfis model capable of utilising five input parameters in determining the volume of sediment deposited in ikpoba dam and the amount not reaching the dam at the time when observations were conducted tagged “sediment-loss-in-transit” (slit). the model design links the input parameters to aggregate the devastating effect of sediment volume intake of the dam which contribute significantly to early loss of storage capacity. sediment evaluation at the dam based on the quantity of soil loss from gully erosion transported through the river channel can serve as guide to the water resource managers, hydraulic engineers, and dam operators in taking appropriate decisions on dam water-sharing routine, dam useful life monitoring, dam rejuvenation efforts, cost-effective sediment remediation method, dam management planning, and so on. 2. materials and methods 2.1. study area the study area captured the upstream part of the ikopba river dam where two prominent gullies, namely, (the university of benin gully and the iguosa-oluku gully) are exiting their massive sediment into the ikpoba river channel within the same catchment in benin city, edo state, southern nigeria. the geographical location of the study area falls within utm zone 31n, 789050.75 me; 705200.22 mn, and 795500.96 me; 715950.26 mn). by koppen classification, benin city has a tropical savanna climate with rainfall intensities reaching up to 2680mm annually in most cases. the rainfall lasts between eight to nine months, starting in march through october at varying degrees, resulting in a substantial amount of overland flow with impactful erosive energy. the average annual temperature at this location was 25.6 ◦c. the benin city region is characterized by the sedimentary formation underlay of what is called the “south sedimentary basin,” with the geology marked by the presence of reddish earth on top, composed of ferruginized or literalized clay sand [34]. the geologic formations in the region are classified into four basic categories: the benin formation; alluvium; drift/topsoil and azagba-ogwashi. the type of soil in this region, which has low cohesive aggregate binding force, is highly susceptible to erosion influence, therefore gully formation is always prevalent. the study area including the immediate catchment is shown in figure 1. 2 oladosu et al. / j. nig. soc. phys. sci. 5 (2023) 1028 3 figure 1. map showing the study area 2.2. the description of the gullies the university of benin gully and the iguosa-oluku gully hereafter referred to as gully (a) and gully (b), respectively, are located in the upper part of the ikpoba dam. although there is no specific documentation on the exact date on which gully (a) began, it started earlier than the year 2005 as captured by google earth historical images. its development has caused some buildings to be completely or partly eroded at the senior staff quarters of the university due to the gradual landslide resulting in gully sidewall collapse. the guest house of the university is presently under threat. figure 2 reveals the impacts of gully erosion on some properties and the immediate surroundings of the invested environment. according to an anonymous resident, the rehabilitation of the benin/lagos expressway in 2014 caused the start of the gully (b). its initiation and subsequent development are tied to the volume of run-off diverted to a drainage system not appropriately designed to convey such an amount. hence, the water created an alternative path across the existing settlement, actively creating a bigger and deeper channel. six years of no visible remedial intervention led to the destruction of more than fifty buildings, including a police station and other utilities. the federal housing estate located close to where the gully extension has reached is now in grave danger of being eroded. figure 3 shows the devastation caused by gully erosion (left) and table 1. construction information of ikpoba dam s/no name of dam: ikpoba remark 1 type of dam earth fill 2 water production per pump day 34080 m3 3 catchment area 120 km2 4 crest level height 35 m (a.m.s.l) 5 dam length 610 m 6 active storage capacity 1.5 x 106 m3 7 reservoir surface area 1.07 x 106m2 8 service spillway length 60 m 9 emergency spillway length 4 m 10 water supply capacity 90,000 m3/day 11 average monthly discharge 31.9m3/s 12 average annual run-off 0.9285 x 109m3 13 population at design 1.0 million 14 first impoundment year 1975 15 commission year 1987 source: [38] the massive soil loss deposited along the ikpoba river course (right). 2.3. description of ikpoba river and dam the ikpoba river originated from the oluku settlement area in an extension of benin city’s western highland towards the northern and north-eastern parts [35]. from the identified source, the river flows from east to west; reverses its course and meanders through utekon before changing direction to the southern and eastern banks through the following axes (ekosodin, ugbowo, okhoro, and new benin). the river has some level of interactions with the exited sediment from the gullies under consideration. the ikpoba dam is an earth dam that is located along the river reach between okhoro and teboga. it is a small dam according to the classification of the international commission on large dams [36]. the construction work of the dam began in 1977 and was commissioned in 1987. the benin-owena river basin of nigeria manages the dam in conjunction with the edo state urban water board. the geological terrain is tertiary and the foundation is pile-based. four stations are present before reaching the dam’s head [34-37]. these are (okhoro, midpoint, low-lift pump, and ekiuwa). according to [38], the dam has a length of 610 meters. table 1 provides useful information on the physical characteristics and parameters of the dam. 2.4. gully data acquisition and preparation a trimble m3 dr 3” total station and a trimble juno 3b handheld gnss receiver were used for data collection. existing ground controls on sites were subjected to an integrity test and found to be stable before proceeding to observations. the cross-sectional design was done in autocad civil 3d for 2017, 2018, and 2019, respectively. samples are presented in figure 4. the volume of each gully segment at chainage 20m apart was computed by the average area of the up and downstream cross sections multiplied by their respective segment lengths. 3 oladosu et al. / j. nig. soc. phys. sci. 5 (2023) 1028 4 figure 2. gully (a) and its negative impacts on settlement figure 3. devastating impact and soil loss from gully b the cumulative sum of all sectional volumes represents the total sediment volume contributed by gullies. see table 2. 2.5. bathymetric data acquisition and preparation the bathymetric surveys of the reservoir were conducted in 2017, 2018, and 2019, respectively. the procedure involved the coupling of the 15-hp yamaha engine to the fiber boat and pushing it away from the bank to gain enough depth to allow for the attachment of the transducer and the mounting of the gnss receiver firmly to their respective positions. the position fixing was done in rtk mode using the hi-target gnss receiver while depths were measured simultaneously with the aid of the hi-target marine hd-max echo sounder powered by a 12-volt battery. the appropriate bar checks were observed against the standard marks at 1 m, 2 m, and 3 m for consistency and the eradication of false depth records. this check was done before and after the bathymetric surveys. twenty-five transect lines (25) and two longitudinal cross lines (2) were traversed. the output from the rtk-gnss receiver are precise 3-d (x, y, z) coordinates with obtained accuracies in the order of 0.02m for horizontal and 0.05m for vertical, respectively. corrections for pitch roll and heave were applied at the software interface on board to get the corrected depths. during the different campaigns, water level monitoring was performed by planting a levelling staff at the bank of the river, and readings were taken with the aid of nikon automatic ac-2s leveling instrument before and after the bathymetric surveys. for the three campaigns, the average water level recorded at the beginning of work was 3.360m and at the end of work, it was 3.361m. the difference gave 0.001m. this result showed that the water at the dam was non-tidal and relatively calm throughout the time of observations. note, that water level measurement may not pose a problem with the rtk technique, but when it is taken, it can be used for verification or validation purposes. we computed the volume of sediment accumulated in the dam using equation 1 [39]. table 2, contains the technical parameters of the hi-target marine hd-max echo sounder, while table 3 provides the summary of the total volume of sediment accumulated in the dam for the three years investigated. figure 5 shows 4 oladosu et al. / j. nig. soc. phys. sci. 5 (2023) 1028 5 figure 4. sample of cross-sectional design for gullies table 2. summary of calculated volume of sediment from gullies year gully id vol. loss (m3) cumulative vol. loss (m3) total vol. (m3) 2017 256791.548 000000.000 2018 a 268363.430 525154.978 2019 277543.350 802698.328* 802698.328 2017 375478.830 000000.000 2018 b 414810.690 790289.520 2019 475275.954 1265565.474* 1265565.474 grand total 2068263.802 2068263.802 2068263.802 note: **are the only final yearly volumes considered in column 4 for to get the grand total the water level monitoring effort at the dam’s location. rannua = v1 − v f t (1) where rannual, represents the annual mean reservoir sedimentation volume in (mm3/year); vi, refers to the initial reservoir volume in (mm3); v f , signifies the final reservoir volume in (mm3); t, is the number of years since dam had been operated. t v u = ± √ a2 + (b × d)2 (2) 2.6. the anfis architecture the input data, the hidden layers, and the output layer are the three fundamental components of the anfis architecture. the anfis model type used in this study is the sugeno, with five layers. what necessitate its use was because of its compactness and efficient computational capability [42]. instead of working with linguistic variables on the consequent part as in the mamdani model, the tsk model [41-45] uses rules as a function of input variables to represent the consequent parts. its success is due mainly to the ease of generating a set of system equations for the consequent parts, the parameters of which are simple to estimate using traditional optimization methods. however, the interpretation of the obtained rules is somewhat difficult, which shows their principal shortcoming. the tsk learning algorithm consists of two processes, the forward and the backward stage. the forward phase goes through the five layers to be discussed after in text while the backward stage fine tune the weights of a neural network by using the error rate previous epoch [24]. in a first-order sugeno fuzzy model, a typical fuzzy rule statement takes the form of say where: a and b are fuzzy sets in the antecedent, is a crisp function in the consequent. the five layers are simplified further to briefly explain what takes place at each layers of the anfis model. (i) layer-1 this layer is the first, and is very critical to the overall process. equations 3 or 4 represent what takes place at this phase [42]. o1,i = µai(x) for i = 1, 2 (3) or o1,i = µ(bi−2)(y) for i = 3, 4 (4) where: x or y -represents the input taken by node i, ai or bi−2 -signifies a form of linguistic descriptions (e.g. high, low 5 oladosu et al. / j. nig. soc. phys. sci. 5 (2023) 1028 6 figure 5. bathymetric survey and water level monitoring exercise at ikpoba dam table 3. technical parameters of hd-max echo sounder s/no (a) technical parameters 1 cpu speed: 1.6g*2 2 ram: 2gb 3 memory space: 16gb ssd 4 display screen size: 17” 5 display resolution: 1280*1024 6 starting time: < 40s 7 high frequency emitted from the probe: 200khz 8 point-positioning precision: <2.5m (built-in gps function) 9 input voltage: 10 30v 10 average power consumption: <40w 11 operating temperature: 0 50°c (b) bathymetric accuracy 12 horizontal = 0.506m, at 95% confidence level. 13 vertical = 5m + (-0.30m) at 95% confidence level. note: we applied the formula provided by the international hydrographic organisation, [40] for the determination of horizontal and vertical uncertainties in depth measurement. the maximum depth obtained during sounding was 6.0m, while a and b are constants provided for in the formula. to compute total horizontal uncertainty, we used, thu = 5m + 5/100 of depth. equation 2 was used to compute the total vertical uncertainty, tvu. the results are contained in the last two rows of table 3. and so on) assigned to node i, o1,i -represents the membership function of fuzzy set ai which signifies the extent to which the input x or y under consideration satisfies the quantifier ai. µai(x) and µbi−2(y) can accommodate any fuzzy membership function assumed to spread between 0 and 1. for example, if the bell-shaped membership function is adopted, µai(x) follows the equation 5 µai (x) = 1 1 + [( x−ci ai )2] bi , i = 1, 2 (5) 6 oladosu et al. / j. nig. soc. phys. sci. 5 (2023) 1028 7 table 4. summary of calculated volume of sediment in dam year ikpoba dam annual sed. vol. (m3) cumul. sed. vol. (m3) comp. bed-load sed. vol. (m3) 1987 base year n/a 000000.000 2017 1st observation 217336.704 217336.704 2018 2nd observation 222790.642 440127.346 2019 3rd observation 400000.000 840127.346* grand total 840127.346 840127.346 840127.346 note:*is the only one considered under column 4 for the ground total where: ai, bi, and ci in equation 5 are the premise parameters set of the generalized bell shape membership function (mf). the current study adopted the gaussian membership function (mf) defined by equation 6. µai (x) = ex p − ( x − ci ai )2 (6) where: ai and ci in equations 6 are the premise parameters set of the gaussian mf. changing the values of these parameters would cause a corresponding change in the generalize bell-shape behaviour and the gaussian shape that will make the fuzzy set “ai” to exhibit various forms of membership functions. in this layer, parameters are commonly called premise parameters [4245]. (ii) layer-2 at this layer, each anfis node represents a fixed node with an output signal as the product of the contributions of the incoming signals. o1,i = wi = µai (x) ∗µ(bi−2) (y) , f or i = 1, 2 (7) each node output in this context is a representation of a rule’s firing strength capacity. the node function in this layer will typically be any other t-norm operator that can execute fuzzy (and). (iii) layer-3 this layer has a fixed node labeled n. here, the ith node computes the ratio of the ith rule’s firing strength to the sum of all rules’ firing strengths. the outputs of this layer are the normalized firing strengths, represented as follows: o3, i = wi = wi w1 + w2 : f or i = 1, 2. (8) (iv) layer-4 every node i contained in this layer represents an adaptive node with a node function: o4, i = wi fi = wi( pi x + qiy + ri), (9) where: the anfis in this layer represents a normalized firing strength inherited from layer-3, and the quantities represented as: (pi, qi and ri) are the parameters set of this node. parameters here are termed consequent [4145]. (v) layer-5 this layer is an anfis with a fixed node label and a single node that sums all incoming signals to produce the overall output: o5, i = ∑ i wi fi = ∑ i wi fi∑ i wi = the overall out put (10) figure 6. anfis architecture. source: [41] these five steps simplify a functional takagi-sugeno fuzzy model for an adaptive network design. the backward stage is a useful process in estimating the database; this consists of the parameters of the membership functions in the antecedent part and the coefficients of linear equations in the consequent part. the least-squares method assists greatly in parameter learning by utilizing standard fuzzy reasoning to anticipate the outcome of the tsk model in the prediction phase [4145]. figure 6 shows the anfis architecture. 2.7. description of anfis input parameters the parameters used for the anfis model contain the five most important site-specific variables that are either observed, measured, or determined through laboratory analysis for the study area. rainfall: rainfall data were collected using two tipping bucket rain gauges located within the catchment of the ikpoba river close to where gully a and gully b exit their sediment into the river channel in conjunction with rainfall data obtained from the nigerian meteorological agency (nimet). the collected rainfall data was used for validation. the data was resampled and gridded to 220 to comply with the gully crosssection. volume: for the three years specified earlier, gully a and gully b had 109 and 111 cross-sectional segments, adding up to 220 for which volumes of sediment were calculated and cumulated to derive the cumulative volume of sediment deposited. slope: the slope length was determined as run over the rise at each gully cross-section. the depth of gullies taken at each cross-sections varied from 6.0 m to 11.23 m. particle size: particle size was obtained from soil samples taken at 1m to 3m from 20 borehole points dug with a hand 7 oladosu et al. / j. nig. soc. phys. sci. 5 (2023) 1028 8 huger at the two gully sites. sieve analysis was carried out at the university of benin civil engineering laboratory. velocity: the velocity was measured manually at pre-defined sections along the gullies on rainy days. the widths were measured with 100 m graduated steel tape, and the upper and lower boundaries of each section were defined with wooden pegs wrapped with caution tape. this information was recorded and kept for further use. on two different rainy days, a floating material (cork) was released from the gully’s head and the time taken for it to travel between the flags was recorded with a stopwatch. five observers were stationed in succession to record time with a stopwatch until the process was completed. to find the velocity in (m/s2), we divided the distance traveled by the average travel time and multiplied the result by a chosen correction factor of 0.9 which is mostly adopted for rivers or flowing water whose bottom consists of smooth mud, and sand, or bedrock. the summary of the description of the anfis input parameters in terms of their minimum and maximum limits and their respective membership functions is presented in table 5. each parameter has three (3) membership functions. 2.8. preliminary investigation on input parameters ordinary least squares (ols) regression models on sediment volume retained all covariates with p<0.05, explaining 93.8% variability in the dataset. assessment of the multicollinearity related to the dataset using the variance inflation factor was found not to pose a problem (vif<5) obtained throughout for covariates is sufficient [46-49]. 2.9. data normalisation avoiding data normalisation may lead to a complex situation in the training process and non-convergence of the algorithm. the min-max normalisation technique that scales the variables in the interval between (0, 1). the data was randomised (the purpose is to reduced bias and make the output more reliable). the data was further divided into training and testing datasets after normalisation. after making a good number of data sharing attempts by trial and error, the choice of 75% was made and used for training, while the remaining 25% was used for testing. this sharing ratio gave the optimal result and accuracy compared to others where less or greater value of sharing resulted in large error. the training data performed the role of the anfis training process and the generation of the fuzzy rule-based systems (frbs), whereas, the testing data served the purpose of verifying the accuracy and effectiveness of the trained anfis model. for efficient anfis processing, the input dataset used was processed in a data frame in form of a matrix (m × n), where m is the number of instances, n is the number of variables, while the last column in the matrix represents the output. tables 6 and 7 are excerpts from the normalized training and testing data respectively. 2.10. training process the process began by obtaining a training data set (input/output data pairs) and checking of the data sets. two vectors are used to train the anfis system: the input vector and the output vector. anfis training rules used was a hybrid learning, which combines the gradient descent and the least-squares method. the training proceeds by determining the fuzzy sets and the number of sets for each input variable and the shape of their membership function. all the training data passes through the neural network, to adjust the input parameters so as to find the relationships between input/output, and to minimize errors propagation. the parameters associated with each membership function kept changing throughout the learning process and a threshold value for the error between the actual and the desired output was determined. if the error is larger than the set threshold value, then the premise parameters are updated using the gradient descent method. the consequent parameters are found using the least-squares method. the process is terminated when the error becomes less than the threshold value. we used the checking data set to compare the model with the actual system. the input node received signals from each of the five predictor nodes, connected through intervening hidden nodes. above each line is the displayed of the respective synoptic weights. the blue circles indicate bias, corresponding to the intercept in the conventional regression model. the variable “sediment volume” represents the output neuron. the network converged when the error reached 0.190791, after 417 steps. the mathematics involved in this process is shown in equations 11 and 12. figures 7 and 8 show the ann network topology and the flow diagram of the ann training process, respectively. wi j (t + 1) = wi j (t) + ηδpiop j (11) δpiwi j (t + 1) = wi j (t)+ηδpiop j+α [ wi j (t) − wi j (t − 1) ] (12) where: weight coefficient in step t+1, from neuron i to neuron j weight coefficient in step t, from neuron i to neuron j : learning coefficient : difference between desired output and network output in neuron p of layer j : output of neuron p of layer j : momentum coefficient : weight coefficient in step t-1, from neuron i to neuron j. 2.11. model validation metrics in this work, model validation was carried out using the mean squared error (mse), the mean absolute error (mae), and the correlation coefficient (r). for simplicity of use, ease of training, and better performance, the rectified linear unit (relu) activation function was used. it is a piecewise linear function that will output the input directly if it is positive, otherwise, it will output zero. equations 13 15 are the statistical models adopted in model validation. rms e = √ ∑n i=0(di − yi) 2 n (13) r = ∑ (yi−ȳ)(di−d̄) n √∑ i (di−d̄)2 n √∑ i (yi−ȳ)2 n (14) 8 oladosu et al. / j. nig. soc. phys. sci. 5 (2023) 1028 9 table 5. brief description of input parameters gully a (109 c.s) gully b (111 c.s) = 220 parameters min. max. min. max. mf rainfall (mm) 55.85 104.49 167.80 215.70 3 slope-length (m) 29.89 71.99 30.85 63.27 3 depth (m) 6.45 11.23 4.70 9.61 3 velocity (m/s) 0.51 1.84 0.44 1.86 3 particle size (µm) 28.77 41.69 34.85 47.90 3 gullies sed.vol. x 104 (m3) 5651 8473 8652 13370 3 note: c.s = cross-section, mf = membership function table 6. excerpt from normalised training data index rainfall(mm) slope(mm) depth(mm) velocity(m/s) p.size(µm) volume(m3) 139 0.952312 0.325011 0.398463 0.514887 0.60261 0.673596 98 0.003783 0.286023 0.533673 0.78711 0.373132 0.220773 112 0.785692 0.205318 0.206067 0.396444 0.645918 0.756984 144 0.979885 0.646603 0.443222 0.099204 0.763262 0.900655 126 0.864056 0.298648 0.32029 0.365815 0.673003 0.817199 209 0.869185 0.089084 0.251222 0.281811 0.753548 0.860469 35 0.269768 0.842347 0.693181 0.1207 0.446288 0.236462 123 0.818387 0.622698 0.445846 0.925704 0.610277 0.618824 177 0.819464 0.238152 0.499117 0.182115 0.509774 0.612393 212 0.840235 0.554263 0.499378 0.074061 1 1 table 7. excerpt from normalised test data index rainfall(mm) slope(mm) depth(mm) velocity(m/s) p.size(µm) volume(m3) 7 0.214163 0.958774 0.722309 0.520943 0.491623 0.271775 109 0.074786 0.497832 0.675136 0.925770 0.395535 0.190251 140 0.883963 0.322317 0.279409 0.314403 0.804152 0.706681 74 0.136978 0.710343 0.793520 0.452986 0.266164 0.117517 205 0.725127 0.308324 0.531787 0.103523 0.842459 0.869096 111 0.753561 0.242328 0.000000 0.000000 0.548667 0.693910 187 0.812660 0.211266 0.441200 0.466538 0.485644 0.623276 100 0.152822 0.205663 0.491521 0.274293 0.426886 0.261896 189 0.892676 0.359164 0.323424 0.951899 0.597544 0.580190 88 0.237729 0.535238 0.651543 0.947341 0.349382 0.198406 mae = 1 n j=1∑ n |yi − ŷi| (15) where: di : desired output of i-th data yi : network output of i-th data n: number of dataset d̄ : mean of desired output ȳ : mean of network outputs rmse: root mean square error mae: mean absolute error r: correlation coefficient the summary of the investigated neural network algorithms presented in table 5 shows that: i.) all models have two hidden layer-nodes (5, 3) with relu activation function ii.) the learning rule used for model 1 (nn-1) is resilient backward propagation. iii) levenberg algorithm was adopted for model 2 (nn-2), while iv) backward propagation algorithm was adopted for model 3 (nn-3). a model that meets all the required evaluation criteria would be the desired model. therefore, the model with the least rmse and mae errors (approaching 0) and the coefficient of determination ”r” (tending to 1) obtained from nn-3 as highlighted in table 8 is the most acceptable network result chosen. the predictive capability of the best model has an rmse of 0.0423 with an r2 of 0.947. table 9 shows the random points to demonstrate normalised and de-normalised for the training and testing datasets. table 9 presents the comparison of the actual test data and their predicted values based on the fitted training model. the 9 oladosu et al. / j. nig. soc. phys. sci. 5 (2023) 1028 10 figure 7. ann training network topology table 8. summary of network algorithms training test model hidden layer nodes activation function (f) learning rule mae rmse r mae rmse r nn-1 2 8 relu resilient backpropagation (rprop+) 0.755 0.578 0.903 0.785 0.594 0.916 nn-2 2 8 relu leverberg algorithm 0.644 0.546 0.924 0.687 0.548 0.919 nn-3 2 8 relu backpropagation (backprop) 0.0456 0.059 0.944 0.048 0.0423 0.947 remaining 25% of input parameters from the test data were fed into the trained anfis model to predict the output. it can be observed from the table that the predicted values were close to the original test data. the error rarely exceeds 10%, so the model was able to predict with an acceptable accuracy. the regression plot in figure 9 (left) displays the network outputs concerning the training and test data. for a perfect fit, the data should fall along a 45-degree line, where the network outputs are equal to the targets. for this particular problem, the fit is reasonably good for all data sets, with a trained coefficient of determination (r2) value in each case of 0.947. figure 9 (right) depicts the anfis training versus validation plot for the test dataset. a close overlap implies a high predictive power of the trained model on the new dataset. i.e., the predicted (purple 10 oladosu et al. / j. nig. soc. phys. sci. 5 (2023) 1028 11 table 9. the predictive power of the model on test data normalised de-normalised randomized index actual predicted actual predicted %error 7 0.2718 0.2676 7748.541 7716.455 0.414 109 0.1903 0.1923 7119.224 7135.055 -0.222 140 0.7067 0.8422 11105.789 12152.114 -9.421 74 0.1175 0.1527 6557.752 6829.093 -4.138 205 0.8691 0.7942 12359.551 11781.385 4.678 111 0.6939 0.6133 11007.203 10385.319 5.650 187 0.6233 0.5879 10461.952 10189.188 2.607 100 0.2619 0.2948 7672.287 7926.644 -3.315 189 0.5802 0.6580 10129.345 10729.688 -5.927 88 0.1984 0.2140 7182.175 7302.460 -1.675 186 0.6273 0.7331 10492.945 11310.069 -7.787 137 0.7870 0.8188 11725.892 11971.350 -2.093 32 0.1499 0.2065 6807.912 7244.302 -6.410 71 0.2234 0.2255 7375.058 7391.463 -0.222 210 0.7040 0.6087 11084.715 10349.643 6.631 line) has 94.7% predictive accuracy as reported from the value of r2. the attained training error for the test model is obtained as 0.0423. 3. results and discussions 3.1. sediment deposit in dam the total volume of sediment contributed for three years by gullies was 2068263.802 m3. similarly, the calculated volume of bed-load sediment gained by the dam was 840127.346 m3 (refer to tables 2 and 4). here, we are faced with using the model to reproduce the amount of sediment deposited in the dam. mathematical relationship: vgully − vt ransit = vdam (16) based on the preamble at the start of this sub-section. let, 2068263.802vgully = 840127.346vdam therefore, vdam = ( 2068263.802 840127.346 ) ×γ ⇒ vdam = 2.4618vgullyγ (17) where: γ is the smoothing parameter or correction factor with value ranging as (0 < γ < 1). %error = vdampredicted vdam × 100% (18) where: vdampredicted refers to the expected volume of sediment obtainable at the dam that would correspond to the volume computed and vdam is the final volume calculated at the dam’s end. the smoothing parameter is the correction term applied to correct unmeasured external factors such as channel gradient, channel roughness, or corrosion. it helps in computing the volume of sediment in the dam for every combination of hyperparameters specified by evaluating against a specified (dam volume constant of 840127.346 m3). the appropriate value of the correction factor can either be learned by trials and error or through cross-validation. this study adopted a cross-validation sequence combining (0.1 and 0.2 step size of 0.005) for hyperparameters. table 10 showed the percentage error and the predicted volume of the dam at each iteration of table 10, reveals the predicted volume of sediment by the anfis model. a grid search predicted value of 842895.8547 m3 obtained at 0.165 yielded the best smoothing value for predicting volume of sediment present in the dam with a percentage error of -0.3295344%. the original volume of sediment computed at the dam was 840127.346 m3 being used as constant all through. initially, the error rate decreases as increases until it attains an optimal value of 0.165. by iterating beyond this value, the error rate starts increasing in the opposite direction. 3.2. sediment loss-in-transit sediment loss-in-transit per-unit volume is the difference between the calculated amount of sediment exited (deposited) from the gullies and the amount of sediment gained by the dam. this concept forms the basis of the proposed model. equation 18 expresses another form of equation 15, obtained by making the volume of sediment loss-in-transit the subject of the formula of the relation. the deduction from the model shows that it is capable of estimating the computed volume of sediment loss-in-transit by using equation 19. vt ransit = vgully − vdam (19) total volume of soil loss (sediment) from gully = 2068263.802 m3= vgully 11 oladosu et al. / j. nig. soc. phys. sci. 5 (2023) 1028 12 figure 8. flow diagram of anfis training process total volume of bed-load sediment gained by dam = 840127.346 m3= vdam total volume of soil loss-in-transit (difference) = 1228136.456 m3= vt ransit note: the difference is the volume of sediment regarded as ”loss-in-transit,” which could not be accounted for at the dam’s end during field campaigns. therefore, we are faced with the task of using the model to predict the volume of (sediment) lossin-transit given an array or matrix of input parameters. the proposed anfis model was trained to reproduce the table 10. percentage error and predicted volume of dam at each iteration of γ γ vdam (m3) vdampredicted (m3) %error 0.100 840127.346 510845.9726 39.19422157 0.105 840127.346 536388.2712 36.15393265 0.110 840127.346 561930.5698 33.11364373 0.115 840127.346 587472.8684 30.0733548 0.120 840127.346 613015.1671 27.03306588 0.125 840127.346 638557.4657 23.99277696 0.130 840127.346 664099.7643 20.95248804 0.135 840127.346 689642.0629 17.91219912 0.140 840127.346 715184.3616 14.8719102 0.145 840127.346 740726.6602 11.83162127 0.150 840127.346 766268.9588 8.791332352 0.155 840127.346 791811.2575 5.751043431 0.160 840127.346 817353.5561 2.710754509 0.165 840127.346 842895.8547 -0.3295344** 0.170 840127.346 868438.1533 -3.369823334 0.175 840127.346 893980.452 -6.410112256 0.180 840127.346 919522.7506 -9.450401177 0.185 840127.346 945065.0492 -12.4906901 0.190 840127.346 970607.3479 -15.53097902 0.195 840127.346 996149.6465 -18.57126794 0.200 840127.346 1021691.945 -21.61155686 ** optimum smoothing parameter volume of sediment loss-in-transit at 10 randomly selected data points from the test data for (predicted gully volume and predicted dam volume) each, respectively. the results generated by the model for the predicted gully volume, the predicted dam volume, and the volume of sediment loss-in-transit including the final summation for each of the last three column variables are presented in table 11. from the 11 randomly selected points tested, the was calculated using the mathematical relationship of equation 15. hence, for a predicted gully volume of 97318.44 m3, a calculated (predicted) volume of 39530.46 m3 would be deposited in the dam. then, predicted volume loss-in-transit is therefore given as (97318.44 39530.46) m3 = 57787.98 m3. this shows that the proposed model is capable of generation output for every instance where predicted gully sediment volume and dam’s sediment volume exist. 4. conclusion this work was carried out to examine the capability of the anfis hybrid model to predict sediment volume supplied from upstream by two prominent gullies through the ikpoba river channel and the amount of sediment accumulated (gained), as much as can be accounted for at the ikpoba dam’s end. threeyear consecutive field campaigns at the gullies’ location and dam’s end enabled us to prepare, refine, normalise, and modify the input parameters for optimum model performance. we first applied the anfis model to reproduce the volume of sediment deposited in the dam as calculated and proposed a mathematical relationship that incorporated the anfis model 12 oladosu et al. / j. nig. soc. phys. sci. 5 (2023) 1028 13 figure 9. regression (left) and validation (right) plots table 11. predicted volume loss-in-transit from a random selection of 10 data points index rainfall (mm) slope (mm) depth (m) velocity (m/s) p. size (µm) volume (m3) predicted gully volume (m3) predicted dam volume (m3) volume loss-intransit (m3) 100 80.280 38.545 7.908 0.827 36.934 7672.287 7926.644 3219.779 4706.865 186 202.963 38.118 6.522 1.412 41.182 10492.945 11310.07 4594.116 6715.953 189 198.573 45.008 6.811 1.791 40.199 10129.345 10729.69 4358.367 6371.321 210 176.937 37.126 5.928 1.210 39.141 11084.715 10349.64 4203.994 6145.649 159 193.554 39.714 6.411 1.463 38.102 9951.850 10079.82 4094.394 5985.429 13 84.328 53.774 8.771 1.823 38.317 7544.943 7750.151 3148.088 4602.063 131 203.851 57.883 8.819 1.505 45.722 11620.630 12271.57 4984.675 7286.895 205 171.784 42.867 8.170 0.584 44.885 12359.551 11781.39 4785.563 6995.822 51 78.476 60.745 9.439 1.320 36.152 7370.700 7215.381 2930.866 4284.515 52 82.163 64.956 9.576 1.656 39.972 8256.199 7904.086 3210.616 4693.470 sum 97318.44 39530.46 57787.98 to generate output in form of a computed volume of sediment loss-in-transit as a function of the normalised gully sectional sediment loss and dam’s sediment accumulation. our investigations showed that the anfis model could estimate sediment deposited in the dam effectively while, at the same time, it was able to determine the volume of sediment loss-in-transit using the established mathematical equation. the best ann model with the least error and the most accurate predictive result was selected and presented. however, we made some assumptions. for example, we assume no silt (sediment) escapes from the dam by any means, (that is, the dam traps all the sediment behind it). the combined volume of sediment exited from the two gullies was treated as stagnant to be able to arrive at a numerical value. the volume of sediment deposited in the dam is treated as being localized (i. e. confined); such that numerical values can be obtained for location-based sediment volume actualisation. for these reasons, sediment transport equations were not incorporated. furthermore, rather than converting the estimated sediment volume to weight equivalent, we left it in meters cubed to save processing time and computational space. 4.1. recommendations according to our discoveries in this work, we recommend the following: due to the fast rate of siltation, there is an urgent need to dredge the dam. at present, sediment impact has rendered the ikpoba dam a failed infrastructure, hence we propose a holistic desilting or dredging for the rejuvenation of the dam to return it back to its 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[49] r. m. o’brien, “a caution regarding rule of thumb for variance inflation factor”, qual quant. 41 (2007) 673. 15 j. nig. soc. phys. sci. 5 (2023) 1083 journal of the nigerian society of physical sciences theoretical investigation of diameter effects and edge configuration on the optical properties of graphdiyne nanotubes in the presence of electric field t. m. j. abdulkadhima,∗, s. a. a. alsaatia, m. h. shinenb acollege of basic education, university of babylon, babylon 51002, iraq bcollege of science, university of babylon, babylon 51002, iraq abstract in this research, the structural, electronic and optical properties of the armchair (ant) and zigzag (znt) graphdiyne nanotubes (gdy-nt) with different diameters were studied based on density functional theory (dft). the computations were done using siesta code, based on linear combination of localized atomic orbitals (lcao) method and the generalized gradient approximation (gga). the results from the band structure analysis show that all these nanotubes are semiconductors with direct band gap at gamma point. the band gap of the nanotubes is clearly dependent on the nanotube diameter, and by increasing the nanotube diameter, the band gap is decreased. optical properties such as dielectric function; absorption coefficient, optical conductivity and refractive index were examined and calculated for all samples. the results show that all these functions have an inverse relationship with the nanotube diameter and a direct relationship with the band gap. the effect of applying the external electric field with intensity of 0.1 v/å, 0.2 v/å in the direction of x-axis (perpendicular to the nanotube axis) on the structural and electronic features of these nanotubes has been studied and calculated. doi:10.46481/jnsps.2023.1083 keywords: graphdiyne nanotubes, electric field, band gap, optical properties, siesta code, density functional theory article history : received: 22 september 2022 received in revised form: 15 november 2022 accepted for publication: 26 november 2022 published: 02 march 2023 © 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: k. sakthipandi 1. introduction unlike natural structures of carbon that often have two stable allotropes of diamond and graphite, and sp3 and sp2 hybridized carbon atoms, respectively, a wide range of synthetic carbon allotropes have been synthesized or theoretically predicted. this has made the present day become known as the ∗corresponding author tel. no: +964 7817977099, +964 7816022376 email address: taqi.mohammed@uobabylon.edu.iq (t. m. j. abdulkadhim) era of carbon allotropes [1]. as the most prominent and most notable achievements, the discovery of fullerenes [2], carbon nanotubes [3], and graphene [4-6] represents the prominent and novel types of zero-, oneand two-dimensional carbon structures that have the carbon structures with sp2 hybridization. in addition, the tendency of carbon to form the three types of sp1, sp2 and sp3 hybridization states allows for the creation of multiple combinations of carbon allotropes by linking the carbon atoms with different hybridizations [7-9]. one of the most interesting families of carbon allotropes introduced in recent years is the graphyne family [8]. this car1 abdulkadhim et al. / j. nig. soc. phys. sci. 5 (2023) 1083 2 bon allotrope has a flat structure with a one atom thick singleplate lattice that is made by replacing some = c = c = bonds in graphene with -c≡cacetylene bonds. these structures have two non-equivalent types of carbon atoms: sp2 hybridized carbon atoms with three identical bonds and sp hybridized carbon atoms with two identical bonds [10]. in graphdiyne, a member of the graphyne family, there are also two sp and sp2 hybrids of carbon atom [11]. the sp2 hybridized carbon atoms form a hexagonal ring, which are connected together by di-acetylene sp connections. compared to graphene, graphdiyne has many interesting properties such as uniformly distributed holes, strong π bonding, controllable electronic features, lower density, high electrical conductivity excellent hardness and high temperature resistance that these properties are due to the sp and sp2 hybrids and the natural holes in its structure [11]. these plane carbon lattices with high number of π bond, density lower than that of graphene, and modifiable electronic properties are the promising materials in the nano-electronics industry. it is also believed that various forms of graphyne with multiple functionalities can compete with the older carbon structures such as fullerenes, carbon nanotubes or graphene and respond positively to the increasing need for carbon nanostructures. 2. computational method the atomic structure of nanotubes has been investigated by density functional theory (dft) approach and using siesta computational code based on lcao method and the gga-pbe approximation for the exchange-correlation functional of e-e interaction [12-13]. brillouin zone (bz) division was done by monkhorst–pack method. the optimum number of k points selected for the armchair graphdiyne nanotubes was 1 × 1 × 7 and for the zigzag graphdiyne nanotubes, it was 1 × 1 × 11. in the computations, the optimal cutoff energy used for the armchair graphdiyne nanotubes was 300 ry and for the zigzag graphdiyne nanotubes, it was 400 ry. a large enough vacuum layer (10 å and 12 å for the armchair and zigzag nanotubes, respectively) is considered for all nanotubes to avoid interacting with adjacent structures. all these nanotubes are relaxed until the total atomic force value reaches less than 0.01ev/å. the optimized lattice constant obtained for the armchair and zigzag nanotubes were 16.71å and 9.46å, the dimeter of znt2, znt3, znt4, ant2, ant3 and ant4 after relaxation is obtained equal to 10.25, 15.56, 20.81, 6.42, 9.08 and 12.04, respectively. in order to perform the optical computations, the energy range in which the optical properties are investigated was selected from zero to 40ev. in the optical computations, the spinorbit interaction is neglected because it will not have much impact on the results. in this section, one of the important parameters is the kind of the optical polarization. in all structures studied, the polarized light and its propagation along the x direction were considered. the optical properties such as the real part of the dielectric function, absorption coefficient, refractive table 1. the optimized bond lengths of the znt and ant graphdiyne nanotubes after relaxation optimized bonding length (å) bonding type 1.40 =c − c ≡ 1.43 =c = c = 1.35 = c − c ≡ 1.24 − c ≡ c −− index and reflection coefficient were calculated using the imaginary part of the dielectric function and the kramers–kronig relations [14]. in the final step, the efficacy of applying the electric field in the direction of x-axis, with the sizes of 0.1 v/å, 0.2 v/å on the structural and electronic features of the zigzag and armchair graphdiyne nanotubes was studied. 3. results and discussion 3.1. investigating the structural properties similar to carbon nanotubes, graphdiyne nanotubes are produced by rolling the graphdiyne sheet (plate). similar to carbon nanotubes, the denomination (n, m) can be used for graphdiyne nanotubes. however, unlike carbon nanotubes, (n, n) represents the zigzag type and (n, 0) represents the armchair type. nanotube maker software was used to obtain the coordinates of points forming nanotubes [14]. in the study, the zigzag (2, 2), (3, 3), (4, 4) and armchair (2, 0), (3, 0), (4, 0) graphdiyne nanotubes were investigated. figures 1 and 2 show the structure of the znt and ant nanotubes examined after optimization, respectively. the bond lengths of nanotubes after the structural relaxation are presented in table 1. the values obtained are in agreement with the results of other researchers [15]. the length of carbon-carbon bonds in the hexagonal ring and the di-acetylene bonds are not the same, indicating the existence of different hybrids in the carbon bonds of graphdiyne nanotubes. this difference leads to more structural flexibility for graphdiyne nanotubes compared to carbon nanotubes [16-17]. another important difference between graphdiyne nanotubes and carbon nanotubes is that gdy-nt have the uniformly distributed holes in their walls, which facilitates the electron transport in the nanotube wall and it is important for the applications such as hydrogen storage. 3.2. electronic properties of gdy-nt the band structure of the znt and ant graphdiyne nanotubes is shown in figures 3 and 4, respectively. the fermi energy was selected at the zero point. all these gdy-nt are semiconductors with the direct band gap at gamma point of the first bz. the values of band gap and diameter of the zigzag and armchair nanotubes are presented in table 2. similar to carbon nanotubes, the diameter of graphdiyne nanotubes can be obtained from the following relationship: d = a p √ n2+nm+m2 2 abdulkadhim et al. / j. nig. soc. phys. sci. 5 (2023) 1083 3 figure 1. structure of the zigzag graphdiyne nanotubes after relaxation that a is the lattice constant of gdy-nt [5]. as shown in these figures, because of the quantum confinement effect, the band gap value of the znt and ant nanotubes is obviously dependent on the nanotube diameter, and decreases with increasing the nanotube diameter. in addition, the armchair nanotubes have a smaller diameter and the larger band gap than the zigzag nanotubes. figures 5 and 6 show the density of the total states (dos) of the znt and ant graphdiyne nanotubes, respectively. the dos at fermi energy is zero, which confirms the semiconducting behavior of gdy-nt. examining the band structure and the dos, we observe that as the diameter of the gdy-nt increases, the band gap decreases and the dos increases. 3.3. optical properties of nanotubes 3.3.1. dielectric function the complex dielectric function describes the optical properties of solids. this function is used to describe the crystal response to electromagnetic fields, which depends on the band structure of crystal [18]. in figures 7 and 8, the diagrams of the real and imaginary parts of the dielectric function figure 2. structure of the armchair graphdiyne nanotubes (double wall) after relaxation table 2. the value of band gap and diameter of the znt and ant graphdiyne nanotubes nano tube diameter (å) band gap (ev) structure 10.25 0.65 znt2 15.56 0.55 znt3 20.81 0.50 znt4 6.42 0.95 ant2 9.08 0.65 ant3 12.04 0.55 ant4 figure 3. band structure of the znt graphdiyne nanotubes of the graphdiynenanotubes are plotted for the polarized incident light in the direction of x-axis (perpendicular to the axis of graphdiynenanotubes), respectively. the imaginary part of the dielectric function partly reflects the actual transfers between occupied and unoccupied states. we also know that the interband transition is due to excitation at the absorption edges, and the intra-band transition is the result of volumetric plasmons (absorption by free electrons). therefore, in diagram 7, the relatively high increase in energy values less than 5ev in the 3 abdulkadhim et al. / j. nig. soc. phys. sci. 5 (2023) 1083 4 figure 4. band structure of the armchair graphdiyne nanotubes figure 5. dos of the zigzag gdy-nt imaginary part of the dielectric function, as well as the almost constant behavior at energies above 30ev confirm these absorption approaches. comparison among different nanotubes with various diameters shows that the highest dielectric function intensity is related to agdynt (2, 0) nanotube which has the smallest diameter and the largest gap, and the lowest dielectric function intensity is related to zgdynt (4, 4) nanotube that has the largest diameter and the smallest gap. the results show that the dielectric function intensity is inversely related to the diameter of the nanotubes and there is a direct relationship between the dielectric function intensity and the band gap. figure 6. dos of the armchair gdy-nt figure 7. real part of the dielectric function of graphdiynenanotubes for the polarized light in the direction of x-axis 3.3.2. optical absorption the absorption spectrum of gdy-nt with different diameters is illustrated in figure 9. the figure shows the permitted optical transitions of the electron between the empty states of the conduction band and the full states of the valence band. the threshold energy for transition in the absorption spectrum corresponds to the gap size in the band structure of graphdiynenanotubes and it means that the electrons are excited and make a transient by receiving the least energy greater than the band gap value. the comparison between different nanotubes with various diameters also shows that the highest absorption is related to 4 abdulkadhim et al. / j. nig. soc. phys. sci. 5 (2023) 1083 5 figure 8. imaginary part of the dielectric function of graphdiynenanotubes for the polarized light in the direction of x-axis figure 9. the diagram of absorption spectra of graphdiynenanotubes with different diameters agdynt (2, 0) nanotube which has the smallest diameter and largest gap, and the lowest absorption is related to zgdynt (4, 4) nanotube which has the largest diameter and the smallest gap. the results show that the optical absorption coefficient is inversely related to the diameter of the nanotubes and is directly related to the band gap. 3.3.3. optical conductivity figure 10 shows the changes of optical conductivity of graphdiynenanotubes with different diameters compared to the incident photon energy. as can be seen, the diagram of optical conductivity is proportional to the imaginary part of the dielectric function. generally, at the energies that present in the imaginary part of the peak dielectric function, these peaks are also seen in the real part of the optical conductivity. in this spectrum, the conductivity increases up to 20ev and then decreases, and at energies with the highest absorption, we have the optical conductivity maximum. the results show that the optical conductivity has an inverse relationship with the diameter of the nanotubes and a direct relationship with the band gap. 3.3.4. reflection the reflection diagram of gdy-nt with different diameters is plotted in figure 11. according to this diagram, we find that the magnitude of the reflection is generally very small at the energies above 6ev. we also have the maximum reflection at the figure 10. the diagram of optical conductivity of gdy-nt with different diameters for the polarized light in the direction of x-axis figure 11. reflection diagram of graphdiynenanotubes with different diameters energies where the highest absorption occurs. here, a comparison of the reflection coefficient of the nanotubes with different diameters shows that the reflection coefficient is inversely related to the diameter of the nanotubes and directly related to the band gap. 3.3.5. refractive index the value of the refractive index at zero energy is called the static refractive index, which is equal to the static dielectric constant. the static refractive index of the znt and ant graphdiynenanotubes with different diameters is reported in table 3. if, by increasing the frequency, the refractive index increases, this behavior is called the normal dispersion, which is the usual behavior of all transparent materials. in areas where the slope of the diagram of refractive index is negative, it is related to absorption. in this area, when passing through the material, longer wavelength light breaks more than short wavelength light, which this behavior is called the anomalous dispersion. the diagram of refractive index and extinction coefficient of gdynt with different diameters are shown in figures 12 and 13, respectively. as you can see in these diagrams, the refractive index reaches its lowest value at the energies between 4ev and 6ev, so the reflection increases, which is consistent with figure 13. 5 abdulkadhim et al. / j. nig. soc. phys. sci. 5 (2023) 1083 6 table 3. static refractive index of graphdiynenanotubes ant2 ant3 ant4 znt2 znt3 znt4 1.68 1.64 1.5 1.62 1.64 1.57 figure 12. diagram of refractive index of gdy-nt with different diameters figure 13. the diagram of extinction coefficient of gdy-nt with different diameters 3.4. effect of applying the external electric field in the final step, the electric field along the x-axis with the sizes of 0.1v/å and 0. 2v/å was applied to the znt and ant graphdiynenanotubes with different diameters. the geometric structure and band structure of the graphdiyne nanotubes were compared in the absence and presence of the external electric field. the results show that by applying the electric field in the direction of the x-axis and with different sizes to the ant2 nanotube, its geometric structure and band structure is not changed (figure 14). for the ant3 nanotubes, as shown in figure 15, there is little change in the energy bands in the presence of the electric field of 0.1v/å, and only slight shift (about 0.02 to 0.05ev) is seen in the energy bands. in addition, the size of the band gap is not changed. but by applying the electric field of 0.2v/å, the band structure is completely changed and the size of band gap increased (figure 15). in addition, by increasing the external electric field, the diameter of ant3 nanotube is also increased. figure 16 shows the band structure of ant4 nanotubes in the presence and in the absence of the external electric field. in the presence of the electric field of 0.1v/å, the size of band gap is not changed. but greater shift is seen in the energy bands comfigure 14. comparison of the band structure of ant2 nanotubes in the absence (e=0) and presence of electric field (e=0.1 and 0.2v/å) along the x-axis figure 15. comparison of the band structure of ant3 nanotubes in the absence (e=0) and presence of electric field (e=0.1 and 0.2v/å) along the x-axis pared to the ant3 nanotube (about 0.1ev). by applying the electric field of 0.2v/å, the band structure is completely changed and the size of band gap increased. the diameter of ant4 nanotube is also increased by applying the electric field. the results are summarized in table 4. the band structure of znt2 nanotubes in the absence and presence of the external electric field is compared in figure 17. as can be seen in the figure, by applying the electric field of 0.1v/å, there is little change in the energy bands. the size of the band gap and the nanotube diameter also show a slight 6 abdulkadhim et al. / j. nig. soc. phys. sci. 5 (2023) 1083 7 table 4. the size of band gap (eg), shift of energy bands and diameter (d) of the armchair graphdiyne nanotubes with different diameters in the presence of the external electric field in the direction of x-axis and with the sizes of 0.1v/å and 0.2v/å shift (ev) d (å) eg (ev) structure e=0.2 e=0.1 e=0.2 e=0.1 e=0 e=0.2 e=0.1 e=0 0 0 5.8 5.8 5.8 6.42[3,4] 0.78 0.78 0.78 0.95[3,4] ant2 structure changed 0.020.05 9.2 9.07 9.04 9.08[3,4] 0.87 0.66 0.66 0.65[3,4] ant3 structure changed 0.1 12.94 11.94 11.65 12.04[3,4] 0.65 0.57 0.57 0.55[3,4] ant4 figure 16. comparison of the band structure of ant4 nanotubes in the absence (e=0) and presence of electric field (e=0.1 and 0.2v/å) along the x-axis decrease. as the external electric field intensity increases, the band structure is changed and the band gap is also decreased, but the nanotube diameter is slightly increased. the band structure of znt3 nanotube in the presence and in the absence of the external electric field is plotted in figure 18. applying the electric field of 0.1v/å, there is little change in the energy bands. the size of the band gap and the nanotube diameter also show a slight decrease. as the intensity of the external electric field increases, there is a slight shift in the band structure (0.04ev). the size of the band gap and the nanotube diameter is also slightly decreased. in figure 19, the band structure of znt4 nanotube is plotted in the absence and presence of the external electric field. by applying the electric field of 0.1v/å, a shift of about 0.02ev is seen in the energy bands. the band gap value and the nanotube diameter also show a slight decrease. applying the electric field of 0.2v/å, the geometric structure and band structure of the nanotubes are completely changed. as shown in figure 20, the shape of zn4 nanotube has become oval and the size of its band gap has decreased in the presence of e= 0.2v/å. the results are summarized in table 5. given that the electric field is applied perpendicular to the axis of nanotubes, the atoms in the wall of the nanotubes have figure 17. comparison of the band structure of znt2 nanotubes in the absence (e=0) and presence (e=0.1 and 0.2v/å) of the electric field along the x-axis the different potentials. the larger the nanotube diameter, the potential difference is also greater. therefore, we expect that the geometric structure and band structure of nanotubes that are larger in diameter undergo more changes compared to nanotubes that are smaller in diameter, which this is consistent with the results obtained. the znt4 nanotube, which has the largest diameter among all nanotubes investigated, exhibits the most changes in the geometric structure and band structure [19-20]. no research was conducted to study the effect of applying the external electric field on gdy-nt for comparison. the results on other carbon nanostructures show that the effect of applying the external electric field on the armchair graphdiynenanoribbons with various widths is different. small width does not change the size of band gap. in larger widths (n> 13), by increasing the electric field intensity, the size of band gap is decreased. by applying the electric field to the (10, 5) carbon nanotube in the direction of x-axis, the size of band gap is first increased and then decreased [21-22]. 4. conclusion in the study, the zigzag (2, 2), (3, 3), (4, 4) and armchair (2, 0), (3, 0), (4, 0) graphdiyne nanotubes were investigated. the results from the investigating the carbon-carbon bond lengths in graphdiyne nanotubes are in agreement with those of other researchers on graphdiyne nano-ribbons and nanotubes. all these nanotubes are semiconductors with direct band gap at gamma point of the first brillouin zone. because of the quantum confinement effect, the value of band gap of the znt and 7 abdulkadhim et al. / j. nig. soc. phys. sci. 5 (2023) 1083 8 table 5. the size of band gap (eg), shift of energy bands and diameter (d) of the zigzag graphdiyne nanotubes with different diameters in the absence and presence of the external electric field along the x-axis and with the sizes of 0.1v/å and 0.2v/å shift (ev) d (å) eg (ev) structure e=0.2 e=0.1 e=0.2 e=0.1 e=0 e=0.2 e=0.1 e=0 structure changed 0.0 10.33 10.286 10.287 10.25[3,4] 0.42 0.63 0.64 0.65[3-5] znt2 0.04 0.0 15.6 15.542 15.541 15.56[3,4] 0.53 0.54 0.55 0.55[3,4] znt3 structure changed 0.02 12.4*24.93 20.803 20.808 20.81[3,4] 0.34 0.50 0.51 0.50[3,4] znt4 figure 18. comparison of the band structure of znt3 nanotubes in the absence (e=0) and presence (e=0.1 and 0.2v/å) of the electric field along the x-axis figure 19. comparison of the band structure of znt4 nanotubes in the absence (e=0) and presence (e=0.1 and 0.2v/å) of the electric field along the x-axis ant nanotubes is obviously dependent on the nanotube diameter, and by increasing the nanotube diameter, it decreases. in addition, the armchair nanotubes have a smaller diameter and the larger band gap compared to the zigzag nanotubes. as the diameter of the nanotubes increases, the band gap decreases and the dos increases around the fermi level. this means that the electrons require less energy to transition from the valence band to the conduction band. the optical properties of all nanotubes above-mentioned were calculated and investigated using siesta package and based on the kramers–kronig relations. dielectric function, absorption coefficient, optical conductivity, reflection, refractive index and extinction coefficient of all samples were calcufigure 20. geometric structure of zigzag graphdiyne nanotube (znt4) in the presence of the electric field of 0.2v/å after relaxation lated. the results show that all these functions are inversely related to the diameter of the nanotubes and are directly related to the band gap, so that the highest intensity is related to the armchair (2, 0) nanotubes and the lowest intensity is to the zigzag (4, 4) nanotubes. in addition, the effect of external electric field on the geometric structure and band structure of the ant and znt graphdiyne nanotubes with different diameters is also studied. the results show that by applying the electric field of 0.1 v/å along the xaxis to the armchair and zigzag graphdiyne nanotubes, there is little change in the geometric structure and band structure. but, in the presence of the electric field of 0.2 v/å, the band structure of the zigzag gdy-nt is changed and the size of band gap also dramatically decreased, with the most change seen in the znt4 nanotube, which becomes oval. in the presence of the electric field of 0.2 v/å in the armchair nanotubes, the size of band gap and diameter of ant3 and ant4 nanotubes are increased and the band structure of these nanotubes also completely changed. references [1] a. hirsch, “graphene is probably the only system where ideas from quantum field theory”, nat mater 9 (2010) 868. 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[20] l. yu-pin, t. li-gan, t. chuen-horng, l. ming-hsien & l. feng-yin, “effect of vacancy defect on electrical properties of chiral single-walled carbon nanotube under external electrical field”, chin. phys. b 20 (2011) 017302. [21] e. p. onokare, l. o. odokuma, f. d. sikoki, b. m. nziwu, p. o. iniagh & j. c. ossai, “physicochemical characteristics and toxicity studies of crude oil, dispersant and crude oil– dispersant test media to marine organism”, journal of the nigerian society of physical sciences 4 (2022) 427. [22] i. l. ikhioya, eli danladi, o. d. nnanyere, a. o. salawu, “influence of precursor temperature on bi doped znse material via electrochemical deposition technique for photovoltaic application”, journal of the nigerian society of physical sciences 4 (2022) 502. 9 j. nig. soc. phys. sci. 3 (2021) 89–95 journal of the nigerian society of physical sciences investigation of groundwater contamination from akanran open waste dumpsite, ibadan, south-western nigeria, using geoelectrical and geochemical techniques j. o. cokera,∗, a. a. rafiub, n. n. abdulsalamc, a. s. ogungbed, a. a. olajidea, a. j. agbelemogea adepartment of physics, olabisi onabanjo university, ago iwoye, ogun state, nigeria bdepartment of physics, federal university of technology, minna, niger state, nigeria cdepartment of physics, university of abuja, nigeria ddepartment of physics, lagos state university , nigeria abstract in recent times, large waste is produced especially in an urban area due to population with careless handling which calls for worries. hence, the study determines the effect of akanran dumpsite on the groundwater quality for drinking and domestic purposes. it employs the electrical resistivity and geochemical methods. the wenner configuration was adopted with constant electrode separation ranging from 5 to 25 m to acquire five profiles within and outside the dumpsite and processed using diprowin 4.01 software. three (3) soil and water samples were collected and analysed. the 2-d pseudosection revealed a very low resistivity value of 9.2 ohm-meter and is suspected to be leachate infiltration which migrates to a depth of 7 m. the results of soil analysis show that clay ranges between 9.61 18.8 %., silt between 9.27 – 19.7 % and an average bulk density of 1.48 (relatively high for a sandy loam) which suggests that infiltration of the leachate is minimal. the ph of the water sample analysis obtained is 6.9, suggesting acidic concentrates in the groundwater of the study area. however, this ph value for drinking water is within the permissible level of 6.5 – 8.5 indicating that the groundwater in the study area is suitable for drinking and also for other purposes. a nitrate level of 2.56 mg/l in the water sample falls within 50.0 mg/l, and nitrite level of 0.09 mg/l which is moderate when compared to the permissible level limit of 0.20 mg/l. the concentration of heavy metals in a hand-dug well sample from akanran dumpsite are zn (1.81 mg/l), cu (0.38 mg/l), cr (0.003 mg/l) which are within the permissible level limit and pb (0.21 mg/l) which recorded high metal concentration which may suggest that the dumpsite contain waste metals which may leach down the soil. in conclusion, the groundwater in the area of the survey is safe for drinking, domestic purpose and agricultural use; and there is high toxicity level and possible leaching of lead. doi:10.46481/jnsps.2021.166 keywords: contaminants, geochemical, leachate, resistivity, dumpsite article history : received: 11 february 2021 received in revised form: 10 may 2021 accepted for publication: 13 may 2021 published: 29 may 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: o. j. abimbola ∗corresponding author tel. no: +2347032261146 email address: coker.joseph@oouagoiwoye.edu.ng (j. o. coker) 1. introduction to get rid of waste generated, many dumpsites around the country have been overfilled and causing environmental pollution. contamination of groundwater is a major issue in a dumpsite, as leachate may infiltrate the aquifer. the quantity and 89 coker et al. / j. nig. soc. phys. sci. 3 (2021) 89–95 90 the concentration of the toxic fluid (leachate and gases) produced depend on the age of the landfill site, types and amount of soil vegetation cover, the geology of the area, amount of precipitation, types of wastes, and the inflow of groundwater [1]. leachate from open dumpsites and landfills usually contain chemical and biological constituents [2]. the leachate flows downward and outward from the dumpsite and this downward flow may threaten springs at the periphery of the landfill [3]. as leachate moves downward, it mixes with the groundwater in between the pore-spaces within the soil, and this moves through the groundwater’s path as plumes of contaminated groundwater. leachate generated in a dumpsite is always associated with a high concentration of ions which results in very low resistivities of host rocks or formation [4]. therefore, the direct current resistivity method, induced polarization, and very low-frequency electromagnetic method are parts of the methods that could be used to effectively map leachate plume zones [5]. water pollutants come not only from urban and municipal wastewater discharges, but also from non-point sources, some of which are not perceived as such. most of the non-point sources have been initially recognised as such by groundwater experts [6] who realized that soil (urban or rural) was an important means of transporting pollution to ground and surface water through complex interactions [7]. various hazards are associated with the waste dumpsites, e.g. surface water contamination, groundwater contamination, bad smell or odour, release of greenhouse gases, accidental hazard caused by fire, slope instability, loss of vegetation, soil contamination and bird-hit [8]. similarly, many of the above mentioned hazards were associated with akanran dumpsite, such as bad odour, loss of vegetation and electronic waste. escherichia coli, better known as e. coli, are a common and diverse group of bacteria found in food, the environment, and in the intestines of both people and certain warm-blooded animals. while e. coli has a bad reputation, the truth is that most strains of e. coli are actually harmless. and some strains are essential to good health, according to the centers for disease control and prevention (cdc); e. coli produces vitamin k and vitamin b12, and maintains a protective space in your gut for other beneficial bacteria. resistivity imaging is relatively less expensive, non-invasive, and has proved very useful in environmental and engineering problems such as investigation of landfill sites and detection of bedrock contamination [9]. the integrated geophysical methods including the 2-d resistivity survey, vlf-em, and seismic survey have provided major tools in environmental studies [10]. leachate from the dumpsite always polluted the groundwater, most especially the hand-dug wells situated near the dumpsites, thereby putting the health of the people under threat. the impact of solid waste disposal on the groundwater within the vadose and saturated zone of two dumpsites namely abaeku and ajakannga in ibadan metropolis was investigated by [11] using vertical electrical sounding (ves) to determine the depth and thickness of subsurface layers. the result from the ves data gives a qualitative lithology of topsoil and clayey soil (aquitard) which has a low resistivity zone ¡ 100 ohm-m, underlain by higher resistive basement. a geophysical investigation involving 2d resistivity survey on aba eku dump site in southwestern nigeria was carried out by ganiyu et al. [12] with a view to map the conductive leachate plume and extent of migration of leachate plumes in the subsurface for possible groundwater contamination. the 2d resistivity survey was carried out using wenner array configuration of electrode spacing distance ranging from 5 – 25 m. the inverse resistivity models of the subsurface from 2d imaging revealed low resistivity value less than 10 ωm suspected to be leachate infiltration. the extent of leachate migration was up to a depth of about 4 m, an indication that the underlying layer has a greater risk of contamination by leachate and decomposed solid waste materials. in this study, electrical resistivity and geochemical studies were carried out on an open waste dumpsite in aba-eku (akanran), ibadan, to ascertain the impact of leachate on the groundwater system. the specific objectives are; to examine the health implication of the leachate generated in the landfill on the inhabitants living near the study area through geochemical analyses of water and soil samples around the study area and to evaluate the groundwater conditions for drinking and domestic purposes. 1.1. location and the geologic settings of the study area ibadan is africa’s largest indigenous settlement located in southwestern nigeria with a population of over 2.6 million people according to the 2006 national census survey. ibadan lies approximately on longitude 3o 35 ′ to 4o 10 ′ e and latitude 7o 20 ′ to 7o 40 ′ n at about 145 km northeast of lagos. in ibadan city, solid waste generation is on the increase. the city currently generates about 1,618,293 kg of solid waste daily with about 10 % of this evacuated by the oyo state waste management authority [12] the two local climates experience in ibadan are the dry season from november to february and the rainy season which runs from march to october, and the temperature ranges between 21 oc and 35 oc. ibadan is a part of the basement complex of the geological setting of southwestern nigeria (fig. 1). the basement complex rocks consist of few intrusions of porphyries of the jurrasic age, the metamorphic rocks of precambrian age, and granites. aba eku is mainly underlain by the quartzite and quartz schist (fig. 1). the common rocks in many parts of ibadan are the weathered regoliths. in ibadan, over 75 % of the rocks are banded gneiss while augen gneisses and quartzites share the remaining in about equal percentages [13, 14]. 2. materials and method the 2-d resistivity technique is widely used in the investigation of contaminants at both the saturated and unsaturated zones [16]. to take a measurement, current was injected through the outer current electrode pair into the ground which generated a potential difference measured by the inner potential electrodes. 90 coker et al. / j. nig. soc. phys. sci. 3 (2021) 89–95 91 figure 1. a generalized geological map of ibadan showing akanran dumpsite [15]. two dimensional (2d) electrical resistivity surveys were carried out within the dumpsite using campus tigre model resistivity meter. four (4) profiles (p1, p2, p3, and p5) were mapped out within the refuse dumpsite and a control profile (p4) using the wenner array configuration. fig. 2 shows the data acquisition map for akanran dumpsite. the electrode separation distance for each traverse ranged from a = 5 m to a = 25 m with a station interval of 5 m. the traverse length ranged from 100 m to 150 m. the wenner array has the strongest signal strength when compared with other arrays and its geometric factor is 2πa [12]. the acquired field data was processed using the diprofwin software. the software iterated the data before giving out the final resistivity structure. soil influences the rate of leachate migration hence; in this study, soil samples were taken from the dumpsite and subjected to sieve analysis to determine its textural characteristics and physical properties. the in-situ soil samples were collected at three different points within the study area and another three samples were collected outside the dumpsite serving as the control. at each sample point, two samples were collected from 020 cm and 20-40 cm depths using a soil auger and a cylindrical core of diameter 5 cm and height 5 cm. the collected soil samples were allowed to pass through 2.00 mm sieve to remove any plant materials after been air-dried and a modified boyoucos hydrometer method described by grossman and reinsch [17] was used to determine the properties of sand, silt, and clay. the soil textural class was estimated using a textual triangle. bulk density [17] was evaluated by using equations 1, 2 and 3 pb = ms(g) vb(cm3) (1) where pb is the soil bulk density (gcm−3); ms = the mass of oven-dried soil (g) and vb = the volume of the soil/cylindrical core (vb = πr2h; r being the internal diameter and h the height figure 2. map of akanran dumpsite showing the distributions of the 2d electrical resistivity profiles, water sample, and soil sample points. of the cylindrical core). total porosity (tp) is: tb = [ 1 − pp ps ] × 100 (2) where t p = the total porosity (%); pb = the soil bulk density (mgm−3); ps = the soil particle density with a constant value of 2.65mgm−3. according to reynolds [5] and transposed darey’s equation for a vertical flow of liquid: ks = ql ta(∆h) (3) where q = the volume of water that flows through the soil column at equilibrium; a is the cross-sectional area of flow (soil core) through the soil column; t is the time interval (hr.); l is the length of soil column (cm) and ∆h is the hydraulic head (cm). ∆h = l + hw where hw is the head of water above the soil column. 50 g air-dry soil (¡ 2 mm) was weighed into a 100 ml glass beaker and 50 ml distil water added by the volumetric flask. the mixture was done in a glass rod and stirred every 10 minutes. the ph meter was calibrated and the combined electrodes were put in suspension (about 3 cm deep) and read after 30 seconds with one decimal. water samples were collected at three different points and analysed to determine the e. coli, total organic carbon, total dissolved solids, nitrate, nitrite, cod, ph, sodium (na), sulphate (s o4), chromium (cr), copper (cu), lead (pb) and zinc (zn). apparatus which include ph meter with combined electrode, beakers: polyethylene, tfe beakers, plastic, wash bottle, reagents were used. total dissolved solids are given in equation 4 t otal dissolved s olid s = ( mg l ) wt(d+s) − wt(d) v ×1000(4) 91 coker et al. / j. nig. soc. phys. sci. 3 (2021) 89–95 92 where wt(d+s) = weight of dish plus solids (mg), wt(d) = weight of dish before use (mg), s = solids (mg) and v = volume of water sample used for measurement (ml). 0.5-1.0 g air-dry soil (0.15 mm) was weighed into a 300 ml calibrated digestion tube, 3 ml concentrated hno3 added, and swirled, and then the tubes were placed in the rack. the heavy metals such as cu, zn, fe, mn, pb, cr, cd, ni, and co were determined using atomic absorption spectrophotometer. all statistical analyses were conducted using the replicated data collected for water and soil samples. using the duncan multiple range test (dmrt) at 5 % probability, the mean of soil samples was differentiated. 3. results and discussion the inverted 2-d resistivity pseudosections are hereby presented in figures 3 to 7. profile 1 (fig. 3) shows the layers of the subsurface, the topsoil has a low resistivity values ranging between 3 to 11 ωm at depths 0 to 5 m. figure 3. 2-d electrical resistivity pseudosection for profile 1. the thin weathered layer which underlies the first layer has true resistivity value between 22 76 ωm indicating clayey material. the last layer has resistivity values between 76 373 ωm at depths 10 – 25 m. although between the distances of 25 m 45 m exists a depression to a depth of about 16 m which could be a fractured zone. the uneven nature of the top of the basement rock is due to uneven weathering around the area investigated. figure 4 presented profile 2 indicating extensive contamination of topsoil (regolith) from the ground surface to 5 m as the resistivity of the topsoil is less than 100 ωm over the horizontal extent. an intermediate resistivity layer occurred at the intermediate depth zone between 5 and 10 m indicating the presence of lateritic clay at the topsoil (regolith) with the resistivity ranging from 100 – 141 ωm. a thick layer with a comparatively high resistivity above 423 ωm was also observed below the layers of low to intermediate resistivity rocks. bedrock resistivity is higher 400 ωm at shallow depth. the contaminant here is limited to the topsoil because the basement is close to the surface so; the plume cannot migrate to the subsurface but can flow within the topsoil and migrate to another location. figure 5 shows the inverted 2d electrical resistivity section for profile 3 conducted north-western part of the dumpsite showed that the topsoil is extensively contaminated at a horizontal distance 10 45 m with resistivity values lower than 22 figure 4. 2-d electrical resistivity pseudosection for profile 2. ωm to a depth of about 2 m. there is evidence of low resistive region inferred as clayey material at the horizontal distances of 11 90 m, at a depth of 5 m to 6 m, and according to urish [18], high contaminant is suspected. an intermediate zone with resistivity range from 55 to 200 ωm at a depth of 6 and 15 m which is part of the weathered zone but it is lateritic the contaminant plume may not be able to seep through it because of its impermeable nature. the resistivity value of 232 ωm was observed at a depth of 20 m which indicates fresh rock material without any structural pattern like faults. figure 5. 2-d electrical resistivity pseudosection for profile 3. figure 6 shows profile 4 the control, with a low resistivity value between 11 ωm and 20 ωm at the surface with a thin depth at a horizontal distance of 30 – 45 m and 105 m to 125 m. the uppermost layer is underlain with resistivity range values from 50 to 170 ωm at a depth of 4 to 7 m which is part of the weathered zone but it is lateritic. the compacted lateritic clay outcrop at the surface between 55 m and 110 m as observed in the field during data acquisition. the compacted lateritic clay will not allow the plume to percolate through it to affect the subsurface beyond the surface layer. any plume around this area will flow to nearby streams or to another region where it can percolate to the subsurface. a very high resistivity value of 400 ωm and above, at a depth of 12 m was recorded and interpreted as a fresh rock material. this is so because, this profile is 200 m far away from the dumpsite unlike other profiles that are within the dumpsite. the contaminant at profile 2 is limited to the topsoil; a high contaminant is suspected at profile 3, in profile 5, the topsoil has very low resistivity values due to contamination from the plume that seeps out of the waste and saturated it. this shows a significant water quality variation with increasing distance from the dumpsite. 92 coker et al. / j. nig. soc. phys. sci. 3 (2021) 89–95 93 figure 6. 2-d electrical resistivity pseudosection for profile 4. figure 7 presented profile 5 with resistivity values between 16 – 40 ωm at the topsoil. the topsoil has very low resistivity values due to contamination from the plume that seeps out of the waste and saturated it. underlies this layer is a zone with resistivity range from 50 to 200 ωm at a depth of 5 to 9 m which is part of the weathered zone but it is lateritic. the compacted lateritic clay serves as seals for the subsurface aquiferous zone if exist around the area and will not allow percolation of the plume to the subsurface. a very high resistivity value of 240 ωm, at a depth of 10 m was recorded and interpreted as a fresh rock material. resistivity distributions observed at greater depth are high when compared with low resistivity values observed at the surface. the uneven surface of the basement rock is as a result of weathering which has affected the hard rock. figure 7. 2-d electrical resistivity pseudosection for profile 5. results of the geochemical method are discussed below starting with the presentation of soil analysis as shown in table 1. 3.1. discussion on geochemical analysis 3.1.1. soil sample analysis the results of the sieve analysis show that the dominant lithology was sand. as depicted in table 1, the soil at akanran on-the-spot (onsite) has 66.6 % of sand while the soil at location away from the site (offsite) which serves as the control has 81.1 % of sand indicating an abundance of loosely soil particles. the silt content for onsite is 14.6 % while the offsite is 9.27 %. the clay content shows 18.8 % onsite and 9.61 % offsite, a relatively low content interpreted as low protective capacity which confirmed easy migration of the plumes/leachate. other properties include soil ph, total porosity, bulk density, and hydraulic conductivity. the mean ph value of the soil samples for onsite is 6.4 (greater than that of the control) near the values of the work of ewemoje et al. [19] with ph values ranging between 6.5 and 7.5. the mean ph value recorded for offsite in this work is 5.9. as the sand becomes increasingly clayey with depth at the offsites which stand for the control, fluids in the intergranular pores spaces are displaced, thereby increasing the bulk density with an average value of 1.62 g/cm3 indicating a reduction in permeability from the topsoil down the depth but at the onsite, the bulk density is lower with average values of 1.37 g/cm3. therefore, with 9.61 18.8 % clay, 9.27 – 19.7 % silt, and an average bulk density of 1.48 (relatively high for a sandy loam) one can say that infiltration of the leachate will be minimal. the soil porosity at the onsite and offsite of the dumpsite ranges from 39 to 48.2 % which could be considered high but, that of the offsites (control) is higher. the hydraulic conductivity of 2.58 cm/hr at the onsite and also 3.0 cm/hr at the offsites are considerably low but, that of the control is higher because of the distance is far away from the dumpsite. 3.1.2. water sample analysis table 2 shows the results of the presentation of water analysis. the concentration of enterococci levels (e coli) (cfu/100 ml) in the water samples of akanran dumpsite is shown in table 2 (0.45 ml) which exceeds the permissible level (20) limit of 0.00 ml and indicating the presence of an anomaly in the body of water at minimal concentrations but, for the control sample, the value is less (0.03 ml). the ph of the water sample analysis obtained is 6.9, suggesting acidic concentrates in the groundwater of the study area. however, this ph value fall within the permissible level (20) of 6.5 – 8.5 for potable water shows that the groundwater in the study area is suitable for drinking although the control sample value is 6.45. the total organic carbon (toc) with a value of 2.14 mg/l is less than the control sample value of 0.67 mg/l and nis, 2007 recommended value of 5.00 mg/l. the total dissolved solid (tds) is 224.1 mg/l. while that of the control sample is half of the value (112 mg/l) and the permissible level (20) limit of 500 mg/l, hence the water is suitable for drinking and also for domestic purposes (fig. 8). nitrate level in the study area is 2.56 mg/l and it leaches into the groundwater in the form of nitrate. nitrate is not a threat to health issues unless it is reduced to nitrite of high concentrations but in this study, nitrate levels in the water samples are within the permissible level (20) limit of 50.0 mg/l and the nitrite level was 0.09 mg/l (greater than the control sample value of 0.03) which is moderate when compared to the nis permissible level (20) limit of 0.20 mg/l. this suggests that the groundwater in the area of the survey is safe. the chemical oxygen demand (cod) of 167.1 mg/l is higher than the permissible level (20) of 150 mg/l and does pose a minimal threat. but, the control sample value of 92.0 mg/l is less than the permissible level and is very safe water quality in the control sample area because, it is far from the dumpsite location. 93 coker et al. / j. nig. soc. phys. sci. 3 (2021) 89–95 94 table 1. results of soil analysis [treatments with the same letter(s) in each property do not differ significantly (p < 0.05); ns indicates not significant at a 5 % level. cv %: coefficient of variation]. treatment ph sand silt clay bd tp ks (1:1 h2o) (%) (%) (%) g/cm3 (%) (cm/hr) akanran onsite 6.4 66.6 14.6 18.8 1.37 48.2 2.58 akanran offsite 5.9 81.1 9.27 9.61 1.62 39.0 3.00 depth 0 2 cm 6.3 ns 70.5 ns 14.2 ns 15.3 ns 1.48 ns 44.0 ns 2.36 ns 20 40 cm 6.2 70.9 13.6 15.5 1.48 44.2 2.16 cv% 4.0 3.0 20.4 12.5 1.2 1.5 24.6 table 2. results of water analysis [lsd: least significant difference]. test elements hand dug control lsd permissible (mg/l) wells value (p < 0.05) level (20) e. coli (cfu/100 ml) 0.45 0.03 1.61 0.00 total organic carbon 2.14 0.67 3.06 5.00 total dissolved solids 224.1 112 314.9 500.0 nitrate (no3) 2.56 1.22 2.90 50.0 nitrite (no2) 0.09 0.03 0.14 020 chemical oxygen demand 167.1 92 217.9 < 150.0 ph 6.9 6.45 1.2 6.5 8.5 sodium (na) 3.44 2.43 1.98 200.0 sulphate (s o4) 4.88 3.35 1.71 100.0 zinc (zn) 1.81 1.24 1.34 3.00 copper (cu) 0.38 0.08 0.57 1.00 lead (pb) 0.21 0.02 0.76 0.01 chromium (cr) 0.003 0.001 0.006 0.005 figure 8. the mean values for tds, nitrate, cod, sodium, and sulphate against nis values. the concentrations of sodium ion (na+) and sulphate ion (s o2−4 ) for akanran dumpsite is 3.44 and 4.88 mg/l and relatively low when compared to their permissible level limit of 200 and 100 mg/l, respectively [20]. sodium saturated soils are not healthy to plant growth according to davis and de wiest [21]. the concentration of heavy metals in well sample from akanfigure 9. the mean values for e-coli, toc, nitrite, ph, zinc, copper, lead and chromium against nis values. ran dumpsite were zn (1.81 mg/l), cu (0.38 mg/l), pb (0.21 mg/l), cr (0.003 mg/l) (fig. 9). results obtained for lead shows a high concentration value above the recommended value of 0.01 mg/l [21]. according to usda [22], heavy metals occur naturally, but rarely at toxic levels. the above listed heavy metals concentration values are below the recommended values of the standard organisation of nigeria (son) [20], including the 94 coker et al. / j. nig. soc. phys. sci. 3 (2021) 89–95 95 control sample values except for lead metal although, the control sample value of 0.02 mg/l is approximately the same with the permissible level without any hazard. the high concentration of lead in the study area could be due to the presence of waste containing fuel and engine oil in the dumpsite. 4. conclusion the effect of akanran dumpsite on groundwater quality for drinking and domestic purposes is minimal. the results of the 2-d resistivity survey give resistivity value of 9.2 ωm indicating the presence of leachate. the extent is within the topsoil, an indication that the underlying layer has a greater risk of contamination by leachate and decomposed solid waste materials. however, the geochemical results also shown possible contamination with time. the ph of the water sample analysis obtained is 6.9, suggesting acidic concentrates in the groundwater of the study area. however, this ph value for drinking water is within the permissible level of 6.5 – 8.5 indicating that the groundwater in the study area is suitable for drinking, domestic purpose and agricultural use; and there is high toxicity level and possible leaching of lead. hence, the combined techniques gave a better understanding of the study area than using a single investigative method. it is hereby recommended that a periodic study of the dumping site be done annually as the age of dumpsite affects significantly the quantity of leachate formed. it is also recommended to know the effect of depth on leachate concentration and chemical constituents of water samples to ascertain the nature of pollutants. acknowledgments the authors hereby acknowledged the people of akanran village for their support during the field work. the authors also wish to thank the referees for the positive enlightening comments and suggestions, which have greatly helped us in making improvements to this paper. references [1] g. j. farquhar, “leachate: production and characterization” canadian journal of civil engineering, 16 (1989) 317. [2] w. j. schneider, “hydrological implications of solid waste disposal” u.s. geological survey, (1970) 12. [3] m. a. oladunjoye, a. i. olayinka & s. a. amidu, “geoelectrical imaging at an abandoned waste dumpsite in ibadan, south-western nigeria” journal of applied sciences, 11 (2011) 755. [4] e. martinho & f. almeida, “3d behavior of contamination in landfill site using 2d resistivity/ip imaging. case studies in portugal” env. geol. 49 (2006) 1070. [5] j. m. reynolds, “an introduction to applied and environmental geophysics” john wiley & sons ltd (1997). [6] r. a. foster, “getting to grips with groundwater pollution protection in developing countries” national resour. forum 10 (1986) 51. [7] b. jimenez, “coming to term with nature: water reuse new paradigm towards integrated water resources management encyclopedia of biological, physiological and health sciences”, water and health, vol.ii: life support system, (2009) 398. [8] h. yadav, p. kumar & v. p. singh, “hazards from the municipal solid waste dumpsites” a review. proceedings of the 1st international conference on sustainable waste management through design (2018) 336. [9] j. r. yoon, k. lee, b. d. kwon & w. s. han, “geoelectrical surveys of the nanjido waste landfill in seoul, korea” env.geology, 43 (2003) 654. [10] i. b. osazuwa & n. k. abdullahi, “2d electrical resistivity and induced polarization investigation at an open solid waste dumpsite: a case study from kaduna, north central nigeria” journal of environmental hydrology 16 (2008) 1. [11] ngsa, “nigeria geological survey agency” nigeria (2010). [12] s. a. ganiyu, b. s. badmus, o. a. idowu, m. a. oladunjoye & o. t. olurin, “2d electrical resistivity imaging investigation of open dump site in basement complex formation” the african review of physics 10 (2015) 0033. [13] o. a. okunlola, o. c. adeigbe & o. o. oluwatoke, “compositional and petrogenetic features of schitose rocks of the ibadan area, south-western nigeria” earth sciences research journals 13 (2009) 29. [14] j. o. coker, “vertical electrical sounding (ves) methods to delineate potential groundwater aquifers in the akobo area, ibadan, south-western, nigeria” journal of geology and mining research 4 (2012) 35. [15] b. s. badmus, v. c. ozebo, o. a. idowu, s. a. ganiyu & o. t. olurin, “physico-chemical properties of soil samples and dumpsite environmental impact on groundwater quality in south western nigeria” african review of physics 9 (2014) 103. [16] a. o. ohwoghere, f. i. chiyem & h. o. nwankwoala, “2d resistivity electrical imaging of unsaturated and saturated zones for crude oil spillage at agbarha in ughelli area of delta state nigeria” technical report (2014). [17] r. b. grossman & t. g. reinsch, “bulk density and linear extensibility” soil science society of america (2002) 201. [18] d. w. urish, “the practical application of surface electrical resistivity to the detection of groundwater pollution” j. ground water 21 (1983) 144. [19] t. a. ewemoje, o. e. ewemoje & s. p. majolagbe, “urbanisation effects on surface and groundwater resources: an assessment of approved dumpsite in ibadan, nigeria”. an asabe meeting presentation (2017) doi: 10.13031/aim.201701388. [20] standard organisation of nigeria (son), nigerian standard for drinking water quality. nis 554 (2007) 30. [21] s. davis & r. de wiest, “hydrogeology” john wiley & sons, new york (1996). [22] usda diagnosis and improvement of saline and alkali soils u.s. salinity laboratory staff, government printing office, washington, d.c. (2000). 95 j. nig. soc. phys. sci. 3 (2021) 172–180 journal of the nigerian society of physical sciences performance analysis of photovoltaic systems using (ramd) analysis anas sani maihullaa,∗, ibrahim yusufb adepartment of mathematics, sokoto state university, sokoto bdepartment of mathematical sciences, bayero university, kano abstract the primary aim of this present study is to examine how reliability, availability, maintainability, and dependability (ramd) are used to describe the criticality of each sub-assembly in gridconnected photovoltaic systems. a transition diagram of all subsystems is produced for this analysis, and chapman-kolmogorov differential equations for each variable of each subsystem are constructed using the markov birth-death process. both random failure and repair time variables have an exponential distribution and are statistically independent. a sufficient repair facility is still available with the device. the numerical results for reliability, maintainability, dependability, and steady-state availability for various photovoltaic device components have been obtained. other metrics, such as mean time to failure (mttf), mean time to repair (mttr), and dependability ratio, which aid in device performance prediction, have also been measured. according to numerical analysis. it is hypothesized that subsystem s4, i.e. the inverter, is the most critical and highly sensitive portion that requires special attention in order to improve the efficiency of the pv device plant. the findings of this research are very useful for photovoltaic system designers and maintenance engineers. doi:10.46481/jnsps.2021.194 keywords: photovoltaic system, mean time to repair, mean time to failure. article history : received: 05 april 2021 received in revised form: 26 june 2021 accepted for publication: 30 june 2021 published: 29 august 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: w. a. yahya 1. introduction the dependability of photovoltaic systems and their components/elements such as solar cells, pv modules, electrical storage systems, inverters, regulators, etc is a critical problem in production efficiency and financially competitive photovoltaic installations. a product’s reliability can be described as the likelihood of system completing its tasks within a given time frame under stipulated conditions. the consistency of a product is calculated on this basis, with the reliability principle be∗corresponding author tel. no: +234(0) email address: anasmaihulla@gmail.com (anas sani maihulla ) ing used in nearly all fields of engineering, including preventive maintenance of structures and their components. pv systems have become a common solution for residential houses and other autonomous applications due to numerous reward schemes and local market conditions in many european countries, as well as around the world. ramd is regarded as one of the most important fields for increasing profitability. ramd modeling can help to improve safety and environmental efficiency, both of which are essential factors in every industry. therefore, it becomes important to evaluate each and every part or subsystem of the system in order to execute safety and environmental performance. many re172 maihulla & yusuf / j. nig. soc. phys. sci. 3 (2021) 172–180 173 searchers have presented appreciable works in this regard. researchers such as; huffman [1], hamdy et al. [2] and graaff et al. [3] investigated on significant issues concerning the reliability, and safety of pv systems, ram study of large-scale grid-connected solar-pv systems is conducted using a range of reliability approaches. zhang et al. [4] and hu et al. [5] conduct reliability block diagram (rbd) and fault tree analysis (fta). in the sense of an fta, the physical structure is translated into a logical diagram, with each block representing a device part. the failure rate is the only thing that describes each block. the overall system’s reliability is calculated by the failure rates of each sub-assembly, so any failure is critical. failure rates are frequently assumed to be constant. gahlot et al. [6] study stochastic analysis of a two units’ complex repairable system with switch and human failure using copula approach. also recently, chiacchio [7] investigated dynamic performance evaluation of photovoltaic power plant by stochastic hybrid fault tree automaton model. ramd technique was used by saini et al. [8] and goyal et al. [9] to define the most vulnerable aspect of serial processes such as evaporation systems in the sugar industry and water treatment plants. this study deconstructs the efficiency indices of the power generation system using stp. the power system was studied using simple probability theory principles and the markovian birth-death process. as a markov process proceeds from one stage to the next. van casteren et al. [10] used the weibull markovian approach to conduct consistency checks on electrical power systems. eti et al. [11] explored a basic issue in order to maximize resource distribution in a thermal power plant. the operational availability assessment was carried out in order to improve the system’s effectiveness. ebeling [12] proposed many methods for testing the reliability and maintainability of devices with differing failure and repair rates. s. gupta et al. [13] studied simulation modeling and analysis of complex system of thermal power plant. as a case study tsarouhas et al. [14] investigated the reliability, availability, and maintainability of a strudel production line using the best suited allocation of failure and repair rates. carazas et al. [15] proposed a framework for evaluating gas turbine power plant efficiency indicators. the study of reliability assessment of system having two subsystem in series configuration attended by human operator using gumbel-hougaard family copula was carried out by singh et al. [16]. singh [17] analyzed the performance of a system with imperfect switch consisting of two subsystem arranged in series through copula approach. yusuf et al. [18] focus on reliability analysis of computer system configured as series-parallel system having three dissimilar subsystem via copula approach. raghav et al. [19] dealt with reliability prediction and performance evaluation of distributed system consisting with similarity software and server architecture using joint probability distribution via copula approach. to address the issues raised in the previous literature on the reliability of grid-connected pv systems, this paper provides a full comprehensive ram analysis for all sub-assemblies of grid-connected solar pv systems with a low reliability grid, taking into account failure details and repair interval (period of identification and replacement of the pv system). furthermore, the aim of this paper is to describe the criticality of each sub-assembly of grid-connected pv systems in terms of reliability. the scope of this paper has also been broadened to establish the best probability density function for the failure rate of each solar-pv device subassembly. the rest of this paper is structured as follows; section 2 captures materials and methods. section 3 provides ramd indices for pv system. section 4 focuses on discussion of the results and the paper is concluded in section 5. 2. materials and methods the ramd analysis was used in this study. ramd is one of the most significant fields for rising profitability. ramd modeling aids in the improvement of safety and environmental performance, both of which are critical in any industry. in this study the ramd analysis has been performed on pv system plant. pv system mainly consists of five components namely pv modules, controller, batteries, inverter and distribution board. all components are arranged in series configuration. the brief description of the subsystems is given below: 1. subsystem r (solar module): there are two units of solar panel which is connected to the following unit in parallel. one is operational while the other one is on standby mode. failure of the two units leads to system failure. 2. subsystem s (charge controller): there is one unit of charge controller which is connected to the following unit in series. failure of this unit leads to system failure. 3. subsystem t (battery): this subsystem has two units of batteries which is connected in parallel to another subsystem. one is operation while the other one is in standby mode. this unit’s failure causes complete system failure. 4. subsystem u (inverter): it consists of one unit of inverter. this unit’s failure causes complete system failure as it is connected to the following unit in series. 5. subsystem v (distribution board): it consists of one unit of distribution board. this unit’s failure causes complete system failure as it is connected to the following unit in series. the ramd analysis was focused on failure rates, and the model data in this article was established theoretically. 2.1. assumptions 1. each subsystem’s failure and repair rates are distributed exponentially. 2. the failure and repair rates are statistically independent of one another. 3. there are no concurrent faults in the subsystem. 4. there are plenty of maintenance and replacement options. repairmen are still present in the factory, and the restored machine performs as well as new. 173 maihulla & yusuf / j. nig. soc. phys. sci. 3 (2021) 172–180 174 table 1. failure and repair rates of component of the photovoltaic system subsystem failure rate (ω) repair rate (β) s 1 ω1= 0.001 β1 = 0.5 s 2 ω2= 0.002 β2 = 0.7 s 3 ω3= 0.003 β3 = 0.9 s 4 ω4= 0.004 β4 = 1.1 s 5 ω5= 0.005 β5 = 1.3 2.2. notation r, s , t , u, and v represent states under which the subsystem is operating at maximum ability. r, s, t, u, and v reflect the conditions in which a subsystem has broken. 3. ramd indices for pv system 3.1. ramd indices for pv modules figure 1. transition diagram pv modules d dt p0 (t) = −ω1 p0 + β1 p1 (1) d dt p1 (t) = − (β1 + ω1) p1 + ω1 p0 + β1 p2 (2) d dt p2 (t) = −β1 p2 + ω1 p1 (3) under steady state, equations (1) (3) yield p1 = β1 ω1 p2 (4) substituting equation (4) into (1) under steady state we have: p2 = ω21 β21 p0 (5) substituting (5) into (3) under steady state we have p1 = ω1 β1 p0 (6) using normalization condition p0+ p1+ p2= 1 it follows that: p0 = β21 β21+ω 2 1 +β1ω1 (7) 3.2. ramd indices for subsystem r figure 2. transition diagram for subsystem r d dt p0 (t) = −ω2 p0+ β2 p1 (8) d dt p1 (t) = −β2 p1+ω2 p0 (9) under steady state equations (8) and (9) reduces to: p1 = ω2 β2 p0 (10) using normalization condition p0+ p1= 1 it follows that p0= β2 β2+ω2 (11) 174 maihulla & yusuf / j. nig. soc. phys. sci. 3 (2021) 172–180 175 table 2. variation of reliability of subsystems with time time (in days) rs1 (t) rs2 (t) rs3 (t) rs4 (t) rs5 (t) rsys (t) 0 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 30 0.92774 0.91393 0.90032 0.88692 0.87372 0.59155 60 0.86071 0.83527 0.81058 0.78663 0.76338 0.34994 90 0.79852 0.76338 0.72979 0.69768 0.66698 0.20701 120 0.74082 0.69768 0.65705 0.61878 0.58275 0.12246 150 0.68729 0.63763 0.59156 0.54881 0.50916 0.07244 180 0.63763 0.58275 0.53259 0.48675 0.44486 0.04285 210 0.59156 0.53259 0.47951 0.43171 0.38868 0.02535 240 0.54881 0.48675 0.43171 0.38290 0.33960 0.01500 270 0.50916 0.44486 0.38868 0.33960 0.29671 0.00887 300 0.47237 0.40657 0.34994 0.30120 0.25924 0.00525 330 0.43823 0.37158 0.31506 0.26714 0.22650 0.00310 360 0.40657 0.33960 0.28365 0.23693 0.19790 0.00184 table 3. variation of maintainability of subsystem with time. time (in days) ms1(t) ms2(t) ms3(t) ms4(t) ms5(t) ms ys(t) 0 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 30 0.77687 0.95021 0.98889 0.99752 0.99945 0.72778 60 0.95021 0.99752 0.99988 0.99999 1.00000 0.94773 90 0.98890 0.99988 1.00000 1.00000 1.00000 0.98876 120 0.99752 0.99998 1.00000 1.00000 1.00000 0.99750 150 0.99945 0.99999 1.00000 1.00000 1.00000 0.99944 180 0.99988 1.00000 1.00000 1.00000 1.00000 0.99988 210 0.99997 1.00000 1.00000 1.00000 1.00000 0.99997 240 0.99998 1.00000 1.00000 1.00000 1.00000 0.99998 270 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 300 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 330 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 360 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 figure 3. transition diagram for subsystem r 3.3. ramd indices for subsystem t d dt p0 (t) = −ω3 p0 + β3 p1 (12) d dt p1 (t) = − (β3 + ω3) p1 + ω3 p0 + β3 p2 (13) d dt p2 (t) = −β3 p2 + ω3 p1 (14) under steady state, equation (1) (3) yield p1 = β3 ω3 p2 (15) substituting 4 into 1 under steady state we have: p2 = ω23 β23 p0 (16) substituting 5 into 3 under steady state we have p1= ω3 β3 p0 (17) using normalization condition p0+ p1+ p2= 1 it follows that: p0= β23 β23+ω 2 3 +β3ω3 (18) 175 maihulla & yusuf / j. nig. soc. phys. sci. 3 (2021) 172–180 176 table 4. ramd indices for the photovoltaic system ramd indices ofsubsystems pv moduless 1 charge crs 2 battery banks 3 inverters 4 distributionboard pv system reliability e−0.0025t e−0.003t e−0.003t e−0.004t e−0.0045t e−0.0175t maintainability 1 − e−0.05t 1 − e−0.1t 1 − e−0.15t 1 − e−0.2t 1 − e−0.25t 1 − e−0.75t availability 0.76923 0.97087 0.97668 0.98039 0.98232 0.70246 mtbf 400 333.33 285.71 250 222.22 1491.27 mttr 20 10 6.67 5 4 45.67 dependability 0.95729 0.97308 0.97867 0.98153 0.98328 0.87985 dependability ratio 20 33.33 42.86 50 55.56 table 5. variation in system reliability as a result of pv module failure rate variation pv system pv system pv modules pv modules time (in days) ω1 = 0.0011 ω1=0.0012 ω1 = 0.0011 ω1=0.0012 0 1.00000 1.00000 1.00000 1.00000 30 0.61693 0.62438 0.96754 0.96464 60 0.38060 0.38985 0.93613 0.93053 90 0.23480 0.24341 0.90574 0.89763 120 0.14486 0.15198 0.87634 0.86589 150 0.08937 0.09489 0.84790 0.83527 180 0.05513 0.05925 0.82037 0.80574 210 0.03401 0.03700 0.79374 0.77724 240 0.02098 0.02310 0.76797 0.74976 270 0.01295 0.01442 0.74304 0.72325 300 0.00799 0.00901 0.71892 0.69768 330 0.00493 0.00562 0.69559 0.67301 360 0.00304 0.00351 0.67301 0.64921 3.4. ramd indices for subsystem u d dt p0 (t) = −ω4 p0+ β4 p1 (19) d dt p1 (t) = −β4 p1+ω4 p0 (20) under steady state equations (8) and (9) reduces to: p1 = ω4 β4 p0 (21) using normalization condition p0+ p1= 1 it follows that p0= β4 β4+ω4 (22) 3.5. ramd indices for subsystem v d dt p0 (t) = −ω5 p0+ β5 p1 (23) d dt p1 (t) = −β5 p1+ω5 p0 (24) under steady state equations (8) and (9) reduces to: p1 = ω5 β5 p0 (25) using normalization condition p0+ p1= 1 it follows that p0= β5 β5+ω5 (26) using the following equations r (t) = ∫ ∞ t f (x) d x reliability function (27) availability = li f e time t otal time = li f e time li f e time + repair time = mt t f mt t f + mt t r (28) 176 maihulla & yusuf / j. nig. soc. phys. sci. 3 (2021) 172–180 177 table 6. variation in system reliability as a result of charge controller failure rate variation pv system pv system charge controller charge controller time (in days) ω2 = 0.0013 ω2=0.0014 ω2 = 0.0013 ω2=0.0014 0 1.00000 1.00000 1.00000 1.00000 30 0.62251 0.62064 0.96175 0.95887 60 0.38752 0.38520 0.92496 0.91943 90 0.24123 0.23907 0.88959 0.88161 120 0.15017 0.14838 0.85556 0.84535 150 0.09348 0.09209 0.82283 0.81058 180 0.05820 0.05715 0.79136 0.77724 210 0.03623 0.03547 0.76109 0.74528 240 0.02255 0.02202 0.73198 0.71462 270 0.01404 0.01367 0.70398 0.68523 300 0.00877 0.00848 0.67706 0.65705 330 0.00544 0.00526 0.65116 0.63002 360 0.00339 0.00327 0.62625 0.60411 table 7. variation in reliability of system due to variation in failure rate of battery bank pv system pv system battery bank battery bank time (in days) ω3 = 0.0015 ω3=0.0016 ω3 = 0.0015 ω3=0.0016 0 1.00000 1.00000 1.00000 1.00000 30 0.62414 0.62625 0.95600 0.95313 60 0.36455 0.39219 0.91393 0.90846 90 0.24783 0.24561 0.87372 0.86589 120 0.15567 0.15382 0.83527 0.82531 150 0.09778 0.09633 0.79852 0.78663 180 0.06142 0.06033 0.76338 0.74976 210 0.03858 0.03778 0.72979 0.71462 240 0.02423 0.02366 0.69768 0.68113 270 0.01522 0.01482 0.66698 0.64921 300 0.00956 0.00928 0.63763 0.61878 330 0.00606 0.00581 0.60957 0.58978 360 0.00377 0.00364 0.58275 0.56214 table 8. variation in reliability of system due to variation in failure rate of inverter pv system pv system inverter inverter time (in days) ω4 = 0.0017 ω4=0.0018 ω4 = 0.0017 ω4=0.0018 0 1.00000 1.00000 1.00000 1.00000 30 0.60230 0.63192 0.95028 0.94743 60 0.36277 0.39932 0.90303 0.89763 90 0.21850 0.25233 0.85813 0.85044 120 0.13160 0.15945 0.81546 0.80574 150 0.07926 0.10076 0.77492 0.76338 180 0.04774 0.06367 0.73639 0.72325 210 0.02875 0.04024 0.69977 0.68523 240 0.01732 0.02543 0.66498 0.64921 270 0.01043 0.01607 0.63192 0.61508 300 0.00628 0.01562 0.60050 0.58275 330 0.00378 0.00642 0.57064 0.55211 360 0.00228 0.00405 0.54227 0.52310 m (t) = 1 − e(− −t mt t r ) maintainability function (29) mt bf = ∫ ∞ 0 r (t) dt = ∫ ∞ 0 e−θt = 1 θ (30) mtbf = mean time between failure mt t r = 1 β mean time to repair (31) 177 maihulla & yusuf / j. nig. soc. phys. sci. 3 (2021) 172–180 178 table 9. variation in system reliability due to variation in delivery board failure rate pv system pv system distribution board distribution board time (in days) ω5 = 0.0019 ω5=0.002 ω5 = 0.0019 ω5=0.002 0 1.00000 1.00000 1.00000 1.00000 30 0.63954 0.63763 0.94460 0.94176 60 0.40902 0.40657 0.89226 0.88692 90 0.26158 0.25924 0.84282 0.83527 120 0.16729 0.16530 0.79612 0.78663 150 0.19700 0.10540 0.75201 0.74082 180 0.06843 0.06721 0.71035 0.69768 210 0.04376 0.04285 0.67100 0.65705 240 0.02799 0.02732 0.63381 0.61878 270 0.01790 0.01742 0.59870 0.58275 300 0.01145 0.01111 0.56553 0.54881 330 0.00732 0.00708 0.53420 0.51685 360 0.00468 0.00452 0.50459 0.48675 figure 4. transition diagram for subsystem u β= repair rate,ω = failure rate d = ξ ω = mt bf mt t r dmin = 1 − ( 1 d − 1 ) ( e− ln d d−1 − e− d ln d d−1 ) 4. ramd analysis system reliability rsys (t) = rs1 (t)×rs2 (t)×rs3 (t)×rs4 (t)×rs5 (t) = e−(ω1 +ω2 +ω3 + ω4 )t (32) system availability asys = as1 × as2 × as3 × as4 × as5 (33) figure 5. transition diagram for subsystem v system maintainability msys(t) = ms 1(t) . ms2(t). ms3(t) . ms4(t). ms5(t) = ( 1 − e−β1 (t) ) × ( 1 − e−β2 (t) ) × ( 1 − e−β3 (t) ) × ( 1 − e−β4 (t) ) × ( 1 − e−β5 (t) ) = 1 − e−(β1 +β2 +β3 +β4 +β5 )t (34) system dependability dmin(sys) = dmin(s1)×dmin(s2)×dmin(s3)×dmin(s4)×dmin(s5)(35) system dependability dmin(sys) = dmin(s1) × dmin(s2) × dmin(s3) × dmin(s4) dmin = 1 − ( 1 d − 1 ) ( e− ln d d−1 − e− d ln d d−1 ) d = β ω = mt bf mt t r d1 = β1 ω1 = 0.05 0.0025 = 20 178 maihulla & yusuf / j. nig. soc. phys. sci. 3 (2021) 172–180 179 d2 = β2 ω2 = 0.1 0.003 = 33.33 d3 = β3 ω3 = 0.15 0.0035 = 42.86 d4 = β4 ω4 = 0.2 0.004 = 50 d5 = β5 ω5 = 0.2 0.004 = 55.56 dmin(s1) = 1 − ( 1 20 − 1 ) ( e− ln 20 20−1 − e− 20 ln 20 20−1 ) dmin(s1) = 0.95729 dmin(s2) = 1 − ( 1 33.33 − 1 ) ( e− ln 33.33 33.33−1 − e− 33.33 ln 33.33 33.33−1 ) dmin(s2) = 0.97308 dmin(s3) = 1 − ( 1 42.86 − 1 ) ( e− ln 42.86 42.86−1 − e− 42.86 ln 42.86 42.86−1 ) dmin(s3) = 0.97867 dmin(s4) = 1 − ( 1 50 − 1 ) ( e− ln 50 50−1 − e− 50 ln 50 50−1 ) dmin(s4) = 0.98153 dmin(s5) = 1 − ( 1 55.56 − 1 ) ( e− ln 55.56 55.56−1 − e− 55.56 ln 55.56 55.56−1 ) dmin(s5) = 0.98328 dmin(sys) = 0.95729 × 0.97308 × 0.97867 ×0.98153 × 0.98328 dmin(sys) = 0.87985 system availability asys =  β21 β21 + ω 2 1 + ω1 β1  × ( β2 β2 +ω2 ) ×  β23 β23 + ω 2 3 + ω3 β3  × ( β4 β4 +ω4 ) × ( β5 β5 +ω5 ) asys = ( (0.05)2 (0.05)2 + (0.025)2 + (0.05) × (0.0025) ) × ( 0.1 0.10 + 0.0030 ) × ( (0.05)2 (0.05)2 + (0.0035)2 + (0.15)(0.0035) ) × ( 0.20 0.20 + 0.0040 ) ( 0.25 0.25 + 0.0045 ) asys = 0.76923 × 0.97087 × 0.97668 × 0.98039 × 0.98232 asys = 0.70246 5. discussion to reflect the effect of system factors, we established formulations for availability, reliability, maintainability, and dependability for each subsystem of the model under consideration. table 2 shows that after 300 days of operation, the pv device plant’s reliability stays 0.00525, however the distribution board’s reliability is significantly poor (0.25924) across all subsystems. this sensitivity analysis reveals that the distribution board’s reliability of the system requires special attention and close monitoring. as a result, machine designers must develop a maintenance plan. tables 59 showed how the reliability behavior of different subsystems improved over time and with varying failure rates. 6. conclusion in this paper, an analytical analysis for a case was performed to obtain the reliability metrics of the various subsystems and procedure as presented in table 1. tables 2 and 3 demonstrate the findings for the reliability and maintenance habits of various subsystems, respectively. table 4 summarizes the remaining ramd phases. based on this study, it is hypothesized that subsystem s4, i.e. the inverter, is the most critical and highly sensitive portion that requires special attention in order to improve the efficiency of the pv device plant. hence it is concluded that machine designers must develop a maintenance plan for the inverter to avoid total breakdown of the whole system and some techniques can be opted to enhance the system reliability. this research 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[19] d. raghav, d. rawal, i. yusuf, r. h. kankarofi & v. singh, “reliability prediction of distributed system with homogeneity in software and server using joint probability distribution via copula approach”, reliability: theory & applications, 16 (2021) 217.https://doi.org/10.24412/1932-2321-2021-161-217-230 180 j. nig. soc. phys. sci. 2 (2020) 160–165 journal of the nigerian society of physical sciences original research l-stable block hybrid numerical algorithm for first-order ordinary differential equations b. i. akinnukawea,∗, k. o. mukab adepartment of mathematics, university of lagos, lagos, nigeria bdepartment of mathematics, university of benin, benin city, nigeria abstract in this work, a one-step l-stable block hybrid multistep method (bhmm) of order five was developed. the method is constructed for solving first order ordinary differential equations with given initial conditions. interpolation and collocation techniques, with power series as a basis function, are employed for the derivation of the continuous form of the hybrid methods. the discrete scheme and its second derivative are derived by evaluating at the specific grid and off-grid points to form the main and additional methods respectively. both hybrid methods generated are composed in matrix form and implemented as a block method. the stability and convergence properties of bhmm are discussed and presented. the numerical results of bhmm have proven its efficiency when compared to some existing methods. keywords: second derivative, stability, hybrid, block, collocation techniques article history : received: 25 may 2020 received in revised form: 13 july 2020 accepted for publication: 14 july 2020 published: 01 august 2020 c©2020 journal of the nigerian society of physical sciences. all rights reserved. communicated by: j. o. kuboye 1. introduction most mathematical models are formulated using ordinary differential equations (odes) of different orders. however, mathematical models that are based on odes of order one occur in several engineering, applied sciences and economics problems. some of the odes have been proven not to have closed-form solution hence the need to develop numerical methods to provide approximate solutions to these problems. consider the initial value problem of the form y′(x) = f (x, y(x)), y(x0) = y0 (1) several numerical methods have been developed by different researchers to circumvent the dahlquist’s order barrier theo∗corresponding author tel. no: email address: iakinnukawe@unilag.edu.ng (b. i. akinnukawe ) rem [1 3] that restricted the use of linear multistep methods of high order to solve (1). some of such researchers are enright [4], akinnukawe and okunuga [5], gear [6], okunuga [7], cash [8], akinfenwa et al. [9, 10], adesanya et al. [11], ijezie and muka [12] to mention but a few. high-order a-stable and l-stable numerical methods are developed by incorporating off-step points, additional stages and/or employing higher differentiation of the solution (lambert [13]). generating approximate solutions for (1) in instances where they exist and are unique but are insoluble via analytical means is very important because they help in the analysis and validation of models in which they evolve from. one of the objectives of developing numerical schemes for solving (1) is to obtain methods with wider stability regions, better convergent rates and computational efficiency. 160 akinnukawe & muka / j. nig. soc. phys. sci. 2 (2020) 160–165 161 in this article, continuous hybrid methods (chm) of order 5 are derived through interpolation and collocation techniques (onumanyi et al. [14]). both methods incorporate off-step points and the second derivative of the scheme to produce the discrete hybrid methods which are composed and implemented as a block method (bhmm) to simultaneously produce approximation at nodal points. the bhmm has the advantage of being self-starting, l-stable in nature and possesses high accuracy because it implemented as a block method (see [15], [16], [17]). 2. derivation of bhmm this section describes the derivation of the k-step hybrid multistep method of the form: k∑ j=0 α jyn+ j + αvyn+v = h k∑ j=0 β j fn+ j + hβv fn+v + h 2φkgn+k (2) where h, n and v = 12 are the step size, grid index and offstep point respectively while α j, β j, j = 0, 1, ..., k and φk are parameters to be determined uniquely. an approximate solution to (1) by the interpolating function y(x) = 4k+1∑ j=0 b jt j (3) where b j, j = 0, 1, ..., 4k + 1 are the unknown coefficients. imposing condition for the construction of the proposed class of methods are: y(xn+r ) = yn+r, r = 0, v (4) y′(xn+r ) = fn+r, r = 0(v)k (5) y′′(xn+r ) = gn+r, r = k (6) equations (4) (6) will lead to a system of 4k + 2 equations. these equations are solved simultaneously to obtain b j and the values of b j are substituted into (3) to form the continuous hybrid method (chm) expressed in the form y(t) = k−v∑ j=0 α j(t)yn+ j + h k∑ j=0 β j(t) fn+ j + h 2φk(t)gn+k (7) where t = x−xn+k−1h and α j(t), β j(t), j = 0(v)k and φk(t) are the continuous coefficients. the main scheme is generated evaluating chm (7) at xn+k while the additional scheme is generated from the second derivative of (7) at xn+v of the form h2y′′(t) = k−v∑ j=0 α j(t)yn+ j + h k∑ j=0 β j(t) fn+ j + h 2φk(t)gn+k (8) the developed main and additional schemes are combined and implemented simultaneously as a block hybrid multistep method bhmm for the numerical integration of ivps (1). following the steps discussed above, the continuous hybrid method (chm) for k=1 is equation (7) y(t) = k−v∑ j=0 α j(t)yn+ j + h k∑ j=0 β j(t) fn+ j + h 2φk(t)gn+k where  α0(t) α 1 2 (t)  = [ 1 0 −48023 128023 −120023 384230 0 48023 −128023 120023 −38423 ]  t0 t1 t2 t3 t4 t5  (1)  β0(t) β 1 2 (t) β1(t)  =  0 1 −13123 265 23 − 224 23 68 23 0 0 −12823 464 23 − 504 23 176 23 0 0 1923 − 89 23 128 23 − 52 23   t0 t1 t2 t3 t4 t5  (2) [ φ1(t) ] = [ 0 0 − 746 17 23 − 26 23 12 23 ]  t0 t1 t2 t3 t4 t5  (3) interpolating (7) at x = xn+1 to generate the main method at k = 1 becomes yn+1 = 7yn 23 + 16yn+ 12 23 + h fn 23 + 8h fn+ 12 23 + 6h fn+1 23 − h2gn+1 46 (9a) equation (8), the second derivative of (7) for k = 1 is h2y′′(t) = k−v∑ j=0 α j(t)yn+ j + h k∑ j=0 β j(t) fn+ j + h 2φk(t)gn+k where  α0(t) α 1 2 (t)  = [ −96023 768023 −1440023 768023960 23 − 7680 23 14400 23 − 7680 23 ] t0 t1 t2 t3 (4)  β0(t) β 1 2 (t) β1(t)  =  − 262 23 1590 23 − 2688 23 1360 23 − 256 23 2784 23 − 6048 23 3520 23 38 23 − 534 23 1536 23 − 1040 23   t0 t1 t2 t3  (5) 161 akinnukawe & muka / j. nig. soc. phys. sci. 2 (2020) 160–165 162 [ φ1(t) ] = [ − 7 23 102 23 − 312 23 240 23 ]  t0 t1 t2 t3  (6) interpolating(8) at x = xn+1/2, the additional method at k = 1 becomes h2gn+ 12 = 240yn 23 − 240yn+ 12 23 + 31h fn 23 + 64h fn+ 12 23 + 25h fn+1 23 − 4h2gn+1 23 (9b) the discrete hybrid methods (9a)and (9b) together forms the one-step block hybrid multistep methods (bhmm). the bhmm can be presented in a matrix block form as a(1)y$ = a(0)y$−1 + hb(1) f$ + hb(0) f$−1 + h 2c(1)g$ (10) where y$ =  yn+ 12 yn+1  ; y$−1 =  yn− 12 yn  ; f$ =  fn+ 12 fn+1  ; f$−1 =  fn− 12 fn  ; g$ =  gn+ 12 gn+1  the 2 by 2 matrices a(0), a(1), b(0), b(1), c(1), d(1) of the bhmm (9) are defined as follows a(1) =  240 23 0 − 16 23 1  a(0) =  0 24023 0 723  b(1) =  64 23 25 23 8 23 6 23  b(0) =  0 3123 0 123  c(1) =  −1 − 423 0 − 146  3. analysis of bhmm 3.1. order and error constant of the method following lambert [13] and fatunla [18], a method was proposed for finding the order p and error constant wp+1 of the block method (9) by first expanding y−, f− and g− functions by taylors series expansion about x and then comparing the coefficients of h. it is established from our calculation that one-step block hybrid multistep method have order and error constants as p = (5, 5)t and wp+1 = ( 13 44160 , 1 66240 ) t respectively where t is transpose. 3.2. zero stability a numerical method is said to be zero-stable if the roots r j, j = 1, 2, . . . , n of the first characteristic polynomial ρ(r) satisfies |r j| ≤ 1, j = 1, . . . , n and those roots with |r j| = 1 is simple (see lambert [3]). applying the above conditions to the derived block method, the first characteristic polynomial ρ(r) = 0 is calculated as ρ(r) = det(ra(1) − a(0)) = 240 23 r(r − 1) the bhmm is found to be zero-stable since ρ(r) = 0 satisfies |r j| ≤ 1, j = 1, 2. 3.3. convergence according to henrici [19], a numerical method converges if it is consistent and zero-stable. since bhmm (9) is of order 5 > 1, then it is consistent and we have established earlier that the method satisfies the conditions of zero-stability. therefore, the block method (9) converges. 3.4. stability of bhmm applying the bhmm to the test equation y′ = λy, λ ≤ 0 we obtain y$ = q(z)y$−1, z = λh where q(z) is the amplification matrix given by q(z) = a(0) + zb(0) a(1) + zb(1) + z2c(1) q(z) has eigenvalues (ζ1,ζ2) = (0,ζ2). the dominant eigenvalue ζ2 is the stability function with real coefficient as ζ2 = 10.4348 + 4.17391z + 0.652174z2 + 0.0434783z3 10.4348 − 6.26087z + 1.69565z2 − 0.26087z3 + 0.0217391z4 the stability function is used to plot the region of absolute stability (ras) of the bhmm (see figure 1). the proposed method bhmm is l-stable since the ras covers the entire left plane of the graph (a-stable) and the limit of the stability function ζ2 is zero as z →∞ 162 akinnukawe & muka / j. nig. soc. phys. sci. 2 (2020) 160–165 163 figure 1. stability region of block hybrid multistep method 4. numerical results the following problems are considered to examine the accuracy and computational efficiency of bhmm. the computations were carried out using mathematica 9.0 software. the following acronyms are used: • es = exact solution • esdmm = error in sdmm • emhirk = error in mhirk • ebhmm = error in bhmm • eesdmm = error in esdmm problem 1: consider the non-linear ivp y′1 = λy1 + y 2 2 y′2 = −y2 with initial conditions y1(0) = −1 λ + 2 , y2(0) = 1, where λ = 104 and the exact solution of the ivp is given as y1(x) = − e−2x λ + 2 , y2(x) = e −x in table 1, the numerical results obtained using bhmm with h = 10−1 compares favourably with mhirk method [10] with h = 10−1 and is superior to that of hojjati et al. [20] that used h = 10−4. problem 2: consider the following stiff problem arising from chemical kinetic reactions in a chemistry experiment. y′1 = −0.013y1 − 1000y1y2 − 2500y1y3 y′2 = −0.013y1 − 1000y1y2 y′3 = −2500y1y3 with initial conditions y1(0) = 0, y2(0) = 1, y3(0) = 1 the computed result at x = 2.0 from the new method bhmm is compared with those of ismail and ibrahim (esdmm [21]), hojjati et al. sdmm [20], akinfenwa et al. mhirk [10]. the step length h = 0.0125 was used for mhirk method and the new method bhmm and compared with ismail and ibrahim esdmm [21], hojjati et al. sdmm [20] with step length h = 0.001. the result obtained with the new method bhmm is superior to others. see table 2 for the numerical results. problem 3: consider the non-linear system y′1 = −2y1 + y2 + 2sin x y′2 = 998y1 − 999y2 + 999(cos x − sin x) with initial conditions y1(0) = 2, y2(0) = 3, 0 ≤ x ≤ 10 with analytical solution given as y1(x) = 2e −x + sin x, y2(x) = 2e −x + cos x 163 akinnukawe & muka / j. nig. soc. phys. sci. 2 (2020) 160–165 164 table 1. comparison of methods for problem 1 x yi es esdmm[20] emhirk[10] ebhmm 3 y1 −0.2478257e − 06 2.478147e − 11 3.06450e − 15 5.00564e − 16 y2 0.4978707e − 01 2.471093e − 06 3.07825e − 10 5.02813e − 11 5 y1 −0.4539085e − 08 3.450217e − 14 9.35475e − 17 1.52787e − 17 y2 0.6737946e − 02 2.304573e − 08 6.94326e − 11 1.13414e − 11 10 y1 0.3059023e − 12 3.456372e − 18 8.49412e − 21 3.75372e − 20 y2 0.3059023e − 04 3.150734e − 10 9.35666e − 13 1.52836e − 13 table 2. comparison of methods for problem 2 x yi es eesdmm[21] esdmm[20] emhirk[10] ebhmm 2.0 y1 −0.361693316929e − 5 0.82e − 10 0.52e − 13 0.110e − 13 2.919e − 15 y2 0.9815029948230 0.61e − 05 0.19e − 08 0.220e − 08 5.586e − 10 y3 1.018493388244 0.57e − 05 0.63e − 08 0.220e − 08 5.584e − 10 the numerical results of problem 3 are shown in table 3 using step size h = 10−3. the derived method integrated the problem efficiently that the numerical solution is close to that of the analytical solution. table 3. the absolute error for problem 3 x yi es ebhmm 0.25 y1 1.805005525 4.50751e − 14 y2 2.526513988 4.84057e − 14 0.5 y1 1.692486858 9.85878e − 14 y2 2.090643881 9.81437e − 14 1.0 y1 1.577229867 9.45910e − 14 y2 1.276061188 9.54792e − 14 2.0 y1 1.179967993 1.68310e − 13 y2 −0.145476270 1.68365e − 13 4.0 y1 −0.720171217 2.21378e − 13 y2 −0.617012343 2.23044e − 13 6.0 y1 −0.274457993 1.01363e − 13 y2 0.9651277911 1.01474e − 13 8.0 y1 0.9900291719 1.93401e − 13 y2 −0.1448291085 1.94650e − 13 10.0 y1 −0.5439303110 6.10623e − 13 y2 −0.8389807292 6.09068e − 13 problem 4: a stiff system of initial value problems y′1 = −8y1 + 7y2 y′2 = 42y1 − 43y2 with initial conditions y1(0) = 1, y2(0) = 8, x ∈ [0, 15] with exact solution given as y1(x) = 2e −x − e−50x, y2(x) = 2e −x + 6 − e−50x problem 4 was integrated using the step size of h = 10−4 to aid in comparing with other methods in literature as shown in tables 4 and 5. it is discovered that bhmm has better accuracy than the others compared with. table 4. numerical results for problem 4 x yi es ebhmm 3 y1 9.9574136 × 10−2 2.68577 × 10−13 y2 9.9574136 × 10−2 2.65843 × 10−13 6 y1 4.9575044 × 10−3 1.68580 × 10−14 y2 4.9575044 × 10−3 1.80611 × 10−14 9 y1 2.4681961 × 10−4 7.57646 × 10−15 y2 2.4681961 × 10−4 5.43191 × 10−15 12 y1 1.2288424 × 10−5 2.10193 × 10−15 y2 1.2288424 × 10−5 2.54783 × 10−15 15 y1 6.1180464 × 10−7 2.29273 × 10−14 y2 6.1180464 × 10−7 1.87085 × 10−14 5. conclusion a one-step l-stable block hybrid multistep method (bhmm) of order five was developed via interpolation and collocation techniques. the bhmm has the advantage of being self-starting and its l-stable in nature as displayed in figure 1. the block method possesses high accuracy as shown in tables (1) − (5) where it was compared with some existing methods. the method satisfies the zero-stability, consistency and convergence conditions. bhmm has proved efficient for solving first-order initial value problems. 164 akinnukawe & muka / j. nig. soc. phys. sci. 2 (2020) 160–165 165 table 5. a comparison of absolute errors of methods for problem 4 x methods error |y(tn) − yn| 5 bhmm 3.1999 × 10−14 tdbdf [22] 1.5472 × 10−2 tdmm [23] 1.5476 × 10−2 10 bhmm 4.642 × 10−15 tdbdf [22] 9.0808 × 10−5 tdmm [23] 9.0808 × 10−5 15 bhmm 1.8709 × 10−14 tdbdf [22] 6.1186 × 10−7 tdmm [23] 6.1186 × 10−7 acknowledgments the authors wish to thank the referees and editor for the comments and suggestions to make this paper a success. references [1] g. g. dahlquist, “a special stability problem for linear multistep methods”, bit 3 (1963) 27. 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[23] a. k. ezzeddine & g. hojjati, “third derivative multistep methods for stiff systems”, international journal of nonlinear science 14 (2012) 443. 165 j. nig. soc. phys. sci. 3 (2021) 1–11 journal of the nigerian society of physical sciences analysis of an hiv hcv simultaneous infection model with time delay a. s. hassan∗, n. hussaini department of mathematical sciences, faculty of physical sciences, bayero university, kano, nigeria. abstract a novel mathematical delay model for simultaneous infection of hiv and hepatitis c virus is formulated and dynamically analyzed. basic properties of the model are established and proved. basic reproductive threshold is systematically calculated as the maximum of three subthreshold parameters. a disease free equilibrium is determined to be globally asymptotically stable for all values of the delay when the threshold is less than unity. however, when the threshold is greater than one, endemic equilibrium emerged which is shown to be locally asymptotically stable for any length of delay. although the delay has no effect on stabilities of equilibria points, however, it is found to reduce the infectivity of the viruses as the length of the delay is increased. epidemiological interpretations of the results and numerical simulations illustrating them are given. doi:10.46481/jnsps.2021.109 keywords: delay, hiv/hcv simultaneous infection, reproduction number, equilibrium, stability. article history : received: 28 may 2020 received in revised form: 27 june 2020 accepted for publication: 18 july 2020 published: 27 february 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: t. latunde 1. introduction around 2.75 million people who have human immunodeficiency virus (hiv) are coinfected with hepatitis c virus (hcv) globally and and on average hiv-infected individuals are six time more likely to get hcv infection than hiv-uninfected [1]. worldwide, hiv and hcv are public health challenges which affected several populations. across the continents, there are about 37 million and 115 people infected with hiv and chronic hcv infections, respectively. numerous mathematical models have been used to explore theoretical aspects of the transmission dynamics for superinfection (strains or diseases never co-exist in a host) [2-4] and for co-infection (diseases can co-exist in the host) [5-15]. most of ∗corresponding author tel. no: +2348036289110 email address: hashitu.mth@buk.edu.ng (a. s. hassan ) the co-infection models, for simplicity, do not allow simultaneous infection [5]. owing to the growing empirical evidence of simultaneous infection (see, for example, [16-20]), which is now a major public health concern [16] and affects the epidemiology [21] and evolution [22] of infectious diseases, zhang et al. [14, 15, 24] recently developed models which include simultaneous infection. these models are only suitable to completely curable diseases such as influenza. alizon [5] studied the effect of co-infection where parasites compete. of recent, some mathematical models for hiv/aids and hcv separately, are formulated and analyzed to explore impacts of the two diseases, see for example [25-28]. in dhutta and gupta [25], a mathematical model is proposed for the dynamics of hiv/aids with incorporation of weak cd4+ t cells. they computed the local stability of the infection-free and infection equilibria for the model when the 1 hassan & hussaini / j. nig. soc. phys. sci. 3 (2021) 1–11 2 valuable reproduction number is less than and greater than one respectively. furthermore, using lyapunov’s second method and the geometric approach, they define novel conditions for the global stability of the equilibria. further, maimuna [27] formulated a mathematical model for the spread of hiv with an anti-retroviral therapy (art) intervention. the author established that dynamics of the model with no art depends on the basic reproduction number. however, sensitivity analysis revealed that the basic reproduction number decrease with increasing number of infected humans who follow the art treatment. jia et al [26], constructed a dynamic model for hcv transmission and prevalence based on the reported data from china and determined the most influential parameters to evaluate the effectiveness of control measures. finally, in [28], the authors applied ideas from the ecology of infectious diseases to model the transmission of hcv in a population of injection drug users. the authors suggested that modelling hcv as an indirectly transmitted infection facilitates a more nuanced understanding of disease dynamics. in addition, sensitivity analysis of parameters on the value of r0 was carried out and parameters related to an interaction with the environmental reservoir are found to be more influential. time delay in epidemiology is mostly incorporated to represent developmental stages, incubation period or wining of immunity [29, 30]. it is an important component that cannot be neglected as it affects the dynamics of models causing instabilities of equilibria resulting in hopf bifurcations [32, 31, 33]. in addition, an important threshold value (the basic reproduction number), as can be seen later in this article, is expressed in terms of time delay [34, 30, 35]. many delay models of hcv/hepatitis b virus (hbv) have been presented in the literature to explore the effects of delay as incubation period. banerjee et al. [36] presented an intracellular time delay model for hepatitis c virus which has shown that the delay doesn’t affect the stability of steady-state, however, destabilized the infected equilibrium with resulting in hopf bifurcation. gourley et al. [37] considered the dynamical properties of hbv delayed model with standard incidence formulation. they realized that the infected steady-state, expressed in terms of delay, is globally stable regardless of the time delay length. in [33], zhao and xu presented a delay hcv model using beddigton-deangels formulation and obtained global stabilities of equilibria when the threshold parameters satisfy certain conditions. in this study, we begin by developing a model using delay differential equation for hiv and hcv co-infection which includes simultaneous transmission from person-to-person. unlike in [5, 14, 15, 24], we introduce delay in order to capture the incubation periods of both hiv and hcv infections. in this research, the model is presented in section 2 while the analysis is given in section 3 . in section 4, we present illustrative graphs for the dynamical properties of the model. lastly, conclusion and summary of our findings are reported in section 5. 2. model formulation the total human population at time t, n(t), is divided into four different classes of healthy (susceptible, (s (t))), hiv-only infected individuals (ih (t)), hcv-only infected individuals (ic (t)), individuals simultaneously infected with both hiv and hcv (ihc (t)), so that n(t) = s (t) + ih (t) + ic (t) + ihc (t). the susceptible population (s (t)) is increased by the recruitment of people into the community at a constant rate π (all newly-recruited individuals are assumed to be susceptible to both infections) and by recovery from hcv (at a rate ψ). the population is decreased by infection with hiv only (at a rate λh ) or hcv only (at a rate λc ) or simultaneous infection with hiv and hcv (at a rate λhc ) (see, for instance, [19, 17] and references therein for the biology). however, the infections with hiv and hcv do not occur as soon as the infectives get into contact with the susceptible individuals. rather, there are time lapses or delays, representing the latent periods, in which the infectives progress to fully infectious people. thus we have ih (t − τ1)e−µτ1 representing the hiv only infectives that can only infect after the elapse of τ1, (the latent period for hiv) and e−µτ1 , the survival probability over the latent period. similarly, ic (t −τ2)e−µτ2 represent the hcv only infectives that can infect after the elapse of τ2, (the latent period for hcv), e−µτ2 is the survival probability for natural death over the latent period. lastly, ihc (t − τ3)e−µτ3 stand for the hiv and hcv infectives that can infect after the elapse of τ3, (the latent period for simultaneous infection of hiv and hcv), while e−µτ3 is the survival probability over the latent period. thus, λh = βh [ ih (t −τ1)e−µτ1 + ηihc (t −τ3)e−µτ3 ] n(t) , λc = βc [ ic (t −τ2)e−µτ2 + ηihc (t −τ3)e−µτ3 ] n(t) and λhc = βhc ihc (t −τ3)e−µτ3 n(t) . (1) in (1), βh , βc and βhc represent the effective contact rates for hiv, hcv and simultaneous hiv/hcv infections, respectively. the parameter η > 1 accounts for the assumed increase in infectiousness of dually-infected individuals due to high immunosuppression caused by the simultaneous infection of hiv and hcv, compared to singly-infected with hiv or hcv. for mathematical simplification we assume that τ1 = τ2 = τ3 = τ. the susceptible population further decreased by natural death (at a rate µ; natural death occurs in all compartments at the same rate). ds dt = π + ψic − (λh + λc + λhc ) s (t) −µs (t). (2) the population of individuals infected with hiv-only (ih ) is generated by hiv infection following the effective contact (at the rate λh ) and by dually-infected individuals (ihc ) recovered from hcv (at a rate κψ; where the modification parameter 0 < κ < 1 models the reduced likelihood of dually-infected 2 hassan & hussaini / j. nig. soc. phys. sci. 3 (2021) 1–11 3 person to recover from hcv, in comparison to those infected with hcv only). it is decreased by hcv infection at rate λc , hiv-related death at rate δc and natural death. dih dt = λh s + κψihc − θ1λc ic − (µ + δh )ih. (3) the population of individuals infected with hcv-only (ic ) is generated by hcv infection (at the rate λc ). it is decreased by hiv infection at rate λh ), recovery from hcv at rate ψ, hcv-related death at rate δh and natural death. dic dt = λc s − θ2λh ic − (ψ + µ + δc )ic. (4) the class of individuals simultaneously infected (ihc ) is generated by hiv and hcv simultaneous infections at rate λhc , by hiv-only infected individuals (ih (t)) who are hcv infected at rate θ1λc and by hcv-only infected individuals (ic (t)) who are hiv infectious at rate θ2λh , where the parameter θi ≥ 1 i = 1, 2 captures the fact that singly-infected individuals have weak immune-system (due to illness) compared to wholly-susceptible individuals. the population suffers death due to the co-infection of both diseases (at a rate δhc ). dihc dt = λhc s + θ1λc ih + θ2λh ic − (κψ + µ + δhc )ihc (5) the following system of equations describes the model based on the aforementioned assumptions and notations while the parameters are presented in table 1: ds dt = π + ψic (t) − (λh + λc + λhc ) s (t) −µs (t), dih dt = λh s (t) + κψihc (t) − θ1λc ih (t) − k1 ih (t), dic dt = λc s (t) − θ2λh ic (t) − k2 ic (t), dihc dt = λhc s (t) + θ1λc ih (t) + θ2λh ic (t) − k3 ihc (t), (6) with initial data s (t) = φ1(t) ≥ 0, ih (t) = φ2(t) ≥ 0, ic (t) = φ3(t) ≥ 0, ihc (t) = φ4(t) ≥ 0, for t ∈ [−τ, 0], (7) where φ1(t), · · · ,φ4(t) are continuous functions on the interval [−τ, 0] and k1 = µ + δh, k2 = ψ + µ + δc, k3 = κψ + µ + δhc . 2.1. basic properties in this part, we present results for basic qualitative properties of the model as follows. theorem 1. the solution (s (t), ih (t), ic (t), ihc (t)) of the model (6), with initial data (7), exists for all time, t > 0 and is unique. furthermore, the solution is nonnegative for all t > 0. proof. for existence and uniqueness of solution, we start as follows: let x(t) = (s (t), ih (t), ic (t), ihc (t)). the system (6), can be represented as dx dt = f (t, xt), where xt(θ) = x(t + θ), f is lipschitz and continuous in x. it follows from theorems 2.1, 2.3 in [38] that the model (6) has a unique solution (s (t), ih (t), ic (t), ihc (t)) satisfying the initial data (7). the positivity of solution is proved by the method of contradiction as shown below: suppose the conclusion is not true. under the given initial function, there exists a time, t1 ∈ [0, +∞) where s (t) will changes sign, at least once, so that s (t) > 0 for all t ∈ (0, t1), s (t1) = 0 and s (t) < 0 for t > t1. furthermore, ih (t) > 0, ic (t) > 0, ihc (t) > 0 for t ∈ (0, t1); or a time t2 such that ih (t) > 0 for all t ∈ (0, t2), ih (t2) = 0 when s (t) > 0, ic (t) > 0, ihc (t) > 0 for t ∈ (0, t2); or a time t3 such that ic (t) > 0 for all t ∈ (0, t3), ic (t3) = 0 when s (t) > 0, ih (t) > 0, ihc (t) > 0 for t ∈ (0, t3); or a time t4 such that ihc (t) > 0 for all t ∈ (0, t4), ihc (t4) = 0 when s (t) > 0, ih (t) > 0, ic (t) > 0 for t ∈ (0, t4). for the first case, consider the first equation in (6), thus ds dt (t1) = π + ψic (t1) − (λh + λc + λhc ) s (t1) −µs (t1), = π + ψic (t1), hence s (t1) = s (θ) ∫ t 0 exp (π + ψic (t1))du, > 0. this is a contradiction to the earlier assumption that s (t1) = 0. therefore, t1 doesn’t exists, hence s (t) > 0 for all t > 0. similarly for the second argument, consider the second equation in (6): dih (t2) dt = λh s (t2) + κψihc (t2) − θ1λc ih (t2) − k1 ih (t2), = λh s (t2) + κψihc (t2), hence ih (t1) = ih (θ) ∫ t 0 exp (λh s (t2) + κψihc (t2))du, > 0. this also contradicts the earlier assumption that ih (t2) = 0. therefore, there is no such time t2 exists. hence ih (t) > 0 for all t > 0. similar arguments can be applied to other two equations to show that ic > 0 and ihc > 0 for all t > 0. � theorem 2. the biologically-feasible region ω is positivelyinvariant for the model (6), where ω = { (s, ih, ic, ihc ) ∈ r4+ : s + ih + ic + ihc ≤ π µ } . 3 hassan & hussaini / j. nig. soc. phys. sci. 3 (2021) 1–11 4 table 1: table 1: description of model (6) parameters parameter interpretation π recruitment rate of susceptible humans by birth ψ recovery rate of hcv λh hiv-only infection rate λc hcv-only infection rate λhc hiv and hcv simultaneous infection rate τ1, τ2 latent period for hiv, hcv respectively βh contact rate for hiv-only infection βc contact rate for hcv-only infection βhc contact rate for hiv and hcv simultaneous infection η > 1 assumed increase in infectiousness of dually infected to single infection 0 < κ < 1 modification parameter for reduced dually infected to single infection µ natural death rate in all classes δh, δc, δhc hiv, hcv, hiv/hcv induced death rates θ1 ≥ 1,θ2 ≥ 1 modification parameters for singly-infected to wholly susceptibles proof. adding the equations in (6) gives dn(t) dt = π −µn(t) − [δh ih (t) + δc ic (t) + δhc ihc (t)], (8) so that, dn(t) dt ≤ π −µn(t), (9) with positivity of ih, ic, ihc . it follows from (9), using gronwall lemma [39], that n(t) ≤ n(0)e−µ(t) + π µ [ 1 − e−µ(t) ] . so that, if n(0) ≤ π µ , then n(t) ≤ π µ . therefore, ω is positively invariant. � 3. existence and stability of equilibria at equilibrium, equations in (6) are equated to zero. in the absence of diseases, the dfe is obtained to be e0 = (s ∗, i∗h, i ∗ c, i ∗ hc ) = ( π µ , 0, 0, 0 ) . (10) it should be noted that at equilibrium, ih (t−τ) = ih (t) = i∗h = 0, ic (t −τ) = ic (t) = i∗c = 0, and ihc (t) = ihc (t −τ) = i ∗ hc = 0. however, when there are diseases in the community, ih (t − τ) = ih (t) = i∗∗h , ic (t − τ) = ic (t) = i ∗∗ c and ihc (t) = ihc (t − τ) = i∗∗hc . hence the following gives the implicit values for the endemic equilibrium: e∗∗ = (s ∗∗, i∗∗h , i ∗∗ c , i ∗∗ hc ), (11) where s ∗∗ = π + ψi∗∗c λ∗∗h + λ ∗∗ c + λ ∗∗ hc + µ , i∗∗h = λ∗∗h s ∗∗ + κψi∗∗hc θ1λ ∗∗ c + k1 , i∗∗c = s ∗∗λ∗∗c θ2λ ∗∗ h + k2 , i∗∗hc = λ∗∗hc + θ1λ ∗∗ c i ∗∗ h + θ2λ ∗∗ h i ∗∗ c k3 , λ∗∗h = βh e−µτ [ i∗∗h + ηi ∗∗ hc ] n∗∗ , λ∗∗c = βc e−µτ [ i∗∗c + ηi ∗∗ hc ] n∗∗ , and λ∗∗hc = βhc e−µτi∗∗hc n∗∗ . (12) the local asymptotic stability of a given equilibrium of the model (6), is determined by linearizing the system about the equilibrium point [40, 41], and showing that all the roots of the transcendental polynomial have negative real parts. thus linearizing the system (6) about the following variables: s (t), ih (t), ic (t), ihc (t), s (t −τ), ih (t −τ), ic (t −τ), ihc (t −τ) yields  dŝ (t) dt dîh (t) dt dîc (t) dt dîhc (t) dt  = j  ŝ (t) îh (t) îc (t) îhc (t) ŝ (t −τ) îh (t −τ) îc (t −τ) îhc (t −τ)  , (13) where ŝ (t) = s (t) − s ∗, îh (t) = ih (t) − i∗h , îc (t) = ic (t) − i ∗ c , îhc (t) = ihc (t) − i∗hc , j = [ j1 j2 ] , and j1 =  a1 a2 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15  , 4 hassan & hussaini / j. nig. soc. phys. sci. 3 (2021) 1–11 5 j2 =  0 −βh e −µτs n − βc e−µτs n − e−µτ(βhη+βcη+βhc )s n 0 βh e −µτs n − θ1βc e−µτih n e−µτ(βhηs−θ1βcηih ) n 0 −θ2βh e −µτic n βc e−µτs n e−µτ(βcηs−θ2βhηic ) n 0 θ2βh e −µτic n θ1βc e−µτih n e−µτ(βhc s +θ1βcηih +θ2βhηic ) n  , with a1 = a2 − (λh + λc + λhc + µ), a2 = (λh + λc + λhc ) s n , a3 = −(βhη + βcη + βhc − λh − λc − λhc ) s n , a4 = −λh s n + λh + θ1λc ih n , a5 = −λh s n − θ1λc + θ1λc ih n − k1, a6 = −λh s n + θ1λc ih n , a7 = a6 + βhη s n − θ1βcη ih n + κψ, a8 = a9 + λc, a9 = −λc s n +θ2λh ic n , a10 = a9+θ2λh, a11 = a9+βcη s n−θ2βhη ic n , a12 = λhc s n +λhc−θ1λc ih n −θ2λh ic n , a13 = −λhc s n +θ1λc−θ1λc ih n − θ2λh ic n , a14 = λhc s n + θ2λh − θ1λc ih n − θ2λh ic n and a15 = −λhc s n −θ1λc ih n + θ1βcη ih n + θ2βhη ic n + βhc s n + θ2λh ic n − k3. assuming x̂(t) = dezt, where x̂(t) = (ŝ , îh, îc, îhc ), is a solution with d = (d1, ...d4), be constants and z is a complex number. equation (13) can be express as:[ j1 + e −zτj2 ] ĵ = 0, where ĵ = [d1ezt d2ezt d3ezt d4ezt]t and 0 is 4 × 1 zero matrix. therefore for nonzero solution, the transcendental equation at any equilibrium is given by: det(j1 + e −zτj2) = 0. (14) however, before we determine the stabilities of equilibria, it is important to calculate the basic reproduction number of the model, denoted by r0. it can be recall that, susceptible humans acquire hiv, hcv infections and hiv-hcv simultaneously following effective contact with infected individuals with these diseases. it follows that, the number of hiv infectives generated by an infected human (near the dfe) is given by the product of the infection rate ( βhn∗ ), the probability that an infected human survives natural death (e−µτ) and the average duration of stay in the infected hiv class ( 1k1 ). hence, the average number of hiv infections generated by infected individual is given by rh = βh e−µτs ∗ k1 n∗ . (15) similarly, the number of hcv infections generated by an infected individual with hcv (near the dfe), is given by the product of the infection rate ( βcn∗ ), the probability that an infected human survived natural death (e−µτ) and the average duration of stay in the infected hcv class ( 1k2 ). therefore, the average number of hcv infected generated by infectious individual is given by rc = βc e−µτs ∗ k2 n∗ . (16) furthermore, the number of hiv-hcv simultaneous infections generated by an infected individual with the two diseases (near the dfe), is given by the product of the infection rate of infected hiv-hcv ( βhcn∗ ), survival probability (e −µτ) and the average duration of stay in the class ihc ( 1 k3 ). thus, the average number of hiv-hcv infections generated by infected individual is given by rhc = βhc e−µτs ∗ k3 n∗ . (17) finally, the maximum of the three sub thresholds in (15), (16) and (17) gives the value of r0, i.e. the average number of new infections generated by infected individuals (with hiv, hcv and hiv-hcv) is given by r0 = max { βh e−µτ k1 , βc e−µτ k2 , βhc e−µτ k3 } , = max{rh,rc,rhc}, (18) noting that at dfe, s ∗ = n∗ = π µ . it is worth stating here that, the formulation of r0 in (18) is derived from the procedure in [12] used in a model that have more than one strains/diseases, to be the maximum of various sub thresholds for the infections. moreover, similar approaches are also applied in [46, 42]. 3.1. local asymptotic stability of dfe at dfe, we claim the stability properties as follows: theorem 3. the dfe e0, is locally asymptotically stable (las) in ω whenever r0 < 1 for all τ ≥ 0 and unstable if r0 > 1. proof. at dfe, the transcendental equation (14) is reduced to g(z) = (z + µ)(z + k1 −βh e −τ(z+µ)) (19) ×(z + k2 −βc e −τ(z+µ)) (20) ×(z −βhc e −τ(z+µ) + k3) = 0. (21) expanding equation (19), we have g(z) =z4 + p1z 3 + p2z 2 + p3z + p4 = e−zτ[q1z 3 + q2z 2 + q3z + q4], (22) where p1 = µ + k1 + k2 + k3, p2 = [µk1 + (µ + k1)k2 + µ + k1 + k2]k3(1 −rhc ), p3 = [(µ + k1)k2 + µk1 + µk1 k2]k3(1 −rhc ), p4 = µk1 k2 k3(1 −rhc ), q1 = (βh + βc + βhc )e −µτ, q2 = [µ(k2βh + βc ) + k1βc (1 −rh )e −zτ + βh + βc ](1 −rhc )k3e −µτ, q3 = [µk1βc (1 −rh ) + (k2 + 1)βh + k2βh (1 −rc )e −zτ](1 −rhc )k3e −µτ, q4 = µe −µτ[k1βc + k2βh (1 −rc )e −zτ]k3(1 −rhc ). first we determine the stability of e0 when τ = 0. from (19), when τ = 0, it is obvious that the eigenvalue −µ is negative, while βh − k1,βc − k2 and βhc − k3 are negative whenever rh < 1,rc < 1 and rhc < 1 or in general if r0 < 1. hence, e0 at τ = 0 is stable when r0 < 1. 5 hassan & hussaini / j. nig. soc. phys. sci. 3 (2021) 1–11 6 further, we determine the stability of e0 when τ > 0. here we investigate whether there is a root z = iy of (19), y ∈ r+, that may cross the imaginary axis and cause stability switches. substituting z = iy in (19) or equivalently in (22), we find y that must satisfy g1(y) =| p1(iy) | 2 − |q1(iy) | 2, (23) where p1(iy) = (y4 − p1iy3 − p2y2 + p3iy + p4), q1(iy) = (−q1iy3 − q2y2 + q3iy + q4). however, (22) is implicit in e−zτ, therefore the process of separating real, imaginary parts and squaring (23) may not lead to explicit expression in y. without loss of generality, one can see from (19) that g(0) = µk1 k2 k3[(1 −rh )(1 −rc )(1 −rhc )] > 0 if r0 < 1 and g(y) = +∞ as y → ∞. therefore, when τ > 0, equation (19) has no positive root on (0, +∞), hence the dfe is las for all τ ≥ 0 if r0 < 1 and unstable if r0 > 1. the biological implication of this result is that the diseases can be completely eliminated whenever the initial population is close to the basin of attraction of the dfe. � 3.2. global asymptotic stability of dfe to ensure complete eradication of the diseases in the population irrespective of the population size started with, we present and prove the following result. theorem 4. the dfe e0, is globally asymptotically stable (gas) in ω whenever r0 < 1 for all delay τ ≥ 0. proof. the global properties can be established using a comparison theorem [23] and the approach in [43]. the equations of the three infected components in model (6) can be expressed using matrix-vector form as dih (t) dt dic (t) dt dihc (t) dt  = (f −v1)  ih (t −τ) ic (t −τ) ihc (t −τ)  −v2  ih (t) ic (t) ihc (t)  − ( 1 − s n )  βh e−µτ 0 βhη 0 βc e−µτ βhη 0 0 βhc   ih (t −τ) ic (t −τ) ihc (t −τ)  , (24) where f =  βh e−µτ 0 0 0 βc e−µτ 0 0 0 βc  , v1 =  0 0 −κψ 0 0 0 0 −θ1λc −θ2λh  and v2 =  k1 + θ1λc 0 0 0 k2 + θ2λh 0 0 0 k3  are the matrices for new infection terms and transitions respectively. since s ≤ n at every time t in ω and by letting v = v1 + v2, it follows from (24) that dih (t) dt dic (t) dt dihc (t) dt  ≤ (f −v)  ih (t) ic (t) ihc (t)  . (25) it can be shown that the spectral radius of fv−1 gives the value of r0 in (18). furthermore, since the eigenvalues of the matrix (f − v) have negative real parts if r0 < 1 [43, 44], consequently, the linearized differential system (25) is stable whenever r0 < 1. as a result, lim t→∞ (ih (t), ic (t), ihc (t)) → (0, 0, 0). (26) therefore, from the first equation in (6) and substituting ih (t) = ic (t) = ihc (t) = 0 from (26), for any t, it follows that ds dt = π −µs (t), (27) so that s (t) = π µ . (28) thus, lim t→∞ (s (t), ih (t), ic (t), ihc (t)) = ( π µ , 0, 0, 0 ) . (29) by lasalles invariance principle [45] e0 is the largest invariance set in ω, hence is gas whenever r0 < 1. � 3.3. stability analysis of endemic equilibrium here, due to the complexity of transcendental equation, we give conditions for stability of any unique endemic equilibrium (ee), whenever it exists. from (12) the transcendental equation (14) is simplified to g2(z)= det  z − a1 −a2 + βh e −µτs n −a2 + βc e −µτs n −a3 −a4 z − a5 − βh e −µτs n −a6 θ1βc e −µτih n a7 −a8 −a9 + θ2βh e −µτic n z − a10 βc e −µτs n a11 a12 −a13 θ2βh e −µτic n −a14 − θ1βc e −µτih n z − a15  = 0, = p2(z) + q2(z)e −zτ = 0, (30) where p2(z) and q2(z) are polynomials of degree 4 and 3 respectively, with real coefficients and hence have no common imaginary roots whenever r0 > 1. furthermore, substituting z = iy, as purely imaginary root, we define g2(y) =| p2(iy) | 2 − |q2(iy) | 2, (31) which is a polynomial of degree 8 whose leading coefficient is positive. suppose (31) has at least zero positive root, then the following stability result for e∗∗ can therefore be stated theorem 5. if the polynomial g2(y) has (a) no positive root, then e∗∗ is locally asymptotically stable for all delays τ ≥ 0 whenever r0 > 1. 6 hassan & hussaini / j. nig. soc. phys. sci. 3 (2021) 1–11 7 (b) at least one simple positive root, then as delay increases there will be n ∈ z+ number of stability switches for fixed parameter values and the endemic equilibrium e∗∗ is locally asymptotically stable for 0 ≤ τ < τ∗ if r0 > 1, here, τ∗ = cot−1 ( − p2req2re +p2imq2im −p2req2im +p2imq2re ) y and the subscripts represents the real and imaginary parts of p2 and q2. 3.3.1. interior endemic equilibrium here, we consider one of the interior equilibria with simultaneous infection of hiv and hcv only. in the absence of hiv and hcv only infections, equation (12) will be reduced to e∗∗3 = (s ∗∗, 0, 0, s ∗∗(rhc − 1)), where s ∗∗ = πrhc e−µτβhc (rhc − 1) + µrhc , (32) provided rhc , e−µτβhc µ+e−µτβhc . similarly, the transcendental equation (30) is simplified to be g2(z) = p2(z) −q2(z)e −zτ, where p2(z) = z 2 − ( p1 + p4)z + (p1 p4 − p2 p3); q2(z) = −p5z + p1 p5 − p3 p5, (33) with p1 = − µk23 (rhc − 1) 2 e−µτβhc −µ; p2 = − µk3(rhc − 1) e−µτβhc ; p3 = µk23 (rhc − 1) 2 e−µτβhc ; p4 = − µk3(rhc − 1) e−µτβhc − k3; p5 = k3. when τ = 0, (33) yields g2(z) = z 2 − ( p1 + p4 + p5)z + ( p1 p4 − p2 p3) + ( p1 p5 + p3 p5), (34) so that after simplification, we have − ( p1 + p4 + p5) = k3(rhc − 1) βhc × [1 + k3(rhc − 1)] + µ, ( p1 p4 − p2 p3) + (p1 p5 + p3 p5) = δhc k 2 3 (rhc − 1) 2 + µk3(rhc − 1). (35) from (35), it can be seen that all the coefficients of g2(z) are positive whenever rhc > 1. therefore, g2(z) is stable (all the roots have negative real parts). furthermore, if τ > 0, we let z = iy be the root of g2(z) in (34) then from (31), separating real, imaginary parts and squaring gives f2(y) = y 4 + (−2a22 + a 2 11 − a 2 33)y 2 + a222 − a 2 44 = 0, (36) where a11 = p1 + p4, a22 = p1 p4 − p2 p3, a33 = p5 and a44 = p1 p5 + p3 p5. here, − 2a22 + a 2 11 − a 2 33 = k23 (rhc − 1) 4 e−2µτβ2hc + 2µk3(rhc − 1)2 e−µτβhc + 2k33 (rhc − 1) 2 e−2µτβ2hc + 2µk33 (rhc − 1) e−µτβhc + k23 [i ∗∗2 hc − 1] s ∗∗2 + µ2 + k3, (37) and a222 − a 2 44 = (a22 − a44)(a22 + a44), = [ δhc (rhc − 1)2 e−µτβhc + k3(rhc − 1) + µ + µk3 ] [ k3(rhc − 1)2 e−µτβhc + µk3(rhc − 1) e−µτβhc + 2µk3 ] . (38) from (37) and (38), if rhc > 1, so also i∗∗2hc > 1, therefore f2(y) in (36) will have no positive root y that can cross the imaginary root as the delay is increased. hence, the interior equilibrium e∗∗3 is absolutely stable for all delay, τ ≥ 0. therefore, we claim the following result: theorem 6. the endemic equilibrium e∗∗3 , (32), is locally asymptotically stable for any delay, τ ≥ 0 when rhc > 1. 3.4. analysis for impact of delay the impact of the time delay is analyzed using threshold approach to show whether increase (decrease) of the delay will have effect on the infectivity of the two diseases in the community. we use the threshold quantity, r0, basic reproduction number, r0 = max{rh,rc,rhc}. (39) as a function of the delay τ. since r0 = 1 is a threshold value for the infection in the population, we determine the critical delay value τcrit above (below) which the diseases have effect as follows: βhβcβhc e−3µτ k1 k2 k3 = 1, e−3µτ = k1 k2 k3 βhβcβhc , τ = τcrit = ln ( k1 k2 k3 βhβcβhc ) −3µ . (40) 7 hassan & hussaini / j. nig. soc. phys. sci. 3 (2021) 1–11 8 0 500 1000 1500 0 0.5 1 1.5 2 2.5 x 10 4 time (days) p op ul at io ns o f s , i h + i c + i h c s i h +i c +i hc (a) 0 500 1000 1500 0 5000 10000 15000 time (days) p op ul at io ns o f s , i h , i c , i h c s i h i c i hc (b) 0 500 1000 1500 0 5000 10000 15000 time (days) p op ul at io ns o f s , i h , i c , i h c s i h i c i hc (c) 0 500 1000 1500 0 0.5 1 1.5 2 x 10 4 time (days) p op ul at io ns o f s , i h , i c , i h c s i h i c i hc (d) figure 1: numerical simulations of model (6), with parameter values from table 2 with (a) the gas of dfe, with βh = 0.002 = βc, βhc = 0.0025, τ = 3. in (b) and (c) βh = 0.002 = βc, βhc = 0.0025, τ = 1 and τ = 10 respectively (d) βh = 0.04 = βc, βhc = 0.045 and τ = 50. table 2: baseline values for the parameters of model eq. 6 parameter baseline values references π 300 day−1 assumed ψ 6.8 × 10−5 day−1 [46] βh variable assumed βc variable assumed βhc variable assumed η 2 assumed κ 0.013 assumed µ 170 day −1 [46] δh 9.12 × 10−4 day−1 [47] δc 9.5 × 10−5 day−1 [46] δhc 9.32 × 10−4 day−1 assumed θ1, θ2 2, 2 assumed 8 hassan & hussaini / j. nig. soc. phys. sci. 3 (2021) 1–11 9 0 200 400 600 800 1000 0 100 200 300 400 500 time (days) in fe ct ed h iv p op ul at io n 20 40 60 200 300 400 (a) 0 500 1000 1500 0 0.5 1 1.5 2 x 10 4 time (days) p op ul at io ns o f s , i h , i c , i h c s i h i c i hc (b) 0 500 1000 1500 0 2000 4000 6000 8000 10000 time (days) c um ul at ve n ew c as es τ=1 τ=15 τ=25 τ=35 (c) figure 2: simulations of model (6), with parameter values as shown in table 2, in (a) showing no oscillation with βh = 0.02 = βc, βhc = 0.025, τ = 30 (b) displaying the interior equilibrium, with βh = 0.0002 = βc, βhc = 0.025, rhc = 1.62, while in (c) the effect of time delay on the infectivity of the viruses with βh = 0.04 = βc, βhc = 0.045, using time delays τ = 1, 15, 25 and τ = 35. the rate of change of r0 with respect to τ is given by ∂r0 ∂τ = ∂ ∂τ max (rh,rc,rhc ) , = ∂ ∂τ ( βhβcβhc e−3µτ k1 k2 k3 ) , = −3µ βhβcβhc e−3µτ k1 k2 k3 . (41) since the derivative in (41) is negative, it indicates that r0 decreases with increase of the time delay τ and vice varsa. this implies that, increasing the delay beyond the threshold value, with fix parameter values, will results in decreasing the number of infected individuals with the two diseases and those that are simultaneously infected. this result can be summarized as proposition 1. the increase in time delay in the model (1) will reduces the infectivity of the two diseases whenever the delay is greater than the threshold value τcrit. 4. numerical simulations numerical experiments are conducted to illustrate the dynamics as described in theoretical results and display the impact of delay in the model. figure 1 (a) depicts the global asymptotic stability for the dfe using parameter values from table 2, except for βh = βc = 0.002, βhc = 0.0025, τ = 3 and different initial conditions so that r0 = 0.16. it can be seen that the susceptible and total populations of infectives converges to π µ and zero asymptotically, respectively. in figures 1 (b)–(d), the local stability of endemic equilibrium is displayed using values from table 2 except for in (b) τ = 1, βh = βc = 0.02, βhc = 0.025, (c) τ = 10 βh = βc = 0.002, βhc = 0.0025, and (d) τ = 30 , βh = βc = 0.04, βhc = 0.045, as proved in theorem 5(a), so that r0 = 1.60, r0 = 1.42 and r0 = 1.45 respectively. in figure 2 (a), it is shown that the length of delay (τ = 30) doesn’t cause any oscillation in the stability of the endemic equilibria as observed in some delay models. figure 2 (b), illustrates the stability if one of the interior equilibrium, e∗∗3 using parameter values in table 2 while βh = βc = 0.0002, βhc = 0.025, so that rhc = 1.62. in this case, one or simultaneous infection will occur. the effect of delay is illustrated in figure 2 (c) to support proposition 1 in which increasing the delay, τ from 1, 15, 25 to 35, caused remarkable decrease in the total number of infected individuals with hiv and hcv. 5. concluding remarks in this work, a delay model of simultaneous infection of hiv and hcv is formulated and dynamically analyzed. the novelty (results) of the research is summarized and discussed below: (i) basic properties (existence, boundedness and positivity of solution)of the model are stated and proved as theorems 1 and 2. 9 hassan & hussaini / j. nig. soc. phys. sci. 3 (2021) 1–11 10 (ii) the basic reproduction threshold is systematically obtained as the maximum of subthreshold values of the individual viruses; a threshold parameter above which the viruses will persist in the population. (iii) a disease free equilibrium is found to be globally asymptotically stable when the basic reproduction threshold is less than unity for any length of time delay. under this case, the diseases will die out in the population whenever the basic reproduction number is brought and maintained below one irrespective of the initial populations started with. this is shown in figure 1(a). (iv) however, if the basic reproduction threshold is greater than unity, endemic equilibrium exists which is shown to be locally asymptotically stable for all values of the incubation period (delay) as shown in figures 1(b – d). this implies that whenever the initial population is within the basin of attraction of the endemic equilibrium and the basic reproduction number is greater than one, the diseases will persist in the population no matter the length of delay. (v) increasing the length of time delay is observed to be decreasing the number of cumulative infectious people when the delay is above a critical value. it is worth remarking here that although the time delay has no effect on the stability of equilibria however, it do affect the infectivity of the viruses as shown in the formulation of basic reproduction number. the implication of this result is, if the incubation period can be extended by say intervention, the number of infected people will be reduced. 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[47] z. mukandavire, w. g.chiyaka, “asymptotic properties of an hiv/aids model with a time delay”, journal of mathematical analysis and applications 330 (2007) 933. 11 j. nig. soc. phys. sci. 4 (2022) 1050 journal of the nigerian society of physical sciences structural, optical and electrical properties of v2o5 thin films by spray pyrolysis method a. sherin fathima, i. kartharinal punithavthy∗, s. johnson jayakumar, a. rajeshwari, a. sindhya, a. muthuvel department of physics, t.b.m.l college (affiliated to bharathidasan university, tiruchirapalli-620024), porayar, tamil nadu-609307, india. abstract vanadium pentoxide (v2o5) and indium tin oxide (ito) coated v2o5 thin films were prepared by hydrothermal technique. the influence of different molarity of v2o5 on the electrical properties of ito-v2o5 films was investigated. the films were found to be polycrystalline and to belong to a v2o5 orthorhombic crystal system based on their x-ray diffraction (xrd) analysis. it was confirmed by fourier transform infrared (ftir) spectrum that v2o5 functional groups formed a v-o bond. the optical band gaps were found to increases with increasing molarity in the range 2.18 – 2.89 ev. scanning electron microscopy (sem) analysis shows crumbled paper-like morphology in the sheet-like layer arrangement. the dielectric current (dc) electrical conductivity was studied as function of molarity which indicated the semiconducting nature. the morphological and structural studies show enhanced results for 4% sample which makes it a viable candidature for optical and electrical applications. doi:10.46481/jnsps.2022.1050 keywords: v2o5 thin films, optical properties, surface morphology. spray pyrolysis, electrical properties. article history : received: 09 august 2022 received in revised form: 14 october 2022 accepted for publication: 17 october 2022 published: 27 november 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published articles title, journal citation, and doi. communicated by: k. sakthipandi 1. introduction transition metal oxides, especially vanadium compounds gain immense interest due to their special physical and chemical properties, making them technological useful for nanoscale device application [1-7]. a thin film of transition metal oxides has been a wide platform of research in recent years due to their application in technological aspects. thin films of v2o5 laid footprints in the sectors of gas sensors, solar cells, and batteries [8-9]. vanadium combines with oxygen to form a variety of compounds with varying structural, optical, and chemical properties [10]. ∗corresponding author tel. no: +91-9442422539 email address: profpunithavthy@gmail.com, profpunithaphysics@gmail.com (i. kartharinal punithavthy) various phases of vanadium oxides possess different properties based on their structural arrangement of atoms. the different phases like v2o5, v2o3, vo2 and vo exhibit different properties. the different phases of vanadium oxides can be achieved through the modifications in the deposition process as well as due to the modification in the post-annealing line process. although there exist several phases of vanadium oxides, vo2 and v2o5 have involved the consideration of the researchers, due to their thermochromic behaviour and electrochromic properties respectively. one of the thermodynamically stable phases of vanadium oxides is vanadium pentoxide thin films, which are also used in smart windows, optical filters, and surfaces with adjustable emittance for temperature control [11]. 1 fathima et al. / j. nig. soc. phys. sci. 4 (2022) 1050 2 in recent years, nanostructured vanadium oxide has been attracted considerable attention due to its organic and physical properties and their countless prospective for applications in catalysis [12] electrochromic devices [13] sensors [14] electrochemistry [15] photocatalytic activities [16] and spintronic devices [17]. v2o5, one of the more stable vanadium oxide phases, has a distinctive set of features and is one of the compositions of vanadium oxide that may be formed [18]. this substance is also employed as an intercalation chemical due to its two-dimensional layer structure. since the discovery of the reversible electrochemical lithium-ion intercalation in v2o5 [19]. due to its low cost, simplicity, synthesis, abundance and high energy density, vanadium pentoxide has received a lot of attention as a potential cathode material for rechargeable lithium-ion batteries. it is a typical embolism compound with a covered crystal structure that allows for the reversible intercalation and extraction of a wide range of atomic and molecular species. the energy storage and the charge/discharge rate are the most crucial factors for electrochemical pseudo capacitor applications. to obtain a high charge/discharge rate, a higher surface range and simple charge conveyance are needed [20]. v2o5 xerogel and aerogel, which together offer large surface range, have been travelled for numerous applications. shenzhen deng et al. recently reported electrochemical performance of vanadium oxide-based cathodes in aqueous zinc -ion batteries [21]. an ultrasound assisted synthesis of v2o5 nanoparticles for photocatalyst and antibacterial activity was reported by karthik et al [22]. the hydrothermal method used by muhammad rafique et al. for degradation of rhb dye uses highly efficient and visible light driven ni doped v2o5 photocatalysts [23]. using ar/o2 gas mixtures, benmoussa et al. successfully fabricated v2o5 thin films on indium tin oxide coated glasses [24]. in the present work, the deposition procedure of v2o5 films on the fused substrate was carried out first. then the v2o5 thin film deposition on the ito coated fused silica substrate was investigated. the prepared thin film was characterized by the xrd, ft-ir, sem with edax and uv-visible spectroscopy. the electrical studied is measured with the v2o5 thin film deposited ito coated fused silica substrate as a working film. 2. experimental procedure the thermal evaporation of pure v2o5 powder (purity 99.99 percent acquired from merck) from an electrically heated molybdenum boat held at 1823 k in a vacuum better than 8 x 10−6 torr was used to creating thin coatings of v2o5 on ito substrates. the experimental films were deposited using a hind high vacuum coating machine. the final pressure was generated using a diffusion pump supported by a rotary pump. ito substrate had been thoroughly cleaned and the appropriate covers were mounted on a copper holder that was set up on a tripod inside the bell jar. the distance between the source and substrate was set in cm. the glow discharge was started to further ionically clean the substrates in the vacuum chamber after achieving the ultimate vacuum of 5 x 10−6 torr and the desired substrate temperature in the chamber and permitted to enter the system. the substance inside the boat evaporated when the figure 1: flow chart for the synthesis of v2o5 thin films through spray pyrolysis method figure 2: xrd patterns of (a) pure v2o5 (b-c) 2 and 4 % ito coated v2o5 thin film power was applied, and the resulting vapours’ reaction with the oxygen gas resulted in the deposition of a coating on the sub2 fathima et al. / j. nig. soc. phys. sci. 4 (2022) 1050 3 figure 3: (a-c) sem image (pure, 2 and 4 %) (d) edax of 2% molarities v2o5 thin films figure 4: surface occupancy plots of thin films (a) pure v2o5, (b) v2o5 at 2% and (c) 4% molarities figure 5: ftir spectra of (a) pure v2o5 and (b-c) v2o5 with 2 and 4 % of molarities strate. using an optical pyrometer, the boat’s temperature was measured during the deposition process. following the maintefigure 6: optical absorption spectra of (a) pure v2o5 thin films and (b-c) v2o5 thin films with 2 and 4 % of molarities figure 7: band gap energy of (a) pure v2o5 thin films and (b-c) v2o5 thin films with 2 and 4 % of molarities nance of the substrates at the necessary deposition temperature, the molybdenum boat with the v2o5 powder was positioned. when the boat’s temperature reached around 1823 k, the shutter covering the substrates was opened, and it remained open during the film deposition process. fig 1 shows the chart for the synthesis of v2o5 thin films through spray pyrolysis method. 2.1. characterization in the present study, jasco v-670 spectrophotometer was used to record optical spectra in the range of 300-800 nm. the crystalline phase structures of the products were examined and studied by x-ray diffractometer (riakgu mini flexell) operated at a 40 kv and current 30 ma with cukα (λ = 1.5406 å). the interaction of functional groups in the synthesized nanoparticles was investigated using a fourier transform infrared (ft-ir) spectrometer (bruker; rfs 27). 3. results and discussions 3.1. xrd analysis the structural analysis of ito-coated v2o5 thin films with two different molarities was carried out by xrd as shown in 3 fathima et al. / j. nig. soc. phys. sci. 4 (2022) 1050 4 figure 8: electrical studies of pure and ito (2 and 4%) coated v2o5 thin films table 1: structural parameters of pure and ito (2 and 4%) coated v2o5 thin film s. no sample average crystallite size ( d) x10−9m average dislocation density (δ) x1014m2 average microstrain ( ε) x10−3m 1 pure 32.0 1.0087e+15 0.00113 2 2% 29.5 1.14061e+15 0.00122 3 4% 26.8 1.38471e+15 0.00134 table 2: vibrational assignments of thin films (pure v2o5 thin films, v2o5 with 2, 4% of molarity) s/n vibrational assignments (cm−1) inference references a b c 1 421 421 421 metal oxide stretching vibrations [30] 2 1412 1412 1412 ch2 bending vibration of carbon chain [33] 3 3199 3199 3199 oh-stretching vibration [32] table 3: absorption and band gap of pure and 2 and 4 % ito coated v2o5 thin tilms composition absorbance bandgap (ev) pure v2o5 362.8 2.18 2% 361.3 2.42 4% 360.5 2.89 fig 2. the diffraction peaks at 2θ = 21.50◦, 30.71◦, 38.41◦ and 45.7◦correspond respectively to reflections from planes (101), (400), (211) and (002) indicating the formation of orthorhombic crystal structure of v2o5 which is agreed with the jcpds card no.: jcpds no.: 89-2482 [25]. other peaks are due to the ito coated glass substrate. as can be seen in all xrd spectra, the peaks are enhanced with different molarity as a sign of improved crystallinity (fig 2). rathika et al. [26], synthesized v2o5 thin films deposited by the spray pyrolysis method and reported a similar orthorhombic structure [26]. the obtained high intense peak at 35.5o corresponds to plane values (201) respectively. the average crystallite size can be calculated from the debye scherrer equation [27-28]. d = kλ β cos θ (1) where λ is wavelength (1.54 å), k is scherrer’s constant, β is full width at half maximum (fwhm) of the diffraction spectra, and θ is diffraction angle. due to the ionic radius of v2o5, the average crystallite size of the thin films decreases from 32 to 26.8 nm with increasing molarities. with an increase in molarity percentage, the minute peaks disappear in comparison to pure v2o5. furthermore, together with v2o5 peaks, vo2 peaks are also observed in the xrd pattern of the undoped film, which may due to high deposition temperature [29]. as molarities increase, the average dislocation density and average microstrain of thin films are linearly increased. 3.2. sem with edax surface morphology of prepared v2o5 thin films with different molar concentrations (2 & 4%) coated on ito substrate. all the micrographs are taken at different magnifications. figure 3 (a-c) shows the stacked layers of v2o5 with a brick-like arrangement with the whole agglomeration of the particles undefined in the pure v2o5 thin film. but under 2 percent of 4 fathima et al. / j. nig. soc. phys. sci. 4 (2022) 1050 5 table 4: electrical parameters of pure and ito (2 and 4%) coated v2o5 thin films sample resistivity (ρ)(ω cm) conductivity (σ)(ω cm)−1 mobility (µ)(cm2/vs) pure 2.1x10−3 4.61x102 2 x101 2% 6.81x10−4 1.42x103 4.24x101 4% 2.08x10−3 4.75x102 3.9 x101 molarity, the thin film shows rough morphology with uniform aggregated grain structure. on further increases in molar percentage, the material forms crumbled paper-like morphology in the sheet-like layer arrangement. the increased molar concentration forms a smooth arrangement on comparing the obtained results of 4 % and can be utilized for further characterization and application. however, at 2% molarity, the crystal grain boundaries become blurred again, indicating crystal dissolution. as a result, it was proposed that the optimum molarity for better crystal quality of v2o5 films is between 2 and 4%. increased molarity may result in significant loss of oxygen concentration as well as several key elements of the film, deteriorating the crystal structure. high molarity, in fact, increases the rapid diffusion of v2o5 into the substrate layers, which can easily modify the characteristics of the transparent electrode. the presence of significant amounts of vanadium and oxygen on the deposited film was confirmed by elemental composition analysis of the film using edax in 2% molarity of ito coated v2o5 thin film (fig 3d). according to edax analysis, no further impurity contamination was discovered on the film. 3.2.1. surface occupancy plots figure 4 (a-c) depicts surface occupancy plots (sop) of thin films. using image j software, the surface occupancy plot of the deposited thin films was plotted. furthermore, the agglomeration was increased as traced in fig 4(b) as a result of the increased particle accumulation density, as evidenced by scanned sem images. the interactive 3d image as seen in fig 4 (c) had sheet-like particles uniformly distributed over the scanning surface area. 3.3. functional group analysis figure 5 shows the ft-ir spectrum of pure v2o5 produced at two different mole percentages (2% and 4%). the absorption of ir in molecular vibrations further confirms the presence of functional groups in thin films. a vibrational frequency of 421 cm−1 represents the stretching vibration of the metaloxygen link [30-31]. the strong, intense peak at 3199 cm-1 is caused by water molecules stretching vibration o-h bonds [32]. an initial precursor contained nitrate ions, as evidenced by the peak at, 1412 cm−1 caused by nitrate stretching vibration (no3−) [33]. 3.4. uv-visible analysis a uv-vis spectrum of samples of pure v2o5 at two different molarities is shown in fig 6, and a band gap calculation is shown in fig 7. as the molarity rises, the absorbance falls. transmittance drops abruptly between 300 nm and 800 nm due to absorption of most incoming photons. the fig 6 shows the region of fundamental absorption edge for ito coated v2o5 thin films with absorbance values of 364.8, 361.3, and 360.5 at pure, 2% and 4% molarities. as crystalline size decreases, the absorption edge shows a red shift in wavelength, which indicates a shift on the lower energy side [34-35]. ito-coated v2o5 thin films with different molarities have different optical properties depending on their microstructure and growth conditions. the optical band gap energies are 2.18, 2.42 and 2.89 ev for pure and ito coated v2o5 thin films at different molarity, respectively. according to the brus equation [36], the band gap energies increase with increasing molarities. as a result of quantum size effects [37], the bandgap widening in v2o5 thin films is even greater at low growth temperatures, where grain sizes are relatively small (≤50 nm). moreover, the mean crystal dimension decreases, resulting in a smaller band gap due to quantum size effects [38]. a vacancy in the lattice of v2o5 thin films creates lower absorption bands in the optical spectrum due to oxygen vacancies between two v-o layers. the disordered atomic arrangement causes this oxygen vacancy. 3.5. electrical studies at room temperature, the hall-effect was used to measure the transport parameters. fig. 8 depicts the alteration of electrical transport characteristics as a function of films. vanadium ions exhibit various semiconducting properties as a result of their various oxidation states, which is explained by the process of a 3d unpaired electron hopping from a v4+ to a v5+ ion. it is evident from fig. 8 that the thin film’s conductivity rises as a result of the creation of o vacancies, which is what is referred to as a low mobility n-type semiconductor. doping has a significant impact on the material’s mobility overall. the mixed-phase of v2o5 may be the cause of the low electrical conductivity of the undoped film in this instance. for 2% of the molar concentration, the resistivity and mobility of the thin films increase gradually and then decrease for 4% of the molar concentration, while the conductivity values are inversely proportional to the resistivity phenomenon, i.e., it decreases (for 2% of molar concentration) and then increases (for 4% of molar concentration). 4. conclusion spray pyrolysis method was successfully used to prepare v2o5 and ito coated v2o5 nanoparticles. the xrd pattern depicts the peaks of the orthorhombic v2o5 thin films that were prepared. the average crystallite size of pure v2o5, doped with 2 and 4% molarities, is 32, 29 and 26 nm, respectively. the size of the crystallites reduces as the molar percentages 5 fathima et al. / j. nig. soc. phys. sci. 4 (2022) 1050 6 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developed to provide new approaches. non-intrusive load monitoring (nilm) is an approach that was originally developed to allow the occupants of a room to identify the contribution of each appliance to the total electricity consumption of the room through a single point measurement device. the aim is to provide customers with information that will enable them to act as “consum’actors”, i.e., people who undertake to change their electricity consumption habits for an objective cause. the progress of artificial intelligence in its various forms (machine learning, big data, internet of things) have greatly contributed to increase the interest of nilm among researchers in different fields. indeed, some of them are adapting this concept to research areas such as water, transport, health, the environment and agriculture. in this context, applications in these fields have been developed to show the potential and benefits of using this approach. in addition to presenting non-intrusive load monitoring (nilm) in its general framework, this article presents the interests and applications of this approach in various fields. doi:10.46481/jnsps.2023.1208 keywords: non-intrusive load monitoring (nilm), power consumption, interests, applications article history : received: 25 march 2023 received in revised form: 06 may 2023 accepted for publication: 10 may 2023 published: 27 may 2023 c© 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: o. j. oluwadare 1. introduction the upheavals caused by climate change, the decline in fossil fuel resources and the increase in global energy demand are some of the reasons why we need to review the way we manage and consume electrical energy. to respond to these worrying challenges, solutions have been proposed, including demandside management (dsm). according to the french agency for the environment and energy management (ademe), dsm ∗corresponding author tel. no: +237 693 42 80 13 email address: wtokam@univ-lome.tg (leonce w. tokam ) refers to the grouping of energy saving actions implemented for the end consumer and not for the energy producer. in many countries around the world, few energy saving actions are promoted. this creates an imbalance between supply and demand that needs to be addressed. faced with this concern, which has major economic implications, technological solutions have been proposed to help reduce this imbalance and thus reduce co2 emissions. previous studies have shown that when energy supply companies communicate information on their consumption patterns to end-users of electricity, either directly or indirectly, the impact is that energy savings of over 12% are observed [1]. however, capitalizing on technological advances 1 tokam & ouro-djobo / j. nig. soc. phys. sci. 5 (2023) 1208 2 such as artificial intelligence and machine learning is helping to extend this type of action. for many scientists, the context is the development and improvement of existing solutions that can address these concerns. one of the solutions being exploited is load monitoring and specifically non-intrusive load monitoring (nilm), which is being used to reduce electricity consumption in households and buildings of all kinds. from the beginning, the nilm concept was limited to the field of electricity. the results obtained as well as the improvements regularly proposed, have greatly contributed to the renewed interest that is currently observed. the applications developed have aroused the interest of researchers in other fields to adapt this concept to their different fields of research. for the latter, it is a question of presenting the great advantage that nilm offers, through its applications in fields such as water, the environment, transport (maritime and rail), health (teleconsultation and assistance to elderly people living alone) and agriculture. in the rest of the paper, we briefly present the context and the different approaches to load monitoring, the general framework for the implementation of nilm, the interests and applications of nilm in the fields where it has been exploited, and finally a conclusion. 2. load monitoring, context and approaches real-time feedback on the electricity consumption of a house or other building to end consumers reflects the desire to influence consumption patterns, with the intention of improving them. the aim is to reduce energy demand by limiting over-consumption and waste. to achieve this, it is essential to have the load signatures (measurable electrical data) that can be detected by the installed measuring device(s). basically, it is a matter of monitoring electricity consumption by appropriate technological means. in energy management, load monitoring refers to the process of identifying and acquiring measurements of electricity consumption of the loads of an electrical system [2–4]. inspired by technological advances and particularly by machine learning methods, load monitoring is attracting a lot of interest [5]. depending on the approach used and the results sought, it can be intrusive or non-intrusive. 2.1. intrusive load monitoring intrusive load monitoring (ilm) is needed when it is necessary to go inside a house or other premise to install a measuring device for each appliance present. the measuring device can be a wireless sensor, a digital meter, or any other object that fulfils this role. this approach is laborious as it requires the installation. depending on the level of intrusion, three types of intrusive load monitoring can be distinguished [6,3]: type 1: metering device are installed at the circuit breaker and only measure the consumption of one part of the house. type 2: the metering devices are placed at the outlets, so as to monitor one or more appliances at the same time. type 3: a measuring device is installed on each appliance. despite the accurate measurement results, this approach is not very cost-effective for the reasons mentioned above. 2.2. non-intrusive load monitoring in contrast to the intrusive approach, non-intrusive load monitoring (nilm) refers to the process by which the overall electricity consumption of a building is identified without intrusion from a single point of measurement, and then the detailed consumption of each device is provided [2,7,8]. the nonintrusive approach is more advantageous than the intrusive one, as it requires only one sensor to be installed, very low maintenance, in addition to being affordable, and non-intrusive to the user [9]. moreover, it represents an interesting alternative to the intrusive approach, as stated by verma et al. [4] in their work, in addition to encouraging many researchers to adapt it to various research domains to obtain new results. for g. w. hart [10], who pioneered the concept of non-intrusive load monitoring, identifying the contribution of each appliance to the total electricity consumption of a house was the beginning of energy savings. the operation of the nilm integrates four (4) steps respectively: data acquisition, event detection, feature extraction and load identification. based on machine learning (ml) and artificial intelligence (ai) techniques, the nilm system identifies variations in voltage and current signals, which can be compared to fingerprints or signatures of each device. this identification is only possible thanks to dedicated algorithms belonging to the supervised, semi-supervised and unsupervised learning algorithm families [11]. 3. general framework and concept of non-intrusive load monitoring in nilm, each step performs a specific task, so that the desired objective (energy disaggregation) is easily achieved. the success of the process of disaggregating the total energy consumed into individual load consumptions provides information on the level of accuracy and performance of the chosen feature extraction and learning methods. in the following, we present the contribution of each step in the nilm process. 3.1. data acquisition the energy consumed in an electrical installation is counted by an electrical meter and therefore represents a data that can be exploited in several ways. in the nilm concept, data acquisition or recording is essential, in that the sampling frequency determines the types of information [12]. for example, at low frequencies, the low sampling rate considerably limits the range of devices that can be detected, and the choice of scheme (steady state, transient or non-conventional) also has an influence. in some nilm systems, transient characteristics of loads or noise generated by devices may be required at high sampling rates to obtain accurate measurements [13]. the choice of electrical characteristics to be recorded depends on the frequency class ; high frequency when the sampling frequency is between 10 and 200 khz, medium frequency when it is between 50 hz and 10 khz, and low frequency for frequencies between 1 and 50 hz [14]. as a reminder, smart meters belong to the low frequency range and are suitable for low sampling rate collections [15]. 2 tokam & ouro-djobo / j. nig. soc. phys. sci. 5 (2023) 1208 3 3.2. event detection this step corresponds to the detection of switches (on/off, steady state to transient state and vice versa) that occurs during the operation of the devices. these detected switches provide information on the variations in energy consumption during operation (source). in the literature, two approaches to non-intrusive load monitoring have been developed depending on the algorithm to be used. these are event-based non-intrusive load monitoring, which requires an event detection step during its implementation as discussed by some authors in their work [16,17], and non-event-based load monitoring [18]. in the case of the first approach, the performance of the algorithms to be used is crucial. during the feature extraction and classification (learning) phase, they allow to limit the number of false commutations likely to appear and affect the performance of the algorithm. in contrast to the previous approach, the non-event-based nilm does not include an event detection step. here, hidden markov models (hmm), as well as their variants that we present in section 3.4.3, are usually used as a learning algorithm. this algorithm has the advantage of processing each global signal sample (power, voltage, current, etc.), and in the end identifying each device [19–20]. it should be added that, for devices that obey on/off, multi-state and continuous operation, respectively, it is easier to detect them. on the other hand, continuously variable device (cvd) devices are more complicated to detect, as their energy consumption is arbitrarily modified. in the literature, different types of devices commonly encountered in energy disaggregation are presented [1,13,17]. these include devices for: type i: these are appliances with only two operating states (on/off) and consuming constant energy throughout their operation, such as an incandescent light bulb, electric kettle, toaster or microwave. these appliances are called resistive appliances because the reactive energy they give off is almost zero. type ii: this category includes finite state machines (fsm) or multi-state machines, because they have a finite number of distinct states that can be repeated. in addition, the state transitions that take place during operation are identified using variations in energy consumption over a period of time. appliances such as electric cookers, refrigerators with automatic defrosting, washing machines, variable speed fans are considered to belong to this family. type iii: loads belonging to this group are called continuously variable devices (cvd). they are characterized by electronic devices that vary their power consumption over time, without a fixed number of states. for these devices, it is difficult to easily allocate their energy consumption. examples include dimmers, electric drills and split air conditioners. figure 1 shows the operating states of the first three types of appliances. type iv: appliances that can be operated at a constant rate throughout the day, such as tv receivers, smoke detectors and telephone sets, belong to this group. the continuous nature of the operating state of devices belonging to the last group cannot be represented in figure 1, but it can be done in a pictorial way. in view of this non-exhaustive inventory, it should be recalled that the main challenge of nilm remains the disaggregation of the energy consumed by each of the loads belonging to these categories, both separately and simultaneously, at regular or different time intervals. figure 1. illustration of the operating status of the devices in the various groups 3.3. features extraction once the detection of the event is done, the essential features of the event are extracted for the identification of the device that caused it [21]. feature extraction plays a crucial role in learning device signatures [1]. furthermore, these features can be classified into three categories: steady state, transient and non-conventional. 3.3.1. steady state a steady state is said to exist when the characteristics of the load remain unchanged for a certain period of time. these are characteristics such as : real (p), reactive (q), apparent (s) power, root mean square (rms), power factor (pf), frequency (f) and time domain characteristics, voltage-current paths, voltage noise, which can be used for load identification [1,3]. in the steady state, the load signature represents the information found in the analysis between two consecutive operational steady states [22]. in steady state, the extraction of the characteristics is done at low frequency, as this favors the detection of small variations that occur continuously, due to the nature of the loads. moreover, this is much easier than a momentary indication. it should be added that in the transient scheme, the steady state signatures are the set satisfying the zero-loop sum constraint, where the sum of power changes in any cycle of state transitions is zero. however, recall that some device signatures may overlap due to their similar characteristics, thus making load identification and disaggregation complex. to solve this difficulty, v-i trajectories are used, in order to distinguish device events [1]. figure 2 shows the different schemes, depending on the features to be extracted and trained in the nilm algorithms. 3.3.2. transient state when a device is switched on, an event (transient state) occurs for a period of time before the stable state indicating the operation of the device. it can be observed in both (on/off) and multi-state devices [23]. a transient is characterized by a 3 tokam & ouro-djobo / j. nig. soc. phys. sci. 5 (2023) 1208 4 figure 2. categorization of nilm systems based on features extraction schemes relaxation time, a damping rate or a quality factor. it usually occurs when there is a change in the system. in the steady state, when two or more loads are in similar power ranges, event detection becomes less and less accurate [1]. the use of transients can overcome this shortcoming, by accurately identifying the events that take place. to make this possible, sophisticated, high-cost, high-accuracy equipment adapted to the circumstances is recommended [24]. the features (transient power, inrush current, transient voltage noise, transient v-i paths and higher order harmonics) extracted for examination are highly sampled (from khz to mhz) [1,3,25,26]. 3.3.3. non-conventional features the non-conventional regime includes characteristics that do not belong to the steady state or the transient regime. this group includes a combination of conventional, contextual, behavioural or indicator features of electrical devices [24]. these include features such as start time, end time, peak time, time of day, frequency of appliance use, current signal eigenvalues, light detection and temperature, which are used to add additional information to the conventional features [1,3,23]. 3.4. load identification and inference this is the next step after features extraction. during the load identification phase, the signatures extracted from the overall power signal of the loads are analyzed, with a view to labelling the loads [15,23]. also, to learn the detected signatures, learning algorithms are used, while the deductions of the load states are performed using inference algorithms, following the observed aggregate power data [15]. the algorithms used in this stage of the nilm belong to either the supervised, semisupervised or unsupervised families of learning algorithms. indeed, supervised learning algorithms require large amounts of data (datasets or dataset) on the characteristics of the individual loads extracted, in order to train the algorithm to classify the loads in operation [3,27]. in addition, this data needs to be labeled, so that the algorithm correlates the input and output data perfectly during the training phase. for example, when the algorithm is trained to deduce whether the signal shape is that of a coffee maker, with supervised learning, it creates a label for each signal used in the training data, indicating whether the signal is that of a coffee maker in operation or not. semi-supervised learning algorithms use a set of labeled and unlabeled data. in order to learn successfully, they need to train small amounts of labeled data [3,23]. these algorithms fall between supervised ones using labeled data and unsupervised ones not using labelled data. it has been shown that the use of unlabeled data, in combination with labeled data, significantly improves the quality of learning. furthermore, labelling data sometimes requires the intervention of a human user. however, when datasets become very large, this operation can be tedious. in this case, semi-supervised learning, which requires only a few labels, is of obvious practical interest. as an example, co-learning is an example of semi-supervised learning, where two classifiers learn a set of data, but each using a different, ideally independent, set of features. if the data are objects to be classified according to their nature, one might use size and the other shape for example. with unsupervised learning, the algorithms are trained on unlabeled data. they scan the data sets for significant connections. in addition, in the context of non-intrusive load monitoring, pre-processing of unlabeled input data is not useful, 4 tokam & ouro-djobo / j. nig. soc. phys. sci. 5 (2023) 1208 5 thus favoring nilm applications operating in real time [15]. furthermore, klemenjack et al. [27] add that unsupervised algorithms must both perform load disaggregation and detection of connected devices in the circuit they monitor. however, there are discrepancies in the assessment of this form of learning. according to tezde et al. [23], although unsupervised learning is suitable for nilm, it is more difficult to achieve satisfactory results than supervised learning. moreover, it is up to the consumer to define (label) which device and state should belong to the clusters formed during learning. however, in the research that is being done, nilm is making remarkable progress, thanks to the construction of much more reliable and inexpensive unsupervised learning models, as reported in [5,28]. 3.4.1. supervised learning methods in their implementation, supervised learning methods make use of labeled training data, and the detected signals are used to establish load signatures [1]. however, these methods can be classified into two approaches [3,23]: optimization approaches: learning problems are evaluated as optimization problems. that is, using the learning experience, they help to optimize the performance criteria by comparing the features extracted from loads to the stored features of loads in a database, in order to obtain the closest possible match. in their work, authors such as [29,30] have used genetic algorithms (ga) as learning methods. in the literature, zoha et al. [5] point out that apart from pattern recognition approaches, optimization approaches are also suitable learning methods for load disaggregation. pattern recognition-based approaches: in this approach, patterns are recognized by machine learning algorithms. pattern recognition is defined as the classification of the data extracted from patterns and/or their previously acquired representations. in sum, solving pattern recognition problems is solving classification problems. in the literature, there are many algorithms that have been used to achieve this. these include support vector machine (svm) algorithms [16,31–36], k-nearest neighbours (k-nn) for load classification based on power signals [37–41], decision trees [42–44], bayes classifiers [45– 46 ], artificial neural networks [47–49], and hidden markov methods [50,51] which have been shown to be capable of introducing temporal and state change information. however, hidden markov methods (hmm) can be used as a supervised or unsupervised approach, depending on the results sought. 3.4.2. semi-supervised learning methods these are methods that combine small amounts of labeled data with large amounts of unlabeled data during training [52,53]. it is a special case of weak supervision. in the literature, few authors have used these methods, although they present interesting performances, as demonstrated by barsim et al. [52] in their work, where they use self-training as a learning tool to solve the energy disaggregation problem. they point out that semi-supervised learning tools have the ability to reduce the labelling effort required, by providing a learning disaggregation system whose performance gradually increases as it observes more unlabeled aggregate measures. according to li et al. [53], semi-supervised learning algorithms are a good alternative to supervised learning algorithms that require the labelling of all the loads connected in the power system, in addition to the fluctuations of the main power supply that must be trained. thus, using graph-based multi-label classification, they manage to obtain better results than the state of the art on some datasets such as redd (reference energy disaggregation data set), blued or ampds. 3.4.3. unsupervised learning methods as mentioned earlier, unsupervised learning methods do not need labeled data to run their algorithms. this makes them suitable for nilm applications running in real time. furthermore, the current trend is that due to their low operating cost and reliability, they are being further developed and improved [15]. these authors add that in the literature, unsupervised learning methods can be classified into three subgroups, namely: unsupervised approaches that use unlabeled data in training to build a database of devices; unsupervised approaches that use labeled data from known buildings to build load models that are then used to disaggregate the energy consumed in unknown buildings; and finally unsupervised approaches that do not require training prior to energy disaggregation. there are authors in the literature who have used unsupervised learning methods for various reasons. for example, k-means algorithms have been used by yang et al. [54] to analyze the clustering formed by matching after detection of events (on/off) of appliances and by buddhahai et al. [55] to perform partitioning of load data by clusters in number of power states, which helped to identify the power state of appliances such as water heater, air conditioner with accuracy and f-score values above 89%. abubakar et al. [13] and gopinath et al. [1] mention in their research that expectation maximization has been used as an unsupervised learning algorithm to detect unknown load states. to identify on/off states of devices gopinath et al.[1] adopt the concept of hidden events and for real and active power consumption the concept of observed events, which together with the transition matrix and initial states will give good accuracies. in order to improve the results previously obtained with hidden markov models (hmm), different variants have been explored, such as factorial hmm (fhmm), conditional fhmm (cfhmm), factorial hidden semi markov model (fhsmm), conditional factorial hidden semi markov model (cfhsmm) [1,23]. from the observation made, it appears that the cfhsmm performs better than the hmm variants for energy disaggregation and that the cfhmm is the second best in its performance. an explanation is given for these hmm variants, in order to better understand them: 1. fhmm: this is the extension of hmm where several hidden state variables are used instead of one hidden state variable. 2. cfhmm: this is a variant used to represent the state sequence. however, it requires the addition of features such as time and sensor measurements. 5 tokam & ouro-djobo / j. nig. soc. phys. sci. 5 (2023) 1208 6 3. fhsmm: this is an extension of fhmm that uses an alternative probability density function for the occupation time of the devices. 4. cfhsmm: is the combination of fhsmm and cfhmm. at the end of the load identification and learning process, the overall aggregate consumption of the loads is decomposed into individual consumptions of the loads installed in the power system. thus, using detailed information obtained from each load, the consumer is informed of what each load represents in the total consumption. 3.5. non-intrusive load monitoring dataset datasets are databases consisting of naturally or synthetically obtained energy consumption measurements (as they reduce costs, increase time savings by eliminating corrupted data and missing values [23]). they are the result of long measurement campaigns in households in various european and north american countries. in addition, these accessible datasets are also used to train the different algorithms of the learning methods mentioned above [27]. to be tested and improved, in order to prove their performance when called upon, the energy disaggregation algorithms need training data from the available energy consumption datasets. the datasets are intended to provide researchers with real-world energy consumption data, so that during the training phase, real-world energy consumption data achieves very good match rates and high levels of accuracy. the reference energy disaggregation data set (redd) is the first publicly available energy dataset released by the massachusetts institute of technology (mit) in 2011 [56]. this dataset aggregates consumption data from six (6) households in the united states obtained at low and high frequencies over relatively short time periods. following the development of redd, a number of datasets have emerged, including the plug-level appliance identification dataset (plaid) that gao et al. [57] presented in in their paper, which is a 30 khz sampled and labeled dataset containing 1876 individually measured device from 17 different appliance types of 330 different makes and models, with 1314 simultaneous operating records from 13 appliance types (i.e. measurements obtained when several appliances were active simultaneously). united kingdom recording domestic appliancelevel electricity (uk-dale) presented by kelly et al. [58] is an open-access dataset for households with data recorded at high frequency (16 khz) describing the overall and actual consumption situations of individual appliances. other examples include blued, greed, ampds and ampds2. 4. interests and applications of nilm nilm is a concept that was first developed by george william hart of the massachusetts institute of technology (mit) in the early 1980s to monitor and evaluate the number and type of loads and their individual energy consumption. despite the revolution that this technology heralded at the time of its inception, it was not a great success, due to its complexity (very large statistical data to be processed, limited computing power of computers). however, the rise of artificial intelligence, as well as improvements in computer performance, have greatly contributed to the renewed interest in nilm among researchers. figure 3 shows the number of publications per year for the research topic ”non-intrusive load monitoring” in the sciencedirect database. also, the continuous discoveries and improvements proposed over the years have led to adaptations of the concept to other research areas figure 3. number of publications on non-intrusive load monitoring from the sciencedirect database in addition to the characteristics of non-intrusive load monitoring (nilm) to provide information on the status of appliances, the amount of electricity consumed per appliance or information on the electricity consumption habits of the occupants, it also has potential in areas other than electricity. its non-intrusive nature gives it an originality that adapts to the specificities of each field, where it can play an essential role in the search for expected solutions. in electricity, nilm has seen a lot of progress in terms of improving the performance of learning algorithms. in practice, there are mobile applications that allow detailed monitoring of electricity consumption, so that electricity bills can be reduced. for the same authors, nilm has enabled some researchers to develop applications that detect illegally connected loads in households and buildings, as well as the presence or absence of people. the aim is to identify respectively the theft of electrical energy that may occur and cause overconsumption [3]. in another case, the development of the internet of things and smart homes has led to the creation of new applications aimed at enabling consumers to save energy [12,59]. nilm toolkits (nilmtk) have also been developed to monitor loads during operation [60]. the daily use of water in a household, a building, or an agglomeration varies according to factors such as age, income level, lifestyle or geography. thus, to considerably reduce overconsumption and waste, communication actions aimed at informing customers about their consumption patterns and habits should be encouraged for common satisfaction. real-time feedback is an action that many water supply companies need to adopt in order to limit water wastage. inspired by the concept of non-intrusive load monitoring developed in electricity, kim et al. [61] have adapted it to water consumption monitoring by proposing an easy-to-install calibration system. the system uses wireless vibration sensors installed in the pipes to inform 6 tokam & ouro-djobo / j. nig. soc. phys. sci. 5 (2023) 1208 7 users about the amount of water used, when and where it is used. in addition, its non-intrusive and self-contained nature requires no plumbing expertise for installation. in a similar vein, schantz et al. [62] carried out similar work to the above. it was found that the designed system is accurate and its use by households on a large scale would contribute to its popularization and to the achievement of large water savings. in the transport sector, the very high level of use of means of transport means that the prevention of failures, which can occur during multiple journeys, should be taken into account and integrated into maintenance systems. to this end, some researchers have deemed it relevant to integrate nilm into warships, freighters or cruise ships, in order to present its usefulness and the great interest it brings. thus, nation et al. [63] have in their work used nilm for the detection and isolation of faults in automated systems on board ships, showing that with nilm, as well as the event (on/off) of the ship’s grey water discharge pump. in the same vein, lindahl et al. [64] have looked at the detection of faults on board ships through non-intrusive load monitoring. according to them, the use of nilm through its applications contributes to improving the health of ships by monitoring engine room equipment and providing early detection of faults that can cause irreversible damage and inestimable losses. there are also some applications of non-intrusive load monitoring in rail transport. in this respect, mariscotti [65] presented the interest of the efficiency of nilm in ac railways. indeed, considering only the identification and extraction steps of the electrical characteristics (current, voltage and harmonics) of the nilm, the performances obtained through classification and clustering methods show that the nilm is not as relevant as in households, due to the fact that the rolling stock obeys regulations dictated by very strict standards. in a completely different area, batra et al. [66] used nilm to develop a technique that predicts household size, occupancy, income and age. in the health sector, a non-intrusive anomaly detection monitoring system has been developed for the elderly to report interference with daily activities [67]. the developed device is based on the concepts of nilm, as mentioned by these authors. the study was motivated by the authors’ desire to provide facilities to improve their health care and well-being by making them more independent. later, the same authors alcalá et al. [68], then proposed a paper in which they examined the home care monitoring system through the data of smart meters installed in these elderly people. later, dai et al. [69] developed in the context of telehealth and for elderly people with reduced mobility and dementia, a model for recognizing patterns of activities of daily living (adl) based on the nilm. for the specific case of nilm applied to adl, the demand response is able to deliver real-time information to the involved parties. however, there is still a threat to the privacy of users. indeed, the identification of user activities is based on user consumption profiles. there are still some flaws in this system, notably the exposure of private habits. on this aspect, gong et al. [70] presented a succinct plan to preserve user privacy for demand in smart grids. the developed and proposed scheme allowed consumers to protect their identity by participating in the demand response program, but offered consumers the possibility to publish their identity under certain scenarios, such as legal disputes. in agriculture, pest diseases are a huge obstacle to increasing crop yields. they result in lower harvests, leading to huge losses of income. the case of soybean cultivation is a clear example, where in some research centers it is difficult to identify the symptoms and types of pest diseases in soybean cultivation [71]. in order to provide suitable solutions, simunjutak et al. [71] have developed an application that helps researchers to identify soybean pests by classification. although the accuracy of the results depends on the amount of data trained, such applications should be improved and disseminated. the search for effective solutions to climate change remains an ongoing challenge, and innovative solutions combining artificial intelligence and big data are now common. to this end, kee et al. [72] conducted a study on the impact of non-intrusive load monitoring on co2 emissions in malaysia. the results of the study revealed that by making use of energy efficiency practices in daily electricity consumption, overall co2 emissions would be reduced by 10.2% in the country. furthermore, by incorporating energy efficiency practices into daily electricity consumption, overall co2 emissions would be reduced by 10.2% in the country. furthermore, by integrating non-intrusive load monitoring into these consumption practices, the rate of co2 emission reduction would increase from 45% to just over 60% by 2030. in conclusion, the authors state that the application of energy efficiency practices based on nilm is a valuable benchmark for the development and adoption of energy efficiency policies to control co2 emissions. in sum, the applications presented in this work reflect the interest that many fields of research and activities would have in integrating artificial intelligence, machine learning and big data for well-functioning solutions that incur reduced operational costs. 5. conclusion given the energy and climate challenges, it is clear that more effort needs to be made to use solutions that meet the requirements of the established difficulties. energy management is one such solution. developing ways to drastically reduce the observed wastage would help to reduce the carbon footprint, the adverse effects of which the world feels on a daily basis. to this end, the concept of non-intrusive load monitoring (nilm) was first developed to allow household occupants to identify how much electricity each appliance consumed as part of the overall household consumption. advances in artificial intelligence have led to major revolutions that have generated growing interest, so that the concept of non-intrusive load monitoring is no longer confined to the electricity sector alone, but also to other sectors such as transport, medicine, water management and environmental protection. the applications developed in this area show the great interest in extending it and proposing new ways of managing and controlling activities. in addition, progress is continually being made to improve the performance of learning algorithms dedicated to nilm, as well as all other essential 7 tokam & ouro-djobo / j. nig. soc. phys. sci. 5 (2023) 1208 8 points contributing to its performance. despite the fact that it is considered as an alternative to be popularized on a large scale to encourage energy savings, and that it also has many advantages such as its affordable cost, its installation at a single measuring point, and the absence of intrusion into the household for its operation, it still remains for many researchers an immature technology that needs to be perfected. acknowledgments the authors express their profound gratitude to the world bank for funding this research through cerme (centre d’excellence régional pour la maitrise de l’electricité). tanks also the referees for the positive enlightening comments and suggestions, which 10 have greatly helped us in making improvements to this paper references [1] r. gopinath, mukesh kumar, c. prakash chandra joshua & k. srinivas, “energy management using non-intrusive load monitoring techniques – state-of-the-art and future research directions”, sustainable cities society 62 (2020) 102411. https://doi.org/10.1016/j.scs.2020.102411. 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[72] k-k. kee, y. s. lim, j. wong & k. h. chua, “impact of nonintrusive load monitoring on co2 emissions in malaysia”, bulletin of electrical engineering and informatics 10 (2021) 1803. https://doi.org/10.11591/eei.v10i4.2979. 9 j. nig. soc. phys. sci. 5 (2023) 1075 journal of the nigerian society of physical sciences collocation method for the numerical solution of multi-order fractional differential equations g. ajileyea, a. a. jamesb,∗ adepartment of mathematics and statistics, federal university wukari, taraba state, nigeria. b department of mathematics and statistics, american university of nigeria, yola, adamawa state, nigeria. abstract this study presents a collocation approach for the numerical integration of multi-order fractional differential equations with initial conditions in the caputo sense. the problem was transformed from its integral form into a system of linear algebraic equations. using matrix inversion, the algebraic equations are solved and their solutions are substituted into the approximate equation to give the numerical results. the effectiveness and precision of the method were illustrated with the use of numerical examples. doi:10.46481/jnsps.2023.1075 keywords: keywords: differential equation, fractional derivatives, approximate solution, power series. article history : received: 17 september 2022 received in revised form: 24 may 2023 accepted for publication: 25 may 2023 published: 11 june 2023 © 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: tolulope latunde 1. introduction in the fields of mathematics, physics, chemistry, and engineering, differential and integral equations involving fractions are of the utmost significance. the use of functional equations, such as ordinary and partial differential equations, is typical when applying mathematics to the modeling of problems arising in the real world. in the early 1900s, italian mathematician vito volterra came up with a whole new sort of equation that came to be known as integro-differential equations in order to investigate the phenomenon of population expansion. in these types of equations, one or more derivatives of the function whose value is unknown is placed under the integral sign. integro-differential equations can be found in a ∗corresponding author tel. no: +2348077092831 email address: adewalejames974@gmail.com (a. a. james ) variety of mathematical formulations of physical phenomena. additionally, these equations can be found in the modeling of certain phenomena in the fields of science and engineering. for instance, the equations of kinetics that support the kinetic theory of rarefied gases, plasma, radiation transmission, and coagulation are some examples. [1]. some of the numerical solution of fractional differential equations developed in the literature include: perturbed collocation method [2], adomian decompositions method by [3-5], collocation method by [6-9], chebyshevgelerkin method [10], bernoulli matrix method [11], differential transform method [12], pseudospectral method [13], bernstein polynomials method [14, 15], the mellin transform approach [16]. [17] utilized a numerical approach based on the boubaker polynomial to generate approximate numerical solutions to the multi-order fractional differential equations. their decision was to use an operational matrix for fractional integration based on boubakar polynomi1 ajileye & james / j. nig. soc. phys. sci. 5 (2023) 1075 2 als. collocation approach for the computational solution of fredholm-volterra fractional order of integro-differential equations was presented by [18]. they solved the problem by first obtaining the linear integral form of it and then transforming it into a system of linear algebraic equations by making use of conventional collocation points. in this research, the collocation method is utilized to solve multi-order fractional differential equations of the form dβy(x) = n∑ j=0 q j(x)d α j y(x) + h(x) (1) subject to the initial condition y( j)(a j) = λ j, j = 0, 1, ..., n − 1, n ∈ n β > αn, (2) where y(x) is the unknown function, dα j and dβ are the caputo’s derivative, h(x) is the force known -prior. q j(x) is the known function, a j and λ j are known constants. 2. basic definitions in this section, we present certain definitions and fundamental ideas of fractional calculus for the purpose of the formulation of the problem that has been presented. definition 2.1: the caputo derivative with order α > 0 of the given function f (x), x ∈ (a, b) is defined as c x d α a y(x) = 1 γ(m −α) ∫ x a (x − s)m−α−1y(m)(s)d s (3) where m − 1 ≤ α ≤ m, m ∈ n, x > 0 definition 2.2: let (an) , n ≥ 0 be a sequence of real numbers. the power series in x with coefficients an is an expression y(x) = a0+a1 x+a2 x 2 +a3 x 3 +· · ·an x n = n∑ n=0 an x n = φ(x) a(4) where φ(x) = [1 x x2 · · · xn ], a = [a0 a1 · · · an ]t then y(x, n) = xna, n = 0(1)n, n ∈ z+ definition 2.3: standard collocation method (scm). this method is used to determine the desired collocation points within an interval. i.e [a,b] and is given by xi = a + (b − a)i n , i = 1, 2, 3, ...n (5) definition 2.4: let y(x) be a continuous function, then 0 i β x ( c 0 d β xy(x) ) = y(x) − n∑ k=0 y(k)(0) k! xk (6) where m − 1 < β < 1 definition 2.5: let p(s) be an integrable function, then 0 i β x ( p(s)) = 1 γ(β) ∫ x 0 (x − s)β−1 p(s)d s (7) definition 2.6: the riemann -liouville derivative of order α > 0 with n − 1 < α < n of the power function f (t) = t p−α is given by dαt p = γ( p + 1) γ( p −α + 1) t p−α (8) 3. mathematical background in this part, we create a collocation approach for numerically solving multi-order fractional differential equations utilizing power series polynomials as the basis function. lemma (3.1) (integral form) let y(x) be a solution to (1) subject to (2), the integral form is y(x) = w(x) + n∑ j=0 1 γ(m j −α j) 1 γ(β) (9) × ∫ x 0 (x − s)β−1q j(s) [∫ s 0 (s − t)m j−α j−1y(m j )(t)dt ] d s where w(x) = n∑ k=0 y(k)(0) k! xk + 1 γ(β) ∫ x 0 (x − s)β−1h(s)d s proof. multiply equation (1) by 0 i β x (.) gives 0 i β x ( dβy(x) ) = 0 i β x  n∑ j=0 q j(x)d α j y(x) + 0 iβx (h(x)) (10) using (6) on equation (10) gives y(x) = n∑ k=0 y(k)(0) k! xk + 0 i β x  n∑ j=0 q j(x)d α j y(x)  (11) applying equations (3) and (7) to equation (11) gives y(x) = n∑ k=0 y(k)(0) k! xk + 1 γ(β) ∫ x 0 (x − s)β−1 (12) ×  n∑ j=0 q j(x) 1 γ(m j −α j) ∫ s 0 (s − t)m j−α j−1 y(m j )(t)dt  d s substituting equation (4) into equation (12) gives y(x) = n∑ k=0 y(k)(0) k! xk + 1 γ(β) ∫ x 0 (x − s)β−1 (13) ×  n∑ j=0 q j(x) 1 γ(m j −α j) ∫ s 0 (s − t)m j−α j−1 dm j dtm j (φ(t)) dta  d s 3.1. method of solution collocating at xi in equation (13) gives y(xi) = w(xi) + n∑ j=0 1 γ(m j −α j) 1 γ(β) ∫ xi 0 (xi − s) β−1q j(s) × (∫ s 0 (s − t)m j−α j−1 dm j dtm j (φ(t)) dt ) d s a (14) where w(xi) = n∑ k=0 y(k)(0) k! xk + 1 γ(β) ∫ x 0 (x − s)β−1h(s)d s 2 ajileye & james / j. nig. soc. phys. sci. 5 (2023) 1075 3 simplifying equation (14) gives φ(xi)a = w(xi) +  ∑n j=0 1 γ(m j −α j) 1 γ(β) ∫ xi 0 (xi − s)β−1q j(s) × (∫ s 0 (s − t)m j−α j−1 dm j dtm j (φ(t)) dt ) d s  a (15) factorizing the values of a from equation (15) gives φ(xi)− ∑n j=0 1 γ(m j −α j) 1 γ(β) ∫ xi 0 (xi − s)β−1q j(s) × (∫ s 0 (s − t)m j−α j−1 dm j dtm j (φ(t)) dt ) d s  a = w(xi) (16) equation (16) can be in the form v (xi)a =w(xi) (17) where v (xi) = φ(xi)− n∑ j=0 1 γ(m j −α j) 1 γ(β) ∫ xi 0 (xi − s) β−1q j(s) (∫ s 0 (s − t)m j−α j−1 dm j dtm j (φ(t)) dt ) d s (18) and a = [a0 a1 · · · an ]t multiplying both sides of equation (17) by v−1(xi) gives a =v−1(xi)w(xi) (19) lemma (3.2) let y(x) be approximated by (11) and let l(x) = 0 i β x  n∑ j=0 q j(x)d α j y(x)  (20) if q j(s) = sp j, then l(x; n) = γ(n + 1)γ(n −α j + p j + 1) γ(n −α j + 1)γ(β + n −α j + p j + 1) × x β+n−α j +p j i a (21) proof. applying equation (3) and (7) into equation (20) gives 0 i β x  n∑ j=0 q j(x)d α j y(x)  = n∑ j=0 1 γ(m j −α j) 1 γ(β) ∫ xi 0 (x − s)β−1q j(s) [∫ s 0 (s − t)m j−α j−1y(m j )(t)dt ] d s (22) substituting (8) into (22) gives = n∑ j=0 1 γ(m j −α j) 1 γ(β) ∫ x 0 (x − s)β−1s p j (23) [∫ s 0 (s − t)m j−α j−1 ( γ(n + 1) γ(n − m j + 1) tn−m j ) dt ] d s a let s − t = (1 − v)s, then t = vs =⇒ dt dv = s =⇒ dt = sdv, substituting into (23) gives = n∑ j=0 γ(n + 1) γ(m j −α j)γ(n − m j + 1) 1 γ(β) ∫ x 0 (x − s)β−1s p j (24) [ s n−α j ∫ 1 0 (1 − v)m j−α j−1v n−m j dt ] d s a simplifying (24), we get l(x; n) = γ(n + 1)γ(n −α j + p j + 1) γ(n −α j + 1)γ(β + n −α j + p j + 1) xβ+n−α j +p j a(25) lemma (3.3) let y(t) be approximated by (9), let c(x) = 0 i β x (h(x)) (26) if h(s) = sm, then c(x) = γ(m + 1) γ(β + m + 1) xβ+m proof. applying equation (7) into (26) gives 0 i β x (h(x)) = 1 γ(β) ∫ x 0 (x − s)β−1h(s) d s substituting for h(s) gives = 1 γ(β) ∫ x 0 (x − s)β−1 smd s let x − s = (1 − u)x, s = ux =⇒ d s du = x =⇒ d s = xdu. c(x) = γ(m + 1) γ(β + m + 1) xβ+m (27) lemma (3.4) let y(x) be the solution of (1) and (2) then the numerical result gives y(x) = φ(xi)v −1(xi) w(xi) (28) where v (xi) = γ(n + 1)γ(n −α j + p j + 1) γ(n −α j + 1)γ(β + n −α j + p j + 1) x β+n−α j +p j i and w(xi) = − n∑ k=0 y(k)(0) k! xki + γ(m + 1) γ(β + m + 1) xβ+mi 3 ajileye & james / j. nig. soc. phys. sci. 5 (2023) 1075 4 proof. approximate solution of equation (17) is y(x) = φ(x) a from equation (19) a =v−1(xi) w(xi) where v (xi) = γ(n + 1)γ(n −α j + p j + 1) γ(n −α j + 1)γ(β + n −α j + p j + 1) x β+n−α j +p j i + br+n+1γ(r + 1) (σ + n + 1) γ(β + r + 1) xβ+r + γ(r + σ + n + 2) (σ + n + 1) γ(β + r + σ + n + 2) xβ+r+σ+n+1 substituting for a in the approximate solution gives the numerical result y(x) = φ(xi)v −1(xi) w(xi) 4. convergence analysis in this section, we establish the convergence of the method by substituting the approximate solution into equation (3.0) yn (x) = w(x) + n∑ j=0 1 γ(m j −α j) 1 γ(β) (29) × ∫ x 0 (x − s)β−1q j(s) [∫ s 0 (s − t)m j−α j−1y (m j ) n (t)dt ] d s subtracting (9) from (29) gives en (x) = yn (x) − y(x). hence |en (x)| ≤ 1 γ(β) ∫ x 0 (x − s)β−1 n∑ j=0 1 γ(m j −α j) q j(s)∣∣∣∣∣∣ [∫ s 0 (s − t)m j−α j−1 en (t)dt ] d s ∣∣∣∣∣∣ therefore ‖en (xi)‖∞ ‖en (t)‖∞ ≤ 1 γ(β) ∫ xi 0 (x − s)β−1∣∣∣∣∣∣∣∣  n∑ j=0 1 γ(m j −α j) q j(s) (∫ s 0 (s − t)m j−α j−1dt ) ∣∣∣∣∣∣∣∣ d s 5. numerical examples in this section, we considered two numerical examples to evaluate the effectiveness and clarity of the method. a maple 18 program is used to perform the computations. let yn(x) and y(x) be the approximate and exact solutions respectively. errorn = |yn(x) − y(x)| . example 5.1. [2] consider multi-order fractional differential equation . d1.5y(x) = −x−1 d0.5y(x) − x0.5y(x) + f (x) with this condition y ′ (0) = y(0) = 0 and exact solution y(x) = x3 − x2 f (x) = [ 6x ( γ(3.5) + γ(2.5) γ(2.5)γ(3.5) + x2 6 ) − 2 ( γ(2.5) + γ(1.5) γ(1.5)γ(2.5) + x2 2 )] x0.5 solution 1. comparing with equation (1.1) and equation (1.2), β = 1.5,α = 0.5 using n = 4 for illustration, and applying equation (6) gives y(x) = w(x) − 1 γ(1 − 0.5) 1 γ(1.5) ∫ x 0 (x − s)1.5−1 s−1 (30)[∫ s 0 (s − t)1−0.5−1 γ(n + 1) γ(n − 1 + 1) tn−1dt ] d s a − 1 γ(1.5) ∫ x 0 (x − s)1.5−1 s0.5 (sn) d s a where w(x) = n∑ k=0 y(k)(0) k! xk + 1 γ(1.5) ∫ x 0 (x − s)1.5−1 f (s)d s substituting (4) into equation (30) gives φ(x) a = w(x) − 1 γ(1 − 0.5) 1 γ(1.5) ∫ x 0 (x − s)1.5−1 s−1 (31)[∫ s 0 (s − t)1−0.5−1 γ(n + 1) γ(n − 1 + 1) tn−1dt ] d s a − 1 γ(1.5) ∫ x 0 (x − s)1.5−1 s0.5 (sn) d s a where w(x) = n∑ k=0 y(k)(0) k! xk + 1 γ(1.5) ∫ x 0 (x − s)1.5−1 f (s)d s equation (31) can be in the form τ(x)a =w(x) (32) where τ(x) = φ(x) + 1 γ(1 − 0.5) 1 γ(1.5) ∫ x 0 (x − s)1.5−1 s−1[∫ s 0 (s − t)1−0.5−1 γ(n + 1) γ(n − 1 + 1) tn−1dt ] d s + 1 γ(1.5) ∫ x 0 (x − s)1.5−1 s0.5 (sn) d s 4 ajileye & james / j. nig. soc. phys. sci. 5 (2023) 1075 5 collocating at x4 = [ 1 4 2 4 3 4 1 ] and substituting the initial conditions gives τ(xi) ∗a = f (xi)∗ (33) where τi (x) ∗ = 0.0000000000 0.0000000000 0.0000000000 0.0000000000 0.0000000000 0.0470157986 0.2239605405 0.1797773130 0.0750069286 0.0274673600 0.1880631945 0.5184447788 0.7543711011 0.6176863534 0.4482932221 0.4231421877 0.9539764130 1.8295669120 2.1838653930 2.3438648990 0.7522527781 1.6010791410 3.5816739880 5.5056804110 7.7368811360 0 1 0 0 0 1 0 0 0 0  f (x)∗ = [0.0000000000 − 0.1047703844 − 0.1366847478 0.3542984804 1.9240064220] we now solve for the unknown values a making use of matrix inversion results in equation (33): y4 = ( 1.365574320288940 × 10−14 − 2.546407529280260 × 10−12 x −0.999999999275360x2 + 0.999999998413614x3 + 6.644427230639850 × 10−10 x4 ) example 5.2. [2] consider multi-order fractional differential equation . d1.5y(x) + 1 x d0.5y(x) + x 1 2 y(x) = + f (x) with this condition y ′ (0) = y(0) = 0 and exact solution y(x) = −x3 + x2 f (x) = [ 2 ( γ(2.5) + γ(1.5) γ(1.5)γ(2.5) + x2 2 ) − 6x ( γ(3.5) + γ(2.5) γ(2.5)γ(3.5) + x2 9 )] x 1 2 solution 2. comparing with equation (1.1) and equation (1.2), β = 1.5,α = 0.5 using n = 4 for illustration, applying equation (6) gives y(x) = w(x) − 1 γ(1 − 0.5) 1 γ(1.5) ∫ x 0 (x − s)1.5−1 s−1(34)[∫ s 0 (s − t)1−0.5−1 γ(n + 1) γ(n − 1 + 1) tn−1dt ] d s a − 1 γ(1.5) ∫ x 0 (x − s)1.5−1 s0.5 (sn) d s a table 1: exact, approximate and absolute error values for example 1 x exact our methodn=4 errorn=4 error [2]=4 0.0 0.00000000000 1.36557432000e-14 1.3656e-14 5.9232e-13 0.1 -0.900000000e-2 -0.89999999950e-2 5.0000e-12 2.6668e-10 0.2 -0.320000000e-1 -0.31999999980e-1 2.0000e-11 9.6994e-10 0.3 -0.630000000e-1 0.06299999997000 3.0000e-11 1.9604e-09 0.4 -0.960000000e-1 -0.95999999980e-1 2.0000e-11 3.0781e-09 0.5 -0.12500000000 -0.1250000000000 0.0000000 4.1532e-09 0.6 -0.14400000000 -0.1439999999000 1.0000e-10 5.0056e-09 0.7 -0.14700000000 -0.1470000000000 0.0000000 5.4456e-09 0.8 -0.12800000000 -0.1280000001000 1.0000e-10 5.2733e-09 0.9 -0.810000000e-1 -0.81000000160e-1 1.6000e–10 4.2787e-09 1.0 0.00000000000 -2.3555727690e-10 2.3555e-10 2.2421e-09 where w(x) = n∑ k=0 y(k)(0) k! xk + 1 γ(1.5) ∫ x 0 (x − s)1.5−1 f (s)d s substituting (4) into equation (34) gives φ(x) a = w(x) − 1 γ(1 − 0.5) 1 γ(1.5) ∫ x 0 (x − s)1.5−1 s−1(35)[∫ s 0 (s − t)1−0.5−1 γ(n + 1) γ(n − 1 + 1) tn−1dt ] d s a − 1 γ(1.5) ∫ x 0 (x − s)1.5−1 s0.5 (sn) d s a where w(x) = n∑ k=0 y(k)(0) k! xk + 1 γ(1.5) ∫ x 0 (x − s)1.5−1 f (s)d s equation (35) can be in the form τ(x)a =w(x) (36) where τ(x) = φ(x) + 1 γ(1 − 0.5) 1 γ(1.5) ∫ x 0 (x − s)1.5−1 s−1[∫ s 0 (s − t)1−0.5−1 γ(n + 1) γ(n − 1 + 1) tn−1dt ] d s + 1 γ(1.5) ∫ x 0 (x − s)1.5−1 s0.5 (sn) d s collocating at x4 = [ 1 4 2 4 3 4 1 ] and substituting the initial conditions gives τ(xi) ∗a = f (xi)∗ (37) where τi (x) ∗ = 0.5000000000 2.3817583340 1.9118819450 0.7976779168 0.2921077680 0.7071067812 1.9493225120 2.8363918970 2.3224651180 1.6855566990 0.8660254038 1.9524590850 3.7444893700 4.4696155660 4.7970791030 1.0000000000 2.1283791670 4.7612638900 7.3189233350 10.2849485700 1 0 0 0 0 0‘ 1 0 0 0  f (x)∗ = [ 1.1142040280 0.5139267798 -0.7251261978 -2.5576594460 0 0 ] we now solve for the unknown values a making use of matrix inversion results in equation (37); y4 = ( 1.428190898877800 × 10−12 − 2.777014174171200 × 10−10 x +1.000000001121990x2 − 1.000000001716120x3 + 6.387779194483300 × 10−10 x4 ) 5 ajileye & james / j. nig. soc. phys. sci. 5 (2023) 1075 6 table 2: exact, approximate and absolute error values for example 2 x exact our methodn=4 errorn=4 error [2]=4 0.0 0.00000000000 1.428190899000e-12 1.4281908990000e-12 3.7782e-12 0.1 0.00900000000 0.00899999998200 1.8000000000000e-11 2.4706e-09 0.2 0.03200000000 0.03199999997000 3.0000000000000e-11 1.4306e-08 0.3 0.06300000000 0.06299999997000 3.0000000000000e-11 3.9585e-08 0.4 0.09600000000 0.09599999999000 1.0000000000000e-11 78988e-08 0.5 0.12500000000 0.12499999990000 1.0000000000000e-10 1.2980e-07 0.6 0.14400000000 0.14399999990000 1.0000000000000e-10 1.8590e-07 0.7 0.14700000000 0.14699999980000 2.0000000000000e-10 2.3778e-07 0.8 0.12800000000 0.12799999970000 3.0000000000000e-10 2.7253e-07 0.9 0.08100000000 0.08099999952000 4.8000000000000e-10 2.7386e-07 1.0 0.00000000000 -3.6122208060e-10 3.6122208060000e-10 2.2207e-07 6. discussion of results in this section, we discuss the numerical results obtained by applying the derived numerical method to the solved examples. we observed from the result obtained for example 1 as shown in table 1 that the approximate solution at n=4 gives y4(x) = 1.365574320288940 × 10−14 − 2.546407529280260 × 10−12 x+ 1.000000001121990x2 − 1.000000001716120x3 + 6.387779194483300 × 10−10 x4. the numerical result almost converges to the exact solution and produces extremely small errors. this demonstrated that our method outperformed the proposed method by uwaheren et al (2020). the results of the numerical example 2 in table 2 shows the approximate solution at n=4 as y4(x) = 1.428190898877800 × 10−12 − 2.777014174171200 × 10−10 x + 1.0000001121990x2 − 1.0000001716120x3 + 6.387779194483300 × 10−10 x4. the numerical result converge to the exact solution and give better result than the method proposed by uwaheren et al (2020) at the same value of n. this shows that the numerical method developed is consistent and converges faster. 7. conclusion in this paper, a new numerical method was developed for solving multi-order fractional differential equations with initial conditions using collocation method. the numerical method derived is consistent, efficient and reliable and easy to compute. maple code was used to implement the developed method. solved numerical examples show that the method is reliable and suitable for these kind of problems. we also compare our absolute errors with uwaheren et al. 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[18] g. ajileye, a. a. james, a. m. ayinde & t. oyedepo, “collocation approach for the computational solution of fredholm-volterra fractional order of integro-differential equations”, j. nig. soc. phys. sci. 4 (2022) 834. 6 j. nig. soc. phys. sci. 3 (2021) 105–115 journal of the nigerian society of physical sciences extension of admm algorithm in solving optimal control model governed by partial differential equation k. a. dawodu∗ department of mathematical sciences, federal university of technology akure, nigeria abstract this paper presents an algorithm for the numerical solution of the optimal control model constrained by partial differential equation using the alternating direction method of multipliers (admm) accelerated with a parameter factor in the sense of nesterov. the admm tool was applied to a partial differential equation-governed optimization problem of the one-dimensional heat equation type. the constraint and objective functions of the optimal control model were discretized using the crank-nicolson and composite simpson’s methods respectively into a derived discrete convex optimization form amenable to the admm. the primal-dual residuals were derived to ascertain the rate of convergence of the model for increasing iterates. an existing example was used to test the efficiency and degree of accuracy of the algorithm and the results were favorable when compared the existing method. doi:10.46481/jnsps.2021.159 keywords: optimal control model, alternating direction method of multipliers, partial differential equation constraint, crank-nicolson, composite simpson’s, primal-dual. article history : received: 22 january 2021 received in revised form: 05 may 2021 accepted for publication: 10 may 2021 published: 29 may 2021 ©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: t. latunde 1. introduction optimal control problems (ocp) involving partial differential equations (pde) are derivations arising from solid mechanics, fluid dynamics, engineering, economics and sciences amongst others. most of which are formulations into second order performance measures governed by pde constraints of the ellipticc type. the classical analytic techniques of the hamilton -jacobi -bellman (hjb) or euler lagrange(e-l) may proof difficult because of the complexity of the formulated continuous nonlinear models. betts and campbell in [1] worked ∗corresponding author tel. no: email address: dawodukazeem@futa.edu.ng (k. a. dawodu ) on the "first discretized then optimize" technique, a novel approach in obtaining a discretized optimization problem with finite number of (large) vectors. ghobadi et al. in [2] extended this approach to the one-dimensional heat equation and enumerated a lot of advantages when compared with the historical "first optimize then discretized" approach. these include (i) the circumvention of the need to establish certain properties of the problem such as the active constraint interval and the number of touch points a priori and (ii) its amenability to well-structured discretization schemes and optimization solvers for inequality constrained problems. the work of [2] was shrouded with conditional stability problem due to the explicit discretization method used. however, this draw105 dawodu / j. nig. soc. phys. sci. 3 (2021) 105–115 106 back of conditional stability will be circumvented in this research with the use of the implicit method of higher order in the discretization and the admm extended in obtaining the optimal solutions with better accuracy of results and faster rate of convergence. moreover, many authors have such as olotu and dawodu in ([3, 4, 5]) have worked extensively on proportional and multidelay systems using the "first discretized then optimize" approach. however, he and yuan in [6] and he et. al in [7] worked on block-convex programming problem synonymous to discretized continuous distributed systems, though the implementation on pde-based optimal control problems were not explored. 2. statement of problem considering a one-dimensional bar connected to a heating (cooling) device at its both ends with known initial and lowerbound conditions but with unknown boundary conditions, then the temperature functions on the boundaries are considered to be the control variables (ui (t ), i = 1,··· , q ) as in [1]. the heat equation is then given as; ∂f (x , t ) ∂t = ∂ 2 f (x , t ) ∂x 2 , 0 ≤ t ≤ t, l1 ≤ x ≤ l2 (1) where f (x , t ) is the temperature at position x and at time t . it is then imperative to have the temperature profile f (x , t ) at each point of the bar that does not go below a given specific temperature function given by g (x , t )(also known as a lower bound function), during the time interval. setting the temperature of the bar very high makes sure the lower-bound condition (inequality constraint) below is satisfied. f (x , t ) ≥ g (x , t ) for all x and t (2) however, the objective of this modeling is to choose an appropriate temperature profile such that the energy consumed (objective function) is at minimum by reducing overheating, that is, to minimize the cost of energy consumed while keeping the temperature of the bar above the lower-bound profile. the appropriate choice of the objective function can be compared to the temperature profile of the bar with the lowerbound profile as closely as possible expressed mathematically as ‖f ((x , t ), y ) − g (x , t )‖. assuming the temperature profile is the approximation of the temperature of the bar provided the heat equation and initial condition are satisfied. and suppose the squared values of all the temperature in time t and space x is expressed as f 2(x , t ), then the objective function to be minimized is the integral of f 2(x , t ) in time and space. then the optimization model of the heat-transfer problem can be mathematically expressed as min u, f ∫ t 0 ∫ l2 l1 f 2(x , t ) d xd t + ∫ t 0 [p u21 (t ) + q u22 (t )]d t , s.t ∂2 f (x , t ) ∂x 2 = ∂f (x , t ) ∂t , f (x , t ) > g (x , t ), f (x , 0) = f0(x ), (3) f (l1, t ) = u1(t ), f (l2, t ) = u2(t ), t ∈ [0, t ], x ∈ [l1, l2], where x ∈ r, u ∈ r and f0(x ) is a known initial function of temperature at the initial time (t = 0). while u1(t ) and u2(t ) are the temperatures at the two boundary points, l1 and l2 respectively, of the bar whose optimal values will be obtained. 3. materials and methods the continuous non-linear model (problem) formulated above can now be solved using the approach of "first-discretizedthen-optimize" to obtain a quadratic optimization problem with finite number of (large) variables. the implicit discretization methods will be used for the discretization of the continuoustime ocp because its (i) stability is unconditional,(ii) time step-length can be chosen independent of the spatial steplength and (iii) coefficient matrix remains well-conditioned. 3.1. discretization of the constraints in the transformation of the continuous time pde constraints, the second-order crank-nicolson method derived from the averaging of the forward difference method at the j th step in time t and the backward-difference at the ( j + 1)th step in t with truncation error h2 ∂2 f ∂t 2 (xi ,µj ) + o(h2) will be used. and given that the grid points are equally-distributed for both the spatial interval (length of bar)[l1, l2] and the time interval [0, t ] with step-lengths δ and h respectively, computed as δ = (l2 − l1)/n and h = t /n . then the number of intervals n from the discretization in space and n from the discretization in time will generate (n +1) and (n +1) number of grid points respectively. in generating the numerical sequence, let xi = l1 + iδ be the i th point in space (for i = 0, 1, 2, . . . , n) and t j = j h be the j th point in time (for j = 0, 1, 2, . . . , n ), provided the usual differentiability conditions are satisfied, then the crank-nicolson (averaged difference) discretizations scheme in time and space with local truncation error of order o(δ2 + h2) expressed below as; ∂f (x , t j ) ∂t = f ′t (x , t j ) ' fi , j +1 − fi , j h , (4) 106 dawodu / j. nig. soc. phys. sci. 3 (2021) 105–115 107 ∂2 f (x , t j ) ∂x 2 = f ′′x (xi , t j ) + f ′′x (xi , t j +1) 2 , ' 1 2 [ fi −1, j − 2 fi , j + fi +1, j δ2 + fi −1, j +1 − 2 fi , j +1 + fi +1, j +1 δ2 ] (5) where fi , j ' f (xi , t j ) and uk , j ' uk (t j ) for i = 0, 1, 2, . . . , n, j = 0, 1, 2, . . . , n −1 and k = 1, 2. the discretization of the pde constraint function using eqns. (4) and (5), given that f (0, t j ) = f0, j = u1, j , f (xn , t j ) = fn, j = u2, j and f (x j , 0) = 0 yields; fi , j +1 − fi , j h = 1 2 [ fi −1, j − 2 fi , j + fi +1, j + fi −1, j +1 − 2 fi , j +1 + fi +1, j +1 δ2 ] , (6) for i = 1, 2,··· , (n −1) and 0 ≤ j ≤ n −1. substituting θ = h/δ2 and collecting like-terms gives the recurrence formula below; −λfi −1, j −λfi −1, j +1 + 2(λ− 1) fi , j + 2(λ+ 1) fi , j +1 −λfi +1, j −λfi +1, j +1 = 0. (7) setting i = 1 and f0, j = u1, j yields; −θu1, j −θu1, j +1 + 2(θ− 1) f1, j + 2(θ+ 1) f1, j +1 −θ f2, j −θ f2, j +1 = 0 for 0 ≤ j ≤ n − 1. (8) for j = 0 given f1,0 = f2,0 = u1,0 = 0 yields; −λu1,1 + 2(θ+ 1) f1,1 −θ f2,1 = 0 (9) for j = 1 yields; −θu1,1 −θu1,2 + 2(θ− 1) f1,1 + 2(θ+ 1) f1,2 −θ f2,1 −θ f2,2 = 0 (10) for j = k (2 ≤ k ≤ n − 1); −θu1,k −θu1,k+1 + 2(θ− 1) f1,k + 2(θ+ 1) f1,k+1 −θ f2,k −θ f2,k+1 = 0. (11) when eqn. (11) is expanded for i = 1 and 1 ≤ j ≤ n − 1 gives the matrix equation below:  −θ 0 . . . . . . . . . 0 −θ −θ 0 . . . . . . ... 0 −θ −θ 0 . . . ... ... . . . . . . . . . . . . ... ... . . . . . . . . . . . . 0 0 . . . . . . 0 −θ −θ     u1,1 u1,2 u1,3 ... ... u1,n   +   2(θ+ 1) 0 . . . 0 2(θ− 1) 2(θ+ 1) . . . ... 0 . . . . . . ... ... . . . 2(θ− 1) 2(θ+ 1)     f1,1 f1,2 ... f1,n   +   −θ 0 0 . . . 0 −θ −θ 0 . . . ... 0 . . . . . . . . . ... ... . . . . . . . . . 0 0 . . . 0 −θ −θ     f2,1 f2,2 ... ... f2,n   =   0 0 ... ... 0   , (12) compactly written as lu1 +ĝ f1 + lf2 = 0, (13) where f1 = [ f1,1, f1,2, .., f1,n ], f2 = [ f2,1, f2,2, .., f2,n ], u1 = [u1,1, u1,2, ..., u1,n ] t and 0 are column vectors of dimension n ×1 while l and ĝ are bi-diagonal square-matrices of n × n dimensions with respective entries described below. l =   −θ : 1 ≤ i ≤ n and j = i −θ : 2 ≤ i ≤ n and j = i − 1 0 : otherwise (14) and ĝ =   2(θ+ 1) : 1 ≤ i ≤ n and j = i 2(θ− 1) : 2 ≤ i ≤ n and j = i − 1 0 : otherwise (15) set i = 2; this then gives the recurrence equation below; −θ f1, j −θ f1, j +1 + 2(θ− 1) f2, j + 2(θ+ 1) f2, j +1− θ f3, j −θ f3, j +1 = 0 for 0 ≤ j ≤ n − 1. (16) for j = 0; and equating f1,0 = f2,0 = f3,0 = 0 yields, −θ f1,1 + 2(θ+ 1) f2,1 −θ f3,1 = 0 (17) for j = 1; −θ f1,1 −θ f1,2 + 2(θ− 1) f2,1 + 2(θ+ 1) f2,2 −θ f3,1 −θ f3,2 = 0 for 0 ≤ j ≤ n − 1. (18) for j = 2; −θ f1,2 −θ f1,3 + 2(θ− 1) f2,2 + 2(θ+ 1) f2,3 −θ f3,2 −θ f3,3 = 0 for 0 ≤ j ≤ n − 1. (19) ... ... ... ... ... ... ... ... ... ..., for j = n − 1; −θ f1,n −1 −θ f1,n + 2(θ− 1) f2,n −1 + 2(θ+ 1) f2,n −θ f3,n −1 −θ f3,n = 0 for 0 ≤ j ≤ n − 1. (20) the matrix equation for eqns. (17) to (20) is then given as, lf1 +ĝ f2 + lf3 = 0. (21) 107 dawodu / j. nig. soc. phys. sci. 3 (2021) 105–115 108 for i = 2, 3,··· , n − 2 and 1 ≤ j ≤ n − 1 will then be expressed below as: lfi −1 +ĝ fi + lfi +1 = 0 (22) for i = n − 1, 1 ≤ j ≤ n − 1 and setting u2 = fn gives; lfn−2 +ĝ fn−1 + lu2 = 0. (23) where fi = [ fi ,1, fi ,2,··· , fi ,n ]t ∈ rn ×1, u2 = [u2,1, u2,2..., u2,n ]t ∈ rn ×1, l and ĝ are described in eqns. (14) and (15) respectively. combining eqns. (13), (22) and (23) gives the matrix formulation below:   l 0 0 0 ... ... ... ... 0 l     u1 u2   +   ĝ l 0 . . . 0 l ĝ l . . . ... 0 . . . . . . . . . 0 ... . . . . . . . . . l 0 . . . 0 l ĝ     f1 f2 ... ... fn−1   =   0 0 ... ... 0   , (24) compactly written as, cu + d f = 0, (25) where c is a block-matrix of dimension (n − 1)n × 2n , d is a square block-matrix of dimension (n − 1)n × (n − 1)n , u is a column vector of dimension 2n × 1 while f and 0 are column vectors of dimension (n − 1)n × 1. u1(t j ) and u2(t j ) are the control variables for which gives access to the heating (cooling) resources, while the temperature variables f (xi , t j ) of the other parts of the bar are computed based on the heattransfer equations. 3.2. discretization of the boundaries given fi , j ≥ g (xi , t j ) for i = 1, 2,··· , n − 1 and j = 1,··· , n , then u1, j ≥ g (l1, t j ) and u2, j ≥ g (l2, t j ) for j = 1, 2,··· , n . therefore, u1, j + u2, j ≥ g (l1, t j ) − g (l2, t j ) for j = 1, 2,··· , n . (26) for j = 1; (u1,1 + u2,1) − z1 = g (l1, t1) + g (l2, t1) = g 1 (27) for j = 2; (u1,2 + u2,2) − z2 = g (l1, t2) + g (l2, t2) = g 2 (28) ... ... ... ... ... ... ... ... ... ... for j = n ; (u1,n + u2,n ) − zn = g (l1, tn ) + g (l2, tn ) = g n (29) combining eqns. (27) to (28) generates the matrix-structure below:     1 0 ··· 0 0 1 . . . ... ... . . . . . . 0 0 ··· 0 1     1 0 ··· 0 0 1 . . . ... ... . . . . . . 0 0 ··· 0 1       u1,1 ... u1,n u2,1 ... u2,n   −   z1 z2 ... ... zn   =   g 1 g 2 ... ... g n   . (30) equation (30) is compactly written as; mu − z = g , (31) where i ∈ rn ×n is the identity matrix, m = [i , i ] ∈ rn ×2n is the augmented double identity matrices, z = [z1, z2,··· , zn ]t ∈ rn ×1 is the positive slack-variable vector, u = [u1,u2]t = [u11, u12, .., u1n , u21, u22, .., u2n ]t ∈ r2n ×1 is the unknown control-variable vector and g = [g 1, g 2,··· , g n ]t ∈ rn ×1 is the known coefficient vector. 3.3. discretization of objective function in the discretization of the continuous-time objective function(performance measure) both in space and time, the composite simpson’s rule and the right-riemann rule will be used respectively as stated below:∫ l2 l1 f (x , t ) d x ' 2δ 3 [ n/2−1∑ i =1 f (x2i , t ) + 2 n/2∑ i =1 f (x2i −1, t ) ] + δ 3 [ f (x0, t ) + f (xi , t ) ] (32)∫ t 0 f (x , t ) d t ' h n∑ j =1 f (x , t j ) (33) then the discretized objective function of the optimal control problem in eqn. (3) using eqns. (32) and (32) is expressed below as; min u, f ∫ t 0 [ 2δ 3 j1 + δ 3 j2 + j3 ] d t (34) 108 dawodu / j. nig. soc. phys. sci. 3 (2021) 105–115 109 where, j1 = n/2−1∑ i =1 f (x2i , t ) + 2 n/2∑ i =1 f (x2i −1, t ), (35) j2 = f (x0, t ) + f (xn , t ), (36) j3 = p u21 (t ) + q u22 (t ). (37) discretizing eqn. (34) using (32) and collecting like-terms gives; ' min u, f [ 4hδ 3 n∑ j =1 n/2∑ i =1 f 22i −1, j + 2hδ 3 n∑ j =1 n/2−1∑ i =1 f 22i , j + ( p + δh 3 ) n∑ j =1 u21, j + ( q + δh 3 ) n∑ j =1 u22, j ] , ' min u, f [ n∑ j =1 [ 4hδ 3 ( f 21, j + f 23, j + .. + f 2n−1, j ) + 2hδ 3 ( f 22, j + f 24, j + .. + f 2n−2, j ) + m1u21, j + m2u22, j ]] (38) where m1 = ( p + δh3 ) and m2 = ( q + δh3 ) . recall that f 20, j = f 2(l1, j ) = u21 (t j ) = u2i , j and f 2n, j = f 2(l2, j ) = u22 (t j ) = u22, j , setting the values of j = 1, 2, ..., n in eqn. (38) yields the matrix operator. for j = 1; 4hδ 3 ( f 21,1 + f 23,1 + .. + f 2n−1,1) + 2hδ 3 ( f 22,1 + f 24,1 + .. + f 2n−2,1) + m1u21,1 + m2u22,1 for j = 2; 4hδ 3 ( f 21,2 + f 23,2 + .. + f 2n−1,2) + 2hδ 3 ( f 22,2 + f 24,2 + .. + f 2n−2,2) + m1u21,2 + m2u22,2 ... ... ... ... ... ... ... ... for j = n ; 4hδ 3 ( f 21,n + f 23,n + .. + f 2n−1,n ) + 2hδ 3 ( f 22,n + f 24,n + .. + f 2n−2,n ) + m1u21,n + m2u22,n . the matrix formation of the above equations is then written as follows: j (f,u ) = 1 2 f t q f + 1 2 u t hu , (39) where, q =   8 3 hδi 0 0 . . . 0 0 43 hδi 0 . . . ... ... . . . ... 0 ... ... 0 ... . . . ... 0 . . . ... . . . 8 3 hδi   , (40) h = ( 2m1 i 0 0 2m2 i ) (41) f ti = [ fi ,1, fi ,2,··· , fi ,n ] ∈ r1×n for i = 1, 2, ..., (n −1) with f t =[ f t1 , f t 2 ,··· , f tn−1 ] ∈ r1×(n−1)n . similarly, u t k = [ uk ,1, uk ,2,··· , uk ,n ] ∈ r1×n for k = 1, 2 with u t = [ u t1 ,u t 2 ] ∈ r1×2n . however, the matrices q ∈ r(n−1)n ×(n−1)n and h ∈ r2n ×2n in the compactified convex quadratic eqn. (39) are real, symmetric and positivedefinite with the diagonal coefficients described below as follows: q =   8 3 hδi : (k − 1)n + 1 ≤ i ≤ k n and j = i , for k = 1, 3, ..., (n − 1) 4 3 hδi : (k − 1)n + 1 ≤ i ≤ k n and j = i , for k = 2, 3, ..., (n − 2) 0 : otherwise (42) and h =   2m1 : 1 ≤ i ≤ n and j = i 2m2 : n + 1 ≤ i ≤ 2n and j = i 0 : otherwise (43) combining eqns. (25), (31) and (39) gives, in a matrix form, the compact convex quadratic optimization problem with linear constraints as expressed below:-  min j = 12 u t hu + 12 f t q f s.t : cu + d f = 0, mu ≥ g . (44) the above re-formulated discrete model is a well-structured, quadratic optimization problem whose objective function is convex and separable. while the constraint function is linear and coupled which made the model amenable to the admm. however, the mixed-inequality constraint model is re-formulated into an equality constraint form by introducing non-negative slack variables z (º 0) into the model. 3.4. stability and feasibility of the model with the assumption that the solution of the pde satisfies the usual differentiability conditions, the second order crank nicholson implicit scheme, with local truncation error of order o(h +δ2), was deplored for the discretization of 109 dawodu / j. nig. soc. phys. sci. 3 (2021) 105–115 110 the pde constraint to avoid the conditional stability and feasibility associated with explicit discretization methods. this stability of the implicit discretization method reflects in the consistency and statbility of the coefficient matrix operators. however, the composite simpson’s scheme, used for the discretization of the objective function, has truncation error of order o(h4) with degree of accuracy (precision) of three. this helps to improve the accuracy and rate of convergence of the model, hence an improvement over the trapezoidal used in [1]. 3.5. sparsity of the model the very suitable approach for ascertaining the sparsity of the model is to compactify it as studied by kameswaran and biegler in [8]. the set of optimization eqns. (45) was then re-structured in a compact form as stated below. min j (u , f ) = ( u t f t ) ( h 0̂ 0̂t q ) ( u f ) s.t ( c d m 0̄ ) ( u f ) ≥ ( 0∗ g ) , compactly written as, min j (w ) = w t p w s.t v w ≥ ḡ , (45) where, w = (u f )t , p = ( h 0̂ 0̂t q ) , v = ( c d m 0̄ ) and ḡ = ( 0∗ g ) . the matrices h and q are the two coefficient sub-matrices of the objective functions while c and d are the two submatrices of the constraint; both correspondent to those of the variables u and f . the sizes of the augmented matrices, showing the number of zero and non-zero entries are expressed in the table 1 below. the sparsity ratio is defined here as the proportion of the zero entries in comparison with the size of the matrix. obj. coef. constr. coef. j w size (n + 3)n × (n + 3)n n n × (n + 1)n zeros (n + 3)n 3n(2n − 1) + 3 − 4n non(n + 3)n + n n (n n + n − 6)+ zeros (n n + 3n − 1) (3n − 3 + 4n ) table 1: sparsity table for the compactified model 3.6. positive definiteness and convexity of the matrix operator definition 1. (sylvester criterion): a square (n × n) matrix is positive definite if it is real, symmetric with all the eigenvalues (λi > 0; i = 1, 2··· , n) or the leading principal minors (mi > 0; i = 1, 2··· , n) strictly positive. definition 2. (corollary to sylvester criterion): if the principal diagonal entries ai i are the only nonzero entries of a square (n×n) matrix such that ai j = 0 for i , j ; then the leading principal minors (mi > 0; i = 1, 2··· , n) are given by mi = ∏i j =1 a j j for i = 1, 2,··· , n. the constructed matrix-operator p for the derived quadratic function (45) is an augmentation of the real symmetric matrices q and h in (40) and (41) respectively whose principal diagonal entries are all strictly positive. therefore, p is symmetric and positive definite by definitions 2 and 3 since all the leading principal minors are all strictly positive. this property is pertinent to guarantee convexity and minimal solution of the quadratic function. 4. admm algorithm formulation in the formulation of the admm, the derived update formulas will be accelerated with an accelerator variant referred to as the relaxation factor. the results will then be extendable to the discretized pde governed ocp (44). the associated augmented lagrangian functional is expressed as lρ(u , f, z ,ξ1,ξ2) = 1 2 u t hu + 1 2 f t q f + l+(z )+ ρ 2 ∥ cu + d f +ξ1 ∥22 + ρ 2 ∥ mu −g − z +ξ2 ∥22 (46) with the m-admm iterations stated thus: u k+1 ← arg min u { 1 2 u t hu + ρ 2 ∥ cu + d f +ξk1 ∥22 + ρ 2 ∥ mu −g − z +ξ2 ∥22 } , (47) f k+1 ← arg min f { 1 2 f t q f ++ρ 2 ∥ cu + d f +ξk1 ∥22 } , (48) z k+1 ← arg min z {ρ 2 ∥ mu −g − z +ξ2 ∥22 } (49) s.t z ≥ 0, ξk+11 = ∂ξ1 l(u k+1, f k+1, z k+1,ξk1 ) (50) ξk+12 = ∂ξ2 l(u k+1, f k+1, z k+1,ξk2 ) (51) where µ ∈ r(n−1)n and λ ∈ rn are lagrange multipliers, ρ > 0 is the penalty parameter, ‖.‖2 denotes the euclidean (spectral) norm of a vector (matrix) argument, ξ1 = µ/ρ ∈ r(n−1)n and ξ2 = λ/ρ ∈ rn are the scaled dual variables, z ∈ rn is the 110 dawodu / j. nig. soc. phys. sci. 3 (2021) 105–115 111 introduced slack vector and l+(z ) is the indicator function for the non-negative orthants defined as l+(z ) = 0 for z ≥ 0 and l+(z ) = +∞ otherwise. as the algorithm runs through the (k +1)-th iteration, the primal variables u k and f k as well as the dual variables µk and λk are updated, hence the need to derive the update formulas. applying the karush-kuhntucker (kkt) optimality conditions on the augmented lagrangian functional (46) for the sequential minimization of the variables requires updating all the critical variables as indicated in the admm iterations. moreover the update formulas of the variables u k and f k will be accelerated using the gauss-seidel relaxation factor α ∈ [0, 2] as clearly illustrated in the works of nesterov [9] and ghadimi et. al [10]. 4.1. derivations of the update formulas applying the karush-kuhn-tucker (kkt) optimality conditions on the augmented lagrangian function in eqn. (46) for the sequential minimization of the variables requires updating all the critical variables as indicated in the admm iterations. u-update: setting ∂u l(u , . k ) = 0 yields; u k+1 = ρb1 m t b2 (u − update). (52) where, b1 = [ h +ρ(c t c + m t m ) ]−1 b2 = (g + z k −ξk2 ) −c t (d f k +ξk1 ) f-update: setting ∂f l(u k+1, f, .k ) = 0 and replacing cu k+1, to relax the algorithm, with h(u k+1, f k ,α) = αcu k+1 + (1 −α)d f k (53) to yield f k+1 = −ρb3b4 (f − update). (54) where, b3 = [ q +ρ(d t d ) ]−1 b4 = [ αcu k+1 + (1 −α)d f k +ξk1 ] z-update: setting ∂z l(. k+1, z , .k ) = 0 and relaxing it with h(u k+1, f k+1,α) yields; z k+1 = max { 0,−z ∗ } (55) where z ∗ = [ αcu k+1 + (1 −α)d f k+1 −g ] (56) updating the scaled-dual variables ξ1, ξ2 yields ξk+11 = ξk1 +cu k+1 + d f k+1, (57) ξk+12 = ξk2 + mu k+1 −g − z k+1, (58) 4.2. convergence analysis considering an optimization problem analogous to the form in (45) above;  min u,y f1(u) + f2(y ) + l+(z ) s.t h1(u) + h2(y ) = 0 g 1(u) − z = 0 (59) where x ∈ rn , u ∈ rm while f1 : rn → r ∪ {∞}, f2 : rm → r ∪ {∞} and f3 : rn → r ∪ {∞} are convex functions. the augmented lagrangian formulation is then stated thus: lρ(u, y,µ,λ) = f1(u) + f2(y ) + l+(z ) +µt ( h1(u) + h2(y ) ) + ρ 2 ∥ h1(u) + h2(y ) ∥22 + λt ( g 1(u) − z ) + ρ 2 ∥ g 1(u) − z ∥22 (60) where µ ∈ rn and λ ∈ rm are lagrange multipliers, ρ > 0 is the penalty parameter, ‖.‖2 denotes the euclidean (spectral) norm of a vector (matrix) argument, ξ1 = µ/ρ ∈ rn and ξ2 = λ/ρ ∈ rm are the scaled dual variables, z is the introduced slack vector and l+(z ) is the indicator function for the nonnegative orthants. from duality theory, it is well-known that if u∗, y ∗,µ∗ and λ∗ are the primal and dual variables that minimize the lagrangian function lρ, then every pair (u k , y k ) in the convex set, lρ(u ∗, y ∗,µ∗λ∗) ≤ lρ(x k , y k ,µ∗λ∗) (61) . let f k = f1(x k )+ f2(y k ) be the objective function value at the k -th iterate (x k , y k ) and f ∗ = f1(x∗) + f2(y ∗) be the optimal function value. let r k+11 = h1(uk )+h2(y k ) and r k+12 = g 1(uk )− z k be the residuals of the convex equality constraints in the k th iteration. the objective of this research is now to prove the following theorems. definition 3. (dual updates): at optimality, the update equations for the dual variables from the augmented lagrangian in (60) are given as µk+1 = µk + ρ [ h1(u k+1) + h2(y k+1) ] , (62) λk+1 = λk + ρ [ g 1(u k+1) − z k+1 ] , (63) definition 4. (lyapunov function): let v k be a non-negative quantity for the algorithm such that v k − v k+1 decreases in each iteration by an amount that depends on the sum of the norms of the residuals and on the change in slack variable z over one iteration. then the quantities are defined as; (i ) v k = 1 ρ ||µk −µ∗||22 + 1 ρ ||λk −λ∗||22 + ρ||z k+1 − z∗||22 (64) (i i ) ρvq ≤ v k − v k+1 ≤ v 0 (65) 111 dawodu / j. nig. soc. phys. sci. 3 (2021) 105–115 112 where vq = ∞∑ k=0 (||r k+11 ||22 + ||r k+12 ||22) + ||z k+1 − z k ||22 see proof in boyd and vandenberghe [11]. theorem 1. given the convex linearly constrained control problem (59) with the optimal dual and primal variables for the problem µ∗, λ∗, u∗, y ∗ and z∗. then the following residual terms (i ) s k1 = ρ [ ∂ht1 (u ∗) ( h2(y ∗) − h2(y k ) ) − ∂g t1 (y ∗) ( z∗ − z k )] (66) (i i ) s k2 = ∂f t2 (y ∗) + ∂h2(y ∗)µ∗ (67) that converge to zero for a given penalty parameter ρ > 0. proof : from the augmented lagrangian function (60), uk+1 minimizes the sub-differential lρ(u, y k ,µk ,λk ) with respect to u. hence lρ(u, y k ,µk ,λk ) with respect to u at uk is expressed as 0 ∈∂f1(uk+1) + ∂ht1 (uk+1)µk + ∂ht1 (uk+1) ( h1(u k ) + h2(y k ) ) + ∂g t1 (uk+1)λk + ∂g t1 (uk+1) ( g 1(u k+1) − z k ) (68) substituting for µk and λk from eqns. (62) and (63) into (68) yields, 0 ∈∂f1(uk+1) + ∂ht1 (uk+1) [ µk+1 − ρ [ h1(u k+1) + h2(y k+1) ]] ︸ ︷︷ ︸ µk + ∂ht1 (uk+1) ( h1(u k ) + h2(y k ) ) + ∂g t1 (uk+1) [ λk+1 − ρ [ g 1(u k+1) − z k+1 ]] ︸ ︷︷ ︸ λk +∂g t1 (uk+1) ( g 1(u k+1) − z k ) (69) ∂f1(u k+1) + ∂ht1 (uk+1)µk+1 + ∂g t1 (uk+1)λk+1 = ρ [ ∂ht1 (u k+1) ( h2(y k+1) − h2(y k ) ) −∂g t1 (y k+1) ( z k+1 − z k )] . (70) if uk+1 = u∗ minimizes the lagrangian functional, then eqn. (70) becomes lρ(x ∗, y ∗,µ∗λ∗) =ρ [ ∂ht1 (u ∗) ( h2(y ∗) − h2(y k ) ) −∂g t1 (y ∗) ( z∗ − z k )] , and converges to zero for any u ∈ ω (convex) which completes the proof for (i). qed � similarly, y k+1 minimizes the sub-differential lρ(uk+1, y,µk ,λk ) with respect to y such that 0 ∈ ∂f2(y k+1) + ∂ht2 (y k+1)µk + ∂ht2 (y k+1) ( h1(u k+1) + h2(y k+1) ) , 0 ∈ ∂f2(y k+1) + + ∂ht2 (y k+1) [ µk+1 − ρ [ h1(u k+1) + h2(y k+1) ]] ︸ ︷︷ ︸ µk + ∂ht2 (y k+1) ( h1(u k+1) + h2(y k+1) ) . this yields the equation below and completes the proof for (ii) ∂f2(y k+1) + ∂ht2 (y k+1)µk+1. (71) hence, y k+1 minimizes the convex function y 7→ f2(y ) + ∂ht2 (y )µk+1. (72) the alternating direction method of multipliers algorithm can now be applied to the derived optimization problem that was obtained from the discretized pde. by extension, the following vanishing dual residual obtained as the iterations get to optimality is stated below. s k1 = ρ [ c t d (f k+1 − f k ) − m t (z k+1 − z k ) ] . (73) algorithm: extension of admm algorithm in solving pde constrained optimal control problem input parameters and operators ρ, α, m , g , c , d initialize u 0, f 0, z 0, ξ01 = µ0/ρ, ξ02 = λ0/ρ set h º 0, q º 0 (symmetric and positive definite) for k = 0, 1, 2,··· , do | compute u k+1 using eqn. (52) | compute f k+1 using eqn. (53) | compute z k+1 using eqn. (54) | update ξk+11 and ξk+12 using eqns. (57) and (58) respectively. | until convergence according to eqn. (73) end (for) return u k+1, f k+1, z k+1, j ∗, ξk+11 , ξ k+1 2 112 dawodu / j. nig. soc. phys. sci. 3 (2021) 105–115 113 5. numerical experiments considering a pde constrained optimal control described heat transfer model written as follows: min u, f ∫ 5 0 ∫ π 0 f 2(x , t ) d xd t + ∫ 5 0 [u21 (t ) + 2u22 (t )]d t , s.t ∂2 f (x , t ) ∂x 2 = ∂f (x , t ) ∂t , f (x , t ) > sin(x ) sin( πt 5 ) − 0.7, f (x , 0) = sin(x ) − 0.7, f (0, t ) = u1(t ), f (π, t ) = u2(t ), t ∈ [0, 5], x ∈ [0,π], this section illiterates, via a number of computational experiments, the effects of different penalty parameter selection on the number of iterations required to reach the stopping criteria on the above problem as posed by ghobadi et. al in [2] and betts and campbell in [1]. the experiment involves selecting the termination (stopping) criteria ²r e l = 10−k and ²ab s = 10−k , k = 3, 4, 5, as relative and absolute tolerances respectively for the convergence of the algorithm. the algorithm was implemented with a matlab subroutine running on a pentium v computer with 4.0 gb ram and 1.7 ghz cpu. the discretization step in time is largely smaller that of the spacial step to keep it feasible; though the implicit cranknicholson numerical scheme used for the discretization of the pde keeps it unconditionally stable. below are the results of the experiment; table 2 above shows the numbers of iter1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 0 100 200 300 400 500 it e ra ti o n s relaxation factor (α) tol. = 10−4 tol. = 10−5 tol. = 10−6 effects of tolerance on varying relaxation factors figure 1: the effects of tolerance on varying relaxation factors 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 20 40 60 80 100 120 140 160 180 it er a ti o n s penalty parameter (ρ) tol. = 10−4 tol. = 10−5 tol. = 10−6 effects of tolerance on varying penalty parameters figure 2: the effects of tolerance on varying penalty parameters 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 50 100 150 200 250 300 it er a ti o n s penalty parameter (ρ) α = 1.2 α = 1.6 α = 2.0effects of α on varying ρ with tol. = 10−4 figure 3: the effects of relaxation factor on varying penalty parameters h = 0.20 h = 0.10 h = 0.05 α 10−4 10−5 10−4 10−5 10−4 10−5 1.20 235 294 192 240 164 210 1.40 86 128 669 107 56 93 1.60 52 82 41 67 32 58 1.80 37 60 28 49 22 41 2.00 28 47 21 38 16 32 table 2: effects of tolerance & time-steps h on varying relaxation factors (ρ = 0.1, δ = 1/3) ations for the varying relaxation parameters α when the tolerance is fixed for a particular step size in time h and penalty parameter ρ. the step sizes were only kept for h = 0.20, h = 0.10 and h = 0.05. for a fixed choice of the relaxation factor α = 2.0, tolerance tol . = 10−3 and step-size ρ = 0.1, the dual sum ||r k+13 ||2 = ||r k+11 + r k+12 ||2, the distances ||r k+11 ||2 = ||µk −µ∗||2 and ||r k+12 ||2 = ||λk −λ∗||2 were plotted against the iteration as shown below in figures 4, 5 and 6 respectively. the convergence of the dual residual sum follows same as the monotonically decreasing quantity; that is ||r k+13 ||2 ≤ v k . 0 20 40 60 80 100 120 0 0.2 0.4 0.6 0.8 1 iterations || r k + 1 3 || 2 ||rk3||2 (dual sum) α = 2.0 ρ = 0.1 tol. = 10−3 figure 4: the response of dual residual sum to iteration table 3 above shows the varying penalty parameters ρ for same fixed α and various values of time step size h. the iteration was plotted against the dual residual ||s k ||2 and the primal variables (||u k+1||2, ||f k+1||2, p k ) and the distance ||z k − z ∗||2 for a fixed choice of the relaxation factor α = 2.0, tolerance tol . = 10−3 and step-size ρ = 0.1. it was however observed, as expressed in figures 7, 8 and 9 respectively that the residuals of the algorithms decrease after subsequent it113 dawodu / j. nig. soc. phys. sci. 3 (2021) 105–115 114 0 20 40 60 80 100 120 0 0.01 0.02 0.03 0.04 0.05 iterations || µ k − µ ∗ || ||µk − µ∗||2 (primal-1) α = 2.0 ρ = 0.1 tol. = 10−3 figure 5: the response of first dual variable (µ) to iteration 0 20 40 60 80 100 120 0 0.2 0.4 0.6 0.8 1 iterations || λ k − λ ∗ || ||λk − λ∗|| α = 2.0 ρ = 0.1 tol. = 10 −3 figure 6: the response of second dual variable (λ) to iteration erations. this then indicates convergence of the algorithm. 0 20 40 60 80 100 120 0 0.2 0.4 0.6 0.8 1 1.2 x 10 −4 iterations s k ||sk||2(dual) α = 2.0 ρ = 0.1 tol. = 10−3 figure 7: the response of dual residuals (sk ) to iteration table 4 below shows the sparsity of the compact form of the model for varying and fixed step-sizes in time and space respectively; with the optimal results obtained at few iterations of the admm. it was observed that the level of sparsity increases for reducing step sized which consequently reduces the computational burden in the iterative process. table 5 h = 0.20 h = 0.10 h = 0.05 ρ 10−4 10−5 10−4 10−5 10−4 10−5 0.10 91 124 111 149 184 245 0.20 39 85 48 66 79 107 0.40 19 30 25 38 43 65 0.60 15 22 18 29 32 51 0.80 20 38 13 23 25 43 1.00 84 156 13 20 19 35 table 3: effects of tolerance & time-steps h on varying penalty parameters (α = 1.6, δ = 1/3) 0 20 40 60 80 100 120 0 1 2 3 4 5 iterations || u k || , || f k || , p k ||uk|— ||f k|| pk figure 8: the response of primal variables to iteration figure 9: the effects of iterations on the distance norm ||z k − z ∗||2 shows the results of the problem for the varying values of the parameters and step size with the optimal results obtained at few iterations of the admm. the results obtained on matlab subroutine were compared with those of ghobadi et al. in [2] for various time grid points (n ) ranging from n = 1000 to 8000 ran on mosek software. table 6 below shows the optimal objective functional values generated at varying numbers n of time grid points (tgp) for fixed number of gridpoints in space kept at n = 10. the results in [2] when implemented on mosek is same as that obtained by the admm at a shorter central processing unit (cpu) time though with higher iteration cycles due to high rate of convergence; using the tolerance of t ol . = 10−4. this clearly shows that the rate of convergence of the proposed algorithm is higher than that of mosek. 6. conclusions the extension of the admm to solving heat transfer problem as an example of pde-based optimal control problems (ocp) with equality-inequality constraints was well studied. the pde-based ocp was transformed into a discretized convex quadratic optimization problem amenable to the admm. h = 0.20 h = 0.10 h = 0.05 (n,n) (25,10) (50,10) (100,10) non-zero 325 650 1300 zero 105300 421850 1688700 size 105625 422500 1690000 sparsity 99.69% 99.85% 99.92% non-zero 883 1783 3583 zero 67867 273217 1096417 size 68730 275000 1100000 sparsity 98.74% 99.35% 99.67% table 4: sparsity from the compactification of the problem model with δ = 1/3 114 dawodu / j. nig. soc. phys. sci. 3 (2021) 105–115 115 h = 0.50 h = 0.10 10−4 10−5 10−4 10−5 ||u || 0.476610 0.476655 0.476600 0.476627 ||f || 0.245101 0.245661 0.239198 0.239730 ||j ∗|| 0.476611 0.476713 0.476611 0.476659 iter.(k ) 70 96 33 49 time 0.09282 0.11094 0.08385 0.11052 (n , n) (10,4) (10,4) (25,4) (25,4) table 5: summary of computational results at optimum parameters (ρ∗ = 1.60, α∗ = 2.00, δ = 1/3) obj. values iterations k (time/sec) n [2] proposed [2] proposed 1000 0.4741987 0.475758 16(0.26) 112(0.11) 2000 0.4741986 0.475781 18(0.43) 120(0.32) 4000 0.4744534 0.476010 18(0.84) 132(0.47) 8000 0.4745807 0.476620 24(2.04) 140(1.02) table 6: comparison of computational results at optimum the implicit crank nicolson method and the 3rd order simpsons rule were used for the discretization of the constraints and cost functional respectively with high feasibility, unconditionally stability and well-conditioned matrices. the admm algorithm converges faster with higher level of accuracy, low processing time and low memory requirements, when compared with that of ghobadi et. al [2]. moreover, this approach can as well be applied to other pde-based optimization problems like that of wave equations or problems involving integral (entropy) functions. acknowledgments i am grateful to prof. montaz ali of the university of the witwatersrand, sa for introducing the admm algorithm to me and for his helpful comments in improving the presentation of this article. references [1] j. t. betts & s. l. campbell, “discretize then optimize, in mathematics for industry: challenges and frontiers", siam (2005). [2] k. ghobadi, n. nedialkov & t. terlaky, on the discretize then optimize approach", preprint for industrial and systems engineering, 2009 [3] o. olotu & k. a. dawodu, “on the discretized algorithm for optimal proportional control constrained by delay differential equation", journal of math. theory & modeling 3 (2013), 157. [4] o. olotu, k. a. dawodu & t. t. yusuf, "algorithm for solving multidelay optimal control problems using modified alternating direction method of multipliers", journal of applied & comp. mathematics 8 (2019) 445. [5] o. olotu & k. a. dawodu, "numerical solution to one –dimensional multi-delay system using the modified alternating direction method of multipliers", journal of namp 50 (2019) 107. [6] b. he & x. yuan, "block-wise alternating direction method of multipliers for multiple-block convex programming and beyond", siam journal of comp. mathematics, 691 (2015), 145. [7] b. he, x. hong & x. yuan, “on the proximal jacobian decomposition of alm for multiple-block separable convex minimization problems and its relationship to admm", journal of scientific computing 66 (2016) 1204. [8] s. kameswaran & l. t. beigler, “advantages of nonlinear programmingbased methodologies or inequality path constrained optimal control problem a numerical study", siam journal on scientific comp., 30 (2008) 957. [9] y. e. nesterov, “a method for solving the convex programming problem with convergence rate o(1/k 2 )", dokl. akad. nauk sssr 269 (1983), 543. [10] e. ghadimi, a. teixeira, i. shames and m. johansson , "optimal parameter selection for the alternating direction method of multipliers (admm): quadratic problems", linnaeus center, ee royal institute of technology, 60 (2014) 644. https://doi.org/10.1109/tac.2014.2354892 [11] s. boyd l. vandenberghe, convex optimization, cambridge university press ny, usa 2004. 115 j. nig. soc. phys. sci. 2 (2020) 1–6 journal of the nigerian society of physical sciences original research analysis of a stochastic optimal control for pension funds and application to investments in lower middle-income countries tolulope latundea,∗, opeyemi odunayo esana, joseph oluwaseun richarda, damilola deborah darea adepartment of mathematics, federal university, oye-ekiti, nigeria abstract one of the major problems faced in the management of pension funds and plan is how to allocate and control the future flow of contribution likewise the proportion of portfolio value and investments in risky assets. this work considers the management of a pension plan by means of a stochastic dynamic programming model based on merton’s model. the model is analysed such that the conditions of optimal contribution and investment in risky assets are determined and sensitized. the case study of nigeria, ghana, kenya is considered for various periods in the model simulation. thus, the volatility condition obtained is used to estimate the efficiency of some important parameters of the model. keywords: pension, stochastic, control, risky asset, parameters article history : received: 26 november 2019 received in revised form: 01 january 2020 accepted for publication: 18 january 2020 published: 19 january 2020 c©2020 journal of the nigerian society of physical sciences. all rights reserved. communicated by: b. j. falaye 1. introduction pension is described as a sum of money paid regularly to a person who has come to the end of his normal working life, or an annuity paid regularly as a benefit due to his retirement (who no longer work because of age, disablement etc.). a pension plan is a method for a prospective retiree to transfer part of his or her current income stream towards a retirement income. pension plans are classified into two categories: the defined benefit (db) and defined contribution(dc) scheme. the db pension plan is mostly preferred by pension members due to his ability to bear the risk to the pension fund manager, while on the other hand dc pension fund plan on the other hand, is more popularly preferred by the pension fund managers due to its ability to transfer the risk to the pension fund members. the ∗corresponding author tel. no: +2348032801624 email address: tolulope.latunde@fuoye.edu.ng (tolulope latunde) last payment (benefit) given to the retiree is based on how the investment performed. in dc, the contributions are said to be fixed and the benefits depend on profits generated on the benefits of assets. the risk obtained from the fund management is born by recipients. it is not at all like the defined benefit plan whereby the benefits are typically identified with last salary cadre and the risk associated with financing is supposed by the sponsoring agent. the utilization of classical instruments as portfolio theory is reviewed [1-3]. the world bank has historically classified every economy as low, middle, or high income. it now further specifies countries as having low-, lower middle-, upper-middle -, or highincome economies. the world bank uses gross national income (gni) per capita, in current u.s. dollars converted by the atlas method of a three-year moving average of exchange rates,as the basis for this classification. it views gni as a broad measure and the single best indicator of economic capacity and progress. the world bank used to refer to low-income and 1 latunde et al. / j. nig. soc. phys. sci. 2 (2020) 1–6 2 middle-income economies as developing economies; in 2016, it chose to drop the term from its vocabulary, citing a lack of specificity. instead, they now refer to countries by their region, income, and lending status. the classification of countries is determined by two factors: a country gni per capita, which can change with economic growth, inflation, exchange rates, and population. revisions to national accounts methods and data can also influence gni per capita. classification threshold: the thresholds are adjusted for inflation annually using the special drawing right (sdr) deflator. mics have a combined population of five billion, or over 70% of the world’s seven billion people, hosting 73% of the world’s economically disadvantaged. representing about one-third of global gdp, middle-income countries (mics) are a major engine of global economic growth. three major countries namely: nigeria, ghana and kenya would furthermore be considered as a case study. looking at the report which classifies nigeria as a lowerincome country it says in sub-saharan africa, middle-income countries (mics) – with a gross domestic income (gdi) per capita between us$1,026 and us$12,475 – are divided into upper-middle-income countries (umis) and lower middle-income countries (lmics). it is noted that nigeria falls under lowermiddle-income countries. as a nation, nigeria is a high-income country and is very rich, however, when the earnings of its average citizens is considered, it could be seen as a low-income one and this is because a large percentage of its population is poor. nigeria is indisputably a low-income country, the larger percentage of citizenry and invariably the workforce earn around 60$ per month at minimum which may not be sufficient to cater for basic needs. ghana, at a per capita income of about $1,820, is an mic, but this classification masks wide gaps in infrastructural and human development. ghana’s position is at the lowest ebb of the lower-middle-income countries, what this means is that the country has only but transited by a small margin from a low-income to a middle-income economy status. many development indicators are still in the state of a low-income country, yet to reflect ghana’s new status as an mic. while it is agreed that the mics are a very diverse group by size, population, income and development levels, ghana’s current position is too precarious for comfort. rather than becoming complacent, government and ghanaian at large must use ghana’s lower-middle-income status as launchpad to propel the economy to the next level. african countries like angola, gabon, and botswana have already risen to upper-middle-income status and share membership with the likes of malaysia and south africa. such rising must be ghana’s aspiration. kenya is the economic, financial, and transport hub of east africa. kenya’s real gdp growth has averaged over 5% for the last decade. since 2014, kenya has been ranked as a lower middle-income country because its per capita gdp crossed a world bank threshold. while kenya has a growing entrepreneurial middle class and steady growth, its economic development has been impaired by weak governance and corruption. although reliable numbers are hard to find, unemployment and underemployment are extremely high, and could be near 40% of the population. in 2013, the country adopted a devolved system of government with the creation of 47 counties, and is in the process of devolving state revenues and responsibilities to the counties. agriculture remains the backbone of the kenyan economy, contributing one-third of gdp. about 75% of kenya’s population of roughly 48.5 million work at least part-time in the agricultural sector, including livestock and pastoral activities. over 75% of agricultural output is from small-scale, rain-fed farming or livestock production. tourism also holds a significant place in kenya’s economy. in spite of political turmoil throughout the second half of 2017, tourism was up 20%, showcasing the strength of this sector. inadequate infrastructure continues to hamper kenya’s efforts to improve its annual growth so that it can meaningfully address poverty and unemployment. kenya has also successfully raised capital in the global bond market issuing its first sovereign bond offering in mid-2014, with a second occurring in february 2018. however, there is a need to consider the utilisation of pension funds allocation of these mics on assets such that the wealth of the countries can be optimized. the dynamic model can be used to calculate optimal asset allocation as it also takes change in the climate investment such as change in expected risk and returns, this model is also important for managing pension fund asset.the theory of risk and return is practically referred to as the income that was established in addition to any change in the market price of the investment. carton articulated that returns are very core to any pension funds since it was shared among several members to the normal contributions [4]. stochastic model is considered because it allows us to investigate fully the dynamics of the fund through time, the analysis and control of pension fund dynamics are getting increasingly important as members start to pay more attention to the security of promised benefit and as sponsoring employers become more concerned about the timing and stability of cash flows and this method. merton pioneered the stochastic optimal control for solving continuous problems in asset management [2]. the hamilton-jacobi-bellman (hjb) equation is a common method used by many kinds of research in solving problems from the dynamic programming under the real-world probability measure, [5]. several authors have laid down analysis related to the stochastic control approach such as using stochastic dynamic programming to analyse the financial risk in a defined contribution (dc) pension scheme under gaussian interest rate models by attempting is to find an optimal investment strategy, [6-11]. likewise, several types of research have worked on utilising the mathematical approach in solving problems associated with pension funds management, [11-14]. however, some experts have carried out researches on the sensitivity analysis to ascertain the behaviour of parameters in the formulation of the model such as in control problem of management of assets, transportation problems, pension schemes and so on. [15-18] thus, we investigate herein the application of the dynamic model for pension fund as optimization system that will ensure appropriate standard of living before and after retirement 2 latunde et al. / j. nig. soc. phys. sci. 2 (2020) 1–6 3 by discovering its explicit solutions using parameter sensitivity analysis and illustrating the solutions by utilizing a process which is reformed to an asset-liability model (alm) as in [3]. 2. model formulation the dynamic model for pension fund management is reviewed herein and analysed [3]. the following variables and parameters used represented in tables 1 2 are functions of time t. parameters r, φ, σ, g, i ,µ are constant. table 1: definition of variables variables definition xt wealth(portfolio market value) pt pension payments ct contributions ht investment in the risky asset at market value of the risky asset. table 2: definition of parameters µ contribution growth rate φ risk premium (positive) r risk free rate g pension growth rate σ volatility of the risky asset i discount rate source: nigeria, ghana and kenya national pension commission (pencom), 2016-2017 annual reports. the two types of pension funds aggregated pension fund in this work where their growth rate g and µ respectively we have: dpt = gpt dct = µct (1) we consider a situation of two types of assets consisting: a riskfree asset with return, r; and a risky asset with price at which supports the brownian motion, [3]. risk-free rate is assumed constant; considering (c∗,h∗) as the policy of optimal control and xm to represent the maximum wealth, we shall look into the two types of pension funds (benefit and contribution) of the financial management that forms the aggregated pension fund dat/at = (φ + r)dt + σdwt (2) where r > 0 and diffusion σ > 0 are constant, wt represents the standard brownian motion defined. suppose contributors do not wish to settle for higher contributions either now or in the nearest period which as its effect on their discount rate. we dominated the psychological discount rate i, under the above conditions and assumptions, a reasoning pension fund manager will be willing to minimize: v = ∫ ∞ 0 ex p(−i s)c2s d s (3) but the optimal policy has to fulfil the given constraints: -payments of pensionpt -value of xt then we try to obtain the policy (ct,ht) where e(v) is minimized under the given constraints. given that ct represents the contribution and pt represents the pensions paid per time (assuming continuous discounts). the variable which is represented by x(t), t ≥ 0, is the adapted process that represents the aggregate of the wealth or the value of pension fund at time t. it is supposed that the activities start on the pension fund at period where t = 0 with wealth x0 ≥ 0. the wealth process is described by the equation: dxt = ht xt dat at + (1 + ht)xtrdt + (ct − pt)dt (4) the first term of the equation on the right is due to the risky asset and the second is due to the riskless asset and third represents the flow with respect to the balance of subscriptions (contributions) and payment of pensions (benefit). the above equation can be re-written by substituting equation (2) into (4) dxt = [rxt + ht xt + ct pt]dt + ht xtdwt (5) and assumption is made that 0 = in f ( e−itc2 + v′t + (rx + φhx + c − p)v ′ x + gpv ′ p + 1 2 v′′x,x x 2h2σ2 ) (6) using the equations above we have: 2r − i −φ2/σ2 > 0 (7) suppose i = r. this assumption becomes r −φ2/σ2 > 0 ≡ ω (8) by solving equation (1) (5) we obtain a solution 2r − i − φ2/σ2 > 0 , that is,by a sufficient high porfolio volatility (σ > φ/ √ r). suppose v(t,x,p) represents the value function of the formulated problem, the with bellman’s equation, we have 0 = in f ( eitc2 + v′t + (rx + φhx + cp)v ′ x + gpv ′ p + 1 2 v′′x,x x 2h2σ2 ) (9) x > 0 and h > 0 been a constraints, since the polynomial function of h and c in the bracket is satisfied by the optimal policy (h∗,c∗). φxv′x + v ′′ x,x x 2hσ2 = 0 (10) 2eitc + v′x = 0 (11) 3 latunde et al. / j. nig. soc. phys. sci. 2 (2020) 1–6 4 table 3: numerical values relative with respective countries country period r φ σ g ω nigeria 2016-2017 10.14 6.77 × 105 6.12 × 105 12.0 8.9163 ghana 2016-2017 21.46 2.12 × 107 6.39 × 107 15.7 21.3499 kenya 2016-2017 8.51 2.30 × 107 3.16 × 107 10.2 7.9802 table 4: risk free rate r vs volatility condition ω r ω 5 3.7763 10 8.7763 15 13.7763 20 17.7763 that is to say h∗ = φv′x v′′x,x xσ2 (12) c∗ = − v′x 2e−it (13) substituting the (11) and (12) in equation (8) gives: − 1 4 ei tv′2x + v ′ t + (rx − p)v ′ x + gpv ′ p − φ2 2σ2 v′2x v′′x,x = 0 (14) suppose v (t, x, p) = e−it f(x, p) (15) thus, the equation that satisfies f becomes − 1 4 f′2x + (rx − p)f ′ x + gpf ′ p − φ2 2σ2 f′x2 f′′x,x = 0 (16) equation (16) is homogeneous at variable y = x/p for f(y). let f(x, p) = p2 f (x/p) (17) then the differential equation satisfying f is given by − 1 4 f ′2 + (2g − i) f + ((r − g)y − 1) f ′ − φ2 2σ2 f ′2 f ′′ = 0 (18) we derive the solution by solving problem of the form: f (k) = ak2 + bk + c (19) identify a, b and c and find f (k) = (2r − i − φ2 σ2 )(y − 1 r − a )2 (20) we obtain the final value function by substituting equation (17) into equation (15), i.e., f into v, and by conjecture : v (t, x, p) = e−it ( 2r − i −φ2/σ2 )( x − p/(r − g) )2 (21) table 5: risk premium φ vs the portfolio volatility condition ω φ ω 1.0 × 105 10.1133 2.0 × 105 10.0332 3.0 × 105 9.8997 4.0 × 105 9.7128 table 6: volatility of the risky asset σ vs the portfolio volatility condition ω σ ω 1.0 × 106 9.6817 2.0 × 106 10.0254 3.0 × 106 10.0890 4.0 × 106 10.1114 this domain xm ≤ x shows the optimal policy to be zero contribution and there exists in the portfolio, no risky asset. by (10), (11) and (18), an optimal policy expression is given: h∗ = {(2r−i−φ2/σ2)(xm−x) 0 ifx ≤ xmifxm ≤ x (22) c∗ = {(2r−β−λ2/σ2)(xm−x) 0 ifx ≤ xmif xm ≤ x (23) the condition σ > φ √ r attest the constraints ( they have been applied, by [7] in the context of dc pension schemes) on h and c to be satisfied. 3. numerical simulations world bank classifies nigeria, ghana and kenya amongst others as lower-middle-income countries in africa. as a result, we select the three counties to utilise the pension data for the model verification and analysis which was obtained via maple software. we utilize the recent records of national pension commission(pencom) for each country such as their inflation, stock market prices, and interest rates. we also consider the volatility of the risky asset, risk-free rate, pension growth rate, risk premium and the volatility condition. tables 3-6 represent the computational results derived from the available data. 4. discussion the portfolio volatility 2r i (φ/σ)2 > 0 need not to be met, since the investments do not achieve a convincing returnto-risk ratio. 4 latunde et al. / j. nig. soc. phys. sci. 2 (2020) 1–6 5 figure 1: graph of the volatility of risk-free rate r against volatility condition ω figure 2: graph of risk premium φ against volatility condition ω figure 3: graph of the volatility of the risky asset σ against volatility condition ω however, by using the nigeria pension portfolio as a case study, the behaviour of some parameters in the model formulation is examined such that the effect of changes and estimation on each parameter on the derived volatility condition is determined and represented in the figures 1 3 below plotted using excel. in this work, we applied a dynamic model of pension management to real-life situations where a volatility condition was obtained alongside the optimal controls to determine the behaviours of the parameters of the model. the behaviour of some parameters of the model such that the effect of changes of the estimation of the values of each parameter on the derived volatility condition is determined and represented using tables and graphs. from figure 1, we determined that the higher the volatility of the risk-free rate r, the higher the volatility condition ω. also from figure 2, we determined that the lower the risk premium φ, the higher the volatility condition ω. from figure 3, we also determined that the higher the volatility of the risky asset σ, the higher the volatility condition ω. hence, we can verify that the volatility condition ω implies that the risk premium can be justified by high volatility. this justifies some logical reasoning aiding decision making in pension management. 5. conclusion in this work, we applied a dynamic model of pension management to real-life situations where a volatility condition was obtained alongside the optimal controls to determine the behaviours of the parameters of the model. this justifies a logical some logical reasoning aiding decision making in pension management. in future works, stochastic analysis of such models can be carried out considering a wider range of data depending on countries from different continents. likewise, a more analytic approach of the sensitivity analysis can be considered using more sophisticated software in solving and representing data and graphs of the new results. references 1. r. c. merton “lifetime portfolio selection under uncertainty: the continuous time case”, review of economics and statistics, 51 (1969) 247. 2. r. c. “merton optimal consumption and portfolio rules in a continuous time model”,journal of economic theory 42 (1971) 373. 3. j. f. boulier, e. trussant & d. florens “a dynamic model for pension funds management”, proceedings of the 5th afir international colloquium 1 (1995) 361. 4. b. carton, measuring organizational performance:an exploratory study, a dissertation submitted to the graduate faculty of the university of georgia, 2004. 5. m. assellaou, hamilton jacobi bellman approach for some applied optimal control problems, mathematics ensta paris tech, 2015. 6. t. e. duncan & b. pasik-duncan “a direct method for solving stochastic problems”, communications in information systems 12 (2012) 1. 7. e. vigna & s. haberman “optimal investment strategy for defined contribution pension schemes”, insurance: mathematics and economics 28 (2001) 233. 5 latunde et al. / j. nig. soc. phys. sci. 2 (2020) 1–6 6 8. p. battocchio & f. menoncin “optimal pension management in a stochastic framework”, insurance: mathematics and economics 34 (2004) 79. 9. f. menoncin & e. vigna “mean-variance target-based optimisation for defined contribution pension schemes in a stochastic framework”, insurance: mathematics and economics 76 (2017) 172. 10. a. cairns & g. parker “stochastic pension fund modelling”,insurance mathematics and economics 21 (2000) 43. 11. b. o. osu “the price of asset-liability control under tail conditional expectation with no transaction cost”, british journal of mathematics and computer science 1 (2011) 129. 12. a. cairns, d. blake & k. dowd “stochastic lifestyling: optimal dynamic asset allocation for defined-contribution pension plans”, journal of economic dynamic and control 30 (2006) 843. 13. g. deelstra, m. grasselli, & k. pierre-françois “optimal investment strategies in the presence of a minimum guarantee”, insurance: mathematics and economics 33 (2003) 189. 14. r. gerrard, s. haberman & e. vigna “optimal investment choices post retirement in a dened contribution pension scheme”, insurance: mathematics and economics 35 (2004) 321. 15. m. latkovic, & i. liker “sensitivity analysis of accumulated savings in a defined contribution pension system”, financial theory and practice 33 (2000) 431. 16. t. latunde, o. m. bamigbola & y. o. aderinto, “sensitivity of parameters in an optimal control model of the electric power generating system”, ilorin journal of computer science and information technology (iljcsit) 1 (2016) 54. 17. t. latunde & o. m. bamigbola “parameter estimation and sensitivity analysis of an optimal control model for capital asset management”, advances in fuzzy systems, (2018) 1. https//doi.org/10.1155/2018/4756520 18. t. latunde, j. o. richard, o. o. esan & d. d. dare “sensitivity of parameters in the approach of linear programming to a transportation problem of an optimal control model for capital asset management”, journal of the nigerian society of physical sciences 1 (2019) 116. 6 j. nig. soc. phys. sci. 3 (2021) 17–25 journal of the nigerian society of physical sciences dynamics of toxoplasmosis disease in cats population with vaccination idris babaji muhammada, salisu usainib,∗ adepartment of mathematics ,faculty of science, bauchi state university, gadau p.m.b65,gadau, nigeria bdepartment of mathematics , kano university of science and technology, wudil, p.m.b. 3244, kano, nigeria abstract we extend the deterministic model for the dynamics of toxoplasmosis proposed by arenas et al. in 2010, by separating vaccinated and recovered classes. the model exhibits two equilibrium points, the disease-free and endemic steady states. these points are both locally and globally stable asymptotically when the threshold parameter rv is less than and greater than unity, respectively. the sensitivity analysis of the model parameters reveals that the vaccination parameter π is more sensitive to changes than any other parameter. indeed, as expected the numerical simulations reveal that the higher the vaccination rate of susceptible cats the smaller the value of the threshold rv (i.e., increase in π results in the decrease in rv), leading to the eradication of toxoplasmosis in cats population. doi:10.46481/jnsps.2021.141 keywords: toxoplasmosis, vaccination, cats, sensitivity index article history : received: 31 october 2020 received in revised form: 24 november 2020 accepted for publication: 05 january 2021 published: 27 february 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: t. latunde 1. introduction the causative agent of toxoplasmosis is toxoplasma gondii (t. gondii), which is a protozoan of the order eucoccidioroda and the family sarcocystidae. t. gondii is characterized as an intracellular parasite that lives in the host cell by regulating vital processes to acquire nutrients, guaranteeing its survival and thus evading the host immune system [12]. it is an infectious pathogen which enters its host through ingestion of either the oocysts, the trachyzoite, or tissue cysts (bradyzoites) from contaminated water, soil, or infected meat. the three major types of disease reservoirs of t. godii are: domestic and wild cats, soil and water contaminated with feces (non-living reservoirs) and ∗corresponding author tel. no: +2348064237334 email address: salisu.usaini@kustwudil.edu.ng (salisu usaini ) bradyzoites in tissue cysts of intermediate hosts (animal reservoirs) [25]. the intermediate hosts of t. gondii include sheep, goats, rodents, cattle, swine, chicken and birds. the modes of t. gondii transmissions include ingestion of oocysts that excrete in cats feces for which tissue cysts develop through exposure to cat litter or soil and water. this is the most common and well known modes of transmission to other animals. t. gondii can also be transmitted via consumption of t. gondii tissue cysts in raw or undercooked meats, unpasteurized milk and consumption of oocysts in foods infected by contaminated fomites. this is called the food born transmission. another route of transmission is congenital toxoplasmos (transplacental transmission) in which a mother infect her offspring in uterus through the blood during pregnancy. a pregnant women with acute infections can transmit to the fetus and cause severe illness such as mental retardation, blindness, and epilepsy [12, 25]. it was re17 muhammad & usaini / j. nig. soc. phys. sci. 3 (2021) 17–25 18 ported in a recent study in dogs that sexual transmission of t. gondii in canine species is possible [25]. treatment of toxoplasmosis include sulfonamides against murine toxoplasmosis, combined therapy with sulfonamides and pyrimethamine as the standard therapy for toxoplasmosis in humans, spiramycin during pregnancy to reduce transmission of the parasite from mother to fetus, clindamycin especially for patients allergic to sulphonamides [12]. a number of mathematical epidemic models for the transmission dynamics of toxoplasmosis parasites with or without vaccination were proposed in the literature. in 2008, aranda et al. [6], proposed a simple mathematical model to study the dynamics of toxoplasmosis disease in the population of colombia. numerical simulations of the model using available data reveals the effect of some (hygienic actions, educations programs, more testing and treatments) strategies for the control of toxoplasmosis parasite. this model seems to be the first mathematical model for the transmission dynamics of toxoplasmosis disease in the human population. in the following year, an epidemic model of toxoplasmosis disease in human and cat populations was presented in [9]. the analysis of the model indicated that toxoplasmosis disease persistence or extinction is determined by the threshold parameter, r0. moreover, they showed by numerical simulations the importance of cats vertical transmission to the dynamics of the infection. in fact, the higher the vertical transmission the higher the proportions of infected cats and infected humans at the endemic state. thus, the vertical transmission could be an important mechanism that favours the maintenance of the virus areas with low human densities. another epidemiological model of toxoplasmosis in a cat population with a continuous vaccination schedule was proposed in [2]. indirectly, the model considers the infection of prey through the oocyst shedding by cats. they proved that the global dynamics and disease outcome are solely determined by the basic reproduction number, r0. furthermore, numerical simulations of the model suggest that the continuous vaccination program is more effective than the removing of oocysts in the environment since the increase of the latter in the environment to feasible values is not enough to reduce the threshold parameter, r0 to a value less than unity. the spatial spread of toxoplasmosis parasite was also considered later on. in 2012, a spatial model for the spread of toxoplasma gondii through a heterogeneous predatorprey system was proposed in [16]. they considered some relevant toy models due to the complexity of the proposed model. then they proved the existence and local stability of a persistent steady state for the underlying predatorprey model systems. a spatial mathematical model for the reproduction dynamics of the toxoplasma gondii parasite in the definitive host felis catus (cat) was presented in [10]. both asexual and sexual reproduction processes of the t. gondii parasite were incorporated in the model. some numerical results showed that variations in the transition and loss rates do not produce significant changes in the reproduction, propagation and creation of new populations. on the other hand, with either low or high consumption of oocysts from the environment by the cat, the involved populations are always reproduced. then they spread by all over epithelial cells and subsequently are expelled to the environment through the cat feces. ferreira et al. [11], studied the dynamical behaviours of both deterministic and spatial models of toxoplasmosis disease in cat and human populations. they showed that the deterministic model exhibits a trans-critical bifurcation and the system has no periodic orbits inside the positive octant. the global asymptotic stability of the endemic equilibrium point of the model was proven in the first octant. moreover, they carried out the local stability analysis for the spatially homogeneous equilibrium points of the reaction diffusion model. in the restricted region (the first octant), the global stability of both the disease-free and endemic states of such a model were established. in the last decade, a hybrid model for the spread of toxoplasmosis between two cities of colombia in cat and human populations was proposed by peña et al. in [4]. the model was developed using system dynamics (sd) and geographic information systems (gis). the analysis of the model indicates that the disease persistence continues in the first city, even after transported to another community. this demonstrated that these diseases should be treated through cooperative mechanisms between communities. a deterministic mathematical model of interaction between toxoplasma gondii transmission dynamics and host immune responses was developed and presented in [1]. the local asymptotic stability analysis of the model equilibria and numerical simulations were carried out to understand the transmission dynamics and the impact of different functional responses. in fact, the holling type ii functional response enhances the effect or cells of hosts immune response. in a recent development, an epidemic mathematical model of toxoplasmosis disease was presented in [15] to study the dynamic behaviour of the parasite for controlling the epidemic. the existence and stability analysis of each equilibrium point of the model was carried out. they constructed a function of two important parameters of the model namely the controlled rate and the rate of infected births which influences the disease spread. the bifurcation analysis of each region was carried out from each function of such parameters. three regional conditions were obtained through this analysis, showing the dynamics of the toxoplasmosis epidemic of these two important parameters with each interpretation of the bifurcation region. the widespread of co-infection by plasmodium species and toxoplasma gondii in humans mostly in the tropical region motivated ogunmiloro to study the co-infection dynamics of malaria-toxoplasmosis in the mainly human and feline susceptible host population in [18]. the analysis showed that malaria-toxoplasmosis infection free equilibrium is locally and globally stable if the basic reproduction number is less than unity. while the co-infection occurs when the basic reproduction number is greater than unity. furthermore, sensitivity analysis of the model parameters indicated the need for proper optimum medical strategies to reduce and contain the spread of malaria-toxoplasmosis co-infections. in 2010, a deterministic mathematical model for the transmission dynamics of toxoplasmosis disease in cats population under vaccination was proposed in [2]. implicitly, their model considers the infection of prey through the oocyst shedding by 18 muhammad & usaini / j. nig. soc. phys. sci. 3 (2021) 17–25 19 cats in the environment. the analysis of the model shows that the global dynamics and disease outcome are completely determined by the basic reproduction number, r0. moreover, the numerical results reveal that the effectiveness of the continuous vaccination program is more than the removing of oocysts in the environment. this is for the fact that an increase of the latter in the environment to feasible values is not enough to control disease persistence by reducing the threshold parameter, r0 to a value less than one. it is instructive to note that in their model the natural immunity and vaccine protection were assumed to be the same and provide permanent immunity. therefore, they merged the vaccinated cats and non vaccinated recovered cats in the same class. we extend their model here by separating vaccinated cats from non vaccinated recovered cats to have two classes. this is done for the fact that there is increasing clinical evidence which differentiate between these two immunities against toxoplasmosis in cats [5, 7, 8, 21]. as in [2], we show that the disease outcome, persistence or extinction is determined by the vaccinated reproduction number rv. the sensitivity analysis of the model parameters indicates that the vaccination parameter π is more sensitive to changes than any other parameter of the model. furthermore, numerical simulations show that the higher vaccination rate of susceptible cats reduces the value of the threshold parameter rv the more. this is also similar to the result presented in [2] that vaccination program is more effective than the removal of oocysts in the environment. we organise the remaining part of the paper as follows: in section 2, model description is provided, section 3 presents the stability analysis of the model equilibria while sensitivity analysis of the model parameters is presented in section 4. finally, we discuss the results and give a concluding remark in section 5. 2. model formulation to extend the model proposed by arenas et al. [2], we separate the compartment of vaccinated cats from that of recovered (vaccinated) cats. thus we divide the total population of cats to four compartments of susceptible (s(t)), cats that may become infected after an effective contact with oocysts, infective (i(t) with t. gondii, vaccinated (v (t)), and non vaccinated recovered r(t). using the following notations and hypotheses as presented in [2], we have the required model (1). the rates of transfer between the four classes are shown in figure 1. 2.1. basic assumptions i. o(t) repreaent the number of oocysts in the environment at time t, ii. the birth rate (δ) of cats is assumed to be equal to the cats natural death rate µ, iii. the non vaccinated susceptible cats transits to an infective compartment after an effective contact with oocysts at the rate β. thus, β depends on the probability of an effective contact with oocysts, τr μi μ0o μs �i αi μv μrγv πs δi βso δ(s+v+r) s i v r o figure 1: schematic diagram of model (1). iv. vaccinated susceptible cat s (t) transits to the vaccinated compartment (v (t)) at a rate π and infected cats to recovered compartment r(t) at a rate α, v. number of oocysts, o(t) at time t determine the number of infected cats i(t), vii. µ0 is the death rate of the oocysts which can be modified by the removal of oocysts from the environment, vi. it is assumed that there is vertical transmission in the infected cats population (i.e., transmission from mother to fetus), viii. vaccinated cats do not shed oocysts and acquire permanent immunity so that they move to recovered compartment r(t). the recovered cats can be re-infected with t.gondii but do not shed oocysts again and so, they move to susceptible class at the rate τ, ix. the susceptible cats s (t) have the same probability of becoming infected (i.e., homogeneous mixing is assumed) x. κ > 0 is the rate of appearance of new oocysts from the infected cats into the environment, xi. the parameter γ is the progression rate from vaccinated class to the recovered compartment. ds dt = µv + µr −βs o −πs + τr, di dt = βs o −αi, dv dt = πs −µv −γv, dr dt = αi −µr −τr + γv, do dt = κi −µ0o, (1) with n(t) = s + i + v + r = 1. in the first equation of model (1), the rate of change of the susceptible cats population is increased by per capita birth rate 19 muhammad & usaini / j. nig. soc. phys. sci. 3 (2021) 17–25 20 table 1: parameter description parameter description value source β effective contact rate 0.5254 wu and pei π vaccination rate of susceptible 0.2 arena et al. α hbv induced death rate 0.5 arenas et al. µ0 death rate of oocysts 7 100000 wu and pei. µ natural birth rate 0.652 wu and pei τ immunity rate 0.452 assumed κ rate of appearance of oocyst 120 wu and pei γ progression rate of vaccinated cats 0.3 assumed δ of susceptible, vaccinated and recovered classes, the waning immunity after recovery at the rate τ. it decreases by vaccination at the rate, π, natural death µ and effective contact at the rate β between the oocysts and the healthy cats. the rate of change of infected cats is increased by effective contact of the susceptible cats and the oocysts at the rate β, with birth rate δ of the infected cats. while it decreases by contracting the disease at the rate α. the third equation of model 1 is increased by vaccinating susceptible cats at the rate π and decreases by natural death µ and the rate at which they move to recovered class γ. similarly, the rate of change of recovered cats is increased at the per capita rates of recovery of infected and vaccinated cats α and γ, respectively. it is reduced by natural death at the rate µ and wane of immunity from the recovered class at the rate τ. finally, the rate of change of oocyst with time is increased by shedding the virus at the rate, κ and reduced by the natural death of oocysts at the rate µ0. using r(t) = 1 − s − i − v , the system of equations (1) becomes. ds dt = (µ + τ)(1 − s − i) −τv −βs o −πs, di dt = βs o −αi, dv dt = πs −µv −γv, do dt = κi −µ0o (2) theorem 1. model (1) is positively invariant and attractive in the biologically feasible region ω0 = { (s, i, v, o) ∈ r4+ : 0 < s ≤ (µ + τ)(µ + γ) (µ + γ)(µ + π + τ) + πτ , 0 6 o 6 κ µ 0 } . proof. using the first and fourth equations of system (2), we have ds dt ≤ (µ + τ)(µ + γ) − [(π + µ + τ)(µ + γ) + τπ]s and do dt ≤ κ−µ0o. then ds dt + [(π + µ + τ)(µ + γ) + τπ]s ≤ (µ + τ)(µ + γ), so that s (t) ≤ (µ + τ)(µ + γ) (π + µ + τ)(µ + γ) + τπ and o(t) ≤ o(0)e−µ0 t + κ µ 0 (1 − e−µ0 t), with o(t) ≤ κ µ 0 if o(0) < κ µ 0 as t approaches ∞. thus ω is positively invariant. moreover, if s (0) > (µ + τ)(µ + γ) (π + µ + τ)(µ + γ) + τπ and o(0) > κ µ 0 then the solution either enters ω in finite times or s (t), o(t) approaches (µ+τ)(µ+γ)(π+µ+τ)(µ+γ)+τπ , κ µ 0 asymptotically [2]. hence the region attract all solutions in r4+. thus it is sufficient to consider the dynamics of the model (2) in ω since it is mathematically well-posed. 3. model analysis 3.1. disease free equilibrium (dfe) in the absence of infection (i.e., when all the infected compartments of the model (2) are empty), we have the disease free equilibrium point of the model denoted by p0. then p0 = (s 0, i0, v0, o0) = ( (µ + τ)(µ + γ) (π + µ + τ)(µ + γ) + τπ , 0, πs 0 µ + γ , 0 ) . for us to establish the linear stability of equilibria, we need a threshold parameter usually called the basic reproduction number. this threshold parameter is defined as the number of secondary infections produced by one infective when introduce into the susceptible population [13]. the next generation operator method can be used to compute such a threshold parameter as presented in [20]. moreover, when there is a vaccination such a parameter is called a vaccinated reproduction number (see for instance, [22]). the method consists of two different matrices, the matrix of the new infection terms and that of the transition terms denoted by g and m, respectively. thus 20 muhammad & usaini / j. nig. soc. phys. sci. 3 (2021) 17–25 21 g = [ βs 0 0 0 ] , m = [ 0 α µ0 −κ ] . then gm−1 = [ κβ(µ+τ)(µ+γ) αµ0 [(µ+π+τ)(µ+γ)+τπ] κβ(µ+τ)(µ+γ) (µ+π+τ)(µ+γ)+τπ 0 0 ] hence, vaccinated reproductive number rv is the largest eigenvalue of the matrix gm−1. therefore rv = ρ(gm −1) = κβ(µ + τ)(µ + γ) αµ0[(µ + π + τ)(µ + γ) + τπ] . 3.1.1. local stability analysis of dfe theorem 2. the dfe, p0 of model (2) is locally asymptotically stable on ω if rv < 1 and is unstable whenever rv > 1. proof. the jacobian matrix of system (2) evaluated at the df e is as follows j(p0) =  −(µ + π + τ) −(µ + τ) −τ −βs 0 0 −α 0 βs 0 π 0 −(µ + γ) 0 0 κ 0 −µ0  . then, interchanging row-3 and row-4 and using row operations, the matrix j(p0) becomes j(p0) =  −(µ + π + τ) −(µ + τ) −τ −βs 0 0 −α 0 βs 0 0 κ 0 −µ0 π 0 −(µ + γ) 0  . divide row 4 by π, then new row 4 is r4 = (π + τ + µ)r4 + r1 and so j(p0) =  −(µ + π + τ) −(µ + τ) −τ −βs 0 0 −α 0 βs 0 0 κ 0 −µ0 0 −(µ + τ) − [ (µ+γ)(µ+π+τ) π + τ ] −βs 0  . thus the first two eigenvalues of j(p0) are λ1 = −(µ + π + τ) and λ2 = − [ (µ + γ)(µ + π + τ) π + τ ] . while the remaining two eigenvalues are for the following 2×2 sub-matrix of j(p0). j0(p0) = [ −α βs 0 κ −µ0 ] . now, the trace and the determinant of this matrix are respectively given by tr(j(p0)) = −(α + µ0) < 0 and det(p0) = αµ0 − κβ(µ + τ)(µ + γ) (µ + γ)(µ + π + τ) + τπ = αµ0 ( 1 − κβ(µ + τ)(µ + γ) αµ0[(µ + γ)(µ + π + τ) + τπ] ) = αµ0(1 −rv). thus the determinant is greater than zero if rv < 1. hence, all the four eigenvalues of j(p0) have negative real parts and so, the dfe, p0 of the system (2) is locally asymptotically stable whenever rv < 1. 3.1.2. global stability of the dfe using lyapunov principle method in conjunction with lasalle’s invariant principles [19], we establish the global asymptotic stability of the dfe as presented in the following theorem. theorem 3. if rv 6 1, then the dfe, p0 of model (2) is globally asymptotically stable on ω. proof. we define a lyapunov function by h(i, o) = i(t) + β(µ + τ)(µ + γ) µ0[(µ + π + τ)(µ + γ) + πτ] o(t). then, the time derivative of v along the solutions of system (2) is ḣ(i, o) = i̇(t) + β(µ + τ)(µ + γ) µ0[(µ + π + τ)(µ + γ) + πτ] ȯ(t) = βs o −αi + β(µ + τ)(µ + γ) µ0[(µ + π + τ)(µ + γ) + πτ] (κi −µ0o) = βo ( s − (µ + τ)(µ + γ) (µ + π + τ)(µ + γ) + πτ ) + α(rv − 1)i = βo(s − s 0) + α(rv − 1)i. then, ḣ 6 0 since s 6 s 0 and rv ≤ 1, with v̇ = 0 if and only if s = s 0 and rv = 1. hence, v is a lyapunov function on ω and so, s → (µ+τ)(µ+γ)(µ+π+τ)(µ+γ)+πτ as t →∞. therefore, the maximum invariant set in {(s, i, o, v ) ∈ ω0 : ḣ 6 0}, is a singleton set p0. it follows from lasalle’s invariant principles [19] that p0 is globally asymptotically stabile when rv 6 1. 3.2. endemic equilibrium (ee) in the presence of disease infection, the infected compartments (i, o) are non empty and so, the disease invades the population. let pe = (s e, ie, oe, ve) be an endemic equilibrium point of model (2), then setting the right hand side of model (2) to zero we obtain after algebraic manipulations. s e = αµ0 κβ , ie = αµ0[(µ + π + τ) + τπ](rv − 1) κβ(µ + γ)(α + µ + τ) , ve = απµ0 κβ(µ + γ) , oe = αµ0[(µ + π + τ) + τπ](rv − 1) βµ0(µ + γ)(α + µ + τ) . hence, pe is the unique endemic equilibrium point of system (2) which exist if and only if rv > 1. 21 muhammad & usaini / j. nig. soc. phys. sci. 3 (2021) 17–25 22 3.2.1. local stability analysis of the eep theorem 4. the eep, pe of model (2) is locally asymptotically stable in the interior of ω if rv > 1. proof. evaluating the jacobian matrix of system (2) at pe gives j(pe) =  −a −(µ + τ) −αµ0 κ −τ b −α αµ0 κ 0 π 0 0 −µ 0 κ −µ0 0  , with a = (µ+π+τ+βoe) and b = βoe. then we use elementary row operations to obtain j(pe) =  −a −(µ + τ) −αµ0 κ −τ 0 −[ αab + (µ + τ)] αµ0 κ [ ab − 1] 0 0 −(µ + τ) −αµ0 κ −( aµ π + τ) 0 κ −µ0 0  . thus the eigenvalues of j(pe) are λ1 = −a λ2 = −( aµ π + τ), λ3 = −[ αa b + (µ + τ)], and λ4 = − µ0 κ (µ + τ + α). therefore, all the four eigenvalues are real and negative and so, the eep, pe is locally asymptotically stable if rv > 1. . 3.2.2. global stability of endemic equilibrium theorem 5. if rv > 1, then there exist an endemic equilibrium point pe and it is globally asymptotically stable in the interior of ω. proof. : given that rv > 1, then the existence of eep is guaranteed (see section 3.4). consider the lyapunov function below: l(s, i, o, v ) = s − s e − s e ln ( s s e ) + i − ie − ie ln ( i ie ) + v − ve − ve ln ( v ve ) + o − oe − oe ln ( o oe ) , the time derivative of l along the solution curve of the system (2) yields l̇ = ( 1 − s e s ) ṡ + ( (1 − ve v ) v̇ + ( 1 − ie i ) i̇ + ( 1 − oe o ) ȯ = ( 1 − s e s ) [(µ + τ)(1 − s − i) −τv −βs o −πs ] + ( 1 − ie i ) (βs o −αi) + ( (1 − ve v ) [πs − (µ + γ)v ] + ( 1 − oe o ) (κi −µ0o) (3) but at endemic state, we have µ + τ = (µ + τ)(s e + ie) + τve + βs eoe + πs e αie = βoes e, πs e = (µ + γ)ve, κie = µ0oe. (4) then using (4) in (3), we obtain alter algebraic manipulations l̇ = (µ + τ)s e) ( 2 − s s e − s e s ) + πs e ( 2 − s s e − s e s ) + (µ + τ)ie ( 1 − s e s ) − (µ + τ)i ( 1 − s e s ) + τve ( 1 − s e s ) −τv ( 1 − s e s ) + βoes e ( 1 − s e s ) + βos e ( 1 − s s e ) + βos ( 1 − ie i ) + βoes e ( 1 − i ie ) + πs ( 1 − v ve ) + πs e ( 1 − ve v ) + κi ( 1 − oe o ) + κie ( 1 − o oe ) . (5) since s ≤ s e, i ≤ ie, v ≤ ve and o ≤ oe, then (5) becomes l̇ ≤ (µ + τ)s e) ( 2 − s s e − s e s ) + πs e ( 2 − s s e − s e s ) + βoes e ( 2 − s s e − s e s ) + βoes e ( 2 − i ie − ie i ) + πs e ( 2 − v ve − ve v ) + κie ( 2 − o oe − oe o ) = s e(µ + τ + π + βoe) ( 2 − s s e − s e s ) + βoes e ( 2 − i ie − ie i ) + πs e ( 2 − v ve − ve v ) + κie ( 2 − o oe − oe o ) . (6) it follows from arithmetic-geometric inequality that( 2 − s s e − s e s ) ≤ 0, ( 2 − i ie − ie i ) ≤ 0,( 2 − v ve − ve v ) ≤ 0, ( 2 − o oe − oe o ) ≤ 0, and so, l̇ ≤ 0. then we conclude this theorem using similar argument as in the proof of theorem 3. 4. sensitivity analysis one can observe from section 3, that the disease persistence or otherwise in a community is determined by the threshold parameter rv. moreover, the uniqueness properties of both disease-free (p0) and endemic (pe) equilibria indicate that there is an exchange of stability between p0 and pe when rv = 1 (i.e., p0 undergoes a transcritical bifurcation at rv = 1, see for instance [23, theorem 4]). however, it is instructive to investigate the model parameter that has greatest effect on this threshold parameter, thereby having same effect on the toxoplasmosis disease persistence. to achieve that, we can apply sensitivity indices to measure the relative change of a state variable with a changing parameter [17, 23]. then using the definition in [17], the normalised forward sensitivity index with respect to each of the parameters of rv is 22 muhammad & usaini / j. nig. soc. phys. sci. 3 (2021) 17–25 23 figure 2: sensitivity analysis of the reproductive number rv, showing changes in the parameters (a) π, (b) β, (c) κ and (d) µ0. parameter values used are as presented in table 1. the following. υ rv κ = ∂rv ∂κ × κ rv = +1 υ rv β = ∂rv ∂β × β rv = +1 υ rv µ = ∂rv ∂µ × µ rv = πµ[αµ0(µ + γ) + τ][αµ0(µ + π + τ)(µ0 + γ) + πτ] (µ + τ)[αµ0(µ + γ)(µ + π + γ) + πτ]2 υ rv τ = ∂rv ∂τ × τ rv = πτ[αµ0(µ + γ) −µ][αµ0(µ + π + τ)(µ0 + γ) + πτ] (µ + τ)[αµ0(µ + γ)(µ + π + γ) + πτ]2 υ rv γ = ∂rv ∂γ × γ rv = γπτ[αµ0(µ + γ)(µ + π + τ) + πτ] (µ + γ)[αµ0(µ + γ)(µ + π + γ) + πτ]2 υ rv α = ∂rv ∂α × α rv = − κβµ0(µ + τ)(µ + γ)2(µ + π + τ) [αµ0(µ + γ)(µ + π + τ) + πτ]2 υ rv µ0 = ∂rv ∂µ0 × µ0 rv = − αµ0(µ + γ)(µ + π + τ) αµ0(µ + γ)(µ + π + τ) + πτ υ rv π = ∂rv ∂π × κ rv = − π[αµ0(µ + γ) + τ] αµ0(µ + γ)(µ + π + τ) + πτ (7) figure 3: sensitivity analysis of the reproductive number rv, showing changes in the parameters (e) γ, (f) µ, (g) τ and (h) α. parameter values used are as presented in table 1. it can be observed from equation (7) that most of the expressions for the sensitivity indices are complex and so, we evaluate these indices at the parameter values presented in table 1. the 23 muhammad & usaini / j. nig. soc. phys. sci. 3 (2021) 17–25 24 table 2: sensitivity indices of rv to parameters for the toxoplasmosis model, evaluated at the parameter values presented in table 1. the parameters are ordered from most sensitive to least one. parameter sensitivity index 1. π −644.49655 2. β +1.00000 3. κ +1.00000 4. µ0 −1.00000 5. γ 0.94230 6. µ 0.58798 7. τ −0.58650 8. α −0.00003 resulting sensitivity indices of rv to the eight model parameters from the most sensitive to the least one are shown in table 2. we infer from figures 2 and 3 that the threshold parameter rv increases with increasing values of β,κ,µ and τ. while it decreases with increasing values of π,µ0 and α. as expected the vaccination parameter π is more sensitive to changes than any other parameter of the model. 5. discussion and conclusion in this note, we extend the deterministic model for the transmission dynamics of toxoplasmosis in cats population with vaccination by separating the vaccinated and the recovered compartments. the stability analysis of the model equilibria are carried out and we show that the unique disease-free equilibrium as well as the unique endemic state of the model are asymptotically stable globally under certain condition. that is, the dfe, p0 is stable globally asymptotically when rv < 1, while the eep, pe is globally asymptotically stable if rv > 1. in order to complete our investigation, we use the sensitivity indices as presented in [17] to determine the most sensitive model parameter. as expected, the vaccination parameter, π is the most sensitive to changes than any other parameter of the model. moreover, considering the complex nature of the expressions for the sensitivity indices in equation (7), we evaluate these indices at the parameter values presented in table 1. then, we order them from the most sensitive to the least one as presented in table 2. furthermore, the numerical simulations in figures 2 and 3 reveal how the threshold parameter rv varies with varying values of each of the model parameters. in fact, rv increases with increasing values of the parameters β,κ,µ and τ and decreases with increasing values of π,µ0 and α. in conclusion, the sensitivity analysis indicates that the best way to contain the spread of toxoplasmosis is either to increase the vaccination rate of susceptible cats or to increase the death rate of the oocysts by environmental sanitation. it can be observed from figure 2 that vaccination is the best control strategy than the removal of the oocysts. this is similar to the result presented in [2]. references [1] a. gautam & a. 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[25] t. dubie, g. terefe, m. asaye & t. sisay , “toxoplasmosis: epidemiology with the emphasis of its public health importance” 2 (2014) 097. 25 j. nig. soc. phys. sci. 3 (2021) 201–208 journal of the nigerian society of physical sciences optimal representation to high order random boolean ksatisability via election algorithm as heuristic search approach in hopeld neural networks hamza abubakara,b,∗, abdu sagir masanawac, surajo yusufb, g. i. boakuc aschool of mathematical sciences, universiti sains malaysia bdepartment of mathematics, isa kaita college of education, p.m.b 5007, dutsin-ma, katsina, nigeria cdepartment of mathematical sciences, federal uniersity dutsin-ma, katsina, nigeria abstract this study proposed a hybridization of higher-order random boolean ksatisfiability (ranksat) with the hopfield neural network (hnn) as a neuro-dynamical model designed to reflect knowledge efficiently. the learning process of the hopfield neural network (hnn) has undergone significant changes and improvements according to various types of optimization problems. however, the hnn model is associated with some limitations which include storage capacity and being easily trapped to the local minimum solution. the election algorithm (ea) is proposed to improve the learning phase of hnn for optimal random boolean ksatisfiability (ranksat) representation in higher order. the main source of inspiration for the election algorithm (ea) is its ability to extend the power and rule of political parties beyond their borders when seeking endorsement. the main purpose is to utilize the optimization capacity of ea to accelerate the learning phase of hnn for optimal random k satisfiability representation. the global minima ratio (mr) and statistical error accumulations (sea) during the training process were used to evaluate the proposed model performance. the result of this study revealed that our proposed ea-hnn-ranksat outperformed abc-hnnranksat and es-hnn-ranksat models in terms of mr and sea. this study will further be extended to accommodate a novel field of reverse analysis (ra) which involves data mining techniques to analyse real-life problems. doi:10.46481/jnsps.2021.217 keywords: hopfield neural network, election algorithm, boolean satisfiability, random ksatisfiability article history : received: 01 may 2021 received in revised form: 14 june 2021 accepted for publication: 24 july 2021 published: xx xxx 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: t. latunde 1. introduction the central motivation of many types of research carried out in the field of, machine learning, decision science and artificial neural networks (anns) is the large structure of their training stages. many anns’ complex training stage provide a powerful ∗corresponding author tel. no: email address: zeeham4u2c@yahoo.com (hamza abubakar ) mechanism for solving optimization problems such as decisionmaking or classification tasks. neural networks are used extensively in many fields of study that originated in mathematical neurobiology. this network attempts to simulate human brain capabilities. it has been utilized since the last decade as a theoretically sound alternative to traditional statistical models. the classification based on the neural networks model becomes efficient when applied in a hybrid framework with many forms of predictive models [16]. the inception of neural networks 201 abubakar et al. / j. nig. soc. phys. sci. 3 (2021) 201–208 202 (anns) was initiated as variety of capable networks that act as a useful tool for specific tasks such as recognition problem [17], data mining [18], forecasting problems [21], prediction problems[14]. there have been many recent developments in an attempt to assemble different structures in refining the existing ann models by integrating them with proficient searching techniques to intensify the quality of the standalone ann framework [1]. hopfield neural network (hnn) is a type of recurrent artificial network that mimics the operating capacity of human memory. the ability of hnn to manage nonlinear complex patterns by its training and testing capability is particularly useful for interpreting complex real-life computer science communities [8]. in this work, a novel and powerful heuristics search technique is known as election algorithms has been explored for effective training of hopfield neural network model. election algorithms has been utilized in solving various mathematics and engineering benchmark problems which yield successful performance in both convergence rate and better identification of global optima. we can infer that the application of employing election algorithms is endless; from industrial planning, scheduling, decision making and machine learning. election algorithms have been utilized in [11] successfully for predicting groundwater level [12]. recently, an election algorithm has been proposed in [20] to accelerate the learning phase of hopfield neural network for random 2satisfiability. the performance of election algorithms in solving the travelling salesmen problem has been investigated in [5]. although election algorithm has been applied in various optimization areas, it recorded tremendous achievement in searching for the optimal solution to various combinatorial optimizations problems. in our work, election algorithms has been applied on the trainig phase of hopfield network model in accelerating the training process for optimal representation (global minimum) solutions to high order logic (ranksat for k ≤ 3). however, we did not come across any work that combined the global and local searching capacities of election algorithms in enhancing the training phase of hopfield neural network in searching for optimal representation of high order logic programming (ranksat). therefore, a new hybrid computational technique has been proposed by integrating hybrid election algorithm in the learning process of hopfield neural network for optimal ranksat logical representation in achieving a efficiency, robustness and better accuracy. the contributions of this work include; (1) upgrading ran2sat to ran3sat; (2) implementing the new logical rule in the hopfield neural network (ranksat-hnn);(3) incorporating election algorithm (ea) in the training phase of the hopfield neural network (hnn) for optimal random ksatisfiability and (4) performing comparison of the election algorithm with other state-of-the-art searching methods in hopfield neural network for random ksatisfiability representation. the present work investigate the effectiveness of the election algorithms in hopfield neural network for ranksat will be explored by comparing its performances to the state-of-the-art model. by developing an effective intelligent working model based on artificial neural networks, the proposed hybrid computational model will be useful to various computational science and optimization communities by proving an alternative approach in doing computation. the structure of this work is as follows. in section 2, we present the proposed methods involving the mapping of ranksat as a new logical rule in hopfield neural network and a heuristic search method in the hopfield neural network learning phase via election algorithms. section 3 reports the models’ performance evaluation metrics. the experimental results of their discussions are reported in section 4. the last section concludes the paper. 2. materials and methods 2.1. random ksatisfiability (k ≤ 3) propositional satisfiability logic can be perceived as a logical rule that consists of clauses that contain literals or variables. ranksat belongs to the family of non-systematic boolean logical clauses in which each logical clause involves a random number of literals [13]. the general description for the formulation franks at is presented in equation (1) as follows. franks at = t ∧ i=0 c(3)i j ∧ i=0 c(2)i d ∧ i=0 c(1)i (k ≤ 3) (1) where t, j, d ∈ [1, 2, ..., n], t > 0, j > 0 and d > 0. the clause fran3s at is defined as a ranksat(k ≤ 3)forc (k) i clauses described in equation (2) as follows. c(k)i =  (αi ∨βi ∨ψi) , i f k = 3 (αi ∨βi) , i f k = 2 τi , i f k = 1 (2) where the variables αi ∈ [αi,¬αi],βi ∈ [ βi,¬βi ] ,τi ∈ [τi,¬τi],ψi ∈[ ψi,¬ψi ] are defines literals and their negation in logical clauses respectively. in this work, we refer to fr it as a conjunctive normal form (cnf) formula in which the logic clauses in ranksat logic are chosen uniformly, independently among all 2x ( d + j + t y ) without replacement a logic clauses of lengthx. we refer to the optimal mapping y (fx) → [1, −1] as a logical interpretation expressed as 1 for (true) and -1 otherwise. logically, there are many formulations to ranksat logical clauses, one of the formations taking into account k ≤ 3 is presented as follows. franks at = (τ1 ∨¬τ2 ∨τ3) ∧ (β1 ∨β2 ∨¬β2) ∧¬α1 (3) according to equation (1),franks at comprises of ci (3) = (τ1 ∨¬τ2 ∨τ3), c2 (2) = (¬β1 ∨β2)and c1 (1) = ¬α1. therefore, the outcome of in equation (3) is satisfied franks at = −1 if (τ1,τ2,τ3,β1,β2,β2,α1) = (1, 1, 1, 1, 1, 1) 3 clauses are satisfied ( c(3)i , c (2) 1 , c (1) 2 ) . this research franks at will be embedded in hopfield neural network via election algorithm as a search technique in the proposed model.franks at logic will modify the networks to represent the true structure or behaviour of the data involved since hnn is a non-symbolic platform, franks at serve as a symbolic mode representation that can be combined with these networks. 202 abubakar et al. / j. nig. soc. phys. sci. 3 (2021) 201–208 203 2.2. random ksatisfiability in hopfield neural network hopfield neural network (hnn) belongs to the family of a recurrent neural network that mimics the human biological brain [15]. its structure consists of interconnected neurons and a powerful feature of content addressable memory that is crucial in solving various optimization and combinatorial tasks[13]. the network is composed of organised neurons of which is portrayed by a variable called the ising variable. in bipolar recognition, the neurons in discrete hnn are used whereby s i ∈ [1,−1]. the fundamental overview of neuron state activation in hopfield neural networks is shown in equation (4) below, s j = { 1 , i f ∑ j wi js j > ς −1 , otherwise (4) where wi j is the synaptic weight vector from starting from j neuron to i neuron. we defineds i as the state of the neuron i in hnn and ς is the predefined value. the value of ς = 0.001 has been specified in [6],[3],[20] to certify that the network’s energy decreases to zero. the synaptic weight connection in the discrete hnn contains no connection with itself, the synaptic connection from one neuron to other neurons is zero. the model holds symmetrical features in terms of architecture. hnn model has similar intricate details to the ising model of magnetism as described in [15]. s i → sgn [hi(t)] (5) where hi is the local field that connects all neurons in hnn. the sum of the field is induced by each neuron state as follows, hi = n∑ k n∑ j wi jks js k + n∑ j wi js j + wi (6) the task of the local field is to evaluate the final state of neurons and generate all the possible 3-sat induced logic that was obtained from the final state of neurons. one of the most prominent features of the hnn network is the fact that it always converges. the generalized fitness function franks at regulates the synaptic combinations in hnn and franks at is presented as follows. efranks at = n n∑ i=1 v∏ j=1 ti jk (7) where v and n n are the number variables and the number of neurons generated in franks at respectively. we defined the inconsistency of franks at representation as follows. ti j =  1 2 ( 1 − s ρ ) , i f ¬ρ 1 2 ( 1 + s ρ ) , otherwise (8) the value ofefranks at is proportional to the value of “inconsistencies” of the logical clauses. the rule for updating the neural state is, s i (t + 1) = { 1 , hi ≥ 0 −1 , hi < 0 (9) equation (9) represents the lyapunov energy function in hnn for 3rd order logic. hfranks at = − 1 3 ∑n i=1 ∑n j,k ∑n k=1,i,k w (3) i jk s is js k − 1 2 ∑m i=1,i, j ∑m j=1 w (2) i j s is j − ∑m i=1,i, j w (1) i s j (10) equation (10) has been applied to classify whether a solution is a global or local minimum energy. the hnn will generate the optimal assignment when the induced neurons state achieved global minimum energy. there are limited works to combine hnn and ea as a single computational network. thus, the robustness of election algorithm manage in improving the training process in hopfield model. consequently, the quality of the final neuronal state can be maintained according to equation (11) as utilized in [6],[4],[3],[20] as follows.∣∣∣hfranks at − hminfranks at ∣∣∣ ≤ ξ (11) where ξis the pre-determined tolerance value. the value ξ = 0.001 was taken in [6],[4],[3],[20],[19],[2][10-11,18-21]. if the franks at logical representation embedded in hnn does not satisfy the criteria state in equation (11) then the final state of the neurons have been trapped in the wrong pattern. 2.3. election algorithm as heuristic search in hopfield learning phase the election algorithm (ea) is an iterative population-based algorithm that provides a varieties of solutions in the search space. the local search function have been partitioned into search spaces. the optimization procedure is inspired based on the voting process in human society [11],[10]]. generally, the population of an individual is divided into two parties which later carry out a series of operations such as initialization, eligibility stages, advertisement and alliance. these stratgiies are resulted in a stronger searching technique. the primary goal of ea is to encourage candidates to coverge on a global minimum solution (best solution) to optimzation problem [10]. optimization process based on standard hopfield model has a high probability of becoming caught at a suboptimal solution with the number of neurons firing into the network [3],[9],[7]. various metaheuristics algorithms, such as electin algorithm, have been purposefully used in hnn to improve its searching capabilities and to solve the issue of premature convergence before achieving the global optimal, which will maximize the number of fulfilled clauses during the network’s training process. because of its ability to merge local search into a partitioned search area, ea used mechanisms such as a constructive marketing approach, a negative campaign, and an alliance to maximize the entire searching space. the main steps of the procedure in the ea-hnn-ranksat model considering k ≤ 3 is presented from stage 1 to 5 as follows: stage 1:initialization the main component of any search and optimization is to find the right solution in terms of the problem’s variable’s parameter. it is created an array of vector parameters to be optimized. a population of this size npop as a starting point of an individual τv = [ τi,τ2,τ3, ....,τnpop ]t for each solution were 203 abubakar et al. / j. nig. soc. phys. sci. 3 (2021) 201–208 204 generated. each solution is distributed within the vector boundary range depending on the variable as follows, τv = λ min v + τ1 ∗ ( λmaxv −λ min v ) (12) where v ∈ n and τv described the location of the vth supporter the nvar −− dimensional search space and npop described to the number of search representative. a random number with a uniform distribution is defined as τ1 ∈ [0, 1]. this problem searches for the optimal ranksat clauses. stage 2:eligibility measurement everyone’s eligibility(fitness) function is measured based on the franks at clauses in using as follows. fτv = t∑ i=1 c(3)i + j∑ i=1 c(2)1 + d∑ i=1 c(1)i (13) where fτv denoted as eligibility of each person in search spaces τv, c (k) i is the clause in franks at and t, j, d ∈ [1, n] are the total number of logical clauses in franks at logic. stage 3: creating an initial population a population of persons (individuals) pc o f npop search agent was employed in ea. each solution represents the eligibility (fitness) of a candidate ( eδc ) and the general political parties (pc ) represent search space. individuals’ populations are splitting population of individual npop into political parties (pc ) is part of the ea policy. each pc includes the contender (δc) and their followers(τv)which served as search agents in the solution space. the δc together with their τv form some pc . they τv are divided δc based on eδc in which the initial number τv of a δc is proportionate to eδc . according to [10] the τv of a δc, are identified by normalizing eδc computed as follows. αi = ∣∣∣eδci ∣∣∣ = ∣∣∣∣∣∣ eδc − max(i)∑ k ∈ δc eδk − max(i) ∣∣∣∣∣∣ (14) where i = { δc j| j ∈ δc } , eδci , αi defined the eligibility of candidate and normalized eligibilities of the candidate δci respectively and δc designated the initial number of candidates in the solution space. the number of an individual to serve as initial candidates δci was modelled as follows. δci = αr npop (15) the original number of δτvi is modelled as follows. δτvi = npop −δci (16) then, we randomly select δτvi of the τv and add to δci which form an pc in the search space. stage 4: campaign strategy: ’ this stage is modelled from step 1 to step 3 as follows step 1: positive advertisement (ϑ) in the modelling of ea ϑ. we select τv a position that is in form of a variable of δc in the search space. the intention of sampling the random numbers is to choose a τv1 location to be replaced by a new voter τv2 . the method for determining the selection rate λτ ∈ [0, 1]. the number of variables transferred to the τv by the δc is defined according to the following. ψτ = λτs c (17) the random choice variables in the problem space that must be replaced are signified by ψτ. the selection rate is defined by λτ. s c signifies the total number of δc in the problem space. to transfer τv to another δc , the eligibility distance coefficient (ed ) was used model as follows, ed = 1∥∥∥∥eδci − eδτvi ∥∥∥∥ + 1 (18) where eδci is utilized to described the eligibility of δci and eδτvi has been used to represent the eligibility of δτv i voter in the search space. step 2: negative advertisement (ω) ea algorithm uses negative campaign operation (ω)as a search mechanism in their opposition movement, to fascinate members of other parties this leads to the marginalized parties’ revival and deterioration of progress in the following ways. ω = τv t = { τvi, r j ≤ ϕ τv j, r j > ϕ, (19) where r j ∈ [0, 1] and ϕ is the negative advertisement constant in ea. step 3: coalition strategy(cl) in election algorithm, two or more parties come together for a new party, sharing the same ideas and aims in space to find solutions; confederates if they have the same ideas; ea, two or more parties may sometimes come together for a new party, possessing the same ideas and aims in space to find solutions. as a result, some applicants are exiting the advertisement with a new nominee dubbed ”leader,” while the candidate who withdrew from the election arena is dubbed ”follower.” to rule the optimum solution search in the search space, ea employs a coalition operation. by building trial vectors using elements of existing party candidates in the solution space and improving the search space, the coalition strategy can effectively gather information about effective party mergers. τ̂ vi = τv1 + λτ(τv3 −τv2 ) (20) where τ̂ vi defined as a coalition parameter of political parties and λτ ∈ [0, 1] served as a scaling factor and i is an index of current solution during the coalition process(cl). in ea, a population of solution vectors is randomly created at the start. in election algorithm, each fresh solution achieved will compete with a united party in the search space. stage 5:stopping condition (election day) at the start of ea, a population of the optimization algorithm is generated at random. in the election algorithm, each new solution will participate in the search space with a unified group [11]. 204 abubakar et al. / j. nig. soc. phys. sci. 3 (2021) 201–208 205 figure 1. implementation of ranksat 2.4. simulation procedure implementation of the neuro-heuristic searching method of ranksat in hnn. the program’s main task is to find the best ”model” that find the optimal occurrences of random-ksat. both logical variables and clauses were initially randomized. simulations were executed by manipulating a different number of neurons complexity ranging from 10 ≤ n n ≤ 110. the simulation has been conducted on ranksat as a logical clause in hnn according to the flowchart in figure 1. 3. performance the efficiency of performance of the ea-hnn-ranksat model has been quantified based on global minimum ratio (mr), statistical error measurement based on the sum of square error (sse) and mean absolute error (mae) as well as model computational time (ct ) presented in equation (20) to equation (23) respectively. mr = 1 ab t∑ i hfranks at (21) s s e = d∑ i=1 ( f n n − h d ) 2 (22) mae = d∑ i=1 1 n | fn n − hd| (23) where fn n and hd characterized the hnn the output and the target output values respectively, d categorized the number of permutations in hnn. 4. results and discussion table 1. global minimum ratio for ranksat logic nn ea es abc 10 1 1 1 20 1 1 1 30 1 1 1 40 1 1 0.9999 50 1 1 1 60 1 1 0.9997 70 1 nil 1 80 1 nil 1 90 1 nil 0.9996 100 1 nil 1 110 1 nil 0.9899 table 1 and figure 2 until figure 4 displayed the performance of ranksat in hnn considering the third order of k ≤ 3 in terms of mr, errors accumulations and time-consuming during the program execution. in our experiments, we explore the performance of the proposed training approach using a different number of neurons10 ≤ n n ≤ 110. the general trend of the model performance indicates a massive increase in errors accumulation and cpu time considering the complexity of the neuron fired to hnn in searching for optimal ranksat representation. the increasing trend in error behaviour shows the complexity of the neuron states of ranksat which proved to be an np problem [7],[9]. according to sse and mae in figure4 and fig 5 respectively in measuring the hnn performance during the learning phase in searching for correct optimal assignment to ranksat logical clauses, the proposed method, ea-hnn-ranksat, was able to achieve efranks at = 0, with lower statistical errors accumulation than abc-hnnranksat. this may be due to the multiple optimization layers involve in ea which has a better screening stage in searching for optimal assignment leading to efranks at → 0in fewer iterations than abc-hnn-ranksat.this explores the optimal capacity of ea in lowering the complexity of the hnn searching of ranksat logical towards error accumulation by reducing the number of iterations in the optimization process. the optimal behaviour of the hopfield neural network models based on mr has been recorded in table 1. the efficiency of the election algorithm (ea) are observed in comparison with other metaheuristics search approaches in hopfield neural network for ranksat representation leading to efranks at → 0. table 1, ea-hnn-ranksat and es-hannranksat can retrieve more accurate neural assignment that leads to the best global solution during the training process, 205 abubakar et al. / j. nig. soc. phys. sci. 3 (2021) 201–208 206 figure 2. sse for ranksat in hnn. figure 3. mae for ranksat in hnn while in abc-hnn-ranksat model, some neural states were stuck at n n = 60, n n = 60, n n = 90, n n = 110 a suboptimal local solution,but still managed to achieved closed to 92% success. meanwhile, es-hnn-ranksat can only accommodate n n ≤ 60, as the model exceeds the running time threshold, the neuron complexity increases, which is particularly problematic in the case of inconsistent interpretation ¬franks at . the core objective of incorporating metaheuristics in artificial neural network is to increase the flow of learning process by reducing the sensitivity of the neuros complexity, allowing the neurons to progress into relaxation and recovery stages sucessfully. the ea-hnn-ranksat model was able to recovered a much more appropriate final configuration that leads to a global minimum solution. this confirms the robustness and higher efficiency in neuro-searching embedded by ea to strengthen the hnn learning process for ranksat logical representation. if the mr of the model hnn network approaches one after the computing cycle, all solutions generated in the network have achieved global minimum energy[19]. it can be observed from figure 3 and figure 4 that the learning errors measured in terms of sse and mae increase massively as the neurons passed n n ≤ 20. figure 2, presents the trend of performance based on sse measure. the high accumulation of sse was demonstrated by es-hnn-ranksat. eahnn-ranksat accumulated lower error with close to 98% figure 4. cpu time for hnn performance. accuracy. ea-hnn-ranksat performs well as the number of neurons reaches a certain threshold. n n ≥ 20 the rapid increase in error was noticed. a similar pattern is reported in figure 3, which evaluate the models’ performance based on mae error. it revealed that ea-hnn-ranksat accumulated lower mae than es-hnn-ranksat with higher mae accumulation. however, the performance of abc-hnn-ranksat and ea-hnn-ranksat displayed similar trend. in general, eshnn-ranksat has the lowest results, with over 45 per cent of searches failing. the sse and mae values for hnn-ranksat searching behaviour have been observed to increase rapidly as the network’s neurons become more complex. compared to ea-hnnranksat and es-hnn-ranksat, the proposed ea-hnnranksat was able to obtain efranks at → 0 with a lower error accumulation. this is due to the ea scanning process involving multiple optimization surfaces that have a better filtering point in the state space, enabling the selection procedure to get excellent performance leading to efranks at → 0 in fewer iterations. in comparison to es-hnn-ranksat and abchnn-ranksat, ea-hnn-ranksat registered lower sse and mae, as per error analysis in figures 2 and 3. this study looked into the robustness of ea in decreasing the resistance of the hnn to error occurrence by lowering the number of iterations to the bare minimum. consequently, in es-hnnranksat, the study of mr, sse, and mae abruptly end at n n = 60. this may be attributed to the ineptness of the learning method used in es-hnn-ranksat, which can’t handle ambiguity. as a result of several fluctuations by neurons, the solutions were crafted at a sub-optimal solution (wrong pattern). it is clear that ea-hnn-ranksat agreed with abc-hnnranksat but outperformed es-hnn-ranksat in searching optimal representation ranksat logical representation. figure 4 displayed the hnn-ranksat models according to their running time during the implementation cycle. the proposed ea-hnn-ranksat was able to execute n n = 90 within 513.23 seconds faster than abc-hnn-ranksat which produce n n = 90 in about 763.02 seconds. the conventional es-hann-ranksat model was able to withstandn n ≤ 60 in 1026.53 seconds. examining the cpu’s use patterns in figure 4, it could be observed that, the ranksat logical clauses are 206 abubakar et al. / j. nig. soc. phys. sci. 3 (2021) 201–208 207 becoming complicated and complex, the searching for global solutions to the ranksat logical clause required more effort which subsequently required more execution time to achieve. the search capacity in es-hnn-ranksat demands more time in to search 30 ≤ n n ≤ 60neurons. the ea-hnn-ranksat and abc-hnn-ranksat are similar in their running time to 10 ≤ n n ≤ 80neurons. however, ea-hnn-ranksat was slightly faster than abc-hnn-ranksat at the initial and final searching process. this is because more neurons are reqyured during the training phase to allow network to migrate through the energy level and best on optimal solutions. in other words, as the number of neurons grew, the number of errors accumulated less in ea-hnn-ranksat, which subsequently reduce the cpu time comsumption. in this case, es-hnn-ranksat required further iterations to find the best solution that corresponds to efranks at = 0, which took more cpu time. in the context of the ranksat logical representation k ≤ 3, the robustness of integrating ea to promote the training phase of hnn can be seen. even during the learning process, ea’s stochastic scanning behaviour expands the structure hnn for appropriate ranksat representation. consequently, the eahnn-ranksat model’s composition would reflect the diversification of the final neural states. as a result, the ea-hnnranksat, where the chance of obtaining diversified franks at solutions is much higher, the solutions are dynamically swapped. as a result, ea-hnn-ranksat will produce a greater variety of franks at logical clauses that are achievable leading to efranks at = 0. the essence of hnn-ranksat, on the other hand, will present difficulties in the event of conflicting assignment. as the number of neurons grew throughout the trial, the integration of ea in hnn dealt systematically with the higher learning complexity. this paper investigates the robustness and efficacy of ea’s local quest and global approach. the robustness of the local and global search capability involved in ea, which serves as the learning mechanism in hnn, is linked to the success of ea-hnn-ranksat in exploring the global solution. the robustness of the local and global search capability contained in ea, which serves as the learning mechanism in hnn, is linked to the success of ea-hnn-ranksat in hunting for the global solution franks at . at the early stages of ea, where the number of neurons is limited, the local search potential has a major impact. this paper investigates how to improve the control parameters in ea to improve the learning process and achieve optimum efranks at = 0 logical representation. at the outset, a candidate selection optimization operator is needed to speed up the process of choosing the most qualified candidate to act as a leader (solution). multiple optimization layers employed by ea-hnn-ranksat to diversify the approach space and increase the searching capability in a specific area [11]. the positive advertisement is the first optimization layer in ea, which optimizes among candidates in a specific political party in the search space. the negative advertisement is another layer that allows other candidates from another party to take the supporter from their party. the coalition’s policy has the potential to have a huge effect in attracting the largest number of people who support the party’s manifesto in seeking global solutions. within a fair period, this process would provide a collaborative candidate of equal fitness [12],[10]. because of these features in the ea, the hybrid proposed model can reduce the number of iterations an hnn is needed during the learning process by ensuring that there is a minimal amount of error accrued after the experimentation. the ea-hnn-ranksat model’s systemic solution search space can make the local and global search processes easier to achieve global solutions. the hybrid approach will effectively search for the optimal solution in all of the described spaces thanks to the partitioning mechanism of the search space. finally, the ea-hnn-ranksat model has a shorter computing time because of its campaign and alliance mechanisms, which systematically improve the unified party’s chances of victory in a decent period. 5. conclusion a hybrid approach was proposed in this work, in which the election algorithm (ea) was combined with a hopfield neural network (hnn) to perform ranksat representation as a new logical law. the proposed hybrid ea-hnn-ranksat model can be conclusively shown to be a robust heuristic methodology that is effective in improving or following preferred assignments, even in clauses of high difficulty, based on the findings presented based on experimental simulations performed. this is related to the ea process’s stronger optimization layers, which speed up hnn’s learning process in looking for an optimal ranksat assignment of greater eligibility. it was revealed that the ea-hnn-ranksat model was able to complete the searching process slightly faster than abc-hnn-ranksat and es-hnn-ranksat. notwithstanding, all of the hnn models under consideration produced excellent results when it came to representing ranksat logic in hnn and computing the global solution to efranks at = 0 within the confines of a reasonable cpu timeline. as a result, in terms of mr, mae, sse, and cpu power, ea-hnn-ranksat faced less numerical strain during the training phase than other models. finally, other metaheuristics approach such as firefly algorithm, dronefly algorithm etc will be hybridized with the hopfield neural network (hnn) to enhance its computational process. we will also explore other np problem such as a 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[21] a. wanto, a. p. windarto, d. hartama & i. parlina, “use of binary sigmoid function and linear identity in artificial neural networks for forecasting population density”,ijistech (international journal of information system & technology) 1 (2017) 43. 208 j. nig. soc. phys. sci. 1 (2019) 103–115 journal of the nigerian society of physical sciences original research investigating the performance of point to multipoint microwave connectivity across undulating landscape during rainfall w. nwankwoa, k. e. ukhureborb,∗ asoftware engineering/cyber physical lab, department of mathematics and computer science, edo university iyamho, p.m.b. 04 auchi, edo state, nigeria bclimatic/environmental/telecommunication physics unit, department of physics, edo university iyamho, p.m.b. 04 auchi, edo state, nigeria abstract one of the most debated issues surrounding wireless connectivity is performance especially under different topographic and climatic scenarios. performance has a direct relationship with throughput measured in terms of how well a given wireless connectivity provides consistent services over a given period compared to the wired alternative. research has shown that wireless connectivity is constrained by significant physical components such as topography, weather conditions, propagation frequency, and distance. it is commonplace to see notable vendors of wireless network products make claims as to how their technologies are designed to remedy any signal degradation that may arise from the aforementioned physical elements. this paper is aimed at evaluating the performance of a point to multipoint connectivity using ubiquiti’s 5.8 ghz point to multipoint base stations deployed within a landscape marked by series of undulating highlands and lowlands. in this experiment, a base station node is established with connectivity to two other nodes of same specifications with one node as the destination radio whereas the other acts as the control which is located on a table land. the nodes were separated by triangular distances of 3 km and network connectivity was maintained over thirty days during periods of rainfall. packets sent and received across each node was carefully recorded. the results from the analysis showed that packet losses to and from the control node was significantly lower than that of the other node under same weather conditions. keywords: landscape, microwave, nodes, performance, signal article history : received: 10 october 2019 received in revised form: 27 october 2019 accepted for publication: 29 october 2019 published: 14 october 2019 c©2019 journal of the nigerian society of physical sciences. all rights reserved. communicated by: o. j. abimbola 1. introduction the technological breakthrough and development experienced in the field of wireless communication would not have been a success if references are not made to early scientists who did a lot of research in the field of wireless communications. several inventions, theories and principles have been established in order to generate background foundation towards enormous advancement in the field of wireless communication [1]. ∗corresponding author tel. no: +2348035383194 email address: ukhurebor.kingsley@edouniversity.edu.ng (k. e. ukhurebor ) these technologies continued to develop for over a century as people began to unravel the intricacies of telecommunications [1, 2, 3]. the development from 1g to 4g and now into lte and beyond to 5g is a milestone. the dramatic development was not just experienced at an instance, but was seen as an improvement over an existing technology, a breakthrough in the field, and a transition from one to another with marked elemental upgrades. the evolution of mobile communication cannot be well understood if proper description of mobile communications, is not reviewed in respect of the overall technology, speed, frequency and system in numeric generations such as 3g, 4g or 5g [1]. each generation has unique technologies 103 nwankwo and ukhurebor / j. nig. soc. phys. sci. 1 (2019) 103–115 104 that define it, and explains the differences throughout the evolution of mobile communications and what we can expect from the future generations of these technologies. wireless networks have some constraints to a wider option in enterprise network design and implementation. such constraints include: attenuation, latency, interference, distortion of signals, as well as the effect of weather conditions specifically rainfall on performance and signal propagation [4-18]. but with consistent advancement in technology, some mechanisms have been integrated into high-end radios to help mitigate some of these constraints. however, there are still some debatable issues as to the performance of wireless networks under extreme weather conditions and unusual landscapes [10, 11, 12]. the aim of this study is to investigate the performance of a 5 ghz point-to-point and point-to-multipoint radios across undulating landscape during rainfall. we intend to achieve this aim by understudying the geography of the study area, which according to previous studies [1-12], have remarkable topographic characteristics with emphasis on the distribution of weather conditions and signal propagation. we would attempt to determine the propagation and monitor the behaviour of radio signals within the 5 ghz frequency band from one point to another having regard to the undulating lowlands and highlands during rainfall. appropriate comparison on performance of same radio specifications under normal weather conditions on lowlands would also be considered in this study. 2. review of related works indisputably, movement of vegetation structures introduces an adverse environment for high frequency radio wave propagation [10]. hashim et al., [19], examines a series of vegetation scattering measurement campaigns during various wind conditions. their measurements were divided into controlled and outdoor environments. the controlled environment measurements were conducted in an anechoic chamber at 0.9,2.0,12.0 and 17.0 ghz., while the outdoor measurements were carried out at 1.8 ghz, as well as recording of a transmitted signal originating from an existing digital cellular system (dcs-1800) base station. their results were presented in terms of first-order and second-order statistics. from their analysis, the received signal behaviour was highly wind dependent, especially when the environment is changing from calm to a windy condition. the signal fastfading was found to be rician distributed and an empirical model of the k-factor variation over wind speed were also presented in their study. in the study carried out by perras et al., [20], they compared the various temporal characteristics of radio channels for a broad range of frequencies, including 2.45,5.25,29.0 and 60.0 ghz, in various vegetation and weather conditions. a considerable number of data points, were in excess of 1.9 billion at 500 samples per second (equivalent to 45 days) was collected and analysed for three particular types of channels (foliated deciduous trees, non-foliated deciduous trees and coniferous trees). the radio channels were statistically analysed and the resulting probability density functions (pdfs) and cumulated density functions cdfs) were compared with existing models. furthermore, wind speeds and rain precipitation were correlated with the power samples, this was done in order to consider rf propagation dependencies. second order statistics was derived including level crossing rate (lcr) and average facie duration (afd). the power profile was analysed for spectral components. the frequency characteristics of the rf propagation channel were also evaluated. they also presented the channelspecific rf propagation attributes. pellet et al., [21], used a fixed terrestrial broadband wireless system such as multipoint microwave distribution system (mmds) which is a cost-effective solution for cable coverage to the immediate surrounding rural area. the wireless system was operated with the existing cable headend and the same subscriber-end cable modem. the system works well with clear line-of-sight transmit/receive antennas. however, in nearline-of-sight transmission where a few foliated trees block the line-of-sight, signal distortion were experienced, especially under conditions of high wind. they argued, that the motion of the trees was responsible for the huge and rapid signal fading. their measurements were taken on fixed wireless paths blocked with a few trees in the vicinity of the receive antenna. fading characteristics of a 6 mhz channel centred at about 2.6 ghz were provided. the fades were mostly flat across the band but with some frequency selective fading. fading rates under windy conditions ranged from 0.5 to 2.0 f ades/second. the slope of the fades occasionally reached 50 db/second. dal bello et al., [22], carried out their studies on propagation in an urban forested park area, their aim was to investigate the statistical nature of the time fading for frequencies ranging from 0.9 ghz to 1.8 ghz, as well as to examine the range dependence and the base station height gain. they used a received signal of around 30 s intervals for a stationary mobile to design the distribution functions for the fading. according to them, the distribution could be approximated by a rician distribution, whose k-factor was found to depend on transmitter height. cuiñas et al., [23], reported that the presence of vegetation in the radio channel could affect the coverage areas of cellular mobile phone systems. they argued that the various components of a tree have influence in the performance of the radio system. although, the trunk is commonly in a fixed and stable location, the leaves could be in continuous movement as a result wind. accordingly, the time variation which can be correlated with the wind speed and direction, could strongly modify the attenuation and scattering effects of the trees on the radio channel performance. their study presents the results (both long-term and short-term) of a measurement campaign of scattering and attenuation effects of isolated trees, under the action of artificial wind of different controlled speeds and directions. they show that the median effect of the presence of a tree is the induction of higher attenuation in the shadow areas and a new distribution of the scattering pattern all around the specimen. but, the wind on the leaves forces their movement and an increment in the time variation around the median received power values. zennaro et al., [24], revisited the issue of link quality in 104 nwankwo and ukhurebor / j. nig. soc. phys. sci. 1 (2019) 103–115 105 wireless sensor networks (wsn). they studied the temporal and energy characteristics of a 24 ghz sensor network in an outdoor environment using different values of output power and sampling period. they analysed battery behaviour in motes placed at different distances and reported that farther motes have a shorter battery life. they suggested that when deployed in the real world, the sampling periods of sensor networks be adjusted according to distance to normalize battery lifetime and a more accurate energy-aware routing protocol be developed. according to zhao et al., [25], wireless sensor networks promise fine-grain monitoring in a wide variety of environments. many of these environments (like indoor environments or habitats) can be harsh for wireless communication. from a networking perspective, the fundamental aspect of wireless communication is the packet delivery performance (the spatio-temporal characteristics of packet loss and its environmental dependence). these factors would severely have impact on the performance of data acquisition from these networks. their study was centred on a systematic medium-scale (up to sixty nodes) measurement of packet delivery in three different environments (an indoor office building, a habitat with moderate foliage and an open parking lot). their results would have interesting implications for the design and evaluation of routing and mediumaccess protocols for sensor networks. puccinelli and haenggi [26], reported that multipath fading severely contributes to the unreliability of wireless links, causing fairly huge deviations from link quality predictions based on path loss models. accordingly, its impact on wireless sensor networks is considerable. although, analytical models provide a probabilistic description, multipath fading is a deterministic phenomenon. moreover, in the case of static nodes, fading is time-invariant. they illustrated its spatial nature with experimental evidence obtained using lower-end sensing node hardware. they also show the limitations of the supposed immunity of wideband radios to multipath fading in indoor deployments. the works of ukhurebor et al., [27] and ukhurebor and umukoro [28] dealt extensively on the effects of some essential meteorological variables such as temperature, relative humidity and mean sea level pressure on the ultra high frequency (uhf) and very high frequency (vhf) radio signals. they stated that the radio signals from both the uhf and vhf television stations were directly proportional to the temperature, inversely proportional to the relative humidity and no defined pattern of proportionality with the mean sea level pressure. they argued that, the radio signals from the uhf television station were seen to be mostly affected by these weather variables. these effects according to them were more pronounced during the months with high relative humidity compared with the months with lower relative humidity. according to [4]-[12], signal attenuation caused by rainfall, is a major challenge to microwave satellite communication especially at frequencies above 10 ghz. in several occasions, they cause signals unavailability. rainfall attenuation predictions have become one of the vital considerations while setting up a satellite communication link [4] [12]. in their respective studies, rainfall attenuation models, cumulative distribution curves and other analytical tools for successful prediction of rain attenuation were presented. basically, this study focussed on comprehensive evaluation of the performance and operation of 5 ghz microwave radios during propagation and transmission of signals across undulating land areas taking into consideration the effect of weather in such circumstances [29]. apparently, the results obtained offered some additional and novel insight on the effects of topography on the propagation of signals under certain weather conditions even where high frequency radios are employed. 3. materials and methods the hierarchical approach to network design and implementation was adopted in this study. the hierarchical approach stipulates that a network be structured in such a way as to categorize similar functions into a layer and separate each layer from other layers, while ensuring communication among the different layers [4] [12]. each layer focuses on specific functions. the advantages of the hierarchical approach are notable, which include: i. enables the network planner or designer to identify and make proper choice of the right equipment for the design and implementation of each layer. ii. ensures proper evaluation of features that makes up the layers. iii. suitability for network designs of varying sizes and requirements. in a hierarchical network the entire network is segmented into discrete layers. each layer, or tier, provides specific functions that define its role within the overall network. this helps to optimize and select the right network hardware, software and features to perform specific roles in the network. three main layers are recognized: access, distribution, and core layers respectively. 3.1. hardware i. three units of ubiquiti rocket m5 point to multi-point (ptmp) base stations. the specification of the radios is: 5−5.83 ghz spectrum, hi-power 2×2 mimo tdma airmax, powersupply: 24 v , 1 a poe. ii. router (huawei 1200 series). iii. switch (s5700 8-port). iv. cat-6 shielded twisted pair cable (40 m crimped cables). v. three computer systems with same specifications: hp probook 6450b, intel r coretm i5 cpu a©2.40 ghz, running microsoft windows 10 professional operating system. vi. infinix s4 smart phone with android 9.0, gps coordinate and elevation apps respectively. 3.2. software i. wireshark 3.0.5(network packet analyzer) ii. enterprise network simulation platform (ensp) iii. microsoft visio 2010 iv. ibm spss v.24 v. ubiquiti unifi controller software 105 nwankwo and ukhurebor / j. nig. soc. phys. sci. 1 (2019) 103–115 106 3.3. study area this study was conducted in iyamho community, a small town in etsako west local government of edo state, nigeria. iyamho is host to the fast-growing model university, edo university iyamho. it has approximately 5,000 inhabitants who are mostly rural dwellers. however, since the establishment of the university, the community is rapidly transforming to a beautiful semi-urban centre following the influx of workers, visitors and students. consequent upon the foregoing, the town is witnessing an upsurge of modern communication infrastructure amidst other notable capital infrastructure such as police barracks, community health clinics and private-owned businesses. in the area of communication, there is relatively a heavy presence of mobile network base stations which have given rise to increased growth in the use of the internet. it is presumably estimated that 55 % of the population are active users of the internet. geographically, iyamho is located on latitude 7.07 o n and longitude 6.27 o e with an elevation of about 188 m above sea level [16, 17, 18]. like other tropical areas of southern nigeria, iyamho enjoys two seasons often categorized as rainy and dry seasons. it enjoys a savannah vegetation. according to ukhurebor et al., [17], its topography is marked with undulating and table lands. figure 1 shows the aerial view of the community. figure 1: aerial view of iyamho, etsakowest of edo state, nigeria (source: google earth). 3.4. design of experiment the experiment involved three locations which were marked as follows: primary station (the base station) is located at the faculty of science, edo university iyamho. the primary station has an elevation of about 45 m above sea level. its location provided by coordinate 7.15174 o n and 6.30098 o e. the experiment is modelled as shown in figure 2. remote station 1 (the control station) is located at the administrative building in the main campus of edo university iyamho. the control station has an elevation of about 188 m above sea level. the remote station 2 (the test station) was deployed in an undulating landscape, located in iyamho town). the elevation is 88 m above the sea level. 3.5. method of data collection data collection was done using wireshark 3.0.5 installed on all the participating computers. connectivity between the base station and the two remote stations located 3 km away from the base station was continuously monitored. packets from the two remote stations were captured simultaneously from april to september, 2019 over a period of 30 days during rainfall. the failure rates from the base station to the remote stations were recorded. the failure and success rates at which packets were received from both remote stations were recorded under the same weather conditions. 3.6. network modelling the following tools were used for network modelling and simulation: huawei enterprise network simulation platform (ensp); huawei ap6510dn series wireless 5 ghz radios, huawei s700 series switch, and a hp probook. figure 3 shows the topology of the model. a class c network with network address of 192.168.1.0 and subnet mask 255.255.255.0 was used. the base station is connected to highly efficient switch, while the other remote stations were connected directly to a hp notebook of same specifications. figure 2: diagrammatic description of experimental design. the essence of using same computer specification across the three locations was to ensure that no extraneous factors were introduced into the network in respect of performance [29]. presumably, the use of same computer specification with same configuration and applications installed which are operating within same physical conditions would help ensure uniformity in performance. following the modelling, a 64k ping flood was sent from the base to the remote stations (remote 1 and remote 2) for three consecutive hours. the same ping flood was sent from the two remote access points to the base over same period. wireshark was used to capture the frequency of the packets sent and the packets received. 3.7. system requirements and physical configuration the physical configuration involves coupling the ubiquiti rocket m5 radios together with the sector antennae which were positioned at the three locations. for the base station, a huawei 106 nwankwo and ukhurebor / j. nig. soc. phys. sci. 1 (2019) 103–115 107 figure 3: topology of the network. s700 series switch was used to connect the hp notebook and the access point. figure 4-6 show the various configurations of the base station (base) and the remote stations (remote 1 and remote 2) respectively. figure 4: configuration of the base station radio. 3.8. data capture with wireshark as mentioned earlier, data capture on the physical network was done using wireshark installed on the three hp notebooks. prior to capture on physical network, a simulation was done using ensp and wireshark on one of the hp notebooks. the result of the simulation did not yield any departure as regards traffic sent to and packets received from the remote stations. in the physical network, 64k frames were flooded to the remote stations from the base station for over a period of 30 days rainy period. it should be noted that the period was only during the rainy cloudy weather. 4. results and discussion table 1 to 6 show the statistics of the packets relayed and received by the base station from the remote stations. note that the remote stations are tagged; remote 1 and remote 2. as previously stated, we used remote 1 as a control station while remote 2 was the test station. figure 7 shows the comparative analysis of the packet losses to both remote 1 and remote 2. on the other hand, figure 8 shows the relationship between the duration of rainfall and packet losses. the charts are created using data from table 5. figure 5: configuration of the control radio (remote 1). figure 6: configuration of the test radio (remote 2). figure 7: comparative analysis of packet losses from base station to remote stations. figure 9 shows a stacked area chart. the area chart summarized the relationship between the lost frames at both remote 1 and remote 2 respectively. in this experiment, three radios were used. the base station was used as the reference point to coordinate the point to multipoint connections. two remote stations were employed with one as the control radio and the second as the test radio. from the charts in figures 7-9, it is evident that regardless of the duration of rainfall or cloudy weather conditions, data losses across the radio located on undulating paths are higher than that located on plane paths. it may be submitted that notwithstanding 107 nwankwo and ukhurebor / j. nig. soc. phys. sci. 1 (2019) 103–115 108 table 1: traffic states on base station . day period of rainfall (minutes) packets sent from base to remote 1 packets received from remote 1 1 15 7005 5600 2 55 3575000 3479154 3 23 112876 102876 4 18 926547 725467 5 75 2367234 1803467 6 10 2136 1159 7 33 82345 61234 8 46 234654 223456 9 32 436788 394659 10 16 64763 59874 11 42 2348765 1863534 12 26 104566 80001 13 31 934525 802211 14 22 932442 823426 15 39 120001 103736 16 4 2034 1108 17 18 789686 723456 18 34 100870 90711 19 53 2346000 2012664 20 8 1264 1023 21 24 109827 90736 22 24 101453 90534 23 21 204332 183454 24 32 2364758 2002374 25 13 23487 21763 26 23 102876 98234 27 10 1726 1407 28 22 841234 80023 29 13 2305 1600 30 11 1656 1432 108 nwankwo and ukhurebor / j. nig. soc. phys. sci. 1 (2019) 103–115 109 table 2: traffic on base station computer. day period of rainfall (minutes) packets sent from base to remote 2 packets received from remote 2 1 15 7005 4600 2 55 3575000 3179152 3 23 112876 92176 4 18 926547 525167 5 75 2367234 1503467 6 10 2136 1102 7 33 82345 60024 8 46 234654 200856 9 32 436788 312259 10 16 64763 50174 11 42 2348765 1023434 12 26 104566 60020 13 31 934525 801241 14 22 932442 811606 15 39 120001 100123 16 4 2034 1003 17 18 789686 702356 18 34 100870 80034 19 53 2346000 2002344 20 8 1264 0865 21 24 109827 90502 22 24 101453 89534 23 21 204332 153454 24 32 2364758 2001234 25 13 23487 21456 26 23 102876 98234 27 10 1726 959 28 22 841234 80023 29 13 2305 1200 30 11 1656 1325 109 nwankwo and ukhurebor / j. nig. soc. phys. sci. 1 (2019) 103–115 110 table 3: traffic from remote 1 to base station. day rainfall time (minutes) packets received by base packets sent to base station 1 15 5523 5600 2 55 3455000 3479154 3 23 100876 102876 4 18 700547 725467 5 75 1727234 1803467 6 10 1140 1159 7 33 61200 61234 8 46 200654 223456 9 32 380788 394659 10 16 57763 59874 11 42 1858765 1863534 12 26 78956 80001 13 31 800525 802211 14 22 800442 823426 15 39 100001 103736 16 4 1100 1108 17 18 700686 723456 18 34 89487 90711 19 53 1986000 2012664 20 8 980 1023 21 24 889827 90736 22 24 90045 90534 23 21 180332 183454 24 32 1934758 2002374 25 13 19487 21763 26 23 92876 98234 27 10 1200 1407 28 22 789234 80023 29 13 1400 1600 30 11 1043 1432 110 nwankwo and ukhurebor / j. nig. soc. phys. sci. 1 (2019) 103–115 111 table 4: traffic from remote2 to base station. days rainfall time (minutes) packets received by base station packets sent to base station 1 15 4002 4600 2 55 3375000 3179152 3 23 75876 92176 4 18 486547 525167 5 75 1367234 1503467 6 10 950 1102 7 33 59345 60024 8 46 194654 200856 9 32 296788 312259 10 16 44763 50174 11 42 164876 1023434 12 26 50456 60020 13 31 724525 801241 14 22 782442 811606 15 39 90080 100123 16 4 950 1003 17 18 659686 702356 18 34 73387 80034 19 53 1896000 2002344 20 8 820 0865 21 24 88582 90502 22 24 80245 89534 23 21 162332 153454 24 32 1834758 2001234 25 13 18487 21456 26 23 92876 98234 27 10 800 959 28 22 72123 80023 29 13 1105 1200 30 11 1100 1325 111 nwankwo and ukhurebor / j. nig. soc. phys. sci. 1 (2019) 103–115 112 table 5: statistics on packets losses to remote stations. days rainfall time (minutes) packet loss to remote 1 packet loss to remote 2 1 15 1405 2405 2 55 95846 395848 3 23 10000 20700 4 18 201080 401380 5 75 563767 863767 6 10 977 1034 7 33 21111 22321 8 46 11189 33798 9 32 42129 124529 10 16 4889 14589 11 42 485231 1325331 12 26 24565 44546 13 31 132314 133284 14 22 109016 120836 15 39 16265 19878 16 4 926 1031 17 18 66230 87330 18 34 10159 20836 19 53 333336 343656 20 8 241 399 21 24 19091 19325 22 24 10919 11919 23 21 20878 50878 24 32 362384 363524 25 13 1724 2031 26 23 4642 4642 27 10 319 767 28 22 761211 761211 29 13 705 1105 30 11 224 331 112 nwankwo and ukhurebor / j. nig. soc. phys. sci. 1 (2019) 103–115 113 table 6: statistics on packet delivery to remote stations. days rainfall time (minutes) packet sent packet received by remote 1 packet received by remote 2 1 15 7005 5600 4600 2 55 3575000 3479154 3179152 3 23 112876 102876 92176 4 18 926547 725467 525167 5 75 2367234 1803467 1503467 6 10 2136 1159 1102 7 33 82345 61234 60024 8 46 234654 223456 200856 9 32 436788 394659 312259 10 16 64763 59874 50174 11 42 2348765 1863534 1023434 12 26 104566 80001 60020 13 31 934525 802211 801241 14 22 932442 823426 811606 15 39 120001 103736 100123 16 4 2034 1108 1003 17 18 789686 723456 702356 18 34 100870 90711 80034 19 53 2346000 2012664 2002344 20 8 1264 1023 0865 21 24 109827 90736 90502 22 24 101453 90534 89534 23 21 204332 183454 153454 24 32 2364758 2002374 2001234 25 13 23487 21763 21456 26 23 102876 98234 88234 27 10 1726 1407 959 28 22 841234 80023 80023 29 13 2305 1600 1200 30 11 1656 1432 1325 113 nwankwo and ukhurebor / j. nig. soc. phys. sci. 1 (2019) 103–115 114 figure 8: relationship between duration of rainfall and packet losses to remote stations. figure 9: stacked area chart showing the relationship between lost frames from the two remote stations. the line of sight existing between two geographically located radios, the impact of rainfall is very likely to intensify when radios are placed on terrains with highly undulating surfaces. 5. conclusion following the data capture and subsequent statistical analysis, we have made the following conclusions: i. signal strength and propagation of 5 ghz radios placed on undulating terrains are severely affected by deteriorating or adverse weather conditions. ii. though the cause of the manifest deviation obtained are not very 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[29] ecc report 192 “the current status of dynamic frequency selection (dfs) in the 5 ghz frequency range”, (2014). 115 j. nig. soc. phys. sci. 3 (2021) 216–223 journal of the nigerian society of physical sciences assessment of excess gamma dose exposure level in typical nigeria commercial building materials distribution outlets a. m. aserea,∗, t. o. owolabia, b. d. alafea, o. p. alabia, m. b. alimia adepartment of physics and electronics, adekunle ajasin university, akungba akoko, ondo state, nigeria abstract the gamma dose rate exposure levels from different brands of building materials at commercial distribution stores/shops in two major cities in ondo state, nigeria, were measured using a well calibrated inspector 1000 scintillator detector. the results showed that the different brands of building materials which are “corrugated iron sheet, aluminum roofing sheets, conduit pipes, paints, cement, pvc pipes, wash hand basin, bath tub, water closet, kitchen zinc, asbestos, floor tiles, wall tiles, bullet proof door, binding wire, rings and rods, red bricks, galvanized pipes, copper pipes, water tanks” contributed excess annual effective doses of 0.332 msv/y and 0.311msv/y to store keepers in ikare akoko and akure cities respectively. the indoor and outdoor annual effective dose of each of the investigated two cities are correlated using simple linear regression equations. the results of the modeling and experiment show that annual effective dose received by the occupants of these shops/stores was about 12 % higher than what could be received in a typical natural radiation environment in the two cities because the building materials acts as a source of radiation indoor. the research indicated that the typical habit of using poorly ventilated and confined space as stores/shops by the sales men might subject them to internal exposure through inhalation of radon gas and its short-lived decay products. implementation of the developed equations would definitely promote rapid determination of outdoor annual effective dose through indoor annual effective dose and ultimately save time and other valuable resources. doi:10.46481/jnsps.2021.188 keywords: building material, ikare akoko, akure, indoor and outdoor gamma exposure, building material shop/store, annual effective dose. article history : received: 31 march 2021 received in revised form: 28 june 2021 accepted for publication: 30 june 2021 published: 29 august 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: w. a. yahya 1. introduction building materials are any materials which are used for construction purposes, such as materials for dwellings, offices and industrial premises. all building materials that are derived from soils and rocks always contain trace amount of natural radionuclides from potassium, uranium, and thorium series, and the radionuclides that are created as their radioactive decay chains [1, 2, 3]. building materials may contain elevated concentration ∗corresponding author tel. no: +234(0)8067226208 email address: (a. m. asere ) of any of the primordial radionuclides or radioactive elements as they occur in nature. elevated concentrations of natural occurring radioactive materials (norm) are often found in certain geological resources [4, 5]. human activities such as that of the building materials manufacturers, exploit these resources and may lead to significantly enhanced concentrations of radionuclides which may also enhance the potential for exposure to natural occurring radioactive materials in these products [6]. human exposure to natural radiation from building materials is both internal and external. the external exposure is 216 asere et al. / j. nig. soc. phys. sci. 3 (2021) 216–223 217 caused by gamma radiation that is released from the building material as a consequence of the radioactive decay of the natural radionuclides present in them while the internal exposure is mainly caused by alpha-particle due to inhalation of radon daughters from the decay of radium-226 which is released from building materials to indoor air [7, 8]. long time exposure to nuclear particles emissions from building materials may lead to several health issues [9]. the type of building material and where it is sourced from can determine the amount of radionuclides in the material since building materials reflect their geologic formation and origin [10, 11]. background radiation is a measure of the ionizing radiation present in the environment at a particular location which is not due to deliberate introduction of radiation sources. background radiation originates from a variety of sources, both natural and artificial. these include both cosmic radiation and environmental radioactivity from naturally occurring radioactive materials as well as manmade fallout from nuclear weapons testing and nuclear accidents [12]. some past researches have reported measurements of gamma dose exposure levels in various building materials in nigeria [13, 14, 15, 16]. their studies indicated the particular brand of building materials that will present elevated levels of gamma dose exposure. [14] suggested that the controls of radioactivity of building materials could be based on a lower dose criteria in the range of 0.3 − 1.0 msv/y, if it is judged that this is desirable and will not lead to impractical control. this research further stated that the dose criteria used for the control of radioactivity should be defined as the excess exposure caused by the building materials. thus, this study determined the gamma dose exposure from different brands of typical building materials commonly found and being used in nigeria. using the commercial stores/shops outlets in ondo state, southwestern nigeria as a study area, the research was geared toward determining the extent at which the gamma dose exposure level inside the commercial building materials stores were in excess to gamma dose exposure in the local background environment. the major practice of the owners of these shops outlets is to display samples of their goods outside the shops for advertisement and for customers to see while the bulk of their goods are inside the shop or another warehouse. the shop owners and sale representatives either sit outside or inside the shops or warehouse to attend to customers all day long. most of these shops have one very small window or no window at all. these confined and poorly ventilated business workplaces expose all the small, medium and large scale business entrepreneurs to the effect of the radioactive emissions from the building materials. with 80 % of their time spent indoor and 20 % spent outdoor on daily basis, the occupants of these shop outlets may be receiving significant gamma dose from both the indoor and outdoor exposure unaware and without taken preventive measures. 2. study area two major cities in ondo state, southwestern nigeria were used for the research study. the two cities are akure and ikare akoko. akure city is the state capital of ondo state, nigeria. it lies on latitude 7◦ 25′ north of the equator and longitude 5◦ 19′ east of the greenwich meridian. it stands on the altitude of about 370 meters above sea level. the climate is hot and humid, influenced by rain bearing monsoon winds from the ocean and dry northwest wind from the sahara desert. the rainy season varies from april to october with rainfall of about 1524 mm per year. the temperature varies from 28◦c to 31◦c with a mean annual relative humidity of about 80 %. akure is a fast growing city and has its population increased tremendously over the years due to many factors. the city is the largest commercial center in ondo state with several markets, industries and higher institutions. the research was carried out in major areas where building materials were being sold in large quantities, such as arakale road, nepa road, oja oba market and several other shops outlets for building materials. ikare akoko city is about 100 km from akure. the city is the current headquarter of akoko north-east local government area in ondo state. it is located at an elevation of about 462 meters above sea level. its coordinates are 7◦ 31′ north and 5◦ 45′ east. it covers an estimated area of about 30 km2 . the topography is that of a hilly environment, gently undulating. ikare akoko falls within the rainforest zone where leaves are ever green during the rainy season and capable of shedding leaves during the dry season. ikare akoko is a chief commercial city in ondo state with several industrial base and markets. the research was carried out around oja oba market and other markets which have several shop outlets for building materials that are sold to customers from within and around the city. the two cities lie within the migmatite gneiss-quartzite complex of southwestern nigeria, generally referred to as the undifferentiated gneiss migmatite complex [17, 18]. the major types of rocks found in the area are the granite gneiss, migmatite, and charnokite. igneous rock such as granite had been identified to contain high levels of radionuclides [3]. 3. measurement procedure the detector used for this research was a high performance well calibrated sodium iodide handheld inspector 1000 spectrometer, figure 3. the instrument is specially designed for environmental screening and field measurements application requiring dose and count rate measurement. it has an energy range of 50 kev to 3 mev. the dose measured correspond to the equivalent dose on the depth of the human tissue (h ∗ 10). the gamma dose rate measurements were performed by placing the detector at a height of one meter above the ground surface with the probe facing downward. three consecutive readings were taken in a location for six minutes each [12] and the average recorded to represent the value for the particular location. the instrument was carried from one building material store to 217 asere et al. / j. nig. soc. phys. sci. 3 (2021) 216–223 218 another and the procedure was repeated in each case. the indoor measurements were performed inside the shops and warehouse where the materials are kept while the outdoor measurements were taken where the materials are displayed outside the stores. the background radiation were measured on the roads and distanced from where the building materials are displayed. the distance of each warehouse from the road varied from location to location. it ranged from a distance of 3 m to about 10 m. figure 1. inspector 1000 spectrometer the total number of shops covered in this research were 32 shops in ikare akoko and 55 shops in akure city. the type of building materials in the two cities and the number of shops selling a particular building materials are shown in the table of values. a total of 19 different building materials types were found in the markets. 4. results and discussions 4.1. gamma dose rate the gamma dose rate values are direct measurements value from the instrument. tables 1 and 2 present the results of both indoor and outdoor gamma exposure dose rate levels in ikare akoko and akure city. the tables present the range of values, the mean gamma dose values and background dose rate levels for each type of building material at various stores. the background dose measured on the roads and distance from where the building materials were being sold showed the environmental dose in the local typical environment outside the building of the shop/store. in ikare akoko, the gamma dose rate background values ranges from (0.045 ± 0.01 to 0.065 ± 0.01) µsvh−1 with a mean value of 0.052 µsvh−1. in akure, the background gamma dose rate value ranges from (0.048 ± 0.01 to 0.063 ± 0.01) µsvh−1 with a mean value of 0.059 µsvh−1. in ikare akoko city, for the building materials kept inside the stores, the indoor gamma dose rate ranges from (0.08 to 0.140) µsvh−1 with a range of mean value from (0.093 ± 0.01 to 0.128 ± 0.01) µsvh−1 while for the materials displayed outside, the outdoor gamma dose rate ranges from (0.05 to 0.010) µsvh−1 with a range of mean value between (0.065 ± 0.01 to 0.094 ± 0.01) µsvh−1. the mean dose rate value from all the building materials available in the stores was 0.104±0.01 µsvh−1 for materials kept indoor and 0.079 ± 0.01 µsvh−1 for materials displayed outdoor. in akure city, for the building materials kept inside the stores, the indoor gamma dose rate ranges from (0.08 to 0.110) µsvh−1 with a ranges of mean value between (0.076 ± 0.01 to 0.115 ± 0.01) µsvh−1 while for the materials displayed outside, the outdoor gamma dose rate ranges from (0.05 to 0.09) µsvh−1 with a range of mean value between (0.062 ± 0.01 to 0.093 ± 0.01) µsvh−1. the mean dose rate value from all the building materials available in the stores was 0.099±0.01 µsvh−1 for materials kept indoor and 0.077±0.01 µsvh−1 for materials displayed outdoor. 4.2. annual effective dose the estimation of exposure scenario for the sales men/women using the shop outlets was based on total exposure from all the building materials listed. this was because there was no shop or warehouse where a single material was being sold; they sell all or more than half of the building materials listed. the mean absorbed dose rate values for each building material indoor and outdoor were used to calculate the annual effective dose. assuming that exposure was uniformly distributed throughout the year, the occupancy time of 3444 hy−1 was used based on working for a maximum of 12 hours daily for 6 days in a week and 52 weeks in a year. the occupancy factor of 0.8 for indoor and 0.2 for outdoor was also used. he = dtf (1) where he is the annual effective dose (ms vy−1 ), d is the mean dose rate value (µsvh−1 ), t is the time of the year (hy−1 ) and f is the occupancy factor. table 3 present the values of annual effective dose in ikare akoko and akure respectively. the estimated value of annual effective dose due to background dose mean value that could be received outside in a local typical environment in the two cities where measurements were made was 0.039 ms vy−1 in ikare akoko and 0.044 ms vy−1 in akure. for ikare akoko, the estimated value of annual effective dose from the building materials stores varies from (0.29 to 0.383) ms vy−1 indoor with a mean value of 0.312 ms vy−1 and from (0.049 to 0.070) ms vy−1 outdoor with a mean value of 0.059 ms vy−1. the total mean annual effective dose value for the occupants of these shops who rotate their sitting position from inside to outside and vice versa for daily typical routine throughout the year was estimated to be 0.371 ms vy−1. for akure, the estimated value of annual effective dose from the building materials stores varies from (0.230 to 0.357) ms vy−1 indoor with a mean value of 0.298 ms vy−1 and from (0.045 to 0.069) ms vy−1 outdoor with a mean value of 0.057 ms vy−1. the total mean annual effective dose value for the shop keepers who either stayed outside or inside to attend to customers for a 218 asere et al. / j. nig. soc. phys. sci. 3 (2021) 216–223 219 typical daily routine business activity throughout the year was estimated to be 0.355 ms vy−1. in the two cities, the annual effective dose due to exposure from the building materials, was about 12 % higher than the natural background annual effective dose. 4.3. excess gamma dose in both cities the background dose values were less than the outdoor dose rate values and both were less than the indoor dose rate values. if the dose criteria used for control of radioactivity of building materials should be defined as the excess exposure caused by building materials; that is, the background gamma dose from natural radionuclides in the local typical environment need to be subtracted from the gamma dose from building materials [15]. the excess annual effective dose was obtained by subtracting the background annual effective dose obtained in the typical local environment from the total annual effective dose obtained from the total indoor and outdoor exposure from the commercial building materials distribution outlets as shown in table 4. it could be deduced that the excess gamma radiation dose originating from these building materials increased the annual effective dose received by the sales men or store keepers from either by staying indoors or outdoors with the building materials by 0.332 ms vy−1 in ikare akoko and 0.311 ms vy−1 in akure. when gamma dose are limited to levels below 1 ms vy−1, controls could be based on a lower dose criteria in the range 0.3 − 1.0 ms vy−1, if it is judged that this is desirable and will not lead to impractical control [15]. in this research, the excess exposure caused by these building materials were 0.311 and 0.332 ms vy−1, which are in the range 0.3 − 1.0 ms vy−1. therefore, for low dose criteria for control of radioactivity of building materials, control could be based in the range 0.3−1.0 ms vy−1. figure 2. regression function showing the relationship between indoor and outdoor gamma dose for ikare city figure 3. regression function showing the relationship between indoor and outdoor annual effective dose for ikare city figure 4. regression function showing the relationship between indoor and outdoor gamma dose for akure city figure 5. regression function showing the relationship between indoor and outdoor annual effective dose for akure city 4.4. correlation cross-plot between indoor and outdoor radiation figure 4.3 presents the correlation cross-plot between indoor and outdoor gamma dose for ikare city while figure 4.3 219 asere et al. / j. nig. soc. phys. sci. 3 (2021) 216–223 220 table 1. the range, mean value of indoor and outdoor gamma dose rate and background dose rate level in ikare akoko s/n building materials no of shops indoor dose (µ sv/h) outdoor dose (µ sv/h) background dose (µ sv/h) range mean ± sd range mean ± sd 1 iron corrugated sheets 7 0.080 0.120 0.104± 0.01 0.070-.110 0.083± .01 0.061± 0.01 2 aluminum roofing sheets 7 0.0800.120 0.105± 0.01 0.070-.110 0.070±0.03 0.058± 0.01 3 conduit pipes 10 0.0900.120 0.104± 0.01 0.060-.100 0.082± .01 0.054± 0.01 4 paints 9 0.0900.130 0.112± 0.01 0.070-.110 0.082±0.01 0.054± 0.01 5 cements 5 0.1200.140 0.128± 0.01 0.080-.100 0.094± .01 0.062± 0.01 6 pvc pipes 11 0.0900.130 0.112± 0.01 0.060-.100 0.088± .01 0.055± 0.01 7 wash hand basin 6 0.0800.090 0.093 ± 0.0 0.060-.100 0.075±0.02 0.04 ± 0.01 8 bath tub 8 0.0800.110 0.099± 0.01 0.060-.100 0.080±0.01 0.049± 0.01 9 water closet 7 0.0800.120 0.101± 0.01 0.060-.100 0.083±0.02 0.050± 0.01 10 kitchen zinc 10 0.0800.130 0.103± 0.01 0.060-.100 0.083±0.01 0.052± 0.01 11 asbestos 7 0.0800.110 0.094± 0.01 0.050-.090 0.071±0.01 0.049± 0.01 12 floor tiles 8 0.0800.120 0.094± 0.01 0.060-.080 0.069±0.01 0.048± 0.01 13 wall tiles 7 0.0800.110 0.093± 0.01 0.060-.080 0.071±0.01 0.046± 0.01 14 bullet-proof door 2 0.0900.100 0.095± 0.01 0.060-.070 0.065± 0.01 0.045± 0.01 15 binding wire, rings & rods 3 0.1100.130 0.127± 0.01 0.080-.090 0.087±0.01 0.053± 0.01 16 red bricks 2 0.1000.120 0.110± 0.01 0.080-.090 0.085±0.01 0.050± 0.01 17 galvanized pipes 2 0.0900.120 0.105± 0.02 0.070-.090 0.080± 0.01 0.050± 0.01 18 copper pipes 2 0.0900.120 0.105± 0.02 0.060-.090 0.075± 0.02 0.055± 0.01 19 water tanks 5 0.0800.110 0.094± 0.01 0.060-.090 0.074± 0.01 0.05 ± 0.01 table 2. the range, mean value of indoor and outdoor gamma dose rate and background dose rate level in akure s/n building materials no of shops indoor dose (µ sv/h) outdoor dose (µ sv/h) background dose (µ sv/h) range mean ± sd range mean ± sd 1 iron corrugated sheets 12 0.07-0.13 0.0967±0.02 0.05-0.09 0.0717±0.01 0.0525±0.01 2 aluminum roofing sheets 14 0.08-0.13 0.0986±0.01 0.05-0.09 0.0736±0.01 0.0554±0.01 3 conduit pipes 11 0.08-0.13 0.1012±0.02 0.06-0.11 0.07818±0.01 0.0618±0.01 4 paints 15 0.07-0.13 0.0927±0.02 0.05-0.09 0.0707±0.01 0.0533±0.01 5 cements 12 0.09-0.14 0.1192±0.02 0.07-0.12 0.0933±0.01 0.0717±0.01 6 pvc pipes 21 0.07-0.13 0.1010±0.02 0.05-0.10 0.0819±0.1 0.0624±0.01 7 wash hand basin 9 0.08-0.13 0.1078±0.01 0.06-0.09 0.0829±0.01 0.0633±0.01 8 bath tub 15 0.08-0.13 0.0993±0.01 0.07-0.09 0.082±0.01 0.0627±0.01 9 water closet 13 0.08-0.12 0.1008±0.09 0.06-0.08 0.0792±0.01 0.0546±0.01 10 kitchen zinc 17 0.08-0.12 0.1006±0.01 0.05-0.09 0.0688±0.01 0.0542±0.01 11 asbestos 19 0.06-0.13 0.0879±0.02 0.04-0.09 0.0715±0.02 0.0563±0.01 12 floor tiles 27 0.07-0.12 0.0933±0.01 0.06-0.09 0.0738±0.01 0.0567±0.01 13 wall tiles 17 0.08-0.12 0.0971±0.01 0.06-0.10 0.0789±0.01 0.0476±0.01 14 bullet-proof door 2 0.09-0.10 0.095±0.01 0.07-0.08 0.075±0.01 0.0550±0.01 15 binding wire, rings & rods 13 0.06-0.09 0.06-0.09 0.05-0.07 0.0615±0.01 0.0492±0.01 16 red bricks 4 0.10-0.13 0.10-0.13 0.05-0.08 0.065±0.01 0.0475±0.01 17 galvanized pipes 4 0.09-0.13 0.09-0.13 0.07-0.11 0.0875±0.01 0.055±0.01 18 copper pipes 4 0.09-0.13 0.09-0.13 0.06-0.11 0.0825±0.02 0.050±0.01 19 water tanks 13 0.08-0.13 0.08-0.13 0.06-0.11 0.0823±0.01 0.0538±0.01 220 asere et al. / j. nig. soc. phys. sci. 3 (2021) 216–223 221 table 3. indoor and outdoor annual effective dose in ikare akoko and akure respectively ikare akoko akure s/n building materials indoor dose ( msv/y) outdoor dose (msv/y) indoor dose (msv/hy) outdoor dose (msv/y) 1 corrugated iron sheets 0.312 ± 0.01 0.062 ± 0.01 0.2896±0.02 0.0537±0.01 2 aluminum roofing sheets 0.314 ± 0.01 0.052 ± 0.03 0.2899±0.01 0.0551±0.01 3 conduit pipes 0.312 ± 0.01 0.061 ± 0.01 0.3031±0.02 0.0585±0.01 4 paints 0.335 ± 0.01 0.061 ± 0.01 0.2777±0.02 0.0529±0.01 5 cements 0.383 ± 0.01 0.070 ± 0.01 0.3570±0.02 0.0699±0.01 6 pvc pipes 0.335 ± 0.01 0.066 ± 0.01 0.3025±0.02 0.0613±0.1 7 wash hand basin 0.279 ± 0.01 0.056 ± 0.02 0.3229±0.01 0.0621±0.01 8 bath tub 0.297 ± 0.01 0.060 ± 0.01 0.2974±0.01 0.0614±0.01 9 water closet 0.303 ± 0.01 0.062 ± 0.02 0.3019±0.01 0.0593±0.01 10 kitchen zinc 0.309 ± 0.01 0.062 ± 0.01 0.3013±0.01 0.0516±0.01 11 asbestos 0.282 ± 0.01 0.053 ± 0.01 0.2633±0.02 0.0535±0.02 12 floor tiles 0.282 ± 0.01 0.052 ± 0.01 0.2795±0.01 0.0553±0.01 13 wall tiles 0.279 ± 0.01 0.053 ± 0.01 0.2908±0.01 0.0591±0.01 14 bullet-proof door 0.285 ± 0.01 0.049± 0.01 0.2845±0.01 0.0456±0.01 15 binding wire, rings & rods 0.380 ± 0.01 0.065 ± 0.01 0.2303±0.01 0.0461±0.01 16 red bricks 0.329 ± 0.01 0.064 ± 0.01 0.3444±0.01 0.0487±0.01 17 galvanized pipes 0.314 ± 0.02 0.060 ± 0.01 0.3295±0.02 0.0655±0.01 18 copper pipes 0.314 ± 0.02 0.056 ± 0.02 0.2896±0.02 0.0618±0.02 19 water tanks 0.282 ± 0.01 0.055 ± 0.01 0.3064±0.01 0.0616±0.01 1 corrugated iron sheets 0.312 ± 0.01 0.062 ± 0.01 0.2896±0.02 0.0537±0.01 2 aluminum roofing sheets 0.314 ± 0.01 0.052 ± 0.03 0.2899±0.01 0.0551±0.01 3 conduit pipes 0.312 ± 0.01 0.061 ± 0.01 0.3031±0.02 0.0585±0.01 4 paints 0.335 ± 0.01 0.061 ± 0.01 0.2777±0.02 0.0529±0.01 5 cements 0.383 ± 0.01 0.070 ± 0.01 0.3570±0.02 0.0699±0.01 6 pvc pipes 0.335 ± 0.01 0.066 ± 0.01 0.3025±0.02 0.0613±0.1 7 wash hand basin 0.279 ± 0.01 0.056 ± 0.02 0.3229±0.01 0.0621±0.01 8 bath tub 0.297 ± 0.01 0.060 ± 0.01 0.2974±0.01 0.0614±0.01 9 water closet 0.303 ± 0.01 0.062 ± 0.02 0.3019±0.01 0.0593±0.01 10 kitchen zinc 0.309 ± 0.01 0.062 ± 0.01 0.3013±0.01 0.0516±0.01 11 asbestos 0.282 ± 0.01 0.053 ± 0.01 0.2633±0.02 0.0535±0.02 12 floor tiles 0.282 ± 0.01 0.052 ± 0.01 0.2795±0.01 0.0553±0.01 13 wall tiles 0.279 ± 0.01 0.053 ± 0.01 0.2908±0.01 0.0591±0.01 14 bullet-proof door 0.285 ± 0.01 0.049± 0.01 0.2845±0.01 0.0456±0.01 15 binding wire, rings & rods 0.380 ± 0.01 0.065 ± 0.01 0.2303±0.01 0.0461±0.01 16 red bricks 0.329 ± 0.01 0.064 ± 0.01 0.3444±0.01 0.0487±0.01 17 galvanized pipes 0.314 ± 0.02 0.060 ± 0.01 0.3295±0.02 0.0655±0.01 18 copper pipes 0.314 ± 0.02 0.056 ± 0.02 0.2896±0.02 0.0618±0.02 19 water tanks 0.282 ± 0.01 0.055 ± 0.01 0.3064±0.01 0.0616±0.01 presents the cross-plot showing annual effective dose for the same city. the equation inscribed in figure 4.3 shows the relationship between indoor and outdoor gamma dose for ikare city. the equation was obtained from simple linear regression using the experimentally acquired data. the equation is characterized with high degree of precision as measured from its root mean square error as well as the mean absolute error. similar 221 asere et al. / j. nig. soc. phys. sci. 3 (2021) 216–223 222 table 4. excess annual effective dose (msvy -1 ) in ikare akoko and akure respectively ikare akure outdoor indoor total background excess outdoor indoor total background excess 0.059 0.312 0.371 0.039 0.331 0.057 0.298 0.355 0.044 0.311 equation relating the indoor and outdoor effective annual dose for the same ikare city is presented in figure 4.3. figure 4.3 and figure 4.3 respectively presents the correlation cross-plot between indoor/outdoor gamma dose and annual effective dose for akure city. implementation of equations would definitely enhance quick determination of outdoor gamma dose and annual effective dose for every known indoor radiation. 5. recommendations in order to limit the level of exposure, shop/storekeepers should stay outside their shops more than they stay inside because the indoor radiation dose is more than the outdoor radiation dose. in the two cities, the indoor exposure to gamma dose was 84 % of the total dose. the gamma exposure could be reduced by improving the ventilation in their stores, and it is advisable for them to spend less time, say about 10 hours or less a day for 6 days in a week for 50 weeks in a year, in order to reduce their level of exposure. however, because of confined space and very poor ventilation inside these shops/stores, these sales men may be subjected to high internal exposure by inhalation of radon and its short-lived progeny. exposure to low doses of gamma radiation in these shops may not cause immediate health effect but it is a minor contributor to overall cancer risk because the risk increases as the dose increases. it is therefore advisable for nigerians that are involved in this kind of business to put good ventilation of their stores into consideration. the practice of having one small window or no window at all in these shops increases their level of internal exposure to high radon gas inhalation. 6. conclusions the gamma dose rate exposure levels in typical commercial building materials distribution outlets in ikare akoko and akure, ondo state, nigeria, has been carried out. the results showed that the total average annual effective dose received by the store keepers from both indoor and outdoor exposure to radionuclides emission from building materials were 0.371 msv/y and 0.355 msv/y respectively which is below the international accepted limit of 1 msv/y for the general public. the excess gamma radiation dose originating from these building materials increased the background annual effective dose with a value a little above 0.3 msv/y in both cities. references [1] a. chandrasekaran, r. ravisankar, a. rajalakshmi, p. eswaran, p. vijayagopal & b. venkatraman, “assessment of natural radioactivity and function of minerals in soils of yelagiri hills, tamilnadu, india by gamma ray spectroscopic and 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(1976) 41. 223 j. nig. soc. phys. sci. 3 (2021) 42–47 journal of the nigerian society of physical sciences an order four continuous numerical method for solving general second order ordinary differential equations f. o. obarhua∗, o. j. adegboro department of mathematical sciences, the federal university of technology, akure, nigeria abstract continuous hybrid methods are now recognized as efficient numerical methods for problems whose solutions have finite domains or cannot be solved analytically. in this work, the continuous hybrid numerical method for the solution of general second order initial value problems of ordinary differential equations is considered. the method of collocation of the differential system arising from the approximate solution to the problem is adopted using the power series as a basis function. the method is zero stable, consistent, convergent. it is suitable for both non-stiff and mildly-stiff problems and results were found to compete favorably with the existing methods in terms of accuracy. doi:10.46481/jnsps.2021.150 keywords: numerical scheme, continuous hybrid method, zero stability, linear and nonlinear article history : received: 05 january 2021 received in revised form: 31 january 2020 accepted for publication: 04 february 2021 published: 27 february 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: b. j. falaye 1. introduction we consider the second order ordinary differential equation (odes) y′′ = f (x, y, y′), y(µ) = ω0, y ′(µ) = ω1 (1) equation (1) occur virtually every areas of physical or biological process in connection with numerous problems that are encountered in various aspects of everyday life. it is well conceived that this type of equation can either be solved directly or solved by reducing to system of first order differential equations before applying different methods available to solve the resulting system of first order odes chan et al. [1], gholamtabar and parandin [2]. ∗corresponding author tel. no: +2348062488066 email address: obarhuafo@futa.edu.ng (f. o. obarhua ) various linear multistep methods with different order of accuracy have been developed for the solution of 1 which varies from discrete linear multistep methods to continuous linear multistep methods. lambert and watsan, [3] reported that linear multistep methods generally are more efficient in terms of accuracy with weak stability properties for a given number of evaluation per step and suffered the disadvantage of requiring additional starting values and special procedures for changing step length h. it is also good to note that continuous linear multistep methods have advantages over the discrete methods in such a way that they give better error estimation, they provide a simplified form of coefficients for further evaluation at different points, and they provide solutions at all interior points within the interval of integration saravi and mirrajei, [4], kayode and awoyemi, [5], golbabai and arabshahi [6]. among the first methods developed are first derivative methods that are implemented in predictor-corrector mode, and taylor series expan42 obarhua adegboro / j. nig. soc. phys. sci. 3 (2021) 42–47 43 sion are adopted to provide the starting values. the identified setbacks of the predictor-corrector methods are; they are very costly to implement and reduced order of accuracy of the predictors. recently, authors have proposed different methods of higher order differential equations to improve on the existing setbacks. such improved methods are kayode and adeyeye, [7, 8] and kayode and obarhua, [9, 10]. they independently proposed linear multistep methods of higher order of accuracies and same order of main predictors and the correctors and hence improved significantly the accuracies of the methods. this work proposed an accurate continuous numerical hybrid method for direct solution of initial value problems of odes. the derived method is capable to handle stiff, mildly stiff, nonlinear and engineering problems modeled as a second order initial value problem of odes. 2. derivation of the method we define the general power series approximation solution in the form y(x) = (c+i)−1∑ j=0 a j x j (2) y′(x) = (c+i)−1∑ j=1 ja j x j−1 (3) y′′(x) = (c+i)−1∑ j=2 j( j − 1)a j x j−2 (4) equating (4) with (1) gives f (x, y, y′) = (c+i)−1∑ j=2 j( j − 1)a j x j−2 (5) equation (2) is interpolated at xn+i, i = 2, 5 2 and (5) is collocated at xn+c, c = 0(1)3. therefore, interpolation and collocation equation at the selected grid and offstep points give rise to system of equations which can be express in matrix form 1 xn+2h x2n+2h x 3 n+2h x 4 n+2h x 5 n+2h 1 xn+rh x2n+rh x 3 n+rh x 4 n+rh x 5 n+rh 0 0 2 6xn 12x2n 20x 3 n 0 0 2 6xn+h 12x2n+h 20x 3 n+h 0 0 2 6xn+2h 12x2n+2h 20x 3 n+2h 0 0 2 6xn+3h 12x2n+3h 20x 3 n+3h   a0 a1 a2 a3 a4 a5  =  yn+2 yn+r fn fn+1 fn+2 fn+3  (6) gaussian elimination method is then applied to solve equation (6) to obtain the unknown coefficients a′j s which is then substituted into (2). continuous system is obtained after some algebraic simplifications. applying transformation t = 1h (x − xn+k−1) , k = 3, t = (0, 1] in obarhua [11], the continuous coefficients are obtained as follows α2 = − ( −rh + th + 2h h(r − 2) ) αr = th h(r − 2) β0 = h5 360 ( −3t5 + 10t3 + 8t + 3tr4 − 24tr3 + 62tr2 − 56tr h3 ) β1 = − h5 120 ( −3t5 − 5t4 + 20t3 − 72t + 3tr4 − 19tr3 + 22tr2 + 44tr h3 ) β2 = h5 120 ( −3t5 − 10t4 + 10t3 + 60t2 + 48t + 3t4r4 h3 −14t4r3 + 2t4r2 + 4tr h3 ) (7) the first derivatives of equation (7) are α′2 = − 1 (r − 2) α′r = 1 (r − 2) β′0 = h5 360 ( −15t4 + 30t2 + 3r4 − 24r3 + 62r3 − 56r + 8 h3 ) β′1 = − h5 120 ( −15t4 − 20t3 + 60t2 + 3r4 − 19r3 + 22r2 + 44r − 72 h3 ) β′2 = h5 120 ( −15t4 − 40t3 + 30t2 + 120t + 12t3r4 − 56t2r3 + 8t3r2 h3 +4r + 48 h3 ) β′3 = − h5 360 ( −15t4 − 60t3 − 60t2 + 3r4 − 9r3 + 8t3r2 + 16t3r + 8 h3 ) (8) evaluating equation (7) and (8) at t = 1 yield the discrete order continuous numerical scheme yn+3 = − 1 360(r − 2) ( −60h2r5 fn+2 − 360yn+2 − 630h 2 fn+2 − 30h2 fn − 3h 2r5 fn+3 + 9h 2r5 fn+2 − 9h 2r5 fn+1 + 3h 2r5 fn + 15h2r4 fn+3 − 20h 2r3 fn+3 − 180h 2r3 fn+1 + 90h 2r3 fn+2 − 180h2r2 fn + 110h 2r3 fn − 30h 2r4 fn + 75h 2r4 fn+1 − 60h2r4 fn+2 + 360ryn+2 + 291h 2r fn+2 + 360yn+r − 360h 2 fn+1 + 38h2r fn+3 + 444h 2r fn+1 + 127h 2r fn ) (9) 43 obarhua adegboro / j. nig. soc. phys. sci. 3 (2021) 42–47 44 its first derivative is given as y′n+3 = − 1 360h(r − 2) ( 9h2r5 fn+2 + 9h 2r5 fn+1 + 3h 2r5 fn + 90h 2r3 fn+2 − 60h2r4 fn+2 + 110h 2r3 fn − 180h 2r3 fn+1 − 180h 2r2 fn − 30h2r4 fn + 75h 2r4 fn+1 − 3h 2r5 fn+3 − 20h 2r3 fn+3 + 15h2r4 fn+3 − 254h 2 fn+3 − 858h 2 fn+2 − 282h 2 fn+1 − 46h2 fn + 360yn+r + 405h 2r fn+2 + 405h 2r fn+1 + 135h 2r fn+3 + 135h2r fn − 360yn+2 ) (10) the values of r is taken in the interval r ∈ (2, 3) to obtain a particular discrete hybrid method. for the purpose of testing the properties of equation (9), the value of r is taken to 52 to have yn+3 = 2yn+ 52 −yn+2+ h2 384 (33 fn+3+83 fn+2−25 fn+1+5 fn)(11) with its first derivative given by hy′n+3 = 2yn+ 52 − 2yn+2 + h 5760 (2047 fn+3 +3069 fn+2 − 999 fn+1 + 203 fn) (12) 3. implementation of the method (11) in order to implement the implicit linear one-point discrete scheme (11) and its derivative (12), the symmetric explicit schemes and their derivatives are also developed by the same procedure for the evaluation of yn+3 and y′n+3 contained in fn+3 in the main scheme (11) and its derivative (12). yn+3 = 2yn+ 52 −yn+1 + h2 240 (66 fn+ 52 −10 fn+2 +5 fn+1− fn)(13) and its first derivative as hy′n+3 = −2hyn+ 52 + 2hyn+1 + h2 3600 ( −4094 fn+ 52 + 1920 fn+2 − 655 fn+1 + 129 fn ) (14) other explicit schemes were also generated to evaluate other starting values using taylor series expansions to evaluate the values for yn+i, y′n+i as yn+i = yn+( jh)y ′ n+ ( jh)2 2! fn+ ( jh)3 3! ( ∂ fn ∂xn + y′n ∂ fn ∂yn + fn ∂ fn ∂y′n ) +o(h4)(15) and y′n+i = y ′ n+( jh) fn+ ( jh)2 2! ( ∂ fn ∂xn + y′n ∂ fn ∂yn + fn ∂ fn ∂y′n ) +o(h4)(16) 4. stability analysis 4.1. region of absolute stability in other to investigate the periodic stability properties of the numerical methods for solving the initial-value problem equation 1 and the interval of periodicity, lambert and watson [3] introduced the following scalar test problem as y′′ = −λ2y (17) based on the theory developed in lambert and watson [3], when multistep method l∑ j=0 α jyn+ j = h 2 l∑ j=0 θ j fn+ j, (18) is applied to the scalar test equation (17), a difference equation of the form l∑ i=0 (αi + h 2θi)yn+i = 0 (19) is obtained, where h = ph, h is the step length and yn is the computed approximation to y(x0 + nh), n = 0, 1, 2, . . . then, we have following definitions. definition. (see konguetsof and simos, [12]) numerical method (19) has an interval of periodicity (0, h20 ), if ∀ h 2 ∈ (0, h20 ), qi, i = 1(1)l satisfy |q1| = |q2| = 1, |q j| ≤ 1, j = 3(1)l. (20) definition. following [3], a numerical method is p-stable if its interval of periodicity is (0, ∞). therefore, we obtain the interval of periodicity of the new method, which is equal to (0, -2.4) and the stability domain of the method is as shown in figure 1. figure 1: the stability domain of the new method 4.2. order and error constant of the method the method proposed by lambert (1973) in olanegan et al. [13] is adopted in this paper, with linear operator: k∑ j=0 α jyn+ j = h 2 k∑ j=0 β j fn+ j (21) 44 obarhua adegboro / j. nig. soc. phys. sci. 3 (2021) 42–47 45 we associate the linear operator l with the scheme and defined as l{y(x), h} = k∑ j=0 [α jy(x + jh) − h 2β jy ′′(x + jh)] (22) where α0 and β0 are both non-zero and y(x) is an arbitrary function which is continuous and differentiable on the interval [a, b]. expanding the form y(x) and y′′(x) in taylor series and comparing coefficients of h, we obtained ∆[y(x); h] = c0y(x) + c1hy ′(x) + · · · + c ph pyp(x) + c p+1h p+1yp+1(x) + c p+2h p+2yp+2(x) + · · · (23) the method (11) and its associate linear difference operator (13) are said to have order p if c0 = c1 = ··· = cp+1 = 0 and cp+2 , 0. the value cp+2 is called error constant. therefore, in this paper, the method (11) is of order 4 and the error constant cp+2 = − 21 2560 or −8.2031 × 10−3 and its derivative (12) is of order 4 and error constant cp+2 = − 35 1536 or −2.2786 × 10 −2. 4.3. consistency of the method a numerical method is said to be consistent if the following conditions are satisfied 1. the order of the method must be greater or equal to 1, p ≥ 1. 2. ∑k j=0 α j = 0 3. ρ(r) = ρ′(r) = 0 4. ρ′′(r) = 2!σ(r) where ρ(r) and σ(r) are the first and second characteristics polynomial of our method. according to adesanya et al. [14], the first condition is a sufficient condition for a method to be consistent. since our method is of order 4 then it is consistent. 4.4. zero stability since |zi| = |0, 0, 1| ≤ 1 the method is zero stable. 4.5. convergence a method is said to be convergent if and only if it is consistent and zero stable, hence our method is convergent. 5. numerical examples the method is applied to solve the following linear and nonlinear second order initial value problems of ordinary differential equations directly without reduction to system of first order equations. problem 1: y′′ = y′, y(0) = 0; y′(0) = −1; h = 0.1 theoretical solution: y(x) = 1 − ex this problem has been used in kayode and adeyeye [8] of order six to check the behavior of the methods. table 1 shows the absolute errors of the methods for the purpose of comparison. the obtained results in the table give the good performance of the proposed method. problem 2: y′′ + 1001y′ + 1000y = 0, y(0) = 1; y′(0) = −1; h = 0.05 theoretical solution: y(x) = e−x the problem 2 was solved by anake [15] of order 4. the new method was applied to solve it for the purpose of comparison. the results are shown in table 2. problem 3: y′′ = 100y′, y(0) = 1; y′(0) = −10; h = 0.01 theoretical solution: y(x) = e−10x table shows the absolute errors at different points of the integration interval when h = 0.01was solved by awari [16] of order five. the new method was applied to solve it for the purpose of comparison. the results show that the proposed method performed better than awari [16]. problem 4: y′′ − x(y′)2 = 0, y(0) = 1; y′(0) = 12 ; h = 0.003125 theoretical solution:y(x) = 1 + 12 ln [ 2+x 2−x ] we have solved this problem with the new method and the results have been compared with kayode and adeyeye [7] of order six shown in table 5. problem 5: resonance vibration of a machine a stamping machine applies hammering forces on metal sheets by a die attached to the plunger moves vertically up and down by a fly wheel spinning at constant set speed. the constant rotational speed of the fly wheel makes the impact force on the sheet metal, and therefore the supporting base, intermittent and cyclic. the bearing base on which the metal sheet is situated has a mass, m = 2000kg. the force acting on the base follows a function: f(t) = 2000 sin(10t), in which t − time in seconds. the base is supported by an elastic pad with an equivalent spring constant k = 2 × 105 n/m. determine the differential equation for the instantaneous position of the base y(t) if the base is initially depressed down by an amount 0.1m. solution: the mass-spring system above is modeled as differential equation as: the bearing base mass = 2000kg spring constant k = 2 × 105 n/m. force (ma) on the metal sheet = m d 2 y dt2 = my ′′ i.e. ma = my′′ = 2000 sin(10t); where a = y′′initial conditions on the system are y(t0) = y0; dy dt |t=o = y ′(t0) = y′0; t0 = 0, y′0 = 0.1 therefore, the governing equation for the instantaneous position of the base y(t) is given by my′′ + ky = f(t); y(t0) = y0, y ′(t0) = y ′ 0 2000y′′ + 2 × 105y = 2000 sin 10t, y′(0) = 0, y(0) = 0.1 theoretical solution: y(t) = 110 cos 10t + 1 200 sin 10t− t 20 cos 10t 6. conclusion an order four continuous numerical method for solving general second order ordinary differential equations is proposed and applied to solve directly without reducing to system of first 45 obarhua adegboro / j. nig. soc. phys. sci. 3 (2021) 42–47 46 table 1 x 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 error in [8] 8.17(-7) 3.10(-6) 6.57(-6) 1.14(-5) 1.79(-5) 2.64(-5) 3.72(-5) 5.06(-5) 6.72(-5) error in (11) 4.25(-8) 7.47(-8) 1.52(-7) 2.45(-7) 3.54(-7) 5.31(-7) 7.37(-7) 9.73(-7) 1.31(-7) table 2 x 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 error in [15] 0.10(-9) 0.20(-9) 0.28(-9) 0.34(-9) 0.39(-9) 0.43(-9) 0.45(-9) 0.4(-9) error in (11) 2.00(-10) 3.15(-10) 2.74(-10) 5.44(-10) 7.53(-10) 2.76(-10) 1.18(-10) 1.76(-10) table 3: absolute errors at different points of the integration x 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 error in [16] 1.15(-7) 3.65(-7) 6.05(-7) 8.50(-7) 1.10(-6) 1.36(-6) 1.45(-6) 1.59(-6) error in (11) 1.29(-8) 3.01(-8) 5.04(-8) 9.32(-10) 1.40(-7) 1.90(-7) 2.58(-7) 3.32(-7) table 4: absolute errors for numerical example 4 x 0.0063 0.0094 0.0125 0.0188 0.0250 0.0313 0.0375 0.0437 0.0500 error in (11) 0.00(0) 0.00(0) 2.81(-14) 2.36(-13) 8.73(-13) 1.91(-12) 2.97(-12) 5.21(-12) 7.55(-12) table 5: comparison of errors with [7] x error in kayode and adeyeye [7] error in the new method (11) 0.0063 4.831(-11) 0.0000(00) 0.0094 3.382(-9) 0.0000(00) 0.0125 1.580(-8) 2.819(-14) 0.0156 4.333(-8) 1.709(-13) 0.0188 9.391(-8) 2.362(-13) table 6: the new derived method was applied to solve this problem modeled as a second order (ivps) and it was seen from the results in the table that the method are useable in engineering field. x exact solution computed solution error 0.01 0.099404629653415691 0.099404630038381694 3.849660(-10) 0.02 0.097958005773976925 0.097958006644224049 8.702471(-10) 0.03 0.095207162458893865 0.095207165387981033 2.929087(-9) 0.04 0.091970827382988077 0.091970831862903016 4.479915(-9) 0.05 0.087961427477332363 0.087961431552930208 4.075598(-9) 0.06 0.082363909854646533 0.082363917430066838 7.575420(-9) 0.07 0.076833743309093400 0.076833753917924741 1.060883(-9) 0.08 0.069604876901833215 0.069604894375183718 1.747335(-9) 0.09 0.062811758617177721 0.062811776980930309 1.836375(-9) 0.10 0.055536073981512724 0.055536101603349465 2.762184(-8) order ordinary differential equations. the method is very flexible and easy to develop and may be applied to solve kinds of second order initial value problems as can be seen in the numerical examples. the method gives a high accuracy when compared the numerical results to the exact solution and a very good performance compared with existing methods in the literature. references [1] r. p. k. chan, p. leone, & a. tsai, “order conditions and symmetry for two-step hybrid methods”, int. j. comp. math. 81 (2004) 1519. [2] s. gholamtabar & n. parandin, “numerical solutions of second-order differential equations by adam bashforth method”, american j. of engi. research 3 (2014) 318. 46 obarhua adegboro / j. nig. soc. phys. sci. 3 (2021) 42–47 47 [3] j. d. lambert & a. watsan, “a symmetric multistep method for periodic initial value problem”, journal of the institute of mathematics and its applications 18 (1976) 189. [4] m. f. saravi & s.r. mirrajei, “numerical solution of linear ordinary differential equations in quantum chemistry by clenshaw method”, world acad. sci. eng. technol. 49 (2009) 1051. [5] s. j. kayode and d.o. awoyemi, “a multiple derivative collocation method for fifth order ordinary differential equations”, j. math. stat. 6 (2010) 60. [6] a. golbabai & m. m. arabshahi, “smart numerov generalized alternating group explicit (snage-pr (2)) scheme for 2-point p.v. problem”, world appl. sci. j. 8 (2010) 267. [7] s. j. kayode & a. adeyeye, “ a 3-step hybrid method for direct solution of second order initial value problems”, aust. j. of basic and applied sciences 5 (2011) 2121. [8] s. j. kayode & a. adeyeye, “two-step two-point hybrid methods for general second order differential equations” afri. j. of math. & compt. sci. res. 6 (2013) 191. [9] s. j. kayode & f. o. obarhua, “continuous y− f unction hybrid methods for direct solution of differential equations”, intern. j. of diff. eq. 12 (2013) 37. [10] s. j. kayode & f. o. obarhua, “3-step y-function hybrid methods for direct numerical integration of second order ivps in odes”, theo. math. & appl. 5 (2015) 39. [11] f. o. obarhua, “an efficient two-step symmetric hybrid block method for solving second order initial value problems of ordinary differential equations”, int. j. of res. and sci. innovation 6 (2019) 200. [12] a. konguetsof & t. e. simos, “an exponentially fitted and trigonometrically fitted method for the numerical solution of periodic initial-value problems”, computers and mathematics with appl. 45 (2003) 547. [13] o. o. olanegan, d. o. awoyemi, b. g. ogunware & f. o. obarhua, “continuous double-hybrid point method for the solution of second order ordinary differential equations”, int. j. of adv. sci. & tech. res. 5 (2015) 549. [14] a. o. adesanya, m. r. odekunle & a. a. james, “order seven hybrid methods for the solution of first order ordinary differential equations”, canada journal on science and engineering mathematics 3 (2012) 154. [15] t. a. anake, continuous implicit hybrid one-step methods for solutions of initial value problems of general second order ordinary differential equations, ph.d thesis, covenant university, ota, nigeria, 2011. [16] y. s. awari, “derivation and application of six-point linear multistep numerical method for solution of second order initial value problems”, iosr journal of mathematics 7 (2013) 23. 47 j. nig. soc. phys. sci. 5 (2023) 1514 journal of the nigerian society of physical sciences the efficiency of the k-l estimator for the seemingly unrelated regression model: simulation and application oluwayemisi oyeronke alabaa,b,∗, b. m. golam kibriab adepartment of statistics, university of ibadan, nigeria bdepartment of mathematics and statistics, florida international university, miami, fl, usa abstract this paper considers the ridge feasible generalized least squares estimator (rfglse), ridge seemingly unrelated regression rs ur and proposes the kibria-lukman k ls ur estimator for the parameters of the seemingly unrelated regression (sur) model when the regressors of the models are collinear. a simulation study was conducted to compare the performance of the three different types of estimators for the sur model. different correlation levels (0.0, 0.1, 0.2, · · · , 0.9) among the independent variables, sample sizes replicated 10000 times and contemporaneous error correlation (0.0, 0.1, 0.2, · · · , 0.9) among the equations were assumed for the simulation study. the efficiency of the three (rfglse, rs ur, and k ls ur estimators for sur, when the predictors are correlated, was investigated using the trace mean square error (tmse). the results showed that the k ls ur estimator outperformed the other estimators except for a few cases when the sample size is small. doi:10.46481/jnsps.2023.1514 keywords: k − ls ur; multicollinearity; ridge feasible generalized least squares estimator; seemingly unrelated regression; trace mean square error article history : received: 22 april 2023 received in revised form: 21 may 2023 accepted for publication: 24 may 2023 published: 14 june 2023 c© 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: b. j. falaye 1. introduction one of the most ingenious and groundbreaking research in econometrics is combining different single equations into a system of equations to improve the efficiency of the parameter estimation [1], [2]. the seemingly unrelated regression (sur) model with m equations and t observations is given as, y = xβ + ε (1) ∗corresponding author tel. no: +2348052172452 email address: oluwayemisioyeronke@yahoo.com (oluwayemisi oyeronke alaba )  y1 ... ym  mn × 1 =  x1 · · · 0 ... 0 · · · xm  mn × ∑ ki  β1 ... βm ∑ ki × 1  ε1 ... εm  mn × 1 (2) where yi is an nm × 1 vector of observations on the ith response variable, xi is a fixed mn× ∑ ki matrix of explanatory variables, βi is a ∑ ki × 1 vector of unknown regression parameters, εi is an nm×1 vector of disturbances such that cov(ε) = e [ε′ε]⊗in, e(ε) = 0. the ordinary least squares (ols) estimator is widely used to estimate the unknown regression parameters one equation 1 alaba & kibria / j. nig. soc. phys. sci. 5 (2023) 1514 2 at a time since each equation is a classical regression. the sur estimator simultaneously captures different regression equations. however, the efficiency gain in sur is premised on the high level of contemporaneous correlation between or among each classical regression equation. notable works on the efficiency gained in sur which takes cognizance of the contemporaneous correlation of error terms in the joint equations include [3], [4], [5], [6], [7], [8], [9], [10], [11] among others. the generalized least squares estimator (glse) is used to estimate the variance-covariance matrix of the disturbances in sur model. it is a statistical fallacy to assume that the relationship between or among explanatory variables plays an insignificant effect on the error structure of the model. the severity of the correlation levels among the predictors can affect the efficiency and sensitivity of the estimators [12], [13]. hence, the variance of the estimator is inflated, unreliable inference and the confidence interval due to multicollinearity is wider which may increase the probability of a type-ii error in hypothesis testing of unknown parameters [14]. numerous research on single equation models when the problem of multicollinearity is inherent are available in literature. notable works include [15], [16], [17], [18], [19], [20], [21], [22] among others. studies on multicollinearity on systems of equations of regression models are still lacking or scarce in literature. however, a notable exception is [23], [24]. recently, shrinkage estimators such as ridge regression estimators have gained attraction among researchers such that quite a number of exciting estimators emerged [17], [25], [26]. [19] proposed the k-l estimator to tackle the correlated regressors problem for the classical linear regression model, which outperformed both the generalized least squares estimator (glse) and ridge feasible generalized least squares estimator (rfglse). the objective of this paper is to develop an estimator which is suitable for the joint modelling of the k-l estimator, kibria-lukman seemingly unrelated regression k−ls ur estimator when the predictors are correlated as well as compare the newly developed estimator with the existing ridge seemingly unrelated regression estimator (rs ur) and ridge feasible generalized least squares estimator (rfglse). the organization of the paper is as follows: the estimators and their trace mean square error (tmse) expressions are given in section 2. a simulation study is presented in section 3. to illustrate the findings of the paper, real-life data are analysed in section 4. the paper ends with some concluding remarks in section 5. 2. statistical methodology the glse is given as β̂gls e = ( x′ ( σ −1 ⊗ i ) x )−1 x′ ( σ −1 ⊗ i ) y (3) the ridge parameter estimator is an important tool when explanatory variables are correlated in classical linear regression analysis. the ridge estimation technique was pioneered by [15] and extended to the sur model by [25], [26], [27]. the ridge estimator for glse is given as: β̂rgls e = [ x ′ (∑ −1 ⊗i ) x + g ]−1 x ′ (∑ −1 ⊗i ) y = [ x ′ ψ−1 x + g ]−1 x ′ ψ−1y where ψ = ∑ −1 it , g = ki∑ ki β̂rgls e = [ i∑ ki + ( x ′ ψ−1 x )−1 g ]−1 β̂gls e (4) where g is a k × k matrix of non-negative elements characterizing the estimator. to circumvent the problem that σ is unknown, a ridge feasible generalized least squares estimator (rfglse) is given as: β̂rfgls e = [ x ′ ( s −1 ⊗ i ) x + g ]−1 x ′ ( s −1 ⊗ i ) y β̂rfgls e = [ i∑ ki + ( x ′ ψ̃−1 x )−1 g ]−1 β̂fgls e (5) where β̂fgls e = ( x ′ ( s −1 ⊗ i ) x )−1 x ′ ( s −1 ⊗ i ) y and ψ̃−1 = s −1 ⊗ i from (1), given λ as the diagonal matrix of the eigenvalues and ψ a matrix whose columns are eigenvectors of x∗ ′ x∗ of the systems of equations, sur. the canonical version of (1) is defined as y∗ = z∗α∗ + e∗ (6) where z∗ = x∗ψ,α∗ = ψ ′ β and z∗ ′ z∗ = ( ψ ′ x∗ ′ x∗ψ ) = λ the glse of sur for α∗ is; α̂∗gls = ( z∗ ′ z∗ )−1 z∗ ′ y∗ (7) bias ( α̂∗gls ) = e ( α̂∗gls −α ∗ ) (8) v ar ( α̂∗gls ) = e ( α̂∗gls −α ∗ ) e ( α̂∗gls −α ∗ )′ = σ2 ( ψ ′ x∗ ′ x∗ψ )−1 (9) = σ2λ−1 the ridge regression estimator is; α̂rs ur = ( z∗ ′ z∗ + ψ∗rψ∗ ′ )−1 z∗ ′ y∗ (10) bias ( α̂∗rs ur ) = e ( α̂∗rs ur −α ∗ ) (11) using(4)and(7) v ar ( α̂∗rs ur ) = [ x ′ (∑ −1 ⊗i ) x + kip ]−1 x ′ (∑ −1 ⊗i ) x [ x ′ (∑ −1 ⊗i ) x + kip ]−1 = ( λ + kip )−1 λ ( λ + kip )−1 (12) where λ = x′(σ−1 ⊗ i)x t ms e ( α̂∗rs ur ) = σ2 ( ip + kλ−1 )−1 λ−1 ( ip + kλ−1 )−1 + ( −k ( λ + kip )−1) α∗α∗ ( −k (λ + kik) −1 )′ (13) 2 alaba & kibria / j. nig. soc. phys. sci. 5 (2023) 1514 3 figure 1. estimated tmse when ρεm = 0.1 following [19], we define the k-lsur estimator as follows: α̂∗k ls ur = [( ip + kλ −1 )−1 ( ip − kλ −1 )] α̂∗ (14) e ( α̂∗k ls ur ) = ( ip + kλ−1 )−1 ( ip − kλ−1 ) e (α̂∗) = ( ip + kλ−1 )−1 ( ip − kλ−1 ) α̂∗ (15) the k-l estimator is an unbiased estimator when k = 0 t ms e ( α̂∗rs ur ) = σ2 ( ip + kλ−1 )−1 ( ip − kλ−1 ) λ−1 ( ip − kλ−1 )′ (( ip + kλ−1 )−1)′ + [( ip + kλ−1 )−1 ( ip − kλ−1 ) − ip ] α∗α∗ ′ [( ip + kλ−1 )−1 ( ip − kλ−1 ) − ip ]′ (16) 3. numerical analysis a simulation study is considered to compare the performance of the estimators in this section. it consists of two parts (i) simulation study (ii) discussion of results. 3.1. simulation study the monte carlo experiment was performed by generating data according to the following algorithm. 1. generate the explanatory variables from mv n3(0, σx) 2. set the true values of β to (1, 1, 1, 1)′ figure 2. estimated tmse when ρεm = 0.4 table 1. description of variables, equations, observations and contemporaneous correlations factors symbol design no of equations m 3 no of observations t 20, 30, 50, 100 correlation among the explanatory variables ρx 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 contemporaneous correlation between corresponding errors among the equaions ρς 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 3. the variance-covariance is given as ∑ =  σ211 σ 2 12 σ 2 13 σ221 σ 2 22 σ 2 13 σ231 σ 2 32 σ 2 33  =  1 0.6 0.8 1 0.7 1  such that y1 : n ( x1β1,σ211 ) y2 : n ( x2β2,σ222 ) y3 : n ( x3β3,σ233 ) 4. simulate the vector random error from mv n3(0, σe) 5. for a given x structure, transform the original model to the canonical form. 6. compute the trace mean squared error of βk ls ur,βrfgls e, and βrs ur 7. repeat the above step 10000 times. 3 alaba & kibria / j. nig. soc. phys. sci. 5 (2023) 1514 4 table 2. estimated tmses for the different methods when ρεm = 0.1, 0.2 and 0.3 at n = 20 n = 20 ρεm = 0.1 ρεm = 0.2 ρεm = 0.3 ρxi x j k ls ur rfgls rs ur k ls ur rfgls rs ur k ls ur rfgls rs ur 0.0 1.9671 31.4793 1078.181 2.9993 34.9221 34.9869 4.3360 33.2312 210.5745 0.1 2.2637 33.5409 40.7652 3.3615 34.4638 39.1302 4.7297 33.9507 179.0863 0.2 2.5411 33.5427 38.5592 3.7682 34.7496 47.0992 5.2687 36.4858 71.1576 0.3 2.8784 32.8007 49.5184 4.2931 35.8924 42.8646 7.7218 37.2722 53717.79 0.4 3.4011 33.7350 53.2065 5.2779 35.8924 75.3289 64.0115 37.3879 113.8371 0.5 4.0056 3.6415 2.5955 6.0753 37.7050 216.6385 8.4646 39.2840 208.3292 0.6 4.9863 3.5757 1.0886 214.7618 38.7659 295.7213 14.9161 40.7934 741.0087 0.7 7.5773 36.6885 1137.117 11.5006 40.7506 462.7717 345.8575 43.9871 1399.8543 0.8 20.4227 42.1282 1948.9526 45.0750 45.1791 16809.71 55.2401 51.3216 3303.2105 0.9 31.9145 51.3059 441.2463 979.4674 59.5731 27157.19 822.7517 70.6499 4580.6889 table 3. estimated tmses for the different methods when ρεm = 0.1, 0.2 and 0.3 at n = 30 n = 30 ρεm = 0.1 ρεm = 0.2 ρεm = 0.3 ρxi x j k ls ur rfgls rs ur k ls ur rfgls rs ur k ls ur rfgls rs ur 0.1 1.1817 32.6906 32.3140 1.4956 32.0123 31.5930 2.3545 33.3188 32.6339 0.2 1.1929 29.9211 29.4511 1.6637 32.7880 32.3673 2.1419 30.0010 32.0582 0.3 1.3553 22.4464 33.2031 1.8878 32.9291 32.5181 2.4342 34.7959 34.2291 0.4 1.5684 30.1094 29.5395 2.1817 31.1496 30.5211 2.8013 33.1929 32.2543 0.5 1.8712 20.1180 31.9485 2.6034 32.1032 31.3159 3.3439 33.6351 32.5898 0.6 2.3131 17.2247 16.8764 3.2240 32.7284 47.2943 4.1401 34.5030 36.8375 0.7 6.9799 31.2066 382.9524 4.3324 33.5406 47.9402 84.9530 35.5565 42.7001 0.8 4.6595 33.2122 1786.0465 6.7134 35.9082 166.2044 19.7855 36.5740 1303.1226 0.9 9.9985 37.0700 3186.9899 16.5210 42.0564 13172.89 528.8485 45.9672 3383.9887 table 4. estimated tmses for the different methods when ρεm = 0.1, 0.2 and 0.3 at n = 50 n = 50 ρεm = 0.1 ρεm = 0.2 ρεm = 0.3 ρxi x j k ls ur rfgls rs ur k ls ur rfgls rs ur k ls ur rfgls rs ur 0.1 0.6101 30.9197 30.7837 0.8323 17.4569 17.3667 1.0599 29.7809 29.5481 0.2 0.6797 32.7346 32.6091 0.9276 32.2903 32.1370 1.1776 33.7307 33.5492 0.3 0.7701 32.2188 32.0525 1.0484 31.8266 31.6524 1.3189 18.5846 18.3629 0.4 0.8906 30.1703 29.9848 1.2107 31.6182 31.4187 1.5282 31.8069 31.5339 0.5 1.0613 30.2766 30.0612 1.4362 19.7875 19.4246 1.8204 34.1597 33.8706 0.6 1.3208 33.5028 33.3809 1.7953 33.5506 33.2716 2.2618 34.4465 34.0654 0.7 1.7521 32.6755 32.2909 2.3778 34.2552 33.8549 3.0007 34.2756 33.6832 0.8 2.6126 34.5632 33.9872 3.5386 35.1982 34.4567 4.4997 36.0475 34.9669 0.9 5.1513 22.4060 20.8302 7.1461 38.2472 36.3301 9.1181 40.2009 38.1831 the simulation results were presented in tables 2 to 13 for ρεm = (0.1, 0.2, 0.3),ρεm = (0.4, 0.5, 0.6) and ρεm = (0.7, 0.8, 0.9) respectively. in addition, some results are illustrated graphically in figures 1 to 4 for ρεm = 0.1, 0.4, 0.7 and 0.9 respectively. we considered r software to conduct the simulation study [27], [28]. 4. discussion of results from tables 2 to 13 and figures 1 to 4, we can see that the proposed k ls ur estimator uniformly dominate the rfgls and rs ur estimator except when the sample size is small (n = 20). also, an increase in the ρxi xi increases the estimated tmse values of the estimators. obviously, the tmse values were significantly large at sample size n = 20 when ρεm = 0.7, 0.8 and 0.9 than other corresponding tmse values with n = 30, 50, 100 for ρεm = {0.0, 0.1, 0.2, · · · , 0.9} and n = 20 4 alaba & kibria / j. nig. soc. phys. sci. 5 (2023) 1514 5 table 5. estimated tmses for the different methods when ρεm = 0.1, 0.2 and 0.3 at n = 100 n = 100 ρεm = 0.1 ρεm = 0.2 ρεm = 0.3 ρxi x j k ls ur rfgls rs ur k ls ur rfgls rs ur k ls ur rfgls rs ur 0.1 0.2958 31.1069 31.0652 0.4001 32.4019 32.3509 0.5034 32.0166 31.9597 0.2 0.3294 32.9002 32.8577 0.4439 31.5425 31.4946 0.5560 31.9488 31.8862 0.3 0.3730 31.0733 31.0145 0.5013 31.7451 31.6767 0.5259 30.9504 30.8669 0.4 0.4317 32.7167 32.6697 0.5793 32.3633 32.3027 0.68903 31.7897 31.7127 0.5 0.5141 30.5062 30.4347 0.6888 30.4318 30.3485 0.85712 30.8029 30.6942 0.6 0.6373 17.8189 17.7757 0.8544 32.3865 32.3051 1.06185 32.3417 32.2352 0.7 0.8467 32.4540 32.3723 1.1325 32.6137 32.4948 1.4077 33.2381 33.0755 0.8 1.2616 30.9976 30.7480 1.6856 31.4613 31.1143 2.0931 32.4275 31.9626 0.9 2.5207 32.4311 31.6849 3.3763 33.1247 32.1534 4.1961 33.8402 32.4072 table 6. estimated tmses for the different methods when ρεm = 0.4, 0.5 and 0.6 at n = 20 n = 20 ρεm = 0.4 ρεm = 0.5 ρεm = 0.6 ρxi x j k ls ur rfgls rs ur k ls ur rfgls rs ur k ls ur rfgls rs ur 0.0 9.7560 36.2392 256.4695 716.2386 35.4935 406.7923 12.0147 40.1432 308.325 0.1 6.4118 36.2199 459.8962 10.6846 37.8609 658.1879 12.6002 38.8413 253.003 0.2 7.7878 36.0814 193.1198 9.9320 39.0815 1772.816 739.9870 42.2980 171.349 0.3 14.6262 39.5822 284.7212 11.7001 40.6640 251.5625 27.9862 43.0574 692.950 0.4 16.2656 39.2694 248.8817 68.8660 42.2592 860.8801 21.2453 45.0407 477.386 0.5 12.6930 42.5478 2629.349 20.1253 45.4033 22608.84 25.9884 47.7564 4292.16 0.6 37.4753 43.3222 286.5887 78.1266 47.4747 252.3527 76.0018 51.3518 581.977 0.7 91.8338 48.2507 769.2762 186.599 53.0107 1751.206 12291.26 57.8010 7708.11 0.8 408.0553 57.7729 6626.982 482.442 65.6280 94973.63 7459.204 72.5054 1255.63 0.9 27135.45 82.4109 3004.294 8687.57 96.2975 18763.57 98272.26 96.1662 2682.17 table 7. estimated tmses for the different methods when ρεm = 0.4, 0.5 and 0.6 at n = 30 n = 30 ρεm = 0.4 ρεm = 0.5 ρεm = 0.6 ρxi x j k ls ur rfgls rs ur k ls ur rfgls rs ur k ls ur rfgls rs ur 0.0 2.1482 33.8490 33.4394 2.58122 33.6536 32.9963 3.0486 34.1725 33.4501 0.1 2.4351 29.8788 47.1569 2.83488 30.7532 29.9543 3.3416 31.2885 32.5218 0.2 2.2597 34.6346 33.5944 3.1033 31.2482 30.7619 3.5688 32.5834 31.1876 0.3 2.9817 35.3758 34.6432 3.5219 36.6019 35.7699 4.0414 36.7851 35.7306 0.4 3.4258 32.9304 32.1480 4.0364 33.0953 32.6963 4.6313 33.6982 48.0887 0.5 4.1175 37.4169 36.4355 4.7903 34.6699 35.0898 5.5344 34.1078 1134.971 0.6 5.0357 35.1816 266.3296 5.9218 35.9237 35.6257 7.0024 36.8212 38.8586 0.7 6.8262 37.1196 504.2674 8.0604 38.0785 683.5997 9.21488 39.7283 2209.564 0.8 12.1846 39.4336 26492.285 13.0759 41.5627 775.9910 16.8529 43.1572 330.2718 0.9 23.4235 52.9490 1302.4404 36.8351 56.4241 6609.2648 44.5576 57.7298 2350.237 for ρεm = {0.0, 0.1, 0.2, · · · , 0.7}. the tmse values become larger at n=20 as ρεm increases from 0.7 to 0.9. the tables and figures consistently showed that the proposed k ls ur estimator performs better than the rfgls and rs ur estimators when there exist moderate to high correlation among the regressors. the plots of tmse against various ρεm in figures 1 to 4 showed that as the value of ρ increases, tmse also increase, while as the sample size increases, the tmse value decreases. the preferred tmse values were at n =100 for ρεm = 0.1 for k ls ur estimator. k ls ur produced the smallest tmse values when ρxi xi range from 0.1 to 0.9, such that the corresponding tmse values for k ls ur estimator increase as the ρxi xi increase. however, it was noted that the ideal and smallest tmse value for k ls ur estimator occurred at ρεm = 0.1 and ρxi xi = 0.1. concisely, k ls ur also produced small tmse values at ρεm = 0.1 and ρxi xi = 0.1, but not smaller in comparison to the one produced by k ls ur estimator. con5 alaba & kibria / j. nig. soc. phys. sci. 5 (2023) 1514 6 table 8. estimated tmses for the different methods when ρεm = 0.4, 0.5 and 0.6 at n = 50 n = 50 ρεm = 0.4 ρεm = 0.5 ρεm = 0.6 ρxi x j k ls ur rfgls rs ur k ls ur rfgls rs ur k ls ur rfgls rs ur 0.0 1.2231 31.8356 31.6180 1.4567 33.0326 32.7876 1.6850 30.8215 30.4527 0.1 1.2845 29.9504 29.6901 1.51162 30.2718 29.9303 1.7267 31.5870 31.1975 0.2 1.4195 32.2654 32.0162 1.6617 31.4414 31.1212 1.8969 31.7036 31.3102 0.3 1.5985 33.1219 32.8367 1.8689 33.8384 33.5117 2.1299 33.8781 33.4787 0.4 1.8389 32.0315 31.6715 2.1469 33.3783 32.9510 2.4424 32.6150 32.0562 0.5 2.1859 32.7642 32.3262 2.5474 32.0638 31.4425 2.9026 33.2641 32.5604 0.6 2.7172 34.1213 33.5881 3.1823 34.3878 33.7234 3.6293 35.1303 34.3261 0.7 3.6180 34.9418 34.1512 4.1749 35.4996 34.5134 4.7830 36.0520 34.80777 0.8 5.4192 36.9573 35.5262 6.3468 38.1684 36.3989 7.2398 38.6846 36.4323 0.9 11.0051 41.8592 41.3861 12.9492 43.3830 53.5260 15.1569 44.8944 47.0988 table 9. estimated tmses for the different methods when ρεm = 0.4, 0.5 and 0.6 at n = 100 n = 100 ρεm = 0.4 ρεm = 0.5 ρεm = 0.6 ρxi x j k ls ur rfgls rs ur k ls ur rfgls rs ur k ls ur rfgls rs ur 0.0 0.5570 31.1110 31.0429 0.65712 31.2067 31.1265 0.7562 32.5784 32.4877 0.1 0.6064 31.5343 31.4591 0.7069 31.4484 31.3653 0.80516 31.0438 30.9381 0.2 0.6666 32.7558 32.6821 0.7744 32.4927 32.4042 0.88786 32.4789 32.3799 0.3 0.7477 31.0310 30.9287 0.8670 30.8659 30.7475 0.9813 30.4286 30.3006 0.4 0.8616 32.4530 32.3617 0.9960 32.7987 32.6845 1.1274 32.8786 32.7485 0.5 1.0175 17.1745 17.1026 1.18127 31.1289 30.9470 1.3339 31.6014 31.3899 0.6 1.2645 32.7726 32.6332 1.4586 32.5574 32.3776 1.6516 32.0510 31.7438 0.7 1.6723 33.5980 33.3833 1.9303 33.4664 33.1906 2.1856 33.9971 33.6736 0.8 2.4884 32.3812 31.7566 2.8805 32.7172 31.9339 3.2131 19.5499 18.9774 0.9 4.9937 34.7593 32.9295 5.7754 35.4817 33.2136 6.5181 36.4971 33.78511 table 10. estimated tmses for the different methods when ρεm = 0.7, 0.8 and 0.9 at n = 20 n = 20 ρεm = 0.7 ρεm = 0.8 ρεm = 0.9 ρxi x j k ls ur rfgls rs ur k ls ur rfgls rs ur k ls ur rfgls rs ur 0.0 1.5970 4.0118 1.5448 27.0078 41.4233 161.6998 20.9443 44.5509 2102.610 0.1 13.3597 41.6366 7274.501 19.8636 43.4407 1096.485 45.4412 44.9649 366.5848 0.2 14.5438 42.4626 301.5057 21.2077 45.1824 402.2686 22.9364 46.0099 133.9393 0.3 15.8899 45.4626 125.4716 30.3843 47.2784 878.4280 109.9613 49.0023 644.4463 0.4 953.5130 47.2117 1997.0092 577.5404 49.5397 939.3170 1.1322 5.2701 2.0398 0.5 30.2281 51.3641 6113.3947 34.0842 53.2688 1525.4125 4787.2699 56.5041 2354.758 0.6 96.7198 53.3738 8236.2489 1440.019 57.9711 403.6016 481.5750 61.4689 10141.04 0.7 44014.384 62.7783 864.9972 9420.9595 67.5711 1731.6622 7184.6716 71.3247 34749.18 0.8 21669.976 79.6336 26152.861 47515.817 84.3908 36052.59 5962.1447 91.7812 717.1957 0.9 75800.340 127.1244 1039.1268 12340.321 137.575 1158.7834 50556.851 148.5709 9359.699 sequently, this implies that k ls ur is the preferred estimator when considering sur model when the regressors of the model are collinear among the three considered estimators. the strength of collinearity among the considered variables for mv n3(0, σe), m = 3 with varied for sample sizes 20 to 100 for k ls ur estimator relies on when n = 100 with ρεm = 0.1 and ρxi xi = 0.1. this implies that the strength and type of dependency among the explanatory variables affect the performance of each of the three estimators of k ls ur, rfgls and rs ur as t increases, such that ρεm and ρxi xi decreases (or relatively low), the tmse values of k ls ur decreases. this implies that there is a gain in the efficiency of the k ls ur estimator. 6 alaba & kibria / j. nig. soc. phys. sci. 5 (2023) 1514 7 table 11. estimated tmses for the different methods when ρεm = 0.7, 0.8 and 0.9 at n = 30 n = 30 ρεm = 0.7 ρεm = 0.8 ρεm = 0.9 ρxi x j k ls ur rfgls rs ur k ls ur rfgls rs ur k ls ur rfgls rs ur 0.1 3.6831 32.4952 31.1870 4.0739 32.6618 30.9791 4.4489 32.9211 31.8359 0.2 4.0185 34.1073 32.5706 4.4305 33.9127 31.9412 4.8275 34.5264 32.4991 0.3 4.5332 37.4486 36.2957 5.0095 37.3701 35.8343 5.4564 38.6045 37.1221 0.4 5.1828 33.9950 40.1823 5.8834 34.7873 35.0340 6.2366 34.9937 36.4678 0.5 6.1682 36.0594 35.1923 6.8139 36.9741 98.5351 7.4014 37.2651 653.2791 0.6 7.6710 36.6365 59.8780 8.4420 38.3879 42.2726 9.1107 3.9449 1.0753 0.7 41.4184 40.5272 92.5242 11.4445 42.0848 4231.299 12.5689 42.5401 83.8831 0.8 28.8063 44.9286 137.4677 1023.891 46.0243 332.3192 23.1223 47.7619 11443.84 0.9 60.4966 62.9849 156.5118 701.8199 65.4741 234.6494 7.8273 6.8693 1.3251 table 12. estimated tmses for the different methods when ρεm = 0.7, 0.8 and 0.9 at n = 50 n = 50 ρεm = 0.7 ρεm = 0.8 ρεm = 0.9 ρxi x j k ls ur rfgls rs ur k ls ur rfgls rs ur k ls ur rfgls rs ur 0.1 1.9390 30.7288 30.2195 2.1375 32.1671 31.6753 2.3270 32.5277 31.9061 0.2 2.1207 31.4594 30.9896 2.7655 32.6984 31.8318 2.5422 34.9202 34.2994 0.3 2.3833 34.6569 34.1233 2.6170 34.4649 33.9328 2.8471 35.1956 34.5160 0.4 2.7287 33.9790 33.2775 2.9985 33.0041 32.2088 3.2564 33.0878 32.1194 0.5 3.2429 33.4776 32.5970 3.5620 33.9245 32.9419 3.8698 34.2694 33.1379 0.6 4.0373 34.8706 33.8102 4.4264 34.1091 32.6144 4.8349 36.7864 35.4188 0.7 5.3456 36.8501 35.2405 5.8171 37.0795 35.3813 6.3794 38.0211 35.8226 0.8 8.0694 39.0196 36.2068 8.8507 39.8438 36.5279 9.5831 39.7160 35.8735 0.9 16.8842 46.8999 215.3601 29.3585 48.1159 97.7010 51.5108 49.3468 56.2438 table 13. estimated tmses for the different methods when ρεm = 0.7, 0.8 and 0.9 at n = 100 n = 100 ρεm = 0.7 ρεm = 0.8 ρεm = 0.9 ρxi x j k ls ur rfgls rs ur k ls ur rfgls rs ur k ls ur rfgls rs ur 0.1 0.8994 31.9839 31.8691 0.9882 31.6234 31.4959 1.0733 31.7436 31.5857 0.2 0.9785 33.1136 32.9996 1.0742 32.6911 32.5621 1.1653 32.2650 32.1150 0.3 1.0918 31.3878 31.2122 1.1969 31.2668 31.0811 1.2964 31.9378 31.7250 0.4 1.2536 32.6015 32.4556 1.3732 32.8539 32.6776 1.4871 32.2467 32.0410 0.5 1.4785 31.6940 31.4353 1.6218 31.1869 30.8941 1.7562 32.0679 31.7361 0.6 1.8292 31.9719 31.6237 2.0088 32.4433 32.0269 2.1661 32.3290 31.8782 0.7 2.4185 34.1733 33.7924 2.6589 34.4667 34.0165 2.8793 34.3942 33.8972 0.8 3.5981 33.4160 32.3067 3.9354 33.7854 32.4953 4.2864 34.0114 32.6416 0.9 7.2537 36.9970 33.7463 7.9236 37.7464 34.1527 8.6199 38.7249 34.4544 table 14. autocorrelation test (durbin-watson test) equation (model) test statistic p-value brazil 2.3800 0.2846 india 1.9160 0.0658 indonesia 2.4776 0.4547 south africa 1.9669 0.1225 turkey 1.5431 0.0032 5. application to further illustrate the results of the theoretical part of this paper, we consider the dataset and structural model by table 15. heteroscedasticity test (breusch-pagan test) equation (model) test statistic p-value brazil 12.4250 0.1332 india 11.0220 0.2005 indonesia 6.0900 0.6372 south africa 5.6808 0.6829 turkey 3.8404 0.8712 [12]. the study considered the foreign direct investment on 7 alaba & kibria / j. nig. soc. phys. sci. 5 (2023) 1514 8 table 16. multicollinearity test (variance inflation factor) equation deflator current account capital growth final consumption import of goods personal remittance total reserves export of goods brazil 56.241 23.344 2.144 6.652 11.143 20.487 46.461 24.510 india 23.798 13.750 1.741 2.497 101.771 2.438 23.185 66.900 indonesia 118.769 12.219 3.552 4.588 42.317 3.214 72.550 124.779 south africa 29.423 39.671 4.278 10.284 109.568 10.157 49.739 45.688 turkey 21.052 206.371 3.548 5.611 258.126 4.557 15.662 224.422 table 17. specification test (regression equation specification error test) equation (model) test statistic p-value brazil 0.0591 0.943 india 1.0576 0.3913 indonesia 0.98417 0.4148 south africa 0.52381 0.6113 turkey 1.0114 0.4059 table 18. shrinkage parameter estimators for the life study k ls ur rfgls rs ur tmse 35.00 40.52786 40.5278 figure 3. estimated tmse when ρεm = 0.7 some economic and financial variables over the period of twenty years between 2001 and 2019. the ”fragile five” countries (cited in [12]) include turkey (tur), south africa (zaf), brazil (bra), india (ind), and indonesia (idn). hence, m = 5 blocks, with measurements of t = 20 years per equation. the raw data were extracted online from the world bank indicators. the structural sur model adopted is given as: figure 4. estimated tmse when ρεm = 0.9 f diit = f (gro, df, cab, fc, i ps, prr, t r, egs ) fdii = β0i + β1igro + β2i d f + β3icab + β4i fc + β5i i ps +β6i pr + β7it r + β8i egs + ei (17) i denotes countries (i= tur, zaf, bra, ind, idn). fdii is the foreign direct investment of the ”fragile five” countries, gro is the gdp per capital growth, df is the gdp deflator, cab is the current account balance, fc is the general government final consumption expenditure, ips is the imports of goods and services, prr is the personal remittances received, tr is the total reserves and egs is the exports of goods and services. the assumptions in each equation and joint model were put into consideration. normality, homoscedasticity, multicollinearity and serial correlation of the error term were examined. the results are available in tables 14 to 18. the null hypothesis for the durbin-watson test is that errors are random and independent. a significant p-value in this test rejects the null hypothesis that the time series is not autocorrelated. table 5 suggests a rejection of the null hypothesis for turkey at the significance level, that is, p-value = 0.0032 ≺0.05. this implies that each equation satisfied the assumption of non-autocorrelation. 8 alaba & kibria / j. nig. soc. phys. sci. 5 (2023) 1514 9 the null hypothesis for the breusch-pagan test is that there is no homoscedasticity (that is, there is presence of heteroscedasticity). since, the p-value for each of the ”fragile five” country is greater than 0.05. we did not reject the null hypothesis in each equation, so the assertion of homoscedasticity in each equation is satisfied. the variance inflation factor (vif) makes it possible to measure how many times the variance of the regression coefficients will be for multicollinear data than for orthogonal/ canonical data. if vif�10 this indicates multicollinearity. concisely, deflator, current account, total reserves; import and export of goods posed to be problematic (that is, multicollinearity problem exist) as their vif values were strictly greater than 10, while others also call for little concern. this implies that the problem of multicollinearity exists in the equations. therefore, the k ls ur estimator will be ideal to solve the problem. the regression equation specification error test is meant for testing the exogeneity of explanatory variables. the null hypothesis for the test is that there is no correlation between the error term and the explanatory variables or that linearity exist in the functional form of the regression model, that is, e [εi | xi] = 0. since, the p-value for each of the ”fragile five” country is greater than 0.05 in table 14, this suggests that there is no correlation between the error term and the explanatory variables for all the countries (turkey, brazil, india, indonesia, and south africa). the shrinkage parameter estimator is designed for model reduction for each of the m equations in order to absolve only significant explanatory variables in line with their number of observations. the shrinkage estimator makes it possible to standardize each reduced equation. from table 18, the shrinkage parameters of the k ls ur estimator via the tmse produced the smallest tmse value of 35.00 compared to rfgls and rs ur that gave higher tmse of 40.52786 and 40.5278 respectively. this makes the k ls ur estimator to possess a higher efficiency gain than the two other estimators. summarily, k ls ur = 35.00 indicates that the estimator outperforms rfgls and rs ur. 6. conclusion the seemingly unrelated regression model is an exciting and celebrated model when the error structure of joint models are correlated. we considered the ridge feasible generalized least squares estimator, ridge seemingly unrelated regression and the kibria-lukman k ls ur estimator for estimating the parameters of the seemingly unrelated regression model when the regressors of the model are collinear. findings from this comparative study revealed that k ls ur estimator is a better alternative when compared with ridge and ridge feasible generalized least squares, which are often used to tackle the problem of multicollinearity in single and joint equations respectively. simulation and real-life data were used for assessment. note that we have considered one of many possible estimators of the ridge parameter k. the conclusion of the paper may change if we consider different values of k and such possibility is under the current investigation. acknowledgements this paper was partially completed while the first author was visiting prof. b. m. golam kibria, florida international university in the month of april-may, 2021. availability of data and codes: available on request. references [1] a. zellner, “an efficient method of estimating seemingly unrelated regression equations and tests for aggregation bias”, journal of the american statistical association 7 (1962) 48. 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[22] g.a. shewa, & f.i. ugwuowo, “a new hybrid estimator for linear regression model analysis: computations and simulations”, scientific african 19 (2023) e01441. [23] w. b. yahya, s. b. adebayo, e. t. jolayemi, b. a. oyejola & o. o. m. sanni, effects of non-orthogonality on the efficiency of seemingly unrelated regression (sur) models, interstat journal, 2008. [24] a. lawal & o. o. alaba, “exploratory analysis of some sectors of the economy: a seemingly unrelated regression approach” african journal of applied statistics 6 (2019) 649. [25] v. k. srivastava, & d. e. a. giles, seemingly unrelated regression equations, marcel dekker, inc., new york (1987). [26] m. alkhamisi, g. khalaf & g. shukur, “some modifications for choosing ridge parameters” communication in statistics theory and methods 35 (2006) 2005. [27] m. r. abonazel, “a practical guide for creating monte carlo simulation studies using r”, international journal of mathematics and computational science 4 (2018) 18. [28] s. o. adams, d. a. obaromi, & a. a. irinews, “goodness of fit test of an autocorrelated time series cubic smoothing spline model” j. nig. soc. phys. sci. 3 (2021) 191. 10 j. nig. soc. phys. sci. 4 (2022) 27–33 journal of the nigerian society of physical sciences mechanical evaluation and minerals phases identification of fine and coarse okelele block clay composites for furnace lining application yusuf olanrewaju saheed, mufutau abiodun salawu∗, aderemi babatunde alabi department of physics, university of ilorin, ilorin, nigeria abstract the suitability of fine and coarse okelele clays as refractory raw materials for furnace lining application was investigated. the clay samples were crushed and pounded with a mortar and pestle to a particle size of 20 microns. 230 g each of fine clay was mixed with 50 mls of water inside a bowl and stirred thoroughly to form homogenous plastic paste. 10 g, 15 g, 25 g, 35 g and 45 g of coarse clay were added respectively to the 230 g of homogenous fine clay paste in different container. the fine and coarse clays composites weighing 240 g, 245 g, 255 g, 265 g and 275 g were respectively put in a mold of dimension 3 x 5 x 6 cm and air dried for 7 days. the samples were fired at temperature of 1200 oc for five hours using carbolite furnace. after cooling, the fine and coarse clay composites of 240 g and 245g were broken by the heat and composites blocks 255 g, 265g and 275g were hardened and remove for compressive test analysis. the fine and coarse clays were characterized using x-ray diffractometer pw 1830 for minerals phases’ identification. the result of xrd shows that the clay was majorly composed of quartz and kaolinite with the traces of other minerals such as smectile, illite/mica, albite, jarosite, gypsum and pyrite. the kaolinite contains aluminum silicate (al2o3·2sio2) and quartz has the silicon and oxygen atoms. the compressive strength test result judged the 275 g fire block of clays composite the best with the maximum force breaks of 7652 n with deflection of 3.734 mm and young modulus of 212 n/mm2 for the time to failure of 22 seconds. the results proved that okelele clays are suitable as refractory material for furnace lining application. doi:10.46481/jnsps.2022.252 keywords: okelele clays, kaolinite, quartz, refractory materials article history : received: 13 june 2021 received in revised form: 14 october 2021 accepted for publication: 19 october 2021 published: 28 february 2022 c©2022 journal of the nigerian society of physical sciences. all rights reserved. communicated by: s. j. adebiyi 1. introduction nigeria is rich with abundant mineral resources but these resources have not been sufficiently explored and used. clay is a naturally occurring material composed of layered structures ∗corresponding author tel. no: email address: salawu.ma@unilorin.edu.ng; abideen2004@gmail.com (mufutau abiodun salawu ) of fine-grained minerals which reveal the property of plasticity at appropriate water content and permanently hard when fired [1]. clay as a mineral that consist of silica (sio2), alumina (al2o3), water (h2o), and other impurities are aluminosilicate, mostly answerable for its thermal property of refractoriness which applicable in the manufacturing of several refractory products. it is earthen and soil with intricate inorganic blend, whose structure diverges and generally depends on the environmental and geographical position [2]. 27 saheed et al. / j. nig. soc. phys. sci. 4 (2022) 27–33 28 high demand for refractory materials for furnace building and other related high temperature processes is enormous. nigeria spends more than 2.27 billion naira yearly on the importation of refractories for industrial application [3]. the application of clay composites as a refractory material depends severally on its thermal property of refractoriness, chemical composition, mechanical and physical properties [2], [5-13]. refractory materials are inorganic materials containing the mixtures of oxides obtained from naturally occurring minerals capable of withstanding very high temperature conditions without cracking, deforming, softening or change in composition [3]. the good characteristic of a refractory is to provide basic thermal properties, support winding (electric resistance) and be able to hold solid or liquid metals without entering into any undesirable chemical reaction with them. thus refractory materials are characterized by the ability to withstand the heat, chemical attack, abrasion, impact, and shock caused by thermal stresses. the clays used for furnace linings in metallurgical industries are classified as refractory clays. however, the degree of refractoriness and plasticity of any clay material is often influenced by the amount of the impurities contained in them [13]. the mechanical properties of different particle sizes of some impurities for some specific application had been investigated [14]. chanchanga, bida, suleja and zungeru clays deposits have better refractory and physical properties when compared with imported ones [3]. some local clay deposits in other part of nigeria have also been investigated with good results. some of the clay deposits investigated for refractory application includes but not limited to dukku clay deposit in gombe state, onibode, ibamajo, ijoko in ogun state and are in ekiti state [15]. the characterization of otukpo clay in benue state was also reported [16]. the economic circumstance in nigeria as at today has necessitated for the inward sourcing of locally available raw materials across the country for domestic and industrial applications. due to the aforementioned economic needs and the fact that the okelele clay deposit in ilorin, kwara state is only used for local pottery by old women living around the area and building bricks by local bricklayer. the minerals phases’ identification and refractory properties of this particular clay deposit needs to be investigated. 2. materials and method the fine and coarse clay samples were collected from a deposit in okelele, ilorin east local government area of kwara state. the molding iron bar, mortar and pistol, electronic weighing balance, ht 4/28 carbolite gero muffle furnace machine located at geology department, university of ilorin, (0-3000 ◦c), xfs300 testometric compression test machine located at agricultural biotechnology laboratory, department of biotechnology engineering, university of ilorin and pw 1830 x-ray diffractometer located at the department of geology, university of ibadan were used in this work. the clay samples were crushed and pounded with a mortar and pestle to a particle size of 20 microns. 230 g each of fine clay was mixed with 50 mls figure 1. shows the broken and unbroken fired block of clays after firing of water inside a bowl and stirred thoroughly to form homogenous plastic paste. 10 g, 15 g, 25 g, 35 g and 45 g of coarse clay were added respectively to the 230 g of homogenous fine clay paste. the fine and coarse clays composites weighing 240 g, 245 g, 255 g, 265 g and 275 g were respectively put in a mold of dimension 3 x 5 x 6 cm and air dried for 7 days. the samples were fired at temperature of 1200 oc for 12 hours using ht 4/28 carbolite gero muffle furnace (0-3000 ◦c). after cooling, the fine and coarse clay composites of 240 g and 245g were broken by the heat and composites 255 g, 265g and 275g were hardened and sound like a glass when tapped. figure 1 shows the fabricated broken and unbroken fired block of clays after firing. 2.1. x-ray diffractometer (xrd) analysis xrd was used to identify the phase of minerals constituents of the clays. the fine and coarse clays were separately crushed and milled to fine particles and put in test tubes. the samples were subjected to x-ray using the philips pw 1830 x-ray diffractometer with a cu-anode at the “university of ibadan” ibadan, oyo state. after the x-ray characterization of the samples, mineral peaks were identified using xpert high score plus software. the background and peak positions were identified and based on the peak positions and intensities; a search-match routine was performed. 2.2. compression test the unbroken fired block of clays (255g, 265g and 275g) were subjected to mechanical compression test at the civil engineering laboratory of the university of ilorin, ilorin kwara state, to show how these materials deform (elongate, compress, twist) or break as a function of applied load, time, temperature and other conditions. the mechanical test was performed using xfs300 testometric compression test machine. the capacity of this machine is 10,000 pounds (tension and compression). the samples of the given clay material took a rectangular shape which is unreformed (with no permanent strain or residual stress), or original shape. 28 saheed et al. / j. nig. soc. phys. sci. 4 (2022) 27–33 29 figure 2. testometric compression test machine used figure 3. cracked block of clay during compression test 3. results and discussion figure 4 and 5 show the xrd patterns of fine and coarse clays. the debye scherer equation was employed for the estimation of grain sizes of fine and coarse clays. grain size g = kh βcosθ (1) where k is the debye scherer constant (0.94) h = 1.56 x10−10m = 0.156nm β = (fwhm) full width at half maximum (radians) θ = peak positions (radians) the estimated grain sizes and mineral constituents of fine and coarse clays are shown in table 1 and table 2. tables 1 and 2 give the results for the minerals phase identification for the fine and coarse okelele clays. the kaolinite and quartz are dominance in the mineral phase identifications for the fine and coarse okelele clays composites. kaolinite which is also called china clay, is the best refractory clay type and will not soften below 1750 ◦c. kaolinite clays possessed little plasticity due to their large clay particles. the kaolinite contains figure 4. xrd pattern of fine clay figure 5. xrd pattern of coarse clay al2o3·2sio2. the pure kaolinite can be found at the site of its parent rock (primary clay) and when it has not been mixed with impurities, its refractoriness is great. the quartz is a very hard crystalline mineral mostly found in nature contained the silicon and oxygen atoms. quartz is the most conventional source of silica to be used for refractory production. the refractory made from silica (silica refractory bricks) possesses excellent thermal shock resistance at specific temperature range. the compressive strength test on 255 g, 265 g and 275 g fire blocks of clay composites were carried out to investigate the load carrying capacity of the fire blocks under compression using compression testing machine. this is important to determine the compressive strength of fire blocks for its suitability as furnace lining. the materials behaviours under a load were determined. the maximum stress a material can withstand over a period under a load (constant or progressive) was determined to a break (rupture) or to a limit. these results are shown in table 3, 4 and 5. 29 saheed et al. / j. nig. soc. phys. sci. 4 (2022) 27–33 30 table 1. estimated grain sizes and mineral constituents of fine clay peak no. 2theta (rad) fwhm grain size (nm) constituents 1 4.43 2.187 3.844597803 smectile 2 8.46 5.038 1.672251355 illite/mica 3 12.24 7.12 1.186799367 kaolinite 4 13.56 5.097 1.660000572 albite 5 15.38 3.167 2.677013045 illite/mica 6 19.47 5.272 1.616958333 clay mineral 7 20.43 2.187 3.90360038 quartz 8 23.46 5.038 1.703266297 kaolinite 9 26.24 7.12 1.211663863 kaolinite 10 27.56 5.097 1.697242735 quartz 11 28.38 3.167 2.736431091 albite 12 30.47 5.272 1.651722605 illite/mica 13 31.32 2.348 3.716247528 albite 14 32.12 7.257 1.204777898 illite/mica 15 34.04 4.328 2.030195653 illite/mica 16 36.58 2.039 4.339821991 clay mineral 17 38.433 2.147 4.144207588 quartz 18 40.465 5.035 1.778423867 kaolinite 19 42.245 7.123 1.264497493 quartz plus kaolinite 20 46.567 5.027 1.819527192 quartz 21 48.382 3.164 2.91109195 quartz 22 50.473 5.278 1.75982879 quartz plus kaolinite 23 54.436 2.157 4.380159907 illite/mica 24 55.467 5.034 1.885640471 quartz 25 60.245 7.125 1.363317463 quartz 26 62.562 5.077 1.936374021 kaolinite figure 6. force (n) against deflection (mm) of 255 g fine and coarse fire block clay composites the 255 g block has the force break of 2632 n and deflection break at 4.343 mm. the time to failure is 26.133 seconds for the young modulus of 174.476 n/mm2 among other parameters (table 3). the 265 g block has the force break of 1439 n and deflection breaks at 4.671 mm. the time to failure is 28.1 seconds for the young modulus of 94 n/mm2 (table 4) while the 275 g block has the force break of 7652 n and deflection breaks at 3.734 mm. the time to failure is 22 seconds for the figure 7. force (n) against deflection (mm) of 265g fine and coarse fire block of clay composites maximum young modulus of 212 n/mm2 (table 5). generally, the 275 g block of fire clays composites requires the maximum break force and has the maximum young modulus relatives to blocks 255 g and 265 g of clays composites under study. the 275 g block of fire clay composites will be better for furnace lining application than the 255 g and 265 g blocks. figures 6, 7 and 8 shows the plots of force (n) against de30 saheed et al. / j. nig. soc. phys. sci. 4 (2022) 27–33 31 table 2. estimated grain sizes and mineral constituents of coarse clay peak no 2theta (rad) fwhm grain size (nm) constituents 1 12.501 0.025 338.0839019 interstratified illitesmectile 2 17.45 0.037 229.7356597 gypsum 3 19.86 0.128 66.63776651 kaolinite 4 20.941 0.136 62.82444633 jarosite 5 21.165 0.164 52.11724848 jarosite 6 21.464 0.0172 497.1759627 jarosite 7 23.622 0.141 60.87648083 kaolinite 8 24.901 0.137 62.8043888 kaolinite 9 24.02 0.12 71.58228586 quartz 10 26.242 0.113 76.34585664 kaolinite 11 28.501 0.026 333.40756 kaolinite 12 34.45 0.038 231.4836125 microcline 13 35.86 0.124 71.21558276 pyrite 14 36.941 0.133 66.60274151 kaolinite 15 37.165 0.162 54.71585907 pyrite 16 38.464 0.017 523.4384171 kaolinite plus interstratified illitesmectile 17 39.529 0.022 405.8083969 kaolinite 18 40.43 0.033 271.3138386 quartz 19 41.821 0.122 73.72312076 kaolinite 20 42.937 0.134 67.37494247 quartz plus kaolinite 21 45.178 0.161 56.52152987 quartz 22 48.426 0. 170 54.14907336 quartz 23 50.643 0.144 64.5478197 quartz plus kaolinite 24 51.936 0.13 71.88749106 quartz 25 55.04 0.134 70.70014627 kaolinite 26 56.228 0.12 79.38153044 quartz 27 60.52 0.022 442.1458637 interstratified illitesmectile 28 62.439 0.03 327.4856536 kaolinite table 3. compressibility analysis of 255g block of clay test no def. @ break (mm) def. @ l.o.p. (mm) def. @ peak (mm) def. @ yield (mm) force @ break (n) force @ l.o.p. (n) force @ peak (n) 1 4.343 2.087 4.186 2.303 2631.700 2911.200 9924.000 test no force @ yield (n) strain @ break (%) strain @ l.o.p. (%) strain @ peak (%) strain @ yield (%) stress @ break (n/mm2) stress @ l.o.p. (n/mm2) 1 3469.000 7.896 3.795 7.611 4.187 2.056 2.274 test no stress @ peak (n/mm2) stress @ yield (n/mm2) time to failure (secs) time to peak (secs) youngs modulus (n/mm2) tangential modulus @ 0.000 n/mm2 (n/mm2) secant modulus 0.000 to 0.000 n/mm2 (n/mm2) 1 7.753 2.710 26.133 25.187 174.476 4.727 flection (mm) for the 255 g, 265 g and 275 g fire blocks of clay composites respectively. figure 9 compares the behaviours of the three fire blocks together. the plot reveals the maximum load of the fire blocks at respective deflection (mm). our interest in these plots is to investigate and compares the maximum load fire block clays composites can withstand. the 275 g block has the maximum compressive strength and young modulus of 7652 n and 212 n/mm2 respectively making it better than the 255 g and 265 g blocks for furnace lining application. 31 saheed et al. / j. nig. soc. phys. sci. 4 (2022) 27–33 32 table 4. compressibility analysis of 265 g block of clay test no def. @ break (mm) def. @ l.o.p. (mm) def. @ peak (mm) def. @ yield (mm) force @ break (n) force @ l.o.p. (n) force @ peak (n) 1 4.671 2.164 3.628 2.273 1438.900 847.400 3851.000 test no force @ yield (n) strain @ break (%) strain @ l.o.p. (%) strain @ peak (%) strain @ yield (%) stress @ break (n/mm2) stress @ l.o.p. (n/mm2) 1 1038.600 8.493 3.935 6.596 4.133 1.022 0.602 test no stress @ peak (n/mm2) stress @ yield (n/mm2) time to failure (secs) time to peak (secs) youngs modulus (n/mm2) tangential modulus @ 0.000 n/mm2 (n/mm2) secant modulus 0.000 to 0.000 n/mm2 (n/mm2) 1 2.735 0.738 28.100 21.845 93.892 21.094 table 5. compressibility analysis of 275 g block of clay test no def. @ break (mm) def. @ l.o.p. (mm) def. @ peak (mm) def. @ yield (mm) force @ break (n) force @ l.o.p. (n) force @ peak (n) 1 3.734 2.349 3.132 3.132 7652.000 2964.300 9658.000 test no force @ yield (n) strain @ break (%) strain @ l.o.p. (%) strain @ peak (%) strain @ yield (%) stress @ break (n/mm2) stress @ l.o.p. (n/mm2) 1 9658.000 6.789 4.271 5.695 5.695 4.270 1.654 test no stress @ peak (n/mm2) stress @ yield (n/mm2) time to failure (secs) time to peak (secs) youngs modulus (n/mm2) tangential modulus @ 0.000 n/mm2 (n/mm2) secant modulus 0.000 to 0.000 n/mm2 (n/mm2) 1 5.390 5.390 22.443 18.842 212.102 6.752 figure 8. force (n) against deflection (mm) of: 275g fine and coarse fire block of clay composites figure 9. force (n) against deflection (mm) of different fabricated fine and coarse fire blocks of clay composites 32 saheed et al. / j. nig. soc. phys. sci. 4 (2022) 27–33 33 4. conclusion in this research work, okelele fine and coarse clays have been characterized to establish their potentials for furnace lining application. the maximum compressive strength and young modulus as demonstrated by 275 g block clay are 7652 n and 212 n/mm2 at firing temperature of 1200 oc. the results of compressive strength analysis, mineral phase’s identification and ability to withstand higher firing temperature of 1200 oc proved that, okelele fine and coarse fire block of clays meet the needed criteria for use as refractory raw materials. references [1] j. b. mokwa, s. a. lawal, m. s. abolarin & k. c. bala,“characterization and evaluation of selected kaolin clay deposits in nigeria for furnace lining application”, nigerian journal of technology (nijotech), 38 (2019) 936. 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[16] e. e. nnuka & agbo j. e, evaluation of the refractory characteristics of otukpo clay deposit, n.s.e, technical transaction 35 (2000) 32. 33 j. nig. soc. phys. sci. 5 (2023) 1392 journal of the nigerian society of physical sciences theoretical study on 10c elastic scattering cross sections using different cluster density distributions and different potentials sunday d. olorunfunmia,∗, a. bahinib, adenike s. olatinwoa adepartment of physics & engineering physics, obafemi awolowo university, ile-ife 220005, osun state, nigeria bithemba laboratory for accelerator based sciences, somerset west 7129, south africa abstract elastic scattering cross sections are a fundamental aspect of nuclear physics research, and studying the cross sections of various nuclei can provide important insights into the behavior of nuclei. in this study, the elastic scattering cross sections of 10c projectile by 27al, 58ni, and 208pb target nuclei are analyzed. the aim of this study is to investigate the cluster structure of 10c and the sensitivity of the elastic scattering cross sections to different potentials. to achieve this objective, the double folding optical model and a simple cluster approach are used to analyze the cross sections. the real part of the optical potential is obtained by folding two different effective interactions, michigan-3-yukawa (m3y) and jeukennelejeune-mahaux (jlm), with four different cluster density distributions of the 10c nucleus: 6be + α, 9b + p, 8be + p + p, and α + α + p + p. the imaginary part is taken to be a woods-saxon phenomenological form. the sensitivity of the elastic scattering cross sections to different potentials is assessed by comparing the results obtained using different potentials. the cluster structure of 10c is validated by comparing the theoretical results with experimental data. the results show that the cross sections are sensitive to the choice of potential used and that the cluster structure of 10c is validated. the theoretical results show reasonable agreement with the experimental data. doi:10.46481/jnsps.2023.1392 keywords: elastic scattering, density distribution, optical model, cluster model. article history : received: 11 february 2023 received in revised form: 25 april 2023 accepted for publication: 10 may 2023 published: 21 may 2023 c© 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: o. j. oluwadare 1. introduction there has been significant interest in studying reaction mechanisms involving weakly bound neutronand proton-rich nuclei, especially because of their astrophysical importance or applications [1, 2, 3, 4, 5, 6]. one of such weakly bound nuclei that is of interest is the unstable proton-rich 10c, which exhibits a three-cluster structure and can decay into 6be + α, 9b + p, and ∗corresponding author tel. no: +2349042713841 email address: sundayolorunfunmi@gmail.com (sunday d. olorunfunmi) 8be + p + p channels with binding energies of 3.821, 4.006, and 5.101 mev, respectively [7, 8]. curtis et. al., [9] studied the break up reaction of 10c and concluded that the proton-rich nucleus can also decay by α + α + p + p channel. the elastic scattering angular distributions of 10c + 27al at incident energy of 29.1 mev have been measured and theoretically analyzed using optical model potential constructed from the combination of real são paulo potential (spp), imaginary woods-saxon potential and complex polarization potentials [10]. the study concluded that the inclusion of the volume and surface complex polarized potentials is needed in order to 1 olorunfunmi et al. / j. nig. soc. phys. sci. 5 (2023) 1392 2 successfully describe the data. these polarized potentials account for fusion and direct coupling. these distributions were recently analyzed by aygun [11] using double folding optical potential with the real potential constructed by folding m3y effective interaction with relativistic mean-field density distribution of 10c and woods-saxon (ws) imaginary potential. a new set of global potential was obtained for the carbon isotope. yang et. al., [7] measured the quasi-elastic scattering angular distributions of 10c + 208pb at 226 and 256 mev. very recently, the elastic scattering cross sections of the same reaction at 66 mev was measured by linares et. al., [8]. the data were analyzed and compared with the results of optical model calculations performed using the spp nucleus-nucleus interaction. again, the same data at 256 mev was further analyzed in ref. [11] using optical-model based double folding potentials. recently, guimaraes et al., [12] measured the elastic scattering of 10c on 58ni target nuclei at incident energy of 35.3 mev with the purpose of studying the coupling effect in the reaction. they analyzed the measured data using microscopic approach within the framework of coupled-channels (cc) and coupled-reaction channels (crc) models. results of the two models could not describe the data satisfactorily. in order to improve the description of the data they performed continuumdiscretized coupled-channels (cdcc) calculations. in cdcc calculations the 10c nucleus was assumed to decay by two channels: 9b + p and 6be + α. in the end they were able to achieve a fair agreement between the theoretical calculations and experimental data, but the need for a more realistic theoretical calculations was emphasized. consequent upon this, aygun [13] carried out a comprehensive theoretical analysis of this reaction using different potentials and simple cluster model. the study considered 6be + α, 9b + p, and 8be + p + p cluster configurations for 10c, and found that the 6be + α configuration describe the data better then 9b + p, and 8be + p + p cases. it was recommended that the cluster structure of 10c be evaluated in the analysis of elastic scattering reactions of 10c with other target nuclei. in this present study, the elastic scattering cross section of 10c projectile nucleus from 27al, 58ni and 208pb target nuclei are calculated using the complex optical model potential with folded real part and phenomenological woods-saxon imaginary part. the real part is constructed by folding two different effective nucleon-nucleon (nn) interactions m3y and jlm with four different cluster density distributions of 10c nucleus. here, we aim to study the structure effect in 10c + nucleus reactions via different choice of simple cluster density distributions of 10c projectile nucleus and to investigate the sensitivity of 10c + nucleus elastic scattering cross sections to different effective nn interactions. 2. theoretical formalism 2.1. the optical model potential the theoretical calculations were performed using the optical model of the form: table 1: the parameters of two-parameter fermi (2pf) density distributions for the 27al, 58ni, and 208pb target nuclei. nucleus c (fm) z (fm) ρ0 (fm−3) reference 27al 2.840 0.569 0.2015 [14] 58ni 4.094 0.540 0.1720 [15] 208pb 6.620 0.551 0.1600 [14] u(r) = v coul(r) − v (r) − iw(r) , (1) where v coul(r) is the coulomb potential, v (r) is the real potential and w(r) is the imaginary potential. the coulomb potential is defined as: v coul(r) =  1 4π�o zp zt e2 r if r ≥ rcoul 1 4π�o zp zt e2 2rcoul ( 3 − r 2 r2 coul ) if r ≤ rcoul (2) with rcoul = 1.25(a (1/3) p + a (1/3) t ) , (3) where zp(t) and ap(t) are the proton and mass number of the projectile (target) nuclei, respectively. in refs. [16, 17], the coulomb potential was used in a theoretical investigation of the half-life of certain nuclei. the real potential v (r) is obtained by using the double folding potential given as: v (−→r ) = nr ∫ d−→r1 ∫ d−→r2ρp( −→r )ρt( −→r )υnn( −→r12) , (4) where −→r12 = [ −→r − (−→r1 − −→r2 ) ] , nr is the normalization constant, υnn is the effective nn interaction, and ρp (ρt) is the density distribution of the projectile (target). in this work, four different cluster density distributions of the projectile nucleus are considered, and each is introduced in the following section. the density distributions of target nuclei are obtained by using the two-parameter fermi (2pf) density ρ(r) = ρ0 1 + exp( r−cz ) , (5) where ρ0 is the maximum density (central density) of the nucleus and the fermi-distribution parameters c and z describe the half-density radius and the diffuseness, respectively. their numerical values are listed in table 1. two forms of effective nn interaction υnn are considered, namely, m3y and jlm. these two interactions are presented in the next section. the imaginary potential w(r) is taken in the woods-saxon form: w(r) = w0 1 + exp( r−ri (a 1/3 p +a 1/3 t ) ai ) , (6) where w0, ri and ai represent the potential depth, the reduced radius, and the diffuseness parameter, respectively. 2 olorunfunmi et al. / j. nig. soc. phys. sci. 5 (2023) 1392 3 figure 1: real folded potential for 10c + 27al at 29.1 mev, 10c + 58ni at 35.3 mev and 10c + 208pb at 66 mev, using m3y and jlm effective interactions with 6be + α, 9b + p, 8be + p + p, and α + α + p + p cluster density distribution of 10c. 3 olorunfunmi et al. / j. nig. soc. phys. sci. 5 (2023) 1392 4 figure 2: elastic scattering angular distributions for 10c + 27al at incident energy of 29.1 mev obtained using m3y and jlm effective interactions with 6be + α, 9b + p, 8be + p + p, and α + α + p + p cluster density distribution of the 10c. the experimental data are taken from ref. [10]. 2.2. effective nucleon-nucleon interaction the two different forms of effective nn interactions considered in the present study are fully described elsewhere [18, 19, 20]. as such, only salient details are provided here. 2.2.1. m3y interaction the density-independent m3y interaction is derived by bertsch et al., [19] and parameterized according to satchler and love [20] as follows: υm3ynn (r) = 7999 exp(−4r) 4r − 2134 exp(−2.5r) 2.5r − 276 [ 1 − 0.005 elab ap ] δ(r) , (7) where elab and ap are the laboratory energy and mass number of the projectile, respectively. the first and the second terms represent the direct part while the third term represents the exchange part of the interaction potential. 2.2.2. jlm interaction the jlm potential derived by jeukenne, lejeune and mahaux [18] was obtained in a brueckner-hartree-fock (bhf) approximation from the reid soft-core nn interaction. the figure 3: same as figure 2 but for 10c + 58ni at incident energy of 35.3 mev. the experimental data are taken from ref. [12]. isoscalar component of the complex jlm interaction has the form [18]: υjlmnn (s,ρ, e) = g(s)v0(ρ, e) + ig(s)w0(ρ, e) , (8) where g(s), v0 and w0 are the radial dependence factor, the real component and the imaginary component of the effective interaction, respectively. in this study, only the real part of the jlm effective interaction is considered and discussed, the imaginary part is replaced with the woods-saxon potential (eq. 6). the density and energy dependence of the real part of jlm interaction is parametrized as follows [18]: v0 = 3∑ i, j=1 ai jρ i−1 e j−1 . (9) the values of the coefficient ai j are taken from ref. [18]. in this study, the local-density approximation (lda) is considered using the arithmetic average approach as prescribed in ref. [21]: ρ = (ρp(r1)ρt (r2)) 1/2 , (10) where the local density is evaluated at each position of the interacting nucleons. the radial dependence factor is taken to be a single-gaussian shape [22] g(s) = (t √ π)−3 exp(−s2/t2) (11) with t = 1.2 fm. 4 olorunfunmi et al. / j. nig. soc. phys. sci. 5 (2023) 1392 5 figure 4: same as figure 2 but for 10c + 208pb at incident energy of 66 mev. the experimental data are taken from ref. [8]. 2.3. cluster density distribution of 10c nucleus the present work considers a simple cluster model, which simply involves adding together the densities of the constituent cluster nuclei. as an example, the density for 10c, is expressed as, ρ10c = ρ6 be + ρα, implying that the clusters are overlapping at the same point inside the nucleus. such an oversimplified approach may still have some merit since in a more realistic calculation, one would have to consider an overlap of the cluster nuclei using the appropriate jacobi coordinates, which, for the case of three or four clusters can turn out to be very complicated for the present investigation. the cluster model density distribution has been used successfully to analyze elastic scattering cross sections of unstable projectile nuclei (see, e.g., refs. [23, 24, 25, 26, 27, 28]. four different forms of cluster density distributions of the projectile nucleus are considered in this study and each is presented in the following section. 2.3.1. 6 be + α system firstly, the 10c nucleus is taken to be a cluster of 6be and α nuclei. hence, the density distribution of 10c takes the form: ρ10c = ρ6 be + ρα . (12) the são paulo (sp) density distribution [29], is used for the density of 6be and parametrized as follows: ρi6 be(r) = ρ0i ( 1 + exp( r − ri ai ) )−1 , (i = n, p) , (13) figure 5: same as figure 2 but for 10c + 208pb at elab = 226 mev . the experimental data are taken from ref. [7]. where rn = 1.49n 1/3 − 0.79, rp = 1.81z 1/3 − 1.12 , (14) and an = 0.47 + 0.00046n, ap = 0.47 − 0.00083z . (15) here, rn(rp) and an(ap) represent the half-density radius and surface thickness parameter of neutron (proton), while z and n are proton and neutron numbers, respectively. the α density is taken to be [20] ρα = 0.4229 exp(−0.7024r 2) . (16) 2.3.2. 9 b + p system another cluster density of 10c considered in this study is given by ρ10c = ρ9 b + ρp , (17) where the density distribution of 9b is given in eq. 13, and that of proton is taken to be [30, 31] ρp = (βπ) −3 exp(−r2/β2) , (18) where β is adjusted to reproduce the rms radius value of 10c. 5 olorunfunmi et al. / j. nig. soc. phys. sci. 5 (2023) 1392 6 figure 6: same as figure 2 but for 10c + 208pb at incident energy of 256 mev. the experimental data are taken from ref. [7]. 2.3.3. 8 be + p + p system the density distribution of 10c can also be evaluated as the sum of 8be, p and p densities ρ10c = ρ8 be + ρp + ρp , (19) where the density distributions of 8be and p are given in eqs. 13 and 18, respectively. 2.3.4. α + α + p + p system the last density distribution form of 10c considered here is obtained from the addition of density distributions of α, α, p and p ρ10c = ρα + ρα + ρp + ρp , (20) where the density distributions of α and p are given eqs. 16 and 18, respectively. the approach of obtaining nuclear density as a sum of the densities of the clusters has been used in refs. [23, 24]. 3. method of calculation the first step in calculating elastic scattering cross section is to obtain the complex total optical potential. in the present study, the real part of the total optical potentials is calculated via the double folding approach as expressed in eq. 4 using figure 7: normalization constant, nr versus incident energy, elab obtained for 10c + 27al at 29.1 mev, 10c + 58ni at 35.3 mev and 10c + 208pb at 66, 226 and 256 mev, using m3y and jlm effective interactions with 6be + α, 9b + p, 8be + p + p, and α + α + p + p cluster density distribution of 10c. the dashed curves are to guide the eye. m3y and jlm effective n n interactions with 6be + α, 9b + p, 8be + p + p, and α + α + p + p cluster density distribution of the 10c. the folded potentials are obtained using the computer code dfpot [32]. the elastic scattering cross sections of 10c are evaluated with the computer code ptolemy [33, 34]. the code takes as input the obtained folded potential to represent the real part of the optical potential while the imaginary part of the potential is taken in the usual phenomenological form as expressed in eq. 6. these potential are used to analyze experimental data of 10c + 27al (at 29.1 mev) [10], 10c + 58ni (at 35.3 mev) [12] and 10c + 208pb (at 66, 226 and 256 mev) [7, 8]. in order to reduce the number of fitting parameters, the imaginary reduced radius, ri and diffuseness parameter ai are fixed at 1.3 and 0.4 fm, respectively. finally, in order to assess the quality of agreement between the calculated results and experimental data, a search on nr and wi was carried out using the usual reduced chi-square approach [20] χ2 = n−1 n∑ k=1 [ σcal(θk) −σex(θk) ∆σex(θk) ]2 , (21) where σcal(θk) and σex(θk) are the calculated and experimental cross sections, respectively, ∆σex(θk) is the experimental error, 6 olorunfunmi et al. / j. nig. soc. phys. sci. 5 (2023) 1392 7 table 2: real normalization parameter, depth of the imaginary potential (wi), real and imaginary volume integrals (jr and ji), total reaction cross sections (σr) and χ2/n values for the elastic 10c scattering on 27al, 58ni, and 208pb target nuclei. the imaginary radius ri and diffuseness ai are fixed at 1.3 and 0.4 fm, respectively. system elab potential cluster nr wi jr ji σr χ2/n (mev) (mev) (mevfm3) (mevfm3) (mb) 10c+27al 29.1 m3y 6be+α 1.0 10.5 416.610 50.729 779.86 0.42 9b+p 1.0 33.5 416.365 161.856 904.88 0.94 8be+p+p 1.0 30.5 416.783 147.355 894.90 0.91 α+α+p+p 1.3 10.5 416.610 50.729 775.29 0.37 jlm 6be+α 1.0 55.5 345.483 268.139 967.45 0.56 9b+p 0.6 55.5 536.506 268.139 910.88 0.61 8be+p+p 0.6 50.5 524.859 243.982 951.23 0.68 α+α+p+p 1.1 40.5 182.643 195.669 910.88 0.41 10c+58ni 35.3 m3y 6be+α 0.7 52.5 415.575 186.713 469.23 2.33 9b+p 0.3 52.5 415.017 186.713 469.61 2.23 8be+p+p 0.4 50.5 415.146 179.600 476.74 2.75 α+α+p+p 0.5 50.5 415.559 179.600 465.91 2.35 jlm 6be+α 1.0 40.0 325.934 142.257 535.16 2.39 9b+p 1.0 40.0 500.371 142.257 535.15 2.42 8be+p+p 1.0 40.0 489.754 142.257 535.16 2.44 α+α+p+p 1.0 40.0 176.553 142.257 535.16 2.42 10c+208pb 66.0 m3y 6be+α 1.2 300.5 406.2207 595.5052 662.66 16.87 9b+p 1.0 250.5 402.3044 496.4195 672.49 20.50 8be+p+p 1.0 250.5 402.7722 496.4195 664.92 19.79 α+α+p+p 1.2 300.5 407.7502 595.5052 648.46 15.80 jlm 6be+α 1.0 450.5 353.997 892.76 708.24 15.56 9b+p 1.0 380.5 428.733 754.042 701.56 16.78 8be+p+p 1.0 380.5 424.329 754.042 699.67 16.88 α+α+p+p 1.0 450.5 287.292 892.763 699.28 15.61 226.0 m3y 6be+α 1.0 30.0 384.199 59.451 3054.40 3.60 9b+p 1.0 30.5 380.620 60.422 3087.70 1.94 8be+p+p 1.0 25.5 380.867 50.534 3046.90 2.39 α+α+p+p 1.0 40.5 385.806 80.26 3108.2 5.48 jlm 6be+α 1.0 50.5 666.353 100.0765 3182.6 1.65 9b+p 0.7 40.5 211.937 80.259 3138.7 1.66 8be+p+p 0.5 40.5 445.607 80.259 3138.1 1.98 α+α+p+p 1.1 50.5 1040.987 100.0766 3168.8 2.11 256.0 m3y 6be+α 1.0 30.0 384.203 59.451 3164.4 1.46 9b+p 0.6 30.0 381.022 59.451 3163.7 1.33 8be+p+p 0.6 32.5 381.016 64.406 3184.6 1.36 α+α+p+p 1.0 30.5 386.112 60.442 3154.1 3.47 jlm 6be+α 0.6 40.5 1103.551 80.259 3240.2 1.06 9b+p 0.4 40.5 401.396 80.259 3245.1 1.25 8be+p+p 0.5 40.5 240.516 80.259 3238.0 1.22 α+α+p+p 0.8 40.5 1681.423 80.259 3234.5 1.65 and n is the number of data points. an average value of 10% error is used as uncertainty on all the experimental data used in this study. the real (jr) and imaginary(ji) volume integrals are computed using the expressions jr(e) = 4π ap at ∫ v (r, e)r2dr , (22) 7 olorunfunmi et al. / j. nig. soc. phys. sci. 5 (2023) 1392 8 table 3: reduced reaction cross sections, σm3yre and σ jlm re , obtained in this work using m3y and jlm potentials, respectively, compared with σ spp re obtained from spp in refs.[7, 8, 10]. system elab ere cluster σm3yre σ jlm re σ spp re (mev) (mev) (mb) (mb) (mb) 10c+27al 29.1 1.402 6be+α 29.353 36.414 36.736 [10] 9b+p 34.058 34.285 8be+p+p 33.683 35.803 α+α+p+p 29.181 34.285 10c+58ni 35.3 1.079 6be+α 12.925 14.741 9b+p 12.935 14.741 8be+p+p 13.132 14.741 α+α+p+p 12.833 14.741 10c+208pb 66.0 1.034 6be+α 10.151 10.850 14.474 [8] 9b+p 10.302 10.747 8be+p+p 10.186 10.718 α+α+p+p 9.934 10.712 226.0 3.541 6be+α 46.791 48.755 48.685 [7] 9b+p 47.301 48.083 8be+p+p 46.676 48.073 α+α+p+p 47.615 48.544 256.0 4.011 6be+α 48.476 49.638 50.079 [7] 9b+p 48.466 49.713 8be+p+p 48.786 49.604 α+α+p+p 48.319 49.550 and ji(e) = 4π ap at ∫ w(r, e)r2dr , (23) respectively. 4. result and discussion the real part of the optical potentials is calculated for the reactions 10c + 27al at 29.1 mev, 10c + 58ni at 35.3 mev, and 10c + 208pb at 66, 226 and 256 mev using the double folding model (eq. 4) with m3y and jlm effective n n interactions, and four 10c cluster structure densities viz. 6be + α, 9b + p, 8be + p + p, and α + α + p + p. typical calculated folding potentials (with nr = 1) for the reactions 10c + 27al at 29.1 mev, 10c + 58ni at 35.3 mev and 10c + 208pb at 66 mev are shown in figure 1. it can be seen that the m3y potentials obtained differ primarily in depth and shape from the jlm potentials. furthermore, for the m3y potential, the folding potentials computed using the α + α + p + p configuration are observed to be deeper than those obtained from the three other 10c cluster density distributions. on the other hand, the use of the 9b + p cluster density results in potentials with a shallower depth compared to the other cluster configurations. in contrast, for the jlm potential, the 9b + p cluster configuration produces a deeper potential compared to the other cluster configurations. the elastic scattering cross sections of the reactions under investigation are calculated using folded real potential and woods-saxon imaginary potential with the four different forms of cluster densities for 10c. the results of the calculations are compared with appropriate experimental data and are shown in figures 2 6. the parameters that give good agreement with experimental data, the real volume integral, jr, imaginary volume integral, ji and the reaction cross sections, σr for all the reactions considered are presented in table 2. it can be seen from the figures, and values of nr and χ2/n in table 2 that the cross section obtained using m3y and jlm interactions give almost the same quality of fit to experimental data. however, the values of σr obtained for the jlm interaction are higher than that of m3y. the elastic scattering cross sections of 10c + 27al at incident energy 29.1 mev are investigated for 6be + α, 9b + p, 8be + p + p, and α + α + p + p cluster densities of 10ca, and the results are shown in figure 2. the results obtained using m3y n n interactions are shown in the top panel while the bottom panel displays the results obtained with jlm interaction. an excellent agreement between calculated results and data is observed. also, the results show similar shape for the different cluster configurations. figure 3 shows the elastic scattering cross sections of 10c + 58ni at 35.3 mev obtained using the aforementioned set of cluster densities for 10c and the two effective interactions m3y and jlm. again, the theoretical results agree reasonably well 8 olorunfunmi et al. / j. nig. soc. phys. sci. 5 (2023) 1392 9 figure 8: reduced reaction cross sections, σre, from the present work and other results from refs. [7, 8, 10] with respect to the reduced incident energy, ere. the curves are to guide the eye. with the experimental data. finally, the theoretical results of elastic scattering cross sections of 10c from 208pb at 66, 226 and 256 mev using df model with the four different cluster densities the two effective interactions are shown and compared with experimental data in figures 4 6. overall, good agreement is obtained between experiment and theory, except at 66 mev where theoretical results underestimates experimental ones in the angular region 115◦ 140◦. similar discrepancy was reported for the same data in ref. [8]. the increase of the elastic cross sections at these backward angles was attributed to coupling to excited states in the projectile nucleus and target nucleus, and this was not captured in the calculations presented in this study. the renormalization factor nr is usually applied to folding potential in order to assess the performance of the df model in describing a nuclear reaction [20]. the values of nr used for the different cluster density configurations and the different interactions considered in this study are presented in table 2 and figure 7. in general, it can be seen that the m3y and jlm interactions show the need for almost the same renormalization factor nr. also, the values of nr are mostly or close to unity, except for the cases of 9b + p, and 8be + p + p in 10c + 58ni (with m3y) and 10c + 208pb at 256 mev (with jlm) where the potentials are heavily reduced (nr = 0.3 to 0.5). in figure 7, figure 9: modulus of the scattering matrix |s l| versus the versus the orbital angular momentum l obtained for 10c + 27al at 29.1 mev, 10c + 58ni at 35.3 mev and 10c + 208pb at 226 mev, using m3y and jlm effective interactions with 6be + α, 9b + p, 8be + p + p, and α + α + p + p cluster density distribution of the 10c. the nr values obtained for both the m3y and jlm interactions are plotted versus the incident energy elab. one observes from this plot that for both interactions the 6be + α cluster density generally give nr value closer to unity than other cluster configurations. also, from this figure, one sees that in the case of 10c + 58ni reaction using m3y potential, the potential is strongly reduced almost for all the cluster configurations. this might be due to the presence of other reaction mechanisms not considered in our calculations, such as the inclusion of the 2+ excited state of the 10c nucleus, as mentioned in ref. [12]. the real volume integrals jr values obtained for both the m3y and jlm interactions as well as the corresponding reaction cross sections σr for each cluster configuration are presented in table 2. one can see that the difference in the value of the reaction cross section for the various cluster configurations considered under the same potential is not more than 3%, except for the reaction 10c + 27al under the m3y potential where the difference is as high as 13%. in general, the cluster configuration 9b + p gives slightly higher value of σr compared to the other three cluster configurations. furthermore, in other to compare the reaction cross sections of the different potentials and densities with each other and with data from literature, a reduction method is used. the reduced reaction cross section 9 olorunfunmi et al. / j. nig. soc. phys. sci. 5 (2023) 1392 10 and reduced energy are given as follows [35] σre = σr (a1/3p + a 1/3 t ) 2 , (24) and ere = ec.m. × a1/3p + a 1/3 t zpzt , (25) where ec.m. is the incident energy in the center of mass frame, and zp(t) is the proton number of the projectile (target) nucleus. the reduced energy and reaction cross section values for the different potentials are listed in table 3 and shown in figure 8. the black filled circle represents the results obtained by yang et al., [7], linares et al., [8] and aguileral et al., [10], using spp potential. it can be seen from the lower panel of figure 8 that the results obtained with jlm potential agree reasonably well with that of spp potential for all the reactions, with the exception of 10c+208pb at 66 mev where the spp gives higher reaction cross section ( see table 3). in the upper panel of the figure, it can be seen that the σre obtained for m3y potentials are consistently lower than that reported for spp. in general, as shown in table 3 and figure 8, one observes that for the m3y effective interaction the cluster configuration 9b + p gives slightly higher value of σre compared to the other three cluster configurations. for the case of jlm interaction, the cluster configuration 6be + α gives slightly higher value of σre compared to the other three cluster configurations. the last parameter presented in table 2 is the reduced chisquare value χ2/n. it can be seen that, in general, the χ2/n values are small, which is an indication of good agreement between the calculated and experimental cross sections. figure 9 shows the plot of the magnitudes of partial-wave scattering (s-matrix) elements |s l| versus the orbital angular momentum l, calculated for reactions 10c + 27al at 29.1 mev, 10c + 58ni at 35.3 mev and 10c + 208pb at 226 mev, using the m3y and jlm potentials. in this figure, we see that the value of |s l| ≈ 0 at small l and increases rapidly as l becomes larger, and finally reaches unity. the value of |s l| indicates the level of absorption. for example, it is known that |s l| = 1 for elastic scattering means no absorption. furthermore, it has been suggested that total absorption happens when the transmission coefficient (1-|s l|2) equals zero [36]. it can be seen from figure 9 that the values of l obtained for both m3y and jlm potentials, increase with increasing target mass number. furthermore, the range of the values of l required for |s l| to rise from 0 to 1 is slightly higher for jlm than for m3y potential. lastly, it is observed that |s l| does not show significant sensitivity to the project cluster densities under the same potentials. 5. conclusions a systematic analysis of elastic scattering cross sections of 10c + 27al, 10c + 58ni, and 10c + 208pb reactions has been performed within the framework of the double-folding optical model. this is with the view to investigating the nuclear structure of the 10c nuclei via the simple cluster model as well as study the sensitivity of 10c elastic scattering cross sections to different effective n n interactions. the real part of the optical potential is constructed by folding two different effective nn interactions m3y and jlm, with the density 10c. a cluster model density distribution is assumed for 10c and four different cluster configurations are considered, namely, 6be + α, 9b + p, 8be + p + p, and α + α + p + p. a phenomenological ws form is used for imaginary part. for the reactions considered in this study, the values of the reduced radius ri, and the diffuseness parameter ai of the ws potential are fixed at 1.3 and 0.4 fm, respectively, while the depth w0 is adjusted to fit the data. a comparative study of the four cluster configurations for 10c shows that the results obtained with 6be + α, 9b + p, 8be + p + p and α + α + p + p cluster configurations describe the experimental data quantitatively well. however, in terms of nr, σr and χ2/n, the 6be + α and 9b + p cluster configurations yield better description of the experimental data than the other cluster configurations. also, it is clear that any cluster density distribution for 10c can be compensated by the parameters w0 and nr . this further confirms the cluster structure of 10c nucleus. in addition, a study of the effect of effective n n interactions on the elastic scattering cross sections of the reactions considered in this study reveals that the jlm interactions is as good as the popular m3y interactions in terms of their agreements with experimental data. however, the jlm potential gives higher σr value than the m3y potentials. the theoretical calculations from both m3y and jlm effective interactions with cluster density distributions of 10c (6be + α, 9b + p, 8be + p + p, and α + α + p + p) provide good description of the experimental data for all the reactions considered in this work. furthermore, this study highlights the importance of considering the overlap of cluster nuclei using appropriate jacobi coordinates in a more realistic calculation. future research can focus on developing more sophisticated models to account for the complex structure and dynamics of three or 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long-running political and civil disputes have lowered the nation’s ability to do so. dengue has spread to seven states nationwide since the outbreak began on august 8th. a poor prognosis could result from the increased probability of coinfection with malaria and/or chikungunya, which complicates case management [1-8]. fractional calculus and nonlinear equations have found applications in a vast variety of seemingly unrelated sectors of science and engineering over the last decade. epidemics, acoustics, biology, electromagnetics, engineering, dengue is a significant public health concern in tropical and subtropical regions. 1 guma et al. / j. nig. soc. phys. sci. 5 (2023) 1464 2 it is transmitted by mosquitoes of the species aedes aegypti and aedes albopictus. four distinct serotypes can induce dengue illness. when a person infected with one serotype recovers, he or she develops complete immunity to that serotype but only partial and temporary immunity to the other [9-11]. in this paper, the fractional-order dengue epidemic model is investigated. examining the stability of equilibrium locations given are the numerical solutions for this model. we like to contend that fractional-order equations are more appropriate than integer-order equations for modeling memory-sensitive biological, economic, and social systems (generally complex adaptive systems). the adams-bashforth algorithm was used to solve and simulate the differential equation system. the format is as follows: the fractional model is introduced in section 1. section 2 provides fundamental definitions; section 3 is devoted to the formulation model. section 4 focuses on the caputo derivative of the model while sections 5 and 6 include stability analysis and numerical analysis. section 7 is the conclusion. 2. preliminaries definition 2.1. the riemann-liouville fractional integral operator of order α > 0 for a function y(τ) is given by [12] : dαy(t) := 1 γ(n −α) ∫ t 0 (t −τ)n−α−1yn(τ)dτ. = in−αyn(t), t > 0. (1) definition 2.2. for y�h1(0, t), t > 0, t > 0,α�(0, 1] then the cf fractional operator [12] is given by. cf 0 d α t y(t) := b(α) 1 −α d dt ∫ t 0 y(τ) exp ( −α t −τ 1 −α ) dτ.0 < α < 1 (2) in this expression b(α) satisfies the condition b(0) = b(1) = 1. definition 2.3. the fractional integral of order α of a function f is defined as [13] iγt y(t) := 2(1 −γ) (2 −γ)m(γ) y(t) + 2(1 −γ) (2 −γ)m(γ) ∫ t 0 y(τ)dτ.t ≥ 0, 0 < α < 1 (3) definition 2.4. caputo derivative of order 0 ≤ n − 1 < α < n with the lower limit zero for a function y(τ) is given by [14] : iαy(t) := 1 γ(α) ∫ t 0 (t −τ)α−1y(τ)dτ.t > 0 (4) definition 2.5. the mittag-leffler function is a generalization of the exponential function. this function can be expressed as follows: eα(t) = ∞∑ k=0 tk γ(αk + 1) (5) definition 2.6. for y�h1(0, t), t > 0, t > 0,α�(0, 1] then the ab fractional operator [12] y(t) in the riemann-liouville is given by: ab 0 d α t y(t) := b(α) 1 −α d dt ∫ t 0 y(τ)eα ( α 1 −α (t −τ)α ) dτ.0 < α < 1. (6) in this expression b(α) satisfies the condition b(0) = b(1) = 1. definition 2.7. for y�h1(0, t), t > 0, t > 0 then the ab fractional operator [12] y(t) in the caputo sense is given by: ab 0 d α t y(t) := b(α) 1 −α ∫ t 0 dy(τ) dτ eα ( α 1 −α (t −τ)α ) dτ.0 < α < 1 (7) in this expression b(α) satisfies the condition b(0) = b(1) = 1. definition 2.8. let 0 < α < 1 and the fractional cf derivative is expressed as: cf 0 d α t y(t) = h(t). (8) 3. the model formulation in this part, we get the mathematical formulation of the dengue fever infectivity model, which is based on a number of propositions, including the following: the total number of humans (nh) and mosquitoes (nm) is assumed to be constant, the birth and mortality rates are assumed to be equal, the births in mosquito and human populations in each class are assumed to be the same, each individual in the population is likely to have the same number of mosquito bites, and the infected mosquito is likely to bite each component of the data provided. the variables used in the dengue fever illness model are listed in table 1. the preceding model can be interpreted mathematically as a host-vector interaction model, which is the following fractional differential model:  ds h dt = µh nh − ( bβmh ih nh + µh ) s h, dih dt = bβmh s h ih nh − (γh + µh) ih, drh dt = γh ih −µhrh, ds m dt = µm nm − ( bβmh ih nh + µm ) s m, dim dt = bβmh ih nh s m −µm im (9) we extend the model (9) employing the newly proposed caputo-fabrizio-caputo; fractional derivatives with variable order α(t). 4. the caputoderivative to the fractional model introducing the notion of fractional derivative in the sense of riemann-liouville to reformulate the dynamics of the classical model (9) in terms of fractional derivatives, we apply the caputo derivative to the dengue fever model (9). the fractional 2 guma et al. / j. nig. soc. phys. sci. 5 (2023) 1464 3 table 1. explanation of the components of the infectivity model: variable description nh(t) total number of humans (constant ≈ 44 million) s h(t) susceptible humans ih(t) infected humans rh(t) recovery humans s m(t) susceptible female mosquitoes im(t) infected female mosquitoes parameter description µ(h) human mortality rate for every person µ(m) corresponding value for the mosquitoes γ(h) recovery rate of the humans b the biting rate β(m) the likelihood that human to mosquito transfer may occur β(h) the likelihood that mosquito to human transfer may occur dengue fever is obtained by replacing the classical derivative by the operator c0 d α(t) t : c 0 d λ 0 s h = θ1 − (θ2 + θ3) s h, c 0 d λ 0 ih = θ2s h − (θ4 + θ3) ih, 0 d λ 0 rh = θ4 ih − θ3rh, (10) c 0 d λ 0 s m = θ6 − (θ2 + θ5) s m, c 0 d λ 0 im = θ2s m − θ5 im, where θ1 = µh nh,θ2 = bβmh ih nh = bβhm ih nh , θ3 = µh,θ4 = γh,θ5 = µm,θ6 = µm nm, (11) with the initial conditions: s h(0) = c1, ih(0) = c2, rh(0) = c3, s m(0) = c4, im(0) = c5. (12) 5. stability analysis here, we discuss this epidemiological model stability. the equilibrium points for system (10) and the jacobian matrix are given by: j =  −bβmh ih nh −µh −bβmh s h nh 0 0 0 bβmh im nh − (γh + µh) 0 0 bβmh sh nh 0 γh −µh 0 0 0 −bβmh s m nh 0 −µm − bβhm ih nh 0 0 bβmh s m nh 0 bβhm ih nh −µm  (13) the disease-free equilibrium point is given as ( 44909351, 0, 0, 168000, 0 ), and the endemic equilibrium table 2. the values of the initial value of model and the parameters are given by real data: variable description values nh(t) total number of humans constant ≈ 44 million s h(0) susceptible humans 3326 ih(0) infected humans 482 rh(0) recovery humans 0 s m(0) susceptible female mosquitoes 117600 im(t) infected female mosquitoes parameter range of values references nh 44909351 constant nm 168000 estimated µh 0.0000031 estimated µm 0.1 [16] γh 2-10 days – [1/3] [16] b 0.7 [16] βmh 0.36 [16] βhm 0.36 [16] points (−3533,−8499,−2833,−132, 30017). the jacobian matrix at disease free equilibrium point is given as follows: j =  −0.0000038 0.000000026 0 0 0 0.000000045 −0.11 0 0 0.000000026 0 0.01 −0.1 0 0 0 −0.0000009 0 −0.100000026 0 0 0.0000009 0 0.000000038 −0.1  (14) the eigenvalues corresponding to matrix j are: σ1 = −0.0000003,σ2 = −0.0000003,σ3 = 0.000003,σ4 = −0.01,σ5 = −0.0000003. (15) the system is stable because all the eigenvalues are negative. the system is unstable at the endemic equilibrium point because some eigenvalues are positive. 5.1. the basic reproduction number: the expected value of the secondary infections rate per time unit is denoted by r0, and the basic reproduction number is a baseline metric in epidemiology. based on the fractional model of equation (10), we have two infected classes ih(t), im(t). cf 0 d λ 0 ih = θ2s h − (θ4 + θ3) ih (16) cf 0 d λ 0 im = θ2s m − θ5 im. (17) we assume that the infection risk rate is a constant β(t) taking the maximum value β(0) = β0 and the minimum value β (t∗) = β∗. let x = (ih, im).and rewrite the system of equation (7) for the susceptible and infected classes in the general from d x dt = f (x) − v(x), (18) 3 guma et al. / j. nig. soc. phys. sci. 5 (2023) 1464 4 where f (x) = [ 0.252 0.036 0.252 1.262 ] , v(x) = [ 0.00000026 0 0.000009 0 ] . (19) now the jacobian of f (x) and v(x) of the disease-free equilibrium point is f = [ 0.000000026 0 0.0000009 0 ] (20) v = [ 160.66 0 0 0.1 ] . (21) therefore v−1 = [ 0.006 0 0 0.1 ] . (22) by using eq. (11), we have r0 = ρ ( fv−1 ) = 10, (23) 6. numerical analysis: we do a numerical simulation of the fractional model of dengue fever outbreaks in sudan, epidemiological numerical analysis, and stability. by using real data in sudan and evaluating various possibilities by increasing and/or decreasing the model parameters’ values, we obtained a numerical simulation of the fractional dengue fever model. the euler approach is employed to generate the numerical results for the model (10). we used the values of numerous parameters from different credible sources to compute numerical results. 6.1. adams-bashforth applied on fractional order system this section presents an explanation of the numerical procedure that will be used by us to get the phase pictures of the fractional-order system (10). the adams-bashforth approach described by garrappa in his review paper [17] is the one that we use. the solution to the fractional differential system (10) can be described as follows: φ (t, x1) = θ1 − (θ2 + θ3) s h, ϕ (t, x2) = θ2s h − (θ4 + θ3) ih, θ (t, x3) = θ4 ih − θ3rh, (24) β (t, x4) = θ6 − (θ2 + θ5) s m, ρ (t, x5) = θ2s m − θ5 im, as well as point tn; eq. (6) may be recast in the following ways in accordance with a numerical strategy known as the adamsbashforth method: x (tn) = x(0) + h n κ̄(a)n φ(0) + n−1∑ j=1 κ (a) n− jφ ( t j, x1 j ) + κ (a) 0 φ ( t j, x p 1v ) y (tn) = y(0) + h a ᾱ(a)n ϕ(0) + n−1∑ j=1 κ (a) n− jϕ ( t j, x1 j ) + x(a)0 ϕ ( t j, x p 1n ) figure 1. the size of susceptible human with time. z (tn) = z(0) + h a κ̄(a)n θ(0) + n−1∑ j=1 x(a)n− jθ ( t j, x1 j ) + x(a)0 θ ( t j, x p 1n ) (25) w (tn) = z(0) + h a κ̄(a)n β(0) + n−1∑ j=1 x(a)n− jβ ( t j, x1 j ) + x(a)0 β ( t j, x p 1n ) v (tn) = z(0) + h a κ̄(a)n ρ(0) + n−1∑ j=1 x(a)n− jρ ( t j, x1 j ) + x(a)0 ρ ( t j, x p 1n ) . the discretization parameters are specified as follows ( h is the step size): π̄(a)n = (n − 1)n − na(n −α− 1) γ(2 + α) . (26) when n = 1, 2, . . ., we set the parameters as follows: κ (a) 0 = 1 γ(2 + α)n κ(a)n = (n − 1)n+1 − 2na+1 + (n + 1)a+1 γ(2 + α) . (27) we get an approximation of the functions in our model: φ ( t, x1 j ) = θ1 − (θ2 + θ3) s h j φ ( t, x2 j ) = θ2s h j − (θ4 + θ3) ih j φ ( t, x3 j ) = θ4 ih j − θ3rh j (28) φ (t, x4 j) = θ6 − (θ2 + θ5) s m j φ ( t, x5 j ) = θ2s m j − θ5 im j. the discretization technique that is going to be discussed in this part provides a lot of benefits. the procedure is reliable and guarantees convergence, and the matlab implementation is both practical and simple. 7. results and discussion we present numerical results assuming the initial values of the model are: s h(0) = 3326, ih(0) = 482, rh(0) = 0, s m(0) = 117600, im(0) = 5600 4 guma et al. / j. nig. soc. phys. sci. 5 (2023) 1464 5 figure 2. the size of infectious human with time. figure 3. the size of recovered human with time. figure 4. the size of mosquitoes with time. figure 5. the size of infectious mosquitoes with time. figure 1 show the determine the property of a fractional order models. figure 2 demonstrate the behavior of susceptible human with t. from figure 2, it is clear that susceptible human group increases with t. figure 3 shows the behavior of infectious human with t. from figure 3, it is clear that infectious people group decreases with time. figure 4 demonstrate the behavior of recovery human with t. from figure 4, it is clear that recovery human group increases with t. figure 5 demonstrate the behavior of susceptible mosquitoes with time. figure 6 demonstrates the behavior of infectious mosquitoes with time. based on the frequency of dengue fever in sudan, the fundamental reproduction number indicates that one sick individual can spread the disease to up to ten additional people. 8. conclusion in order to numerically resolve a fractional-order outbreak of dengue disease in sudan, a computational technique built on the adams-bashforth approach has been applied in this study. the illustrations in this paper demonstrate how parameter values and fractional derivatives both continually affect the result. from numerical and stability analysis, it is clear that the fractional-order dengue fever model depends on its parameters at a given time t. as a result, the parameter values are crucial in determining how many people recover from their illness and how many contract it. the suggested method works well to demonstrate the behavior of the solution over a lengthy time horizon, which is useful for precisely forecasting the dengue virus outbreak in sudan. in the future, we want to solve several novel fractional models, such as those in [17,20], and compare them to other numerical approaches [21,25]. references [1] m. a. abdoon & f. l. hasan, “advantages of the differential equations for solving problems in mathematical physics with symbolic computation”, mathematical modelling of engineering problems 9 (2022) 268. [2] m. abdoon, f. hasan & n. taha, “computational technique to study analytical solutions to the fractional modified kdv-zakharov-kuznetsov equation”, abstract and applied analysis 2022 (2022). [3] a. ahmed, y. ali, b. elmagboul, o. mohamed, a. elduma, h. bashab, a. mahamoud, h. khogali, a. elaagip & t. higazi, “dengue fever in the darfur area, western sudan”. emerging infectious diseases. 25 (2019) 2125. [4] a. ahmed, a. elduma, b. magboul, t. higazi & y. ali, “the first outbreak of dengue fever in greater darfur, western sudan”, tropical medicine and infectious disease. 4 (2019) 43. [5] a. ahmed, i. dietrich, a. labeaud, s. lindsay, a. musa & s. weaver, “risks and challenges of arboviral diseases in sudan: the urgent need for actions”, viruses 12 (2020) 81. [6] a. atangana, & d. baleanu, “new fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model”, arxiv preprint arxiv:1602.03408 (2016). 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[19] m. abdoon, f. hasan & n. taha, “computational technique to study analytical solutions to the fractional modified kdv-zakharov-kuznetsov equation”, abstract and applied analysis 2022 (2022). [20] s. kumar, s.ghosh, m. jleli & s.araci, “a fractional system of cauchyreaction diffusion equations by adopting robotnov function”, numerical methods for partial differential equations, 38 (2022) 470. [21] h.ramos & m. rufai, “a new one-step method with three intermediate points in a variable step-size mode for stiff differential systems”, journal of mathematical chemistry (2022) 1. [22] m. duromola, a. momoh, m. rufai & i. animasaun, “insight into 2step continuous block method for solving mixture model and sir model”, international journal of computing science and mathematics 14 (2021) 347. [23] h. ramos & m. rufai, “an adaptive one-point second-derivative lobattotype hybrid method for solving efficiently differential systems”, international journal of computer mathematics, 99 (2022) 1687. [24] a. qazza, m.abdoon, r.saadeh & m.berir, “a new scheme for solving a fractional differential equation and a chaotic system”, european journal of pure and applied mathematics , 16 (2023) 1128. [25] f. e. guma, o. m. badawy, a. g. musa, b. o. mohammed, m. a. abdoon, m. berir, s. y. salih,“risk factors for death among covid-19 patients admitted to isolation units in gedaref state, eastern sudan: a retrospective cohort study”, journal of survey in fisheries sciences 10 (2023) 712. 6 j. nig. soc. phys. sci. 3 (2021) 144–147 journal of the nigerian society of physical sciences antimicrobial activity of green synthesized tri-metallic oxide ni/cr/cu nanoparticles sathish kumar ka,∗, g. r. venkatakrishnanb, r. rengarajb, p. k. gayathria, g. lavanyaa, d. hemapriyaa adepartment of chemical engineering, sri sivasubramaniya nadar college of engineering, kalavakkam, chennai, tamil nadu, india bdepartment of electrical & electronics engineering, sri sivasubramaniya nadar college of engineering, kalavkkam, chennai, tamilnadu, india abstract the tri-metallic oxide ni/cr/cu nanoparticles (nps) were synthesized using coriander sativum extract as the reducing agent. the precursors namely cuso4.5h2o, ni(no3)2·6h2o and cr (no3)3·9h2o were used for the green synthesis. further, the prepared nps were characterized using ultraviolet-visible (uv-vis) spectroscopy and x-ray diffraction (xrd) studies. its antimicrobial property against two fungal and two bacterial species was determined by measuring the respective zone of inhibition (zoi) in well diffusion method. a dose dependent inhibition was observed in all the four species of pathogens including escherichia coli, staphylococcus aureus, aspergillus flavus and penicillium sp. this antimicrobial property of tri-metallic oxide nps may be utilized in the field of medical research, pharmaceutical industries and environmental sciences. doi:10.46481/jnsps.2021.237 keywords: tri-metallic oxide nps, uv-vis, xrd, antimicrobial activity, zone of inhibition. article history : received: 23 may 2021 received in revised form: 21 june 2021 accepted for publication: 01 july 2021 published: 29 august 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: e. edward anand 1. introduction the common methods to synthesize of multi metallic nps includes laser ablation, ultrasonication, sol-gel, hydrothermal, co-precipitation and green synthesis [1,2]. the physical and chemical methods involve higher cost and are not eco-friendly. on the other hand, use of green synthesis methods offer several advantages such as cost-effectiveness, simplicity, and environmentally friendliness. the green sources used for synthesizing ∗corresponding author tel. no: email addresses: sathishkumark@ssn.edu.in (sathish kumar k ), gayathri.kothandaraman@gmail.com. (p. k. gayathri), lavanya17026@chemical.ssn.edu.in; hemapriya17021@chemical.ssn.edu.in (d. hemapriya ) nps are micro-organisms and plants [3-6]. the choice of these sources as bioreducing agents should be delicately considered in the green synthesis approach [7,8]. on comparing, the choice of plants is less expensive than micro-organisms. coriander sativum is an important medical plant belonging to umbelliferae family comprising several bio-active compounds such as flavonoid, phenol, terpenoid, tannin and glycosides with high level of in vitro radical scavenging activity [9,10]. these bio-active compounds were believed to play a major role in the synthesis of metal oxide nps. due to the distinctive properties of bimetallic nps, it has attracted the attention of research community [11,12]. for various applications, the most suitable methods were to fabricate multi-metallic nps followed by core-shell and/or alloy forma144 sathish et al. / j. nig. soc. phys. sci. 3 (2021) 144–147 145 tion. extensive research has been established on traditional single metallic methods. exploration tri-metallic nanostructures have been initiated recently [13] since an elaborate study with respect to its application is still at its infancy. the greatest obstacle for scientists trying to eradicate dangerous pathogens such as bacteria, moulds, yeast and viruses has been growing resistance to antibiotics. multi-metallic nps have been heavily studied against these dangerous pathogens as feasible therapeutic and diagnostic methods [14,15]. an effective approach to overcome these issues is to synergistically engineer these nps with various formulations, such as the creation of multi-metallic composites, to strengthen current vulnerabilities. the antimicrobial resistance has long been of concern to researchers and the use of metal oxide nps has increased the interest in nanomaterial researchers. hence in the present study, a green method to synthesize tri-metallic oxide nps comprising nickel (ni), copper (cu), and chromium (cr) using the coriander sativum extract will be developed. subsequently, its antimicrobial efficacy against two bacterial species and two fungal species will be assessed. this investigation of antimicrobial activity of tri-metallic oxide nps could be encouraging for a pharmaceutical application. 2. materials and methods 2.1. materials required chemicals including cuso4·5h2o, ni(no3)2·6h2o, cr (no3)3·9h2o were purchased from srl india. double distilled water was used for the preparation of chemical reagents.. all antibacterial and antifungal media were purchased from himedia. laboratory grade solvents were used for our study. the bacterial and fungal species were obtained from mtcc. 2.2. collection of plant coriander sativum leaves were used for our study. the collected leaves were washed with tap and double distilled water and finally dried. the leaves were crushed and grounded to fine powder. the extract was prepared using 5 g of fine powder in 100 ml distilled water for 15 min at 400w. it was filtered using whatman no.1 filter paper and filtrate was used as a reducing agent for green synthesis of tri-metallic oxide nps. 2.3. green synthesis of tri-metallic oxide nps the method involves simultaneous reduction of precursor salts cuso4·5h2o, ni(no3)2·6h2o and cr(no3)3·9h2o with the c. sativum extract [16]. typically, 0.01m of salt solution comprising all three metals was mixed with the leaf extract in 250 ml erlenmeyer flask. the temperature of the reaction mixture was maintained 40◦c for 45 min in a water bath. after completion of the reaction, mixture was cooled to achieve room temperature. the mixture was centrifuged at 3000 rpm for 15 min and washed several time with distilled water. finally the mixture was washed with ethanol to remove impurities. the sediment comprising the tri-metallic nps were stored in vacuum oven at 45◦c to maintain its stability and integrity. 2.4. characterization of nps the synthesized tri-metallic oxide nps were characterized using uv-vis spectroscopy and x-ray diffraction (xrd). the composition of the ni/cr/cu nps were investigated using a empyrean x-ray diffractometer, operated at 45 kv, 40 ma, with cukα1 radiation (wavelength λ=1.5406 å) and copper filter. the x-ray diffractogram was recorded in the 2θ range from 10◦ to 80◦ at scanning steps of 0.026◦. an uv-vis spectrum was measured between 190-800 nm using jasco spectrophotometer to determine the characteristic peaks of the synthesized nps. 2.5. antimicrobial activity the antimicrobial activity of tri-metallic nps was evaluated against two bacterial and two fungal species by well diffusion method [17,18]. 1 ml of the selected microbial species was inoculated over the entire agar surface. a well with 5 mm diameter was bored aseptically, and 50µl of the nps solution at desired concentration was introduced into the well. the inoculated plates were incubated under suitable conditions depending upon the test microorganism. the details of the medium, species and incubation conditions are provided below. 2.5.1. antibacterial activity mueller hinton agar plates were prepared and inoculated with standardized bacterial strains comprising e. coli and s. aureus in individual plate. five different wells were bored on the agar plates where, four different concentrations of the nps were added in four separate wells and control (amoxicillin) was added in the other well. the plates were incubated (37◦c/24 hours) and observed for zone of inhibition (zoi). 2.5.2. antifungal activity sabouraud dextrose agar plates were prepared and inoculated with standardized fungal strains of a. flavus and penicillium sp. five different wells were bore on the agar plates where, four different concentrations of the nps were added in four individual wells and control (fluconazole) was added in the other well. the plates were then incubated at 25◦c for 72 hours and observed for zone of inhibition (zoi). 2.6. statistical analysis all results of antimicrobial activity are expressed only numerically as mean ± standard deviation (sd) without any analysis. 3. results and discussions 3.1. synthesis and characterization of tri-metallic nps initially, the extract and the precursor solution looked tanned brown colour. after the reduction reaction, the solution turned dark brown in colour. after centrifugation and drying, a dark brown powder was collected. the uv-vis absorption of the c. sativum extract was compared with the collected tri-metallic oxide nps. the tri-metallic oxide nps showed three characteristic peaks at 261 nm, 426 nm and 564 nm (figure 1) that correspond to tri-metallic oxide ni/cr/cu nps reported earlier 145 sathish et al. / j. nig. soc. phys. sci. 3 (2021) 144–147 146 figure 1. uv-visible spectrum of tri-metallic oxide ni/cr/cu nps figure 2. powder xrd of tri-metallic oxide ni/cr/cu nps [16]. similar results were obtained in monometallic nps synthesized using individual metallic precursors [19–21]. the ni, cu, and cr nps tend to show characteristic xrd peaks at 40-50◦ and 50-60◦ [19–21] individually. a combination of peaks was observed in xrd (figure 2) and new peak at 20◦ characteristic to that of graphene nanomaterials was also observed. this may be due to the reduction reaction of phytochemical that is present in the leaf extract of c. sativum. using scherrer’s equation, the crystallite size of the tri-metallic oxide nps was calculated as 17.72 nm from the xrd. in the present study, the cumulative peaks of all the three metal were exhibited in both uv-vis and xrd. this shows that tri-metallic oxide nps comprising ni, cr and cu were synthesized. 3.2. antibacterial activity the antibacterial activity of the tri-metallic nps was determined against e. coli and staph. aureus by measuring the zoi after incubation (37◦c/ 24 h) (table 1). the observed results are presented in figure 3. it can be seen that the tri-metallic nps exhibited dose-dependent antibacterial activity. that is, the zone of inhibition increased proportionally with the concentration of nps. among the tested species, the tri-metallic nps showed better antibacterial activity against e. coli than staph. aureus. there are reports that discussed the antibacterial activity of these metals. in a study by o. dlugosz et. al., copfigure 3. agar plates showing zoi against (a) e. coli and (b) staph aureus. control amoxicillin figure 4. agar plates showing zoi against (a) a. flavus and (b) penicillium sp. control fluconazole per showed improved antibacterial activity against e. coli and staph. aureus with respect to increasing size. but its effect reduced when it is combined with silver. on the other hand, it exhibited inhibition against broader spectrum of microbes. in another study, the bimetallic nps made of gold and silver showed that the size and the antimicrobial property are inversely proportional to each other [15]. there is only one study which discussed similar antibacterial property of similar tri-metallic nps wherein all the three metals showed synergistic antimicrobial effect [16]. hence as per the earlier studies and the present results, it can be understood that the antibacterial property of the metallic nps was due to the size and the charge of the active surface area. 3.3. antifungal activity the antifungal activity of the tri-metallic nps was determined against a. flavus and penicillium sp. by a similar method to that of antibacterial activity. the measured zoi and its respective plates are provided in table 1 and figure 4. it was observed that the organism showed resistance towards the trimetallic nps at 125 µg/ml concentration. but as the nanoparticle’s concentration increased, the diameter of the zone of inhibition also increased. thus, the dose and inhibition were directly proportional to each other. in earlier studies, the bimetallic nps showed good activity towards fungal species candida sp. and b. cinerea [22,23] due to their external charge and size. the tri-metallic nps showed better antifungal activity against penicillium sp. than a. flavus that can be attributed towards even 146 sathish et al. / j. nig. soc. phys. sci. 3 (2021) 144–147 147 table 1. zone of inhibition (mm) measured for each microbes across the concentration gradient. all the values are given as mean value. s no nps concentration (µg/ml) e. coli (mm) staph aureus (mm) a. flavus(mm) penicillium sp. 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[22] al-dhabaan fa, shoala t, m ali aa, alaa m, abd-elsalam k, abdk, ”chemically-produced copper, zinc nanoparticles and chitosanbimetallic nanocomposites and their antifungal activity against three phytopathogenic fungi”, int j agric technol. 13 (2017) 753. [23] j. a. gutiérrez, s. caballero, l. a. dı́az, m. a. guerrero, j. ruiz & c. c. ortiz cc, ”high antifungal activity against candida species of monometallic and bimetallic nanoparticles synthesized in nanoreactors”, acs biomater sci eng. 4 (2018) 647. 147 j. nig. soc. phys. sci. 3 (2021) 190 journal of the nigerian society of physical sciences corrigendum to “simulation and optimization of lead-based perovskite solar cells with cuprous oxide as a p-type inorganic layer [j. nig. soc. phys. sci. 1 (2019) 72–81, https://doi.org/10.46481/jnsps.2019.13]” eli danladia,d,∗, m. y. onimisia, s. garbab, r. u. ugbea, j. a. owolabia, o. o. igea, g. j. ibeha, a. o. muhammedc adepartment of physics, nigerian defence academy, kaduna, nigeria bdepartment of chemistry, nigerian defence academy, kaduna, nigeria cdepartment of physics, bayero university, kano, nigeria ddepartment of physical sciences, greenfield university, kaduna, nigeria doi:10.46481/jnsps.2021.332 article history : received: 06 august 2021 accepted for publication: 13 august 2021 published: 29 august 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: b. j. falaye the original content of this published article has an error in the name of one of the co-authors eli danladi which was written incorrectly as d. eli. we highly regret this. ∗corresponding author tel. no: +2348063307256 email address: danladielibako@gmail.com (eli danladi ) 190 j. nig. soc. phys. sci. 1 (2019) 143–146 journal of the nigerian society of physical sciences original research optimization of method and components of enzymatic fuel cells s. i. akinsola∗, a. b. alabi, m. a. soliu, t. akomolafe department of physics, university of ilorin, kwara state, nigeria. abstract enzymatic fuel cells produce electrical power by oxidation of renewable energy sources. an enzymatic glucose biofuel cell uses glucose as fuel and enzymes as biocatalyst, to convert biochemical energy into electrical energy. the applications which need low electrical voltages and low currents have much of the interest in developing enzymatic fuel cells. the cell was constructed using three different materials with different electrodes (bitter leaf and copper electrodes (bcu), bitter leaf and carbon electrodes (bc) and water leaf and carbon electrodes (wc)). the short circuit current and open circuit voltage were measured in micro-ampere (µa) and milli-volt (mv ) respectively at 30minutes interval over the period of 12hours (from dawn to dusk). the results which show that fuel cells constructed using bitter leaf with carbon electrode has the highest open circuit voltage, short circuit current and generated power of 162.8 mv , 1.65 µa and 268.62 nw respectively at 720 mins is obtained from the plots generated by the use of microsoft excel. the results show that all short circuit currents, voltages and powers generated increases with time and this is as a result of the exposure to solar radiation during the period of taking the measurement. keywords: enzymes, fuel cells, power generation. article history : received: 16 september 2019 received in revised form: 02 november 2019 accepted for publication: 10 november 2019 published: 30 december 2019 c©2019 journal of the nigerian society of physical sciences. all rights reserved. communicated by: w. a. yahya 1. introduction the international energy agency (iea) explains renewable energy resources as those derived from natural processes and replenished at a faster rate than they are consumed. the iea definition of renewable energy includes the following sources: electricity and heat derived from solar, wind, ocean, hydropower, biomass, geothermal resources, and biofuels and hydrogen derived from renewable resources. a fuel cell is like a battery that generates electricity from an electrochemical reaction. both batteries and fuel cells convert chemical energy into electrical energy and also, as a by-product ∗corresponding author tel. no: +2348166602544 email address: siakinsola711@gmail.com (s. i. akinsola ) of this process, into heat. a fuel cell uses an external supply of chemical energy and can run indefinitely, as long as it is supplied with a source of hydrogen and a source of oxygen (usually air). the source of hydrogen is generally referred to as the fuel and this gives the fuel cell its name, although there is no combustion involved. oxidation of the hydrogen instead takes place electrochemically in a very efficient way. during oxidation, hydrogen atoms react with oxygen atoms to form water; in the process electrons are released and flow through an external circuit as an electric current [8]. biological fuel cells convert the chemical energy of carbohydrates, such as sugars and alcohols, directly into electrical energy with the aid of a biological agent, i.e. a single strain or a consortium of microorganisms or isolated microbial enzymes [6]. they use the available substrates from renewable sources and convert them into harmless 143 akinsola et al. / j. nig. soc. phys. sci. 1 (2019) 143–146 144 by-products with simultaneous production of electricity [3]. figure 1: working principle of a fuel cell [4]. in fuel cells, a substance with reduction properties (fuel) is oxidized at the anode, while the produced electrons are transferred, via an external circuit, to a suitable electron acceptor molecule, or oxidant (such as oxygen) at the cathode [5]. there are two types of biological fuel cells, namely microbial fuel cells (mfcs) and enzymatic fuel cells (efcs). microbial biofuel cells employ living cells to catalyze the oxidation of the fuel, whereas enzymatic biofuel cells use enzymes for this purpose. although they have lower power densities, the current advantage to microbial biofuel cells is that they typically have long lifetimes (up to five years) and are capable of completely oxidizing simple sugars to carbon dioxide [2]. 1.1. enzymatic fuel cell enzymatic fuel cell is a specific type of biofuel cell which uses enzymes as a catalyst to oxidize its fuel, rather than precious metals [4]. enzymes have a complex structure comprising proteins. the electron-transferring unit of the enzyme, namely the apoenzyme and cofactor, is deeply buried inside its complex structure. enzymatic fuel cells are compatible with immobilization and wiring, and consequently, can offer high current densities, especially when used in concentration form [6]. 2. materials and method 2.1. materials the materials and equipment used in the development of the enzymatic biofuel cell are: glucose oxidase (ec 1.1.3.4, gox) from waterleaf (talinum triangulare) and bitter leaf (vernonia amygdalina), dglucose, digital multimeter, micro-ammeter, distilled water, agar, salt (nacl), copper, carbon, potassium dihydrogen phosphate (k h2 po4), disodium hydrogen phosphate (na2 hpo4), beaker, stirrer, the cooking gas, soldering iron, pvc pipe, transparent plastic containers, jumper wire, paper tape, glue. figure 2: schematic diagram of an enzymatic fuel cell [1]. 2.2. methodology preparation of buffer solution the solution of phosphate buffer ph 7.0 was prepared by dissolving 0.50 g of anhydrous disodium hydrogen phosphate (na2 hpo4), and 0.35 g of potassium dihydrogen phosphate (k h2 po4) in sufficient water to produce 1000 ml. construction of the separator (ion electrolyte membrane). in constructing the polymer, 9 grammes of agar and 11.6 grammes of sodium chloride was measured and mixed with 10 ml of distilled water inside a beaker with the help of a stirrer. it was placed on the bunsen burner to heat for about 5 minutes and then allowed to cool. it was poured inside the pvc pipe blocked at one end (the pvc pipe was blocked at one to avoid leakage of the solution), then leave it for some minutes to totally cool to give a solidified form which stayed fixed inside the pvc pipe and serves as the separator. construction of the cell an enzymatic biofuel cell consists of an anode and a cathode chambers. the two chambers were constructed using two transparent four-sided plastic containers. a hole was drilled at one-side of each container by the use of a soldering iron. the two chambers were separated by an ion exchange membrane prepared from a solution of agar and salt (nacl) inside a pvc pipe. the pvc pipe containing the polymer was inserted in the drilled holes of the two containers using glue to make it tight and to prevent leakage and air. the anode chamber was then filled with the d-glucose which serve as the fuel, glucose oxidase which serves as the enzyme, catalase which is the oxidant, phosphate buffer ph 7.0 which helps in maintaining the ph, temperature and the concentration, and an electrode (carbon) while the cathode chamber was filled with distilled water and an electrode. the ion exchange membrane serves as a separator of the two chambers and also facilitates the transfer of protons 144 akinsola et al. / j. nig. soc. phys. sci. 1 (2019) 143–146 145 and electrons from the anode to the cathode. a tiny hole was drilled in the covers of the two containers for the passage of a probe each. the electrode (carbon) was immersed deeply inside each of the container with the use of one end of the probes leaving the other end as the positive and negative terminal of the cell. then the probes were sealed to the covers of the container to make it fixed and to prevent air. figure 3: construction image. measurements the open circuit voltage and the current of the enzymatic fuel cell were measured in millivolts (mv ) and micro ammeter (µa) respectively at 30 minutes interval over a period of 12 hours (from dawn to dusk). micro ammeter was used because the current generated was very small. 3. results and discussion the short circuit current and open circuit voltage were measured in micro-ampere (µa) and millivolt (mv ) respectively at 30 minutes interval over the period of 12 hours. the measurements were made for the three cells constructed, i.e. fuel cell using bitter leaf and copper electrodes (bcu), using bitter leaf and carbon electrodes (bc) and using water leaf and carbon electrodes (wc). figures 4, 5, and 6 show the variation of the current, voltage and power measured with time from the three cells respectively. from figures 4, 5 and 6 above, it was observed that the current generated by the fuel cells ranges from 0.72−1.32 µa, 0.78− 1.65 µa and 0.60−1.24 µa for fuel cell constructed using bitter leaf with copper electrodes, bitter leaf with carbon electrodes and water leaf with carbon electrodes respectively, while the voltage generated by the cells ranges from 34.2−116.8 mv, 51.3− 162.8 mv and 11.7 − 90.81 mv for fuel cell constructed using figure 4: current measurements for the three fuel cells. figure 5: voltage measurements in the three fuel cells. figure 6: power measurements for the three fuel cells. bitter leaf and copper electrodes, bitter leaf with carbon electrodes and water leaf with carbon electrodes respectively. the fuel cell constructed using bitter leaf with carbon electrodes generated the highest open circuit current and voltage, followed by that constructed using bitter leaf with copper electrodes, while that constructed using water leaf with carbon electrodes generated the least open circuit current and voltage. the low spikes in the graph shows the moment with minimal solar irradiation or cloudy weather. the current and voltage generated by the fuel cells increases steadily from the initial point (at time 0 second). as shown in figure 4, the current gain of the fuel cells per time is not as much as the voltage gain per time of the cells shown in figure 5 as the voltage gain of the cells shows a steeper slope than the current gain. this shows that the potential difference between the electrodes and the rate of transfer of electron through separator is lowest in the fuel cell constructed using water leaf with carbon electrodes but highest in the fuel cell constructed using bitter leaf with carbon electrodes. this might owe to the fact that electron transport mechanism which takes place with the used electrodes favour this performance. generated electrons during oxidation reaction should reach the anodic electrode to 145 akinsola et al. / j. nig. soc. phys. sci. 1 (2019) 143–146 146 enter into the power circuit. the higher current generated by the bitter leaf cell as compared with the other cell could be as a result of the concentration of electrons in the used bitter leaf which will in turn has high electrons content reaching the electrode and hence generating highest electric current, compared to cells with other type of leaf. the carbon electrode used could also be responsible for the increased in electron mobility than the other type of electrode used. carbon has graphite as one of its allotrope, which could have a work function lower than copper and hence eject electrons from it faster than copper. the cell constructed using bitter leaf with copper lies between the two. the power generated by the fuel cells follows a similar trend with the current and voltage generated. theoretically, the results obtained from this experiment could be modeled using the equation used for treating rc circuit, which is suggesting the fuel cells working like a capacitor. vc(t) = vs ( 1 − e−t/rc ) , (1) the quantity rc is the time constant of the system. equation 1 describes when the capacitor is charging while equation 2 is for when it is discharging. vc(t) = vse −t/rc, (2) although this is subject to further studies. 4. conclusion in this work, three types of enzymatic fuel cells have been constructed and the open circuit current and voltage generated by the fuel cells were measured in the interval of 30 minutes over the period of 12 hours. the results show that fuel cells constructed using bitter leaf with carbon electrode has the highest maximum open circuit voltage of 162.8 mv and the highest maximum short circuit current of 1.65 µa and highest maximum power of 268.62 nw at time 720 mins. the construction with the lowest maximum open circuit voltage of 90.8 mv , the lowest maximum short circuit current of 1.24 µa and the lowest maximum power of 112.592 nw at time 720 mins is waterleaf with carbon. acknowledgments we thank the referees for the positive enlightening comments and suggestions, which have greatly helped us in making improvements to this paper. references [1] d. x. benetton, s. srikanth, y. satyawali, k. vanbroekhoven & d. pant, “enzymatic electrosynthesis: an overview on the progress in enzyme electrodes for the production of electricity, fuels and chemicals”, journal of microbial & biochemical technology 007 (2013) doi: 10.4172/1948-5948.s6-007. [2] s. d. minteer, b. y. liaw & m. j. cooney, “enzyme-based biofuel cells”, current opinion in biotechnology 18 (2007) 228. [3] y. mohan, k. s. manoj muthu & d. das, “electricity generation using microbial fuel cells”, international journal of hydrogen energy 33 (2007) 423. [4] m. patil, p. pawar, m. a. k. kerawalla & p. goswami, “enzymatic biofuel cells a progress review”, journal of chemical, biological and physical sciences 6 (2016) 39. [5] d. sell, “bioelectrochemical fuel cells”, biotechnology 10 (2001). [6] a. k. shukla, p. suresh, s. berchmans & a. rajendran, “biological fuel cells and their applications”, current science 87 (2004) 455. [7] y. song, v. penmasta & c. wang, “recent development of miniatured enzymatic biofuel cells”, biofuel’s engineering process technology (2011). [8] http://www.fuelcelltoday.com/analysis/patents/2011/2011-patent-review (assessed june, 2019) 146 j. nig. soc. phys. sci. 3 (2021) 209–215 journal of the nigerian society of physical sciences implementation of modified differential evolution algorithm for hybrid renewable energy system g. r. venkatakrishnana,∗, r. rengaraja, k. k. sathishb, r. k. dinesha, t. nishantha adepartment of electrical and electronics engineering, sri sivasubramaniya nadar college of engineering, kalavakkam, chennai bdepartment of chemical engineering, sri sivasubramaniya nadar college of engineering, kalavakkam, chennai abstract a hybrid renewable energy system, which could provide a reliable energy alternative for conventional battery systems is implemented in this paper. the primary requirement is that the hybrid energy system should be cost-effective while meeting the energy demand. the hybrid system is implemented to a area located in uttarakhand, india using solar photovoltaic cells to supply power during hot and humid conditions and using wind turbine generators to supply power during windy conditions. the wind turbine generators and photovoltaic cells are used in a combined manner along with the diesel generators as in case if it fails to meet the demand. in order to meet the requirements, modified differential evolution (de) algorithm is being implemented. moreover, the effectiveness of the performance is evaluated by comparing the results obtained from modified de with other optimization algorithms. in comparison with other optimization algorithms, results indicate that the implementation of the modified de algorithm helps in obtaining the best cost effective solution for the system along with meeting the energy demand. doi:10.46481/jnsps.2021.240 keywords: hybrid systems, evolutionary algorithms, renewable energy, differential evolution article history : received: 28 may 2021 received in revised form: 22 june 2021 accepted for publication: 18 july 2021 published: 29 august 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: e. etim 1. introduction complexity and size of the power system is increasing day by day, owing to increasing population and demand of electricity. hence, a continuous and reliable supply of electricity is the most important requirement of industries and other fields. in the modern integrated power system, due to unaccommodating requirements various challenging problems are imposed on the electrical utilities. thus, a proper planning of the power system using the optimal operation technique is required at a ∗corresponding author tel. no: email address: venkatakrishnangr@ssn.edu.in (g. r. venkatakrishnan ) high level. the power system when it is optimally operated determines the best system state by considering all kinds of constraints to which it is subjected. the different considerations by which the power system can be operated optimally are either the economics of operation or system security or emission of fossil fuel plant. since this consideration makes a conflict over one another, a compromise between these is required to operate the power system optimally. among the different consideration, the ideology of using the available power efficiently among the public and government, the economic analysis of power system has gained more importance when compared to others. hence the importance and the implementation of renewable energy resources come into existence. in order to save fossil fuels and to reduce the harmful greenhouse gasses emissions, re209 venkatakrishnan et al. / j. nig. soc. phys. sci. 3 (2021) 209–215 210 newable energy resources must be exploited at a greater extent. hence, the sustainability of such a system will be questioned as these renewable resources are not found throughout the year around. therefore, it is better to implement a hybrid system in such a case where if one part of the system falls back, there are other resources to help meet the demand. now, we would have to have a proper optimization of the given resources to attain a maximum efficiency and at a minimum cost. although there a lot of iterative approaches in existence, the best way to optimize them would be the implementation of the evolutionary algorithms. the reasons for the implementations of the evolutionary algorithms than its counterparts as well as the advantages of the modified de algorithm over other algorithms are discussed as follows. abbaas azarpour et al have discussed the importance of renewable energy resources in their paper [1]. more emphasis is given over the ill-effects caused by these renewable energy resources in addition to their advantages over conventional sources of energy such as fossil fuels, etc. the effects caused by each renewable energy source are explained in this paper. the increasing usage of fossil fuels has lead to its depletion and thus the advantages of the usage of solar energy to countries that lie in the equator have been studied by ho soonmin[2,3]. the importance of wind energy was being highlighted by tarang agarwal et al[4]. the various issues and challenges faced by a country while installing a standalone wind turbine system was being put forward in this paper. the various issues includes grid related issues, design issues, location issues, and its impact on the environment. now, when all these renewable energy sources are considered individually, they have their drawbacks. for instance, the solar plants have certain disadvantages, such as high installation costs, and the peak output would not be obtained during the cloudy and windy days. the biomass plant’s maximum output reduces during the lower temperature—similarly, the wind turbines become faulty during very high or during very lowspeed applications. so, to relinquish the drawbacks of one of the sources, renewable resources are used in a combined manner. renewable resources when combined, for instance, if wind and solar plant combine, then during cloudy days, the wind turbine’s output increases. during the hot and humid climates, solar plants produce a peak output. thus, the drawback of one of the resources is compensated in the other while using hybrid renewable energy resources. various types including hybrid wind-solar energy system, hybrid wind-solar diesel energy system, hybrid wind-diesel energy system, hybrid solar-diesel energy system and various other hybrid systems are being discussed in [5]. a small brief summary about the various types are being discussed. then, small introductions about the various types of optimization algorithms that can be included in these systems are being summarized. the paper concludes by saying that with hybrid renewable energy systems, it is possible to achieve better results, at lower cost and hence save resources. the combination of wind as well as solar energy systems with battery as a backup source was being implemented to various test case systems in [6] and [7]. particle swarm optimization (pso) method was being implemented to achieve the power demand individually for every particular house from a hybrid renewable energy system was being implemented [8]. a hybrid renewable energy system (hres) combining pv, wind as well as biomass energy systems was being implemented [9]. based on minimal cost of energy, the optimal configuration for the hres system is obtained. the optimal solution is being obtained using genetic algorithm (ga) and pso and the end results were being tabulated. a new approach was being implemented to reduce the dependence on diesel generators by an optimal combination of the solar, and wind system with the backup of a battery storage system being implemented [10]. a comprehensive review consisting sizing and optimization of hybrid pv-wind energy system was being studied and discussed thoroughly by hadi nabipour-afrouzi et al [11]. conventional methods became less used owing to the fact that these methods face a lot of disadvantages when compared to the modern optimization algorithms. conventional methods were satisfactorily successful for purely continuous variable optimal power flow, whereas that is not the situation in for real life problems as there exists several discrete control variables. over the past few decades, many new evolutionary algorithms have been developed to overcome the conventional methods. different evolution (de) algorithm is one of the evolutionary algorithms which has been implemented in solving problems in different engineering fields. the de algorithm has been implemented to a standard ieee 30 system with 6 thermal plants and 2 wind farms [12]. in [13-21], de algorithm is used to optimize the parameters in different power system problems. but the success of de algorithm depends on the choice of parameters like population size np, mutation rate fand crossover rate crvaries its performance (searching accuracy and convergence speed). hence, in [22], an improved version of de algorithm is implemented to solve the above complex non-linear problems. in this paper, that modified de algorithm is being used to optimize the various parameters of the hres system in [23], and the objective function is to minimize the final system cost. it helps identify the best possible combination of the hybrid system components which meets the demand as well as possessing the minimum cost. the end results are a final tabulation comparing the results obtained from ga, pso and modified de. figure 1 depicts the block diagram of chosen system. 2. modelling of hybrid system 2.1. pv system the hourly power output of the pv array is calculated using eqn. (1) as follows: ptpv = fpv ∗ ypv ∗ ( it is ) (1) where, ptpv is the hourly power output of the pv array in kw, fpv is the pv derating factor, ypv is the pv array capacity in kw, it is the global solar radiation incident on the pv array and is is equal to 1 kw/m2 210 venkatakrishnan et al. / j. nig. soc. phys. sci. 3 (2021) 209–215 211 figure 1. block diagram of hybrid system 2.2. wind turbine generator the wind turbine generator’s power output is calculated using eqn. (2) as v (h hub) v (h anem) = ( h hub h anem )α (2) where, h anemis the anemometer height in meter (m), h hubis the hub height in meters (m), v (h anem)is the speed of the wind at the anemometer height in meters per second (m/s),v (h hub) is the speed of the wind at the hub height in meters per second (m/s), and α is the power law exponent. once the wind speed is adjusted accordingly, the wind turbine’s output power is calculated using eqn. (3) as ptwt g =  0 0 ≤ v ≤ vcut−inand v ≥ vcut−out( a ∗ v3 ) + (b ∗ prated ) vcut−in ≤ v ≤ vrated prated vrated ≤ v ≤ vcut−out (3) where the constants a and b in the above formula are a given by the following equations a = prated( v3rated − v 3 cut−in ) (4) b = vcut−in( v3rated − v 3 cut−in ) (5) where vcut−in, vcut−out, and vrated are the cut in, cut out and rated speeds of the wind turbine generator respectively in m/s and prated gives the rated output power of the wind turbine generator unit in kw. 2.3. storage battery the storage battery is modeled using the following equation ptbatt = ∑ ptpv−actual + ∑ ptwt g−actual − loadt ηinv (6) where ηinv is the inverter efficiency, and load t is the load during the time unit, kw. the equation indicates the kw power flow through the battery system. the modes of operation of the battery are dependent on the battery’s state of charge (soc). the state of charge can be calculated using the following equation: s oct+1 = s oct [ 1 − ( σ 24 )] + ptbatt ∗ l (t) ∗ηbatt ebatt (7) where ηbatt represents the efficiency of the battery, σ gives the self discharge rate of the battery, l(t) is the length of tth time unit and ebatt is the energy rating of battery storage. in addition, the battery charging and the discharging operation is constrained by the following equations ptbatt.cmax ≤ p t batt ≤ p t batt.dmax (8) where ptbatt.cmax = (s ocmax − s oct) ∗ ebatt is the maximum battery charge power and ptbatt.dmax = (s oct − s ocmin) ∗ ebatt is the maximum battery discharge power 2.4. converter a converter helps in transferring power flow from the ac terminus to dc terminus. the efficiency of the inverter (ηinv), is assumed constant and is taken as 90%. the equation abiding the power flow within the converter is given by equation (9). ptinout = ( ptpv−actual + p t wt g−actual + p t batt ) ∗ηinv (9) pv modules selected is a 36 cell polycrystalline (pv-mf 100ec4) rated at 1kw. the wtg used is ampair 3kw, 48v dc type. the cut in, cut out and the rated speed of the wind turbines is given in table 2. the batteries are rated at 6v, 360 ah (2.16 kwh). thus 8 batteries of the mentioned rating are arranged in series to be capable of producing 17kwh of electricity. the dc bus voltage is fixed at 48v. a detailed overview of components is summarized as a table in [15]. 3. formulation of optimization function the objective function of the given hres system along with other constraints is given below: ct otal = min ∑ ni { ri ai ∗ r0(1 + r0) m (1 + r0) m − 1 + om ∗ ri ai } + costreliability,i (10) i = pv, wtg, batt, conv costreliability = cens ∗ eens (11) subject to: ∑ ptpv−actual + ∑ ptwt g−actual + ∑ ptbatt + ens t = loadt ηinv (12) 211 venkatakrishnan et al. / j. nig. soc. phys. sci. 3 (2021) 209–215 212 figure 2. hourly load profile ≤ ni ≤ n s ocmin ≤ s oc ≤ s ocmax where ni represents the number of the ith component, ri gives the power output capacity of the ith component, ai gives the unit cost of the ith component(in rs/kwh), r0 is the annual interest rate of each component in percentage, m is component life time in years and om gives the running and service cost in percentage. eens is the expected energy not supplied(in kwh/year) and its cost is given by cens . ptpv−actual, p t wt g−actual, p t batt, ens t, loadt are the pv power, wind power, battery power, energy not supplied and system load respectively at any period t. n is the maximum number of the ith component. the value for cens is fixed at rs336/kwh. s ocmin and s ocmax are the minimum and maximum soc value of the storage battery respectively. its values are fixed at 40% and 100% respectively. 3.1. description of study area 3.1.1. location the brief description of the seven villages and its location which is considered in this paper is being summarized in a tabular format in [15]. 3.1.2. load profile on the basis of the energy requirements and the energy consumptions in the given research area, a year is divided into four seasons and its duration and peak energy requirements have been briefly discussed and converted into a tabular form in [15]. from the given data, a seasonal hourly load profile is randomly generated. the load data in figure 2. 3.1.3. solar radiation the solar radiation data for the study area has been taken from [23]. the total solar insolation for almora district is between 6.6kwh/m2/day and 3.39 kwh/m2/day and the insolation for all the months throughout the year has been tabulated in [15]. hourly solar radiations from the available data have been generated using a random reference profile. the calculated values are then plotted as shown in fig. 3. figure 3. hourly solar radiation profile figure 4. hourly wind speed data 3.1.4. wind potential the study area is said to have an average wind profile between 2 and 15 m/s. the wind speed data was measured at almora district was obtained from [23]. in [15], the mean monthly wind speed for the given study area has been taken from. hourly wind speeds from the available data have been synthesized using random reference values. the plot given in figure 4 gives the hourly wind speeds for the whole year around. 4. implementation of modified differential algorithm this section provides an overall view or the steps involved in the modified de algorithm which is proposed by authors in [23]. the different steps involved in the modified de algorithm is explained as follows: 4.1. initialization of parameter vectors xi,g is used to represent initial population vector which is initialized by x ji,g = x j min + rand ( x jmax − x j min ) , j = 1, 2, ..., d i = 1, 2, ...., n, (13) 212 venkatakrishnan et al. / j. nig. soc. phys. sci. 3 (2021) 209–215 213 where the number of decision variables is given by d, x jmin and x jmax are the minimum and maximum value of the decision variable j. 4.2. mutation the mutation strategies implemented in this algorithm are: mi,g = xbest,g + f ( xv1,g − xv2,g ) (14) mi,g = xv1,g + f ( xv2,g − xv3,g ) (15) where v1,g , v2,g and v3,g are randomly generated exclusive integers within nsuch that v1,g , v2,g , v3,g , i. here, fis the mutation factor which lies between 0 and 1.2. xbest,g is the best individual target vector corresponding to best fitness value. 4.3. crossover in de, u ji,g is the trial vector which is produced using the following equation: u ji,g =  v j i,g x ji,g if rand ji ≤ cr or j = jrand otherwise (16) where cris the crossover rate between 0 and 1 and jrand is randomly chosen integer within d. 4.4. selection the selection scheme used in this algorithm is given by xi,g+1 = { ui,g xi,g if f ( ui,g ) ≤ f ( xi,g ) otherwise (17) where f ( ui,g ) and f ( xi,g ) are the fitness values of ui,g and xi,g . 4.5. terminating condition fixing the number of generations gmax as 100 is used as the stopping criterion in this algorithm. adaptation techniques for fand cris used in this algorithm to eliminate the disadvantage of randomly choosing it. 4.6. adaption techniques for mutation rate,f the self adaptation technique used in this algorithm is given by fg+1 =  max { lm, 1 − ∣∣∣∣ fmaxfmin ∣∣∣∣} max { lm, 1 − ∣∣∣∣ fminfmax ∣∣∣∣} if ∣∣∣∣ fmaxfmin ∣∣∣∣ < 1, otherwise, (18) where lm = 0.4, fminand fmaxare the minimum and maximum fitness values respectively. 4.7. adaptation technique for crossover rate, cr the self-adaptation scheme used in the depas algorithm is given by: cri,g+1 = { rand1 ci,g rand2 ≤ τ2 otherwise (19) in this algorithm, it is assumed thatτ2 = 0.1 a flowchart to explain the process of modified de is given in figure 5. 5. results the modified de algorithm is applied to the hres system problem in the almora district in uttarakhand using matlab. to analyze the betterment of the de algorithm, it is compared with pso and ga. the various control parameters involved in the three optimization algorithms are being summarized in table 1. the crossover rates in the ga are taken to be in the range between 0.6 and 1.0 whereas the mutation rates are taken to be less than 0.1. the acceleration constants in the pso is taken in such a manner that c2>c1 which is generally advantageous for complex nonlinear integral problems. the crossover rates and the mutation rates are first fixed at 0.5 for both respectively. then for every iteration, the constants are iterated and different values are obtained. the best optimized results are then obtained for the crossover rate and the mutation rate. the convergence characteristic for the three optimization algorithms namely ga, pso and modified de for the modeled hres system is shown in figure 6 respectively. the abscissa for the convergence plot gives the combined cost for all the components. the perpendicular axis gives the total number of iterations to reach the global minimum in each of the situations. since the computational time for the modified de algorithm is slightly higher than the rest of the algorithms, the number of iterations was limited to 600, whereas the numbers of iterations were extended up to 1000 for both pso as well as ga. a close look at the convergence characteristics indicates that the modified de produces faster results (obtaining the global minimum) when compared to other optimization algorithms. so although the computational time is slightly higher, the faster response helps reduce the computational complexity leading to less computer resource usage. the best optimal solutions from the three algorithms are briefed in table 2. 6. conclusion in this paper, an attempt has been made to implement the modified de algorithm for solving hybrid renewable energy systems problem. the developed methodology was tested for a remote society in uttarakhand, india and the obtained results were compared with those obtained from ga as well as with pso. results reveal that the modified de evolutionary algorithm outperforms the genetic algorithm as well as the particle swarm optimization algorithm. a hybrid system consisting of 6 1 kw pv panels, 8 3kw wtg, a 20kw converter and 10 battery strings each of 17kw produces the most optimal solution for the given study area. the developed objective function substantiates the capital cost along with the running and service cost as well as the lifetime of all the components while also including the force outage rate for the wind turbine generators and the soc condition for the battery. the system can be further extended by including multi – objective function for optimization by including the transmission and line losses in the system. it also can be extended to optimize time varying data. 213 venkatakrishnan et al. / j. nig. soc. phys. sci. 3 (2021) 209–215 214 figure 5. flow chart of modified de algorithm table 1. control parameters for different algorithm s/n optimization technique n control parameters 1. genetic algorithm 20 pm = 0.07, pc = 0.85 2. particle swarm optimization 30 c1 = 2, c2 = 3 3. modified differential evolution 380 f = 0.5, cr = 0.5 table 2. results obtained from algorithms optimization algorithm no. pv panels no. wtg no. of battery strings converter (kw) optimized cost is rupees ga 8 7 11 20 737694.48 pso 8 7 10 20 713703.416 modified de 6 8 10 20 667960.321 figure 6. convergence characteristics of different algorithms acknowledgements the authors are very grateful to anonymous referees for their contributions and suggestions. references [1] a, azarpour, s. suhaimi, g. zahedi & a. bahadori, “a review on the drawbacks of renewable energy as a promising energy source of the future”, arabian journal for science and engineering 38 (2013) 317. 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[23] vivekananda parvatiya krishi anusandhan sansthan, indian council of agricultural research (icar) laboratory, almora, uttarakhand, india (2002). 215 j. nig. soc. phys. sci. 3 (2021) 256–261 journal of the nigerian society of physical sciences influence of silicon nanoparticle on the electrical properties of heterostructured cdte/cds thin films based photovoltaic device a. a. faremia,∗, s. s. oluyamob, k. d. adedayob, y. a. odusoteb, o. i. olusolab adepartment of physics, federal university, oye-ekiti, nigeria bfederal university of technology, akure, nigeria abstract this paper presents the influence of silicon nanoparticles at the interface of heterostructured cadmium telluride and cadmium sulfide thin films based photovoltaic device with improved electrical parameters leading to tremendous improvement in cds/cdte thin film based solar cells performance. the films of cdte, cds and si were electrodeposited using electrodeposition technique to form a heterostructured cdte/si/cds/fto. the films respective structural properties were also examined using x-ray diffractometer (xrd) before forming a heterostructured material. the heterostructured cdte/si/cds/fto and the structure without the inclusion of silicon nanoparticle were examined using electrometer for the extraction of electrical parameters such open circuit voltage (voc ), short circuit current density (js c ), and fill factor (ff). although a large body of experimental results are available to date on the optoelectronics properties of the materials. however, there is relatively low research studies or works on the electrical properties of the materials. therefore, we formed heterostructured based photovoltaic device and characterized the structure to determine useful electrical properties. the value obtained for voc , js c and ff are 418 mv, 25 ma/cm2 and 0.72 which are indicative of pin holes free semiconductor materials and no leakage path emerging from high-grade materials used in the deposition of heterostructured cdte/si/cds. doi:10.46481/jnsps.2021.267 keywords: silicon nanoparticles, electrodeposition technique, heterostructured based photovoltaic, electrical properties article history : received: 23 june 2021 received in revised form: 23 july 2021 accepted for publication: 01 august 2021 published: 29 august 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: e. edward anand 1. introduction the use of p-type cdte, n-type cds and transparent conducting substrates such as fluorine doped tin oxide and indium doped tin oxide for the formation of heterostructured cdte/cds based solar cells have received considerable attention because of their notable performance in energy conversion devices [13]. the suitability of cds thin film as a good window layer ∗corresponding author tel. no: +2347033712587 email address: abass.faremi@fuoye.edu.ng (a. a. faremi ) material capable of providing excellent receptive surface to absorber layer materials such as p-type cadmium telluride (cdte), p-type copper indium selenide (cuinse2), etc is owned to its high electron affinity [3]. despite the potential of cds in the formation of heterojunction based solar cells, there is need to treat its surface for better receptive surface which offer better improvement on the electrical parameters of the photovoltaic solar cells [4-6]. cdte thin film as one of the primary candidates in the field of photovoltaic technology has gained worldwide prominence owning to its near ideal bandgap energy of 1.45 ev for the absorption of photons [6]. interdiffusion of tel256 faremi et al. / j. nig. soc. phys. sci. 3 (2021) 256–261 257 lurium as a dopant in cdte has been reported as a hindrance to the performance of cdte based devices [7]. however, a lot of improvement have been made to prevent such as an interdiffusion such as post-heat treatment of the films surface in the presence of cadmium chloride or oxygen [6], [8], [9]. silicon thin film as a leading and the most building block for electronics has been report being a useful material to set an enabling environment for cdte/cds based solar cells [7]. we therefore use silicon nanoparticles as a preventive measure to interdiffusion of tellurium since we aimed at forming heterojunction of cdte/cds. the films of cdte, cds and si have been prepared by various techniques, including sputtering, spraypyrolysis, chemical bath deposition, sol-gel, close space sublimation, ion implantation and electrodeposition [1], [2], [10-12]. among the various techniques available for the synthesis of these materials, electrodeposition and sol-gel techniques have been reported as viable techniques over other processes because of their simplicity in term of material growth [13-15]. however, in a typical sol-gel technique, the chosen reagents for a thin film need to undergo series of hydrolysis and polymerization reactions to form a solution bath [16]. the series of process require in the formation of solution bath make the technique to be more less cost-effective than electrodeposition technique. the technique has also gained research attention due to its possibility of scaling down bulk silicon to either thin film or nanoparticles [17], [18]. in our study, we employed electrodeposition in the synthesis of the heterostructured cdte/si/cds because of its relatively simple, easy, robust, easily scalable, bath selfpurification and economically viable for large area production of photovoltaic devices [4], [19], [20]. 2. materials and method the film of cds sourced from the solution of high-grade cadmium sulphate (3cdso4.8h2o, 99%) and thiourea (cs(nh3)2, 99%) without further purification was electrodeposited at a cathodic potential of 1400 mv on a thoroughly degreased conducting fto/substrate of 3.2 by 3 cm2 in dimension. after deposition, the material was subjected to curing at substrate temperature of 400o c in 20 minutes and further purified in deionized water so that the surface could provide a perfect receptive surface for the other materials since we aimed at forming a heterostructured device. silicon powder of 2 grams by weight was dispensed in 500 ml beaker containing 400 ml of deionized water and magnetically stirred for 2hrs. the electrolytic bath of silicon powder was electrodeposited potentiostatically at cathodic potential of 1000 mv on the deposited cds. the structure of si/cds/fto was also further cured at the same substrate temperature and washed thoroughly to remove possible undissolved particles on the surface to enable proper adherent of cdte film sourced from the solution of high-grade cadmium sulphate (3cdso4.8h2o, 99%) and tellurium dioxide (teo2, 99%). the electrolytic bath of cdte, cds and si was adjusted and controlled using ammonium solution and ph probe. the ph value of the bath was 2.0 since the employed technique is favourable in acidic medium. deposition took place at bath temperature of 80 oc in 60 minutes using working electrode with carbon as counter electrode. the films of cdte, cds, si, the heterostructure of cdte/si/cds/fto and structure without the inclusion of silicon was also fabricated. the structure was examining for their structural and electrical properties using xrd and electrometer probe. the particle crystallite size and the electrical parameters of the electrodeposited structure were extracted using equation 1-5 d = kλ βcosθ (1) where λ is x-ray wavelength in a ◦ , k is a scherrer constant (k=0.9), β is measured at half-maximum of the diffraction peak and θ is the bragg’s angle and d is the particle size or crystalline size. η= pmax pin = vm im pin (2) where pmax is the maximum power out, pin is the maximum power input (a solar simulator made from a 25 w incandescent light bulb was employed for this. the distance between the bulb and the surface of the cells was approximately 9.2 cm, which good enough for maximum radiance without significant heat transfer. the radiance intensity of the simulator used, considering sample distance from source as 9.2 cm was estimated at 220 w/m2.), vm is the maximum voltage power and im is the maximum current power f f= pmax voc isc = vm im voc isc (3) where ff is the fill factor, voc is the open circuit voltage and is c is the short circuit current the conversion efficiency can also be estimated by combining equation 1 and 2 to form equation 3 η= f f×voc×isc pin (4) it is important to note the relationship between jsc and isc as the two parameters are often used. jsc= isc a (5) where a is the surface area of the heterostructured cdte/si/cds which is estimated as 0.000096 m2. 3. result and discussion 3.1. structural characterization of cdte, cds thin films and si nanoparticle figure 1 reveals eight basic peaks (111), (210), (211), (220), (222), (321), (411) and (420) with their corresponding diffraction angles 22.71o, 28.01o, 30.00o, 37.00o, 48.80o, 50.00o, 57.88o and 60.24o. the different peaks as observed in the xrd pattern of cdte, cds and si indicated structural transformation from single phase to polycrystalline. there is no noticeable broad hump in the structures of polycrystalline cdte, cds and si. figure 2 reveals similar structural transformation with six basic 257 faremi et al. / j. nig. soc. phys. sci. 3 (2021) 256–261 258 figure 1. xrd plot for cadmium telluride thin film prepared using 1400 mv cathodic deposition technique, with peaks characteristics of polycrystalline cadmium telluride figure 2. xrd plot for cadmium sulfide thin film prepared using 1400 mv cathodic deposition technique, with peaks characteristics of polycrystalline cadmium telluride peaks (111), (220), (311), (400), (420) and (422), as a function of diffraction angles 20.97o, 34.01o, 41.08o, 50.00o, 54.68o and 58.72o. in figure 2 and 3, three prominent peaks that’s (111), (220) and (311) are observed which holds potentiality to polycrystalline cadmium sulfide (cds) thin films and silicon (si) nanoparticles [21], [22]. peak 111 as a dominant peak in both figures 1, 2 and 3, revealed the preferred orientation of the structures being the most intense peak which is also an indication of particles crystallization (highly and randomly oriented) and zinc blende structure. the full width at half maximum of cdte cds and si were obtained by broadening most intense peak through a gaussian fitting which was used to determine the particle crystallite size using sherer’s equation. the estimated particle crystallite size of cdte, cds and si are 11.98, 14.08 and 89 nm respectively. the estimated particle crystallite sizes of both cdte, cds and si agreed with the previous work [21], [22], [24], [25]. figure 3. xrd plot for electrodeposited silicon nanoparticles prepared using 1000 mv cathodic potential, with peaks characteristics of polycrystalline silicon figure 4. sem micrograph of electrodeposited cdte thin film 3.2. morphological characterization of cdte, cds thin films and si nanoparticle the electrodeposited materials are uniformly distributed on the substrate and the grains as depicted in figure 4-6 have spherical shape with faceted edges. the uniformity of materials reveals the significance of cathodic deposition technique over anodic technique [26]. however, the nucleation cdte, cds and si on the well degreased fto/substrate show uniform distribution of grains and relatively little agglomeration. such a uniformity in the grains distribution could reducing electron trapping in the lattice site resulting to improved electrical parameters [22], [23]. the little agglomeration in the sem micrographs of the materials agrees with previous works [3], [4], [26], [27]. 3.3. electrical characterization of heterostructured cdte/si/cds and cdte/cds in order to characterize the electrical behaviour at the interface of the heterostructured devices, it is necessary to carry out i-v characterization under illumination and dark conditions which helps to harness the potentials of heterostructured photovoltaic devices [28], [29]. figure 7 revealed the electrical parameters 258 faremi et al. / j. nig. soc. phys. sci. 3 (2021) 256–261 259 figure 5. sem micrograph of electrodeposited cds thin film figure 6. sem micrograph of electrodeposited si thin film of cdte/si/cds/fto based photovoltaic measured under illumination condition using electrometer. the fill factor as an important photovoltaic parameter which characterized the grade and the curve fitness of the device revealed the potential of the structures. the slight change in the electrical parameters as depicted in figure 8 was as result of measurement condition, dark. despite the variation in the parameters, there is tremendous improvement in the electrical conductivity of the device. such improvement could be attributed to overcoming the barrier potential or potential hill using forward bias voltage. figure 9 and 10 revealed the electrical parameters of the device without the inclusion of the silicon nanoparticle. the injection of silicon nanoparticles in the structure also aided the device electrical parameters such as open circuit voltage (voc), short circuit current density (jsc), and ll factor (ff) of the heterostructure devices as shown in table 1. the structure exhibits pin-diode like characteristic due to the injection of silicon nanoparticles which serve as an intrinsic material between the p-type cdte and ntype cds. such a structure could act as a good photodetector, high voltage rectifier and radio frequency switches. the high values of the extracted open circuit voltage (voc), short circuit current density (jsc) and ll factor (ff) showed that pinhole free semiconductor materials have been successfully fabricated and their suitability have been confirmed in the electrical properties figure 7. electrical properties of cdte/si/cds/fto based photovoltaic structure measured under illumination condition using electrometer figure 8. electrical properties of cdte/si/cds/fto based photovoltaic structure measured under dark condition using electrometer figure 9. electrical properties of cdte/cds/fto based photovoltaic structure measured under illumination condition using electrometer of the heterostructured cdte/cds based photovoltaic. 259 faremi et al. / j. nig. soc. phys. sci. 3 (2021) 256–261 260 table 1. quantitative electrical parameters of heterostructured based photovoltaic device heterostructured cdte/si/cds light dark voc (mv) js c (ma/cm2) vmax (mv) imax (ma) ff voc (mv) j s c (ma/cm2) vmax (mv) imax (ma) ff 418 25.02 395 17.17 0.72 367 12.85 0.339 9.25 0.69 heterostructured cdte/cds light dark voc (mv) js c (ma/cm2) vmax (mv) imax (ma) ff voc (mv) js c (ma/cm2) vmax (v) imax (ma) ff 398 18.58 362 13.10 0.66 338 9.55 0.297 6.33 0.61 figure 10. electrical parameters of the fabricated cdte/si/cds and cdte/cds under illumination and dark conditions 4. conclusion the potential of electrodeposition technique in the fabrication of heterostructure based photovoltaic has made a significant improvement in the electrical properties of the cdte/si/cds and cdte/cds configurations. the probe into electrical behaviour at the interface of the heterostructured based device has revealed the successive development of photovoltaic devices. the injection of silicon into the structure also offer better and improved photovoltaic parameters leading to high value in both open circuit voltage and short circuit current density. such increase in the photovoltaic parameters are indicative of high-grade reagents used which have also resulted to no leakage path and pinhole free compound semiconducting materials. the structural analysis of the films revealed structural transformation from single phase to polycrystalline which is an evidence that we have successfully fabricated polycrystalline cdte, cds and si acknowledgments the authors wish to acknowledge the financial support of tertiary education trust fund (tetfund) of nigeria through federal university oye-ekiti and the 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[29] s. krishnan, g. sanjeev, m. pattabi & x. mathew, “electrical properties of rf sputtered cdte/cds thin film solar cells”, open fuels and energy science journal 2 (2009) 110. https://doi: 10.2174/1876973x00902010110. 261 j. nig. soc. phys. sci. 4 (2022) 20–26 journal of the nigerian society of physical sciences perovskite tetragonality modeling for functional properties enhancement using newtonian search based support vector regression computational method peter chibuike okoyea, samuel ogochukwu azib, taoreed o. owolabia,∗ aphysics and electronics department, adekunle ajasin university, akungba akoko, 342111, ondo state, nigeria. bdepartment of physics, university of benin, benin city, edo state, nigeria. abstract tetragonality occurs as a result of stretching the crystal structural lattice of perovskite along one of its lattice vectors such that the three axes are mutually perpendicular with two of the axes having equal lengths. this tetragonality distortion easily triggers functional properties such as pyroelectricity, ferroelectricity, capacitance, and piezoelectricity among others, while synthesizing functional ceramics for a particular application. this work addresses and circumvents the challenges of experimental stress involved in functional ceramics synthesis by developing a newtonian search-based support vector regression (gsb-svr) model for perovskite tetragonality prediction using dopants concentration and ionic radii as the model predictors. the performance of the proposed gsb-svr model is compared with the existing kelvin & rick model and better performance of 35.82% improvement based on mean absolute percentage error (mape) and 36.44% improvement based on mean absolute error (mae) is obtained. the influence of lanthanides and zirconium incorporation on functional ceramics on the material tetragonality is also modeled by the developed gsb-svr model. the metal in the lanthanide series considered includes lanthanum (la), praseodymium (pr), neodymium (nd), and samarium (sm). the obtained variation in their tetragonality follows the same trend as their variation in atomic numbers. maximum distortion occurs between concentrations of 0.05 and 0.1, and each of the examined tetragonality distortions has a parabolic tetragonality distortion variation. titanium and zirconium dopants were incorporated into the crystal lattice structure of pb0.9ba0.1(zrxti1−x)o3 and pb0.9ba0.1(zrx−1tix)o3. the tetragonality distortion in pb0.9ba0.1(zrxti1−x)o3 was observed to be minimum while pb0.9ba0.1(zrx−1tix)o3 perovskite show maximum tetragonality distortion. the observed tetragonality distortion can be utilized to enhance the functional properties of perovskite. the precision of the developed model, its easily fetched predictors, and its pre-laboratory ability to effectively and efficiently model the perovskite tetragonality are of high importance in tailoring and enhancing functional properties of materials for desired applications. doi:10.46481/jnsps.2022.248 keywords: support vector regression; tetragonality; distortion; perovskite; newtonian based gravitational search algorithm. article history : received: 10 june 2021 received in revised form: 10 august 2021 accepted for publication: 09 december 2021 published: 28 february 2022 c©2022 journal of the nigerian society of physical sciences. all rights reserved. communicated by: s. j. adebiyi ∗corresponding author tel. no: +234(0)8067226208 email address: owolabitaoreedolakunle@gmail.com (taoreed o. owolabi ) 1. introduction the science of materials is basically centered on dealing with relationships between structure and property, with materials’ composition being an important processing parameter [1]. perovskites belong to a large structural family of com20 p. c. okoye et al. / j. nig. soc. phys. sci. 4 (2022) 20–26 21 pounds with crystal structures that are similar to calcium titanate (catio3). they are chemically represented as abx3, where a and b are cations with a having a larger atomic radius than b and x is an anion which can be an oxide or a halogen. the presence of (xb2a4) coordination, cuboctahedral coordination (ax12), and octahedral (bx6) coordination shells is defined by assuming that the number of bonds between cation-anion and anion-cation are equal in any particular structure of a given perovskite. [2]. perovskite compounds are extensively applied in modern devices because their variability in composition and structure produces several important properties such as ferroelectricity, superconductivity, piezoelectrics, pyroelectrics, and spin-dependent transport [3, 4, 5, 6]. these important functional properties of perovskite can be easily induced by tetragonality distortion. tetragonality occurs as a result of stretching a cubic lattice along one of its lattice vectors such that the three axes are mutually perpendicular and two of the axes have equal lengths. the tetragonal unit cell is characterized by a four-fold symmetry axis around which the atoms align with their initial positions by rotating the unit cell at an angle of 90◦. designing materials with special properties such as ferroelectrics, piezoelectrics, high-temperature superconductors, high-k capacitors, and pyroelectrics for many applications premises on the precise measurement of tetragonality distortion [6, 7]. this work aims to address the challenges associated with experimental stress in functional ceramics synthesis through the development of a newtonian search-based intelligent model that allows for the prediction of tetragonality in perovskites using dopants concentrations and ionic radii of the perovskite constituents as the model predictors. aside from the superiority of the proposed hybrid gravitational searched-based support vector regression (gsb-svr) model as compared with the existing model in the literature [1], the developed gsb-svr model can be implemented using easily fetched descriptors and foster pre-laboratory modeling. support vector regression (svr) is a mathematical learning theory derived from vapnik-chervonenskis theory of statistical learning [8]. svr makes use of a collection of training data that includes predictor variables and their respective experimental targets to adequately construct a predictive model. with the aid of only predictors, the strength of svr extends to prediction of future data. because this model does implement input data probabilistic distribution for obtaining exact data, it is referred to as a nonparametric process. using kernel function, the input data is reorganized and mapped into feature space where high precision modeling and simulation are performed. support vector regression (svr) gains wider applicability as a data mining algorithm to solve both linear and nonlinear problems [9]. hyper-parameters of the svr algorithm play crucial roles in actualizing a robust model as they are optimized using optimization algorithm that is based on newtonian mechanics. rashedi, nezamabadi-pour, and saryazdi [10] introduced gravitational search algorithm based on gravity in 2009. it is an algorithm that is inspired by newton’s second law of motion as well as the law of gravity. it relies on the universal assumption of recognizing three kinds of mass namely: active/ passive gravitational mass and inertial mass [11]. in gsa, every agent is considered to be an object and the mass of each agent determines the individual performance of that agent. each object possesses four pieces of information namely: passive/active gravitational mass, position, and inertial mass; all masses obey newton’s second law and the law of gravity [12]. gravitational force compels the objects to attract one another and also makes every object with a lighter mass to be attracted in the direction of objects with greater masses [10] which represents a more promising solution [13]. the performance of every object within the algorithm is measured based on its mass. better solutions are represented by heavier masses and they move more slowly than lighter masses [14]. 2. gravitational search algorithm gravitational search algorithm (gsa) is a type of swarm algorithm created from newton’s second law of motion and the law of gravity [10]. the basic mathematical principle of the gravitational principle is: f = g m1 m2 r2 . (1) f is force of gravitation, g is constant of gravitational, the masses of the objects are denoted by m1 and m2, and r is the separation between objects. in this algorithm, objects are considered as agents and every object’s performance is evaluated from their individual masses. the gravitational force makes objects to attract one another, forcing all objects to gravitate in the direction of objects with higher masses. this is because objects with heavy masses have better fitness values and provide better solutions to problems and their movement is not as fast as the objects with lighter masses [15]. each object in the gsa is identified by its position, its inertial mass (mii), active gravitational mass (mai), and passive gravitational mass (mpi). gsa search process starts by creating a population of n individuals at random in the search space. we begin the algorithm by defining the position of the jth agent as: x j = ( x1j . . . x d j . . . x n j ) for j = 1, 2, . . . , n (2) the space dimension of the problem is n and xdj illustrates agent’s jth position with dth dimension. force acting on mass j from mass k at a given time t is defined, according to newton’s gravitational theory, as: fdjk(t) = g(t) m j(t) × mk(t) r jk(t) + � ( xdk (t) − x d j (t) ) , (3) where mass of agent j is m j, mk is mass of agent k, g(t) is constant of gravitation at time t, � is a small constant and the euclidian distance r jk(t) between j and k agents is defined as: r jk(t) = ‖x j(t), xk(t)‖2 (4) in a d dimension, the sum of forces acting on j agent is assumed to a weighted sum of the dth component of forces of other agents calculated at random. 21 p. c. okoye et al. / j. nig. soc. phys. sci. 4 (2022) 20–26 22 the total force is given as: fdj (t) = n∑ k=1,k,i randk f d jk (5) randk is a random number in the interval [0, 1]. according to newton’s law of motion, the acceleration (adj (t)) of the agent j at time t in dth direction is defined by: adj (t) = fdj (t) mii(t) (6) where m is the inertial mass of the jth agent. the sum of an agent’s current velocity and current acceleration determines its next velocity, given, respectively, as: xdj (t + 1) = x d j (t) + v d j (t + 1) (7) vdj (t + 1) = rand j × v d t (t) + a d j (t) (8) where vdt (t) and x d j (t) represents the agent’s velocity and position respectively. rand is a uniform random variable in the interval [0, 1] and is used to give a randomized feature to the search. by reducing the randomly initialized constant of gravitation g with time, the search accuracy is controlled. this is to say that g is a function of time t and the initial value g is given as: g(t) = g(g0, t) (9) gravitational and inertial masses are computed using fitness evaluation. the efficiency of an agent in relation to the solution it represents is dependent on the heaviness of the mass of the agent. in other words, heavier masses will not move as fast as the lighter masses. the masses are updated using m j(t) = fit j(t) − wors(t) best(t) − worst(t) , (10) and mi(t) = m j(t)∑n k=1 mk(t) (11) with the following assumptions: mai = mpi = mii = mi, i = 1, 2, . . . n (12) where fit j(t) is the fitness value of agent j at time t, worst(t) and best(t) are defined as d in best(t) = min k∈1,...n fitk(t) (13) and worst(t) = max k∈1,...n fitk(t), (14) respectively, for a minimization problem addressed in this contribution. 2.1. support vector regression in 1995,vapnik and co-workers proposed the support vector machine (svm) [8] which is a tool obtained from statistical learning theory for carrying out classification and regression tasks. svm was initially developed for solving classification problems but it later evolved into solving regression problems [16]. several implementations of svm have been achieved in different areas of research after it was proposed [17, 18]. svm is therefore a universal term that can be subdivided into support vector classification (svc) and support vector regression (svr) [19]. svc employs only one slack variable while svr uses two slack variables. both svc and svr employ very similar algorithms, the difference is the number of slack variables and types of variables they predict. in general, linear regression with svr is defined as f (x,∝) = 〈w, x〉 + b (15) where w ∈ k and b ∈ r . svr algorithm’s main purpose is to find w and b in a way that � is not exceeded in every training dataset. to achieve this, vector w must be minimal and equation (15) must be flat. minimization of the euclidean norm ‖w‖2 by means of a transformation to a convex optimization problem is needed to flatten equation (15) as shown in equation (16) below. minimize 1 2 ‖w‖2 subject to  y j − 〈 w, x j 〉 − b ≤ �〈 w, x j 〉 + b − y j ≤ � (16) constraints that may prevent the possibility of the convex optimized problem in equation (16) are added by the introduction of slack variables (ξ j and ξ∗j ). new optimization problem is presented as minimize 1 2 ‖w‖2 + c j∑ j=1 ( ξ j + ξ ∗ j ) subject to  y j − 〈 w, x j 〉 − b ≤ � + ξ j〈 w, x j 〉 + b − y j ≤ � + ξ∗j ξ j,ξ ∗ j ≥ 0 (17) where c is regularization or penalty factor. performance generalization of svr model is determined by regularization factor c, the epsilon parameter �, and the kernel option, which are carefully selected user-defined parameters. the regularization factor balances the difference between complexity of model and error tolerance in training data, with greater errors being equivalent to lower values of c and vice versa. the number of support vectors in the insensitive zones is regulated by the epsilon parameter, while input data transformation to a feature space of higher dimension is governed by the kernel option [20]. this study used gravitational search algorithm to choose the parameters discussed above. 22 p. c. okoye et al. / j. nig. soc. phys. sci. 4 (2022) 20–26 23 3. methodology and the employed computational strategies this section contains the proposed hybrid model’s computational approach as well as an overview of the dataset used. 3.1. description of the dataset using experimental ionic radii data extracted from the literature [1], the proposed hybrid gravitational searched support vector regression (gsb-svr) model was developed. 3.2. computational details of the proposed hybrid model this research paper develops a robust model with optimal functionality resulting from the combination of gsa and svr algorithms. in this hybrid model, all the computational tasks were implemented on the matlab programming environment. before starting modeling and simulation, existing data was randomized in order to facilitate equal distribution of data points and a more effective computation when the data is split into 80% and 20% training and testing set respectively. with the help of the developed real svr algorithm codes, support vectors were created with existing training data points, and the generalization ability of the generated support vectors is corroborated by a testing set of data. gsa is aimed at finding the best hyper parameter values of svr for excellent generalization of the developed model when applied to a collection of data not used in the training process. the penalty or regularization factor (c), epsilon (�), and the kernel option (σ)) of the optimum kernel functions are the svr hyper-parameters optimized by gsa in each population. the search for optimal values of hyper-parameters starts with initialization of gsa agents which requires putting n agents in a search space, with each agent encoding svr hyper-parameters. following the initialization of the number of agents, every agent in the population was used to train svr algorithm with the training set of data, and their fitness was evaluated with the testing set of data. each agent’s mass was calculated using equation (11). equations (5) and (6) were used to compute the total gravitational force and acceleration of every agent. the computation of position and velocity of each agent was carried out and repeated until maximum iteration is reached using equations (7) and (8). as a result, hyper parameters of svr were optimized with gsa and the gsa-svr model was developed. the optimal hyper-parameter value obtained from an agent with the highest fitness value at maximum iteration is then used to train the gsa-svr model. the following are the procedures for the developed hybrid gsb-svr model. step i: randomization of data and partitioning: for even distribution and more efficient computation during modelling, the available data is randomized. the data is partitioned into two sets, 80% for training and 20% for testing. step ii: choose a kernel function from the listed options. step iii: gsa agent initialization: fill a search space with n agents while each agent encodes svr hyper-parameters. (penalty factor, epsilon and kernel option). compute each agent’s fitness in the following way: (a) train svr algorithm with each agent and a chosen kernel function. the number of svr trained algorithm is equal to the initial number of agents. using the training dataset, determine the performance measuring parameters (mean absolute percentage error (mape) and mean absolute error (mae)) for each trained algorithm. (b) compute mape and mae using the testing set of data to determine each trained algorithm’s generalization capacity. (c) compute the fitness of mape and mae for each agent, choose and save the trained algorithm with the minimum value. step iv: use equation (11) to estimate the mass of every agent with the assumptions contained in equation (12). step v: calculate the overall force of gravity and acceleration for each agent using equation (5) and equation (5). step vi: determine the velocity and position of the agent. that is vdj (t + 1) = rand j × v d t (t) + a d j (t) xdj (t + 1) = x d j (t) + v d j (t + 1) step vii: the velocity and position of the agents are updated until maximum iteration of 100 is attained. step viii: repeat step iii-step vii with a different kernel function. step ix: save the optimum kernel function and hyperparameters for future implementation. 4. results and discussion this section presents the outcomes of the developed gsbsvr model with inclusion of results of the implemented optimization algorithm. comparison of estimates of the developed gsb-svr with the existing model is also presented. 4.1. optimization of svr hyper-parameters optimization of svr hyper-parameters is presented in figure 1. the effect of the number of agents in enhancing exploration and exploitation capacities of newtonian optimization algorithm is presented in the figure. figure 1: convergence of gsb-svr model with number of iteration at different initial number of agent 23 p. c. okoye et al. / j. nig. soc. phys. sci. 4 (2022) 20–26 24 local convergence was attained when the number of the agent was set at ten. the local solution can be attributed to weak exploitation due to limited number of agents exploiting the search space. exploitation capacity becomes enhanced as the number of agents was increased to thirty while the search space is well explored at this value. increase in the number of agents above thirty results into high level of complexity within the search space since larger number of agents are exploring the search space and each agent experience strong gravitational pull which further weakens exploration and exploitation capacities of the algorithm. optimum hyper-parameters were attained when the number of agents exploring the space was set at thirty as presented in figure 1. other investigated factors influencing the optimization strength of the implemented population-based algorithm include the initial values of the gravitational constant, maximum number of iteration and parameter alpha that influences the gravitational pull between the agents. 4.2. comparison of the performance of the present and existing model performance comparisons between the present and existing (kelvin & rick) model are presented in figure 2 and figure 3, respectively on the basis of mean absolute percentage error (mape) and mean absolute error (mae). figure 2: comparison of the mean absolute percentage deviation of the developed gsb-svr model with existing kelvin & rick model [1]. figure 3: comparison of the mean absolute error of the developed gsb-svr model with existing kelvin & rick model [1]. on the basis of mape, the developed gsb-svr model has a better performance than the existing tolman & ubic model with 35.82% improvement in performance while 36.44% improvement in performance was attained while comparing the developed gsb-svr model with the existing kelvin & rick [1] model using mae as performance evaluator. the observed performance enhancement of the developed gsb-svr model over the existing model can be attributed to strong mathematical foundation of the implemented svr algorithm coupled with the excellent global solution searching capacity of the hybridized gravitational search algorithm. 5. investigating the influence of lanthanides on tetragonality of pb1−3xl2x(ti)o3 perovskite the significance of lanthanides inclusion on pb1−3xl2x(ti)o3 perovskite is presented in figure 4. the metal in the lanthanide series considered include lanthanum (la), praseodymium (pr), neodymium (nd) and samarium (sm). the obtained variation in the tetragonality of the investigated perovskite follows similar trend of the variation in atomic number as we move from lanthanum to samarium. the tetragonality distortion variation of each of the investigated tetragonality distortion shows a parabolic trend with maximum distortion between the concentrations of 0.05 to 0.1. this observed behavior of the tetragonality distortion coupled with the precision of the developed gsb-svr model strongly indicates the tendency of the developed model in enhancing functional properties of perovskite as can be easily induced by the tetragonality distortion. 24 p. c. okoye et al. / j. nig. soc. phys. sci. 4 (2022) 20–26 25 figure 4: influence of lanthanides (l)con tetragonality of pb1−3xl2x(ti)o3 perovskite . 6. influence of titanium dopants on the tetragonality distortion of pb0.9ba0.1(zrxti1−x)o3 perovskite incorporation of titanium and zirconium dopants into crystal lattice structure of pb0.9ba0.1(zrxti1−x)o3 and pb0.9ba0.1(zrx−1tix)o3 perovskites, respectively using the developed gsb-svr model is presented in figure 5. the tetragonality distortion in pb0.9ba0.1(zrxti1−x)o3 was observed to be minimum when pb0.9ba0.1(zrx−1tix)o3 perovskite suffers maximum tetragonality distortion. tetragonality in other perovskite compounds can be investigated using the developed model purposely to reveal the functional properties needed for a particular application. figure 5: effect of zirconium on tetragonality of pb0.9ba0.1(zrxti1−x)o3 perovskite. 7. conclusion this work presents hybrid gravitational search algorithm and support vector regression for modeling the tetragonality distortion of perovskite compounds using the ionic radii and the concentration of dopants as descriptors to the model. the performance of developed gsb-svr model is compared with the existing kelvin & rick model, the developed model is found to perform better than the existing model in terms of the measured absolute percentage error (mape) as well as mean absolute error (mae). the 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(2015) 70, doi: 10.1007/978-1-4302-5990-9. 26 j. nig. soc. phys. sci. 3 (2021) 477–483 journal of the nigerian society of physical sciences performance study of n-grams in the analysis of sentiments o. e. ojo∗, a. gelbukh, h. calvo, o. o. adebanji ainstituto politécnico nacional, natural language and text processing laboratory, centro de investigacion en computación, cdmx, mexico abstract in this work, a study investigation was carried out using n-grams to classify sentiments with different machine learning and deep learning methods. we used this approach, which combines existing techniques, with the problem of predicting sequence tags to understand the advantages and problems confronted with using unigrams, bigrams and trigrams to analyse economic texts. our study aims to fill the gap by evaluating the performance of these n-grams features on different texts in the economic domain using nine sentiment analysis techniques and found more insights. we show that by comparing the performance of these features on different datasets and using multiple learning techniques, we extracted useful intelligence. the evaluation involves assessing the precision, recall, f1-score and accuracy of the function output of the several machine learning algorithms proposed. the methods were tested using amazon, imdb, reuters, and yelp economic review datasets and our comprehensive experiment shows the effectiveness of n-grams in the analysis of sentiments. doi:10.46481/jnsps.2021.201 keywords: n-grams, machine learning, deep learning, sentiment analysis article history : received: 3 july 2021 received in revised form: 13 september 2021 accepted for publication: 14 september 2021 published: 29 november 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: j. ndam 1. introduction machine learning and deep learning architectures are the bane of many natural language processing (nlp) research works. to address a variety of tasks, including sentiment analysis, several machine learning and deep learning architectures have been proposed. uncommon and unfamiliar words used in information and knowledge exchange can influence different aspects of life including marketing, education, governance, etc. as an integral part of the internet, the digital media platforms facilitates meaningful information and knowledge exchange with a list of other network users. data collection and reviews, with diverse ∗corresponding author tel. no: +525560590794 email addresses: olumideoea@gmail.com (o. e. ojo ), gelbukh@gelbukh.com (a. gelbukh), hcalvo@cic.ipn.mx (h. calvo), olaronke.oluwayemisi@gmail.com (o. o. adebanji) views and opinions about events, is gaining more impact and fast becoming an attraction for researchers and generating significant computational challenges. effective wide-ranging mining of information from text helps to discover useful knowledge of vital significance. computers can detect, interpret and produce the sentiments (or tags) of a text, thereby improving government and private companies’ operations, recognizing possible threats, minimizing crime, and improving public services. the main objective of this research is to observe the efficacy of unigrams, bigrams and trigrams as characteristics of a word sequence and to predict a tag for sentiment classification. the target is to extract words or phrases behind the tags and to use the machine learning and deep learning methods to classify the data whilst measuring the accuracy of the classification. using n-grams to study the opinion of people, we will be able to see their strength by tagging them. as they contribute to conceptual 477 o. e. ojo et al. / j. nig. soc. phys. sci. 3 (2021) 477–483 478 characterization, we use machine learning and deep learning models on the text. in machine learning, two major techniques are adopted: supervised learning [1, 2, 3, 4] and unsupervised methods [5]. supervised methods have a training dataset with manually defined tags, and they learn the characteristics that match the tags from the training data. gelbukh and kolesnikova [2, 6] developed methods that allowed the automatic sorting of word combinations into pre-established categories relating to automated collocation classification. gambino and calvo [7] considered the use of text-learned opinions using nlp techniques to distinguish tags. on the other hand, unsupervised systems are more flexible across different kinds of texts and domains. different machine learning algorithms have been used in past works [6, 2, 4], and also neural network models [1] to gain the knowledge of how to predict the sentiments of text [3, 8]. pre-trained models are very helpful in classifying text and other nlp activities. to extract the keywords behind the text’s feelings, the models will be pre-trained on the training data and the accuracy of prediction of the models will be measured, registered and compared. the remaining part of the paper is organized as follows: section 2 deals with the background and relevant works, section 3 describes the approach used in this study, while section 4 shows the features we experimented. sections 5 and 6 provides the information about the machine learning and deep learning algorithms used and our experimental findings with the discussion of results in section 7. section 8 gives conclusion about the work. 2. background and related work different works have been carried out in the field of sentiment analysis [1, 7, 3, 4]. the social media and other digital media platforms, as an accepted means of communication, has flourished thereby aiding intelligence gathering and information dissemination. machine learning techniques have shown good results in analysing sentiments in text [6, 2, 4] and other tasks such as part of speech recognition (pos) [9], named entity recognition (ner) [10], etc. linear statistical models, such as random-field (crf) and hidden markov (hmm) fields, are nlp approaches used for sequence tagging with a long history of excellent performance. however, adapting these models to new tasks in new domains or languages is challenging. the combination of categorical grammar, annotation, acquisition of lexicons and semantic networks was used by pekka et al. [11] to analyze the feelings of the text and to define the tags of the text. they investigated how the overall phrase structured data and domain-specific language usage could aid in the detection of semantic orientations in financial and economic news. in [12], the use of syntactic n-grams (sn-grams) to incorporate syntactic knowledge into machine learning algorithms proved successful. sn-grams were utilized as a baseline for authorship identification, replacing standard n-grams of words, pos tags, and characters. gbc knn lrm nba dtc xgb rfc mlp svm models 70 75 80 85 % ac cu ra cy unigram bigram trigram figure 1. distribution of n-grams accuracy scores in the models for the first dataset dtc gbc knn rfc xgb nba mlp lrm svm models 50 55 60 65 70 75 80 85 % ac cu ra cy unigram bigram trigram figure 2. distribution of n-grams accuracy scores in the models for the second dataset [13] have used text-cnn for extraction of text features with lstm architecture in addition to the unimodal input functions. in a multimodal sentiment analysis task, they explored and analyzed the performance of three deep-learning-based architectures and recorded their results. on a social media data baseline, [14] explored the efficiency of deep neural network models of different complexity based on character n-grams. the training was done with augmented data and pseudo-labeled samples, and the accuracy result was enhanced. [15] also used classifiers to predict sentence tags using an objective function to infer similarity between sentences. a new objective function was used to train many classifiers to make predictions at the instance level, promoting smoothness of inferred instance-level labels while keeping group-level label constraints in place. 478 o. e. ojo et al. / j. nig. soc. phys. sci. 3 (2021) 477–483 479 table 1. precision, recall, f1 score and accuracy of the classifiers trained on the first dataset. model n-grams precision recall f1 score accuracy logistic regression unigram 0.85 0.69 0.72 80.5% bigram 0.81 0.59 0.58 74.4% trigram 0.86 0.57 0.54 73.6% support vector machine unigram 0.88 0.81 0.83 87.1% bigram 0.79 0.70 0.72 79.4% trigram 0.78 0.64 0.65 76.4% naive bayes unigram 0.84 0.72 0.75 81.7% bigram 0.85 0.54 0.48 71.6% trigram 0.85 0.51 0.43 69.8% gradient boosting unigram 0.35 0.50 0.41 69.3% bigram 0.35 0.50 0.41 69.3% trigram 0.35 0.50 0.41 69.3% decision tree unigram 0.79 0.77 0.78 81.7% bigram 0.66 0.64 0.65 71.6% trigram 0.67 0.61 0.61 72.1% random forest unigram 0.90 0.77 0.80 85.5% bigram 0.79 0.64 0.66 76.9% trigram 0.82 0.60 0.59 74.9% k-nearest neighbors unigram 0.75 0.72 0.73 78.4% bigram 0.70 0.65 0.66 74.6% trigram 0.62 0.63 0.62 66.2% xgboost unigram 0.85 0.75 0.77 83.3% bigram 0.80 0.61 0.62 75.6% trigram 0.75 0.58 0.56 73.1% multi-layer perceptron unigram 0.84 0.81 0.82 85.5% bigram 0.78 0.74 0.76 80.7% trigram 0.76 0.62 0.62 75.1% dtc gbc xgb knn rfc lrm mlp svm nba models 50 55 60 65 70 75 80 % ac cu ra cy unigram bigram trigram figure 3. distribution of n-grams accuracy scores in the models for the third dataset 3. approach to recognize patterns and regularities in data, the machine learning and deep learning algorithms use learned patterns to predict new observations. we pre-processed the text data before we applied the different learning algorithms on the text gbc dtc rfc xgb nba knn lrm svm mlp models 50 55 60 65 70 75 80 % ac cu ra cy unigram bigram trigram figure 4. distribution of n-grams accuracy scores in the models for the fourth dataset data. we tokenized the data into words and n-grams and generated a vocabulary of all the special n-grams that occurred in the document. using the term frequency-inverse document frequency (tf-idf) technique, data features were rescaled. we used supervised learning and we compared the results. for our work, we chose to filter out uncommon, non-informative 479 o. e. ojo et al. / j. nig. soc. phys. sci. 3 (2021) 477–483 480 table 2. precision, recall, f1 score and accuracy of the classifiers trained on the second dataset. model n-grams precision recall f1 score accuracy logistic regression unigram 0.86 0.85 0.85 85.5% bigram 0.76 0.75 0.75 75.5% trigram 0.63 0.57 0.51 57.0% support vector machine unigram 0.87 0.86 0.86 86.5% bigram 0.75 0.75 0.75 75.0% trigram 0.63 0.57 0.51 57.0% naive bayes unigram 0.86 0.84 0.84 84.5% bigram 0.73 0.71 0.70 70.5% trigram 0.65 0.57 0.51 57.0% gradient boosting unigram 0.79 0.79 0.78 78.5% bigram 0.73 0.71 0.70 71.0% trigram 0.72 0.58 0.50 58.0% decision tree unigram 0.78 0.78 0.78 78.0% bigram 0.68 0.67 0.66 66.5% trigram 0.73 0.56 0.46 56.0% random forest unigram 0.82 0.82 0.82 82.0% bigram 0.75 0.73 0.72 73.0% trigram 0.69 0.55 0.45 55.0% k-nearest neighbors unigram 0.80 0.79 0.78 78.5% bigram 0.76 0.55 0.43 54.5% trigram 0.53 0.51 0.41 51.0% xgboost unigram 0.82 0.82 0.82 82.0% bigram 0.71 0.64 0.61 64.0% trigram 0.75 0.51 0.36 51.0% multi-layer perceptron unigram 0.85 0.84 0.84 84.5% bigram 0.75 0.74 0.74 74.5% trigram 0.64 0.57 0.52 57.5% content by extracting n-grams to make the algorithms more intelligent. we identified some common techniques used in recent studies [6, 2, 4, 16], namely decision tree classifier (dtc), gradient boosting classifier (gbc), naive bayes algorithm (nba) and random forest classifier (rfc). others are k-nearest neighbors (knn), extreme gradient boosting (xgb), support vector machines (svm), logistic regression model (lrm) and the multi-layer perceptron (mlp) classifier. for sentiment analysis, these models have been extensively tested, and provided accurate results when working with various dataset types. the words were patterned for parsing, such that every n-gram consists of n terms and are tagged accordingly. the accuracy of these methods often differ widely in validation, ranging from using small samples to a wide array of tagged data. 4. experiments the data used for this analysis consist of a collection of four related economic and financial market reviews selected from multiple texts that have been tagged with positive, negative and neutral classes. these four datasets, extracted from different digital media platforms, have been selected because they contain explicit economic sentiments from which the machine and deep learning algorithms can learn. we have used the reuters dataset in pekka et al.[11], containing subjective sentences from economic review, and the imdb, amazon, and yelp datasets in kotzias et al.[15], which contains text sentences from reviews of products, movies, and restaurants. the first dataset contains reviews and tags for products sold on amazon.com while the second dataset contains the sentiment dataset for imdb movie reviews. the third and fourth datasets have a collection of texts about economic and restaurant reviews respectively. the text were splitted into training and testing data. using the training set, the machine and deep learning algorithms were trained to understand, extract and evaluate subjective information from the data with n-grams as features. basically, after fitting the training data to the models, we used the various models to predict the tags of the test data. using the training set to train the algorithm, we translated the data into numeric form, while the test set was used to evaluate the performance of the machine and deep learning models. the machine and deep learning algorithms learnt from the training data, passing the features and tags as parameters. the models predicted the outcomes, while the precision, accuracy and f1 score were obtained using the n-gram features within the model. to keep a list of the word vectors, we transformed the text array into a tf-idf function matrix and a vocabulary was created. a 480 o. e. ojo et al. / j. nig. soc. phys. sci. 3 (2021) 477–483 481 table 3. precision, recall, f1 score and accuracy of the classifiers trained on the third dataset. model n-grams precision recall f1 score accuracy logistic regression unigram 0.80 0.80 0.80 80.0% bigram 0.66 0.66 0.66 66.0% trigram 0.55 0.53 0.48 53.0% support vector machine unigram 0.82 0.82 0.82 82.0% bigram 0.67 0.67 0.66 66.5% trigram 0.54 0.53 0.48 52.5% naive bayes unigram 0.84 0.82 0.82 82.5% bigram 0.70 0.62 0.59 62.5% trigram 0.60 0.53 0.43 53.0% gradient boosting unigram 0.64 0.64 0.63 63.5% bigram 0.56 0.55 0.54 55.0% trigram 0.54 0.52 0.44 52.0% decision tree unigram 0.63 0.63 0.63 63.0% bigram 0.60 0.59 0.58 59.0% trigram 0.57 0.52 0.42 52.0% random forest unigram 0.75 0.74 0.74 74.5% bigram 0.61 0.59 0.59 59.5% trigram 0.56 0.54 0.48 53.5% k-nearest neighbors unigram 0.73 0.73 0.73 73.0% bigram 0.69 0.69 0.68 68.5% trigram 0.51 0.51 0.46 50.5% xgboost unigram 0.72 0.71 0.71 71.5% bigram 0.51 0.51 0.47 51.0% trigram 0.48 0.49 0.38 49.5% multi-layer perceptron unigram 0.81 0.81 0.81 81.0% bigram 0.66 0.66 0.65 65.5% trigram 0.52 0.51 0.39 50.5% machine and/or deep learning algorithm can then directly be used on the encoded vectors. the classification and evaluation of the different meanings of the text was carried out and we compared them to each other. the n-grams offered an indication of the words that could affect the tags of the text. we extracted the n-gram distribution such as unigrams, bigrams, and trigrams for use in the different models, thereby making the learning algorithms more intelligent for proper prediction. we applied the machine and deep learning algorithm on the text for classification and the accuracy score for all models used in the experiment were calculated. 5. results and discussion in this study, we present a performance review of special n-gram based evaluation of a sequence labeling task using different learning algorithm. we introduced machine learning and deep learning techniques to analyze the sentiments in the data for better and faster decision making, and we were able to compare the output of the techniques implemented, thus adding to the state-of-the-art literature on tasks of sentiment analysis. these algorithms were applied on the datasets to predict the tags and to classify it accordingly using the n-grams features. the performance of the n-grams in the different machine and deep learning approach was calculated using the overall accuracy measurement. for a comparative performance evaluation of each system in terms of predicting the tags correctly, we present the results for the nine methods used for precision, recall, accuracy and f1-score calculation. tables 1-4 shows the model classification values of the models and figures 1–4 depicts the distribution of the values on a line graph. the macro-averaged f1, recall, precision and accuracy scores of the various models used are shown in tables 1-4. the findings indicate that the svm and the mlp models generally improved the effectiveness of the classification. the results also reveals that the dtc, gbc, knn and the xgb failed to perform well in the classification task. in the comparative analysis, using the different methods of machine learning and n-gram approaches on the datasets, results were better compared and the effectiveness of the n-gram features were recorded. the n-gram features gave a very good performance for all learning algorithms with the unigrams performing better than the bigrams and trigrams in the classification task. the svm, lrm, rfc, nba and the mlp models are the most reliable for all of the n-gram features. in the first dataset (see figure 1), rfc, mlp, svm had maximum scores among the models. the svm, lrm, and mlp models gave the highest output for all n-gram functions on the second dataset (see figure 2). on the third dataset, nba, svm and mlp are with the highest results 481 o. e. ojo et al. / j. nig. soc. phys. sci. 3 (2021) 477–483 482 table 4. precision, recall, f1 score and accuracy of the classifiers trained on the fourth dataset. model n-grams precision recall f1 score accuracy logistic regression unigram 0.80 0.80 0.79 79.5% bigram 0.68 0.68 0.67 67.5% trigram 0.57 0.53 0.45 53.0% support vector machine unigram 0.81 0.81 0.80 80.5% bigram 0.69 0.69 0.69 69.0% trigram 0.59 0.54 0.46 54.0% naive bayes unigram 0.75 0.75 0.75 75.0% bigram 0.70 0.64 0.61 64.0% trigram 0.63 0.51 0.36 51.0% gradient boosting unigram 0.64 0.64 0.63 63.5% bigram 0.60 0.59 0.59 59.5% trigram 0.50 0.50 0.37 50.0% decision tree unigram 0.66 0.66 0.66 66.0% bigram 0.59 0.58 0.57 58.0% trigram 0.57 0.51 0.38 51.0% random forest unigram 0.72 0.72 0.72 72.0% bigram 0.64 0.63 0.63 63.0% trigram 0.55 0.52 0.41 51.5% k-nearest neighbors unigram 0.77 0.76 0.76 76.0% bigram 0.69 0.67 0.66 66.5% trigram 0.45 0.48 0.38 48.5% xgboost unigram 0.73 0.72 0.72 72.5% bigram 0.59 0.56 0.53 56.5% trigram 0.55 0.52 0.41 51.5% multi-layer perceptron unigram 0.81 0.81 0.81 81.0% bigram 0.68 0.68 0.68 68.0% trigram 0.59 0.54 0.45 53.5% (see figure 4) while on the fourth dataset, svm, lrm and mlp performed best on the test dataset (see figure 4). the models used with the feature classification techniques shows the effectiveness of n-grams for sentiments tagging and the most reliable methods of classification. 6. conclusion an important problem in the analysis of sentiments is being able to determine the contextual labels or tags of words and phrases. we addressed this problem in this study by successfully introducing various machine learning and deep learning approaches to produce the labels or tags of economics and financial reviews text using n-grams as features. modeling was performed using different pre-processing techniques in texts, converting the text into vectors, and applying various machine learning and deep learning techniques on the different datasets. the use of multiple classifiers in this analysis led to a better evaluation efficiency than any individual classifier. the findings recorded in this study suggests that the support vector machine and multi-layer perceptron neural networks were the best options for achieving successful results, because they efficiently and effectively classify the sentiment tags behind the sentence in the text. the unigram model, which is an n-gram analysis representation at low level, has a greater predictive potential compared to the bigram and trigram models. while high-level n-gram representations account for the complexities of the human language, their use in predicting consumers’ choices is less efficient than low-level n-gram representations in these economic reviews. acknowledgments the work was done with partial support from the mexican government through the grant a1-s-47854 of the conacyt, mexico and grants 20211784, 20211884, and 20211178 of the secretarı́a de investigación y posgrado of the instituto politécnico nacional, mexico. the authors thank the conacyt for the computing resources brought to them through the plataforma de aprendizaje profundo para tecnologı́as del lenguaje of the laboratorio de supercómputo of the inaoe, mexico. the authors also wish to thank malo pekka and his colleagues for sharing the financial phrasebank dataset [11] and the university of california, irvine machine learning repository of databases for the datasets. 482 o. e. ojo et al. / j. nig. soc. phys. sci. 3 (2021) 477–483 483 references [1] h. gómez-adorno, i. markov, g. sidorov, j. posadas-durán, m. a. sanchez-perez, & l. chanona-hernandez, “improving feature representation based on a neural network for author 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[14] s. t. aroyehun & a. gelbukh, “aggression detection in social media: using deep neural networks, data augmentation, and pseudo labeling”, proceedings of the first workshop on trolling, aggression and cyberbullying (trac-1) santa fe, usa (2018) 90. [15] d. kotzias, m. denil, n. de freitas & p. smyth, “from group to individual labels using deep features”, kdd 2015 (2015) 597. [16] v. athanasiou & m. maragoudakis, “a novel, gradient boosting framework for sentiment analysis in languages where nlp resources are not plentiful: a case study for modern greek”, algorithms 10 (2017) 34. 483 j. nig. soc. phys. sci. 1 (2019) 1–5 journal of the nigerian society of physical sciences original research the delineation of groundwater and geotechnical parameters within marmara area of chikun local government of kaduna state, nigeria n. kurea,∗, i. h daniela, c. g. afuwaia, e. j. adoyia, i. a. bellob adepartment of physics, kaduna state university, kaduna, nigeria bdivision of agricultural colleges, ahmadu bello university, zaria, nigeria abstract geophysical investigation for engineering studies was carried out around faith academy, marmara village, kaduna within the basement complex of north-western nigeria. six (6) vertical electrical sounding (ves) were established using schlumberger configuration. the geoelectric section revealed five subsurface layers defined by the topsoil, which comprises humus, clay and laterite; followed by weathered layer which comprises of sandy soil, fractured layer which constitutes of coarse grain sands and gravels and fresh basement which is porphyritic granite. the resistivities and thicknesses of the topsoil range from 47 − 4212ωm and 2 − 5m respectively. the weathered/fractured basement has average thickness of 42m with resistivity ranging from 350 − 774ωm. the areas found to meet the conditions of high basement resistivity and shallow depth to basement, are found to be at ves a1, a2, a3, a4 and a6. hence, ves a1 have been found to be the competent zones within the study area, and are good for the construction of high-rise buildings, roads and bridges. ves a3, a4 and a6 are areas that do not meet the conditions for construction of a high-rise buildings. however, ves a3, a4 and a6 appears to be viable for groundwater exploration with aquifer thickness ranges between 18 − 42m and depth ranges of 6 − 10m, while the average depth to fresh basement is 43m. keywords: geotechnical, resistivity, chikun, topsoil/laterite, aquifer article history : received: 13-03-2019 received in revised form: 26-03-2019 accepted for publication: 27-03-2019 published: 30-03-2019 c©2019 journal of the nigerian society of physical sciences all rights reserved. communicated by: o. j. abimbola 1. introduction the development of technology has made the quest for water for all purpose in life to drift from ordinary search to prospecting to steady and reliable subsurface or groundwater from boreholes. in nigeria, presently, borehole has rescued the citizenry from acute shortage of water. the geoelectric method has been found to be very reliable for groundwater studies over the years. the geotechnical evaluation of subsurface condition of a site is necessary in generating ∗corresponding author tel. no: 09036041268 email address: nicodemus.kure@kasu.edu.ng (n. kure ) relevant data inputs for the design, and construction of structures and boreholes [1]. the search for groundwater dates back to history. this is because, of all the natural resources, water permeates perhaps must deeply into the all aspect of our life. thus, water is essential to life. its presence or lack of it determines, to great extent, the nature of the natural environment in which we live and majority of our economic activities depend on it. the development and utilization of groundwater resource therefore becomes very imperative [2, 3]. due to the collapse of bridges, buildings and other structures in nigeria, there are needs to provide information on the subsurface sequence and structural disposition necessary for the 1 kure et al. / j. nig. soc. phys. sci. 1 (2019) 1–5 2 buildings. detailed geophysical and geotechnical investigation provides such information to provide adequate solution to prevention against future occurrence. furthermore, lack of proper soil investigation and interpretation of sites conditions amongst the causes that may pose danger to the existing structures in nigeria. the importance of geological exploration methods, as means of exploring the groundwater and zones for engineering structures such as high rising building and bridges cannot be over emphasized. methods including vertical resistivity sounding (ves), seismic refraction, horizontal profiling (wenner) and very low frequency (vlf) have been successfully used to explore for groundwater [4, 5]. the vertical electrical sounding is a direct current resistivity tool of geophysics for the determination of the subsurface of a place. it has been used extensively for the determination of the aquifer potential for the drilling of boreholes. these days, it is used for in-depth geotechnical studies to determine the suitability of a site for the building of heavy structures including high rise buildings, bridges, and stadia. hence, the method can be used in the study of subsurface layering of a place, to provide information that will help in determining the nature of soil at the site and its stratification, selecting the type and depth of foundation problems (e.g. expansive soil and collapsible soil) and determining the depth and nature of bedrock, if and when encountered. in this work, d.c resistivity method-considered to be the quickest and the most economic technique, using veswill be used to provide information on the zone. 2. theory the theoretical study of the earth resistivity methods is to consider the case of completely homogenous isotropic medium. the equation which gives the potential due to a single point source of current at surface can be deduced from two basic equations; ohm’s law and divergence condition. the ohm’s law is given as: ∆ · j = 0 (1) the potential due to a single point source of current at earth surface is given as: u = ρi 2πr (2) where ρ denotes the earth density, i is the current, 2π is a constant, and r is resistance. the current i, is passed through current electrode c1, c2 as shown in figure below (figure 1): the potential difference at m is um = iρ 2π [ 1 r1 − 1 r2 ] . (3) the potential difference at n is un = iρ 2π [ 1 r3 − 1 r4 ] . (4) figure 1. general four electrode configuration for resistivity measurement. the difference in potential at m and n is given as ∆u = um−un un = iρ 2π [ 1 r3 − 1 r4 − 1 r3 + 1 r4 ] . (5) hence, the resistivity is given by ρa = ∆u i [ 2π { 1 r3 − 1 r4 − 1 r3 + 1 r4 }] . (6) let k = 2π { 1 r3 − 1 r4 − 1 r3 + 1r4 } , therefore, ρa = kr (7) where, k is a geometric factor and r = ∆u/i. 3. methodology 3.1. geology of the study area marmara is a settlement of chikun local government area of kaduna state, nigeria. it is located geographically between latitude 10◦3′ and 10◦50′n and longitude 6◦4′ and 7◦5′e as shown in figure 2. chikun local government area of kaduna state geographically is in the north-west part of nigeria which is part of west africa zone. these unconsolidated materials are known to reflect some dominant hydrologic properties, and the highest groundwater yields in basement complex area found in areas of thick overburden overlying fractured zones and are characterized by relatively low resistivity [6]. groundwater occurrence in the area could be grouped into three; the weathered/fractured basement complex, newer basalts and river alluvium. climate and vegetation of study area has a typical savannah climate with distinct wet and dry seasons, the wet season ranges from april to october, while the dry season between end october to march, the average annual rainfall for kaduna is 300mm. rainfall generally reaches its peak in august and it has a characteristic mean annual temperature of 29◦c in march/april. the dominant river kaduna controls the course of most of the rivers and streams in the area [7] 2 kure et al. / j. nig. soc. phys. sci. 1 (2019) 1–5 3 figure 2. map of the study area showing all the ves stations 3.2. materials and method the basic requirement used for the study is ohmega resistivity meter which displays resistance value digitally as computed from ohm’s law. the ohmega resistivity meter is powered by a 12.5 volts, d.c. power source. other materials that accompany the equipment are, two measuring tapes, four hammers, thirty steel electrodes and four connecting wires (cables) for current and potential electrodes and a global positioning system (gps). generally four-electrode array are used at the surface, one pair for introducing current into the earth and the potential difference these established are measured in the vicinity with the second electrode pair. large numbers of electrode arrangement have been used for resistivity exploration[5]. any of these can be used for lateral variation of resistivity or vertical variation of resistivity as a function of the current depth. in resistivity method, current are driven into the ground. any variation of subsurface resistivity (ρ) alter the current flow which in turn affects the distribution of electric potentials. the potential, these established, are measured at the surface. deviation of these pattern of potential difference from uniform earth represent the geological features of resistivity exploration. 3.3. choice of electrode configuration the choice of a configuration for prospecting is dependent on a number of factors amongst which are the type of investigation required, the terrain of the area of the investigation, and the position of the suspected geological structure. in vertical electrical sounding (ves) measurement, the centre of the electrode spread remains fixed but the separation of electrodes is progressively increased, hence choice of schlumberger configuration (figure 1), this configuration is therefore more cost effective since it saves time and manpower[1]. applying equations 5 and 6 to the above array gives k = 2π/ { 1 a − b − 1 a + b − 1 a + b + 1 a − b } k = π/ { 1 (a − b) − 1 a + b } k = π/ { a2 2b − b 2 } , (8) therefore, ρa = π/ { a2 2b − b 2 } ∆/i (9) where, • a = separation between current electrode at the centre of the configuration, • b = separation between potential electrode at the centre of the configuration, and • ρa= apparent resistivity of the earth for a schlumberger array. this last expression gives the approximate relationship between the apparent resistivity ρa, and the approximate resistivity value term usually given by terrameter during field measurement. the coefficient of ∆u/i is the geometric factor which is characteristics of the spread used. the electrical resistivity data was acquired with abem terrameter sas 300 along with other equipment cables and electrodes. table 1. resistivity range of some rock types (compiled from [2, 4, 9, 10]) rock type resistivity (ω − m) topsoil/clay/silt 65 200 laterite/indurated laterite 45 800 weathered basement 2 – 220 fractured basement 218 520 fresh basement < 1000 3.4. data interpretation the interpretation of the data was done using a computer program (res1d version 1.00.07 beta). to interpret the data, some initial value parameters were found from the observed field curve and fed into the program. the initial value parameters were taken such that each points of maxima, minima and inflection indicate the existence of layer boundaries. the program uses these values to design a model curve, comparing it with the observed field resistivity curve. where the modelled curve and the observed field curve are not in agreement, the initial value parameters were altered until the best agreement between the modelled curve and the observed field curve is obtained. at the point when the two curves were in best agreement and minimum error, the layer resistivity and thickness were recorded[8]. the geoelectric sections show the average variations of resistivity and thickness values of layers within the depth penetrated in the study area at the indicated sections. generally, the sections revealed two to four subsurface layers: topsoil/laterite, indurated laterite, weathered and fresh basement layers. 3 kure et al. / j. nig. soc. phys. sci. 1 (2019) 1–5 4 figure 3. typical resistivity curve. 3.5. data processing vertical electrical sounding (ves) data was collected on six (6) ves points using the schlumberger configuration. the first stage in any interpretation of apparent resistivity sounding curves is to note the curve shape [10, 11]. before applying a more complicated method of interpretation, it is useful to consider a rough idea. the data collected in the study area were reduced and computed. the computed apparent resistivity values were then fed into a computer geosoftware (resid version 1.00.07 beta) for resistivity curve interpretation. the interpreted data was used to generate the final model geoelectric parameters presented in figure 3 which shows a typical example of resistivity curve. 3.6. resistivity values table 2 shows the resistivity range of geoelectrical layer compiled from some previous works around the study area. table 2. resistivity values of some rock types basement area (compiled from [1, 2, 3, 9, 10]) soil and rock types resistivity value (ωm) fadama loam 30 90 sandy clay and sandy silty 100 200 sand and gravel laterite 10 150 weathered basement 20 200 fractured basement 500 1000 fresh basement > 1000 4. results and discussion results obtained indicates that the terrain is highly variable in thickness and resistivity. according to alao[11] and fadele et al. [12], the engineering competence of the subsurface can be qualitatively evaluated from the layer resistivity, since the higher the layer resistivity value, the higher the competence of the layer. the final model geoelectric parameters along the six ves stations were used for the preparation of the geoelectric/geologic figure 4. shows the geoelectric and geologic section of the study area. sections for one of the profiles (figure 4). ves a1 is recommended for the construction of high-rise buildings, this is because the depth to basement is approximately 25m. hence, the subsurface layers at a1 are not favorable for groundwater exploitation. ves a3, a4 and a6 are not recommended for highrise buildings. however, surface layers appear to be highly favorable for groundwater exploration which has the highest aquifer thickness of 46m at a depth of 48m. this is because the thickness of the aquifer at this point ranges from 18m to 42m with a depth of approximately 6m to 10m respectively. according to aboh [1], the areas underlain with low bed rocks resistivity (< 1000ωm) in figure 4 could indicate fractures or faults. hence, are less competent. based on the resistivity values of the different geoelectric layers and the various geologic units obtained, the study has identified ves stations a1, a2 and a5 as competent zones. these regions were chosen due the fact that, there was no indication of geologic trend such as fracture or fault that could cause structure failure. fadele, et al. [12], dogara, et al. [13] and alao [11], noted that the seasonal variation in the saturation of clay which causes ground movement could cause by clay swells and shrinkages; these factors could contribute to structural defect of building but the presence of consolidated laterite in the study area beneath the clayey topsoil which extends beyond 6.5m, could reduce the danger posed by clay formation to large buildings. however, the topsoil resistivity of the chosen zones is characterized with low resistivity and thin in thickness, which may not be good enough to support massive buildings. hence, this superficial cover is loose humus/loam and clay formation at the topsoil in some part of the study area; thus need to be totally excavated in any future structural and engineering works. in summary, the interpreted data shows that there are no indications of any major trends such as fracture or faults in the area chosen as competent zones (a1, a2 and a5), that could aid building subsidence. 4 kure et al. / j. nig. soc. phys. sci. 1 (2019) 1–5 5 5. conclusion the investigation shows the geo-electric and geologic sections derived from the study area which has revealed five subsurface layers, namely: lateritic topsoil followed by clayey/ sandy/ silty, layer, weathered basement and fractured/ fresh basement rocks. a broad section of the area was classified as competent for civil works. there are no indications of any major linear structure such as fracture, faults or voids that could aid building subsidence in the study area. however, the seasonal variation in the saturation of clay which causes ground movement could cause havoc in building construction due to the sensitiveness of soils to moisture loss or gain (swells and shrinkages) in ves a3 and a5. considering the presence of consolidated laterite which extends beyond 3.0 m could reduce the danger posed by clay formation to large buildings as well as tarred roads. in summary, the interpreted data shows that there are no indications of any major trend such as fracture, cavities, voids or faults that could aid building subsidence in the area identified as competent zones. other probable factors such tree roots, organic deposits, superficial cover that could cause foundation defects can therefore be eliminated by total evacuation of the topsoil. the lithologies thickness and depth are as follows: topsoil ranges from 1.0m − 1.8m; indurated laterite ranges from 1.9m−6.8m; weathered/fractured basement has infinite thickness. the areas found to meet the conditions of high basement resistivity and shallow depth to basement, are found to be at ves a1, a2, a3, a4 and a5. hence, a1 have been found to be the competent zones within the study area, and are good for the construction of highrises, roads and bridges. ves a3, a4 and a6 those not meet the conditions for construction of highrises building. however, ves a3, a4 and a6 areas appears to be very viable for groundwater exploration with aquifer thickness ranges from 18m−42m with depth ranges 6m−10m. and the average depth to fresh basement is 43m. references [1] h. o. aboh, “geotechnical characterization of surface materials in kaduna area, kaduna state”, zuma journal of pure and applied science 4 (2002) 23. [2] c. o. ajayi & m. hassan, “the delineation of aquifer overlying the basement complex in the western part of kubani basin of zaria nigeria”, journal of mining and geology 26 (1990) 117. [3] n. kure, h. o. aboh, r. jimoh, o. a. joseph & i. h. daniel, “the delineation of potential groundwater aquifers within basement complex in abu zaria, nigeria”, british journal of applied science and technology 19 (2017) 1. [4] e. m. shemang, electrical depth sounding of selected well sites within the kubani river basin, zaria, nigeria, unpublished msc thesis, physics department, abu zaria, 1990. [5] o. koefoed, “an appropriate method of resistivity sounding interpretation”, geophysical prospecting 24 (1979) 617. [6] m. o. olorumfemi, s. a. fasuyi, “aquifer types and geoelectric/hydrogeologic characteristics of part of central basement terrain of nigerian (niger state)”, journal of earth science 16 (1993) 309. [7] p. o. udo, physical geography of nigerian, heineman education publishers, 1982. [8] k. m. lawal, the application of reduction to the pole technique for the interpretation of aeromagnetic data of zaria area, north central nigeria, 1998. [9] n. k. abdullahi, e. e. udensi, a. iheakanwa & b. e. eletta, “geoelectrical method applied to evaluation of groundwater potential and aquifer protective capacity of overburden units”, british journal of applied science and technology 4 (2014) 2024. [10] m. w. telford, l. p. geldart, r. e. sheriff & d. a. keys, applied geophysics, cambridge university press, london, 1970. [11] j. o. alao, delineation of the interfacial configuration in a section of millennium city, kaduna, unpublished msc. thesis, physics dept. kaduna state university, kaduna, 2017. [12] s. i. fadele, s. b. jatau, & n. o. patrick, “geophysical engineering investigation around makiyaye village, shika area within the basement complex of north-western nigeria”, international journal of engineering research and applications (2012) 1143. [13] m. d. dogara, j. alao, h. abdullahi, j. ezekiel, j. george, & r. a. ahammed, “delineation of the geotechnical parameters within the kaduna refining and petrochemical corporation layout”, world journal of applied physics 2 (2017) 36. 5 j. nig. soc. phys. sci. 3 (2021) 181–188 journal of the nigerian society of physical sciences analysis of indoor radon level and its health risks parameters in three selected towns in port harcourt, rivers state, nigeria orlunta aloysius ndubisia,∗, briggs-kamara margaret apaemia, sigalo friday barikpea, iyeneomie tamunobereton-aria adepartment of physics, faculty of science, rivers state university, nkpolu oroworukwo, port harcourt, rivers state, nigeria. abstract analysis of indoor radon level and its health risk parameters has been carried out in borikiri (bt), diobu (dr), and rebisi (rb) towns in port harcourt, rivers state, nigeria. a pocket sized corentium arthings digital radon detector meter was used to record the indoor radon concentration levels. the geographical coordinates were recorded using a hand-held geographical positioning system (gps) for the various sample points. a total of ten houses were measured for each town making a total of 30 sample points for the three communities. the results obtained show that for borikiri town, the concentration level varied from 30.7100 − 19.9800 bqm−3 with an average of 11.32 ± 2.59 bqm−3. the annual absorbed dose varied from 7.7478 − 1.1202 msv/yr with a mean value of 2.59 ± 0.65 msv/yr while the annual equivalent dose rate varied from 0.829 − 0.336 msv/yr with an average of 0.69±0.16 msv/yr the excess life time cancer risk calculated for seventy years (70yrs) varied from 6.510−0.941 with an average of 2.45 ± 1.71. the results of the indoor concentration level for diobu town ranged from 37.74 − 5.9200 bqm−3 with a mean value of 12.95 ± 2.91 bqm−3. the annual absorbed dose for the area ranged from 9.5214 − 1.1494 with an average of 3.26 ± 0.73 msv/yr, the annual equivalent dose rate varied from 0.694−0.359 with a mean of 0.78±0.8, the excess life time cancer risk calculated for seventy years ranged from 8.000 − 1.725 with a mean of 2.91 ± 0.61. the indoor concentration level for rebisi town ranged from 12.9500 − 4.0700 bqm−3 with an average of 8.55 ± 1.00, the annual absorbed dose ranged from 3.2671 − 1.0268 msv/yr, the annual equivalent dose rate varied from 0.784 − 0.269 with an average of 0.52 ± 0.06, the excess life time cancer risk of 2.745 − 0.863 with an average of 1.82 ± 0.21. the results of the indoor concentration levels, the annual absorbed dose and the annual effective dose rate are all below the icrp safe limit. however, the results of the excess life time cancer risk are all higher than the icrp safe standard limit of 0.029 × 10−3 . doi:10.46481/jnsps.2021.203 keywords: indoor radon, annual absorbed dose, annual effective dose, excess life cancer risk article history : received: 20 april 2021 received in revised form: 10 may 2021 accepted for publication: 09 july 2021 published: 29 august 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: w. a. yahya 1. introduction radon is a radioactive gas which is colourless, odourless and tasteless. it is predominantly found in rock samples, bedrock formations, soil and ground water all over the world, [1]. radon222 is the immediate decay product of radium226 during the ∗corresponding author tel. no: +234(0)8060700503 email address: aloyndubisi@gmail.com (orlunta aloysius ndubisi ) decay series of uranium-238 which is the source of radon-222. it has a half-life of 3.81 days. radon-222 decays to polonium298 with the release of an alpha particle and consequently to other progenies of radon or radon daughters. radon has been classified by the international agency for research on cancer [2] as a carcinogenic gas. the world health organization [3] also classify radon as 181 orlunta et al. / j. nig. soc. phys. sci. 3 (2021) 181–188 182 the second highest cause of lung cancer after cigarette smoking. it was estimated that radon-222 causes between 3% to 14% of all lung cancers. radon is the leading cause of lung cancer among non-smokers. it is the cause of about fifteen percent of all lung cancer cases throughout the world [4]. people are exposed to high radon level in small houses than in bigger apartments [5]. the entry routes of indoor into the houses includes, the door, the windows, cracks on the walls, sinks, basements, floors etc. the health effects as a result of higher exposure to radon222, is radiation induced. inhalation is the main route through which humans are exposure to radon radiation. the dose contribution to radon may be small, but the inhalation of the progenies of radon which have very short half-life can deposit nonhomogenously in the human respiratory track and irradiate the bronchial epithelium which is usually very harmful. two of the radon progenies (daughters) polonium-214 and polonium-218 release the highest amount of alpha radiation dose to the lungs [6]. when these radioactive particles settle in a person’s lung they can cause damage to the mucosa linings of the lungs. a prolonged exposure to radon-222 radiation can also lead to series of damage to the pulmonary mucosa which could result to lung cancer. an individual’s average radiation come from the decay products of radon-222 such as, polonium-218 and polonium-214. these products of radon-222 are in the solid form. they could be attached or unattached to the surface of aerosols, dusts and smoke particles. they can adhere deeply or stick to the lungs where they irradiate and penetrate the cells of the mucosa membranes, bronchi and pulmonary tissues. the ionizing radiation which emanate from the radioactive decay of radon-222 begins the process of carcinogenesis in the human lungs. it is estimated that about 90% of the radon progenies can attach to airborne particles [7]. also, the unattached fraction of the particles which constitutes about 10% has a higher rate of deposition and is more efficient in delivering doses to the sensitive cells of the lungs [8]. the more the concentration of radon in homes the greater the risk of lung cancer due to radon exposure. the risk of lung cancer increases for every100bq/m 3 increase in radon concentration, according to the world health organization (who, 2016). however, there is no thresholds value below which radon exposure carries no risk of lung cancer. therefore it is important that every country should set up a national reference level “as low as reasonably achievable” alara [9]. the reason for carrying out radon measurements in these towns is because radon-222 concentration level has not been done within port harcourt local government. therefore, we wanted to find out the radon-222 concentration level and compare the results with international set standards. also, measurement of radon-222 concentration level in these areas will provide baseline studies and literature for other researchers. it is well known fact that these areas are densely populated with building styles that are poorly ventilated therefore, it was necessary to find out if the residents of these towns are exposed to higher dose of radon-222 from their environment. additionally, there has been incident of lung cancer cases within these areas in recent times even though not officially reported, therefore it was important to confirm if it was as a result of high level of radon exposure. figure 1. map of study area figure 2. contour map of borikiri town 2. materials and methods the radon detector used in this work is airthings corentium digital radon detector designed in oslo norway in the year 2019. it is calibrated to measure radon-222 in picocurie per litre (pci/l). the airthings corentium digital detector can be used to monitor radon concentration levels for a minimum period of 48 hours, for daily, weekly, monthly and yearly. the radon meter works on the principle that the radon diffuses into a detection chamber. as the atom decay they emit energetic alpha particles. the energetic alpha particles are detected by a silicon photodiode. the alpha particle generates a small signal current when it hits the photodiode. by the use of a low power 182 orlunta et al. / j. nig. soc. phys. sci. 3 (2021) 181–188 183 table 1. radon levels in borikiri town (bt) and the geographical coordinates of the sample points. s/n location gps coordinates radon levels (pci/l) borikiri town (bt) latitudes longitudes 1.0 bt1 n4◦49 ′ 55.09524 ′′ e7◦0 ′ 8.298720 ′′ 0.21 2.0 bt2 n4◦44 ′ 50.30664 ′′ e7◦2 ′ 37.05684 ′′ 0.18 3.0 bt3 n4◦44 ′ 48.28200 ′′ e7◦2 ′ 40.37400 ′′ 0.83 4.0 bt4 n4◦50 ′ 11.16600 ′′ e6◦59 ′ 49.2420 ′′ 0.54 5.0 bt5 n4◦45 ′ 24.35400 ′′ e7◦2 ′ 3.384000 ′′ 0.12 6.0 bt6 n4◦44 ′ 47.92800 ′′ e7◦2 ′ 36.49200 ′′ 0.24 7.0 bt7 n4◦44 ′ 45.58200 ′′ e7◦2 ′ 36.76800 ′′ 0.27 8.0 bt8 n4◦45 ′ 41.24400 ′′ e7◦1 ′ 42.47400 ′′ 0.15 9.0 bt9 n4◦45 ′ 34.58400 ′′ e7◦1 ′ 38.63400 ′′ 0.21 10.0 bt10 n4◦44 ′ 37.01838 ′′ e7◦1 ′ 37.78428 ′′ 0.37 table 2. radon levels in diobu residential area (dr) and the geographical coordinates of the sample points. s/n location gps coordinates radon levels (pci/l) diobu residential area (dr) latitudes longitudes 1.0 dr1 n4◦47 ′ 55.956 ′′ e6◦59 ′ 22.740 ′′ 0.22 2.0 dr2 n4◦49 ′ 9.1560 ′′ e6◦59 ′ 32.280 ′′ 0.31 3.0 dr3 n4◦48 ′ 4.5550 ′′ e6◦59 ′ 15.906 ′′ 0.16 4.0 dr4 n4◦54 ′ 58.524 ′′ e6◦59 ′ 49.692 ′′ 0.27 5.0 dr5 n4◦55 ′ 8.3860 ′′ e6◦59 ′ 44.119 ′′ 1.02 6.0 dr6 n4◦48 ′ 20.874 ′′ e6◦59 ′ 16.542 ′′ 0.31 7.0 dr7 n4◦47 ′ 33.690 ′′ e6◦59 ′ 32.598 ′′ 0.24 8.0 dr8 n4◦47 ′ 35.592 ′′ e6◦59 ′ 31.740 ′′ 0.46 9.0 dr9 n4◦48 ′ 18.100 ′′ e6◦59 ′ 09.500 ′′ 0.29 10.0 dr10 n4◦48 ′ 35.7282 ′′ e6◦59 ′ 16.11168 ′′ 0.43 table 3. radon levels in rebesi town (rb) and the geographical coordinates of the sample points. s/n location gps coordinates radon levels (pci/l) rebesi town (rb) latitudes longitudes 1.0 rb1 n4◦49 ′ 19.53522 ′′ e6◦57 ′ 36.53706 ′′ 0.35 2.0 rb2 n4◦49 ′ 13.3944 ′′ e6◦57 ′ 32.43258 ′′ 0.19 3.0 rb3 n4◦49 ′ 12.84108 ′′ e6◦57 ′ 33.7956 ′′ 0.27 4.0 rb4 n4◦49 ′ 7.63932 ′′ e6◦57 ′ 29.95014 ′′ 0.16 5.0 rb5 n4◦49 ′ 9.0387 ′′ e6◦57 ′ 24.7416 ′′ 0.31 6.0 rb6 n4◦49 ′ 2.71242 ′′ e6◦57 ′ 22.00974 ′′ 0.12 7.0 rb7 n4◦49 ′ 9.59052 ′′ e6◦57 ′ 17.23656 ′′ 0.24 8.0 rb8 n4◦49 ′ 4.3086 ′′ e6◦57 ′ 57.75876 ′′ 0.33 9.0 rb9 n4◦49 ′ 3.9673 ′′ e6◦57 ′ 25.41666 ′′ 0.11 10.0 rb10 n4◦49 ′ 7.37353 ′′ e6◦57 ′ 41.99568 ′′ 0.23 amplifier stage, the signal current is converted into a large voltage signal. the maximum amplitude of the voltage signal is detected and sampled by an analogue to digital converter (adc). the amplitude of this signal is proportional to the energy of the alpha particle that hit the photodiode. the brain of the monitor is a micro-controller which registers the time and the energy of every detected particle. this information is used to calculate the mean radon concentration for, daily, weekly, monthly and yearly periods. indoor radon concentration measurement was carried out in three towns in port harcourt using the arthings digital radon detector while the geographical position system (gps) was used for the measurement of the geographical coordinates of the sample points. a total of 30 sample points were measured for the three towns, borikiri (bta) ,diobu (dr) and rebisi (rb) making 10 samples each. the arthings corentium digital radon detector was placed in the dwelling room at a height of about 50cm above the ground 183 orlunta et al. / j. nig. soc. phys. sci. 3 (2021) 181–188 184 table 4. computed values of annual effective dose and annual equivalent dose rate of borikiri town s/n s/pts crn (pci/l) crn (bq/m3) drn (msv/y) aedr (msv/y) elcr×10−3 1.0 bt1 0.21 7.7700 1.9603 0.470 1.6470 2.0 bt2 0.18 6.6600 1.6802 0.403 1.4117 3.0 bt3 0.83 30.7100 7.7478 1.859 6.5096 4.0 bt4 0.54 19.9800 5.0407 1.210 4.2351 5.0 bt5 0.12 4.4400 1.1202 0.269 0.9411 6.0 bt6 0.24 8.8800 2.2403 0.538 1.8823 7.0 bt7 0.27 5.5500 1.4002 0.336 1.1764 8.0 bt8 0.15 7.7700 1.9603 0.470 1.6470 9.0 bt9 0.21 13.6900 3.4538 0.829 2.9019 10.0 bt10 0.37 7.7700 1.9603 0.470 1.6470 average 0.31 ± 0.07 11.32 ± 2.59 2.59 ± 0.65 0.69 ± 0.16 2.40 ± 0.55 table 5. computed values of excess lifetime cancer risk for ages of ; 70(std), 60, 50, 40 and 30 yrs of borikiri town s/n s/pts elcr ×10−3 70 yrs(std) elcr ×10−3 60 yrs elcr ×10−3 50 yrs elcr ×10−3 40 yrs elcr ×10−3 30 yrs 1.0 bt1 1.647 1.411 1.176 0.941 0.706 2.0 bt2 1.412 1.210 1.008 0.807 0.605 3.0 bt3 6.510 5.578 4.649 3.719 2.789 4.0 bt4 4.235 3.629 3.024 2.420 1.815 5.0 bt5 0.941 0.807 0.672 0.538 0.403 6.0 bt6 1.882 1.613 1.344 1.075 0.807 7.0 bt7 2.118 1.815 1.512 1.210 0.907 8.0 bt8 1.176 1.008 0.840 0.672 0.504 9.0 bt9 1.647 1.411 1.176 0.941 0.706 10.0 bt10 2.902 2.487 2.072 1.658 1.243 average 2.45 ± 1.71 2.10 ± 1.47 1.75± 1.22 1.40 ± 0.98 1.05 ± 0.73 table 6. computed values of, annual effective dose and annual equivalent dose rate and risk of diobu residential area (dr) s/n s/pts crn (pci/l) crn (bq/m3) drn (msv/y) aedr (msv/y) 1.0 dr1 0.22 8.1400 2.0536 0.493 2.0 dr2 0.31 11.4700 2.8937 0.694 3.0 dr3 0.16 5.9200 1.4935 0.358 4.0 dr4 0.27 9.9900 2.5204 0.605 5.0 dr5 1.02 37.7400 9.5214 2.285 6.0 dr6 0.31 11.4700 2.8937 0.694 7.0 dr7 0.24 8.8800 2.2403 0.538 8.0 dr8 0.46 17.0200 4.2939 1.031 9.0 dr9 0.29 10.7300 2.7071 0.650 10.0 dr10 0.43 8.1400 2.0536 0.493 average 0.37 ± 0.08 12.95 ± 2.91 3.26 ± 0.73 0.78 ± 0.78 and about 150cm from both the window and door and 25cm from the walls, [10] the detector was kept for a period of two days (48hrs.) in a given dwelling before relocating to another house or home. the windows and doors were kept closed throughout the period of the measurement to ensure that the indoor air is not distorted to achieve accuracy within the period of 48 hours. 2.1. radon risk parameters 2.1.1. annual absorbed dose(drn) from radon concentration the annual absorb dose calculated from the radon concentration is the magnitude of energy delivered in a tissue from exposure to ionizing radiation within a specified period. the physics of the equation employed in the computation of the annual absorbed dose rate according to the report of united nation’s scientific committee on the effects of atomic radiation 184 orlunta et al. / j. nig. soc. phys. sci. 3 (2021) 181–188 185 table 7. computed values of excess lifetime cancer risk for ages of; 70(std), 60, 50, 40 and 30 yrs of diobu residential area (dr) s/n s/pts elcr ×10−3 70 yrs(std) elcr ×10−3 60 yrs elcr ×10−3 50 yrs elcr ×10−3 40 yrs elcr ×10−3 30 yrs 1.0 dr1 1.725 1.479 1.232 0.986 0.739 2.0 dr2 2.431 2.083 1.736 1.389 1.042 3.0 dr3 1.255 1.075 0.896 0.717 0.538 4.0 dr4 2.118 1.815 1.512 1.210 0.907 5.0 dr5 8.000 6.855 5.713 4.570 3.428 6.0 dr6 2.431 2.083 1.736 1.389 1.042 7.0 dr7 1.882 1.613 1.344 1.075 0.807 8.0 dr8 3.608 3.092 2.576 2.061 1.546 9.0 dr9 2.274 1.949 1.624 1.299 0.975 10.0 dr10 3.372 2.890 2.408 1.927 1.445 average 2.91± 0.61 2.50± 0.52 2.08± 0.43 1.66± 0.35 1.25 ± 0.26 table 8. computed values of annual effective dose and annual equivalent dose rate of rebisi town (rb) s/n s/pts crn (pci/l) crn (bq/m3) drn (msv/y) aedr (msv/y) 1.0 rb1 0.35 12.9500 3.2671 0.784 2.0 rb2 0.19 7.0300 1.7736 0.426 3.0 rb3 0.27 9.9900 2.5204 0.605 4.0 rb4 0.16 5.9200 1.4935 0.358 5.0 rb5 0.31 11.4700 2.8937 0.694 6.0 rb6 0.12 4.4400 1.1202 0.269 7.0 rb7 0.24 8.8800 2.2403 0.538 8.0 rb8 0.33 12.2100 3.0804 0.739 9.0 rb9 0.11 4.0700 1.0268 0.246 10.0 rb10 0.23 8.5100 2.1470 0.515 average 0.23 ± 0.03 8.55 ± 1.00 2.16 ± 0.25 0.52 ± 0.06 table 9. computed values of excess lifetime cancer risk for ages of 70(std) 60, 50, 40 and 30 yrs of rebisi town (rb) s/n s/pts elcr ×10−3 70 yrs(std) elcr ×10−3 60 yrs elcr ×10−3 50 yrs elcr ×10−3 40 yrs elcr ×10−3 30 yrs 1.0 rb1 2.745 2.352 1.960 1.568 1.176 2.0 rb2 1.490 1.277 1.064 0.851 0.638 3.0 rb3 2.118 1.815 1.512 1.210 0.907 4.0 rb4 1.255 1.075 0.896 0.717 0.538 5.0 rb5 2.431 2.083 1.736 1.389 1.042 6.0 rb6 0.941 0.807 0.672 0.538 0.403 7.0 rb7 1.882 1.613 1.344 1.075 0.807 8.0 rb8 2.588 2.218 1.848 1.479 1.109 9.0 rb1 0.863 0.739 0.616 0.493 0.370 10.0 rb10 1.804 1.546 1.288 1.031 0.773 average 1.82 ± 0.21 1.55 ± 0.18 1.29 ± 0.15 1.04 ± 0.12 0.78 ± 0.09 [7, 11] is given as: drn(ms v/y) = crn.d.h.t.f (1) where crn = the the measured concentration of indoor radon222 in bqm−3, f = radon-222 equilibrium indoor factor of (0.4), t = the indoor-222 occupancy time 7000 hr, (0.8×24 hr× 365), h = indoor radon-222 occupancy factor (0.4), and d = dose conversion factor (9.0 × 10−6 ms v/hr per bqm−3). 2.1.2. annual effective dose rate (aedr) from radon concentration the annual dose rate was calculated by applying a tissue and radiation weighting factor [12, 13]. the inhalation dose equation is given as: aedr(ms v/y) = drn.wr.wt (2) where drn = indoor radon concentration, wr = radiation 185 orlunta et al. / j. nig. soc. phys. sci. 3 (2021) 181–188 186 figure 3. contour map of diobu residential area (dr) figure 4. contour map of rebesi village (rb) figure 5. average indoor radon level of sample locations and icpr reference 1 weighting factor for alpha particles (20), and wt = tissue weighting factor for the lung (0.12). 2.1.3. excess life time cancer risk(elcr) from radon concentration the excess life time cancer risks (elcr) is the potential carcinogenic effects, from the calculation based on probability of cancer induced incidence in a population. it indicates the figure 6. annual absorbed dose of sample locations and icrp standard figure 7. annual effective dose rate (aedr) and icrp standard figure 8. excess life time cancer risk (elcr) of sample locations with world standard chances of contracting a cancer from the exposure from radiation or toxic chemical substances for a specific life time. according to [14] the excess life time risk was calculated from the equation: eclr = aedr × dl × rf (3) where aedr = annul effective dose rate, dl = average duration of life (70 years), rf = risk factor (0.05) 186 orlunta et al. / j. nig. soc. phys. sci. 3 (2021) 181–188 187 statistical analysis the mean values and standard deviation for the radon concentrations was computed using equations (4) and (5). this can also be computed with the aid of a histogram and contour maps showing the spatial distribution of radon concentration and its health risk parameters which will also be plotted with the aid of statistical package for social sciences (spss) version (22) and surfer-8 contour map software. µx = ∑n i x n (mean value) (4) s.d. = √∑n i (xiµx) 2 n − 1 (5) 3. results and discussion indoor radon concentration and their geographical coordinates for the three borikiri, diobu and rebisi are presented in tables 1, 2 and 3, respectively. tables 4 – 9 represent the computed values of annual effective dose and annual equivalent dose rate and the values of excess lifetime cancer risk for ages of: 70(std), 60, 50, 40 and 30 yrs of borikiri , diobu and rebesi towns respectively. from the results obtained, the indoor radon concentration level for the towns as 30.7100 − 19.9800 bq/m3, 37.7400 − 5.9200 bq/m3 and 12.9500 − 4.0700 respectively. the annual absorbed dose for the thre communities varied as 7.7478−1.1202 msv/yr, 9.5214 − 1.1494 msv/yr and 3.2671 − 1.0268 msv/ yr with averages of 0.69 ± 0.16m sv/yr, 3.26 ± 0.73 msv/yr and 2.16 ± 0.25 msv/yr. the annual equivalent dose rate ranged from 0.829−0.336 msv/yr, 0.694−0.359 msv/yr, 0.784−0.269 msv/yr with mean values of 0.69 ± 0.16 msv/yr, 0.78 ± 0.78 msv/yr, 0.52 ± 0.06 msv/yr. the excess life time cancer risk calculated for seventy years (70yrs) varied from 6.510 − 0.941, 8.000 − 1.725, 2.745 − 0.863 with mean values of 2.45±1.71, 2.91±0.61, 1.82±0.21. the measured values of the indoor radon concentration and the calculated values of the annual absorbed dose, annual equivalent dose rate are all lower than the set standard by the international commission on radiological protection (icrp). the calculated values of the excess life time cancer risk calculated for life time of 70yrs, 60 yrs, 50 yrs, 40 yrs and 30 yrs are all higher than the set limit by international commission on radiological protection (icrp). however, there are no observed cases of lung cancer epidemic in these areas. the contour maps of the study area are plotted in figures 2, 3, and 4, respectively. the red colour on the contour map represents area of high radon concentration while the green colour represents area of very low radon distribution. also, the values of the indoor concentration levels, annual absorbed dose, annual equivalent dose rate and the excess life time cancer risk are plotted in the bar-chart of figures 5, 6, 7, and 8, respectively. from the result obtained, there is no appreciable difference between the different study areas, this is due to the similarities in the building design, materials used in the building, ventilation rate of the living houses and the life style of the people. these results obtained are comparable to the indoor radon concentration level for different types of buildings in covenant university, nigeria by [15]. the mean radon concentration for the three different building types glass, brick house and basement house were 14.96 bqm−3, 10.74 bqm−3, and 144,6 bqm−3, respectively. however, the concentration level for the basement house was higher than the recommended value by the international commission on radiological protection (icrp). also, the indoor radon level concentration measured by [16] in obasfemi awolowo university ile-ife are similar to the indoor radon level measured in borikiri, (bt), diobu (dr) and rebisi (rb) towns. the radon level obtained from the offices varied from 0.0 − 5.3 bqm−3. the average value obtained was 0.9 pci/l. similarly, the health risk parameters, the annual absorbed dose, the annual equivalent dose rate and the excess life time cancer risk agreed with the calculated values in borikiri, diobu and rebisi towns. the indoor radon level concentration measured at borikiri, diobu and rebisi towns are also comparable to the work done earlier by [17] in okrika local government area in rivers state, nigeria. the average concentration level was 19.36±bqm -3 for mud house, while the overall average indoor radon concentration was 11.37±3.28 bqm−3 for okirika local government. also, the overall average of the annual absorb dose was 3.36 ± 0.26 msv/yr, the mean annual effective dose rate was 0.15 ± 0.42 msv/yr, these values are all lower than the icrp set standard. however, the overall excess life time cancer risk calculate from the indoor radon concentration level was (0.52± 0.15)×10−3 and all above the world standard of 0.029 × 10−3. 4. conclusion the analysis of indoor radon concentration level and its health risk parameters has been done in borikiri, diobu and trans-amadi towns in port harcourt, rivers state, nigeria using the corentium arthings radon digital detector. the concentration levels measured for the three towns were all below the action level of 200 − 600 bq/m3 given by the international commission on radiological protection (icrp). however, the excess life time cancer risk for the three towns were all higher than the world average. it was also observed that there was no appreciable difference in the indoor radon concentration level between the three towns borikiri, diobu and rebisi respectively. this is because of the similarities in the building styles, materials used for the buildings, the life style of the people and the ventilation methods. references [1] t.j ojo and i. ajayi, “outdoor radon concentration in the township of adoekiti nigeria”, journal of atmospheric pollution 3 (2015) 21. [2] iarc cd-rom, globocan 1: cancer incidence and mortality worldwide in 1990, lyon, international atomic energy agency press (1998). 187 orlunta et al. / j. nig. soc. phys. sci. 3 (2021) 181–188 188 [3] who. who handbook on indoor radon: “a public health perspective” world health organization, geneva, (2009). [4] who “handbook on indoor radon” world health organisation (2009). [5] unscear “effects and risks of ionizing radiations. new york: united nations scientific committee on the effects of atomic radiation” (2000). [6] ncrp (u.s. national council on radiation protection and measurement) ”measurement of “radon and radon daughters in air”, ncrp report no97, isbn 0-913392-97-9, bethesda (1988). [7] icrp “protection against radon-222 at home and at work,” international commission on radiological protection publication 65: ann icrp 23 (1993). [8] unscear, “sources and effects of ionizing radiation”, united nations scientific committee on the effects of atomic radiation (1993). [9] a.w.k. yeun, the “as low as reasonably achievable” (alara) principle: a brief historical overview and a bibliometric analysis of the most cited publications radioprotection (2019). [10] w. s. kent, and s. e. william “geochemical and ray characterization of pennsylvanian black shales: implications for elevated home radon levels in vanderburgh county, indiana”, journal of environmental radioactivity 148 (2015) 162. [11] unscear. sources and effects of ionizing radiation. volume ii: effects. unscear (2000) report. united nations scientific committee on the effects of atomic radiation, report to the general assembly, with scientific annexes. united nations sales publication e.00.ix.4. united nations, new york, (2000). [12] icrp “recommendations of the international commission on radiological protection” icrp publication 60 ann. icrp 21 (1991) 3. [13] icrp “lung cancer risk from radon and progeny, statement on radon” international commission on radiological protection publication115.annal icrp 40 (2010). [14] m. r. usikalu, c. a. onumejor, j. a. achuka, a. akinpelu, m. omeje and t. a. adagunodo“ monitoring of radon concentration for different building types in covenant university, nigeria,” cogent engineering 7 (2020) 1759396 [15] s. a. sokari, “estimation of radiation risks associated with radon within residential buildings in okrika, rivers state, nigeria”, asian journal of physical and chemical sciences 6 (2018) 1. [16] o. t. afolabi, t.e. deborah, b. bosun, a. f benjamin., e. t. james and b. o. babakayode ‘radon level in a nigerian university campus ’. bmc res notes 8 (2015) 677 [17] s. a. sokari, “estimation of radiation risks associated with radon within residential buildings in okrika, rivers state, nigeria”, asian journal of physical and chemical sciences 6 (2018) 12. 188 j. nig. soc. phys. sci. 5 (2023) 1222 journal of the nigerian society of physical sciences tuning the optical properties and some surface structure of cd-o thin film electrodeposited by two-electrode: an effect of cobalt incorporation rafiu a. busaria,b,∗, ezekiel omotosob, lukman o. animasahunb,c, saheed a. adewinbid, emmanuel o. adewumib, comfort t. famorotib, bidini a. taleatub, adeniyi y. fasasie adepartment of physics, nigerian army university, biu, nigeria bdepartment of physics and engineering physics, obafemi awolowo university, ile-ife 220005, nigeria cdepartment of physics, electronics and earth sciences, fountain university, osogbo, nigeria ddepartment of physics, osun state university, osogbo 210001, nigeria ecentre for energy research and development, obafemi awolowo university, ile-ife 220005, nigeria abstract the tuning of optical and dielectric parameters, structural and microstructural properties of cdo synthesized via a solution growth two-electrode cell arrangement under ambient environment, with the incorporation of co ion into its matrix was investigated. the energy band gaps of the films was estimated in the range of 1.69 ev ≤ eg ≤ 1.96 ev. the extinction coefficient, k for all the samples decreases as the incident photon energy increases. the films exhibit considerably high optical conduction across the photon energy with estimated power of 1013 (ωm)−1. the elemental composition of the samples was determined using the energy dispersive x-ray spectrometry technique. the micrograph images from scanning electron microscopy technique shown that the films are polycrystalline and well-adhered to the substrates with their crystal grains evenly dispersed across the substrates’ surface. the x-ray diffraction analysis confirmed that the deposited films are of polycrystalline in nature. the films show preference for orientation along the (111) plane. doi:10.46481/jnsps.2023.1222 keywords: polycrystalline; solar cell; electrochemical deposition; direct band gap; transparent conducting oxide article history : received: 23 november 2022 received in revised form: 31 january 2023 accepted for publication: 14 february 2023 published: 02 may 2023 c© 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: edward anand emile (phd) 1. introduction in the fields of microelectronics, optical coatings, integrated optics, superconductors, and other technologies, thin film technologies have been crucial [1, 2]. the use of ii-vi group n-type ∗corresponding author tel. no: +2348064244378 email address: rafiu.busari@naub.edu.ng, rafiuwaleb@gmail.com (rafiu a. busari ) semiconducting materials as well as the synthesis of different compositions of these materials have grown tremendously in recent years. this is due to their scientific importance and a wide range of potential uses in optoelectronic devices and applications. many scientist and researchers are interested in the binary, ternary as well as the quaternary forms of these materials because of the possibility of controlling, tuning, and modifying their physical, optical, and electrical properties [3]. deposited 1 busari et al. / j. nig. soc. phys. sci. 5 (2023) 1222 2 oxide thin films such as those of sn, cd, zn, ti, and so on and the incorporation of appropriate dopant impurities, have formed a range of useful synthetic transparent conductors applicable in solar cells, optoelectronic, and energy storage devices [1, 2, 4]. cadmium oxide (cdo), a reddish powder is a transparent conducting oxide. it is a semiconducting material of the iivi group that is n-type, having a direct energy band gap of around 2.0 ev. each atom in the cubic structure of this substance is surrounded by six other atoms with the opposite electrical charge. the material exhibits a cubic crystal structure [4,5]. cdo nanostructures, even without doping have a low ohmic resistance, a considerably high optical transmittance in the solar spectrum’s visible range, and a moderate refractive index. these features have potential uses in optoelectronic field such as transparent electrodes, photovoltaic cells, photodiodes, phototransistors as well as gas sensors [4]. the crucial factors in these applications includes specific surface area, particle size, and porosity. up to recent time, different cdo nanostructures, including nanowires, nanoneedles, nanobelts, nanoclusters, nanorods, and nanoparticles have so far been synthesized via different chemical and physical techniques. these include; vapour phase transport, spray pyrolysis, sputtering, templateassisted, mocvd, solvothermal and so on. by calcining hydroxyl and carbonate-containing cadmium compound precursor at 500 oc in air for two hours, guo et al. were able to prepare porous cdo nanowires that were around 100 µm long and 120 nm in diameter [6]. cdo nanotubes was fabricated by lu et al. using thermal evaporation of cd powder under moderate pressure and temperature of 500 oc [7]. the use of intricate precursor materials, drawn-out experimental procedures, expensive and sophisticated equipment and atmosphere controlled heating regimes have all compounded the aforementioned methods. in the present work, we investigated the effects of co ion dopant on the optical, structural and microstructural performance of two-electrode electrodeposited cdo thin film. this synthesis method is a simple, cost effective, scalable and easy to operate technique. its ability to produce materials for a variety of uses, including optical mirrors, photosensitive devices, integrated circuits, colour television screens, magnetic disks, and more, makes it distinctive [2,8,9]. 2. experimental procedure 2.1. materials in the experiment, we employed the use of cadmium chloride (cdcl2; 98%), sodium hydroxide (naoh; 99.99%), cobalt chloride (cocl2; 99.99%), glass pieces coated with indium-tin oxide (ito) as substrates with surface resistivity of 15/ sq.meter and (20 × 15 × 0.7) mm dimensions in the deposition process. distilled water was used as the solvent. all the reagents used in this experiment were analytical grade (sigma-aldrich). the ito substrates were cleaned as reported in our previous work [8]. following a thorough cleaning with a cotton-bud and soap solution, the substrates were rinsed under running water and ultrasonically treated with distilled water, acetone, and methanol for 10 minutes each at a temperature of 40 oc. after cleaning, the substrates were dried at a moderate temperature in an open furnace. all of these actions were taken to clean the substrates’ surface of any potential impurities. 2.2. samples preparation deposition of cdo and co-doped cdo thin films were done by a two-electrode set-up arrangement. the cell involves a glassy-carbon piece and a conductive substrate (ito) as counter and working electrodes, respectively. for the deposition of undoped cdo (tag: x1), an electrolyte was made from the mixture of 20 ml of 0.2 m; cdcl2 and 20 ml of 0.1 m; naoh and the resultant was fed to the cell. the electrodes were then fastened to the holder already connected to d. c. source and a digital multi-meter with sensitive current and voltage modes selected to achieve stability in power supply according to the reports in [2,8]. the experiment was conducted at a cathodic potential of 1.9 v. the sample growth was monitored through the multi-meter as the current degrades. optimum growth of the sample x1 was then achieved after 25 min of deposition at ambient temperature. for the growth of co-doped cdo samples, electrolyte formed for sample x1 was modified by adding 10 ml of 0.05 m of cocl2 for the deposition of sample x2 and 10 ml of 0.1 m of cocl2 for the position of sample x3, respectively. after the optimum growth is obtained, all the three samples were taken out of the bath, distillated, and then dried for 5 minutes at 120 oc in an open furnace to eliminate any remaining water content and other potential adsorbed surface contaminants [8]. all the deposited films were annealed in air with constant temperature of 350 oc for 1 hour. this process was done to make sure the films were as crystalline as possible and to position the particles in the right equilibrium sites. 2.3. samples characterization an x-ray diffraction via the xpertpro diffractometery, panalytical bv with wavelength of 1.5406 å and cukα as radiation source was used to analyze the materials’ structural composition. surface micrograph of the samples was obtained from scanning electron microscopy (sem) (zeiss ultra-plus 55). the films’ elemental composition was investigated using energy dispersive x-ray spectrometry technique. samples’ optical characteristics were analysed on a double beam spectrophotometer (shimadzu uv-1800). 3. results and discussion 3.1. chemical composition energy dispersive x-ray spectroscopy was used to analyze the pure and co-doped samples’ chemical composition, as shown in the fig. 1. figure 1(a) shows the spectrum of the undoped sample with peaks of cadmium and oxygen, while figure 1 (b) shows the spectrum of the doped sample with peaks of cadmium, cobalt and oxygen. the obtained results supported the thin film deposition of cdo and co-doped cdo. 2 busari et al. / j. nig. soc. phys. sci. 5 (2023) 1222 3 figure 1: edx spectrum of (a) cdo thin film; (b) co-cdo thin film figure 2: sem images of (a) pure cdo, (b) co-cdo @ 0.05 m and (c) co-cdo @ 0.1 m 3.2. morphological studies morphological investigation plays important roles in appraising the films’ nature and their surface properties. the micrograph images of the electrodeposited cdo and co-cdo films analyzed by sem are shown in the fig. 2. it is seen from the micrograph images of the deposited samples, a smooth and homogeneous surface morphology. fig. 2 (a) revealed the distribution of sphere-like with fused rice-like particles across the substrate’s surface. it is seen from the images that the surface characteristic of the cdo film particles changes as of co ion concentration rises. the incorporation of the co-impurity is also seen to have introduced crack and void into the host’s structure. these obvious varying characteristics are due to the enormous variation ∼ 32% in ionic radius between cadmium with 0.095 nm and cobalt with 0.065 nm. co’s atomic radii is 125 pm, while cd’s is 151 pm, respectively. the development of morphologies may be concerned with various atomic radii and electronegativities of the dopant ions, which have an impact on the thermodynamically stable growth process of the free surfaces of cdo thin film crystal faces [10]. figure 3: x-ray diffraction pattern of the cdo and co-doped cdo thin films 3.3. structural analysis to learn more about the cdo film’s crystallinity and crystallite size, an x-ray diffraction technique was conducted. the x-ray diffraction patterns for pure and cdo doped co ion thin films are shown in fig. 3. the patterns’ many peaks suggest that they are polycrystalline cubic (fm-3m) cdo thin film structure with lattice parameter of ∼ 0.474 nm, which is near the value of 0.46948 nm for cdo [11]. using the astm standard, the identified peaks in the patterns can be indexed to the (111), (220) and (311) planes [12]. in a polycrystalline material, the crystallites typically involve differential crystallographic orientation among their neighbours. in relation to some chosen frame of reference, this preference orientation may be dispersed randomly. in the present work, the films exhibit a preferential orientation along the (111) diffraction plane. the texture coefficient tc(hkl) measured from the samples was used to appraise the possibility of the preference orientation. the tc(hkl) has been examined from the x-ray diffraction data using the expression below [12,13]: t c(hkl) = i(hkl)/ io(hkl) n−1r ∑ nr i(hkl)/io(hkl) , (1) where i(hkl) is intensity of the (hkl) plane, io(hkl) is the reference corresponding powder (astm standard), while nr is referred to the reflection number. from the definition, it shows that the deviation of texture coefficient from unity signifies the growth’s preferred orientation. the debye-scherer’s equation was used to evaluate the crystallite size (d) of the samples [14]. d = 0.94 λ βcosθ , (2) where λ is the wavelength of the cuk radiation, β is the diffraction peak’s fwhm, and while θ is the bragg’s diffraction angle. the estimated crystallite size and some other structural data are presented in table 1. it shows that the d values of the films were dependent on the dopant molar concentration. 3 busari et al. / j. nig. soc. phys. sci. 5 (2023) 1222 4 the micro-strain and dislocation density of the deposited films were also calculated from the following relations [14]: microstrain (ε) = βcosθ 4 , (3) δ = 1 d2 . (4) 3.4. optical characterisation 3.4.1. optical absorption and transmission studies optical absorption study plays important roles in the understanding of various semiconductor and non-metallic materials’ structural band nature [15,16]. uv-visible spectrum has been utilized to determine the absorption coefficient (α), energy band gap (eg), extinction coefficient (k), refractive index (n), optical conductivity (σopt), and other parameters for all the deposited samples. fig. 4(a) represents the absorbance spectra of the pure and co-doped cdo thin films. it is evident from the spectra that the samples absorption strength is generally dependent on the dopant concentration. it is seen that sample x3 has the highest absorption power across the wavelength region. inset is the variation of the percentage optical transmission with wavelength, within the wavelength range of 200 – 900 nm. it is observed that as the dopant concentration increases, the samples’ percentage transmission decreases across the wavelength region from around 93 to 57 %. similar results have also been reported in [10,13]. it is also worthy to point out that the films transmission across the wavelength region strongly dependent on the film’s thickness. the absorption coefficient (α) was estimated via the beerlambert relation in equations (5 and 6) using the absorbance (a) and transmittance (t) data [16,17]. i = ioe −∝t, (5) ∝ = ln (io/i) t = − ( lnt t ) = 2.303a t , (6) where i and io are instantaneous and initial intensities of the photon and while t is the film’s thickness. the films’ thickness was determined by the gravimetric weight difference technique using the relation below [18]: t = m ad , (7) where m is mass of the prepared sample, a is area of the laminated sample on the substrate and d sample’s density. the estimated values of m, a, d and t are presented in table 1. the plot of absorption coefficient (α) as a function of incident wavelength within the range 200 – 900 nm is shown in the fig. 4(b). the value of α is considerably high for all the samples up till around 300 nm and at higher wavelength as seen from the figure, the value of α is observed to be nearly invariant for all the samples. this behaviour may be attributed to the lattice deformation or internal electric fields as owing to strain [16]. figure 4: (a)uv-vis absorbance spectra of the thin films (inset: variation of optical transmission with wavelength). (b) optical absorption coefficient against incident wavelength of the deposited films 3.4.2. energy band gap the samples energy band gap was determined from the relation [8,19]: ∝ h = a(h − eg) n, (8) where a, v, h and n are the empirical constant, frequency of the incident light, planck’s constant and power factor, respectively. thin film power factor depends on transition nature and which may have values from 12 , 3 2 , 2 or 3 representing direct allowed, direct forbidden, indirect allowed or indirect forbidden. value of n for the present work is 32 for determining the energy band gaps from the optical absorption data. by estimating the band gaps, we extrapolate the linear portion of the plot of (αhν)2 against hν at when (αhν)2 = 0. this gives the value of energy band gap as shown in the fig. 5. the obtained direct allowed band gaps for the samples x1, x2 and x3 are 1.96, 1.69 and 1.71 ev, respectively. the obtained energy band gaps for pure and doped cdo films are in good agreement with the literature [10, 13, 20]. it is observed that the film’s energy band 4 busari et al. / j. nig. soc. phys. sci. 5 (2023) 1222 5 table 1: some estimated parameters from xrd pattern and thickness of the deposited samples tag a (å) d (nm) ε(×10−4) δ(×10−4)(lines/nm) m (mg) a (cm2) d (g/ml) thickness, t(nm) x1 4.83 38.74 8.27 6.66 0.15 1.5 8.15 120 x2 4.74 41.82 7.51 5.72 0.26 1.5 8.53 200 x3 4.65 43.68 7.08 5.24 0.31 1.5 8.53 240 figure 5: tauc’s plot for the cdo and co-cdo thin films gap narrows as co ion concentration rises. the structural and morphological changes, as well as the fact that co ions have been known to introduce some energy levels in the cdo band gap close the valence band edge, are attributed for the reduction in the optical band gap of cdo after co-doping [10]. 3.4.3. extinction coefficient extinction coefficient (k) is crucial in determining a variety of optical measurements, particularly those that are connected to the dielectric constants and light wave’s absorption in a medium. the percentage of light lost due to scattering and absorption per unit of the penetrating medium is measured by the extinction coefficient, k value [3,21]. extinction coefficient, k for all the samples was calculated from the relation [16]: k = ∝ λ 4π . (9) fig. 6 shows the dependence of k upon the energy of the photon. it is depicted that k decreases as the incident photon energy increases and vice versa. this behaviour represents a strong absorption. it is seen from the figure that sample x1, the pure cdo has the lowest magnitude across the incident photon energy region of around2.5 × 10−3. 3.4.4. refractive index another significant optical constant is refractive index (n), which is critical in choosing the right materials to create optical devices or to use in optical applications. it provides details on the polarization, phase velocity, and local fields of light in the figure 6: extinction coefficient, k upon the photon energy material. in the present study, the value of n was derived from the relationship shown below [16]. n = 1 t s + ( 1 t s − 1 )1/2 , (10) where ts represents percentage transmission coefficient. fig. 7 illustrates the dependence of refraction of the deposited samples upon the photon energy. for all samples, it can be seen that the refractive index keeps increasing at the higher energy with the least value of around 0.4 for the pure sample and 0.7 for the doped sample. it is important to notice that n values were practically unchanged at lower incident photon energies. the matching of the incident photon frequency and plasma frequency values could be the cause of the behaviour described above. 3.4.5. optical conductivity optical band gap, refractive index, absorption coefficient, incident photon frequency, and extinction coefficient are the major parameters that determined the optical conductivity (σopt) of any material. in the present study, the value of optical conductivity was calculated from: σopt = αnc 4π , (11) where c and α are the speed of light and optical absorption coefficient, respectively. the behaviour of optical conductivity of the deposited films is shown in fig. 8. it is seen that the samples have considerably high values of optical conduction with the least value around 1 × 1013 (ωm)−1. the obtained results showed that the films exhibit excellent optical conductivity [16]. the optical conductivity of the material has been found to 5 busari et al. / j. nig. soc. phys. sci. 5 (2023) 1222 6 figure 7: variation of refractive index against the incident photon energy for the cdo and doped cdo figure 8: optical conductivity of the films against the incident photon energy increase with the increase in photon energy. the observed increase in the optical conductivity can be attributed to increase in density of localized states in the energy band [21]. introduction of co-dopant in the matrix of cdo have been seen to effectively enhance the conductivity of the material. 3.4.6. high frequency dielectric constant high frequency dielectric constant is estimated in the present study by using [3]. n2 = ε∞ − 1 4π2εo ( e2 c2 ) ( nopt m∗ ) λ2, (12) where the parameters, e is the electronic charge, nopt is the number of the carrier concentration, c denotes speed of light, εo is permittivity of free space and m* represents effective mass of the charge carriers. plot of square of refractive index, n2 against the square of wavelength, λ2 yields a straight line. extrapolating this line intercepts the n2-axis (at λ2 = 0). the obtained values after extrapolation correspond to high frequency dielectric constant, ε∞. fig. 9 revealed the representation of n2 against λ2. it is observed that the obtained values for all the samples are dependent on the dopant molar concentration. figure 9: representation of n2 against λ2 for the determination of ε∞ of the deposited thin films figure 10: variation of dielectric constants (a) real part; (b) imaginary part of the thin films 3.4.7. complex dielectric constant another crucial characteristic of materials that have both real, εr and imaginary, εi parts is the complex dielectric con6 busari et al. / j. nig. soc. phys. sci. 5 (2023) 1222 7 stant. this is related εr and εi parts according to the following ε∗ = εr + iεi, (13) where the real part of dielectric constant, εr gives information about the dispersion of light waves that propagates inside the material medium and it’s also characterize for slowdown of this light as it travels through the materials while the energy absorption by an electric field resulting from dipole motion is represented by the imaginary part, εi. the εr and εi can be calculated using the following formulae [3,16]: εr = n 2 − k2, (14a) εi = 2nk. (14b) fig. 10(a and b) illustrates the relationship between the deposited films’ εr and εi values and the incident wavelength. the two graphs are seen to be relatively similar. it is also observed that εr of dielectric constant is greater than that of εi (εr >> εi). this can be attributed to εr strongly depend on the refractive index as n >>k. likewise, the values of εi depends on the values of k. therefore the small value of k reduces the values of product, nk. the figures also revealed that the sequentially rising in the real and imaginary parts of the dielectric constant can be associated with increase in the dopant molar concentration. 4. conclusion an home-grown, cost-effective and environmental friendly two-electrode electrodeposition technique has been successfully used to grow polycrystalline thin film of cdo and co-cdo on indium tin oxide coated glass substrates at ambient temperature. results from edx analysis revealed all the expected elements. microstructural sem images as corroborated by the xrd analysis revealed that the deposited samples are of polycrystalline cubic structure of cdo thin films. the average particle size of the films increases as the molar concentration of the co-dopant increases. optical and all the studied dielectric parameters showed that the samples’ properties are greatly dependent on co ion concentration. the optical band gap tuning was observed with variation in molar concentration of the dopant in the lattice of cdo. the study has successfully revealed a novel solution growth route of tuning the physical properties of cdo thin films by incorporation of co ion dopant. acknowledgment the authors gratefully appreciate the efforts of members of staff at the physics department, university of pretoria, south africa for the assistance on xrd and sem characterization. references [1] r. l. mishra, a. k. sharma & s. g. prakash, “gas sensitivity and characterization of cadmium oxide (cdo) semi conducting thin film deposited by spray pyrolysis technique”, digest journal of nanomaterials and biostrutures 4 (2009) 511. 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abstract this paper presents the mellin transform for the solution of the fractional order equations. the mellin transform approach occurs in many areas of applied mathematics and technology. the mellin transform of fractional calculus of different flavours; namely the riemann-liouville fractional derivative, riemann-liouville fractional integral, caputo fractional derivative and the miller-ross sequential fractional derivative were obtained. three illustrative examples were considered to discuss the applications of the mellin transform and its fundamental properties. the results show that the mellin transform is a good analytical method for the solution of fractional order equations. keywords: caputo fractional derivative, fractional calculus, mellin transform, miller-ross fractional derivative, riemann-liouville fractional derivative, riemann-liouville fractional integral article history : received: 22 june 2019 received in revised form: 24 august 2019 accepted for publication: 18 september 2019 published: 30 december 2019 c©2019 journal of the nigerian society of physical sciences. all rights reserved. communicated by: t. latunde 1. introduction the use of fractional derivatives and integrals provides a powerful tool for incorporating history due to its non-local nature [1]. fractional calculus has gained much attention due to the fact that fractional differential equations provide an excellent instrument for the description of many practical dynamics phenomena emanating from mathematical finance, economics, biology, physics, just to mention a few, see ref. [2]. the numerical solution of differential equations of integer order has for a long time been a standard topic in numerical and computational mathematics. however, in spite of a large number of recently formulated applied problems, the state of the art ∗corresponding author tel. no: +2348067032044 email address: sunday.fadugba@eksu.edu.ng (sunday emmanuel fadugba) is far less advanced for fractional order differential equations [3]. for various methods of solving fractional differential equations see; [4-9], just to mention a few. in this paper a survey of the applications of the mellin transform in the solution of fractional calculus of different flavours is presented. it covers the widely known flavours, such as riemann-liouville fractional derivative, riemann-liouville fractional integral, caputo fractional derivative and the miller-ross sequential fractional derivative. the rest of the paper is organized as follows. section two presents some definitions of terms. section three presents the main results for the mellin transform of fractional operators of different flavours. in section four, three illustrative examples were presented. section five concludes the paper. 138 fadugba / j. nig. soc. phys. sci. 1 (2019) 138–142 139 2. preliminaries definition 1 let f (t) be locally lebesgue integrable over (0,∞). the mellin transform of f (t) denoted by f(ω) is defined as f(ω) = m( f (t),ω) = ∫ ∞ 0 f (t)tω−1dt (1) where ω ∈ c, ω is complex, such that a1 < <(ω) < a2. the mellin transform (1) exists if the function f (t) is piecewise continuous in every closed interval [a, b] ⊂ (0,∞) and∫ 1 0 | f (t) | ta1−1dt < ∞, ∫ ∞ 1 | f (t) | ta2−1dt < ∞ (2) definition 2 let f (t) be integrable with fundamental strip (x, y). if b is such that x < b < y and f(ω = b + it) = m( f (t); ω) is integrable, the inverse mellin transform is defined as f (t) = 1 2πi ∫ b+i∞ b−i∞ f(ω)t−ωdω, t ∈ (0,∞) (3) the integration contour in (3) is called the bromwich contour. the fundamental properties of the mellin transform are presented in the following results. theorem 1. let f (t) be mellin transformable function defined on r+. if differentiation under integral sign is permitted, then (a) m(td f (t); ω) = f(ω + d) (b) m( f (t) ∗ g(t); ω) = f(ω)g(1 −ω) (c) m(tλ ∫ ∞ 0 τµ f (τt)g(τ)dτ; ω) = f(ω + λ)g(1 −ω−λ + µ) (d) m( f n(t); ω) = γ(1−ω+n) γ(1−ω) f(ω− n) (e) m(tn f n(t); ω) = (−ω)n f(ω) (f) ( d dt )n m( f (t); ω) = ∫ ∞ 0 f (t)(log t)ntω−1 = m( f (t)(log t)n; ω− 1) (g) m (∫ t 0 f (τ)dτ; ω ) = − 1 ω f(ω + 1) for more details see refs. [3, 10, 11], just to mention a few. definition 3 a real function f (t), t > 0 is said to be in space cµ, µ ∈ r if there exists a real number ρ > µ, such that f (t) = tρ f1(t) where f (t) ∈ c(0, 1), and it is said to be in space cnµ if f n ∈ cµ, n ∈ n definition 4 the riemann-liouville fractional derivative operator of order α > 0 of a function f (t) is defined as 0 d α t f (t) = 1 γ(n −α) ∫ t 0 (t −τ)n−α−1 f (τ)dτ, α ∈ (n − 1, n) (4) definition 5 the riemann-liouville fractional integral operator of order α ≥ 0 of a function f ∈ cµ, µ ≥ 1 is defined as jα f (t) = 0 d −α t f (t) = 1 γ(α) ∫ t 0 (t −τ)α−1 f (τ)dτ,α > 0,τ > 0 (5) the following result gives some of the properties of the riemannliouville fractional operator. theorem 2. if α,β ∈ r+, then (a) 0 dαt [0 d −β t f (t)] =0 d α−β t f (t) (b) jαjβ f (t) = jα+β f (t) (c) jαjβ f (t) = jβjα f (t) (d) j0 f (t) = f (t) definition 6 the caputo fractional derivative of the function f ∈ cn −1, n ∈ n is defined as c 0 d α t f (t) = 1 γ(n −α) ∫ t 0 (t −τ)n−α−1 f n(τ)dτ, α ∈ (n − 1, n] (6) the properties of the caputo fractional derivative were summarized in the following result. theorem 3. if α ∈ (n − 1, n], n ∈ n, f ∈ cn −1,µ > −1, then (a) jα[c0 d α t f (t)] = f (t) − ∑n−1 j=0 f j(0) ( t j j! ) (b) c0 d α t [j α f (t)] = f (t) definition 7 the miller-ross sequential fractional derivative is defined as adσnt ≡ ad αn t ad αn−1 t ...ad α1 t (7) adσn−1t ≡ ad αn−1 t ...ad α1 t (8) where σn = n∑ j=1 α j, α j ∈ (0, 1], ( j = 1, 2, ..., n) (9) 3. mellin transform of fractional operators of different flavours this section presents the main results generated for the mellin transform of fractional operators of different flavours. 3.1. mellin transform of the riemann-liouville fractional derivative and integral operators the mellin transform of the riemann-liouville fractional derivative and integral operators are given in the following results. theorem 4. let f (t) be mellin transformable function, where 0 ≤ n − 1 < α < n, t > 0, then the mellin transform of the riemann-liouville fractional derivative is obtained as m(0 d α t f (t); ω) = γ(1 −ω + α) γ(1 −ω) f(ω−α) (10) proof: take 0 ≤ n−1 < α < n, t > 0, where α is the order. from the definition of the riemann-liouville fractional derivative in (4), one can write that 0 d α t f (t) = 1 γ(n −α) ∫ t 0 (t −τ)n−α−1 f (τ)dτ = dn dtn 0 d−(n−α)t f (t) (11) 139 fadugba / j. nig. soc. phys. sci. 1 (2019) 138–142 140 taking the mellin transform of (11) yields m(0 d α t f (t); ω) = ∫ ∞ 0 dn dtn h(t)tω−1dt = ∫ ∞ 0 hn(t)tω−1dt (12) with h(t) = 0 d −(n−α) t f (t) (13) by means of the derivative property of the mellin transform and simplifying further, (12) becomes m(0 d α t f (t); ω) = n−1∑ j=0 γ(1 −ω + j) γ(1 −ω) [ 0 d α− j−1 t f (t)t ω− j−1 ]∞ 0 + γ(1 −ω + α) γ(1 −ω) f(ω−α) (14) equation (14) can be written in a simplified form as m(0 d α t f (t); ω) = [ 0 d α−1 t f (t)t ω−1 ]∞ 0 + γ(1 −ω + j) γ(1 −ω) f(ω−α) (15) for α ∈ (0, 1). since the first term in the rhs of (15) becomes zero, (10) is established. this completes the proof. theorem 5. let f (t) be mellin transformable function, where 0 ≤ n − 1 < α < n, t > 0, then the mellin transform of the riemann-liouville fractional integral is obtained as m(jα f (t); ω) = m(0 d −α t f (t); ω) = γ(1 −ω−α) γ(1 −ω) f(ω + α) (16) proof: by means of variable transformation τ = tρ, dτ = tdρ, the riemann-liouville fractional integral operator given by (5) as jα f (t) = 1 γ(α) ∫ t 0 (t −τ)α−1 f (τ)dτ,α > 0,τ > 0 yields jα f (t) = tρ γ(α) ∫ ∞ 0 f (tρ)g(ρ)dρ (17) where g(t) = { (1 − t)α−1, 0 ≤ t < 1 0, t ≥ 1 (18) the mellin transform of (18) is the euler beta function of the form m(g(t); ω) = b(α,ω) = γ(α)γ(ω) γ(α + ω) (19) by means of (17) and (19), the mellin transform of the riemannliouville fractional integral operator is obtained as m(jα f (t); ω) = 1 γ(α) f(ω + α)b(α, 1 −ω−α) (20) with b(α, 1 −ω−α) = γ(α)γ(1 −ω−α) γ(1 −ω) (21) hence, (16) is established. 3.2. mellin transform of the caputo fractional derivatives the following result gives the mellin transform of the caputo fractional derivatives. theorem 6. let f (t) be mellin transformable function, where 0 ≤ n − 1 < α ≤ n, t > 0 and α is the order, then the mellin transform of the caputo fractional derivatives is given by m(c0 d α t f (t); ω) = γ(1 −ω + α) γ(1 −ω) f(ω−α) (22) proof: the caputo fractional derivative given by (6) can be written as c 0 d α t f (t) = j n−αdnt f (t) = 1 γ(n −α) ∫ t 0 (t −τ)n−α−1 f n(τ)dτ (23) let l(t) = f n(t) (24) taking the mellin transform of (23) yields m(c0 d α t f (t); ω) = m(j n−αdnt f (t)) = γ(1 −ω− n + α) γ(1 −ω) l(ω + n −α) (25) where l(ω + n −α) = n−1∑ j=0 γ(1 −ω− n + α + j) γ(1 −ω− n + α)[ f n− j−1(t)tw+n−α− j−1 ]∞ 0 + γ(1 −ω + α) γ(ω−α) f(ω−α) (26) simplifying (25) further and for α ∈ (0, 1), yields m(c0 d α t f (t); ω) = γ(α−ω) γ(1 −ω) [ f (t)tω−α ]∞ 0 + γ(1 −ω− n + α) γ(1 −ω) f(ω−α) (27) if the function f (t) and <(ω) are such that all substitutions of the limit t = 0 and t = ∞ in (27) equals zero, then (22) is established. hence, this completes the proof. 3.3. mellin transform of the the miller-ross sequential fractional derivatives the mellin transform of the miller-ross sequential fractional derivatives is given in the following result. theorem 7. by means of induction, the mellin transform of the miller-ross sequential fractional derivative is given by m(0 d σn t f (t); ω) = γ(1 −ω−σn) γ(1 −ω) f(ω−σn) (28) 140 fadugba / j. nig. soc. phys. sci. 1 (2019) 138–142 141 proof: for n = 2, setting h(t) = 0 d β t f (t) and by means of the property of the mellin transform, one gets m(0 d σ2 t f (t); ω) = [ 0 d α2 t h(t)t ω−1 ]∞ 0 + γ(1 −ω + α2) γ(1 −ω) h(ω−α2) (29) solving (29) further and rearranging terms yields m(0 d σ2 t f (t); ω) = [ 0 d σ2−1 t f (t)t ω−1 ]∞ 0 + γ(1 −ω + α2) γ(1 −ω) [ 0 d σ1−1 t f (t)t ω−α2−1 ]∞ 0 + γ(1 −ω + σ2) γ(1 −ω) f(ω−σ2) (30) in general, by means of induction, the mellin transform of the miller-ross sequential fractional derivative is obtained as m(0 d σn t f (t); ω) = n∑ i=1 γ(1 −ω + σn −σi) γ(1 −ω) × [ 0 d σi−1 t f (t)t ω−σn−σi−1 ]∞ 0 + γ(1 −ω + σn) γ(1 −ω) f(ω−σn) (31) since the first term of (31) yields zero, therefore (31) reduces to (28). this completes the proof. 4. applications this section presents three illustrative examples. 4.1. application 1 solve the following problem via the mellin transform t2 f 2(t) + t3 f 3(t) = δ(t − a) (32) by applying the mellin transform to both sides of (32), one obtains m((t2 f 2(t) + t3 f 3(t)); ω) = m(δ(t − a); ω) (33) using the property of the mellin transform of the form m(tα f α(t); ω) = γ(ω + α) γ(ω) f(ω) (34) and the fact that m(δ(t − a); ω) = aω−1 (35) equation (34) becomes γ(ω + 2) γ(ω) f(ω) + γ(ω + 3) γ(ω) f(ω) = aω−1 (36) thus, f(ω) = aω−1γ(ω) γ(ω + 2) + γ(ω + 3) (37) applying the inverse mellin transform to (37) by using complex inversion integral to cover the function f (t) explicitly as the solution of (32) f (t) = 1 2πi ∫ b+i∞ b−i∞ γ(ω) γ(ω + 2) + γ(ω + 3) aω−1t−ωdω (38) 4.2. application 2 solve the following problem via the mellin transform t 3 2 f 3 2 (t) + t 5 2 f 5 2 (t) = (log t)αδ(t − a) (39) taking the mellin transform of (39) yields m((t 3 2 f 3 2 (t) + t 5 2 f 5 2 (t)); ω) = m((log t)αδ(t − a); ω) (40) once again, using (34) and the fact that m((log t)αδ(t − a); ω) = aω−1(log a)α (41) (40) becomes γ ( ω + 32 ) γ(ω) f(ω) + γ ( ω + 52 ) γ(ω) f(ω) = aω−1(log a)α (42) rearranging terms in (41), one gets f(ω) = aω−1(log a)αγ(ω) γ ( ω + 32 ) + γ ( ω + 52 ) (43) once again, applying the inverse mellin transform to (43) by using complex inversion integral to cover the function f (t) explicitly as the solution f (t) = 1 2πi ∫ b+i∞ b−i∞ γ(ω) γ ( ω + 32 ) + γ ( ω + 52 ) ×aω−1(log a)αt−ωdω (44) equation (40) is the required explicit solution of (39). 4.3. application 3 consider the solution of the caputo fractional derivative of the form 1∑ k=0 tα+k c0 d α+ky(t) = f (t) (45) subject to the initial conditions y(0) = y′(0) = 0, y(∞) = y′(∞) = 0 (46) via the mellin transform. applying the mellin transform to both sides of (45), yields γ(1 −ω)(1 −ω−α) γ(1 −ω−α) y (ω) = f(ω) (47) therefore, γ(1 −ω−α)f(ω) γ(1 −ω)(1 −ω−α) = y (ω) (48) equation (48) can be written in a compact form as f(ω)h(1 −ω) = y (ω) (49) where h(1 −ω) = γ(1 −ω−α) γ(1 −ω)(1 −ω−α) (50) 141 fadugba / j. nig. soc. phys. sci. 1 (2019) 138–142 142 in the same manner, h(ω) = γ(ω−α) γ(ω)(ω−α) (51) equation (51) can be split as follows h(ω) = a(ω)b(ω) (52) a(ω) = γ(ω−α) γ(ω) (53) and b(ω) = 1 ω−α (54) respectively. the inverse mellin transform of (53) and (54) are obtained as a(t) =  t−α(1−t)α−1γ(α) , t ∈ (0, 1)0, t > 1 (55) and b(t) = { t−α, t ∈ (0, 1) 0, t > 1 (56) respectively. the inverse mellin transform of (51) is obtained as h(t) = ∫ ∞ 0 a ( t τ ) b(τ) dτ τ (57) from (55) and (56), it follows that h(t) = ∫ 1 t a ( t τ ) b(τ) dτ τ , t ∈ (0, 1) (58) and that h(t) = 0, t > 1 (59) the function h(ω) has a double pole at ω = α and ordinary poles at ω = α − n − 1, n ∈ z+. by means of the residue theorem of the form 1 2πi ∫ δω f (ω)dω = k∑ j=0 res( f,ω j),ω ∈ c one obtains h(t) = ∫ pτ h(ω)t−ωdω = 1 2πi ∫ b+i∞ b−i∞ h(ω)t−ωdω = ∑ [res h(ω)t−ω]ω=α + ∞∑ n=0 [res h(ω)t−ω]ω=α−n−1 (60) simplifying (60) further, yields h(t) = t−α γ(α) (ln t + ξ(α) − φ) + ∞∑ n=0 (−1)ntn−α+1 (n + 1)γ(n + 2)γ(α− n − 1) (61) where φ is the logarithmic derivative of the gamma function and the euler constant ξ = 0.577215. the solution of the caputo fractional differential equation (45) subject to (46) is obtained y(t) = ∫ ∞ 0 f (tτ)h(τ)dτ (62) equation (62) is known as the mellin convolution of the functions f (t) and h(t). 5. conclusion this paper presents the solutions of fractional order equations in the domain of the mellin transform. the mellin transform of the riemann-liouville fractional derivative, riemannliouville fractional integral, caputo fractional derivative and the miller-ross sequential fractional derivative were obtained. moreover, three fractional order equations were solved via the mellin transform. hence, the solutions obtained were recovered by means of the mellin transform inversion formula. references [1] a. a. kilbas, h. m. srivastava & j. j. trujillo, “theory and applications of fractional differential equations”, (2006) elsevier, amsterdam, netherland. [2] a. a. elbeleze, a. kilicman and b. m. taib, “homotopy pertubation method for fractional black-scholes european option pricing equations using sumudu transform”, mathematical problems in engineering, 2013 (2013) 1. [3] i. podlubny, fractional differential equations. an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, 198 (1999), mathematics in science and engineering, academic press. [4] s. e. fadugba, “closed-form solution of generalized fractional blackscholes-like equation using fractional reduced differential transform method and fractional laplace homotopy perturbation method”, international journal of engineering and future technology 16 (2019) 13. [5] s. kumar, a. yildirim, y. khan, h. jafari, k. sayevand & l. wei, “analytical solution of fractional black-scholes european option pricing equation by using laplace transform”, journal of fractional calculus and applications 2 (2012) 1. [6] ch. friedrich, “relaxation and retardation functions of the maxwell model with fractional derivatives”, rheologica acta 30 (1991) 151. [7] r. l. bagley & r. a. calico, “fractional order state equations for the control of viscoelastically damped structures”, j. guidance 14 (1991) 304. [8] k. s. miller & b. ross, an introduction to the fractional calculus and fractional differential equations, (1993), john wiley and sons inc., new york. [9] k. b. oldham, c. g. zoski, “analogue instrumentation for processing polarographic data”, electroanal. chem. 157 (1983) 27. [10] a. kiliçman, “a note on mellin transform and distributions”, math. comput. appl. 9 (2004) 65. [11] a. erdélyi, w. magnus, f. oberhettinger & f. tricomi, tables of integral transforms, 1 (1954), first edition, mcgraw-hill, new york. 142 j. nig. soc. phys. sci. 3 (2022) 138–145 journal of the nigerian society of physical sciences half-metallic characteristics of the novel half heusler alloys xcrsb(x = t i, zr, hf ) b. e. iyorzor∗, m. i. babalola department of physics, university of benin, benin city, nigeria. abstract ab-initio calculations are performed to examine the structural, mechanical, electronic, magnetic and thermodynamic properties of the half-heusler ternary alloys xcrsb (x = h f , ti, zr). in this study, the spin-polarized density functional theory (dft) method that is spin-polarized with generalised gradient approximation (gga) are used to perform ab-initio calculations to investigate the physical properties of a novel half-heusler ternary alloys xcrsb (x = h f , ti, zr). it was confirmed that the alloys are stable mechanically and exhibit ferromagnetic states (fm). the study reveals that the alloys portray half-metallic character with narrow energy gaps. and it also shows that they have a total magnetic moment of approximately 3µβ. from the formation energy calculation, it shows that the alloys can be synthesized experimentally. also, it was observed that they are mechanically stable. the heat capacities and debye temperatures were also computed and they show high thermodynamic stability. doi:10.46481/jnsps.2021.297 keywords: half-heusler alloys, half-metallic ferromagnet (hmf), spin-polarization, band structure, density of state, quasi-harmonic approximation (qha) article history : received: 08 july 2021 received in revised form: 22 august 2021 accepted for publication: 24 august 2021 published: 28 february 2022 c©2022 journal of the nigerian society of physical sciences. all rights reserved. communicated by: e. a. emile 1. introduction the heusler alloys have been studied extensively in the recent past, and one of the major results is that they are among the best half-metallic ferromagnets (hmf) to achieve 100% spin polarization at room temperature. density functional theory (dft) has proven to be a reliable tool for calculating electronic structures for many decades, this is due to its simple and usefull approach in approximating the ground state functionals of real many-body electrons. the theory is built on the fact that the properties of many-body interacting system can ∗corresponding author tel. no: email addresses: beniyorzor@uniben.edu (b. e. iyorzor ), michael.babalola@uniben.edu (m. i. babalola ) be seen as a functional of the ground state electron density [1]. from the works of mehmood et al [2-3], it was gathered that the half-heusler alloys yzsb (z = cr, mn) and rhcrz (z = s i, ge) exhibit half-metallic properties within the framework (dft) among others. also, the electronic, magnetic and optical properties were equally studied. the field of spintronics is quite different from the conventional electronics in which charges are responsible for transfer of information, whereas in spintronics spins are responsible for the transfer of information. some of these enhancements and advantages are; increasing the speed of the data processors, drop in the consumption of electric power and the increase in integration density [4−6]. the study of half heusler (hh) alloys present interesting and different magnetic occurances, and this have attracted researchers in re138 iyorzor & babalola / j. nig. soc. phys. sci. 3 (2022) 138–145 139 cent time [7−15]. ternary half-heusler compounds are usually denoted by the chemical formula xyz, where x is rare-earth metal, y is transition metal and z is the main group elements in the periodic table. this area has attracted a lot of attention by many researchers for about a decade now. relatively, the hh compounds form a big family of materials with several physical properties and applications especially in the areas of optoelectric semiconductors and spintronics [16 − 18], thermoelectiric semiconductors [19 − 21], piezoelectric semiconductors [22] and topological insulators [23, 24]. considering the half metallic materials, the majority and minority spin-bands exhibit different behaviours. the majority spin band portrays metallic behaviour while the minority spin band displays a semiconductor character with a narrow band gap within the fermi level. and around the fermi level, the gap results in a 100% spinpolarization [25, 26]. since the discovery of the half-metallicity in the (hh) compounds, numerous researchers have done vigorous studies determining the stability of the physical/mechanical, electronic/phonon dispersion and as well as thermodynamic properties using first-principles calculations. amoung them are: zrmnas [27], ptxbi (x = fe, co, mn and ni) [28], zrnipb [29], and xvsb (x = fe, ni, co) [30]. the works of rogl and coworkers [31] provides an extensive review on the mechanical properties of many hh systems, like the xnisn and xcosb where x = t i, zr and hf. they studied the fracture toughness, hardness/elastic moduli and density-porosity of alloys empirically and theoretically. however, nothing has been mentioned about the replacement of co atom with cr atom in the xcosb series. considering the industrial uses of chromium, ranging from hardening of steel, chromium plating, leather tanning, as well as industrial catalysts to pigmentation. therefore, it became imperative in this present study, to predict the physical, mechanical, electronic, magnetic and thermodynamic properties of the novel half-heusler alloys using ab-initio calculations 2. methodology the ab-initio total energy calculations were executed employing the quantum espresso (qe) as applied in the works of giannozi [32]. the projected augumented wave pseudo potential was used in the computation [33]. the exchange interrelationship between electrons, bonding and magnetic features were treated with the gga-pbe approach. the (hh) alloy xyz, has an fcc structure, in space group f − 43m no. 216 with its structurbericht designation of c1 b as recorded in the international table of crystallography. the crystal structures for the three alloys are presented in figure 1. we use the spinpolarized density funtional theory (dft). the energy cut-off value and the k-points are essential flags (in the input-file) in ensuring the accuracy of the calculations [34]. the k-points are used for the structural optimization and static scf calculations for the alloys xcrsb (x = h f ,ti and zr). convergence tests for the energy cut-off and the k-points are carried out before determining their appropriate values for the calculations. the monkhorst pack format with 8 x 8 x 8 framework was used, and the kinetic energy cut-off were fixed to 60ry for hfcrsb figure 1. the crystal structure of xcrsb where x=ti, zr and hf. and zrcrsb, 65ry for ticrsb alloys. the pseudo-atomic configurations used as valence electrons, in this study, are hf 5d2 6s2 , ti 3d2 4s2 , zr4d2 5s2 , cr 3d5 4s1 , and sb 4d10 5s2 5p3 respectively. the (x = h f ,ti and zr), sb and cr occupy the atomic positions 4a(0, 0, 0), 4b( 12 , 1 2 , 1 2 ) and 4c( 1 4 , 1 4 , 1 4 ) respectively. 3. results and discussions 3.1. structural and mechanical properties investigating the structural properties of the ground state configuration of the xcrsb compounds (where x = t i, zr and hf), firstly, the structural/geometry optimisation was carried out by minimizing the total energy of the alloys with respect to the variation of the lattice parameters. the lattice constant a, the bulk modulus b and its pressure derivative b′ were obtained when the energy-lattice parameter was fitted to the birch murnaghan equations of state. these results are presented in table 1 and the obtained energy-lattice curves for the various alloys are shown in figure 2. the three hh alloys in different magnetic states were studied in order to ascertain the true nature of the alloys. the results for the ferromagnetic (fm) states, non-magnetic (nm) states as well as for the anti-ferromagnetic (afm) states were presented in fig. 3. it is observed that the (fm) states posses the lowest ground state energy for the three hh alloys, hence they are all ferromagnets. it is the ferromagnetic states of these alloys that have been used to compute the other properties in this work. in solids, the elastic constants and some structural features play vital roles in determining the mechanical stability of the material [35]. for cubic phases, the stability is measured by the following criteria c11 + 2c12 > 0, c11 − c12 > 0, , and c11 > 0 [36]. the alloys are mechanically stable since all the necessary conditions were satisfied, see table 2. the bulk modulus b, young modulus e and shear modulus g are parameters used to quantify the mechanical properties of solids. from table 1, the results of the b and e show that the deformation resistance decreases in trend from hfcrsb to zrcrsb to ticrsb alloys, except for the shear modulus that has a contrary behaviour. an expression to establish the plasticity of a material is given by the bg ratio. the threshold value for distinguishing between ductility and brittleness of materials is about 1.75 [37] 139 iyorzor & babalola / j. nig. soc. phys. sci. 3 (2022) 138–145 140 figure 2. relationship between total energies per unit cell and lattice parameters for hfcrsb, ticrsb and zrcrsb from our results presented in table 2, it is obvious that the alloys are ductile in nature since the values of the b/g ratio are more than 1.75. although, there was no available experimental or theoretical reports on the xcrsb series for comparison, but our results for b, e, and the elastic properties are still within appreciable range when compared with the xcosb series in literatures [31, 38]. it is observed that the values of the mechanical properties obtained for xcosb hh alloys are higher than that of xcrsb obtained in this work. this is due to the fact that the bonding during the hybridization of the d-orbital of cr atom and the d-orbital of x(ti, zr and hf) atom is weak as shown in figures 7 − 9. the present calculated results of zener anisotropy a, shows that the three alloys are anisotropic, since the values are approaching 1, which is an indication of high elastic anisotropy. these results were obtained using the relation, from equation (1), as recorded in [39]. a = 2c44 c11 − c12 (1) another factor that can affect the stability of a material against shear stress is the poisson′s ratio µ. it reveals the nature of the bonding forces in materials. the value range is of the order 0 ¡ ν ¡0.5 [39]. from the results of ν in table 2, it shows that the three alloys are of good plasticity. the cauchy relation is another parameter which expresses the ductility and brittleness of materials. when the value is positive, the material is considered ductile, otherwise it is brittle. the alloys under consideration are ductile as reported in table 2. the formation energy ∆ h of the alloys were investigated to verify whether they can be synthesized experimentally. the calculation was done using equation 2: ∆h = ect − ( ee1b na + ee2b nb + ... ) (2) where ect is the total energy of the alloy, e e1 b and e e2 b are the total energies of the constituent elements respectively, and na represents the number of atom per cell. from the calcultions, the formation energies of xcrsb (x = h f , ti, and zr) are −1.49ev, −1.10ev and −1.0ev respectively. these results confirm that the alloys can be synthesized easily because the enthalpies have negative values. 140 iyorzor & babalola / j. nig. soc. phys. sci. 3 (2022) 138–145 141 figure 3. the total energies per unit cell as a function of lattice constants for the ferromagnetic (fm), antiferromagnetic (afm) and non-magnetic (nm) states of the alloys. 3.2. electronic and magnetic properties from the crystal structure of the (hh) compounds in figure 1, the x(x = t i, zr and hf) atom forms a tetrahedral position with the cr atom as the cation at the center. the zincblende sublattice [crx]3+ formed hybridizes with the [s b]−3 ion. with this configuration the cr atom gains one extra electron added to its initially six valence electrons thereby resulting to seven electrons that occupy thed-orbital. during the filling of the d-orbital, three electrons are left unpaired which constitue the magnetic moment of the alloys. the spin magnetic moment for each of the alloys is approximately 3µ as shown in table 3. this satisfies the spin magnetic moment predicted by the slaterpauling rule [40]. the rule asserts that the total spin magnetic moment per formula unit mtot is given by mtot = |nv − 18| for (hh) compounds, where nv is the sum of electrons at the valence. the sum of electrons at the valence for ti, zr and hf are 4 each while that of cr and sb are 6 and 5 respectively. hence the total number of electrons at the valence for each compounds is 15. the total and partial magnetic moments are presented in table 3. in order to obtain quantitative results to reveal the half metallic character from the electronic band structure calculations, the spin polarization p was computed for the minority and majority bands. from the three alloys, we have 0ev, 0.25ev for the up and down dos at the fermi level respectively. using equation (3) for spin polarization [41], we obtained 100% spin polarization in the alloys. p = dos ( e f (up) ) − dos ( e f (down) ) dos ( e f (up) ) + dos ( e f (down) ) × 100% (3) the electronic properties of the three (hh) alloys have been computed in the form of band structures and are presented in figures 4 − 6. the (hh) alloys xcrsb (x = t i, zr and hf) are found to posses the half-metallic properties since their band structures with the majority spin channels have indirect band gaps while their band structures for the minority spin channels show metallic behaviour. the measured band gaps for ticrsb, zrcrsb and hfcrsb are 0.31ev, 0.49ev and 0.42ev respectively. the pdos for the three compounds have also been computed and are presented in figures 7 − 9. the purpose of the pdos plot is to give insight into the nature of bonding between the orbitals of the individual atoms. the pdos of the three half 141 iyorzor & babalola / j. nig. soc. phys. sci. 3 (2022) 138–145 142 table 1. the structural properties: the a (å), b (gpa), and b′, energy of formation ∆h(ev), band gap near fermi energy eg(ev) and magnetic ground state mg. alloy a b bâ´ ∆h eg mg ticrsb 6.197 84.96 4.43 -1.10 0.31 fm zrcrsb 6.383 95.80 4.65 -1.00 0.49 fm hfcrsb 6.341 99.66 4.75 -1.49 0.42 fm table 2. the mechanical/elastic properties: the c11,c12and c44 (gpa), b / g ratio, e (gpa), poisson’s ratio υ, zener anisotropy a, and the cauchy relation. alloy ticrsb zrcrsb hfcrsb c11 108.80 112.76 128.29 c12 73.02 69.87 85.33 c44 33.70 30.98 32.33 g 27.38 27.17 27.99 b/g 3.10 3.53 3.56 e 74.17 73.58 76.76 ν 0.35 0.35 0.37 a 1.88 1.44 1.51 c12 c44 39.32 38.89 53.00 table 3. the total and partial magnetic moments of the half-heusler alloy xcrsb (x=hf, ti, zr). alloy mx(µβ) mcr (µβ) ms b(µβ) mtot(µβ) ticrsb 0.2899 2.7133 -0.0025 3.0007 zrcrsb 0.1835 2.8653 -0.0386 3.0102 hfcrsb 0.1900 2.7696 0.0001 2.9597 table 4. the specific heat capacity cv , debye temperature θd , zero point energy eo , and debye sound velocity vs. alloy cv(j/nmol.) θd(k) e0(kj/nmol.) vs(m/s) ticrsb 72.33 253 7.11 2302 zrcrsb 72.63 238 6.69 2232 hfcrsb 73.16 209 5.87 1946 heusler alloys are some how similar this is due to the fact that they have similar atoms inhabiting the y and z atomic sites and the three different atoms inhabiting the x atomic site are in the same group. from the minority-spin channels of the three hh alloys, the t2g states of cr atom dominate the region below the fermi energy while the eg states dominate in the region above the fermi energy. since both states are occupied around the fermi energy, hence they show metallic character. the gaps seen around the fermi energy in the majority spin straits for the three half-heusler compounds are due to the crystal field splitting of the d-orbital of cr atom into t2g and eg. due to the exchange interactions between the cr d-orbital and the x(ti, zr and hf) d-orbital, there is depletion of electrons in the majority spin states leading to unoccupied states. this results to the development of half-metallic gap around the fermi energy. we also observed the bonding states between the cr d-orbital and the x(ti, zr and hf) d-orbital just below the fermi energy and the anti-bonding states between the cr d-orbital and the x(ti, zr and hf) d-orbital just above the fermi energy. this observation is in line with observations by other researchers who worked on half heusler alloys. it is also important to investigate the relationship between the magnetic moment and the lattice constant in order to determine the range of lattice constants at which the magnetic moment is preserved. from figure 10, it is observed that the magnetic moment of 3µβ is preserved within the range of 11.2a.u to 14a.u for three hh alloys 3.3. thermodynamic properties in solid state physics, the debye temperature θd is one basic factor in describing phenomena associated with many physical properties, like melting points, lattice vibration, specific heat, thermal expansion etc. the θd is used for delineating high temperature regions from low temperature regions in solids. basically, the θd depends on the elastic constants. the θd can be calculated from the average sound velocity, vm by the expression [42, 43] θd = h kb ( 3n 4πva )1/2 vm (4) vm = [ 1 3 ( 1 v 3t + 2 v 3t )]−1 3 (5) vt = ( 3b + 4g 3ρ ) 1 2 (6) 142 iyorzor & babalola / j. nig. soc. phys. sci. 3 (2022) 138–145 143 figure 4. band structure of ticrsb alloy figure 5. band structure of zrcrsb alloy a = 2c44 c11 − c12 (7) where b, g and ρ are the bulk modulus, shear modulus and densities of the various alloys respectively. the calculated specific heat capacity, debye temperatures, zero point energy and the debye sound velocity for the various alloys are presented in table 4. the debye temperatures computed in this work are in reasonable range when compared with other half heuslers [31, 38]. the debye temperature is related to the strenght of the covalent bonds in solids, that is, the higher the ∆d of a compound indicates a stronger covalent bond which leads to high melting point. also, a strong covalent bond is associated with the hardness of the material 44. the other thermodynamic properties such as the enthalpy h, free energy f , heat capacity at constant volume cv and the entropy s of the alloys are evaluated by using the qha. the results are shown in figure 8. we can see that the cv of the alloys increases swiftly as the temperature increases, such that at very high temperature it tends to reach the dulong-petit terminal point. it was also observed that at very low temperature, cv is proportional to t 3. the entropy figure 6. band structure of hfcrsb alloy figure 7. pdos for ticrsb alloy figure 8. pdos for zrcrsb alloy and enthalpy graphs show that as the temperature increases the values of the properties increased, while the reverse is observed for that of free energy. 143 iyorzor & babalola / j. nig. soc. phys. sci. 3 (2022) 138–145 144 figure 9. pdos for hfcrsb alloy figure 10. total magnetic moment per formular unit as a function of lattice constants for the three alloys. 4. conclusion the physical properties of a novel (hh) compounds xcrsb (where x = h f , ti, and zr) have been investigated using abinitio calculations. the ferromagnetic state is found to be more stabled and favourable than the antiferromagnetic and nonmagnetic states structurally in term of total energy. the alloys are found to be stable from the mechanical stability conditions. also, the b/g ratio and the cauchy-relation reveals that the alloys are ductile. the electronic band structure and the dos calculated reveal that the alloys exhibit half metallic ferromagnetic 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[44] f. bakare, m. i. babalola & b. e. iyorzor, “the role of alloying elements on the structural, mechanical and thermodynamic properties of al3x binary alloy system (x=mg, sc and zr): first principle calculations”, materials research express 4 (2017) 116502. 145 j. nig. soc. phys. sci. 3 (2021) 406–413 journal of the nigerian society of physical sciences characterization and evaluation of human health risk of heavy metals in tin mine tailings in selected area of plateau state, nigeria d. d. bwedea, r. a. wuanab, g. o. egahc,∗, a. u. itodob, e. ogahd, e. a. yerimac, a. i. ibrahimc adepartment of basic sciences, plateau state college of health technology zawan, nigeria bdepartment of chemistry federal university of agriculture makurdi, benue state, nigeria cdepartment of chemical sciences, federal university wukari, taraba state, nigeria ddepartment of chemistry university of jos, jos, plateau state, nigeria abstract tin mining tailings are unprocessed waste materials that overlie an ore which are displaced during mining activities. this research work is aimed at characterizing and evaluating the human health risk of heavy metals in tin mine tailings in zabot (s3) and tafan (s4) districts in barkin ladi local government area of plateau state, nigeria. the samples were characterized using edx-xrf and sem. the concentrations of seven heavy metals (pb, cr, as, ni, cd, cu and zn) were determined in s3 and s4. the results showed that cr, ni, cd, cu and zn were within the usepa permissible limits, except for pb and as with range of (270-300) mg/kg and (40-70) mg/kg respectively for both mining and control sites of s3 and s4. the sem results revealed small particles size with fine porous structure, and rough areas with varying sizes and pores distributed over the surface for s3 and s4 respectively. results of the risk assessment showed that the hazard quotient hq and hi values were greater than 1 indicating high risk. the carcinogenic and non-carcinogenic risks associated with pb, zn, cd, cr, ni and as were evaluated for s3 and s4 for the three exposure pathway and it was found that the mining sites pose more risk than the control and the children were more exposed than the adults. the carcinogenicity of these samples were due to the high hazard quotient for ingestion and dermal exposure pathway. the rtotal results for as, cr, pb and ni for mining site s3 were found to be (1.39 × 102, 2.02 × 10−7, 3.30 × 103 and 8.17 × 10−8), and control site (3.42 × 103, 2.64 × 10−5, 38.30 × 101, 6.90 × 10−8) for as, cr, pb and ni respectively. from the rtotal results as and pb were more than the acceptable threshold, while cr and ni were below the threshold of 1×10−4. for the mining site s4, the rtotal were found to be (5.70×102, 1.82×10−7, 3.63×104 and 9.64×10−9), and the control (1.16×103, 1.71×10−7, 31.1×102 and 1.51×10−8) for as, cr, pb and ni respectively. from the results of the mining and control sites, as and pb rtotal were higher than the acceptable threshold, while cr and ni were below the threshold of 1 × 10−4. doi:10.46481/jnsps.2021.262 keywords: tailings, heavy metals, hazard index, carcinogenic risk and hazard quotient article history : received: 18 june 2021 received in revised form: 16 september 2021 accepted for publication: 17 september 2021 published: 29 november 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: s. j. adebiyi ∗corresponding author tel. no: +2347061097995 email address: egah.godwin@yahoo.com (g. o. egah ) 1. introduction tin mining tailings are wastes fractions of an ore or mineral body which are discarded during mining operations without being processed. tin tailings contains both magnetic minerals 406 bwede et al. / j. nig. soc. phys. sci. 3 (2021) 406–413 407 (iron ore, columbite) and non-magnetic minerals (cassiterite, monozite, zircon sand in large quantity and silica) [1]. they are used in preparation of fertilizer, animal feeds, refractory products, for building roads and backfilling of tailing storage facilities [2]. in the 1960s, nigeria was regarded as one of the world’s leading tin producing country, but production later decreased towards the end of the twentieth century. however, mining activities are still going on in these areas. apart from tin which is the primary target, tin mining generates tin tailings, which are by-product of the ore. the amount of tailings produced ranged from 90-98 % for copper ores and 20-50 % for other minerals [3]. some of the heavy metals found in contaminated tailings are pb, cr, as, zn, cd, cu, ni and hg [4, 5, 6]. heavy metal contamination and their human health threats are some of the serious environmental problems limiting mining activities [7, 8]. with the development of mining, smelting and other industrial activities, heavy metals are increasingly being found in the environment which can pose severe threats to humans and the environment. pollution by heavy metals such as lead (pb), chromium (cr), arsenic (as), nickel (ni), cadmium (cd), copper (cu) and zinc (zn), affects the quality of the ambient air, soil and water bodies which in turn threatens the life of both animals and humans through the food chain [2]. during mining activities huge waste tailing ponds are created which have a high environmental impact on the surrounding ecosystems and populations when used [9, 10]. in order to evaluate the risk posed by tin mine tailings activities to human health, there is need to assess the level of heavy metal pollution in these sites and the rate at which they affects human life. this is based on preliminary studies on waste properties, heavy metals concentration and their relation to the environment as they affect the individuals who participated in these tailings activities and the community [11]. human health risk assessment involves the evaluation of possible human health effect in the contaminated environmental media [12]. the health effect of contaminants on humans depends on the level of exposure, nature of the contaminants and vulnerability of the individual affected [13]. health effects may include risk of cancer, hypertension, acute foetus neurological disorders, organ dysfunction, respiratory difficulties, physical and mental disorder, reduced life expectancy and weakening of the body’s immune system [12]. therefore, this research is aimed at characterizing and evaluating the human health risk of heavy metals in tin mine tailings in zabot and tafan district both represented by s3 and s4 respectively in barkin ladi local government area, plateau state, nigeria. 2. research methodology 2.1. sample location and description this research work was carried out in zabot (s3) and tafan (s4) district of barkin ladi local government area of plateau state, nigeria. it is located between latitude 9051′30′′n and longitude 8048′00′′e. the state is located in the middle belt of nigeria, with an area of 30.91 km (11936 square mile). the state has a population of about three million people in estimate. the name plateau state was given because of its topography with wonderful rock formations. the height of the mountains ranges from 1,200 to 1,829 meters above sea level. mining and subsistence farming are the major occupation of its residents. the sampling locations and points are as shown in figure 1, using the geographical positioning system (gps) to locate zabot (s3) and tafan (s4). figure 1. gps map showing the sites of sample collection. 2.2. sampling collection and preparation the random sampling technique was applied for sample collection with little modification. two samples of 50 g each were taken from mining sites in zabot (s3) and tafan (s4) in barkin ladi local government area of plateau state, with their respective control. the tin mine tailings and soil samples were washed, dried and pulverized into the required particle sizes of 2 mm. pre-treatments which does not alter the chemical composition of the analytes were done to obtain the original concentration of the analyte found in the sample. 2.3. determination soil ph and temperature 1.0 g of soil sample from both mining and control sites were mixed with 100 cm3 of deionized water in 250 cm3 conical flask and stirred using a magnetic stirrer for 10 minutes. the temperature and ph were determined using hanna portable ph meter (model hi8043) and thermometer respectively. the readings were taken in triplicates and the average recorded accordingly. 407 bwede et al. / j. nig. soc. phys. sci. 3 (2021) 406–413 408 2.4. sample characterization 2.4.1. elemental analysis the concentrations of the various heavy metals contained in the samples were determined using energy dispersive x-ray fluorescence spectrometer edx-xrf (minipal4). the xrf analysis was done directly on solid powdered specimen for accurate results with no risk of contamination. sample preparation involves milling of the sample to less than 75 µm in fraction. retsch rs 200 vibratory disk milling machine was used at 1500 min−1 motor speed for 5 minutes. the milled sample was collected in xrf cup and placed in the xrf spectrometer and analyzed for chemical composition. 2.4.2. determination of surface morphology the surface morphology of the tailings was determined using scanning electron microscope (sem-mve016477830). the sputter coater was operated in an argon atmosphere using a current of 6 ma for 3 minutes. 2.5. human health risk assessment parameters the carcinogenic and non-carcinogenic risks were evaluated using the human health risk assessment model for dermal contact, ingestion and inhalation exposure pathways [13]. the health risk assessment is centered on the exposure factors and guidelines handbook of united states environmental protection agency (usepa) [14]. the average daily dose (add) via inhalation (addinh), ingestion (adding) and dermal contact (addderm) for both children and adults were evaluated using equations (1) (3) as adopted from qing et al., [13]. ingestion dose ding−s = cs × ingr × ef × ed × cf bw × at (1) inhalation dose (dinh−s) = cs × inhr × ef × ed bw × at × pef (2) dermal dose (dder−s) = cs × s a × s l × ef × ed × cf bw × at ,(3) where cs is the concentration of the analyte in the tailing from the exposure point (mg/kg), ingr tailing ingestion rate for the receptor (mg/d), inhr soil inhalation rate for the receptor (m3/d), ef exposure frequency (days/year), ed exposure duration (years), pef soil-to-air particulate emission factor (m3/kg), sa skin surface area available for exposure (cm2), sl soil-to-skin adherence factor (mg/cm2/event), bw timeaveraged body weight (kg), at average time of non-carcinogenic and carcinogenic risks (days) and abs dermal absorption factor (dimensionless). the hazard quotient, hazard index and total cancer risk were evaluated using equations (4) – (6) as adopted from man et al., [15]. the hazard quotient is given as: hq = d rfd , (4) where d = dose (ingestion, inhalation or dermal), rfd = reference dose. the hazard index hi is given as: hi = hqing + hqinh + hqderns (5) total cancer risk (rt) is given as: rt = d × s f, (6) where d = dose, sf=slope factor. risk characterization was considered separately for carcinogenic and non-carcinogenic effects [16, 17]. health risks were obtained by comparing the calculated hq, hi and rtotal values with recommended maximum values shown on table 1. 3. results and discussion 3.1. determination soil ph and temperature the results of the ph and temperature obtained from the study area are presented in table 2. the ph and temperature results of the samples are presented in table 2. the ph obtained from the mining and the control sites were (5.23, 5.11) and (7.48, 7.21) for zabot and tafan respectively. the ph results showed that the two mining sites were slightly acidic, while the control sites were slightly alkaline. the soil temperatures were within the range of 29 to 30◦c, which are suitable for plants growth [19]. 4. scanning electron microscopy of tin mines tailings the results of scanning electron microscopy (sem) for tin mine tailing site s3 and s4 are as shown in plates 1 (figure 2) and 2 (figure 3), respectively. figure 2. plate 1: the scanning electron microscope micrograph (sem) of zabot s3. result of s3 on plate 1, showed homogenous small size particles with fine porous structure. while results on plate 2 (s4), showed a micrograph with rough area having different irregular shapes of varying sizes and pores distributed over the surface. the more the number of pores the better the soil aeration [20, 21]. from the two results (plates 1 and 2), it can be seen that the s4 has more pores than the s3. 408 bwede et al. / j. nig. soc. phys. sci. 3 (2021) 406–413 409 table 1. usepa reference doses for non carcinogens and slope factor for carcinogens source: [18] rf ding rf d/mg.kg−1.d−1 rf ddermal sfing sf/kg.d,mg−1 sfinh sfdermal as 3.00e-04 1.23e-04 1.50 e+00 1.51 e+00 3.66 e+00 cr 3.00e-03 2.86 e-05 6.00 e-05 4.2 e+01 cu 4.00 e-02 4.02 e-02 1.20 e-02 pb 1.40 e-03 3.52 e-03 5.25 e-05 8.50 e-03 ni 2.00 e-02 2.06 e-02 5.40 e-03 8.40 e-1 zn 3.00 e-01 3.00 e-01 6.00 e-02 hg 3.00 e-04 8.57 e-05 3.00 e-05 rfd=reference dose and sf=slope factor table 2. ph and temperature of tin mine tailings and control soil sample ph temperature ◦c tin mine site s3 5.23 30 control s3 7.48 29 tin mine site s4 5.11 29 control s4 7.21 30 figure 3. plate 2: the scanning electron microscope micrograph (sem) of tafan (s4). 4.1. heavy metal concentration in tin mine tailings and control sites the heavy metals concentrations of pb, cr, as, zn, cd, ni and cu in tin mine tailings and control sites are presented in table 3. the results on table 3, belong to the heavy metals concentrations of the tin mine tailings and the control site in zabot of barkin ladi local government area. these were determined using energy dispersive x-ray fluorescence spectrometer edxxrf (minipal4). the results showed that the heavy metals concentrations were within the usepa permissible limits of the soil, except for pb and as with range of (270-300) mg/kg and (40-70) mg/kg respectively, which are higher than the usepa permissible limits of 80 and 0.07 mg/kg for both pb and as respectively [22]. the high concentration of as and pb may be traceable to the tin mining activities, farming activities such as application of agricultural chemicals and atmospheric depositions by transport mechanism on the site [19]. this agrees with the work reported by bwede et al., [3]. the extreme concentration of as and pb may be via bioaccumulation in plant which then enters the food chain when these plants are consumed [22]. also, the concentrations of the heavy metals from the mining sites are significantly higher than the control sites which may be due to anthropogenic activities within the environment [23]. zn found in the area may be as a result of its natural abundance in the parent material [24]. this agrees with the work reported by banerjee [25], that iron oxides adsorb some quantities of zn in the lattice structure. 4.2. human health risk assessment of the site the human health risk assessment results for the non-carcinogenic and carcinogenic are shown in tables 4 7 for children and adult respectively. for the non-carcinogenic risk (tables 4 and 5), if the hazard quotient (hq) and hazard index (hi) are greater than 1, then adverse health effects may occur [15, 16]. also for carcinogenic risk level (table 6-7), total cancer risk (rtotal) values greater 1 × 10−4 represents elevated risks, rtotal less than 1 × 10−6 does not pose any significant health risk, and rtotal values between 1 × 10−4 and 1 × 10−3 are generally considered acceptable [26, 17]. 4.2.1. non-carcinogenic risk assessment (ncr) for s3 and s4 the results on tables 4 5, are the non-carcinogenic risk assessment of (pb, zn, cd, cr, ni and as) for s3 and s4. the three human exposure routes considered in this study were ingestion, inhalation and dermal exposure. for mining site s3, values of hq for ingestion and dermal for children are all greater than 1, except for inhalation which are all less than 1. according to huang et al., [16], hq and hi greater than 1 are indication of high cancer exposure risk, while values less than 1 indicates that there are no significant effects. therefore, since the ingestion and dermal values are greater than 1, they have high risk exposure. similar results were observed for adults, except for cr (1.69 × 10−1) which is less than 1 for ingestion. for the control (s3), the results of children for the three exposure pathways are greater than 1, except for cr, pb and ni with values (7.05 × 10−5, 1.32 × 10−4 and 0.60 × 10−6), respectively which are less than 1 indicating no risk [16]. similar results were observed for adults, except for ni (2.10 × 10−6) 409 bwede et al. / j. nig. soc. phys. sci. 3 (2021) 406–413 410 table 3. heavy metals concentration in tin mine tailings and control samples in zabot, and tafan in barkin ladi local government area of plateau state, nigeria in (mg/kg) heavy metal tin mine tailing control usepa (2009) s3 s4 s3 s4 pb 300 1,900 280 270 80 cr 900 1,000 170 1,600 100,000 as 40 60 70 60 0.07 zn nd 200 bdl 180 23,000 cd nd nd bdl bdl 1.7 ni 210 50 70 79 1,600 cu nd nd bdl bdl 3,000 bdl: below detection limits table 4. human health risk assessment of non-carcinogenic hazard of heavy metals for zabot (s3) non carcinogenic hazards s3 group heavy metal hqing hqinh hqdern-s hi tin mine tailing from mine site children as 2.65×106 nc 2.19×105 2.87×106 cr 9.43×106 8.78×10−5 4.84×105 9.91×106 pb 1.9×107 1.67×10−4 5.83×105 1.95×107 ni 3.43×104 0.74×10−6 0.15×103 3.46×104 zn 7.73×104 2.33×10−7 4.43 7.73×103 sum 3.12×107 2.56×10−4 1.29×106 3.23×107 adult as 4.03×105 nc 4.46×104 4.48×105 cr 1.69×10−1 4.44×10−3 1.21×102 1.27×102 pb 2.91×106 0.86×10−4 1.17×105 3.02×106 ni 0.53×104 3.85×10−7 0.03×103 5.33×103 zn 1.18×103 1.06×10−7 8.87 1.19×103 sum 3.35×106 4.53×10−3 1.62×105 3.48×106 soil from agricultural farmland around mine site children as 2.03×106 nc 1.79×105 2.21×106 cr 7.86×106 7.05×10−5 4.02×104 7.72×106 pb 1.50×107 1.32×10−4 4.60×105 1.54×107 ni 2.76×104 0.60×10−6 0.12×103 2.77×104 zn 7.33×103 1.96×106 4.22×101 7.37×103 sum 2.49×107 1.96×106 6.79×105 2.54×107 adult as 3.37×105 nc 3.71×104 3.71×104 cr 0.68×103 5.56×103 0.51×105 5.72×104 pb 2.29×106 0.66×104 0.92×105 2.88×106 ni 4.22×103 2.10×10−6 0.02×103 4.24×103 zn 1.12×104 1.43×10−7 8.43×10−3 1.12×104 sum 2.64×106 1.22×104 1.80×105 2.99×106 nc: not calculated 410 bwede et al. / j. nig. soc. phys. sci. 3 (2021) 406–413 411 table 5. human risk assessment of non-carcinogenic hazard of heavy metals in tafan (s4) non carcinogenic hazards s4 group heavy metal hqing hqinh hqdern−s hi tin mine tailing from mine site soil from agricultural farmland around mine site children as 1.80×10−8 2.70×101 2.19×104 2.87×106 cr 9.43×106 2.54×101 4.84×105 9.91×106 pb 1.9×107 1.67×104 5.83×105 1.95×107 ni 3.45×104 0.74×10−6 0.15×103 3.46×104 zn 7.73×104 2.33×107 4.43×101 7.73×103 sum 2.85×107 2.33×107 1.09×10−6 3.23×107 adult as 4.03×105 5.48×107 4.46×104 4.48×105 cr 5.63×101 7.24×103 1.21×102 1.27×102 pb 2.91×106 0.86×10−6 1.17×105 3.02×106 ni 0.53×104 3.85×107 0.03×103 5.33×103 zn 1.18×103 1.06×107 8.87×105 1.19×103 sum 3.32×106 1.45×108 1.62×105 3.47×106 children as 2.03×106 2.20×101 1.79×105 2.21×106 cr 7.86×106 2.11 4.02×104 7.72×106 pb 1.50×102 4.64×107 4.60×105 1.54×107 ni 5.51×102 2.76×104 0.12×103 2.77×104 zn 7.33×103 1.96×107 4.22×101 7.37×103 sum 2.49×107 6.60×107 6.79×105 2.32×107 adult as 3.37×105 nc 3.71×104 3.74×105 cr 2.03×103 5.56×103 0.51×105 5.72×104 pb 2.29×106 0.66×10−4 0.92×105 2.38×106 ni 4.22×103 2.10×106 0.02×103 4.24×103 zn 1.12×104 1.43×10−7 8.43×10−3 1.12×104 sum 2.65×106 2.11×106 1.80×105 2.83×106 nc: not calculated table 6. individual carcinogenic risk at the site zabot (s3) group heavy metal carcinogenic risk s3 rtotal ring−s rinh−s rdern−s tin mine tailings from mine site as 1.17 × 102 1.14 × 10−8 22.18 1.39 × 102 cr nc 1.64 × 10−7 nc 2.02 × 10−7 pb 3.30×103 nc nc 3.30 × 103 ni nc 8.17 × 10−8 nc 8.17 × 10−8 sum 3.41 × 103 2.57 × 10−3 22.18 16.41 × 106 soil from agricultural farmland around mine site as 7.61×102 2.12×10-8 1.60×101 3.42×103 cr nc 2.64×10−5 nc 2.64×10−5 pb 2.16 ×101 nc 1.67×101 38.30×101 ni nc 6.90×10−8 nc 6.90×10−8 sum 78.26×102 2.64×10−5 32.70×101 3.80×109 nc: not calculated 411 bwede et al. / j. nig. soc. phys. sci. 3 (2021) 406–413 412 table 7. individual carcinogenic risk at the site zabot (s3) group heavy metal carcinogenic risk s3 rtotal ring−s rinh−s rdern−s tin mine tailings from mine site as 4.55×102 4.16×10−8 115.14 5.70×102 cr nc 1.82×10−7 nc 1.82×10−7 pb 3.63×104 nc nc 3.63×104 ni nc 9.64×10−9 nc 9.64×10−9 sum 36.76×103 2.33×10−4 115.14 9.64×109 soil from agricultural farmland around mine site as 1.14×103 3.46×10−8 2.4×101 1.16×103 cr 1.71×10−7 nc nc 1.71×10−7 pb 2.05×102 nc 106.11 31.1×102 ni nc 1.51×10−8 nc 1.51×10-8 sum 1.34×103 4.97×10−2 130.11×102 4.27×103 nc: not calculated and zn (1.43 × 10−7) that are less than 1 and pose no risk [16]. also, the hi values were all greater than 1, indicating significant effects. the summation of hq for ingestion in the mining and control sites are (3.12×107 and 3.35×106), and (2.49×107 and 2.64 × 106) for children and adults respectively. from the results, it is observed that the mining site poses higher risk than the control site, and the children are at higher risk than the adults due to their high values [27]. similar results were reported by ngole-jeme and fantke [28] for, studies on ecological and human health risks associated with abandoned gold mine tailings contaminated soil. the non-carcinogenic risk (ncr) results for tafan (s4) are presented in table 5. for mining site, the hq values for children were found to be greater than 1, except for as and ni with values (1.80×10−8 and 0.74×10−6) via ingestion and inhalation respectively. according to man et al., [15], hq greater than 1 is an indication of high risk exposure. for the adults, the hq were found to be greater than 1, except for pb with value 0.86×10−6 which is less than 1 indicating significant and no significant effect respectively [28]. for the control site (s4) results for children (table 5), it was observed that both the hq and hi values were all greater than 1 representing significant effects [16]. the adults result for as, cr, pb, ni and zn also showed that hq and hi values were all greater than 1, except for zn which has values of 1.43 × 10−7 and 8.43×10−3 for inhalation and dermal less than 1, indicating cancer and non-cancer risk respectively [15]. the summation of hq for ingestion in the mining and control sites are (2.85×107 and 3.32×106), and (2.49×107 and 2.65×106) for children and adults respectively. in terms of population group for ncrs, it is observed that the mining site pose more risk than the control site, and the children are at higher risk than the adults due to their high values [28]. from the results for the three different exposure pathways of metals for children and adults, the contribution of hq is in the order of ingestion greater than dermal and dermal greater than inhalation for as, cr, pb, ni and zn in the studied mining and control areas for s4. 4.2.2. carcinogenic risk assessment for s3 and s4 the carcinogenic risks associated with as, cr, pb and ni were evaluated as presented in table 6. from the s3 results, the rtotal for mining site were found to be (1.39 × 102, 2.02 × 10−7, 3.30×103 and 8.17×10−8), and control site (3.42×103, 2.64× 10−5, 38.30 × 101, 6.90 × 10−8) for as, cr, pb and ni respectively. for carcinogenic risk (table 6), total cancer risk (rtotal) values greater than 1×10−4 represents elevated risks, rtotal less than 1 × 10−6 represents no significant health risk, and rtotal values between 1 × 10−4 and 1 × 10−3 are generally considered acceptable [26, 17]. from the mining and control results, it was observed that the values of the rtotal for as and pb were more than the acceptable threshold, while cr and ni were below the threshold representing elevated risks and no significant health risk respectively [17]. similar results were observed by narsimha and haike [26] for, studies on distribution, contamination, and health risk assessment of heavy metals in surface soils from northern telangana, india. from results of the mining site s4 (table 7), the rtotal were found to be (5.70×102, 1.82×10−7, 3.63×104 and 9.64×10−9), and the control were found to be (1.16 × 103, 1.7110−7, 31.1 × 102 and 1.51 × 10−8) for as, cr, pb and ni, respectively. from the results of the mining and control sites, it is observed that the values for as and pb rtotal were higher than the acceptable threshold, while cr and ni were below the threshold. these results showed that as and pb pose elevated risks, while cr and ni does not pose any significant health risk [26, 17]. furthermore, values from the mining site indicate more pollution than the control. 5. conclusion the results of the analysis of this research work showed that both the mining and control sites in zabot and tafan in barkin ladi l.g.a and environs contains certain concentration of heavy metals (pb, ni, zn, cr, cd, as and cu) in different fractions. the surface morphology of the tin mine tailings through the use of scanning electron microscopy (sem) 412 bwede et al. / j. nig. soc. phys. sci. 3 (2021) 406–413 413 technique revealed homogenous sized particles with fine porous structure, and rough area having different irregular shapes of varying sizes and pores distributed over the surface for both s3 and s4 respectively. from the hazard quotient (hq) and human health risk derived from carcinogenic and non-carcinogenic hazards for adults and children, it is observed that the mining site pose more risk than the control site, and children are at higher risk than the adults due to their high values. from 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[27] c. s. qu, z. w. ma, j. yang, y. liu, j. bi & l. huang, “human exposure pathways of heavy metals in a lead-zinc mining area, jiangsu province”, china. plos one 7 (2012) 11 [28] v. m ngole-jeme & p. fantke, “ecological and human health risk associated with abounded gold mining contaminated soil”, review metal journal 42 (2017) 100. 413 j. nig. soc. phys. sci. 3 (2021) 189 journal of the nigerian society of physical sciences corrigendum to “numerical simulation of copper indium gallium diselenide solar cells using one dimensional scaps software [j. nig. soc. phys. sci. 3 (2021) 48–58, https://doi.org/10.46481/jnsps.2021.133]” c. o. lawania, g. j. ibeha, o. o. igea, eli danladia,b,∗, j. o. emmanuela,c, a. j. ukwenyaa,1, p. o. oyedared adepartment of physics, nigerian defence academy, kaduna, nigeria bdepartment of physical sciences, greenfield university, kasarami, kaduna, nigeria cdepartment of basic science and general studies, federal college of forestry mechanization, kaduna, nigeria ddepartment of science laboratory technology, federal polytechnic ede, osun state, nigeria doi:10.46481/jnsps.2021.333 article history : received: 06 august 2021 accepted for publication: 13 august 2021 published: 29 august 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: b. j. falaye the original content of this published article has an error in the name of one of the co-authors eli danladi which was written incorrectly as d. eli. we highly regret this. ∗corresponding author tel. no: +2348063307256 email address: danladielibako@gmail.com (eli danladi) 189 j. nig. soc. phys. sci. 3 (2021) 278–281 journal of the nigerian society of physical sciences ridge estimation’s effectiveness for multiple linear regression with multicollinearity: an investigation using monte-carlo simulations o.g. obadinaa, a. f. adedotunb,∗, o. a. odusanyac adepartment of mathematical sciences, olabisi onabanjo university, ago-iwoye, ogun state, nigeria bdepartment of mathematics, covenant university ota, ogun state, nigeria cdepartment of mathematics, d.s adegbenro (ict) polytechnic, itori, ogun state, nigeria abstract the goal of this research is to compare multiple linear regression coefficient estimation technique with multicollinearity. in order to quantify the effectiveness of estimations by the mean of average mean square error, the ordinary least squares technique (ols), modified ridge regression method (mrr), and generalized liu-kejian method (lkm) are compared with the average mean square error (amse). for this study, the simulation scenarios are 3 and 5 independent variables with zero mean normally distributed random error of variance 1, 5, and 10, three correlation coefficient levels; i.e., low (0.2), medium (0.5), and high (0.8) are determined for independent variables, and all combinations are performed with sample sizes 15, 55, and 95 by monte carlo simulation technique for 1,000 times in total. as the sample size rises, the amse decreased. the mrr and lkm both outperformed the ols. at random error of variance 10, the mrr is the most suitable for all circumstances. doi:10.46481/jnsps.2021.304 keywords: monte-carlo, multicollinearity, regression model, ridge estimation, simulations article history : received: 16 july 2021 received in revised form: 30 september 2021 accepted for publication: 05 october 2021 published: 29 november 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: t. latunde 1. introduction multiple linear regression (mlr), a widely used and wellknown statistical technique, is now applied in a variety of fields [1,2]. this method is a statistical strategy that predicts the values of a response by combining many predictors (independent variables). mlr’s purpose is to identify the optimal model for describing the linear relationship between predictor and response variables. after obtaining the best subsets of predictors, ∗corresponding author tel. no: +2348055711272 email address: adedayo.adedotun@covenantuniversity.edu.ng (a. f. adedotun ) mlr’s main objective is to estimate coefficients, find the most fitting estimates, and minimize errors. the least squared approach has long been used as a tool for estimating. it is a common and acceptable approach. however, this technique has a multicollinearity constraint, which is a major mlr roadblock. literatures on ridge regression are concerned with the problem of finding a better substitute to the least square estimator. common methods in dealing with multicollinearity include but not limited to ridge regression [3]. estimation procedures are obtained using some specific assumptions such as the random vector ε should be independent and identically distributed random variables. but when these assumptions are violated, these methods do not yield the desirable results and leads to problems 278 obadina et al. / j. nig. soc. phys. sci. 3 (2021) 278–281 279 such as heteroscedasticity and autocorrelation [4]. li, & yang [5,7] suggested jackknifed modified ridge estimator (jmre) and it was shown that it superior to other models. giacalone et al, [7] define multicollinearity as a condition were regressor variables in multiple linear regression model is almost linearly dependent. this condition causes the variance of least squares estimator tends to be large and the estimator becomes unstable. hence, this condition will result in a reduced explanation of the result of the regression model and ridge regression is used to address these difficulties. ref. [7] introduce the lpmin method to determine and address the multicollinearity problem. the major advantage of the approach is that it produces more efficient estimates of the model’s parameters than the ordinary least squares method. ref. [8] proposed a new collinearity diagnostic test. monte carlo simulation study conducted to compare the existing and proposed tests. it is based on coefficient of determination and adjusted coefficient of determination on auxiliary regression of regressors while ref. [9] examined estimators which are resistant to the combined problems of multicollinearity and nonnormal disturbance distributions. can the ridge estimator and some robust estimation technique be combined to produce a robust ridge regression estimator? an algorithm that uses the α-level estimation method to evaluate the parameters of the ridge fuzzy regression model was proposed by ref. [10]. parameter bias, type i and type ii error, and variance inflation factor (vif) values produced by multiple regressions with two, four, and six predictors under various multicollinearity circumstances were examined [11]. multicollinearity is not linked to type i error, but it does increase type ii error, according to the findings. multicollinearity appears to enhance the variability in parameter bias while resulting in overall parameter underestimate. vif is also increased by collinearity. increasing the number of predictors, on the other hand, interacts with multicollinearity in all diagnostics to exacerbate difficulties. an extended conventional semi-parametric partial linear regression model was introduced [12]. the effectiveness of the proposed method is then illustrated through two numerical examples including a simulation study. they also compared with some common fuzzy multiple regression models with fuzzy predictors and fuzzy responses. in the method of ridge regression, a constant bias ridge k was added to x′x matrix. this work illustrates the use of restricted ridge regression method in disabling the multicollinearity in regression model. the method was developed by using the prior information of the parameter β [13]. 2. materials and methods ridge regression is suited to deal with the problem of multicollinearity, especially when the predictors are highly correlated. in 1970, [14] proposed the ridge regression estimator, which included a scalar multiplication, the product of a positive real number and the identity matrix, within the inverse component of the least square estimator. this yielded more accurate ridge parameter estimations than least square estimates, and its variance and mean square errors are frequently lower than least square estimates. the following is a comparison of the three approaches for estimating multiple linear regression coefficients: least squares method (ols), modified ridge regression method (mrr), and liu kejian method (lkm): 2.1. ordinary least square method (ols) the best linear unbiased estimator is a method for estimating multiple linear regression coefficients that is unbiased and has the least variation of estimation (blue). the estimated value is expressed as β̂ols = ( x′x )−1 x′y (1) where x is the n × p predictors matrix, y is the n × 1 observation vector, β̂ols is the vector of coefficients estimates. the mean square error of β̂ols is σ2tr (x′x) −1. 2.2. modified ridge regression (mrr the ridge estimation for multiple linear regression coefficients is β̂ridge = (x ′ x + kin) −1 x′y, k > 0, (2) where β̂ridge is p × 1 ridge estimator, k is a positive real number also known as a constant bias ridge, in is an identity matrix of size n. the approximation of the linear regression coefficients is more accurate and closer to the real values when historical data is used with the ridge regression approach. the modified ridge regression (mrr) approach is as follows: β̂mrr = ( x ′ x + kin )−1 (x ′ y + k j) (3) where j is p × 1 historical observation vector, j = ( ∑p i=1 β̂ols p )1, 1 is p × 1 vector of ones where every element is equal to one. from equation (3), β̂mrr = β̂ols when k = 0 the estimation of k is considered using the following two cases: 1. σ2 is known, k̂ =  pσ2(̂ βols −j )′(̂ βols −j ) −σ2 tr(x′x)−1 , i f (̂ βols − j )′ (̂ βols − j ) −σ2tr(x′x)−1 > 0 pσ2(̂ βols −j )′(̂ βols −j ), otherwise 2. σ2 is unknown, k̂ =  pσ̂2(̂ βols −j )′(̂ βols −j ) −σ̂2 tr(x′x)−1 , i f (̂ βols − j )′ (̂ βols − j ) − σ̂2tr(x′x)−1 > 0 pσ̂2(̂ βols −j )′(̂ βols −j ), otherwise where σ̂2 = ( y−xβ̂ols )′( y−xβ̂ols ) n−p is an unbiased estimator of σ 2. the mean square of β̂mrr is σ̂2tr (( x ′ x + k̂in )−1 (x′x) ( x ′ x + k̂in )−1) + k̂2 (̂ βols − j ) ( x ′ x + k̂in )−2 (̂ βols − j ) 279 obadina et al. / j. nig. soc. phys. sci. 3 (2021) 278–281 280 2.3. generalized liu kejian method (lkm) in the situation of a multiple-relationship between the independent variables, this is a method for estimating the multiple linear regression coefficient. the advantages of the ridge regression approach and the stein method are combined. the generalized kejian method is the name of this method, and the form of the multiple linear regression coefficient estimator is β̂lk m = ( x′x + in )−1 (x′y + dβ̂ols ) , 0 < d < 1 (4) when d = 1, β̂lk m = β̂ols and β̂lk m = ( x′x + in )−1 (x′y + dβ̂ols ) = ( x′x + in )−1 (x′y + d)̂βols ) = ( in − ( x′x + in )−2(in − d)) β̂ols = ( in − ( x′x + in )−2(in − d)2) β̂ols (5) where d = diag (d1, d2, . . ., dp), 0 < di < 1, i = 1, 2, . . ., p and the estimates of di is d̂i = 1 − σ̂ (λi + 1)√ λiβ̂ 2 ols i + σ̂2 the mean square error of β̂lk m is (in− ∆2) (x′x) −1(in− ∆2)σ2 + ∆2ββ′∆2, where ∆ = (x′x + in) −1(in − d) 2.4. monte carlo simulation a monte carlo simulation scenario with three and five independent variables was developed by [8], the properties were zero mean normally distributed random error of variance 1, 5, and 10, and three correlation coefficients levels; i.e., low (0.2), medium (0.5), and high (0.8) are determined for independent variables, and all combinations are performed with sample sizes 15, 55, and 95 by monte carlo simulation. the steps for carrying out the simulation are: 1. the random error (ε) is simulated as ε ∼ n (0,σ2εin), where σ2ε = 1, 5, 10. 2. an observation matrix, x, is simulated from x ∼ nn(0, in ) with different levels of polynomial relations such that ρ = 0.2, 0.5, 0.8. 3. generate response values of y from the model with multiple linear regression coefficient β. 4. multiple linear regression coefficients are estimated for all methods. the step in 1-3 are repeated 1,000 times in each scenario. 5. then calculate the mean of the mean square error of multiple linear regression, ams e = 11000 ∑1000 i=1 ms e), the method with lowest amse is selected as the best method for the scenario involved. table 1. the best method of all scenarios from 1000-monte carlo simulations predictors σ ρ n=15 n=55 n=90 3 1 0.3 mrr mrr mrr 3 1 0.6 mrr mrr mrr 3 1 0.9 mrr mrr mrr 3 5 0.3 mrr mrr mrr 3 5 0.6 mrr mrr mrr 3 5 0.9 mrr mrr mrr 3 10 0.3 lkm mrr mrr 3 10 0.6 lkm lkm mrr 3 10 0.9 lkm mrr mrr 5 1 0.3 mrr mrr mrr 5 1 0.6 mrr mrr mrr 5 1 0.9 mrr mrr mrr 5 5 0.3 lkm mrr mrr 5 5 0.6 mrr mrr mrr 5 5 0.9 mrr mrr mrr 5 10 0.3 lkm mrr mrr 5 10 0.6 lkm lkm mrr 5 10 0.9 lkm lkm lkm 3. conclusion the optimum multicollinearity mlr coefficients estimation approach for each simulation circumstance is shown in table 1. clearly, the ols is not suited in every application. mrr and lkm are the best approaches for determining mlr coefficients when multicollinearity exists. mrr is appropriate for all sample sizes and data with low predictor correlation degrees and small to moderate error variance. the generalized liu kejian method is well suited to small datasets with a high degree of predictor correlation and a large error variance. lkm outperforms mrr as the number of predictors grows, however the more predictors, the greater the risk of multicollinearity. references [1] k. k. adesanya, a. i. taiwo, a. f. adedotun & t. o. olatayo “modeling continuous non-linear data with lagged fractional polynomial regression”, asian journal of applied sciences 06 (2018) 315. [2] g. ciulla, & a. d’amico “building energy performance forecasting: a multiple linear regression approach”, applied energy 253 (2019) 113500. [3] h. yang, & x. chang “a new two-parameter estimator in linear regression”, communications in statistics theory and methods 39 (2010) 923 doi: 10.1080/0361092090280791. [4] b. m. g. kibria “performance of some new ridge regression estimators”, communications in statistics simulation and computation, 32 (2003) 419. doi: 10.1081/sac-120017499. [5] y. li, & h. yang “on the performance of the jackknifed modified ridge estimator in the linear regression model with correlated or heteroscedastic errors”, communications in statistics theory and methods 40 (2011) 2695. doi: 10.1080/03610926.2010.491589 [6] d. c. montgomery, e. a. peck e & g. g. vining “introduction to linear regression analysis”, (united states: john wiley & sons) 2001. [7] m. giacalone, d. panarello, & r. mattera “multicollinearity in regression: an efficiency comparison between l p-norm and least squares estimators”, quality & quantity 52 (2018) 1831. [8] m. i ullah, m. aslam, s. altaf, & m. ahmed “some new diagnostics of multicollinearity in linear regression model”, sains malaysiana 48 (2019) 2051. 280 obadina et al. / j. nig. soc. phys. sci. 3 (2021) 278–281 281 [9] a. f. lukman, k. ayinde, & a. s. ajiboye “monte carlo study of some classification-based ridge parameter estimators”, journal of modern applied statistical methods 16 (2017) 24. [10] s. h. choi, h. y. jung, & h. kim, “ridge fuzzy regression model”, international journal of fuzzy systems 21 (2019) 2077. [11] m. r. lavery, p. acharya, s. a. sivo, & l. xu, “number of predictors and multicollinearity: what are their effects on error and bias in regression?”, communications in statistics-simulation and computation 48 (2019) 27. [12] m. g. akbari, & g. hesamian “a partial-robust-ridge-based regression model with fuzzy predictors-responses”, journal of computational and applied mathematics 351 (2019) 290. [13] f. a. o rumere, s. m. soemartojo & y. widyaningsih, “restricted ridge regression estimator as a parameter estimation in multiple linear regression model for multicollinearity case”, journal of physics: conference series 24 (2021) 1725. [14] a. e. hoerl & r. w. kennard “ridge regression: biased estimation for nonorthogonal problems”, american society for quality 12 (1970) 44. 281 j. nig. soc. phys. sci. 1 (2019) 62–71 journal of the nigerian society of physical sciences original research a class of block multi-derivative numerical techniques for singular advection equations m. o. ogunniran∗ department of mathematical sciences, osun state university osogbo, nigeria abstract the integration of some differential equations is hard to acquire because of the presence of singular point(s) in these equations. these equations are best solved by some unique technique. multi-derivative techniques have a long history of powerful integration of such equations yet till date, a couple of class of this technique has been explored for integrating partial differential equations. this work centers around the development, analysis and implementation of a class of multi-derivative technique on partial differential equations. the approaches were effectively analyzed and were turned out to be consistent, stable and convergent. numerical outcomes got likewise demonstrated the approximation quality of the technique over existing techniques in literature. keywords: advection equations, singular point, multi-derivative, conservation law, off-step point article history : received: 21 april 2019 received in revised form: 03 june 2019 accepted for publication: 04 june 2019 published: 13 july 2019 c©2019 journal of the nigerian society of physical sciences. all rights reserved. communicated by: t. latunde 1. introduction considering an equation which is known as the linear advection equation for a quantity u(x, t): ∂u ∂t + v(x, t) ∂u ∂x = 0 (1) where v(x, t) is known, this type of equation in (1) is classified as first order hyperbolic partial differential equation. equation (1) is termed as a conservation law since it expresses conservation of mass, energy or momentum under the conditions for which it is derived, i.e. the assumptions on which the equation is based. ∗corresponding author tel. no: +2347039104606 email address: muideen.ogunniran@uniosun.edu.ng (m. o. ogunniran ) in physical phenomenon, v(x, t) is linear or flow velocity. although equation (1) is possibly the simplest partial differential equation, this simplicity is deceptive in the sense that it can be very difficult to integrate numerically due to the presence of singularity, a distinctive feature of first order hyperbolic partial differential equations. equation (1) will have to be supplemented with an initial condition: u(x, 0) = u0(x) (2) and if necessary boundary condition(s): u(a, t) = u1(t), u(b, t) = u2(t). x ∈ [a, b] (3) one of the most popular techniques applied to the solution of advection equation is finite difference method [1,2]. in recent times, the finite element and the finite volume methods were 62 m. o. ogunniran / j. nig. soc. phys. sci. 1 (2019) 62–71 63 also introduced. many special methods have also been developed to deal with singularly perturbed problems. many authors [3,4,5,6] are devoted to singularly perturbed problems of ordinary differential equation and the authors discussed the situation and width of boundary layer(s) and give some effective numerical algorithms. in the work of xianyl [7], a linear hybrid variable method was developed for the solution of advection equations and an efficient numerical treatment of non-linearities in the advection-diffusion-reaction equations was presented by utku et al. [8]. an extensive work was done by gaetamo and silvia [9] on the fractional diffusion-advection equations for fluids and plasmas. adrian [10] presented a generalization of prefactored compact schemes for the solution of advection equations. 2. method and analysis the methods were derived by assuming a continuous approximant for un(x) of a multi-step multi-derivative method of the form: un(x) = αn(x)yn + l∑ i=1 hi(βi,0(x) f (i−1) n + βi,v(x) f (i−1) n+v + k∑ j=1 β j,k(x) f (i−1) n+ j ) ≈ y(x) (4) for solving ivp: y′ = f (x, y), y(0) = y0, where f (x, y) is continuous and jth differentiable, yn is an approximation to y(xn), xn = nh, h > 0 and f ( j) m = f ( j)(xm, ym) such that: f (0)(x, y) = f (x, y), f ( j)(x, y) = ∂ f ( j−1)(x, y) ∂x + f (x, y) ∂ f ( j−1)(x, y) ∂y , where k is the step number, l is derivative order, v ∈ (0, k) is an off-step point and j is the derivative order of f (x, y). however, the continuous coefficients βi, j(x), i = 0, v, k, j = 1(1)k were determined. to this end, approximation of the exact solution y(x) was sought by evaluating the function: u(x) = r+ls−1∑ j=0 a j x j (5) where a j, ( j = 0, 1, ..., r + ls−1) are coefficients determined, x j are the basis functions of degree r + ls − 1, l, r and s being the derivative order, interpolating and collocations points respectively. while ensuring that the function (4) corresponds with the analytical solution at the end point xn, the following conditions were imposed on u(x) and its derivatives u(k)(x) to get the coefficients of the desired methods: u(xn+ j) = yn+ j, j = 0 u′(xn+ j) = fn+ j, j ∈ [0, · · · , k] u′′(xn+ j) = f ′n+ j = gn+ j, j ∈ [0, · · · , k] u′′′(xn+ j) = f ′′n+ j = hn+ j, j ∈ [0, · · · , k] ... u(k)(xn+ j) = f (k−1) n+ j , j ∈ [0, · · · , k]  (6) the coefficients obtained were substituted into (5) to obtain the continuous coefficients in (4) which were then evaluated at xn+k to obtain the desired methods. 2.1. one-step, first derivative and two off-step method here, the first derivative evaluation was considered taking into consideration, one step and two off-step points. the procedure is as follows: for k = 1, l = 1 and v = 17 , 2 7 (4) becomes: u(x) = αn(x)yn + h(β1,0(x) fn + β1, 17 (x) fn+ 17 +β1, 27 (x) fn+ 27 + β1,1(x) fn+1) (7) while taking r = 1 and s = 4, (5) and its derivative become: u(x) = 4∑ j=0 a j x j (8) u′(x) = 4∑ j=1 ja j x j−1 (9) imposing the following conditions obtained from (6): u(xn+ j) = yn+ j, j = 0. (10) u′(xn+ j) = fn+ j, j = 0, 1 7 , 2 7 , 1 (11) a system of five algebraic equations with five unknowns were obtained. the 5 × 5 system of equations obtained are given below: a0 = yn (12) a1 = fn (13) a1 + 2 7 a2 + 3 49 a3 + 4 343 a4 = fn+ 17 (14) a1 + 4 7 a2 + 12 49 a3 + 32 343 a4 = fn+ 27 (15) a1 + 2a2 + 3a3 + 4a4 = fn+1. (16) solving the resulting equations gives: a0 = yn (17) a1 = fn+1 (18) a2 = − 23 4 fn + 49 6 fn+ 17 − 49 20 fn+ 27 + 1 30 fn+1 (19) a3 = 35 3 fn − 49 2 fn+ 17 + 196 15 fn+ 27 − 7 30 fn+1 (20) a4 = − 343 40 fn+ 27 + 49 120 fn+1 − 49 8 fn + 343 24 fn+ 17 (21) a4 = − 49 8 fn + 343 24 fn+ 17 − 343 40 fn+ 27 + 49 120 fn+1 (22) substituting these values in (7) and collecting like terms in yn, fn, fn+ 17 , fn+ 27 , fn+1 yields the continuous method: y(x) = αn(x)yn + h(β0,1(x) fn + β1, 17 (x) fn+ 17 +β1, 27 (x) fn+ 27 + β1,1(x) fn+1) (23) 63 m. o. ogunniran / j. nig. soc. phys. sci. 1 (2019) 62–71 64 where: αn(x) = 1 β1,0(x) = − 49 (2 x−1)4 8 + 35 (2 x−1)3 3 − 23 (2 x−1)2 4 + 2 x − 1 β1, 17 (x) = 343 (2 x−1) 4 24 − 49 (2 x−1)3 2 + 49 (2 x−1)2 6 β1, 27 (x) = −343 (2 x−1) 4 40 + 196 (2 x−1)3 15 − 49 (2 x−1)2 20 β1,1(x) = 49 (2 x−1)4 120 − 7 (2 x−1)3 30 + 1/30 (2 x − 1) 2  (24) evaluating (23) at xn+1 yields the one-step, first derivative main method possessing two off-grid points: yn+1 = yn + h 24 (19 fn − 49 fn+ 17 + 49 fn+ 27 + 5 fn+1) (25) also, evaluating (23) at xn+ 17 and xn+ 27 yields the additional method as: yn+ 17 = yn + h 5880 (335 fn + 595 fn+ 17 − 91 fn+ 27 + fn+1) yn+ 27 = yn + h 21 ( fn + 4 fn+ 17 + fn+ 27 ) (26) the one-step, first derivative and two off-grid points method is presented in block form as follows: yn+1 = yn + h 24 (19 fn − 49 fn+ 17 + 49 fn+ 27 + 5 fn+1) yn+ 17 = yn + h 5880 (335 fn + 595 fn+ 17 − 91 fn+ 27 + fn+1) yn+ 27 = yn + h 21 ( fn + 4 fn+ 17 + fn+ 27 ) (27) 2.2. two-step, first derivative and two off-step method here, a new member of the methods is obtained. the procedure is as follows: for k = 2, l = 1, and v = 17 and 9 7 , (4) becomes: u(x) = αn(x)yn + h(β1,0(x) fn + β1, 17 (x) fn+ 17 +β1, 97 (x) fn+ 97 + β1,1(x) fn+1 + β1,2(x) fn+2) (28) taking r = 1 and s = 5, (5) and its derivative become: u(x) = 5∑ j=0 a j x j (29) u′(x) = 5∑ j=1 ja j x j−1 (30) imposing the following conditions obtained from (6): u(xn+ j) = yn+ j, j = 0. (31) u′(xn+ j) = fn+ j, j = 0, 1 7 , 1, 9 7 , 2. (32) the two-step, first derivative and two off-grid point method is presented in block form as follows: yn+2 = yn + h(− 53 135 fn + 2401 2340 fn+ 17 + 11 4 fn+1 + 24012700 fn+ 97 + 227 975 fn+2) yn+ 17 = yn + h( 3391 52920 fn + 10579 131040 fn+ 17 − 41 8820 fn+1 + 439 151200 fn+ 97 − 79 382200 fn+2) yn+1 = yn + h(− 257 1080 fn + 14749 18720 fn+ 17 + 127 180 fn+1 − 5831 21600 fn+ 97 + 113 7800 fn+2) yn+ 97 = yn + h(− 3141 13720 fn + 11259 14560 fn+ 17 + 59136860 fn+1 − 747 5600 fn+ 97 + 11421 891800 fn+2)  (33) 2.3. two-step, second derivative and one off-step method here k = 2, l = 2, and v = 17 is the chosen off-step point. the new approximation now becomes: u(x) = αn(x)yn + h ( β1,0(x) fn + β1, 17 (x) fn+ 17 +β1,1(x) fn+1 + β1,2(x) fn+2 ) +h2 ( β2,0(x) fn + β2, 17 (x)gn+ 17 +β2,1(x)gn+1 + β2,2(x)gn+2 ) (34) where g(xn+ j) = f ′(xn+ j) = u′′(xn+ j), j = 0, · · · , k. setting r = 1 and s = 4, the interpolating function now becomes: u(x) = 8∑ j=0 a j x j. (35) obtaining the first and second derivatives, we have: u′(x) = ∑8 j=1 ja j x j−1 u′′(x) = ∑8 j=2 j( j − 1)a j x j−2. (36) imposing the following conditions as obtained from (6): u(xn+ j) = yn+ j, j = 0 (37) u′(xn+ j) = fn+ j, j = 0, 1 7 , 1, 2. (38) u′′(xn+ j) = gn+ j, j = 0, 1 7 , 1, 2. (39) thus, the required method is given in block method as: yn+2 = − 31h2 gn+2 845 + 16h2 gn+1 45 + 19208h2 gn+ 17 7605 + 7 5 h2 gn + 781h fn+2 2197 + 4 9 h fn+1 − 470596h fn+ 17 19773 + 25h fn + yn yn+ 17 = − 15997h2 gn+2 66805808160 − 33203h2 gn+1 3557705760 − 516419h2 gn+ 17 250417440 + 565619h2 gn 395300640 + 191117h fn+2 173695101216 + 9703h fn+1 304946208 + 3016667h fn+ 17 39862368 + 5308663h fn 79060128 + yn yn+1 = − 43h2 gn+2 81120 − 197h2 gn+1 4320 + 333739h2 gn+ 17 730080 + 101h2 gn 480 + 539h fn+2 210912 + 919h fn+1 2592 − 19176787h fn+ 17 5694624 + 385h fn 96 + yn (40) 64 m. o. ogunniran / j. nig. soc. phys. sci. 1 (2019) 62–71 65 3. analysis of methods 3.1. order and error constant in this section, analysis of the methods using taylor’s series method for the derivation of order and error constant of the methods were carried out. following lambert (1991) and fatunla (1991), the linear difference operator l associated with the method is defined by: l[y(x); h] = k∑ i=0 αi(x)y(x + ih) − l∑ i=1 hi(βi,0(x)y (i)(x + ih) +βi,v(x)y (i)(x + vh) + k∑ j=1 β j,k(x)y ( j)(x + jh)) (41) where y(x) is an arbitrary function, continuously differentiable on [a, b]. recall that; while expanding y(x + ih) and its derivatives y( j)(x + ih) as taylor’s series about x, and collecting terms gives: l[y(x); h] = c0y(x) + c1hy (1)(x) +· · ·+ cqh qy(q)(x) +· · ·(42) where cq, q = 0, 1, 2, · · · are the constant coefficients are given as follows: c0 = 1 −αn c1 = k −βi,0 −βi,v − ∑k j=1 β j,k ... cq = kq q! − vq−1 (q − 1)! βi,v − kq−1 + 1 (q − 1)! ∑k j=1 β j,k  (43) according to henrici (1962), the main method (4) has order p if: l[y(x); h] = o(hp+1), (44) where: c0 = c1 = · · · = cp = 0, cp+1 , 0. therefore, cp+1 is the error constant and cp+1hp+1(xn) is the principal local truncation error at point xn. 3.2. consistency and zero-stability 3.2.1. consistency according to lambert (1991), a numerical scheme is said to be consistent if the order p of the method is such that p > 1. 3.2.2. zero stability a numerical method is said to be zero-stable if no root of the first characteristic polynomial has modulus greater than one that is if every root with modulus one is simple (lambert, 1991). a linear multi-step method is said to be zero-stable if no root of the first characteristic polynomial defined by: ρ(r) = det[ra(0) − a(1)] (45) where a(0) is an identity matrix of dimension k, a(1) is a matrix of dimension k and satisfy |rs| ≤ 1 and every root of |rs| = 1, s = 1(1)k, has multiplicity not exceeding as h → 0. in what follows, the k-step, m-derivative block method is written as a matrix finite difference equation of the form: a(0)yµ+1 = a (1)yµ + h[b (0) fµ+1 + b (1) fµ] +h2[c(0)gµ+1 + c (1)gµ] + · · · + hm[z(0)zµ+1 + z (1)zµ] (46) where m is the derivative order. yµ+1, yµ, fµ+1, fµ, gµ+1, gµ, · · · , zµ+1, zµ are column matrix determined from the derived methods. a(0), a(1), b(0), b(1), c(0), c(1), · · · , z(0), z(1) are coefficients matrices of the derived methods. to this end, the zero stability of the derived methods are discussed below. for method (27), the matrix difference equation is: a(0)yµ+1 = a (1)yµ + h[b (0) fµ+1 + b (1) fµ] (47) where: yµ+1 = (yn+ 17 , yn+ 27 , yn+1) t yµ = (yn−17 , yn− 27 , yn) t fµ+1 = ( fn+ 17 , fn+ 27 , fn+1) t fµ = ( fn− 17 , fn− 27 , fn) t a(0) =  1 0 0 0 1 0 0 0 1  , a(1) =  0 0 1 0 0 1 0 0 1  b(0) =  595 5880 − 91 5880 1 5880 4 21 1 21 0 − 49 24 49 24 5 24  , b(1) =  0 0 3355880 0 0 121 0 0 1924   (48) zero-stability is a concept concerned with the limiting factor of the difference equation as h → 0. therefore, the difference equation is reduced to: a(0)yµ+1 = a (1)yµ (49) in which the first characteristic polynomial: ρ(r) = 0. (50) the solution of equation (50) corresponds to equation (45), thus solving (50) with parameters obtained in (48) gives the roots of r. ρ(r) = det r  1 0 0 0 1 0 0 0 1  −  0 0 1 0 0 1 0 0 1   = 0 (51) ρ(r) = det   r 0 −1 0 r −1 0 0 r − 1   = 0 (52) from (52) we have: r2(r − 1) = 0 r = 0, 0, 1. (53) 65 m. o. ogunniran / j. nig. soc. phys. sci. 1 (2019) 62–71 66 table 1: table of orders and error constants of the derived multi-derivative methods method evaluating points order error constant xn+1 4 −31 35280 = −8.79e − 04 27 xn+ 17 4 −1 246960 = −4.05e − 06 xn+ 27 4 −1 1512630 = −6.61e − 07 xn+2 5 2 1575 = 1.27e − 03 33 xn+ 17 5 2491 282357600 = 8.82e − 06 xn+1 5 53 117600 = 4.51e − 04 xn+ 97 5 − 39447 94119200 = 4.09e − 04 xn+2 8 157 38896200 = 4.04e − 06 40 xn+ 17 8 162133 1025046183571200 = 1.58e − 10 xn+1 8 307 1244678400 = 2.47e − 07 since no root of the characteristic equation violates the condition: |rs| ≤ 1, s = 1(1)m and m is the highest power of the characteristics equation ρ(r) = 0. it is concluded that method (27) is zero-stable. table 2: table of roots of the characteristic equations for the derived multiderivative methods method absolute value of roots, |rs| 27 0,0,1. 33 0,0,0,1. 40 0,0,1. 3.3. convergence stating without proof and following the fundamental theorem of dahliquist as reported by henrici (1962) for a linear multi-step method. 3.3.1. theorem 1 the necessary and sufficient conditions for a numerical method to be convergent are that it is consistent and zero-stable. here, it is sufficient to say that the derived methods are convergent since they are all consistent and zero-stable. 3.4. linear stability the linear stability properties of the derived methods are determined by expressing them in a form applicable to the test problem: y′ = λy, for which y′n = λyn, y ′′ n = λ 2 yn, · · · , y (n) n = λ nyn., λ < 0 (54) to yield: yµ+1 = m(z)yµ, z = hλ (55) where the amplification matrix m(z) is given by: m(z) = (a(0) − zb(0) − z2c(0) − ·· ·− zlz(0))−1(a(1) +zb(1) + z2c(1) + · · · + zlz(1)). (56) where z = hλ. the matrix m(z) has eigenvalues ξ1,ξ2, · · · ,ξm = 0, 0, · · · ,ξm, where the dominant eigenvalue ξm is the stability function r(z) which is a rational function with real coefficients, m is the order of r(z). the stability function and plot for each of the derived methods are as given below: figure 1: stability region of method (27) figure 2: stability region of method (33) 66 m. o. ogunniran / j. nig. soc. phys. sci. 1 (2019) 62–71 67 figure 3: stability region of method (40) 4. numerical examples and results 5. computational details in the implementation of the derived methods, a system of nonlinear equations must be solved in order to obtain the desired approximation. to solve these nonlinear systems, a newton-krylov solver, nsoli.m or a modified newton solver, nsold was used and compared numerical results with standard and existing methods in literature. it is important to point that the numerical methods were programmed via matlab 9.2 version on a personal computer with the following specifications: • system namehp pavilion x360 convertible • processorintel(r) core(tm) i3-7100u cpu @ 2.40ghz • installed memory (ram)8.00gb • system type64-bits operating system, x64-based processor • operating systemwindows 10 moreso, computational experiments were done with software optimization and only the points of emphases were shown. 5.0.1. example 1 advection equation where v(x, t) is constant: ∂u ∂t = −v ∂u ∂x u(x, 0) = exp(− x 2 l2 ) exact solution:u(x, t) = exp(− (x−vt) 2 l2 ) v = 0.5, l = 10, x ∈ [0, 1], t ∈ [0, 1] (57) is considered. 5.0.2. example 2 an advection equation where v(x, t) = 1 + x2 1 + 2xt + 2x2 + x4 ∂u ∂t = −v(x, t) ∂u ∂x u(x, 0) = exp(− x 2 l2 ) exact solution:u(x, t) = exp(− 1l2 (x − t 1+x2 )) l = 10, x ∈ [0, 1], t ∈ [0, 1] (58) is also considered. 5.0.3. example 3 the inviscid burgers equation where v(x, t) = u(x, t) ∂u ∂t = −u(x, t) ∂u ∂x u(x, 0) = x exact solution: u(x, t) = x t + 1 x ∈ [0, 1], t ∈ [0, 1]. (59) 5.0.4. example 4 a nonlinear advection equation with perturbation term: � ∂u ∂t + u ∂u ∂x = f (x, t), (x, t) ∈ d = [0, 1],×[0, 1] u(0, t) = �t, u(x, 0) = x, (60) where f (x, t) = �2 +�t + x, u(x, t) = �t + x is the true solution, � = 10−4 is also considered. 6. tables of results and comparison in this section, the standards methods of upwind, lax method, the method of wang et al. [11] and the method were employed successfully for solving the advection equations. the numerical results of the proposed method show some superiority over other methods in-terms of accuracy and reliability. moreover, the method is also effective for solving the equation with singular terms and other nonlinear singular perturbation problems. wang et al. [11] used a perturbation method and reproducing kernel method in solving a class of singularly perturbed partial differential equation. from table 3 above, it could be observed that the method shows high computational strength as it compares favourably well with some known standard methods reported. 1 from the table above, it could also be observed that the method shows high computational strength as it compares favourably well with some known standard methods reported. 2 the superiority of the proposed methods could be established in the table above. from the table above, it could also be 1(x, t) is the coordinate of x and t, u(x, t) is the exact solution, um is the upwind method, lm is the lax method, method represents the solution using the method and relative error=| u(x, t) method |/u(x, t). percentage error = relative error × 100%. cpu time is the method’s computational time. 2(x, t) is the coordinate of x and t, u(x, t) is the exact solution, um is the upwind method, lm is the lax method, method represents the solution using the method and relative error=| u(x, t) method |/u(x, t). percentage error = relative error × 100%. cpu time is the method’s computational time. 67 m. o. ogunniran / j. nig. soc. phys. sci. 1 (2019) 62–71 68 table 3: numerical comparison for example 1 for v(x, t) = 0.5, ∆x = 0.1, ∆t = 0.05, l = 10 (x, t) u(x, t) um lm method (27) relative error percentage error (%) (0.1000,0.1000) 0.9999750003 0.9999453153 0.9998539643 0.9999937500 1.87502 e -05 0.0019 (0.2000,0.2000) 0.9999000050 0.9998334259 0.9996180921 0.9999000050 0.00000 0.00 (0.3000,0.3000) 0.9997750253 0.9996699630 0.9993277560 0.9996000800 1.74985 e -04 0.017 (0.4000,0.4000) 0.9996000800 0.9994562666 0.9989757869 0.9991004049 4.99875 e -04 0.050 (0.5000,0.5000) 0.9993751953 0.9991927826 0.9985937720 0.9984012793 9.74525 e -04 0.097 (0.6000,0.6000) 0.9991004049 0.9988796763 0.9981806744 0.9975031224 1.59872 e -03 0.16 (0.7000,0.7000) 0.9987757500 0.9985170174 0.9977971450 0.9964064722 2.37218 e -03 0.24 (0.8000,0.8000) 0.9984012793 0.9981048482 0.9974581631 0.9951119854 3.29456 e -03 0.33 (0.9000,0.9000) 0.9979770489 0.9976432089 0.9972837348 0.9936204364 4.36544 e -03 0.44 (1.0000,1.0000) 0.9975031224 0.9971321487 0.9973810576 0.9919327166 5.58435 e -03 0.56 cpu time na 0.460000s 0.430000s 0.399600s table 4: numerical comparison of example 2 for ∆x = 0.1, ∆t = 0.05, l = 10 (x, t) u(x, t) um lm method (33) relative error percentage error (%) (0.1000,0.1000) 0.9999900991 1.000150539 1.000294258 1.0005001250 5.10031 e -04 0.051 (0.2000,0.2000) 0.9999230799 1.000304064 1.000350801 0.9999000050 2.30767 e -05 0.0023 (0.3000,0.3000) 0.9997523243 1.000452525 1.000336322 0.9996000800 1.52282 e -04 0.015 (0.4000,0.4000) 0.9994484280 1.000569182 1.000136052 0.9991004049 3.48215 e -04 0.035 (0.5000,0.5000) 0.9990004998 1.001112699 0.9998403420 0.9984012793 5.99820 e -04 0.060 (0.6000,0.6000) 0.9984130253 0.9930245386 0.9993584603 0.9975031224 9.11349 e -04 0.091 (0.7000,0.7000) 0.9977006342 1.041218330 0.9987797082 0.9964064722 1.29714 e -03 0.13 (0.8000,0.8000) 0.9968829170 0.9006435880 0.9980268384 0.9951119854 1.77647 e -03 0.18 (0.9000,0.9000) 0.9959804757 1.111721077 0.9972063456 0.9936204364 2.36956 e -03 0.24 (1.0000,1.0000) 0.9950124792 0.9510926234 0.9963208653 0.9919327166 3.09520 e -03 0.31 cpu time na 0.510000s 0.370000s 0.250000s table 5: numerical comparison of example 3 for, ∆x = 0.1, ∆t = 0.05 (x, t) u(x, t) um lm method (33) relative error percentage error (%) (0.1000,0.1000) 0.09090909091 0.0904875000 0.0904875000 0.09090909091 1.000000 e -16 0.00 (0.2000,0.2000) 0.1666666667 0.1653231744 0.1653231744 0.1666666667 1.000000 e -16 0.00 (0.3000,0.3000) 0.2307692308 0.2283156714 0.2283156715 0.2307692308 1.000000 e -16 0.00 (0.4000,0.4000) 0.2857142857 0.2821173696 0.2821192653 0.2857142857 1.000000 e -16 0.00 (0.5000,0.5000) 0.3333333333 0.3289352242 0.3286585068 0.3333333333 1.000000 e -16 0.00 (0.6000,0.6000) 0.3750000000 0.3543939265 0.3694124359 0.3750000000 1.000000 e -16 0.00 (0.7000,0.7000) 0.4117647059 0.6111136606 0.4055658801 0.4117647059 1.000000 e -16 0.00 (0.8000,0.8000) 0.4444444444 0.00861594345 0.4381714651 0.4444444444 1.000000 e -16 0.00 (0.9000,0.9000) 0.4736842105 38.46493835 0.4681540744 0.4736842105 1.000000 e -16 0.00 (1.0000,1.0000) 0.5000000000 0.1220675888 0.4964220517 0.5000000000 1.000000 e -16 0.00 cpu time na 0.350000s 0.320000s 0.281250s observed that the method shows high computational strength as it compares favourably well with some known standard methods reported. 3 from table 6 above, it could also be observed that the method shows some superiority in-terms of accuracy when compared with the method of wang et al. [11] and also compares favourably 3(x, t) is the coordinate of x and t, u(x, t) is the exact solution, um is the upwind method, lm is the lax method, method represents the solution using the method and relative error=| u(x, t) method |/u(x, t). percentage error = relative error × 100%. cpu time is the method’s computational time. well with some known standard methods reported. 4 7. summary and discussion of results this work contains a new class of multi-derivative method. the multi-derivative method was considered up to the second 4(x, t) is the coordinate of x and t, u(x, t) is the exact solution, um is the upwind method, lm is the lax method, method represents the solution using the method and absolute error=| u(x, t) method |. cpu time is the method’s computational time. 68 m. o. ogunniran / j. nig. soc. phys. sci. 1 (2019) 62–71 69 table 6: numerical comparison for example 4 for v(x, t) = u(x, t), ∆x = 0.01, ∆t = 0.01, � = 10−4 (x, t) u(x, t) um lm wang et al. [11] method (40) absolute error (0.0000,0.0000) 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 (0.0100,0.0100) 0.01000100000 0.01009999010 0.01009999010 0.01000000000 0.01000010000 9.000000 e -07 (0.0300,0.0300) 0.03000300000 0.03089993849 0.03089993849 0.03000000000 0.03000030000 2.700000 e -06 (0.0500,0.0500) 0.05000500000 0.05249984036 0.08002383910 0.05000000000 0.05000050000 4.500000 e -06 (0.0600,0.0600) 0.06000600000 0.06359977227 0.1648091770 0.06000000000 0.06000060000 5.400000 e -06 (0.0800,0.0800) 0.08000800000 0.08639959684 0.4698987551 0.08000000000 0.08000080000 7.200000 e -06 (0.1000,0.1000) 0.1000100000 1.009904485 0.8868283796 0.1000000000 0.1000010000 4.500000 e -06 cpu time na 0.400000s 0.171875s na 0.078125s table 7: error comparison for example 1 for v(x, t) = 1.0. n adrain (2019) pc4 scheme adrain (2019) pc4 scheme proposed method 40 6.43092 e -03 1.81277 e -04 5.58435 e -04 60 9.73768 e -04 2.78696 e -07 3.36544 e -09 80 2.71484 e -04 5.66109 e -09 3.24956 e -10 100 1.06781 e -04 2.87180 e -09 1.33524 e -10 figure 4: graph of exact solution of example 1 derivative (k = 2). the method was developed using some selected off-step points in its formation (this is to improve its accuracy), evaluation such as collocation and interpolation of these points were carried out and a system of algebraic equations was obtained for each value of k together with number of off-step points considered. these equations were solved to obtain the unknown parameters. the interpolating function is then evaluated at the selected collocation points to obtain a block form of this class of methods. these methods were then analyzed and were found to be stable and convergent. table 3-7 show the numerical computational results on problems advection equations. extensive comparison was done with existing and standard methods as shown in the literature. these methods were demonstrated on some examples and the superiority of the derived methods over these existing methods was established. it is worthy to say that the derived methods exhibit stronger computational strength. also, figures 4-11, in pairs, figure 5: graph of numerical solution of example 1 show the graphical comparison of derived methods with exact for the advection problems considered. 8. conclusion multi-derivative techniques are numerical tools designed for approximating singular problems. these methodologies use the evaluations of several derivatives at some prior points. due to the characterized multi-step nature of some class of these methods, additional methods of evaluation called the predictor are always required in computation of previous values. this study produces a block form of these methodologies which need no predictions and thereby taking care of this drawback. multiderivative techniques have a long history of improvement in integrating ordinary differential equations, but to date, a few class of these techniques have been explored in the integration of partial differential equations. this research work explored 69 m. o. ogunniran / j. nig. soc. phys. sci. 1 (2019) 62–71 70 figure 6: graph of exact solution of example 2 figure 7: graph of numerical solution of example 2 figure 8: graph of exact solution of example 3 figure 9: graph of numerical solution of example 3 figure 10: graph of exact solution of example 4 figure 11: graph of numerical solution of example 4 the possibility of multi-derivative methods in integrating singular advection equations. from the computational results of this study, it could be observed that the derived methodologies solve satisfactorily the advection problems. 70 m. o. ogunniran / j. nig. soc. phys. sci. 1 (2019) 62–71 71 acknowledgment i thank the referees for the positive enlightening comments and suggestions, which have greatly helped us in making improvements to this paper. references [1] a. balzano, “mosquito: an efficient finite difference scheme for numerical simulation of 2d advection equation”, international journal for numerical methods in fluids 32 (1999) 2. 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[7] z. xianyl, ”linear hybrid-variable methods for advection equations”, advances in computational mathematics, journal of differential equations, conf 23 (2016) https://doi.org/10.1007/s10444-018-9647-z. [8] e. utku, s. murat and k. huseyin, “efficient numerical treatment of nonlinearities in the advection-diffusion-reaction equation”, international journal of numerical methods for heat and fluid flow, http://doi.org/10.1108/hff-02-2017-0198. [9] z. gaetamo and p. silvia, “on the fractional diffusion-advection equation for fluids and plasmas”, fluid 4 (2019) 62. [10] s. adrian, “a generalization of prefactored compact schemes for the solution of adevection equations”, arxiv:1902.04429v1 math.na (2019). [11] y. wang, h. yu, g. tan, and s. qu, “solving a class of singularly perturbed partial differential equations by using the perturbation method and reproducing kernel method”, hindawi abstract and applied analysis (2014) 5. 71 j. nig. soc. phys. sci. 4 (2022) 16–19 journal of the nigerian society of physical sciences heterogeneous catalyzed synthesis of biodiesel from crude sunflower oil selvaraju sivamani∗, marwan ahmed sulieman al aamri, aseela musalem awad anthroon jaboob, azeezah mohammed masoud kashoob, layal kamall abdullah al-hakeem, mouna salim mhaad said almashany, muna ahmed mohammed safrar engineering department, university of technology and applied sciences (salalah college of technology), salalah, oman abstract biodiesel is the fatty acid alkyl esters that are used as the substitute for petro-diesel. the aim of the present work is to optimize biodiesel production from plant based non-edible crude oils with methanol using heterogeneous catalyst by transesterification for its commercialization. the factors affecting the biodiesel production from plant oils are volume of feedstock (oil), volume of alcohol (excess reactant), quantity of calcium hydroxide (catalyst), reaction time, temperature, and agitation speed. transesterification is the reaction between acid and alcohol to produce ester in the presence of alkali catalyst. in this work, transesterification was carried out between crude sunflower oil and methanol in the presence of calcium hydroxide as catalyst. finally, the maximum conversion of 85.6% was achieved at the optimum process parameters of 2 l of crude sunflower oil, 0.3 l methanol, 23 g of calcium hydroxide, reaction time of 24 h, temperature of 65 ◦c, and agitation speed of 100 rpm. the results showed that crude sunflower oil could serve as potential renewable substrate for biodiesel commercialization. doi:10.46481/jnsps.2022.230 keywords: crude sunflower oil, calcium hydroxide, transesterification, biodiesel article history : received: 16 may 2021 received in revised form: 12 january 2022 accepted for publication: 27 january 2022 published: 28 february 2022 c©2022 journal of the nigerian society of physical sciences. all rights reserved. communicated by: b. j. falaye 1. introduction biodiesel is an alternative fuel for clean combustion from domestic and renewable resources. the fuel is a mixture of alkyl esters of fatty acids made from pure edible vegetable oils, pure non-edible vegetable oils, waste vegetable oils, crude vegetable oils, algal oils, animal fats or recycled fats [1]. biodiesel can be used in its pure form with little or no modification in existing diesel engines, wherever applicable. biodiesel is easy to use, biodegradable, non-toxic and essentially free of sulphur and aromatics. it is generally used as ∗corresponding author tel. no: email address: sivmansel@gmail.com (selvaraju sivamani ) a petro-diesel additive to reduce particulate, carbon monoxide, hydrocarbon and pollutant concentrations in diesel vehicles [2]. when used as an additive, the resulting diesel fuel may be referred to as b5, b10 or b20. this is the percentage of biodiesel that is mixed with petro-diesel. for example, b5 means that biodiesel and petro-diesel are mixed in the ratio of 5:95 [3]. biodiesel is produced by the process in which oils are combined with alcohol (ethanol or methanol) in the presence of a catalyst (alkali or heterogeneous) to form ethyl or methyl esters. ethyl or methyl esters of biomass can be mixed with conventional diesel fuel or used as a pure fuel (100% biodiesel) [4]. biodiesel can be made from vegetable oil, animal fat, or microalgae oil. the three basic ways of producing biodiesel 16 sivamani et al. / j. nig. soc. phys. sci. 4 (2022) 16–19 17 from oils and fats are as follows [5]: 1. direct transesterification of oils or fats catalyzed by alkali or heterogeneous catalyst (if free fatty acid (ffa) < 2.5%) 2. esterification of oils or fats by acid catalysis and then by transesterification (if ffa > 2.5%) 3. conversion of oils and fats to its fatty acids and then to biodiesel the most common feedstock of edible oils used to produce biodiesel are soybean oil [6], rapeseed oil [7] and palm oil [8], which accounts for the bulk production of global biodiesel. other raw materials may come from non-edible sources such as jatropha [9], mustard [10], flax [11], and hemp [12]. animal fats, including sebum [13], lard [14], yellow fat [15], chicken fat [16] and fish oil [17] derivatives, may contribute to a small percentage of biodiesel production in the future, but their supply is limited and inefficient to raise animals for their fat. biodiesel can be mixed with petro-diesel in any proportion to produce a biodiesel blend, or it can be used in a pure form. like petro-diesel, biodiesel works with the diesel engine with auto-ignition and essentially requires little or no engine modification, as biodiesel has similar properties to diesel [18]. it can be stored as a petro-diesel and therefore does not require a separate infrastructure. the use of biodiesel in conventional diesel engines results in a significant reduction in the emission of unburnt hydrocarbons, carbon monoxide and particulates. currently, a large number of biodiesel production plants around the world are functioning to full capacity, and a large number are under construction or designed to meet growing global demand [19]. literature studies show that abundant scientific reports are available on production of biodiesel from pure edible vegetable oils [6-8], pure non-edible vegetable oils [9-12], waste vegetable oils [20-22], algal oils [23-25], and animal fats [13-17]. but, only limited literature is available on biodiesel production from crude vegetable oils [26-27], and recycled fats [28-29]. hence, the present work focuses on the optimization of process parameters for production of biodiesel from plant based nonedible crude oils with methanol using heterogeneous catalyst by transesterification for its commercialization. 2. materials and methods 2.1. materials crude sunflower oil was generously provided by omani vegetable oils & derivatives company (llc), raysut, oman. all the chemicals used in the work are of analytical grade and the products of vwr international. double distilled water was used in this study. 2.2. methods a known volume of crude sunflower oil (2 l) was mixed with known volume of methanol and known quantity of calcium hydroxide and mixed well for certain time and temperature with mixing. after the time is completed, the mixture was allowed to settle for 8 h and the bottom glycerol layer was removed and acidulated with phosphoric acid to separate glycerol, sodium phosphate and methanol. sodium phosphate is reacted with water to produce phosphoric acid and recycled back. from the top layer, the mixture of biodiesel and water with little quantity of methanol was washed to remove excess methanol present, and dried to remove moisture from biodiesel. finally, ffa of oil and biodiesel were measured and the percentage conversion was calculated using the equation as given below: % conversion = (f f a in oil − f f a in biodiesel) f f a in oil x100 ffa was estimated following the procedure as follows [30]: standard solvent was prepared by mixing 25 ml diethyl ether and 25 ml 95% ethanol, and titrated against 0.1 n koh using 1 ml of 1% phenolphthalein solution as an indicator. 5 g of oil was dissolved in 50 ml of standard solvent in a 250 ml erlenmeyer flask. the contents are titrated against 0.1 n koh using few drops of phenolphthalein as an indicator. the end point is the appearance of pink color that lasts for 15 s. then, ffa was calculated using the equation as below: free f atty acid value (mg koh/g oil) = t itre value x normality o f koh x 28.05 mass o f oil in g the effect of methanol (0.1-0.5 l) was studied by fixing volume of oil, mass of calcium hydroxide, time, temperature and agitation speed at 2 l, 23 g, 24 h, 65 ◦c and 100 rpm respectively. the effect of catalyst (13.8-32.2 g) was studied by fixing volume of oil, volume of methanol, time, temperature and agitation speed at 2 l, 0.3 l, 24 h, 65 ◦c and 100 rpm respectively. the effect of time (16-32 h) was studied by fixing volume of oil, volume of methanol, mass of calcium hydroxide, temperature and agitation speed at 2 l, 0.3 l, 23 g, 65 ◦c and 100 rpm respectively. the effect of temperature (55-75 ◦c) was studied by fixing volume of oil, volume of methanol, mass of calcium hydroxide, time and agitation speed at 2 l, 0.3 l, 23 g, 24 h and 100 rpm respectively. the effect of agitation speed (50-150 rpm) was studied by fixing volume of oil, volume of methanol, mass of calcium hydroxide, time and temperature at 2 l, 0.3 l, 23 g, 24 h and 65 ◦c respectively. 3. results and discussion figure 1(a) shows the effect of volume of methanol on percentage conversion at constant values of volume of oil of 2 l, mass of calcium hydroxide of 23 g, reaction time of 24 h, temperature of 65 ◦c and agitation speed of 100 rpm. volume of methanol varied from 0.1 to 0.5 l. volume of methanol was selected based on the molar ratio of oil to methanol. from the literature, molecular weight of oil is 292 g/mol. in transesterification reaction, oil is the limiting reactant and methanol is the excess reactant. according to the reaction stoichiometry, 1 mole of oil reacts with 3 moles of methanol to produce biodiesel and glycerol. but, in practice, moles of methanol required to produce 17 sivamani et al. / j. nig. soc. phys. sci. 4 (2022) 16–19 18 figure 1. effect of (a) volume of methanol (b) mass of calcium hydroxide (c) reaction time (d) temperature and (e) agitation speed on percentage conversion biodiesel is more than the stoichiometric requirement. percentage conversion initially increased with increase in volume of methanol. after 0.3 l of methanol, conversion decreased. this is due to the fact that high content of methanol reduces the quality of biodiesel and hence, decrease in conversion was observed [31]. figure 1(b) shows the effect of mass of calcium hydroxide on percentage conversion at constant values of volume of oil of 2 l, volume of methanol of 0.3 l, reaction time of 24 h, temperature of 65 ◦c and agitation speed of 100 rpm. mass of calcium hydroxide varied from 13.8 to 32.2 g. mass of calcium hydroxide was selected based on the mass ratio of catalyst to oil. in transesterification reaction, oil is the limiting reactant and methanol is the excess reactant and alkali or heterogeneous materials as catalyst. according to the standard operating procedure, mass ratio between catalyst and oil should be between 0.5 and 2% (w/w). percentage conversion initially increased with increase in mass of calcium hydroxide. after 23 g of calcium hydroxide, conversion decreased. this is due to the fact that catalytic poisoning occurs at higher concentration of catalyst [32]. figure 1(c) shows the effect of reaction time on percentage conversion at constant values of volume of oil of 2 l, volume of methanol of 0.3 l, mass of calcium hydroxide of 23 g, temperature of 65 ◦c and agitation speed of 100 rpm. reaction time varied from 16 to 32 h. reaction time was selected based on the literature. in transesterification, reaction should be carried out between oil, methanol and catalyst for certain period of time. percentage conversion initially increased with increase in time. after 24 h, conversion decreased [33]. figure 1(d) shows the effect of temperature on percentage conversion at constant values of volume of oil of 2 l, volume of methanol of 0.3 l, mass of calcium hydroxide of 23 g, time of 24 h, and agitation speed of 100 rpm. reaction temperature var18 sivamani et al. / j. nig. soc. phys. sci. 4 (2022) 16–19 19 ied from 55 to 75 ◦c. reaction temperature was selected based on the boiling point of alcohol. in transesterification, reaction should be carried out between oil, methanol and catalyst at certain temperature. percentage conversion initially increased with increase in temperature. after 65 ◦c, conversion decreased. this is due to the fact that above 65 ◦c, loss of methanol is more which leads to the decrease in conversion [34]. figure 1(e) shows the effect of agitation speed on percentage conversion at constant values of volume of oil of 2 l, volume of methanol of 0.3 l, mass of calcium hydroxide of 23 g, time of 24 h, and temperature of 65 ◦c. agitation speed varied from 50 to 150 rpm. the effect of agitation speed on percentage conversion is not significant [35]. 4. conclusion the aim of the present work was to optimize biodiesel production from plant based non-edible crude oils with methanol using heterogeneous catalyst by transesterification for its commercialization. finally, the maximum conversion of 85.6% was achieved at the optimum process parameters of 2 l of crude sunflower oil, 0.3 l methanol, 23 g of calcium hydroxide, reaction time of 24 h, temperature of 65 ◦c, and agitation speed of 100 rpm. the results showed that crude sunflower oil could serve as potential renewable substrate for biodiesel commercialization. acknowledgments we would like to acknowledge the management of the university of technology and applied sciences, oman for their kind cooperation and support in completing the project. references [1] y. chisti, ”biodiesel from microalgae”, biotechnol. adv. 25 (2007) 294. [2] j. van gerpen, ”biodiesel processing and production”, fuel process. technol. 86 (2005) 1097. [3] h. fukuda, a. kondo & h. noda, ”biodiesel fuel production by transesterification of oils”, j. biosci. bioeng. 92 (2001) 405. [4] l. c. meher, d. vidya sagar & s. n. naik, ”technical aspects of biodiesel production by transesterification—a review”, renew. sust. energ. rev. 10 (2006) 248. [5] j. m. marchetti, v. u. miguel & a. f. errazu, ”possible methods for biodiesel production”, renew. sust. energ. rev. 11 (2007) 1300. [6] w. cao, h. han & j. zhang, ”preparation of biodiesel from soybean oil using supercritical methanol and co-solvent”, fuel 84 (2005) 347-351. [7] s. saka & d. kusdiana, ”biodiesel fuel from rapeseed oil as prepared in supercritical methanol”, fuel 80 (2001) 225. [8] m. i. al-widyan & a. o. al-shyoukh, ”experimental evaluation of the transesterification of waste palm oil into biodiesel”, bioresour. technol. 85 (2002) 253. 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[14] m. berrios, m.c. gutiérrez, m.a. martı́n & a. martı́n, ”application of the factorial design of experiments to biodiesel production from lard”, fuel process. technol. 90 (2009) 1447. [15] m. canakci & j. h. van gerpen, ”comparison of engine performance and emissions for petroleum diesel fuel, yellow grease biodiesel, and soybean oil biodiesel”, trans. asae 46 (2003) 937. [16] v. f. marulanda, g. anitescu & l. l. tavlarides, ”investigations on supercritical transesterification of chicken fat for biodiesel production from low-cost lipid feedstocks”, j. supercrit. fluids 54 (2010) 53. [17] r. yahyaee, b. ghobadian & g. najafi, ”waste fish oil biodiesel as a source of renewable fuel in iran”, renew. sust. energ. rev. 17 (2013) 312. [18] y. shimada, y. watanabe, a. sugihara & y. tominaga, ”enzymatic alcoholysis for biodiesel fuel production and application of the reaction to oil processing”, j. mol. catal., b enzym. 17 (2002) 133. 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[29] o. andersen & j. weinbach, ”residual animal fat and fish for biodiesel production. potentials in norway”, biomass bioenergy 34 (2010) 1183. [30] s. sadasivam & a. manickam, biochemical methods for agricultural sciences. wiley eastern limited, 1992. [31] a. s. ramadhas, s. jayaraj & c. muraleedharan, ”biodiesel production from high ffa rubber seed oil”, fuel 84 (2005) 335. [32] m. balajii & s. niju, ”a novel biobased heterogeneous catalyst derived from musa acuminata peduncle for biodiesel production–process optimization using central composite design”, energy convers. manag. 189 (2019) 118. [33] j. zhang, s. chen, r. yang & y. yan, ”biodiesel production from vegetable oil using heterogenous acid and alkali catalyst”, fuel 89 (2010) 2939. [34] s. sivamani, s. selvakumar, k. rajendran & s. muthusamy, ”artificial neural network–genetic algorithm-based optimization of biodiesel production from simarouba glauca”, biofuels 10 (2019) 393. [35] o. s. stamenković, m.l. lazić, z. b. todorović, v. b. veljković & d. u. skala, ”the effect of agitation intensity on alkali-catalyzed methanolysis of sunflower oil”, bioresour. technol. 98 (2007) 2688. 19 j. nig. soc. phys. sci. 3 (2021) 267–277 journal of the nigerian society of physical sciences groundwater quality assessment using multivariate analysis and water quality index in some saline fields of central nigeria n. d. umara, o. v. omononab,∗, c. o. okogbuec adepartment of geology, federal university of lafia bdepartment of geology/geophysics, alex ekwueme federal university, ndufu alike iko cdepartment of geology, university of nigeria abstract the groundwater of awe-keana saline fields central nigeria was studied to investigate physicochemical processes that influence its groundwater chemistry and quality and hence determine its quality for drinking and irrigation purposes. twenty groundwater samples were collected from hand dug wells and borehole for the purpose of identifying the hydrochemical characteristics and assessing the quality of groundwater of the awe-keana saline fields. principal component analysis was performed to identify the hydrochemical controlling processes while water quality index (wqi) was used to determine the overall quality of the water samples. multiple regression analysis however, revealed the parameter(s) that impact the overall water quality the most. the results showed that the chemical compositions of the groundwater of the area is influence by weathering of host rocks, salinity and anthropogenic activities. four hydrochemical facies were deciphered (ca− mg− hco3, na− k − hco3, na− k −cl−s o4, and ca − mg − cl − s o4) and this revealed the diversity in the chemical controlling processes that yield different facies. two clusters of water groups were identified from cluster analysis, namely, groundwater characterized with very high salinity, high nitrate contamination and high ca, cl, na, and hco3 ionic concentrations and groundwater with high mg, k, and s o4 ionic concentrations. saturation indices in relation to different minerals showed that precipitation and dissolution processes gave rise to the concentrations of different ions in the groundwater. water quality assessment showed that about 85 % of the groundwater of the area is unsuitable as drinking water but, generally suitable for irrigation. multiple regression analysis revealed that no3 ion among the hydrochemical parameters measured was observed to be the major pollutant in groundwater of the study area. doi:10.46481/jnsps.2021.183 keywords: awe-keana, multivariate analysis, saline field, water quality, water quality index article history : received: 17 march 2020 received in revised form: 17 june 2021 accepted for publication: 16 july 2021 published: 29 november 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: o. j. abimbola 1. introduction the awe – keana saline fields is located in parts of the central benue trough (cbt), nigeria. the occurrence of saline groundwater as springs, lakes and ponds is common in parts of ∗corresponding author tel. no: +(234)8036145840 email address: victor.omonona@funai.edu.ng (o. v. omonona) the sedimentary basin. different hypothesis has been given to explain the origin of saline water in the trough, however, the theories postulated by [1] are widely acceptable. the saline groundwater in the southern and central sections of the benue trough is frequently associated with tectonic elements such as intrusive and mineralized veins. prominent outcrops which commonly support local table salt industries are found in the cbt. these industries serve as sources of employment to the 267 umar et al. / j. nig. soc. phys. sci. 3 (2021) 267–277 268 locals who engage in the table salt production business. the area is also known for agricultural activities, mainly crop production. domestic and agricultural water supply in the central benue trough is largely through groundwater sources. groundwater supports human existence on earth therefore, its quantity and quality are very important with regard to drinking, irrigation and industrial water supplies. the world over, there is decline on the available of high-quality groundwater for human consumption [2]. this quality can be impacted significantly by land usage, geology and anthropogenic activities. groundwater is preferred to surface water because it is readily available throughout the year as surface water in the area usually becomes unavailable during the long dry season. besides, it is cheaper to access groundwater than exploring for and exploiting surface water. groundwater is also more potable and not easily contaminated as surface water but is more difficult to remediate a contaminated groundwater source. however, advances in the physicochemical characteristics of groundwater with respect to drinking and agricultural purposes has not been prioritized, as water from these hand-dug wells and boreholes are put to use without any quality consideration and hence its health implications. the determination of groundwater quality for human consumption is important for the wellbeing of the increasing population [3]. it is important therefore to establish those quality criteria for human health and food security considering the growing population and increased agricultural activities in the study area. it is thus pertinent to know the possible sources of contaminants and quality of groundwater supply for various purposes in area. [4] identified natural processes as the controlling factors of the hydrochemistry of groundwater chemistry while, anthropogenic contaminations, natural mineralisation and cation exchange as factors controlling the hydrochemistry. groundwater quality for drinking and agricultural or other purposes has been studied by [5, 6 and 7]. water quality index (wqi) have been identified as an important technique for define groundwater quality and its potability. [2, 6 and 8] used the wqi to classify groundwater for drinking. previous works on groundwater quality for drinking and agricultural or other purposes around the benue trough are available likewise studies on the hydrochemical characteristics of groundwater [5, 6 and 7] but no hydrochemical work has been carried out around the saline field of the awe-keana region. this present study seeks to investigate physicochemical processes that influence the groundwater chemistry and quality, decipher the underlying contaminants imparting the quality of groundwater of the area and hence determine its quality for drinking and irrigation purposes. 2. the study area 2.1. location awe-keana lies within latitudes 8◦06 ′ n to 9◦ 09 ′ n and longitudes 8◦ 47 ′ e to 9◦ 09 ′ e. the area forms part of the saline fields in central nigeria (figure 1). it is generally a lowland area with few scattered hills with elevation that ranges from 115 165 m above mean sea level. the river keana and river tunga, which are major tributaries of the river benue constitute the main drainage system of the area. the climatic condition is made up of two major and distinct seasons: a wet season and a dry season. the former lasts from may to october while the latter lasts from november to april. the mean annual rainfall varies between 1000 and 1500 mm and the relative humidity between 60 and 80 %. the average annual humidity and temperature are 70 % and 28.50 ◦c respectively [9]. figure 1. google earth view of the awe-keana area showing the sample collection points. 2.2. geology and hydrogeology in the middle benue trough, six upper cretaceous lithogenic formations (asu river group, ezeaku formation, keana formation, awe formation, awgu formation and lafia sandstone) comprise the stratigraphic succession (figure 2). the asu river group comprises of limestones, shales, micaceous siltstones, mudstones and clays [10, 11 and 12]. these are overlain by the cenomanian-turonian ezeaku formation which deposition marked the beginning of marine transgression in the late cenomanian which took place in a presumably shallow marine coastal environment. the sediments are made up mainly of calcareous shales, micaceous fine to medium friable sandstones and beds of limestones which are in places shelly. the keana formation resulted from the cenomanian regression which deposited fluviodeltaic sediments. the formation consists of cross-bedded, coarse grained feldspartic sandstones, occasional conglomerates, and bands of shales and limestones towards the top. massive outcrops occur at keana, azara and daudu. this was followed by the awe formations which was deposited as passage 268 umar et al. / j. nig. soc. phys. sci. 3 (2021) 267–277 269 (transitional) beds during the late albian early cenomanian regression. its typical sections occur around the town of awe, where [10] estimated the thickness to be about 100 m. the formation consists of flaggy, whitish, medium to coarse grained calcareous sandstones, carbonaceous shales and clays. the late turonian-early santonian coal-bearing awgu formation lies conformably on the awe formation. the deposition of the awgu formation marked the end of marine sedimentation in this part of the benue trough. the formation is made up of bluish-grey to dark-black carbonaceous shales, calcareous shales, shaley limestones, limestones, sandstones, siltstones, and coal seams. the major outcrop of the coal-bearing awgu formation is at the bank of river dep in shankodi, 7 km to the west of the village of jangwa. the post-folding campano-maastrichtian lafia formation ended the sedimentation in the middle benue trough, after which widespread volcanic activities took over in the tertiary. it is lithologically characterized by ferruginized sandstones, red, loose sands, flaggy mudstones, clays and claystones. outcrops and sections of the lafia formation occur in and around the town of lafia, and along the bank of river amba on the lafiadoma road. figure 2. stratigraphic succession in the middle benue trough (obaje, 2009). the awe-keana saline field of the cbt is underlain by the following geological sequence: the asu river group (marine), the ezeaku (marine), the keana/awe formation (continental) the awgu formation (marine) and the lafia sandstone (continental). detailed discussions on the geology of the central benue trough were presented by many authors notably [10, 11 and 12]. awe-keana brine fields are known to have very strange and difficult hydrogeological situations. these conditions arise from the fact that most of the potential aquifers are either limited in extent, thinly developed with consistent clay and shale interbeddings or even highly indurated that only the development of secondary voids created by fractures, joints and solutions channels can attract hydrogeological interest. the stratigraphic sequence (table 1) shows that the study areas are made up of alternate shale and sandstone horizons which are suspected to correspond to the sources of the saline and freshwater respectively. ezeaku, keana and awe formation aquifers are the main aquifer units in study area [13]. the uppermost aquifer is the sandstone member of the ezeaku formation which is composed of series of shale-limestone and sandstone beds. keana formation aquifer is composed of more heterogeneous, massive and predominantly fine, coarse and pebbly sandstone beds. keana formation is a good aquifer but is limited in extent which renders it unproductive for groundwater exploration. however, keana together with ezeaku formations form a very thick productive aquifer when encountered in a borehole. awe formation aquifer is the lowest aquifer and is composed of series of shale and porous sandstone beds and is highly productive. however, the presence of salt in it renders it unfavourable for groundwater exploration as the water from wells tapping the aquifer around old awe town (tsohon gari) show high saline concentration because of the out-cropping brinebearing awe formation [14]. 3. materials and methods twenty (20) groundwater samples were collected from shallow hand dug wells and boreholes in the study area. sampling was done early in the morning before water abstraction commenced by the residents of the study area. the study area was gridded into ten quadrants for even distribution of sample locations and two samples were collected from each quadrant at an arbitrary distance. at each location, two water samples were collected and filtered in situ through a 0.45 µm membrane filter, collected in labelled 125 ml bottles and kept under cold condition for the detection of anions and cations. samples for cation analytes were acidified to a ph less than 2 with concentrated nitric acid in water solution at 0.15 % concentration by weight. transient physicochemical parameters (temperature, total dissolved solids, electrical conductivity and redox potential) were measured immediately in the field using portable t/ph/ec /tds meter (h19813-5 model) and portable eh meter while the cations and anions were collected in a fresh new high-density polyethylene (hdpe) bottles for laboratory analysis. the concentrations of the cations were determined using icp-es/icpms at acme analytical laboratories ltd, canada, while the anions were analyzed using hach dr 2000 spectrophotometer at unicef assisted water supply and sanitation laboratory, ibadan, nigeria. [15] quality assurance programme was employed to check the validity of the field measurements and laboratory analytical results. quality control measures applied in269 umar et al. / j. nig. soc. phys. sci. 3 (2021) 267–277 270 table 1. hydrostratigraphic units of rocks in the study areas (modified from offodile, 2002). epoch age geologic formation rock unit aquifers santonian-campanian volcanic cretaceous maastrichtian lafia formation fine to coarse grained, friable and feldspartic sandstone, brownish at top and whitish at depth. coniacian awgu formation greybedded shale with occasional sandstone bed and limestone. late turonianearly turonian ezeaku formation thick calcareous shales, micaceous and fine to medium grained sandstones. late cenomanian keana formation crystalline fine, coarse and pebbly sandstone. aquifer early cenomanian awe formation flaggy, whitish, medium to coarse grained feldspartic sandstones, calcareous sandstones, limestone interbedded with carbonaceous shale. aquifer early cretaceous mid-late albian asu river group marine shales, clays siltstones and mudstones. pre cambrian basement complex and meta-sediments crystalline rock cluded the use of blank samples and estimation of the cation–anion ratios [16, 17, 18 and 19]. saturation index of calcite, dolomite, halite, gypsum, siderite, and hematite was estimated using ph-reeqc 3.3.8-11728. wqi was calculated according to [3, 20 and 21 and was used to classify groundwater quality into the different degree of quality on the overall quality of water for human consumption [22 and 23]. statistical analyses were performed using statgraphics centurion xvi.i which includes (mean, minimum, maximum, coefficient of variation, and standard deviation). pearson’s correlation matrix, principal components analysis (pca), cluster analysis (ca), and multiple regression analysis. pca and ca were carried out using standardized data. standardization of the data for the statistical tests was done in order to resolve the effect of differences in the units of measurements and large variations between the data units. pca were performed on the correlation matrix of the data and the number of extracted principal components was based on a minimum eigen value of 1.0. ca was carried out based on ward’s method and the squared euclidean distance metric mode. the clusters defined by the ca were based on the similarity in chemical compositions of the various water sampling stations [24]. multiple regression analysis was conducted on the data to determine the parameter(s) that most influenced the character of the wqi. wqi values were the dependent variable while the concentrations of the parameters used in the estimation of wqi were the independent variables. the model was fitted upon the forward stepwise selection procedure. 4. results and discussion 4.1. physicochemical characteristics the summary of the results of the hydrochemical parameters along with the who standard for drinking water are presented in table 1. the ph of the groundwater samples ranges from 5.52 to 7.72, with a mean value of 6.46. low ph is predominant around awe area and could be attributed to the presence of humic shale found around that area, while the low ph around keana could attributed to photosynthetic processes. water with low ph (below 6.5) is unacceptable for drinking purposes, as such water is reported to cause acidosis [25 and 26]. the redox potential (eh) value ranges from 8 to 281 mv with an average value of 175.3 mv. it was observed that eh has negative correlations with all the physicochemical parameters except co3 and s o4. the positive correlation coefficient values of co3 and s o4 with eh may be attributed to the high tendency of c and s to participate in redox reactions because of their variable oxidation states. eh has significant negative correlation coefficient values (-0.519) with cl and salinity. this shows that high salinity and chloride concentration in groundwater in the area is favoured under a low redox condition (an anoxic environment). electrical conductivity (ec) values ranged from 56.1 to 1059 µ s/cm with a mean value of 591.45 while total dissolved solids (tds) ranged from 37.58 to 709.53 ppm with a mean value of 396.27. the higher variations in the ec value are partly as a result of the wide variation in the water table and saturated zone in the area and partly due to the effect of increased temperature and pressure with depth which increased the rate of reaction and dissolution of ions. a linear relationship occurs between ec and tds (correlation coefficient of 1.0). this imply that 270 umar et al. / j. nig. soc. phys. sci. 3 (2021) 267–277 271 tds is the dominant factor in the ec of the study area. about 65 % of the groundwater samples from the area have ec values above the stipulated guideline value while 40 % of the water samples have tds values above the guideline value [24]. salinity values of the groundwater samples ranged from 0.011 to 15.510 ppt with a mean value of 0.885. at awe area, high salinity is confined to the southwest zone (salinity value ¿ 15 ppt was measured). however, in keana, high salinity is spread throughout the area (figure 2). 4.2. cation and anion: concentrations and relationships the concentration of sodium varied ranges from 4.22 to 3024 ppm, potassium from 0.84 to 79.18 ppm while calcium and magnesium varied from 3.63 to 137.30 ppm and 0.96 to 48.66 ppm respectively. magnesium concentration in all the groundwater studied is below the guideline value of 50 ppm while 25 % of the water samples have calcium concentrations above the [24] guideline value of 75 ppm. bicarbonate concentrations varied from < 1 ppm (below detection limit) to 888.2 ppm, chloride concentration ranges from 5.99 to 8584.95 ppm while sulphate concentration range from 2 to 60 ppm (table 2). generally, there is high variability in the concentrations of the major ions which could be the results of differences in the geologic units and lithogenic processes of the aquifers, recharge rate variability, and anthropogenic factors. the following cation-anion pairs with correlation coefficients greater than 0.5 (p < 0.005) (table 3) predominate in the groundwater: ca − hco3 (0.7342), mg − hco3 (0.7537), na − hco3 (0.7923), na − cl (0.7107), ca − no3 (0.5281), na − no3 (0.5751), mg−no3 (0.5748), k−s o4 (0.5407), and mg−s o4 (0.5526). the high correlation coefficient values of these ionic pairs suggest that they may have come from the same source(s) and or have been produced by the same process(es). 4.3. hydrochemical facies groundwater in the study area was characterized using piper (1944) trilinear diagram. [27 and 28] classifications were used to classify the groundwater of the study area into different hydrochemical facies. from the diagram (figure 3), four (4) different hydrochemical facies namely ca − mg − hco3, na − k − hco3, na − k − cl − s o4, and ca − mg − cl − s o4 (in the order of geochemical evolution) were deciphered. the ca−mg−hco3 facies (constituting 65 %) is the most dominant facies in the area and it reflects water from recharge zone and prevalence of rock weathering. the ca−mg−cl−s o4 ranked second in abundance (constituting 25 %) and it reflects groundwater of reverse ion exchange. ca − mg − cl − s o4 facies is a mixed water type with ca − mg − hco3 and na − k − cl − s o4 as the two end members. the na − k − hco3 facies constituted 5 % of the total facie types. this facie reflects base ion exchange; exchange of ca and mg by na and k in the initial ca − mg − hco3 facies. the na − k −cl − s o4 facies (5 %) reflects water formed as a result of evaporation or mixing with seawater or saline water. a geochemical evolution model may be defined here, with water facies beginning with the ca − mg − hco3 to na − k − hco3, then na−k−cl−s o4 and again from ca−mg−hco3 to ca − mg − cl − s o4. figure 3. piper (1944) diagram showing the different hydrochemical facies. 4.4. sources of ions gibbs diagram (figure 4) revealed that rock weathering is the dominant physicochemical process controlling the chemistry of the groundwater in the area. table 4 presents the saturation indices (si) in relation to different minerals. the table reveals, that all the groundwater samples were undersaturated (< 0) with respect to halite, gypsum, and siderite. therefore, the concentrations of na and cl in the groundwater is attributed to the dissolution of halite while s o4 and total fe (fe2+ and fe3+) is attributed to the dissolution of gypsum and siderite respectively. eleven of the groundwater samples constituting 55 %, were undersaturated with respect to both calcite and dolomite, and the remaining nine samples (constituting 45 %) were oversaturated with respect to calcite and dolomite. ca and mg concentrations may be considered to be as a result of the dissolution of calcite and dolomite respectively. it was observed that the samples from awe area were characterized more with undersaturation of calcite and dolomite while, those from the keana area were characterized more with oversaturation with respect to calcite and dolomite. the oversaturation with respect to calcite and dolomite is evidenced by the presence of limestone deposits around the keana segment. no limestone quarries exist around the awe segment. it is observed that throughout the entire study area, the lateritic cap is very rich in iron and hence the groundwater is super oversaturated with hematite. the high concentration of no3 (from 0.32 to 90.23 mg/l) detected in some parts of the awe-keana saline fields, indicated 271 umar et al. / j. nig. soc. phys. sci. 3 (2021) 267–277 272 table 2. summary statistics of the hydrochemical data. parameter unit average standard deviation coefficient of variation minimum maximum who (2008) limit ph nil 6.46 0.54 8.38 5.52 7.72 6.5 8.5 eh mv 175.3 76.35 43.55 8 281 ec µs/cm 591.46 349.52 59.09 56.1 1059 500 tds ppm 396.28 234.18 59.09 37.59 709.53 500 salinity ppt 0.89 3.44 389.04 0.01 15.51 cl ppm 489.92 1905.97 389.04 5.99 8584.95 250 hco3 ppm 285.24 253.83 88.99 0 888.2 500 co3 ppm 36.60 44.45 121.46 0 120.00 po4 ppm 1.64 1.80 109.49 0 5.11 no3 ppm 27.76 28.45 102.47 0.32 90.23 45 s o4 ppm 24.50 19.36 79.04 2 60.00 250 k ppm 28.19 30.60 108.54 0.84 79.18 na ppm 339.00 890.14 262.58 4.22 3024 ca ppm 58.76 38.78 66.02 3.63 137.30 75 mg ppm 18.20 13.73 75.40 0.96 48.66 50 table 3. pearson’s correlation matrix. ca cl co3 ec eh hco3 k mg na no3 ph po4 salinity s o4 tds ca 0.4881 0.2007 0.7536 -0.330 0.7342 0.6257 0.6865 0.676 0.5281 -0.056 0.4797 0.4881 0.4229 0.7536 cl 0.4881 -0.188 0.3167 -0.519 0.5583 0.3572 0.2976 0.7107 0.5123 -0.085 -0.069 1 0.0259 0.3167 co3 0.2007 -0.188 0.2407 0.3795 0.1655 0.2401 0.4752 -0.258 0.164 0.001 -0.059 -0.188 0.8708 0.2407 ec 0.7536 0.3167 0.2407 -0.133 0.5123 0.5766 0.4375 0.4194 0.4302 -0.267 0.253 0.3167 0.4697 1 eh -0.330 -0.519 0.3795 -0.133 -0.426 -0.385 -0.134 -0.742 -0.266 -0.042 -0.193 -0.519 0.1637 -0.133 hco3 0.7342 0.5583 0.1655 0.5123 -0.426 0.3251 0.7537 0.7973 0.8299 -0.243 0.3658 0.5583 0.2974 0.5123 k 0.6257 0.3572 0.2401 0.5766 -0.385 0.3251 0.3769 0.4956 0.1127 0.1209 -0.004 0.3572 0.5407 0.5766 mg 0.6865 0.2976 0.4752 0.4375 -0.134 0.7537 0.3769 0.4191 0.5748 0.1711 0.2519 0.2976 0.5526 0.4375 na 0.676 0.7107 -0.258 0.4194 -0.742 0.7973 0.4956 0.4191 0.5751 -0.102 0.2689 0.7107 -0.016 0.4194 no3 0.5281 0.5123 0.164 0.4302 -0.266 0.8299 0.1127 0.5748 0.5751 -0.303 0.2829 0.5123 0.2477 0.4302 ph -0.056 -0.085 0.001 -0.267 -0.042 -0.243 0.1209 0.1711 -0.102 -0.303 -0.33 -0.085 -0.017 -0.267 po4 0.4797 -0.069 -0.059 0.253 -0.193 0.3658 -0.004 0.2519 0.2689 0.2829 -0.33 -0.069 -0.081 0.253 salinity 0.4881 1 -0.188 0.3167 -0.519 0.5583 0.3572 0.2976 0.7107 0.5123 -0.085 -0.069 0.0259 0.3167 s o4 0.4229 0.0259 0.8708 0.4697 0.1637 0.2974 0.5407 0.5526 -0.016 0.2477 -0.017 -0.081 0.0259 0.4697 tds 0.7536 0.3167 0.2407 1 -0.133 0.5123 0.5766 0.4375 0.4194 0.4302 -0.267 0.253 0.3167 0.4697 correlation coefficient > 0.5 are in bold. 272 umar et al. / j. nig. soc. phys. sci. 3 (2021) 267–277 273 table 4. saturation indices in relation to calcite (caco3), dolomite (camg(co3)2), gypsum (cas o4 · 2h2o), halite (nacl), hematite (fe2o3) and siderite (feco3) minerals. (nd => not determined) location caco3 camg(co3)2 cas o4 · 2h2o nacl fe2o3 feco3 awe 1 -0.17 -0.45 -2.62 -5.42 nd nd awe 2 -0.28 -0.63 -2.65 -3.35 nd nd awe 3 -2.33 -4.69 -3.81 nd 15.42 -0.36 awe 4 -2.44 -4.81 -3.43 nd 7.88 -1.08 awe 5 -0.58 -1.25 -2.82 -7.16 13.50 -0.28 awe 6 2.10 3.68 -2.10 -7.75 14.37 -13.17 awe 7 2.10 4.02 -2.18 -7.08 13.46 -13.40 awe 8 -0.11 -0.32 -2.46 -6.90 17.00 -0.08 awe 9 -1.46 -3.44 -3.14 -8.52 11.99 -0.98 awe 10 -0.98 -2.27 -2.75 -6.46 15.20 -0.45 keana 11 1.90 3.49 -1.92 -6.75 15.22 -11.27 keana 12 1.78 3.42 -2.11 -7.09 17.49 -7.47 keana 13 2.04 4.12 -2.11 -6.44 14.70 -12.11 keana 14 1.72 3.09 -2.12 -6.75 14.90 -12.47 keana 15 1.72 3.15 -2.22 -6.72 15.96 -10.58 keana 16 1.73 3.69 -2.54 -7.19 15.48 -12.94 keana 17 -1.46 -3.16 -3.76 -8.70 17.80 0.18 keana 18 2.00 3.78 -2.16 -6.59 16.43 -9.15 keana 19 -2.52 -5.22 -4.29 -9.12 16.41 -0.28 keana 20 -1.97 -4.24 nd -8.38 20.56 -1.68 table 5. saturation indices in relation to calcite (caco3), dolomite (camg(co3)2), gypsum (cas o4 · 2h2o), halite (nacl), hematite (fe2o3) and siderite (feco3) minerals. pc 1 pc 2 pc 3 ca 0.335818 0.293805 0.135665 cl 0.360345 -0.105957 -0.0336973 eh 0.307956 -0.0532982 -0.234097 hco3 0.374573 -0.0145737 0.00042887 k 0.17127 -0.28981 0.591508 mg 0.150976 0.554345 0.035744 na 0.390315 -0.0936606 -0.00266831 no3 0.302649 -0.060511 -0.0534841 ph -0.0458886 0.518323 -0.336216 salinity 0.360345 -0.105957 -0.0336973 s o4 -0.273386 0.0726313 0.454219 tds 0.139542 0.458819 0.501138 eigen value 5.57428 1.84474 1.0301 % of variance 46.452 15.373 8.584 cumulative % 46.452 61.825 70.409 table 6. table of principal component scores. location pc 1 pc 2 pc 3 awe 1 3.99 -0.24 0.135665 awe 2 8.50 -0.85 -0.14 awe 3 0.43 0.53 0.37 awe 4 0.02 1.63 -0.60 awe 5 -1.62 -0.50 -1.32 awe 6 0.17 -0.80 -1.44 awe 7 -0.76 0.02 -0.69 awe 8 -1.09 -0.52 -1.08 awe 9 -0.93 0.58 -0.32 awe 10 -0.97 0.22 -0.37 keana 11 -1.90 -1.67 1.65 keana 12 -1.46 -1.54 -0.01 keana 13 -0.90 3.32 -0.31 keana 14 -0.90 -1.31 1.08 keana 15 -1.66 -1.01 -1.29 keana 16 0.31 -0.90 -0.41 keana 17 -0.45 1.18 0.99 keana 18 -0.90 -1.22 1.88 keana 19 0.16 1.08 1.72 keana 20 -0.052 2.20 0.18 273 umar et al. / j. nig. soc. phys. sci. 3 (2021) 267–277 274 table 7. water quality index (wqi) classification for water of awe-keana area. location wq1 value type of water awe 1 694.83 water unsuitable for drinking purposes awe 2 4181.90 water unsuitable for drinking purposes awe 3 127.50 poor water awe 4 93.72 good water awe 5 313.06 water unsuitable for drinking purposes awe 6 552..58 water unsuitable for drinking purposes awe 7 510.35 water unsuitable for drinking purposes awe 8 379.89 water unsuitable for drinking purposes awe 9 163.31 poor water awe 10 403.33 water unsuitable for drinking purposes keana 11 356.69 water unsuitable for drinking purposes keana 12 291.05 water unsuitable for drinking purposes keana 13 413.45 water unsuitable for drinking purposes keana 14 466.34 water unsuitable for drinking purposes keana 15 332.60 water unsuitable for drinking purposes keana 16 559.42 water unsuitable for drinking purposes keana 17 132.49 poor water keana 18 378.37 water unsuitable for drinking purposes keana 19 32.64 excellent keana 20 48.51 excellent table 8. multiple regression analysis. parameter estimate standard error statistic p-value constant −4.7676 × 10−9 1.4758 × 10−5 −3.2303 × 10−4 0.9997 ca 1.32267 3.880021 × 10−7 3.40875 × 106 0.0000 cl 0.3968 4.66743 × 10−9 8.50147 × 107 0.0000 hco3 0.1984 7.19455 × 10−8 2.75764 × 106 0.0000 mg 1.984 9.96068 × 10−7 1.99183 × 106 0.0000 no3 2.20444 4.67962 × 10−7 4.71074 × 106 0.0000 s o4 0.3968 4.7444 × 10−7 8.36355 × 105 0.0000 tds 0.1984 4.99819 × 10−8 3.96943 × 106 0.0000 274 umar et al. / j. nig. soc. phys. sci. 3 (2021) 267–277 275 that unconfined aquifers predominate in the area. it was observed further that, nitrate concentrations did not relate linearly with well depth. most of the nitrate-contaminated groundwater sources are located very close to farmlands and areas with very poor sanitary control and protection (this was from reports obtained during the field investigations and the interaction with the people of the locality). increased nitrate concentrations in the groundwater of the area could be attributed to wastes emanating from fertilized crop fields and runoffs and discharges from livestock. figure 4. gibbs diagrams of the hydrochemical data of a) keana and b) awe. 4.5. statistical analysis cluster analysis (figure 5) grouped the various sampling stations (20 locations) into two clusters, based on the similarities among the chemical parameters involved in the groundwater quality. cluster 1: groundwater characterized with very high salinity, high nitrate contamination and high ionic (ca, cl, na, and hco3) concentrations and cluster 2: groundwater with high ionic (mg, k, and s o4) concentrations. table 4 presents the results of principal component (pc) including component-loading matrix, eigen values, percentage variance and total cumulative variance and cumulative percentage. three components were extracted that accounted for 70.41 % of the total variance. pc 1 describes 46.45 % of the total variance and has high positive component-loadings on eh, salinity, ca, na, cl, no3, and hco3. pc 1 could be attributed to weathering and leaching of the host rocks, salinity and nitrate pollution factors. pc 2 has high positive component-loadings on ph, mg, and tds and accounts for 15.37 % of the total variance. pc 2 may therefore, be said to reflect tds factor. the dissolution and migration of the ions are strongly influenced by ph. pc 3 accounted for 8.58 % of the total variance and has high positive loadings on k, s o4 and tds, but high negative loadings on ph. this pc reflects the effects of sulphate minerals dissolution. principal component score (pcs) loadings on the groundwater sample locations are presented in table 5. pcs loadings at and above ±1.0 were considered significant controlling processes on the sample locations. from the table, pc 1 reflect the effect of weathering and leaching of host rocks, salinity and nitrate pollution has high impacts in the groundwater chemistry of sample locations 1, 2, 5, 8, 11, 12 and 15. the chemical composition and chemistry of groundwater of sample locations 4, 5, 11, 12, 13, 14, 15, 17, 18, 19, and 20 are strongly influenced by tds, while those of sample locations 5, 6, 8, 11, 14, 15, and 18 are strongly influenced by sulphate minerals dissolution. figure 6 presents the distributions and spatial relationships between sampling point loading characteristics. figure 5. cluster analysis dendrogram showing the different water classes. figure 6. principal component score scatterplot for the awe-keana area. 4.6. groundwater quality evaluation of the groundwater quality for drinking and other uses was done through water quality index (wqi). the wqi technique was first introduced by [29] and has been employed by different authors [2, 30, 31, 32 and 33]. the results of the calculated wqi are presented in table 6. four classes of water quality were defined, namely, “excellent 275 umar et al. / j. nig. soc. phys. sci. 3 (2021) 267–277 276 water” (10 %), “good water” (5 %), “poor water” (15 %) and “water unsuitable for drinking purposes” (70 %). the results showed that groundwater in the area is generally of very poor quality for drinking purposes. the results of the analysis (table 7) revealed that nitrate, mainly from nitrate-rich-fertilizers and domestic wastes emanating from septic storages, magnesium and calcium released from dissolutions of dolomite, and calcite minerals in that order are the principal parameters contributing to the pollution and contamination of groundwater of the area. the suitability of the groundwater for irrigation purposes on the other hand, was evaluated from the [34] wilcox (ussl, 1954) diagram. from the diagram (figure 7), groundwater from the area plotted within the following fields, low sodium alkali hazard-low salinity hazard (c1-s1), medium salinity hazardlow sodium alkali hazard (c2-s1), high salinity hazard -low sodium alkali hazard (c3-s1) and high salinity hazard-medium sodium alkali hazard (c3-s2). most of the groundwater samples plotted within the c1-s1 field and hence, can be used for irrigation. however, those that plotted c3-s2 should be used with caution for irrigation. figure 7. wilcox diagram for groundwater of the area. 5. conclusion hydrochemical characteristics of groundwater from the awekeana saline field, cbt nigeria have been studied and different indices used to assess the quality of the groundwater. results of the hydrochemical analysis revealed that the groundwater in the area is slightly acidic to neutral and that anoxic conditions prevailed during the period the groundwater formation. the concentrations of the major ions showed high variability which could be attributed to differences in the underlying geologic materials, prevailing lithogenic processes, recharge rates, and anthropogenic factors. four hydrochemical facies were deciphered, namely, ca−mg−hco3 (65 %); ca−mg−cl−s o4 (25 %); na − k − hco3 (5 %) and na − k − cl − s o4 (3 %) in that order. the prevalence of the ca − mg − hco3 could be attributed to the increased action of co2 on h2o, low waterrock interactions rate, and low residence time. different sources and processes were contributed by the concentration of the various ions in the groundwater. prominent controlling processes include dissolution and precipitation of calcite and dolomite, precipitation of hematite and dissolution of gypsum, halite and siderite. cluster analysis identified two types of sample locations cluster based on their chemical composition similarities (cluster 1: groundwater characterized with very high salinity, high nitrate contamination and high ionic (ca, cl, na, and hco3) concentrations and cluster 2: groundwater with high ionic (mg, k, and s o4) concentrations) while principal component analysis revealed three hydrochemical processes controlling the groundwater chemistry (weathering and leaching; dissolution and migration of ions and dissolution of sulphate minerals). evaluation of the groundwater quality for drinking and other domestic purposes showed that 85 % of the groundwater sources examined are not potable while only 15 % is suitable for drinking and other domestic purposes. no3 was found to be the most paramount parameter impairing the quality of groundwater in the area. the groundwater is generally suitable for irrigation particularly for halophytes but not for the cultivation of glycophytes. acknowledgments we thank the referees for the positive enlightening comments and suggestions, which have greatly helped us in making improvements to this paper. references [1] k. uma, “the brine fields of the benue trough, nigeria: a comparative study of geomorphic, tectonic and hydrochemical properties” journal of african earth sciences 2 (1998) 261. 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[33] ussl. diagnosis and improvement of saline and alkali soils. usda agricultural handbook (1954) no.60, washington dc. 277 j. nig. soc. phys. sci. 4 (2022) 54–58 journal of the nigerian society of physical sciences current understanding of the equatorial e × b drift velocities in the african sector: a short review b. o. adebesina,b,c,∗, j. o. adeniyia,b, i. a. adimula1, s. j. adebiyia,b, s. o. ikubannia,b, b. j. adekoyae a climate action sdg 13 group (space weather and environment), landmark university, nigeria. b department of physical sciences, landmark university, p.m.b 1001, omu-aran, kwara state, nigeria. c landmark university centre for research, innovation and discoveries (lucrid), nigeria. ddepartment of physics, university of ilorin, ilorin, nigeria e department of physics, olabisi onabanjo university, ago-iwoye, ogun state, nigeria abstract a short review of the pattern and morphology of the equatorial plasma drift velocities, particularly during the evening-time pre-reversal enhancement (pre) period in the african region had been presented. the seasonal pre peak values across some locations in the west-african region were considered and compared with other sectors of the world. while most plasma drift observations in the african region were calculated from ionosonde measurements, the observations from other sectors involved direct measurement from satellite and the incoherent scatter radar (isr) observations. the importance of the pre in ionospheric electrodynamics was highlighted, the better in the use of either the virtual or real heights of the f-layer in inferring vertical drift velocities were enumerated, revealing the strengths and weakness of each method. the general observations revealed that pre peak magnitude is commonly weaker in the african region in comparison with the american/peruvian and indian sectors, seasonal and solar activity dependent, and could be higher during either magnetic quiet or disturbed activity than when both magnetic activity conditions are combined. the first work to present a regional pre model around the african equatorial ionization anomaly region (adebesin et al model) was mentioned. the relevance of the e × b drift in quantifying the daytime equatorial electrojet (eej) current was also discussed. doi:10.46481/jnsps.2021.327 keywords: vertical e x b drift; ionosphere, plasma; pre-reversal enhancement (pre); african sector. article history : received: 31 july 2021 accepted for publication: 04 february 2022 published: 28 february 2022 c©2022 journal of the nigerian society of physical sciences. all rights reserved. communicated by: o. j. abimbola 1. introduction equatorial vertical plasma drifts (v p) explain the extent in terms of speed to which the ionospheric f-layer is moving vertically with respect to time. v p plays a significant factor in describing and specifying many ionospheric models across the globe. it has also been found to be a significant factor in ∗corresponding author tel. no: +234(0)8058051253 email address: adebesin.olufemi@lmu.edu.ng (b. o. adebesin ) the manifestation of plasma bubbles and equatorial spread-f [1, 2, 3, 4] through the action of the evening-time pre-reversal enhancement (pre) activity. the pre is the evening-time enhanced e x b drifts driven by the eastward electric field., and consequently lifts the ionosphere. the lifting interrupts the ionospheric density stability through the rayleigh-taylor mechanism. vertical drifts had also been useful in the description of electron density response to solar eclipses, thus making the e x b drift an essential factor for ionospheric f2layer redistribution during eclipse period [5]. the vertical 54 b. o. adebesin et al. / j. nig. soc. phys. sci. 4 (2022) 54–58 55 drift velocities had been widely investigated in most sectors of the world including the peruvian/american sector [6], and indian sector [7, 8, 9], with little done in the african sector [10, 11, 12, 13, 14, 15]. the space-borne drift measurements through the use of the incoherent scatter radar (isr) have been well documented over jicamarca and millstone hill and had been the major source of data for modelling ionospheric vertical plasma drift [16]. data from the jicamarca isr and aee satellite had been employed by [16] to find a first global empirical model for different solar activity conditions and seasons in the f-region, and subsequently injected into the iri-2007 model. other empirical models of repute include [17] using ionosonde observations, [18] and [19] obtained from rocsat-1 measurements, and [20] from champ satellite. all of these models and many more are predicted/forecasted outside the african sector. the first work to report modelling of the vertical plasma drift velocity in the african region, to the best knowledge of the authors, is the work of [14]. the adebesin et al model used ground-based ionosonde data observations from ouagadougou (geographic lat. 12.4◦n, long. 358.6◦e and geomagnetic lat. 0.59◦n, long. 71.5◦e) spanning 1966-1998. the model was used in inferring a regional empirical model between the evening-time drift and the solar activity index. the model presents a strong potential for obtaining vp magnitude around the evening/nighttime period, especially in the african equatorial ionization anomaly (eia) region. the mathematical relation that describes the model as a function of the solar activity condition is given by the expression: pre = 0.158rz + 1.495 (1) while satellite and isr measurements had been widely used in the investigation of the e x b drift in the other sectors of the world, the ground-based ionosonde/digisonde had been the major source of data acquisition for inferring drift pattern in the african sector. the contribution of real-time ionospheric measurements from ionosondes/digisondes are useful parameters in space weather study and forecasting [21]. the estimation of the vertical plasma drift from ground-based ionosonde measurements in the african sector had been achieved majorly by either computing the time rate of change of the f2-layer height of the peak electron density d(hmf2)/dt, or by that of the virtual height of reflection d(h′f)/dt [13, 22, 23, 24]. the limitation inherent in the description of vp using ground-based measurement is well documented in [12]. consequently, a review of both the quantitative and qualitative characteristics of previous works done on ionospheric drift pattern in the equatorial african sector will add to our understanding on its morphology when compared with other sectors of the world. from this point forward, all observations reported are in the equatorial african sector, except otherwise specified. 2. variations in drift pattern obtained from ground-based ionosonde/digisonde real height and virtual height measurements observations involving inferring vertical plasma drift from both the height of the peak electron density (hmf2) and virtual height of the f-layer (h′f) methods are well established. argument on the best v p inferring parameter between the two (h′f and hmf2) is a continuous scientific debate. however, works that describe both methods in a single paper are scarce, even from other sectors. [15] describes the morphology of v p obtained from each method of measurement (d(hmf2)/dt) and d(h′f)/dt) using digisonde data obtained from ilorin (geographic 8.5◦n, 4.5◦e) in the west african region during a year of low sunspot activity. for better interpretation, the result was compared with those obtained from isr observations (which serves as the reference drift condition) spanning the same study period. they reported a pre highest magnitude around 19 lt for the entire seasons with hmf2-vp while that of h′f−v p also recorded the maximum at 19 lt (except for december solstice and september equinox, which peaked at 18 lt). the nighttime downward reversals peak values range from −2 to −14 m/s and −4 to −14 m/s respectively for the h′f−v p and hmf2−v p observations. further, the pre peak magnitude across the entire seasons ranges from 4−14, 3−14 and 2−14 m/s respectively for the isr-vp, hmf2-vp and h′f − v p. a correlation percentage of above 83% was recorded for the isr-vp versus hmf2-vp and isr-vp versus h′f −v p pairs of observations between 1620lt during equinoxes, but lower for solstices (¡ 40%). table 1 depicts the intra-hour bin time correlation values for the ground-based ionosonde versus isr observations revealing the correlation magnitudes (r) expressed in percentage, between the pairs for different time intervals. while the pre magnitude of h′f − v p compares well with isr-vp during equinoxes, that of hmf2-vp is equitable to isr-vp during solstices. bearing in mind that both methodologies are governed by the same mechanism at night, it was concluded that any of the two can be used as long as the 300 km threshold magnitude condition suggested by bertoni [25] and [26] is fulfilled, else chemical modification may be necessary. see also [27] for more discussion on this condition. 3. seasonal variations in the pre peak magnitude across equatorial stations in africa and comparison with other sectors. a summary of the evening-time maximum pre magnitude of the vertical plasma drift velocity across various stations in africa (and other sectors), including the type of methodology involved in inferring the drift, as well as solar activity conditions are presented in table 2. the table revealed that during the combined magnetic disturbed and quiet conditions in the african region, the maximum pre magnitude for june solstice, december solstice, and equinox respectively generally spans around 7.6-12.0 m/s, 8.0-15.2 m/s and 8.0-20.0 m/s (from the pre magnitudes presented by [14, 15, 28]). it was observed that the maximum peak limit for each of these seasons during 55 b. o. adebesin et al. / j. nig. soc. phys. sci. 4 (2022) 54–58 56 table 1: seasonal correlation coefficient (r) between isr-vp versus h′f − v p and isr-vp versus hmf2-vp for different nighttime interval periods (** are periods without data for the isr observation) time interval parameter mar. equinox sept. equinox jun. solstice dec. solstice 16-19 lt isr-vp vs h′f − v p 83% ** 99% 33% isr-vp vs hmf2-vp 99% ** 78% 88% 19-21 lt isr-vp vs h′f − v p 88% 87% 68% 93% isr-vp vs hmf2-vp 94% 90% 86% 99% 21-06 lt isr-vp vs h′f − v p 67% 53% 87% 76% isr-vp vs hmf2-vp 72% 29% 82% 58% this condition of combined quiet and disturbed magnetic condition (12, 15, and 20 m/s) can be boosted either during only magnetic disturbed or quiet condition as shown from the pre peak values reported by [10, 29, 30]. besides, pre peak values are mostly higher during magnetic quiet period relative to disturbed activities. [31] had submitted that quiet magnetic activities compare reasonably with the thremospheric ground state owning to small rate of atomic oxygen (o+) during the time. the separate magnetic quiet or disturbed during june solstice is associated with high pre peak values comparable to the values during equinoxes (i.e., [31]). further, pre peak values are lesser during low solar activities when compared with high solar activity magnitudes, establishing solar dependence of pre. this study also revealed that on the average, the pre peak value obtained from h′f is higher than that inferred from hmf2 for the entire seasons from the work of [15]. this higher value associated with pre obtained from h′f measurements may be connected to the disappearance of both the dand e-layer of the ionospheric region at night, thereby leaving the f-layer active. during this period, the f-layer bottomside time rate of change is more accurate. conversely for the f-layer peak height (hmf2), the exact height may not be obtained accurately owning to merging of all layers during its measurement. it has been observed by [18] that pre peak magnitude in the american region is higher than those obtained from other sectors including the african sector. this statement was confirmed from table 2. the table showed that the pre peak magnitude during the equinoxes is higher in the american sector in comparison to the african region (with the exception of the one by [15]; using isr observations during low sunspot condition). the situation is same for the december solstice period. it is also worth noting that the evening time pre peak presents inverse linear relationship with the electron ionization gradient parameter defined by the height rate of change of the f2-layer peak electron density, dn/dh [35]. 4. measuring the electrojet current from ground-based vertical drift measurements of the diverse relevance of the equatorial vertical plasma drift velocities, its practical relation to physical events like the equatorial ionization anomaly, fountain effect, and spread-f are outstanding as these phenomena can alter the reliability of communication and navigation systems. the equatorial spreadf phenomena originates within the e-layer current system, in form of an intense ionospheric daytime eastward current, and narrowed within 3◦ dip latitude; often referred to as the equatorial electrojet (eej). several studies have been done on the electrojet strength using different approaches, methodologies, and tools. the electrojet current is triggered by local intensified ionospheric currents as well as physical structures flowing at the dip equator with high current strengths at daytime period leading to variation in solar quiet (sq) signature. while the relationship between the measurement of the difference between the horizontal magnetic field component at an equatorial and an off-equatorial station, at daytime, had been used to obtain the equatorial vertical plasma drift [36], the quantification of the daytime eej current with the plasma drift in the american and indian longitudes had also been established [7, 36, 37]. while [38] and [39] had presented novel observation of the eej current in the african sector, the relationship between electrojet current and the e x b drift were not considered. based on the authenticated outcome of e x b drift velocities eej relationship around daytime hours obtained by [40] and [7] for the american and indian sectors respectively, [24] attempted same in the african region, using data from ilorin, being the first of such study in the african sector. magdas and ionosonde inferred measurements at ilorin were used during solar minima year. the results obtained revealed that a maximum morning time representative parameter defined by the expression ew = [d(∆h)/dt]max at daytime, representing the east-west electric field in electrojet current matches considerably with the e × b drift, and therefore can serve as a representation/proxy parameter in depicting the magnitude of the vertical drift in the morning hour. the linear correlation between the two parameters (ew and plasma drift) is 0.94 and observed during low solar activity. [7] recorded a correlation coefficient of 0.66 for a high solar activity condition in the indian sector, while [36] reported a correlation value of 0.87 at the american sector. the good relationship between the two parameters is because an increase in the ew magnitude implies an intensification of the eia, which subsequently reduces the plasma density around the equator; thereby rapidly increasing the plasma density along the crest regions. the plasma density reduction over the magnetic equator further diminishes the effect of ion drag on neutrals, thus enhancing the zonal wind prior to sunset. these processes create a large eastward electric field raising the amplitude of the post-sunset e × b drift of the f-layer. 56 b. o. adebesin et al. / j. nig. soc. phys. sci. 4 (2022) 54–58 57 table 2: seasonal pre peak magnitudes in the african and other sectors source data instrument / source sector/station solar activity equinox (m/s) solstices (m/s) march september june december [28] ionosonde / h′f median africa (korhogo) descending phase of solar cycle (sc) 22 12.6 8.7 8.0 12.0 africa (ouagadougou) 15.2 11.0 7.4 10.9 africa (dakar) 10.0 7.7 7.9 8.0 [29] ionosonde/ h′f mean africa moderate 40.1 30.1 31.2 30.2 [14] ionosonde / h′f mean africa (ouagadougou) sc 20. 19661976 18.6 16.3 10.7 10.6 africa (ouagadougou) sc 21. 19761986 19.5 19.0 11.3 13.2 africa (ouagadougou) sc 22. 19861996 19.1 18.0 10.1 14.7 africa (ouagadougou) 1966-1998 18.1 15.8 12.1 12.9 [15] digisonde / h′f mean africa (ilorin) low 12.7 12.8 2.2 13.3 digisonde / hmf2 mean africa (ilorin) low 9.2 3.8 0.3 8.0 [11] ionosonde / hmf2 mean africa (ouagadougou) high 17.1b 16.1 14.1 [30] ionosonde / h′f2 median africa (ibadan) high 33.6b 26.9 26.7 [10] ionosonde / h′f mean africa (ibadan) quiet high 33.0 28.0 42.0 28.0 africa (ibadan) disturbed low a 18.0 23.0 23.0 [15] isr peruvian (jicamarca) low 13.6 9.6 a 6.4 [32] isr peruvian (jicamarca) high 32.0b 14.0 28.1 [16] isr and ae-e satellite peruvian (jicamarca) 1968-1992, 1977-1979 45.0b 35.0 38.1 [33] hf sounder indian (kodaikanal) high 26.4b 25.0 20.8 [34] vhf radar peruvian (jicamarca) high 50.0b 17.0 38.1 ainsignificant magnitude bequinoctial average values 5. conclusion and recommendation a review of the morphology of vertical plasma drift in the african region was presented. reviewed literatures capturing drift velocities in the west-african region including ibadan (nigeria), ilorin (nigeria), ouagadougou (burkina faso), dakar (senegal), and korhogo (cote d’ivorie), were used, and compared with other observations especially during the pre period in the peruvian/american sector. the observations include periods of high-, moderate-, and low-solar activities, as well as 11-year and descending phases of solar activities. it also includes different magnetic activity conditions. the various observations point to the fact that the magnitude of the drift pattern obtained from ground-based ionosondes in the african region is lesser than those obtained from other sectors. however, most ionospheric satellite observational studies have revealed exciting longitudinal pattern in equatorial region electrodynamics. the peak of these variations in terms of irregularities is severe in the african sector owning to the better alignment between the geomagnetic and geodetic equators, whereas in the american/peruvian sector, the geomagnetic equator dips, presenting fairly huge disparity between the geomagnetic and geodetic equator. this could have been responsible for the large excursion in the pre. it has been observed by fejer et al. 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[40] d. anderson, a. anghel, j. l. chau, & o. veliz, “daytime vertical e x b drift velocities inferred from ground-based magnetometer observations at low latitudes”, space weather 2 (2004) s11001, doi: 10.1029/2004 sw000095. 58 j. nig. soc. phys. sci. 3 (2021) 250–255 journal of the nigerian society of physical sciences preliminary investigation of microplastic as a vector for heavy metals in bye-ma salt mine, wukari, nigeria b. n. hikona,∗, g. g. yebpellaa, l. jafiyab, s. ayubaa adept. of chemical sciences, faculty of pure and applied sciences, federal university wukari, taraba state, nigeria bdept. of chemistry, faculty sciences, federal university gashua, yobe state, nigeria abstract this study is aimed at the preliminary investigation of microplastics as carrier of heavy metals pollution in surface sediment. heavy metals concentration was determined by faas while microplastics characterization was analysed by atr-ftir spectrophotometer. the results obtained showed high level of lead (pb) concentrations which ranged from 21.37−32.80 mg/kg across the sampling sites while cd has the least concentration between 0.04 − 0.80 mg/kg. the concentration of pb and cd were above the usepa permissible limit in sediment. the following absorption bands; 2978.19, 1728.28 and 1458.23 cm−1 with the functional groups; c-h stretch, c=o stretch and ch2 bend indicates the presence of ethylene vinyl acetate (eva) in site s2 and s4 respectively. other microplastics found in the sampling sites are nylon, nitrile, polycarbonate and poly propylene. this indicates that there is identical distribution of the microplastics in the sampling sites. the quantities of microplastics isolated ranged from 8.11−8.16 g across the sites. aquatic organisms fed on these polymeric materials because of their unique appearance. hence, heavy metals adsorption will lead to higher concentrations on microplastics which could be ingested and lead serious complication in their intestine. doi:10.46481/jnsps.2021.259 keywords: microplastics, sediment, aquatic, heavy metals, functional group. article history : received: 17 june 2021 received in revised form: 26 july 2021 accepted for publication: 23 august 2021 published: 29 august 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: edward anand emile 1. introduction water has faced series of challenges ranges from point and non point sources of pollution and these challenges have remained the same from time immemorial, the nature of pollution has evolved and lengthened over time. freshwater biotas all over the planet earth are being endangered by both old and new form of pollutants. plastic debris are found in seas, oceans and large body of festered water worldwide [1, 2] mostly constituted by organic pollutants such as microplastics. [3, 4] ∗corresponding author tel. no: +234(0)8065374951 email address: babahikon@fuwukari.edu.ng (b. n. hikon ) defines microplastics as plastic constituent part smaller than 5 mm in size. the word “micro plastic” differs from upper limit of 0.5 mm to 5 mm (universally used), and a lesser limit of 1 m often used for practical purposes. however, groups of microplastics such as biofilms may penetrate and contribute to the sequestration of microplastics and metals in sediments [5, 6]. microplastics are ingested by fish and other aquatic organisms when feeding in sediments. once ingested, it results to problems like pseudo satiation, obstruction of the intestine, endocrine disorder through percolated plasticizers and contamination by adhered pollutants can arise [7, 8]. one of the factors that sway the ingestion of the microplastics is “colour”. some sea debris has the colour that resemble that of their prey which 250 hikon et al. / j. nig. soc. phys. sci. 3 (2021) 250–255 251 entice raiders when ingested causes severe damage to the respiratory system of the aquatic organisms [9]. however, recent research by [10] showed that metal sorption kinetics fouling with organic and inorganic matter over time, increases the surface area and generating anionic active sites of the microplastics for the adsorption of metals from sediment [11]. the hydrophobic nature of the microplastics attract persistent organic pollutants and accumulate heavy metals like cadmium (cd), copper (cu), iron (fe), silver (ag), manganese (mn), aluminum (al) zinc (zn) and lead (pb), which could lead to a greater bio-accessibility of metals when aquatic animals consume micro plastics [12]. this research is aimed at isolation and characterization of microplastics in sediment as carrier of heavy metals in an aquatic system. 2. materials and methods 2.1. study area the study area covers bye-ma, former salt mine pond under chonku ward but presently hospital ward wukari local government of taraba state, nigeria. presently, the pond is used for fishing. it is located between longitudes 7◦51′0′′ north and 9◦47′0′′ east of the greenwich meridian. wukari local government area is situated in the southern part of taraba state and it is about two hundred kilometers away from jalingo the state capital. the local government is bounded by plateau state in the north, benue state in the southwest. it has an area of about 4308 km2 (1663 sq mi). figure 1. showing the study area (bye-ma pond) 2.2. sample collection about four sampling sites were randomly mapped out for sediment samples collection, and were designated as s1, s2, s3 and s4 representing sites 1, 2, 3 and 4 respectively. the sampling sites were 50 m apart. each site was further subdivided into four giving a total of sixteen sites. a 100 g each of the sediment samples were collected from the sampling sites in the month of march, 2020. the samples were collected with the aid of a stainless steel hand trowel at the depth of 0−15 cm into clean glass container that was previously washed in 1% hcl and transported to the laboratory. the sediment samples from each sub sites were bulked together and mixed thoroughly to achieve homogeneity of the representative sample. 2.3. sample preparation the samples were air-dried in the laboratory for two days, manually sorted out debris of large size above 5 mm mesh size. a set of four sieves with mesh sizes 8, 5, 1 and 0.3 mm was obtained from soil science laboratory for the separation of particles size. sediment samples were then sieved mechanically to obtain a fraction of 0.3 mm. the sediment samples were divided into two portions (for heavy metal determination and isolation of microplastics) and are stored in glass bottles at room temperature until ready for further analysis. 2.4. extraction, isolation and characterization of microplastics from sample matrix sediments samples collected on the 0.3 mm sieve are subjected to wet peroxide oxidation (wpo) in the presence of a fe(ii) catalyst to digest labile organic matter. a 6 g of salt (nacl) to the mixture to increase its density of the aqueous solution according to [13]. microplastics were floated on the surface of the solution, were filtered, dried and manually sorted out and characterized with atr-ftir spectrophotometer: model 630 agilent tech usa. figure 2. micro plastics displayed on filtration apparatus 2.5. determination of total mass of micro plastics an empty vial was weighed and labelled a, the identifiable micro plastics was transferred to the vial and then reweighed b. the mass of the isolated micro plastics c was determined by subtracting the mass of a from b (formula: b − a = c). this procedure was repeated for all sediment samples [13]. 251 hikon et al. / j. nig. soc. phys. sci. 3 (2021) 250–255 252 figure 3. micro plastics on sieve 0.3 mm. figure 4. retained debris/macroplastics on sieve 8 mm 2.6. determination heavy metal concentrations in microplastics method of heavy metals determination was adopted from [10] with little modifications. a 1.0 g of dried 0.3 mm size fraction of the microplastics sample were weighed into a beaker and digested with 25 ml mixture of analytical grade acids hno3:hcl in the ratio 3:1. the digestion was performed at a temperature of about 90◦c for 30 minutes in a fume cupboard until clear solutions was obtained. digested samples were allowed to cooled, filtered into a 100 ml volumetric flask, and made up to the mark with deionized water. digests were analyzed by flame atomic absorption spectrometry (faas, spectra aa 50, varian). triplicate determinations were made. the actual concentrations of heavy metals were calculated from the formula below: conc.(mg/kg) = conc.(mg/l) weight of sample digested ×dilution volume(1) 3. result and discussion 3.1. accumulation of heavy metals on microplastics the result obtained from the determination of heavy metals; lead (pb), cadmium (cd), zinc (zn) and copper (cu) in micro plastics from four different samples were shown in table 1. pb and cu have the highest concentration level in all the samples while cadmium has the least concentration and ranged from 0.040.80 mg/kg compared to other metals. the high concentration of pb which ranged from 21.37 − 32.80 mg/kg could be attributed to the following activities along the sampling sites; panel beating, automobile repairs, and discharge of lead from paints factories, lead acid accumulator cells and other dried cells. these wastes are released into water bodies by run off and atmospheric deposition. this agreed with the work of [14] who reported that, high level of heavy metals in sediment is associated with anthropogenic activities. the results presented in this study noticeably indicate a high affinity of metals in solution to microplastics in the sediment. the concentrations of zn and cu in all the sites are within the permissive level by the united state environmental protection agency (usepa). this concentration level is by far less than the permissible limit for zinc concentration in sediment which range from 50 − 300 and 20 mg/kg. zn present in the area could be as a result of its natural abundance, its association with cadmium and as a result of mechanical abrasion of crushing/grinding [15]. cadmium concentrations in all the sites were above the [16] permissible limit of 0.03 − 0.3 mg/kg in sediment. cadmium is emitted to air by mines, metal smelters and industries using cadmium compounds for alloys, batteries, pigments and in plastics. all the concentrations of lead were above the permissible limit of 2 − 20 mg/kg as stated by [16]. [12, 10] and environmental monitoring [17] have indicated that microplastics accumulate metals in aquatic environment. metal ions or complexes interact directly with the charged or neutral sites of the surface of the microplastic, and co-precipitate with or sorption onto hydrous oxides [12]. 3.2. mass of micro plastics figure 5 shows the quantities of microplastics recovered from each of the sampling sites after peroxide oxidation. s1 has the highest level of microplastics (8.16 g), followed by s4 which has 8.14 g and s3 with 8.13 g while s2 has 8.11 g respectively. the quantity and colors of microplastics normally draws the attention of aquatic organisms. [9] reported that aquatics animals sees plastic materials as prey and when ingested it could results to complications in their digestive system. 3.3. characterization of microplastics table 2 below showed the types and the various absorption bands of polymer materials identified in s1. micro polymeric constituents discovered are polycarbonate and nylon (polyamide). nylon is one of the common polymer material found in storage sack, thread for shoe sawing etc. plastics aptitude to adsorb other pollutants makes them a potential trajectory for transferring other pollutants to the aquatic ecosystems, such as heavy metals. both plastics and these pollutants are very difficult to degrade in the environment [18, 19]. the absorption spectrum of s1 displayed in table 2 above, showed that polycarbonate was detected within the following absorption bands: 678.97 cm−1, 1519.96 cm−1, 1712.85 cm−1 252 hikon et al. / j. nig. soc. phys. sci. 3 (2021) 250–255 253 table 1. mean concentration of heavy metals (mg/kg) heavy metals / sample locations pb cd zn cu s1 30.79 ± 0.006 0.05 ± 0.006 0.32 ± 0.006 1.33 ± 0.003 s2 24.68 ± 0.003 0.04 ± 0.002 1.03 ± 0.006 1.45 ± 0.002 s3 21.37 ± 0.002 0.80 ± 0.003 0.55 ± 0.006 1.32 ± 0.003 s4 32.80 ± 0.007 0.50 ± 0.007 0.55 ± 0.006 1.73 ± 0.003 usepa 2002 2 − 20 0.003 − 0.3 50 − 300 20 table 2. ftir result for s1 absorption bands (cm−1) range functional groups polymer type 678.97 630 − 750 c=o bending polycarbonate1519.96 1500 − 1550 aromatic ring stretch 1712.85 1706 − 1730 c=o stretching 1519.96 1500 − 1550 n-h bend nylon (polyamide) 3248.23 3200 − 3550 n-h bend 2985.91 2800 − 3000 c-h stretch 2862.46 2800 − 3000 c-h stretch 3340.82 3584 − 3700 c=o stretch figure 5. quantity of microplastic in each sampling sites and 1519.96 cm−1 with the functional groups: c=o bending, aromatic ring stretch, c=o stretching and n-h bend respectively while nylon (polyamide) was detected within the bands: 2985.91 cm−1, 2862.46 cm−1 and 3340.82 cm−1 with the following functional groups: c-h stretch, c=o stretch and n-h bend. nylons are made from organic (carbon based) found in natural materials such as coal or petroleum, it can also be got from renewable materials called zytel. polycarbonates are made from the condensation of carbonic acid and bisphenol a. these materials after usage are disposed into water ways. because of their non degradable nature they remain in the environment and are finally deposited in soil, water and eventually accumulate in the sediment. the results obtained for ftir analysis in table 3 showed that only ethylene vinyl acetate (eva) polymers are predominant in sample s2. ethylene vinyl acetate (eva) has the following absorption bands and functional groups: 2978.19 cm−1, 1728.28 cm−1 and 1458.23 cm−1 respectively (c-h stretch, c=o stretch and ch2 bend). the result of microplastics characterization for sample s3 as showed in table 4 indicates that absorption occurred at the following frequencies: 2916.47 cm−1, 2244.91 cm−1, 1458.23 cm−1 and 941 cm−1 with functional groups: c-h stretch, c≡n stretch, c=c stretch and =c-h str. this absorption band and functional groups represent nitrile polymeric constituent. consequently, absorption bands at 2916.47, 1458.23 and 941.29 cm−1 which give the following functional groups (c-h stretch, ch2 bend, c-h bend and ch3 bend) indicates the presence of poly propylene in the sample. the absorption bands obtained for sample s4 as displayed in table 5 above showed that ethylene vinyl acetate (eva) is present in the sample. hence, it indicates that the absorption band of ethylene vinyl acetate (eva) in s2 (table 3) correspond with that of s4 in table 5. this indicates that there is equal distribution of the microplastics debris in the sampling sites. 4. conclusion in the present study, a preliminary assessment of microplastics pollution in the surface sediments from bye-ma salt mine pond was obtained. the results of flame atomic absorption spectrometer and ftir showed that microplastics are carriers of heavy metals since considerable concentrations of these metals, pb and cu were determined from the plastics materials. however, the polymer materials discovered are nylon, nitrile, 253 hikon et al. / j. nig. soc. phys. sci. 3 (2021) 250–255 254 table 3. ftir result for s2 absorption bands (cm−1) range functional groups polymer type 2978.19 2800 − 3000 c-h stretch ethylene vinyl acetate (eva)1728.28 1706 − 1730 c=o stretch 1458.23 1430 − 1470 ch2 bend table 4. ftir result for s2 absorption bands (cm−1) range functional groups polymer type 2916.47 2800 − 3000 c-h stretch nitrile2264.91 2260 − 2320 c≡n stretch 941.29 900 − 950 =c-h str 2916.47 2800 − 3000 c-h stretch poly propylene1458.23 1430 − 1470 ch2 bend 941.29 900 − 950 ch3 bend table 5. ftir result for s2 absorption bands (cm−1) range functional groups polymer type 2924.18 2800 − 3000 c-h stretch ethylene vinyl acetate (eva)2862.46 2800 − 3000 c-h stretch 1735.99 1706 − 1730 c=o stretch 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ic3@smart production, university of applied sciences wildau, germany abstract refrigeration applications consume a significant share of total electricity demand, with a high indirect impact on global warming through greenhouse gas emissions. modern technology can help reduce the high power consumption and optimize the cooling control. this paper presents a case study of machine-learning for controlling a commercial refrigeration system. in particular, an approach to reinforcement learning is implemented, trained and validated utilizing a model of a real chiller plant. the reinforcement-learning controller learns to operate the plant based on its interactions with the modeled environment. the validation demonstrates the functionality of the approach, saving around 7% of the energy demand of the reference control. limitations of the approach were identified in the discretization of the real environment and further model-based simplifications and should be addressed in future research. doi:10.46481/jnsps.2023.1244 keywords: refrigeration system, reinforcement learning, optimized control. q-learning article history : received: 09 november 2022 received in revised form: 16 january 2023 accepted for publication: 16 january 2023 published: 24 february 2023 © 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: b. j. falaye 1. introduction the united nations (un) adopted the agenda for sustainable development for 2030, which entails 17 sustainable development goals (sdg) [1]. an important topic regarding these goals is the energy consumption through cooling applications. cooling processes in germany for example, contribute for around 14% of the german electric power consumption [2]. finding novel and innovative approaches is necessary in order to reduce and optimize the energy consumption regarding cooling processes like heating ventilation and air conditioning (hvac). as a result, our society could benefit from lower energy costs, ∗corresponding author tel. no: +49 3375508442 email address: constantin.falk@th-wildau.de (constantin falk) reduced emission of co2 and reduced peak loads for electricity providers [3]. a promising industry 4.0 technology that could help reach the sdgs is decision-making processes like machine learning [1]. one such machine learning method, namely reinforcement learning (rl), most of all approaches based on q-learning, showed promising results [4, 5]. due to its simplicity, q-learning is one of the most popular reinforcementlearning algorithms [6]. yet there is hardly any research on the implementation of q-learning in hvacs control concerning industrial settings. thus, the objective of this paper is to provide a case study on a q-learning based control strategy for an industrial chiller, which provides cooling capacity for two warehouses. a real hvac serves as a model for the environment. 1 falk et al. / j. nig. soc. phys. sci. 5 (2023) 1244 2 2. related work for the development of a suitable rl based control strategy for industrial hvac, this section aims at providing an overview of scientific contributions in recent years. concerning the efficient control of cooling applications, many methods and models have been developed. hovgaard et al. developed an economicoptimizing model predictive control (mpc) scheme that reduces operating costs by utilizing thermal storage capabilities [7, 8]. applying their proposed mpc controller on a simulation of a supermarket refrigeration system, it was shown that potentially savings up to 9-32 % could be achieved using thermal storage capacities in combination with prediction of varying loads and energy prices. in [9], a mpc with a least square tracking error criterion to solve and optimize a power balancing problem with flexible thermal storage units was presented. their power balancing aggregator takes regulating power prices into account and they showed that the constraints and objectives for each unit are satisfied. in [10], a multiple nonlinear regression (mnr) model is used to predict the hourly cooling load in a library. they found that the mnr improved the accuracy of the predicted load, which is important for saving energy while operating the hvac. several works have compared rl to conventional control strategies like chiller-priority or storage-priority control and mpc [11, 12]. henze et al. stated that the rl controller does not rely on physical models of building energy systems and the environment [11]. it worked without the need for prediction and it was able to tune itself to the actual system, while complex learning tasks prolong the learning time. beghi et al. addressed this issue partially in [13], in which a rl controller acted as a supervisory system and determined the setpoints for a local controller. they used domain knowledge to speed up the convergence of the learning algorithm. schreiber et al. also addressed the issue of long training time, as well as in their following work [14, 15]. they trained a deep q-learning agent in advance for 105 interactions with a simulation of an admixing heater and then used it to control an injection heater and a throttle cooler. they found that after approximately 4200 interactions of online training the controller improved visibly and as a result the training time could be reduced. liu et al. used three different techniques, namely randomly initialized q-learning, asynchronous q-learning, deep q-learning in order to improve the learning speed and performance [4]. in addition, they stated that the q-learning controller performs better than conventional control strategies (chiller-priority) but is still outmatched by the mpc scheme. overall, a large part of the literature deals with the implementation of different rl control schemes for commercial buildings. in regards to industrial applications zhang et al. dealt with improving the efficiency of a refrigerating system by combining rl and a coarse model [16]. the coefficient of performance (cop) of the refrigeration system is used as the cost function. they found that the proposed algorithm exceeded the conventional conversion efficiency of the refrigeration system from the viewpoint of the average, although showing larger fluctuations. li et al. used a deep reinforcement learning (drl) framework to optimize a cooling application in an industrial setting [17]. other approaches as in [18] are multiagent drl, which is shown to reduce thermal energy costs of heating, ventilation, and air conditioning (hvac), compared to a heuristic (hs) and rule-based scheme (rs) with an on/off policy. chen et al. showed in a simulation, that a qlearning based control strategy could achieve up to 23 % lower hvac energy consumption, compared to a rule-based heuristic control [19]. qiu et al. applied the mentioned rl algorithm for optimal chiller control for an office building [20]. they showed in their following work [21], that a q-learning controller is also able to generate data with which data-driven chiller models can be trained and enhance the accuracy, generalization and robustness. besides applying the algorithm in question on hvac, in [5] guo et al. proposes a q-learning based rl controller for a district cooling energy plant which could achieve energy savings up to 8 %. the literature shows that the algorithm in question is more often applied for the air conditioning in office buildings. the authors suspect that because of the economic and safety risks that come with it, the literature concerning the investigation and use of q-learning for the control of chiller in the industrial setting is scarce. therefore, the objective of this work is presenting a case study with particular emphasizes on q-learning for the control of a chiller for two industrial storage chambers. 3. model 3.1. principles according to kaelbling et al. reinforcement learning is based on an agent interacting with a given environment, which has a discrete set of states s [22]. the agent is able to perform an action a from a discrete set of actions a, in order to change the state s of the environment. as a result, the agent will receive a reward r, both potentially negative as well as positive, which is typically a boolean or real number. the agent’s overall goal is to find a policy π, which maps states to actions in order to maximize the reward in the long run. rl is an iterative process, which can be distinguished by value iteration or policy iteration [6]. the investigated approach is characterized by value iteration with the objective of finding the optimal value function. according to [22] the optimal value function is defined as v∗(s) = max a (r(s, a) + γ ∑ s′∈s t (s, a, s′)v∗(s′)),∀s∈s, (1) where γ →[0,1] is the discount factor, which defines whether the agent should favor immediate rewards (γ=0) over long term rewards (γ=1). t(s,a,s’) is the state transition function, which is the probability of making a transition from one state s to a next state s’ using the action a. r(s, a) is the reinforcement function, which specifies the expected instantaneous reward as a function of the current state and action. kaelbling et al. stated that v∗(s) asserts that the value of a state s is the expected instantaneous reward plus the expected discounted value of the next state v∗(s′) using the best available action [22]. as a result given the optimal value function, the optimal policy can be 2 falk et al. / j. nig. soc. phys. sci. 5 (2023) 1244 3 specified as π∗(s) = argmax a (r(s, a) + γ ∑ s′∈s t (s, a, s′)v∗(s′)). (2) according to vazquez et al. rl can be further differentiated between the model-based and model-free approach [6]. they stated that q-learning is the most used model-free rl algorithm worldwide, due to its simplicity. kaelbling et al. stated that in the model-free approach a markov decision process (mdp) model is not known [22]. according to white et al. additionally to the states and actions, a mdp model consist of t(s,a,s’) and r(s, a) [23]. white et al. indicated that a model is mdp if the transitions from one state to another depends only on the current state and are therefore independent of any previous states or agent actions [23]. kaelbling et al. argued that the agent has to therefore obtain the optimal policy without knowledge of the mentioned mdp-model [22]. this means that the agent has to directly interact with the environment in order to obtain and process information to produce the optimal policy. q-learning was first proposed by watkins et al. in with the objective to determine an optimal policy π∗ without knowing the mdp model [24]. to achieve this task, a matrix of states by actions is built, in which the state-action values or q-values are initialized. these q-values are updated each iteration by using the q-function. vasquez et al. stated that the updating rule is defined as q(s, a) := q(s, a) + α(r + γ max a′ q(s′a′) − q(s, a)) (3) where α ∈ [0,1] is the learning rate, which defines to what degree new knowledge overrides old knowledge [6]. for α=0, no learning occurs and for α=1 all prior knowledge is overriden. in order to update the q-value, the difference between the previous q-value q(s,a) and the discounted next maximum q-value q(s’,a’) is formed and added to the reward r. this difference is then multiplied with the learning rate α and added to the previous q-value. updating the q-value results in the enviroment entering a new state, in which the agent has to select the next action. for this selection a trade-off between exploration and exploitation, which is also called action-selection is made. vazquez et al. state, that in the exploration phase the agent chooses actions at random, inversely in the exploitation phase the action with the highest q-value is chosen and that for managing this trade-off the ε-greedy policy can be applied [6]. it consits of taking the action with the greatest q-value with the probability (1-ε), and selecting a random action with the probability ε. overall the q-values converge to the optimal q-value q∗(s, a) , which, as shown in [25], results in convergence to the optimal policy. reinforcement learning approaches are dependant on a learning environment. in some situations, the agent’s intended use cases may fit as learning environment, for example when an agent learns playing a video game. however, in most use cases an agent should learn the consequences of its own possible actions in a simulated environment for mainly two reasons: firstly, to avoid harm or damage of entities in the real-word and, secondly, to accelerate the learning process itself. on the other figure 1. case study hand, the biggest drawback of a simulation, however, is the need to create an environment that well represents real-world conditions, as this constitutes a non-negligable cost and time factor. 3.2. case study this work presents a case study for a q-learning based hvac controlling strategy. figure 1 shows a schematic representation on how such a controller could be realized. according to this representation the agent interacts with an environment, which entails the hvac and both warehouses w1 and w2, through its actions and feedback in the form of states and rewards. through its actions it is able to manipulate the screw compressor speed, cold-water feed-pump as well as the selection of which warehouse should be cooled. through a feedback loop the agent is informed about the states of the environment and based on the resulting parameters a reward is calculated and fed back to the agent. 3.3. environment an ammonia-based vapor compression refrigeration system (vcrs) composed of three open screw compressors, refrigerant, cold-waterand cooling-water circuit is adopted as case system. this system is used as a reference for tuning the model parameters based on real plant data. the cold water circuit supplies refrigeration energy to two separate warehouses, cooling them to different target temperatures. the refrigerant condensates whether in a water cooled condensator or in an air-water condensator, depending on the outdoor temperature and if heating of one of the warehouses is necessary. if heating of one of the warehouses is required, the water-circuit heat-exchanger is 3 falk et al. / j. nig. soc. phys. sci. 5 (2023) 1244 4 used, while the heat pump can be switched on if not enough heat is available. the condensed refrigerant is fed from both condensers to the separator via throttle valves. if heat energy is required, the heat pump uses the cooled, liquid refrigerant to return it to the separator in a gaseous state. an additional deep freezing circuit powered by two reciprocating compressors utilizes the cold provided by the cold-water circuit. both water circuits contain a water-glycol mixture, wherein the ciculation is realized by two pumps each. two of the three screw compressors and all four water pumps are speed-regulated by use of frequency converters. matlab with the simulink-package has been used, to build a computerized model of the given system, since simulink is easy to use and well suited for creating physical simulation. following [19], the system is simplified into the following equation 4, while the structure of the model can be traced in a simplified version from figure 2, where the input represents all external variables or constants which are not conditioned by the system itself, while not every block uses each of the available values. q̇total = q̇en + q̇r + q̇hx (4) q̇total represents the total heat gain for the perspected room and is composed of the environmental heat gain q̇en, the solar radiation q̇r and the cooling energy from the air conditioning q̇ac. the environmental heat gain is the thermal energy gained through the building envelope and is the sum of the heat gain of all walls of the considered warehouse. it depends on the conductance of the wall hi, surface area ai of each external wall i and on the temperature difference between the inside and outside environments (tout − tin). q̇en = n∑ i=0 hi ai(tout − tin) (5) the heat gain through solar radiation depends on the actual radiation per m² and the surface area of each external wall/ceiling of the warehouses that are exposed to the sun over a day. depending on the position of the sun, the solar radiation per wall is determined approximately [26]. the hvac itself gets simulated by a combination of regressions and simplifications. the cooling energy q̇ac gets neared with a second degree polynomial regression utilizing the compressors revolutions-per-minute (rpm), cooling-water and coldwater feed pumps rpm as well as the condenser inlet temperature and evaporator inlet temperature, based on data of the real plant. the model is allowed to cool only one of two warehouses simultaneously. however, this limitation is also present at the real plant, so that the warehouses must always be cooled one after the other. utilizing the cooling energy q̇ac together with the evaporator inlet temperature tei and the mass-flow rate of the water-glycol mixture in the evaporator ṁ f e leads to the evaporator outlet temperature teo = tei − q̇ac ṁ f e · cp (6) where ṁ f e is approximated by use of a second-degree polynomial regression using the rpm of the feed-pump [27, p. 356f.]. for reasons of simplicity, the specific heat capacity of water cp is used for the water-glycol mixture throughout this work and the heat loss to the environment is neglected. the resulting teo is used for the calculation of the heat gain in the heat exchangers in the warehouses, which also requires the corresponding warehouse temperatures tw1 and tw2 as well as m f e. furthermore, the chilled water either flows into w1 or into w2, depending on the actual set warehouse to cool. in this way, one of the warehouses always receives m f e = 0 and the corresponding heat-flow is also q̇hx = 0. however, the heat flow is given by q̇hx = k · a · ∆tm (7) where k represents the heat transfer coefficient of the heat exchanger, while a is the surface and ∆tm the logarithmic temperature difference in the heat exchanger. another simplification was made, by using the water output temperature from the warehouse’s heat exchanger as evaporator inlet temperature tei. in the same way, as ṁ f e only flows into one the warehouses, tei is used from the active warehouse to lose the loop. the heatflow of the heat exchanger gets summed up with the other heatflows for the corresponding warehouse to a total heat-flow. using an estimated constant warehouse cooling capacity cw and the total heat-flow leads to the warehouse temperature difference ∆tw in a simulation step with a given duration. for simplification purposes, only the heat-flows and temperatures are displayed in figure 2. the estimation of the electrical power ptotal utilizes the rpm of all compressor stages, feedpumps, the outside temperature, teo and the condenser outlet temperature tco. tco is the only input of ptotal which is not constant, an input variable of the model or already calculated. the calculation is similar to teo and is accordingly based on the condenser inlet temperature tci, the mass-flow rate of coolingwater ṁ f c and the condenser heat-flow q̇c . while ṁ f c is also regression based on the actual rpm of the water feed-pumps, q̇c is simplified the sum of q̇ac and ptotal. tci itself is also a regression-based value, which is approximated by ptotal and the outside temperature. summing up the heat flows leads to the total heat flow of the respective warehouse as shown in equation 4, with which the resulting temperature difference can be calculated for a considered time step. therefore, both warehouses utilize an estimated thermal capacity cw as constant input. 4. validation 4.1. simulation the model presented in subsection 3.3 gets simulated in matlab’s simulink environment. thus, the agent, current state detection and the corresponding selection of the action based on the policy take place externally. to find a subset of states s t which represents the most important system conditions, the most important influencing variables needs to be discretized. the actual state gets defined by the evaporator-inlet temperatur 4 falk et al. / j. nig. soc. phys. sci. 5 (2023) 1244 5 figure 2. simplified functionality of the model tei, condenser inlet temperature tci, both warehouse temperatur tw1 and tw2, outside temperature tout and the actual sun intensity psun ∈ [0, 1]. the state space along s t =  tei tci tw1 tw2 tout psun  (8) with three discretization steps for each variable can be seen in equation 8, while the action space is at =  nc ncwp w  =  0; 3000; 6000 0; 2944; 5888 w1; w2 (9) the action space includes the screw compressor speed nc, cold-water feed-pump speed ncwp and the selection of the warehouse w to be cooled. the q-table represents the complete state-action-space following q = s × a. it is worth noting, that the actual cardinality of action-statepairs directly depends the time of convergence to an optimal policy. increasing the number of states or actions to increase the achievable policy precision, simultaneously increases the time required to achieve convergence [13]. thus, the presented cardinality is a tradeoff between precision and processing time, with the available computing power having a corresponding effect on speed. because the agent initially has to observe the environment, with the look-up table q(s, a) initialized to zero, it’s interactions are divided into explorationand exploitation-mode. from earlier experiments it can be deduced that the learning rate α should be initialized with 0.7 in exploration-mode and decreases to 0.125 in exploitation. the discount factor γ is set to 0.7 to provide the agents with a solid foresight. following the εgreedy policy introduced in subsection 3.1, ε= 0.9 in explorationmode, such that a random action is chosen 9 times out of 10 in the current state. in exploitation-mode, on the other hand, ε= 0.1. both, α and �, are not set to 0 to ensure that it is possible to react to possible changes in the environment even during operation. the reward rt represents the system performance and depends on the system’s power consumption ptotal as well as on the actual warehouse temperatures tw. different states use different formulas to calculate rewards, with the respective conditions, derived from beghi et al., defined in equation 10 and the reward calculation in equation 11 [13]. 5 falk et al. / j. nig. soc. phys. sci. 5 (2023) 1244 6 figure 3. agent’s rewards in exploration and exploitation mode c1 = {tw|tw < twmin ∧ tw > twmax} c2 = {tw|tw ≤ twmin + ∆t set ∧ tw ≥ twmax − ∆t set} c3 = {tw|twmin + ∆t set < tw < twmax − ∆t set} (10) rt w =  −1000 − 50 · ∆tmin ; if c1 100 − 50 · ∆t set ∆tmin ; if c2 100 ; if c3 rt p = − j · ptotal rt = rt w1 + rt w2 + rt p (11) a separate reward rt w is calculated for both warehouse temperatures and depends if the actual tw whether is outside the specified warehouse temperature, within the safety margin of limiting temperatures or within all limits. furthermore, ∆tmin specifies the minimal deviation from the warehouse limits twmin and twmax, while ∆t set is an user-defined safety margin that shall not be exceeded as this might cause unintended operating conditions or even damage to the system. temperatures outside the given limits will be always rewarded negative, while temperatures in the safety margin are tolerated and rewarded depending on the distance to the limits. temperatures inside all limits are always rewarded constantly good to avoid influencing the agent’s temperature set-point. rewards of the actual power consumption rt p are always negative, while it’s influence on the total reward can be adjusted by the weight factor j. in general, one simulation step represents 10 minutes. to speed up the exploration of state-actions, the simulation gets parallelized, such that multiple environments are simulated simultaneously step-by-step, while the q-table is updated after every step with updates from all environments. the parameters of the environments are instantiated randomly with orientation to the discretization steps. as a consequence, it is possible for the system to start outside the specified limits in some environments. 4.2. results to ensure comparability of the results, a reference 2-stepcontroller has been implemented, similar to the real plant, and simulated on the presented model. the 2-step-controller works like a real plant, by cooling one room after the other using the cold-water feed-pumps if necessary and only turning on the compressors, if the water temperature tei is too high. after 1500 steps of exploration, the agent switches into exploitation mode and uses the filled q-table to control the plant properly. figure 3 displays the averaged reward earned by the parallel working agents per simulation step. it can be seen, that in exploration mode the rewards are very unsteady with a conspicuousness in the recurring reward peaks. this is due to the fact that all environments have been re-instantiated after a simulated day, so they are mostly within the given system limits (see equation 11) and accordingly receive rewards for staying within the boundaries at the beginning. while exploring the environment, most agents maneuver the plant out of the target range, earning negative rewards. it can be seen, that even in exploitation-mode averaged rewards from all environments are always negative. this can be attributed to the big impact of the actual power consumption on the rewards, since the weight factor was instantiated as j = 4. it is also noticeable that the average rewards achieved decreased over the further steps. this indicates that some agents utilized the margin of the temperature limits, which in turn can be attributed to agents acting too short-sightedly. figure 4 displays the average electrical energy consumption of the refrigeration system in different simulation environments. therefore, a set of 30 simulation environments were instantiated and controlled on the one hand by the q-learning agent and on the other hand by the reference 2-step controller. the set of environments includes all seasons with different weather conditions and outdoor temperatures. in about 10 simulated days, the 2-point controls consumed on average about 450 kwh more than the q-learning based controls. the course of the difference over the simulated period can be seen in the line chart. 6 falk et al. / j. nig. soc. phys. sci. 5 (2023) 1244 7 figure 4. comparison of averaged energy consumption with (left) the average energy consumption and (right) the difference between the 2-step-control and the q-learning approach in summary, the refrigeration plant controlled by the q-learning approach consumes an average of 5.765 kwh in the 250 hours (1500 steps), while that controlled by the 2-step controller requires 6.203 kwh. this results in energy savings of about 7 % using the q-learning approach compared to the conventional control. the results demonstrate that the chosen approach is functioning, although performance can still be improved. especially the consistent compliance with the temperature limits should be focused by more far-sighted control in a follow-up version. however, the approach also shows a lot of potential, as the environmental model created so far does neither simulate load behaviour in warehouses nor weather forecasts. a promising approach to predict selected meteorological parameters was proposed in [28] and could be considered in future research. these are recurring characteristics or dependencies, which a rl-approach can learn very well and derive optimal control behaviors accordingly. furthermore, the approach could be improved by extending the action space to increase the agent’s scope, while also requiring greater computational power. another variant could be to apply a deep-q-learning algorithm, as already successfully implemented by yu. et al. in a commercial building, where no states or actions would have to be discretized [29]. 5. conclusion in this paper, we conducted a case study to examine potentials in energy efficiency of q-learning in cooling applications. for this purpose, a model based on a real refrigeration plant including warehouses and environmental influences was implemented and used as training environment for the q-learning algorithm. the simulations cover different environmental influences typical for german weather conditions including different seasons. the controller is able to perform actions with predetermined parameters and receives a reward as feedback as well as the new state of the system, which allows the controller learning the behavior of the system directly through the performed actions. the results confirm that q-learning is a suitable approach to derive a control policy to optimize energy consumption. however, it was also shown that constraints such as discretized state and action spaces or simplified models limit the possibilities of q-learning. future works should investigate the benefits of reinforcement learning on recurrent load behaviors compared to conventional controls, as well as identify potential benefits of deep q-learning application on refrigeration systems. acknowledgements this research was supported by the ministry for economic affairs, labour and energy, state of brandenburg (no: 80237615). we would also like to 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[29] k. h. yu, y. a. chen, e. jaimes, w. c. wu, k. k. liao, j. c. liao, k. c. lu, w. j. sheu & c. c. wang, “optimization of thermal comfort, indoor quality, and energy-saving in campus classroom through deep q learning”, case studies in thermal engineering 24 (2021) 100842. appendix a, at set of actions ai surface area of the external wall a action a′ next action cw warehouse cooling capacity cp capacity of water hi conductance of the wall j weight factor k heat transfer coefficient of the heat exchanger ṁ f e mass-flow rate of the water-glycol mixture in the evaporator ṁ f c mass-flow rate of cooling-water nc screw compressor speed ncwp cold-water feed-pump speed ptotal electrical power psun sun intensity q(s, a) q-value q∗(s, a) optimal q-value q̇ac cooling energy from air conditioning q̇c conderser heat-flow q̇en enviromental heat gain q̇hx heat flow q̇r solar radiation q̇total total heat gain r(s, a) reinforcement function r, rt reward rt p rewards of the actual power consumption rt w separate reward s, s t set of states s state s’ next state t(s,a,s’) transition function tci condenser inlet temperature tco condenser outlet temperature tei evaporator inlet temperature teo evaporator outlet temperature tin inside temperature tout ouside temperature tw warehouse temperature twmin, twmax warehouse temperature limits v∗(s′) value function w warehouse α learning rate γ discount factor ∆t set safety margin ∆tm logarithmic temperature difference in the heat exchanger π policy π∗ optimal policy 8 introduction related work model principles case study environment validation simulation results conclusion j. nig. soc. phys. sci. 3 (2021) 287–291 journal of the nigerian society of physical sciences fabrication and characterization of a dye-sensitized solar cell using natural dye extract of rosella (hibiscus sabdariffa l.) as photosensitizer a. a. willoughbya, a. o. sogea,∗, o. f. dairoa, o. d. olukannib, e. u. durugboc, w. s. michaela, t. a. adebayod aredeemer’s university, faculty of natural science, department of physical sciences, gbongan-osogbo expressway, akoda ede, osun state, nigeria, 232102 bredeemer’s university, faculty of basic medical sciences, department of biochemistry, gbongan-osogbo expressway, akoda ede, osun state, nigeria, 232102 credeemer’s university, faculty of natural science, department of biological sciences, gbongan-osogbo expressway, akoda ede, osun state, nigeria, 232102 dredeemer’s university, faculty of natural science, department of chemical sciences, gbongan-osogbo expressway, akoda ede, osun state, nigeria, 232102 abstract the relatively low energy conversion efficiency of dye-sensitized solar cells (dsscs) is a key challenge hindering the commercialization of the solar cell. the photochemical performance of the dye used as a photosensitizer for the dssc greatly determines the efficiency of the solar cell. this study demonstrates the suitability of dye extracted from rosella (hibiscus sabdariffa l.) flowers as a photosensitizer for a dssc. the natural dye was extracted using the acid water extraction method and was characterized using ftir spectroscopy and uv–vis spectrophotometry.the absorption spectra of the dye were examined to determine the aptness of the dye as a photosensitizer indsscs. the ir absorption spectra of the extracted dye confirmed both amine and hydroxyl compounds as functional groups in the natural dye, which established the suitability of the dye as a photosensitizer in dsscs.the uv-vis absorption spectra of the natural dye within the visible region illustrate that the aqueous extract from rosella flowers has stable absorption of visible light, thus validating the natural dye as a good candidate for photosensitizer in a dssc. the fabricated dssc delivered a short-circuit current of 5 µa and an open-circuit voltage of 0.637 v. doi:10.46481/jnsps.2021.346 keywords: dye-sensitized solar cells, rosella flowers, natural dye, energy conversion efficiency, photosensitizers, absorption spectra article history : received: 13 august 2021 received in revised form: 24 september 2021 accepted for publication: 25 september 2021 published: 29 november 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: e. etim 1. introduction renewable energy sources can meet the growing energy demand and are poised to supply the world with a reliable, environmentally friendly, and sustainable energy system [1]. dyesensitized solar cells (dsscs), being a type of photovoltaic ∗corresponding author tel. no: +2347069381694 email address: sogea@run.edu.ng, ayosoge@gmail.com (a. o. soge ) (pv) device, have gained widespread attention in recent years due to their comparative advantages over silicon pv cells – easy fabrication procedures, low manufacturing cost, and compatibility with flexible substrates [2]. besides, dsscs have a high potential for industrial-scale manufacturing due to their production using established scalable manufacturing methods [3, 4, 5, 6, 7]. additionally, dsscs display higher performance under lowand indoor-light conditions than the other pv technologies [8]. the expanded option of designing dsscs using 287 a. a. willoughby et al. / j. nig. soc. phys. sci. 3 (2021) 287–291 288 various materials coupled with flexibility in shape, colours, and transparency, has qualified dsscs for potential applications in pv windows and textiles [5, 9, 10]. the low photoelectric conversion efficiency and instability of the solar cells have been identified as the major challenges impeding the commercialization of dsscs [8]. although the highest theoretical efficiency for dsscs has been calculated to be 32% [11], the maximum efficiency reported in the literature is about 14.3% as measured by kakiage et al. in 2015 [12]. the authors utilized carboxy-anchor organic dye (leg4) and alkoxysilyl-anchor dye (adeka-1) as co-sensitizers for the photoelectrodes together with a cobalt-based electrolyte in fabricating the devices [12]. extensive research is being undertaken to investigate the factors that regulate the performance of dssc to ultimately enhance its efficiency [2]. for instance, research interests in this field have focussed on optimizing the redox couple [13] and dye absorbance [14], modifying semiconductors with a wide band-gap as photo-electrodes [15, 16] and developing high-performance counter-electrodes [17]. moreover, natural and commercial synthetic dyes have been reported as a good replacement for transition metal coordination compounds (rutheniumpolypyridyl complexes) which are regarded as effective sensitizers but contain a costly heavy metal that posed environmental challenges [18]. besides, the process of synthesizing the complexes is complicated and expensive [19]. the advantages of natural dyes include their abundance in nature, environmental friendliness, and affordability.on the other hand, synthetic organic dyes, such as coumarin, xanthene, phthalocyanine, and cyanine dyes exhibit poor performance in dsscs owing to weak anchorage with tio2 film and low uvvis absorption capability [2]. however, these synthetic organic dyes are inexpensive and easy to prepare [20]. the energy conversion efficiency of a dssc is determined mainly by the absorption spectrum of the dye couple with its anchorage to the tio2 surface [21]. hence, the dye used as a sensitizer in a dssc plays a prominent role in the performance of the solar cell [19]. to improve the performance of dsscs with natural dye as a sensitizer, several researchers have investigated a wide variety of plants, fruits, and flowers to determine their effectiveness as sensitizers in dsscs. for instance, sudhakaret et al. [22] fabricated dsscs using extracted dye from bauhinia purpurea l. flower as a sensitizer. according to the experimental results, the as-prepared solar cells delivered a conversion efficiency of 0.63%, with an open-circuit voltage of 0.7 v, short-circuit current density of 1.4ma/cm2,and a fill factor of 0.64%. reda et al. [23] investigated anthocyanin extracts from mulberry (morus alba l.) fruit and akenchira (striga hermonthica(delile) benth.) flower as natural photosensitizers with a tio2 semiconductor. anthocyanin raw pigments extracted from m. alba fruits and s. hermonthicaflowers under acidic conditions delivered incident photon to current conversion efficiencies (ipces) greater than 15%. however, both the ipces and solar energy conversion efficiency of the dsscs were enhanced using acidified ethanol for dye extraction. the ipces obtained for the dye extracted from m. alba and s. hermonthica were 27% and 17%, respectively, while conversion efficiencies of the dsscs fabricated using dyes extracted from m. alba and s. hermonthica were 0.420% and 0.304%, respectively. abdou et al. [19] compared the performance of dsscs fabricated using three different dyes as photosensitizers rosella (hibiscus sabdariffa l.) flowers, remazole red rb-133,and merocyanin-like dye-based on 7-methyl coumarin. the highest conversion efficiency of 0.27% was obtained with the natural dye extracted from rosella flowers. remazole red rb-133 and merocyanin-like dye delivered lower conversion efficiencies of 0.14% and 0.001%, respectively. additionally, senthil et al. [24] investigated the photochemical characteristics of two natural dyes extracted from eugenia jambolana l. and delonix regia and reported efficiencies of 0.55% and 0.317 %, respectively. in a similar study, chang and lo [25] prepared dsscs using dyes extracted from pomegranate leaves and mulberry fruits as photosensitizers. the dsscs fabricated with the dye mixture delivered the best conversion efficiency of 0.722% while those of pomegranate leaves and mulberry fruits offered relatively lower conversion efficiencies of 0.597% and 0.548%, respectively. gomez-ortiz et al. [26] prepared solar cells using tio2 and zno nanostructured, mesoporous films sensitized with annatto, bixin, and norbixin dyes extracted from achiote seeds (bixaorellana l.). best results were obtained for bixin-sensitized tio2 solar cell with conversion efficiency up to 0.53%. this study investigates the performance of rosella (hibiscus sabdariffa l.) as a photosensitizer in a dssc. additionally, the ir and uv-vis absorption spectra of the natural dye are examined to validate the suitability of the dye as a photosensitizer in the fabrication of a dssc. 2. materials and methods dye was extracted from rosella (h. sabdariffa) flowers (figure 1) using the acid water (h2s o4 ) extraction method. 2.1. preparation the dye extract was prepared by grinding 12 g of dried rosella flowers in 500 ml of acid water with ph = 5.7. the extract was filtered and centrifuged separately to remove any solid residue and stabilized at ph = 5.7 by adding an aqueous solution of 0.1 m hcl. the half-time deactivation of the dye solution was over 12 months provided they are correctly stored at 4 ◦c in a refrigerator [27]. the natural dye extracted from the rosella flowers was characterized using t60 uv-vis spectrophotometer (pg electronics, uk) by measuring their uv absorbance. additionally, a fourier transform infrared (ftir) spectroscopy analysis was carried out on the natural dye to determine its chemical compositions using ftir-8400s spectrophotometer (shimadzu scientific instruments inc., japan). 288 a. a. willoughby et al. / j. nig. soc. phys. sci. 3 (2021) 287–291 289 figure 1: rosella (h. sabdariffa) flowers . the fabrication of dsscs normally involves using materials with properties that typically increase the performance and efficiency of the cells. this includes a porous film of nanocrystalline semiconductor such as titanium dioxide film, an electrolyte, dye sensitizers, and conductive glasses as electrodes. the electrolyte was prepared using 10 ml ethylene glycol, 0.127 g iodine, and 0.83 g potassium iodide. the electrolyte solution was stirred thoroughly until no grains of iodine and potassium iodide were visible. the photo-anode comprising fluorine-doped tin oxide glass substrates annealed with a porous tio2 was synthesized by absorbing a small amount of the dye extract on the tio2 layer. the electrode was left for about 10 minutes for complete absorption of the dye, then rinsed first using deionized water and then a final rinse with ethanol (which acts as a drying agent to remove any present water). afterward, the film was dried in an oven at 60 ◦c for 5 minutes. with this approach, the dye extends the photoanode’s spectral sensitivity, thereby enabling the collection of lower energy photons. the counter-electrode (or cathode) was prepared by coating the conductive side of the fluorine-doped tin oxide glass with graphite after wiping the surface with ethanol. the dssc was assembled by placing the tio2-coated anode facing upward, and the conductive side of the graphite-coated cathode faced the tio2 film. the electrodes were positioned slightly offset to allow enough space for electrical contacts and were held together using binder clips, as shown in figure 2. the electrolyte was introduced into the space between the electrodes by capillary action. the open-circuit voltage and short-circuit current of the fabricated dsscs were determined under a laboratory condition using a multimeter with a flash lamp (ctl – rl020) with an intensity less than 100 mw/cm2 illumination. 3. results and discussion the ftir result showing the infrared (ir) spectra of the extracted dye from rosella flowers is presented in figure 3. a broad absorption typical of the hydroxyl group was observed at figure 2: assembled dye-sensitized solar cells showing the electrodes being held together with binder clips. 3421.83 nm for the natural dye. the second absorption band was detected at 2359.02 nm (figure 3). this relatively narrow absorption is attributed to the c-h stretch of a terminal alkyne or acetylenic compound. additionally, the prominent absorption band for the dye at ∼ 3430 nm is broad and indicates the presence of hydroxyl and n-h amine/amide groups. furthermore, the results show the presence of amide i (1600– 1800 cm−1) and amide ii (1470–1570 cm−1) groups. amide i groups are products of c-o bonds while amide ii groups are associated with n-h bending typically of amino acids. amino acids are protein constituent which functional groups readily increase electron transfer process in dssc layer. they achieve this by acting as acids in strong bases (donating protons) and as bases (accepting protons from strong acids) [28]. thus, the ir spectra of the extracted dye confirmed the presence of amine and hydroxyl groups in the natural dye which implies that the dye is a suitable photosensitizer for dsscs. figure 3: infra-red spectra of the extracted dye from rosella (h. sabdariffa) flowers showing the ir transmission percentage (%t) versus wavelength in nm. the uv-vis absorption spectra for the aqueous extract of rosella (h. sabdariffa) flowers are illustrated in figure 4. absorbance is a measure of uv light absorbed by the dye. the higher the value, the more a particular wavelength is being absorbed. many organic compounds undergo electronic 289 a. a. willoughby et al. / j. nig. soc. phys. sci. 3 (2021) 287–291 290 figure 4: uv-vis absorption spectra of rosella (h. sabdariffa) flowers . changes owing to shorter wavelengths and higher energy radiation in the uv (200-400 nm) and visible (380-800 nm) ranges of the electromagnetic spectrum [29]. the purple colour of the dye extracted from rosella flowers (h. sabdariffa) could be attributed to both the absorption peak at 520 nm lying within the green-cyan region (500 – 565 nm) and the absorption peaks at 230 and 290 nm falling within the uv region (200 – 400 nm). the uv-vis absorption spectra of the natural dye within the visible region (380 – 800 nm) illustrates that the aqueous extract from rosella flowers has stable absorption of visible light, implying that the dye is a good candidate for photosensitizer in dsscs. moreover, the fabricated dssc with the natural dye extracted from rosella flowers as a photosensitizer, demonstrated promising photoelectrochemical performances delivering shortcircuit current of 5 µa and an open-circuit voltage of 0.637 v using a liquid-state electrolyte composed of peo-ethylene glycol, iodine, potassium iodide blend. 4. conclusion this work reports an investigation on the suitability of the natural dye extracted from rosella (h. sabdariffa) flowers as a photosensitizer in a dssc. the ir spectra of the extracted dye confirmed both amine and hydroxyl groups in the natural dye suggesting that the dye is a suitable photosensitizer for dsscs. the uv-vis absorption spectra of the natural dye within the visible region (380 – 800 nm) demonstrate that the dye extracted from rosella flowers has stable absorption of visible light, 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[29] excited electronic states: electronic spectroscopy of molecules (3rd february 2016). https://chem.libretexts.org/@go/page/44497 (accessed on 13th september 2021). 291 j. nig. soc. phys. sci. 5 (2023) 1435 journal of the nigerian society of physical sciences numerical investigation of nonlinear radiative flux of non-newtonian mhd fluid induced by nonlinear driven multi-physical curved mechanism with variable magnetic field k. m. sannia,∗, a. d. adesholab, t. o. aliub adepartment of mathematics, comsats university islamabad, chak shahzad road, islamabad, 44000 bdepartment of mathematics and statistics, kwara state university malete abstract this paper discusses two-dimensional heat flow of an incompressible non-newtonian hydromagnetic fluid over a power-law stretching curved sheet. the energy equation of the flow problem considers a radiative flux influenced by viscous dissipation and surface frictional heating. lorentz force and joule heating are taken in the consequence of applied variable magnetic field satisfying solenoidal nature of magnetism. the governing equations are reduced to boundary-layer regime using dimensionless quantities and the resulting pdes are converted into odes by suitable similarity variables. the flow fields; velocity and temperature are computed numerically by implementing keller-box shooting method with jacobi iterative technique. error analysis is calculated to ensure solutions’ convergence. interesting flow parameters are examined and plotted graphically. results show that velocity is increased for large number of fluid rheology and opposite effects are recorded for increasing curvature, lorentz force, and stretching power. flow past a flat and curved surfaces are substantial in validation of this present work. doi:10.46481/jnsps.2023.1435 keywords: power-law, cross fluid, radiation, dissipation, mhd, curved surface, joule heating. article history : received: 01 march 2023 received in revised form: 09 may 2023 accepted for publication: 10 may 2023 published: 11 june 2023 c© 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: o. adeyeye 1. introduction several studies that addressed heat transfer problems in thermal engineering have been documented over the years. attentions have been given towards heat flow and thermal energy exchange through a physical system that involve heat mechanisms like convection, conduction as well as radiation. this is due to undisputed applications of heat generation, heat conversions, and heat transfer of energy by phase changes in ∗corresponding author tel. no: email address: annikennie13@gmail.com (k. m. sanni ) various devices including condensers, boilers, and evaporators. however, fluids flow accompanied by thermal energy over stretchable materials have important applications in many industrial processes which are not limited tocooling of metallic sheets, liquified and plastic film, glass blowing, paper production, drawing of plastic film, glass fiber, food processing, polymer sheet extrusion from dye among several others. nevertheless, evolution of boundary-layer study of heat transfer for nonnewtonian fluids tremendously plays a vital role in processing of material and manufacturing products. non-newtonian fluids have been extensively investigated in account for the dynamic behaviours of most natural, biological, and industrial fluids. 1 sanni et al. / j. nig. soc. phys. sci. 5 (2023) 1435 2 so far there is no unique model that exhibits all the characteristics of non-newtonian fluids. in response to this observation, various rheology models have been established with certain precisions and limitations. nonlinear stretching terms such as power-law, exponential, and quadratic have extensively examined to have greater impacts on flow problems physically in view of real-life scenarios. these forms of stretching exhibition are more challenging in exploring the physical significance of flow situations. literature on this subject is so enormous to be discussed here hence, the present investigation concentrates only on recent published articles. hayat et al. [1] examined the heat transport influenced by cattaneo-christov heat flux of a stagnation point flow induced past a nonlinear stretching surface. they concluded that non-fourier heat flux reduces the heat flow as compared with fourier expression. rout and mishra [2] discussed the heat transfer of a hydromagnetic nanofluid over an unsteady stretching surface. turkyilmazoglu [3] investigated heat transfer of a micropolar fluid flow past a stretching porous sheet. he presented a unique close form solution which satisfied the entire physical parameters of the problem. zeeshan et al. [4] documented the flow of viscous ferro-fluid induced by a stretching surface with magnetic dipole effect and thermal radiation. their results showed that increased magnetic parameter diminished the velocity field and enhanced the temperature distribution. the numerical solution of carreau nanofluid flow induced by a nonlinear stretching porous surface is carried out by mohamed et al. [5] in which the impact of rheology parameter of the fluid showed kinetic differences between pseudo-plastic and dilatant nanofluid. prasnnakumara et al. [6] examined the flow of mhd sisko nanofluid with nonlinear radiation over nonlinear stretching sheet whereas collocation method is applied by feroz et a. [7] to analyze the melting heat of sisko fluid past a moving sheet. lund et al. [8] investigated dual solution of hydromagnetic williamson fluid flow with slip condition. khalil et al. [9] presented a numerical treatment of mhd casson fluid flow with thermal stratification and dual convection. the heat transport of mhd power-law fluid by virtue of lie-group analysis is provided by mohamed et al. [10]. hayat et al. [11] carried out numerical simulation of mhd cross fluid past a stagnation point stretching flat sheet with energy equation under certain flow conditions. khan et al. [12] examined heat flow of a cross fluid along axisymmetric channel in a radially stretched plate. effects of mixed convection and thermal radiative heat flux of a cross fluid over a stretching plane surface are investigated by manzur et al. [13]. for more relevant studies for non-newtonian fluids in the light of ellis fluid, eyring-powell fluid, maxwell fluid, sutterby fluid, second and third grade fluids, and etc., interested readers are referred [14-26]. the fluid flow induced by curved stretching sheet has attracted the interest of researchers in the last decade because of its applications in engineering of stretchable curved materials. initiated by the fundamental paper of sajid et al. [27], abbas et al. [28] included heat transfer analysis in an electrically conducting newtonian fluid flow over stretching curved surface. the results provided in these articles have been recently revised by sanni et al. [29, 30] with the accurate magnetic field included in the momentum equations. the results are modified having satisfied solenoidal magnetic property, ∇.b = 0, as well as the geometry. the consequences of internal heat generation in a flow of nanofluid induced by stretched curved surface is studied by saba et al. [31]. hayat et al. [32] offered the impacts of homogeneous-heterogeneous reaction on electrically conducting micropolar fluid past a curved stretching sheet whereas in an unsteady permeable curved sheet, saleh et al. [33] analyzed the shrinking and stretching flow scenario. the flow of mhd viscous fluid is documented by naveed et al. [34] under dual solution effects with shrinking curved sheet likewise sanni et al. [35] documented the viscous fluid flow caused by a nonlinear power-law driven curved surface. nadeem et al. [36] worked on magneto nanofluid over a stretching curved surface in the presence of variable viscosity and carbon nanotubes. studies from the aforementioned literature revealed that absolutely, no investigation of heat transport of non-newtonian cross fluid over a power-law stretching curved surface has been addressed. therefore, the prime objective of this study is to investigate heat transfer of a cross fluid under certain flow conditions. it is worth mentioning that nonnewtonian rheological model of some fluids like sisko fluid, bingham fluid, and power-law fluid can be recovered with definite constraints being met from cross fluid viscosity equation. for instance, it turns sisko model if ξ � ξ0, bingham model taking n = 1, and power-law model for ξ � ξ0 and ξ � ξ∞ [37]. characterizing parameters, n and γ, of the fluid curve fitting are more important due to strong flow prediction at low and high shear rates. in addition, applied variable magnetic field is considered due to solenoidal nature of magnetism and to elucidates its consequences on the lorentz force and surface frictional heating. this work is prepared in sections that is the basic equations for the flow momentum, problem description accompanied by the heat transport, numerical computations, and results and discussion. special cases of newtonian fluid for flat and curved surfaces remain vital to authenticate the present findings. the results obtained are significantly useful in thermal engineering and manufacturing processing of stretchable sheets. 2. model description consider rheological cross fluid model and navier-stokes equation of an incompressible fluid with body force as [37]: s = η∞ + (η0 −η∞) [ 1 1 + (γγ̇)n ] a1, (1) ρ dv dt = −p + ∇.s + f, (2) ∇.v = 0, (3) where s is the extra stress tensor with η0 and η∞ being low and infinite shear viscosity shear rate, n, γ, and γ̇ = √ trace(a1 )2 2 2 sanni et al. / j. nig. soc. phys. sci. 5 (2023) 1435 3 figure 1: problem geometry represent fluid behavioural index, material constant, and strain rate tensor respectively. fluid density denoted by ρ, velocity field v = (v, u, 0), p symbolizes the pressure, f the body force, and ∇ = êr ∂ ∂r + ês r r+r ∂ ∂s , in which êr and ês are unit vector in radial, r− and axial, s− directions. presenting 2d curvilinear coordinates system (r, s, 0), the first rivlin-erickson tensor a1 = ∇v + (∇v)t is given by a1 = [ 2 ∂v ∂r ∂u ∂r + r r+r ∂v ∂s − u r+r ∂u ∂r + r r+r ∂v ∂s − u r+r 2r r+r ∂u ∂s + 2v r+r ] . (4) 3. problem formulation the steady 2d flow of an incompressible electrically conducting fluid driven by nonlinear curved stretching sheet of radius r is considered. the magnetic field b(r) = rb0r+r êr is variably imposed satisfying the solenoidal magnetic property and perpendicular to flow direction. a resistive force under the effect of magnetic reynolds number yields f = j × b. electrical field varnishes as e ≈ 0 induced a negligible magnetic field implying that the current density j = σ(v × b). then we get f̄ = (−σ rb0 r + r u, 0, 0), (5) where b0 is the strength of magnetic field and σ gives conductivity of the fluid. physical problem geometry is represented in fig. 1. utilizing eqs. (1) (5), governing flow equations can be expressed as following ∂v ∂r + r r + r ∂u ∂s + v r + r = 0 (6) v ∂v ∂r + ru r + r ∂v ∂s − u2 r + r = − 1 ρ ∂p ∂r −n1 [ s rs ∂ ∂r ( 1 1 + (γγ̇)n ) + rs ss r + r ∂ ∂s ( 1 1 + (γγ̇)n )] +n2 [ 1 r + r ∂ ∂r ( rs rr r + r ) + r r + r ∂s rs ∂s − s ss r + r ] (7) v ∂u ∂r + ru r + r ∂u ∂s + uv r + r = − r ρ(r + r) ∂p ∂s −n1 [ s rr ∂ ∂r ( 1 1 + (γγ̇)n ) + rs sr r + r ∂ ∂s ( 1 1 + (γγ̇)n )] +n2 [ 1 r + r ∂ ∂r ( rs sr r + r ) + r r + r ∂s ss ∂s + s sr r + r ] − σr2 b2o ρ(r + r)2 u, (8) such that n1 = γn η0 n 2ρ (γ̇) n−2 2 and n2 = η0 ρ (1 − γn(γ̇) n 2 ). the corresponding stress components s rr , s rs, and s ss are given as s rr = 2 ∂v ∂r (9) s rs = ∂u ∂r + r r + r ∂v ∂s − u r + r (10) s ss = 2r r + r ∂u ∂s + 2v r + r . (11) relevant boundary conditions associate with this problem are u(0)|r=0 = as m, v(0)|r=0 = 0, (12) u(∞)|r→∞ = 0, ∂u(∞) ∂r |r→∞ = 0, (13) in which a is a constant (l1−mt−1) with l being the surface length, and m the stretching power. employing suitable scale together with dimensionless characteristics of the form u∗ = u u∞ , v∗ = vl u∞δ , s∗ = s l , r∗ = r δ , r∗ = r δ , p∗ = p ρu2w , b = η0nu 2n∞ δ2n . (14) after using eq. (14), eqs (6)-(8) give the boundary-layer equations as u2 r + r = ∂p ∂r , (15) v ∂u ∂r + ru r + r ∂u ∂s + uv r + r = − 1 ρ − σr2 b20 ρ(r + r)2 u, −n3 ( (n + 1) ∂2u ∂r2 − n − 1 r + r ∂u ∂r + n − 1 (r + r)2 u ) + η0 ρ ( ∂2u ∂r2 + 1 r + r ∂u ∂r − u (r + r)2 ) , (16) in which n3 = nbγn ( ∂u ∂r − u r+r )n whereas eq. (6) is identically satisfied. existing models of newtonian fluid for both flat and curved surface are recovered in the limiting state as r −→ ∞ at zeroth pressure, p = 0 and material constant γ = 0 (see refs. 20,21,27). relevant similarity variables are calculated as follow u = asmh′(ξ),ξ2 = r2 aρs(m−1) η0 , p = a2 s2 n(ξ), (17) γ2 = r aρs(m−1) η0 , m2 = σb20a 2 η0 , res = aρs(m+1) η0 , (18) 3 sanni et al. / j. nig. soc. phys. sci. 5 (2023) 1435 4 v = −a r r + r √ η0 s(m−1) ρa [( m + 1 2 ) h(ξ) + ξ ( m − 1 2 ) h′(ξ) ] , (19) where h(η) denotes the flow stream, h′(η) represents the flow speed, res the reynolds number, and u∞ the ambient velocity. by virtue of eqs. (17)-(19), eqs. (15) and (16) become (h′)2 ξ + γ = n′(ξ) (20) 2mγ ξ + γ n(ξ) = h′′′ + h′′ ξ + γ − h′ (ξ + γ)2 + γ ξ + γ [( m + 1 2 ) hh′′ − m(h′)2 ] −nn4 [ (n + 1)h′′′ + (1 − n)h′′ ξ + γ + (n − 1)h′ (ξ + γ)2 ] − m2γ2h′ (ξ + γ)2 + γ (ξ + γ)2 ( m + 1 2 ) hh′, (21) subject to h(0)|ξ=0 = 0, h ′(0)|ξ=0 = 1, h′(∞)|ξ=∞ = 0, h ′′(0)|ξ=∞ = 0. (22) in these equations, m2 = σb 2 o a 2 η0 represents the hartmann number and we = a2γres the weissenberg number whereas n4 = we ( h′′ − h ′ ξ+γ ) . in the limit of γ approaches infinity at n = 0, equations (∀m ≥ 0) of fluid flow past a flat surface is recovered and it substantiates the accuracy of present model. h′′′+ ( m + 1 2 ) hh′′−m(h′)2 = (weh′′)n(n+1)h′′′−m2h′,(23) utilizing eq. (20) in eq. (21), the pressure term varnishes and we get h′′′′ + 2h′′′ ξ + γ − h′′ (ξ + γ)2 + h′ (ξ + γ)3 + γ ξ + γ [( m + 1 2 ) hh′′′ − ( 3m − 1 2 ) h′h′′ ] − γ (ξ + γ)3 ( m + 1 2 ) hh′ + γ (ξ + γ)2 [( m + 1 2 ) hh′′ − ( 3m − 1 2 ) (h′)2 ] = nn4 [ (n + 1)h′′′′ + h′′′ ξ + γ + (n − 1)h′′ (ξ + γ)2 − (n − 1)h′ (ξ + γ)3 ] + nnn4( h′′ − h ′ ξ+γ ) [(n + 1)(h′′′)2 − 2nh′′h′′′ ξ + γ + 2nh′h′′′ (ξ + γ)2 + (n − 1)(h′′)2 (ξ + γ)2 ] − nnn4( h′′ − h ′ ξ+γ ) [ (2n − 2)h′h′′ (ξ + γ)3 + (n − 1)(h′)2 (ξ + γ)4 ] + m2γ2 (ξ + γ)2 ( h′′ − h′ ξ + γ ) . (24) 4. heat transfer analysis under the present flow conditions, the heat transport by virtue of energy equation follows ρc p dt dt = ∇(k∇t ) + φ + ∇.q + qo, (25) where φ = 2s (s 2rr + s 2 rs + s 2 ss) is viscous dissipation term, qo = 1 σ (j̄ × j̄) the ohmic heating term and c p, t , k, and q represent fluidheat capacity, temperature, thermal conductivity, and ohmic heating respectively. by rosseland expression, radiative flux q̄ gives q̄ = − 4σ∗ 3γ∗ ∂t 4 ∂r (26) such that γ∗, σ∗ are mean spectral absorption coefficient and stefan-boltzmann constant respectively. it worth nothing that for small heat kinetics, eq. (26) can be linearized. however, this investigation is subjected to substantial high nonlinear radiation. the radiative heat flux is now expanded as q̄ = − 16σ∗ 3γ∗ t 3 ∂t ∂r (27) incorporating eq. (27) into eq. (25), the temperature boundary layer region becomes v ∂t ∂r + ru r + r ∂t ∂s = π(1 + rnt 3) ∂2t ∂r2 + 3πrnt 2 ( ∂t ∂r )2 + π r + r ∂t ∂r + σr2 b20u 2 ρc p(r + r)2 + η0 ρc p ( ∂u ∂r − u r+r )2 1 + [ γ ( ∂u ∂r − u r+r )]n . (28) t → t∞ and t = tw referred as the ambient and surface temperature respectively whereas π = k ρc p denotes fluid thermal diffusivity and rn = 16σ ∗ 3γ∗k being the radiation parameter. require boundary conditions t (0)|r=0 = tw(0) = 0, t (0)|r→∞ → t∞. (29) using an existing similarity quantity in non-dimensional form θ(ξ) = t − t∞ tw − t∞ , (30) invoking eqs. (17) (19) and (30) in eq. (28) we get (1 + rn)[1 + (θw − 1)θ]3 pr θ′′ = 3rn(1 − θw)[1 + (θw − 1)θ]2 pr (θ′)2 − θ′ pr(ξ + γ) − γ(m + 1)hθ′ 2(ξ + γ) − ec ( h′′ − h ′ ξ+γ )2 1 + [ we ( h′′ − h ′ ξ+γ )]n = γ2w(h′)2 pr(γ + ξ)3 (31) 4 sanni et al. / j. nig. soc. phys. sci. 5 (2023) 1435 5 corresponding boundary conditions become θ(0)|ξ=0 = 1,θ(∞)|η=∞ = 0, (32) in which pr = η0c p k , θw = tw t∞ , ec = u 2 ∞ c p (tw−t∞) , rd = 16σ ∗t 3∞ 3k∗k0 , and w = σb20 u 2 ∞a 2 k(tw−t∞) are known as prandtl number, temperature parameter (θw ≥ 1), eckert number, radiation parameter, and ohmic heating parameter respectively. and br = prec the brinkman number. it worth mentioning that radiative flux arises when θw > 1. in response to industrial significance of this work from engineering point of view, physical quantities such as surface drag force and rate of heat transfer are calculated as csk = τrs|r=0 1 2 ρu 2 w , nu = sq̄ k(t − tw) , (33) where uw = asm, τrs|r=0 = η0( ∂u∂r − ur+r ) 1+[γ( ∂u∂r − ur+r )] n |r=0 , and q̄ = −k ∂t ∂r |r=0. utilizing this expression together with eqs. (15)-(19) into eq. (33), we obtain − 1 2 r 1 2 escsk = h′′(0) − 1 γ 1 + [ we ( h′′(0) − 1 γ )]n (34) and r − 1 2 es nu = −θ ′(0). (35) these equations are indispensable in thermal engineering and flow control. 5. computations approach analytic approach to solve the boundary value problems, bvps obtained in eqs. (24) and (31) subjected to eqs. (22) and (32) remains a task and open for future investigation. this section presents a numerical method. firstly, bvps initializes into initial value problem, ivps using h = s1, h ′ = s2, h ′′ = s3, h ′′′ = s4,θ = s5,θ ′ = s6, (36) after effecting eq. (36), eqs. (24) and (31) are linearised into first order of the form  s′1 s′2 s′3 s′4 s′5 s′6  =  s2 s3 s4 −2βs4 + β2 s3 −β3 s2 −βγ [( m+1 2 ) s1 s4 + βγ ( 3m−1 2 ) s2 s3 ] −(we)n(s3 −βs2)n[(n + 1)s′4 + βs4 + (n − 1)β 2 s3 − (n − 1)β3 s2] −n(we)n(s3 −βs2)n−1[(n + 1)s24 − 2nβs3 s4 + 2nβ 2 s2 s4 +(n − 1)β2 s23 − (2n − 2)β 3 s2 s3 + (n − 1)β4 s22] +m2γ2β2(s3 −βs2)s6 − 1 1+rnd31 [ γβpr ( m+1 2 ) s1 s6 + 3rn(θw − 1)d21 s 2 6 + βs6 ] − 1 1+rnd31 ( prec(s3−s2 )2 1+[we(s3−s2 )]n + wγ2β2 s22 )  , (37) subject to s1(0) s2(0) s3(0) s4(0) s5(0) s6(0)  =  0 1 z1 z2 1 z3  (38) where d1 = 1 + (θw − 1)θ and β = 1 ξ+γ whereas z1, z2, and z3 are initial missing conditions. employing keller-box shooting technique together with jacobi iterations [38]. let w′ = w1, w ′ 1 = w2, w ′ 2 = w3, ..., w ′ k−1 = h(ξ, w1, w2, w3, ..., wk−1), (39) wk+1i − w k i−1 δh = (w1) k i−1/2, (w1)k+1i − (w1) k i−1 δh = (w2) k i−1/2, ... (wn−1)k+1i − (wn−1) k i−1 δh = h2(ξ k i−1/2, w k i−1/2, (w1) k i−1/2, ..., (wn−1) k i−1/2) (40) implementing eq. (40) explicitly, we can write (δh)−1(sk+1i − s k i−1) = −0.5((s1) k i + (s1) k i−1), sk+10 = 0, (δh)−1((s1) k+1 i − (s1) k i−1) = −0.5((s2) k i + (s2) k i−1), (s1) k+1 0 = 1, (δh)−1((s2) k+1 i − (s2) k i−1) = −0.5((s3) k i + (s3) k i−1), (s2) k+1 0 = z1, (s4) k+1 i − (s4) k i−1 = −δh[2β((s4) k i + (s4) k i−1) 5 sanni et al. / j. nig. soc. phys. sci. 5 (2023) 1435 6 −β2((s3) k i + (s3) k i−1)] −δh[((s3) k i + (s3) k i−1) −β((s2) k i + (s2) k i−1)] n a1 −δh[((s3) k i + (s3) k i−1) −β((s2) k i + (s2) k i−1)] n−1 a2 −δhβ3((s2) k i + (s2) k i−1) + δhm 2γ2β2 a5 −δhγβ ( m + 1 2 ) a3 + δhγβ ( 3m + 1 2 ) a4 (s3) k+1 0 = z2, (s5) k+1 i − (s5) k i−1 = −0.5δh((s6) k i − (s6) k i−1), (s5) k+1 0 = 1, (s6) k+1 i − (s6) k i−1 = − δh 1 + rnd21 a6, (s6) k+1 0 = z3, where zi for i = 1(1)3 are missing initial conditions. these constants are randomly selected using matlab solver. their choices of values are changed and the accuracy is determined with corresponding end point. the iterations loop with the step size (δh = 0.01) until the solutions are found satisfying the criterion |‖w(n+1)‖2 −‖wn‖2| < �. 6. error analysis theorem: let w1 = w1(h, h1, h2, h3), and w2 = w2(θ,θ1, h, h1, h2) be differentiable functions, the maximum error reached by the keller-box shooting technique together with jacobi iterations for eq. (24) is bounded. proof: discretising this equation following the keller-box method with jacobi iterative scheme, one can write hn+1i − h n i−1 δh + (h1) n i−1/2 = 0, (h1)n+1i − (h1) n i−1 δh + (h2) n i−1/2 = 0, (h2)n+1i − (h2) n i−1 δh + (h3) n i−1/2 = 0, (h3)n+1i − (h3) n i−1 δh + (w1) n i−1/2 = 0 (41) in which the exact scheme can be written as hei − h e i−1 δh + (h1) e i−1/2 = 0, (h1)ei − (h1) e i−1 δh + (h2) e i−1/2 = 0, (h2)ei − (h2) e i−1 δh + (h3) e i−1/2 = 0, (h3)ei − (h3) e i−1 δh + (w1) n i−1/2 = 0, (42) subject to solution error at any grid point of the form (e1) n i = h n 1 − h e 1 , figure 2: (a) to (e). effects of γ, m, m, n, and we on f (ξ) (e2) n i = (h1) n i − (h1) e i , (e3) n i = (h2) n i − (h2) e i , 6 sanni et al. / j. nig. soc. phys. sci. 5 (2023) 1435 7 figure 3: (a) to (e). effects of γ, m, m, n, and we on f ′(ξ) (e4) n i = (h3) n i − (h3) e i . (43) figure 4: (a) to (e). effects of γ, m, m, n, and we on θ(ξ) by virtue of mean value theorem, we get w1(h n i , (h1) n i , (h2) n i , (h3) n i ) − w1(h e i , (h1) e i , (h2) e i , (h3) e i ) 7 sanni et al. / j. nig. soc. phys. sci. 5 (2023) 1435 8 figure 5: (a) to (e). effects of pr, ec, rn, w, and θw on θ(ξ) = (ē1) n i .∇w1(z1, z2, z3), (44) in which z1 = h n i + ε1(h2) n 1, z2 = (h1) n i + ε2(e2) n i , z3 = (h2) n i + ε3(e3) n i , (45) zi�[0, 1] for i = 1(1)3, and (ē1)ni = [(e1) n i , (e2) n i , (e3) n i ], whereas convergence error equations are define as (e1) n+1 = (e1) n i−1 + δh(e1) n i−1/2, (e2) n+1 i = (e2) n i−1 + δh(e2) n i−1/2, (e3) n+1 i = (e3) n i−1 + δh(ê1) n i−1/2∇w1, (46) from eq. (46), once can obtain the inequalities |(e1) n+1 i | ≤ |(e1) n i−1| + δh|(e2) n i−1/2|, |(e2) n+1 i | ≤ |(e2) n i−1| + δh|(e3) n i−1/2|, |(e3) n+1 i | ≤ |(e3) n i−1| + δh|(ê1) n i−1/2.∇w1|. (47) setting ∇w1 = [w̄ 11 , w̄ 2 1 , w̄ 3 1 ], eq. (47) can be written as |(e3) n+1 i | ≤ |(e3) n i−1| + δh|σ 3 j=1(e j) n i−1/2ŵ j 1|, ≤ |(e3) n i−1| + δhς 3 j=1|(e j) n i−1/2w̄ j 1|, (48) such that |(e3) n+1 i | ≤ |(e3) n i−1| + δh|σ 3 j=1(e j) n i−1/2ŵ j 1|, ≤ |(e3) n i−1| + δhς 3 j=1|(e j) n i−1/2w̄ j 1|. (49) the maximum error is estimated in the form (e1) n i = maxi=1(1)n|(e1) n i |, (e2) n i = maxi=1(1)n|(e2) n i |, (e3) n i = maxi=1(1)n|(e3) n i |, (ē)n = max[maxi=1(1)n (e1 = 1(1)n) n i ], (50) where n denotes the number of nodes and eqs. (47) and (48) can be expressed as en+11 ≤ e n 1 + δhe n 2 + m̄1o(δh) 2, en+12 ≤ e n 2 + δhe n 3 + m̄2o(δh) 2, ēn+1 ≤ (1 + 4δhς4j=1|w̄ j 1|)ē n + m̄3o(δh) 2. (51) evaluating n = 0, 1, and n in the above expression, one can get ē1 ≤ (1 + 4δhς4j=1|w̄ j 1|)ē 0 + m̄3o(δh) 2, ē2 ≤ (1 + 4δhς4j=1|w̄ j 1|) 2ē0 +[1 + (1 + 4δhς4j=1|w̄ j 1|)]m̄3o(δh) 2, ēn ≤ (1 + 4δhς4j=1|w̄ j 1|) nē0 +[1 + (1 + 4δhς4j=1|w̄ j 1|) + ..., +(1 + 4δhς4j=1|w̄ j 1|) n−1]m̄3o(δh) 2. (52) taking nth term series sum, we obtain ēn ≤ (1 + 4δhς4j=1|w̄ j 1|) nē0 8 sanni et al. / j. nig. soc. phys. sci. 5 (2023) 1435 9 +  [1 + 4δhς4j=1|w̄ j 1|] n 4δhς4j=1|w̄ j 1|  m̄3o(δh)2, ≤ (1 + 4δhς4j=1|w̄ j 1|) nē0 +exp(4δhς4j=1|w̄ j 1|)m̄3o(δh) 2. (53) utilizing eq. (53) into eq. (52), one can get en1 ≤ (1 + δh)[(1 + 4δhς 4 j=1|w̄ j 1|) nē0 +exp(4(n − 1)δhς4j=1|w̄ j 1|)m̄3o(δh) 2] +m̄1o(δh) 2, (54) en2 ≤ (1 + δh)[(1 + 4δhς 4 j=1|w̄ j 1|) nē0 +exp(4(n − 1)δhς4j=1|w̄ j 1|)m̄3o(δh) 2] +m̄2o(δh) 2. (55) equations (54) and (55) express the maximum error bounds of eqs. (22) and (24). in similar ways, maximum error can be obtained in view of eqs. (31) and (32) (see ref. [40]). 7. results and graphical analysis this section presents the graphical illustrations of the flow quantities that is; the stream function h(ξ), velocity h′(ξ), and temperature θ(ξ) against characterizing parameters. figures 2(a, b, and c) show the behaviour of stream function in response to curvature parameter γ, magnetic field m, and nonlinear stretching power m. in these graphs, the stream function decreases for increasing these parameters which means that the flow trajectory can be minimized either by reducing the curvature or enhancing the lorentz force as well as stretching power. on the other hand, in figs. 2(d and e) the flow trajectory is enhanced by increasing the rheology parameters (n and we) of the fluid. this means that fluid resistance to flow reduces and the fluid behaves more likely as shear thinning. figure 3(a) shows that the velocity h′(ξ) decreases steadily by increasing radius of curvature (as the surface becomes flat). the velocity graphs plotted in figs. 3(b and c) elucidate reductions in flow velocity as a consequence of opposing lorentz force due to applied high magnetic field. this observation substantiates the impact of the lorentz force and stretching power m and confirms the above conclusion made for the flow trajectory. in other words, the flow velocity can be regulated with the aid of geometry parameters that is radius of curvature, magnetic field, and stretching power. to investigate the rheology effects nfluid power-law behavior and weweissenberg number, figures 3(d and e) are plotted. in these figures, the velocity increases for increasing n and we. physically, shear thinning (low viscosity) of the fluid shows less resistance to flow and invariably enlarges the momentum boundary-layer. figure 4 is given to illustrates the influence of nonlinear radiation θw > 1 on temperature profile as compared to linear radiation θw = 1. in fig. 4(a), we observe that increasing radius of curvature γ decreases the temperature profile θ(ξ) as well as thermal boundary layer region. this observation on the other hand shows that, heat flows more quickly over curved surface which can be controlled by value of γ. a striking difference is obvious when θw = 1.5 as compared with θw = 1. effect of increasing magnetic field on temperature is displayed in fig. 4(b). the thermal characteristics is increased due to opposing lorentz force. a usual contribution in thermal engineering study with reason being that the force induced heat from the curved surface to the fluid. the effect of stretching power on temperature profile is plotted in fig. 4(c). this graph shows that increasing m shrinks the thermal region as well as associated thermal boundary layer. this observation infers that either any side of the thermal spectrum can also be controlled by virtue of stretching power of the fluid velocity. figures 4(d and e) elucidate the impacts of fluid parameters (n and we) on temperature profile. in both graphs, the thermal kinetic energy decreases for large fluid power-index and weissenberg number. this observation implies that shear thinning fluid absorbed more heat due to weak or free viscous bond. figure 5 maintains the usual interpretations of the various physical parameters involved in this problem. figure 5(a) shows a decrease in temperature profile as a consequence of lowering thermal conductivity of the fluid for large pr. figure 5(b) indicates a positive response of viscous heating in enhancing the thermal energy of the fluid by increasing ec. this observation is more significant when θw = 1.5. the effect of radiation on temperature profile is given in fig. 5(c). as habitual source of heat, increasing rn enlarges the thermal boundary layer thickness and temperature profile. the effect of ohmic heating from surface to the temperature field is given in fig. 5(d). in this graph, the thermal characteristics increase slightly for increasing w due to additional heat generated at the surface to the fluid and invariably enhances fluid temperature. finally, the impact of temperature difference on temperature as well as thermal boundary layer thickness is substantiated in fig. 5(e) showing a positive influence of nonlinear radiation. in other words, increasing temperature difference θw parameter expands the temperature and associated boundary thickness. table 1 compares the present results of heat transfer with published work in the limiting case when n = 0, rn = 0, m = 0, w = 0, ec = 0, and we = 0. it can be seen that the results in the literature are special cases of the present study. table 2 presents the impact of various parameters on the skin friction coefficient and rate of heat transfer for the present problem. 8. concluding remark modelling of nonlinear radiative heat transfer of an electrically conducting cross-fluid under variable applied magnetic field has been studied. governing equation is conducted in the presence of viscous dissipation and ohmic heating. the fluid motion is induced by power-law stretching velocity over curved surface mechanism.the behaviours of flow fieldvelocity as well as temperature against interesting parameters are summarized. 1. the flow stream patterns, velocity, and associated boundary-layer thickness are increased for large number 9 sanni et al. / j. nig. soc. phys. sci. 5 (2023) 1435 10 table 1: comparison of nusselt number, −θ′(0) with published results pr hammad wang khan et al. g and s ijaz et al. present [17] [18] [12] [21] [14] results 0.2 0.139100 0.169100 0.164037 0.139100 0.196550 0.196502 0.7 0.453900 0.453900 0.418299 0.453900 0.454446 0.454369 20 3.353900 3.353900 3.256030 3.353900 3.359500 3.353902 70 6.462200 6.462200 6.366620 6.462200 6.462290 6.462200 table 2: numerical values of the skin-friction coefficient and rate of heat transfer for n = 2. parameters θw = 1 θw = 1.5 γ m m ec pr rn we w re0.5s csk −θ(0) −θ(0) 5 0.3 0 0.2 2 0.9 2 0.5 0.24141 0.48065 0.26879 0.4 0.24029 0.47929 0.26750 0.5 0.23888 0.47758 0.26590 1 0.22868 0.66205 0.25535 2 0.21036 0.82918 0.22628 10 0 0.1 0.24128 0.50082 0.28532 0.2 0.48188 0.27561 0.3 0.46295 0.26590 0.2 0.48188 0.33336 3 0.46474 0.38631 5 2 0.5 0.23888 0.55891 0.35957 0.7 0.51446 0.30477 1 0.49760 0.41772 0.22763 1.5 0.32994 0.45326 0.24932 10 0.5 0.24128 0.48188 0.27561 1.5 0.20253 0.13530 2 0.06285 0.06495 of fluid rheology n and weissenberg number we. opposite effects are observed for increasing geometry curvature γ, magnetic field m, and stretching power m. that is, the flow stream together with the velocity and associated boundary-layer thickness are decreased. 2. the temperature profile together with thermal boundarylayer are reduced for large curvature and stretching power. 3. increase in magnetic field enhances the thermal region and consequently enlarges its thermal boundary-layer. 4. existing interpretation of thermal profiles for eckert number ec, prandtl number pr, radiation rn parameter, and ohmic heating parameter w, are well 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[38] yasir n & shoaib arif m, “keller-box shooting method and its application to nanofluid flow over convectively heated sheet with stability and convergence. num heat transfer, b: fundamentals”, 76 (2019) 180. 9. appendix a1 = (we) n (n + 1) (s4)k+1i − (s4)ki−1 δh + β((s4) k i + (s4) k i−1)  + (we)n(n − 1)β2((s3) k i + (s3) k i−1) − (we)n(n − 1)β3((s2) k i + (s2) k i−1) a2 = n(we) n(n + 1)((s4) k i + (s4) k i−1) 2 − 2n2β(we)n((s3) k i + (s3) k i−1)((s4) k i + (s4) k i−1) + 2n2β2(we)n((s2) k i + (s2) k i−1)((s4) k i + (s4) k i−1) + n(n − 1)(we)nβ2((s3) k i + (s3) k i−1) 2 − n(2n − 2)(we)nβ3((s2) k i + (s2) k i−1)((s3) k i + (s3) k i−1) + (n − 1)β4((s2) k i + (s2) k i−1) a3 = ((s1) k i + (s1) k i−1)((s4) k i + (s4) k i−1) a4 = ((s2) k i + (s2) k i−1)((s3) k i + (s3) k i−1) a5 = (((s3) k i + (s3) k i−1) −β((s2) k i + (s2) k i−1)) a6 = γβpr ( m + 1 2 ) ((s1) k i + (s1) k i−1)((s6) k i + (s6) k i−1) + 3rn(θw − 1)d 2 1((s6) k i + (s6) k i−1) 2 + β((s6) k i + (s6) k i−1) + prec(((s3)ki + (s3) k i−1) −β((s2) k i + (s2) k i−1)) 2 1 + [we(((s3)ki + (s3) k i−1) −β((s2) k i + (s2) k i−1))] n + wγ2β2((s2) k i + (s2) k i−1) 2 11 j. nig. soc. phys. sci. 4 (2021) 446–454 journal of the nigerian society of physical sciences food and environmental degradation as causative agents of honey bee colonies decline: mathematical model approach kabir lere najiba,b, adam shitu hassana,∗ adepartment of mathematical sciences, college of physical and pharmaceutical sciences, bayero university kano, p.m.b. 3011, kano, nigeria bdepartment of mathematics, faculty of science, kaduna state university abstract in this research, a new compartment model of honey bee population is developed to study the effects of gradual change of food availability as a result of environmental degradation on bee population growth and development. the model is proved to be mathematical well posed and a non-trivial equilibrium point is shown to exist and asymptotically stable under certain conditions. the model predicts a critical threshold environmental degradation rate above which the population size of bees decline and subsequently collapsed. low environmental degradation and high food availability leads to stable bee population while the reverse leads to honey bee population collapse. global sensitivity analysis is conducted to determine the most sensitive parameters of the model that can lead to colony collapse disorder. numerical simulations are conducted to illustrate the results. doi:10.46481/jnsps.2021.283 keywords: honey bees, colony collapse disorder, environmental degradation, global sensitivity analysis article history : received: 30 june 2021 received in revised form: 04 september 2021 accepted for publication: 14 september 2021 published: 29 november 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: t. latunde 1. introduction honey bees are social insects that live in colony and each member of the colony has its role. in each colony there is a queen, hundreds of male drones and thousands of female worker bees. honey bees feed on flower pollen, and they also store nectar in their hives to feed young ones during winter. the life cycle of bee followed the incomplete metamorphosis starting from eggs. the eggs hatched to larvae which are sealed in the cell with wax cap by the worker bees where they developed into pupae. when fully developed, new adult worker bee emerged from the comb. it takes 21 days for a new worker bee to emerge ∗corresponding author tel. no: +23480246098108 email address: hashitu.mth@buk.edu.ng (adam shitu hassan ) from egg [1]. the survival of bee at brood stage of development depends on the number of worker bees in the hive [2]. honey bee colonies exhibit seasonal population growth and decline [3]. a colony lies dormant in winter as bees remain in the hive to maintain hive temperature above 100c [1]. in spring, population grow, then remain high in summer and decrease in autumn towards winter [1]. the queen is the only member of the colony that lays eggs and can deposit up to 2000 eggs per day in summer and its lifespan ranges between 3-5 years [4, 5]. it is fed and groomed by the worker bees and only leaves the hive to mate with the drones or to swarm (start new colony). the worker bees are female with undeveloped reproductive organs. they have pollen baskets on their hind legs and antennal cleaners on their forelegs. there are about 20,000 to 60,000 worker bees in a colony depending on the time of the year [1]. 446 najib & hassan / j. nig. soc. phys. sci. 4 (2021) 446–454 447 the young worker bees are in the hives and part of their duty is to clean the hive, feed the queen, nurse the brood, convert nectar into honey and store it in the hive as food for members of the colony [6]. the old worker bees are the ones responsible for foraging but sometimes return to hive to take care of the brood when there are insufficient hive bees. worker bees live up to 4-5 weeks in the summer and up to six month in the winter [5]. drones are the male bees and their only duty in the colony is to mate with the queen that takes place outside the hive and die immediately. they are slightly smaller than the queen and larger than the worker bees. honey bees play vital role in pollination of flowers in plant communities which is responsible for food production [7]. studies have shown the key role played by honey bees in agricultural fields in seed and crop productions [8, 9, 10]. since 2006 there is increase in reported number of colony collapse disorder (c.c.d) cases [11, 12]. colony collapse disorder is the sudden disappearance of honey bee colony. the most common symptom of c.c.d is the absence of worker bees from the hive, in which worker bees leave for foraging and they never returned [13]. causes of c.c.d can be attributed to a large extent on environmental degradation factors which include excessive use of pesticides and insecticides (used to fight against honey bee pests), predators and diseases[14]. some of these neonicotinoid family and eutrophication of floral resources associated with anthropogenic, which have been established to have negative impact on honey bees including colony collapse [15, 16, 17]. since floral diversity is required to ensure the supply of nectar and pollen to colony [18], bees travel long distance in search of food and in such case, they are exposed to dramatic temperature fluctuations which can cause stress and death [13]. other factors of environmental degradation that contribute to c.c.d includes, radiation, climate change, global warming, geomagnetic disturbance [19, 16], air pollution [20], predation by other insects [21] and electromagnetic radiation from the sun [22]. mathematical models play fundamental role in the study of biological phenomena and dynamics of such natural occurrences providing useful insight towards better understanding of the situation. degrandi-hoffman in [23], developed a mathematical model that studied the laying rate of the queen bee. in their work, the model was divided into two components: the component that study the number of eggs laid by the queen each day and proportion of eggs that developed into drones and workers. the second component of the model track the development of eggs to adult bee. in [24], schmickl and crailsheim studied an age structured honey bee population model using difference equations. their model was an extension of the one in [23]. divison of labour and nutrients allocation in a colony is considered in their model. as a fellow up to causes of colony collapse disorders, of recent, some mathematical models were proposed, amongst which is the one by [25]. in [25], a model of honey bee colony collapse due to the contamination of forager bees in regions contaminated with pesticides was developed. the model attributed c.c.d. in terms of increased homing failure by contaminated forager bees. furthermore, petric et al. [26] investigated the interplay between nosema ceranae infections and increased forager losses due to exposure to agricultural pesticides as leading factors for colony failure. on the side of diseases as causative agent of hive bee c.c.d., dénes and ibrahim [27] formulated a deterministic model that simultaneously modeled the spread of virus by varroa mites among the hive bees population. they found three reproduction numbers as thresholds for global stability of four possible equilibria and later determined the key parameters that caused the collapse or survival of the bee colony. in torres and torres [28], a mathematical model was developed of a honey bee colony with varroa destructor mites as parasitic disease. the model simulations revealed important parameters that determined the survival of colony. comper and eberl [29] formulated yet another mathematical model for a colony of honey bees infected with a parasite nosema ceranae. numerically, they analyzed effects of the parasite infection, food storage dynamics and implications towards colony survival and annual honey yield under treatment. on another cause of c.c.d. in honey bee colony, khoury et al. in [30] presented a model that shows how death rate of forager bees leads to colony collapse. the model predicted a critical threshold forager death rate beneath which colonies regulate a stable population size. this model was improved in [31], by adding stored food in hive and brood compartments. it indicated how interactions of food and forager mortality affect the colony dynamics. according to the model in [30], the quantity of hive stored food is proportional to the amount of foragers in the colony. this is true for a healthy bee colony with abundant food resources in its surrounding but for a colony in an environment where its resources are facing deleterious disturbance, the dynamics may not be the same. in our model we put into consideration pollen availability in the surrounding of honey bee colonies in modeling the food collection by foragers. it was reported in [32] that availability of pollen in agricultural landscapes is essential for the successful growth and reproduction of honey bee colonies. field experiment also revealed that pollination of wild flowers and crops may be threatened by the widespread decline of pollinators (honey bees) [33]. in view of this, we extended the models in [30, 31] by adding a compartment to represent the food collected by foragers from the surrounding environment and its subsequent degradation due to environmental factors. 2. model formulation our model is extension of the ones in [30, 31]. we add a compartment that models the food in bee colony surrounding to investigate how gradual change in the environment will affect the dynamics of bee colony. the components of the model are divided into five compartments. these are the brood class (b) which refers to the bee in either egg, larva or pupa stage of development. broods that developed into adult bees living in the hive are denoted by (h). bees that transit into foraging from the hive bees formed the forager class given as (f). the food stored in the hive collected by foragers is denoted by class ( f ). the nectar and pollen in the environment constitute the food in the colony surrounding is represented by (e). therefore, the 447 najib & hassan / j. nig. soc. phys. sci. 4 (2021) 446–454 448 equations of the model as illustrated in the schematic diagram in figure 1, are given below: 2.1. model equations b′ = l ( f 2 b2 + f 2 ) ( h h + v ) −φb, (1) h′ = φb − h [ αmin + αmax ( b2 b2 + f 2 ) −σ ( f f + h )] , (2) f′ = h [ αmin + αmax ( b2 b2 + f 2 ) −σ ( f f + h )] − mf, (3) f ′ = cf ( e2 e2 + d2 ) −γa(h + f) −γb b, (4) e′ = λe f ρ + f −µe. (5) the model satisfies the nonnegative initial conditions b(0) = b0, h(0) = h0, f(0) = f0, f (0) = f0, e(0) = e0.(6) equation (1) describes the rate of change of brood with time. the parameter l represents the laying rate of eggs by the queen. holling type iii function is used in equation (1) to show how the survival of brood depends on the amount of stored food in the hive. mechelis-menten function is used to show the impact of hive bees to the brood survival as it was used in [31]. the parameter b controls the rate at which recruitment into foraging decreases as stored food increase. the parameter v determine how rapidly the survival rate of brood tends to one as h increases. the parameter φ in equation (1) is the rate at which the eggs are developing into young adult bees. equation (2) describes the rate of change of hive bees (h) with time. the hive bee population is increasing by number of adult bees that emerged pupation and is decreasing by number of young bees that are recruited into foraging. hive bees become foragers at rate αmin while αmax governs the strength of the effect that low food stores has in the transition to foragers. parameter σ is the rate at which forager bees are returning back into the hive when there are insufficient hive bees to take care of the brood. equation (3) represents the rate of change of forager bees with time. the population of foragers is increasing by the number of hive bees that are recruited into foraging and it is decreasing by the number of foragers that transit back into the hive due to social inhibition of the bee colony and foragers that die while foraging at rate m. equation (4) represents the rate of change of stored food in the hive with time. the quantity of stored food is increasing by the amount of food collected per forager from the surrounding. the stored food compartment is modelled in such a way that food gathering by foragers depends on availability of food in the surrounding. saturation function is used to model the effect of food density in the colony surrounding on rate of food collection by foragers. the parameter c is the amount of food collected per forager and d is the half saturation constant. the hive stored food is decreasing by the quantity of food consumed by foragers, hive bees and broad at rates γf , γh and γb, respectively. here, we assume that the food consumed by foragers and hive bees by γa, to be the average of γf and γh , as their ratio in the hive equilibrates quickly [31]. equation (5) represents the rate of change of food in the bee colony surrounding with time. we use a mechelis-menten function to show the impact of foragers in the production of food through pollination. the parameter λ is the food collection rate by foragers in the surrounding while ρ is the half saturation constant of the forager in the environment. the food in colony surrounding is decreasing due to environmental degradation at rate µ. 2.2. basic properties of the model the basic properties of the model, which include existence, uniqueness, boundedness and stabilities of equilibria can now be studied. since the model is representing living organisms, without loss of generality, we assume that all variables and parameters are nonnegative. to prove the existence, uniqueness and boundedness of solution, we claim the following result: theorem 1. the solution (b(t), h(t), f(t), f (t), e(t)) of model equations (1)-(5) with initial conditions b(0) = b0, h(0) = h0, f(0) = f0, f (0) = f0, e(0) = e0 exists, and is bounded in ω = { (b, h, f, f, e) : t = b + h + f ≤ max{ l m , t0}, f ≤ f0 + α̃ ∫ t to t (s)d s, e ≤ e0e (λ−µ)t } . proof. for existence and uniqueness of solution, the system of equations (1)-(5), can be expressed as dx dt = g(t, x), where g is lipschitz continuous function while x(t) = (b(t), h(t), f(t), f (t), e(t)). it follows from the existence and uniqueness theorem in [34] that the system of equations (1)-(5) has a unique solution ( b(t),h(t),f(t),f(t),e(t)) for all time t > 0 given the initial conditions. to show boundedness, from system equations (1)-(5), let t = b + h + f, where t is the total number of bees. from equations (1)-(3), we get dt dt = l ( f 2 b2 + f 2 ) ( h h + v ) − mf, ≤ l − mt. t ≤ l m − ( l m − t0)e −mt. using standard comparison theorem [35], we get t ≤ max { l m , t0 } . from equation (4) we have d f dt = cfg −γa(h + f) −γb b, d f dt ≤ cf −γ(b + h + f). 448 najib & hassan / j. nig. soc. phys. sci. 4 (2021) 446–454 449 social inhibition γhh h γbb γff food availability affects broad survival mf λ cg food collection by forager h f f b φb eclosion hive bees number affect broad survival µe e hr figure 1. schematic diagram of the model of honey bee with environmental degradation but f ≤ t , d f dt ≤ (c −γ)t, where γ = min{γa,γb}, let α̃ = (c −γ), from fundamental theorem of calculus, f ≤ f0 + α̃ ∫ t to t (s)d s. from equation (5), de dt = λe f ρ + f −µe, de dt ≤ λe −µe, e ≤ e0e (λ−µ)t. hence the solution is bounded in ω. this ends the proof of theorem 1. 2.2.1. existence and stability of equilibrium when the food in both hive and surroundings increase with bounds, these approaches the non-trivial equilibrium which can be obtained by setting the right hand sides of system (1)-(5) to zero. thus, after cumbersome computations, we obtained the equilibrium as: (7) where j = (σ+m−α( f ∗))+ √ (α( f ∗)−σ−m)2 +4mα( f ∗) 2α( f ∗) and α( f ∗) = αmin + αmax b2 b2 + f ∗2 . the non-trivial equilibrium is real and positive if, λ > µ and µ < λlmρ+l . combining these conditions, we obtained the critical or threshold environmental degradation value µcrit as µcrit = min { λ, λl mρ + l } . (8) the bee colony will therefore collapse if µcrit > min { λ, λlmρ+l } . theorem 2. the non-trivial equilibrium of system (1)-(5) given in (7) is locally asymptotically stable (las) when λ > µ and µ < λlmρ+l . proof. for the local stability of equilibrium point (7), we need to compute all the eigenvalues ω for the characteristics polynomial p(ω), with respect to the jacobian matrix j∗ of the system evaluated at (7). computing the jacobian of the model equations (1)-(5) at (7) gives, j∗ =  −φ l f∗2 v (h∗+v)2 ( f∗2 +b2 ) 0 h ∗b2 (h∗+v)( f∗2 +b2 )2 0 φ −(αmin + αmax b2 ( f∗2 +b2 ) −σ f ∗2 (f∗+h∗)2 ) σ h ∗2 (f∗+h∗)2 σ 2αmax f∗2 h∗b2 (b2 + f∗2 )2 0 0 (αmin + αmax b2 ( f∗2 +b2 ) −σ f ∗2 (f∗+h∗)2 ) −(σ h ∗2 (f∗+h∗)2 + m) −(σ 2αmax f∗2 h∗b2 (b2 + f∗2 )2 ) 0 −γb −γa ce∗ e∗2 +d2 −γa 0 2cf∗e∗d2 (e∗2 +d2 )2 0 0 λρe∗ (ρ+f∗)2 0 λf ∗ ρ+f∗ −µ  449 najib & hassan / j. nig. soc. phys. sci. 4 (2021) 446–454 450 figure 2. simulation of the model equations using parameter values from table 1 when all conditions are satisfied (µ < λ and µ < λlmρ+l , µ = 0.044,λ = 0.07 and λlmρ+l = 0.046.) figure 3. numerical simulation of the model depicting the effect of environmental degradation on honey bee colony using parameter values in table 1 except for µ = 0.043, 0.046 and 0.047. let c1 = l f ∗2 v (h∗+v)2 ( f ∗2 +b2 ) , c2 = h∗b2 (h∗+v)( f ∗2 +b2 )2 , c3 = αmin + αmax b2 ( f ∗2 +b2 ) , c4 = σ f∗2 (f∗+h∗)2 , c5 = σ h∗2 (f∗+h∗)2 , c6 = σ 2αmax f ∗2 h∗b2 (b2 + f ∗2 )2 , c7 = ce∗ e∗2 +d2 , c8 = 2cf∗e∗d2 (e∗2 +d2 )2 , c9 = λρe∗ (ρ+f∗)2 , c10 = λf∗ ρ+f∗ . thus, the characteristic polynomial p(ω) is defined as p(ω) = |ωi − j| = ∣∣∣∣∣∣∣ ω + φ −c1 0 −c2 0 −φ ω + (c3 − c4 ) −c5 −c6 0 0 −(c3 − c4 ) ω + (c5 + m) c6 0 γb γa −(c7 −γa ) ω −c8 0 0 −c9 0 ω− (c10 −µ) ∣∣∣∣∣∣∣ = 0,(9) where i is a 5 × 5 identity matrix. figure 4. numerical simulation of the model depicting the effect of environmental degradation on food stored in bee colony and surrounding using parameter values in table 1 except for µ = 0.043, 0.044, 0.045. evaluating the determinant in (9), we get p(ω) = ω5 + (φ + m + µ + c5 + c3 − c10 − c4)ω 4 + (φ m + φσ3 + γbc2 + c3µ + c3m + c4c10 + φ c5 + c5µ + mµ + c6c7 + φµ− c4µ− c4m −φ c1 − c5c10 − mc10 − c3c10 −φ c10)ω 3 + (γbc2µ + c6c7µ−φ c5c10 − c4mµ−φ c4m −γbc2σ4 + φγac2 −γbc2σ10 + φ mµ + φ c4c10 − c6c7c10 + c9c6c5 + φ c3µ−φ mc10 + γbc2m + γbσ1c6 −φ c4µ + φ c5µ + γac6m − c3mc10 −φ c1m + c4mc10 + γbc2c5 −φ c3c10 + φ c3m + φ c1σ10 −φ c1µ + φ c6c7 + c3mµ−φ c1c5 −φ c4 + γbc2c3)ω 2 + (γbσ2c4c10 −γac6mc10 −φγac2c10 −φ c6σ7c10 −φ c3mc10 −φ c4mµ−φ c1c5µ−φ c1mµ −φ c1c6c7 −φ c3c2c7 −φ c4c2γa −γbc2c5c10 −γbc2mσ10 −γbc2c3c10 −γbc1c6c10 −γbc2σ4µ−γbc2c4m + γac6mµ + φγac2c5 + φγac2µ + φ c6c7µ + φ c9c6c5 + φ c3mµ + φ c4mc10 + φγac6m + φ c1σ5c10 + φ c1mc10 + φ c1c6γa + φ c3c2γa + φ c4c2c7 + φγaσ2m + γbc1c6µ + γbc1c6m + γbc2c5µ + γbc2mµ + γbc2c3µ + γbc2c3m)ω + (φ c4c2γac10 +φ c4c9c2c5 +φγac6mµ+φ c1c6c7c10 +φ c1c6γaµ+φ c3c2c7c10 + φ c3c2γaµ + φ c4c2c7µ + φγac2c5µ + φγac2mµ + γbσ1c6mµ +γbc2c3mµ+γbc2c4mc10−φ c4σ2γaµ−φγac6mc10−φ c1c6c7µ −φ c1c6γac10−φ c1c9c6c5−φ c3c2c7µ−φ c3c2γac10−φ c3c9c2c5 −φ c4c2c7c10 −φγac2c5c10 −φγac2mc10 −γbc1c6mc10 −γbc2σ3mc10 −γbc2c4mµ) = 0. hence to show las for (7), we need to establish that all the five roots of p(ω) are strictly less than 0. however, due to high dimensionality of p(ω), and the algebraic complexity of parameters involved, the analysis of stability by linearization becomes cumbersome analytically. thus numerical approach is used to 450 najib & hassan / j. nig. soc. phys. sci. 4 (2021) 446–454 451 show the stability of the non-trivial equilibrium point (7) in section 3. 3. numerical simulations in this section, numerical simulations are conducted to illustrate analytical and other results. furthermore, we present sensitivities for effects of varying certain parameters in the model as they affect the bee colony and food in the hive and environment. figure (2) shows numerical solutions for the non-trivial equilibrium using the parameter values in table 1. in particular, λl mρ+l = 0.046 > µ = 0.044 to satisfy the conditions of theorem 2. using the initial conditions as used in [31], one can observe that bees and food are asymptotically converging to the non-trivial equilibrium point (7) as stated in theorem 2. in figure 3, numerical simulations of the model are displayed using three different values of µ = 0.043, 0.046 and 0.047 to show the effect of environmental degradation on honey bee colony. the simulation is run for 400 days using 33, 0000 initial total number of bees in the colony. it can be observed from figure (3) that when µ = 0.043 (mild effect), the bees survived throughout the 400 days. however, when the environment degradation parameter µ is increased to 0.046, the bees population start to decline after 200 days and the colony collapsed in about 360 days. furthermore, when the environment degradation parameter µ is increased to 0.047, the bees population start to decline in just 100 days and the colony collapsed (c.c.d. occurred) in about 270 days. figure 4, depicts the numerical simulations of the model using three different values of µ = 0.043, 0.044, 0.045 to show the effect of environmental degradation on food stored in bee hive and environment. the simulation is run for 500 days with 1.8 × 104kg of initial total food. it can be observed from the figure that when µ = 0.043 (mild effect), the food grows in abundance. however, when the environment degradation parameter µ is increased to 0.044, the food decline drastically. similarly, when µ is increased to 0.045, the food start to decline and exhausted after about 270 days. the fluctuations of total food observed in figure 4 translate the seasonal variation in the environment. in order to confirm the effect of forager death on bee colony as found in [31], in figure 5, we illustrates the numerical simulation of the model using different forager death rate. as can be observed from the figure, when m = 0.1 and 0.2 (low death rates), the bees population increase. however, when the death rate of forager m is increased to 0.3, at the initial stage there is fluctuations in the total number of bees. subsequently, after 100 days, the number turns to zero (c.c.d. occurred). this result is in conformity with the one presented in [31]. 3.1. global sensitivity analysis from previous numerical simulations, we tested the variation of two parameters (µ, m) as they affect bee colony and total food in the surrounding. in order to investigate further the influence of other parameters, in this section, we conduct figure 5. simulations of the model equations to illustrate the effect of death rate of forager bees on the colony using parameter values in table 1 except for m = 0.1, 0.2, 0.3. global sensitivity analysis on seven parameters that are associated with either foraging or food consumption to determine the extent of contributions on this matter. these parameters are c (rate of food collection per foragers), αmin (rate of transition from hive to foraging), αmax( rate of transition from hive to foraging in the absence of food), m (foragers death rate), σ ( rate of inhibition), γa (rate of stored food consumption by foragers and hive bees), γb (rate of stored food consumption by the brood), λ (food collection factor by foragers from surrounding) and µ (rate at which food within the hive environment is degrading). using the method of partial rank correlation coefficients (prcc), as presented by [37] and successfully implemented in [38, 39], we carry out the global sensitivity analysis of mentioned parameters on food stored in the colony and that in the environment. the main objective is to determine the most influential parameters that contribute to colony failure as a result of food scarcity in the hive and surrounding environment. to compute the prcc values, we use the matlab r2017b with ranges of parameters involved as given in table 1, varied with ±50%, divided into 1000 sample sizes. the parameter with prcc value far away from zero, on absolute term indicates is more influential compared to one closer to zero. the results of global sensitivity analysis are presented graphically and summarized quantitatively in figures 6, 7 and table 2, respectively. in figure 6, global sensitivity analysis of aforementioned parameters on food stored in hive is displayed. it can be observed that µ and λ are the most influential parameters followed by m and c in that order. figure 7 presents the prcc simulations of parameters on food in the colony surrounding. the results indicate that food resources in colony surrounding is also most sensitive to µ and λ, followed by m and c respectively. it is worth noting that in all the results of sensitivity analysis shown in figures 6 and 7, as summarized in table 2, the parameter µ, which is the environment degradation rate, is the 451 najib & hassan / j. nig. soc. phys. sci. 4 (2021) 446–454 452 table 1. description of parameters, values and sources parameter description value range reference l egg laying rate of the queen 2000 eggs/day [31] σ rate of social inhibition 0.75/day [0.375 − 1.125] [31] v brood maintenance coefficient 5000 bees/day [31] φ rate of eclosion 0.11 bees/day [31] αmin rate of transition from hive to foraging 0.25bees/day [0.125 − 0.375] [31] αmax rate of transition from hive bee to foraging in the absence of food 0.25 bees/day [0.125 − 0.375] [31] ρ saturated constant of foragers in environment 5000 bees/day assumed c rate of food collection par forager 0.091g/day [0.045 − 0.136] [31] λ food collection factor by foragers from the colony surrounding 0.07/day [0.035 − 0.105] [36] d saturated constant of food in surrounding 5000g/day assumed γa rate of stored food consumption per hive and forager bee 0.007g/day [0.0035 − 0.0105] [31] γb rate of stored food consumption/brood 0.018g/day [0.009 − 0.027] [31] b food dependent term 500g [31] m forager death rate 0.2bees/day [0.1 − 0.3] [31] µ environmental degradation rate 0.044 [0.022 − 0.067] assumed table 2. prcc values for relevant parameters of the model for foods in the hive and environment parameter range prcc values for prcc values for food in hive food in environment λ [0.035 − 0.105] 0.69 0.70 γb [0.009 − 0.027] -0.18 -.08 γa [0.0035 − 0.0105] -0.17 -0.07 µ [0.022 − 0.067] -0.65 -0.78 c [0.045 − 0.136] 0.41 0.30 m [0.1 − 0.3] -0.51 -0.52 αmin [0.125 − 0.375] 0.18 0.08 αmax [0.125 − 0.375] 0.08 0.08 figure 6. sensitivity analysis of selected parameters on food stored in hive. figure 7. sensitivity of selected parameters with food in hive surrounding. 452 najib & hassan / j. nig. soc. phys. sci. 4 (2021) 446–454 453 overall most sensitive parameter in reciprocal manner. 4. discussion and concluding remarks in this research, we extended the models in [30, 31] by incorporating a compartment of food in the surrounding bee colony environment to determine the effect of environmental degradation on bee colony collapse. mathematical analysis shows that the model is well posed and has a non-trivial equilibrium which is determined to be locally asymptotically stable under certain conditions. the model suggests that the availability of food in the colony surrounding and also within the hive have strong influence on colony growth and development. when environmental degradation rate µ is high, the quantity of food in the surrounding and hive reduces asymptotically to zero. consequently, the number of bees also reduce to zero at higher environmental degradation. however, at lower rates of µ, the amount of food in the hive and surrounding become abundant so also the number of bees. this agree with the assertion in [31] that in an environment with abundant of food, a healthy bee colony will continue to accumulate food into the hive until they become limited by storage space, seasonal change or dearth in forage. furthermore, it is observed that sudden change of environmental degradation rate has a complex effect on the bee population. when the environment degradation is low, bees will accumulate enough food into their hives and any sudden increase will not affect the bee colony since there is enough stored food. when the rate is suddenly decreased it takes a while before the colony population start to increase and so also the stored food. this has been shown by experiment in [40], when food is provided to a starved colony it takes time for the bees to return to their normal way of foraging. it is worth reporting here, the implications and impacts of our results as par the ecological and other issues are concerned. as shown from the results, high rate of environmental degradation will leads to colony collapse due to non-availability of food in the hive and surroundings. this will directly affects food/fruits production as a results of absence of pollinators (bees). in addition, this will affects the ecosystem of the surrounding due to absence of shrubs and trees. from our results, again, we can address the colony collapse by choosing appropriate critical parameter values responsible for c.c.d. to further promote the bee colony development, from agricultural point of view, by planting abundant flowering plants with minimum proximity to the colonies so as to reduce the risk of foragers going far in search of food. from governments sides, sponsored bills and laws should be enacted to stop or reduce the usage of chemicals in controlling pests or parasites in farming. appropriate measures should also be taken by government to monitor the transportation of bees across various locations to avoid spread of diseases amongst the colonies. as a further work, our model can be extended by incorporating spread of diseases/pests in the colony or seasonal variation in food supply or climatic changes. acknowledgement we are grateful to the unanimous reviewers for reading the manuscript and providing useful suggestions which enhance the quality of our 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[40] d. j. schulz, z. y. huang & g. e. robinson, “effects of colony food shortage on behavioral development in honey bees”, behav. ecol. sociobiol. 42 (1998) 295. 454 j. nig. soc. phys. sci. 5 (2023) 1029 journal of the nigerian society of physical sciences first principles calculation of half metallic proprieties of qcras (q=hf, ti and zr) m. i. babalola∗, b. e. iyorzo, s. o. ebuwa department of physics, university of benin, nigeria abstract the structural, electrical, magnetic, mechanical, and thermodynamic properties of some novel half-heusler alloys qcras(q=hf, ti and zr) are investigated using first principles calculations. the results show that the three half heusler alloys are half metals and they can find application in spintronics industries. they possess magnetic moment of 3µb. the mechanical properties shows that they are mechanically stable. the b/g ratio of the three half-heusler alloys show that they are ductile in nature and the poisson’s ratio reveal that the plasticity of ticras and zrcras are higher than that of hfcras. the debye temperature and average sound velocity of zrcras is observed to be higher than the other two alloys. this implies that the thermal conductivity of zrcras is the highest. doi:10.46481/jnsps.2023.1029 keywords: half heusler, half-metallic gap, electronic band structure, mechanical properties article history : received: 03 september 2022 received in revised form: 01 november 2022 accepted for publication: 08 november 2022 published: 21 january 2023 c© 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: k. sakthipandi 1. introduction for many years now, the search for functional materials has been on the increase in the scientific community. examples of functional materials include thermoelectric materials [1], piezoelectric materials [2], optoelectric materials [3], spintronic materials [4] etc. of all materials that have been found and synthesized till date, the heusler alloys and its family have been one of the most widely studied material [5-7], this is because they can be easily synthesized and have multifunctional capability which are very useful in technological industries. the heusler family with this multifunctional capabilities include the full heusler alloy [8], quaternary heusler alloy [9], half heusler ∗corresponding author tel. no: +234 8060797955 email address: michael.babalola@uniben.edu (m. i. babalola) alloy [10], binary heusler alloy [11], and inverse heusler alloy [12]. interestingly, different characteristics of these heusler family can be predicted just by knowing their valence electron count [13]. some years back, gautier and coworkers predicted over 300 half heusler alloys [14], ever since then, more half heusler alloys have been investigated till date. half heusler alloys having valence electrons greater or less than 18 can be said to be ferromagnetic, hence such half heusler allloys could possess half-metallic properties which is useful in spintronics industries. one of the properties sough after in spintronics is the half-metallic property. the half-metallic property is described as a situation whereby a material having metallic nature in one spin channel and having semiconducting properties in another spin channel. this concept was discovered by de groot and co-workers in 1983 [15]. half-metals possess 1 babalola et al. / j. nig. soc. phys. sci. 5 (2023) 1029 2 100% spin-polarization at the fermi level, thereby making it possible for them to be used as a spin valve to enhance giant magneto-resistance [16], spin injector electrode in tunnel magneto-resistance [17] and current perpendicular to plane giant magneto-resistance (cpp-gmr) in spintronics [18]. they also find application in spin torque devices [19] and magnetic tunnel junction (mtj) devices [20]. half-heusler alloys uvw(xyz) with half-metallic character are usually described as having the u atomic position occupied by one of the followings: main group 1 or 2 element, the rare earth metals and a transition metal [13]. the v atomic position is occupied a transaction metal which is less electropositive compared to u. the w atomic position is occupied by elements from a main group 3, 4 and 5. in this work, we use ab initio calculation to explore the half-metallic, mechanical and thermodynamic properties of the novel half heusler alloys qcras (q=hf, ti and zr). the remaining portion of this work is divided into the following sections: section 2 covers the computational specifics, section 3 covers the discussion and results, and section 4 concludes with a summary. 2. computational details first principles spin-polarized density functional theory (spdft) calculation has been used to perform the ground state properties of qcras(q=hf, ti and zr) alloys. a projected augmented wavefunction (paw) type of the generalized gradient approximation (gga) which is the choice of exchange correlation as implemented in the quantum espresso code[21] is used. the valence electron configurations of hf(4f146s25d2), ti(4s23d2),cr(3d54s1), as(4d25s2) and zr(5s24d2) are used. an optimized value of 70ry for the plane-wave basis set of the kinetic energy cutoff and a 9×9×9 k-point mesh are used to determine the structural parameters for the three half heusler alloys. optimizaion was carried out for various values of k-point starting with 4x4x4, 5x5x5, . . . 15x15x15. at the end of the calculation, the difference between the last k-point and the others were observed. the difference between the energies of k-point 9x9x9 and the last k-point fell within the acceptable range of 0.1mev.convergence threshold for all selfconsistency calculation is set at 10−6 ry/atom. spin-polarised was taken care of by introducing nspin which activates the magnetism in the compounds. magnetic spins were allocated to the atoms of the transition metals using starting magnetization. duriing band calculation, we used spin component to distinguish between spin up and spin down components. the thermo pw package is used to compute the thermodynamic and mechanical properties [22]. 3. results and discussion 3.1. structural properties a cubic structure with mgagas type c1b structure forms during the crystallization of half heusler alloys (uvw). they have space group of f-43m (no 216). the half heusler stucture is seen as a combination of zincblende and rocksalt structure. wyckoff locations 4b(1/2,1/2,1/2), 4c(1/4,1/4,1/4), and 4a figure 1. unit cell of qcras (q=ti, hf and zr) hh alloys (0,0,0) are occupied by the u, v, and w atoms, respectively. the unit cell of the half-heusler alloy is shown in fig. 1. the structural parameters such as the equilibrum lattice constant, bulk modulus and pressure derivative are computed by fitting the total energies versus lattice constant curve with the murnaghan equation of state with the results presented in table 1. fig. 2 shows the ferromagnetic state and non magnetic state of the three half-heusler alloys, and from the graph, their ferromagnetic state posses the lowest ground state energies. this implies that the three half-heusler alloys are ferromagnetic in nature. the lattice constant of the three hh alloys satisfy the condition ti<zr<hf, this could be as a result of the atomic mass satisfying the same condition. the half heusler alloys qcras(q=hf, ti and zr) are novel and have no experimental and theoretical result to compare with. nevertheless, the lattice constants of qcras(q=hf, ti and zr) when compared with results of qcrbi(q=hf, ti and zr) [23] are within reasonable range. the formation energies of the three alloys have also been computed using eq. 1 as shown in table 1. qcras(q=hf, ti and zr) half heusler alloys all have negative the formation energies of the three half-heusler alloys have negative values and these indicate that they can be synthesized in experimentally. e f (qcras) = et (qcras) − (e(q) + e(cr) + e(as)) (1) the ground state energies of each element are represented as e(q), e(cr) and e(as)), while et (qcras) is the sum of the energies in the hh compounds. the formation energies for each hh are displayed in table 1. the compounds can be synthesized experimentally if the formation energies have a negative value. 3.2. magnetic and electronic properties the magnetic and electronic properties of the three half heusler alloys qcras (q=hf, ti and zr) have been computed 2 babalola et al. / j. nig. soc. phys. sci. 5 (2023) 1029 3 table 1. computed lattice parameters a, bulk moduli (b), pressure derivatives (b’), magnetic states mg magnetic moment (mm), and the formation energy e f of qcras (q=hf, ti and zr compounds a(å) b(gpa) b’ mg mm(µb) e f (ev) ticras 5.87 116.80 6.20 fm 3.012 -1.58 5.81 128.90 4.33 nm zrcras 6.10 103.50 4.28 fm 3.026 -1.44 6.02 117.80 4.30 nm hfcras 6.07 108.50 4.61 fm 2.988 -1.62 5.98 122.10 4.16 nm figure 2. total energies of xcras (x=hf, ti, and zr) against lattice constants in ferromagnetic (fm) and non-magnetic (nm) states (a)hfcras, (b) ticras and (c) zrcras using the spin-resolved dft technique. the total magnetic moment that results from the contributions of the local moments of each atom in the hh alloys is shown in table 1. the mm obtained in table 1 are in agreement with the mm predicted by slater-pauling rule. m=|vn -18|, where vn is the valence electrons count (vec) per formula unit and m is the total magnetic moment per formula unit, gives the slater-pauling rule. the valence electrons for each element are: hf ( 5d26s2), ti(3d24s2), zr( 4d25s2) cr( 3d54s1) and as(4s24p3). the value of vn for each hh alloy is 15, hence the predicted mm from slater-pauling rule is 3µb. the magnetic moment computed are approximately 3µb. the idea behind the magnetic moment can be explained from the crystal field splitting of the d-orbital of the atom occupying the v atomic position (the cr element) [24]. the cr atom with the as atom form a tetrahedral position thereby hybridizing to form [cras]4−. the resulting ion interact with q(=hf, ti and zr)4+ ion during which electrons are exchanged. the cr− ion gains one extra electron during this process thereby making it possess seven electrons. the seven electrons are the shared among the triplet and doublet states leaving three unpaired electrons which constitute the total mm of the compounds. figures 3, 4, and 5 display the electronic band structure for both spin channels along with the corresponding partial density of states (pdos) for each alloy. it is evident from the pdos that the three hh alloys are halfmetallic with their spin up channels being semiconductors and figure 3. hfcras band structures for both spin channels figure 4. ticras band structures for both spin channels their spin down channels being metals. these results suggest that the three compounds are half-metals. the highest occupied molecular orbit (lowest occupied molecular orbit) homo (lumo) for hfcras, ticras and zrcras are 12.8334ev (12.6114ev), 12.1927ev (11.8789ev) and 12.7092ev (12.3076ev) respectively. the band gap of hfcras, ticras and zrcras are 0.222ev, 0.3138ev and 0.4016ev respectively. 3 babalola et al. / j. nig. soc. phys. sci. 5 (2023) 1029 4 figure 5. zrcras band structures for both spin channels table 2. computed elastic constants ci j , shear moduli g, young moduli e, b/g ratio with the poisson’s ratio v of qcras (q=hf, ti and zr) compounds hfcras ticras zrcras c11 (gpa) 108.62 105.18 116.00 c12 (gpa) 116.61 101.07 104.19 c44 (gpa) 11.78 19.61 11.79 e(gpa) 16.15 24.69 26.08 g(gpa) 5.47 8.51 8.93 b/g 19.8 13.7 11.6 v 0.6385 0.4508 0.4597 3.3. mechanical properties for qcras (q=ti, zr, and hf), the parameters that make up the mechanical properties are calculated and are shown in table 2. these parameters include the elastic constants, the young’s modulus, the shear modulus, and poisson’s ratio. the elastic constants typically related to cubic materials are c11, c12 and c44. they describe the response of materials to external forces. the cubic structure’s mechanical stability criteria are specified as c11 >0, c44 >0, c11 >c12 and c11+2c12 >0, c11c12 >0. once this criteria is satisfied then the material is said to be mechanically stable. according to the results in table 2, zrcras has the strongest resistance to linear and shear deformation when compared to ticras and hfcras. the b/g ratio of the three half-heusler alloys are greater than the critical value of 1.75, it means the three alloys are ductile. the poisson’s ratio determines a compound’s plasticity. the critical value for metals and their alloys is 0<v<0.5. the lower the value, the greater the plasticity. ticras and zrcras fall within the criteria and they have better plasticity compared to hfcras. 3.4. thermodynamic properties the results of the thermodynamic properties of qcras (q=ti, zr, and hf) are shown in table 3 and figure 6. the constant volume heat capacity of the three compounds are similar while the debye temperature of zrcras has the highest value implying that it has higher thermal conductivity than the other half heusler alloys. the zero-point energy of the three alloys are table 3. computed heat capacities cv at 300k, zero point energies eo, debye temperatures θd, and the sound velocity vav of the hh alloys compounds cv (j/kmol) eo (kj/nmol) θd (k) vav(m/s) hfcras 74.7 2.263 80.616 717.68 ticras 74.7 2.218 79.004 682.86 zrcras 74.6 2.643 94.152 843.28 figure 6. thermodynamic characteristics of qcras (q=hf, ti, and zr), showing the heat capacity, the entropy, the internal energy, and the gibb’s free energy in (a), (b), (c), and (d), respectively close and it gives information about the vibrational energies of materials at absolute zero temperature. the heat 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[24] b. e. iyorzor & m. i. babalola, “half-metallic characteristics of the novel half heusler alloys xcrsb (x= ti, zr, hf)”, journal of the nigerian society of physical sciences 3 (2022) 138. 5 j. nig. soc. phys. sci. 5 (2023) 1154 journal of the nigerian society of physical sciences degradation of pet nanoplastic oligomers at the novel phl7 target:insights from molecular docking and machine learning c. e. durua,∗, c. e. enyohb, i. a. duruc, m. c. enedohd a surface chemistry and environmental technology (scent) research group, department of chemistry, imo state university, owerri, pmb 2000, imo state, nigeria. bgraduate school of science and engineering, saitama university, 255 shimo-okubo, sakura-ku, saitama city, saitama 338-8570, japan. c department of chemistry, federal university of technology owerri, pmb 1526, imo state nigeria. d department of chemistry, imo state university, owerri, pmb 2000, imo state, nigeria. abstract the versatility of polyethylene terephthalate (pet) as a material with numerous applications in the food industry and its recalcitrance to chemical and microbial degradation has recently made it an environmental nuisance. in this study, we applied computational methods to ascertain the dependence of pet nanoplastic (np) degradation on the chain length of the oligomer. the binding affinities of the nps on the novel enzyme polyester hydrolase leipzig 7 (phl7) were used to relate their ease of degradation at the enzyme active site. the results revealed that the binding affinity of pet nps at the enzyme target decreased from -5.2 kcal/mol to -0.8 kcal/mol, with an increase in pet chain length from 2.18 nm to 5.45 nm (2-5 pet chains). the binding affinities became positive at chain lengths 6.54 nm (6 pet chains) and above. these findings indicated that pet np degradation at this enzyme’s active site is most efficient as chain length decreases from 5-2 units and is not likely to occur at longer pet chains. a feedforward artificial neutral network (ann) analysis predicted that the energy of the pet nps is a very important factor in its degradation. doi:10.46481/jnsps.2023.1154 keywords: polyethylene terephthalate; nanoplastic; polyester hydrolase leipzig 7; binding affinity; artificial neutral network. article history : received: 28 october 2022 received in revised form: 05 january 2023 accepted for publication: 10 january 2023 published: 27 january 2023 c© 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: k. sakthipandi 1. introduction plastics are synthetic materials of long carbon chains that can be formed into different shapes when still molten and then transform into a slightly elastic solid form. nanoplastics (nps) are particles ranging from 1 to 1000 nm, unintentionally produced from the degradation and manufacturing of plastic ∗corresponding author tel. no: +2348037131739 email address: chidiedbertduru@gmail.com (c. e. duru ) objects [1]. because they may pass through biological membranes, nanoplastics have a higher potential for danger than microplastics [2, 3]. nps can be ingested by various creatures [4, 5], thereby raising concerns about possible bioaccumulation and biomagnification. there is increasing evidence that marine creatures consume nps, some evidence of translocation outside the stomach, and much less evidence of transfer between trophic levels. many studies have also demonstrated the exposure and toxicity of nps to human health [6, 7]. 1 c. e. duru et al. / j. nig. soc. phys. sci. 5 (2023) 1154 2 plastic degradation is the change in the polymer’s shape, colour, tensile strength, and molecular weight under the influence of chemicals, light, heat, or applied force. though plastic degradation represents the failure of the polymer to perform a required service, the process can be useful as it concerns the recycling of polymer wastes to reduce the environmental pollution they can cause [3]. these days, the most popular methods for getting rid of plastic garbage in the environment include landfilling, burning, and mechanical and chemical recycling [8]. landfilling is the main method of getting rid of plastic trash in most countries, especially impoverished ones because it is straightforward and inexpensive. however, accumulating plastic trash has consumed a substantial amount of space. while burning plastic garbage may assist reduce the need for landfill space and provide thermal energy, we also need to consider the environmental effects of secondary pollutants, including dioxins, carbon monoxide, and nitrogen oxides formed during the incinerating process. even though mechanical recycling has grown to be the most popular method for recovering thermoplastic wastes, the bulk of recovered materials significantly degrade after a few processing cycles, limiting their economic value. the success of chemical recycling on the other hand depends on the accessibility of the processes and the effectiveness of the catalysts [9]. currently, there are reports that natural enzymes can catalyze the hydrolysis of micro and nanoplastics as an alternative to chemical processes [10, 11]. the ester bond linkages in pet can be hydrolyzed by various esterases like petase [12], cutinase [13], and lipase [14]. the extent of hydrolysis of pet by these enzymes is quite low and yields the monomeric units mono-2-hydroxyethylterephthalate (mhet) and bis(2-hydroxyethyl)terephthalate (bhet) [15]. these monomers can further be degraded into ethylene glycol and terephthalic acid, the initial reactants used for their formation. the high recalcitrant nature of pet waste is a major bottleneck in its hydrolysis by enzymes. factors such as hydrophobicity, the crystallinity of pet, low accessibility, and structure usually limit enzyme function, thus making depolymerization very difficult [16]. therefore, finding more effective and sustainable methods for breaking down pet plastic and other polymers could have significant benefits in terms of reducing pollution and increasing the recyclability of plastic waste. in this study, we applied computational techniques [17] for the first time in studying the ease of hydrolysis of different pet nanoplastic chains by a microbial enzyme. molecular docking of some pet nanoplastic oligomers was performed at the active site of the novel enzyme polyester hydrolase leipzig 7 (phl7). the binding affinities of the oligomers at the active site of the enzyme were used to estimate the chain lengths at which pet hydrolysis is most effective. also, artificial neural network (ann) was used to identify ligand-dependent factors responsible for the degradation process. 2. computational methods 2.1. preparation of pet nanoplastic oligomers polyethylene terephthalate (pet) nanoplastic (np) chains ranging from 1-10 polymer units were designed in chemdraw and saved in mdl sdfile fileformat [18]. they were optimized using open babel in python prescription (version 0.8), which converted them to their most stable structures using merk molecular force field 94 (mmff94). the optimized structures (table 1) were used as small molecules in the study. 2.2. identification and preparation of enzyme target the 3d x-ray crystallographic structure of the novel enzyme polyester hydrolase leipzig 7 (phl7) with identity 7nei [19] was retrieved from the protein data bank (pdb). the chain a of the enzyme was used as target to study the effect of chain length on pet np hydrolysis. removal of the interfering crystallographic water molecules and minimization of the protein was done using ucsf chimera 1.14 [20, 21, 22]. 2.3. molecular docking studies site-directed docking of the pet np chains was performed on the active site of the enzyme with autodock vina in pyrx software version 0.8 [23]. the amino acids at the active site were selected and toggled on the enzyme surfaces in the pyrx software. the specific site on the receptor was set using the grid box with dimensions:center x: 22.249, center y: – 1.869, centerz: – 22.211, and size x: 23.795, size y: 14.112, sizez: 15.463. at the end of the molecular docking, the binding poses of the enzyme-ligand complex were generated, and their scoring results were also created. theinteractions between the enzymenp complex were visualized using bioviadiscovery studio 4.5 [24]. 2.4. artificial neural network (ann) analysis computational models with numerous processing layers may learn data representations at various abstraction levels [25, 26]. the current study employed a feedforward artificial neural network (ann) made up of many perceptron layers (with threshold activation). backpropagation is a supervised learning method that the ann uses during training. a minimum of three layers of nodes make up this ann: the input layer, the hidden layer, and the output layer. each node, except the input nodes, is a neuron that employs a nonlinear activation function. the network information for the ann is summarized in table 2. the input layers include the determined variables from the pet nps oligomers, including energy, molecular mass (mm), and chain length (cl). the anns used 70 % of the input data to train the model, while 30 % was used for testing the model. the ann had two hidden layers based on a hyperbolic tangent activation function. the dependent variable or binary classifications are contained in the output layer. the ann will examine the output layer’s dependent and the input layer’s independent 2 c. e. duru et al. / j. nig. soc. phys. sci. 5 (2023) 1154 3 table 1: optimized np chains used for the study np chain length structure chemical formula molecular mass minimized energy (ha) np1 c10h10o5 210.18 76.01 np2 c20h18o9 402.35 195.80 np3 c30 h26 o13 594.52 418.82 np4 c40 h34 o17 786.69 577.84 np5 c50 h42 o21 978.86 633.86 np6 c60 h50 o25 1171.02 995.98 np7 c70 h58 o29 1363.19 1427.46 np8 c80 h66 o33 1555.36 1159.54 np9 c90 h74 o37 1747.53 2118.52 np10 c100 h82 o41 1939.70 2415.53 3 c. e. duru et al. / j. nig. soc. phys. sci. 5 (2023) 1154 4 table 2: network information for the anns input layer covariates 1 energy 2 molecular mass (mm) 3 chain length (cl) number of unitsa 3 rescaling method for covariates standardized hidden layer(s) number of hidden layers 2 number of units in hidden layer 1a 1 number of units in hidden layer 2a 1 activation function hyperbolic tangent output layer dependent variables 1 binding affinity number of units 1 rescaling method for scale dependents adjusted normalized activation function hyperbolic tangent error function relative error sum of squares a excluding the bias unit variables during training to see how they relate to one another. the hidden layer’s nodes include mathematical functions that define the relationships. once the connections have been established, the testing data will be used to validate them. error functions, such as the relative error (re) and the sum of squares error (sse) shown in equations (1) and (2), would be used to verify and evaluate how well an ann model predicts the output. figure 1: structure of pet re = ∣∣∣∣∣ baa − bapbap ∣∣∣∣∣ × 100 (1) s s e = ∑ (bap − baa) 2 , (2) where (ba)p is the estimated value of the binding affinity in kcal/mol by ann model, (ba)a is the experimental value of the binding affinity in kcal/mol. table 3: chain lengths and binding affinity of nps chains at the phl7 active site np oligomers chain length (nm) binding affinity (kcal/mol) np1 1.09 5.4 np2 2.18 5.2 np3 3.27 5.1 np4 4.36 3.8 np5 5.45 0.8 np6 6.54 23.2 np7 7.63 38.5 np8 8.72 51.4 np9 9.81 137.8 np10 10.90 139.8 figure 2: plot of pet chain length against binding affinity at phl7 active site 4 c. e. duru et al. / j. nig. soc. phys. sci. 5 (2023) 1154 5 figure 3: fitting of np oligomers at the enzyme binding pocket (a) np1 (b) np2 (c) np3 (d) np4 (e) np5 (f) np6 figure 4: anns for predicting pet nps degradation based on its propertiesenergy, molecular mass (mm), and chain length (cl) figure 5: linear correlation of predicted binding affinity and actual values from molecular docking 5 c. e. duru et al. / j. nig. soc. phys. sci. 5 (2023) 1154 6 figure 6: the most important property for nps degradation by phl7 from the ann prediction 3. results and discussion polyethylene terephthalate is a semi-crystalline polymer produced from the reaction of terephthalic acid and ethylene glycol. each monomer unit (figure 1) has a physical length of about 1.09 nm and a molecular weight of ≈ 200 [27]. it is a thermoplastic material with excellent chemical resistance, melt mobility, and spinnability. the bacterial strain ideonellasakaiensis 201-f6 was recently found to exhibit a rare ability to grow on pet as a major carbon and energy source [28]. a novel enzyme, polyester hydrolase leipzig 7 (phl7), isolated from a compost metagenome, can completely hydrolyze amorphous pet films within hours has been freshly reported by sonnendecker and coworkers [19]. we have shown in this study the binding affinities of modeled pet np oligomers at the phl7 active site, and the results are given in table 3. the pet monomer, which served as control in this study, had a binding affinity of 5.4 kcal/mol at the enzyme target and was closely followed by np2 and np3 with binding affinities of -5.2 kcal/mol and 5.1 kcal/mol respectively. these results suggested that the degradation of pet at this enzyme target would be most efficient at 2.18 nm and 3.27 nm chain lengths. the drop in the binding affinity from -3.8 kcal/mol for np4 to – 0.8 kcal/mol for np5 indicated a reduction in the hydrolyzing ability of the enzyme as the chain length increased from 4.36 nm to 5.45 nm. positive binding affinity values, which increased steadily, were obtained with np6-np10 oligomer units. this implied that the hydrolysis of pet within the range of 6.54 nm chain length and above was not feasible with this enzyme. therefore, pet polymer materials should be within 5.45 nm and below for efficient binding and hydrolysis at the phl7 active site. a plot of the binding affinity of pet np oligomers as a function of np chain length is shown in figure 2. as chain length increased, three notable degradation characteristics of pet at the enzyme target were observed. there is a reactive stage at np2-np5 where hydrolysis was feasible, an intermediate nonreactive stage at np6-np8 where enzyme action is highly limited, and a recalcitrant stage at np8 and above where total recalcitrance of the np was manifest. the 3d views of enzyme-np interactions showed that the pet trimer (np3) is the maximum chain unit that can fit perfectly in the enzyme binding cavity (figure 3). this fitting becomes increasingly difficult from np5 and above. the recalcitrant nature of pet polymer and the difficulty in its hydrolysis could therefore be reduced by subdividing it to less than 5.45 nm chain lengths which have good fitting in the enzyme pocket. the ann for the degradation of pet nps oligomers by phl7 based on the intrinsic properties of the oligomers is shown in figure 4. the network involved two hidden layers, which processed the input variables. the output results from this network were compared with the actual binding affinity values from the molecular docking by linear regression (figure 5). the results showed that the anns could predict the binding affinity with high accuracy with r2 of 0.985. the ann was further checked using error models, which gave small errors for sse (0.045) and re (0.028). the results indicated that the energy of the oligomers was the most important property for its degradation, with 100 % normalized importance (figure 6). the energy of the oligomers increases with an increase in their sizes. the high energy of the longer oligomers reduced their binding affinity at the active site, which makes their degradation difficult at the enzyme target. efficient degradation would therefore occur at a lower energy of the nps, which is obtainable at chain lengths between np2-np4. 4. conclusion the molecular docking of pet np oligomers ranging from 1.09 nm 10.90 nm at the phl7 active site was performed and their binding affinities were used to determine the effect of chain length on the degradation of pet. binding affinities of the nps decreased from -5.2 kcal/mol to -0.8 kcal/mol, as pet chain length increased from 2.18 nm to 5.45 nm (2-5 pet chains). at chain lengths of 6.54 nm (6 pet chains) and above, the binding of the pet nps on the enzyme was nonspontaneous, as was seen in the resulting positive binding affinity values obtained with these chains. artificial neural network analysis revealed that structural energy is the major determinant factor in the pet degradation process. references [1] j. gigault, a. t. halle, m. baudrimont, p. 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[28] g. j. palm, l. reisky, d. böttcher, h. muller, e. a. p. michels, m. c. walczak, l. berndt m. s. weiss, u. t. bornscheuer & g. weder, “structure of the plastic-degrading ideonella sakaiensis mhetase bound to a substrate”, nature communications 10 (2019) 1717. 7 j. nig. soc. phys. sci. 5 (2023) 1364 journal of the nigerian society of physical sciences an empirical study on anomaly detection using density-based and representative-based clustering algorithms gerard shu fuhnwia,∗, janet o. agbajeb, kayode oshinubic, olumuyiwa james peterd,∗∗ adepartment of computer science, montana state university, montana, usa. bdepartment of mathematical science, montana technological university, montana, usa. cschool of informatics, computing and cyber systems, northern arizona university, arizona, usa. ddepartment of mathematical and computer sciences & department of epidemiology and biostatistics, school of public health, university of medical sciences, ondo, nigeria. abstract in data mining, and statistics, anomaly detection is the process of finding data patterns (outcomes, values, or observations) that deviate from the rest of the other observations or outcomes. anomaly detection is heavily used in solving real-world problems in many application domains, like medicine, finance , cybersecurity, banking, networking, transportation, and military surveillance for enemy activities, but not limited to only these fields. in this paper, we present an empirical study on unsupervised anomaly detection techniques such as density-based spatial clustering of applications with noise (dbscan), (dbscan++) (with uniform initialization, k-center initialization, uniform with approximate neighbor initialization, and k-center with approximate neighbor initialization), and k-means−− algorithms on six benchmark imbalanced data sets. findings from our in-depth empirical study show that k-means−− is more robust than dbscan, and dbscan++, in terms of the different evaluation measures (f1-score, false alarm rate, adjusted rand index, and jaccard coefficient), and running time. we also observe that dbscan performs very well on data sets with fewer number of data points. moreover, the results indicate that the choice of clustering algorithm can significantly impact the performance of anomaly detection and that the performance of different algorithms varies depending on the characteristics of the data. overall, this study provides insights into the strengths and limitations of different clustering algorithms for anomaly detection and can help guide the selection of appropriate algorithms for specific applications. keywords: outliers, noise points, ann, k-means−−, dbscan, dbscan++ article history : received: 22 january 2023 received in revised form: 18 march 2023 accepted for publication: 22 march 2023 published: 19 april 2023 c© 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: b. j. falaye 1. introduction anomaly detection is the process of finding data patterns (outcomes, values, or observations) that deviate from the rest of ∗corresponding author tel. no: +1 931 266 5067 ∗∗corresponding author tel. no: +234803 356 0280 email addresses: gerard.shufuhnwi@student.montana.edu (gerard shu fuhnwi ), peterjames4real@gmail.com (olumuyiwa james peter ) the other observations or outcomes. anomaly detection is heavily used in solving real-world problems in many application domains like medicine, cybersecurity [1], fraud detection [2], networking, transportation, and military surveillance for enemy activities, but not limited to only these fields, as anomaly detection is classified under deep learning which is applicable in all fields such as in mathematics and statistics [3]. these deviating outcomes or observations are referred to as anomalies (outliers, 1 fulafia mis typewriter doi:10.46481/jnsps.2023.1364 fuhnwi et al. / j. nig. soc. phys. sci. 5 (2023) 1364 2 deviants, discordant observations, exceptions, surprises, or abnormalities) in different application domains [4]. anomaly detection algorithms can be categorized into two types: supervised and unsupervised. supervised anomaly detection involves training a model on labeled data, where anomalies are explicitly labeled or defined. on the other hand, unsupervised anomaly detection involves identifying anomalies without using explicit labels or prior knowledge about the data. unsupervised methods are more commonly used in practice, as labeled data for training supervised models may be scarce or difficult to obtain. various techniques can be used for anomaly detection, including statistical methods, clustering, nearest-neighbor methods, and machine learning algorithms such as support vector machines, decision trees, and neural networks [5]. with the increase in the volume and complexity of data, traditional rulebased approaches have proven to be insufficient in detecting anomalies. as a result, clustering-based techniques have gained popularity as reliable methods for identifying anomalies. in this regard, density-based and representative-based clustering algorithms have been widely used in anomaly detection due to their ability to identify clusters of data points dissimilar to the majority of the data. there have been many approaches to solving anomaly detection problems, with the unsupervised algorithms being the most widely usedâ because the techniques involve training the model with unlabeled data. clustering is the process of grouping a set of observations or data points into multiple groups so that observations within a group or cluster have high similarity but dissimilar to observations from the other clusters. clustering-based techniques fall under a class of unsupervised anomaly detection techniques that operate on the output of the clustering algorithm and thus turn out to be much faster in general. the clustering based techniques can be grouped into the following categories: representative-based techniques, densitybased techniques and hierarchical-based techniques. throughout the research community, a lot of work has been done to detect anomalies using clustering-based techniques, see the work of [6-16]. in this paper, we present what is (to the best of our knowledge) the first attempt of an empirical study on anomaly detection using k-means−−, dbscan, and dbscan++ using data sets from different domains with varying proportions of outliers. this paper aims to evaluate the performance of representative-based and density-based clustering algorithms detecting anomalies, their computational efficiency, and the effect of varying algorithm parameters on their performance. our goal is to find out how these methods perform on different data sets with regards to the following evaluation metrics: f1 score, false alarm rate, jaccard coefficient, and adjusted rand index including the run time of these algorithms on the data sets. finally, and most importantly, the above mentioned techniques all have the tendency of finding noise points (anomalies or outliers) and assigning labels to them as noise points. although the main goal is to evaluate the effectiveness of density-based clustering algorithms like dbscan, dbscan++ and representative-based clustering algorithm like k-means−−. our approach can also provide guidance on how to evaluate and analyze these clustering techniques in solving anomaly detection problems. also, this method can be used to overcome one of the main challenges of anomaly detection techniques, which is accurate representative labels for normal and abnormal instances, which is a major concern. to overcome this challenge in most anomaly detection problems, our approach can be used as a pre-labeling technique and then apply supervised anomaly detection techniques to solve anomaly detection problems. overall, our empirical results demonstrate the potential of density-based clustering and representativebased clustering and provide valuable insights for future research in this field. the rest of this paper is structured as follows: in section 2, we briefly give a description of the algorithms used in this paper. section 3, analyses the empirical evaluation, where we review data sets used, evaluation metrics description, variations in evaluation metrics, results, and result discussion. section 4 covers the conclusion and future directions. 2. methods this section presents the anomaly detection techniques used in this paper. these anomaly detection techniques are: kmeans−−, and two versions of density-based spatial clustering of application with noise (dbscan and dbscan++). 2.1. k-means−− k-means−− [17] is a representative-based clustering technique, which is an extension of the k-means algorithm. kmeans−− is a more computationally efficient version of the kmeans algorithm, which achieves this by updating the cluster centroids more incrementally. in the k-means−− algorithm, the centroid of each cluster is initialized as the mean of a randomly selected subset of data points rather than as a randomly selected data point, as in the original k-means algorithm. then, for each iteration of the algorithm, k-means−− updates the centroids by considering only the data points that belong to the cluster being updated rather than all the data points in the data set. this results in faster convergence and improved scalability, particularly for large data sets. the pseudo-code of the k-means−− is shown in algorithm 1. we implemented the k-means−− in python using the pseudo-code in algorithm 1, since the implementation was not available in sklearn. 2.2. density-based spatial clustering of applications with noise (dbscan) dbscan is a density-based clustering technique capable of finding arbitrarily shaped clusters. dbscan proceeds by computing the empirical densities for each sample point and then designating points whose densities are above a threshold as core points. then, a neighborhood graph of the core points is constructed, and the clusters are assigned based on the connected components. the pseudo-code of dbscan [18] is shown in algorithm 2. we used the sklearn implementation of dbscan in python. still, this implementation could not be 2 fuhnwi et al. / j. nig. soc. phys. sci. 5 (2023) 1364 3 figure 1: k-means−− pseudo-code faster with large data sets due to memory consumption, scalability, parameter tuning, noise sensitivity, and difficulty handling high-dimensional data since it uses the kdtree to build the nearest neighbor tree. 2.3. dbscan ++ dbscan++ [19] is an extension to the dbscan, which runs much faster, more efficient, and less sensitive to hyperparameter settings. we couldn’t find any python implementation of the dbscan++, so we used the pseudo-code provided [19] by using the kdtree scipy in the implementation. this implementation didn’t allow us to run it on large data sets due to parameter tuning, noise sensitivity, and difficulty handling high-dimensional data because of the limitation of the kdtree implementation. the pseudo-code is shown in algorithm 3. there were two initialization methods mentioned in this paper. 1. uniform initialization 2. k-center initialization uniform initialization was implemented by uniformly sampling m number of points from the given data set. we only ran kdtree queries for m sampled data points, after running the queries we developed the core point set and then we created the neighbourhood tree by adding edges to points in the radius of the core points. k-center initialization was more complicated than the uniform initialization, we had to implement part of the greedy kcenter clustering algorithm for this initialization. as mentioned in the paper [19], k-center initialization should run faster than dbscan algorithm on the same hyper-parameters, but this did not happen in our implementation, k-center initialization took considerable amount of time even though the time complexity was o(mn). to improve the performance of this initialization, we vectorized the computation and used a slightly better algorithm mentioned in geometric approximation algorithms [20]. we saw a massive improvement in initialization time but still, it took longer than dbscan algorithm. this happened because in both dbscan and dbscan++, we had to build the figure 2: dbscan pseudo-code figure 3: dbscan++ pseudo-code kdtree which takes the same time if the input data set is the same. whereas, in k-center initialization, we had to run the initialization algorithm to pick the m points. although this m number of points are less than the total number of points, the speed up gain from running m number of queries against running queries for all the points does not exceed the time taken to run the k-center initialization. we suspect that if this was implemented in c++, there might be a difference in result. since the paper did not mention about any implementation details we cannot be certain about this. 2.4. dbscan and dbscan++ on approximate nearest neighbour (ann) since we could not run dbscan or dbscan++ on large data sets, we moved on to implementing the approximate nearest neighbour on dbscan and dbscan++ algorithms. we used a python library annoy [21], which wrapped a c++ implementation of approximate nearest neighbour tree using python. we saw a massive speed up in creating the nearest neighbour tree after implementing this. this library did not allow us to query the points given in a radius ball what it allowed us to do was to get the approximate k-nearest neighbours, this posed a challenge for us because we need to query the points given in a ε− radius ball. so what we did was to query 2 ∗ minpts number of points for each core point selection query, and check if there are more than minpts number of points that has less than eps distance to the queried point. this allowed us to reduce the uncertainty of not picking all the points in the ε− radius ball. since the minpts is a small value, going through 2∗minpts was not affecting the performance of the algorithm. 3 fuhnwi et al. / j. nig. soc. phys. sci. 5 (2023) 1364 4 after completing the implementation of these algorithms, we ran experiments on the given data sets. since we are interested in detecting anomalies, we plotted histograms of ground truth labels of each data sets. then we decided what class labels are normal instances (expected behavior or pattern of the system or data being analyzed) and what class labels are noise instances (data points or events that deviate significantly from the expected or normal behavior). some of the data sets already had document explaining what class labels can be identified as normal instances and what class labels are noise labels. generally, if a class label had a less frequency, we picked them as noise labels. after running the algorithms, we modified the ground truth and cluster label arrays to only contain two class labels. 0 if a data point is a normal instance and 1 if data point is a noise instance. we did this because we are only interested in detecting normal and abnormal instances, we no longer care whether we have the right number of clusters as a result. then we ran different assessment metrics on both the ground truth and the labels obtained from these algorithms. explanation of these results are mentioned in the empirical evaluation section. also, we made a small modification to dbscan algorithm hoping to solve the problem of detecting small outlier clusters. what we did was we added a threshold parameter to the dbscan algorithm where it will check the size of clusters before assigning the cluster label, if the size of cluster is smaller than the given threshold, it was marked as a noise cluster. we only made this change to the dbscan on an approximate nearest neighbor (ann) implementation and we tested this on the shuttle data set. this has a very small change, but we got extremely good results for shuttle data set. this part was done as an extension to what we already did. we could not test this algorithm for all the data sets. it was only tested on the shuttle data set because it contained small outlier clusters and it was a large data set. we will explain the results in the discussion section. 3. empirical evaluation 3.1. data sets we perform our experiments on six data sets from uci machine learning repository [22]. the data sets description and distribution of the classes is shown on the figures and table below: we had to change the data sets proposed due to the fact that dbscan was unable to process large data sets. 3.2. evaluation metrics four evaluation metrics were used to assess the validity of the results of this experiments. 1. false alarm rate 2. f-score (weighted) 3. jaccard coefficient 4. adjusted rand index table 1: dbscan results on chosen data sets data set #points #dim #outliers outlier % pima 768 8 268 35 cardio 1831 21 176 9.60 wine 129 13 10 7.70 glass 214 9 9 4.20 breastw 683 9 239 35 shuttle 43500 (36752) 9 2644 7.19 false alarm rate is the ratio of number of incorrectly labelled noise instances that were normal instances in ground truth over total number of noise instances predicted. the fmeasure of a cluster is the harmonic mean of the precision and recall values of a cluster. we took the weighted f-measure values of each cluster as the final f − score. [23] fi = 2 1 preci + 1recalli = 2 ∗ preci ∗ recalli preci + recalli f = k∑ i=1 wi ∗ fi where wi is the weight of the cluster the jaccard coefficient measures the fraction of true positive point pairs, but after ignoring the true negatives. it is defined as follows: [23] jaccard = t p t p + f n + f p before we use these metrics, we converted the ground truth and cluster labels to two classes containing normal and outlier classes. this helped us to focus more on noise prediction results rather than looking at cluster predictions. the adjusted rand index assessment is included but was not use in interpreting the result. 3.3. empirical results presented in the form of tables 3.4. discussion of results the factor parameter in dbscan++ is the number of point that were quarried by the algorithm. l and k parameters in the k-means−− results indicates the outlier parameter and cluster number parameter. far means false alarm rate, ars means adjusted rand score and js means jaccard score. each data set’s result will be described separately and will make conclusions based on the entire results. for each data set, we experimented with various parameters, and we included a range of parameters and their results in the empirical result tables. the density-based algorithms (dbscan and dbscan++) did not perform well on the pima data set. we only included the parameter and best possible result for the pima data set. in most cases, it either recognized all the points as normal instances or outliers. we believe this was as a result of the high percentage of outliers in the pima data set, which 4 fuhnwi et al. / j. nig. soc. phys. sci. 5 (2023) 1364 5 figure 4: breast cancer and cardiotography data sets class distribution figure 5: pima and wine data set class distribution figure 6: glass data set class distribution is 35% as shown in table 1. we could not find the balance between the minpoints and epsilon values that can distinguish the clusters and the noise points. this is because this data set has different densities points at different levels. when we ran kmeans−− on pima data set, we got slightly better results (table 4), and we had a smaller false alarm rate value of 0.4216 as opposed to 0.651 obtained from dbscan algorithm. in the case of dbscan or dbscan++ tests, more than half of the detected noise points are false positive values indicating that pima data set does not consist of a density-based cluster structure. low f-score of 0.185 in both dbscan and dbscan++ tests indicates that quality of instances detected as normal, and outlier cluster are low. jaccard coefficient of 0.349 indicates that false negatives and false positive value pairs are high compared to true positive pairs. when we look at the k-means−− results of pima data set, we have better jaccard coefficient and f1 score 0.4068 and 0.7057 respectively (table 4). even though k-means−− results are better than dbscan and dbscan++ results, the results are not accurate enough. then we employed statistical techniques such as dimensionality reduction (pca) on the pima data set before using the above mentioned algorithms, which resulted in better results. we first normalize the data, and then applied pca on the pima data set. we then ran dbscan and dbscan++ algorithms on the data set after selecting the best 7 components from the results. for the results refer the table a5. we had to increase the min points parameters to 270 − 290 range to get better results. this is because the data set only contains one major cluster and all the other points are considered outliers. there are 268 outliers in the data set, thus we had to bring the minpoints to 270 range to exclude outliers from the result. this resulted in better outputs. we could achieve a false alarm rate of 0.45 at eps = 0.35 with minpts = 270 and f-score of 0.69 on dbscan algorithm. we could achieve similar results for dbscan++ on both initial5 fuhnwi et al. / j. nig. soc. phys. sci. 5 (2023) 1364 6 table 2: dbscan results on chosen data sets data sets parameters evaluation measures min pts epsilon f1 score false alarm adjusted rand index jaccard pima 20 5 0.1805 0.651 0 0.349 cardio 10 3 0.88 0.58 0.31 0.26 10 4 0.88 0.43 0.19 0.14 10 5 0.88 0.15 0.17 0.12 wine 4 25 0.91 0.58 0.46 0.41 4 30 0.95 0.41 0.664 0.58 4 35 0.97 0.23 0.83 0.76 4 38 0.98 0.16 0.88 0.83 4 72 0.95 0.14 0.65 0.54 glass 8 0.9 0.8736 0.8085 0.206 0.1915 8 1 0.8767 0.8043 0.213 0.1957 8 1.2 0.86 0.875 0.11 0.11 8 2 0.9 0.88 0.08 0.08 breastw 10 2.9 0.9508 0.119 0.8098 0.8745 10 4 0.9636 0.072 0.8579 0.9027 10 5 0.82 0.05 0.429 0.54 10 5.5 0.74 0.03 0.26 0.36 20 5 0.9225 0.0483 0.7143 0.7912 table 3: dbscan++ k-center results on data sets data sets parameters evaluation measures min pts factor epsilon f1 score false alarm adjusted rand index jaccard pima 10 0.2 0.9 0.1805 0.651 0 0.349 10 0.35 2 0.6797 0.3396 0.1177 0.4136 20 0.2 5 0.1805 0.651 0 0.349 cardio 10 0.5 5 0.8835 0.1538 0.1788 0.1222 10 0.5 4 0.88 0.42 0.2 0.15 10 0.5 3 0.88 0.59 0.3 0.27 wine 4 0.3 25 0.84 0.72 0.25 0.27 4 0.3 30 0.9 0.6 0.43 0.4 4 0.3 35 0.92 0.54 0.51 0.45 4 0.3 38 0.97 0.28 0.78 0.71 4 0.3 72 0.95 0.14 0.65 0.54 10 2 1.5 0.0112 0.9225 0 0.0775 glass 8 0.5 0.4 0.41 0.94 -0.04 0.05 8 0.5 0.9 0.87 0.8 0.2 0.19 8 0.5 1 0.87 0.8 0.21 0.19 8 0.5 1.2 0.86 0.87 0.11 0.11 8 0.5 2 0.8885 0.8485 0.1499 0.1351 breastw 10 0.5 1.9 0.8812 0.2578 0.5723 0.7422 10 0.5 2.9 0.9509 0.1218 0.8098 0.875 10 0.5 4 0.96 0.07 0.85 0.89 10 0.5 5 0.82 0.05 0.42 0.54 10 0.5 5.5 0.74 0.03 0.26 0.36 20 1 4 0.9607 0.0794 0.847 0.8958 20 1 5 0.9634 0.0542 0.8576 0.9008 izations at 0.5 factor values. then we ran the k-means−− on the data set (dimensionally reduced), we got better results compared to dbscan results. refer the table a4. it shows at k = 1 and l = 268, we get the lowest false alarm rate of 0.45 and highest f1-score of 0.68 for this data set. however, on both algorithms there are considerable amount of false positive noise predictions. and the jaccard coefficients of both algorithms for this data set is low as well, which indicates that there are false 6 fuhnwi et al. / j. nig. soc. phys. sci. 5 (2023) 1364 7 table 4: k means −− results on data set data sets parameters evaluation measures k l iteration epsilon f1 score false alarm adjusted rand index jaccard pima 2 268 10 0.2 0.7057 0.4216 0.1614 0.4068 cardio 2 176 10 0.05 0.9312 0.358 0.5536 0.4728 2 176 20 0.05 0.9345 0.3409 0.5734 0.4915 3 176 20 0.05 0.9421 0.3011 0.6201 0.5371 wine 2 10 10 0.2 0.9845 0.1 0.8753 0.8182 2 10 10 0.05 0.9845 0.1 0.8753 0.8182 glass 2 9 10 0.05 0.9252 0.8889 0.0658 0.0588 breastw 2 239 10 0.2 0.9444 0.0795 0.7879 0.8527 2 239 10 0.05 0.9239 0.1088 0.716 0.8038 2 239 10 0.35 0.9444 0.0795 0.7879 0.8527 3 239 10 0.05 0.9209 0.113 0.706 0.797 table 5: dbscan result on ann using shuttle data set data set parameters evaluation measures minpts eps factor f-score false alarm rate ars jaccard score shuttle 10 4.5 1 0.88758301 0.775670841 0.146802475 0.130299252 10 4.8 1 0.89255751 0.757217848 0.149587292 0.126857143 10 5 1 0.89574317 0.740279938 0.150095945 0.123035363 10 5.3 1 0.90068313 0.695989651 0.162924769 0.126344086 10 5.5 1 0.90195679 0.679245283 0.162242457 0.123463687 10 5.8 1 0.90298582 0.659681475 0.158728371 0.118332848 10 6 1 0.90418584 0.628742515 0.156338474 0.11362248 10 6.8 1 0.90566123 0.566360053 0.153339959 0.107317073 10 7 1 0.90567793 0.550143266 0.149568631 0.103698811 10 9 1 0.90520426 0.504621072 0.135841986 0.091875214 10 10 1 0.90441125 0.507936508 0.126851763 0.085517241 10 28 1 0.89643437 0.655462185 0.041910848 0.029285714 10 28.5 1 0.89647958 0.65106383 0.042092565 0.029317125 table 6: result of shuttle data set on k−means −− data set parameters evaluation measures k outliers iterations f1 score far ars js time shuttle 1 2500 50 0.96890 0.08634 0.72703 0.62157 55 1 2644 50 0.97034 0.09082 0.74045 0.63810 45 1 2700 50 0.97089 0.09704 0.74599 0.64520 35 2 2500 50 0.97169 0.07394 0.75167 0.65113 149 2 2644 50 0.97226 0.08608 0.75763 0.65910 128 2 2700 50 0.96234 0.25512 0.68565 0.58323 56 negative pairs in the result. slightly higher f-scores indicates that quality of the clustering is much better. the second data set we experimented on was the cardio data set, which also performed well on k-means−− algorithm. this data set contains about 9.60% outliers. on dbscan, cardio data set gives about 0.8 − 0.9 f-score, which means that the quality of clusters is high. note that we used weighted fscore. false alarm rate of both dbscan and dbscan++ algorithms were around 0.15, indicating a low false positive noise points in the predicted labels. jaccard coefficient results of the dbscan algorithm produced less-than-ideal results, with 0.12 where dbscan++ gave around 0.15. this happened because of high false negative value pairs. we know that we have low number of fp due to lower false alarm rate, so we can conclude that we have low jaccard coefficient because of false negative pairs, which means dbscan could not identify when two points are in different groups in the data set. it should be noted that dbscan++ has a higher f-score and a lower false alarm rate just like dbscan. since we took the weighted f-score and there are higher number of normal instances in the cardio data set, we can conclude that normal point prediction accuracy is high. we can also conclude that true positives of predicting 7 fuhnwi et al. / j. nig. soc. phys. sci. 5 (2023) 1364 8 table 7: result of shuttle data set using modified dbscan on ann data set parameters evaluation measures minpts eps factor threshold f1 score far ars js #noise pts shuttle 10 4.5 1 0.2 0.61831 0.87032 -0.00768 0.12968 27136 10 4.8 1 0.2 0.93744 0.50524 0.56876 0.49476 9603 10 5 1 0.2 0.96127 0.37524 0.70772 0.62476 8210 10 5.3 1 0.2 0.96998 0.31289 0.76610 0.68711 7797 10 5.5 1 0.2 0.97280 0.29058 0.78586 0.70942 7660 10 5.8 1 0.2 0.97900 0.23716 0.83098 0.76284 7392 10 6 1 0.2 0.98369 0.19243 0.86654 0.80757 6875 10 6.8 1 0.2 0.98868 0.14016 0.90574 0.85984 6648 10 7 1 0.2 0.98988 0.12682 0.91537 0.87318 6584 10 9 1 0.2 0.99274 0.09359 0.93872 0.90641 6444 10 10 1 0.2 0.99313 0.08834 0.94191 0.91103 6414 10 28 1 0.2 0.89643 0.65546 0.04191 0.02929 263 10 28.5 1 0.2 0.89648 0.65106 0.04209 0.02932 260 normal instances as normal instances is high with this result. k-means−− result of the cardio data set has best f-score of 0.94 and 0.30 false alarm rate (lowest of k-means−− tests for cardio data set) and jaccard coefficient of 0.5 (table 4). f-score and jaccard coefficients are better than the density-based results, although we had higher false alarm rate than the density level results, which indicate that out of the noise points predicted, there were high number of false positives. however, the jaccard coefficient was high for this test, indicating that false negatives pairs are low in the k-means−− result. as a result of these high f-score values, we can conclude that if the quality of two clusters are high, then true positive numbers should be high as well. then this higher jaccard coefficient should have come from the low false negative pairs. this indicates that k-means−− was good at identifying pairs of points that are in different clusters but it was not good at identifying some normal instances as normal instances. our third data set was the wine data set. both densitybased and representative-based algorithms performed well on this data set. dbscan and dbscan++ algorithms gave best f-scores around 0.98 and best false alarm rates around 0.14 and best jaccard coefficients of 0.83 (table 2, table 3 and table a1). note that the best jaccard coefficient came from uniform initialization of dbscan++. the best value for the dbscan++ k-center was 0.71. k-means−− algorithm gave the best results for this data set in terms of all the assessment metrics. f-score of 0.98, false alarm rate of 0.14 and jaccard coefficient of 0.81 (table 4). glass data set was our fourth data set. on both types of algorithms (representative-based and the density-based), they were able to identify the noise instances as noise instances but there were lot of false positives. both algorithms types gave more than 0.8 f-score, indicating that the clustering is of high quality. but both types of algorithms had very high false alarm rates, which means algorithms classified normal instances as noise instances. both result types had low jaccard coefficients as well, which occurred due to high false positives and false negatives. this occurred due to classifying pairs of normal instances in two different clusters. this shows that both algorithms could not identify the anomalies correctly. the fifth data set is the breastw data set, which has around 35% of outliers. both types of algorithms performed very well on this data set. both had very low false alarm rates and high fscores, and jaccard coefficients, which indicates that algorithms were able to predict the anomalies accurately. breastw data set has a gaussian-based and density-based cluster structure, which helped the algorithms to identify the cluster structures more accurately. however, we could not run our dbscan or dbscan++ implementations on large data sets because of the kdtree limitations. as a result, we used an approximate nearest neighbour library to query the nearest neighbours. we tested this on shuttle data set, which has 43500 data points. there are 7 ground truth class labels in the data set. class label 1 has the highest frequency, all the other classes has lower frequencies compared the class 1. we removed data with class label 4 and considered all other classes except class 1 as outliers. the important thing about this data set is that its outliers are in small clusters. for example, class 2, 3, 5, 6, 7 are outlier classes. if those outlier classes have different densities, such as lower densities, density-based algorithms cannot detect those outliers by tweaking the minpoint and epsilon parameters. please refer table 5. with eps = 9, we had the lowest false alarm rate, and then it increases. this is because outliers form small clusters and density-based algorithms cannot find it by tweaking the parameters due to breaking of cluster structure. out of dbscan and dbscan++ algorithms, dbscan on ann performed well, this is because we are only querying a part of data points to find the core points. thus, some core points that are identified in the dbscan are no longer identified as a core point, thus we would get a higher false positive noise points, which is why we get high false alarm rate for the uniform and k-center initialization (refer tables a2 and a3). we also ran this on k-means−− algorithm, and it gave us excellent results on this data set. not only it took less time, but the resultant clusters were of higher quality. in table 6, we have very low false alarm rates and high 8 fuhnwi et al. / j. nig. soc. phys. sci. 5 (2023) 1364 9 f-score. we even changed the input k value to the algorithm and checked the results, even if we input a higher cluster value, we still get the correct numbers of outliers, with good results. we even changed the range of outlier numbers input to the parameter and algorithm seem robust even if we slightly increase or decrease the number of outliers. next, we modified the dbscan on ann algorithm as mentioned in 2.4 then we ran the algorithm on shuttle data set. we got extremely good results (refer table 7). we changed the threshold to 0.2 because the shuttle data set only has 1 class and it takes about 80% of the data. we ran the experiments on a wide range of eps values from 4.5 to 28.5. the results we obtained were better. we got best f-score of 0.99 and false alarm rate of 0.88, and jaccard score of 0.91. this accuracy is better than the k-means results. the best thing about this is that we only need to know the percentage of outliers in the data set. we do not need the number of clusters in the data to get a better result. however, we ran this on normalized and dimensionality reduced pima data set, hoping to see better results, but we did not obtain better results, but they were very close to dbscan results. thus, we did not include the results in this paper. one of the interesting observation that was identified in the results is that k-center initialization of dbscan takes more time to run than the normal dbscan instance on the same parameters, even for a small factor value. this contradicts with the results shown in [19], where they showed that k-center initialization runs faster than normal dbscan. however, we did not see this in our results. it seems that speed up gained from running fewer kdtree queries does not compensate the time that it takes to initialize k-center points. we also implemented a slightly better k-center initialization algorithm mentioned in [20] and improved the calculations by vectorizing. still, time taken to run the dbscan++ on k-center is greater than dbscan on same parameters. however, dbscan++ on uniform initialization ran faster than all the other algorithms. 4. conclusions and future directions from the results obtained from these experiments, we can conclude that k-means−− is a more robust algorithm than dbscan or dbscan++ algorithms in terms of time and performance, especially when data sets have small outlier clusters with different densities. density-based algorithms struggle to find the outliers, especially if the small outlier clusters have higher densities than the normal clusters, it becomes challenging to tweak the dbscan hyper-parameters. k-means−− algorithm seems more robust in this case; however, we need to know the number of clusters and outlier percentage beforehand to get better results. nonetheless, k-means−− has shown to be robust to slight changes in input parameters. refer tables a4 and 6. the modification of dbscan algorithm with approximate nearest neighbour implementation worked very well in terms of time. we could also improve the k-center initialization a little bit more by paralleling the k-center initialization, which can be a good future direction in terms of improving the running time. although k-means−− is robust, we can create synthetic data sets that would not work very well on k-means−− by adding non-gaussian-shaped clusters and adding noise points. the problem with density-based algorithms to find noise points is that it is hard for the density-based algorithms to identify small outlier clusters, but we can change this by modifying the dbscan algorithm. we need to add another parameter (say “t”) that will act as a threshold for determining a small outlier cluster. at the end of the dbscan algorithm, when we go through the connected components, we need to check the number of nodes in these connected components, if the fraction of number of nodes in these connected components is less than this threshold, we can identify these nodes as a outlier cluster. by making this modification, we can overcome this weakness in density-based algorithms. we already made this change and tested on a data set with oultlier clusters which resulted in extremely good results. however, we could not run this modified algorithm on all the data sets because of time limitations, we believe testing this modified algorithm will be a good future direction. the weakness of k-means−− is that we need to have an understanding about the cluster structure to get accurate result, but we believe by making this change to dbscan we could have a robust algorithm than the k-means−−. another proposed change will be to run in polynomial time given that we only have to go through the connected components to find the size of it; if we improve this graph data structure we should be able to this in constant time. these are some good future directions that we can use. we still believe density-based algorithm should be more powerful than representative-based methods, but we need to make some modifications to these algorithms to make it better. and also find how these algorithms can perform against unsupervised learning (like isolation forest) [24] and semi-supervised learning (one-class support vector) [25] based outlier detection methods as well. also, we should look at how k-nearest neighbour-based methods perform against these algorithms. another future work area would be to find how we can use time series data on density-based algorithm to find the outliers. acknowledgments we thank the reviewers for the positive enlightenment, comments, and suggestions, which have greatly helped us in making improvements to this paper. references [1] s. m. shagari, d. gabi, n. m. dankolo & n. n. gana, “countermeasure to structured query language injection attack for web applications using hybrid logistic regression technique“, journal of the nigerian society of physical sciences 4 (2022) 832. https://doi.org/10.46481/ jnsps.2022.832 [2] c. l. udeze & i. e. eteng & a. e. ibor, “application of machine 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[25] l. m. manevitz & m. yousef, “one-class svms for document classification”,journal of machine learning research 2 (2011) 139. 10 fuhnwi et al. / j. nig. soc. phys. sci. 5 (2023) 1364 11 table a1: dbscan++ uniform results on the dataset dataset parameters evaluation measures min pts factor epsilon f1 score false alarm adjusted rand index jaccard pima 10 0.2 0.9 0.1805 0.651 0 0.349 10 0.35 2 0.6797 0.3396 0.1177 0.4136 20 0.2 5 0.1805 0.651 0 0.349 cardio 10 0.5 3 0.88 0.6 0.3 0.27 10 0.5 4 0.88 0.46 0.2 0.15 10 0.5 5 0.88 0.17 0.18 0.12 wine 4 0.3 25 0.78 0.78 0.14 0.21 4 0.3 30 0.9 0.6 0.43 0.4 4 0.3 35 0.89 0.61 0.41 0.38 4 0.3 38 0.97 0.23 0.83 0.76 4 0.3 72 0.98 0.16 0.88 0.83 glass 8 0.5 0.4 0.55 0.93 -0.02 0.06 8 0.5 0.9 0.83 0.85 0.13 0.15 8 0.5 1.2 0.87 0.8 0.21 0.19 8 0.5 2 0.86 0.87 0.11 0.11 breastw 10 0.5 2.9 0.94 0.12 0.79 0.84 10 0.5 4 0.96 0.07 0.84 0.89 10 0.5 5 0.84 0.06 0.49 0.6 10 0.5 5.5 0.77 0.05 0.33 0.44 20 1 4 0.9607 0.0794 0.847 0.8958 20 1 5 0.9634 0.0542 0.8576 0.9008 table a2: result of shuttle data set on dbscan++ on ann with uniform initialization dataset parameters evaluation measures minpts eps factor f-score far ars js #noise recnzd shuttle 10 4.5 0.1 0.7798 0.87893 0.05142 0.10694 13234 10 4.8 0.1 0.7947 0.87076 0.06308 0.11277 12169 10 5 0.1 0.7961 0.87803 0.05530 0.10488 11582 10 5.3 0.1 0.8028 0.88439 0.04908 0.09699 10638 10 5.5 0.1 0.8134 0.86978 0.06712 0.10918 10154 10 5.8 0.1 0.8186 0.87152 0.06558 0.10584 9673 10 6 0.1 0.8188 0.87662 0.05956 0.10065 9395 10 6.8 0.1 0.8180 0.89827 0.03413 0.08005 8597 10 7 0.1 0.8261 0.87291 0.06435 0.10194 8553 10 9 0.1 0.8272 0.89449 0.03868 0.08094 7713 10 10 0.1 0.8285 0.90223 0.02960 0.07354 7398 10 28 0.1 0.8226 0.92128 0.00768 0.05852 7255 10 28.5 0.1 0.8203 0.92319 0.00551 0.05739 7575 11 fulafia mis typewriter appendix a. fuhnwi et al. / j. nig. soc. phys. sci. 5 (2023) 1364 12 table a3: result of shuttle data set on dbscan++ on ann with kcenter initialization dataset parameters evaluation measures minpts eps factor f-score far rs js #noise pts rcgzd shuttle 10 4.5 0.1 0.63363 0.92146 0.00109 0.07327 20717 10 4.8 0.1 0.66854 0.92519 0.00127 0.06839 18291 10 5 0.1 0.68213 0.93147 -0.00188 0.06170 16989 10 5.3 0.1 0.70901 0.93335 -0.00383 0.05880 14934 10 5.5 0.1 0.72200 0.93444 -0.00510 0.05716 13929 10 5.8 0.1 0.73162 0.93805 -0.00852 0.05329 13025 10 6 0.1 0.73937 0.94119 -0.01173 0.04999 12320 10 6.8 0.1 0.75174 0.95081 -0.02170 0.04069 11051 10 7 0.1 0.75317 0.95335 -0.02429 0.03839 10832 10 9 0.1 0.75998 0.96095 -0.03254 0.03151 10029 10 10 0.1 0.76010 0.96276 -0.03435 0.02998 9941 10 28 0.1 0.75842 0.98063 -0.05189 0.01525 9609 10 28.5 0.1 0.75853 0.98061 -0.05190 0.01526 9603 table a4: result of pima data set on k-means−− after running pca and selecting 7 components dataset parameters evaluation measures k lterations l (outliers) f1 score far ars js type pima 1 50 250 0.6559 0.4880 0.0899 0.3282 k-means−− 1 50 268 0.6823 0.4552 0.1247 0.3744 k-means−− 1 50 275 0.6923 0.4436 0.1393 0.3923 k-means−− 2 50 250 0.6743 0.4600 0.1147 0.3525 k-means−− 2 50 268 0.6979 0.4328 0.1487 0.3958 k-means−− 2 50 275 0.6534 0.4982 0.0851 0.3407 k-means−− 3 50 250 0.7163 0.3960 0.1815 0.4114 k-means−− 3 50 268 0.6823 0.4552 0.1247 0.3744 k-means−− 3 50 275 0.6689 0.4764 0.1053 0.3609 k-means−− 12 fuhnwi et al. / j. nig. soc. phys. sci. 5 (2023) 1364 13 table a5: result of dbscan and dbscan++ algorithms on pima dataset after running pca with 7 components dataset parameters evaluation measures minpts eps factor f-score far rs js type pima 270 0.5 1 0.5377 0.5806 0.0068 0.0455 dbscan 270 0.4 1 0.6566 0.4425 0.1001 0.2812 dbscan 270 0.35 1 0.6901 0.4554 0.1342 0.4064 dbscan 270 0.3 1 0.1805 0.6510 0.0000 0.3490 dbscan 270 0.2 1 0.1805 0.6510 0.0000 0.3490 dbscan 270 0.1 1 0.1805 0.6510 0.0000 0.3490 dbscan 280 0.5 1 0.5430 0.5455 0.0110 0.0524 dbscan 280 0.4 1 0.6617 0.4350 0.1066 0.2899 dbscan 280 0.35 1 0.6973 0.4532 0.1441 0.4330 dbscan 280 0.3 1 0.1805 0.6510 0.0000 0.3490 dbscan 280 0.2 1 0.1805 0.6510 0.0000 0.3490 dbscan 280 0.1 1 0.1805 0.6510 0.0000 0.3490 dbscan 290 0.5 1 0.5456 0.5294 0.0131 0.0559 dbscan 290 0.4 1 0.6684 0.4301 0.1145 0.3046 dbscan 290 0.35 1 0.6821 0.4767 0.1207 0.4469 dbscan 290 0.3 1 0.1805 0.6510 0.0000 0.3490 dbscan 290 0.2 1 0.1805 0.6510 0.0000 0.3490 dbscan 290 0.1 1 0.1805 0.6510 0.0000 0.3490 dbscan 270 0.5 0.5 0.5433 0.5676 0.0095 0.0554 initialization.uniform 270 0.4 0.5 0.6613 0.4513 0.1030 0.3006 initialization.uniform 270 0.35 0.5 0.6937 0.4540 0.1391 0.4185 initialization.uniform 270 0.3 0.5 0.1805 0.6510 0.0000 0.3490 initialization.uniform 270 0.2 0.5 0.1805 0.6510 0.0000 0.3490 initialization.uniform 270 0.1 0.5 0.1805 0.6510 0.0000 0.3490 initialization.uniform 280 0.5 0.5 0.5692 0.4348 0.0323 0.0903 initialization.uniform 280 0.4 0.5 0.6692 0.4439 0.1126 0.3175 initialization.uniform 280 0.35 0.5 0.6859 0.4724 0.1262 0.4487 initialization.uniform 280 0.3 0.5 0.1805 0.6510 0.0000 0.3490 initialization.uniform 280 0.2 0.5 0.1805 0.6510 0.0000 0.3490 initialization.uniform 280 0.1 0.5 0.1805 0.6510 0.0000 0.3490 initialization.uniform 290 0.5 0.5 0.5451 0.5641 0.0104 0.0586 initialization.uniform 290 0.4 0.5 0.6770 0.4242 0.1250 0.3239 initialization.uniform 290 0.35 0.5 0.6833 0.4755 0.1226 0.4491 initialization.uniform 290 0.3 0.5 0.1805 0.6510 0.0000 0.3490 initialization.uniform 290 0.2 0.5 0.1805 0.6510 0.0000 0.3490 initialization.uniform 290 0.1 0.5 0.1805 0.6510 0.0000 0.3490 initialization.uniform 270 0.5 0.5 0.5458 0.5526 0.0116 0.0588 initialization.kcentre 270 0.4 0.5 0.6613 0.4513 0.1030 0.3006 initialization.kcentre 270 0.35 0.5 0.6821 0.4762 0.1207 0.4420 initialization.kcentre 270 0.3 0.5 0.1805 0.6510 0.0000 0.3490 initialization.kcentre 270 0.2 0.5 0.1805 0.6510 0.0000 0.3490 initialization.kcentre 270 0.1 0.5 0.1805 0.6510 0.0000 0.3490 initialization.kcentre 280 0.5 0.5 0.5468 0.5610 0.0113 0.0619 initialization.kcentre 280 0.4 0.5 0.6618 0.4518 0.1034 0.3025 initialization.kcentre 280 0.35 0.5 0.6183 0.5280 0.0469 0.4359 initialization.kcentre 280 0.3 0.5 0.1805 0.6510 0.0000 0.3490 initialization.kcentre 280 0.2 0.5 0.1805 0.6510 0.0000 0.3490 initialization.kcentre 280 0.1 0.5 0.1805 0.6510 0.0000 0.3490 initialization.kcentre 290 0.5 0.5 0.5468 0.5610 0.0113 0.0619 initialization.kcentre 290 0.4 0.5 0.6612 0.4550 0.1021 0.3036 initialization.kcentre 290 0.35 0.5 0.1805 0.6510 0.0000 0.3490 initialization.kcentre 290 0.3 0.5 0.1805 0.6510 0.0000 0.3490 initialization.kcentre 290 0.2 0.5 0.1805 0.6510 0.0000 0.3490 initialization.kcentre 290 0.1 0.5 0.1805 0.6510 0.0000 0.3490 initialization.kcentre 13 j. nig. soc. phys. sci. 3 (2021) 154–158 journal of the nigerian society of physical sciences virtual screening of selected natural products as human tyrosinase-related protein 1 blocker chidi edbert durua,∗, ijeoma akunna durub, chiagoziem wisdom chidieberea asurface chemistry and environmental technology (scent) research unit, department of chemistry, imo state university, owerri, imo state, nigeria bdepartment of chemistry, federal university of technology owerri, imo state nigeria. abstract many researchers have widely explored the need to replace the harmful compound hydroquinone in skin-lightening creams with more skin-friendly compounds that can give similar results. some compounds from the plant kingdom have been shown to possess human tyrosinase inhibitory action with no adverse effect on the skin. in this study, the virtual screening of glabridin, kojic acid, arbutin, niacinamide, ascorbic acid, salicin, lactic acid, glutathione, azelaic acid, linoleic acid, glycolic acid, acclaimed to possess this activity as well as the synthetic compound hydroquinone, as human tyrosinase-related protein 1 inhibitor was investigated using computational methods. site-directed docking was performed at the binding pocket on the enzyme carrying the cocrystallized ligand tropolone. the binding affinity of salicin (−6.7 kcal/mol), α-arbutin (−6.3 kcal/mol), glutathione (−6.2 kcal/mol), ascorbic acid (−5.7 kcal/mol), and niacinamide (−5.7 kcal/mol) were higher than that of the cocrystallized ligand tropolone (−5.5 kcal/mol) and the synthetic skin lightening compound hydroquinone (−4.8 kcal/mol). α-arbutin and glutathione also interacted with similar amino acids units as hydroquinone, suggesting that they followed the exact mechanism of action. these findings strongly corroborate the claim that these natural products could inhibit melanin production and may serve to replace hydroquinone in skin lightening creams. doi:10.46481/jnsps.2021.253 keywords: human tyrosinase-related protein 1; skin lightening; hydroquinone; tropolone; salicin. article history : received: 16 june 2021 received in revised form: 2 july 2021 accepted for publication: 10 july 2021 published: 29 august 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: e. etim 1. introduction the pigment melanin majorly impacts the colour of the human skin, hair, and eye. skin bleaching also known as skin whitening, is applying chemical substances to the skin to make the skin lighter by altering the nature of melanin concentration in the skin [1]. it can also be regarded as the gradual change of the human skin from dark to fair by applying soaps, herbs, chemicals, fade creams, etc., which are strong enough to slow down the function of melanin [2]. between 25-80% ∗corresponding author tel. no: +234(0)8037131739 email address: chidiedbertduru@gmail.com (chidi edbert duru ) of asians and africans use skin-lightening products to change their skin colour. about 75% of nigerian women and between 52-67% of senegalese women apply skin bleaching products [3]. a survey conducted in pretoria, south africa, reported that 35% of women in this area apply these products [4]. the three melanogenic enzymes, tyrosinase (tyr), tyrosinase-related proteins (tyrp1), and tyrosinase-related proteins (tyrp2), are required for the biosynthesis of melanin [5]. difficulties with producing these enzymes in pure form have hampered the understanding of their activity and the effect of mutations that cause albinism and pigmentation disorders [6]. studies suggest that the tyrp1 enzyme may be responsible for stabilizing 154 duru et al. / j. nig. soc. phys. sci. 3 (2021) 154–158 155 tyrosinase and determining the shape of melanosomes, which are the structures in melanocytes where melanin is produced. melanin helps to absorb the uv radiations from the sun and make it fit for people living in the tropical climate of africa. many skin lightening products contain hydroquinone (about 2%), which inhibits melanin production in the skin. studies have shown that this compound has adverse effects such as cancer of melanocytes (melanin cells), contact dermatitis, skin irritation, and exogenous ochronosis mostly rampant in dark-skinned people [7, 8, 9]. exogenous ochronosis results when the skin is exposed to sun rays over a period of time causing an irregular blue black staining on the skin and nails. it affects the melanin in the skin by inhibiting the polymerization of amino acid tyrosine through oxidation [10]. a resolution to ensure the regulation of the formulation and distribution of beauty products, especially bleaching creams, was recently passed by senators in the federal republic of nigerian [11]. this action was taken to protect nigerians against the numerous harmful skin whitening cream and soap formulations sold in the market. several tyrosinase inhibitors have been tested in cosmetics and pharmaceuticals for preventing excess production of melanin in epidermal layers. natural products like glabridin, kojic acid, arbutin, niacinamide, ascorbic acid, salicin, lactic acid, glutathione, azelaic acid, linoleic acid, and glycolic acid have been reported to inhibit the action of melanin in the skin [12]. these studies posited that these compounds effectively inhibit melanin formation without any associated cytotoxicity to melanin cells. reports on applying computational techniques to validate the integrity of these findings are scarce in literature. in the present study, the inhibitory potentials of these natural products on the human tyrosinase-related protein 1 (tyrp1) were studied in silico as a validation of these claims. their binding affinity on this enzyme was determined and compared with hydroquinone in the bid to understand their mode of action and identify possible candidates that could replace this synthetic compound in skin lightening products. 2. computational methods 2.1. identification and preparation of ligands the 3d structure-data files (sdf) of the selected natural products and hydroquinone were identified and downloaded from the pubchem database. they were minimized in pyrx virtual screening tool, using universal force field at 200 steps followed by their conversion to autodock ligands (pdbqt). these files were then used for the docking analysis. 2.2. receptor preparation the chain a of the human tyrosinase-related protein 1 (pdb id: 5m8o) with resolution 2.50 å was identified from literature and used as a target for the study. the interfering crystallographic water molecules and cocrystallized ligand were removed, and minimization of the energy of the protein was then done using ucsf chimera 1.14 [13, 14, 15]. the protein was minimized at 300 steepest descent steps at 0.02 å. the conjugate gradient steps were ten at 0.02 å and ten update intervals. gasteiger charges were also added using dock prep to get a good structure conformation [16]. figure 1. crystal structure of human tyrosinase-related protein 1 (trp1) showing binding site 3. validation of the docking protocol the docking protocol was validated to determine the accuracy and reliability of the docking results. it aimed to accurately reproduce the binding pore and the molecular interactions of the cocrystallized ligand in the protein structure. the native ligand of the x-ray protein was downloaded from the pubchem database and minimized in pyrx virtual screening tool. the ligand was then docked into chain a of tyrp1’s active site using auto dock vina in pyrx. the docked complex was superimposed with the x-ray resolved crystal tyrp1 downloaded from pdb bearing the cocrystallized ligand to generate the root mean square deviation (rmsd) value in pymol. the rmsd value ranging from (0-2) å is appropriate for docking and indicates that the protocol could be used to determine the inhibition of the protein by the other small molecules [17]. 3.1. docking studies the multiple ligand docking of the compounds on tyrp1target was done with autodock vina in pyrx software version 0.8 [18, 19]. site-directed docking was performed at the binding site of the cocrystallized ligand tropolone. the center grid box was set to the dimension center x : −10.018, center y : −1.078, center z : −23.140, and size x : 19.309, size y : 21.549, size z : 19.534. the binding affinity of the compounds was determined in terms of their binding free energy values (∆g). 3.2. analysis of protein-ligand interactions hydrogen bonding and other hydrophobic interactions between the protein-ligand complex of the compounds were visualized using biovia discovery studio 4.5. 155 duru et al. / j. nig. soc. phys. sci. 3 (2021) 154–158 156 table 1. binding affinity values of the natural products on the human tyrp1 enzyme compound pubchem cid structure ∆g (kcal/mol) glabridin 124052 -3.6 kojic acid 3840 -5.5 α-arbutin 158637 -6.3 niacinamide 936 -5.7 ascorbic acid 54670067 -5.7 salicin 439503 -6.7 lactic acid 612 -4.1 azelaic acid 2266 -5.0 glutathione 124886 -6.2 glycolic acid 757 -3.8 linoleic acid 5280450 -5.3 hydroquinone 785 -4.8 tropolone 10789 -5.5 4. results and discussion the docking protocol validation was carried out to assure the deployed docking tools accurately give correct binding interactions between the receptor and the natural products investigated in this study. the docked complex reproduced the original pose of the native ligand (tropolone) with an rmsd value of 1.105 å. the binding affinity values of the natural products on the human tyrp1 are summarized in table 3. the binding affinity of salicin (−6.7 kcal/mol), α-arbutin (−6.3 kcal/mol), glutathione (−6.2 kcal/mol), ascorbic acid (−5.7 kcal/mol), and niacinamide (−5.7 kcal/mol) were higher than table 2. binding interactions of the natural products on the human tyrp1 enzyme compound protein-ligand interactions number of hydrogen bonds interacting residues tropolone 1 his381; thr391; ser394 hydroquinone 1 his381; thr391 salicin 4 arg321; arg374; asn378; gln390; thr391 that of the cocrystallized ligand tropolone (−5.5 kcal/mol) and the synthetic lightening compound hydroquinone (−4.8 kcal/mol). salicin is an alcoholic β-glucoside usually extracted from willow bark. it has been used to treat hyperpigmentation by altering the formation of melanin pigment [20]. α-arbutin is commonly extracted from berries and has been reported to reduce skin pigment production. studies show that α-arbutin is safer to apply on the skin than hydroquinone [21]. glutathione is an antioxidant found in meat and many vegetables like garlic, onion, carrot, potatoes, melon, spinach, etc. studies have shown that glutathione causes skin whitening by direct inhibition of tyrosinase enzymes [22]. ascorbic acid, otherwise known as vitamin c is a powerful antioxidant found mostly in citric fruits. it inhibits tyrosine conversion to melanin by tyrosinase and prevents the damaging of the skin by ultraviolet radiation. it improves the appearance of the skin, thereby reducing the rate of aging [23]. niacinamide, also known as nicotinamide, is a form of vitamin b3 found in meat, fish, nuts, mushrooms, and to a lesser extent in some vegetables. it is a skin-lightening compound that inhibits melanosome transfer from melanocytes to keratinocytes [24, 25]. the interactions of the potent natural product compounds with the human tyrp1 enzyme are shown in table 2. the co-crystallized ligand tropolone associated with his381, thr391, and ser394 forming pi-pi stacked, carbon-hydrogen, and hydrogen bond interactions. salicin (thr391), α-arbutin (his381; thr391), glutathione (his381; thr391; ser394), ascorbic acid (ser394), niacinamide (ser394), and the synthetic com156 duru et al. / j. nig. soc. phys. sci. 3 (2021) 154–158 157 table 2 continued α−arbutin 2 glu216; tyr362; arg374; his381; thr391 glutathione 5 arg374; his381; thr391; ser394 l-ascorbic acid 4 his192; tyr362; his377; asn378; ser394 niacinamide 3 ser394; tyr362 pound hydroquinone (his381; thr391) also interacted with one or more of these amino acids in the enzyme using similar interactive forces. the hydroxyl (−oh) and the carbonyl (c = o) functional groups were the predominant moieties that interacted with the amino acid residues in the enzyme. the binding of αarbutin and glutathione on the human tyrp1 enzyme involved his381 and thr391, which are the two amino acids that bind hydroquinone at this site. this indicated that aside from having a better binding affinity than hydroquinone at this site, they also inhibited melanin production following a similar mechanism as this synthetic compound. the observed number of hydrogen bond interactions between tyrp1 enzyme and salicin (4), αarbutin (2), glutathione (5), ascorbic acid (4), and niacinamide (3) were more than the number in the tropolone (1) and hydroquinone (1) interactions with this enzyme. hydrogen bond interaction between small molecules and protein sites give the molecules better stability at the binding pockets of the proteins [26]. all the natural product inhibitors studied formed more hydrogen bonds at the binding site of tyrp1than hydroquinone and indicated that they were more stable at this site than the synthetic inhibitor. 5. conclusions the in silico validation of the ability of some natural products claimed to possess tyrosinase inhibitory action was performed. salicin, α-arbutin, glutathione, ascorbic acid, and niacinamide gave a higher binding affinity to the human tyrosinase 1 enzyme than the cocrystallized ligand tropolone used as control as well as the synthetic skin lightening compound hydroquinone. α-arbutin and glutathione inhibited melanin production following a similar mechanism as hydroquinone and were also more stable at the enzyme site. these findings implicate these natural products as dermatologically active and skin-friendly ingredients that could be used as replacements for hydroquinone in skin bleaching creams. acknowledgements the authors are grateful to chemsolvers research and computational laboratory studio, owerri, nigeria, for assisting in the in silico study. references [1] e. darj, j.j. infanti, b.m. ahlberg & j. okumu, “the fairer the better? use of potentially toxic skin bleaching products”, afri health sci 15 (2015) 1074. 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[26] c.e. duru, i.a. duru & a.e. adegboyega “in silico identification of compounds from nigella sativa seed oil as potential inhibitors of sars-cov-2 targets”, bull natl res cent 45 (2021) 57. https://doi.org/10.1186/s42269-021-00517-x 158 j. nig. soc. phys. sci. 5 (2023) 1386 journal of the nigerian society of physical sciences experimental and computational studies of the corrosion inhibitive effects of zingiber officinale rhizomes on mild steel corrosion in acidic solutions c. b. adindua,∗, s. c. nwanonenyib, c. b. c. ikpaa adepartment of chemistry, imo state university, p.m.b 2000 owerri, imo state, nigeria bdepartment of polymer and textile engineering, federal university of technology, p.m.b.1526, owerri, nigeria abstract the study investigates the anticorrosion potentials of zingiber officinale (zo) on mild steel induced in 1.0 m hcl and 0.5 m h2so4 acid solution respectively using structural characterization (gas chromatography-mass spectroscopy, gc-ms and fourier transform infrared spectroscopy, ftir) and electrochemical (electrochemical impedance spectroscopy, eis and potentiodynamic polarization, pdp) techniques respectively and theoretical simulations. the structural characterization was performed to identify chemical constituents and functional groups present in the plant extract whereas electrochemical techniques and theoretical computations were used to examine the anticorrosion potentials of the extract and validate the experimental results. the gc-ms result revealed the presence of twenty-three (23) compounds within the extract and out of which three (1-(1,5-dimethyl-4-hexenyl)-4-methyl-, dodecanoic acid and 9-octadecenoic acid (z)-2-hydroxy-1-(hydroxymethyl)ethyl ester) were selected for computational simulation and the results of ftir revealed the presence of the following functional groups (o–h, c=c, c=o, c–c and c–h) in the zo extract. the results of eis revealed that extract of zo exhibited corrosion inhibition efficieny of 82.7% and 93.6 % for mild steel in 1 m hcl and 0.5 m h2so4 solution respectively at maximum inhibitor concentration of 1000 mg/l for mild steel. also, pdp results revealed that zo extract functioned as mixed inhibitor because both the anodic and cathodic reaction process was altered. the quantum chemical calculation results revealed that 9octadecenoic acid (z)-2-hydroxy-1-(hydroxymethyl) ethyl ester had a good energy gap (∆e) compared to other two compounds, indicating its better adsorption interaction with the metal surface in sulfuric acid environment. this was further confirmed by its good adsorption energy of -355.55 kcal/mol with mild steel surface in h2so4 environment compared with -167.81kcal/mol in hcl environment from the molecular dynamic simulation. doi:10.46481/jnsps.2023.1386 keywords: zingiber officinale, solvation molecules, corrosion, molecular dynamic simulation, inhibition efficiency article history : received: 06 february 2023 received in revised form: 19 may 2023 accepted for publication: 12 june 2023 published: 23 august 2023 © 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: k. sakthipandi ∗corresponding author tel. no: +234 7030676855 email address: blessingojiegbe@yahoo.com ( c. b. adindu ) 1. introduction corrosion of component tools and parts made of metallic alloys in different service environments and its control is a vital research area and requires adequate attention due to its utmost importance in industrial and engineering sectors. thus, 1 adindu et al. / j. nig. soc. phys. sci. 5 (2023) 1386 2 purified metals together with their alloys possess some inherent features and exhibit outstanding performances in different service environment to ensure sustainable economy. corrosion of metals is an unpreventable phenomenon caused by unfavourable interactions between the metal substrate and its environment surrounding and the interaction is electrochemical in nature involving anodic and cathodic process. metals and their alloys are indispensible materials for the design and fabrications of water treatments plants, heat exchangers, dye-bath, drill bits, food processing equipment, hollow pipes for fluid transportation, distribution and storage, etc. filter membrane, etc. however, it is observed over time that scales of inorganic materials or products develop on the internal walls of these components and reduced the efficiency or productivity. on the other, removing these unwanted products via acid cleaning or acidizing process with dilute mineral acids (hydrogen chloride and sulphuric acid) is ideal but significant corrosion effects are introduced on the metal surface on the process. the resultant effects of metal corrosion are enormous; it causes tremendous damage to life-span and integrity of industrial tools of metallic component especially in oil and gas sectors and other chemical processing industries [1-3], creates pollution problems which poses great health challenge to man and his environment and results to waste of economic resources in routine corrective and preventive maintenance and material loss. thus, identifying the possible solution to mitigate these problems associated with metal corrosion is imperative and highly efficient corrosion inhibitors from organic sources together with computational modeling techniques possess all it will take to achieve this feat. many researchers in the field of corrosion of metals and their inhibition have reported the use of several methods to combat the problem of corrosion of metals in different service environment [4]. one of such methods is the use of material regarded as corrosion inhibitor and it is not a complex technique neither does it requires much training during application. corrosion inhibitor is a material capable of reducing the corrosive effect of any corrodent system when added in a little amount or quantity. many corrosion inhibitors in use during metal corrosion and its control are sourced from either organic or synthetic background with little or no modification. chromates, some imidazole based inhibitors, etc. are typical examples of corrosion inhibitor from synthetic sources with effective performance but the application of these inhibitors are currently restricted due to their toxicity and other environmental issues. over time, the use of extracts from various plant leaves, bark, root, etc. has received detailed attention as anti-corrosion material. some of these natural materials particularly plant materials, proteins, biopolymer and amino acids have been reported to function as effective corrosion preventing additives [510]. plant materials are viewed to contain rich chemical compounds that can be extracted by simple methods and some of these chemicals are known to resemble the synthetic organic corrosion retarding compounds and have been proven to function as effective as their synthetic counterparts. these materials have at different proportions hetero atoms which contain sulphur, nitrogen and oxygen in a conjugation as well as double bond and aromatic ring in their chemical structures [11-20]. it is believed that these functional groups within the chemical structure of inhibitor play significant roles in corrosion, adsorption and inhibition. zingiber officinale (ginger) plays a very important role in medicinal science and it has high medicinal value due to its antifungal, anti-bacterial and other nutritional benefits. thus, it has been in use in many parts of the world as remedy and cure for many known diseases [21-22]. this study is aimed at investigating the corrosion inhibitive effects of zo ethanol extract on mild steel corrosion in acidic solutions using experimental and theoretical studies respectively. the essence of employing the computational study in the study is to unravel the correlation existing between the inhibitor efficiency and their electronic molecular structures. furthermore, the work seeks to widen the application of ginger as eco-friendly corrosion inhibitor for mild steel protection [23-25]. 2. experimental section 2.1. preparation of materials the metal used for the research was mild steel low carbon grade with the following composition: 0.05%c, 0.36%p, 0.03%si, 0.6%mn and 98.96%fe. the metal coupons were cut to 3 x 3 x 0.14 cm using mechanical cutter, cleaned, polished with fine emery paper (1000 and 1200 grade) degreased and prepared as earlier described in our previous work [26]. zinbiber officinale rhizomes tuber used was obtained from harvested samples kept at agricultural farm store in imo state polytechnic umuagwo, imo state and identified at crop science laboratory, federal university of technology owerri, imo state. the zo tuber foreskin was washed thoroughly with water to remove dirts, pealed to remove the thick back, sliced into pieces, dried thoroughly, pulverized to fine powder particle, stored in an air tight container and kept for corrosion studies. the prepared zo sample (50 g) was introduced in 500 ml of absolute ethanol in a beaker and allowed to stand for 72 h and kept in an aerated condition. the mixture was thoroughly filtered with filter paper and the solution obtained was used to prepared inhibited solution for corrosion studies. the stock blank of 1 m hcl and 0.5 m h2so4 acid solution respectively was prepared using dilution serial principle whereas the inhibited solutions were prepared by introducing 200 and 1000 mg of zo extract respectively into 1 l of 1 m hcl and 0.5 m h2so4 acid solution, respectively. the prepared zo extract sample was concentrated with rotary evaporator and subjected for functional group examination and analysis in a fourier transform infrared (ftir) spectrophotometer (nicolet magna-ir 560 model)[28]. also, the prepared zo powdered sample was subjected to gas chromatography-mass spectrophotometer, gc-ms 2 adindu et al. / j. nig. soc. phys. sci. 5 (2023) 1386 3 (19091s-433ui model) for examination and analysis. the hp5ms ultra inert 0◦c 325◦c: with dimension 30 m × 250 µm × 0.25 µm was used for the gc-ms analysis, at the gas chromatography section, the oven temperature was 50 oc, held for 120 secs raised to 180 oc at a rate of 50 oc/min the final temperature was 325 oc with injection volume of 10 µl and 7.3614 psi pressure. the carrier gas was helium with flow rate of 0.97414 ml/min, for mass spectrometer part. the solvent was methanol and the scanning was at 40-650 m/z the result obtained was compared with nist mass spectral library search program. 2.2. electrochemical experiments (a) electrochemical experiments were done using direct current voltammetry (versastat 400) advanced electrochemical workstation. the corrosion cell is made of cylindrical glass and contains three conventional electrodes namely [7]; working electrode (mild steel coupon), graphite rod (counter electrode) and reference electrode (saturated calomel electrode). the working electrode, a mild steel coupon of dimension 1.5 cm × 1.5 cm was affixed in polytetrafluoroethylene (ptfe) rods using epoxy resin in a way that just one of its surfaces of area 1 cm 2 was exposed to test solution under evaluation. the electrochemical workstation, central processing system, monitor and electrolytic cell terminals were connected tightly with luggin capillaries. open circuit potential (ocp) was performed for the test solution to achieve steady state potential at 1800 secs and 303 k before the eis and pdp experiments commenced. the measurements were performed at the end of 1800 secs in an aerated environment and stagnant test solution. polarization curves were determined from the scanning electrode potential of -250 mv to +250 mv versus corrosion potential (ecorr) at a scan rate of 0.33 mv/s [29] while linear polarization segments of cathodic and anodic curves were extrapolated to determine corrosion current densities (icorr) using v3 studio software. electrochemical impedance measurements were undertaken at corrosion potential (ecorr) at a frequency range of 10 m hz to 100 khz and amplitude of perturbation of 5 mv. electrochemical data were analyzed using the zsim win 3.10 software modeling package. the experimental measurements (i.e both eis and pdp) were repeated three different times to ensure reproducibility of results. eis measurement was performed to monitor the effect of thin film of the active inhibitor components adsorbed on the metal surface while pdp measurement was performed to proof evidence of effect of inhibitor on cathodic and anodic partial reactions and this assists to classify whether the inhibitor acts as cathodic, anodic or mixed type inhibitor. 2.3. theoretical computations the results of corrosion studies revealed that ethanol zo extract retarded the corrosion of mild steel in the different acid solution medium studied and theoretical computations was used to validate the experimental results. the theoretical studies provide evidence of chemical parameters within the inhibitor structures that are responsible for the corrosion inhibition and also give information on the adsorption energy between the corrosion inhibitor species and metal surface. molecular geometries and electron distributions influencing corrosion inhibition properties of some materials known had been previously evaluated using density functional theory [30 32]. structures of the most three (3) active constituents of the ginger rhizomes were selected from the gc-ms results and modeled. the theoretical calculation was done with material studio software accelrys (7.0 version) within the framework of density functional theory (dft) tools (quantum chemical computation and molecular dynamic simulation) from the molecular point of view. 3. results and discussion 3.1. sample characterization 3.1.1. ftir result the ftir characterization was performed to ascertain the functional groups present in the extract of zo which may have facilitated its adsorption on the mild steel surface. the result obtained is presented in figure 1. the prominent peaks observed in the figure include; a strong and broad hydrogen bonded (o – h) stretching band at 3272.6 cm-1, the strong band at 1640.0 cm-1 may be assigned to c = c and c = o stretching vibrations which may be due to the presence of conjugation in the chemical constituents of the zo powder and the band at 1408.6 cm-1 which corresponds to the c – h bending bands of ch2 and ch3 groups and the absorption bands at 1043.7 – 1084.7 cm-1 which may be assigned to c – o functional group [34]. the ftir result revealed that the extract contain functional groups which may facilitate the effectiveness of the extract to adsorb and interact with the mild steel surface. as a result, protection of the metal surface may be as a result of the adsorption of these functional groups present in the zo powder [35]. figure 1: ftir image of zo powder 3.1.2. gc-ms results the active phytochemical components of zo [36] were revealed in the gas chromatography-mass spectroscopy result presented in table 1 and figures 2 & 3. the peaks were identified by comparing their retention times with those found in (nist-ms) library. compounds with less than 4 % area 3 adindu et al. / j. nig. soc. phys. sci. 5 (2023) 1386 4 peak were not included in the presented result. benzene, 1(1,5-dimethyl-4-hexenyl)-4-methyl-, dodecanoic acid and 9octadecenoic acid (z)-2-hydroxy-1-(hydroxymethyl) ethyl ester were selected for the computational studies. 3.2. electrochemical results potentiodynamic polarization (pdp) and electrochemical impedance spectroscopy (eis) experiments were carried out to understand the effect of zo extract on the corrosion of mild steel in acidic environments. the concentrations of the zo extract chosen for the experiment were 200 and 1000 mg/l respectively and the plots of ocp obtained in 1.0 m hcl and 0.5 m h2so4 environments are presented in figure 4 (a & b). 3.2.1. electrochemical impedance spectroscopy results impedance experiments were undertaken on mild steel in the selected acid solutions in the presence and absence of zo inhibitor at two concentrations. the nyquist and bode plots respectively for mild steel corrosion in 1 m hcl and 0.5 m h2so4 acid solution respectively are presented in figures 5 (a & b) and 6 (a & b) respectively while the electrochemical parameters from the eis experiments are presented in table 2. the nyquist plot can be seen to show a single depressed semicircle in the region of high frequency for all the systems studied equivalent to one time constant found in the bode graphs. in the nyquist plot, the high frequency that intercepts with the real axis in named the solution resistance (rs) whereas the low frequency that intercept with the real axis is called the charge transfer resistance (rct) [37]. the impedance results were fitted into an equivalent circuit model rs (qdlrct) to get the impedance data presented in table 2 and qdl is the double layer capacitance. this equivalent circuit has been previously used to model impedance result [38-39] for mild steel corrosion in acidic media. the solution resistance in the equivalent circuit was shorted by a constant phase element (cpe) which was subsequently connected parallel to the charge transfer resistance (rct). the pure capacitor was replaced in the equivalent circuit to account for the deviations from ideal dielectric behavior which may arise from the dissimilar nature of the electrode surfaces. the electrochemical impedance response of the constant phase element is given in equation (1): zcpe = q −1 (jω−n) . (1) here q is the cpe constant, n is the cpe exponent, j an imaginary number which is equivalent to (-1)1/2 and ω is the angular frequency in s-1 (ω = 2π f ), f is the frequency in hz. the inhibition efficiencies presented in table 2 from the impedance response were estimated according to the equation (2): i e% = ( 1 − rct, bl rct, in ) × 100, (2) where rct,bl represents charge transfer resistance in the absence of the inhibitor and rct, in represents charge transfer resistance when the inhibitor was added. the results presented in table 2 shows that rct values increased with increase in the concentration of the zo inhibitor whereas the qdl values decreased as the zo concentration increased, the former action indicated that anticorrosion process is experienced whereas the later showed that the zo extract actually adsorbed on the mild steel surface. 3.2.2. potentiodynamic polarization (pdp) results the mechanisms of the partial anodic and cathodic half corrosion reactions as well as the effect of zo on either reactions were studied with the aid potentiodynamic polarization experimental techniques. figure 7 shows the pdp plots for mild steel corrosion in (a) 1 m hcl solution and (b) 0.5 m h2so4 acid solutions with and without zo inhibitor and table 3 shows the pdp parameters for mild steel corrosion in the presence and absence of the inhibitor. the result in table 3 shows that the addition of the inhibitor reduced the anodic and also the cathodic half corrosion reactions while shifting the corrosion reaction potential (ecorr)towards the negative value as well as reducing both the anodic and cathodic current densities in both acidic environments. these effects showed that zo functioned as a mixed type corrosion inhibitor for mild steel in both acidic solutions. the corrosion inhibition efficiencies (ie%) reported for zo were estimated from the current densities without (icorr,bl) and with (icorr,inh) zo inhibitor according to the equation (3): ie% = ( 1 − icorr, inh icorr, bl ) × 100. (3) the (i e%) values shown in table 3 are remarkably high indicating that zo is a good anti-corrosion material. 3.3. theoretical computational results 3.3.1. quantum chemical computation: exact experimental identification of contributions of the individual constituents of the bulk extract to the overall corrosion inhibition has been hindered as a result of the complex nature of the plant extract, we therefore rely on quantum chemical computations as well as molecular dynamic simulations to ascertain the individual contributions of three selected major constituents of zo extract to the experimentally observed corrosion inhibition effects. the molecular structures of benzene, 1-(1,5-dimethyl-4-hexenyl)-4-methyl(α-curcumene), dodecanoic acid (lauric acid) and 9-octadecenoic acid (z)-, 2-hydroxy-1-(hydroxymethyl)ethyl ester present in the zo were optimized with the density functional theory electronic program dmol3 utilizing a mulliken population analysis [40, 41). electronic condition for the simulation are restricted spin polarization by the dnd basis set and the perdew–wang (pw) local correlation density functional, further calculation was achieved to obtain the quantum chemical parameters such as the lowest unoccupied molecular orbital energy (elumo) the highest occupied molecular orbital energy (ehomo), energy gap (∆e), absolute hardness (η), chemical softness (σ), the fraction of electrons that were transferred from inhibitor molecule to the metal surface (∆n) and fukui functions. to mimic the extraction medium, these calculations were done in the presence of ethanol as the solvated medium using lda and pwc functional. 4 adindu et al. / j. nig. soc. phys. sci. 5 (2023) 1386 5 table 1: some of the active components of zo from the gc-ms result peak no. retention time name of compound molecular weight (g/mol) molecular formular area percent 11 16.6262 benzene, 1-(1,5-dimethyl-4hexenyl)-4-methyl202.3352 c15h22 4.1746 13 17.0368 1,3-cyclohexadiene, 5-(1,5dimethyl-4-hexenyl)-2-methyl-, (zingiberene) 204.3511 c15h24 4.6564 18 19.6874 dodecanoic acid (lauric acid) 200.32 c12h24o2 5.414 23 23.8033 tetradecanoic acid 228.37 c14h28o2 5.2783 35 27.7678 n-hexadecanoic acid 256.40 c16h32o2 4.6996 41 29.9013 9,17-octadecadienal, (z)264.40 c18h32o 5.0745 55 32.8447 9-octadecenoic acid (z)-, 2hydroxy-1-(hydroxymethyl)ethyl ester 356.50 c21h40o4 16.4469 figure 2: gc-ms microgram of zo powder 5 adindu et al. / j. nig. soc. phys. sci. 5 (2023) 1386 6 figure 3: mass spectra and molecular structures of the some zo constituents table 2: electrochemical impedance parameters for mild steel corrosion with and without zo system rs (ω cm2 ) rct (ω cm2) n qdl (µf cm -2) ie% 1 m hcl 1.67 104 0.88 91.7 200 mg/l zo 1.78 286 0.89 38.9 63.6 1000 mg/l zo 1.81 602 0.89 22.3 82.7 0.5 m h2so4 2.87 8.4 0.84 258.8 200 mg/l zo 3.16 40 0.89 153.6 79 1000 mg/l zo 3.26 132 0.89 61.5 93.6 table 3: potentiodynamic polarization parameters for mild steel corrosion with and without zo inhibitor system ecorr (mv vs sce) icorr (µa/cm2) ba (mv dec-1 ) bc (mv dec-1 ) ie% 1 m hcl -471.3 649.4 126.3 168.5 200 mg/l zo -477 224.5 112.7 123.2 65.4 1000 mg/l zo -474.3 109.6 104.8 116.9 83.1 0.5 m h2so4 -500 2940 136.7 208.5 200 mg/l zo -506 1476 123.5 194.7 49.8 1000 mg/l zo -509 756.9 114.3 186.2 74.3 6 adindu et al. / j. nig. soc. phys. sci. 5 (2023) 1386 7 figure 4: graph of open circuit potential for mild steel in (a) 1 m hcl (b) 0.5 m h2so4 in the presence and absence of zo. the simulation results including the optimized structure, frontier orbitals (homo and lumo orbital), the fukui indices (f+ and f-) and the electron density are shown in figure 8 while the quantum chemical parameters such as homo energy (ehomo), lumo energy (elumo), energy gap, (∆e = (elumo (ehomo), electron charge transfer (∆n), absolute hardness (η), absolute electronegativity (χ) and absolute softness(η) are presented in table 4. high values of ehomo indicate the ability of an inhibitor to donate electron the unoccupied d-orbital of the metal. similarly, low value of the energy gap shows that the energy needed to remove the lowest occupied orbital will be small indicating that the corrosion retardation will be high. accordingly, f-values show the reactivity relating to electrophilic attack while the corresponding f+ values indicate the reactivity in relation to nucleophilic attack or the ability of the metal to attract electron. the ionization potential (i) and electron affinity (a) are, respectively, related to the homo and lumo energies as shown in equations (4) and (5): i = −ehomo (4) a = −elumo. (5) the quantification of the electron charge transfer (∆n) from the electron rich inhibitor to the surface of the metal was done according to equation (6): ∆n = χmetal −χinh 2 (ηmetal + ηinh) , (6) figure 5: nyquist and bode plots for mild steel corrosion without and with the zo extract in 0.5 m h2so4 where χmetal and χinh /represent the absolute electronegativity of the metal and inhibitor respectively whereas ηmetal and ηinh are the absolute hardness of metal and inhibitor respectively. the tabulated values of ∆n were accordingly computed using theoretical values of 7 ev/mol for χmetal and θ ev/mol for ηmetal [41]. it has been previously reported that values of ∆n relate well with binding energies of corrosion inhibitors with high values favoring stronger adsorption of the inhibitor on the metal surface [42]. the values of η, χ and σ reported in table 4 are computed according to equations (7)-(9): η = i − a 2 (7) χ = i + a 2 (8) σ = 1 η . (9) accordingly our obtained results fall within the range of values already reported for some organic corrosion inhibitors 7 adindu et al. / j. nig. soc. phys. sci. 5 (2023) 1386 8 figure 6: (a) nyquist and (b) bode plots for mild steel in 1.0 m hcl in the presence and absence of zo extract [42]. the low value of ∆e obtained for 9-octadecenoic acid (z)-, 2-hydroxy-1-(hydroxymethyl)ethyl ester suggests that it must have interacted more covalently with the metal surface than the other constituents studied showing that it might have contributed more to the observed inhibiting property exhibited by the extract. 3.3.2. molecular dynamic simulation results molecular dynamic simulation was performed on the three compounds previously selected using the forcite quench tool of material studio. the molecular structures were first subjected to geometric optimization using a maximum iteration of 1000 and energy of 0.01 kcal/mol. modeling was achieved using the condensed phase optimized molecular potentials for atomistic simulation studies (compass) force field including smart algorithm. the temperature was maintained at 298k using the andersen thermostat having nve microcanonical figure 7: potentiodynamic polarization plots for mild steel corrosion in (a) 1 m hcl and (b) 0.5 m h2so4 solutions ensemble, 1fs step time and 5ps total simulation time were used and the number of steps were 5000. the fe (110) slab was chosen for the simulation with a simulation box of dimension 30 × 25 × 29 å and periodic boundary conditions to simulate representative section of the interface [36].the simulation box consist of fe (1 1 0), electrolytic systems [(h2o, h3o+, cl-) and (h2o, h3o+ and so4-)] and inhibitor molecules. the configuration of the lower layer of the fe slab was constraint to the lager position, although the other degrees of freedom were relaxed before the optimization of the fe (1 1 0) was done which was further increased to 12 × 8 supercell. quenching was done every 250 steps. the results of interaction mechanism between the inhibitor and the metal surface in the presence of the following molecules h2o, h3o+, cland so4performed with molecular dynamic simulation are presented in figure 9, figure 10 and table 4. 8 adindu et al. / j. nig. soc. phys. sci. 5 (2023) 1386 9 figure 8: electronic properties of benzene, 1-(1,5-dimethyl-4-hexenyl)-4methyl,dodecanoic acid and 9-octadecenoic acid (z)-2-hydroxy-1(hydroxymethyl)ethyl ester the results in figure 8 and 9 show that the inhibitor maintained a flat-lying adsorption orientation on the surface of the metal as a result of the delocalization of the electron density all over the molecule. this type of orientation is known to improve close contact between the inhibitor and the metal surface. in other to study the effect of the different acid corrodent on the adsorption energy, the solvation particles h2o, h3o+, cl-) and h2o, h3o+ and so4-)] were packed into the simulation box using the amorphous cell tool of the material studio 7.0 modeling and simulation software. the level of interaction between each inhibitor molecule and the metal surface was quantified by evaluating the energy of interaction energy equation (10): interaction energy = etotal− ( efe+solvation particles − einhibitor ) ,(10) where etotal, efe+solvent particles and einhibitor corresponds to the total energy of the fe (110)/efe+solvent particles couple, energy of the fe slab in addition to the solvent particles and the energy of the inhibitor molecule respectively, the solvation particles refer to either h2o, h3o+, cl-) and h2o, h3o+ and so4-, a negative value of the interaction energy is an indication that the molecule possess a stable adsorption structure, all the molecules studied showed negative values which accounts for 9 adindu et al. / j. nig. soc. phys. sci. 5 (2023) 1386 10 table 4: calculated quantum chemical properties for the most stable configurations of the selected zo constituents quantum chemical parameter benzene, 1-(1,5dimethyl-4-hexenyl) -4-methyldodecanoic acid 9-octadecenoic acid (z)-, 2-hydroxy-1(hydroxymethyl) ethyl ester ehomo (ev) -5.0347 -5.7252 -3.3152 elumo (ev) -1.7278 -2.0098 -2.9711 energy gap (∆e) 3.3069 3.7154 0.3440 electron charge transfer (∆n) 1.0943 0.8431 11.2051 absolute hardness (η) 1.65375 1.8577 0.1721 absolute electronegativity (χ) 3.3813 3.8675 3.1432 absolute softness (σ) 0.6048 0.5383 5.8106 interaction energy (kcal/mol) hcl -117.82 -172.46 -167.81 h2so4 -167.98 -145.35 -355.55 figure 9: depiction of the side and top views revealing suitable configurations for adsorption of zo constituents on fe (110) surface in h2so4 solution the inhibition efficiencies obtained in the experimental results.. the large interaction energy (-355.5kcal/mol) observed for 9octadecenoic acid (z)-, 2-hydroxy-1-(hydroxymethyl)ethyl ester 0.5 m h2so4 mirrors its ability to adsorb better on the metal surface than the other constituents which agrees with the quantum chemical computation result. 4. conclusion the corrosion inhibition study revealed that the ethanol extract of zingiber officinale efficiently inhibited the corrosion of figure 10: depiction of the side and top views revealing suitable configurations for adsorption of zo molecules on fe (110) surface in 1.0 m hcl acid solution mild steel induced on 1.0 m hcl and 0.5 m h2so4 acid solution respectively. the pdd results showed that zingiber officinale is a mixed type corrosion inhibitor for mild steel in both acidic solutions while eis results revealed that zingiber officinale adsorbed on the surface of mild steel indicating the evidence of metal protection. also, gc-ms results revealed that extract of zingiber officinale contains twenty-three (23) compounds of which three (3) compounds namely; benzene, 1(1,5-dimethyl-4-hexenyl)-4-methyl-, octadecanoic acid and 9octadecenoic acid (z)-, 2-hydroxy-1-(hydroxymethyl)ethyl ester were used for computational studies. quantum chemical calculation results 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[42] a. e. founda, a. h. el-ashalany, a. f. molouk, n. s. elsheikh & a. s abousalem, “experimental and computational chemical studies on the corrosion inhibitive properties of carbonitrile compounds for carbon steel in aqueous solutions”, science representative 11 (2021) 21672, https: //doi.org/10.1038/s41598-021-00701-z 12 j. nig. soc. phys. sci. 3 (2021) 340–343 journal of the nigerian society of physical sciences preparation and characterization of zinc metal soap from shea butter (vitellaria paradoxa) o. amosa,∗, t. e. odetoyeb, d. s. ogunniyib adepartment of industrial chemistry, federal university, lokoja, p.m.b 1154, lokoja, nigeria bdepartment of chemical engineering, university of ilorin, p.m.b 1515, ilorin, nigeria abstract zinc metal soaps are of great importance in the manufacture of personal care products and other industrial applications. variations in the soaps and their properties are usually due to the type of oil used in the synthesis. shea butter (vitellaria paradoxa), being a valuable industrial raw material, was investigated for the synthesis of zinc metal soap. locally obtained shea butter was characterized, refined and used to synthesize metal soap of zinc which was characterized. the zinc soap produced exhibited an off-white appearance, ph of 7.8, non-foaming, and no free alkalinity present. the functional groups in the soap were confirmed by ftir. doi:10.46481/jnsps.2021.238 keywords: zinc metal soap, shea butter, fitr. article history : received: 2 august 2021 received in revised form: 7 october 2021 accepted for publication: 8 october 2021 published: 29 november 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: e. a. emile 1. introduction shea butter is a common fat extracted from the fruit of the african shea tree (vitellaria paradoxa) that is available in many african countries in the wild [1]. the shea tree fruits are usually collected by women while the fat is obtained traditionally by boiling the kernel in water and scooping the solidified fats after cooling. the butter is also obtained by screw press. it usually presents yellow colour in the raw form with the processed versions being ivory or whitish in colour [2]. shea butter consists mainly of stearic and oleic fatty acid moieties in its triglycerides [3]. it is widely used in cosmetics as a moisturizer, salve, lotion and medicinal purposes [4]. shea butter is edible and used as an ∗corresponding author tel. no: +2348030677482 email address: olusegun.amos@fulokja.edu.ng, olusegunamos@yahoo.com (o. amos ) alternate butter in chocolate production [1]. it has been reported as a potential raw material for biodiesel production [5]. metal soaps are alkaline-earth or heavy-metal long chain carboxylates that are insoluble in water but soluble in non-aqueous solvents. metal soaps can be represented with a general formula (rco 2 ) 2 m where m can be zn, cd, pb, ba, ca, co, cu, al, fe, e.t.c. and r is a linear or branched alkyl group [6]. metal soaps can be produced using any of the following processes: double decomposition (metathesis), direct reaction of carboxylic acid with metal oxides, hydroxides and carbonates and direct reaction of metals with molten fatty acid [7]. metallic soaps have been found to be useful in various fields. metal carboxylates are significant in the production of poly vinyl chloride (pvc) which is a vital commercial polymer [8]. metal carboxylates have been used as green initiators for the ring-opening alternating copolymerization of cyclic anhydride and epoxide [9]. metal soaps from nonconventional oils can 340 o. amos et al. / j. nig. soc. phys. sci. 3 (2021) 340–343 341 also be utilized as paint additives, stabilizer for pvc and for inhibition of corrosion in metals and used as grease [8],colouring vanishes, cosmetics and textiles [10]. the metal soaps of lead, manganese, cobalt and zinc soaps are used in paints to accelerate drying while copper soaps exhibit fungicidal properties. soaps of zinc, iron, nickel, cobalt and chromium have been applied for the production of water proofing leather and canvas. metal soaps are also important in the studies of degradation in paints, their formation resulting in chemo-mechanical damage in historical oil paintings have been reported [11]. the sizes of the metal soap nucleus have been found to influence undesirable metal soap crystallization and growth in historical oil paintings [11, 12]. hence, there is a need for more research to be directed towards the production and properties of more metal soaps from various oil sources for more suitable applications. moreso, there is a growing demand for metal soaps as polymer additives [13]. in pharmaceutical field, zinc stearate is a component of the familiar facial or dusting powder and serves as an antiseptic emollient when incorporated with petrolatum. zinc stearate is also useful as an activator in the accelerated vulcanization of rubber. utilization of shea butter in the production of metallic soap is a potential means of promoting agro-industrial economy of the african countries where the shea trees abound. the aim of this study is to investigate the preparation of zinc metal soaps from shea butter as a potential industrial raw material for metal soap production. based on the review of earlier literature, works on zinc soap of shea butter is scarcely reported. 2. materials and methods 2.1. source of material the shea butter (vitellaria paradoxa) was purchased from kabba central market in kogi state, nigeria. 2.2. refining of shea butter a portion of the shea butter was refined via sequential steps of degumming, neutralization and bleaching in accordance to the methods described in literature [14, 15] with little modification of washing the oil several times with hot water during neutralization process to ensure complete removal of the foot. 2.3. characterization of the butter the physical and chemical properties of the shea butter sample were analyzed according to astm standards [16] for the determination of density, specific gravity, acid value, iodine value and saponification value. the functional groups in the shea butter were determined by the infrared spectrophotometer while the fatty acid constituents of shea butter were also recorded on the agilent gas chromatograph-mass spectrometer (model 7890a gc system,5675c inert msd with triple axis detector). figure 1: saponification of oil to form sodium soap figure 2: precipitation of zinc soap 2.4. synthesis of zinc metal soap metathesis, which is the mostly used method involves two reactions. firstly, the aqueous caustic is reacted with fatty acid and subsequently, the reaction of the formed sodium salt with the inorganic salt of the desired metal as shown in figures 1 and 2. the saponification reaction is usually carried out at high temperature and in the presence of excess water to dissolve the glycerol formed in the reaction mixture. consequently, this disallowed the reactions between glycerol (by-product) and the product of metal soap. the zinc metal soap was prepared by metathesis in aqueous alcoholic solution as summarized in figure 3, where r in figures 1 and 2 is stearic acid. figure 3: summary of zinc metal soap formation 9.2 g of refined shea butter was dissolved in 10 ml of ethanol at a temperature of 70◦c followed by the addition of 5 ml of 5m naoh solution. to this mixture, 20 ml of 1.86m znso4 solution was slowly added with continuous stirring. upon addition of the solution, the precipitation of the metal soap commenced [8] with little modification). the metal soap that was precipitated was then filtered off and washed with 30 ml hot water (80◦c) and air dried at room temperature for 48 h. the colour of the metal soap was noted and the yield was calculated. 2.5. characterization of the zinc metal soap synthesized the properties of the zinc metal soap prepared were determined for ph, free caustic alkalinity, foaming power and stability[8]. as stated by osmond [17] that ftir are highly effective for the characterization of metal soaps, the spectra were 341 o. amos et al. / j. nig. soc. phys. sci. 3 (2021) 340–343 342 also recorded on fourier transform infra-red spectrophotometer. 3. results and discussions 3.1. properties of shea butter the melting point of the crude shea butter was 32◦c while its cloud point was found to be 24◦c. table 1 shows the results obtained for the properties of both the crude and refined shea butter. the values obtained for shea butter, both refined and unrefined shown in table 1. when compared with the values obtained by chiboret al [18],a high acid values of 4.48 and 5.32 for the crude and refined butters respectively indicated that the oil sample contained more free fatty acids thus indicating its exposure to rancidity. the values depend on the method of extraction as the acid values obtained by solvent extraction and cold press methods were 3.50 and 9.54 mg koh/g respectively [19]. the saponification values obtained for the crude and refined samples were 122 mg/koh and 150 mg/koh, respectively. these values were smaller to the one reported by abdul-hammed for the solvent extracted (172.2mg/koh) and cold press extracted (185.7 mg/koh) methods [20, 21, 22, 23]. a high saponification value shows that the oil is suitable for industrial use, especially for soap making [24]. the desirably low iodine number of shea butter is indicative of the rich saturated fatty acids contents of the oil, which ensure stability against oxidation and rancidity of food prepared from the oil. it also promotes desirable properties for shortening and margarine production. 3.1.1. physicochemical properties of prepared zinc metal soap figure 4 shows the metal soap produced. the properties of the prepared zinc metal soap are as shown in the table 2. the appearance of the zinc metal soap prepared is as shown in figure 4. the metal soap was milky/off-white in colour as a characteristic of zinc salts. zinc salts are not expected to be coloured since it is not a transition metal. zinc metal soap being white can be applicable in white paints as pigments and paint driers [8]. the ph of the metal soap was 7.8 which is neutral in nature and thereby making it safe to handle. the yield of 6.38 38 g (98%) is a good one and also a signal of high profitability if table 1: physicochemical properties of crude and refined shea butter sample properties crude oil refined oil density (g/cm3) 0.69 0.69 specific gravity 0.90 n.d acid value 4.48 5.32 iodine value 3.04 4.568 saponification value 122 150 n.d – not determined figure 4: prepared zinc metal soap sample. table 2: the properties of the zinc metal soap prepared metal soap colour weight (g) ph foam stability free caustic alkalinity zinc metal soap milky 6.38 7.8 no foam nil industrially exploited. . the metal soap produced was a mixture of fatty acid carboxylates of zinc, mainly zinc stearate since the major fatty acid components of the shea butter was stearic acid there was no free caustic alkalinity present in the synthesized d metal soap which indicated that there are no alkalinities left in the soap. the zinc metal soap prepared has poor detergency properties due to its hydrophobic property. figure 5: ftir of refined shea butter. figures 5 and 6 showed the infrared spectra of the refined shea butter and that of zn metal soap in region 4000 to 500 cm−1 for both samples respectively. from the spectra, the absorption band at 2926 cm−1 was the asymmetry vibration of ch2 stretch, 342 o. amos et al. / j. nig. soc. phys. sci. 3 (2021) 340–343 343 figure 6: ftir spectra of zinc metal soap. while that at 2850 cm−1 was due to the symmetry vibration of ch2 stretch. the c = o stretch of triglyceride linkage was seen at 1742 cm−1. the small absorption band observed at 1669 cm−1 was the characteristic carbonyl stretching band of ester linkage (c = o). it was observed that the absorption at 1710 cm−1 due to c = o of free fatty acids in the shea butter (figure 5) has disappeared in the metal soap (figure 6). the c − h absorption of bending vibration of ch2 and ch3 bands was observed at 1470 cm−1. the bands of ch2 bending vibrations were also observed at 1381 and 1109 cm−1. the absorption due to c − o ester bonds were observed at 992, 830 and 1097 cm−1 the ch2 rocking vibration can be clearly seen at 722 cm−1. the ftir spectra confirmed the functional groups fatty acid carboxylates of zinc produced. 4. conclusion in this study, the preparation of zinc metal soap from shea butter was successfully done using the double decomposition (metathesis) method. the shea butter metal soap produced exhibited desirable characteristic metal soap colour, hydrophobic and free caustic alkalinity properties. ftir spectra confirmed the functional groups of the soap. successful production of metallic soap from shea butter establishes its potential use as an industrial raw material in addition 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[24] r. bacalogulu & m. fisch, “degradation and stabilization of poly(vinyl chloride) 11. kinetics of the thermal degradation of poly (vinyl chloride)”, journal of polymer degradation and stability 45 (1994) 325. 343 j. nig. soc. phys. sci. 3 (2021) 239–249 journal of the nigerian society of physical sciences an investor’s investment plan with stochastic interest rate under the cev model and the ornstein-uhlenbeck process edikan e. akpanibaha,∗, udeme o. inib adepartment of mathematics and statistics, federal university otuoke, bayelsa, nigeria bdepartment of mathematics and computer science, niger delta university, bayelsa, nigeria abstract the aim of this paper is to maximize an investor’s terminal wealth which exhibits constant relative risk aversion (crra). considering the fluctuating nature of the stock market price, it is imperative for investors to study and develop an effective investment plan that considers the volatility of the stock market price and the fluctuation in interest rate. to achieve this, the optimal investment plan for an investor with logarithm utility under constant elasticity of variance (cev) model in the presence of stochastic interest rate is considered. also, a portfolio with one risk free asset and two risky assets is considered where the risk free interest rate follows the ornstein-uhlenbeck (o-u) process and the two risky assets follow the cev process. using the legendre transformation and dual theory with asymptotic expansion technique, closed form solutions of the optimal investment plans are obtained. furthermore, the impacts of some sensitive parameters on the optimal investment plans are analyzed numerically. we observed that the optimal investment plan for the three assets give a fluctuation effect, showing that the investor’s behaviour in his investment pattern changes at different time intervals due to some information available in the financial market such as the fluctuations in the risk free interest rate occasioned by the o-u process, appreciation rates of the risky assets prices and the volatility of the stock market price due to changes in the elasticity parameters. also, the optimal investment plans for the risky assets are directly proportional to the elasticity parameters and inversely proportional to the risk free interest rate and does not depend on the risk averse coefficient. doi:10.46481/jnsps.2021.172 keywords: o-u process, stochastic interest rate, optimal investment plan, legendre transform, cev model article history : received: 13 march 2021 received in revised form: 27 may 2021 accepted for publication: 07 june 2021 published: 29 august 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: t. latunde 1. introduction in the study of the optimal investment plan for any financial institution and considering the fact that the economy of most countries and even the financial markets are presently in serious crisis due to the suffocating impact of the novel covid-19 pandemic, the role of stochastic volatilities cannot be over emphasized as it plays a significant role in the behaviour of the ∗corresponding author tel. no: +2347030811189 email address: edemae@fuotuoke.edu.ng (edikan e. akpanibah ) stock market price due to its unstable nature resulting from various information obtainable from the market. in order to take a seemingly right decision while investing in risky assets such as stock, the stochastic volatility models become necessary to understand the random nature of the stock market price. in this paper, we consider the cev model which is one of the stochastic volatility models used in describing the behaviour of the stock market price. this model was first used in [2] and degenerate into a constant volatility model called the geometric brownian motion (gbm) when its elasticity parameter equals zero. one significant property of this model is its ability to capture the 239 akpanibah & ini / j. nig. soc. phys. sci. 3 (2021) 239–249 240 implied volatility skew. several works have been done on utility maximization under the cev model, some of which include [3] [5]. in [6], the reinsurance problem and optimal investment under the cev model was studied. [1], studied the optimal investment problem with taxes, dividend and transaction cost using different utility functions under the cev process. [7, 8] solved the optimal investment problem for a defined contribution (dc) pension plan with return of premiums clauses under different assumptions and assumed that the stock market price follows the cev process; they used the game theoretic approach and solved the resultant extended hamilton jacobi bellman equation for the optimal investment plan. the optimal investment plan with stochastic interest rate under gbm has been studied by some authors; these include [9], who studied the investment portfolio under stochastic interest rate for a case of protected dc fund. also, [10, 11], used stochastic interest rate to obtain optimal investment plan for a dc plan. in [12, 13], the optimal investment plan with the vasicek interest rate was studied while [14] [16] solved some optimization problems for the optimal investment plan when the interest rate is of affine type. the optimal investment plan with the cev process under several assumptions has been studied by different authors when the risk free interest rate are constant as seen in the above literatures except for [17] [19], who investigated the optimal control strategies under the cev model with stochastic interest rate. in [17], the optimal investment plan of an insurer with stochastic interest rate under the cev model was studied using logarithm utility, they assumed that the risk free interest rate follows the coxingersoll-ross (cir) process and used the power transformation, change of variable and asymptotic approach to find the asymptotic solution of the optimal investment plan. in [18], the expected utility of an investor with exponential utility function was maximized using the legendre transformation and asymptotic expansion method to find a closed form solution of the optimal investment plan. they pointed out that authors find it difficult to combine the cev model and stochastic interest rate in determining investment strategies due to the complexity of the resultant optimization problem. also, they pointed out that in real life applications, interest rates are usually not constant but fluctuating in nature and the volatility of the interest rate generate some market risks; that is to say, when this risk are not considered, we are under estimating the effect of this risk emanating from this interest rate which is critical in influencing the prices of different assets available in financial market. in [19], the optimal investment plan for an investor with exponential utility under the modified cev model was studied. in their work, the risk free rate follows the o-u process. most of the literatures mentioned above used the (cir) model, vasicek model, affine model etc. to model their interest rate, however very few used o-u process. according to [21], we know that it is more accurate to use o-u process in modelling both interest rate and stock market price since it reflects the fluctuation of the interest rates and asset prices. also, the o-u process is closer to the change in interest rate. based on this important characteristic of the o-u process and considering the unstable nature of the financial market at this present time, we are motivated to develop a robust investment plan for an investor with logarithm utility which takes into considerations fluctuations in interest rate as well as the volatility of the stock market prices. we do this by studying the optimal investment plans of an investor exhibiting the crra and whose risky assets and risk free interest rate are modelled by the cev process and the o-u process respectively. furthermore, the legendre transformation and asymptotic technique are used to determine asymptotic solutions of the optimal investment plan. we also give some numerical simulations to explain our results. the main difference between our work and that of [18] is that we will consider an investor with logarithm utility where the investor’s behaviour toward risks are not influenced by the risk aversion coefficient instead of the exponential utility which depend on the risk aversion coefficient. also, investment is done with three assets instead of two assets and the risk free interest rate follows the o-u process instead of the cir process. the rest of the paper is outlined as follows; section 2 described the financial models of the three assets and the risk free rate. section 3 gives the definition of the value function, derives the corresponding hamilton jacobi bellman equations by maximum principle and application of legendre transform to the hjb equation. section 4 provides closed-form solutions of the optimal investment plan for the three assets under crra utility function. section 5 and 6 present numerical simulations of the effects of some sensitive parameters on the optimal investment plan and discussions. section 7 concludes the work. 2. the financial market model consider a portfolio comprising of one risk free asset and two risky assets in a financial market which is open continuously for an interval t ∈ [0, t ] where t is the expiration date of the investment. let {b0 (t) , b1 (t) ,b2 (t) : t ≥ 0} be standard brownian motion defined on a complete probability space (ω, f, p) where ω is a real space and p is a probability measure and f is the filtration which represents the information generated by the three brownian motions. let st (t) denote the price of the risk free asset at time t and the model is given as follows ds0 (t) s0 (t) = r(t)dts0 (0) = s0 > 0 (1) where r(t) is the short interest rate process and driven by the stochastic differential equation dr (t) = θ(µ0 − r (t))dt −δdb0(t), r (0) = r0 > 0 (2) where θ, µ0 and δ are positive real numbers see [19] [21]. let s1 (t) and s2 (t) denote the prices of two different stocks which are described by the cev model and the dynamics of the stock market prices are described by the stochastic differential equations at t ≥ 0 as follows ds1 (t) s1 (t) = µ1dt + σ11s β 1 (t) db1 (t) + σ12s β 1(t)db2(t) (3) ds2 (t) s2 (t) = µ2dt + σ21s β 2 (t) db1 (t) + σ22s β 2(t)db2(t) (4) 240 akpanibah & ini / j. nig. soc. phys. sci. 3 (2021) 239–249 241 where µ1 and µ2 are appreciation rate of the two risky assets, σ11s β 1, σ12s β 1, σ21s β 2and σ22s β 2 are instantaneous volatilities and form a 2 × 2 matrix σ = {σmn}2×2 such that σσ t is positive definite and β < 0 represent elasticity parameter. (see [20] for details). note if β = 0, the stock market price is modelled by gbm. 3. optimization problem 3.1. wealth formulations and hjb equation let ϕ be the optimal portfolio strategy and we define the utility attained by the investor from a given state x at time t as lϕ (t, r, s1, s2, x) = (5) eϕ [ u (x (t )) ∣∣∣∣∣∣ r (t) = r,s1 (t) = s1, s2 (t) = s2,x (t) = x ] , where t is the time, r is the risk free interest rate and x is the wealth. the objective here is to determine the optimal portfolio strategies ϕ∗ = (ϕ0, ϕ1, ϕ2) and the optimal value function of the investor l (t, r, s1, s2, x) given as l (t, r, s1, s2, x) = sup ϕ∗ lϕ∗ (t, r, s1, s2, x) (6) such that lϕ∗ (t, r, s1, s2, x) = l (t, r, s1, s2, x) . (7) let x (t) be the investor’s wealth at time t and the differential form associated with the fund size is given by dx (t) = x (t) ( ϕ0 ds0 (t) s0 (t) + ϕ1 ds1 (t) s1 (t) + ϕ2 ds2 (t) s2 (t) ) . (8) substituting (1), (3) and (4) into (8), we have dx (t) = x (t)  (ϕ1 (µ1 − r) + ϕ2 (µ2 − r) + r) dt + ( ϕ1σ11s β 1 (t) + ϕ2σ21s β 2 (t) ) db1 + ( ϕ1σ12s β 1 (t) + ϕ2σ22s β 2 (t) ) db2 x (0) = x0  ,(9) where ϕ0, ϕ1 and ϕ2 are the optimal investment plans for the risk-free asset and the two risky assets respectively, such that ϕ0 = 1 −ϕ1 −ϕ2. from [19], applying the ito’s lemma and maximum principle, the hamilton jacobi bellman (hjb) equation which is a nonlinear pde associated with (9) is obtained by maximizing lϕ∗ (t, r, s1, s2, x) subject to the investor’s wealth as follows lt + µ1 s1ls1 + µ2 s2ls2 + 1 2as 2β+2 1 ls1 s1 + 12bs 2β+2 2 ls2 s2 + cs β+1 1 s β+1 2 ls1 s2 +rxlx + θ (µ0 − r)lr + 12 δ 2lrr + dδρs β+1 1 ls1 r + eδρsβ+12 ls2 r +supϕ1,ϕ2  ( 1 2aϕ 2 1 s 2β 1 + cϕ1ϕ2 s β 1 s β 2 + 12bϕ 2 2 s 2β 2 ) lxx ((µ1 − r) ϕ1 + (µ2 − r) ϕ2)lx + ( dδρϕ1 s β 1 + eδρϕ2 s β 2 ) lxr( as2β+11 ϕ1 + cϕ2 s β+1 1 s β 2 ) lxs1 + ( +cϕ2 s β+1 1 s β 2 + bϕ2 s 2β+1 2 ) lxs2   = 0(10) differentiating (10) with respect to ϕ1 and ϕ2, we obtain the first order maximizing condition for equation (10) as ϕ∗1 =  − [ csβ1 (µ2−r)−bs β 2 (µ1−r) ] x(ab−c2)s2β1 s β 2 lx lxx − s1 lxs1 xlxx − (bd−ce)δρ x(ab−c2)sβ1 lxr lxx  (11) ϕ∗2 =  − [ csβ2 (µ1−r)−as β 1 (µ2−r) ] x(ab−c2)sβ1 s 2β 2 lx lxx − s2 lxs2 xlxx − (ae−cd)δρ x(ab−c2)sβ2 lxr lxx  (12) substituting (11) and (12) into (10), we have lt + µ1 s1ls1 + µ2 s2ls2 + 1 2as 2β+2 1 ls1 s1 + 12bs 2β+2 2 ls2 s2 + cs β+1 1 s β+1 2 ls1 s2 +θ (µ0 − r)lr + 1 2 δ 2 jrr + dδρs β+1 1 ls1 r +eδρsβ+12 ls2 r + 1 2 ( f sβ1 s β 2 − g s2β1 − h s2β2 ) l2x lxx +rxlx − (µ1 − r) s1 lxlxs1 lxx − (µ2 − r) s2 lxlxs2 lxx −δρ ( k1 (µ1−r) sβ1 + k2 (µ2−r) sβ2 ) lxlxr lxx − 1 2as 2β+2 1 l2xs1 lxx − 1 2bs 2β+2 2 l2xs2 lxx − 1 2 k3ρ 2δ2 l2xr lxx −csβ+11 s β+1 2 lxs1lxs2 lxx  = 0(13) where a = σ211 + σ 2 12, b = σ 2 21 + σ 2 22, c = σ11σ21 + σ12σ22, d = σ11 + σ12,e = σ21 + σ22 f = 2c(µ1−r)(µ2−r) (ab−c2) ,g = b(µ1−r) 2 (ab−c2),h = a(µ2−r) 2 (ab−c2), k1 = (bd−ce) (ab−c2), k2 = (ae−cd) (ab−c2) .k3 = (bd2 +ae2−2cde) (ab−c2) (14) from [20], we assumed that the optimal investment plan for risky assets’ prices are known based on the assumption that µ1ϕ ∗ 1 + µ2ϕ ∗ 2 = n (15) where n is a constant. substituting (11) and (12) into (15), we derive an expression for 1 sβ1 s β 2 as 1 sβ1 s β 2 =  ab−c2 c(2µ1µ2−µ1 r−µ2 r) ×  nxlxx lx + bµ1 (µ1−r) (ab−c2)s2β1 + µ1 s1 lxs1 lx + (bd−ce)µ1δρ (ab−c2)sβ1 lxr lx + aµ2 (µ2−r) (ab−c2)s2β2 +µ2 s2 lxs2 lx + (ae−cd)µ2δρ (ab−c2)sβ2 lxr lx   (16) substituting (16) into (13), we have lt + µ1 s1ls1 + µ2 s2ls2 + [r + α1] xlx + 12as 2β+2 1 ls1 s1 + 1 2bs 2β+2 2 ls2 s2 +csβ+11 s β+1 2 ls1 s2 + 1 2 δ 2lrr + dδρs β+1 1 ls1 r +eδρsβ+12 ls2 r + 1 2 ( α2 s −2β 1 +α3 s −2β 2 ) l2x lxx +θ (µ0 − r)lr + ( α4 − (µ1 − r) s1 ) lxlxs1 lxx + ( α5 − (µ2 − r) s2 ) lxlxs2 lxx + ( α6 s −β 1 +α7 s −β 2 ) lxlxr lxx − 1 2as 2β+2 1 l2xs1 lxx − 1 2bs 2β+2 2 l2xs2 lxx − 1 2 k3ρ 2δ2 l2xr lxx −csβ+11 s β+1 2 lxs1lxs2 lxx  = 0(17) 241 akpanibah & ini / j. nig. soc. phys. sci. 3 (2021) 239–249 242 α1 = f n(ab−c2) 2c(2µ1µ2 −µ1r −µ2r) ,α2 = fbµ1 (µ1 − r) (ab−c2) 2c(2µ1µ2 −µ1r −µ2r) −g, α3 = faµ2 (µ2 − r) (ab−c2) 2c(2µ1µ2 −µ1r −µ2r) −h, α4 = fµ1 s1 (µ2 − r) (ab−c2) 2c(2µ1µ2 −µ1r −µ2r) , α5 = fµ2 s2 (µ2 − r) (ab−c2) 2c(2µ1µ2 −µ1r −µ2r) , α6 = (bd−ce)fµ1δρ 2c(2µ1µ2 −µ1r −µ2r) − k1δρ (µ1 − r) , α7 = (ae−cd)fµ1δρ 2c(2µ1µ2 −µ1r −µ2r) − k2δρ (µ2 − r) 3.2. legendre transformation and dual theory the differential equation obtained in (17) is a non linear pde and is somehow complex to solve. in this section, we will introduce the legendre transformation and dual theory and use it to transform the non linear pde to a linear pde. let f : rn → r be a convex function for z > 0, define the legendre transform n (z) = max x { f (x) − zx} , (18) the function n(z) is the legendre dual of the function f (x). (c.f. [22]) since f (x) is convex, from theorem 3.2 and [21], the legendre transform for the value function l (t, r, s1, s2, x) can be defined as follows l̂ (t, r, s1, s2, z) = (19) sup l (t, r, s1, s2, x) − zx | 0 < x < ∞} 0 < t < t where l̂ is the dual of l and z > 0 is the dual variable of x, the value of x where this optimum is achieved is represented by g (t, r, s1, s2, z) , such that g (t, r, s1, s2, z) = (20) inf x ∣∣∣∣ l (t, r, s1, s2, x) ≥ zx + l̂ (t, r, s1, s2, z)} 0 < t < t.  from (20), the function g and l̂ are very much related and can be referred to as the dual of l and are related thus, l̂ (t, r, s1, s2, z) = l (t, r, s1, s2, g) − zg. (21) where g (t, r, s1, s2, z) = x, lx = z, g = −l̂z. (22) differentiating (21) with respect to t, r, s1, s2 and x lt = l̂t, ls1 = l̂s1 ,ls2 = l̂s2, lr = l̂r, , lrx = −l̂rz l̂zz , ls1 r = l̂s1 r − l̂s1 zl̂rz l̂zz , ls2 r = l̂s2 r − l̂s2 zl̂rz l̂zz , lxx = −1 l̂zz , ls1 s1 = l̂s1 s1 − l̂2s1 z l̂zz ,ls2 s2 = l̂s2 s2 − l̂2s2 z l̂zz , lx = z, ls1 x = −l̂s1 z l̂zz ,ls2 x = −l̂s2 z l̂zz lrr = l̂rr − l̂2rz l̂zz (23) at terminal time t , we define the dual utility in terms of the original utility function u (x) as û (z) = sup u (x) − zx | 0 < x < ∞} , and g (z) = sup?{x|u(x) ≥ zx + û (z) . as a result l̂ (t, r, s1, s2, z) = l (t, r, s1, s2, g) − zg. g (z) = (u′)−1 (z) , (24) where g is the inverse of the marginal utility u and note that l (t, r, s1, s2, x) = u(x). at terminal time t , we can define g (t, r, s1, s2, z) = in f x>0 x ∣∣∣∣ u (x) ≥ zx + l̂ (t, r, s1, s2, z)} and l̂ (t, r, s1, s2, z) = sup x>0 { u (x) − zx} so that g (t, r, s1, s2, z) = (u ′)−1 (z) . (25) substituting (23) into (11), (12) and (17), we have l̂t + µ1 s1l̂s1 + µ2 s2l̂s2 + [r + α1] gz + 12as 2β+2 1 l̂s1 s1 + 1 2bs 2β+2 2 l̂s2 s2 + 12 δ 2l̂rr + dδρs β+1 1 l̂s1 r + eδρsβ+12 l̂s2 r − 1 2 ( α2 s −2β 1 + α3 s −2β 2 ) z2l̂zz − 1 2 δ 2 ( 1 − k3ρ2 ) l̂2zr l̂zz + ( α4 − (µ1 − r) s1 ) zl̂s1 z + (α5 − (µ2 − r) s2) zl̂s2 z +csβ+11 s β+1 2 l̂s1 s2 + θ (µ0 − r)l̂r + ( α6 s −β 1 + α7 s −β 2 ) zl̂rz  = 0.(26) ϕ∗1 =  − [ csβ1 (µ2−r)−bs β 2 (µ1−r) ] x(ab−c2)s2β1 s β 2 zl̂zz − s1l̂s1 z x − (bd−ce)δρ x(ab−c2)sβ1 l̂rz  (27) ϕ∗2 =  − [ csβ2 (µ1−r)−as β 1 (µ2−r) ] x(ab−c2)sβ1 s 2β 2 zl̂zz − s2l̂s2 z x − (ae−cd)δρ x(ab−c2)sβ2 l̂rz  . (28) 242 akpanibah & ini / j. nig. soc. phys. sci. 3 (2021) 239–249 243 from equation (22), differentiating (26), (27) and (28) with respect to z, we have gt + (α4 + rs1) gs1 + (α5 + rs2) gs1 − (r + α1) g − (r + α1)z gz + 1 2as 2β+2 1 gs1 s1 + 12bs 2β+2 2 gs2 s2 + cs β+1 1 s β+1 2 gs1 s2 + 1 2 δ 2grr +θ (µ0 − r) gr + dδρs β+1 1 gs1 r +eδρsβ+12 gs2 r + 1 2 ( α2 s −2β 1 +α3 s −2β 2 ) z2gzz + ( α2 s −2β 1 +α3 s −2β 2 ) z gz + ( α4 − (µ1 − r) s1 ) zgs1 z + ( α5 − (µ2 − r) s2 ) zgs2 z + ( α6 s −β 1 + α7 s −β 2 ) (gr + zgrz) − 1 2 δ 2 ( 1 − k3ρ2 ) ( 2gr grz gz − gzz g2r g2z )  = 0.(29) ϕ∗1 =  [ csβ1 (µ2−r)−bs β 2 (µ1−r) ] x(ab−c2)s2β1 s β 2 z gz + s1 x gs1 + (bd−ce)δρ x(ab−c2)sβ1 gr  (30) ϕ∗2 =  [ csβ2 (µ1−r)−as β 1 (µ2−r) ] x(ab−c2)sβ1 s 2β 2 z gz + s2 x gs2 + (ae−cd)δρ x(ab−c2)sβ2 gr  (31) where, l (t, r, s1, s2, z) = u(z) and u(z) is the marginal utility of the investor. next, we proceed to solve (29) for g considering an investor with logarithm utility, then substitute the solution into (30) and (31) for the optimal investment plan using change of variable and asymptotic method. 4. investor’s optimal investment plan under logarithm utility here, we consider an investor with utility function exhibiting constant relative risk aversion (crra) different from the one in [18] where the investor exhibited constant absolute risk aversion (cara). since our interest here is to determine the optimal investment plan for the investor with crra utility, we choose the logarithm utility function similar to the one in [20, 21]. from [1, 21], the logarithm utility function is given as u(x) = lnx, x > 0 from (25), g (t, r, s1, s2, z) = (u ′)−1 (z) = 1 z (32) next, we conjecture a solution to (21) similar to the one in [21] with the form:{ g (t, r, s1, s2, z) = 1 z [e (t, r, s1) + f (t, r, s2)] + h(t) e (t, r, s1) = 1 2 , f (t, r, s2) = 1 2 , h(t ) = 0, (33)  gt = 1 z [et + f t] + ht, gs1 = 1 z es1 , gs2 = 1 z fs2, gs1 s1 = 1 z es1 s1, grs1 = 1 z [ers1 + f rs1 ] gs2 s2 = 1 z fs2 s2, gz = − 1 z2 [ e + f ] , gzz = 2 z3 [ e + f ] , gs1 z = − 1 z2 es1 , gs2 z = − 1 z2 fs2, grs2 = 1 z [ers2 + f rs2 ], gr = 1 z [er + f r ], grr = 1 z [err + f rr ] , grz = − 1 z2 [er + f r ] (34) substituting (34) into (29), we have [ht − (r + α1)h] + 1z  et + µ1 s1es1 + 12as2β+21 es1 s1 +dδρsβ+11 es1 r + θ (µ0 − r) er + 1 2 δ 2err  + 1z  ft + µ2 s2 fs2 + 12bs2β+22 fs2 s2 +eδρsβ+12 f s2 r + θ (µ0 − r) fr + 1 2 δ 2 frr   = 0(35) splitting (35) we have{ ht − (r + α1)h = 0 h(t ) = 0 (36)  et + µ1 s1es1 + 1 2as 2β+2 1 es1 s1 + dδρs β+1 1 es1 r +θ (µ0 − r) er + 1 2 δ 2err = 0 e (t, r, s1) = 1 2 (37)  ft + µ2 s2 fs2 + 1 2bs 2β+2 2 fs2 s2 + eδρs β+1 2 f s2 r +θ (µ0 − r) fr + 1 2 δ 2 frr = 0 f (t, r, s2) = 1 2 (38) solving equation (36) for h, we obtain h = 0 (39) g (t, r, s1, s2, z) = 1 z (40) to prove the proposition above, we attempt to solve equations (37) and (38) by stating and proofing the following lemmas the solution of equation (37) is given as e (t, r, s1) = u (t, r, m) = uα (t, r, m) = 1 2 , where uα (t, r, m) = u1 (t, r, m) + √ αu2 (t, r, m) + αu3 (t, r, m) and u1 (t, r, m) = 1 2 , u2 (t, r, m) = 0, and u3 (t, r, m) = 0. assume{ e (t, r, s1) = u (t, r, m) , m = s −2β 1 e (t, r, s1) = 1 2 , (41) then et = ut, es1 = −2βs −2β−1 1 um , es1 s1 = 2β (2β + 1) s −2β−2 1 um + 4β 2 s−4β−21 umm, er = ur, err = urr, ers1 = −2βs −2β−1 1 urm  (42) 243 akpanibah & ini / j. nig. soc. phys. sci. 3 (2021) 239–249 244 substituting (42) into (37), we have ut − 2µ1βmum + aβ (2β + 1) um +2β2amumm + θ (µ0 − r) ur + 12 δ 2urr − 2βdδρ √ murm  = 0 (43) we can rewrite (43) as (v1 + v2 + v3) u = 0 (44) where v1 = [ θ (µ0 − r) ur + 1 2 δ2urr ] u (45) v2 = [ ut + β ( a (2β + 1) −2µ1m ) um + 2β 2 amumm ] u (46) v3 = [ −2βdδρ √ murm ] u (47) next, we follow the approach in [18] by applying the asymptotic expansion method to solve the problem in (44). assume that the volatility follows a slow fluctuating process, we attempt to find an asymptotic solution of (44) by a following slow-fluctuating process rαto replace (2), in which 0 < α � 1 is a small positive parameter: drα (t) = θ(µ0 − rα(t))dt −δdb0(t) (48) substituting (48) into (44) and also replacing µ0−rα(t) by α(µ0− r), we have( αv1 + v2 + √ αv3 ) uα = 0 (49) next, we conjecture a solution for (49) as follows uα (t, r, m) = ( u1 (t, r, m) + √ αu2 (t, r, m) +αu3 (t, r, m) ) . (50) substituting (50) into (49), ( αv1 + v2 + √ αv3 )  u1 (t, r, m) + √ αu2 (t, r, m) +αu3 (t, r, m)  = 0 simplifying the above equation, we arrive at v2u1 (t, r, m) + [ v2u2 (t, r, m) +v3u1 (t, r, m) ] √ α + [ v1u1 (t, r, m) + v2u3 (t, r, m) +v3u2 (t, r, m) ] α  = 0. this implies that{ v2u1 (t, r, m) = 0 u1 (t, r, m) = 1 2 (51) { v2u2 (t, r, m) + v3u1 (t, r, m) = 0 u1 (t, r, m) = 1 2 , u2 (t, r, m) = 0 (52) { v1u1 (t, r, m) + v2u3 (t, r, m) + v3u2 (t, r, m) u1 (t, r, m) = 1 2 , u2 (t, r, m) = 0 u3 (t, r, m) = 0 (53) to obtain the solution of (50), we solve (51), (52) and (53). recall from (45), (46) and (47), equation (51), (52) and (53) can be expressed as ( u1t + β ( a (2β + 1) −2µ1m ) u1m + 2β2amu1mm ) = 0 u1 (t, r, m) = 1 2 (54)  u2t + β (a (2β + 1) − 2µ1m) u2m +2β2amu2mm − 2βdδρ √ mu1rm = 0 u2 (t, r, m) = 0 (55)  u3t + β (a (2β + 1) − 2µ1m) u3m + 2β2amu3mm +θ (µ0 − r) u1r + 1 2 δ 2u1rr − 2βdδρ √ mu2rm u3 (t, r, m) = 0 (56) let u1 (t, r, m) = q0 (t, r) + mq1 (t, r) (57) and u1t = q0t + mq1t, u1m = q1 , u1mm = 0 (58) substituting (58) in (54), we have q0t + βa (2β + 1)q1 = 0,q0 (t, r) = 1 2 (59) q1t − 2µ1mq1 = 0,q1 (t, r) = 0 (60) solving (60), we have q1 (t, r) = 0 (61) putting (61) into (59) and solving the resultant equation, we have q0 (t, r) = 1 2 (62) hence from (54) u1 (t, r, m) = q0 (t, r) + mq1 (t, r) = 1 2 (63) next, we solve (55), by assuming a solution of the form u2 (t, r, m) = ( q2 (t, r) + m 1 2 q3 (t, r) + mq4 (t, r) +m 3 2 q5 (t, r) ) (64) and  u2t = q2t + m 1 2 q3t + mq4t + m 3 2 q5t , u2m = 1 2 m − 1 2 q3 + q4 + 3 2 m 1 2 q5 u2mm = − 1 4 m − 3 2 q3 + 3 4 m − 1 2 q5, u1rm = 0  (65) substituting (65) into (55), we have{ q2t + β (2β + 1)aq4 = 0, q2 (t, r) = 0 (66) { q3t −µ1βq3 + 3 2 β (3β + 1)aq5 = 0, q3 (t, r) = 0 (67) { q4t − 2µ1βq4 = 0, q4 (t, r) = 0 (68) 244 akpanibah & ini / j. nig. soc. phys. sci. 3 (2021) 239–249 245{ q5t − 3µ1βq5 = 0, q5 (t, r) = 0 (69) solving (66), (67), (68) and (69) with their boundary conditions, we have q2 (t, r) = q3 (t, r) = q4 (t, r) = q5 (t, r) = 0, hence from (64) u2 (t, r, m) = ( q2 (t, r) + m 1 2 q3 (t, r) +mq4 (t, r) + m 3 2 q5 (t, r) ) = 0 (70) next, we attempt to solve (56), by assuming a solution of the form u3 (t, r, m) = ( q6 (t, r) + m 1 2 q7 (t, r) + mq8 (t, r) +m 3 2 q9 (t, r) + m2q9 (t, r) ) (71) u3t = q6t + m 1 2 q7t + mq8t + m 3 2 q9t+m2q10t , u3m = 1 2 m − 1 2 q7 + q8 + 3 2 m 1 2 q9+2mq10 u2mm = − 1 4 m − 3 2 q7 + 3 4 m − 1 2 q9 + 2q10, u1r = 0, u1rr = 0, u2rm = 0  (72) substituting (72) in (56), we have q6t + β (2β + 1)aq8 = 0,q6 (t, r) = 0 (73) q7t −µ1βq7 + 3 2 βa (3β + 1)q9 = 0,q7 (t, r) = 0 (74) q8t − 2µ1βq8 + 2βa (4β + 1)q10 = 0,q8 (t, r) = 0 (75) q9t − 3µ1βq9 = 0,q9 (t, r) = 0 (76) q10t − 4µ1βq10 = 0,q10 (t, r) = 0 (77) solving (73), (74), (75), (4,45) and (77), we have q6 (t, r) = q7 (t, r) = q8 (t, r) = q9 (t, r) = q10 (t, r) = 0. therefore, (71) reduces to u3 (t, r, m) = q6 (t, r) + m 1 2 q7 (t, r) + mq8 (t, r) + m 3 2 q9 (t, r) + m 2 q9 (t, r) = 0 (78) hence, from (48), (63), (70) and (78), we have e (t, r, s1) = uα (t, r, m) = u1 (t, r, m) + √ αu2 (t, r, m) + αu3 (t, r, m) = 1 2 the solution of equation (38) is given as f (t, r, s2) = v (t, r, `) = vα (t, r, `) = 1 2 where vα (t, r, `) = v1 (t, r, `) + √ αv2 (t, r, `) + αv3 (t, r, `) and v1 (t, r, `) = 1 2 , v2 (t, r, `) = 0, and v3 (t, r, `) = 0 assume{ f (t, r, s2) = v (t, r, `) , ` = s −2β 2 f (t, r, s2) = 1 2 (79) then ft = vt, fs2 = −2βs −2β−1 2 v` , fs2 s2 = 2β (2β + 1) s −2β−2 2 v` + 4β 2 s−4β−22 v``, fr = vr, frr = vrr, frs2 = −2βs −2β−1 2 vr`  (80) substituting (80) into (38), we have[ vt − 2µ2β`v` + bβ (2β + 1) v` + 2β2b`v`` +θ (µ0 − r) vr + 1 2 δ 2vrr − 2βeδρ √ `vr` ] = 0 (81) we can rewrite (81) as (m1 + m2 + m3) v = 0 (82) where m1 = [ θ (µ0 − r) vr + 1 2 δ2vrr ] v (83) m2 = [ vt + β (b (2β + 1) − 2µ2`v`) v` +2β2b`v`` ] v (84) m3 = [ −2βeδρ √ `vr` ] v (85) substituting (48) into (82) and replacing µ0 −rα(t) by α(µ0 −r), we have,( αm1 + m2 + √ αm3 ) vα = 0 (86) next, we conjecture a solution for (86) as follows vα (t, r, `) = v1 (t, r, `) + √ αv2 (t, r, `) + αv3 (t, r, `) (87) substituting (87) into (86), we have( αm1 + m2 + √ αm3 ) (v1 (t, r, `) + √ αv2 (t, r, `) + αv3 (t, r, `) ) = 0. simplifying the above equation, we arrive at( m2v1 (t, r, `) + [m2v2 (t, r, `) + m3v1 (t, r, `)] √ α + [m1v1 (t, r, `) + m2v3 (t, r, `) + m3v2 (t, r, `)] α ) = 0. this implies that{ m2v1 (t, r, `) = 0 v1 (t, r, `) = 1 2 , (88) { m2v2 (t, r, `) + m3v1 (t, r, `) = 0 v1 (t, r, `) = 1 2 , v2 (t, r, `) = 0 , (89) { m1v1 (t, r, `) + m2v3 (t, r, `) + m3v2 (t, r, `) v1 (t, r, `) = 1 2 , v2 (t, r, `) = 0 v3 (t, r, `) = 0 .(90) to obtain the solution of (87), we solve (88), (89) and (90). recall from (83), (84) and (85), equation (88), (89) and (90) can be expressed as{ v1t + β (b (2β + 1) − 2µ1`) v1` + 2β2b`v1`` = 0 v1 (t, r, `) = 1 2 , (91) 245 akpanibah & ini / j. nig. soc. phys. sci. 3 (2021) 239–249 246 v2t + β (b (2β + 1) − 2µ2`) v2` +2β2b`v2`` − 2βeδρ √ `v1r` = 0 v2 (t, r, `) = 0 , (92)  v3t + β (b (2β + 1) − 2µ2`) v3` + 2β2b`v3`` +θ (µ0 − r) v1r + 1 2 δ 2v1rr − 2βeδρ √ `v2r` v3 (t, r, `) = 0 . (93) let v1 (t, r, `) = r0 (t, r) + `r1 (t, r) (94) and v1t = q0t + `r1t, u1` = r1 , u1`` = 0. (95) substituting (95) in (91), we have r0t + βb (2β + 1)r1 = 0,r0 (t, r) = 1 2 (96) r1t − 2µ1`r1 = 0,r1 (t, r) = 0 (97) solving (97), we have r1 (t, r) = 0. (98) putting (98) into (96) and solving the resultant equation, we have r0 (t, r) = 1 2 . (99) hence from (94) v1 (t, r, `) = r0 (t, r) + `r1 (t, r) = 1 2 (100) next, we solve (92), by assuming a solution of the form v2 (t, r, `) = ( r2 (t, r) + ` 1 2 r3 (t, r) +`r4 (t, r) + ` 3 2 r5 (t, r) ) (101) and v2t = r2t + ` 1 2 r3t + `r4t + ` 3 2 r5t , v2` = 1 2 ` − 1 2 r3 + r4 + 3 2 ` 1 2 r5 v2`` = − 1 4 ` − 3 2 r3 + 3 4 ` − 1 2 r5, u1r` = 0  (102) substituting (102) into (92), we have r2t + β (2β + 1)br4 = 0,r2 (t, r) = 0 (103) r4t − 2µ2βr4 = 0,r4 (t, r) = 0 (104) r5t − 3µ2βr5 = 0,r5 (t, r) = 0 (105) r3t −µ2βr3 + 3 2 β (3β + 1)br5 = 0,r3 (t, r) = 0 (106) solving (103), (104), (105) and (106) with their boundary conditions, we have r2 (t, r) = r3 (t, r) = r4 (t, r) = r5 (t, r) = 0, hence from (101) v2 (t, r, `) = ( r2 (t, r) + ` 1 2 r3 (t, r) + `r4 (t, r) +` 3 2 r5 (t, r) ) = 0.(107) next, we attempt to solve (93), by assuming a solution of the form v3 (t, r, `) = r6 (t, r)+` 1 2 r7 (t, r)+`r8 (t, r)+` 3 2 r9 (t, r)+` 2 r9 (t, r)(108) v3t = r6t + ` 1 2 r7t + `r8t + ` 3 2 r9t+` 2r10t , u3` = 1 2 ` − 1 2 r7 + r8 + 3 2 ` 1 2 r9+2`r10 v2`` = − 1 4 ` − 3 2 r7 + 3 4 ` − 1 2 r9 + 2r10, v1r = 0, v1rr = 0, v2r` = 0  (109) substituting (109) in (93), we have r6t + β (2β + 1)br8 = 0,r6 (t, r) = 0, (110) r7t −µ2βr7 + 3 2 βb (3β + 1)r9 = 0,r7 (t, r) = 0, (111) r8t −2µ2βr8 + 2βb (4β + 1)r10 = 0,r8 (t, r) = 0,(112) r9t − 3µ2βr9 = 0,r9 (t, r) = 0, (113) r10t − 4µ2βr10 = 0,r10 (t, r) = 0. (114) solving (110), (111), (112), (4,82) and (114), we have r6 (t, r) = r7 (t, r) = r8 (t, r) = r9 (t, r) = r10 (t, r) = 0. therefore (108) reduces to v3 (t, r, `) = r6 (t, r) + ` 1 2 r7 (t, r) + `r8 (t, r) (115) + ` 3 2 r9 (t, r) + ` 2 r9 (t, r) = 0. finally, substituting (100), (107) and (115) into (87), we have f (t, r, s2) = vα (t, r, `) (116) = v1 (t, r, `) + √ αv2 (t, r, `) + αv3 (t, r, `) = 1 2 . from lemma 4, 4 and equation (39), proposition 4 is proved. the optimal investment plans are given as ϕ∗1 = 1 x2 sβ2 s 2β 1  (( σ221 + σ 2 22 ) (µ1 − r) s β 2 − (σ11σ21 + σ12σ22) (µ2 − r) s β 1 ) (σ211 + σ 2 12)(σ 2 21 + σ 2 22) − (σ11σ21 + σ12σ22) 2  ϕ∗2 = 1 x2 sβ1 s 2β 2  ( (σ211 + σ 2 12) (µ2 − r) s β 1 − (σ11σ21 + σ12σ22) (µ1 − r) s β 2 ) (σ211 + σ 2 12)(σ 2 21 + σ 2 22) − (σ11σ21 + σ12σ22) 2  ϕ∗0 = 1 −ϕ ∗ 1 −ϕ ∗ 2 recall that equation (22) and (23) are given as ϕ∗1 = [ csβ1 (µ2 − r) −bs β 2 (µ1 − r) ] x ( ab−c2 ) s2β1 s β 2 z gz + s1 x gs1 + (bd−ce) δρ x ( ab−c2 ) sβ1 gr ϕ∗2 = [ csβ2 (µ1 − r) −as β 1 (µ2 − r) ] x ( ab−c2 ) sβ1 s 2β 2 z gz + s2 x gs2 + (ae−cd) δρ x ( ab−c2 ) sβ2 gr 246 akpanibah & ini / j. nig. soc. phys. sci. 3 (2021) 239–249 247 figure 1. time evolution of the optimal investment plan ϕ∗0, ϕ ∗ 1, and ϕ ∗ 2. from proposition 4 and equation (22) we have g (t, r, s1, s2, z) = 1 z and g = x. differentiating g with respect to r, s1, s2, z and substitute into equation (30) and (31), we have ϕ∗1 = 1 x2 sβ2 s 2β 1   ( σ221 + σ 2 22 ) (µ1 − r) s β 2 − (σ11σ21 + σ12σ22) (µ2 − r) s β 1  (σ211 + σ 2 12)(σ 2 21 + σ 2 22) − (σ11σ21 + σ12σ22) 2  ϕ∗2 = 1 x2 sβ1 s 2β 2   ( σ211 + σ 2 12 ) (µ2 − r) s β 1 −(σ11σ21 + σ12σ22) (µ1 − r) s β 2 ( σ211 + σ 2 12 ) ( σ221 + σ 2 22 ) −(σ11σ21 + σ12σ22) 2  ϕ∗0 = 1 −ϕ ∗ 1 −ϕ ∗ 2 i. recall that from equation (33), j (t, r, s, z) = w (t, r, s) + v(t, r, s)lnz from proposition 4 and 4, the proof is completed. from lemma 4, we observed that our result is similar to the one in [3] but the difference between our result and theirs is that their interest rate is a constant while ours is stochastic. 5. numerical simulations in this section, some numerical simulations are presented to study the effect of some sensitive parameters on the optimal investment plans under logarithm utility. to achieve this, the following data will be used unless otherwise stated; table 1. parameters σ11 σ12 σ21 σ22 µ1 values 1.0 0.9 0.85 0.80 0.25 µ2 x r (0) s1 (0) s2 (0) β t 0.2 1 0.1 1.5 1.2 −1 3 figure 2. time evolution of ϕ∗0 with different elasticity parameter β. figure 3. time evolution of ϕ∗1 with different elasticity parameter β. figure 4. time evolution of ϕ∗2 with different elasticity parameter β. 6. discussion in this section, the effects of some parameters on the optimal investment plans are examined. in figure 1, the simulation of optimal investment plans for the three assets are given against time; the graph shows that at the initial time, the investor will invest more in the risk free asset and suddenly reduce it and then increase it continually till the expiration of investment. also, the investor will invest less in the two risky assets and suddenly 247 akpanibah & ini / j. nig. soc. phys. sci. 3 (2021) 239–249 248 figure 5. the effect of the risk free interest r on ϕ∗0. figure 6. the effect of the risk free interest r on ϕ∗0. increase it and then reduced it continually till the expiration of investment. the behavior of the optimal investment plans in figure 1 is due to o-u process used in modeling the risk free interest rate which reflects a fluctuation in interest rate. from this result, we can see that o-u process enable the investor to know when there is a change in interest rate for the risk free asset thereby adjusting the optimal investment plan of the different assets to suit the market condition at each point in time. the implication of this is that each time the risk free interest rate appreciate significantly, the investor will likely to invest more in the risk free asset while reducing that of the two risky asset in a way that he or she will maximize their portfolio. furthermore, we observed that the investor prefers investing more in stock 1 than stock 2; this is because the appreciation rate of stock 1 is higher than the appreciation rate of stock 2, thereby making stock 1 more attractive than stock 2 all other factors remaining constant. figure 2, presents simulation of ϕ∗0 against time with different values of the elasticity parameter β. the graph shows that as β reduces, the optimal investment plan ϕ∗0 for the risk free asset increases. conversely, figures 3 and 4, present simulations of the optimal investment plan against time with different values of the elasticity parameter β; the graphs show that as β reduces, the optimal investment plans for the two risky assets decrease. this is because a more negative elasticity parameter β gives rise to an increased probability of a large adverse movement in the stock market prices which lead to an increased expected decrease in volatility that comes with a lower β. therefore, the investor is advised to reduce the fraction of his or her investment in the two risky assets thereby hedging the volatility risk that comes with a decreased value of β. also, we observed that from equations (3) and (4), the stocks market prices are increasing functions of the appreciation rate µ1 and µ2 and a decreasing function of the instantaneous volatility as the elasticity parameter decreases. this implies that as β decreases, the risky assets become less attractive thereby leading to decrease in the optimal investment plan of the two risky assets. figure 5 and 6 give the analysis of the effect of the risk free interest rate on the optimal investment plan; in figure 5, the optimal investment plan for the risk free asset increases as the interest rate r increases. it is observed that the proportion invested in risk free assets increases faster at early of investment and very slow as retirement age gets closer; this implies that at the early stage of investment, the investor may not be interested in taking more risk but as retirement age draw closer, he invest more in the two risky assets with the aim of maximizing his portfolio. on the contrary, figure 6 shows that the optimal investment plan for the risky assets decreases with increase in risk free interest rate r. this can be attributed to two factors; first, investors are naturally attracted to high interest rate and if this happens, they will love to invest more in risk free assets for higher returns thereby reducing investment in risky assets. secondly, considering (µ1 − r) in ϕ∗1 where µ1, is the appreciation rate of stock 1 and r is the risk free interest rate. we observed that as r increases, (µ1 − r) decreases. hence, from figure 6, we observed that as the interest rate increases more than the appreciation rate, the investor invests less in stock 1 and when the interest rate is less than the appreciation rate, the investor invests more in stock 1. so, in general, we observed that all other factors being constant, the investor will choose his portfolio depending on the value of the risk free interest rate and the corresponding appreciation rates of the risky assets. finally, from proposition 4, we observed that optimal investment plan for an investor with logarithm utility is not affected by the risk averse coefficient unlike the result in [18] where the optimal investment plan depends on the investor’s behaviour towards risk. 7. conclusion in conclusion, this work studied the optimal investment plan for an investor with logarithm utility under the constant elasticity of variance model in the presence of stochastic interest rate modelled by the ornstein-uhlenbeck process. we developed a portfolio consisting of three assets (risk free asset and two risky assets). the legendre transformation and dual theory with asymptotic expansion technique was used to find closed form solutions of the optimal investment plans. also studied were the effects of some parameters on the optimal investment plans using numerical simulations. we observed that when the risk free asset is modelled by the o-u process and the two risky asset are modelled by the cev process, the optimal investment plan for the three assets give a fluctuation effect, showing that 248 akpanibah & ini / j. nig. soc. phys. sci. 3 (2021) 239–249 249 the investor’s behaviour in his investment pattern changes at different time intervals due to some information available in the financial market such as the fluctuations in the risk free interest rate, appreciation rates of the price of the risky assets and the volatility of the stock market price due to changes in the elasticity parameters. also, we observed that the investor’s behavior toward risk in the case of logarithm utility does not depend on the risk averse coefficient as in the case of [18] but depend on the factors mentioned above. acknowledgements the authors are very grateful to anonymous referees for their careful and diligent reading of the paper and helpful suggestions. this research has not received any sponsorship. references [1] d. li, x. rong & h. zhao, “optimal investment problem with taxes, dividends and transaction costs under the constant elasticity of variance model”, transaction on mathematics 12 (2013) 243. 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[22] m. jonsson & r. sircar, “optimal investment problems and volatility homogenization approximations”, modern methods in scientific computing and applications 75 (2002) 255. 249 j. nig. soc. phys. sci. 5 (2023) 1368 journal of the nigerian society of physical sciences approximate solution of space fractional order diffusion equations by gegenbauer collocation and compact finite difference scheme k. issa∗, a. s. olorunnisola, t. o. aliu, a. d. adeshola department of mathematics and statistics, kwara state university, malete, kwara state, nigeria. abstract in this paper, approximation of space fractional order diffusion equation are considered using compact finite difference technique to discretize the time derivative, which was then approximated via shifted gegenbauer polynomials using zeros of (n − 1) degree shifted gegenbauer polynomial as collocation points. the important feature in this approach is that it reduces the problems to algebraic linear system of equations together with the boundary conditions gives (n + 1) linear equations. some theorems are given to establish the convergence and the stability of the proposed method. to validate the efficiency and the accuracy of the method, obtained results are compared with the existing results in the literature. the graphical representation are also displayed for various values of β− gegenbauer polynomials. it can be observe in the tables of the results and figures that the proposed method performs better than the existing one in the literature. doi:10.46481/jnsps.2023.1368 keywords: caputo fractional derivative; stability; fractional diffusion equation; gegenbauer polynomials; compact finite difference method (cfdm). article history : received: 18 february 2023 received in revised form: 26 april 2023 accepted for publication: 10 may 2023 published: 22 may 2023 © 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: oluwatimilehin oluwadare 1. introduction in the last decade, significant interest has been given to the study of fractional calculus due to its application in applied mathematics. some of its application can be traced to modeling of some physical phenomena such as communication, robotics, transportation systems, finance, signal processing, genetic algorithms and the damping visco-elasticity as discussed in [1–4]. in recent time, the theory and applications of fractional calcu∗corresponding author tel. no: +2348036554437 email address: issa.kazeem@kwasu.edu.ng (k. issa ) lus have expanded enormously, its first appearance was in a letter written by gottfried wilhelm leibniz in 1695, as discussed in [5]. one of the popular areas of fractional calculus is the space fractional order diffusion equations (sfodes). several researchers have studied sfodes and applied different methods to find the approximate solution, [6, 7] employed finite difference method (fdm) to discretize the time derivative, then used gegenbauer polynomial as approximating polynomial, [8] developed tau method approach to find an approximant to sfodes, [9] employed second kind shifted chebyshev polynomials for solving sfodes, [10] proposed generalized shifted chebyshev polynomials for solving space fractional optimal 1 issa et al. / j. nig. soc. phys. sci. 5 (2023) 1368 2 control problems, new aspects of fractional biswas-milovic model with mittag-leffler law was reported in [11]. [12] employed riesz derivative to solve high-order solvers. [13] solved fractional rayleigh-stokes model in their paper titled ”numerical solution of fractional rayleigh-stokes model arising in a heated generalized second grade fluid”. [14] investigate nonlinear variable order fractional reaction-diffusion equation with mittag-leffler kernel. [15] developed lines and splines methods for solving sfodes. [16] employed spline method combined with richardson extrapolation to solve sfodes, [17] discussed shifted legendre spectral method for fractional-order multi-point boundary value problems, [18] analysis the initialboundary value problems for the linear time-fractional diffusion equations with a uniformly elliptic spatial differential operator of the second order and the caputo type time-fractional derivative acting in the fractional sobolev spaces, [19] proposed spatio-temporal homogenization function method for solving 1d fourth-order fractional diffusion-wave equations. [20] investigate caputo-fabrizio and fractal fractional derivative operator in the of studied hiv/aids fractional-order model is studied, [21] studied the fractional-order covid-19 epidemic model using laplace homotopy analysis method and [22] investigated the numerical solution of sfodes using chebyshev collocation method of the fourth kind and compact finite difference scheme. the main aim of this paper is to extend the work reported in [22] by introducing shifted gegenbauer polynomials as approximating polynomial un (x, t ) for solving sfodes using shifted gegenbauer collocation method to find the solution, since its solution generalizes the results of some other orthogonal polynomials such as legendre, jacobi polynomials pα,βn (x) of certain kinds with α = β and shifted chebyshev polynomials of certain kinds. in this paper, we want to employ compact finite difference method (cfdm) to discretize the time derivative, then the discretized sfodes will be collocated at (n − 1) points. these equations together with the boundary conditions generate (n + 1) system of linear equations. this paper is organized as follows. in section 2, we define caputo fractional order derivative, review some notable orthogonal polynomials, state the problem under consideration and some useful lemmas from the literature are recalled (see 2.1,2.2,2.3). shifted gegenbauer interms of differential operator is presented in section 3, we present the formulation of the scheme in section 3.1. we analyze the convergence and the stability of the proposed method in section 3.2, in section 4 numerical examples are presented with computational results and graphical representation to show the efficiency and the accuracy of the proposed method, concluding remarks are given in section 5. 2. preliminaries 2.1. caputo fractional order derivative operator the caputo fractional derivative operator dω of order ω is given as dω f (t) = 1 γ(n −ω) ∫ t 0 f (i)(t) (t − x)ω+1−i d x, i − 1 < ω < i, i ∈ n (1) where ω > 0 is the order of the derivative. the linearity property of the caputo fractional derivative is: dω(λ f (t) + µg(t)) = λdω f (t) + µdωg(t), (2) where, λ & µ are constants. the following results are obtained: dωt j =  0, for j ∈ n0 , j < dωe γ( j+1) γ( j+1−ω) t j−ω, for j ∈ n0 , j ≥ dωe (3) where dωe is the smallest integer greater than or equal to ω. 2.2. some of orthogonal polynomials some of the notable orthogonal polynomials ψ(t) are defined here: ψ(t) is an orthogonal polynomial with respect to the weight function ω(t) in an interval [a, b], if the inner product of ψ(t) is zero. that is, 〈ψm(t),ψn(t)〉 = ∫ b a ω(t)ψm(t)ψn(t)dt =  0, m 6= n λn, m = n (4) some of these orthogonal polynomials ψi(t) are: 2.2.1. legendre polynomials an orthogonal polynomial ψm(t) defined in an interval [−1, 1] is said to be a legendre polynomial if the weight function ω(t) = 1. the recurrence relation of a legendre polynomial is given as: pk+1(u) = 2k + 1 k + 1 upk(u) − k k + 1 pk−1(u), k ≥ 1, (5) with p0(u) = 1, p1(u) = u. 2.2.2. chebyshev polynomials the prominent one listed with their respective weight functions ω(t) in the interval [−1, 1] are found in [23] as: ψm =  tm(t) = cos(mt), ω(t) = 1 √ 1 − t2 , um = sin (m + 1) t sin (t) , ω(t) = √ 1 − t2, vm = cos ( m + 12 ) t cos ( t 2 ) , ω(t) = √ 1 + t 1 − t , wm = sin ( m + 12 ) t sin ( t 2 ) , ω(t) = √ 1 − t 1 + t . (6) 2 issa et al. / j. nig. soc. phys. sci. 5 (2023) 1368 3 the inner product for the third and fourth kinds chebyshev polynomials in the interval [−1, 1] are defined as: 〈ψm,ψn〉 =  〈vm, vn〉 = ∫ 1 −1 √ 1 + t 1 − t vm(t)vn(t)dt = 0, m 6= nπ, m = n , 〈wm, wn〉 = ∫ 1 −1 √ 1 − t 1 + t wm(t)wn(t)dt = 0, m 6= nπ, m = n . (7) while the fifth kind was reported in [14, 24]. 2.2.3. shifted gegenbauer polynomials gegenbauer polynomials c(β)i (t) defined in the interval [−1, 1] with respect to weight function ω(t) = (1 − t2)(β− 1 2 ) can be determined using c(β)i (t) = i∑ j=0 (−1) jγ(2β + 2i − j)γ(β + 12 ) (i − j)! γ(2β)γ( j + 1)γ(i − j + β + 12 ) ti− j. (8) the recurrence relation is given as c(β)i (t) = 1 i [ 2(i + β− 1)tc(β)i−1(t) − (i + 2β− 2)c (β) i−2(t) ] , i ≥ 2, (9) where cβ0 (t) = 1, c β 1 (t) = 2βt. to use this polynomial in the interval t ∈ [a, b] a shifted gegenbauer polynomial is defined by introducing the variable λ = 2t−(a+b)b−a . hence, the shifted gegenbauer polynomial in t is obtained as: c(β)∗i (t) = 1 i [ 2(i + β− 1) ( 2t − (a + b) b − a ) c(β)∗i−1 (t) (10) − (i + 2β− 2)c(β)∗i−2 (t) ] , i ≥ 2 where c(β)∗0 (t) = 1, c (β)∗ 1 (t) = 2β ( 2t−(a+b) b−a ) . the analytic form of the shifted gegenbauer polynomial c(β)∗i (t) is given as: c(β)∗i (t) = i∑ j=0 (−1) jγ(2β + 2i − j)γ(β + 12 ) (i − j)! γ( j + 1)γ(i − j + β + 12 )γ(2β) ti− j. (11) the orthogonality condition is 〈c(β)∗m (t), c (β)∗ n (t)〉 = ∫ 1 0 ( t − t2 )(α− 12 ) c(β)∗m (t)c(β)∗n (t)dt (12) =  0, for m 6= n π21−4αγ(n+2α) n![γ(α)]2 (n+α) , for m = n let f (x) be a square integrable function in [0, 1], expressing it in terms of the shifted gegenbauer polynomials as follows: f (x) = ∞∑ i=0 αic (β)∗ i (x), (13) where αi is defined as: αi = i! [γ(β)]2(i + β) π21−2βγ(i + 2β) ∫ 1 −1 (1 − x2)β− 1 2 f ( x + 1 2 ) cβi (x)d x, (14) or, αi = n! [γ(β)]2(i + β) π21−4βγ(i + 2β) ∫ 1 0 (x − x2)β− 1 2 f (x)c(β)∗i (x)d x, (15) (n + 1) terms of c(β)∗n (x) are considered in the approximation. then eq. (13) becomes f (x) = n∑ i=0 αic (β)∗ i (x). (16) 2.3. statement of the problem the problem under consideration and its initial and boundary conditions are: ∂u(x, t) ∂t = a(x) ∂ωu(x, t) ∂xω +b(x, t) 0 < x < ς, 0 ≤ t ≤ υ, 1 < ω < 2 (17) u(x, 0) = g(x), 0 < x < ς (18) u(0, t) = µ0(t), 0 ≤ t ≤ υ u(ς, t) = µ1(t), 0 ≤ t ≤ υ (19) eqs. (18) and (19) are initial and boundary conditions respectively. when ω = 2, eq. (17) becomes a classical diffusion equation, that is: ∂u(x, t) ∂t = a(x) ∂2u(x, t) ∂x2 + b(x, t). (20) here, we try to reduce the fractional derivative to a system of ordinary differential equations by utilizing the shifted gegenbauer polynomials and then solve the resulting system of equation using compact finite difference technique. we recall some lemmas to justify our finding let η denote an open bounded domain in r2 space and l2(η) represents a hilbert space with the inner product 〈p(x), q(x)〉 = ∫ η p(x)q(x)d x, (21) together with the euclidean norm ‖u(x)‖= 〈u(x),υ(x)〉2, and the sobolev space as h s(η) = u�l2(η), dsu d xs �l2(η) suppose that b(x) > 0 for 0 ≤ x ≤ 1 lemma 2.1. for any p, q ∈ h β 2 (η), 〈a d β x p, q〉 = 〈a d β 2 x p,x d β 2 b q〉, 〈x d β b p, q〉 = 〈x d β 2 b p,a d β 2 x q〉, for 1 < β < 2, 〈a d β x p,x d β b p〉 = cos(βπ)‖a d β x p‖ 2= cos(βπ)‖x d β b p‖ 2, f orβ > 0 3 issa et al. / j. nig. soc. phys. sci. 5 (2023) 1368 4 lemma 2.2. for the functions f (x) and 〈a d β x f (x)〉�h β(η), ∃ δτ > 0 3 ‖ f (x) + δ 2 b(xn)a d β x f (x)‖≤ ‖ f (x)‖, f or 1 < β < 2 see [22] for details. 3. shifted gegenbauer cfdm here, we employed the formula for fractional derivative f (x) derived in [7]: assume ω > 0, now using caputo linearity property, we have: dω fn (x) = n∑ i=0 αi d ω ( c(β)∗i (x) ) . (22) by linearity property and eq. (3) we obtain: dω ( c(β)∗i (x) ) = 0, i = 0, 1, . . . ,dωe− 1,ω > 0 (23) dω ( c(β)∗i (x) ) = i∑ j=0 (−1) jγ(2β + 2i − j)γ(β + 12 ) (i − j)! γ( j + 1)γ(i − j + β + 12 )γ(2β) (24) × dωxi− j, i ≥ dωe applying eq. (3) to eq. (24), we obtain: dω ( c(β)∗i (x) ) = i−dωe∑ j=0 (−1) jγ(2β + 2i − j)γ(β + 12 ) γ( j + 1)γ(i − k + β + 12 )γ(2β)γ(i + 1 − j −σ) (25) × xi− j−ω substituting (25) into (22) to obtain: dω(gn (x)) = (26) n∑ i=dωe i−dωe∑ j=0 αi (−1) jγ(2β + 2i − j)γ(β + 12 ) γ( j + 1)γ(i − j + β + 12 )γ(2β)γ(i + 1 − j −σ) xi− j−ω let ni, j = (−1) jγ(2β + 2i − j)γ(β + 12 ) γ( j + 1)γ(i − j + β + 12 )γ(2β)γ(i + 1 − j −σ) (27) simplifying eq. (26) gives dw( fn (x)) = n∑ i=dωe i−dωe∑ j=0 αi ni, j x i− j−ω. (28) see [7] for for more details 3.1. formulation of the scheme solving eq.(17) based on cfdm, using gegenbauer collocation, given two integers m, n > 0 and two mesh points τm−1 = (m − 1)δτ f or m = 1, 2, ..., m + 1, δτ = t m . introducing taylor’s expansion for the time-discretizing that is: ∂u(xn, tm) ∂t = δτu(xn, tm) + δτ 2 ∂2u(xn, tm) ∂t2 + 0(δτ2) (29) where δτu(xn, tm) = umn −u m−1 n δτ , substituting eq.(29) into eq.(3) δτu(xn, tm)+ δτ 2 ∂2u(xn, tm) ∂t2 +0(δτ2) = a(xn) ∂wu(xn, tm) ∂xw +b(xn, tm) (30) differentiating eq.(17) ∂2u(x, t) ∂t2 = a(xn)δτ ∂wu(xn, tm) ∂xw + δτb(xn, tm) (31) substituting eq.(31) in eq.(30) gives: δτu(xn, tm) = a(xn) ∂wu(xn, tm) ∂xw + b(xn, tm) (32) − δτ 2 [ a(xn)δτ ∂wu(xn, tm) ∂xw + δτb(xn, tm) ] + · · · simplifying eq.(32) gives umn − u m−1 n = δτ 2 a(xn) ∂wumn ∂xw + δτ 2 a(xn) ∂wum−1n ∂xw + δτ 2 (bmn + b m−1 n ) (33) + rn(x)(δτ)3 where,rn(x) is the truncation term, let u(xn, tm) = u mn u mn − δτ 2 a(xn) ∂wu mn ∂xw = u m−1n + δτ 2 a(xn) ∂wu m−1n ∂xw (34) + δτ 2 (bmn + b m−1 n ) + r n(x)(δτ)3, to obtain full discrete form, it is required to approximate the caputo derivative in ∂ w u mn ∂xw . the approximant un (x, t) is constructed using the gegenbauer collocation technique. un (x, t) = n∑ k=0 uk(t)c (β)∗ k (x) (35) using eqs. (34),(35) and the roots of shifted gegenbauer polynomial of degree n − 1, we obtain the simplified form as n∑ k=0 uk(t)c (β)∗ k (xn) − δτ 2 a(xn) n∑ k=dwe k−dwe∑ i=0 uk(t)nk,i x k−i−w = n∑ k=0 um−1k c (β)∗ k (xn) + δτ 2 a(xn) n∑ k=dwe k−dwe∑ i=0 uk(t)nk,i x k−i−w + δτ 2 (b(xn, tm) + b(xn, tm−1)) (36) 4 issa et al. / j. nig. soc. phys. sci. 5 (2023) 1368 5 with the boundary conditions: u(0, t) = n∑ n=0 (−1)nγ(n + 2β) n! γ(2β) un(t) = µ0, u(1, t) = n∑ n=0 γ(n + 2β) n! γ(2β) un(t) = µ1 (37) eqs. (36) and (37) gives (n +1) system of linear equations, then obtain un, n = 0, 1, · · · , n. to get the initial condition u 0n of eq.(36), the initial condition of the problem u(x, 0) combining eq. (15) will be used to obtain u 0n . 3.2. convergence and stability analysis in this section, we consider convergence and stability of the proposed method. let η denote an open bounded domain in r2 space and l2(η) represents a hilbert space with the inner product 〈p(x), q(x)〉 = ∫ η p(x)q(x)d x, (38) together with the euclidean norm ‖u(x)‖= 〈u(x),υ(x)〉2, and the sobolev space as h s(η) = u ∈ l2(η), dsu d xs ∈ l2(η) suppose that b(x) > 0 for 0 ≤ x ≤ 1 the following lemmas are required to investigate the stability and convergence of the method. lemma 3.1. let u j ∈ h1(η), j = 1, 2, ..., j be the solution of equation (34)and u 0 be the initial condition, then the following inequality holds ‖u j‖≤ ‖u 0‖+ max 0≤τ≤n δτ 2 (‖b ji‖+‖b j−1 i ), (39) where, u j = u(x, t j) proof: by mathematical induction on j. for j = 1 in eq. (34), we obtain u 1− δτ 2 a(xi)a d ω x u 1 = u 0 + δτ 2 a(xi) a d ω x u 0 + δτ 2 (b1 +b0), (40) multiplying eq. (40) by u 1 and integrating on η, we get: ‖u 1‖2− δτ 2 a(xi)〈a d ω x u 1, u 1〉 = 〈u 0, u 1〉 + δτ 2 a(xi)〈a d ω x u 0, u 1〉 + δτ 2 (〈b1, u 1〉 + 〈b0, u 1〉). (41) since cos ( β 2 π ) < 0, using lemma 2.1, we have 〈a d β xu 1, u 1〉 = 〈a d β 2 x u 1,x d β 2 b u 1 〉 = cos ( β 2 π ) ‖a d β 2 x u 1 ‖ 2< 0 (42) since cos( β2 π) < 0, for 1 < β < 2. from the left side of eq. (41), we obtain ‖u 1‖2≤ ‖u 1‖2− δτ 2 a(xi)〈a d β xu 1, u 1〉 (43) using cauchy-schwartz inequality, the right side of eq. (41) becomes ‖〈u 0, u 1〉 + δτ 2 a(xi)〈a d ω x u 0, u 1〉‖ ≤ ‖u 0 + δτ 2 a(xi) a d ω x u 0 ‖‖u 1|≤ ‖u 0‖‖u 1‖ (44) using eqs. (42),(43), and (44), to obtain ‖u 1‖≤ ‖u 0‖+ max 0≤n≤n δτ 2 (‖b ji‖+‖b j−1 i ‖) (45) assume eq. (39) is true for k = 1, 2, ..., j − 1, then ‖u k‖≤ ‖u 0‖+ max 0≤n≤n δτ 2 (‖bki ‖+‖b k−1 i ‖) (46) now, multiplying eq. (34) by u j and integrating on η ‖u j‖2− δτ 2 a(xi)〈a d ω x u j, u j〉 = 〈u j−1, u j〉 + δτ 2 a(xi)〈a d ω x u j−1, u j〉 + δτ 2 (〈b j, u j〉 + 〈b j−1, u j〉) (47) from the previous procedure, eq. (47) becomes ‖u j‖≤ ‖u j−1‖+ max 0≤i≤n δτ 2 (‖b ji‖+‖b j−1 i ‖) theorem 3.2. the approximation method introduced in equation (34) is unconditionally stable. proof let u ji , j = 1, 2, ..., j be the approximate solution obtained in eq.(34) with the initial condition u 0i = u(xi, 0), then the error ε j = u(xi, t j) − u j i satisfies ε j − δτ 2 p(xi) ∂βε j ∂xβ − δτ 2 p(xi) ∂βε j−1 ∂xβ = ε j−1. (48) according to lemma 3.1, ‖ε j‖≤ ‖ε0‖ for, j = 1, 2, . . . , m, hence the proof. 4. numerical experiments in this section, we validate the efficiency and the accuracy of the proposed method (pm) on some selected examples from the literature by comparing the maximum error �n with the existing results in the latest literature. where �n = max 0≤i≤n |u(xi, t ) − un (xi, t )| (49) 5 issa et al. / j. nig. soc. phys. sci. 5 (2023) 1368 6 example 4.1 considering sfode [7, 9, 22] ∂u(x, t) ∂t = γ ( 6 5 ) x 9 5 ∂ 9 5 u(x, t) ∂x 9 5 + 3x2(2x − 1) exp(−t) u(x, 0) = x3(x−1 − 1), u(0, t) = u(1, t) = 0, t > 0. (50) with the closed form solution u(x, t) = x3(x−1 − 1) exp(−t). comparing eq. (50) with (17), we have a(x) = γ ( 6 5 ) x 9 5 , b(x, t) = 3x2(2x − 1) exp(−t), ω = 9 5 for approximation of degree 3, that is n = 3, eqs. (35) and (36) gives u3(x, t) = 3∑ k=0 umk (t)c (β)∗ k (x) (51) um0 c (β)∗ 0 (xn) + u m 1 c (β)∗ 1 (xn) + u m 3 c (β)∗ 3 (xn) + δτ 2 a(xn) [ um2 n2,0 x 0.2 + um3 (n3,0 x 1.2 + n3,1 x 0.2) ] = um−10 c (β)∗ 0 (xn) + u m−1 1 c (β)∗ 1 (xn) + u m−1 3 c (β)∗ 3 (xn) + δτ 2 a(xn) [ um−12 n2,0 x 0.2 + um−13 (n3,0 x 1.2 + n3,1 x 0.2) ] + δτ 2 (b(xn, tm) + b(xn, tm−1)), (52) respectively. at x = x0, collocating at zeros of c (β)∗ 2 (x), we obtain um0 c (β)∗ 0 (x0) + u m 1 c (β)∗ 1 (x0) + u m 3 c (β)∗ 3 (x0) + δτ 2 a(x0) [ um2 n2,0 x 0.2 + um3 (n3,0 x 1.2 + n3,1 x 0.2) ] = um−10 c (β)∗ 0 (x0) + u m−1 1 c (β)∗ 1 (x0) + u m−1 3 c (β)∗ 3 (x0) + δτ 2 a(x0) [ um−12 n2,0 x 0.2 + um−13 (n3,0 x 1.2 + n3,1 x 0.2) ] + δτ 2 (b(x0, tm) + b(x0, tm−1)) (53) we have c(β)∗0 (x0) = 1. let g1 = c (β)∗ 1 (x0), g2 = c (β)∗ 3 (x0), h1 = δτ 2 a(x0)n2,0 x 0.2, h2 = δτ 2 a(x0)(n3,0 x 1.2 + n3,1 x0.2) the equation becomes um0 + u m 1 g1 + u m 3 g2 + u m 2 h1 + u m 3 h2 = um−10 + u m−1 1 g1 + u m−1 3 g2 + u m−1 2 h1 + u m−1 3 h2 (54) at x = x1, we get um0 c (β)∗ 0 (x1) + u m 1 c (β)∗ 1 (x1) + u m 3 c (β)∗ 3 (x1) + δτ 2 a(x1) [ um2 n2,0 x 0.2 + um3 (n3,0 x 1.2 + n3,1 x 0.2) ] = um−10 c (β)∗ 0 (x1) + u m−1 1 c (β)∗ 1 (x1) + u m−1 3 c (β)∗ 3 (x1) + δτ 2 a(x1) [ um−12 n2,0 x 0.2 + um−13 (n3,0 x 1.2 + n3,1 x 0.2) ] + δτ 2 (b(x1, tm) − b(x1, tm−1)). (55) let g11 = c (β)∗ 1 (x0), g22 = c (β)∗ 3 (x0), h11 = δτ 2 a(x0)n2,0 x 0.2, h22 = δτ 2 a(x0)(n3,0 x 1.2 + n3,1 x0.2) eq. (55) becomes um0 + u m 1 g11 + u m 3 g22 + u m 2 h11 + u m 3 h22 = um−10 + u m−1 1 g11 + u m−1 3 g22 + u m−1 2 h11 + u m−1 3 h22. (56) from eq. (37) we obtain um0 (t)− γ(1 + 2β) γ(2β) um1 (t) + γ(2 + 2β) 2γ(2β) um2 (t)− γ(3 + 2β) 6γ(2β) um3 (t) = µ0, (57) and um0 (t) + γ(1 + 2β) γ(2β) um1 (t) + γ(2 + 2β) 2γ(2β) um2 (t) + γ(3 + 2β) 6γ(2β) um3 (t) = µ1. (58) eqs. (54),(56)-(58) can be written as aum = bum−1 + δτ 2 ( bmn − b m−1 n ) , (59) or um = a−1 bum−1 + δτ 2 ( bmn − b m−1 n ) , (60) where a =  1 g1 h1 g2 + h2 1 g11 h11 g22 + h22 1 −γ(1+2β) γ(2β) γ(2+2β) 2γ(2β) − γ(3+2β) 6γ(2β) 1 γ(1+2β) γ(2β) γ(2+2β) 2γ(2β) γ(3+2β) 6γ(2β)  , b =  1 g1 h1 g2 + h2 1 g11 h22 g33 + h22 0 0 0 0 0 0 0 0  , um =  um0 um1 um2 um3  table 1 shows the computational errors at different values of β while figure 1a is the corresponding figure. table 2 display the comparison of the absolute errors at β = 0.5 with the existing results in the literature at β = 0.5. example 4.2 consider the space fractional order diffusion problem [7, 9] ∂u(x, t) ∂t = γ(2.2) 6 x2.8 ∂ 3 2 u(x, t) ∂x 3 2 − x3(1 + x) exp(−t) u(x, 0) = x3 u(0, t) = 0, u(1, t) = exp(−t), t > 0. the exact solution is u(x, t) = x3e−t. table 3 display the comparison of the maximum errors at different values of β and at β = 1 in [7] is equivalent to the results obtained in [9]. figure 2 is the graphical representation of the exact solution and its corresponding approximate solutions at different values of β while figure 3 display the absolute errors at various values of β. 6 issa et al. / j. nig. soc. phys. sci. 5 (2023) 1368 7 figure 1: example 4.1, relationship between exact and approximate solution at n = 3, & t = 1 example 4.3 consider the space fractional diffusion problem [7] ∂u(x, t) ∂t = γ ( 3 2 ) x 1 2 ∂ 3 2 u(x, t) ∂x 3 2 − 2x sin(t + 1) + (x2 + 1) cos(t + 1) u(x, 0) = (x2 + 1) sin(1) u(0, t) = sin(t + 1), u(1, t) = 2 sin(t + 1), t > 0. the exact solution is u(x, t) = (x2 + 1) sin(t + 1) comparison of the maximum errors are display in table 4. figure 4 is the graphical representation of the absolute errors at various values of β. 5. discussion of results and conclusion 5.1. discussion of results table 1 shows the absolute errors of the proposed method for example 4.1 with respect to the values of β at n = 3 and t = 1. the absolute errors of the proposed method is compared with the results in [22],[7] and [9] as shown in table 2. the approximate solutions at β = 1 and β = 1.5 correspond to the table 1: example 4.1, absolute errors for various values of β x β = 0.5 β = 1 β = 1.5 0 0 2.2959 × 10−41 4.4409 × 10−16 0.1 1.0738 × 10−9 1.2324 × 10−9 1.1618 × 10−9 0.2 1.7375 × 10−9 1.9252 × 10−9 1.8596 × 10−9 0.3 2.0553 × 10−9 2.1777 × 10−9 2.1705 × 10−9 0.4 2.0916 × 10−9 2.0890 × 10−9 2.1718 × 10−9 0.5 1.91075 × 10−9 1.7583 × 10−9 1.9406 × 10−9 0.6 1.5770 × 10−9 1.2849 × 10−9 1.5542 × 10−9 0.7 1.1547 × 10−9 7.6791 × 10−10 1.0897 × 10−9 0.8 7.0821 × 10−10 3.0655 × 10−10 6.2437 × 10−10 0.9 3.0187 × 10−10 2.1667 × 10−17 2.3540 × 10−10 1 0 2.1667 × 10−17 8.8818 × 10−16 table 2: exapmle 4.1, comparison of absolute errors x [9] [22] [7] pm 0 0 4.77 × 10−6 0 0 0.1 5.33 × 10−6 3.17 × 10−9 5.46 × 10−6 1.07 × 10−9 0.2 8.06 × 10−6 5.85 × 10−9 8.51 × 10−6 1.74 × 10−9 0.3 8.72 × 10−6 7.97 × 10−9 9.60 × 10−6 2.06 × 10−9 0.4 7.84 × 10−6 9.44 × 10−9 9.18 × 10−6 2.09 × 10−9 0.5 5.96 × 10−6 1.02 × 10−8 7.69 × 10−6 1.91 × 10−9 0.6 3.59 × 10−6 1.01 × 10−8 5.60 × 10−6 1.58 × 10−9 0.7 1.29 × 10−6 9.12 × 10−9 3.33 × 10−6 1.15 × 10−9 0.8 4.32 × 10−7 7.17 × 10−9 1.34 × 10−6 7.08 × 10−10 0.9 4.04 × 10−6 4.16 × 10−9 8.39 × 10−8 3.02 × 10−10 1 0 7.55 × 10−17 0 0 table 3: example 4.2, maximum error for various values of β β [7] pm β = 0.5 2.25 × 10−06 6.30 × 10−09 β = 1 8.83 × 10−06 1.32 × 10−08 β = 1.5 8.35 × 10−06 3.69 × 10−08 7 issa et al. / j. nig. soc. phys. sci. 5 (2023) 1368 8 table 4: maximum error for example 4.3 relative to values of β β [7] pm β = 0.5 1.56 × 10−05 8.23 × 10−09 β = 1 1.36 × 10−05 2.01 × 10−08 β = 1.5 11.05 × 10−05 1.06 × 10−09 figure 2: example 4.2, relationship between exact and approximate solutions at n = 3, & t = 1 figure 3: absolute errors for example 4.2 at n = 3, & t = 1 figure 4: absolute errors for example 4.3 at n = 3, & t = 1 approximate solutions when using second kind shifted chebyshev polynomial u∗j (x) and jacobi ( j, 1, 1, x) (that’s p 1,1 j (x) as an approximation polynomials. tables 2-4 show the comparison of the errors at various values of β relative to the existing results in the literature. figures 1-2 are the comparison of the solution for various values of β, while figures 3 and 4 represent the absolute errors of the proposed method at various values of β. 5.2. conclusion this paper studied space fractional order diffusion equation by proposing a new technique for finding the approximate solution using compact finite difference method to discretize the time derivative, then use shifted gegenbauer polynomials as approximating polynomial. the computational results are plotted and detailed in the tables to justify the accuracy of the propose method. it can be observe in the tables of the results and figures that the proposed method performs better than the existing one in the literature. references [1] j. t. machado, v. kiryakova & f. mainardi, “recent history of fractional calculus”, commun. nonlinear sci. numer. simul., 16 (2011) 1140. 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[24] w. m. abd-elhameed1, & y. h. youssri, “fifth-kind orthonormal chebyshev polynomial solutions for fractional differential equations”, comp. appl. math. 37 (2018) 2897. 9 j. nig. soc. phys. sci. 5 (2023) 1055 journal of the nigerian society of physical sciences modelling the transmission dynamics of omicron variant of covid-19 in densely populated city of lagos in nigeria bolarinwa bolajia,d,∗, u. b. odionyenmab, b. i. omedea,d, p. b. ojiha, abdullahi a. ibrahimc amathematical sciences department, prince abubakar audu university, anyigba, nigeria bmathematics department, federal university of technology, owerri, nigeria cmathematical sciences department, baze university, abuja, nigeria dlaboratory of mathematical epidemiology and applied sciences, prince abubakar audu university, anyigba, nigeria abstract the kernel of the work in this article is the proposition of a model to examine the effect of control measures on the transmission dynamics of omicron variant of coronavirus disease in the densely populated metropolis of lagos. data as relate to the pandemic was gathered as officially released by the nigerian authority. we make use of this available data of the disease from 1st of december, 2021 to 20th of january, 2022 when omicron variant was first discovered in nigeria. we computed the basic reproduction number, an epidemiological threshold useful for bringing the disease under check in the aforementioned geographical region of the country. furthermore, a forecasting tool was derived, for making forecasts for the cumulative number of cases of infection as reported and the number of individuals where the omicron variant of covid-19 infection is active for the deadly disease. we carried out numerical simulations of the model using the available data so gathered to show the effects of non-pharmaceutical control measures such as adherence to common social distancing among individuals while in public space, regular use of face masks, personal hygiene using hand sanitizers and periodic washing of hands with soap and pharmaceutical control measures, case detecting via contact tracing occasioning clinical testing of exposed individuals, on the spread of omicron variant of covid-19 in the city. the results from the numerical simulations revealed that if detection rate for the infected people can be increased, with majority of the population adequately complying with the safety protocols strictly, then there will be a remarkable reduction in the number of people being afflicted by the scourge of the highly communicable disease in the city. doi:10.46481/jnsps.2023.1055 keywords: coronavirus, face masks, contour plots, social distancing, case-detection, data-fitting, stability article history : received: 11 september 2022 received in revised form: 06 march 2023 accepted for publication: 07 march 2023 published: 17 april 2023 c© 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: oluwaseun adeyeye 1. introduction most humans inhabiting the planet earth are currently under the yoke of covid-19 pandemic; mankind is being tormented ∗corresponding author tel. no: +2348034995772 email address: bolarinwa.s.bolaji@gmail.com (bolarinwa bolaji ) once again as it was on planet earth in 1918 when a worldwide pandemic occurred. though the origin of the current pandemic is shrouded in mystery, the pandemic was reported to have first broken out in the city of wuhan in china, in the year 2019, where a number of individuals who fell ill after having patronised a market where sea foods are sold in wuhan. they were admitted to hospitals in december 2019, they were first diag1 bolaji et al. / j. nig. soc. phys. sci. 5 (2023) 1055 2 nosed with pneumonia [1]. as of 22nd of january 2020, 571 cases of individuals infected with the coronavirus disease were confirmed as having contacted the disease in several parts of china [1, 2]. as of 30th january, 2020, there was confirmation of 7734 cases in china while 90 individuals infected with the disease were reported across europe involving 13 countries; america and some parts of asia [3, 1]. it was reported that through-out the world, a case of about 5,656,615 of covid-19 has occurred with 355,355 deaths as at may 27, 2020 [4]. the disease was first reported in nigeria on february 27, 2020 [5], with 8,733 cases resulting in 254 deaths and 5,978 active cases of infection requiring treatment, whereas 2,501 cases of discharged patients were reported by may 27, 2020. as of 23rd july, 2020, a total number of 38,948 cases, with 833 deaths, and 22,887 active number of cases of individuals undergoing treatment in different isolation centres were announced by the nigerian authority as a nationwide situation of the disease while as of the same date, particularly for abuja, one of the epicentres of the disease, 2,957 total number of cumulative cases and 78 new cases, 39 deaths, and 2,042 active cases of covid-19 were announced. however, it is worth mentioning that for a country of over 200 hundred million people, only less than half a million people were tested for the virus, unlike the united states, china, canada and other advanced countries of the world have tested millions of their people for the virus. obviously, hundreds of thousands of cases might have been detected in nigeria if adequate tests were carried out. the expectation is that it will become very clear the correct situation of the disease in nigeria as there is improvement in contact tracing and the rate of testing [6]. when a healthy individual makes contact with an infected person, then there will be human-to-human transmission of the disease by way of inhaling droplets from the atmosphere or making contact with contaminated surfaces [7]. it is revealed through clinical evidence that the range of incubation period of the disease is between 2 to 14 days, in which case, infected people not showing any symptoms of the disease and may not be conscious of having being infected of the deadly disease during this period are capable of transmitting the disease to those around them. coughing, breathing difficulties, and fever, are the symptoms of the disease [8]. it should be noted that there are no antiviral drugs or vaccines that have been discovered for the prevention or management of deadly disease for now [9]. consequently, heavy reliance is on detecting in time and the quarantine of symptomatic individuals as measures for guiding against the spread of the disease. among other strategies to combat the spread of the disease are impositions of lockdown, strict enforcement of safety protocols, avoidance of crowded events, wearing of highly efficacious face masks by individuals while in public spaces, maintenance of a specific distance between individuals [10]. mathematical models are a great way to study the transmission and control of contagious illnesses (see the models in [6,11-17]). a number of researchers have developed mathematical models to examine the transmission and control of coronavirus disease in a number of nations. this include deterministic model without demographic parameters but incorporating non-pharmaceutical control measures through which a study of the transmission dynamics of the deadly disease in nigeria was made [6]. in their findings, they discovered that when the inhabitants of the sub-saharan country can adhere strictly to the covid-19 safety protocol, the affliction associated with the scourge of the highly contagious disease will be adequately mitigated and the curve of the transmission of the disease will be ultimately flattened in no time. in their work, khan and atagana proposed a deterministic model with fractional order derivatives, through which they obtained approximate solution to the system of fractional order differential equations so formulated [18]. likewise, muhammed and atangana formulated a system of fractional order differential equations to gain insights into the dynamics of the transmission of the disease. using real life data obtained from the city of wuhan, they did simulation of their model through which some thresholds as regards the control of the disease in the community were obtained. major recommendation from their work was that scientists must expedite action at obtaining cure for the disease so that humanity can be saved from the scourge of the deadly disease [8]. sowole et al studied the early stage of the disease using regression model to make forecast and projections that will help in the control of the spread of the disease [19]. waku et al proposed an approach on how to estimate the highest number of reproduction numbers of covid-19. they made recommendations has to measures that can be put in place to bring the values of the reproduction numbers down towards effectively controlling the spread of the highly communicable disease in a community [20]. as a result, the primary goal of this research is to develop a system of nonlinear equations that can be used to investigate the extent to which non-pharmaceutical strategies can be used to combat the scourge of coronavirus disease in nigeria. the major aim of this work is the provision of the detailed assessment of the spread of the disease in the city. we shall do this by parameterizing our model, applying the number of cases of infection reported and the number of actively infected individuals in lagos as obtained from the statutory authority officially saddled with such a responsibility. consequently, we will do the quantitative study of the model by using relevant data gathered carefully towards assessing the impact of the afore-mentioned non-pharmaceutical measures such as adherence to safety protocols for covid-19 on the spread of the disease in lagos. the sole purpose of doing this is to be able to make forecasts and projections towards ascertaining the time the curve of the spread of the disease will be flattened. consequently, the scope of this work is formulation of deterministic epidemiological model without demographic parameters in form of system of non-linear differential equations by incorporating the safety strategies to adequately capture the dynamics of the transmission of coronavirus vis-à-vis the safety protocols being enforced by the nigerian health policy makers that are needed to put under control the spread of the disease. while the specific objectives of the study are: (i) to rigorously analyse the model for its basic properties: local and global asymptotic stability with a view to obtaining thresholds needed as necessary and sufficient con2 bolaji et al. / j. nig. soc. phys. sci. 5 (2023) 1055 3 ditions for the effective control of the spread of the disease. (ii) to compute the final size of the epidemic in lagos city, nigeria. (iii) to carry out uncertainty and sensitivity analysis that will help determining the top rank parameters that play dominant role in the dynamics of the spread of the deadly disease with a view to making them available for the health care policy makers to target towards effectively controlling the spread of the disease in nigerian community. (iv) to do data fitting to our model using real life data that were obtained from the relevant nigeria authority for the disease control towards estimating adequate parameter values that will be needed to carry out numerical simulations of the model so formulated. (v) using some parameters values so obtained from data fitting to model initially carried out in addition to other parameters values so obtained from literature, we shall conduct numerical simulation of the model with a view to obtain imitation of current real life realities as it pertains to covid-19 disease. (vi) to develop a predictive tool that will be of help to make forecasts and projections about the dynamics of the disease. (vii) to interprete the plots gotten from the numerical simulations of the model with a view to ascertaining measures that will help in the control of the spread of the disease. (viii) to make adequate recommendations that will be of great help to healthcare policy makers in formulation of policies that will help curtail the spread of the disease and ultimately help put it under control. the following, therefore, is how we organized this article. the formulation of the mathematical modeling of the disease was given in section 2. in section 3, we established the model’s basic features and their stability. following that, in section 4, we ran numerical simulations of the model, fitted pertinent data to the model for the time period under consideration, and presented the numerical results. 2. description of how the model was formulated the total inhabitants of the city at time (t), represented by n(t) has its components being susceptible individuals s (t), exposed latently-infected individuals e(t) , exposed individuals that are latently-infected and quarantined eq(t), infected individuals that are asymptomatic ia(t), infected individuals that are symptomatic is (t), infected individuals who are asymptomatic and symptomatic and who are diagnosed via contact tracing occasioned by testing id(t), infected individuals who are treated after case detection it (t) and those people who recovered from the disease r(t). it is presumed that those in the id(t) class are people in isolation centres and are segregated from the general population. thus, n(t) = s (t) + e(t) + eq(t) + ia(t) + is (t) + id(t) + it (t) + r(t) the susceptible population class s u (t) are made up of all living individuals in the geographical region under consideration, people progressed out of this class at the rate λ = βc (αia +is +ηt it ) n−(eq +id +it ) , where λ is the force of infection, α being the tuning parameter which accounts for the decreased infectiousness in the infected asymptomatic class ia (t) as compared to individuals in the is (t) class; ηt (for ηt ≥ 1) is the risk associated with infectiousness of individuals in the treated class it (t) in comparison with individuals in the classes ia (t) and is (t); and βc is the effective rate of contact with infected person, with the fraction, f , where 0 < f < 1 is for individuals that progressed to latently-infected class e (t) and the remaining fraction (1 − f ) are for the individuals who are latently-infected and progressed to eq(t) class. the progression rate for individuals moving from e (t) to latently-infected quarantined class eq(t) is given by δ while those that progressed from eq(t), do so at the rate qσ and (1 − q) σ to actively infected classes ia (t) and is (t) respectively for 0 < q < 1 while the individuals that progressed from these actively-infected classes to the class of those that are diagnosed of the disease via contact tracing and testing in the class id (t) do so at the rate θ and ψ respectively. individuals that moved from detected infected class progress at the rate γ j while individuals that recovered from actively infected classes ia (t) , is (t) , id (t) and treated class it (t) do so at the rate γa,γi, γ0, and γb respectively. the proposed nonlinear system of differential equations are as follows (see figure 1 for a schematic diagram and table 1 for a summary of the model’s variables and parameters): ds dt = −βc (αia + is + ηt it ) n − ( eq + id + it ) s de dt = f βc (αia + is + ηt it ) n − ( eq + id + it ) s −δe deq dt = (1 − f ) βc (αia + is + ηt it ) n − ( eq + id + it ) s + δe −σeq dia dt = qσeq − (θ + γa) ia dis dt = (1 − q) σeq − (ψ + γ0 + d0) is did dt = θia + ψis − ( γi + dd + γ j ) is dit dt = γ j id − (γb + dt ) it dr dt = γi id + γa ia + γ0 is + γb it (1) the model’s assumptions are listed as follows: (i) since we are dealing with a pandemic in this work, then demographic parameters, natural birth rate, and death rate are excluded from our model in line with similar previous works [6,9,21-14]. 3 bolaji et al. / j. nig. soc. phys. sci. 5 (2023) 1055 4 figure 1. schematic diagram of model (1) with λ = βc (αia +is +ηt it ) n−(eq +id +it ) . (ii) susceptible individuals who made contact with infected individuals diagnosed of covid-19 after testing are quarantined and are in class eq(t) of the model. (iii) in line with clinical evidence, people undergoing treatment are open to being infected with the disease. due to strong antibodies that often kill the virus in healthy individuals before it starts wreaking havoc in the system, there is natural recovery from the infection with the disease. (iv) death rates due to the disease are the same in all infected compartments of the model. our model (1) is an extension of models that have been formulated towards gaining insights into how the novel coronavirus disease is transmitted from one person to another, such as those in [6,18,25-33], by inter alia: (i) introducing a compartment for exposed individuals who come in contact with confirmed infected individuals diagnosed of the disease after contact tracing occasioned by testing; note that works on covid-19 existing in literature such as those in [6,18,31-33] do not include this compartment in their model. (ii) introducing a compartment for infected individuals undergoing treatment of one form or the other in isolation centres. note that this is not obtainable in the models in [6,25,26,33,34]. (iii) introducing parameters to represent covid-19 common safety protocols such as the use of efficacious face masks, maintenance of minimum social distance among individuals while in public, in the force of infections of the model as to adequately capture reality circumstances as it relates to the control of the disease, and these are missing in the models in [18,25,31,33,34]. we reformulated our model (1) by introducing the safety strategies to adequately capture the dynamics of the transmission of coronavirus vis-à-vis the safety protocols being enforced by the nigerian health policy makers to put under control its spread represented by parameters, where ρ stands for the fraction of the inhabitants of the city that maintain the minimum distance, at least one metre apart, required to prevent an infection, so that 0 ≤ ρ ≤ 1, parameter ε represents the proportion of the inhabitants of the city that effectively wear highly figure 2. plot on cumulative number of active cases showing (a) data fitting of the cumulative number of active cases (b) projection on the cumulative number of active cases. efficacious face masks at any time they are in public, so that 0 ≤ ε ≤ 1. another parameter ω stands for the proportion of the inhabitants of the city that effectively indulge frequently in personal hygiene involving periodic use of hand sanitizers and washing of hands with soaps when they interact with surfaces that may be contaminated, so that 0 ≤ ω ≤ 1. when these are incorporated into model (1), we have: ds dt = −βc (1 −ρ) (1 −ε) (1 −ω) (αia + is + ηt it ) n − ( eq + id + it ) s de dt = f βc (1 −ρ) (1 −ε) (1 −ω) (αia + is + ηt it ) n − ( eq + id + it ) s −δe deq dt = (1 − f ) (1 −ρ) (1 −ε) (1 −ω) λs + δe −σeq dia dt = qσeq − (θ + γa) ia dis dt = (1 − q) σeq − (ψ + γ0 + d0) is did dt = θia + ψis − ( γi + dd + γ j ) id dit dt = γ j id − (γb + dt ) it dr dt = γi id + γa ia + γ0 is + γb it (2) note that in this work, the circumstance under consideration is the nigerian government’s strict adherence to covid-19 4 bolaji et al. / j. nig. soc. phys. sci. 5 (2023) 1055 5 table 1. parameters and their interpretation for model (1) parameter interpretation βc effective transmission rate δ rate of progression from exposed latently-infected compartment to exposed quarantine compartment σ progression rate from exposed latently-infected quarantine compartment to infectious state q fraction of new infectious individuals that progressed to infectious asymptomatic and symptomatic infected class. f fraction of infectious individuals that progressed from susceptible class to f exposed classes. ηt the risk of infectiousness of individuals in the treated class it (t) in comparison with individuals in other infectious classes. α modification parameter to account for reduction in infectiousness of individuals in the ia(t) compartment when compared to individuals in the is (t) compartment. γa,γ0,γi,γ j rate of recovery for individuals in infected compartments ia(t),is (t), id(t) and it (t) respectively. ψ rate of detection obtained through contact-tracing and testing for the individuals in infected symptomatic class is (t). θ rate of detection obtained through contact-tracing and testing for the individuals in infected asymptomatic classia(t) ε proportion of the inhabitants of the city that effectively wear highly efficacious face masks at any time they are in public. ρ fraction of the inhabitants of the city that maintain the minimum social distance ω proportion of the inhabitants of the city that effectively indulge frequently in personal hygiene. d0, dd, dt death rates due to the disease for those in the infected class is (t), id(t) and it (t) classes respectively. safety protocols, which includes proper wearing of face masks and maintaining a minimum distance between members of the society, which were vigorously promoted and enforced, particularly in lagos, nigeria’s commercial nerve centre and the epicentre of the pandemic, since the pandemic have been wreaking havoc in the city. towards gaining insights into the dynamical features of the disease so as to combat its spread, we now rigorously analyse our model (2). 3. dynamical features of model (2) the norm in mathematical epidemiology is that when a new mathematical model is formulated to study the transmission dynamics of contagious diseases, such model will be analysed for its dynamical features. consequently, in this section, we now do the qualitative study of the dynamical features of the transmission dynamics of model (2). 3.1. analysis of the model 3.1.1. positivity and boundedness of the model in order for our model to make sense epidemiologically, there is compelling need to prove that for all time(t), all the state variables contained in the model are non-negative and are greater than zero, and there is the need to show that each of the equations in the model is bounded. this is what we do in this section. the following is claimed: theorem 3.1. let us assume that the inceptive data for the model (2) be s (0) ≥ 0, e(0) ≥ 0, eq(0) ≥ 0, ia(0) ≥ 0, is (0) ≥ 0, id(0) ≥ 0, it (0) ≥ 0, r(0) ≥ 0. consequently, solutions( s (t), e(t), eq(t), ia(t), is(t), id(t), it (t), r(t) ) of the model (2) are positive for all time t > 0. proof: let t1 = sup{t > 0 : s (t) > 0, e(t) > 0, eq(t) > 0, ia(t) > 0, is (t) > 0, id(t) > 0, it (t), r(t) ∈ [0, t]}. hence, t1 > 0. by considering the first of the equations in model (2), we have: d sdt = −λs where λ = f βc (1−ρ)(1−ε)(1−ω)(αia +is +ηt it ) n−(eq +id +it ) which can be rewritten as: ∫ d s dt = − ∫ λs , so that s (t1) = s (0) [ − ∫ t1 0 λ (u) du ] > 0. by following the same procedure, it can be shown that e(0) ≥ 0, eq(0) ≥ 0, ia(0) ≥ 0, is (0) ≥ 0, id(0) ≥ 0, it (0) ≥ 0, r(0) ≥ 0. lemma 1. given that region d = {( s (t), e(t), eq(t), ia(t), is(t), id(t), it (t), r(t) ) ∈ r8+ : n(t) ≤ n(0) } . then d is positively-invariant, and it attracts all positive solutions of the model (2). proof. taking the summation of all the equations of the model (2), we have: dn dt = − (d0 is + dd id + dt it ) 5 bolaji et al. / j. nig. soc. phys. sci. 5 (2023) 1055 6 , which when rewritten gives: dn dt ≤ δn, with δ = min (d0, dd, dt ) . thus, ∫ dn dt ≤− ∫ δdt so that n(t) ≤ n(0)e−δt n(t) → n(0),as t → ∞. hence, the domain d is positivelyinvariant. consequently, all the solutions in r7+ is attracted to region d. since the domain d is positively-invariant, it is sufficient to investigate the dynamics of the flow generated by the omicron variant of covid-19 model (2) in d. consequently, the model (2) is both mathematically and epidemiologically well posed. 3.2. the disease-free equilibrium (dfe) of the model (2): its local asymptotic stability by setting each of the expressions on the right-hand-sides of the equations in eqn. (2) to zero, the dfe for coronavirus disease model (2) is obtained, which is given by: ξ0 = ( s ∗(t), e∗(t), e∗q(t), i ∗ a(t), i ∗ s (t), i ∗ d(t), i ∗ t (t), r ∗(t) ) = (n(0), 0, 0, 0, 0, 0, 0, 0) we investigate the linear stability of the dfe using the next generational operator technique on equation (2). the matrix of new infection terms denoted by f and that of transfer terms denoted by v , using the notations in [35] are stated as follows respectively and setting: f =  0 0 f b0α f b0 0 f b0ηt 0 0 (1 − f ) b0α (1 − f ) b0 0 (1 − f ) b0ηt 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  , v =  δ 0 0 0 0 0 −δ σ 0 0 0 0 0 −qσ θ + γa 0 0 0 0 − (1 − q) σ 0 k1 0 0 0 0 −θ −ψ k2 0 0 0 0 0 −γ j γb + dt  , where b0 = βc (1 −ρ) (1 −ε) (1 −ω) , k1 = ψ + γ0 + d0 k2 = γi + dd + γ j. the controlled reproduction number, represented as r0 = ρ ( fv−1 ) is given by: r0 = b0 ( αqσb3 b4 b5 + b1 b2 b4 b5 + ηt γ j (qσθb3 + ψb1 b2) ) σb1 b3 b4 b5 (3) where b1 = θ + γa, b2 = (1 − q) σ, b3 = ψ + γ0 + d0, b4 = γi + dd + γ j, b5 = γb + dt . consequently, from [35] the basic reproduction number of the model (2),r0, is rewritten as: r0 = b0αq b1 + b0 b2 σb3 + b0ηt γ j (qσθb3 + ψb1 b2) σb1 b3 b4 b5 therefore, r0 = b0 [ αq b1 + b2 σb3 + ηt γ j (qσθb3 + ψb1 b2) σb1 b3 b4 b5 ] (4) that is, r0 = ra + rs + rit . the quantity r0 is the addition of the sub-reproduction numbers associated with the number of new coronavirus disease brought about by asymptomaticallyinfected individuals (ra), symptomatically infectious humans (rs ), and detected infectious humans receiving one form of treatment or the other in health care centres ( rit ) . theorem 2: the dfe, ξ0, of the model (2) is locally asymptotically stable (las) if r0 < 1, and unstable if r0 > 1. proof: we analyse the las of the coronavirus model (2) at dfe, ξ0, by evaluating the jacobian matrix which is given by: j (ξ0) =  0 0 0 −b0α −b0 0 −b0ηt 0 0 −δ 0 f b0α f b0 0 f b0ηt 0 0 δ −σ mα m 0 mηt 0 0 0 qσ −b1 0 0 0 0 0 0 b2 0 −b3 0 0 0 0 0 0 θ ψ −b4 0 0 0 0 0 0 0 γ j −b5 0 0 0 0 γa γ0 γi γb 0  with m = (1 − f )b0, b1 = θ + γa, b2 = (1 − q) σ, b3 = ψ + γ0 + d0, b4 = γi + dd + γ j, and b5 = γb + dt . the eigenvalues of the jacobian matrix, j (ξ0), are the solutions of its characteristic polynomial given below: λ6 + a0λ 5 + a1λ 4 + a2λ 3 + a3λ 2 + a4λ + a5 = 0, (5) where a0 = δ + σ + b1 + b3 + b4 + b5, a1 = δσ + (δ + σ) (b1 + b3 + b4 + b5) +b1 (b3 + b4 + b5) + b3 b4 + b3 b5 + b4 b5 − (1 − f ) b0 (αqσ + b2) a2 = δσb1 + b1 b3 (b4 + b5) +b4 b5 (b1 + b3) + (b3 + b4 + b5) × (b1 (δ + σ) + δσ) + (b3 b4 + b3 b5 + b4 b5) (δ + σ) ×b0 (αqσ (1 − f ) (b3 + b4 + b5) +b2 (1 − f ) (b1 + b4 + b5) + δ (αqσ + b2)) a3 = b3 b4 b5 (δ + σ + b1) + (b1 + δσ) (b3 b4 + b3 b5 + b4 b5) +δσb1 (b3 + b4 + b5) + b0 f b2 (b1 (b4 + b5) + b4 b5) −b0 (αqσ (1 − f ) (b3 b4 + b3 b5 + b4 b5) +ηt γ j (1 − f ) (θqσ + ψb2) + b2 (b1 b4 + b1 b5 + b4 b5) + αqσδ (b3 + b4 + b5) + δb2 (b1 + b4 + b5)) a4 = b1 b3 b4 b5 (δ + σ) + δσb1 b3 (b4 + b5) +δσb4 b5 (b1 + b3) + b0 f qσ ( αb4 b5 + θηt γ j ) +b0 f b1 b2 ( b4 b5 + ηt ψγ j ) − b0 (αqσδ (b3 b4 + b3 b5 + b4 b5) + qσ ( αb3 b4 b5 + δθηt γ j ) + ηt γ j (qσθb3 + δψb2) +b1 b2 ( ηt ψγ j + δ (b4 + b5) ) + b2 b4 b5 (δ + b1) ) , a5 = δσb1 b3 b4 b5 (1 − r0) . applying the routh-hurwitz criterion which asserts that all roots of the polynomial (5) have negative real parts if the coefficients of ai are positive, for i = 0, 1, 2, 3, 4, 5. clearly, for a5 > 0 in equation (5); then r0 < 1. therefore, the dfe ξ0 is locally asymptotically stable (las) if r0 < 1. 6 bolaji et al. / j. nig. soc. phys. sci. 5 (2023) 1055 7 note that the quantity r0 represent the average number of secondary infections brought about by a typical actively infected individual introduced into a totally susceptible population where covid-19 safety protocols (control measures) are being implemented [35]. the implication of theorem 2 epidemiologically is that coronavirus disease will be made to die out from the population when r0 < 1, if the model’s initial population sizes are within dfe’s attraction region. to ensure that covid-19 elimination is independent of the initial sizes of the infected individuals in the population, it is necessary to show that the dfe is globally asymptotically stable. 3.3. global asymptotic stability (gas) of the dfe of the model (2) in view of the fact that the linear lyapunov function has found wide application to prove the gas of the dfe [11, 36]. we apply the method as follows via the lyapunov function: l = (b4 b5 (αqσb3 + b1 b2) +ηt γ j (qσθb3 + ψb1 b2) ) e + (b4 b5 (αqσb3 + b1 b2) +ηt γ j (qσθb3 + ψb1 b2) ) eq +σb3 ( θηt γ j + αb4 b5 ) ia +σb1 ( ψηt γ j + b4 b5 ) is +σb1 b3ηt γ j id + σb1 b3 b4ηt it , (6) where b1 = θ + γa, b2 = (1 − q) σ, b3 = ψ + γ0 + d0, b4 = γi + dd + γ j, b5 = γb + dt . with lyapunov derivatives (where a dot represents differentiation with respect to time). l̇ = ( b4 b5 (αqσb3 + b1 b2) + ηt γ j (qσθb3 + ψb1 b2) ) ė + ( b4 b5 (αqσb3 + b1 b2) + ηt γ j (qσθb3 + ψb1 b2) ) ėq +σb3 ( θηt γ j + αb4 b5 ) i̇a + σb1 ( ψηt γ j + b4 b5 ) i̇s +σb1 b3ηt γ j i̇d + σb1 b3 b4ηt i̇t (7) l̇ = ( b4 b5 (αqσb3 + b1 b2) + ηt γ j (qσθb3 + ψb1 b2) ) × ( f βc (1 −ρ) (1 −ε) (1 −ω) (αia + is + ηt it ) s n − ( eq + id + it ) −δe) +( b4 b5 (αqσb3 + b1 b2) + ηt γ j (qσθb3 + ψb1 b2) ) × ( (1 − f ) βc (1 −ρ) (1 −ε) (1 −ω) (αia + is + ηt it ) s n − ( eq + id + it ) +δe −σeq ) + σb3 ( θηt γ j + αb4 b5 ) ( qσeq − b1 ia ) +σb1 ( ψηt γ j + b4 b5 ) ( b2 eq − b3 is ) +σb1 b3ηt γ j (θia + ψis + b4 id) + σb1 b3 b4ηt ( γ j id − b5 ia ) . (8) simplifying the above equation, we have: l̇ = (b4 b5 (αqσb3 + b1 b2) +ηt γ j (qσθb3 + ψb1 b2) ) × ( βc (1 − rho) (1 −ε) (1 −ω) (αia + is + ηt it ) s n − ( eq + id + it ) ) − ασb1 b3 b4 b5 ia −σb1 b3 b4 b5 is −ηt σb1 b3 b4 b5 it (9) note: s (t) ≤ n (t) − ( eq (t) + id (t) + it (t) ) for all time t, so l̇ ≤ (b4 b5 (αqσb3 + b1 b2) +ηt γ j (qσθb3 + ψb1 b2) ) βc (1 −ρ) (1 −ε) (1 −ω) (αia + is + ηt it ) − σb1 b3 b4 b5 (αia + is + ηt it ) (10) l̇ ≤ (αia + is + ηt it ) × (βc (1 − rho) (1 −ε) (1 −ω) × ( b4 b5 (αqσb3 + b1 b2) + ηt γ j (qηt θb3 + ψb1 b2) ) −σb1 b3 b4 b5) (11) l̇ ≤ (αia + is + ηt it ) ×σb1 b3 b4 b5 (b4 b5 (αqσb3 + b1 b2) +ηt γ j (qσηt b3 + ψb1 b2) ) × ( βc (1 −ρ) (1 −ε) (1 −ω) σb1 b3 b4 b5 − 1 ) (12) l̇ ≤ (αia + is + ηt it ) σb1 b3 b4 b5 (r0 − 1) (13) since all the model parameters are non-negative, it follows that from equation (13), l̇ ≤ 0 for r0 ≤ 1 with l̇ = 0 if and only if e = 0, eq = 0, ia = 0, is = 0, id = 0 and it = 0. hence, l in equation (13) is a lyapunov function for the model (2). therefore, by the lasalle’s invariance principle [37], the dfe of the omicron variant of covid-19 model (2) is globally asymptotically stable whenever r0 ≤ 1. 3.4. final size of the epidemic in this section, we derive a relation between the basic reproduction number corresponding to the model (2) and the size of the epidemic. the final size of an epidemic can be defined as the total number of people experiencing infection during an outbreak. it is typically called the attack rate by applied epidemiologist [38]. the final size relation is a relation involving the basic reproduction number and the number of the members of the population that remain in each disease-free compartment over the course of the epidemic [39]. to derive the final size relation, it is necessary to carry out some change of variables of model (2) so as to reduce model (2) into a three dimensional system as observed in [38, 40]. consequently, we say let x ∈ r6+ represents the set of infected compartments, y ∈ r+ represents the susceptible compartment, and z ∈ r+ represents the recovered compartment of model (2). that is, variable y is defined as y (t) = s (t), variable x is defined as: x (t) = ( e (t) , eq (t) , ia (t) , is (t) , id (t) , it (t) )t , and z (t) = r (t) . (14) 7 bolaji et al. / j. nig. soc. phys. sci. 5 (2023) 1055 8 we let d be an m × m diagonal matrix whose diagonal entries σi > 0 are the relative susceptibilities of the corresponding susceptible class: note that if m = 1 thend is a scalar. we let π be an n × m matrix with the property that the(i, j)entry represents the fraction of the jth susceptible compartment that goes into the ith infective compartment on becoming infected [40]. we also let b be an n-dimensional row vector of relative horizontal transmissions, this vector is multiplied by the scalar factor representing infectivity [40]. we use β as the infection rate of the coronavirus model (2), for the general incidence this factor is a function of the total population size and / or infective population size and we write it as β (x, y, z) [40]. by applying the above transformation to the model (2) we have: m = 1, n = 6, b = [ 0 0 α 1 0 ηt ] , d is the scalar (1 −ρ) (1 −ε) (1 −ω) and π =  f 1 − f 0 0 0 0  , ẋ = πdyβ (x, y, z) bx − v x, . y = −dyβ (x, y, z) bx, . z = w x, (15) here, the n × n matrix v describes the transitions between infected compartments as previously define in section 3.2. the k × n matrix w has the property that the (i, j) entry represents the rate at which members of the jth disease compartment go into the ith recovered compartment. the disease-free set {(x, y, z) |x = 0, y ≥ 0, z ≥ 0} is invariant. suppose that the point (0, y0, z0) is referred to as the disease-free equilibrium, then the reproduction number is given as: r0 = β (0, y0, z0) bv −1 πdy0 (16) theorem 3.2: consider the omicron variant of covid-19 model (2), and let s (∞) = limt→∞ s (t). the final size relation of the covid-19 pandemic is given by: ln ( s (0) s (∞) ) ≥ r0 ( s (0) − s (∞) s (0) ) +β ( αq b1 + b2 σb3 + ηt γ j (qσθb3 + ψb1 b2) σb1 b3 b4 b5 ) ( e (0) + eq (0) ) +β ( α b1 + ηt γ jθ b1 b4 b5 ) ia (0) + β ( 1 b3 + ηt γ jψ b3 b4 b5 ) is (0) + βηt γ j b4 b5 id (0) + βηt b5 it (0) . (17) 4. numerical simulations and results in a work of this nature, uncertainties do occur in the estimation of some parameters in the mathematical model representing the transmission dynamics of the novel coronavirus figure 3. (a) effects of varying ε on the number of active cases. (b) effects of varying ψ on the number of active cases. (c) effects of ψ on the number of symptomatic infectious individuals (undetected). (d) effects of θ on the number of asymptomatic infectious individuals (undetected). disease covid-19, there is a need to carry out an analysis of this, together with sensitivity analysis to determine how sensitive some of the parameters in the model are. these is what we do in this section. also, to assess the impact of various control strategies necessary to reduce the transmission burden of the disease, we carry out the numerical simulations of the model. the mode of doing this is by solving numerically the equations in model (2) using matlab ode 45 solver which is based on the famous runge-kutta method of the fourth order. note that the stability of our method is well established in [32]. 4.1. uncertainty and sensitivity analysis we implemented latin hyper-cube sampling (lhs) [41] on the parameters of the model in order to do the estimates of parameters involved in the numerical simulations of the model; while we carry out a partial correlation coefficient (prcc) between values of the parameters in the response function and that derived from the sensitivity analysis for the required (sensitivity) analysis. tens of hundred simulations of the model (2) were run in all, where it was observed as shown in table 2, using the reproduction number r j as the response function that transmission rate βc, the case detection rates for asymptomatic and symptomatic infectious humans, θ and ψ, respectively, the modification parameter accounting for infectiousness of asymptomatic infectious humans, α are the top-ranked parameters that drive the dynamics of model (2). from the prcc results, whereas, βc and α are positively correlated, on the other hand, θ and ψ are negatively correlated. it is pertinent to note that the public health implication of this, is that, a reduction in the transmission rate of omicron variant of covid-19, achieved through strict adherence to safety protocols involving non-pharmaceutical control measures, such as use of face masks, regular washing of hands and keeping of social distancing and effective treatment of symptomatic infectious individu8 bolaji et al. / j. nig. soc. phys. sci. 5 (2023) 1055 9 figure 4. surface plots showing impacts of various covid-19 parameters on r0. (a) surface plot of r0 against ε and ρ. (b) surface plot of r0 against ε and ψ. (c) surface plot of r0 against ε and θ. (d) surface plot of r0 against ε and ω. (e) surface plot of r0 against θ and ψ. (f) surface plot of r0 against ω and ρ. (g) surface plot of r0 against ρ and ψ. (h) surface plot of r0 against ω and θ. (i) surface plot of r0 against ω and ψ. als can be a panacea to effective control of the dreaded disease via reduction in transmission rate. furthermore, when efforts are geared up to increase the detection rates for the asymptomatic and symptomatic infectious individuals towards isolating them for adequate treatment, this will ultimately lead to significant reduction in disease burden of covid-19 in the population under reference. by using the number of individuals in the class of those individuals who are symptomatically infected, ia as the response function, it is the transmission rate, βc , the fraction of exposed individuals who progress to the class of those asymptomatically infected, q, the transition rate out of the class of those exposed quarantined, σ and the detection rate for asymptomatic infectious humans, θ that play dominant role on the dynamics of the dreaded disease. on the other hand, by using as response function, the number of individuals in the class of those individuals who are symptomatically infected, is , we have as driving force in the transmission dynamics of the disease, top-ranked parameters, the transmission rate, βc, the transition rate out of the class of exposed quarantined individuals, σ, case detection rate for symptomatic infected individuals, ψ and the fraction of exposed individuals who progress to the class of those asymptomatically infected, q. furthermore, when we used as response function, the population of those infected individuals that are detected, id, the top-ranked parameters that are dominant in the dynamics of the disease are: the transmission rate, βc, the detection rates symptomatic and asymptomatic infectious individuals ψ and θ, respectively, the transition rate out of the class of exposed quarantined individuals, σ, and the recovery rate for detected infectious humans, γi. finally, by using the total number of people receiving one form of treatment or the other, in class it as the response function, we have effective contact rate 9 bolaji et al. / j. nig. soc. phys. sci. 5 (2023) 1055 10 βc, case detection rate for symptomatic and asymptomatic infected individuals, ψ and θ, respectively, and the transition rate out of the class of exposed quarantined individuals, σ and the recovery rate for detected infectious humans, γb , are the five top-ranked parameters that has strong influence on the dynamics of the disease. in the table 2, are the values of eprcc for model (2) parameters where we used the total number of infected individuals and the reproduction number, r j associated with covid-19 as response functions. written in bold fonts are the parameters which has strong influence on the transmission dynamics of the model with respect to each of the response functions. table 2. parameters and the partial correlation coefficient values for model (2). par represent parameters. par ia is id it r j βc 0.1736 0.2784 0.2617 0.3199 1 δ −0.0436 −0.0592 −0.0661 −0.0065 −0.0342 σ 0.7541 0.7459 0.7849 0.7483 0.0730 q 0.0975 0.0664 −0.0791 0.0551 0.0560 f 0.2553 0.4031 0.3049 0.2674 −0.4443 ηt −0.0758 0.0014 0.0325 −0.0637 −0.0634 α −0.0092 0.0408 −0.0705 −0.0868 −0.0141 γa −0.6450 −0.1000 0.0830 −0.0162 0.0503 γ0 −0.0519 −0.2414 −0.2751 −0.2270 −0.1868 γi 0.1369 −0.0229 −0.1702 −0.1451 0.0223 γ j 0.1361 −0.0281 −0.0566 0.3476 0.0018 γb 0.1420 0.0179 0.1141 −0.1971 0.0242 ψ 0.1526 −0.1121 0.4083 0.3808 −0.1180 θ −0.6935 0.0070 −0.0719 −0.0551 −0.0854 d0 0.0591 −0.2092 −0.2305 −0.0945 0.0732 dd 0.0718 −0.3151 −0.3295 −0.1352 0.1605 dt 0.0710 0.0136 −0.2005 −0.0777 0.1634 ω −0.3589 0.5442 −0.5205 −0.5025 −0.7047 ρ 0.0676 −0.2624 −0.2916 −0.1204 0.1093 ε −0.4225 −0.5466 −0.4913 −0.4080 −0.7483 4.2. starting values of the model’s parameters we now implement our model (2) to examine the disease’s dynamics in nigeria using data from the covid-19 outbreak as reported day-to-day by the nigeria centre for disease control (ncdc). to perform numerical simulations on the model, we extracted data from 1st of december, 2021 to 20th of january, 2022 when omicron variant was first discovered in nigeria. we estimate the parameters in the model by fitting the model with the daily cumulative number of active cases in nigeria obtained from ncdc. the model fitting was performed using the fmincon algorithm in matlab. by following the reasoning of okuonghae and omame [6] on modelling the dynamics of the pandemic in the city of lagos, we estimated some of our parameters to be σ = 1/5.2 per day, α = 0.5 per day, d0 = dd = 0.015 per day and γ0 = γa = 0.013978 per day. on the other hand, we estimated other parameters βc,θ and ψ, by fitting the model (2) with the daily “cumulative number of reported cases and number of active cases”. furthermore, we will estimate the starting number of latently infected and infectious persons who were detected in the city in the period the index case for omicron variant of covid-19 was reported on the 1st december, 2021, that is, e(0), eq(0) and id(0); by taking note that the population of lagos is 14, 368, 332 so that we take this value as the starting point, our initial condition for our simulation, thus setting s (0) = 14, 368, 332 , while we set id(0) = 1 and r(0) = 0, we do this by considering the reported date of the index case. logically, there exists some undetected infected individuals among the city’s population, owing to the fact that extensive spread screening and testing of the population for the disease had not yet begun. similarly, for the remaining state variables, it is necessary to estimate the potential values of other beginning conditions, namely: , e(0), eq(0), ia(0), is (0), and it (0). table 3 shows the values of the other parameters that we used in our simulations. table 3. parameters and their baseline values for model (2) parameter baseline range reference value (/day) (/day) βc fitted − estimated α 0.5 [0, 1] [42, 6] δ 15.2 [ 1 14 , 1 3 ] estimated σ 15.2 [ 1 14 , 1 3 ] [1, 6] q 0.5 [0, 1] [42, 6] ηt 0.5 [0, 1] [42, 6] γi 1 15 [ 1 3 , 1 30 ] [42, 6] γa = γ0 = γb 0.13978 [ 1 3 , 1 30 ] [43, 6]] θ fitted − estimated d0 = dd 0.015 [0.001, 0.11] [28, 6] dt 0.017 [0.002, 0.12] estimated ψ fitted − estimated γ j fitted − estimated 4.3. model fitting for our matlab-based function optimizer, we use the genetic algorithm (ga) for model fitting [44]. the ga approach used in the coding helped us locate the accurate basin of attraction, which provided the starting values of the estimated parameters used in the “fmincon” function in the matlab optimization toolbox. to obtain a more accurate estimate, we adopt a combination of two optimization methods for data fitting, the ga method and the fmincon method in matlab. the implementation of our model fitting is for the period, 1st december 2021 the time that the first case of omicron variant of covid-19 was reported in lagos to 20th january. our choice of the days that the omicron variant of covid-19 was afflicting people of lagos as chosen here is such that we shall be able to capture the disease’s dynamics in lagos and the disease’s rate of transmission. we seek to estimate effective transmission rate, rate of detection through testing and contact tracing and the initial conditions e(0), a(0) and i(0). it is pertinent to note that there could be 10 bolaji et al. / j. nig. soc. phys. sci. 5 (2023) 1055 11 under reporting in the city caused by little tests performed during the period under consideration. furthermore, it should be noted that during this period in the city of lagos, the enforcement of “social distancing” and “use of face masks” while in public were no longer being adequately and strictly observed too. see table 3 for specific values used for simulations. 4.3.1. results of simulations when data and model fitting was done using the cumulative number of reported cases of covid-19 infections we simulate the model using the data pertaining to daily cumulative number of reported cases of infections within the period under review, noting that except otherwise stated all model fitting was done with settings ε = δ = 0 as previously explained. from the plot in figure 2(a), it was observed that model (2) fitted on cumulative number of active cases. the cumulative number of active cases of the disease reaches its pick within fifty days, meaning that starting from 1st december 2021 when the omicron variant of covid-19 was first discovered in nigeria, it reaches its peak by end of january 2022 and start declining from first week of february 2022 till date through to in four hundred days’ time. (here we got the peak of the pandemic in lagos, the epicentre of the disease in nigeria). from figures 3(c) and 3(d), the effect of increase in detection rate for infected individuals was shown revealing an increase in the number of active undetected symptomatic cases with peak period varying between 35 to 42 days from the start of the discovery of the variant of covid-19 in lagos. the number of active cases is expected to reduce after some period as projected in figure 2(b). the effects of ε on the number of active cases over some days is shown in figure 3(a). when ε is small (i.e., ε = 0.1), the cumulative activate cases attained its peak around the first 50 days after which is started showing a monotonic behaviour. as ε increases, the cumulative number of active cases was reducing, attaining peak level within the first 50 days then reducing. this implies that as the proportion of the inhabitants of the city that effectively wear highly efficacious face masks at any time they are in public increases, the cumulative active cases is reducing and after some period of time, the virus is eradicated. we study the effects of ψ on number of activate cases in figure 3(b) and number of infected symptomatic class. increasing the rate of detection through contact-tracing and testing for individuals can reduce the number of active cases as shown in figure 3(b), ensuring that covid-19 (active) cases is eradicated after some days. in a similar reasoning, increasing the rate of detection ψ help in reducing the number of infected symptomatic class (is) as in figure 3(c). this declining number will take some days before the class is can be eradicated. in figure 3(d), the effects of rate of detection, θ in infected asymptomatic class at varying θ. as θ increases the class ia is reducing after some days. the peak is attached within the first 20 days after which the number of class ia shows monotonic behaviour. the basic reproduction number, r0 is a threshold parameter for the stability of the disease and is closely related to an epidemic’s peak and final size, together with the important parameter can help us to gain insights into the dynamics of the disease. we study r0 with other parameters as shown in figure 4. from figure 4(a), as fraction of the inhabitants of the city of lagos that maintain the minimum distance increases, this leads to decrease in the proportion of the individuals that effectively use face masks at any time when they are in public, leading to increase in diseases transmission. in figure 4(b), as the rate of detection of covid-19 in infected symptomatic class increases, the proportion of the individuals that effectively use face masks reduces, resulting to decrease in disease transmission. in fig. 4(c), when there is reduction in transmission as a result of increasing rate of detection of infected asymptomatic class even though the proportion of the individuals that effectively wear face masks reduces. the parameters such as minimum distancing, indulging personal hygiene and increasing rate of detection of class ia can have impact on r0 as shown in figures 4(d) – 4(f). the disease transmission can be reduced when there is either increase in rate of detection of class is , maintaining social distances or individuals indulging frequently in personal hygiene while in public, the pandemic will be gotten rid of as shown in figures 4(g) – 4(i). revelation from figure 4(i) is that if policy makers in lagos city can increase their case detection rate for infected symptomatic class, then the reproduction number of the pandemic can be flattened below unity such that a great decrease in the disease’s burden will be certain. to effectively manage the transmission of the omicron in the city of lagos, we show how different parameters can help to reduce the burden. wearing of face mask in the public can help to reduce the active cases as people have lesser chance of transmitting disease or getting infected. increasing the detection rate in both symptomatic and asymptomatic classes is another way of reducing the number of active cases. then, the surface plots in figs. 4 shows how combination of some model parameters can help to reduce the spread of the virus. 5. conclusion this research study the dynamics of the omicron variant of covid-19 in lagos, the epicentre of the pandemic in nigeria. we assessed the impact of covid-19 safety protocols such as keeping of minimum social distancing among individuals, regular hands washing and use of sanitizers, wearing of face masks among the inhabitants of the city while in public spaces. we calculated the pandemic’s reproduction number using the available data, and we established a forecast tool for the “cumulative number of active cases” using the model developed to capture the disease’s dynamics. from model fitting to data and numerical simulations, the effects of covid-19 safety measures (safety protocols) on the disease’s transmission dynamics in the city was assessed. furthermore, the revelation from this work is that disease transmission can be effectively controlled if detection rate of infected and infectious individuals can be increased, more individuals practice personal hygiene, maintain minimum social distances, as strict observance of these mea11 bolaji et al. / j. nig. soc. phys. sci. 5 (2023) 1055 12 sures have huge impacts on the transmission dynamics of omicron variant of covid-19 (see fig. 4). as contribution to knowledge, in future work on the subject matter, optimal control measures can be incorporated into the model towards procuring more adequate measures to effectively combat the scourge of the deadly disease. consequently, our recommendations towards curtailing the spread of the disease at the community level is that policy makers implement strict measures in order to discover new cases of infection through general screening and testing, in combination with strict observance and compliance with covid-19 safety protocols, 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[44] j. mccall, ‘’genetic algorithms for modelling and optimisation”, j. comput. appl. math. 184 (2005) 205. 13 j. nig. soc. phys. sci. 4 (2022) 9–15 journal of the nigerian society of physical sciences investigations of the elastic moduli of er2o3nps nps doped teo2 − b2o3 − sio2 glasses using theoretical models u. s. aliyua,∗, i. g. geidamb, m. s. ottoa, m. hussainic a department of physics, faculty of science, federal university lafia, nasarawa sate, nigeria bdepartment of physics, faculty of sciences, yobe state university damaturu, nigeria cdepartment of physics, faculty of sciences, umar suleiman college of education gashua, yobe state, nigeria abstract elastic moduli of {[(teo2)0.7 (b2o3)0.3]0.8 (sio2)0.2}1−y (er2o3 nps)y glasses with y = 0.01, 0.02, 0.03, 0.04, 0.05 were studied in this work using the theoretical elastic models. the makishima & mackenzie, rocherulle and bond compression models were employed for the study. in the makishima and mackenzie model, the packing density was calculated from the bulk glass molar weight and the bulk glass density whereas in rocherulle model it is determined as the individual oxides. young, shear and bulk moduli as well as the poisson ratio were calculated for the glasses in the makishima and rocherulle models, while longitudinal, was calculated in addition to young, bulk and shear moduli using the bond compression model. bond per unit volume number (nb), bulk modulus, bulk modulus ratio (kbc/ke), atomic ring size (`) and stretching force constantwere also calculated and presented. the values of the young, bulk and shear moduli obtained from makishima model increased from 52.854 to 55.335 gpa, 35.754 to 39.862gpa and 21.080 to 21.809gpa respectively with er2o3nps composition increase from 1% to 5%..the rocherulle model presented increasing values for young, bulk and shear moduli as 56.910 to 58.432gpa, 41.452 to 44.450 gpa and 22.385 to 22.809 gpa respectively with er2o3nps composition increase from 1% to 5%. the bond compression model presented much higher values of the elastic moduli compared to the experimentally obtained values and showed an increasing trend as the er2o3nps concentration increases. in the glass network, the atomic ring size value decreased from 0.5698 to 0.5091 nm indicating an increase in the close packing of atoms. based on the elastic moduli values presented by all the models, makishima and mackenzie model presented a more reliable data and hence represents the best model for the studied glass system. doi:10.46481/jnsps.2022.222 keywords: erbium oxide, tellurite glass, elastic moduli, poisson ratio, theoretical models article history : received: 05 may 2021 received in revised form: 04 december 2021 accepted for publication: 05 december 2021 published: 28 february 2022 c©2022 journal of the nigerian society of physical sciences. all rights reserved. communicated by: b. j. falaye 1. introduction tellurite based glasses doped with rare earth ions have been used in amorphous silicon solar cells for the improvement of the cell efficiency [1]. these glasses are developed widely in various optical applications in glass sensors, optoelectronics, fibre ∗corresponding author tel. no: +2348039364898 email address: usaltilde@yahoo.com (u. s. aliyu ) optics and light emitting diode (led) [2, 3, 4, 5]. in most cases glasses requiring mechanical strength for special applications are fabricated in combination with sio2as silica combination is known to provide thermal stability, mechanical strength and chemical stability [6, 7, 8, 9]. silica incorporation in glasses improves their optical transparency in both lasing and excitation wavelengths [10]. b2o3 is considered an excellent raw material in combination with other teo2 based glasses for quality 9 aliyu et al. / j. nig. soc. phys. sci. 4 (2022) 9–15 10 improvement in terms of hardness, optical transparency, easy glass fabrication (as it lowers the glass’s melting temperature) and rare earth ion solubility [11]. various researchers in the area of glass science, technology and engineering have been engaged in the study of different glass properties for applications in the areas of communications, optical, laser and other [12, 13, 14]. glass’s application in any technology is affected by either its mechanical strength, optical behaviours, elastic properties, thermal stability or chemical stability [15, 16].because of mechanical strength importance in optical disc, optical fibres medical and dental implants, electronic displays and radiation shielding applications, sio2 are used as substrates to provide such mechanical strengths [17, 18, 19]. elastic moduli of glasses of different compositions have been studied by various researchers over the years. makishima and mackenzie developed a model for the calculation of the elastic moduli of glasses theoretically, in consideration to its importance in the manufacture of strong optical fibres [20, 21]. rocherulle et al. [22] proposed a model that improved the makishima and mackenzie model through the extension of the dissociation energy and packing factors. the bond compression model is based on the consideration of the oxides’ atomic geometry which include their coordination number as well bond length [23]. in this work, the elastic moduli of the er2o3 doped teo2 – b2o3 – sio2 glasses were determined using mackishima and mackenzie, rocherulle and bond compression models. the study of the elastic moduli is important for the definition of its applicability in optical fibre, laser and other optoelectronic applications. the objectives of the study are as follows; 1. to determine the elastic moduli of the glasses under study using theoretical models. 2. to compare the values of the elastic moduli obtained from each model with the others. this work is important considering the importance of the elastic moduli to the application in different areas of technology and the fact that there is no such work carried out on the studied glasses composition. 2. theoretical models this section presents the four theoretical models adopted in this work for the study of the elastic properties of the er2o3 and er2o3 nps doped rice husk silicate borotellurite glass systems. the models used include the makishima-mackenzie model, rocherulle model, bond compression model and the ring deformation model. 2.1. makishima and mackenzie model the makishima and mackenzie model proposed a theoretical approach to determine the elastic moduli of oxide glasses with consideration to to the chemical composition (xi) of the constituting oxide, individual oxides’ packing densities (vi), and their corresponding dissociation energies (gi) [24]. the glass young modulus is expressed in terms of the packing density (v t), and the dissociation energy (gt) as; em = 2vt ∑ i gi xi = 2vtgt (1) from the oxides’ packing density, vtcan be determined as by vt = ( ρ m ) ∑ i vi xi (2) where m = glass molecular weight, ρ = glass density, xi =ithcomponent’s molar fraction (i), and v i is calculated for an oxide (ax oy) as: vi = na (4π/3) ( xr3a + yr 3 o ) , (3) where ro and ra respectively represent the ionic radii of oxygen and cation respectively [23]. according to makishima and mackenzie, bulk modulus (km), shear modulus (gm)and poisson ratio (σm) for oxide glasses on any component are calculated as follows: km = 1.2vt e (4) gm = (3ek/9k − e) (5) σm = (e/2gm − 1) (6) 2.2. rocherulle model a modified expressions of the makishima and and mackenzie model was proposed by rocherulle at al. (1989) [22]. the packing density, vi in the makishima-mackenzie model is replaced with ci which is expressed as follows: ci = na (4πρ/3m) ( xr3a + yr 3 o ) (7) for glasses of polycomponent nature, the ct factor is therefore expressed as follows: ct = ∑ i ci xi (8) ci = ∑ i ρi mi vi xi (9) the young modulus (er ), bulk modulus (kr ), shear modulus (gr ) and the poisson ratio are calculated as in equations (1), (4), (5), and (6) respectively. the basic difference between the two models is that the makishima and mackenzie model take into consideration the bulk density and molecular weight of the glass, while the rocherulle model considers the individual oxides’ density and molecular weights [22]. 10 aliyu et al. / j. nig. soc. phys. sci. 4 (2022) 9–15 11 2.3. bond compression and ring deformation models the theoretical model of bond compression takes into consideration the atomic networking in a material and the bond stretching force constants between them to theoretically estimate the elastic characteristics of the material [25]. for single oxide glass systems, the bulk modulus is obtained as; kbc= nb fr2 9 (10) the expression for a multicomponent oxide glasses is given as: kbc = (ρna/9m) ∑ i (xn f fr 2)i, (11) where f = the average stretching force constant, and nb= the bond number per unit volume and r is the cation-anion bond length. the bond number per unit volume is calculated as: nb = ( n f ρna/m ) (12) n f is the number of bonds per unit glass formula, ρ is the glass density,na is avogadro’s number and m is the glass molecular weight [26]. the stretching force constant ( f ) is deduced for multi-component glass using the expression as reported by [27] as: f= ∑ i ( xn f f ) i∑ i ( xn f ) i (13) the calculations of poisson’s ratio (σbc) and the corresponding average cross link density (nc) of the glasses are deduced as in equations (13) and (14) respectively. σbc = 0.28(nc) −0.25 (14) nc = 1 η ∑ i (xnc nc)i, (15) where nc= oxide (i) cross-link number per cation and nc= cation number per unit glass formula. the total number of cations per unit glass formula for a multicomponent glass system (η) is obtained as: η = ∑ i (xnc)i (16) kbc and σbc are used to calculate the young, shear and longitudinal moduli. the ring deformation model theoretically estimates the atomic ring size in the glass network structure. the model uses the experimental bulk modulus (ke) values and the bending force constant (fb) values to estimate the atomic ring size [28]. in the approximation process, the average stretching force constant is used in place of fb. the following formula is used in the determination of the atomic ring size: ke = 0.0106fb(l) −3.84 (17) the value l represents the atomic ring size and is defined as the diameter of the external ring. the ring perimeter can be determined using l as bond number x bond length divide by π [29]. 3. results and discussions 4. makashima model using the makishima and mackenzie model, the results of the theoretical elastic properties data obtained is presented in this section. figure 1 illustrates the packing density and dissociation energy variation of the {[(teo2)0.7 (b2o3)0.3]0.8 (sio2)0.2}1−y (er2o3 nps)y glass system with increasing concentration of er2o3nps. the packing density value increased from 0.5637 to 0.6003 cm−3. the value increases maybe associated to the increment in the density of the material. this increment is connected to increase in the material compactness and rigidity. it can also be said to be due to increase in the conversion of teo3 to teo4 structural form [30]. the dissociation energy decreased from 46.878 to 46.088 kj/cm3 with the increased er2o3 concentration from 1% to 5%. the decrease in the value maybe due to the introduction of more low dissociation energy er2o3 into the system. it may also relate to the observed increase in the molar volume of the glasses [31]. figure 2 and table 1 presents the elastic moduli of the glass system of {[(teo2)0.7 (b2o3)0.3]0.8 (sio2)0.2}1−y (er2o3 nps)y composition. the young, bulk and shear moduli increased from 52.8537 to 55.3347 gpa, 35.7544 to 39.8619 gpa and 21.0804 to 21.8087 gpa respectively. the increase in the elastic moduli increase with increase in the er2o3 concentration may be due to increase in material compactness associated with increase in the teo4 units’ concentration. the elastic moduli increase is generally associated with rigidity increase [15, 32]. the poisson ratio value for the {[(teo2)0.7 (b2o3)0.3]0.8 (sio2)0.2}1−y (er2o3 nps)y presented in table 1 increased from 0.2536 to 0.2686 with the increase in the er2o3 concentration from 1% to 5%. the poisson ratio increases normally with increase in the crosslink density of a material [33]. according makishima and mackenzie, poisson ratio of a silicate glass should be around 0.25 when the work done after glass deformation involves the compression or stretching of the si-o-si links [21]. 5. rocherulle’s model table 2 presents the packing density, elastic moduli, and the poisson ratio values for the glass system of {[(teo2)0.7 (b2o3)0.3]0.8 (sio2)0.2}1−y (er2o3 nps)ycomposition. the packing density values increased from 0.6008 to 0.6094 when the er2o3 concentration was increased from 1% to 5%. the increase in the packing density might be due to the decrease in the interatomic spaces resulting from the formation of more bridging oxygens (bos) with the formation of more teo4 structural units in the glass network [34]. this can also be due to the formation of more bo3 units and decreasing concentration of bo2o nature containing one (1) non-bridging oxygen (nbo) and two (2) bridging oxygen atoms attacked to the boron atom [35]. 11 aliyu et al. / j. nig. soc. phys. sci. 4 (2022) 9–15 12 table 1: packing density (vt ), dissociation energy (gt ), elastic moduli (em, km and gm) and poisson ratio (σm) for glass system of {[(teo2)0.7 (b2o3)0.3]0.8 (sio2)0.2}1−y (er2o3 nps)ycomposition. y (mol) vt( cm3 mol−1 ) gt (kj/cm3) em (gpa) km (gpa) gm (gpa) σm 0.01 0.5637 46.8784 52.8538 35.7544 21.0804 0.2536 0.02 0.5710 46.6809 53.3064 36.5233 21.2081 0.2567 0.03 0.5755 46.4833 53.4977 36.9424 21.2521 0.2586 0.04 0.5879 46.2857 54.4262 38.3990 21.5333 0.2638 0.05 0.6003 46.0882 55.3347 39.8619 21.8087 0.2686 figure 1: packing density and dissociation energy variation with the molar fraction of er2o3nps in {[(teo2)0.7 (b2o3)0.3]0.8 (sio2)0.2}1−y (er2o3 nps)y glass system figure 2: the variation of the elastic moduli with molar fraction of er2o3 in {[(teo2)0.7 (b2o3)0.3]0.8 (sio2)0.2}1−y (er2o3 nps)y glass system the young and shear moduli increased from 56.9097 to 58.4324 gpa and 22.3445 to 22.8090 gpa, respectively with an increase in the er2o3 molar fraction from 0.01 to 0.05. the decrease in the values may be associated with decrease in the dissociation energy which overrides the observed increase in the packing density. the bulk modulus was found to have increased from 41.4525 to 44.4498gpa. the bulk modulus increase might be due to the increase observed in the packing density of the glasses. the bulk modulus dependence to the packing density is more than its dependence on the dissociation table 2: packing density (ct ), elastic moduli (er, kr, and gr), poisson ratio (σr) for glass systemof {[(teo2)0.7 (b2o3)0.3]0.8 (sio2)0.2}1−y (er2o3 nps)ycomposition. y (mol) ct( cm3 mol−1 ) er (gpa) kr (gpa) gr (gpa) σr 0.01 0.6070 56.910 41.452 22.385 0.2712 0.02 0.6137 57.298 42.198 22.493 0.2737 0.03 0.6205 57.682 42.947 22.600 0.2762 0.04 0.6272 58.060 43.697 22.705 0.2786 0.05 0.6339 58.432 44.450 22.809 0.2809 figure 3: elastic moduli variation with molar fraction of er2o3nps for glass systemof {[(teo2)0.7 (b2o3)0.3]0.8 (sio2)0.2}1−y (er2o3 nps)y composition. energy [22]. this leads to the overriding effect of the packing density on the bulk modulus value than the dissociation energyon the bulk modulus value [36]. the poisson ratio value for the glass system of {[(teo2)0.7 (b2o3)0.3]0.8 (sio2)0.2}1−y (er2o3 nps)ycomposition as shown in table 2 increased from 0.2712 to 0.2809 as the er2o3concentration increased from 1% to 5%. the poisson ratio is a rigidity dependent parameter, as such increase in the poisson ratio maybe due to an increase in the rigidity of the glasses. this is due to the closure of the interstices between atoms in the glass network this allows large structural relaxation with the introduction of the sound waves, which makes the lateral strain to grow larger compared to the longitudinal strain [31]. the increase can also be attributed to the decrease in the iconicity caused by the introduction of more polar er3+ 12 aliyu et al. / j. nig. soc. phys. sci. 4 (2022) 9–15 13 ions in the network [37]. 6. bond compression model and ring deformation model the results of the elastic moduli and the poisson ratio calculated using the bond compression model for the{[(teo2)0.7 (b2o3)0.3]0.8 (sio2)0.2}1−y (er2o3 nps)yglassesare presented in this section. also discussed in this section is the ring deformation model results for the glass system of {[(teo2)0.7 (b2o3)0.3]0.8 (sio2)0.2}1−y (er2o3 nps)y composition. figure 4 and table 3 presents the variation of the bond per unit volume number and the average stretching force constant with molar fraction of er2o3 composition for the glass system of {[(teo2)0.7 (b2o3)0.3]0.8 (sio2)0.2}1−y (er2o3 nps)y composition. the bond number per unit volume increased, decreased and then increased with the increase in the er2o3 concentration. the increase in the value might be due to the increase in the number of teo4 and bo3 structures with decreasing number of teo3 and bo2o structures, indicating an increase in the number of bridging oxygen [31]. it can also be attributed with observed increase in cross-link density [36] as shown in table 3. the decrease in the value maybe also associated with the increase in the glasses’ molar volume resulting from increased interstitial spacing between constituting glass atoms [34]. the value of the average stretching force constant for the studied glasses decreased from 362.5248 to 349.8995 nm−1 as the er2o3 nps concentration increased from 1% to 5%. the decrease in the first order average stretching force constant might be associated with the substitution of oxides of higher stretching force constant; sio2 [25], b2o3 [38] and teo2 [39] with er2o3 [14, 15] having much lower stretching force constant value. the elastic moduli for glass system of {[(teo2)0.7 (b2o3)0.3]0.8 (sio2)0.2}1−y (er2o3 nps)y composition is presented in figure 5 and table 4. the values of the bulk, shear, longitudinal and young moduli ranged from 79.56544 to 81.35993 gpa, 53.76034 to 55.97828 gpa, 151.2532 to 155.9976gpa and 131.636 to 1236.6052gpa respectively with increase in the er2o3 concentration from 1% to 5%. generally, young modulus is related to the bond stretching force constant whereas the shear modulus is related to the bond bending force constants of the constituting oxides of a glass [37].the decrease in the values of the moduli maybe due the introduction of er2o3 with larger cation-anion length (r= 0.225 nm) compared to the substituted teo2 (with r = 199 nm), b2o3 (r=0.138 nm) and sio2 (r= 161) [23, 25, 40]. the decrease reflects the dependence of the elastic moduli on the average stretching force constant which decrease due the introduction of more er2o3 lower stretching force constant [41]. figure 6 presents the variation of the atomic ring size of the glass systemof {[(teo2)0.7 (b2o3)0.3]0.8 (sio2)0.2}1−y (er2o3 nps)ycomposition with er2o3 molar fraction. the atomic ring size value decreased from 0.5996 to 0.5786 nm when the er2o3 concentration was increased from 1% to 5%. the decreased in the value maybe attributed to the increase in density, crosslink density and rigidity of the glasses [37]. this implies the close packing of the material in the glass network [42]. figure 4: bond per unit volume and stretching force constant variation with molar fraction of er2o3nps for glass system of {[(teo2)0.7 (b2o3)0.3]0.8 (sio2)0.2}1−y (er2o3 nps)ycomposition. figure 5: elastic moduli variation with molar fraction of er2o3 for glass system of {[(teo2)0.7 (b2o3)0.3]0.8 (sio2)0.2}1−y (er2o3 nps)y composition. figure 6: atomic ring size variation with molar fraction of er2o3 for glass system of {[(teo2)0.7 (b2o3)0.3]0.8 (sio2)0.2}1−y (er2o3 nps)y composition table 5 presents the elastic moduli (young, bulk and shear moduli) obtained from experiment and theoretical models of makishima & mackenzie, rocherulle and bond compression. the data from makishima & mackenzie model presented a much closer elastic moduli values to the experimental values compare to the rocherulle and bond compression models. the 13 aliyu et al. / j. nig. soc. phys. sci. 4 (2022) 9–15 14 table 3: bond per unit volume number (nb), bulk modulus, bulk modulus ratio (kbc/ke), atomic ring size (`) and stretching force constant (f) for {[(teo2)0.7 (b2o3)0.3]0.8 (sio2)0.2}1−y (er2o3 nps)y glass system. y nb (×1028 m−3) kbc (gpa) ke (gpa) kbc/ke l (nm) f (n/m) nc 0.01 8.4014 79.5728 30.6560 2.5957 0.5828 362.525 2.4289 0.02 8.4360 79.7584 31.1124 2.5636 0.5792 359.322 2.4704 0.03 8.4304 79.5654 33.3348 2.3869 0.5680 357.029 2.5112 0.04 8.5419 80.4765 41.1798 1.9543 0.5360 353.009 2.5516 0.05 8.6507 81.3599 43.1171 1.8870 0.5284 349.900 2.6161 table 4: elastic moduli and poisson ratio for glass system of {[(teo2)0.7 (b2o3)0.3]0.8 (sio2)0.2}1−y (er2o3 nps)y composition. y kbc (gpa) gbc(gpa) lbc(gpa) ebc(gpa) σbc 0.01 79.5738 53.7603 151.253 131.636 0.22429 0.02 79.7584 54.1123 151.908 132.396 0.22334 0.03 79.5654 54.2003 151.833 132.512 0.22243 0.04 80.4765 55.0357 153.858 134.457 0.22154 0.05 81.3599 55.9783 155.998 136.605 0.22016 table 5: comparative presentation of bulk, young and shear moduli obtained from non-destructive ultrasonic spectroscopic analysis (experimental) and makishima & mackenzie, rocherulle and bond compression models y bulk modulus young modulus shear modulus ke km kr kbc ee em er ebc ge gm gr gbc 0.01 30.656 35.754 41.452 79.573 31.801 52.854 56.910 131.64 20.546 21.080 22.385 53.760 0.02 31.112 36.523 42.198 79.758 33.951 53.306 57.298 132.40 21.685 21.208 22.493 54.112 0.03 33.335 36.942 42.947 79.565 34.770 53.498 57.682 132.51 22.436 21.252 22.600 54.200 0.04 41.180 38.399 43.697 80.477 35.412 54.426 58.060 134.46 23.841 21.533 22.705 55.036 0.05 43.117 39.862 44.450 81.360 37.720 55.335 58.432 136.61 25.302 21.809 22.809 55.978 values of the elastic moduli from the elastic models showed an increasing trend just as obtained from the experimental data but demonstrating comparatively very high values from the bond compression model. data obtained from the rocherulle model showed a very much closer relation to the data obtained from the experimental assessment compared to the bond compression model. the increase in the elastic moduli is generally associated with increased rigidity and network connectivity [43, 44]. 7. conclusions makishima and mackenzie, rocherulle and bond compression models were used in this work for the theoretical determination of the elastic moduli of er3+ ions doped silica borotellurite glass system with empirical formula {[(teo2)0.7 (b2o3)0.3]0.8 (sio2)0.2}1−y (er2o3 nps)y with y = 0.01, 0.02, 0.03, 0.04, 0.05. the elastic moduli (young, shear and bulk) increased with increasing concentration of er2o3 in the makishima and mackenzie model. the values of the elastic moduli calculated presented a decreasing value of young and shear elastic moduli with increasing pattern of the corresponding bulk modulus value as the er2o3 concentration was increased. the bond compression model presented rather much higher values of the elastic moduli compared to the other two models with an increasing value variation against the er2o3 molar concentration. this might be due to the fact that, unlike the makishima & mackenzie and rocherulle models which commonly considers densities and dissociation 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[44] r. a. tafida et al., “structural, optical and elastic properties of silver oxide incorporated zinc tellurite glass system doped with sm3+ ions”, mater. chem. phys. 246 (2020) 122801, doi: 10.1016/j.matchemphys.2020.122801. 15 j. nig. soc. phys. sci. 4 (2022) 59–63 journal of the nigerian society of physical sciences effect of time on the syntheses of carbon nanotubes via domestic oven n. kurea,∗, i. h. daniela, n. m. hamidonb, i. i. lakina, b. u. machua, e. j. adoyia adepartment of physics, faculty of science, kaduna state university, kaduna, nigeria binstitute of nanoscience and nanotechnology, universiti putra malaysia, selangor, malaysia abstract in this study, carbon nanotubes (cnts) were synthesized directly on coated silicon substrate via commercial microwave oven at 2.45 ghz for 3 minutes (sample a) and 4 minutes (sample b). the plasma provides the required temperature for catalytic decomposition of carbon source (polyethylene) at 750 ◦c under atmospheric pressure. raman spectroscopy, field emission scanning electron microscopy (fesem), high resolution transmission electron microscope (hrtem), x-ray diffractometer (xrd) techniques are used to characterize the as-synthesized. results indicate that, the calculated carbon quality was found to be 1.01 and 1.02 for sample a and sample b respectively with average diameter range of (6.0 to 10.0) ± 0.5 nm. the high intensity ratio is attributed to the defect mode in the cnts. also, the analysis from fesem shows twisted and randomly oriented structures with an interlayer spacing of about 0.35 nm in the internal structure of most cnts. hrtem further confirmed the interlayer spacing of about 0.35 nm corresponding to fesem result. the crystallinity of the cnts was obtained via x-ray diffraction techniques. lastly, the results indicate sample a and b produces cnts, with sample b having more graphitic structure than sample a due to duration of synthesis process. doi:10.46481/jnsps.2022.355 keywords: plasma, raman, coated silicon, polyethylene, quartz article history : received: 18 august 2021 received in revised form: 18 january 2022 accepted for publication: 19 january 2022 published: 28 february 2022 c©2022 journal of the nigerian society of physical sciences. all rights reserved. communicated by: e. a. emile 1. introduction carbon nanotubes (cnts) are carbon-based nanomaterials which has attracted much interest from scientific community due to its exceptional carbon-carbon bonding which assists carbon to form its allotropes, such include carbon nanofibers; graphites; graphenes and carbon quantum dots [1, 2, 3, 4, 5]. these exceptional properties possess by the nanomaterials tremendously revolutionize the area of carbon-based nanoscience such as electrical, thermal, chemical and mechanical properties [1, 6]. these ∗corresponding author tel. no: email address: kurenicodemus@gmail.com (n. kure ) carbons-based nanomaterials have several potential applications due to is unprecedented properties amongst such are superconductivity, drug delivery, field emission, sensor and super capacitors [7, 8, 9]. researchers developed keen interest on how to synthesize this carbon-based nanomaterial among such methods are arc discharge [6], laser ablation [10] and chemical vapor deposition [11]. however, these techniques of synthesizing cnts are believed to be time consuming and expensive, in order to eliminate that, an attempt was made in this study to synthesize cnts on silicon substrate using commercial microwave oven with operating power of 600 w at 2.45 ghz utilizing the microwave energy [4, 12]. the synthesis utilizes microwave 59 kure et al. / j. nig. soc. phys. sci. 4 (2022) 59–63 60 heating due to its advantages over the conventional methods such as rapid reaction time and volumetric heating which provides the temperature needed for the catalytic decomposition of carbon precursor (polyethylene) [13, 3, 4]. polyethylene has dielectric loss tangent at 2.45 ghz, which makes it a good conducting polymer for the synthesis [4, 14]. 2. materials and methods all materials were used as purchased without further purification. the microwave was modified with cylindrical quartz tube to serve as the reaction chamber (reactor). the experiments were carried-out in tubular reactor (quartz tube) of length 55.4 ± 0.1 cm, with outer and inner diameter of 7.1 ± 0.1 cm and 6.6 ± 0.1 cm, respectively. the carbon precursor of about 100 mg was used and iron (iii) nitrate (fe (no3)3.9h2o) as catalyst [4]. the 1 g of iron (iii) nitrate was dissolved in 50 ml of ethanol, which was impregnated on the two substrates and allowed to dry at ambient temperature. the substrate was placed in oven for 1 hour at 200 ◦c to convert the catalyst into iron oxides [15]. carbon source and substrate were placed inside the reaction chamber and the pressure inside was reduced to 0.81 mbar by rotary vacuum pump. the process of plasma formation and decarbonation was carried out as explained [4]. thermocouple was embedded inside the chamber to measure the temperature of the synthesis process at 750 ◦c for 2 minutes – 3 minutes. a black deposit was found on the substrate surface after the synthesis process [4, 12]. the deposits were allowed to cool down at ambient temperature before characterization [4, 12]. the carbon quality was analyzed using witec alpha 300r confocal raman microscope with laser excitation energy of 532 nm. morphological investigation was carried out using field emission scanning electron microscope (fesem, s3400n hitachi). high resolution transmission electron microscopy (hrtem) images were obtained using a fei tecnai g2 f20 x-twin. x’pert highscores and image j processing software was used to obtained the level of crystallinity and diameter of the as-synthesized. the schematic representation of the experimental setup is illustrated in figure 1. the commercial microwave oven was modified with small diameter of holes, where quartz tube of volume 1860 cm3 was inserted perpendicular to the position of the waveguide inside the microwave cavity. the tube was made vacuum with the help of teflon cap and rotary vacuum pump below atmospheric pressure continually throughout the experiment [3, 4, 12]. 3. results and discussion raman spectrum in figure 2 shows two prominent peaks in the first-order raman scattering at 1336.77 cm−1 as d-band and 1588.50 cm−1 as g-band of the synthesized cnts in sample a. the d-band indicates the disordered carbon atoms and g-band indicates the ordered carbon atoms on the cns [4, 16]. carbon quality can be deduced using the ratio of the defect intensity (id) to graphite intensity (ig ) [11, 13]. the intensity ratio (id/ig ) of the obtained cnts was calculated to be 1.01, figure 1. schematic of the experimental setup figure 2. figure 2 raman spectra of sample a which shows the carbon nanostructures are graphitic and contain amorphous materials as defects. the fesem image shows the growth of twisted and randomly oriented structures. figure 3 (a) (b) shows fesem image of the cnts obtained from plasma catalytic decomposition of carbon precursor on coated substrate. the diameter distributions were calculated from fesem image (figure 3 (a)) using image j image processing software and the obtained data were used to plot the histogram as presented in figure 4. the histogram also shows the highest average diameter range at (6.0to 10.0) ± 0.2 nm. the cnts diameters are in comparison with related works and it is believed that the catalyst particle size has significant effect on the cnts diameter due to interaction between the catalyst and the substrate [4, 17]. the raman shift (wave number) indicates the graphitic nature of the cnts in figure 5, this shows the raman spectra of sample b with first-order raman scattering at 1336 cm-1 and 1604 cm-1 representing d and g-band respectively [18]. the intensity ratio of id/ig of the cnts is calculated to be 1.02 [4, 11, 13], and intensity (id) increment may arise due to the presence of defects from the edges and amorphous carbon remains in the cnts [13]. the raman shift for both samples tends to have higher order degree of vibrations at the range 2500 cm-1 – 3000 cm-1 as the crystal size decreases. 60 kure et al. / j. nig. soc. phys. sci. 4 (2022) 59–63 61 figure 3. fesem image of sample a (a) 100,000 magnification (b) 400,000 magnification figure 4. diameter distribution of sample a the surface morphology of cnts was investigated using fesem. the images from fesem shows surface morphology of the cnts were twisted and randomly oriented structures [13]. as clearly indicated in sample b (figure 6 (a) (b)), the cnts were twisted and randomly oriented structures due to amorphous carbon attributed to the d-band level in raman spectra (figure 5). the arrows indicate remains of catalyst emfigure 5. raman spectra of sample b figure 6. fesem image of sample b (a) 100,000 magnification (b) 150,000 magnification with arrows indicating remain of catalyst bedded inside the nanotubes (figure 6 (b)), due to capillary effect which depend on the catalyst-substrate interaction [13]. the histogram shown in figure 7 was computed based on the data collected from figure 6 using image j, image processing software. this shows the diameter distribution of the obtained cnts with error bars, and most of the obtained cnts have highest average diameter range at (6.0 to 10.0) ±0.5 nm. 61 kure et al. / j. nig. soc. phys. sci. 4 (2022) 59–63 62 figure 7. diameter distribution of sample b the cnts diameters are in comparison with related works and it is understood that the catalyst particle size has significant effect on the cnts diameter [4, 17]. the hrtem analysis was conducted to verify the quality of the as-synthesized. micrograph from hrtem confirms the as-synthesized indeed consist of good graphitic sheets in agreement with raman spectroscopy. the hrtem image shows the samples (a and b) are twisted and randomly oriented structures in agreement with fesem results as depicted in figure 8 (a) (b) respectively. the graphitic sheets were parallel and are separated uniformly of about 0.35 nm in distance between each sheet [4, 19]. the crystalline structures of the as-synthesized were analyzed. diffraction pattern indicate the samples are crystalline in nature, all the samples have sharp peak position at 25.2◦ (indexed as c(002)) attributed to the inter-layer spacing of 0.35 nm which could be assigned to the hexagonal structure of graphite sheets forming the cnts [4, 20, 21]. the peak at 13.77◦, 16.56◦ and 18.33◦ are attributed to the catalyst as shows in figure 6 (a) (b) . the peak at 28.08◦ indicate graphitic nature of the as-synthesized corresponding to the inter-layer spacing of about 0.35 nm layer graphite, as shown in figure 9 (a) (b) [4, 20, 21] . from, figure 9 (b) cnts was clearly seen in sample b than in sample a, may be due to duration of synthesis . therefore, the obtain products indicate presence of hexagonal graphite which correspond to cnts, since they consist of graphitic sheets folded into hollow cylinders, and the distance between cylinders sheets is very close to that of layer graphite [19]. furthermore, the xrd also indicates the level of samples purity. hence, the main elemental compositions in the study samples were carbon, iron and silicon which arises from the carbon precursor, catalyst and subrates respectively. the carbon content in the samples is about 95% purity of the as-synthesized as shown in figure 9 ((a) and (b)), in conformity with research reports [4, 12, 13, 22, 23]. 4. conclusion the experimental setup successfully synthesized cnts. the synthesis pressure and temperature were maintained at 0.81 mbar figure 8. hrtem image at 10 nm on sample (a) a (b) b and 750 ◦c respectively and the reaction was carried out for a period of 3 minutes and 4 minutes for sample a and sample b respectively. the by-products of the reaction were allowed to reach ambient temperature, and then collect for characterization. the raman spectra show the graphitic nature of the as-synthesized, and the defects indicate the presence of amorphous carbon. the quality of the graphite is calculated to be 1.01 and 1.02 in sample a and b respectively. from the fesem images, it shows the formation of twisted and randomly oriented structures. the irregularities in the shape of the structure are attributed to amorphous carbon. the diameter of the as-synthesized is determined by the catalyst size as observes. image j image processing software is used to estimate the diameter distribution, with most of the cnts at (6.0 to 10.0) ± 0.5 nm in both samples. catalyst encapsulated inside the as-synthesized shows the characteristic of cnts obtained via microwave oven [13]. the hrtem analysis further confirms the presence of twisted and randomly oriented structures. the remains of catalyst in the nanotubes tips and bottom were due to capillary action which arises from catalystsubstrate interaction. the xrd analysis shows the crystallinity of the as-synthesized. the diffraction pattern from x’pert highscores software reveals the crystalline nature consists of interlayer spacing of about 0.35 nm close to a typical graphitic sheet. 62 kure et al. / j. nig. soc. phys. sci. 4 (2022) 59–63 63 figure 9. xrd pattern of sample (a) a (b) b characterization analysis indicates that samples are cnts. although, the sample duration of 3 minutes (sample a) shows a cnts structure with little amount of carbon content compared to sample b with 4 minutes duration, which basically shows that time actually has an effect in the formation of a well graphitic cnts. the develop plasma technique is economical, less time consuming in an eco-friendly manner for synthesizing carbonbased nanostructures as compared with related work [4, 12, 13, 21, 22]. we recommend researchers to focus on the optimum condition; the defects zone in the carbon structure may be minimal, and result to even distribution of nanoparticles 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[23] n. kure, z. yunusa, m. n. hamidon, i. h. daniel & i. i. lakin, “synthesis of carbon nanostructures using microwave enhanced chemical vapor deposition and its potential application to ammonia sensing”, physicsaccess 1 (2021) 44. 63 j. nig. soc. phys. sci. 1 (2019) 131–137 journal of the nigerian society of physical sciences original research first-principles calculations of fluorine-doped titanium dioxide: a prospective material for solar cells application a. shamsudeenb, a. shuaibua,∗, s. g. abdua, m. s. abubakara, abdullahi lawalb adepartment of physics, kaduna state university, p. m. b. 2339, kaduna, nigeria bdepartment of physics, federal college of education zaria, nigeria abstract this study focuses on the anatase tio2 doped fluorine to investigate their structural and electronics properties using density functional theory (dft) within generalized gradient approximation (gga) as implemented in quantum espresso (qe). for the anatase tio2 phase the calculated electronic band structures of puretio2 and tio2 doped fluorine nanocrystals are displayed along a high symmetry directions and the energy range of band structure is plotted from 0.0 ev to 3.9 ev , the energy separation between the bottom of the conduction band and the top of valence band occurred at the γ and n points, indicating that anatase t io2 is an indirect band gap material with an approximate value of 2.30 ev energy gap, this value is consistent with previous dft result. when f is added the band structure did not change much because fluorine element doping is conducive to the generation of oxygen holes and enhances the mobility of effective electrons which can enhance the conductivity of the adsorbent substrate and improve the solar cell performance of the fluorine-doped tio2. the band gap value obtained for f doped tio2 was found to be 2.11 ev . the dopant formation energy of fluorine is calculated to be −55.6 ry which is equivalent to −756.5 ev . keywords: dft, t io2, fluorine, electronic properties, solar cells article history : received: 26 august 2019 received in revised form: 10 october 2019 accepted for publication: 11 october 2019 published: 17 december 2019 c©2019 journal of the nigerian society of physical sciences. all rights reserved. communicated by: b. j. falaye 1. introduction engineers and scientists show so much interest in compounds made of transition metal oxides which have wide band gap, very good light absorption capability, very good value of heat formation and other various fascinating properties. this is because these properties present the potential of use in advanced applications such as solar cells [1, 2, 3]. nowadays a lot of investigations are carried out on oxide materials, especially to meet industry requirements. this is because material oxides are used in a lot of art technologies such as magnetic disks, ics and solar cells and their likes. a lot of properties sought ∗corresponding author tel. no: +2348179931945 email address: alhazikara@gmail.com (a. shuaibu ) by researchers and the industry, which is driving more research into the area, include need for economical alternative to current materials, non-toxic and environmentally friendly, wide availability and good electrical, magnetic and or optical properties for use as base materials or doping in optical fibers and similar uses. these demands are partly driven by need to meet energy demand in the future [3, 4, 5]. one of the oxides which attracted a lot of theoretical and experimental research effort is tio2 [6]. this is in order to better understand the optoelectronic properties of tio2. many naturally occurring and engineered polymorphs of t io2 exist. these include anatase, baddeleyite, pyrite, columbite, brookite, fluorite, cotunnite and rutile. three of these (i.e. anatase, brookite and rutile) exist naturally while the rest are manufactured. due to popular demand and remark131 shamsudeen et al. / j. nig. soc. phys. sci. 1 (2019) 131–137 132 able properties of the naturally occurring polymorphs such as rutile, for use in production of catalyst, flat panel displays, optoelectronic devices, sensors, and solar cell technology and so on, many of them are often artificially synthesized. rutile, as a result of its structure, possesses one of the highest stabilities that can be found as compared to others. anatase is also thermodynamically stable up to a temperature of about 800 oc. on another hand brookite has a structure that resembles that of rutile, though with less stability at high temperatures [7, 8]. the structure of anatase is the most suitable for photo catalytic applications. it is one of the reasons for attracting a lot of research interest into the properties of t io2. other properties of t io2, for which it is held in high regards is its use in many ceramic materials to add features and capabilities such as being hydrophilic, photo catalytic, hydrophobic and having antibacterial features [9], this is aside its good chemical stability, nontoxicity and reduced cost. t io2 is also used for air and water purification by taking advantage of its antibacterial capabilities. it is used in pvc fabrics, used to make materials with self-cleaning effects, used for preservation of glass and cultural heritage [10]. it is also used in de-synthesizing photo cells as a common photocathode. the history of t io2 went as far back as 1971 when the work of fushijima and honda revealed the photo electrochemistry of the material while using it as an anode in an electrochemical cell. this application is what gave rise to all the interest that was generated over the years which led to discovery of a lot of its other properties [11]. wang et al.[12] reported that t io2, when used to fabricate ultra-violet (uv) photo-detector, shows more energy efficient operation than other material such as gallium nitride (gan), silicon (s i) and zinc oxide (zno). its wide band gap is what limits the usage of t io2 as a semiconductor directly used for semiconductor component. this is why it is mostly used as a substrate for other semiconductor materials especially in a thin film solar cell technology. because of these multiple uses and benefits that can be derived from t io2, it is obviously of paramount importance, to areas of harnessing solar energy, optoelectronics, capturing visible and near visible light, to unveil the full potentials of this compound and its various forms. this should be done through tuning the properties of the oxides such as its band gap and so on. this need is the drive for theoretical and experimental researches in order to have the comprehensive knowledge regarding its structure, properties and their relationship to its performance. a lot of work, especially experimental work is now available in the research community regarding t io2, but theoretical results of research related to the compilation that are adequate to characterize it are still scarce. currently most of the theoretical works are at the level of investigations which were performed via density function theory (dft). since it gives a simple basis to characterized the quantum properties of various materials and plays an important role in designing many new materials. in addition to many experimental studies of t io2, is seen by many as a favorable material for many important high technological applications, a few important first principle dft studies are reported in previous work. furthermore, the studies discussed and many others, the band gap values of the materials are either overor under-estimated. this results in deducting contradictory nature of the band gaps which highlights the need for more studies. in addition, a lot of the work was done at different side by side of gga and lda, a technique which underestimate the band gap. thus, to better understand the optoelectronic properties of t io2 such as strong light absorption, photocurrent sensitivity to the polarization of light it is essential to use reliable xc approximations to determine its electronic properties. to utilize the maximum spectral range in the solar spectrum, the band gap of t io2 should be tuned, which will broaden the operational optical window of the t io2 -based optoelectronic devices. nowadays, doping is one of the well-approaches to improve the properties of materials. doping can dramatically modify physical and chemical properties of materials [13]. therefore, t io2 is repeatedly doped with different metals and transition elements to tune its band gap and enhance its optical, electrical, and magnetic properties [14, 15, 16, 17, 18]. non-metal doping is another approach used to narrow the band gap; in comparison to metal doping that often forms a donor level in the forbidden band, non-metal doping usually shifts the valence band edge upward [18]. doping with non-metal elements such as f shows an enhancement in the electronic, optical, and magnetic properties of narrow-band gap semiconductor materials [19, 20, 21]. the doping by f has also shown higher thermal stability and conductivity in graphene [22]. the aforesaid reasons motivated us to study the doping by fluorine atom, which can significantly modify/tune the electronic properties t io2. to the best of our knowledge, no theoretical studies on f-doped t io2 exist in literature. the fluorine doping approach in t io2 may open new paths to non-metal elements doping for various other potential applications such as infrared detectors, infrared leds, lasers, transistors, and thermo-photovoltaic systems. 2. computational details 2.1. computational method the calculations are performed on the 2 × 2 × 2 supercells relative to the standard primitive unit cell of fluorine doped anatase t io2 within first principle calculation using quantum espresso simulation package [23]. perdew-burke-ernzerhof generalized gradient approximation (pbe-gga) exchange-correlation potential [24] were used for treating electron-electron effects. for integrals, smearing has been adopted and to be specific maxfessel-paxton smearing method. the brillouin zone integration is performed using monkhorst-pack scheme [25] with 3 × 3 × 2 k-points grids for all the materials. for fluorine doped t io2 structure 2 × 2 × 2 supercells relative to the standard conventional unit cell were used. the supercell consists of twelve numbers of atoms: four titanium atom and eight oxygen atoms. one o atom was replaced by fluorine atom making 0.25 % occupancy by the dopant. the super cell dimensions are kept fixed throughout the calculations, while the atomic positions are fully relaxed for all calculations using broydenfletcher-golfarb-shannon (bfgs) algorithm, until the forces acting on the atoms are below 0.001 ev/å. 132 shamsudeen et al. / j. nig. soc. phys. sci. 1 (2019) 131–137 133 3. results and discussion 3.1. convergence test in any dft calculation using dft code, it is of paramount importance to perform a convergence test calculation before commencing the actual calculation. the results presented below represent the convergence test with respect to plane wave kinetic energy cut-off and k-points mesh for undoped anatase t io2 and fluorine atom doped t io2. 3.1.1. convergence test results of undoped t io2 figure 1: (a) the convergence of total energy with respect to the kinetic energy cut-off. (b) the convergence of the total energy with respect to the k-points grids. 3.1.2. convergence test results of doped fluorine atom. it could be seen from figure 1 that the total energy changes considerably with the kinetic energy cut-off, until at some energy cut-off where it becomes almost stable. in all the two cases, as the total energy decreases, the kinetic energy cut-off increases from 10 ry to 40 ry and becomes almost stable at 40 ry. that is to say, the total energy remains constant with any further increases in the kinetic energy cut-off from 40 ry. this indicates a well-converged energy cut-off. that is why; figure 2: (a) no heading. (b) no heading. 40 ry was used as the plane wave basis set for the kinetic energy cut-off of the two cases. however, figure 2b and figure 5b show the variations of the total energy with respect to the k-points grids. the total energy changes considerably with the number of k-points at a certain points, showing a well-converged value. in all the two cases, the total energy increases from 1×1×2 to 3×3×2 k-point grids and become almost stable at 3 × 3 × 2. as such, 3 × 3 × 2 k-points have been adopted for the three cases. furthermore, monkhorst and pack method of selecting k-points is commonly used in most dft calculations [25]. most of the dft codes provide ways of choosing k-points using monkhorst and pack method. 3.2. structural properties of the undoped t io2 and t io2 doped f. in order to explore the structural properties of the anatase t io2 with space group 14/amd was simulated in figure 3. the anatase structure exhibit tetragonal geometry and have a high symmetry in which six oxygen atoms is surrounded to each t i atom. from table 5, it can be observed that, for 12.5 % substitutional doping case, there are eight symmetrically different approach in which oxygen (o) atom can be replaced by fluorine (f) atom(s). but for this particular study, as shown in tables above, one configuration (d8 0.125) is considered. no structural transition is seen; as such the crystal parameters remained 133 shamsudeen et al. / j. nig. soc. phys. sci. 1 (2019) 131–137 134 table 1: convergence test of t io2 total energy with respect to kinetic energy cut-off of the plane wave of undoped t io2. s/n kinetic energy cut-off (ry) total energy (ry) 1 10 -680.917 2 20 -740.466 3 30 -747.717 4 40 -747.374 5 50 -748.413 6 60 -748.421 7 70 -748.429 table 2: convergence of the total energy with respect to the k-points grids of undoped t io2. s/n number of k-points total energy (ry) 1 1 × 1 × 2 -749.018 2 3 × 3 × 2 -747.670 3 5 × 5 × 2 -747.673 4 7 × 7 × 2 -747.673 5 9 × 9 × 2 -747.672 6 11 × 11 × 2 -747.673 7 13 × 13 × 2 -747.674 table 3: convergence test of total energy with respect to the kinetic energy cut-off. s/n kinetic energy cut-off (ry) total energy (ry) 1 10 -694.517 2 20 -756.204 3 30 -763.468 4 40 -764.135 5 50 -764.174 6 60 -764.183 7 70 -764.191 table 4: convergence of the total energy with respect to the k-points grids of undoped t io2. s/n number of k-points total energy (ry) 1 1 × 1 × 2 -764.697 2 3 × 3 × 2 -763.468 3 5 × 5 × 2 -763.481 4 7 × 7 × 2 -763.478 5 9 × 9 × 2 -763.478 6 11 × 11 × 2 -763.479 7 13 × 13 × 2 -763.480 table 5: configurations for substitutional doping of o by f anatase t io2 material. undoped o o o o o o o o d1 0.125 f o o o o o o o d2 0.125 o f o o o o o o d3 0.125 o o f o o o o o d4 0.125 o o o f o o o o d5 0.125 o o o o f o o o d6 0.125 o o o o o f o o d7 0.125 o o o o o o f o d8 0.125 o o o o o o o f 134 shamsudeen et al. / j. nig. soc. phys. sci. 1 (2019) 131–137 135 figure 3: (a): anatase structure for undoped t io2 and (b) t io2 doped f. the same as that for the undoped system. to find the stability of the structure after fluorine doping, the dopant formation energy of the fluorine atom is estimated using equation 1 [26]. the dopant formation energy in this context simply refers to the energy needed to insert one fluorine atom with a chemical potential µf into the supercell after removing one oxygen atom with chemical potential µf from the same position [27]. e f = edoped − eundoped + µo −µf (1) where, edoped is the total energy of the anatase t io2 material; eundoped is the total energy of the anatase undoped t io2 system; µo is the chemical potential per atom of oxygen bulk crystal; µf is the chemical potential per atom of fluorine bulk crystal. following common practice, the chemical potentials were found as the dft total energy per atom in the bulk systems. the dopant formation energy of fluorine is calculated to be −55.6 ry which is equivalent to −756.5 ev . this serves as the measure of the stability of the doped structure, the lower value of the formation energy signifies the most stable structure. from this value of dopant formation energy, this shows that, the fluorine doped t io2 is stable. 3.3. electronic properties electronic properties calculations are very crucial for describing the optoelectronic properties of solids. the electronic properties investigations of t io2, doped fluorine covers the electronic band structure, density of state (dos) and partial density of state (pdos). the main purpose of the ground state, electronic band structure, dos and pdos calculations in this work is to obtain ks eigenvalues and eigenfunctions as well as useful information about the electronic properties of the concerned materials. to understand the effect of doping on t io2, fluorine (f) was added in the calculations. the calculated electronic band structures of pure t io2 and t io2 doped fluorine thin film are displayed along the eight symmetry (γ → h → n → p → γ → x → m → r) directions and the energy range of band structure is plotted from 0.0 ev to 3.9 ev . the fermi level position on the band structure of these crystals is shown by the zero on the energy scale. pbe exchange correlation potentials is chosen over lda, because in several cases gga-pbe gives more reliable and accurate results for dft electronic calculation. the energy separation between the bottom of the conduction band and the top of valence band occurred at the γ and n points or band structure calculations within pbe which indicate that anatase t io2 is an indirect band gap material with value of 2.30 ev energy gap, this value is consistent with previous dft result. however, the value is smaller than experimental result of 3.21 ev [28] and this effect is the limitation of dft approach due to approximations used in the exchangecorrelation functional. on the other hand, the f-doped t io2 model was established based on the perfect crystal plane model. a fluorine atom replaces one of the o atoms on the surface of the anatase t io2 perfect crystal plane, and f atom combined with t i atoms to form t i − f bonds. it can be seen that the energy band gap did not change much after the fluorine atom being doped and the value obtained was found to be 2.11 ev . furthermore, fluorine element doping is conducive to the generation of oxygen holes and enhances the mobility of effective electrons, which can enhance the conductivity of the adsorbent substrate and improve the solar cell performance of the fluorinedoped t io2. figure 4: band structure of (a) pure t io2 (b) f doped t io2 for more clarification for the nature of the energy gap, we have also study the total density of state (dos) of anatase t io2, and f doped anatase t io2. figure 5 shows a decomposition of the calculated total dos of t io2 and f doped t io2 in bulk forms. for pure t io2 the lowest valence states is dominated by t i-s orbital and o-p orbital while in the cased of f doped t io2 the lowest valence states is dominated by t i-s orbital, o-p orbital and f-p orbital. the t i-d orbital and o-s orbital contribute much high in the conduction band for pure t io2 while t i-d, o-s, o-s orbital as well as f-s orbital contributed slightly higher in the conduction band. the t i-p and o-p orbitals of t io2 contribute a slightly higher in the valence band near the fermi level. for f anatase doped t io2 the main contribution 135 shamsudeen et al. / j. nig. soc. phys. sci. 1 (2019) 131–137 136 table 6: calculated energy gap of t io2 and t io2 doped f with previous first principle calculations and experimental data. material work method band gap eg(ev ) nature pure t io2 our work pbe-gga 2.30 indirect other work pbe-gga[29] 2.14 indirect ev-gga[29] 2.32 indirect pbe-gga[30] 2.30 indirect pbe-gga[31] 2.30 indirect experiment[32] 3.21 indirect f doped t io2 our work pbe-gga 2.11 indirect of valence band near the fermi level is from t i-p orbital and op orbital. the t i-s, o-s and o-p orbitals contributed the most near the fermi level, thus held responsible for the properties of t io2. also, sand p-orbitals of t i atoms, sand p-orbitals of o atoms and s-orbital of f atoms of f doped t io2 compound contributed the most near the fermi level, thus held responsible for the properties. figure 5: total and partial density of state (a) pure t io2 (b) f doped t io2. 4. conclusion a first principle study was used using pbe xc functional in quantum espresso code. the available results were compared with other experimental values. the electronic properties density of state, partial density of state and band gap values were calculated in the ground state properties. it was found that t io2 in anatase phase has indirect band gap with a value of 2.30 ev , this value is consistent with previous dft results. the obtained band gap value is smaller than experimental result of 3.21 ev and this effect is the limitation of dft approach due to approximations used in the exchange-correlation functional. the fluorine doped t io2 model was established based on the perfect crystal plane model. a fluorine atom replaces one of the o atoms on the surface of the anatase t io2 perfect crystal plane, and f atom combined with t i atoms to form t i − f bonds. it can be seen that the energy band gap did not change much after the fluorine atom being doped. furthermore, fluorine element doping is conducive to the generation of oxygen holes and enhances the mobility of effective electrons which can enhance the conductivity of the adsorbent substrate and improve the solar cell performance of the fluorine-doped t io2. the dopant formation energy of fluorine is calculated to be −55.6 r which is equivalent to −756.5 ev . the formation energy serves as the measure of the stability of the doped structure; the lower value of the formation energy signifies the most stable structures [26]. from this value of dopant formation energy, this shows that the fluorine doped anatase t io2 is stable. acknowledgments we thank the referees 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[32] c. pawan, p. basyach & a. choudhury, “structural, optical and photocatalytic properties of t io2/s no2 and s no2/t io2 core-shell nanocomposites: an experimental and dft investigation”, chemical physics 434 (2014) 1. 137 j. nig. soc. phys. sci. 3 (2021) 74–81 journal of the nigerian society of physical sciences synthesis of sno2/cuo/sno2 multi-layered structure for photoabsorption: compositional and some interfacial structural studies l. o. animasahuna,b,∗, b. a. taleatua, s. a. adewinbia,c, h. s. bolarinwab, a. y. fasasid adepartment of physics and engineering physics, obafemi awolowo university, ile ife, nigeria bdepartment of physics electronics and earth sciences, fountain university osogbo, nigeria cdepartment of physics, osun state university, osogbo, nigeria dcenter for energy research and development, obafemi awolowo university, ile ife, nigeria abstract many metal oxide heterostructures have been synthesized as mixed oxides or layered structures for photocatalytic, photodegradation of pollutants and light-harvesting applications. however, in the layered structures the effects of interfacial properties and composition have largely not been explored. hence, the effects of interfacial mixing and diffusion of sandwiched thin cuo layer on optical absorption of as-deposited and heattreated multi-layered structured sno2/cuo/sno2 films were studied. the rbs analysis of the as-deposited films showed the presence of a minute amount of cu in the surface and bottom sno2 layers of the structure. we attributed this to inhomogeneous layer thickness evidenced by very low sn/cu atoms ratio of the cuo layer. however, the thermal treatment of the layered structure led to pronounced interlayer mixing and consequent formation of sno2-cuo solid solutions throughout the layered structure. the layer integrity of the inserted cuo of the as-deposited films was very high and the as-deposited structure was far more optically absorbing. however, the annealed structure showed lesser optical absorption because of the onset of interfacial mixing and improved crystallization. this reflected in the optical bandgap variations of the as-deposited and annealed multilayered structures. the significance of this result is that the multi-layered films possess band narrowing – evidence of increased photon absorption making it a better candidate than pure sno2 oxide for photocatalysis, photodegradation and photodetection applications. it also pointed to the fact that attention must be paid to effects of heat treatments or annealing when inserting an absorbing layer into a photocatalyst or a material meant for photodegradation or any light-harvesting material. doi:10.46481/jnsps.2021.160 keywords: rutherford backscattering spectroscopy, thin films, interfacial mixing, optical absorption, photocatalysis, photodegradation article history : received: 26 january 2021 received in revised form: 12 march 2021 accepted for publication: 26 march 2021 published: 29 may 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: b. j. falaye 1. introduction one of the attendant drawbacks of decades of global reliance on fossil energy sources is the environmental pollutions ∗corresponding author tel. no: email addresses: luqman.animasahun@gmail.com (l. o. animasahun ), afasasi@cerd.gov.ng (a. y. fasasi) caused exploration processes and release of toxic wastes into the environment. prominent among the various techniques that have been used in solving this problem is the use of metal oxides semiconductor heterostructures in photoabsorption and photocatalytic degradation of many pollutants [1-7] and detection of harmful gases [8-11]. metal oxides heterostructures offer 74 animasahun et al. / j. nig. soc. phys. sci. 3 (2021) 74–81 75 several advantages such as easy band engineering, improved charge carrier transport, increased electromagnetic photon absorption and overall better efficiencies over single oxides [1,45,12-15]. many metal oxide heterostructures have been synthesized as mixed oxides or layered structures [2,5,6,13,15,16]. besides, a lot of these heterostructures have been used for other applications such as electrocatalysts for fuel cells, an electron transporting layer in perovskite solar cells, photocatalyst for hydrogen productions [11,15,17,18]. however, despite these observed advantages, the confusion about the actual role of the added second oxides in photocatalysis or gas sensing in many heterostructures is yet to be cleared. the argument has been whether the added second oxide plays catalytic or carrier transport role or aids in photon absorption [17]. one way to gain insight into this problem is to know the relative positions of each constituent in the heterostructure. for instance, in gas sensing, an oxide that is not in contact with the analyte gas could not have performed a catalytic role. hence, we have demonstrated in this work the use of rutherford backscattering spectroscopy to investigate the composition, and thickness, and interfacial mixing of the inserted cuo layer in a sno2 /cuo/sno2 multi-layered film (figure 1). rbs belongs to a cluster of techniques called ion beam analysis. it is a quantitative materials characterization technique that has been excellent for evaluating atomic densities of thin films and compositional depth profiling of blanket and multi-layered structures [19]. it is also capable of measuring quantitatively the interfacial mixing in multi-layered films. all of these parameters are central to the understanding and optimization of the properties of materials. nevertheless, its application to investigate laterally inhomogeneous samples is chiefly unexplored. also, we investigated the effect of copper oxide insertion and interfacial diffusion on the optical bandgap structure of sno2. this is of importance to us because the optical bandgap determines to a very great extent the electromagnetic photon absorption properties of most materials. it predicts effectively whether a material will absorb only in the uv region or it will extend its absorption edge to the visible region of the e-m spectrum. this is of great importance in photocatalysis, photodegradation of pollutants and other light-harvesting applications. besides, we examined the variations in the urbach tail width as a result of thermal treatment to have an insight into the disorder, which could also affect optical absorption, in the multilayer film. we noted that several methods could be used in thickness determination of films [20]. however, most of these methods could not give compositional stoichiometry and depth profiling of the films. our choice of sno2 as the base material is informed by its popularity in photodegradation and gas sensing applications [3,9,10,21-26]. besides, because of its n-type semiconducting nature, cuo – a known p-type material was selected as the heterostructure pair. we used a chemical spray pyrolysis method to obtain uniform and adherent films of the oxides. the spray pyrolysis process is a simple, robust and if properly controlled yields oxide films of high quality at rather low costs. besides, it has the potential to produce thin layers of film on varieties of substrates. 2. materials and methods multi-layered films of sno2 and cuo on soda-lime glass substrate were deposited via spray pyrolysis technique. the spray solutions were prepared from analytical grade tin ii chloride dihydrate salt, copper ii nitrate trihydrate salt, ethanol and distilled water. 0.1 m clear solutions of sncl2.2h2o and cu(no3)2.3h2o were prepared by dissolving appropriate mass of each salt in 50 ml of solvent and followed by vigorous stirring using a magnetic stirrer. the soda-lime substrates were cleaned by washing in dilute hydrochloric acid, ethanol, acetone and distilled water in that order before drying. the deposition of the multi-layered films was carried out using the chemical spray pyrolysis technique. a schematic diagram of the setup is shown in figure 2. in a typical deposition, the substrate temperature was maintained at 350 ± 5 oc while the nozzle-to-substrate distance and tilt angle of the spray gun was fixed at 23 cm and 45o respectively. each layer was deposited by spraying constant volume of the starting solutions for a constant number of passes (to and fro movement of the spray gun). the sequence of the layers is depicted in figure 1. the first/bottom layer is sno2, followed by cuo and the topmost/surface sno2 layer. after deposition, the sample was annealed at temperatures between 400 550 oc for six hours in a tubular furnace. the rbs experiment was performed using the 1.7 mev pelletron tandem accelerator at the centre for energy research & development, obafemi awolowo university, ile-ife, nigeria. for this purpose, 4he2+ ion beam was used as the projectile ions. the scattering angle was 165o and the resolution of the detector was 12 kev. during the process of acquiring the spectra data, the energy of the ion beam used was 2.2 mev. all the measurements were performed at room temperature with current varying between 20 and 60 na at a constant charge of 20 c. all the spectra were ?tted using windows simnra software for the estimation of composition and thickness of the ?lms. the surface morphology was studied using scanning electron microscope at the university kwazulu-nata, durban, south africa. crystal structures of the ?lms were determined through x-ray diffraction studies carried out with an x-ray diffractometer model bruker axs d8 advance. the optical properties were studied using uv–visible – nir spectranet ultraviolet spectrometer model ep2000. 3. results and discussions 3.1. compositional study the simulations of rbs data were done using the windows simnra software. the result of the simulations gave the stoichiometric ratio of elements, compositions of layers and the film thickness of each sample. the rbs spectra are shown in figures (3) and the analysis presented in the table (1). some important features could be inferred from the rbs analysis. the as-deposited sample has its inner/middle layer of cuo solidly intact with a minute amount of sno2. this is reflected in the sn:cu atomic ratio as shown in table 1. this effect is also noticeable in the surface (sno2) and bottom (sno2) layers where 75 animasahun et al. / j. nig. soc. phys. sci. 3 (2021) 74–81 76 figure 1. spray pyrolysis set-up figure 2. multilayered sno2/cuo/sno2 films showing inter-diffusion a small atomic per cent of cu atoms is also present. the presence of the sn atoms in the middle layer and cu atoms in the surface and bottom layers could be attributed to a lot of factors – interfacial mixing, inhomogeneous layer thickness or surface roughness. though these factors are somehow difficult to take into consideration in most iba simulations, recent codes were able to incorporate these factors in rbs analysis [27]. in the heat-treated sample, however, the embedded middle cuo layer seems to have disappeared leaving behind multi-layered films which are composed of essentially solid solutions of the sno2 and cuo phases. this could be seen in the sn:cu ratio of the middle layer and the increase in the number of cu atoms in the surface and bottom layers of the sample. it could be argued that this could also be attributed to any of the aforementioned factors interfacial mixing, inhomogeneous layer thickness or surface roughness however the obvious increase in the amounts indicated the effect of thermal treatment since both samples (as-deposited and annealed) were synthesized under the same conditions and with the same parameters, any observed difference in properties should be attributed to the effect of thermal treatment on the annealed sample. besides, the annealed sample also showed an increase in thickness compared to the as-deposited one. this we attributed to the well-known behaviour of sno2 and most oxides whose grains grow in sizes when heated [28, 29]. the effects of these changes in compositions are pronounced in the optical absorption spectra of these samples. also, it could result in the variations in the number of heterojunctions formed between the grains of the copper oxide and the tin oxide in the structure. the significance of this is that in explaining any observed effects in the applications of the layered films whether in gas sensing, energy harvesting or photocatalysis, attention must be paid to the possibility of compositional inhomogeneities in layers of the films. so, for instance, improved photon absorption could not possibly be the only justification/reason for improved photocatalytic behaviour for a surface that contains two catalytically active oxides. summarily, the power of rbs in detecting a minute amount of interfacial diffusion and mixing is crucial in gaining insight into the understanding of observed effects and possible optimization of materials properties. the surface morphology of the thin film structure of the annealed deposited sno2/cuo/sno2 multilayer is investigated using the sem micrographs presented in figure 4. to adequately assess its surface morphology, the micrographs are represented at various magnifications. the low magnification image shows networks of nanowalls that have grown with a nestlike structure. the nest-like structures were made up of a network of nanowalls that were non-uniform and interconnected. in terms of a vapour-solid model, the formation of such a specific structure has been explained [30]. generally, precursor decomposition in spray pyrolysis involves four real-time important processes residual solvent evaporation, droplet spreading, the formation of precipitate and vaporization, and salt decomposition – which precede and affect the formation of the film when aerosol droplets from the spray gun hit the surface of the heated substrate. the elevated temperature around the substrate caused the aerosol droplets to evaporate before reaching the substrate vicinity. hence, the precipitate formation would occur as early as possible. as soon as the precipitates reached the heated substrate, they were converted to the vapour phase and then undergo a sequential heterogeneous reaction process 76 animasahun et al. / j. nig. soc. phys. sci. 3 (2021) 74–81 77 figure 3. rbs spectrum of (a) as-deposited and (b) annealed (at 500 oc) layered sno2/cuo/sno2 sample table 1. results of the analysis of rbs data showing the compositions, layer thickness and sn/cu atomic ratios of the as-deposited and films annealed at 500 0c layers compositions as-deposited/annealed@500 0c sn/cu ratios as-deposited/annealed@500 0c thickness (nm) as-deposited/annealed surface sn0.1200.84cu0.04 / sn0.1200.81cu0.07 3.00/1.71 380/400 middle sn0.0800.40cu0.52 / sn0.0700.84cu0.09 0.15/0.88 140/230 bottom sn0.1000.88cu0.02 / sn0.0800.87cu0.05 5.00/1.60 500/550 starting with diffusion and adsorption of reactant molecules to the surface of the substrate. these are followed by surface diffusion and incorporation of molecules into the lattice and finally desorption and diffusion of the product molecules from the surface of the heated substrate. for that reason, during the beginning of the growth, a templating/seeding effect occurred directly on the substrate surface due to self-nucleation, following a high substrate temperature (360 oc) regime. after the self – nucleation of the seeds, heterogeneous nucleation and energy of the product molecules favoured the arrangement of product molecules into the observed nest – like morphology. the high density of both sno2 and cuo nuclei on the substrate surface easily coalesced into the growth of sno2/cuo/sno2 nanostructures during the sno2/cuo/sno2 nucleation and growth phase. at higher magnification figure 4(b), interaction between each interface of the films is suspected. for our sno2/cuo/sno2 structure networks, the nanowall heights and widths have been found to be in the submicron region. our parameters led to a classical cvd – like process that has been known to yield high sticking probability and quality films. 3.2. x-ray diffraction analysis the x-ray diffraction patterns of the samples (shown in figure 5) were studied to confirm the crystallinity of the samples and investigate the crystal structures of the as-deposited and heat-treated samples. the analysis of dominant peaks using scherrer’s formula confirmed the presence of polycrystalline sno2 and cuo phases. the sno2 is tetragonal rutile (cassiterite) and cuo is tenorite in nature according to crystallography open database (cod) reference entry no: 96-100-0063 and entry no: 96-101-1195, respectively [13,26]. besides the increased intensities in the peaks of the annealed sample which implies improved crystallization, the xrd experiment did not reveal any new phases of the oxides even after the heat treatments. the orientation of the dominant peak remained the same indicating that the film is textured and corroborating our assertion that the annealing of the multi-layered films only led to interlayer mixing and diffusion of the oxides rather than the formation of a new chemical compound. moreover, the positions of the peaks also agreed well with other reported works [31-33]. 3.3. films’ optical transmittance and band gaps the transmittance data obtained from the uv – vis spectrophotometer at the cerd were analyzed to investigate the samples’ optical energy band structures. the transmittance and tauc’s plots showing the bandgap determination and the effect of annealing temperature are presented in figures 5(a and b). figure 5(a) presents the normalized transmittance of the asdeposited and annealed multi-layered films. the as-deposited film was far more absorbing over the visible region than any of the annealed films. this could be attributed to the high transmittance of sno2 and the absorbing nature of cuo. when the film has not been annealed, the copper layer absorbed the transmitted light by the sno2 layer because the diffusion of the cuo in sno2 is still low. however, after the film was annealed the films become less absorbing because of improved crystallization and onset of diffusion of the absorbing cuo atoms in the 77 animasahun et al. / j. nig. soc. phys. sci. 3 (2021) 74–81 78 figure 4. sem micrographs showing surface features of annealed layered sno2/cuo/sno2 table 2. estimated values of optical bandgap and sub-bandgap sample optical bandgap (ev) urbach energy eu (ev) as-deposited 2.75 1.37 annealed @ 400 oc 3.45 1.92 annealed @ 450 oc 3.47 1.92 annealed @ 500 oc 3.49 1.76 figure 5. xrd pattern of layered sno2/cuo/sno2 sno2 matrix. in addition, adsorbed surface impurities which could have emanated from the deposition process, hinders the transition of light beam and might have been wiped off as a result of the heat treatments. this consequently reduces the absorption effect. the absorption coefficient α was generated from equation (1) using the transmittance data (assuming the films’ reflectance is negligible) and film thicknesses obtained from the rbs analyses. α = 1 d ln 1 t (1) where d is the film thickness and t is the normalized transmittance. optical energy band gaps for allowed direct and indirect transition were evaluated using the tauc’s relation in equation (2) assuming that the valence band and the conduction band are parabolic. (αhυ)1/n = a ( hυ− eg ) (2) where hυ is the photon energy, n is a power factor that determines the nature of transition, a is a band tail constant and eg is the optical bandgap. n will take values of 1/2, 3/2, 2 and 3 for direct allowed, direct forbidden, indirect allowed and indirect forbidden transition respectively. tauc’s method of plotting (αhυ)1/n against hv was adopted to evaluate our deposited films’ optical band gaps. the linear section of the resulting curve was then extrapolated to the photon energy, hv axis which then gives the bandgap energy, eg value. the plots are presented in figure 5(b) and the films’ estimated optical band gaps values are reported in table 2. the results revealed the increase in the bandgap of the annealed film compared to the as-deposited film reflecting the observed features of the transmittance spectra. also, in figure 5(c), it was observed that there is a wide disparity (increase) between the bandgap of the as-deposited film and the film annealed at 400 oc. however, beyond 400 oc the multi-layered films showed a very negligible change in bandgap up to maximum annealing temperature of 550 oc. of particular importance to us is the behaviour of the layered films compared to the well-known properties of pure sno2. for all values of annealing temperature 78 animasahun et al. / j. nig. soc. phys. sci. 3 (2021) 74–81 79 figure 6. figure 6: plots showing (a) transmittance versus wavelength and (b) tauc’s plot for optical bandgap estimation figure 7. plot for estimation of sub-bandgap (urbach energy) used, there was bandgap narrowing compared to the reported values between 3.5 – 4.0 ev [34,35] for pure sno2 even though the annealed films displayed higher band energies. this is evident that our deposited samples exhibit direct allowed photon energy transition. it also corroborates our earlier result showing that annealing has led to increased transmittance as a result of diffusion and/or mixing of the absorbing copper oxide layer and better crystallization. incidentally, the optical bandgaps of the annealed layered films also coincide with the reported values for the copper doped tin oxide and sno2-cuo mixed or composite oxides. this corroborated the rbs analysis that after the heat treatment, the layers of the films essentially become mixed or solid solutions of the two constituent oxides. band narrowing in the multi-layered film can be attributed to its increased photon absorption making it a better candidate than single layer sno2 for photocatalysis, photodegradation and photodetection applications. our results also demonstrated the influence of heat treatments on the material’s properties suitable for these applications. the thermal treatment led to decreased photon absorption compared to the as-deposited film as evidenced in bandgap behaviour. this would have profound effect on the photocatalysis and photodegradation ability of the multilayered films. hence, in designing multilayered structures for light absorption applications, optimum annealing temperature should be chosen such that integrity of each layer would be maintained with little or less interfacial diffusion or mixing. 3.4. urbach energy while it is has been established that optical bandgap gives fundamental pieces of information about band to band absorption properties of materials. however, beyond the lower limit value of bandgap, the structural characteristics of a material such as lattice distortions or disorder are capable of producing extended localized states in the energy gap of such materials leading to sub-bandgap absorption edge commonly called urbach energy, eu. in this lower photon energy range, the spectral dependence of the absorption edge follows the empirical urbach rule given by equation (3) [1,33] where αo and ec are material constants, e is the photon energy and eu denotes the width of the tail of localized states in the bandgap (urbach energy) ∝ (e) = αo exp ( e − ec eu ) (3) the urbach energy eu was estimated by calculating the inverse of the slope of the linear fit obtained by plotting ln(α) against the photon energy e. the graphs are shown in figure 7 and the results are presented in table 2. urbach energy can be seen as a measure of the energetic sharpness of the optical band edge. usually, lower values of eu suggest a high structural quality film or material which also indicates good electronic properties such as high carrier mobility [33]. as shown in table 2, 79 animasahun et al. / j. nig. soc. phys. sci. 3 (2021) 74–81 80 the sub-bandgap for all the films was small indicating the good quality of the films. but most importantly the values of the urbach energy also justify the rbs results that annealing the multi-layered films led to solid-state diffusion in their structure and consequently lattice disorder which resulted in the slight increase in the urbach energies of the annealed. though one would expect that annealing the film should lead to improving crystallization and better organization of the lattice and therefore lower urbach energy, it seems the effect interlayer mixing dominated that of the crystallization and hence the increase in urbach energy. 4. conclusion the thickness and composition of multi-layered sno2 /cuo /sno2 composite thin films were investigated using the rbs method. the rbs analysis of the films showed that the copper oxide layer in between the two sno2 diffused into the tin oxide matrix after annealing with minute amount reaching the surface layer and that the annealed sets of the same composition increased in thickness compared with the as-deposited sample. the bandgap for the as-deposited and annealed films was evaluated. it was discovered that copper oxide addition led to band narrowing and indirect band type as against the direct band type of tin iv oxide. this work gave the evidence of the suitability of copper oxide in narrowing the bandgap of tin oxide which is necessary for visible light photocatalysis and 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[35] l. o. animasahun, b. a. taleatu, h. s. bolarinwa, a. y. fasasi, m. a. eleruja & e. i. obinajunwa, “spray pyrolysis deposition and characterizations of dielectric sno2 thin films”. fountain journal of natural and applied sciences 8 (2019) 11. 81 j. nig. soc. phys. sci. 5 (2023) 1180 journal of the nigerian society of physical sciences optical, dielectric and optoelectronic properties of spray deposited cu-doped fe2o3 thin films a. y. fasasia, e. ajenifujaa,∗, e. osagiea,d, l. o. animasaunc,d, a. e. adeoyeb, e. i. obiajunwaa acentre for energy research & development, obafemi awolowo university, ile-ife, nigeria. b technical university, km 15, lagos ibadan expressway, ibadan, oyo state, nigeria. cdepartment of physics, electronics & earth sciences, fountain university, osogbo, osun state, nigeria. dphysics & engineering physics department, obafemi awolowo university, ile-ife, nigeria. abstract copper-doped hematite thin films were prepared by spray pyrolysis technique using a mixture of ethanol and distilled water precursors. visual observations showed that aqua precursor produced films of less integrity compared with ethanol that produced thin, uniform and transparent yellowish-brown films that adhered well to the substrate. composition and thickness measurements determined by rbs revealed that ethanol precursor produced thinner films of 94.45 and 51.77 nm while aqua precursor produced films of 1,370 and 1,120 nm for undoped and cu-doped fe2o3 respectively. this is an indication that ethanol solutions produced nano-thick films of high integrity. the composition revealed that only the cu-doped fe2o3 deposited by ethanol solution gave composition close to stoichiometric fe2o3 while the others gave non-stoichiometric fe (oh)3. optical characterization carried out using uv-visible spectrophotometer in transmittance mode indicated that the film thickness was directly proportional to the number of passes which is inversely proportional to the transmittance. three bandgap determination methods namely; tauc, absorption fitting spectrum (afs) and davis-mott were employed with the result that tauc and afs gave close direct and indirect bandgap energies (eg) of 3.44 and 1.98 for afs and 3.43 and 2.32 ev for tauc respectively. the urbach tail energy determined was 1,100 mev which is an indication of a broad onset of absorption. the steepness parameter (σ) was found to be 7.83 while the electron-phonon (eph) coupling energy was found to be 0.85 ev. it was also observed that the refractive index (n) was about 15 times greater than the extinction coefficient (k). in the study of the dispersion parameters using single oscillator and sellmier models, the values of the single oscillator energy (eosc), dispersion energy (ed), zero frequency dielectric constant (�o), zero frequency refractive index (no), the average oscillator strength (so), the average oscillator parameter (λo) and the dispersion parameters were determined. all the values of the parameters estimated are of the same order of magnitude with other semiconducting materials. the study showed that cu-doped fe2o3 could be employed as dielectric material as well as in optoelectronic devices. doi:10.46481/jnsps.2023.1180 keywords: thin film, bandgap, urbach energy, refractive index, dispersion parameters, oscillator parameters article history : received: 07 november 2022 received in revised form: 13 march 2023 accepted for publication: 15 march 2023 published: 14 june 2023 c© 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: e. a. emile ∗corresponding author tel. no: +2348057415551 email address: eajenifuja@gmail.com ( e. ajenifuja ) 1. introduction metal oxides possess a broad range of electrical, chemical, and physical properties that are often highly sensitive to changes in their chemical environment. because of these 1 fasasi et al. / j. nig. soc. phys. sci. 5 (2023) 1180 2 properties, metal oxides have been widely studied and specific applications are based on appropriately structured and doped oxides [1]. the enhancement in the gas sensing performance of metal oxides by electron, ultraviolet, and plasma irradiations was due to the modified surface structure [2]. nanocrystalline transition metal oxide particles and films have found considerable interest in recent years because of their special properties, such as a large surface-to-volume ratio, increased activity, special electronic and unique optical properties, as compared to those of the bulk materials. the oxides of transition metals are an important class of semiconductors, which have applications in magnetic storage media, solar energy transformation, electronics, and catalysis [3]. magnetic nanoparticles of iron oxide due to their biocompatibility, catalytic activity, and low toxicity have dragged significant attention for their applications in various fields of medical care such as drug delivery systems, cancer therapy, and magnetic resonance imaging. apart from the biomedical applications, these iron oxide nanoparticles are of technological importance due to their application in many fields including high density magnetic storage devices, ferro-fluids, magnetic refrigeration systems, catalysis, and chemical/ biological sensors [4]. fe and o form a number of phases, e.g., feo (wustite), fe3o4 (magnetite), α-fe2o3 (hematite), and γ-fe2o3 (maghemite). the latter phase is synthetic while the remaining oxides occur in nature. the fe-o phase diagram shows the predominance of the fe2o3 stoichiometry for most temperature and pressure preparation conditions [5, 6]. of all the transition metal oxides, α-fe2o3 has been extensively studied due to its low cost, high chemical stability, nontoxicity, abundance in nature, and biocompatibility [7]. hematite crystals are believed to nucleate and grow from iron (iii) organic complex which through hydrolytic breakdown yield nano-sized ferrihydrite aggregates that eventually transform to afe2o3 through dehydration and rearrangement according to the following equation [8]: fe3+ yield s −−−−→ fe5o8 · 4h2o yield s −−−−→ αfe2o3 (1) certain studies believed that without slight deviation from stoichiometry, iron oxide (α-fe2o3) is a thermodynamically stable oxide of the hexagonal packed crystal structure is considered an insulator with localised fe3+ ions while with a slight deviation from stoichiometry hematite is an n-type indirect semiconductor with bandgap energy commonly considered to be 2.2 ev [9]. some studies have also shown that hematite can exhibit both indirect and direct transitions with indirect and direct bandgap energies around 1.9 and 2.7 ev [4, 7, 10–15]. this range of band gap energy has conferred on hematite the ability to be employed in many electrochemical applications since about 29 % of visible light has energies greater than the hematite band gap (2.2 ev) [16] as a result of which α-fe2o3is capable of absorbing a large portion of the visible solar spectrum (absorbance edge ˜600 nm). with all of these advantages, the usage of α-fe2o3 has been restricted by many anomalies such as higher electron-hole recombination rate leading to poor electrical conductivity with a hole length of 2 – 4 nm, poor mobility, insufficient ionic diffusion rate leading to low specific capacitance and vb positioning (vb is positive with respect to h+/h2 potential) [9, 17–19]. several methods have been used to improve on the limitations of α-fe2o3 for improved solar conversion. enormous efforts were devoted to the modification of the electronic structure of hematite via doping [20], thin film deposition, core-shell structure, and coating. to this end, sn [21, 22], ti [20, 29], si [23, 30], zn [24], mgo and cao [26], ir [27], zr [31], cu [32], ti+al [25], ti+sn [28] –doped and undoped [33, 34] have been employed to dope fe2o3 individually or in combination as photoanodes for water splitting application. on the other hand, cr [35, 36], nd [37], zno [38], and eu [39] have been used to dope α-fe2o3 for improving the magnetic properties. in the area of portable, flexible and wearable electronics and asymmetric supercapacitors undoped α-fe2o3 [40, 41], carbon incorporated α-fe2o3on carbon nanotube [18], αfe2o3/graphene/ink [42], α-fe2o3 on carbon cloth [43], oxygen vacancy engineered α-fe2o3 nanoarrays on carbon cloth [44], α-fe2o3nanorods in ni/fe battery [45], mxene@ αfe2o3 coreshell /carbon cloth [46], si@ α-fe2o3/carbon cloth [47], polypyrole doped a-fe2o3 [48], α-fe2o3@pani coreshell [49], tio2@ α-fe2o3coreshell on array of carbon cloth [50] and cu-doped α-fe2o3 [51] have all been employed to increase the conductivity of α-fe2o3for environmental remediation through waste water treatment, α-fe2o3/ppy for photocatalyst degradation of methylene blue under uv irradiation [52], α-fe2o3in the treatment of oily waste water [53], pure copper and go-doped α-fe2o3 [54] for waste water treatment, cudoped fe @fe2o3 coreshell nanoparticles for removing as(iii) for smelting waste water [55], zn-doped α-fe2o3for photocatalytic degradation of rose bangal dye [56], activated carbon coated α-fe2o3 adsorbent for chlorinated gas treatment [57], ni-doped α-fe2o3for removing toxic metals for aqueous solution [58], y2o3/ α-fe2o3/tio2 [59] and α-fe2o3@ceo2zro2/polygorskite [60] composite catalysts for treatment of organic pollutants have all been used successfully. as gas sensors, pd-doped α-fe2o3 [61], nb-doped α-fe2o3 [62], cu-doped αfe2o3 [63], and tiα-fe2o3 [64] have all been synthesised and employed for lpg, no, nox, ethanol and co gas sensing, respectively. different method of synthesis have been used to produce dope α-fe2o3such as ultrasonic spray pyrolysis [4], dc magnetron and dc pulsed hollow cathode sputtering system [7], silar (10), normal spray pyrolysis [16], flame annealing [21], atmospheric pressure cvd [23], electrodeposition [31], coprecipitation [36, 64, 69], combustion synthesis method [37], gelation and polymerisation [52], sol-gel [54, 56, 70], high energy ball milling [58], pulsed laser deposition [65], filtered arc deposition [67], oxygenplasma assisted mbe [69]. of all the methods of preparation, spray pyrolysis offers the ease of preparing small as well as large area coating of thin films and nanopowders at low cost for different technological applications. moreover, it does not require high quality targets nor 2 fasasi et al. / j. nig. soc. phys. sci. 5 (2023) 1180 3 vacuum at any stage of preparation. the deposition rate and the film thickness can be controlled effectively by changing the spray parameters such as the precursor composition, nozzle to substrate distance, flow rate, and the number of passes. most of the studies on cu-doped α-fe2o3 focussed mainly on the application and not on the detailed optical and dielectric studies needed to determine the areas of applications. this study therefore presents detailed optical and dielectric studies of cudoped α-fe2o3 films deposited using spray pyrolysis technique, uv-visible spectrometry and rutherford backscattering spectroscopy (rbs). 2. experimental procedure the starting solutions were prepared from iron (iii) chloride (fecl3) and copper (ii) chloride (cucl2) purchased from sigma aldrich with ethanol and distilled water as solvents. the salts were analytical grade and used as purchased without further purification. for the aqueous solution, 0.1 m solution of iron (iii) chloride was prepared by dissolving 4.0 g of fecl3 and 2.56 g of cucl2 in 250 ml of distilled water. for ethanol solution, the same mass of fecl3 was dissolved in 250 ml of ethanol while cucl2 was dissolved in a mixture of ethanol: distilled water in the ratio of 200:50 giving four different solutions. the solutions were thoroughly stirred using a magnetic stirrer. the pyrex glass substrates were cleaned in dilute hydrochloric acid, alcohol, and distilled water and dried in an oven before deposition at a substrate temperature of 320 ± 5 0c and nozzle–to–substrate distance of 23 cm. suction–based airbrush and air blast atomization was employed for the depositions. for cu-doped fe2o3 deposition, the fecl3:cucl2 volume ratio of 80:20 was adopted. after deposition, the samples were annealed in a tubular furnace at different temperatures ranging from 350 -500 oc in air. the rbs experiment was performed using the 1.7 mev pelletron tandem accelerator at the centre for energy research & development, obafemi awolowo university, ile-ife, nigeria. for this purpose, 4he2+ ion beam was used as projectile ions. the scattering angle was 165o and the resolution of the detector was 12 kev. during the process of acquiring the spectra data, the energy of the ion beam used was 2.2 mev. all measurements were performed at room temperature with current varying between 20 and 60 na at a constant charge of 20 µc. all the spectra were fitted using windows simnra software for the determination of the composition and the thickness of the films. the optical properties were studied using a stellanet uv-visible spectrophotometer (model ep2000) with wavelength covering 200 to 1100 nm in the transmission mode. 3. results and discussion 3.1. visual observation typical surfaces of the deposited films are shown in figure 1 for doped and undoped films. the surfaces are clean and regular without voids with strong adherence to the substrate. the change in colour due to cu addition is obvious. figure 1: surfaces of the undoped (left) and cu-doped (right) fe2o3 deposited films taken by an optical camera camon 12 pro. 3.2. rbs study determination of the thickness and composition. preliminary results of the rbs analyses of the four films grown on pyrex glass using ethanol and distilled water as solvents for doped and un-doped fe2o3 is presented in figure 2 (a-d). the shape of the spectra clearly showed the effect of the solvent employed in depositing the films. sharp peaks are associated with ethanol while broad peaks are associated with distilled water. this may be due to the film thickness and the degree of structural orderliness. all the films deposited using distilled water for the same number of passes are 15 times thicker than the films deposited using ethanol as shown in table 1. this may also be an indication of the film integrity in that alcohol solutions produced more ordered films than distilled water. the spraying process employed in this study can lead to the formation of films of feoh, feo, fe2o3, or fe3o4. in a pure crystal the ratio of atomic content fe:o is 0.5:0.5 for feo, 0.4:0.6 for fe2o3, 0.429:0.571 for fe3o4 and fe: oh of 0.25:0.75 for fe(oh)3. from the elemental composition of the films, we can deduce that the films are neither feo nor fe3o4. the rbs simulation of the films gave the atomic percentage of fe, o and cu. the results, using fe/o values in table 1, indicate that undoped alcohol and undoped distilled water precursors produce non-stoichiometric films of fe(oh)3 with anion deficiency compensated by excess cation for undoped alcohol precursor and cation vacancies compensated by excess oxygen for undoped aqua precursor. in the case of cu-doped precursors, distilled water also produce non-stoichiometric films of fe(oh)3 but with ca substituting for fe and both anion and cation vacancies are probably compensated by cu. this explains the high content of cu in the film but does not explain the incursion of ca into the film. to explain the presence of ca in the film, the rbs of the substrates for each film were also analysed simultaneously with the film and the result is shown in table 2. it is obvious from the table that ca did not diffuse from the substrate into the film and therefore we may infer that ca got introduced into the film from the distilled water used as a solvent. but there is no presence of ca in the undoped aqua precursor and this led us to believe that cu must have acted as a catalyst in the film formation and coexisted in the film without forming a compound with fe. lastly, for cu-doped ethanol precursor producing film thickness of 51.77 nm, the fe/o ratio of the film gave a value of 0.660 which is very close to 0.667 for pure fe2o3. it can then 3 fasasi et al. / j. nig. soc. phys. sci. 5 (2023) 1180 4 figure 2: rbs spectral of undoped and cu-doped fe2o3 in ethanol (a, b) and undoped and cu-doped fe2o3 in distilled water (c, d). be said that cu acted as a catalyst for the formation of fe2o3 and existed in the film as a solid solution without forming a definite phase in the film. hence, from this point, further analyses will focus solely on cu-doped alcohol precursors. 3.3. transmission analyses the effect of the number of passes on cu-doped fe2o3 is depicted in figure 3a where it can be seen that the increase in the number of passes led to a decrease in transmittance. this trend had been observed in cuo film deposited by spray pyrolysis [71]. two distinct regions can be observed on the spectra with the line of demarcation around 648 nm which corresponds to 1.92 ev. at this wavelength, the transmittance for 40, 60, and 80 passes were 85, 82, and 78% respectively. above this wavelength is region 2 where the transmittance is slowly varying as the wavelength increases. below this wavelength is region 1 where the transmittance is rapidly changing as the wavelength decreases. region 1 is the absorbing region with 80 passes having the highest absorbance which is probably due to the increased thickness of the film as the number of passes increased while region 2 is the transparent region. this result showed that the film is slightly opaque to photon energies less than 1.92ev and that it can serve as an optical window and a good solar absorber material. figure 3b is the comparison of the transmittance of cu-doped fe2o3 films grown on borosilicate glass with 80 passes using deionized water and alcohol. the graph shows that deionised water produced thicker film due to the low transmittance (maximum of 30%) while ethanol produced thinner films of high transmittance (maximum of 80 %). this corroborates the results obtained from rbs. 3.4. absorption coefficient and skin depth 3.4.1. absorption coefficient the absorption coefficient α of a semiconducting material is an important parameter useful in determining how the incoming photon energy interacts with the atoms of the semiconductor. the absorption coefficient was calculated using the relation α = 1 t ln ( 1 t ) (2) since it is a function of the energy of the incoming photon, figure 4a depicts the variation of the absorption coefficient with the wavelength of the interacting photon where it can be seen that the absorption is high at high photon energy (ultraviolet region) and decrease gradually till 700 nm where it becomes invariant at the lowest value of 5.0 × 104 cm-1. this trend has been observed in many films even prepared with different methods [72]. the high value of the absorption coefficient makes the material a good absorber and very useful in optoelectronic applications. 4 fasasi et al. / j. nig. soc. phys. sci. 5 (2023) 1180 5 table 1: thickness (nm) and composition (at. %) of the annealed films from rbs analyses. precursor thickness fe o cu ca fe:o fe/o remark undoped alcohol soln 94.45 nm 27.83 72.17 --------0.278 : 0.723 0.359 fe(oh)3 cu-doped alcohol soln 51.77 nm 39.35 59.62 1.03 ----0.394 : 0.596 0.660 fe2o3 undoped aqua soln 1370 nm 17.85 82.15 --------0.179 : 0.822 0.217 fe(oh)3 cu-doped aqua soln 1120.80 13.56 69.26 4.40 12.77 0.136 : 0.693 0.196 fe(oh)3 table 2: composition (at. %) of the substrate employed for the films film thickness (nm) ca si o fe na al fe2o3/ alcohol 94.45 1.83 28 56 0.52 12.6 0.53 cu-fe2o3/ alcohol 51.77 1.83 28 56 0.52 12.6 0.53 fe2o3/ distilled h2o 1370 1.83 28 56 0.52 12.6 0.53 cu-fe2o3/ distilled h2o 1120.80 1.83 28 56 0.52 12.6 0.53 3.4.2. skin depth on the other hand, when a ray of light passes through thin films, the photon intensity decreases dramatically for many reasons, like the density of the material, refractive index, morphology of the surface, and the film microstructure. the thickness of the film that causes the intensity of the photon to be reduced to 1/e value below the surface of the film is called the skin depth δ which is defined as the inverse of the absorption coefficient δ = 1 α . it is therefore a function of frequency and hence the band gap. fig. 4b is depicting the connection between the skin depth and photon energy of cu-doped fe2o3. the gradual decrease in skin depth can be seen until it gets to a minimum depth of 2.68 x 10-6 cm (26.8 nm) at around 4 ev (λ = 342 nm). this minimum skin depth of 26.8 nm corresponds to the middle of the film. this result is contrary to other studies where it was shown that the skin depth reduced to zero at a certain energy corresponding to the cut-off wavelength λcutoff [72, 73]. this may be because the nature of their films was glassy and contain a mixture of crystalline and amorphous phases whereas the film in this study may be completely crystalline. 3.4.3. bandgap determination for the photon energy range corresponding to the transmittance spectra, the absorption coefficient can be approximated by simple expressions. for example, assuming a parabolic e versus k relationship, the absorption in semiconductors and dielectrics films can be expressed as α = b ( hν− eg )m hν for hν > eg (3) the quantity α is the absorption coefficient, hν denotes photon energy, b is a constant that includes information of the convolution of the valence and conduction band states and on figure 3: (a) effect of the number of passes on the transmittance of cu-doped fe2o3 and (b) comparison of the effect of precursor solvent on the transmittance for 80 passes. the matrix elements of optical transitions which is assumed to be independent of energy, eg is the tauc’s optical gap and b depends on the material composition and deposition method and conditions, m is the order of transition that take values of 1/2, 2, 3/2 and 3 representing direct allowed, indirect allowed, direct forbidden and indirect forbidden transitions, respectively. other approximations such as the absorption fitting spectrum (afs) procedure and david-mott model exist for determining the band gap of semiconducting materials. absorption fitting spectrum [74] uses equation (3) but is expressed 5 fasasi et al. / j. nig. soc. phys. sci. 5 (2023) 1180 6 figure 4: graphical representation of the variation of (a) the absorption coefficient with wavelength and (b) the skin depth with photon energy for cu-doped fe2o3 from ethanol solution. in terms of “λ”. rearranging equation (3) gives αhc λ = b ( hc λ − hc λg )m (4) where for direct transition (m = 1/2) and indirect transition (m = 2), we obtain( α λ )2 = k ( 1 λ − 1 λg ) and ( α λ ) 1 2 = c ( 1 λ − 1 λg ) (5) where λg, h, and c are the wavelength corresponding to the optical bandgap value, planck’s constant, and the velocity of light respectively. davis and mott on the other hand proved that through the in-depth analysis of the matrix elements for transitions between extended states and those between weakly localized states, the absorption coefficient (α) can be expressed through the following equation [75–77] αhν = 4πσmin 3πc (∆e)2 (hν− e2) 3 . (6) this can be re-written as: (αhν) 1 3 = k (hν− e2) , (7) where σmin = ( 2π3 c2 h3 a m2 ) [n (ec)] 2 is the minimum metallic conductivity and e2 = ec − ev. thus plotting (αhν) m against hν for tauc approximation, ( α λ )m against 1 λ for afs procedure and (αhν)m against hν for davis-mott model where m = 1/2 or 2 for direct and indirect electronic transitions in the case of tauc and afs but m = 1/3 or 2/3 for direct and indirect transitions in the case of davis-mott, which actually represent direct forbidden and indirect forbidden transitions respectively, we obtain figure 3. graphical representation of equations (3), (5) and (7) for determining the bandgap is depicted in figure 3 (a-c) and the result is presented in table 3. it can be observed that for direct electronic transition, the three models gave almost equal values of 3.11, 3.44, and 3.43 ev while for indirect electronic transition the disparity in the values is wide, 3.44, 1.98, and 2.32 ev. since it is a known fact that the direct band gap energy is always higher than an indirect band gap energy, the davis-mott model does not reflect this trend and therefore can be neglected. on the other hand, afs and tauc models follow this trend and can lead us to say that cu-doped fe2o3 has an indirect bandgap that lies between 1.98 and 2.32 ev and a direct band gap of 3.43 ev. this can be explained by noting that the initial orbital assignments of the bandgap suggested it was due to an indirect transition of fe3+ d–d origin and that a stronger direct transition involving a charge transfer from an o2p orbital to fe3d did not occur until 3.2 ev [11]. the result indicates that fe2o3 prepared through ethanol solution is an indirect band gap material though many researchers concluded that cu-doped fe2o3 can exhibit both a direct and an indirect bandgap transition [10–15]. table 3: analyses of the different models employed in determining the band gap of cu-doped fe2o3 film. model direct (ev) indirect (ev) davis-mott 3.11 3.44 afs 3.44 1.98 tauc 3.43 2.32 3.4.4. urbach band tail energy at photon energies below the exponential absorption edge, there is a sub-gap absorption that originates from transitions between deep localized electron states in the pseudo-gap and extended states in the conduction and the valence bands. the exponential rise is called the urbach tail and it appears in polycrystalline, partially crystalline, and non-crystalline materials, because of the existence of these localized states [78–80]. the relationship between the absorption coefficient and urbach tail energy is given as α = αoe hν eu , (8) where αo is a constant and eu refers to the energy of the bandtail width (urbach energy). this energy, eu depends slightly on temperature and is often interpreted as the band-tail width owing to the localized states in the forbidden band gap and associated with the perturbation of the non-crystalline and low crystalline materials [81]. linearisation of equation (8) and 6 fasasi et al. / j. nig. soc. phys. sci. 5 (2023) 1180 7 (a) band gap determination using afs model for direct (left) and indirect (right) transition. (b) band gap determination using davis-mott model for direct (left) and indirect (right) transition. (c) band gap determination using tauc’s method for direct (left) and indirect (right) transition. figure 5: band gap determination plotting ln α against hν gives a straight line where the inverse of the slope is the value of urbach energy. ln α = ln αo + hν eu (9) 7 fasasi et al. / j. nig. soc. phys. sci. 5 (2023) 1180 8 figure 6: plot of ln α against hν for urbach energy determination the plot of ln α against hν is presented in figure 6 where the straight line nature is observed. the value calculated for urbach energy from the slope is 1,100 mev. the urbach energy quantifies the steepness of the onset of absorption near the band edge, and hence the broadness of the density of states. since a sharper onset of absorption represents a lower urbach energy, cu-doped fe2o3 has a broad onset of absorption. 3.4.5. steepness parameter there is another expression that relates the absorption coefficient α to the band gap energy eg according to urbach’s assumption [67–69] given by α = β exp [ σ(hν− eo) kbt ] , (10) where β is a pre-exponential factor, kb is the boltzman’s constant, t is the temperature, σ is another optical constant called the steepness parameter, eo is the energy of the transition which depends on the type of transition. if it is a direct transition, eo = eg while for indirect transition eo = eg ± e ph where e ph is the energy due to lattice vibration. since we have agreed that this is a direct transition then we can replace eo with eg in equation (10). simplifying the equation by taking the natural logarithm of both sides gives ln α = ln β + σhν kbt − σeg kbt ln α = ( ln β− σeg kbt ) + σhν kbt (11) we then obtain inαo = [ inβ− σeg kbt ] and hν eu = σhν kbt (12) ⇒ σ = kbt eu (13) where kb = 8.6173 × 10−5 ev.k−1 and t = 300 k (room temperature). therefore, we can calculate the steepness parameter σ for cu-doped fe2o3 to be 7.83 and the electronphonon coupling energy, eph = 2 3σ to be 0.85 ev while αo = 1.03 × 104 cm−1. 4. refractive index, reflectance and extinction coefficient the refractive index which governs the speed of wave propagation, reflectance which indicates the percentage of the wave reflected and the extinction coefficient which shows the propagation loss in a material are important parameters in determining the optical properties of thin films to be employed in optoelectronic devices. in the present study, the reflectance had been determined using the relation: r = 1 + [ t e2.303αt ] 1 2 , (14) where t is the transmittance, α is the absorption coefficient, and t is the film thickness obtained from the rbs study. the refractive index n, the extinction coefficient k and the optical conductivity σ were determined through equations (15) and (16). n = 1 + √ r 1 − √ r (15) k = αλ 4π (16) figure 7: plot of n and k against wavelength for cu-doped fe2o3 graphical representation of equations (15) and (16) against wavelength is shown in figure 7 to understand their variations as photon energy changes. the refractive index showed a high value at high energy and decreased as the wavelength increased and became constant from 700 nm. the extinction coefficient also showed the same trend as the refractive index. both “n” and “k” became constant from 700 nm up to 1000 nm. the figure also indicated that the refractive index is 16 times higher than the extinction coefficient. the dependence of the index of refraction on photon energy can be well described using 8 fasasi et al. / j. nig. soc. phys. sci. 5 (2023) 1180 9 a single-effective-oscillator formulation according to wemple and di-domenico [82, 83] n2 − 1 = ed eosc[ e2osc − (hν) 2 ] (17) by taking the inverse of equation (17), we obtain 1( n2 − 1 ) = eosc ed − (hν)2 ed eosc , (18) where eosc is the single-oscillator energy and ed is the dispersion energy. by plotting ( n2 − 1 )−1 against (hν)2 we obtain a straight line whose intercept equals eosced and the slope is 1 ed eosc . the graph is presented in figure 8 where the values of 4.44 and 6.12 ev were obtained for ed and eosc respectively. the expression (17) holds for photon energies well below eosc. at energies approaching eosc, deviations to the simple law originating from the proximity of the main band-to-band transitions, are measured. furthermore, the knowledge of ed and eosc can be utilized in determining the zero-frequency dielectric constant εo and zero frequency refractive index no through the relation (19), and the values obtained are recorded in table 4. εo = n 2 o = 1 + ed eosc (19) figure 8: plot for the determination of the single oscillator energy (eosc) and the dispersion energy (ed) another way of discussing the dispersion parameters is to employ the long wavelength approximation of the single-term sellmier relation since it retains the physical significance of the oscillator parameters. the relation is given by equation (19), n2 − 1 = s oλ2o[ 1 − ( λo λ )2], (20) where so is the average oscillator strength and λo is the average oscillator parameter. again, linearising equation (20) gives,( n2 − 1 )−1 = 1 s oλ2o − 1 λ2s o . (21) figure 9: graph for determining the average oscillator strength (so) and average oscillator parameter ( λo). plotting ( n2 − 1 )−1 against λ−2 will give a straight line whose intercept ( 1 s oλ2o ) and the slope is ( 1 s o ) can be determined and the graph is presented in figure 9. the dispersion parameter eos o where eo = ~c eλo can also be determined. the values of s o, λo and eo s o determined through this process is tabulated in table 4. 4.1. density of state effective mass ratio the relationship between the refractive index “n” and the wavelength of the incident photon can also be employed to determine the density of state effective mass ratio and the high frequency dielectric constant through equation (22): n2 = ε∞ − 1 4π2εo ( e2 c2 ) ( nc m∗ ) λ2 (22) where ε∞ is the high frequency dielectric constant, “e” is the electronic charge, “c” is the speed of light and “nc/m*” is the density of state effective mass ratio. plotting n2 against λ2 will give a curve where the intercept of the straight part of the curve on the y-axis equals ε∞ and ( nc m∗ ) can be determined from the slope which is 14π2εo ( e2 c2 ) ( nc m∗ ) . with m* = 0.44 mo, the value of nc can also be calculated. these values of ε∞, nc/m* and nc are recorded in table 4. 4.2. dielectric constant the knowledge of “n” and “k” permits the determination of the real, imaginary, and loss tangent of cu-doped fe2o3 films through equations (23)-(25), and the graphical representation is depicted in figure 11. εr = n 2 − k2 (23) εi = 2nk (24) tan δ = εi εr (25) 9 fasasi et al. / j. nig. soc. phys. sci. 5 (2023) 1180 10 figure 10: variation of n2 vs λ2 for the determination of the high frequency dielectric constant ε∞ and the density of state nc figure 11: variation of εr , εi and tan δ with photon wavelength figure 11 shows that εr is far greater than εi since εi is a product of n and k and with smaller values of k, εi will be lower than εr . it can also be observed that εr , εi and tan δ decreased gradually until around 700 nm where they become independent of the photon wavelength. the variation of these three parameters is an indication of photon energy interaction with the free electrons of the cu-fe2o3 matrix. the low loss tangent (tan δ) means that energy dissipation and propagation loss of the photons in the matrix are low indicating that photons are able to pass freely in the matrix which will lead to increased optical mobility and conductivity. 4.3. relaxation time, mobility, resistivity and conductivity the variation of the imaginary part of the dielectric constant εi with wavelength can also be employed to determine the relaxation time τ, optical mobility “µopt”, optical resistivity “ρopt” and optical conductivity “σopt” through the following equations: εi = 1 4π3εo ( e2 c3 ) ( nc m∗ ) . 1 τ λ3 (26) table 4: elemental composition, optical, dielectric and oscillator parameters for cu-doped fe2o3 parameter symbol & unit cu doped fe2o3 iron content fe (at. %) 39.35 oxygen content o2 (at. %) 59.62 copper content (at. %) cu (at. %) 1.03 plasma frequency wp (s-1) 6.85 × 1013 high frequency dielectric constant ε∞ 1.81 film thickness t (nm) 51.77 density of state effective mass ratio nc/m* (m-3.kg-1) 5.1 x 1054 optical carrier charge density nc (m-3) 2.0 ×1024 urbach energy eu (mev) 1,100 relaxation time τ (s) 1.3 ×10-14 optical mobility µ (m2.v-1.s-1) 5.2 ×10-3 optical resistivity ρ (ω.m) 6.0 x 10-4 optical conductivity σ (ω-1m-1) 1.67 ×103 effective single oscillator energy eosc (ev) 6.12 dispersion energy ed (ev) 4.44 effective single oscillator energy band gap ratio eosc/eg 2.64 3.09 zero frequency refractive index no 1.31 zero frequency dielectric constant εo 1.72 average oscillator strength sosc (m-2) 1.93 ×1013 average oscillator wavelength λosc (m) 1.95 x 10-7 average oscillator energy eo (ev) 6.33 x 1018 dispersion parameter eo/so (ev.m2) 3.30 ×105 bandgap eg (ev) 1.98 – 2.32 linear susceptibility χ(1) 5.8 x 10-2 nonlinear susceptibility χ(3) (esu) 1.79 x 10-15 nonlinear refractive index n2 (esu) 5.1 x 10-14 µopt = eτ m∗ (27) ρopt = 1 eµopt nc (28) σopt = 1 ρopt (29) where “e”, “c” and “m*” are the electronic charge, velocity of light and effective mass (0.44 mo) of free charge carriers respectively. the plot of εi against λ3 will produce a curve where the slope of the linear part of the curve equals 14π3εo ( e2 c3 ) ( nc m∗ ) . 1 τ which gave τ to be equal to 1.3 × 10−14s. with this value of τ, 10 fasasi et al. / j. nig. soc. phys. sci. 5 (2023) 1180 11 µopt, ρopt and σopt are determined as tabulated in table 4. the values obtained from this study are of the same order of magnitude with the values obtained for some chalcogenide compounds [84]. figure 12: plot of εr versus λ3 for the determination of the relaxation time. 4.4. plasma frequency the relationship between n2 and λ2 given in equation (22) can also be expressed in another form: n2 = ε∞ −  w2p4πc2  λ2. (30) comparing equation (23) with (30), it can be seen that both equations have the same slope and with the slope of equation (22) determined to be 4.16×109 and equating this value to ( w2p 4πc2 ) which is the slope of equation (30), the value of wρ has been calculated to be 6.85 × 1013 s-1. this value corresponds well with the values obtained by other authors [84]. 4.5. optical nonlinearity nonlinear optic happens in a media where the polarization density p responds non-linearly to the electric field e of the light. the non-linearity is typically observed only at very high light intensities. nonlinear optical phenomena, in which the optical fields are not too large, can be described by a taylor series expansion of the dielectric polarization density (electric dipole moment per unit volume) p(t) at time t in terms of the electric field e(t): p(t) = εo ( χ(1) e(t) + χ(2) e2(t) + χ(3) e3(t) + · · · ) (31) where the coefficients χ(n) are the n-th-order susceptibilities of the medium, and the presence of such a term is generally referred to as an n-th-order nonlinearity. simple semiempirical relation based on generalized miller’s rule allows an estimation of nonlinear susceptibility (χ(3)) and non-linear refractive index (n2) from linear refractive index and/or from the dispersion energy and the energy of effective oscillator of the wemple-di domenico model [84, 85]. the relevant equations are expressed below: χ(1) = ed 4πeso (32) χ(3) = c  ( n2o − 1 4π )4 (33) n2 = 12πχ(3) no (34) where χ(1), χ(3), no and c are the linear susceptibility, nonlinear susceptibility, nonlinear refractive index and a constant which is equal for all materials with a value of 1.7×10−10 esu and it is independent of frequency. the values obtained using equations (32)-(34) are tabulated in table 4. 5. conclusion cu-doped hematite thin films were prepared from a mixture of distilled water and ethanol precursor using spray pyrolysis technique. apart from rbs studies that provided the film thickness and the composition, all other parameters of the films were deduced from the optical transmittance studies. rbs showed that only the cu-doped film prepared from ethanol solution gave the composition very close to fe2o3 with film thickness of 51.77nm. optical transmittance studies of the cu-doped films also showed that the transmittance of the film prepared using distilled water is lower than that of ethanol solution which is an indication that ethanol solution produced nano-thick films of high integrity. cu-doped fe2o3 film is highly absorbing with α greater than 105 cm-1. tauc, afs and davis and mott models were employed to determine the actual value of the bandgap using direct and indirect transition to conclude that cu-doped fe2o3 is an indirect bandgap material with 1.98 ev < eg < 2.32 ev and a direct band gap energy of 3.43 ev. the value of urbach energy gave an indication of a broad onset of absorption. the variations of the refractive index with photon energy or wavelength allowed the estimation of the single-oscillator energy, dispersion energy, average oscillator strength, average oscillator parameter, dispersion parameter, density of state effective mass ratio, high frequency dielectric constant and the plasma frequency. ε∞ > n2o means that free charge carriers contribute in the polarization process within cu-doped fe2o3 thin film. the variation of the imaginary part of the dielectric constant (εi) with photon wavelength led to the estimation of the relaxation time, optical mobility, resistivity and optical conductivity. the 1st and 3rd order nonlinear susceptibility and nonlinear refractive index were estimated. all the values of the parameters estimated were of the same other of magnitude with other semiconducting materials. the values also indicated that cu-doped fe2o3 film could be suitable for nonlinear, electrochemical as well as optoelectronic applications. 11 fasasi et al. / j. nig. soc. phys. sci. 5 (2023) 1180 12 references [1] y. zhu, h. lu, y. lu & x. pan, “characterization of sno2 films deposited by d.c. gas discharge activating reaction evaporation onto amorphous and crystalline-substrates”, thin solid films 224 (1993) 82. 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[85] h. tichá & l. tichý, “semiempirical relation between non-linear susceptibility (refractive index), linear refractive index and optical gap and its application to amorphous chalcogenides”, journal of optoelectronics and advanced materials 2 (2002) 386. 13 j. nig. soc. phys. sci. 5 (2023) 1438 journal of the nigerian society of physical sciences photocatalytic and antibacterial activities of green-mediated khaya senegalensis-silver nanoparticles and oxidized carbon nanotubes a. h. labuloa, a. d. ternab,∗, o. f. oladayoa, h. ibrahima, n. s. tankoa, r. a. ashonibarec, j. d. opeyemia, z. tywabi-ngevad adepartment of chemistry, federal university of lafia, lafia, nasarawa state, nigeria bdepartment of chemistry, federal university of technology, pmb 1526, owerri, imo state, nigeria cdurable crops research department, nigerian stored products research institute, p.m.b. 1489, ilorin, kwara state, nigeria d department of chemistry, faculty of science, centre for rubber science and technology, nelson mandela university, gqeberha, south africa, 6001. abstract this study investigated the photocatalytic and antibacterial activities of plant-mediated silver nanoparticles (agnps) from a medicinal plant extract of khaya senegalensis (k. senegalensis) and oxygen functionalized carbon nanotubes (ocnts), respectively. the cnts were functionalized using acid treatment. the green synthesized agnps from k. senegalensis (ks-agnps) and ocnts were characterized by uv–visible spectroscopy, fourier transform infrared spectroscopy (ftir), transmission emission microscopy (tem), scanning electron microscopy (sem), and x-ray diffraction (xrd). the formation of ks-agnps was confirmed by the uv–vis absorption spectra, which showed an absorption band at 427 nm with a color change from yellow to brown. the morphology of ks-agnps was spherical in shape, with an average particle size of 9.30 nm. the ftir analyses revealed distinctive functional groups, such as, hydroxyl (o-h), amines (n-h), and carbonyl (c-o), which were directly involved in the synthesis and stability of agnps. the xrd spectra was distinctive with five intense peaks at 2θ angles of 38.12°, 44.28°, 64.43°, 77.48°, and 81.54o while ocnts gave intense peaks at 2θ angles of 26.43o, 42.36o, 44.46o, 54.51o, 59.98o, and 77.40o. the photocatalytic property of green synthesized ks-agnps was determined to be 40.7 % higher than that of ocnts when applied for treatment of industrial waste water. the ability of green-mediated ks-agnps to inhibit against gram-positive and gram-negative bacteria was observed to be that gram (-) bacteria (e. coli) was more susceptible to ks-agnps than the gram (+) bacteria (s. aureus), in which case their susceptibility was least in ocnts for both bacteria, respectively. doi:10.46481/jnsps.2023.1438 keywords: khaya senegalensis, silver nanoparticles, carbon nanotubes, photocatalytic, antibacterial activity article history : received: 08 march 2023 received in revised form: 01 may 2023 accepted for publication: 05 may 2023 published: 22 june 2023 © 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: e. a. emile ∗corresponding author tel. no: +2348100049983 email address: ternaaugustine2020@gmail.com (a. d. terna ) 1. introduction new research suggest to possibilities to create antibacterial and photocatalytic compounds that are more effective and less toxic for usage in order to tackle water pollution, which 1 labulo et al. / j. nig. soc. phys. sci. 5 (2023) 1438 2 is the primary cause of infectious diseases [1]. producing eco-friendly nanoparticles follows the principles of green chemistry, in which naturally existing bioactive chemicals in plants serve as reducing and capping agents while also preserving the environment [2]. by adjusting the concentration of plant extract, it is possible to create shapeand size-dependent nanoparticles under regulated conditions [1], [3]. the treatment of industrial wastewater, in particular that is produced by the textile dying process, is a serious environmental concern. the destruction of organic dyes and bacteria by nanoparticles’ catalytic activity makes them potentially useful for the treatment of industrial wastewater [4]. clean, pure drinking water is currently a major issue due to the rise of so many startup companies and the need to address environmental degradation [5]. applications for nanoparticles (nps) in the oxidative degradation of harmful pigments and environmental pollutants seem to be numerous [6]. the enhanced photocatalytic and degrading effectiveness is due to the small, crystalline, particle size, surface clustering, significant optical absorption in the visible range, quick charge transfer, reduced electron-hole charge recombination, and high oxidizing ability of the nanoparticles [7]. nps have been found to have very sporadic antibacterial action against s. aureus and high antibacterial activity against e. coli [1]. the combination of surface area, electrostatic interactions between the anionic dye and cationic surface of the nanocatalyst, and the oxidation of polyphenolic/phenolic chemicals or flavonoids present on the surface of the nanoparticles enhances the photocatalytic degradation process of nanoparticles [8]. moreover, smallersized silver nanoparticles (agnps) have greater surface areas and more active sites available for the degradation of organic dyes via electron transfer [4]. further studies have revealed that the high surface-to-mass ratios and unique activities of silver nanoflowers—which are brought on by the high densities of edges, corners, and stepped atoms on their nano-petals—have enhanced photocatalytic qualities over semi-spherical agnps [9]. agnps made from aqueous leaf extracts have been reported for water remediation [10], as it facilitates the catalytic breakdown of dyes in the presence of visible light [5]. at the moment, bacterial resistance to antimicrobial drugs is the biggest obstacle to treating infectious diseases [10]. agnps may infiltrate bacterial cells and injure them by interacting with dna and other molecules containing phosphorus and sulfur, as silver reacts strongly with these elements. this possibility has been highlighted [10]. the antibacterial effects of agnps on bacteria may be caused by their capacity to bind to the surface of bacterial cell membranes and impair the permeability and respiration of the bacterium cell [10]. in wastewater, gram positive and negative microorganisms are frequently present [4]. it has been demonstrated that gram-negative bacteria are more susceptible to the antibacterial effects of green-mediated agnps than gram-positive pathogens [8]. controlled oxidation is typically used to insert oxygen-containing groups, such as ketone, phenol, lactone, carboxyl group, ether, acid anhydride, etc., to the surfaces of cnts [11]. the photocatalytic activity of the composite photocatalyst is highly associated with the amount of integrated mwcnts [12]. mwcnts have attracted a lot of interest because of their exceptional physical characteristics and adaptable morphologies [13]. as a new class of nanovectors for therapeutic administration, functionalized multi-walled carbon nanotubes (fmwcnts or omwcnts) are demonstrating their effectiveness in the transport and cellular translocation of therapeutic compounds [14], [15], [16]. these f-cnts can distribute drugs to cells or organs and can be made using one or more bioactive ingredients from the class of compounds that includes peptides, proteins, nucleic acids, and pharmaceuticals [17], [18], [19]. according to the number of concentric layers, cnts are divided into single-wall (with a typical diameter of 0.4 – 2 nm) and multi-wall (with more layers) (with a typical diameter of 10 – 100 nm) [14]. the carbon nanotube’s surface is covered in functional groups, which inhibit undesirable absorption and desorption processes [20], [21], [22], [23]. in this instance, undesirable absorption takes the form of biological medium molecule attachment and potential carbon nanotube content distribution [21]. since good aqueous stability is a crucial quality for bio-medical applications, cnts may be more cost-effective and effective than conventional antibiotic therapy because their surfaces can be modified to reduce hydrophobicity and improve adhesion through chemical attachment of proteins and polymers [14]. gram-positive and gram-negative bacteria are both susceptible to cnts’ antibacterial action [24]. the physical manner of bactericidal action and the generation of oxidative stress, which result in cell membrane damage brought on by the cnts, are responsible for this activity [24], [25]. the effects of cnts on infections caused by multidrug-resistant bacteria have been the subject of several in-vitro research [16], [19], [26]. this study was the first to investigate k. senegalensis for the production of silver nanoparticles. in order to compare the effectiveness of the photocatalytic and antibacterial activities of green-synthesized silver nanoparticles (ks-agnps) and functionalized carbon nanotubes (ocnts), respectively, a comparative analysis of both materials’ properties was conducted when applied for treatment of industrial waste water. 2. materials and methods 2.1. collection of plant materials and waste-water samples khaya senegalensis (african mahogany) leaves were collected from lafia, nigeria, in february, 2022. test bacterial species (escherichia coli and staphylococcus aureus) and industrial waste water were procured and sourced from genetics and biotechnology department, akwa ibom state university and eastern obolo oil producing community, akwa ibom, nigeria, respectively. silver nitrate (agno3) (99.80 %) was purchased from merck south africa. multi-walled-carbon nanotubes (outside diameter 8 – 15 nm, length 15 µm) were obtained from times nano. ltd. (chengdu, china). all the reagents and chemicals used in this study were of analytical grade. aqueous solutions were prepared using double-distilled water. 2 labulo et al. / j. nig. soc. phys. sci. 5 (2023) 1438 3 2.2. preparation of plant extract fresh leaves of k. senegalensis (1) were properly cleaned under running tap water followed by washing with distilled water. dried leaves (5 g) were ground into a powder and extracted with double-distilled water (ratio 1:10), the mixture was boiled for 25 min in a water bath at 70 °c. the mixture was then filtered with a whatman paper (no. 14) and kept in a dark bottle in a refrigerator for further analysis [27]. figure 1: khaya senegalensis leaves 2.3. synthesis of silver nanoparticles from plant extract in a typical experiment, 1 mm agno3 was added to 4 ml leaf extract in a 250 ml beaker and stored in a conical flask at room temperature for the reduction of ag+ to ag0. a colour change from dark green to brown visually identifying the formation of agnps [28]. green-mediated agnps were purified after several centrifugation at 20,000 rpm for 30 mins [29]. k. senegalensis synthesized agnps were labeled as ks-agnps and stored in an air-tight plastic sample vial for further use 2. figure 2: reaction pathway for reduction of silver ions to silver nanoparticles using k. senegalensis leaves extract. 2.4. preparation of oxidized carbon nanotubes (ocnts) the ocnts was obtained by acid treatment (oxidizing) of cnts with h2so4/hno3 (3:1, v/v) [30]. 1 ml 0.1 m agno3 solution was then added into 20 ml 4 mg/ml ocnts solutions drop-wisely under assistance of ultra-sonication [31], [32]. the oxidized cnts were successively eluted with deionized water, then by nh4oh, water, hcl and deionized water until the ph of filtrate was stabilized. 2.5. characterization green synthesized ks-agnps’ surface plasmon resonance (spr) peak was identified by observing uv-visible spectra between 300 and 700 nm [33]. the phytochemical functional groups, which are found in the khaya senegalensis leaves extract employed in the capping and bio-reduction of silver ions were identified using perkinelmer spectrum and fourier transform infrared (ftir) (perkinelmer, inc. waltham, massachusetts, usa) spectrometer in the between 400 and 4000 cm-1. xrd analysis was performed using bruker d8 x-ray diffractometer to determine the biosynthesized agnps crystallite size and structure. morphological characteristics (i.e., size and shape) were examined using scanning electron microscope jeol jsm-it 300 (jeol) ltd., tokyo, japan) working at a 30 kv voltage. transmission electron microscopy (tem) images were acquired on tecnai g2 f20s-twin tem (fei ltd., usa). the elemental composition of the green-mediated nps, were observed using energy dispersive x-ray (edx) analysis (jsm-7610f). 2.6. photocatalytic activity ks-agnps and ocnts were used as catalysts in the degradation of industrial waste-water analyzed under sunlight for a period of three days. first, the waste-water was twice diluted with double distilled water and scanned with a uv-vis between 300 700 nm wavelength [34], and the maximum wavelength obtained (590 nm) was recorded. 10 mg of ks-agnps and ocnts were mixed separately with 100 ml diluted waste-water and magnetically stirred for 30 mins in order to attain equilibrium as working solution. first reading was noted after stirring. the dispersion was then exposed to sunlight irradiation and monitored between the hours of 9 am (gmt+1) and 4 pm (gmt+1). 2 ml aliquot of the suspension was taken after one hour interval, filtered and used for photocatalytic degradation evaluation. absorbance value measurement at 580 nm was used to calculate the percentage dye degradation process according to the equation below: decolourization(%) = a0 − at a0 × 100 (1) where: a0 = absorbance of the untreated and at = absorbance of the treated industrial waste water. to ascertain the efficiency of both ks-agnps and ocnts as a biosynthesized catalyst, selected physicochemical properties (i.e., ph, total dissolved solid (tds), total hardness, calcium, magnesium, chloride, biological oxygen demand (bod) and total alkalinity) of the industrial waste water were assessed using standard analytical methods described by american society for testing and materials (astm) [35]. 2.7. antibacterial activity the antibacterial activity of the synthesized ks-agnps and ocnts was investigated against gram-positive (staphylococcus aureus) and gram-negative (escherichia coli) bacteria by the bauer-kirby disk diffusion method [36]. nutrient agar 3 labulo et al. / j. nig. soc. phys. sci. 5 (2023) 1438 4 medium was prepared according to manufacturer’s instructions by autoclaving at 121 oc for 15 mins. after sterilization, the agar was allowed to cool to about 45 oc, after which the molten agar was poured into petri dishes and allowed to solidify. the pathogens to be used as indicator organisms for antibacterial activity were inoculated on the solidified plates by using standard spread plate technique. a cork borer of about 2 mm in diameter was used to bore holes in the plates. both the agnps and ocnts to be tested for antibacterial activity were inoculated in the bored holes using a pipette. for the control, antibiotic disks were placed 2 cm away from the wall of the culture plate. the plates were incubated at 37 °c for 24 hrs, and the zone of inhibition of bacterial growth was used as a measure of susceptibility. 3. results and discussion 3.1. uv-visible spectroscopy the green-mediated reduction of the silver metal ions to ks-agnps was accompanied by a color change yellow to brown confirms the reduction of ag+ ions to ago. this was monitored using uv-vis spectrophotometer. figure 3 shows the surface plasmon resonance (spr) with high band intensities and peaks under the visible spectrum. the ks-agnps showed absorbance bands within the range of 427 nm which correlate with the results reported in other literatures on the green synthesis of agnps [3], [37], [38], [39], [40]. the ability of the plant extracts to act as a stabilizing and capping agent can be connected to the flavonoids and protein present in k. senegalensis aqueous extract [41]. figure 3: uv–visible spectra of green synthesized ks-agnps 3.2. ftir analysis figure 4 shows the spectra of the aqueous extract of k. senegalensis, ks-agnps, and ocnts. the aqueous extract of k. senegalensis showed characteristic peaks at 3300, and 1630 cm-1 which could be attributed to hydroxyl group o–h stretching [42], and c–o stretching in the carbonyl group [43] and the n–h group of amines [44], respectively. the broadband peak at 3263 cm-1 observed in the ks-agnps spectra could be attributed to the overlapping stretching mode of n–h and o–h functional groups [44]. the c–o band at 1030 cm-1 may be assigned to the polyols (i.e., flavones, terpenoids, and carbohydrates) present in the plant extract. the different assignments are in line with those for similar compounds that have been published in other literatures [39], [45], [46]. a broadband peak at 1549 cm-1 most probably corresponds to aromatic and unsaturated structure of c=c bonds was observed for ocnts [47]. however, the acid treatment of cnts led to the incorporation of –cooh group at 1720 cm-1. figure 4: ftir spectra of green synthesized ks-agnps, k. senegalensis leaf extract and ocnts 3.3. xrd studies data obtained from the silver nanoparticles were used to obtain the plot shown in figure 5. the diffraction patterns were also used to obtained the peak positioning, however, a total of five distinct peaks at 2θ with braggs’ reflection values of 38.12o, 44.28o, 64.43o, 77.48o, and 81.54o, which corresponds to (111), (200), (220), (311), and (222) plane of faced centered cubic (fcc) lattice of silver, and compared with the standards powder diffraction card of joint committee on powder diffraction standards (jcpds) silver file n0. 04-0783. the most significant peak from the xrd graph is centered at 2θ = 38.117. this may be the reason why the faced-centered cubic structure of the ks-agnps has grown favorably [1]. in the diffraction patterns obtained in the ocnts xrd spectra, six distinct peaks were observed with reflection values of 26.43o, 42.36o, 44.46o, 54.51o, 59.98o, and 77.40o, corresponding to (002), (100), (101), (004), (103), and (110), plane of lattice respectively. similar result have been reported by [48]. the average particle size was estimated using the debye-scherrer formula as follows: d = kλ β cos θ , (2) where d is the crystallite size of the agnps, λ is the wavelength of the x-ray source (1.54056 a), β is full width at half 4 labulo et al. / j. nig. soc. phys. sci. 5 (2023) 1438 5 maximum (fwhm) of the diffraction peak in radian, k is the scherrer constant that varies from 0.9 to 1, and θ is the bragg angle in radian [49]. the average size was found to be 9.30 nm. one of the structural changes that causes the distortions of crystallographic materials is d-spacing (inter-planar spacing) [50]. the xrd method relies on the broadening of diffracted peaks to quantify dislocation density that happens when atoms in crystal unit cells move away from their ideal positions as a result of numerous lattice defects (strain widening) [51]. figure 5: xrd spectra of ocnts and ks-agnps 3.4. sem and tem analysis sem images (figure 6) shows the morphological character of the silver nanoparticle synthesized by the extract of khaya senegalensis (african mahogany) leaves . the sem image revealed a flake-like morphology for ks-agnps. figure 7 shows the tem images of a mono-dispersed sphericalshaped green-mediated ks-agnps. the interaction between the biomolecules contained in the extract suggests that the ksagnps, which are uniformly distributed and mono-dispersed and have a spherical shape, have an active surface [7]. tem analysis consequently revealed that the average size of the synthesized ks-agnps is 9.30 nm. 3.5. photocatalytic activity of ks-agnps and ocnts activity on industrial waste water agnps and cnts have been used as catalysts to remove halogenated organics from water, including pesticides, heavy metals, and microbes [52]. nanoparticles are particularly effective in water treatment and disinfection to inhibit pathogenic bacteria and viruses due to their demonstrated antibacterial effectiveness [53]. in this study, photocatalytic activity of the bio-synthesized ks-agnps and ocnts was assessed through decolourization of industrial wastewater under solar light after different time intervals (54 hours). the change in color was used to identify rate of pigment degradation. after 55 hours (three days) of exposure, increased decolorization was evident as color of figure 6: ks-agnps sem images figure 7: tem images of green synthesized ks-agnps at (a) 50 nm and (b) 200 nm treated industrial wastewater changed from deep orange to light orange color. a continuous decrease in the absorbance of the treated wastewater obtained using uv spectrophotometer were recorded at time interval. steady absorbance readings obtained at the 55th hour suggested that total degradation was attained (figure 8). the percentage degradation efficiency of the treated wastewater was highest in ks-agnps (86.21 %) while ocnts had (82.14 %). yasmin et al. (2020)[54] recorded close range of value for agnps in earlier studies. in relative comparison, ks-agnps, has proven to have more photocatalytic effect as shown in figure 8, in the degradation of wastewater than ocnts, after 55 hours of exposure. the increased absorption rate of the nano-sized silver particle distribution in ks-agnps compared to ocnts may be the cause of their higher degrading efficiency [55]. more specifically, silver metal nanoparticles’ highly effective catalytic activities are a result of their distinctive properties, such as their extremely small dimensions, high surface-to-volume ratio, high dispersivity, and capacity for electron transfer between the donor and acceptor electron relay systems [56], and because catalysis happens on metal surfaces during degradation, there will be a considerable increase in surface area available, which will increase the catalyst’s effectiveness [57], in this case, ks-agnps. the mechanism of the catalytic photo degradation of the wastewater was proposed for better understanding as shown in figure 9. 5 labulo et al. / j. nig. soc. phys. sci. 5 (2023) 1438 6 figure 8: effect of exposure time on the photocatalytic degradation of industrial wastewater after (a) first day (b) second day and (c) third day figure 9: mechanism of photocatalytic activity of ks-agnps and ocnts the result of photocatalytic degradation of both ks-agnps and ocnts were compared with some results obtained from other literatures as shown in table 1, with a high degradation rate observed in ks-agnps, confirming its excellent photocatalytic activity. sequel to the photocatalytic degradation of the waste water analyzed, selected physicochemical parameters were also subjected to evaluation before and after treatment with ks-agnps and ocnts separately as shown in figure 10. comparison was also made with standard permissible limits recommended by world health organization (who). in general, the evaluation showed that both ks-agnps and ocnts (in same quantity) significantly reduced the parameters analyzed. however, greenmediated ks-agnps greatly reduced in all parameters. decrease in color change of the treated industrial waste water observed can be attributed to organic pollutants adsorption. industrial waste water ph before (10.55) and after treatment with both ks-agnps and ocnts showed that ks-agnps had greater reduction efficiency (7.04) in comparison to ocnts figure 10: plot of physicochemical assessment after treatment with ks-agnps and ocnts: tds – total dissolved solids, th – total hardness, ca – calcium, mg – magnesium, cl – chlorine, bod – biochemical oxygen demand, ta – total alkalinity (9.21). the ph decrease recorded for ks-agnps may be due to waste water anionic species removal. observation from ksagnps was however, within who ph permissible limit of 6.5 – 8.5. the tds of industrial wastewater was 350.05 mg/l, treatment with ks-agnps reduced the tds concentration to 286.09 mg/l while ocnts recorded a decrease of 299.19 mg/l. satisfactorily, both ks-agnps and ocnts reduction were lower than who recommended limit of 1000 mg/l for tds in domestic water. higher efficiency of silver nanoparticles obtained from k. senegalensis (ks-agnps) results from the reduced size of the bio-synthesized silver ion in the water. 3.6. antibacterial assay antimicrobial activity of green synthesized ks-agnps, ocnts, and control antibiotics (gentamicin and ciprofloxacin) were tested against two selected bacteria, escherichia coli (gram negative bacteria) staphylococcus aureus (gram positive bacteria) and investigated. figure 11: inoculated plates with bore holes containing ks-agnps, ocnts and known antibiotics (gentamicin and ciprofloxacin) before incubation for both (a) gram (+) and (b) gram (-) bacteria the antibacterial activity ranged from 0 – 4 mm as observed from the results in tables 2 and 3. ocnts produced the lowest susceptibility to both gram-positive and gram-negative bacteria compared to ks-agnps, which had higher susceptibility. ksagnps had susceptibility between 0.5 3.5 mm while ocnts had susceptibility between 0 – 0.5 mm. the known antibiotic, 6 labulo et al. / j. nig. soc. phys. sci. 5 (2023) 1438 7 table 1: different catalysts and their rate of decolourization of pollutants s/n catalyst pollutants effects reference 1. h3pw12o40/go methyl orange, crystal violet, and congo red 85.62 % methyl orange, 89.78 % crystal violet, and 82.62 % congo red decolourization respectively [58] 2. rambutan peel extractzno nanoparticles methyl orange 83.99 % decolourization [59] 3. terminalia chebula fruits extract-zno nanoparticles rhodamine b 70 % decolourization [60] 4. tinospora cordifolia-cuo nanoparticles methylene blue 80 % decolourization [61] 5. *o-carbon nanotubes industrial waste water (crude oil polluted water) 82.14 % decolourization present study 6. *ks-agnps industrial waste water (crude oil polluted water) 86.21 % decolourization present study gentamicin had the highest susceptibility (4 mm), followed by the green-mediated ks-agnps (3.5 mm), while ocnts had the least. the gram-negative bacteria (e. coli) were more susceptible to ks-agnps than the gram-positive bacteria (s. aureus), this is due to the thickness of the cell wall and cell composition. gram (-) bacteria has thinner peptidoglycan walls compare to gram (+) bacteria which has thicker peptidoglycan wall. however, given that small nanoparticles can easily pass through the cell membrane, susceptibility of ks-agnps is thought to rely on their size and shape [44]. agnps either adhere to the bacterial membrane or impair membrane permeability, which has an antibacterial impact, and they penetrate the cytoplasm and kill bacteria by disrupting their metabolism [62], [63]. additionally, the destruction of bacterial pathogens may also be caused by production of free radicals by the silver nanoparticles [64]. figure 12: antibacterial activity of (a) green-mediated ks-agnps and ocnts compared with (b) antibiotics a study by blasco et al. [65] demonstrated the efficiency of agnps produced through green synthesis against multiple antibiotic-resistant bacterial strains, including staphylococcus aureus, staphylococcus epidermidis, streptococcus pyogenes, klebsiella pneumoniae, and salmonella typhi, which further confirms the results obtained in this study. moreover, agnps can produce pits and gaps that reduce the permeability of bacterial cell membranes, resulting in the death of the bacteria [66], [67], [68]. table 2: results of the antibacterial activity of the green-mediated ks-agnps and ocnts inhibition zone of e. coli bacteria against inhibition zone (mm) condition (sensitive/resistance) ciprofloxacin antibiotics 0 resistance to antibiotics ks-agnps 3.5 sensitive to nanoparticle ocnts 0.5 slightly sensitive table 3: results of the antibacterial activity of the green-mediated ks-agnps and ocnts inhibition zone of s. aureus bacteria against inhibition zone (mm) condition (sensitive/resistance) gentamicin antibiotics 4 sensitive to antibiotics ks-agnps 1.5 sensitive to nanoparticle ocnts 0.5 slightly sensitive 7 labulo et al. / j. nig. soc. phys. sci. 5 (2023) 1438 8 the versatility of carbon-based nanomaterials, particularly ocnts and graphene, has shown considerable promise in biological applications such as tissue engineering, drug delivery, and bio-imaging [69], [70]. therefore, agnps are highly effective due to their clearly defined structures and substantial surface areas. by attaching agnps to the surface of graphene oxide or oxidized cnts, certain potent antibacterial agents have been produced [70,71]. 4. conclusions this study describes the quick and efficient green-mediated synthesis of ks-agnps employing an aqueous k. senegalensis extract as the capping and reducing agent. the color change and uv–vis adsorption confirm the reduction of ag+ to ag0. the tem showed spherical shape with average particle size of 9.30 nm. the ftir results confirmed the biomolecules capping and stability of the ks-agnps. the ks-agnps are stable and have excellent photocatalytic and antibacterial activities in comparison with ocnts. the green produced ks-agnps method is suitable for large-scale manufacturing, quick, affordable, nontoxic, and ecologically beneficial silver nanoparticles. overall, this work presents an effective method for creating smaller silver nanoparticles, which can be employed as antibacterial agents and to treat industrial waste water. acknowledgement the authors acknowledge the federal university of lafia for providing the research environment for this work. authors are also grateful to mrs. rasheedah labulo for proofreading this work. references [1] a. u. rahman, a. u. khan, q. yuan, y. wei, a. ahmad, s. ullah, z. u. h. khan, s. shams, m. tariq & w. ahmad, “tuber extract of arisaema flavum eco-benignly and effectively synthesize silver nanoparticles: photocatalytic and antibacterial response against multidrug-resistant engineered e. coli qh4” journal of photochemistry & photobiology, b: biology 193 (2019) 31, https://doi.org/10.1016/j.jphotobiol. 2019.01.018. 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[71] n. el-desouky, k. shoueir, i. el-mehasseb & m. el-kemary, “synthesis of silver nanoparticles using bio valorization coffee waste extract : photocatalytic flow-rate performance, antibacterial activity , and electrochemical investigation”, biomass conversion and biorefinery 4 (2022) 15, https://doi.org/10.1007/s13399-021-02256-5. 10 j. nig. soc. phys. sci. 1 (2019) 147–155 journal of the nigerian society of physical sciences review review of top notch electrode arrays for geoelectrical resistivity surveys h. e. ohaegbuchua,∗, f. c. anyadiegwua, p. o. odoha, f. c. orjia adepartment of physics, college of physical and applied sciences, michael okpara university of agriculture, umudike, nigeria. abstract the different arrangements of electrodes used in geoelectrical resistivity surveys and measurements are referred to as electrode arrays. in this review, we have revisited most of the widely used electrode arrays as well as the uncommon ones, which are nonetheless, useful in certain situations. this review has provided detailed information about eleven (11) of the top notch electrode arrays employable in our regular resistivity surveys, making it clear that in practice, the arrays that are most commonly used for 2-d imaging surveys are the wenner, dipole-dipole, wenner-schlumberger, pole-pole and the pole-dipole arrays. they have their strengths and weaknesses. they are typically described by their signal-to-noise ratio. their depth of investigation, ability for lateral location of the target and their mapping abilities of horizontal layers or steeply dipping structures among other factors determine which array to adopt. keywords: electrode arrays, wenner, schlumberger, dipole-dipole, pole-dipole, resistivity surveys article history : received: 16 september 2019 received in revised form: 08 january 2020 accepted for publication: 11 january 2020 published: 19 january 2020 c©2019 journal of the nigerian society of physical sciences. all rights reserved. communicated by: o. j. abimbola 1. introduction the purpose of electrical surveys is to determine the subsurface resistivity distribution by making measurements on the ground surface. from these measurements, the true resistivity of the subsurface can be estimated. the ground resistivity is related to various geological parameters such as the mineral and fluid content, porosity and degree of water saturation in the rock. electrical resistivity surveys have been used for many decades in hydrogeological, mining and geotechnical investigations. more recently, it has been used for environmental surveys. the resistivity measurements are normally made by injecting current into the ground through two current electrodes (a or ∗corresponding author tel. no: +2348132954367 email address: eo.henry@mouau.edu.ng (h. e. ohaegbuchu ) c1 and b or c2), and measuring the resulting voltage difference at two potential electrodes (m or p1 and n or p2). from the current (i) and voltage (v) values, an apparent resistivity (ρa) value is calculated. the different arrangements of these electrodes are referred to as electrode arrays. electrode arrays were developed in order to make field measurements more efficient and data interpretation easier. in this review, we will explore eleven (11) state-of-the-art electrode arrays that are employed for resistivity surveys, including the benefits, considerations, and applications of each array method. 2. wenner array 2.1. background the wenner array was invented in 1915 by american physicist frank wenner (1873-1954), who worked at the american bureau of standards. his development of a four-terminal bridge 147 ohaegbuchu et al. / j. nig. soc. phys. sci. 1 (2019) 147–155 148 design consisting of two outer current injection electrodes and two inner potential electrodes equally spaced became known as the “wenner array”’ [1, 2]. 2.2. field procedure the wenner electrode array is the simplest of arrays; in it, the four electrodes —a, m, n, and b— are placed in line and spaced equidistant from each other. the two outer electrodes, a and b, are current electrodes, and the two inner electrodes, m and n, are potential electrodes. with the wenner array, the resistivity of subsurface layers is found by increasing the distance between the electrodes while maintaining the location of the center point of the array [3]. the apparent resistivity measured by this array is given by equation 1. ρa = 2πa ∆v i , (1) figure 1: wenner array (modified from advanced geoscience, inc). 2.3. applications, benefits, & considerations of the wenner array the wenner array is commonly used in profiling for lateral exploration of the ground, like soil testing and sometimes ves for vertical exploration of the ground, like defining horizontal layers [1]. the wenner array is cumbersome when performing ves surveys. since you will have to move all four electrodes for each new measurement, this means a lot of walking when the electrode spacing becomes large. for example, to move all four electrodes from 50 meter electrode spacing to 100 meter electrode spacing, one person must walk a total of 600 meters for one measurement only—and a typical ves survey may comprise around 15 measurements. the alternative is to use more people; for example, one person at each electrode plus the instrument operator. 3. schlumberger array 3.1. background the schlumberger array is named for conrad schlumberger, founder of the modern-day schlumberger oilfield services company and pioneer of electrical methods in the early 1900s. 3.2. field procedure the schlumberger array is an array where four electrodes are placed in line around a common midpoint. the two outer electrodes, a and b, are current electrodes, and the two inner electrodes, m and n, are potential electrodes placed close together [4]. with the schlumberger array, for each measurement the current electrodes a and b are moved outward to a greater separation throughout the survey, while the potential electrodes m and n stay in the same position until the observed voltage becomes too small to measure [5]. at this point, the potential electrodes m and n are moved outward to a new spacing. as a rule of the thumb, the reasonable distance between m and n should be equal or less than one-fifth of the distance between a and b at the beginning. this ratio goes about up to one-tenth or one-fifteenth depending on the signal strength. equation 2 represents the apparent resistivity of the schlumberger array. ρa = π(s2−a2 ) 4 a ∆v i , (2) figure 2: schlumberger array (modified from advanced geoscience, inc). 3.3. applications, benefits, & considerations of the schlumberger array the schlumberger array is commonly used for vertical electrical sounding (ves) for groundwater and aggregate minerals. vertical electrical sounding (ves) using the schlumberger array provides better resolution, and take less time to deploy than the wenner array [6]. the schlumberger array is the best method for ves for practical reasons—it is significantly less labor-intensive than the wenner array. for ves in general, the schlumberger array is at advantage, because most of the time only the outer electrodes a and b need to be moved [4]. a crew of three people is normally enough for ves—with two people moving the outer electrodes, the instrument operator typically moves the inner electrodes the few times they need to be moved. the schlumberger array is also used for mapping or profiling for lateral resistivity changes. the typical profiling is done by larger fixed ab current electrode pairs and moving mn potential electrode pairs between them [5, 7, 8]. it is also done like wenner profiling with fixed four abmn electrode spacing the disadvantage of using profiling, regardless of using the schlumberger or wenner arrays, is that it demands homogeneous horizontal layers, which hardly ever occur naturally. 148 ohaegbuchu et al. / j. nig. soc. phys. sci. 1 (2019) 147–155 149 3.4. potential drawbacks • substantial lengths of cable energized with current at high voltage present a safety hazard. • the schlumberger array is a labour-intensive survey because of the cable lengths required and the movement of the electrodes during the survey. 4. dipole-dipole array 4.1. background the dipole-dipole array was originally used for mineral exploration with the induced polarization (ip) method. the dipoledipole method made it easier to visualize the collected data in the form of 2d handmade contour plots. even though you couldn’t get a perfect image of the ground, it gives you an approximate image which you could interpret the geology from [9]. as previously mentioned, the big change with the dipoledipole array came with automated resistivity and ip imaging (where raw data is collected automatically and inverted (computermodelled) into true ip and true resistivity, which we’ll examine below. 4.2. field procedure a dipole is a pair of oppositely charged electrodes that are so close together that the electric field seems to form a single electric field rather than a field from two different electric poles. on the other hand, when the separation of the two charged electrodes is large enough that the observer detects the electric field from two poles, it’s called a bipole [9]. the dipole-dipole array consists of a current electrode pair a and b and a potential electrode pair m and n, and it offers a way to plot raw data in order to get an idea of a cross-section of the earth. those using the dipole-dipole array look at a measurement value called apparent resistivity, which represents a weighted average of the resistivities under the four electrodes used to take the reading. the apparent resistivity is typically calculated by modern instruments from the geometry between the four electrodes and the injected current and measured potential [10]. the result of a dipole-dipole survey is plotted in a pseudosection. for each measurement, the apparent resistivity data is plotted at the midpoint between the two dipoles and at a depth half the distance between the two dipoles [10]. the value is finally contoured and colourized, which represents a rough and often severely distorted image of the subsurface. so while this array doesn’t provide an actual image of the ground, the data points resulting from these measurements provide an image of the cross section that one can then try to interpret [11]. today, modern inversion software can recalculate all of these apparent resistivity values to “true” resistivity values so that a realistic image of the ground can be created. the primary advantages of the dipole-dipole array are its high resolution and multi-channel capability; it provides a very detailed image instead of providing a “big picture” image like the wenner array. the dipole-dipole array is fast because it supports multiple receiver channel measurements simultaneously (the agi supersting r8 can measure eight simultaneous measurements for each current injection) whereas the wenner array can only collect one data point for each current transmitter injection. additionally, it is relatively simple to perform a survey along profile lines using the dipole-dipole array. the disadvantage of this array is that the dipoles will lose the signal if they’re placed too far apart, thereby decreasing the ability to see deeper into the earth. the expression for the apparent resistivity is as follows: ρa = πn(n + 1)(n + 2)a ∆v i , (3) figure 3: dipole-dipole array (modified from advanced geoscience, inc). 4.3. using the dipole-dipole array with modern resistivity & ip scanning instruments to conduct a survey using the dipole-dipole array, you first place a large number of electrode stakes out with equal spacing between the stakes. for example, you could have 100 electrodes, each spaced one meter apart. if this were the case, you’d have a 99 meter long profile. with spacings at 10 meters apart, you will have a 990 meter long profile. the longer the profile, the deeper you are able to see —at a cost of less detail in the image. after all the stainless steel electrode stakes are installed in the ground and connected with one multi-conductor cable to the receiver instrument, you then start the measuring sequence. the electrodes in positions one and two are your current electrodes, labelled a and b. the electrodes in positions three and four are your potential electrodes, labelled m and n. the instrument takes a reading referred to the mid-point between the current and potential dipoles. once that is complete, the instrument automatically moves the potential electrodes (m and n) to the electrodes in positions four and five. (the electrodes stay in the same place physically, but the instrument connects with a different pair of electrodes). this process repeats—electrodes a and b stay in positions one and two while m and n move out one electrode spacing for each reading. for every step out, the signal becomes weaker. at a certain point, the instrument can no longer take a reading. at this point, the instrument automatically moves the current electrodes a and b out a step—from positions one and two to positions two and three. this cycle continues throughout the 149 ohaegbuchu et al. / j. nig. soc. phys. sci. 1 (2019) 147–155 150 electrode array. using the supersting r8, you can conduct a field survey in roughly 15 minutes (instead of the two or more hours another system may take, which decreases the costs of time in the field). keep in mind that modern equipment and software allow you to perform multiple arrays at one time. using the supersting instrument, you could perform a combination dipole-dipole and schlumberger (gradient) array at once to allow you to see depth and detail even better. 5. pole-dipole array 5.1. background the pole-dipole array has relatively good horizontal coverage, but it has a significantly higher signal strength compared with the dipole-dipole array and it is not as sensitive to telluric noise as the pole-pole array. unlike the other common arrays, the poledipole array is an asymmetrical array and over symmetrical structures the apparent resistivity anomalies in the pseudo section are asymmetrical. in some situations, the asymmetry in the measured apparent resistivity values could influence the model obtained after inversion. one method to eliminate the effect of this asymmetry is to repeat the measurements with the electrodes arranged in the reverse manner. by combining the measurements with the “forward” and “reverse” poledipole arrays, any bias in the model due to the asymmetrical nature of this array would be removed. 5.2. field procedure a pole is a single transmitting electrode, and a dipole is a pair of oppositely charged electrodes that are so close together that the electric field seems to be a single electrode field instead of fields from two different electric poles. the pole-dipole array is similar to the dipole-dipole array, but the pole-dipole array is used when the surveyor needs to see deep within a cross section of the earth [12]. the achievable depth is based entirely on the distance between the two electrodes (the dipole and the pole). the dipole-dipole array, on the other hand, is used to provide a very detailed image of a cross section of the earth—but will lose signal if the dipoles are placed too far apart. like the dipole-dipole array, the poledipole array is most often used for mineral and ore exploration. the apparent resistivity is given as: ρa = 2πn(n + 1)a ∆v i , (4) 5.3. using the pole-dipole array with modern resistivity & ip scanning instruments when conducting a survey using the pole-dipole array, the transmitting remote electrode is moved to infinity. in other words, the transmitter remote electrode should be so far away from the receiver dipole that the instrument does not sense the effect of the remote (pole) electrode, which is often 10 times the distance of the survey area. the reason for this is that the distance between the two electrodes determines the depth of figure 4: pole-dipole array (modified from advanced geoscience, inc). electrical penetration—and explains why the pole-dipole array is popular when you need to look deep into the earth. this is a critical element because if the remote electrode is not far enough away from the dipole, the electrical current from the dipole to the pole will create a concentric circle of electricity (so long as the ground is homogeneous). if there is a geologic deviation in the ground, placing the remote electrode too close to the dipole will create a deviation in this concentric circle—but will still distort the measurements. when correctly configured, you move the potential electrodes (the dipole) to a position near the current electrode (the pole), and take a reading. then you move the dipole one step forward, and take another reading. you continue this until you begin losing signal between the current and the potential electrodes. keep in mind that modern equipment and software allows you to perform multiple arrays at one time. you could use the supersting instrument to perform a pole-dipole array. 6. pole-pole array 6.1. field procedure the four electrodes in the pole-pole array are arranged so that the distance between the transmitter (a&b) dipole and the receiver (m&n) dipole is small in comparison. in fact, the distance ought to be 10% that of the a&b dipole [10]. in a pole-dipole survey, one transmitting electrode is moved away from the dipole to infinity (i.e., 10 times the distance of the survey area) so the instrument does not sense the pole electrode. in a pole-pole survey, one receiver electrode is also moved to infinity, but additionally, one of the potential electrodes is moved to infinity in the opposite direction. in other words, in a pole-pole array, you have a stationary infinity electrode on either side of the survey area. then proceed with your resistivity or induced polarization measurements exactly as you would with a pole-dipole array: move the a and m electrodes to a new location inside the survey area and take a reading with a receiver instrument. the apparent resistivity according to [13] is: ρa = 2πn ∆v i , (5) 150 ohaegbuchu et al. / j. nig. soc. phys. sci. 1 (2019) 147–155 151 figure 5: pole-pole array (modified from advanced geoscience, inc). 6.2. considerations for using the pole-pole array the primary issue with using the pole-pole array is space—a problem that makes it far less common than both the dipoledipole array and the pole-dipole array. the two infinity electrodes are stationary, but you need to place them far out in opposite directions, and you may have to get permission to place them on someone else’s land. you also have to consider hazards like traffic, creeks, or brush when handling the connecting wire. a large mn may pick up plenty of cultural, sp, and telluric noise. for example, if you wanted to conduct a resistivity survey to look for groundwater at approximately 300-400 meters deep, you would need to have two kilometres between your electrodes. this is an enormous area to cover for a survey, and there are numerous things to consider in this circumstance, including the cost, the amount of time it takes to roll out the wire, and the labour necessary. furthermore, you have to be mindful of any physical or geographical hazards. for instance: • you may need permission from the owner of neighbouring land to place your electrode over their boundary. • you could be restricted if the electrode cable needs to cross a busy road. • bodies of water or vegetation may complicate matters. 7. equatorial array 7.1. field procedure also known as the equatorial-dipole array, the equatorial array consists of the two dipoles—a-b and m-n—oriented perpendicular to the survey line. during the sounding process, m-n is moved away from ab along the perpendicular bisector (n). this perpendicular bisector may be regarded as an equator to the poles at a and b; hence the name ”equatorial arrangement.” [12]. 7.2. applications, benefits & considerations the advantage of the equatorial array is that you are able to penetrate even deeper into the ground than the dipole-dipole array (assuming the length along the survey line is the same). this can be particularly advantageous when pulling an array of electrodes behind a vehicle or in limited spaces [14]. figure 6: equatorial array (modified from advanced geoscience, inc). one consideration of the equatorial array is that even though it can see deeper than other arrays, it will lose signal as it is expanded, which limits its use. the equatorial array is used less often than other arrays, but it is still very useful. it can be especially helpful in situations where you are limited by tight space constraints. for example, imagine doing a survey in a parking garage. certainly, you do not have much space between support structures there. using this array, you can see as deep as possible in a limited space. 8. square array 8.1. field procedure the square array is a special case of the equatorial array. the name derives from the electrodes forming the shape of a square—with the four sides of the array being of equal length. this array is used most frequently in determining anisotropy, or the resistivity in different directions of a geological formation. instead of expanding the electrodes, in square array surveys, the square is typically turned 15 degrees around its centre point for each measurement [15]. the apparent resistivity for the square array depends on what abmn geometry (k) you use. below is the generic formula for apparent resistivity to get you started. ρa = k ∆v i , (6) figure 7: square array (modified from advanced geoscience, inc). 151 ohaegbuchu et al. / j. nig. soc. phys. sci. 1 (2019) 147–155 152 figure 8: square array (modified from advanced geoscience, inc). 8.2. applications, benefits & consideration as with the equatorial array, an advantage of the square array is that you are able to penetrate deeper into the ground than when using the dipole-dipole array (assuming the length along the survey line is the same). as mentioned above, it is a good array for determining anisotropy. with the development of 3d arrays, the need for the square array has diminished over the years. it is often not used much in the u.s. any more, though it still has some popularity in other countries. with a 3d array, you could easily see clearer data than what is provided by the square array. that said, the square array is the cheaper option of the two. the square array is usually only used if the field conditions are right and if the survey operator has some base knowledge of fractures. square array resistivity surveys would be a good fit for identifying environmental problems. specifically, preferential flow paths and the way spills will flow and follow fractures [15]. let us say you are a hydrogeologist that needs to survey a rock body with a fracture system. you will want to locate the general fracture direction. taking a number of readings through the square array, you can determine which direction is more conductive and thus the main direction of the fractures. furthermore, if you needed to see deeper to gain more information on fractures in the rock body, you could expand the size of the square and repeat the same procedure. it may, for instance, show that the fracture changes direction at a particular depth. 9. gradient array 9.1. field procedure the gradient array is similar to the schlumberger array. the difference is that the schlumberger array records only the center receiver dipole, whereas the gradient array measures all adjacent dipoles from one transmitter electrode to the other, including the center-most dipole [16]. to take a measurement with the gradient array, a reading is taken of the potential field between two current electrodes side-by-side (which form a dipole). the next reading goes one step to the right and continues down the row of electrodes. this process repeats until you have mapped the whole area you are examining. the lateral changes in the potential field of the gradient array are measured between the transmitter dipole a&b. recording the lateral changes is often referred to as a profiling method [17]. 9.1.1. what is the difference between edge gradient and strong gradient? as mentioned above, the two forms of gradient array that can be used for 2d profiles are the edge gradient and the strong gradient. all measurements on an edge gradient use the ”a” electrode spacing between the transmitters a&b. current electrodes—starting with the b-current electrode—remain at the first electrode until the a-current electrode reaches the last electrode on the 2d line. then, the a-current electrode remains at the last electrode until the b-current electrode reaches the a-current electrode. this gradient array generates a large number of measurements. alternatively, the measurements on a strong gradient is comprised of multiples of eight measurements. these are always taken between two transmitting electrodes with different ”a” electrode spacing like a, 2a, 3a, 4a, etc. figure 9: gradient array (modified from advanced geoscience, inc). 9.2. applications, benefits & considerations profiling methods such as the gradient array are ideal for locating sinkholes, fractures, and other geological changes. the strong gradient array is also great for conductive environments such as on the seafloor to enhance measured signal [18]. a full spread of 56 electrodes with one meter spacing between will give you a very strong signal near the two current electrodes using the edge gradient. (but a very weak signal in the center of your survey area). the strong gradient array is a good choice to improve such weak signal issues. strong gradient produces a strong signal and it is recommended for a large number of electrodes to reduce the number of measurements. this is one of the reasons we recommend the strong gradient for most geophysical surveys—even just as a backup array [17]. historically, these arrays were considered to be impractical, slow, and hard to visualize. before automatic multi-electrode systems, resistivity imaging instruments most often consisted of four electrodes connected to a voltmeter. using the gradient array with this type of system was very time consuming, as it requires someone to take readings and then move the electrodes until the whole survey area was mapped. but with modern multi-electrode resistivity imaging systems, you can automatically take measurements between each dipole for a quick and efficient survey [19]. since gradient arrays, especially the edge gradient array, frequently depend on the first and last electrodes on the survey line, it is highly recommended to closely observe the contact resistances on these electrodes during the entirety of your survey to reduce noisy data. 152 ohaegbuchu et al. / j. nig. soc. phys. sci. 1 (2019) 147–155 153 10. azimuthal method 10.1. field procedure you may have noticed that we said “method” and not “array” at the beginning of this section. that is because the azimuthal method is less of an array and more of an electrode orientation or method—similar to vertical electrical sounding (ves), profiling, or mise-a-la-masse. “azimuth”—in relation to a geophysical survey—is the direction or angle of a particular location compared to north. you can measure azimuth with many different electrode arrays—the wenner array being one of them. with the wenner array—an axial array—you set your current electrodes along the 12 o’clock and 6 o’clock and your inner electrodes on that line at a 10 meter electrode spacing. you then move your electrodes clockwise either 10 degrees or a quarter turn (depending on your preference) and take another reading. as you repeat this, your outer electrodes form a circle with a 30 meter diameter, and your inner electrodes form another circle with a 10 meter diameter. you can increase the wenner electrode spacing to measure deeper [20]. evaluation of this data gives you information about fracture patterns. the wenner “a spacing” is associated with resistivity at different depth levels in the formation below, which is interpreted through what is called a polar diagram or rose diagrams. this diagram uses concentric circles (which represent resistivity value) and a north, south, east, and west axis. if anisotropy is present in the ground, then the polar diagram forms an ellipsoid figure when a line is drawn from each data point, as seen in the image below [2]. figure 10: azimuthal method (modified from advanced geoscience, inc). 10.2. applications, benefits & considerations hydrogeologists can learn a great deal by applying this method to study fracture patterns and how water flows through them. rock formations in the earth are anisotropic—meaning they have different resistivities based on the direction of the resistivity measurement. the changes in directional resistivity are often due to the fracture patterns in the subsurface [21]. so, for example, if a hazardous liquid is spilled, it may be critical to know the direction of the fractures in a particular area to contain the site—and electrical resistivity testing allows for a noninvasive way to “see” these fractures [22]. consider an electrode array rotating around a center point. if there are fractures in the subsurface, at some point the electrode array will be parallel to the fractures. this happens because the fractures hold sediment, soil, clay, and moisture, which creates a more conductive path then the surrounding material. these fractures are termed heterogeneous features. anisotropy is another directional component found in some soils which can have an azimuthal response as well [17]. differentiating these two features requires core samples in many cases. the azimuthal method may exhibit a response which looks like fractures, but in fact be due to a layer called a dipping bed. this is a flat layer of lower resistivity material which angles down away from the surface. this is because azimuthal method is a 1d ves or sounding method and thus cannot image the true shape of the bed or fracture like a 3d image would. 3d imaging is the most reliable way to measure fractures. multiple parallel 2d lines can also be merged into a good 3d model for this same goal [23, 24]. as previously noted, the wenner array is not the only array used with the azimuthal method—and it can actually be prone to issues. there are some who suggest that the square array is more appropriate for azimuthal measurements, as it is the most sensitive for direction [25]. 11. mise-a-la-masse 11.1. field procedure like vertical electrical sounding (ves) and profiling, or the azimuthal method—mise-a-la-masse is more of a method than an electrode array. you may have seen this method also called the “charged body potential method” or the “excitationat-the-mass method”. loosely translated from french, mise-ala-masse means “charged body” [26]. this method was very popular in the 1920’s and 1930’s for searching out ore bodies. in fact, it is still used today for the same reason—among others. if you are looking to use this method, the pole-dipole array is most frequently paired with mise-a-la-masse. to perform this method, attach a current electrode (a) to an exposed portion of your conductive target—such as ore, water, or plumes. because the target is conductive, it acts as a giant electrode. the current electrode (b) is placed at infinity—and the potential electrode (n) is placed at infinity opposite of b. the m electrode is then used to detect the edges of the underground target [27, 28]. 153 ohaegbuchu et al. / j. nig. soc. phys. sci. 1 (2019) 147–155 154 figure 11: mise-a-la-masse array (modified from advanced geoscience, inc). figure 12: mise-a-la-masse array (modified from advanced geoscience, inc). 11.2. applications, benefits & considerations ore body exploration. the mise-a-la-masse method can be useful if an ore body is discovered during drilling. while you may know there is an ore body in the vicinity, you likely do not know its direction or how it extends underground. using misea-la-masse can help you to be more precise in your exploration and excavation. for ore exploration, you would essentially follow the steps we outlined above—though your current electrode (a) would be attached to a wire and lowered into a borehole to attach to an exposed portion of the ore body [28]. detecting pond membrane leakage. mise-a-la-masse can also be used to detect leakage in rubber pond membranes or liners. for this application, it is the same steps as stated above, except you will place your current electrode (a) near the edge of the pond in the water. if there are no holes or leaks in the membrane, the instrument will not be able to send out the requested current when the current is injected into the water. but if a hole or leak is present—even if it is the size of a pinhole—the current from the electrode will find the hole and move toward the infinity electrode (b). this creates a source of electricity in the pond which can easily be located [27]. mise-a-la-masse was popular in the 20’s and 30’s for a reason. it is a relatively easy method to use, and it is a great option for the applications we have listed below. though it is great for a specific use, it is not a method that you will need to use often. 12. bipole-bipole array 12.1. field procedure it is easy to get this array confused with the dipole-dipole since they are similar. in fact, the field from the transmitting electrodes when using the dipole-dipole array is actually the field from a bipole (especially as it pertains to the measurements closest to the transmitting electrodes). the word “bipole” is used instead of “dipole” when the two transmitting electrodes are placed so far apart that the electric field from them can be considered a field from two separate poles. in other words, the distance between the receiver and transmitter dipoles is small in relation to the dipoles themselves [29]. figure 13: bipole-bipole array (modified from advanced geoscience, inc). 12.2. applications, benefits & considerations as mentioned above, the bipole-bipole array is useful for some uncommon 3d geometries. for instance, if you want to install electrodes surrounding a large building, or structure. you would want to use bipole-bipole instead of dipole-dipole because the current and voltage electrodes are going to be far apart. the bipole-bipole array becomes very useful on cross-borehole studies for strong current injection, where generally, dipoledipole fails to do so. it is also useful for some uncommon 3d geometries. though it is also possible to use the bipole-bipole array for any mix of surface and depth electrodes (electrodes installed within the earth materials)—you cannot use this array unless some of the electrodes in your geometry have a negative zvalue. so if using this array, be prepared to install electrodes beneath the surface via boreholes or some other method [29]. 154 ohaegbuchu et al. / j. nig. soc. phys. sci. 1 (2019) 147–155 155 13. conclusion having reviewed the various electrodes designs for geoelectrical resistivity surveys, we conclude by noting that in practice, the arrays that are most commonly used for 2-d imaging surveys are the (a) wenner, (b) dipole-dipole (c) wennerschlumberger (d) pole-pole and (e) poledipole. these arrays are commonly used in resistivity surveys. they have their strengths and their weaknesses. they are typically described by their signal-to-noise ratio. their depth of 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[29] s. szalai, a. koppan & l. szarka, “effect of positional inaccuracies on multielectrode results” 13th european meeting of environmental and engineering geophysics, istanbul, turkey (2007). 155 j. nig. soc. phys. sci. 4 (2022) 75–82 journal of the nigerian society of physical sciences higher order motional resonances spectra of electrons with non-linear axial oscillations in quadrupole penning trap b. m. dyavappa∗ department of physics, government college for women, kolar, karnataka, india abstract experimentally the motional resonances spectra of an electron cloud confined in a quadrupole penning trap, weakly excited with rf field, which is swept from 1 mhz to 1 ghz was examined. the higher order motional resonances of the electron cloud with non-linear axial oscillations show that they result from couplings in the motional degrees of freedom, which characterize the perturbations in the trap, and cause loss of electrons from the trap. the likely presence of perturbative terms of quadrupole potential is identified by studying the motional resonances spectra and hence the multipole terms are calculated. a periodically forced barrel spring with a steel ball at the free end can be regarded as a model of non-linear oscillator and a periodically forced steel ball hung through a flexible rod, which is deflected alternately toward the two magnets can be regarded as a mechanical model of duffing oscillator are developed, in analogy to observed higher order motional resonances of the electron cloud excited with different rf field powers. doi:10.46481/jnsps.2022.371 keywords: quadrupole penning trap, motional resonance spectrum, quadrupole potential, duffing oscillator, non-linear oscillator article history : received: 30 august 2021 received in revised form: 25 october 2021 accepted for publication: 02 january 2021 published: 28 february 2022 c©2022 journal of the nigerian society of physical sciences. all rights reserved. communicated by: w. a. yahya 1. introduction the motional resonances spectra of electrons contain the fundamental eigen frequencies, viz. the modified cyclotron frequency ( f ′c ), magnetron frequency ( fm), axial oscillation frequency ( fz) and their linear combinations in penning trap [1, 2]. these frequencies can be measured by resonant excitation of the motion of electrons with rf-field applied to the trap electrode. the linear combinations of the fundamental frequencies and their harmonics become visible in the spectrum of motional resonances, which also depend on the power of the rf field [3]. the pure cyclotron frequency ( fc) can be obtained ∗corresponding author tel. no: +919110854782 email address: dyavappabm@gmail.com (b. m. dyavappa ) from direct excitation of the side band frequency ( f ′c + fm). in this paper investigations on motional resonances of an electron cloud probed from the radio frequency of 1 mhz to 1 ghz are presented, which helps in characterizing the instabilities of the electron plasma in the trap. the non-linear higher order motional resonances of the electron cloud show that they result from couplings in the motional degrees of freedom that characterize the perturbations in the trap [4, 5, 6]. 2. theory the general form of the potential in penning trap can be written as [7, 8, 9, 10, 11] 75 dyavappa, b. m. / j. nig. soc. phys. sci. 4 (2022) 75–82 76 φ(r,θ) = φ0 n∑ 0 cn ( r r0 )n pn (cos θ) (1) the presence of higher order terms in the potential due to perturbations cause couplings among the motional degrees of freedom of electrons, and result in motional resonances which occur at certain linear combinations of fundamental frequencies in addition to the characteristic fundamental eigen frequencies and their harmonics [7, 8, 9]. thus at these frequencies in motional resonances spectra, we may express the relation among the pure eigen frequencies as [1, 2, 9, 10, 11] n′c f ′ c + nm fm + nz fz = 0 (2) where n′c, nm, nz are integers, there are many possible sets of integer triplets ( n′c, nm, nz ) which can be predicted using mat lab code, but only a few are observed in motional resonances spectra. we know that the eigen frequencies are given by [7, 8, 9, 10, 11] f ′c = fc + √ f 2c − 2 f 2z 2 , fm = fc − √ f 2c − 2 f 2z 2 , (3) fc = f ′ c + fm = qb 2πm let a = 2 ( n′c − nm )2 +4n2z , b = −2(n ′ c +nm)nz, c = 4n ′ cnm,(4) then aω2z − 2bωcωz + cω 2 c = 0 (5) ∴ k = b ± √ b2 − ac a . (6) here k = ωz ωc = fz fc , fc > √ 2 fz, kmax = 1/ √ 2, where k is a constant related to the integers n′c, nm, andnz. for stable trapping condition, ωc > √ 2ωz and kmax = 1/ √ 2 = 0.7071 or 0 < k < 1/ √ 2. one can compute numerically the integer triplet ( n′c, nm, nz ) sets for a given value of k, and hence a list of possible integer triplets for different values of k. motional instabilities may occur at frequencies that satisfy 2 for any particular value of k < 1/ √ 2. table 1: different values of k for different magnetic fields and storage voltages b (10−4t ) k (2 v) k (10 v) 97 0.069878 0.210543 258 0.032437 0.072532 an expression of the potential of the form given by equation (1) in a fourier series restricts the integer triplets as [7, 8, 9, 10] |n′c| + |nm| + |nz| ≤ n (7) this equation indicates that the perturbations at least of order n are present. however all of these perturbations may not cause instabilities. 3. experimental procedure the quadrupole penning trap is designed with threeelectrode infinite hyperboloid revolution of structure, which consists of two end-cap electrodes and a ring electrode. the equation of hyperbola of ring electrode is r 2 r20 − z2 z20 = +1 and that of two similar end cap electrodes are given by r 2 r20 − z2 z20 = −1 . the dimensions of quadrupole penning trap are designed such that r0 = √ 2z0 = 7mm, where r0 is the inner radius of the ring electrode in the radial plane and z0 = 5mm is half of the vertical distance between the two end-cap electrodes. a thoriated tungsten filament is used to generate electrons by thermionic emission through the passage of small 6 v dc current. we generate magnetic field using an electromagnet by passing a dc current of 0 20 a through a pair of coils of wire separated by 11.5 cm. magnetic field is generated by electromagnetic induction and it can be measured using hall probe and magnetic field at the trap centre measured from cyclotron frequency from motional resonances spectra is very accurate is mentioned here. the electrons are trapped in ion trap at a vacuum pressure range of 10−6 to 10−10 torr range created by turbo molecular pump with backup rotary vane pump and ion pump. high vacuum of 10−7 torr can be created and maintained by turbo molecular pump. an ion pump is used to create ultra high vacuum of up to 10−10 torr [11]. the arrangement to measure the motional resonances of trapped cloud of electrons in penning trap is described in this section. the couplings in the motional degrees of freedom of the electron cloud are determined by probing the confined cloud of electrons, through exciting the additional modes of oscillations by external rf field applied through an external antenna as shown in figure 1. we fed rf energy into the trap using a small antenna placed into a narrow hole of the ring electrode and it is isolated from the ring using a ceramic washer. the antenna is basically a co-axial cable of copper wire; the outer insulating cover has been removed partly and inserted into a narrow hole made in the ring electrode. the insulator separating the outer metal sheath and the inner copper wire is not an uhv compatible material, but we have created a vacuum of 10−9 torr without much difficulty, beyond which one may have to use an alternative material. the rf generator switched to sweep mode and is used to feed rf energy into the trap through the antenna. the rf power is kept very low of the order of a few /x to weakly probe the motion of the trapped electron cloud. if the rf power is kept high, then it will resonantly drive the trapped electron cloud in the trap and more electrons are lost. the constant signal strength of resonance absorption signal of electrons in the absence of external excitations is recorded, and the detection cycle is repeated continuously. the constant signal height of electrons serves as a baseline of reference for the motional resonance spectrum measurements. when the electrons are excited by continuously sweeping the rf field, the motional frequencies of electrons respond to the external rf field at an appropriate step of frequency, consequently some of the electrons gain enough energy to escape the trap, consequently the signal strength is reduced and appear as dips, whose strength is di76 dyavappa, b. m. / j. nig. soc. phys. sci. 4 (2022) 75–82 77 figure 1: quadrupole penning trap with antenna to feed rf field rectly proportional to the total number of electrons lost from the trap. the data of signals of electrons are continuously acquired and recorded using a lab view program written in a computer, which yields the desired motional spectrum of trapped electrons. a personal computer installed with lab view program is used for data acquisition and control trap system using a multifunction ni daq card. it controls, monitors and synchronizes four tasks viz. generates the trap potential and timing, controls the electron loading, and controls the rf sweep and data acquisition. the motional resonance control and acquisition (mrca) lab view program runs all the tasks simultaneously and are performed in a times share method and all tasks are synchronized at the beginning. all the four modules of the program and the tasks are described in detail below: trap potential timing sequence:. the ring voltage is timed so that it is suitable to trap and detection, to meet the requirements of motional resonances and the time dependence of the voltage is controlled by the lab view program called motional resonance control and acquisition (mrca) using computer. the time measurement cycle has mainly 6 stages as shown in fig.2 [1, 6, 11, 12]. (i) loading control: the electron loading can be controlled by controlling filament current or the negative bias voltage provided by a dc power supply applied between the filament and the closer end cap. the electron loading is turned off by reducing the bias or brought close to zero. the bias is turned on during the creation time (t1), for a short while of about 100 ms and then turned off till the first detection cycle is over. (ii) sweep control: a radio frequency generator operated in the sweep mode is used for generating rf field which is fed through the antenna. the sweep is done in two frequency ranges, the first range is from 1-100 mhz in steps of 0.1 mhz and the second range is 100-1000 mhz in steps of 1mhz. (iii) data acquisition: the trap potential with the pre-defined time cycle begins; the data is acquired continuously and sampled at a rate of 1000 samples per second using the lab view program developed in computer and the data obtained is plotted in origin. figure 2: trap potential timing sequence, t1: creation time (100 ms), t2: cooling time (10 ms), t3: storage time (120 ms), t4: detection time (60 ms), t5: kick out time (10 ms), t6: waiting time (100 ms). 4. results and discussion the motional resonances spectra of trapped electrons contain the fundamental oscillation frequencies f ′c , fm, fz, in addition some of their linear combinations, satisfying the equation n′c f ′ c + nm fm + nz fz = c and multiples of these frequencies n ′ c f ′ c , nm fm, and nz fz. we have carried out measurements close to the stability limit at magnetic fields of 97g and 258g. (i) rf field power dependence: when the power of rf field is −10dbm then the trapped cloud of electrons is more unstable. more resonances dips are observed due to quick loss of electrons from the trap. on the other hand if the power of rf field is −100dbm then the excitation is very weak and the trapped cloud of electrons are slightly unstable, then some of the higher order axial oscillation resonances dips disappear and the intensities decrease with higher orders of resonances, as loss of electrons from the trap decreases . it can be observed from the figure 3 (i), (ii) and (iv) that the axial oscillation frequency fz and its higher order multiples 2 fz, 3 fz, 4 fz are observed at both the magnetic fields of 97g and 258g. but the intensities of fz, 2 fz, 3 fz and 4 fz decreased with decrease in power of rf field. but the modified cyclotron frequency and the side band frequency are almost remaining unchanged. (ii) storage voltage dependence: on comparing fig.3 (iii) and (i) we can observe that, as the storage voltage is increased, the dips corresponding to axial resonances shift towards the region of higher frequencies of resonances spectra but the dips corresponding to cyclotron frequency and the side band frequency resonances remain unchanged. as the axial frequency and storage voltage are related as fz ∝ √ v0 ( fz = 1 2π √ 4qv0 md2 ) , the shifts in dips corresponding to axial resonances are directly proportional to square root of storage voltages. but the pure cyclotron frequency is independent of storage voltage and hence the dips corresponding to cyclotron frequency and side band frequency ( fc = f ′c + fm) resonances remain unchanged. 77 dyavappa, b. m. / j. nig. soc. phys. sci. 4 (2022) 75–82 78 figure 3: motional resonances spectra showing dips for different magnetic fields, storage voltages and rf field powers of (i) 97g, 10v,-10dbm (ii) 97g, 10v, -100dbm (iii) 97g, 2v, -10dbm, (iv) 258g, 10v, -10dbm [11] (iii) magnetic field dependence: the resonances spectra show the dependence on magnetic fields as observed in figure 3 (i) at 97g and (iv) at 258g. the axial oscillation frequency fz and its higher order multiples 2 fz, 3 fz, 4 fz are observed at both the magnetic fields of 97g and 258g. the axial frequency ( fz ) is independent of magnetic field, but the modified cyclotron frequency and magnetron frequency show dependence on the magnetic field. 5. oscillators models (i) non-linear driven and damped harmonic oscillator model: we can develop a model of non-linear driven and damped harmonic oscillator for cloud of trapped electrons at very low rf field power of -100 dbm as shown in figure 4 [13, 14]. non-linear spring is one that exerts a variable rate of force per unit displacement due to extension. a nonlinear spring has a defined non-linear loaddisplacement function, being equivalent to strain energy absorption rate of the spring. this means that the load applied to the spring is not proportional to the amount of extension produced, based on the spring’s stretching rate as it is variable. a barrel spring’s stretching rate is not a constant and whose force is based on a variable increase per unit distance of travel due to the different outer diameters per coil. the barrel spring’s pitch is highly involved in spring’s force, travel capacities and so having a variation of pitch between spring’s coils will definitely alter springs force, turning this into a non-linear spring [14]. the equation of motion is given by mz̈ + αż − frestore = f0z sin ωt (8) the response at the third harmonic is generated by the cubic component of the restoring force (frestore). for a sinusoidal displacement z = z0 (cos ωt + i sin ωt) = z0eiωt, the restoring force is given by [14] frestore = −mω 2 z z−βz 3 = −mω2z z0e −iωt −βz30e i3ωt.(9) with sinusoidal rf drive at frequency f , this non-linear cloud of electrons will respond at both the drive frequency f , and at the third harmonic of the drive frequency generating the third harmonic 3 f . for non-linearities at the 78 dyavappa, b. m. / j. nig. soc. phys. sci. 4 (2022) 75–82 79 figure 4: the strengths of 3rd harmonics in motional resonances spectra for different magnetic fields, storage voltages and rf field powers of (i) 97g, 10v,-10dbm (ii) 97g, 10v,-100dbm (iii) 97g,2v,-10dbm, (iv) 258g,10v,-10dbm figure 5: magnetic field at the trap centre measured from cyclotron frequency from motional resonances spectra higher powers of zn, the trapped cloud of electrons will generate the nth harmonic at frequency n f . when we drive the simple oscillator with a sinusoidal driving force, figure 6: axial resonance frequency versus square root of storage voltage we get a response at the driving frequency. suppose that we drive a nonlinear system with a frequency fz, the output consists of the original frequency fz and its harmonics 2 fz, 3 fz, 4 fz, etc. the presence of harmonics depends on 79 dyavappa, b. m. / j. nig. soc. phys. sci. 4 (2022) 75–82 80 figure 7: a periodically forced barrel spring with a steel ball at the free end can be regarded as a model of non-linear oscillator the non-linearity in the cloud of electrons. the set of frequencies which are related to each other by integer ratios as fn = n fz is called the harmonic series [15]. (ii) driven duffing oscillator model: we can develop a model of certain forced and damped non-linear oscillator described by non-linear second-order differential equation named after george duffing as duffing oscillator, for high rf field power of -10 dbm as shown in figure 5 [16, 17]. the forced duffing oscillator is a system for the study of chaotic dynamics and development of analytical and experimental techniques for nonlinear systems. the doublewell duffing oscillator model considered here is a narrow ferromagnetic flexible rod whose one end is fixed to a rigid support attached to a driving system and the free end carrying a steel ball, and positioned between two permanent magnets while undergoing lateral forces. this mechanical system has several features of the model duffing oscillator such as a potential for the lateral position of the ball created by the two permanent magnets; nonautonomous forcing resulting in motion of the ball, potentially chaotically between the two wells; and mechanical dissipation within the ball and also from air resistance against the motion of ball [18, 19]. the non-linear oscillations arise from a forced and narrow flexible rod with steel ball-magnets system, moving in the magnetic fields of two permanent magnets. as the flexible rod with steel ball is excited with an ac current, a cubic magnet-position dependent force appears on it. the magnet-flexible rod with steel ball interactions, results in the non-linear equation of motion similar to trapped electrons, this is found to be a forced and damped cubic duffing oscillator moving in a quartic potential [20]. the motion of centre of mass of the cloud of electrons is described by non-linear second-order duffing differential equation [14, 15] mz̈ + αż + mω2z z + βz 3 = f0z sin ωt (10) the symbols α,β, and ω are constants such that α is damping coefficient, β is anharmonic coefficient and ωz is harmonic coefficient. the appearance of the non-linear term βz3 in duffing equation changes the situation comfigure 8: a periodically forced steel ball hung through a flexible rod, which is deflected alternately toward the two magnets can be regarded as a mechanical model of duffing oscillator when the force is very strong [21] pletely. for α > 0, the duffing oscillator can be interpreted as a forced oscillator with a non-linear spring whose restoring force is written as [21] frestore = −αz − βz3. if β > 0, the force is very strong, on the other hand if β < 0, the force is very weak. the potential of timeindependent part of force mω2z z + mβz 3 is [22, 23] v (z) = 1 2 mω2z z 2 + 1 4 mβz4 (11) 6. perturbative multipole terms in quadrupole potential now consider the quadrupole potential in penning trap on expanding equation (1) as φ(r,θ) = φ0 c0 + c1 ( r r0 ) (cos θ) + c2 ( r r0 )2 + 1 3 [cos 2θ− 1] (12) + c3 ( r r0 )3 8 5 [cos 3θ− 3 cos θ] +c4 ( r r0 )4 192 105 [cos 4θ− 80 cos 2θ− 7] . . .∞  the legendre coefficients c2 and c4 represent the quadrupole term and octupole term respectively. here c2 represents the pure quadrupole term which can be normalized to 1, and c4 is the most likely leading term of the perturbation. (i) quadrupole moment co-efficient (c2 ) the quadrupole moment co-efficient c2 can be calculated from axial frequency as [8, 11] ωz = √ 4ev0 (1 + c2) m ( r20 + 2z 2 0 ) , c2 = 4π f 2z m ( r20 + 2z 2 0 ) 4ev0 −1(13) if fz = 10×106 hz and v0 = 0.7v , then c2 = −0.2. from figure 6, c2 value can be calculated from slope 13.4×106 of cyclotron frequency v/s square root of trap voltage fz √ v0 = 13.4 × 106 (14) 80 dyavappa, b. m. / j. nig. soc. phys. sci. 4 (2022) 75–82 81 c2 = π2m ( r20 + 2z 2 0 ) e [ fz √ v0 ]2 − 1 = −0.1 (15) (ii) octupole moment co-efficient (c4) the octupole moment co-efficient c4 can be measured from the width of the axial resonance line. if ez is the energy in axial direction, fz is the axial frequency of oscillations, r = 5 mm is the amplitude of the axial oscillations, we consider r to be the radius of the electron cloud with the identical resonant frequency [9, 11]. ∆ωz ≈ 3 2 c4 ezωz ev0 , c4 ≈ ev0 6πez fz ∆ωz (16) if fz = 10×106 hz and v0 = 0.7 v , then ez = 1 2 meω 2 z e 2 ec = 4.486 × 10−20 j. therefore c4 ≈ 0.032. table 2: c2 and c4 simulation values for trap design and experimental values cn equation trap design values simulation experimental values c2 4π f 2z m(r20 +2z20) 4ev0 − 1 −0.17 −0.2 c4 ev0 6πez fz ∆ωz 0.03 0.032 7. conclusion the non-linear higher order motional resonances of the cloud of electrons are observed, that result from couplings in the motional degrees of freedom and characterize the perturbations in the trap. the likely presence of perturbative terms of quadrupole potential is identified by studying the motional resonances spectra and hence quadrupole moment co-efficient (c2), octupole moment co-efficient (c4) multipole terms are calculated. the resonances spectra show the dependence on magnetic fields, rf field powers and storage voltages. a periodically forced barrel spring with a steel ball at the free end can be regarded as a model of non-linear oscillator and a periodically forced steel ball hung through a flexible rod, which is deflected alternately toward the two magnets can be regarded as a mechanical model of duffing oscillator are developed for the observation of non-linear higher order axial oscillatory motional resonances that consist of the original frequency fz and its harmonics 2 fz, 3 fz, 4 fz, etc and the presence of harmonics depend on the non-linearity in the cloud of trapped electrons, for different rf field excitation powers of -10dbm and -100 dbm, respectively. appendix matlab code for generation of all possible integer triplets [11] for k = 0.02 a1=0.0220; b1=0.0218; a2=0.0220; b2=0.0218; h=1; for nc=0:1:5 for nm=-10:1:10 for nz=-10:1:10 a=2*(nc-nm)^2+4*nz^2; b=-2*(nc+nm)*nz; c=4*nc*nm; l=abs(nz)+abs(nc)+abs(nm); if(a>0) if((b^2-a*c)>=0) k1=(b+sqrt(b^2-a*c))/a; k2=(b-sqrt(b^2-a*c))/a; if ((k1<=0.022) && (k1>0.018))||((k2<=0.022) && (k2>0.018)) char(h, :)= [nc, nm, nz, l, k1]; h=h+1; end end end end end end acknowledgement i acknowledge prof. sharath ananthamurthy, department of physics, bangalore university, bengaluru, for the support to carrying out experiment and bashyam govindan, chennai, india, who helped to develop matlab code for generation of all possible integer triplets for desired values of k and reviewers of jnsps who assisted immensively in improving this article. references [1] p. paasche, t. valenzuela, d. biswas, c. angelescu, & g. werth , “individual and center of mass resonances in the motional spectrum of an electron cloud in a penning trap” eur. phys. j. d 18 (2002) 295. 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[23] m. rafikov, j. m. balthazar, a. m. tusset, & j. braz. “an optimal linear control design for nonlinear systems”, soc. mech. sci. & eng. 30 (2008). 82 j. nig. soc. phys. sci. 3 (2021) 459–468 journal of the nigerian society of physical sciences computational study of some three-step hybrid integrators for solution of third order ordinary differential equations mark i. modebeia, olumide o. olaiyaa,, ignatius p. ngwongwob, adepartment of mathematics programme, national mathematical centre, abuja, nigeria bdepartment of mathematics, federal university of technology, owerri, owerri, nigeria. abstract block of hybrid methods with three off-step points based on collocation technique are presented in this work for direct approximation of solution of third-order initial and boundary value problems (ivps and bvps). these off-step points are formulated such that they exist only on a single step at a time. hence, these points are shifted to three positions respectively in order to obtain three block different integrators for computational analysis. these analysis includes; order of the methods, consistency, stability and convergence, global error, number of functions evaluation and cpu time. the superiority of these methods over existing methods is established numerically on different test problems in literature. doi:10.46481/jnsps.2021.323 keywords: third order ordinary diferential equation, initial value problem, boundary value problems, block method, finite diferences, linear multistep hybrid method article history : received: 29 july 2021 received in revised form: 01 september 2021 accepted for publication: 13 september 2021 published: 29 november 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: t. latunde 1. introduction third-order problems considered in this work are of the form: y′′′ = f (x, y, y′, y′′), x ∈ [a, b] (1) with appropriate initial conditions and boundary conditions respetively. it is assumed that; the function f is continuous in [a, b] × r3. also, as discussed in [1, 2], the existence and uniqueness of the solution of (1) is assumed. the problem in (1) is assumed well posed and the numerical solution is the interest in this work. numerical methods for the solution of (1) with initial conditions or boundary conditions and for third order singularly perturbed boundary value problems are numerous. some include, email address: gmarc.ify@gmail.com, (olumide o. olaiya ) collocation method [3, 4, 5, 6], non-polynomial splines [7], quartic splines [8], adomian decomposition method [9], exponential quartic spline [10], quartic b-spline method [11], and many others. in this work, the collocation technique is employed. this method is found to be flexible and more efficient in that since it approximates the solution of (1) at several intra-points, its block formulations consists of several linear multistep methods which are required for direct solution of (1), such that it overcomes the overlapping of pieces of solutions and it is self starting. in this work we seek the numerical solution of (1) directly with three different three-step continuous hybrid block methods with three off-step points. these methods are developed using the collocation approach. 459 modebei et al. / j. nig. soc. phys. sci. 3 (2021) 459–468 460 2. derivation of the methods here, the derivation of a continuous implicit three mid intra-step hybrid block method is described, for the solution of (1) over the integration interval [a, b], πn ≡ {a = x0 < x1 < · · · < xn−1 < xn = b} with h the constant step-size, h = xi − xi−1, i = 1, 2, . . . , n. for the solution of (1) with initial condition, the method proposed in [12], three off-step points in the interval 0 < xr < xs < xt < 1 are given such that (r, s, t) = ( 38 , 5 8 , 7 8 ). also, an optimized two-step method with three off-step points proposed in [13] is such that r, s, v are in the interval 0 < r, s, t < 2. here, on the interval [xn, xn+3], we consider the following subintervals; [xn, xn+1], [xn+1, xn+2] and [xn+2, xn+3] where the points (r, s, u), (r′, s′, u′) and (r′′, s′′, u′′) are the assigned off-set points in each of the subinterval respectively as xn < xn+r < xn+s < xn+u < xn+1 < xn+2 < xn+3, < xn < xn+1 < xn+r′ < xn+s′ < xn+u′ < xn+2 < xn+3 and < xn < xn+1 < xn+2 < xn+r′′ < xn+s′′ < xn+u′′ < xn+3. where (r, s, u) = ( 14 , 1 2 , 3 4 ), (r ′, s′, u′) = ( 54 , 3 2 , 7 4 ) and (r ′′, s′′, u′′) = ( 94 , 5 2 , 11 4 ). consider the approximation w(x) of y(x) given by the polynomial y(x) ' w(x) = 9∑ i=0 ai x i . (2) and the third derivative y′′′(x) ' w′′′(x) = 9∑ i=3 ρii(i − 1)(i − 2)x i−3 (3) where ai are coefficients to be determined. 2.1. specifications 1 to obtain the method in the interval xn < xn+r < xn+s < xn+u < xn+1 < xn+2 < xn+3, we interpolate (2) at x = xn+i, i = 0, 1, 2 and collocating (3) at the points x = xn+ i4 , i = 0, 1, . . . , 4, 8, 12, a system of 10 equations with 10 unknown ai, i = 0, 1, . . . , 9. this system can be written in matrix form as 1 xn x2n x 3 n x 4 n x 5 n x 6 n · · · x 9 n 1 xn+1 x2n+1 x 3 n+1 x 4 n+1 x 5 n+1 x 6 n+1 · · · x 9 n+1 1 xn+2 x2n+2 x 3 n+2 x 4 n+2 x 5 n+2 x 6 n+2 · · · x 9 n+2 0 0 0 6 24xn 60x2n 120x 3 n · · · 504x 6 n 0 0 0 6 24xn+ 14 60x2 n+ 14 120x3 n+ 14 · · · 504x6 n+ 14 0 0 0 6 24xn+ 12 60x2 n+ 12 120x3 n+ 12 · · · 504x6 n+ 12 0 0 0 6 24xn+ 34 60x2 n+ 34 120x3 n+ 34 · · · 504x6 n+ 34 0 0 0 6 24xn+1 60x2n+1 120x 3 n+1 · · · 504x 6 n+1 0 0 0 6 24xn+2 60x2n+2 120x 3 n+2 · · · 504x 6 n+2 0 0 0 6 24xn+3 60x2n+3 120x 3 n+3 · · · 504x 6 n+3   a0 a1 a2 a3 a4 a5 a6 a7 a8 a9  =  yn yn+1 yn+2 h3 fn h3 fn+ 14 h3 fn+ 12 h3 fn+ 34 h3 fn+1 h3 fn+2 h3 fn+3  where y( j)n+i ' y ( j)(xn+i), fn+i ' f (xn+i, yn+i, y′n+i, y ′′ n+i), on solving the above system and obtaining the coefficients ai’s (not included here) which are then substituted into the polynomial (2), the following formula is obtained: w(x) = 2∑ i=0 αi(x)yn+i + h 3  3∑ i=0 βi(x) fn+i + 3∑ i=1 β i 4 (x) fn+ i4  (4) where the α and β are continuous coefficients (which are large expressions and are not included here, but can be easily obtained with the help of maple software). evaluating w(x) in (4) at the points x = xn+ 14 , xn+ 12 , xn+ 34 and xn+3 and after some simplifications, we obtain the following methods: yn+ 14 = 21yn 32 + 7yn+1 16 − 3yn+2 32 + h 3 ( 34621 fn 8847360 − 18589 f n+ 14 1013760 + 25783 f n+ 12 368640 − 46109 f n+ 34 829440 + 79271 fn+1 1474560 + 3019 fn+2 2949120 − 403 fn+3 18247680 ) yn+ 12 = 3yn 8 + 3yn+1 4 − yn+2 8 + h 3 ( 4997 fn 967680 − 5459 f n+ 14 194040 + 221 f n+ 12 2520 − 341 f n+ 34 4536 + 23099 fn+1 322560 + 3083 fn+2 2257920 − 941 fn+3 31933440 ) yn+ 32 = 5yn 32 + 15yn+1 16 − 3yn+2 32 + h 3 ( 23831 fn 6193152 − 218389 f n+ 14 9934848 + 159311 f n+ 12 2580480 − 343927 f n+ 34 5806080 + 221689 fn+1 4128768 + 36983 fn+2 36126720 − 45137 fn+3 2043740160 ) yn+3 = yn − 3yn+1 + 3yn+2 − h3 ( 149 fn 1890 − 8192 f n+ 14 24255 + 104 f n+ 12 315 + 512 f n+ 34 2835 − 251 fn+1 315 − 983 fn+2 2205 − 115 fn+3 12474 )  460 modebei et al. / j. nig. soc. phys. sci. 3 (2021) 459–468 461 if w′(x) of (4) is evaluated at x = x i 4 , i = 0(1)4, 8, 12, following formulae for approximating the first derivatives are obtained: y′n = − 3yn 2h + 2yn+1 h − yn+2 2h + h 2 ( 139 fn 5670 − 5056 f n+ 14 72765 + 1784 f n+ 12 4725 − 12352 f n+ 34 42525 + 2161 fn+1 7560 + 181 fn+2 33075 − 443 fn+3 3742200 ) y′ n+ 14 = − 5yn 4h + 3yn+1 2h − yn+2 4h + h 2 ( 916957 fn 92897280 − 175423 f n+ 14 2661120 + 1696777 f n+ 12 9676800 − 234391 f n+ 34 1555200 + 887681 fn+1 6193152 + 422549 fn+2 154828800 − 902123 fn+3 15328051200 ) y′ n+ 12 = − yn h + yn+1 h − h 2 ( 73 fn 725760 + 577 f n+ 14 72765 + 121 f n+ 12 4725 + 337 f n+ 34 42525 + 5 fn+1 48384 − fn+2 8467200 + fn+3 119750400 ) y′ n+ 32 = − 3yn 4h + yn+1 2h + yn+2 4h − h 2 ( 958907 fn 92897280 − 1048031 f n+ 14 18627840 + 1702583 f n+ 12 9676800 − 219329 f n+ 34 1555200 + 4451939 fn+1 30965760 + 2958517 fn+2 1083801600 − 128971 fn+3 2189721600 ) y′n+1 = − yn 2h + yn+2 2h − h 2 ( 467 fn 22680 − 8768 f n+ 14 72765 + 1516 f n+ 12 4725 + 14528 f n+ 34 42525 + 533 fn+1 1890 + 1447 fn+2 264600 − 221 fn+3 1871100 ) y′n+2 = yn 2h − 2yn+1 h + 3yn+2 2h + h 2 ( 607 fn 5670 − 6592 f n+ 14 10395 + 7304 f n+ 12 4725 − 10816 f n+ 34 6075 + 7951 fn+1 7560 + 208 fn+2 4725 − 2693 fn+3 3742200 ) y′n+3 = 3yn 2h − 4yn+1 h + 5yn+2 2h − h 2 ( 22889 fn 22680 − 56384 f n+ 14 10395 + 52396 f n+ 12 4725 − 442688 f n+ 34 42525 + 169 fn+1 54 − 44249 fn+2 37800 − 105641 fn+3 1871100 )  similarly, evaluating w′′(x) of (4), at the points x = x i 4 , i = 0(1)4, 8, 12, we obtain the formulae which approximates the second derivatives: y′′n = yn h2 − 2yn+1 h2 + yn+2 h2 − h ( 113 fn 945 + 64 f n+ 14 693 + 232 f n+ 12 315 − 1472 f n+ 34 2835 + 1411 fn+1 2520 − fn+2 90 − 61 fn+3 249480 ) y′′ n+ 14 = yn h2 − 2yn+1 h2 + yn+2 h2 − h ( 23999 fn 645120 − 59981 f n+ 14 388080 + 2921 f n+ 12 3360 − 8929 f n+ 34 15120 + 53261 fn+1 92160 + 16361 fn+2 1505280 − 4967 fn+3 21288960 ) y′′ n+ 12 = yn h2 − 2yn+1 h2 + yn+2 h2 − h ( 145 fn 3456 − 6452 f n+ 14 24255 + 73 f n+ 12 105 − 1564 f n+ 34 2835 + 22973 fn+1 40320 + 1031 fn+2 94080 − 947 fn+3 3991680 ) y′′ n+ 32 = yn h2 − 2yn+1 h2 + yn+2 h2 − h ( 77173h fn 1935360 − 96433h f n+ 14 388080 + 5561h f n+ 12 10080 − 4433h f n+ 34 6480 + 374419h fn+1 645120 + 49123h fn+2 4515840 − 14941h fn+3 63866880 ) y′′n+1 = yn h2 − 2yn+1 h2 + yn+2 h2 − h ( 107 fn 2520 − 6464 f n+ 14 24255 + 64 f n+ 12 105 − 832 f n+ 34 945 − 61 fn+1 126 + 13 fn+2 1176 − fn+3 4158 ) y′′n+2 = yn h2 − 2yn+1 h2 + yn+2 h2 + h ( 374 fn 945 − 56512 f n+ 14 24255 + 584 f n+ 12 105 − 17984 f n+ 34 2835 + 1243 fn+1 360 + 391 fn+2 1470 − 739 fn+3 249480 ) y′′n+3 = yn h2 − 2yn+1 h2 + yn+2 h2 − h ( 5165 fn 1512 − 453952 f n+ 14 24255 + 1792 f n+ 12 45 − 112064 f n+ 34 2835 − 9607 fn+1 630 − 34877 fn+2 17640 − 16573 fn+3 62370 )  2.2. specification 2 to obtain the method in the interval < xn < xn+1 < xn+r′ < xn+s′ < xn+u′ < xn+2 < xn+3, we interpolate (2) at x = xn+i, i = 0, 1, 2 and collocating (3) at the points x = xn+ i4 , i = 0, 4(1)8, 12, a system of 10 equations with 10 unknown ai, i = 0, 1, . . . , 9 in matrix form is given as  1 xn x2n x 3 n x 4 n x 5 n x 6 n · · · x 9 n 1 xn+1 x2n+1 x 3 n+1 x 4 n+1 x 5 n+1 x 6 n+1 · · · x 9 n+1 1 xn+2 x2n+2 x 3 n+2 x 4 n+2 x 5 n+2 x 6 n+2 · · · x 9 n+2 0 0 0 6 24xn 60x2n 120x 3 n · · · 504x 6 n 0 0 0 6 24xn+1 60x2n+1 120x 3 n+1 · · · 504x 6 n+1 0 0 0 6 24xn+ 54 60x2 n+ 54 120x3 n+ 54 · · · 504x6 n+ 54 0 0 0 6 24xn+ 32 60x2 n+ 32 120x3 n+ 32 · · · 504x6 n+ 32 0 0 0 6 24xn+ 74 60x2 n+ 74 120x3 n+ 74 · · · 504x6 n+ 74 0 0 0 6 24xn+2 60x2n+2 120x 3 n+2 · · · 504x 6 n+2 0 0 0 6 24xn+3 60x2n+3 120x 3 n+3 · · · 504x 6 n+3   a0 a1 a2 a3 a4 a5 a6 a7 a8 a9  =  yn yn+1 yn+2 h3 fn h3 fn+1 h3 fn+ 54 h3 fn+ 32 h3 fn+ 74 h3 fn+2 h3 fn+3  on solving the above system and obtaining the coefficients ai’s (not included here), are then substituted into the polynomial (2), the following formula is obtained: w(x) = 2∑ i=0 αi(x)yn+i + h 3  3∑ i=0 βi(x) fn+i + 7∑ i=5 β i 4 (x) fn+ i4  (5) where the α and β are continuous coefficients (which are large expressions and are not included here, but can be easily obtained with the help of maple software). evaluating w(x) in (5) at the points x = xn+ 54 , xn+ 32 , xn+ 74 and xn+3 and after some simplifications, we obtain the following methods: 461 modebei et al. / j. nig. soc. phys. sci. 3 (2021) 459–468 462 yn+ 54 = − 3yn 32 + 15yn+1 16 + 5yn+2 32 − h 3 ( 16453 fn 27095040 + 311963 fn+1 4128768 − 588473 f n+ 54 4515840 + 240025 f n+ 32 1548288 − 361883 f n+ 74 4515840 + 38035 fn+2 2064384 − 45137 fn+3 433520640 ) yn+ 32 = − yn 8 + 3yn+1 4 + 3yn+2 8 − h 3 ( 1097 fn 1354752 + 32509 fn+1 322560 − 2999 f n+ 54 17640 + 401 f n+ 32 1890 − 373 f n+ 74 3528 + 7939 fn+2 322560 − 941 fn+3 6773760 ) yn+ 74 = − 3yn 32 + 7yn+1 16 + 21yn+2 32 − h 3 ( 37607 fn 61931520 + 111511 fn+1 1474560 − 16343 f n+ 54 129024 + 180517 f n+ 32 1105920 − 9869 f n+ 74 129024 + 54527 fn+2 2949120 − 403 fn+3 3870720 ) yn+3 = yn − 3yn+1 + 3yn+2 + h3 ( 71 fn 13230 + 316 fn+1 315 − 4864 f n+ 54 2205 + 3208 f n+ 32 945 − 4864 f n+ 74 2205 + 316 fn+2 315 + 71 fn+3 13230 )  if w′(x) of (4) is evaluated at x = x i 4 , i = 0, 4(1)8, 12, the following formulae for approximating the first derivatives are obtained: y′n = − 3yn 2h + 2yn+1 h − yn+2 2h + h 2 ( 6043 fn 198450 + 13337 fn+1 7560 − 135488 f n+ 54 33075 + 2600 f n+ 32 567 − 83648 f n+ 74 33075 + 110 fn+2 189 − 2693 fn+3 793800 ) y′n+1 = − yn 2h + yn+2 2h − h 2 ( 2573 fn 793800 + 377 fn+1 945 − 23872 f n+ 54 33075 + 332 f n+ 32 405 − 14272 f n+ 74 33075 + 149 fn+2 1512 − 221 fn+3 396900 ) y′ n+ 54 = − yn 4h − yn+1 2h + 3yn+2 4h − h 2 ( 5264363 fn 3251404800 + 6257533 fn+1 30965760 − 399871 f n+ 54 1209600 + 493205 f n+ 32 1161216 − 1789537 f n+ 74 8467200 + 304673 fn+2 6193152 − 128971 fn+3 464486400 ) y′ n+ 32 = − yn+1 h + yn+2 h − h 2 ( fn 25401600 + 23 fn+1 241920 + 263 f n+ 54 33075 + 29 f n+ 32 1134 + 263 f n+ 74 33075 + 23 fn+2 241920 + fn+3 25401600 ) y′ n+ 74 = yn 4h − 3yn+1 2h + 5yn+2 4h + h 2 ( 5265037 fn 3251404800 + 6242651 fn+1 30965760 − 2879903 f n+ 54 8467200 + 2461463 f n+ 32 5806080 − 1870343 f n+ 74 8467200 + 1508483 fn+2 30965760 − 902123 fn+3 3251404800 ) y′n+2 = yn 2h − 2yn+1 h + 3yn+2 2h + h 2 ( 643 fn 198450 + 3047 fn+1 7560 − 22208 f n+ 54 33075 + 2488 f n+ 32 2835 − 12608 f n+ 74 33075 + 97 fn+2 945 − 443 fn+3 793800 ) y′n+3 = 3yn 2h − 4yn+1 h + 5yn+2 2h + h 2 ( 3697 fn 793800 + 1972 fn+1 945 − 27584 f n+ 54 4725 + 27436 f n+ 32 2835 − 244928 f n+ 74 33075 + 24713 fn+2 7560 + 2183 fn+3 56700 ) ,  similarly, evaluating w′′(x) of (4), at the points x = x i 4 , i = 0, 4(1)8, 12, we obtain the formulae which approximates the second derivatives: y′′n = yn h2 − 2yn+1 h2 + yn+2 h2 − h ( 278 fn 1323 + 16091 fn+1 2520 − 35008 f n+ 54 2205 + 488 f n+ 32 27 − 22336 f n+ 74 2205 + 1481 fn+2 630 − 739 fn+3 52920 ) y′′n+1 = yn h2 − 2yn+1 h2 + yn+2 h2 + h ( 23 fn 3528 + 13 fn+1 18 − 1216 f n+ 54 735 + 512 f n+ 32 315 − 1984 f n+ 74 2205 + 169 fn+2 840 − fn+3 882 ) y′′ n+ 54 = yn h2 − 2yn+1 h2 + yn+2 h2 + h ( 2503 fn 387072 + 523829 fn+1 645120 − 5633 f n+ 54 3920 + 9313 f n+ 32 6048 − 4357 f n+ 74 5040 + 41777 fn+2 215040 − 14941 fn+3 13547520 ) y′′ n+ 32 = yn h2 − 2yn+1 h2 + yn+2 h2 + h ( 5491 fn 846720 + 32443 fn+1 40320 − 580 f n+ 54 441 + 1604 f n+ 32 945 − 1964 f n+ 74 2205 + 1601 fn+2 8064 − 947 fn+3 846720 ) y′′ n+ 74 = yn h2 − 2yn+1 h2 + yn+2 h2 + h ( 5843 fn 903168 + 521837 fn+1 645120 − 3155 f n+ 54 2352 + 2671 f n+ 32 1440 − 27127 f n+ 74 35280 + 41113 fn+2 215040 − 4967 fn+3 4515840 ) y′′n+2 = yn h2 − 2yn+1 h2 + yn+2 h2 + h ( 43 fn 6615 + 2021 fn+1 2520 − 64 f n+ 54 49 + 1672 f n+ 32 945 − 1216 f n+ 74 2205 + 59 fn+2 210 − 61 fn+3 52920 ) y′′n+3 = yn h2 − 2yn+1 h2 + yn+2 h2 − h ( 13 fn 1512 − 2113 fn+1 630 + 5440 f n+ 54 441 − 20288 f n+ 32 945 + 5696 f n+ 74 315 − 18619 fn+2 2520 − 2851 fn+3 13230 )  2.3. specification 3 finally, obtaining the method in the interval xn < xn+1 < xn+2 < xn+r′′ < xn+s′′ < xn+u′′ < xn+3, we interpolate (2) at x = xn+i, i = 0, 1, 2 and collocating (3) at the points x = xn+ i4 , i = 0, 4, 8(1)12, a system of 10 equations with 10 unknown ai, i = 0, 1, . . . , 9 in matrix form is given as 1 xn x2n x 3 n x 4 n x 5 n x 6 n · · · x 9 n 1 xn+1 x2n+1 x 3 n+1 x 4 n+1 x 5 n+1 x 6 n+1 · · · x 9 n+1 1 xn+2 x2n+2 x 3 n+2 x 4 n+2 x 5 n+2 x 6 n+2 · · · x 9 n+2 0 0 0 6 24xn 60x2n 120x 3 n · · · 504x 6 n 0 0 0 6 24xn+1 60x2n+1 120x 3 n+1 · · · 504x 6 n+1 0 0 0 6 24xn+2 60x2n+2 120x 3 n+2 · · · 504x 6 n+2 0 0 0 6 24xn+ 94 60x2 n+ 94 120x3 n+ 94 · · · 504x6 n+ 94 0 0 0 6 24xn+ 52 60x2 n+ 52 120x3 n+ 52 · · · 504x6 n+ 52 0 0 0 6 24xn+ 114 60x2 n+ 114 120x3 n+ 114 · · · 504x6 n+ 114 0 0 0 6 24xn+3 60x2n+3 120x 3 n+3 · · · 504x 6 n+3   a0 a1 a2 a3 a4 a5 a6 a7 a8 a9  =  yn yn+1 yn+2 h3 fn h3 fn+1 h3 fn+2 h3 fn+ 94 h3 fn+ 52 h3 fn+ 114 h3 fn+3  thus, solving the above system and obtaining the coefficients ai’s, then substituted into the polynomial (2), the following formula is obtained: w(x) = 2∑ i=0 αi(x)yn+i + h 3  3∑ i=0 βi(x) fn+i + 11∑ i=9 β i 4 (x) fn+ i4  (6) 462 modebei et al. / j. nig. soc. phys. sci. 3 (2021) 459–468 463 where the α and β are continuous coefficients (which are large expressions and are not included here, but can be easily obtained with the help of maple software). evaluating w(x) in (24) at the points x = xn+ 94 , xn+ 52 , xn+ 114 and xn+3 and after some simplifications, we obtain the following methods: yn+ 94 = 5yn 32 − 9yn+1 16 + 45yn+2 32 + h 3 ( 996379 fn 681246720 + 826499 fn+1 12042240 + 97453 fn+2 1376256 + 60029 f n+ 94 1935360 − 97477 f n+ 52 860160 + 247559 f n+ 114 3311616 − 33373 fn+3 2064384 ) yn+ 52 = 3yn 8 − 5yn+1 4 + 15yn+2 8 + h 3 ( 111341 fn 31933440 + 374389 fn+1 2257920 + 14657 fn+2 64512 + 169 f n+ 94 22680 − 533 f n+ 52 2520 + 6007 f n+ 114 38808 − 6721 fn+3 193536 ) yn+ 114 = 21yn 32 − 33yn+1 16 + 77yn+2 32 + h 3 ( 10073 fn 1658880 + 6018419 fn+1 20643840 + 691801 fn+2 1474560 − 10439 f n+ 94 165888 − 21131 f n+ 52 73728 + 154817 f n+ 114 645120 − 492349 fn+3 8847360 ) yn+3 = yn − 3yn+1 + 3yn+2 + h3 ( 115 fn 12474 + 983 fn+1 2205 + 251 fn+2 315 − 512 f n+ 94 2835 − 104 f n+ 52 315 + 8192 f n+ 114 24255 − 149 fn+3 1890 )  if w′(x) of (4) is evaluated at x = x i 4 , i = 0, 4(1)8, 12, following methods for approximating the first derivatives are obtained: y′n = − 3yn 2h + 2yn+1 h − yn+2 2h + h 2 ( 39883 fn 935550 + 132803 fn+1 264600 − 4087 fn+2 945 + 454208 f n+ 94 42525 − 50056 f n+ 52 4725 + 357824 f n+ 114 72765 − 20207 fn+3 22680 ) y′n+1 = − yn 2h + yn+2 2h − h 2 ( 259 fn 48600 + 11833 fn+1 66150 − 4939 fn+2 7560 + 71872 f n+ 94 42525 − 8084 f n+ 52 4725 + 5312 f n+ 114 6615 − 1661 fn+3 11340 ) y′n+2 = yn 2h − 2yn+1 h + 3yn+2 2h + h 2 ( 4423 fn 935550 + 8219 fn+1 37800 + 22 fn+2 189 + 10688 f n+ 94 42525 − 328 f n+ 52 675 + 3008 f n+ 114 10395 − 1361 fn+3 22680 ) y′ n+ 94 = 3yn 4h − 5yn+1 2h + 7yn+2 4h + h 2 ( 106886797 fn 15328051200 + 359414603 fn+1 1083801600 + 14053789 fn+2 30965760 + 60743 f n+ 94 10886400 − 4098743 f n+ 52 9676800 + 5766623 f n+ 114 18627840 − 6451643 fn+3 92897280 ) y′ n+ 52 = yn h − 3yn+1 h + 2yn+2 h + h 2 ( 1103999 fn 119750400 + 3774721 fn+1 8467200 + 192743 fn+2 241920 − 8017 f n+ 94 42525 − 1681 f n+ 52 4725 + 23999 f n+ 114 72765 − 57289 fn+3 725760 ) y′ n+ 114 = 5yn 4h − 7yn+1 2h + 9yn+2 4h + h 2 ( 25105411 fn 2189721600 + 606913043 fn+1 1083801600 + 7056257 fn+2 6193152 − 4098337 f n+ 94 10886400 − 2296823 f n+ 52 9676800 + 6636359 f n+ 114 18627840 − 8237603 fn+3 92897280 ) y′n+3 = 3yn 2h − 4yn+1 h + 5yn+2 2h + h 2 ( 51307 fn 3742200 + 6371 fn+1 9450 + 11197 fn+2 7560 − 23872 f n+ 94 42525 − 556 f n+ 52 4725 + 4544 f n+ 114 10395 − 1063 fn+3 11340 )  similarly, evaluating w′′(x) of (4), at the points x = x i 4 , i = 0, 4(1)8, 12, we obtain the formulae which approximates the second derivatives: y′′n = yn h2 − 2yn+1 h2 + yn+2 h2 − h ( 7999 fn 31185 + 3859 fn+1 2520 − 10109 fn+2 630 + 112576 f n+ 94 2835 − 2488 f n+ 52 63 + 12736 f n+ 114 693 − 25229 fn+3 7560 ) y′′n+1 = yn h2 − 2yn+1 h2 + yn+2 h2 + h ( 1013 fn 83160 + 793 fn+1 4410 − 2231 fn+2 840 + 832 f n+ 94 135 − 1856 f n+ 52 315 + 21568 f n+ 114 8085 − 299 fn+3 630 ) y′′n+2 = yn h2 − 2yn+1 h2 + yn+2 h2 + h ( 8 fn 891 + 8059 fn+1 17640 + 269 fn+2 210 − 3008 f n+ 94 2835 + 88 f n+ 52 315 + 192 f n+ 114 2695 − 55 fn+3 1512 ) y′′ n+ 94 = yn h2 − 2yn+1 h2 + yn+2 h2 + h ( 52169 fn 5806080 + 2062307 fn+1 4515840 + 888467 fn+2 645120 − 39223 f n+ 94 45360 + 319 f n+ 52 1440 + 3149 f n+ 114 35280 − 75403 fn+3 1935360 ) y′′ n+ 52 = yn h2 − 2yn+1 h2 + yn+2 h2 + h ( 11951 fn 1330560 + 128917 fn+1 282240 + 18367 fn+2 13440 − 692 f n+ 94 945 + 23 f n+ 52 63 + 116 f n+ 114 1617 − 1487 fn+3 40320 ) y′′ n+ 114 = yn h2 − 2yn+1 h2 + yn+2 h2 + h ( 573899 fn 63866880 + 2062267 fn+1 4515840 + 59125 fn+2 43008 − 4997 f n+ 94 6480 + 1087 f n+ 52 2016 + 7899 f n+ 114 43120 − 80579 fn+3 1935360 ) y′′n+3 = yn h2 − 2yn+1 h2 + yn+2 h2 + h ( 2239 fn 249480 + 403 fn+1 882 + 3419 fn+2 2520 − 1984 f n+ 94 2835 + 128 f n+ 52 315 + 10432 f n+ 114 24255 + 11 fn+3 270 )  3. analysis of the method 3.1. local truncation error and order the linear difference operators associated with the formula in (4) is of the form l[z(x); h] ≡ w(x) − 2∑ j=0 α jzn+ j + h 3  3∑ j=0 βi(x)z ′′′ n+ j + 3∑ j=1 β j 4 z′′′ n+ j4  (7) same can be formulated for (5) and (24). the taylor series expansion of (7) around x yields l[z(x); h] = c0z(x) + c1hz ′(x) + c2h 2z′′(x) + · · · + c ph pz( p)(x) + o(h(p+1)). (8) where the ci are constants. suppose the first p + 3 terms vanishes, that is c0 = c1 = c2 = · · · = c p+2 = 0 and c p+3 , 0, then l[z(x); h] = c p+3h p+3y(p+3)(x) + o(hp+4) (9) 463 modebei et al. / j. nig. soc. phys. sci. 3 (2021) 459–468 464 here (9) is the local truncation error (lte) for (7) and equivalently for (4) and p is the order. the lte is the amount by which the exact solution of the ode fails to satisfy the corresponding difference operator. the method in (4) (and similarly for (5) and (24)) is said to be consistent if p > 1 (see [14]). the ltes of (2.1) are cyn+1/4 = 2351 332943851520 y(10)(xn)h 10 + o(h11) (10) cyn+1/2 = 941 20808990720 y(10)(xn)h 10 + o(h11) (11) cyn+3/4 = 459 4110417920 y(10)(xn)h 10 + o(h11) (12) cyn+3 = −27 1003520 y(10)(xn)h 10 + o(h11) (13) the method that approximates the first and second derivative in (2.1) and (2.1) respectively can be computed in a similar fashion. the ltes of (2.2) are cyn+5/4 = 4667875 66588770304 y(10)(xn)h 10 + o(h11) (14) cyn+3/2 = 20439 183500800 y(10)(xn)h 10 + o(h11) (15) cyn+7/4 = 5519899 33973862400 y(10)(xn)h 10 + o(h11) (16) cyn+3 = 2781 5017600 y(10)(xn)h 10 + o(h11) (17) the method that approximates the first and second derivative in (2.2) and (2.2) respectively can be computed in a similar fashion. finally, the ltes of (2.3) are cyn+9/4 = 39621879 20552089600 y(10)(xn)h 10 + o(h11) (18) cyn+5/2 = 10239625 4161798144 y(10)(xn)h 10 + o(h11) (19) cyn+11/4 = 5089372651 1664719257600 y(10)(xn)h 10 + o(h11) (20) cyn+3 = 18657 5017600 + o[h]y(10)(xn)h 10 + o(h11) (21) (22) the method that approximates the first and second derivative in (2.3) and (7) respectively can be computed in a similar fashion. the methods have order p = 7. 3.2. zero-stability and convergence a numerical method is zero-stable if the solutions remain bounded as h → 0. following the procedure in [6, 15], to show the zero-stability the block method (2.1)-(2.1) (similarly for (2.2)-(2.2) and (2.3)-(7) respectively) may be rewritten in a form such that y(k) n+ j4 , are on the left hand side. so that the method in matrix form becomes a j0y ( j) τ+1 = a j 1y ( j) τ−1 + h 3(b j1 f ( j) τ+1 + b j 0 f ( j) τ−1), j = 0, 1, 2 (23) as h → 0, (8) becomes a j0y ( j) τ+1 = a j 1y ( j) τ−1, j = 0, 1, 2 (24) where y ( j) τ+1 = ( y( j)1/4, y ( j) 1/2, y ( j) 3/4, y ( j) 1 , y ( j) 2 , y ( j) 3 )t , y j µ−1 = ( y( j) −3/4, y ( j) −1/2, y ( j) −1/4, y ( j) 1 , y ( j) 2 , y ( j) −1/4 )t , a j0 is an 18 × 18 identity matrix given by a j0 =  a10 0 0 0 a20 0 0 0 a30  with a10 = a 1 0 = a 2 0 = i6×6; a j1 =  a11 0 0 0 a21 0 0 0 a31  with a11 = a 2 1 = a 3 1 =  1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0  . the characteristic polynomial of each of the matrix a11 (similarly for a21 and a 3 1) is given as λ 5(λ − 1) = 0. the roots of the characteristic polynomial are λ j = 0, for j = 1, . . . , 5 and λ6 = 1. since the roots of the characteristic polynomial are all zero but one, whose modulus is 1 (see [14]), then the method ((2.1)-(2.1)) is zero stable. the method is thus convergent by the following theorem. theorem 3.1. henrici [16]. a linear multistep method is said to be convergent if it is consistent (with order p ≥ 1) and it is zero-stable. 3.3. computational procedure the methods in (2.1)-(2.1) (equivalently for the methods in (2.2)-(2.2) and (2.3)-(7) respectively) are executed in single block on the interval [xn, xn+3], where n = 0, 3, . . . , n − 3, n, the number of subintervals must be divisible by 3 in order to obtain the last points on the integration interval b = xn . thus, the methods in (2.1)-(2.1) are put in the form z(y) = 0 such that the system is solved by the newtons method of the form y j+1 = y j − (j j)−1z j where y = (y′0, y ′′ 0 , y1/4, y ′ 1/4, y ′′ 1/4, y1/2, y ′ 1/2, y ′′ 1/2, y3/4, y ′ 3/4, y ′′ 3/4, y1, . . . , . . . , y ′′ n ) j is the jacobian matrix of z. the starting values used in the newtons method are the approximations given by the taylor series yn+ j4 = yn + j h 4 y ′ n + 1 2 ( j h4 )2 y′′n + 1 6 ( j h4 )3 fn y′ n+ j4 = y′n + j h 4 y ′′ n + 1 2 ( j h4 )2 fn y′′ n+ j4 = y′′n + j h 4 fn (26) for j = 0(1)4, 8, 12 464 modebei et al. / j. nig. soc. phys. sci. 3 (2021) 459–468 465 æ æ æ æ à à à à ì ì ì ì 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 10-12 10-10 10-8 10-6 cpu time m ax e rr or ì s3hi3 à s3hi2 æ s3hi1 figure 1. efficiency plot for s3hi1, s3hi2 and s3hi3 showing the maximum error against the cpu time. 4. numerical test problems and results numerical problems are presented to show the accuracy of the three specification of methods proposed. this comparison is done based on the maximum error against the machine cost (cpu time). this is to ascertain which of the proposed methods performed favourably and in comparison to other methods in literature. the methods used are denoted as follows: • s3hi1: proposed method described in specification 1; • s3hi2: proposed method described in specification 2; • s3hi3: proposed method described in specification 3. • mae: maximum absolute error obtained in [10]; • s3hbm: 3-step hybrid block method described in [6]; • ostbm: step block method described in [3]; • ts: total nummber of sub intervals ts on [a,b]; • n: number of steps. problem 1.. consider the third-order singularly perturbed boundary value problem discussed in [10] −�y′′′ + y = 81�2 cos(3x) + 3� sin(3t), x ∈ [0, 1] y(0) = 0, y(1) = 3� sin(3), y′(0) = 9� (27) whose exact solution is y(x) = 3� sin(3x). figure 1 shows the efficiency curve of maximum error against the cpu time for the methods s3hi1, s3hi2 and s3hi3. the maximum error are determined for n = 6, 12, 24 and 48 and the cpu time for each n, for s3hi1, s3hi2 and s3hi3 respectively. the curve reveals that s3hi2 performed better in terms of error than s3hi1 and s3hi3 but slightly costly in terms of its cpu time. table 1 confirms the output of the efficiency curve for this problem. the s3hi2 performed better compared to other methods. æ æ æ æ à à à à ì ì ì ì 0.00 0.05 0.10 0.15 10-17 10-15 10-13 10-11 cpu time m ax e rr or ì s3hi3 à s3hi2 æ s3hi1 figure 2. efficiency plot for s3hi1, s3hi2 and s3hi3 showing the maximum error against the cpu time. problem 2.. consider the ivp discussed in [12] y′′′ = 3 sin(x), x ∈ [0, 3] y(0) = 1, y′(0) = 0, y′′(0) = 2 (28) whose exact solution is y(x) = 3 cos(x) + x 2 2 . the figure 2 shows the efficiency curve of s3hi1, s3hi2 and s3hi3 where the maximum error obtained in each care is plotted against the cpu time. the maximum error are determined for n=6,12,24 and 48 and the cpu time for each n, for s3hi1, s3hi2 and s3hi3 respectively. the efficiency curve shows that s3hi3 performed better in terms of error than s3hi1 and s3hi3 which have slightly better cpu time than s3hi2. table 2 shows errors obtain for problem 2 with h = 0.1 using the methods s3hi1, s3hi2, s3hi3 and a method of order 8 in [12]. the table agrees with the efficiency curve which indicate that s3hi1 and s3hi3 have the same performance while s3hi2 performed better when compared to its counterparts and the method of order 7 in [12]. problem 3.. consider the ivp discussed in [5] y′′′ + 4y′ = x, x ∈ [0, 1] y(0) = y′(0) = 0, y′′(0) = 1 (29) whose exact solution is y(x) = 316 (1 cos(2x)) + x2 8 . table 3 shows the maximum error obtained for different values of n at the point xn = b. the results shows the superiority of the proposed methods when the number of steps (subinterval) considered are less than those used in [5]. figure 3 is the efficiency curve showing the different methods and their individual performance in the absolute errors obtained against the cpu time in each case. the methods are relatively the same in terms of errors but s3hi2 exhibits more cpu time for n = 48. problem 4.. consider the nonlinear bvp discussed in [3]. y′′′ + 2e−3y = 4(1 + x)−3, x ∈ [0, 1] y(0) = 0, y′(0) = 1, y′(1) = 0.5 (30) 465 modebei et al. / j. nig. soc. phys. sci. 3 (2021) 459–468 466 table 1. comparison of maximum errors (me) for problem 1 methods � n = 10 n = 20 n = 40 s3hi1 1/16 1.49 × 10−8 1.53 × 10−10 1.87 × 10−12 1/32 7.78 × 10−9 8.02 × 10−11 1.01 × 10−12 1/64 4.22 × 10−9 4.41 × 10−11 5.76 × 10−13 s3hi2 1/16 3.65 × 10−9 1.35 × 10−11 6.47 × 10−14 1/32 2.08 × 10−9 7.59 × 10−12 3.73 × 10−14 1/64 1.31 × 10−9 4.70 × 10−12 2.44 × 10−14 s3hi3 1/16 4.70 × 10−8 2.74 × 10−10 2.46 × 10−12 1/32 2.60 × 10−8 1.48 × 10−10 1.34 × 10−12 1/64 1.57 × 10−8 8.62 × 10−11 7.93 × 10−13 mae in [? ] 1/16 2.32 × 10−4 6.12 × 10−5 1.52 × 10−5 1/32 9.77 × 10−5 2.59 × 10−5 6.45 × 10−6 1/64 3.78 × 10−5 1.00 × 10−6 2.50 × 10−6 table 2. comparison of errors for problem 2 x error in s3hi1 error in s3hi2 error in s3hi3 error in [12] 0.1 1.2458 × 10−16 2.1548 × 10−17 5.1274 × 10−16 4.1078 × 10−15 0.2 2.2580 × 10−15 1.7852 × 10−17 4.4770 × 10−15 1.6875 × 10−14 0.3 1.2358 × 10−15 4.2878 × 10−17 1.2258 × 10−15 5.0848 × 10−14 0.4 5.2145 × 10−15 3.2154 × 10−17 1.5551 × 10−15 1.1779 × 10−13 0.5 3.0125 × 10−15 1.3358 × 10−15 1.6712 × 10−15 2.4081 × 10−13 0.6 2.1258 × 10−15 1.2015 × 10−15 7.1255 × 10−15 4.3709 × 10−13 0.7 1.1125 × 10−14 1.3287 × 10−15 1.0021 × 10−14 7.3708 × 10−13 0.8 7.1258 × 10−14 8.2148 × 10−15 4.2014 × 10−14 1.1662 × 10−12 0.9 2.2158 × 10−14 3.2358 × 10−15 2.1584 × 10−14 1.7587 × 10−12 1.0 1.0012 × 10−14 1.9985 × 10−15 1.8877 × 10−14 2.5466 × 10−12 table 3. comparison of maximum errors for problem 3 s3hi1 s3hi2 s3hi3 method of order 7 in [5] b ts me me me ts me 5 30 1.25 × 10−13 5.58 × 10−13 5.54 × 10−13 46 1.20 × 10−10 45 5.21 × 10−13 1.12 × 10−13 5.02 × 10−13 56 3.69 × 10−11 60 2.35 × 10−13 3.87 × 10−13 1.11 × 10−12 88 2.44 × 10−12 10 60 4.25 × 10−13 1.57 × 10−13 8.41 × 10−12 61 5.54 × 10−09 75 1.22 × 10−13 1.01 × 10−13 6.63 × 10−12 91 5.04 × 10−10 90 8.14 × 10−13 1.87 × 10−13 6.74 × 10−12 136 4.53 × 10−11 15 75 1.66 × 10−12 9.81 × 10−13 3.42 × 10−12 76 2.67 × 10−08 90 2.25 × 10−12 1.77 × 10−13 1.08 × 10−12 110 2.91 × 10−09 105 1.11 × 10−12 2.31 × 10−13 9.52 × 10−11 180 1.52 × 10−10 20 90 2.58 × 10−11 3.27 × 10−13 2.57 × 10−11 91 5.29 × 10−08 105 2.78 × 10−10 1.14 × 10−13 1.21 × 10−10 129 6.54 × 10−09 120 2.36 × 10−11 2.48 × 10−13 1.18 × 10−10 204 4.19 × 10−10 table 4. comparison of maximum errors (me) for example 5 h me in s3hi1 me in s3hi2 me in s3hi3 me in [12] 1 16 5.11541 × 10 −14 2.23457 × 10−14 1.00124 × 10−15 1.03179 × 10−11 1 32 6.55230 × 10 −15 1.20048 × 10−15 5.47895 × 10−15 3.24907 × 10−13 1 64 3.75411 × 10 −16 4.78951 × 10−17 3.47785 × 10−16 1.02789 × 10−14 466 modebei et al. / j. nig. soc. phys. sci. 3 (2021) 459–468 467 æ æ æ æ à à à à ì ì ì ì 0.02 0.04 0.06 0.08 0.10 0.12 10-13 10-12 10-11 10-10 cpu time m ax e rr or ì s3hi3 à s3hi2 æ s3hi1 figure 3. efficiency plot for s3hi1, s3hi2 and s3hi3 showing the maximum error against the cpu time for problem 3 for n = 6, 12, 24, 48. æ æ æ æ à à à à ì ì ì ì ò ò ò ò ô ô ô ô 0.0 0.1 0.2 0.3 0.4 0.5 10-17 10-15 10-13 10-11 10-9 cpu time m ax e rr or ô ostbm ò s3hbm ì s3hi3 à s3hi2 æ s3hi1 figure 4. efficiency plot for s3hi1, s3hi2, s3hi3, s3bhm and ostbm showing the maximum error against the cpu time for problem 4 for n=6,12,24,48. whose exact solution is y(x) = ln(1 + x). figure 4 is the efficiency curve showing the different methods and their individual performance in the absolute errors obtained against the cpu time in each case. from figure 4, one observes that, s3bhm shows good error performance relative to ostem (where ostem is an order 8 method). the proposed s3hi1, s3hi2, s3hi3 performed well in terms of error and time as such, they out performed s3bhm and ostem in terms of accuracy and time cost. example 5.. consider the bvp discussed in [12]. y′′′ + y = (x − 4) sin(x) + (1 − x) cos(x), x ∈ [0, 1] y(0) = 0, y′(0) = 1, y′(1) = −sin 1 (31) whose exact solution is y(x) = (x − 1) sin(x). table 4 shows the maximum error obtained using different step-sizes and compared with the an order 7 method in [12]. it clearly shows that s3hi1, s3hi2 and s3hi3 are almost equivalent in performance but more superior to the method cited. 5. conclusion block of 3 points hybrid method are proposed and applied to solve third-order linear and non linear ivps and bvps in ordinary differential equations. the method are such that the off-step points are shifted between two step points in order to investigate which position is more efficient. all three methods possess the same characteristics, viz-a-viz, order of accuracy and number of functions evaluations. they are found to be consistent, zero stable and convergent as well. they are less ambiguous to derive. it should also be noted that this paper proposes computational comparison of 3 classes of a family of 3 steps linear multistep method. however, less emphases are placed on these methods outperforming existing methods in literature. albeit, they performed favourably well when compared to other methods in literature. references [1] j. henderson & and k. r. prasad, ”existence and uniqueness of solutions of three-point boundary value problems on time scales”, journal nonlinear science 8 (2001) 1. [2] c. p. gupta & v. lakshmikantham, ”existence and uniqueness theorems for a third-order three point boundary value problem”, journal of nonlinear analysis: theory and methods in application 15 (1991) 949. [3] r. k. sahi, s. n. jator & n. a. khan, ”continuous fourth derivative method for third-order boundary value problems”, international journal of pure and applied mathematics 85 (2013) 907. [4] s. n. jator, ”novel finite difference schemes for third order boundary value problems” international journal of pure and applied mathematics 53 (2009) 37. [5] o. adeyeye & z. omar, ”solving third order ordinary differential equations using one-step block method with four equidistant generalized hybrid points”, iaeng international journal of applied mathematics (ijam) 49 (2019) 1. 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[11] h. k. mishra & s. saini, ”quartic b-spline method for solving a singular singularly perturbed third-order boundary value problems”, american journal of numerical analysis 3 (2015) 18. 467 modebei et al. / j. nig. soc. phys. sci. 3 (2021) 459–468 468 [12] ra’ft abdelrahim, ”numerical solution of third order boundary value problems using one-step hybrid block method”, ain shams engineering journal 10 (2019) 179. [13] b. s. h. kashkari & s. alqarni, ”optimization of two-step block method with three hybrid points for solving third order initial value problems”, journal of nonlinear science and application 12 (2019) 450. [14] j. d. lambert, ”computational methods in ordinary differential equations”, john wiley, new york (1973). [15] m. i. modebei, r. b. adeniyi, s. n. jator & h. c. ramos, ”a block hybrid integrator for numerically solving fourth-order initial value problems”, applied mathematics and computation 346 (2019) 680. [16] p. henrici, ”discrete variable methods in ordinary differential equations”, john wiley & sons, new york, usa, (1962). 468 j. nig. soc. phys. sci. 2 (2020) 228–240 journal of the nigerian society of physical sciences original research gps-tec variations over the african low-latitude ionosphere during march 2013 and 2015 geomagnetic storms s. o. ikubannia,∗, s. j. adebiyia, b. o. adebesina, o. s. bolajib,c, b. j. adekoyad, b. w. joshuae, j. o. adeniyia aspace weather group, landmark university, p.m.b. 1001, omu-aran, nigeria bdepartment of physics, university of lagos, nigeria cdepartment of physics,university of tansmania, australia ddepartment of physics, olabisi onabanjo university, ago-iwoye, ogun state edepartment of physics, kebbi state university, aleiro, kebbi state abstract intense geomagnetic storms offer opportunity to understand ionospheric response to space weather events. using total electron content (tec) data from stations along the east african sector, the two most intense storms during the 24th solar cycle, with similarly occurrence season and time were studied. we observe that ionospheric effect during the main phase is not a function of the severity of the storm, whereas the more intense storm shows greater influence on the african ionosphere during the recovery phase. plasma movement within the equatorial ionization anomaly (eia) was evident particularly during the recovery phase, especially during the 2015 event. for both storms, the nighttime/early morning ionospheric effect is more pronounced than the daytime effects across all stations. doi:10.46481/jnsps.2020.112 keywords: disturbance electric field, equatorial ionosphere, geomagnetic storms, gnss article history : received: 17 june 2020 received in revised form: 07 august 2020 accepted for publication: 20 august 2020 published: 15 november 2020 c©2020 journal of the nigerian society of physical sciences. all rights reserved. communicated by: o. j. abimbola 1. introduction energetic mass particles released during solar storms impact the earth’s magnetosphere if earth-directed. this impact leads to perturbations in the various components of the earth’s magnetic field, in what is termed “geomagnetic storms” and subsequently causes rapid and sometimes substantial changes in ionospheric plasma distribution referred to as “ionospheric storms.” ionospheric response to magnetospheric perturbations ∗corresponding author tel. no: +2347032261146 email address: ikubanni.stephen@lmu.edu.ng (s. o. ikubanni ) depends on the driving conditions and ionospheric conductivity [1]. the disturbance of the earth’s magnetic field has varying severity, hence classified into categories. for classification of geomagnetic storms see [2]. there is a large body of literature documenting the ionospheric changes associated with the two recent st. patrick day geomagnetic storms of varying intensity, which occurred in march 2013 and 2015; periods around the solar maxima in solar cycle 24. though, both storms can be classified as intense, the 2015 event was the first storm to make the severe category during the sunspot cycle 24 [3]. the two st. patrick day storms were triggered by interplanetary coronal mass ejection (icme) but of different origins and with differ228 ikubanni et al. / j. nig. soc. phys. sci. 2 (2020) 228–240 229 ent magnitudes [4]. a summary of some of the most recent and significant observations are presented in [5]. in this paper, we limit the review to published observations of ionospheric responses in the low latitude/equatorial regions. hairston et al. [6] analysis of plasma flows, densities, composition, and temperatures obtained from five polar-orbiting defence meteorologial satellite program (dmsp) spacecraft operational during the storm and from the equatorial-orbiting communications/navigation outage forecasting system (c/nofs) spacecraft revealed the influence of the prompt penetration electric field (ppef) and the disturbance dynamo electric field (dd ef); as well as the resulting meridional (vertical) ion flows in the equatorial ionosphere during the march 2015 storm. the disturbed electric fields, which are the ppef and ddef, were decisive during both storms. during disturbed periods, errors in single-frequency global navigation satellite systems (gnss) caused by ionospheric delays due to deviation from the quiettime total electron content (tec) average can be large, leading to loss of data, satellite signal degradation, breakdown in quality of communication links, reduced capacity for navigation and positioning [7]. the ppef, which is usually short-lived, emerges due to the injection of electric field in the solar wind into the ionosphere, during the period of southward turning of the interplanetary magnetic field (imf) [8] and it remarkably depends on local time [9]. for example, the ppef caused a large increase in vertical plasma drift at local evening time, leading to equatorial spread-f irregularities/plasma bubbles and scintillation [10, 11]. imf-bz orientation during sunset can enhance or inhibit post sunset formation of equatorial irregularities; it inhibits if northward (as it occurred during the 2015 storm) and enhances if southward (as it occurred during the 2013 storm) [12]. ray et al. [13] identified occurrence of equatorial spread-f (esf) at the east pacific and the indian sectors due to the time of the southward turning of imf-bz. the ddef, usually long-lived, is a product of the enhanced energy injection into the auroral; which alters the global circulation and the generation of electric fields and currents [9]. the ddef can cause generation of equatorial irregularities around sunrise, by causing large increase in the f2 peak height (hmf2) (as observed during both 2013 and 2015 storms in the 100 o e sector) [12]. also, ddefs exhibit remarkable local time dependence; due to the local time differences between different sectors as documented by [9], it caused intense negative ionospheric storm effects in the asian-australian equatorial sector and positive storm effects in the american equatorial sector during 2015 event. with analysis of data from both hemispheres, kalita et al. [12] attributed the observed hemispherical asymmetry of electron density depletion to the asymmetric expansion of the storm-time thermospheric composition. in another study, nava et al. [13] noted that the largest storm effects occurred at longitudinal sectors that are on the evening/night sides during the time of very large energy inputs. in agreement, the disturbance dynamo was reported to have caused an upward ion drift (reaching 200 ms−1) in the midnightdawn sector, a downward ion drift (reaching 80 ms−1) in the early morning sector and a westward ion drift (reaching around 100 ms−1) near dawn [14]. also, there was increase in the concentration of oxygen ion (o+) from < 60 % of the constituent ion before the storm to between 80 – 90 % during the 2015 storm; and attributed the sudden increase in o+ to the effect of disturbance neutral winds on the post-midnight equatorial ionosphere [14]. the disturbed neutral wind and the disturbed electric field could modify the equatorial ionization anomaly (eia) in terms of the location and magnitude of the two crests with respect to quiet periods as seen in the 2013 storm [15]. another study during the 2015 event[16] suggested the altitudinal dependence of the storm time electrodynamics of the topside ionosphere. observations of the geospace effects of the st. patrick’s day storms over the african sector are very limited; some of the available ones include [17, 18, 19, 20]. bolaji et al. [20] documented observations of equatorial ionospheric irregularities across several sectors, the african sector inclusive, using the rate of change of tec index (roti) as indicator. their work showed longitudinal variation in the strength of the irregularities partly due to the different local time effect of the storminduced drivers. the irregularities decrease eastward from the american sector through the african sector, but not present in the oceania. also, the african equatorial/low-latitude ionosphere, during the 2015 event, experienced positive phases during the main phase and negative phases several hours after the storm commencement and during the recovery phase. the positive ionospheric storm during the mp was attributed to the ppef while ddef was suggestive of the negative phase and suppression of irregularities over all considered stations [19]. amaechi et al.[19] report also showed that the northern eia peak was formed beyond the 15o geographical latitude and the southern peak between the geographical equator and the −5o latitude. the limitation of [4], based on this, is that the southern stations employed all fall beyond the eia crest. for this work, stations including those that fall within the 15 o n and 5 os of the geographical equator have been employed. the stations were also chosen along close longitudes to reduce local time differences. hence, the work is meant to contribute to the discourse on the geospace effects of the two recent st. patrick’s day storms, using datasets from stations that have largely not been investigated for these storms. 2. data and methodology we have employed data from the ace satellite to describe the interplanetary solar wind characteristics that lead to the geomagnetic storms of 17 march 2013 and 17 march 2015. these data covers from the 16 – 20 march for both events and are provided by the national space science data centre (http://nssds cgsfc.nasa.gov/omniweb). these data include the z-component of the interplanetary magnetic field (imf-bz), which describes the orientation of the earth-directed sun’s magnetic field contained in the solar wind that impacts the earth’s magnetosphere, recorded in nanotesla (nt ); the speed of the solar wind, recorded in kilometers per second (km/s); and the interplanetary electric field (ief) measured in millivolts per metre (mv/m). the other complementing information are geomagnetic parameters 229 ikubanni et al. / j. nig. soc. phys. sci. 2 (2020) 228–240 230 obtained from ground-based observing instruments. these include the symmetric disturbance of the horizontal component of the earth’s magnetic field (sym-h), which is the measure of perturbations in the h-component of the earth’s magnetic field due to the influence of ring currents and other current system in the magnetosphere [21]. sym-h is a product of monitoring these perturbations at designated stations around the equatorial/low-latitude regions and recorded in nt; and auroral electrojet (ae) which is observed at designated stations in the polar/high-latitude region and recorded in nt. ionospheric electron density variation in the african lowlatitude region was investigated using derived vertical total electron content (vtec)retrieved from raw observables from six (6) global positioning system (gps) receivers. these work in particular focused on the symmetry of the northern and southern anomaly crests and the plasma reversal at the trough. most of the observables in receiver independent exchange (rinex) format were retrieved from scripps orbit and permanent array center (sopac) (ftp://garner.uscd.edu), while some (such as for mtandika ‘mtdk’ and solar village ‘sola’ stations) were retrieved from unavco database (ftp://data-out.unavco.org/ pub/rinex/obs/).satellite and receiver biases contained in files from center for orbit determination in europe (code) and available at “ftp://ftp.unibe.ch/aiub/code” was also retrieved. tec was obtained from the rinex and code files by applying a procedure developed at the boston college and package in the software “gps v2.9.5”, which is freely available at “http://seemala.blogspot.com/”.using some stations within the african eia, we investigated the tec magnitude at both crests of the eia in relation to the trough, the asymmetry at the two crests and fountain plasma reversal during the storm main and recovery phases. the station names, their codes and geomagnetic coordinates are shown in table 1. figure 1 presents the position of these stations on the african map. for both storm events, five stations within the eia were considered. one station is within the equatorial electrojet (eej) belt and the remaining four seasons fall equally on both sides of the magnetic equator, of the six stations listed in table 1 only four (4) stations have for the 2013 and 2015 storm events considered. zamb has data for the period under consideration in 2013 while mtdk has data for 2015 only; hence, they were respectively adopted as the southern eia crest station for the individual storm events. the deviation of storm-time electron density denoted as tec was quantified as a percentage of the quiet-time reference. this is expressed as presented in equation 1. %∆t ec = ( t ecdisturbed − t ecquiet t ecquiet ) × 100%, (1) where t ecdisturbed is the storm-time tec values and t ecquiet is the quiet-time average tec values. the quiet-time value is the mean from five most quiet days of the month. the selected quietest days were obtained from the german research centre for geosciences (http://www.gfz-potsdam.de/en/section/earthsmagnetic-field/data-products-services/kp-index/qd-days/qddays-since-2010/). the values obtained from employing equation 1 indicates either storm-induced changes when |%∆t ec| exceeds ±25% for at least 3 consecutive hours. positive values indicate enhancements while negative values indicate depletions. figure 1. map of africa showing the stations used for this work. 3. results and discussion 3.1. description of the storms from the left panel of figure 2, the 2013 storm-time interplanetary and geomagnetic parameters show a sharp southward turning of imf-bz to a magnitude of 14 nt around 0900 ut on 17 march (marking the beginning of the geo-effectiveness of the storm) shortly after the shock (sudden positive increase in sym-h). the magnitude of imf-bz in the southward orientation decreases to about zero after 5 hours. it then increases to another peak (∼ 12 nt ) at 1800 ut before gradually decreasing and then turning northward around 2300 ut. the shock occurred around 0600 ut on 17 march and produced a symh value of ∼ 20 nt . the continuous injection of energy into the magnetosphere after the shock led to increase in the ring current and causing a depression in the sym-h to a minimum of −132 nt around 2000 ut on 17 march. the interplanetary electric field, e increases abruptly from around zero at 0700 ut on 17 march to a peak of ∼ 10 mv/m three hours later. the magnitude of e then fluctuates as it decreases reaching approximately zero value at 1400 ut before increasing to another peak of ∼ 7 mv/m at 1800 ut on 17 march. the ae-index increased to 781 nt coinciding with the time of the shock and reached a peak of 1060 nt an hour later; it then decreases and fluctuates before increasing to another peak of ∼ 1800 nt around 1600 ut on 17 march. ae-index remained relatively high for another 7 hours (i.e. until around 2300 ut). from the right panel of figure 2, the 2015 storm-time interplanetary and geomagnetic parameters show a sharp increase in the imf-bz magnitude (∼ 20 nt ) in the northward direction at 0500 ut on 17 march shortly after the shock arrival; and then instantly turned southward. the southward imf-bz reached a peak (∼ 16 nt ) at 0800 ut (marking the beginning of the 230 ikubanni et al. / j. nig. soc. phys. sci. 2 (2020) 228–240 231 table 1. gps stations, countries, and codes as well as their geographical and geomagnetic coordinates . station(country) station code geographic coordinates geomagnetic coordinates lt (hr) mtandika (tanzania) mtdk 7.54 ◦s 36.42 ◦e 17.56 ◦s 107.79 ◦e ut+2:25 lusake (zambia) zamb 15.43 ◦s 28.31 ◦e 16.93 ◦s 77.96 ◦e ut+2:00 malindi (kenya) mal2 3.00 ◦s 40.19 ◦e 6.41 ◦s 111.92 ◦e ut+2:40 eldoret (kenya) moiu 0.28 ◦n 35.29 ◦e 2.68 ◦s 107.72 ◦e ut+2:20 addis-ababa (ethiopia) adis 9.04 ◦n 38.77 ◦e 5.35 ◦n 112.53 ◦e ut+2:35 solar village (saudi arabia) sola 24.26 ◦n 46.51 ◦e 17.71 ◦n 118.16 ◦e ut+3:00 figure 2. some interplanetary and geomagnetic parameters between 16 and 20 march in year 2013 (left); and year 2015 (right). storm’s geo-effectiveness) and fluctuates till until it reached another peak (∼ 18 nt ) around 1400 ut on 17 march). the magnitude of imf-bz in the southward orientation remained substantial until around 2300 ut before decreasing to zero around 0500 ut. the shock occurred around 0500 ut on 17 march and produced a sym-h value of ∼ 55 nt . the continuous injection of energy into the magnetosphere after the shock led to increase in the ring current and causing a depression in the sym-h to a minimum of −232 nt around 2300 ut on 17 march. the interplanetary electric field, e decreases abruptly around 0500 ut and then increases to a peak of ∼ 9 mv/m around 0800 ut on 17 march. it then decreases again to a minimum value of ∼ 5.7 mv/m (at 1100 ut) before increasing again to a peak (∼ 10.6 mv/m) at 1400 ut and remained relatively constant till around 2300 ut before returning back to the pre-storm level few hours into the recovery phase. the ae-index increased to ∼ 780 nt around 0800 ut on 17 march, decreases and then increases to a peak (∼ 1600 nt ) at 1400 ut on 17 march. aeindex remained relatively high for another 9 hours (i.e. until around 2300 ut). 3.2. observations in 2013 during the march 2013 storm, as presented in figure 3, the main phase (mp) of the storm showed daytime changes that are more prominent at stations north of the magnetic equator (sola and adis). the post-sunset/pre-midnight peak in tec during the mp is clearly formed only around the eia trough station of moiu and its southern station of mal2, coinciding with the period of minimum sym-h. figure 3 could not capture distinct events at the beginning of the recovery phase (2000 ut or ∼ 2230lt ), probably due to the low nighttime tec magnitudes. however, with the aid of the quantifier presented in equation 1, figure 4 reveals large enhancement from the beginning of the recovery phase (around 2000 ut on 17 march) and persisted for more than 7 hours, particularly at sola. there were other episodes of persistent depletions and some enhancement, but with smaller magnitude during the other recovery days at sola as well as some episodes of depletions (particularly at midnight) at the southern eia crest regions (such as zamb). generally, there seems to be consistent pattern of premidnight depletions at the eia crest in both hemispheres during the recovery days. for a clearer picture of events around seen in figures 3 and 4, the line plots presented in figure 5 showed the changes between the quiet and disturbed tec along the latitudes in each hour. though, the storm sudden commencement (sc) is around 0600 ut (∼ 0830 lt ), followed by immediate injection of ring current, deviation from quiet-time values (enhancements) started becoming visible around 0800 ut (∼ 1030 lt ), partic231 ikubanni et al. / j. nig. soc. phys. sci. 2 (2020) 228–240 232 figure 3. the superposed plot of quiet-time reference tec (red solid curve) and the observed tec (blue broken curve) during march 16 – 20, 2013 geomagnetic disturbance (the first panel shows the varying sym-h index during the storm). figure 4. the percentage change in disturbed tec with respect to the quiet-time reference value for the march 16 – 20, 2013 storm. ularly in the north of the magnetic equator. the enhancements then extend to the eia trough about an hour later (0900 ut or ∼ 1130 lt ). by 1000 ut (∼ 1230 lt ), the well-known daytime eia trough of tec magnitude had disappeared. instead of a minimum tec value at the trough flanked by two peak values on its either sides, there was a peak value higher than that of the southern eia station. by 1300 ut (∼ 1530 lt ), depletions were beginning to set-in at the southern station and persisted till around 1700 ut. there was no substantial effect until 2000 ut (∼ 2230 lt ) where depletions towards the north and south crests. by 2200 ut (∼ 0030 lt ), the time around which the maximum sym-h depression was observed, a prominent peak compared to the quiet-time value has been formed at the magnetic equator. 232 ikubanni et al. / j. nig. soc. phys. sci. 2 (2020) 228–240 233 figure 5. latitudinal plot of quiet-time tec (red broken curve) and the disturbed tec (blue solid curve) for each hour during the 2013 storm day (17 march). figure 6. same as figure 5, but for 19 march 2013. the recovery phase started just before 2300 ut on 17 march. however, the lack of data at moiu on 18 march could not allow a detail analysis of the latitudinal variation of disturbed tec with respect to the quiet-time values. on 19 march (figure 6), the effect of the recovery phase was seen as slight tec increase at the northern eia station and decrease at the trough beginning from around 0900 ut (∼ 1130 lt). this persisted till 1200 ut (∼ 0230 ut) resulting in a stronger eia during the storm than during quiet time. also, from 2200 and 2300 ut, there was a total disappearance of the eia structure compared to the quiet time morphology. the effect of the recovery phase on the eia structure was also seen on 20 march (figure 7), marked by slight tec increase at the northern eia station and decrease at the trough beginning from around 0800 ut (∼ 1030 233 ikubanni et al. / j. nig. soc. phys. sci. 2 (2020) 228–240 234 lt) and persisted till around 1100 ut (∼ 0130 ut) resulting in a stronger eia during the storm than during quiet time. shortly after, decreases at the north most eia station accompanied by increases at the eia trough station was seen resulting into disappearance of the eia at 1600 ut (∼ 1830 lt). from 1900 (∼ 2100 lt) – 2300 ut (∼ 0100 lt), there was increase north of the equator and decreases at the south, indicating plasma movement from the south to the north without a change in the eia structure from that of the quiet time. 3.3. observations in 2015 during the march 2015 storm, as presented in figure 8, the main phase (mp) of the storm showed changes that are more prominent towards the end of the main phase (1600 ut – 2300 ut) and for a larger part of the recovery phases at all stations. the post-sunset/pre-midnight peak in tec during the main phase is clearly formed within the eia region. figure 9 reveals large changes between 1600 ut (∼ 1830 lt) and 2300 ut (∼ 0130 lt) with enhancement at the northern station of sola, which decreases towards the equator; then followed immediately by depletions at the equator and extended to the southern stations (mal2 and mtdk). however, unlike the 2013 storm, deviations from quiet-time values due to the storm (enhancements) were not observed until around 1700 ut (∼ 1930 lt), particularly in the north of the magnetic equator and the eia trough station. the deviations (depletions) then extend to stations south of the magnetic equator about four hours later (2100 ut or ∼ 2330 lt). the depletions at the southern stations coincided with the beginning of the recovery phase. the first day of the recovery phase (18 march) is characterized by consistent enhancements at all stations between 0300 ut (∼ 0530 lt) and 0700 ut (∼ 0930 lt), brief episode of depletions around 1400 ut (∼ 1630 lt) and longer episode of depletions which extends till the next recovery day (19 march) (2000 ut – 0300 ut) at the southern stations. the depletions lasted more than 12 hours at mtdk with larger amplitudes around 2300 ut (∼ 0130 lt). on other recovery days (19 and 20 march), there were other episodes of enhancements at adis between 1800 ut (∼ 2030 lt) and 2300 ut (∼ 0130 lt) (as well as 18 march) and depletions, but with smaller magnitude, during the other recovery days at sola as well as the southern eia crest regions (such as zamb). these depletions were persistent occurring for the larger part of the days. generally, there seems to be consistent pattern of pre-midnight depletions at the eia crest in both hemispheres during the recovery days, except at adis. for a clearer picture of events around seen in figures 8 and 9, the line plots presented in figures 10 – 14 showed the changes between the quiet and disturbed tec along the latitudes in each hour. first for the storm day (17 march), the storm sc is around 0500 ut (∼ 0730 lt), around same period as that of the 2013 event, followed by immediate injection of ring current. there was no modification of the eia structure, but slight deviations in form of depletions at all stations become visible at the instance of the sc until 0700 ut (∼ 0930 lt) and between 1100 and 1400 ut (∼ 1330 and 1630 lt) although, these deviations cannot be regarded as storm-induced since the |%∆t ec| is less than 25 (see figure 9). however, beginning from around 1600 ut (∼ 1830 lt), there was substantial modification of the eia; there was rapid enhancement at the northern eia crest stations. these led to a reversal of the most prominent peaks. while the peak in the southern hemisphere is greater than that of the north during quiet period, the storm reversed this and caused the peak in the northern hemisphere to be greater. this persisted until 2300 ut (∼ 0130 lt on 18 march). however, there was decrease in the magnitude of the enhancements at the northern stations simultaneously with depletions at the southern stations. on the first day of the recovery phase (18 march), the modification of the eia structure continued with the disappearance of the quiet-time peak at the equator between 0100 and 0200 ut (∼ 0330 and 0430 lt). there were enhancements beginning from 0400 ut (∼ 0630 lt) at all stations and lasted till around 0800 ut (∼ 1030 lt) accompanied by a more prominent peak at the southern crest region. the enhancement persisted until1000 ut (∼ 1230 lt) at the equator, leading to the disappearance of the equatorial trough. the immediate accompanying depletions at the southern stations led to the formation of only one peak (i.e. the northern peak) at 1100 ut (∼ 1330 lt). however, the depletions in the south continued, but with decreased magnitude, until another peak is formed at the equator around 1600 ut (∼ 1830 lt). then enhancements in the north of the equator beginning from 1700 ut (∼ 1930 lt) led to the formation of only one peak (i.e. the northern peak) due to the disappearance of the southern peak. this persisted until 2300 ut. observations during the other recovery days (19 and 20 march) (figures 12 and 13) are similar to observations on the first day. 4. discussion electric field action with the perpendicular northward horizontal component of the earth’s magnetic field around the magnetic equator plays a very important role in plasma movement along the equatorial latitudes. this action accounts for changes in the daytime equatorial morphology of the ionospheric electron density during quiet time. the daytime eastward orientation of the electric field causes upward motion of the plasma to higher altitudes (region of lower recombination rates) over the equator, and then the plasma diffuses along magnetic lines to either sides of the equator. this leads to a trough of plasma density at the equator and two crests at either side. at nighttime, when the orientation of the electric field is westward, plasma moves back to lower altitudes. this may lead to collapse of the eia structure, but this is not the case, as the plasma fountain effect is sustained by slow decay rate of the f-layer plasma during recombination [22, 23]. the pre and/or the slow f-layer plasma decay rate, both occurring in the absence of strong solar control, inhibit plasma movement from the eia crests to the trough [24, 25, 26]. during disturbed time, the ejection of disturbed electric fields (both the ppef and ddef), modifies plasma distribution depending on the local time. for example, tsurutani et al. [22] attributed increase in eia intensity to the unusual increase in 234 ikubanni et al. / j. nig. soc. phys. sci. 2 (2020) 228–240 235 figure 7. same as figure 5, but for 20 march 2013. figure 8. the superposed plot of quiet-time reference tec (red solid curve) and the observed tec (blue broken curve) during march 16 – 20, 2015 geomagnetic disturbance (the first panel shows the varying sym-h index during the storm). the eastward electric field due to ppef. the ppef causes lifting of the ionospheric plasma to unusual altitudes, and the diffused plasma now concentrates at latitudes farther away from the magnetic equator. hence, the eia crests during the disturbance form at latitudes higher than those of the quiet periods while the ongoing/concurrent ionization at lower heights continue to produce plasma that replenish the trough. as a result, the eia structure is maintained, but with the crests at higher latitudes and tec magnitude increasing across all latitudes in the eia region.increase in the f-region electric field during prereversal enhancement (pre) can also maintain or increase the strength of the plasma fountain effect [27, 28]. the ddef, a delayed modification of the electric field acting to suppress the solar quiet (sq) electric field, usually accounts for plasma dis235 ikubanni et al. / j. nig. soc. phys. sci. 2 (2020) 228–240 236 figure 9. the percentage change in disturbed tec with respect to the quiet-time reference value for the march 16 – 20, 2015 storm. figure 10. latitudinal plot of quiet-time tec (red broken curve) and the disturbed tec (blue solid curve) for each hour during the 2015 storm day (17 march). tribution to leads to changes in the eia structure depending on the local time. thus, ddef causes eastward electric field to be westward, thereby weakening the eia structure or causing its totally disappearance and vice-versa. another modification factor to plasma during storm events, particularly the prolonged positive/negative storm effects irrespective of local time is the thermospheric circulation. mendillo [29] noted that changes to o/n2 composition ratio such as increase/decrease in the ratio may drive positive/negative phases respectively. the more prominent in enhancements, particularly at stations north of the equator (sola and adis) during the march 2013 storm’s mp and the post-sunset/pre-midnight peak in tec towards the end of the mp (as observed from figure 3) signals the to the action of the ppef. the large enhancement from 236 ikubanni et al. / j. nig. soc. phys. sci. 2 (2020) 228–240 237 figure 11. same as figure 10, but for 18 march 2015. figure 12. same as figure 10, but for 19 march 2015. the beginning of the recovery phase in the northern and persisted for more than 7 hours, particularly at sola (as seen from figure 4) may be due to the nighttime eastward electric field caused by the ddef. this leads to plasma movement to the low latitudes (i.e. the eia crest) from the equator. the other episodes of depletions during the remaining recovery period are prolonged suggesting the role of changing thermospheric composition. this resonates with [12] who attributed the negative phases observed in the asian sector during the 2013 event to electric field dynamics and changing thermospheric composition. observations from figures 5 – 7 provide hourly consideration of plasma-redistribution during the entire 2013 storm event. the asymmetry in enhancements on either side of the equator between 0800 and 1200 ut (∼ 1030 and ∼ 1430 lt) and followed by the disappearance of the eia structure between 1000 and 1200 ut (∼ 1230 and ∼ 1430 lt) may be due to 237 ikubanni et al. / j. nig. soc. phys. sci. 2 (2020) 228–240 238 figure 13. same as figure 10, but for 20 march 2015. plasma injection from higher latitudes in the northern region. if it were to be due to ppef, it is expected that the eia will be strengthened. however, this was not so, rather, a large increase in the north and a higher magnitude at the trough than the south was observed. for four stations in the south of the eia as documented by [4], there were little or no tec changes during the mp. apart from the continued disappearance of the eia structure, the peak at the north of the eia persisted and accompanied by reduction in tec magnitude at either side of the equator between 1300 and 1500 ut (∼ 1530 and ∼ 1730 lt). this may be explained in terms of reduction in ionization and plasma drifting towards the eia trough. therefore, by 1600 ut (∼ 1830 lt), the eia has returned to its quiet-time structure. at the beginning of the recovery phase (2100 ut / ∼ 2330 lt), the appearance of a peak tec value at the eia trough (deviation from the quiet-time eia structure) was seen an it persisted for about 3 hours. this is due to the ddef which is known to suppress quiet-time eia structure. the other modifications of the eia structure during the 2013 event are on 19 march around 1200 ut (where the structure is strengthened) and 2200 – 2300 ut (where the structure is slightly suppressed) as well as 20 march around 0300 – 0600 ut, 1500 ut and 2200 – 2300 ut (where the structure is suppressed). the short duration of these observations may not be attributable to the storm or could be due to the nature of the storm, where the processes such as the impact of the ddef is short-lived. for example, ddef around 1200 ut (∼ 1430 lt) on 19 march is expected to decrease the influence of the eastward electric field.however, substantial storm-induced changes are expected to persist for more than two hours. therefore, the limitations of observational equipment could not allow for further investigation to identify the mechanism involved. during the 2015 event, prominent observations are towards the end of the main phase (1600 ut – 2300 ut) and for a larger part of the recovery phases at all stations. this is a clear deviation from report by [12] and [30], where a short-lived enhancement was observed around 1200 ut at the asian sector (∼ 1900 lt). the prominent post-sunset/pre-midnight peak in tec during the main phase at all the stations with higher magnitude than the quiet-time value within the eia may be due to ppef. the enhancements between 1600 ut (∼ 1830 lt) on 17 march and 2300 ut (∼ 0130 lt on 18 march) north of the equator, then followed immediately by depletions at the equator and extended to the southern stations on the storm day could be as a result of movement of plasma from higher latitudes in the north across the equator to the south. the absence of daytime enhancements/depletions particularly during the mp may be due to the result of the overlapping actions of ppef and ddef. according to hairston et al. [6], the ddef dominates the ppef for a larger period of mp. the depletions at the southern stations occurring around the beginning of the recovery phase could be attributed to the ddef. however, its interaction with the ppef limits the magnitude of the depletions. ikubanni et al. [4] also reported a 5-hour duration of depletionsat the southern eia stations around the period of minimum sym-h. recall that the [19] had shown that the eia crests in the african sector formed beyond 15o n and in the south just below geographical equator. thus, the eia trough is expected to form midway, where adis is located. hence, the persistent and long-lasting evening/nighttime depletions at stations other than adis, where enhancements were observed, the on all the recovery days could be due to the role of the changing o/n2 ratio. increase in o around the eia trough region leads to enhancement while increase in n2 at higher latitudes causes 238 ikubanni et al. / j. nig. soc. phys. sci. 2 (2020) 228–240 239 depletions. 5. conclusion with data from stations along the eastern corridor of the african sector, taking account the local time differences, we have studied the movement of plasma across the stations. clear distinctions in the responses to the events both during the mp and the recovery phase were observed. the most prominent effect of the 2013 storm was at the beginning of the recovery phase particularly at the eia crest station of sola. for the 2015 event, substantial effects were observed towards the end of the mp as well as throughout the entire recovery phase, extending into hours. the dynamical roles of several factors were discussed in line with the observations.the study of the two storms showed that recovery phases may experience larger amplitudes of depletions/enhancements than the main phase depending on the magnitude of the event and other accompanying processes. improved instrumentation in africa will aid in better investigation of future space weather events on the african geospacer. acknowledgments we thank the referees for the positive enlightening comments and suggestions, which have greatly helped us in making improvements to this paper. the authors also appreciate the free availability of interplanetary and geomagnetic parameters by the national space science data centre (http://nssdscgsfc. nasa.gov/omniweb). the availability of the employed gps rin ex files used at scripps orbit and permanent array center (sopac) (ftp://garner.uscd.edu) and from unavco databases (ftp://data-out.unavco.org/pub/rinex/obs/) is also appreciated. references [1] z. xu, m. d. hartinger, c.r. clauer, t. peek, r. behlke, “a comparison of the ground magnetic responses during the 2013 and 2015 st. patrick’s day geomagnetic storms”, j. geophys. res. space physics, 122 (2017) 036, doi:10.1002/2016ja023338. 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[30] c. jiang, g. yang, j. liu, t. yokoyama, t. liu, t. lan, c. zhou, y. zhang, z. zhao, t. komolmis, p. supnithi & c. y. yatini, “equatorial and low-latitude ionospheric response to the 17-18 march 2015 great storm over southeast asia longitude sector”, j. geophys. res. space physics, 122(2017), https://doi.org/10.1002/2017ja024134. 240 j. nig. soc. phys. sci. 4 (2022) 99–104 journal of the nigerian society of physical sciences juxtaposing vertically transmitted infections (vtis) and the spread of hiv/aids in a typically infection prevalent region in nigeria matthew iseh, anthony usoro, nsisong ekong∗, idara ukpe adepartment of statistics, akwa ibom state university, ikot akpaden, nigeria abstract in this paper, we investigate by putting side by side the rate at which human immuno-deficiency virus/ acquired immune deficiency syndrome hiv/aids spreads through vertical transmission (vt) in a typically infection prevalent region. also, we model this relationship and present the enormity and the likelihood of mother-to-child infections given age, weight, and dosage of drugs taken by the pregnant women. binary logistics regression is the approach employed in the analysis of this work. the method is necessary given the dichotomous nature of the variable under investigation, in this case, the status of the newborns. the result from the binary logistic regression shows that the probability that a child will be infected given that the mother is positive is 0.098235 ≈ 0.1, and this is unlikely to happen given the small probability. this result is in line with the proportion of event count which is ≈ 0.1. other complimentary test results also agree that given the biomarkers (age, weight and drug doses of the mothers) used in the study, the possibility of a newborn acquiring the deadly disease is very minimal. the study revealed that, given these biomarkers, the chances of newborns being infected with hiv/aids through vt is very minimal. hence, there is need for donors and interventionists agencies to redirect attention and interventions to other modes of transmissions. doi:10.46481/jnsps.2022.418 keywords: human immunodeficiency virus (hiv), acquired immune deficiency syndrome (aids), vertically transmitted infections (vtis), akwa ibom state. article history : received: 20 september 2021 received in revised form: 26 november 2021 accepted for publication: 02 january 2022 published: 28 february 2022 c©2022 journal of the nigerian society of physical sciences. all rights reserved. communicated by: w. a. yahya 1. introduction the speedy upsurge of certain diseases notwithstanding the myriads of precautionary and preventive measures taken globally by different quarters to evade the prevalence of such diseases is a thing of great concern. one of such diseases is the human immuno-deficiency virus/ acquired immune deficiency syndrome hiv/aids a disease that is still considered a major global public health issue. so far, it has been with us for ∗corresponding author tel. no: +2347069150513 email address: nsisongekong@aksu.edu.ng (nsisong ekong) the past six decades and has so far claimed not less than 33 million lives globally [9]. according to the same report, it is estimated that 38 million people were living with this disease at the end of 2019 across the globe. upon an establishment that the disease is one of the numerous vertically transmitted infections (vtis), that is, the transmission of from a mother to her child during pregnancy, labour, delivery or breastfeeding, different approaches have been set in motion to curb the mother-to-child transmission and so far, about 85% of either pregnant women or breastfeeding mothers living with the disease have received anti-retroviral therapy (art) which apart 99 iseh et al. / j. nig. soc. phys. sci. 4 (2022) 99–104 100 from reducing the possibilities of transmitting this disease to their newborns, has made the disease manageable thereby enabling the carriers to lead long and healthy lives. unlike other forms of infection, preventing vertical transmission of hiv is more complex, requiring intervention from preconception to the end of breastfeeding. this includes primary prevention of hiv before, during, and after pregnancy; prevention of unwanted or unplanned pregnancies; early initiation of art; retention on art for women with hiv; and retesting women without hiv to detect new infections early. early antenatal care is also important to optimise general care, monitoring of maternal viral load, safe delivery practices, and infant postexposure prophylaxis [10]. though a good number of women within this subgroup have had good access to these therapies and interventions, others are getting it difficult, or at worst, not able to access it at all. several factors may contribute to this vulnerability. it could be legal, religious, illiteracy, biological, social, or sociocultural ([1]; [8]). in the absence of any intervention, transmission rates range from 15% to 45% ([4]; [3]). this rate can be reduced to below 5% with effective interventions during the periods of pregnancy, labour, delivery and breastfeeding. these interventions primarily involve antiretroviral treatment for the mother and a short course of antiretroviral drugs for the baby. they also include measures to prevent hiv acquisition in the pregnant woman and appropriate breastfeeding practices. however, a concerted global effort against this disease has steadily increased the hiv services coverage. apart from the administration of 68% adults and 53% of children living with hiv globally, one of such efforts is the move to eradicate mother to child transmission of the disease. this genuine interest to curb further spread in the west african region also gave birth to nigerian hiv/aids indicator and impact survey (naiis) in 2018 a collaborative project by federal ministry of health, nigeria, university of baltimore, maryland, etc. in attempt to ascertain the prevalence of the disease in the country. the report of this survey as published on naiis website and other media houses showed that akwa ibom state, the major oilproducing state in the country, has the highest number of people living with this virus. though the survey and data collection approach used was best known to the investigators, the report, however, raises some sort of panic amongst the residents of the state. nevertheless, this panic is neither peculiar to akwa ibom state nor nigeria, but to other african sub-region and the world at large. to understand the prevalence trend and other infection dynamics of the disease in this state, a good number of researches have been carried out on this topic. to proffer ways of reducing the mother-to child-transmission (mtct), [2] carried out some findings from an early infant diagnosis program in the south-south region of nigeria (with akwa ibom state inclusive) and concluded that reduction of mtct of hiv is possible with effective prevention of mother-to-child transmision (pmtct) interventions, including improved access to arvs for pmtct and appropriate infant feeding practices. on the same vein, [5] investigated knowledge and practice of prevention of mother-to-child-transmission of hiv among traditional birth attendants practising in the state, and they discovered that the knowledge and practice of pmtct of hiv by the tbas practising in akwa ibom is less than adequate thus increasing the risk of vertical transmission of hiv. the authors suggested an urgent need for the education and training of tbas to improve their knowledge and enhance their practice. on the global scene, many pieces of research are carried out on this topic and they could be found in the literature ([7]; [6]; and others). different from these researches which entirely focus on the preventive measures on tackling mother-to-child-transmission, this paper will employ the technique of logistic regression to give an insight into the proportion of hiv/aids infections attributed to vt in akwa ibom state. based on the data available, the paper will predict the status of newborns given the chosen biomarkers in the pregnant and breastfeeding women. this paper is organised as follows; in section 1, a general introduction is given, section 2 gives an overview of the methodology employed in the data analysis, and section 3 presents the results of the analyses. finally, section 4 concludes with a summary. 2. methodology in the analysis of data related to this research, we will employ the binary logistic regression technique. 2.1. logistic regression analysis regression methods have become an integral component of any data analysis when it comes to deciphering the relationship between a response variable and explanatory variable(s) given that the dataset are discrete in nature. logistic regression model has become the standard of analysis in many fields of studies specifically the medical field. at this point it is pertinent to note that the goal of logistic regression is to find the best fitting and closest, yet reasonable model to describe the affinity or relationship between an outcome that is dependent or response variable and a set of independent otherwise known as predictor or explanatory variables. in statistics, logistic regression, or logit regression, or logit model is a regression model where the dependent variables are categorical. unlike the linear regression analysis which is continuous, it is binary or dichotomous in nature, also the choice of a parametric model and assumptions differs, thus, these features distinguish it from others. it could be noted that once this difference has been accounted for, the procedures employed in an analysis of logistic regression follows the same general principle of that of the linear regression. 2.1.1. the odds consider the linear regression for n-independent variables as follows y = β0 + β1 x1 + β2 x2 + ... + βn xn + ε (1) where y is the response variable, x1, x2, ...,xn are the predictor variables with associated parameters β1 β2, ..., βn respectively. ε is the error term associated with the model. 100 iseh et al. / j. nig. soc. phys. sci. 4 (2022) 99–104 101 the above equation could be considered as binomial case with n-trials concerning the response variable. also, consider the linear regression with three independent variables we have; ŷ = β̂0 + β̂1 x1 + β̂2 x2 + β̂3 x3 + ε (2) in the above equation, x1, x2, and x3 are the biomarkers representing the age of the infected pregnant women, the weight of the pregnant women infected, and the drug dosage taken by the pregnant women infected respectively.. eq (2) is the bernoulli distribution with n = 1 with respect to the dependent variable y , which represent the binary logistic regression with n = 1 trials, where success is ”1” and failure is ”0” also, recall that the probability of success in a bernoulli distribution is p and failure q is given as follows: q = 1 − p recall that, the expectation of a bernoulli distribution, given as e(x) is e(x) = p q (3) for q = 1 − p, eq. (3) becomes e(x) = p 1 − p (4) eq. (3) is called the odds, which is the expected value of a random variable having a bernuoulli distribution. in logistic regression, we estimate an unknown p for any given linear combination of the independent variables, thus, linking the independent variables to the bernoulli distribution gives us the logit(l). this, of course, is the ultimate goal of the logistic regression. thus, we need a function that links the linear combination of variables and the bernoulli distribution together or map the linear combination of variables that could result in any value onto the bernoulli probability distribution with a domain from 0 to 1. this result in taking the natural log of eq.(4), which is the log of the odds. thus this gives; logit(l) = logodd s = ln( p 1 − p ) = ŷ (5) 2.1.2. the binary logistic regression the binary logistic regression model is obtained from the combination of eqs. (2) and (5). so that we have, ln( p 1 − p ) = β̂0 + β̂1 x1 + β̂2 x2 + β̂3 x3 (6) taking the exponent of eq. (6) above, we have p 1 − p = eβ̂0 +β̂1 x1 +β̂2 x2 +β̂3 x3 (7) by simplification, we obtain p̂ = eβ̂0 +β̂1 x1 +β̂2 x2 +β̂3 x3 1 + eβ̂0 +β̂1 x1 +β̂2 x2 +β̂3 x3 (8) eq. (8) gives the estimate of the logistic regression, where x1, x2, and x3 are as defined earlier with their respective estimated parameters β̂1, β̂2, and β̂3, e denotes the exponential function. p is the probability that the child is infected given the mother has the disease. 2.1.3. the odds ratio the odds ratio (denoted as θ) between the odds of two sets of the predictor (say x1 x2). for a continuous independent variable, the odds ratio defined as: θ = p (1−p) ||x = x1 p (1−p) ||x = x2 (9) but the odds of success are as given in (7), 2.1.4. wald test the wald test (also called the wald chi-squared test) is a way to find out if explanatory variables in a model are significant. “significant” means that they add something to the model; variables which add nothing is removed without affecting the model in any meaningful way. the test can be used for a multitude of different models, including those with binary variables or continuous variables. the null hypothesis for the test is: some parametesr are zero in value (not sugnificant). if the null hypothesis is rejected, it suggests that the variables in question can be removed without much harm to the model fit. if the wald test shows that the parameters of some explanatory variables are zero, you can remove the variables from the model. if the test shows the parameters are not zero, you should include the variables in the model. the wald test statistic formula is: wald t est(w j) = β2j s e(β2j ) (10) where; β2j is the square of the linear regression coefficient, s e(β 2 j ) is the square of the standard error of the regression coefficient and is asymptotically distributed as a chi-square distribution with 1 degree of freedom. 2.1.5. the likelihood function for logistic regression (deviance) deviance measures the discrepancy between the current model and the full model. the full model is the model that has n parameters, one parameter per observation. the full model maximizes the log-likelihood function. the full model provides a point of comparison for reduced model, that is, model with fewer than n parameters. comparisons to the full model use the scaled deviance. the contribution to the scaled deviance from each individual data point depends on the model. the likelihood is given in the equation below λ ∗ = −2(l(β0) − l(β1)) (11) where l(β1) is the log-likelihood of the fitted (full) model. l(β0) is the log-likelihood of the (reduced) model specified by the null hypothesis evaluated at the maximum likelihood estimate of that reduced model. 3. data analysis and results r and minitab were used in the analysis of the recorded cases of vertically transmitted hiv/aids in akwa ibom state. 101 iseh et al. / j. nig. soc. phys. sci. 4 (2022) 99–104 102 table 1: 2018 early infant diagnosis (eid) results summaro mnth no. of pw no. of + enrolled not enrolled jan 291 23 16 7 feb 48 15 12 3 mar 193 12 9 3 apr 200 13 11 2 may 89 20 13 7 jun 36 12 10 2 jul 132 33 16 17 aug 96 12 10 2 sep 143 9 7 2 oct 92 5 0 5 nov 250 12 11 1 total 1570 166 115 51 table 2: data count variable value count y 1 166 (event) 0 1404 total 1570 table 3: model summary deviance deviance r-sq r-sq (adj) aic 0.04% 0.00% 1067.35 3.1. data collection data used for this research work are secondary data obtained from the akwa ibom state ministry of health (moh). monthly dataset for the year 2018 was considered. 3.2. data presentation table 1 shows the dataset used for the analysis of this research, pw: pregnant women. +: positive. mnth: month enrolled: number of pregnant women on antiretroviral therapy (art) intervention. not enrolled: number of pregnant women not on antiretroviral therapy (art) intervention. 3.3. results of analyses the data was bernoulli (binary) coded to reflect the status of the respondents. the data were randomly distributed without any prior knowledge about the outcome the data count is as shown in table 2. the outputs are given in table 4: binary logistic regression: y(newborn hiv/aids status-positive (1), negative (0) versus age, weight, dosage (drugs) 3.4. the binary logistic regression function or the logit recall that the logit function is as given below. p = eŷ where ŷ = β̂0 + β̂1 x1 + β̂2 x2 + β̂3 x3 using the above, we have that ŷi = −2.312+0.0037age−0.121weight+0.314dosage (drugs) the probability of the child being infected is computed as follows p = eŷ (1 + eŷ ) for i = 1, the probability of that child being positive given that the mother is negative is given by; p = eŷ 1 + eŷ = 0.108937 1 + e(−2.21699) = 0.098235 thus, the probability that her child will be infected is 0.098235 ≈ 0.1, and this is unlikely to happen given the small probability. this result agrees with the proportion of event count recorded in table 2 as 1661570 ≈ 0.1 3.5. the odds ratio the odds ratio denoted by θ and given in eq. (10) is the odds ratio between two sets of the predictor presented in table 5 below. the result above implies that a unit increase in age increases the odds of a newborn being infected by the mother by a factor of 1.0037. in the same vein, the result shows that a unit increase in weight increases the odds of a newborn being infected by the mother by a factor of 0.8862. and also, a unit increase in dosage of drugs increases the odds of a newborn being infected by the mother by a factor of 1.3691. this follows that age and dosage may increase the odds of a newborn being infected by an infected mother minimally while weight does not yeild such increase at all. 102 iseh et al. / j. nig. soc. phys. sci. 4 (2022) 99–104 103 table 4: coefficients (wgt:weight; dd: drug dosage.) term coef se 95% ci z-val p vif const -2.312 0.503 (-3.298, -1.326) -4.59 0 age 0.004 0.011 (-0.018, 0.025) 0.34 0.733 1 wgt -0.121 0.550 (-1.199, 0.958) -0.22 0.826 1 dd 0.314 0.651 (-0.962, 1.590) 0.48 0.629 1 table 5: odds ratio odds ratio 95% ci age 1.0037 (0.9824, 1.0255) weight 0.8862 (0.3014, 2.6056) dosage (drugs) 1.3691 (0.3821, 4.9056) 3.6. the wald test the wald test is the test of significance for individual regression coefficients in the logistic regression, (recall that in the generalized linear regression analysis we use the t-test statistics). for maximum likelihood estimates, we use eq. (11), and these are z − values given in table 4. using the formula in eq. (10) and the values of β′s, the wald test is computed and the result is as follows, 0.113 for age, 0.0484 for weight and 0.233 for drugs dosage. the smallness of the value of the wald test result for the weight biomarker suggests that the presence of weight may not be necessary for the model. the above indicator informs us to refit the model without the weight; the parameters of the reduced model are presented in table 6; the wald test results for the two biomarkers are 0.116 for age and 0.225 for dosage of drugs. these may not be as strong as should, but it is a better model to be entertained. 3.7. the likelihood ratio test or deviance here, we use the likelihood-ratio test to compare two models. specifically, we are testing h0 : βage = βweight = βdosage = 0 the likelihood ratio test is used to test the null hypothesis (h0) that any subset of the β′s = 0. where the number of β′s = 0 in the full model is p, while the number of the β′s = 0 in the reduced model is r. (r is the model that results when β′s = 0 in the null hypothesis are set to 0). the result is as given in the analysis of deviance in table 7. from the result of our wald test, it is observed that the value of the test for weight was small. at that point, it was suspected that fitting a new model without the variable would give a better fit. due to this, a reduced model with weight was fitted and the deviance analysis carried out. we see that the test statistic that we had just calculated appears in the output. the very big p-value suggests that there is no significant difference between the larger model and the reduced one. the values of the residual deviance for the two models also support the above claim as the two values are almost the same. hence, there is need for the weight biomarker to be removed from the model since it may lead to overfitting. of course, this is also an indication that in investigating the vt infections from positive pregnant women to the newborns, the weight of the women is not necessarily one of the determining factors. an overview of the two fitted models generally suggests that age, weight, and dosage of drugs taken by reactive pregnant women do not imply or contribute to the vertical transmission of the hiv/aids from the mothers to their newborns. 4. conclusion this paper is designed to explore the proportion of hiv/aids infections attributed to vt in akwa ibom state. from the data collected from the akwa ibom ministry of health on the virus and vt related cases, the paper used the binary logistic regression analysis to predict the hiv status of newborns given the chosen biomarkers in the pregnant and breastfeeding women, in this case, age, weight, and drug (dosage) intake. in doing this, the logit function, the odds ratio, the wald test, and the likelihood ratio test were used to explain the adequacy of the fitted model. the results clearly showed that given these biomarkers (age, weight, and dosage of drugs), the rate of spread of the hiv/aids is seldom through vertical transmission, that is, from an infected mother to her newborn. the implication of this finding is that the prevalence rate of the disease in akwa ibom state may be attributed more to other modes of transmission and less to vt. while it is important to continue in the interventions focusing on antiretroviral treatments for the positive mothers, it is more advisable to intervene by creating more awareness on the other modes of transmission. 5. abbreviations hiv human immunodeficiency virus. aidsacquired immune deficiency syndrome. vtis vertically transmitted infections. vt vertical transmision. mtct mother-to-child transmision. pmtct prevention of mother-to-child transmision. naiis nigeria aids/hiv indicator and impact survey. art antiretroviral therapy. 6. availability of data and materials all data used for supporting the conclusions of this article are available from the public data repository at the website https://zenodo.org/record/4243862. 103 iseh et al. / j. nig. soc. phys. sci. 4 (2022) 99–104 104 table 6: coefficient of the reduced model estimate std. error t value pr(> |t|) (intercept) -2.373300 0.418955 -5.665 1.75e-08 age 0.003736 0.010968 0.341 0.733 dosage of drugs 0.308619 0.650987 0.474 0.636 table 7: analysis of deviance table resid. df resid. dev df deviance pr(> chi) 1 1566 1059.3 2 1567 1059.4 -1 -0.048187 0.8265 references [1] e. a. anigilaje, o. j. dabit, b. ageda, s. hwande, t. t. bitto, “the prevalence and predictors of hiv infection among children of mothers who missed prevention of mother to child transmission of hiv interventions in makurdi, nigeria”, j aids clin res. 4 (2013) 249. [2] c. anoje, b. aiyenigba, c. susuki,t. badru, k. akpoigbe, m. odo, s. odafe, o. adedokun, k. torpey and o. n. chabikuli, “reducing motherto-child transmission of hiv: finding from an early infant diagnosis program in south-south nigeria” bmc public health 12 (2012) 184. [3] besser, m. hiv in pregnancy: doing more with less: mothers2mothers (2010). [4] e. m. connor, r. s. sperling, r. gelber, p. kiselev, g. scott, m. j. o’sullivan, r. vandyke, m. bey, w. shearer, r. l. jacobson & e. jimenez, “reduction of maternal-infant transmission of human immunodeficiency virus type 1 with zidovudine treatment”, new england journal of medicine 331 (1994) 1173. [5] u. i. eshiet, & p. g. ekpe, “knowledge and practice of prevention of mother-to-childtransmission of hiv among traditional birth attendants practicing in akwa ibom state, nigeria”, british journal of pharmaceutical research 13 (2016) 10. 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[8] b. o. ogunbosi, r. e. oladokun, o. awolude, b. j. brown, o. a. adeshina, m. kuti, b. taiwo, b. berzins, d. n. kyriacou, e. g. chadwick and k. osinusi, “missed opportunities for prevention of mother-to-child transmission of hiv (pmtct) in ibadan, southwest nigeria”, world j aids 4 (2014) 356. http://dx.doi.org/10.4236/wja.2014.43042 [9] world health organization hiv/aids key facts. www.who.int/newsroom/facts-sheets/detail/hiv-aids, (2020) [10] world health organization. use of antiretroviral drugs for treating pregnant women and preventing hiv infection in infants. https://www.who.int/hiv/pub/mtct/programmatic update2012/en/ (2012) 104 j. nig. soc. phys. sci. 1 (2019) 6–11 journal of the nigerian society of physical sciences original research antimicrobial assessment of different solvent extracts of the root-bark, stem-bark and leaves of acacia ataxacantha linn. o. s. oguntoyea, a. m. okoobohb, o. m. belloc,d,∗, a. h. usmana, s. s. abdussalama, g. v. awololad adept. of applied chemistry, faculty of science, federal university dutsin-ma, katsina state, nigeria. bfederal college of education, pankshin plateau state-nigeria cdept. of chemistry, faculty of science, university of ilorin, kwara state, nigeria. ddept. of industrial chemistry, faculty of science, university of ilorin, kwara state, nigeria. abstract acacia ataxacantha dc. (fabaceae) is one of the most important medicinal plant used traditionally in managing many ailments in nigeria particularly infectious diseases. in this study, the stem-bark, roots and leaves of a. ataxacantha were extracted using hexane, ethyl acetate, methanol and water (from non-polar to polar solvents). these extracts were initially screened to identify the secondary metabolites present. they were then examined against eight bacteria (pseudomonas aeruginosa, escherichia coli, staphylococcus aureus, bacillus subtilis, klebsiella pneumoniae, streptococcus pneumonia, proteus mirabilis, salmonella typhi-murium) and seven fungi (candida albicans, penicillium spp, aspergillus fumigatus, aspergillus flavus, trichophyton megninii, trichophyton. rubrum, salmonella typhi-murium). disc diffusion method was employed to establish the antimicrobial activity of the plant parts. serial dilution method was used to determine the minimum inhibitory concentration (mic), minimum bactericidal (mbc) and fungicidal concentration (mfc). methanol and hexane fractions of the root bark exhibited the highest inhibition zone ranging from 19-21 mm against the tested microorganism hence these were further assessed by determining their mic, mbc and mfc. mic value for the aqueous extract of the plant was 25 mg/cm3 against the fungi used while the methanol fraction was 15 mg/cm3 against the bacteria used. this work suggests that acacia ataxacantha may possess compounds which can be template for acquiring novel drugs which could act against infectious diseases. keywords: acacia ataxacantha, infectious diseases, mic, antimicrobial drugs article history : received: 13-03-2019 received in revised form: 06-04-2019 accepted for publication: 06-04-2019 published: 09-04-2019 c©2019 journal of the nigerian society of physical sciences all rights reserved. communicated by: b. j. falaye 1. introduction over a million and half of deaths associated with human diseases in the tropical countries are from infectious diseases. the cause of death and illness amongst inhabitants in developing countries like west africa can be traced to these infectious ∗corresponding author tel. no: +2348062320327 email address: obello@fudutsinma.edu.ng (o. m. bello ) diseases [1]. bacteria, viruses, parasites or fungi are some of these deadly microorganism, they are pathogen in nature and they are responsible for the alarming and high incidence of infectious diseases [2]. the recent discovery of developing resistance of pathogens to prevailing and current antibiotics is obvious [3]. there is a pressing demand to discover novel compounds with excellent antimicrobial activities because there has been a growing and disturbing increase in the occurrence of new and re-emerging infectious diseases [4, 5]. 6 oguntoye et al. / j. nig. soc. phys. sci. 1 (2019) 6–11 7 many bacterial species already have the genetic make-up to gain and transmit resistance against the current available antibiotics. many clinical challenges experienced around the world are as a result of the resistance posed by pathogens. the growth of this incidence is so alarming that, it has provided access to a new wave of widespread consumption of antibiotics [6-9]. the search for new antimicrobial agents is a serious burden due to the constant emergence of micro-organisms resistant to conventional antimicrobials, associated and adverse side effects of some antimicrobial drugs and the suddenly emergence of some uncommon infections [10, 11]. pathogen resistance to currently used antibiotics increases death incidence, there is serious likelihood of increase in patients visitation and stay in the clinics and hospitals [12]. plant-based therapy has been used against array of diseases for generations hence looking to them for solution against microbes may not be an exception [5,13]. many secondary metabolites of excellent biological activities have been isolated from mother “nature”. nature wherein medicinal plants are found, provides a serious hope a better template in the discovery of new antimicrobial drugs [ 5, 14]. it is obvious from the literature gathered that little work has been done on establishing the phytochemistry of acacia ataxacantha. so far only five compounds have been isolated from the root-bark of this medicinal plant. most biological evaluations were done on the root-bark, very few research was done on the stem bark while none on the leaves [15]. the medicinal plant is renowned traditionally for its antimicrobial uses, so there is a need to further explore, compare and justify using all the different parts i.e. stem bark, leaves and root-bark for antimicrobial use. this will serve as basis for further studies on the phytochemistry of this plant employing spectroscopic techniques to elucidate the active and non-active constituents in the fractions. 2. materials and methods 2.1. plant material different parts of a. ataxacantha were obtained from jos, nigeria in november, 2016. the plant was properly identified by the curator of the herbarium, department of biological science, university of jos, nigeria. 2.2. extraction the root-barks, stem-barks and leaves were air-dried and grounded using wooden mortar and pestle. the powdered sample was subjected to cold extraction (maceration) using hexane and then methanol. the crude methanol extract was suspended in water and washed with ethyl acetate, in that order, to yield water (waf), methanol (mef), ethyl acetate fraction (eaf) and hexane (hef) respectively. 2.3. phytochemical screening standard techniques of screening and detecting secondary metabolites in plants was used [16, 17]. the metabolites tested for were alkaloids, anthraquinones, cardiac glycosides, carbohydrates, flavonoids, saponins, steroids, tannins and terpenes. 2.4. determination of zone of inhibition. the zone of inhibition of a. ataxacantha plant extracts were determined by using selected pathogenic micro-organisms obtained from the national veterinary research institute, (nvri) vom, near jos in plateau state. purity of the clinical isolates employed were verified and they were preserved in nutrient agar. the weight of the plant extracts (0.2 g) were taken, dimethyl sulphoxide (dmso) (10 mls) was used to dissolve the plant extracts to have a fixed concentration of 20 mg/ml. this was the preliminary concentration of the extracts that was employed to test the antimicrobial properties of the plant extracts. the growth medium for the microbes was mueller hinton agar. the medium was prepared according to the manufacturers guidelines were followed in preparing the medium used. furthermore, zone of inhibition of a. ataxacantha plant extracts were evaluated according to the methods used by lalnundanga et al. and mathur [18, 19]. then these plates of the medium were observed for the zone of inhibition of growth of the microbes, the diameter of the zones were taken and recorded in millimetres (mm). 2.5. determination of minimum inhibitory concentration (mic) resistance is a growing menace, mics are often used by laboratories to check resistance, though it is also employed as a means to evaluate the antimicrobial activity (in vitro) of new agents [20]. the mic of the methanol and water crude extracts of the root bark of a. ataxacantha was verified using broth dilution method [21]. the broth labelled “mueller hinton broth” was made by strictly following the manufacturers instructions. the mic of the various solvents fractions of the parts of a. ataxacantha was determined using the modified agar well diffusion method [16, 18, 19, 22]. 2.6. determination of minimum bactericidal concentration (mbc). the minimum bactericidal concentrations (mbcs) of the water and methanol extracts were ascertained by selecting and sub-culturing the tubes that did not show growth during mic determination, into fresh medium which contained no extract. wire loop was used to transfer from these tubes, these were sub-cultured over the surface of extract free nutrient agar, in petri dishes. these were incubated for 24 h at 37 ◦c. the lowest extract concentration from which the organism did not recover and grow on the nutrient agar was documented as mbc. 2.7. determination of the minimum fungicidal concentration (mfc). sabour and dextrose agar (sda) plates were made following the manufacturers instructions. the lowest dilution that did not show growth during mic was inoculated onto the sda plates. these plates were incubated at ambient temperature for 24 hours. the plates were then observed for growth. the lowest extract concentration from which the fungi did not recover and grow on the agar was noted as the minimum fungicidal concentration (mfc). the result is recorded in table 2.0. 7 oguntoye et al. / j. nig. soc. phys. sci. 1 (2019) 6–11 8 2.8. test organisms the biological activities of the extracts were carried out at the national veterinary research institute (nvri), vom, plateau state, nigeria. all the microorganisms used are of clinical strains from the same institute. the choice of these pathogens is based on their implication in human diseases such as typhoid, enteric fever, pneumonia, urinary tract infections, dysentery, wound infections and others. 3. result and discussion 3.1. phytochemical screening phytochemical constituents extracts of a. ataxacantha are shown in table 1. on the whole, coumarins, alkaloids, saponins, tannins, flavonoids, tannins, anthraquinones, cardiac glycosides and triterpenes were identified in all extracts. the ethyl acetate of the root-bark extract gave a poor result for most groups of secondary metabolites investigated. the phytochemical screening reveals that flavonoids and carbohydrates are present in the various extracts. 3.2. diameters of zones of inhibition the bacteria and fungi inhibition produced by a. ataxacantha extracts varied in relation to the type of extract and to the micro-organism strain used. though most of the crude extracts exhibited varying degree of inhibition zones. methanol and hexane extracts of the roots of a. ataxacantha had the highest inhibition zone ranging from 10-19 mm against the tested bacteria used while the ethyl acetate extract of the stem bark and the hexane extract of the roots showed the highest zone of inhibition ranging 05 10 mm against the tested fungi. implying that it is the more likely active fraction. the antibacterial activity of the exhibited by the roots extracts is good though from this study the antifungal activity is not too commendable. the antibacterial activity of the plant must be due to the phytochemicals identified in the extracts. 3.3. mic, mbc and mfcs result 3.3.1. mic this was carried out to determine the minimum concentration of the extracts that can inhibit the growth of the microbes. the mic results showed for only aqueous and methanol extract of the root-bark for the plant, these were chosen because of their interesting activity by the zone of inhibition as table 4 reveal. the extracts showed moderate activity against the bacteria and fungi used. the water extract of the roots exhibited a moderate activity against the fungi with mic of 25 mg/ml while the methanol extract of the same plant showed moderate activity against the bacterial with mic ranging from 25-50 mg/ml. the former showed mic of 25 mg/ml against p. aeruginosa which a gram negative bacteria but the latter showed a better mic against all the tested bacterial. 3.3.2. mbc and mfc determination of mbc and mfc was done to check if microbes evaluated were actually killed by the extracts or their growth was impeded. the results from mbc and mfc showed that none of the extract killed the bacteria after they were subcultured onto agar plate. that is when the results from mic test tubes were sub-cultured, they still showed growth. this means none of the extract was bactericidal to the eight (8) bacteria employed for the study. in the same vein, the results showed no fungicidal effects on the three (3) fungi used in the study. in this study, the preliminary phytochemical screening tests were done for three different parts of the a. ataxacantha (leaves, stem bark and roots), each of the parts into four different polarities (water, methanol, ethyl acetate, hexane). the tests showed that alkaloids, anthraquinones, cardiac glycosides, coumarins, flavonoids, saponins, tannins and triterpenes were present in all extracts as shown in table i. this screening may be useful in the discovery of the bioactive compounds and subsequently may lead to drug discovery and development. in antibacterial and antifungal studies carried out earlier, akapa et al. (2015) investigated the antibacterial and antifungal of the root-bark of a. ataxacantha using different polarities [23], it was reported that ethyl acetate fraction showed the most significant result with mic of 2.5 mg/ml towards b. subtilis, e. coli, s. typhi and k. pneumonia [24, 25]. the antifungal activity of stembark of a. ataxacantha extracts against six strains of aspergillus was determined by amoussa et al. [14] the ethyl acetate extract of the stem bark was reported to significantly inhibits the growth of mycelial and sporulation of aspergillus strains. this study compares favourably with the literature where the stem bark extract (ethyl acetate fraction) of the plant was the most active against the fungi used. looking at the results in table ii and iii, it was found that the stem bark extract (ethyl acetate fraction) showed the most interesting zone of inhibition (10 mm) against candida albican in comparison with the control (0.5 mm), the same extract showed a better antibacterial activity against s. pneumoniae with an inhibition zone of 15 mm, which is better than to the control (10 mm). the ethyl acetate extract of the stem-bark showed an inhibition zone of 5 mm against t. rubrum while the control showed no zone of inhibition. looking at the results in the table iii for antibacterial activity, it was found that the hexane and methanol extracts of the root bark showed the most interesting zone of inhibition (19 mm) against pseudomonas aeruginosa in comparison with the control “cyprofloxacin” (19 mm), the hexane extract of the root-bark showed also an interesting antibacterial activity against s. pneumoniae with an inhibition zone which is equal to 15 mm, which is better than to the control (10 mm). though the ethyl acetate extract of the leaves showed an inhibition zone of 20 mm against s. aureus as same with the control (20 mm). figure 1 showed that methanol extract of the rootbark gave the highest zone of inhibition against s. typhi when compared with the other extracts. ethyl acetate extract of the leaves of a. ataxacantha, methanol and ethyl acetate extracts of the stem-bark displayed similar zone of inhibition against the fungi used. figure 2 revealed the antibacterial activities of 8 oguntoye et al. / j. nig. soc. phys. sci. 1 (2019) 6–11 9 table 1. the phytochemical screening of different parts of a. ataxacantha with solvents of different polarities. h2o : water, meoh : methanol, e.a : ethyl acetate, hex : hexane, +++ = very good, ++ = good, + = fair, = not present s/n constituents leaf extract stem bark extract root-bark extracts h2o meoh e.a hex h2o meoh e.a hex h2o meoh e.a hex 1 alkaloids + 2 saponins + + + +++ +++ ++ + 3 tannins ++ +++ ++ + ++ + +++ 4 flavonoids +++ +++ ++ + ++ ++ ++ ++ 5 carbohydrates ++ ++ ++ + ++ +++ ++ + +++ ++ +++ 6 steroids + ++ ++ ++ + ++ +++ + + ++ 7 anthraquinones ++ + + ++ + 8 cardiac glycosides + + + + + + 9 terpenes ++ + % yield 8.2 17 3 2 2.4 8.8 1 0.68 3 10.4 table 2. diameter of zones of inhibition of fungi (mm). h20: water, meoh: methanol, e.a: ethyl acetate, hex: hexane, values are means of three replicates (n = 3 ± s d). s/n fungi ctrl leaf extract stem-bark extract root-bark extracts h2o meoh e.a hex h2o meoh e.a hex h2o meoh e.a hex 1 candida albican 0.5 0 0 0 08 0 0 10 0 0 06 0 0 2 penicillium spp 0.8 0 0 0 0 0 0 0 0 0 0 0 05 3 a. flavus 10 0 05 08 05 03 03 08 05 03 0 0 05 4 a. fumigatus 12 0 05 05 0 05 05 05 05 05 0 0 05 5 t. megninii 0.3 0 0 0 0 0 0 0 0 0 0 0 05 6 t. rubrum 0 0 0 0 0 04 0 05 0 0 0 0 0 7 s. typhi 12 08 09 10 05 0 10 10 03 0 11 0 0 table 3. diameter of zones of inhibition of bacteria (mm). h2o = water, meoh = methanol, e.a = ethyl acetate, hex = hexane, ketoconazole (an antifungal conventional drug), cyprofloxacin with (an antibacterial conventional drug). values are means of three replicates (n = 3 ± s d). s/n bacteria ctrl leaf extract stem-bark extract root-bark extracts h2o meoh e.a hex h2o meoh e.a hex h2o meoh e.a hex 1 p. aeruginosa 19 0 0 0 0 0 0 0 0 0 19 0 19 2 e. coli 13 0 09 11 10 09 0 11 10 0 09 0 11 3 s. aureus 20 0 10 20 12 0 0 13 0 15 10 0 11 4 bacillus spp 20 10 0 09 10 0 08 07 0 0 08 10 10 5 k. pneumoniae 30 10 15 15 10 10 10 10 05 11 10 0 10 6 s. pneumoniae 10 07 0 0 0 10 07 0 0 0 10 0 15 7 p. mirabilis 13 10 0 09 10 0 0 10 0 0 10 0 07 8 s. typhi 15 10 11 10 10 0 10 10 09 0 10 0 11 the various parts of this medicinal plant. ethyl acetate extract of the leaves gave the highest zone of inhibition against s. aureus 9 oguntoye et al. / j. nig. soc. phys. sci. 1 (2019) 6–11 10 table 4. mic result of the crude methanol and water extracts of the root-bark of acacia ataxacantha. mer: methanol extract of the root, wer: water extract of the root, values are means of three replicates (n = 3 ± s d). s/n test organism bacteria mer (mg/ml) wer (mg/ml) control +ve control ve 1 pseudomonas aeruginosa 200 25 + 2 escherichia coli 50 400 + 3 staphylococcus aureus 50 400 + 4 bacillus spp 100 400 + 5 klebsiella pneumoniae 100 400 + 6 streptococcus pneumonia 25 200 + 7 proteus mirabilis 400 400 + 8 citrobacter spp 50 100 + fungi 1 candida abaccum 25 25 + 2 aspagirus flavus 100 25 + 3 aspagirus funigatus 100 25 + while methanol extract of the root-bark showed a promising activity too. the presence the phytochemical compounds in these active extracts are responsible for the antimicrobial activity. the mic results showed in table iv are for aqueous and methanol extract of the root bark of the plant, these were chosen because of their interesting activity by the zone of inhibition. the mic values for methanol extract of the root was better than the aqueous extracts for bacteria while aqueous extract was better for the fungi tested. aqueous extract showed the mic values of 25 mg/ml against the fungi i.e. candida abaccum, aspagirus flavus, aspagirus funigatus while methanol extract displayed a mic value of 25 mg/ml against only candida abaccum and displayed mic values of 100 mg/ml and 200 mg/ml against both aspagirus flavus and aspagirus funigatus respectively. the methanol extract of the root exhibited a moderate activity against streptococcus pneumonia with mic of 25 mg/ml while the aqueous extract of the same plant showed a very low activity against the same bacterial with mic ranging from 200 mg/ml. mic values for the aqueous extract are relatively low compared to methanol extract. this confirm the reported work of akapa et al. [23] that methanol and chloroform fractions of root-bark had mic values of 5 mg/ml and petroleum ether had 10 mg/ml for all the test organisms [23], though ethyl acetate gave the most significant result with 2.5 mg/ml for b. subtilis, e. coli, s. typhi and k. pneumonia [23, 24]. the result for mbc and mfc tests for both the methanol and water extracts on bacteria and fungi are not significant. 4. conclusion this study on the antimicrobial activity of different fractions of the roots, stembark and leaves of acacia ataxacantha figure 1. antifungal activities of different parts of a. ataxacantha figure 2. antibacterial activity of different parts of a. ataxacantha 10 oguntoye et al. / j. nig. soc. phys. sci. 1 (2019) 6–11 11 linn were carried out employing disc diffusion method. the extracts were examined against eight bacteria (which consist of both + gram and gram bacteria) (p. aeruginosa, e. coli, s. aureus, bacillus spp, k. pneumonia, s. pneumonia, p. mirabilis, s. typhi ) and seven fungi (c. albican penicillium spp, a. flavus, a. fumigatus, t. megninii, t. rubrum, s. typhi). within the use of traditional nigeria medicine, our studies are in agreement with the reputed potency of parts of a. ataxacantha against common bacterial and fungal complaints. the facts in this study confirms the use and potency of a. ataxacantha as an effective antimicrobial plant whose active principles could serve as a potential chemotherapeutic agent. acknowledgments we thank the referees for the positive enlightening comments and suggestions, which have greatly helped us in making improvements to this paper. conflict of interest no conflict of interest funding no funding was received for this study references [1] a. o. amoussa, a. l. lagnik & a. sanni, “acacia ataxacantha (bark): chemical composition and antibacterial activity of the extracts”, international journal of pharmacy and pharmaceutical science 6 (2014) 11. 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[25] k. seo, h. akiyoshi & y. ohnishi, “alteration of cell wall composition leads to amphotericin b resistance in aspergillus flavus”, microbiology and immunology 43 (1993) 1017. 11 j. nig. soc. phys. sci. 3 (2021) 121–130 journal of the nigerian society of physical sciences lattice dynamics and thermodynamic responses of xnbsn half-heusler semiconductors: a first-principles approach o. e. osafile∗, o. n. nenuwe department of physics, federal university of petroleum resources, pmb 1221, effurun, nigeria abstract the need to predict the properties of new half heusler alloys for possible technological applications from first principles cannot be overemphasised. this need is driven by the huge industrial demand for desirable properties under the influence of temperature. in this work, xnbsn (x = co, ir, rh) half-heusler (hh) alloys were investigated for the structural, electronic, elastic, and thermodynamic properties using pbegga exchange-correlation functional as implemented in the quantum espresso computational suite. xnbsn alloys crystallise in the face centred cubic (fcc) structure with space group f-43m and number 216. the lattice structures were optimised, and the lattice constants of conbsn alloy compares well with reports in literature, while irnbsn and rhnbsn are reported on for the first time. xnbsn hh alloys are non-magnetic semiconductors with bandgaps 1.03, 0.72 and 0.78 for conbsn, irnbsn and rhnbsn alloys, respectively. the splitting of the bandgap at the fermi level is dominated by the d-d hybridisation between the d orbitals of nb and co/ir/rh. the negative formation energies obtained from computing the formation enthalpy support the structural stability and possible experimental simulation of the alloys. there is a clear bandgap across the brillouin zones at a frequency of 150 cm−1; rhnbsn exhibits the largest bandgap while the bandgap is almost non-existent in conbsn. non-negative frequencies in the phonon dispersion predict the thermodynamic stability of the alloys. the alloys dulong-petit law is obeyed at a temperature of about 600 k and heat capacity of 74 j mol −1k −1. the phonon dispersion and density of states show that the d orbitals of co and nb are the dominant contributors to the dispersions at both the acoustic and optical modes of the alloys. irnbsn alloy has the least debye temperature (low thermal conductivity), positioning it as the most promising material for possible thermoelectric applications. doi:10.46481/jnsps.2021.174 keywords: density functional theory, density functional perturbation theory, half-heusler compounds, phonon dispersions, thermodynamic properties. article history : received: 15 march 2021 received in revised form: 04 may 2021 accepted for publication: 08 may 2021 published: 29 may 2021 ©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: a. h. labulo 1. introduction heusler alloys are a large family of alloys with properties that can be modified to suit technological applications in several areas of scientific and engineering research. there are ∗corresponding author tel. no: +2347033249124 email address: osafile.omosede@fupre.edu.ng (o. e. osafile ) two main classes of heusler alloys; these are the full and half heusler alloys. albeit, there are several variants arising from the two main classes [1-5]. heusler first announced the full heusler alloys in 1903 [6,7]; it has the x2yz (2:1:1) stoichiometry and generally crystallises in the fcc phase and l21 structure. the half heusler alloy, on the other hand, was discovered in 1983 by de groot et al. [8]. 121 gemanam et al. / j. nig. soc. phys. sci. 3 (2021) 121–130 122 it has the xyz (1:1:1) stoichiometry and crystallises mostly in the fcc phase. the x and y elements are transition or rare earth metals, while the z element is a main group (or a pblock) element. the properties of the heusler alloy can be tuned via doping, strain & stress application, vacancies, impurities, dislocations, and heat treatments to achieve the desired property for technological applications [9-12]. in recent times, the thermoelectric properties of materials became desirable research focus primarily due to their applications in power generation and refrigeration from supposed waste materials. a material is deemed useful for thermoelectric application based on the efficiency, and the efficiency is parametrised by the seebeck coefficient, electrical conductivity, and thermal conductivity. the efficiency is measured relative to the temperature change using the three parameters mentioned above. fermi energy plays a prominent role in the seebeck coefficient and electrical conductivity of the material. the fermi level is defined by the electron and hole concentrations of the alloy, effective mass, and temperature. suggestions in literature indicate that decoupling the seebeck coefficient and electrical conductivity without compromising efficiency can be achieved by employing the transport functions of the alloys [13]. the most efficient materials are expected to exhibit high electrical conductivity, high seebeck coefficient and a relatively low thermal conductivity to minimise thermal losses [14-16]. despite the gains from other materials as thermoelectric materials, their setback positions the half heusler alloys as promising materials in thermoelectric applications. in 2016 huang et al. reported a zt of ≈ 1 in nbcosn, nbfesn and vfesb in the cubic mgagas fcc phase [17]. liu et al. recorded a zt of ≈ 1 and 0.7 for pand n-type materials at a temperature of 978 k respectively by combining 17-electron tifesb and 19-electron tinisb hh alloys, based on the same parent tife0.5ni0.5sb alloy [18]. an experimental report by xia et al. showed that zt = 0.9 was obtained for nb0.83cosb at 1123 k compared to the pristine n-type nbcosb and zt peak of approximately 0.4 at 973 k [19, 20]. also, the p-type fenbsb doped by hf exhibited a zt 0f 1.5 at a temperature of 1200 k putting its performance ahead of other materials at such high temperature [21]. the beauty of the half heusler alloy is that elements can be interchanged, replaced, and doped to achieve properties desired for specific applications. many n-type and p-type half heusler semiconductor materials have been predicted from first-principles calculations and experiments for application as thermoelectric materials. these half heusler alloys have been reported to exhibit substantial te properties comparable to conventional bi2te3 and pbte based materials [22-27]. poon et al. reported that the n-type hf0.6 zr0.4 nisn0.995 sb0.005 hh alloy and p-type hf0.3 zr0.7 cosn0.3 sb0.7 /nano-zro2 composites achieved zt = 1.05 and 0.8 near 900 – 1000 k, respectively [28] . in another work, joshi et al. reported a high zt value of ∼1 at 700 ◦c for a nanostructured p-type nb0.6ti0.4fesb0.95sn0.05 composition [25]. another attraction of the half-heusler alloys is the possibility of tuning the bandgap. the bandgap of a thermoelectric material plays a vital role in defining the scope of application in a thermoelectric device. recall that the dimensionless figure of merit zt is given as z t = σs 2t /κ where σ is the electrical conductivity, s is the seebeck coefficient, t is temperature, and κ is the thermal conductivity. the thermal conductivity is given by the sum of contributions from the electronic carriers (κe l ) and the lattice (κl ) carriers. zt is enhanced when σ and s increase as κ decreases. however, the σ, s & κ parameters are interdependent, such that it becomes a challenge to increase zt. if the thermal conductivity must decrease, the lattice carrier κl is expected to dominate. one way to increase the electrical conductivity σ is to increase the carrier concentration. unfortunately, increasing carrier concentration compromises the seebeck coefficient and increases the electronic carrier so that the thermal conductivity increases. the half heusler semiconductors have come in handy in solving this problem because the tuning of the bandgap (that is, increasing or decreasing the size of the bandgap) can help separate the n-type and p-type carriers as the concentration increases, hence, having a single carrier type without compromising charge mobility. tuning the bandgap also provides the opportunity of reducing the thermopower as the temperature increases. thermopower reduction happens when the bandgap is narrow enough; hence, at an excited state, n-type carriers can cross to the conduction band and p-type carriers to the valence band, resulting in a cancelling effect that creates a balance [29-32]. the present work is focused on investigating the lattice and thermodynamic responses of xnbsn (x = co, rh, ir) halfheusler semiconductors considering the lattice dynamics and phonon dependent parameters in the cubic (f-43m) space groups via first-principles calculations. this work is motivated by suggestions that these 18-valence half heusler semiconductors are promising materials for thermoelectric applications [13]; these alloys are known to have closed-shells nonmagnetic and semiconducting [33]. there are no substantive claims for the mechanical stability of xnbsn alloys; hence, their viability as thermoelectric materials are unsubstantiated. though, mubarak et al. reported on the thermal, electro magnetic and thermoelectric properties of conb1− x tix sn (x= 0, 0.75, 0.5, 1) hh alloy using the dft full-potential linearized augmented plane wave (fp-lapw ) method as implemented in the wien2k code [34], there are no experimental reports to compare with. there are no reports in the literature for lattice dynamics and thermodynamic properties of irnbsn and rhnbsn compounds to the best of our knowledge. the lattice dynamical parameters and phonon properties currently absent in literature play a vital role in understanding the physical properties; hence in this work, we investigate the structural, electronic properties, formation energies, mechanical stability, and lattice dynamical properties of the alloys in detail. the rest of this article is organised as follows, and section 2 reports the computational method while results are presented and discussed in section 3, followed by the conclusion. 122 gemanam et al. / j. nig. soc. phys. sci. 3 (2021) 121–130 123 2. computational method 2.1. structural and electronic properties we have used the fcc primitive unit cell for structural and electronic property calculations with three atoms (x, nb, sn) in this investigation, where x is co, rh, or ir. the first principles investigations are based on the quantum espresso (qe) computational suite [35, 36]. the structural and electronic property calculations are computed within the framework of the density functional theory (dft) implemented in qe. the projector augmented wave (paw ) functional and the generalised gradient approximation (gga) with the perdewburke-ernzerhof (pbe) exchange and correlation between electrons [37-39] was adopted for all calculations. an 8×8×8 grid in the monkhorst-pack scheme [40] was constructed for the electronic structure calculation and convergence of 70 ev kinetic energy cut-off of the plane-wave as the basis function. to ascertain the convergence of the results, we set a 10−12 ry criterion for convergence of the total energy, and a marzarivanderbilt [41] smearing of 0.02 thickness was applied to the alloys. before the computation of the structural and electronic properties, the alloys’ unit-cells were fully relaxed to obtain the equilibrium configuration for the atomic positions. the converged values for the smearing, lattice constant, k-point grid, and atomic positions were inputted for the electronic structure calculation. a denser k-point mesh of (14×14×14) grid was used to calculate states’ electronic density (dos) calculation with a tetrahedra occupation. at equilibrium temperature, the relevant information about the structural behaviour of the alloys was extracted by fitting the result obtained from the total energy calculation to the birch-murnaghan equations of state [42-44]. for the band structure calculation, dense high symmetry k-points were selected along w → l →g → x→w in units of 2p/a. using crystalline structures and densities (xcrysden) package [45] in the fcc crystallised phase for visualisation. 2.2. mechanical and phonon properties after computing the structural and electronic properties of the systems, the elastic, mechanical, thermodynamic, and phonon dispersion properties were determined using the density functional perturbation theory (dfpt). the linear response method in the quasi-harmonic approximation as implemented in the thermo-pw package [46,47] was explored as a post-proc in qe. for convergence, a 4×4×4 uniform supercell was used with symmetry implemented. thermodynamic quantities immediately obtained from the approximation includes the vibrational energy ei = ∑ q ,ν ( nq ,ν + 12 ) }ωq ,ν of the phonon modes where nq ,ν is the number of phonons and ωq ,ν is the phonon frequency. others include the isochoric heat capacities, elastic constant and compliances, bulk modulus, and thermal expansion using voigt, reuss, and hills [48-50] relations. table 1: equilibrium lattice constant a0 (å) and energy (ry) of the three possible configurations of conbsn, rhnbsn and irnbsn half heusler alloys using pbe-gga approximation. compound eg (ry) a0 (å) xyz (0.0, 0.5, 0.25) conbsn -938.227 6.188 rhnbsn -1246.416 6.310 irnbsn -1497.761 6.299 yzx (0.25, 0.0, 0.5) conbsn -938.260 6.225 rhnbsn -1246.457 6.340 irnbsn -1497.839 6.322 zxy (0.5, 0.25, 0.0) conbsn -938.451 5.981 rhnbsn -1246.597 6.214 irnbsn -1497.984 6.246 3. results and discussion 3.1. structural and electronic properties the electronic configuration for co, rh and ir transition metals are (3d7 4s2), (4d8 5s1), and (5d7 6s2) with atomic mass units of 58.3195u, 102.9055u, and 192.217u, respectively. in comparison, nb and sn have electronic configurations of (4d4 5s1) and (5s2 5p2), respectively. the three alloys are 18 valence electron hh non-magnetic semiconductors. fundamentally, there are three possible configurations for hh alloys. we investigated the three possible configurations of the alloys to establish the equilibrium lattice constant and the most stable structural state of the alloys. in table 1, results for the atomic positions, energy and lattice constants for the three configurations of the alloys are presented, and the results for the variation of energy with the lattice constant are represented in figure i. figure i shows that the most stable state of the three alloys is the zxy configuration. the hh alloy crystallises in the c1b structure belonging to the f 43m group with space group number 216. the result shows that irnbsn has the lowest ground state energy of 1497.984 ry. it also exhibits the largest volume while conbsn has the highest ground state energy and the smallest volume; this is expected considering the atomic mass of ir compared to co and rh. the relevant information about the structural behaviour of the alloys was extracted by fitting the result obtained from the energy-volume optimisation calculation to the birch-murnaghan equation of state at 0 k and 0 gpa as given in eqn. (1). e (v ) = e0 + 9v0b 16   [( v0 v ) 2 3 − 1 ]3 b ′ + [( v0 v ) 2 3 − 1 ]2 [ 6 − 4 ( v0 v ) 2 3 ]  (1) the equilibrium lattice constant, total energy, bulk modulus, pressure, and pressure derivatives are documented in table 2. results for the lattice constants are in good agreement with 123 gemanam et al. / j. nig. soc. phys. sci. 3 (2021) 121–130 124 table 2: equilibrium lattice constant a0 (å), bulk modulus b0 (g p a), pressure derivative of the bulk modulus b ′ 0 , formation energy e f (ry), and bandgap eg (ev) of conbsn, rhnbsn and irnbsn half heusler alloys using pbe-gga approximation compared with existing results where available. compound a0 b b’ e f e g conbsn 5.98 5.9783 4 5.9475 4 5.905 5 152.7 158.193 4 170.555 5 4.34 -3.05 -3.083 4 1.03 1.025 5 0.983 4 irnbsn 6.21 153.1 4.37 -2.02 0.72 rhnbsn 6.24 168.6 4.53 -3.189 0.78 figure 1: energy-lattice parameter relationship of (a) conbsn alloy (b) rhnbsn alloy (c) irnbsn alloy with xyz (0.0, 0.5, 0.25), yzx (0.25, 0.00, 0.5), and zxy (0.5, 0.25, 0.0) atomic arrangements. results reported in literature. in addition, we observe that the lattice parameter increases from co 3d to rh 4d and ir 5d electronic configurations, showing that the lattice parameter increases with volume. table 2 also shows results for the formation energy (e f ) of each alloy. the formation energy predicts the possibility of synthesis; robust negative formation energy supports synthesis while positive formation energy does not. the formation energy (ef ) is computed by subtracting the energy (ex, enb, esn) of the constituent elements in the unit cell of the alloy from the total energy (ex n bsn ) of the alloy using eqn. (2). e f = e x n bsn − e x − e n b − esn , (2) x is either co, rh, or ir. the negative formation energy obtained for the three compounds shows they can be synthesised experimentally. the band structure and the partial density of states (pdos) from the electronic structure calculations are presented in figs 2 (a-f ). the bandgaps are 1.02, 0.72 and 0.78 ev for conbsn, irnbsn and rhnbsn alloys, respectively. the band structure shows a denser particle concentration in the valence band in comparison with the conduction band. the compounds display indirect bandgaps between x and w with the conduction band minimum (cbm) at x and the valence band maximum (vbm) at w. in irnbsn, however, the vbm has electron concentration at both l and w. from fig. 2b, the co, ir, rh (d) orbital, nb (d) orbital and the sn (p) orbital contributes most to the bandgap around the fermi level in the valence band. however, in the conduction band, the d orbitals of co and nb dominates, the d-d hybridisation between the orbitals accounts for the splitting that results in the bandgap. high electron concentration is observed in conbsn alloy at 1.9 ev, and the electron concentration can be adduced to the hybridisation of the d orbitals of co and nb below the fermi level (valence band), the concentration at 1.0 ev results from the hybridisation of the d orbitals of co and nb and the p orbitals of nb and sn. the d orbitals of nb and ir/rh contributes strongly to the bandgap in the valence band of irnbsn and rhnbsn, while the p orbitals of ir/rh and sn contributes weakly; the dominant contributor in the conduction band is the d orbital of nb for both alloys. the alloys are all p-type semiconductors. 3.2. elastic and mechanical properties the elastic constants, cij, are essential parameters in describing the behaviour of alloys under applied stress, and it 124 gemanam et al. / j. nig. soc. phys. sci. 3 (2021) 121–130 125 (a) (b) (c) provides information on the elastic stability and rigidity of the alloys. in this section, we study the mechanical stability of the compounds. to do this, we calculated the three independent elastic constants using the thermo-pw package in interface with quantum espresso by the quasi-harmonic approximation. the values of the elastic constants c11, c12, and c44 obtained from the coefficient of the linear term are presented in table 3. the elastic stability of a cubic structure is determined (d) (e) (f ) figure 2: (a) band structure of conbsn (b) partial density of states of conbsn (c) band structure of irnbsn (d) partial density of states of irnbsn (e) band structure of rhnbsn (f ) partial density of states of rhnbsn alloys using the computed equilibrium lattice constants. by the following rules as proposed by born and huang [51]. c11 > 0; c44 > 0 (c11 −c12) > 0 (c11 + 2c12) > 0   (3) these equations are often referred to as the born stability cri125 gemanam et al. / j. nig. soc. phys. sci. 3 (2021) 121–130 126 teria. however, these criteria do not entirely constitute a necessary and sufficient condition for stability in cubic crystalline compounds; therefore, mouhat and coudert stated in their article that a crystalline structure could also be classified as stable if the elastic energy is a positive definite(e > 0,∀ε,0) in the absence of external load in the harmonic approximation [52]; this is denoted mathematically as e = e0 + 1 2 v0 6∑ i , j =1 ci , j εi εj + 0 ( ε3 ) (4) this condition is termed the elastic stability criterion. an alloy must satisfy the necessary and sufficient stability conditions that the matrix c and the eigenvalues of c are favourable [52]. against this background, we present in table 3 the results of the three independent elastic constants of the alloys; the results show that the alloys are stable. other properties calculated from the elastic constants are young’s modulus (y ), the poisson ratio (v), bulk modulus (b) and shear modulus (g). they were evaluated using the voigtreuss-hill (vrh) average approximation. the result of the vrh and relevant derivatives are tabulated in table 4. the derivatives include the pugh’s ratio, anisotropy, vicker’s hardness, and cauchy’s pressure. for crystals, the bulk modulus b defines the ability of the material to resist fracture, while the shear modulus measures resistance to plastic deformation. b and g can be deducted from vrh approximations usingb = 12 (bv + br )g = 12 (gv +gr ); b is the bulk modulus, bv and br are the voigt and reuss bound of the bulk modulus. g is the shear modulus, while gv and gr are the voigt and the reuss bound of the shear modulus. the debye temperatureθd is crucial to predicting the thermal properties; it separates the high-temperature region of a crystal alloy from the low-temperature region. by default, a higher θd defines a material to have a higher thermal conductivity [53]. the debye temperature can be obtained using the average sound velocity and density from the relation h kb [ 3n 4p n aρ m ]1/3 νm where h is planck’s constant, kb is boltzmann’s constant, n is the number of atoms per unit cell, na is avogadro’s number, ρ its density, and m is the molecular weight. νm is the sound velocity at zero pressure. the result θd is recorded in table 4, with conbsn having the highest debye temperature. theθd report by ref. [34] is larger than our report. ref. [34] calculated the three independent (c11, c12 and c44) elastic constants via the stress tensor matrix, under small strains δ using the elastic code by charpin integrated in the wien2k code. the methodology and convergence of the parameters used by the authors probably account for the discrepancy between the results presented. we also compared the debye temperature from this work with results obtained by carrette at al. [54] using machine learning. the anisotropic factor (a), which as opposed to isotropy, describes the properties of the alloys in different directions and is evaluated using the relation 2c44/(c11 −c12). following the rule that any alloy whose value deviates from one (1) is anisotropic, the alloys are all anisotropic, having values of 0.69, 1.18, and 1.09 for conbsn, irnbsn, and rhnbsn alloys, respectively. conbsn deviates most from isotropy. the ratio of stress to strain as given by the young’s modulus (e) measures the material’s stiffness. a stiff material has a high young’s modulus meanwhile a ductile material has a low young’s modulus. e is derived using 9b g3b+g , where the parameters retain their usual meaning. the poisson ratiov measures the absolute value of the ratio of transverse contraction strain to the longitudinal extension strain and is given as 3b−2g2(3b+g ) . since no force or constraint was applied to the crystal surfaces, the poisson ratios predict the stability of the alloys. the poisson’s ratio can also be used as an indicator in bond sorting. for a typical covalently bonded compound, the value of the poisson ratio should be lower than 0.25, while that of a typical ionic compound is nearly 0.25 or more [55]. the values of the ratio show that ionic bonding is preferred and that the materials are ductile. pugh rule (b /g ) [56] defines the brittleness or ductility of a material. pugh proposes that the critical value between brittleness and ductility is 1.75; below this value, the material is brittle, and above it, the material is ductile; results in table 4 predicts the alloys to be ductile. the cauchy’s pressure (cp ) (c12 −c44) predicts that materials with negative values are brittle, while positive values are ductile. a material favouring covalent bonding has a negative cauchy pressure while it is positive for the compounds with dominant ionic bonds [57]. the results in table 4 shows that ionic bonding is favoured in the crystals. considering hardness as a measure of the resistance that a lattice offers to the motion of dislocations, it becomes necessary to predict the hardness of these materials to enable correct recommendations for technological application. some properties that predict the level of hardness of a crystal compound includes the microhardness parameter (h ), kleinman’s parameter (ς), and the lame’s coefficients (λ& µ). the parametersλ and µalso known as the lame constants predict the hardness of a compound. λ, which is the first lame constant, describes the possibility of compressibility in compounds, and the second lame constant µdescribes the shear stiffness of compounds, these constants are obtained using λ = eν(1+ν)(1−2ν) ,µ = e2(1+ν) as proposed by matori et al. [58]. the result obtained for the lame constants is presented in table 5, and further predicts the alloys to be anisotropic since for isotropic systemsλ and µare expected to be equivalent to c12 and c44. also, µit is expected to be equal to the shear modulus b; for all compounds, µit is in agreement with the shear modulus as expected. kleinman parameter ςshows the bending and stretching capability of a compound; it is computed using c11+8c127c11+2c12 . the operational principle of the kleinman capacity is that at a value approaching 1 (0), minimum bond stretching (bending) is attained. the values reported in table 5 shows that the alloys maintain a balance between bond stretching and bending. for a layered crystal, compounds are not expected to be isotropic; therefore, the anisotropy elasticity was calculated. we used the universal anisotropy index au proposed by ranganathan and ostoja-starzewski [59], which accounts for both compression and shear contributions 5 gvgr + bv br − 6, gv , bv , 126 gemanam et al. / j. nig. soc. phys. sci. 3 (2021) 121–130 127 gr , and br are the bulk modulus and shear modulus of voigt & reuss approximations, respectively. if au is not zero, the compound is not isotropic, and the extent of deviation from zero is a measure of the elastic anisotropy of the compound. the universal anisotropy indices are 0.16, 0.03, and 0.009 for conbsn, irnbsn, and rhnbsn half heusler alloys. in this case, conbsn alloy exhibits more anisotropic behaviour, while the others are almost negligible. some authors have proposed relations (vickers hardness and elastic moduli) for computing the microhardness (hv ), also known as vicker’s hardness of a cubic crystal solid; they can be predicted using the following relations: hv = 0.1769g − 2.899 [60], hv = 0.1475g [61], and hv = 0.0607e [61]. 3.3. thermodynamic stability to correctly recommend materials for an experimental procedure, it is essential to ascertain the thermodynamic stability. hence, in this section, we present results and analyse the behaviour of the alloys at low and high temperatures. the calculations were done using the quasi-harmonic approximation (qha) rather than the harmonic approximation. the qha is preferred because the harmonic approximation does not account for thermal expansion and thermal transport as temperature increases; it also does not take phonon interactions into account; hence the phonon lifetime is interpreted as infinite. effective implementation of the qha accommodates the possible change in lattice parameter/volume with temperature. this addition is termed the x-factor in the vibrational helmholtz energy, where x represents the lattice parameter or the volume. the relation for the qha from the vibrational helmholtz energy (f v i b ) is given in eqn. (5) as: f v i b (x , t ) = 1 2 ∑ −→ q ,υ }ω (−→ q ,υ, x ) +kb t ∑ −→ q ,υ i n [ 1 − e x p ( −}ω (−→ q ,υ, x ) kb t )] (5) where q is the wave-vector, the vibrational frequency is ω(q), kb is the boltzmann constant, t is temperature, } is the reduced planck’s constant, and υ is the frequency. we, however, do not consider the effect of pressure on the crystal. the parameters analysed and reported include the debye temperature θd , specific heat capacity at constant volume cυ ( its relation to the dulong-petit law), and the average mean velocity. the debye temperature provides information on the thermal properties of the material at ambient and high temperatures. table 6 presents the longitudinal and transverse sound velocities and average sound velocity. the average sound velocity (υm ) can be calculated using the compressional /longitudinal (vl ) and shear/transverse (vt ) sound velocities obtained from the modulus known as the navier’s equation [62] as υm = [ 1 3 ( 2 v 3 l + 1 v 3t )]−1/3 (6) figure 3: heat capacity cv of conbsn, irnbsn, and rhnbsn hh alloy the longitudinal vl and transverse vt velocities can be obtained using equations (7) and (8) vl = [( b + 4 3 g ) 1 ρ ]1/2 (7) vt = √ g ρ (8) the parameters in eqn. (7) retain their usual meanings, ρ is the density of the alloy. the average sound velocity (debye temperatures) of the alloys are 3.354 km/s (382.44 k), 2.757 km/s (300.93 k), and 2.792 km/s (306.47 k) for conbsn, irnbsn, and rhnbsn alloy, respectively. at 0 gpa and constant volume, the compound does not obey the dulong petit law, as shown in the heat capacity in fig. 3. the heat capacity can be defined by the relation cυ = ∂u∂t |v . the dulong-petit limit is achieved at a temperature of about 600 k and cυ of 74 j mol −1k −1 in agreement with einstein’s contribution to the heat theory at high temperature. the debye temperature and heat capacity behaviour with the dulong-petit law is consistent with other half heusler compounds with similar atomic mass units. 3.4. dynamical stability and phonon dispersions the lattice vibrations and behaviour of the phonon dispersions contribute to establishing phase stability in crystalline solids. we have calculated the phonon dispersion and the vibration of the lattices with temperature increase using the density functional perturbation theory (dfpt) and qha implemented in quantum espresso. the calculations were performed on a 2×2×2 mesh in the first brillouin zone in the reciprocal lattice space, and force constants in real space derived from the computations are used to interpolate between the q-points and to obtain the continuous branches of the phonon band structure [33]. we present the phonon dispersion curves along γ→x→k→γ→l→w→x symmetry directions based on results from a 2 × 2 × 2 cubic supercell. the results compare well with ref. [33] for the phonon dispersions of 127 gemanam et al. / j. nig. soc. phys. sci. 3 (2021) 121–130 128 table 3: elastic constants for the born elastic stability criteria of conbsn, irnbsn, and rhnbsn half heusler compound using the quasi-harmonic approximation. compound c11 c12 c44 (c11−c12) (c11+2c12) conbsn 273.50 301.663 4 82.58 81.173 4 66.27 110.073 4 190.92 438.66 irnbsn 256.72 136.12 71.18 120.60 529.96 rhnbsn 191.54 88.25 56.37 103.29 368.04 table 4: computed values of poisson ratio (v), bulk modulus (b in gpa), shear modulus (g in gpa), pugh’s ratio (b/g), anisotropy (a), young’s modulus (e in gpa), cauchy’s pressure (cp i n g p a), density (ρ in g/cm 3 ), and debye temperature (θ in k) using gga-pbe compared with results from literature where available. compound ν b g b /g θ a e h c p ρ conbsn 0.28 146.22 76.72 1.905 1.403 4 382.44 502.53 4 34357 0.69 0.9983 4 195.91 11.22 16.31 8.41 irnbsn 0.33 176.32 66.61 2.64 300.93 33457 1.18 177.47 7.56 64.94 11.01 rhnbsn 0.31 122.68 54.43 2.25 306.47 39657 1.09 142.25 6.88 31.88 8.72 table 5: lame’s coefficients (λ and µ) in gpa, dimensionless kleinman’s parameter (ς), and the microhardness parameter (h) in gpa obtained using the relations in ref. [60,61] separated by semicolons of conbsn, irnbsn, and rhnbsn half heusler alloys compound λ µ ς hv conbsn 97.39 76.52 0.45 10.67; 11.32; 11.89 irnbsn 129.51 66.71 0.65 8.88; 9.82; 10.77 rhnbsn 88.58 54.29 0.59 6.72; 8.02; 8.63 table 6: results for the voigt-reuss-hill average longitudinal and transverse sound velocities (vl and vt ) in km/s and average debye sound velocity υm in km/s, of conbsn and irnbsn, and rhnbsn alloys using qha compound vl vt υm conbsn 5.4355.9873 4 3.0203.6183 4 3.354 irnbsn 4.906 2.459 2.757 rhnbsn 4.731 2.498 2.792 conbsn alloy. however, up to now, there have no experimental/other theoretical works exploring the lattice dynamics of the compound under zero pressure to compare with results for irnbsn and rhnbsn alloys obtained in this work. the alloys investigated has three atoms in the unit cell; hence, the phonon dispersion spectra presented in fig. 4(ac) displays six (6) optical modes and three (3) acoustic modes with non-negative frequencies. the highest frequency gap between the acoustic modes and the optical mode is seen in rhnbsn, while the gap is almost non-existent in conbsn. the absence of soft (negative) frequencies further ascertains that the compounds are dynamically stable. the optical modes have frequencies 150 cm−1 and 250 cm−1 for the alloys: this put the alloys in the far-infrared region at wavelengths of about 30303 nm. the strongest dispersions occur at γ both in the acoustic and optical modes, giving rise to one longitudinal optical (lo) phonon and two transverse optical (to), we also observe a phonon splitting at γ, and l. lo-to splitting removes the degeneracy between the lo and to phonons at the brillouin zone centre. the lo-to splitting is an essential parameter that aids in evaluating the strength of ionic bonding in a material. in the acoustic mode, the dispersion at γ produces one longitudinal acoustic (la) phonon and two transverse acoustic (ta) phonons at x and l. the d orbital of co and nb contributes to the dispersions both in the acoustic mode and optical modes. this behaviour suggests a strong bonding between the atoms and the absence of rattling. 4. conclusion we have investigated the structural, electronic, mechanical, lattice dynamics, and thermodynamic properties of xnbsn (x = co, ir, rh) half heusler alloys using the gga-pbe density functional theory for structural and electronic properties and the density functional perturbation theory (dfpt) for the phonon and thermodynamic properties. results for the lattice constants and elastic properties for conbsn are in reasonable agreement with results in existing literature. we also computed the formation energy for the materials; the negative formation energies support possible experimental simulation. the alloys are non-magnetic semiconductors. the elastic parameters and the lattice dynamics suggest that the alloys are structurally, mechanically, and thermodynamically stable, as there are no negative frequencies in the phonon dispersions. the materials are ductile and ionic bonding is most favoured. the alloys comply with the dulong-petit law at heat capacities of 74 j/kmol, and temperature 600 k. the phonon frequencies (250 cm−1) translate to the wavelength of the materials being in the far-infrared region with a wavelength of about 30303 nm. irnbsn/conbsn has the lowest/highest θd 128 gemanam et al. / j. nig. soc. phys. sci. 3 (2021) 121–130 129 figure 4: (a) phonon dispersion of conbsn (b) phonon dispersion of irnbsn (c) phonon dispersion of rhnbsn alloys using the density functional perturbation theory. of 300.93 k/382.44 k at zero pressure. this positions irnbsn as a more promising material for thermoelectric applications. acknowledgments we thank the referees for the positive enlightening comments and suggestions, which have greatly helped us in making improvements to this paper. references [1] k. özdogan, i. galanakis, e. ?a?ioglu & b. akta?, "search for half-metallic ferrimagnetism in v-based heusler alloys mn2 vz (z= al, ga, in, si, ge, sn)", journal of physics: condensed matter 18 (2006) 2905. 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[61] a. l. orson, "a simplified method for calculating the debye temperature from elastic constants", journal of physics and chemistry of solids 24 (1963) 909. 130 j. nig. soc. phys. sci. 5 (2023) 1112 journal of the nigerian society of physical sciences a review on quadrant interlocking factorization: w z and w h factorization dlal bashira,∗, hailiza kamarulhailia, olayiwola babarinsab aschool of mathematical sciences, universiti sains malaysia, 11800 pulau pinang, malaysia bdepartment of mathematics, federal university lokoja, kogi state, nigeria abstract quadrant interlocking factorization (qif ), an alternative to lu factorization, is suitable for factorizing invertible matrix a such that det(a) , 0. the qif , with its application in parallel computing, is the most efficient matrix factorization technique yet underutilized. the two forms of qif among others, which are not only similar in algorithm but also in computation, are w z factorization and w h factorization yet differs in applications and properties. this review discusses both the old form of qif, called w z factorization, and the latest form of qif, called w h factorization, with an example and open questions to further the studies between the two factorization techniques. doi:10.46481/jnsps.2023.1112 keywords: quadrant interlocking factorization, w z factorization, w h factorization, lu factorization, linear systems article history : received: 06 october 2022 received in revised form: 14 january 2023 accepted for publication: 14 january 2023 published: 24 february 2023 © 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: s. fadugba 1. introduction over a century, factorization of a square matrix has been the research interest of matrix theorists [1]. many factorization techniques were deployed such as lu , qr and cholesky factorization. lu factorization is a well-known method with high accuracy deployed in enigma machine during world war i. lu factorization was introduced to solve square system of linear equations by inverting a matrix with underlying gaussian elimination procedure [2]. this feature combined with the low computational complexity and partial pivoting techniques makes lu -factorization extremely efficient. without a proper ordering in the matrix, lu factorization may fail to occur. the flaw ∗corresponding author tel. no: +60 163725820 email address: dlaljumabashir@yahoo.com (dlal bashir) can be removed by reordering the rows of b so that the first element of the permuted matrix is nonzero [3]. thus, a proper permutation in rows or columns is sufficient for the lu factorization to be numerically stable [4, 5]. lu factorization is known to be implemented in lapack library to exploit the standard software library architectures [6]. to improve the efficiency of computation during factorization, an alternative technique to lu factorization was developed, named quadrant interlocking factorization qif [7]. factorization of matrix b is difficult to compute and applying different optimization techniques couple with parallelism of contemporary computers makes qif factorization extremely efficient and suitable for parallel computing. among others factorization techniques such as lu , qr and cholesky decomposition, qif proves to be the best factorization algorithm in terms of efficiency, parallelization and accuracy [8, 9]. 1 bashir et al. / j. nig. soc. phys. sci. 5 (2023) 1112 2 2. list of abbreviations and symbols for the readers to understand this review paper, the symbols and abbreviation used in the article are given in the sections as follows: 2.1. list of abbreviations qif quadrant interlocking factorization lu lower and upper triangular matrices q orthogonal matrix lapack linear algebra package pie parallel implicit elimination ge gaussian elimination openmp open multi-processing gpu graphics processing unit cuda compute unified device architecture) edk hw/sw embedded development kit hardware/software p permutation matrix amd advanced micro devices intel integrated electronics. matlab matrix laboratory ggh goldreich-goldwasser-halevi encryption scheme 2.2. list of symbols r the set of real numbers det(b) determinant of matrix b ≥ greater than or equal to ≤ less than or equal to , not equal to bt transpose of matrix b n ∑ i=1 xi the finite sum of spaces n ∏ i=1 xi the finite product of spaces bbc floor of b dbe ceiling of b ||b|| norm of matrix b h hourglass matrix ht (nz) total number of nonzero entries in h ht (z) total number of zero entries in h fm filanz submatrix of h lim n→∞ f (n) limit of a function as n tends to infinity λ lambda g graph v vertex e edge ea(g) energy of a graph |b| modulus of b f2 friendship graph g mixed hourglass graph m(g ) mixed hourglass-adjacency matrix em(g ) mixed energy of a mixed hourglass graph ak number of arcs in g . 3. quadrant interlocking factorization quadrant interlocking factorization (qif ) or butterfly factorization of nonsingular matrix b was coined by evans and hatzopoulos [7]. in 1979, evans and hatzopoulos [10, 11] gave details of the factorization as an alternative to lu factorization and the avoidance of breakdown of qif algorithm. the factorization breaks up matrices to structural forms which are then regrouped and solved as sub-blocks [12]. qif is known for the adaptability of its direct method to solve systems of linear equations. thus, the factorization gives rise to the use of parallel implicit elimination (pie) for the solution of linear system to simultaneously compute two matrix elements (two columns at a time) for parallel implementation, unlike gaussian elimination (ge) which computes one column at a time [13]. the stability of qif comes from the centro-nonsingular matrix (central submatrices are nonsingular) which is far reliable than any other type of factorization [8]. 3.1. w z factorization w -matrix (bow-tie matrix) and z-matrix exists together in w z factorization of nonsingular matrix b [14, 15]. z-matrix and w -matrix are well-known as interlocking quadrant factors of b having butterfly shape of the form w =  1 ◦ • 1 • • ◦ 1 ◦ • • ◦ ◦ 1 ◦ ◦ • • ◦ • 1 • ◦ • • • 1 • • • 1 • ◦ 1   and z =   • • • • • • • • ◦ ◦ ◦ ◦ ◦ • ◦ ◦ ◦ • ◦ • • ◦ • ◦ ◦ ◦ • ◦ ◦ ◦ ◦ ◦ • • • • • • • •   such that b = w z. (1) z-matrix and w -matrix of order n (n ≥ 3) are generally defined as [10, 16] z =   (0,...,0︸ ︷︷ ︸ i−1 ,zi,i,...,zi,n−i+1,0,...,0)t , i = 1,...,b (n+1) 2 c; (0,...,0︸ ︷︷ ︸ n−i ,zi,n−i+1,...,zi,i,0,...,0)t , i =b (n+1) 2 c+ 1,...,n (2) 2 bashir et al. / j. nig. soc. phys. sci. 5 (2023) 1112 3 w =   (1,0,...,0︸ ︷︷ ︸ n−1 ); (wi,1,...,wi,i−1,1,0,...,0︸ ︷︷ ︸ n−2i+1 ,wi,n−i+2,...,wi,n), i = 2,...,b (n+1) 2 c; (wi,1,...,wi,n−i,0,...,0︸ ︷︷ ︸ 2i−n−1 ,1,wi,i+1,...,wi,n), i =b (n+1) 2 c+ 1,...,n−1; (0,...,0︸ ︷︷ ︸ n−1 ,1). (3) in w z factorization, there are bn2−1c ∑ k=1 (n−2k) of 2×2 linear systems to be solved which account for the elements in w -matrix and z-matrix [17]. the direct method to solve the linear systems of qif under the nonsingularity constraint presumed for their determinants solely depends on a conventional method called cramer’s rule. the unique solution (x1,x2,...,xn) provided by cramer’s rule [18] to the system bxi = c, (4) is xi = det(bi|c) det(b) , (5) where, det(b) , 0, b = (bi, j) 1 ≤ i, j ≤ n , xi = (x1,...,xn)t , c = (c1,...,cn)t ; x,c ∈ rn, b ∈ rn×n. bi|c is the matrix obtained from b by substituting the vector column of c to the ith column of b, for i = 1,2,...,n. 3.1.1. w z factorization algorithm for the w z factorization, matrix b is nonsingular. however, if central matrix of b is singular then interchange columns or rows of the matrix by suitable permutation to avoid breakdown of the factorization method, else the factorization breakdown. the establishment of elements in w -matrix (with 1’s in its main diagonal and 0’s in the antidiagonal), column ith and (n−1)th are obtained by solving simultaneous equation via cramer’s rule which requires matrix b to be successfully updated and this update changes matrix b to z-matrix [19, 20]. the matrix update of w z factorization indicates the most time consuming part of the factorization. the steps to obtain z-matrix is as follows: step 1: let b(0) = z(0)for initial update and obtain the first and last rows of z-matrix as b(0)1,1 = z (0) 1,1, b (0) 1,i = z (0) 1,i , b (0) 1,n = z (0) 1,n, b (0) n,1 = z(0)n,1, b (0) n,i = z (0) n,i , b (0) n,n = z (0) n,n, where i = 2,...,n−1. now, we compute w (1) i,1 and w (1) i,n from (n−2) sets of 2×2 linear system in equation (6) of matrix b using cramer’s rule z (0) 1,1w (1) i,1 + z (0) n,1w (1) i,n =−z (0) i,1 z(0)1,nw (1) i,1 + z (0) n,nw (1) i,n =−z (0) i,n . (6) the values of w(1)i,1 and w (1) i,n are put in matrix form as: w (1) =   1 0 ··· 0 0 w(1)2,1 1 ... ... w(1)2,n ... 0 ... 0 ... w(1)n−1,1 ... ... 1 w(1)n−1,n 0 0 ··· 0 1   . step 2: we update matrix b (let b(1) = z(1) for the first update) and compute: z(1) = w (1)z0. (7) 3 bashir et al. / j. nig. soc. phys. sci. 5 (2023) 1112 4 we, therefore, proceed analogously for the inner square matrices of z(1) of size (n−2) and so on. step 3: next, we compute w(k)i,k and w (k) i,n−k+1 from equation (8) by solving its 2×2 linear equations using cramer’s rule, where k = 1,2,..., n2 −1; i = k + 1,...,n−k. z (k−1) k,k w (k) i,k + z (k−1) n−k+1,kw (k) i,n−k+1 =−z (k−1) i,k z(k−1)k,n−k+1w (k) ik + z (k−1) n−k+1,n−k+1w (k) i,n−k+1 =−z (k−1) i,n−k+1. (8) then, we put the values of w(k)i,k and w (k) i,n−k+1 in a matrix form as: w (k) =   1 w(k)k+1,k ... w(k)k+1,n−k+1 ... ... ... w(k)n−1,k ... w(k)n−k,n−k+1 1   . step 4: we further compute for kth such successful steps as: z(k) = w (k)z(k−1). (9) to arrive at the z-matrix, we let z(k) = z. thus, z =   z(0)1,1 z (0) 1,2 ··· ··· ··· z (0) 1,n−1 z (0) 1,n 0 ... ... ··· ... ... 0 0 0 z(k−1)k,k z (k−1) k,n+1−k 0 0 ... ··· ... ... ... ··· ... 0 0 z(k−1)n+1−k,k z (k−1) n+1−k,n+1−k 0 0 0 ... ... ··· ... ... 0 z(0)n,1 z (0) n,2 ··· ··· ··· z (0) n,n−1 z (0) n,n   . a complete one-stage in w z factorization is when z(k−1) is computed. however, the factorization requires b(n−1)2 c stages to compute all the elements of the matrix w and z [10]. after the algorithm of w z factorization is established, the following theorems were put forward: theorem 3.1. factorization theorem [8]. let b ∈ rn×n be a nonsingular matrix that has a unique qif factorization, then b = w z if and only if the submatrices of b are invertible. theorem 3.2. [8]. if b ∈ rn×n is nonsingular matrix, then there exist a row permutation matrix p for qif to be carried out with pivoting such that pb = w z. corollary 3.3. [8]. every symmetric positive definite and strictly diagonally dominant matrix has a qif . theorem 3.4. [21]. let b be nonsingular tri-diagonal diagonally dominant, then its factored z-matrix from qif factorization is also tri-diagonal diagonally dominant. due to its uniqueness, w z factorization exists for every nonsingular matrix often with pivoting [22, 8]. pivoting results in swapping rows or columns in a matrix or by multiplying the matrix with permutation matrix which improves the numerical stability of w z factorization [1]. w z factorization will not fail without pivoting if the matrix is real symmetric, positive definite or diagonally dominant, see [23, 8, 24]. the matrix norm of w z factorization is the frobenius norm given as [25] ‖b−w z‖f = √√√√√   n∑ i=1 n ∑ j=1 |bi, j −wi, j zi, j|  . (10) the numerical accuracy ( −log10 ‖b−w z‖ n·‖b‖ ) of w z factorization based on the relative error depends on the matrix norms [16]. thus, the efficiency of w z factorization depends on an efficacious use of the memory echelon (processors array). if there is no sufficient fast memory, then the processor will create waiting time for the data and thereby reducing its efficiency [26]. 4 bashir et al. / j. nig. soc. phys. sci. 5 (2023) 1112 5 3.1.2. importance of w z factorization w z factorization has been applied in modeling of markov chains aside its parallelization usage [27]. w z factorization offers parallelization in solving both sparse and dense linear system to enhance performance using openmp, gpu, cuda or edk hw/sw codesign architecture [28, 29, 12]. then, yalamov [24] presented that w z factorization is faster on computer with a parallel architecture than any other matrix factorization methods. the factorization has been applied in scientific computing especially in science and engineering, see [30, 31, 27, 32, 33]. the block w z factorization is discussed in [34, 15, 23, 26] where the z-matrix is divided into r2 block each of the size s×s and n = r×s. theorem 3.5. [26] the block w z factorization exists if the matrix b has a strict dominant diagonal. z-matrix of even order (also applicable to odd order) can be partitioned to form structured zsystem of 2×2 triangular block matrices which is defined as zsystem =   [ z1,1 z1,2 z2,1 z2,2 ] if n is even;  z1,1 x1 z1,2 0 x 0 z2,1 x2 z2,2   if n is odd. (11) each block contains n2 × n 2 block size if n is even dimension while additional column vector, x̃, position at n+12 th column of the matrix if n is odd dimension [26]. this column vector, x̃, can be further partitioned into x1, x and x2. then, the schur complement of a matrix block in equation (11) is defined as follows zsystem/z1,1 = z2,2 −z2,1z−11,1 z1,2. (12) theorem 3.6. [26] if the zsystem and the matrix z1,1 are invertible then the matrix ( z2,2 −z2,1z−11,1 z1,2 ) is a lower triangular invertible matrix. 3.1.3. w z factorization versus lu factorization w z factorization proves to be better on intel processors than on amd processors [16]. even though w z factorization and lu factorization have similar computational complexity, the w z factorization still shown to be better than lu factorization (except block lu factorization) irrespective of the version of matlab or the number of processors used [35, 16]. however, for a uniprocessor, w z factorization does not exhibit any advantage over lu factorization since every step performed is in serial [12]. while lu factorization performs elimination in serial with n−1 steps, w z factorization executes components in parallel with n2 steps if n is even or n−1 2 steps if n is odd. w z factorization simultaneously computes two matrix elements (two columns at a time), unlike lu factorization which computes one column at a time [13, 12, 36]. unlike w z factorization, lu factorization is not unique but block lu factorization with higher diagonal blocks gives similar analytic result as w z factorization [37]. for larger matrices, the stimulation time on the multiprocessor show that the w z factorization is faster than lu factorization appears to be 20% for all values of processor [38, 39]. aside not being better than w z factorization on parallel computer or mesh multiprocessors, lu factorization does not account for the property of centrosymmetric matrix in its factors [40]. for sparse matrices, lu and w z factorization generate approximately similar number of non-zero elements [41]. though, lu factorization relies on leading principal submatrices ([ bi j ]k i,k=1 , l = 1,...,n ) whereas w z factorization relies on nonsingular central submatrices ([ b jk ]n+1−l j,k=l , l = 1,..., [ n+1 2 ]) [15]. based on the form of matrices, incomplete w z preconditioning gives better results than the incomplete lu factorization [26]. although some parts of the equation in lu factorization that consist many loops can be parallelized, the w z factorization is uniquely known for its ability to offer parallelization. even if w z factorization and lu factorization are both implemented on openmp, the w z factorization performs better in execution time than lu factorization when the number of thread increases [42, 43]. 3.2. w h factorization demeure [44] first posited the term hourglass matrix in describing a matrix derived from factorizing a square matrix via quadrant interlocking factorization or bowtie-hourglass factorization. it was further elucidated that hourglass matrix is synonymous to z-matrix which can be partitioned into blocks structured z-system [15, 28]. unfortunately, there are changes in structure of z-matrix from w z factorization which depend on the type of matrix (toeplitz, hankel, hermitian, centrosymmetric, diagonally dominant or tridiagonal matrix) being factorized. nevertheless, z-matrix may not always imply hourglass matrix nor their applications are always indistinguishable. consequently, the notion of sameness between hourglass matrix and z-matrix was gradually dropped over time without a cogent reason. recently, babarinsa and kamarulhaili [45] gave meticulous details of hourglass matrix, denoted as (h-matrix), and its quadrant interlocking factorization by restricting the computed entries of the factorization to be nonzero in accordance with the shape of hourglass device. definition 3.1. [45] let h be an hourglass matrix of order n (n ≥ 3), then hourglass matrix is defined as h =   hi, j 1 ≤ i ≤b (n+1) 2 c, i ≤ j ≤ n + 1−i; hi, j d (n+2) 2 e≤ i ≤ n, n + 1−i ≤ j ≤ i; 0 otherwise, (13) where hi, j ∈r are strictly nonzero elements. in other words, hourglass matrix is a nonsingular matrix of order n (n ≥ 3) with nonzero entries from the ith to the (n−i + 5 bashir et al. / j. nig. soc. phys. sci. 5 (2023) 1112 6 1) element of the ith and (n− i + 1) row of the matrix, otherwise 0’s, for i = 1,2,...,bn+12 c. to buttress the shape of hourglass matrix, figure 1 illustrates the structural comparison between the hourglass device and hourglass matrix with nonzero elements denoted with black dots, otherwise 0’s. figure 1. structural comparison between hourglass device and hourglass matrix. the factorization algorithm to obtain hourglass matrix is known as w h factorization. like the factorization of z-matrix, the factorization of hourglass matrix requires w -matrix (see the form of w -matrix in equation 3) to be computed during the factorization process. during w h factorization of nonsingular matrix b, h-matrix exists together with w -matrix, such that b = w h. (14) the matrix norm, numerical accuracy and efficiency of w h factorization algorithm is similar to w z factorization. the factorization of hourglass matrix and z-matrix are quite similar yet the factorization for hourglass matrix restricts the computed entries to be nonzero at every stage during the factorization. however, the qif of hourglass matrix specifies the number of times row-interchange can be done at each stage of the factorization if the computed entries yield zero, else the factorization breakdown. 3.2.1. w h factorization algorithm the sequential steps for the factorization are as follows [45]: step 1: let b = h(0) for initial update and check if the first row ( h(0)1, j ) and last row ( h(0)n, j ) of h(0) contains zero. if h(0)1, j = 0 or h(0)n, j = 0, then use suitable row-interchange in h (0), where j = 1,2,...,n. then, we compute w(1)i,1 and w (1) i,n in equation (15) from matrix h(0) by solving 2×2 system of linear equations via equation 5 h (0) 1,1w (1) i,1 + h (0) n,1w (1) i,n =−h (0) i,1 ; h(0)1,nw (1) i,1 + h (0) n,nw (1) i,n =−h (0) i,n , (15) whenever h(0)n,nh (0) 1,1 −h (0) 1,nh (0) n,1 = 0 use suitable row-interchange to avoid factorization breakdown. then the values of w(1)i,1 and w(1)i,n can be written in w -matrix as: w (1) =   1 0 ··· 0 0 w(1)2,1 1 ··· ... w(1)2,n ... 0 ... 0 ... w(1)n−1,1 ... ··· 1 w(1)n−1,n 0 0 ··· 0 1   . step 2: we, therefore, update matrix h(0) to h(1) for the first update by evaluating its entries as h(1)i, j = h (0) i, j + w (1) i,1 h (0) 1, j + w (1) i,n h (0) n, j, (16) if one of the computed entry h(1)2, j = 0 or h (1) n−1, j = 0 in equation (16), then apply row-interchange in h(1) at h(1)i, j for i = 2,...,n−1 and j = 1,...,n in no more than (n−4) times, else the factorization breakdown. step 3: now, we compute w(k)i,k and w (k) i,n−k+1 from matrix h(k−1) by solving 2×2 linear systems in equation (17) to generalize for every update of h(k) and proceed similarly for the inner square matrices of size (n−2k) and so on. that is, h (k−1) k,k w (k) i,k + h (k−1) n−k+1,kw (k) i,n−k+1 =−h (k−1) i,k ; h(k−1)k,n−k+1w (k) i,k + h (k−1) n−k+1,n−k+1w (k) i,n−k+1 =−h (k−1) i,n−k+1, (17) where k = 1,2,...,bn−12 c; i = k + 1,...,n−k. whenever h(k−1)n−k+1,n−k+1h (k−1) k,k −h (k−1) n−k+1,kh (k−1) k,n−k+1 = 0 use suitable row-interchange to avoid factorization breakdown. then, we put the values w(k)i,k and w (k) i,n−k+1 in a w -matrix of the form w (k) =  1 0 ... ... 1 0 w(k)k+1,k ... ... w(k)k+1,n−k+1 ... ... ... w(k)n−1,k ... ... w(k)n−k,n−k+1 0 1 ... ... 0 1   . step 4: we finally compute for kth steps of h(k)i, j as: h(k)i, j = h (k−1) i, j + w (k) i,k h (k−1) k, j + w (k) i,n−k+1h (k−1) n−k+1, j, (18) where j = k + 1,...,n−k. from equation (18), if one of the computed entries is zero, then apply possible row-interchange in no more than (n−2k) times in h(k−1) and re-factorize, else the factorization breakdown to produce h k (h-matrix). after the successful kth steps we get hourglass matrix (h(k) = h) of the form: 6 bashir et al. / j. nig. soc. phys. sci. 5 (2023) 1112 7 h =   h(0)1,1 h (0) 1,2 h (0) 1,3 ··· ··· ··· ··· ··· h (0) 1,n−2 h (0) 1,n−1 h (0) 1,n 0 h(1)2,2 h (1) 2,3 ··· ··· ··· ··· ··· h (1) 2,n−2 h (1) 2,n−1 0 0 0 h(2)3,3 ··· ··· ··· ··· ··· h (2) 3,n−2 0 0 ... 0 0 ... ... ... ... ... 0 0 ... ... ... ... h(k−1)k,k ··· h (k−1) k,n−k+1 ... ... ... 0 0 0 0 ... ... 0 0 0 0 ... ... ... h(k−1)n−k+1,k ··· h (k−1) n−k+1,n−k+1 ... ... ... ... 0 0 ... ... ... ... ... 0 0 ... 0 0 h(2)n−2,3 ··· ··· ··· ··· ··· h (2) n−2,n−2 0 0 0 h(1)n−1,2 h (1) n−1,3 ··· ··· ··· ··· ··· h (1) n−1,n−2 h (1) n−1,n−1 0 h(0)n,1 h (0) n,2 h (0) n,3 ··· ··· ··· ··· ··· h (0) n,n−2 h (0) n,n−1 h (0) n,n   . if there exists row-interchange at any stage k that yields permutation matrix p, then h = ( w (k−1)p(k−1)w (k−2)p(k−2)...w (2)p(2)w (1)p(1) )−1 b. (19) recall that k = 1,2,...,bn−12 c and that there are b n−1 2 c stages in the factorization. from every successful loops i, j = k + 1,k + 2,...,n−k for each stage, there are (n−2k) simultaneous equations each to be solved in (n−2k) times during the factorization using cramer’s rule. to avoid breakdown at its filanz submatrices (see definition 3.3), there must be row-interchange at every stage of the factorization process. this ensures that the 2×2 submatrix has the least condition number adopting any matrix norm. the overall time of w h factorization algorithm may increase if there is row-interchange at every stage of the factorization process due to the moving and sorting of data in and out of the processor. 3.2.2. on hourglass matrix proposition 3.7. [45] let h, ht (nz) and ht (z) be an hourglass matrix of order n (n ≥ 3), the total number of nonzero entries, and the total number of zero entries in hourglass matrix respectively. then, ht (nz) = n2 + 2n− ∣∣(n + 1) mod 2−1∣∣ 2 (20) and ht (z) = n2 −2n + ∣∣(n + 1) mod 2−1∣∣ 2 . (21) definition 3.2. [45] filanz submatrix, denoted as f 1≤i≤dn−12 e m , is a 2×2 non-singular matrix obtained by taking the first and the last nonzero elements of the ith and (n + 1−i)th row of h-matrix given as f 1≤i≤dn−12 e m =   h(i−1)i,i h(i−1)i,n+1−i h(i−1)n+1−i,i h (i−1) n+1−i,n+1−i   1≤i≤dn−12 e (22) definition 3.3. [45] epicenter element, denoted as h n+1 2 , n+1 2 is the nonzero element located at the intersection of ( n+12 ) row and ( n+12 ) column of hourglass matrix of odd order. proposition 3.8. [45] let det(h) be the determinant of hourglass matrix of order (n ≥ 3) then, det (h) =   dn−12 e ∏ i=1 ∣∣∣∣∣∣ h (i−1) i,i h (i−1) i,n+1−i h(i−1)n+1−i,i h (i−1) n+1−i,n+1−i ∣∣∣∣∣∣ if n is even; h( n+1 2 , n+1 2 ) dn−12 e ∏ i=1 ∣∣∣∣∣∣ h (i−1) i,i h (i−1) i,n+1−i h(i−1)n+1−i,i h (i−1) n+1−i,n+1−i ∣∣∣∣∣∣ if n is odd. (23) the determinant of matrix b can be computed as det(b) = det ( w (k−1) ·p(k−1) ·····w (2) ·p(2) ·w (1) ·p(1)· )−1 h. (24) 7 bashir et al. / j. nig. soc. phys. sci. 5 (2023) 1112 8 where det ( w (k−1) ·····w (2) ·w (1) )−1 = 1 and det ( p(k−1) ·····p(2) ·p(1) )−1 = (−1)pn . but (−1)pn = { 1 if even number of rows are interchanged, −1 if odd number of rows are interchanged. therefore, det(b) = (−1)pn det (h) (25) where pn is the total number of permutation matrix (successful row interchange) occurs in the factorization. partitioning of hourglass matrix of order n (n > 3) into 2×2 block triangular matrices with each block containing bn2c×b n 2c matrices is called hsystem. that is hsystem =   [ h1,1 h1,2 h2,1 h2,2 ] if n is even;  h1,1 x1 h1,2 0 x 0 h2,1 x2 h2,2   if n is odd. (26) where h1,2 = { hi j, 1 ≤ i ≤dn−12 e, b n+3 2 c≤ j ≤ n + 1−i; 0, otherwise. h2,2 = { hi j, bn+32 c≤ i ≤ n, n + 1−i ≤ j ≤d n−1 2 e; 0, otherwise. h1,1 = { hi j, 1 ≤ i ≤dn−12 e, i ≤ j ≤d n−1 2 e; 0, otherwise. h2,1 = { hi j, bn+32 c≤ i ≤ n, b n+3 2 c≤ j ≤ i; 0, otherwise. x1 = { hi j, 1 ≤ i ≤ n−12 , j = n+1 2 . x2 = { hi j, n+3 2 ≤ i ≤ n, j = n+1 2 . x = { hi j, i = n+1 2 , j = i . theorem 3.9. [46] schur complement exists in hsystem only if h-matrix is nonsingular. 3.3. comparison between w z factorization (or z-matrix) and w h factorization (or h-matrix) the w z factorization is possible provided the submatrices of the nonsingular matrix are invertible, while w h factorization does not only depend on the invertibility of the submatrices but also that the elements in the first row and in the last row of its submatrix are nonzero. if assume the entries hi, j is analogous to zi, j , then z-matrix will imply hourglass matrix provided that the computed z(k−1)i, j and z (k−1) n, j are strictly nonzero, for k = 1,2,...,b n−1 2 c. however, the entries of z-matrix are unbound to be nonzero. then it is obvious that it will no longer be an hourglass matrix if one of its strictly nonzero elements is replaced with zero. in general, every h-matrix is a z-matrix but the converse is not true, depicted in figure 2. h-matrix z-matrix figure 2. h-matrix as a subset of z-matrix. 8 bashir et al. / j. nig. soc. phys. sci. 5 (2023) 1112 9 the w z factorization exists for every nonsingular matrix often with pivoting while w h factorization may fail to exist even if the matrix is nonsingular. unlike the factorization of zmatrix, the factorization of an hourglass matrix from a nonsingular matrix may not necessarily be from a symmetric positive definite or diagonally dominant matrix but definitely not from a tridiagonal matrix. w z factorization may work with large dimension sparse matrices whereas w h factorization will not. in hsystem, each block has specific number of zero and nonzero entries, unlike zsystem. 3.3.1. unique properties of hourglass matrix the entries in h-matrix are linearly independent which make the matrix nonsingular. the transpose of hourglass matrix does not retain the shape of the matrix but rather form a bowtie matrix or butterfly matrix. inverse and nth root of hourglass matrix is again hourglass matrix. the minimum order of hourglass matrix is 3 and the matrix cannot be a symmetric. regardless of order of hourglass matrix, the total number of zero entries is even. besides, hourglass matrix has minimum matrix density of 0.5 as lim n→∞ n2+2n−|(n+1) mod 2−1| 2 n2 . 3.3.2. application of w h factorization and hourglass matrix w h factorization has great tendency in usage than w z factorization. first, though with little evidence it has been proposed that the linearly independent columns of hourglass matrix forming the basis vectors of a lattice will make it suitable for lattice-base cryptography, especially in ggh goldreichgoldwasser-halevi encryption scheme, see [47, 46]. the usage of hourglass matrix is expected to be able to reduce the size of bases. this reduction will allow the ggh scheme to be implemented in higher lattice dimension while still being able to be efficient and practical, and the generation of hourglass matrix can be executed in polynomial time. lastly, unlike z-matrix, hourglass matrix has be represented as weighted mixed graph called mixed hourglass graph [48]. we know that a simple graph g = (v,e) is an ordered pair consisting of a set of vertices v = {v1,v2,...,vn} and a set of undirected edges e ={e1,e2,...,en}, no loops nor multiple edges permitted [49, 50]. a mixed graph g = (v,e,a) is an ordered triple consisting of a set of vertices v = v1,v2,...,vn, a set of undirected edges e = {e1,e2,...,en}, and a set of directed arcs a [51]. now, an hourglass graph (butterfly graph) is a planar undirected graph formed by at least two triangles intersecting in a single vertex, especially from 5-vertex graph of two k3’s or from friendship graph f2, see for examples [52, 53, 54]. however, a mixed hourglass graph is a mixed complete graph coined from the name of its mixed adjacency matrix which is obtained from hourglass matrix [55]. definition 3.4. [48] a mixed hourglass graph g = (v,e,a) is an ordered triple consisting of a set of vertices v ={v1,v2,...,vn}, a set of undirected edges e = {e1,e2,...,en}, and a set of directed arcs a. definition 3.5. [55] a mixed hourglass-adjacency matrix m(g ) of a mixed hourglass graph g is the n×n(n≥3) matrix m(g )n×n = (hi, j)n×n defined by m(g ) =   1 if viv j is an edge ; −1 if vi,v j is an arc; 0 otherwise. (27) lemma 3.10. [55] for every mixed hourglass graph g of order n, the number of undirected edges m is m = n−β 2 , (28) where β = ∣∣(n + 1) mod 2−1∣∣. theorem 3.11. [55] let em(g ) be the mixed energy of a mixed hourglass graph g of order n and λi(g ) = {λ1,λ2,...,λn} be the mixed eigenvalues of a mixed hourglass-adjacency matrix m(g ). then em(g ) = n ∑ i=1 |λi(g )|= n−β , (29) where β = ∣∣(n + 1) mod 2−1∣∣. 3.4. numerical example of w z and w h factorization given a dense nonsingular square matrix b of order 6, apply both w z and w h factorization algorithm on the matrix. b =   2 0 2 4 3 −1 5 10 −7 8 11 4 0 −12 9 6 18 1 −13 12 8 −20 14 17 3 1 1 −1 1 4 10 6 9 −13 10 14   . 3.5. w z factorization of matrix b step 1: let b(0)i, j = z (0) i, j , for i, j = 1,...,6. we now compute the set of 2×2 system of linear equations from z (k−1) k,k w (k) i,k + z (k−1) n−k+1,kw (k) i,n−k+1 =−z (k−1) i,k z(k−1)k,n−k+1w (k) i,k + z (k−1) n−k+1,n−k+1w (k) i,n−k+1 =−z (k−1) i,n−k+1. for k = 1,...,bn−12 c= 2. if k = 1 then we have z (0) 1,1w (1) i,1 + z (0) n,1w (1) i,n =−z (0) i,1 z(0)1,nw (1) i,1 + z (0) n,nw (1) i,n =−z (0) i,n . whenever i = 2, then z (0) 1,1w (1) 2,1 + z (0) 6,1w (1) 2,6 =−z (0) 2,1 z(0)1,6w (1) 2,1 + z (0) 6,6w (1) 2,6 =−z (0) 2,6 ⇒ 2w (1) 2,1 + 10w (1) 2,6 = −5 −w(1)2,1 + 14w (1) 2,6 = −4 ⇒  w (1) 2,1 = − 15 19 w(1)2,6 = − 13 38 9 bashir et al. / j. nig. soc. phys. sci. 5 (2023) 1112 10 whenever i = 3, then  z (0) 1,1w (1) 3,1 + z (0) 6,1w (1) 3,6 =−z (0) 3,1 z(0)1,6w (1) 3,1 + z (0) 6,6w (1) 3,6 =−z (0) 3,6 ⇒ 2w (1) 3,1 + 10w (1) 3,6 = 0 −w(1)3,1 + 14w (1) 3,6 = −1 ⇒  w (1) 3,1 = 5 19 w(1)3,6 = − 1 19 whenever i = 4, then  z (0) 1,1w (1) 4,1 + z (0) 6,1w (1) 4,6 =−z (0) 4,1 z(0)1,6w (1) 4,1 + z (0) 6,6w (1) 4,6 =−z (0) 4,6 ⇒ 2w (1) 4,1 + 10w (1) 4,6 = 13 −w(1)4,1 + 14w (1) 4,6 =−17 ⇒  w (1) 4,1 = 176 19 w(1)4,6 = − 21 38 whenever i = 5, then  z (0) 1,1w (1) 5,1 + z (0) 6,1w (1) 5,6 =−z (0) 5,1 z(0)1,6w (1) 5,1 + z (0) 6,6w (1) 5,6 =−z (0) 5,6. ⇒ 2w (1) 5,1 + 10w (1) 5,6 = −3 −w(1)5,1 + 14w (1) 5,6 = −4. ⇒  w (1) 5,1 = − 1 19 w(1)5,6 = − 11 38 therefore, we write the values of w(1)i,1 and w (1) i,n in a matrix form as w (1) =   1 0 0 0 0 0 −1519 1 0 0 0 − 13 38 5 19 0 1 0 0 − 1 19 176 19 0 0 1 0 − 21 38 − 119 0 0 0 1 − 11 38 0 0 0 0 0 1   step 2: we update z0i, j to z 1 i, j by computing the entries as z(k)i, j = z (k−1) i, j + w (k) i,k z (k−1) k, j + w (k) i,n−k+1z (k−1) n−k+1, j ⇒ z (1) i, j = z(0)i, j + w (1) i,1 z (0) 1, j + w (1) i,6 z (0) 6, j. when i = 2 and j = 2,3,4,5, so z(1)2,2 = z (0) 2,2 + w (1) 2,1z (0) 1,2 + w (1) 2,6z (0) 6,2 = 10 + ( −1519 ) (0)+ ( −1338 ) (6) = 15119 z(1)2,3 = z (0) 2,3 + w (1) 2,1z (0) 1,3 + w (2) 2,6z (0) 6,3 = −7 + ( −1519 ) (2)+ ( −1338 ) (9) =−44338 z(1)2,4 = z (0) 2,4 + w (1) 2,1z (0) 1,4 + w (4) 2,6z (0) 6,4 = 8 + ( −1519 ) (4)+ ( −1338 ) (−13) = 35338 z(1)2,5 = z (0) 2,5 + w (1) 2,1z (0) 1,5 + w (3) 2,6z (0) 6,5 = 11 + ( −1519 ) (3)+ ( −1338 ) (10) = 9919 when i = 3 and j = 2,3,4,5, then z(1)2,2 = z (0) 3,2 + w (1) 3,1z (0) 1,2 + w (1) 3,6z (0) 6,2 = −12−( 519 )(0)+ ( − 119 ) (6) =−23419 z(1)3,3 = z (0) 3,3 +w (1) 3,1z (0) 1,3 +w (1) 3,6z (0) 6,3 = 9−( 5 19 )(2)+ ( − 119 ) (9) = 17219 z(1)3,4 = z (0) 3,4 + w (1) 3,1z (0) 1,4 + w (1) 3,6z (0) 6,4 = 6−( 519 )(4)+ ( − 119 ) (−13) = 14719 z(1)3,5 = z (0) 3,5 + w (1) 3,1z (0) 1,5 + w (1) 3,6z (0) 6,5 = 18−( 519 )(3)+ ( − 119 ) (10) = 34719 when i = 4 and j = 2,3,4,5, we have z(1)4,2 = z (0) 4,2 + w (1) 4,1z (0) 1,2 + w (1) 4,6z (0) 6,2 = 12 + ( 176 19 ) (0)+ ( −2138 ) (6) = 16519 z(1)4,3 = z (0) 4,3 + w (1) 4,1z (0) 1,3 + w (1) 4,6z (0) 6,3 = 8 + ( 176 19 ) (2)+ ( −2138 ) (9) = 819 38 z(1)4,4 = z (0) 4,4 + w (1) 4,1z (0) 1,4 + w (1) 4,6z (0) 6,4 = −20 + ( 176 19 ) (4)+ ( −2138 ) (−13) = 92138 z(1)4,5 = z (0) 4,5 + w (1) 4,1z (0) 1,5 + w (1) 4,6z (0) 6,5 = 14 + ( 176 19 ) (3)+ ( −2138 ) (10) = 68919 when i = 5 and j = 2,3,4,5, then z(1)5,2 = z (0) 5,2 + w (1) 5,1z (0) 1,2 + w (1) 5,6z (0) 6,2 = 1 + ( − 119 ) (0)+ ( −1138 ) (6) =−1419 z(1)5,3 = z (0) 5,3 + w (1) 5,1z (0) 1,3 + w (1) 5,6z (0) 6,3 = 1 + ( − 119 ) (2)+ ( −1138 ) (9) =−6538 z(1)5,4 = z (0) 5,4 + w (1) 5,1z (0) 1,4 + w (1) 5,6z (0) 6,4 = −1 + ( − 119 ) (4)+ ( −1138 ) (−13) = 9738 z(1)5,5 = z (0) 5,5 + w (1) 5,1z (0) 1,5 + w (1) 5,6z (0) 6,5 = 1 + ( − 119 ) (3)+ ( −1138 ) (10) =−3919 thus, z(1) =   2 0 2 4 3 −1 0 15119 −443 38 353 38 99 19 0 0 −23419 172 19 147 19 347 19 0 0 16519 819 38 921 38 689 19 0 0 −1419 −65 38 97 38 −39 19 0 10 6 9 −13 10 14   . 10 bashir et al. / j. nig. soc. phys. sci. 5 (2023) 1112 11 step 3: now, we can compute the next set of 2×2 systems of linear equation from the entries in z1i, j . let k = 2, then z (1) 2,2w (2) i,2 + z (1) n−1,2w (2) i,n−1 =−z (1) i,2 z(1)2,n−1w (2) i,2 + z (1) n−1,n−1w (2) i,n−1 =−z (1) i,n−1. whenever i = 3, then z (1) 2,2w (2) 3,2 + z (1) 5,2w (2) 3,5 =−z (1) 3,2 z(1)2,5w (2) 3,2 + z (1) 5,5w (2) 3,5 =−z (1) 3,5 ⇒  151 19 w (2) 3,2 +(− 14 19 )w (2) 3,5 = 234 19 99 19 w (2) 3,2 +(− 39 19 )w (2) 3,5 = − 347 19 ⇒  w (2) 3,2 = 13985 4503 w(2)3,5 = 75563 4503 whenever i = 4, then z (1) 2,2w (2) 4,2 + z (1) 5,2w (2) 4,5 =−z (1) 4,2 z(1)2,5w (2) 4,2 + z (1) 5,5w (2) 4,5 =−z (1) 4,5 ⇒  151 19 w (2) 4,2 +(− 14 19 )w (2) 4,5 = − 165 19 99 19 w (2) 4,2 +(− 39 19 )w (2) 4,5 = − 689 19 ⇒  w (2) 4,2 = 3211 4503 w(2)4,5 = 87704 4503 thus, w (2) =   1 0 0 0 0 0 0 1 0 0 0 0 0 139844503 1 0 75563 4503 0 0 32114503 0 1 87704 4503 0 0 0 0 0 1 0 0 0 0 0 0 1   step 4: we then proceed to update z1i, j to z 2 i, j by computing the entries as z(k)i, j = z (k−1) i, j + w (k) i,k z (k−1) k, j + w (k) i,n−k+1z (k−1) n−k+1, j ⇒ z (2) i, j = z(1)i, j + w (2) i,2 z (1) 2, j + w (2) i,5 z (1) 5, j. when i = 3 and j = 3,4, so z(2)3,3 = z (1) 3,3 + w (2) 3,2z (1) 2,3 + w (2) 3,5z (1) 5,3 = 172 19 +( 13984 4503 )( −443 38 )+( 75563 4503 )( −65 38 ) =− 9557475 171114 z(2)3,4 = z (1) 3,4 + w (2) 3,2z (1) 2,4 + w (2) 3,5z (1) 5,4 = 147 19 +( 13984 4503 )( 353 38 )+( 75563 4503 )( 97 38 ) = 13589845 171114 when i = 4 and j = 3,4, so z(2)4,3 = z (1) 4,3 + w (2) 4,2z (1) 2,3 + w (2) 4,5z (1) 5,3 = 819 38 +( 3211 4503 )( −443 38 )+( 87704 1171 )( −65 38 ) =− 3435276 171114 z(2)4,4 = z (1) 4,4 + w (2) 4,2z (1) 2,4 + w (2) 4,5z (1) 5,4 = 921 38 +( 3211 4503 )( 353 38 )+( 87704 4503 )( 97 38 ) = 13788034 171114 z =   2 0 2 4 3 −1 0 15119 − 413 38 503 38 99 19 0 0 0 −9557475171114 13589845 171114 0 0 0 0 −3435276171114 13788034 171114 0 0 0 −1419 − 65 38 97 38 − 39 19 0 10 6 9 −13 10 14   . thus, b = ( w (2) ·w (1) )−1 ·z w h factorization of matrix b step 1: we check the first and last row of matrix b before the initial update. b(0)1,1 = h (0) 1,1 = 2, b (0) 1,2 = h (0) 1,2 = 0, b (0) 1,3 = h (0) 1,3 = 2, b (0) 1,4 = h (0) 1,4 = 4, b(0)1,5 = h (0) 1,5 = 3, b (0) 1,6 = h (0) 1,6 =−1, b (0) 6,1 = h (0) 6,1 = 10, b (0) 6,2 = h(0)6,2 = 6, b (0) 6,3 = h (0) 6,3 = 9, b (0) 6,4 = h (0) 6,4 =−13, b (0) 6,5 = h (0) 6,5 = 10, b(0)6,6 = h (0) 6,6 = 14. since h(0)1,2 = 0, then we interchange the first row with any other row except the last row. in this case we interchange first row with the fifth row such that the first and last row of the matrix has no zero entry as h(0) =   3 1 1 −1 1 4 5 10 −7 8 11 4 0 −12 9 6 18 1 −13 12 8 −20 14 17 2 0 2 4 3 −1 10 6 9 −13 10 14   . with h(0) having permutation matrix p(1) =   0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1   . we begin to compute the set of 2×2 system of linear equations from h (k−1) k,k w (k) i,k + h (k−1) n−k+1,kw (k) i,n−k+1 =−h (k−1) i,k h(k−1)k,n−k+1w (k) i,k + h (k−1) n−k+1,n−k+1w (k) i,n−k+1 =−h (k−1) i,n−k+1. for k = 1,...,bn−12 c= 2. now, let k = 1 then we have h (0) 1,1w (1) i,1 + h (0) n,1w (1) i,n =−h (0) i,1 h(0)1,nw (1) i,1 + h (0) n,nw (1) i,n =−h (0) i,n . whenever i = 2, then 11 bashir et al. / j. nig. soc. phys. sci. 5 (2023) 1112 12 h (0) 1,1w (1) 2,1 + h (0) 6,1w (1) 2,6 =−h (0) 2,1 h(0)1,6w (1) 2,1 + h (0) 6,6w (1) 2,6 =−h (0) 2,6 ⇒ 3w (1) 2,1 + 10w (1) 2,6 = −5 4w(1)2,1 + 14w (1) 2,6 = −4 ⇒  w (1) 2,1 = −15 w(1)2,6 = 4 whenever i = 3, then  h (0) 1,1w (1) 3,1 + h (0) 6,1w (1) 3,6 =−h (0) 3,1 h(0)1,6w (1) 3,1 + h (0) 6,6w (1) 3,6 =−h (0) 3,6 ⇒ 3w (1) 3,1 + 10w (1) 3,6 = 0 4w(1)3,1 + 14w (1) 3,6 = −1 ⇒  w (1) 3,1 = 5 w(1)3,6 = − 3 2 whenever i = 4, then  h (0) 1,1w (1) 4,1 + h (0) 6,1w (1) 4,6 =−h (0) 4,1 h(0)1,6w (1) 4,1 + h (0) 6,6w (1) 4,6 =−h (0) 4,6 ⇒ 3w (1) 4,1 + 10w (1) 4,6 = 13 4w(1)4,1 + 14w (1) 4,6 =−17 ⇒  w (1) 4,1 = 176 w(1)4,6 = − 103 2 whenever i = 5, then  h (0) 1,1w (1) 5,1 + h (0) 6,1w (1) 5,6 =−h (0) 5,1 h(0)1,6w (1) 5,1 + h (0) 6,6w (1) 5,6 =−h (0) 5,6. ⇒ 3w (1) 5,1 + 10w (1) 5,6 = −2 4w(1)5,1 + 14w (1) 5,6 = 1. ⇒  w (1) 5,1 = −19 w(1)5,6 = 11 2 therefore, we write the values of w(1)i,1 and w (1) i,n in a matrix form as w (1) =   1 0 0 0 0 0 −15 1 0 0 0 4 5 0 1 0 0 −32 176 0 0 1 0 −1032 −19 0 0 0 1 112 0 0 0 0 0 1   step 2: we update h0i, j to h 1 i, j by computing its entries as h(k)i, j = h (k−1) i, j + w (k) i,k h (k−1) k, j + w (k) i,n−k+1h (k−1) n−k+1, j ⇒ h (1) i, j = h(0)i, j + w (1) i,1 h (0) 1, j + w (1) i,6 h (0) 6, j. when i = 2 and j = 2,3,4,5, so h(1)2,2 = h (0) 2,2 +w (1) 2,1h (0) 1,2 +w (1) 2,6h (0) 6,2 = 10 +(−15)(1)+(4)(6) = 19 h(1)2,3 = h (0) 2,3 +w (1) 2,1h (0) 1,3 +w (1) 2,6h (0) 6,3 =−7+(−15)(1)+(4)(9) = 14 h(1)2,4 = h (0) 2,4 + w (1) 2,1h (0) 1,4 + w (1) 2,6h (0) 6,4 = 8 +(−15)(−1)+(4)(−13) = 29 h(1)2,5 = h (0) 2,5 + w (1) 2,1h (0) 1,5 + w (1) 2,6h (0) 6,5 = 11 +(−15)(1)+(4)(10) = 36 when i = 3 and j = 2,3,4,5, then h(1)2,2 = h (0) 3,2 + w (1) 3,1h (0) 1,2 + w (1) 3,6h (0) 6,2 =−12 +(5)(1)+(− 3 2 )(6) = −16 h(1)3,3 = h (0) 3,3 + w (1) 3,1h (0) 1,3 + w (1) 3,6h (0) 6,3 = 9 +(5)(1)+(− 3 2 )(9) = 1 2 h(1)3,4 = h (0) 3,4 + w (1) 3,1h (0) 1,4 + w (1) 3,6h (0) 6,4 = 6 +(5)(−1)+(−32 )(−13) = 41 2 h(1)3,5 = h (0) 3,5 + w (1) 3,1h (0) 1,5 + w (1) 3,6h (0) 6,5 = 18 +(5)(1)+(− 3 2 )(10) = 8 when i = 4 and j = 2,3,4,5, we have h(1)4,2 = h (0) 4,2 + w (1) 4,1h (0) 1,2 + w (1) 4,6h (0) 6,2 = 12 +(176)(1)+(−1032 )(6) =−121 h(1)4,3 = h (0) 4,3 + w (1) 4,1h (0) 1,3 + w (1) 4,6h (0) 6,3 = 8 +(176)(1)+(− 103 2 )(9) = −5592 h(1)4,4 = h (0) 4,4 + w (1) 4,1h (0) 1,4 + w (1) 4,6h (0) 6,4 = −20 +(176)(−1)+(−1032 )(−13) = 1027 2 h(1)4,5 = h (0) 4,5 + w (1) 4,1h (0) 1,5 + w (1) 4,6h (0) 6,5 = 14 +(176)(1)+(−1032 )(10) =−325 when i = 5 and j = 2,3,4,5, then h(1)5,2 = h (0) 5,2 + w (1) 5,1h (0) 1,2 + w (1) 5,6h (0) 6,2 = 0 +(19)(1)+( 11 2 )(6) = 52 h(1)5,3 = h (0) 5,3 + w (1) 5,1h (0) 1,3 + w (1) 5,6h (0) 6,3 = 2 +(19)(1)+( 11 2 )(9) = 141 2 h(1)5,4 = h (0) 5,4 + w (1) 5,1h (0) 1,4 + w (1) 5,6h (0) 6,4 = 4 +(19)(−1)+( 112 )(−13) =− 173 2 h(1)5,5 = h (0) 5,5 + w (1) 5,1h (0) 1,5 + w (1) 5,6h (0) 6,5 = 3 +(19)(1)+( 11 2 )(10) = 77 in h(1) the entries h(1)2, j and h (1) 5, j are nonzero (i.e. h (1) 2,2 = 19, h (1) 2,3 = 14, h(1)2,4 =−29, h (1) 2,5 = 36, h (1) 5,2 = 52, h (1) 5,3 = 141 2 , h (1) 5,4 =− 173 2 , h (1) 5,5 = 77) for j = 2,3,4,5. otherwise apply suitable row interchange in h(0) and re-factorize, else the factorization breakdown. step 3: now, we can compute the next set of 2×2 systems of linear equation from the entries in h1i, j . let k = 2, then h (1) 2,2w (2) i,2 + h (1) n−1,2w (2) i,n−1 =−h (1) i,2 h(1)2,n−1w (2) i,2 + h (1) n−1,n−1w (2) i,n−1 =−h (1) i,n−1. whenever i = 3, then 12 bashir et al. / j. nig. soc. phys. sci. 5 (2023) 1112 13 h (1) 2,2w (2) 3,2 + h (1) 5,2w (2) 3,5 =−h (1) 3,2 h(1)2,5w (2) 3,2 + h (1) 5,5w (2) 3,5 =−h (1) 3,5 ⇒ 19w (2) 3,2 + 52w (2) 3,5 = 16 36w(2)3,2 + 77w (2) 3,5 = 3−8 ⇒  w (2) 3,2 = − 1648 409 w(2)3,5 = 728 409 whenever i = 4, then h (1) 2,2w (2) 4,2 + h (1) 5,2w (2) 4,5 =−h (1) 4,2 h(1)2,5w (2) 4,2 + h (1) 5,5w (2) 4,5 =−h (1) 4,5 ⇒ 19w (2) 4,2 + 52w (2) 4,5 = 121 36w(2)4,2 + 77w (2) 4,5 = 325 ⇒  w (2) 4,2 = 7583 409 w(2)4,5 = − 1819 409 thus, w∗(2) =   1 0 0 0 0 0 0 1 0 0 0 0 0 −1648409 1 0 728409 0 0 7583409 0 1 − 1819 409 0 0 0 0 0 1 0 0 0 0 0 0 1   step 4: we then proceed to update the matrix h(1) to h(2) by computing its entries as h(k)i, j = h (k−1) i, j + w (k) i,k h (k−1) k, j + w (k) i,n−k+1h (k−1) n−k+1, j ⇒ h (2) i, j = h(1)i, j + w (2) i,2 h (1) 2, j + w (2) i,5 h (1) 5, j. when i = 3 and j = 3,4, so h(2)3,3 = h (1) 3,3 + w (2) 3,2h (1) 2,3 + w (2) 3,5h (1) 5,3 = 1 2 +(− 1648 409 )(14)+( 728 409 )( 141 2 ) = 56913 818 h(2)3,4 = h (1) 3,4 + w (2) 3,2h (1) 2,4 + w (2) 3,5h (1) 5,4 = 41 2 +(− 1648 409 )(−29)+( 728 409 )(− 173 2 ) =− 13591 818 when i = 4 and j = 3,4, so h(2)4,3 = h (1) 4,3 + w (2) 4,2h (1) 2,3 + w (2) 4,5h (1) 5,3 = −5592 +( 7583 409 )(14)+(− −1819 409 )( 141 2 ) =− 272786 818 h(2)4,4 = h (1) 4,4 + w (2) 4,2h (1) 2,4 + w (2) 4,5h (1) 5,4 = 1027 2 +( 7583 409 )(−29)+(− −1819 409 )(− 173 2 ) = 294916 818 h =   3 1 1 −1 1 4 0 19 14 −29 36 0 0 0 56913818 − 13591 818 0 0 0 0 −272786818 294916 818 0 0 0 52 1412 − 173 2 77 0 10 6 9 −13 10 14   . the factorization stops since the entries h(2)3, j and h (2) 4, j are nonzero, for j = 3,4. to get the matrix b, we express b as b = ( w (2) ·w (1) ·p(1) )−1 ·h 4. investigation for further studies on w z and w h factorization there are bridging gaps to be uncovered not only between w z factorization and w h factorization but also within z-matrix and hourglass matrix. further research could be carried on the following: 1. if there exists w h factorization for a nonsingular matrix b, then there exists w z factorization. 2. w z factorization does not necessarily imply w h factorization. 3. zsystem is a matrix group of degree 2 over r but hsystem is not. 4. if both hsystem and h1,1 are invertible then ( h2,2 −h2,1h−11,1 h1,2 ) is a lower triangular invertible matrix. 5. if there exists w h factorization for a nonsingular matrix b, then the factorization is unique. 6. let em(g ) be the energy of mixed hourglass graph g and ak be the number of arcs in g . then, does there exist a mixed hourglass graph g of order n (n > 4) satisfying the following? ak = em(g ). acknowledgments this review paper is funded by frgs (fundamental research grant scheme), grant reference number 1/2020/stg06/usm/01/1. references [1] f. bornemann, matrix factorization, springer (2018). 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[55] o. babarinsa & h. kamarulhaili, mixed energy of a mixed hourglass graph, communications in mathematics and applications 10 (2019) 45. http://dx.doi.org/10.26713/cma.v10i1.1143 14 j. nig. soc. phys. sci. 3 (2021) 308–333 journal of the nigerian society of physical sciences a new special 15-step block method for solving general fourth order ordinary differential equations victor oboni ataboa,∗, solomon ortwer adeeb adepartment of mathematics, ahmadu ribadu college, yola, nigeria bdepartment of mathematics, modibbo adama university, yola, nigeria abstract a new higher-implicit block method for the direct numerical solution of fourth order ordinary differential equation is derived in this research paper. the formulation of the new formula which is 15-step, is achieved through interpolation and collocation techniques. the basic numerical properties of the method such as zero-stability, consistency and a-stability have been examined. investigation showed that the new method is zero stable, consistent and a-stable, hence convergent. test examples from recent literature have been used to confirm the accuracy of the new method. doi:10.46481/jnsps.2021.337 keywords: consistency, zero stability, a-stability, implicit block formula article history : received: 11 august 2021 received in revised form: 22 october 2021 accepted for publication: 13 november 2021 published: 29 november 2021 ©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: t. latunde 1. introduction ordinary differential equations (odes) have important applications and are a powerful tool in the study of many problems in the natural sciences and in technology; they are extensively employed in mechanics, astronomy, physics, and in many problems of chemistry and biology. mathematical models in these vast range of disciplines, describe how quantities change. this leads naturally to the language of ordinary differential equations (odes). for instance, newton’s laws in mechanics make it possible to reduce the description of the motion of mass points or solid bodies to solving ordinary differential equations. the computation of radiotechnical circuits or satellite tragetories, studies of the stability of a plane in flight, and explaining the course of chemical ∗corresponding author tel. no: +2348061362898 email address: atabovictor@gmail.com (victor oboni atabo ) reactions are all carried out by studying and solving ordinary differential equations. the most interesting and most important applications of these equations are in the theory of oscillations, physical ship dynamic theory and in automatic control theory. these applied problems in turn produce new formulations of problems in the theory of ordinary differential equations from first, second, third, fourth, fifth, e.t.c, to other higher derivatives in odes conti, graffi and sansone[1]. however, the solutions to some of these varieties of higher order odes problems do not exist explicitly. hence, the need to develop numerical methods in the form of implicit linear multistep methods (lmms) to solve these numerous problems in the form of odes. therefore, in this research paper, we shall consider the general fourth order ordinary differential equation of the form: 308 atabo and adee / j. nig. soc. phys. sci. 3 (2021) 308–333 309 y(iv) = f (x, y, y′, y′′, y′′′), y(a) = �0, y ′(a) = �1, y ′′(a) = �2, y ′′′(a) = �3 (1) where, r × rm × rm → rm and �0, �1, �2, �3 ∈ r. however, interpolation and collocation techniques have been used overtime for developing implicit linear multistep methods using power series as a basis function. such authors as raymond, skwame and adiku [2] formulated a four-step one off-grid block method using interpolation and collocation approach for the solution of fourth derivative ordinary differential equations. two-step hybrid linear multistep block method for solving second, third and fourth order initial value problems of ordinary differential equations directly has also been derived by abolarin, kuboye, adeyefa and ogunware [3]. hermite interpolation polynomial as a basis function has extensively been used also for the formulation of implicit block methods. two point implicit block method of uniform order 6 has been derived for solving fourth-order initial value problems directly by allogmany, ismail, majid and ibrahim [4] using the aforementioned basis function. in addition, in order to avoid order reduction at solving higher order odes, such authors as kuboye [5], jena, mohanty and mishra [6], adoghe and omole [7], omar and kuboye [8], duromola [9], adesanya, momoh, adamu and tahir [10], awoyemi, kayode and adoghe [11], awoyemi [12] and ukpebor, omole and adoghe [13] have solved equation (1) directly. similarly, omole and ukpebor [14] and kuboye, elusakin and quadri [15] have both formulated a 4-point and 5-point hybrids block formulae for the solution of system and linear fourth order initial value problems in ordinary differential equations using power series as a basis function via interpolation and collocation techniques with uniform order 4 and 7 respectively. therefore, we are motivated to improve on jena et al. [6] who have formulated a 9-point block method of a uniform order 12 for the direct solution of equation (1) by proposing a new special 15-step block numerical method for the direct approximation of (1). consequently, in section two, we carry out the formulation of 15-step block formula, section three provides the analysis of basic numerical properties, section four considers numerical examples, its implementation and discussion of results and in section five, the conclusion is drawn. 2. materials and methods of 15-step block method in this section, we shall consider the formulation of a 15step block method for the numerical approximation of equation (1). 2.1. formulation of 15-step block method let us consider the power series approximations of the form: y(x) = k+4∑ j=0 a j x j (2) which approximates equation (1), so that a j′s, j = 0(1)(k+4) are parameters to be found and k is the step-number. thus, the first, second, third and fourth derivatives of equation (2) become: y′(x) = k+4∑ j=0 ja j x j−1 (3) y′′(x) = k+4∑ j=0 j( j − 1)a j x j−2 (4) y′′′(x) = k+4∑ j=0 j( j − 1)( j − 2)a j x j−3 (5) y(iv)(x) = k+4∑ j=0 j( j − 1)( j − 2)( j − 3)a j x j−4 (6) therefore, equating equation (1) and (6) gives: k+4∑ j=0 j( j − 1)( j − 2)( j − 3)a j x j−4 = f (x, y, y′, y′′, y′′′) (7) thus, equation (2), (3), (4) and (5) are then interpolated at xn+i, i = (k−15) and equation (7) is collocated at xn+i, i = 0(1)k to give system of linear equations in the form: cx = d (8) where, c =  1 xn x2n x 3 n x 4 n x 5 n ... x k+4 n 0 1 2xn 3x2n 4x 3 n 5x 4 n ... (k + 4)x k+3 n 0 0 2 6xn 12x2n 20x 3 n ... (k + 4)(k + 3)x k+2 n 0 0 0 6 24xn 60x2n ... (k + 4)(k + 3)(k + 2)x k+1 n 0 0 0 0 24 120xn ... (k + 4)(k + 3)(k + 2)(k + 1)xkn 0 0 0 0 24 120xn+1 ... (k + 4)(k + 3)(k + 2)(k + 1)x k n+1 . . . . . . . . . . . . . . . . . . . . . . . . 0 0 0 0 24 120xn+k · · · (k + 4)(k + 3)(k + 2)(k + 1)x k n+k  , x =  a0 a1 a2 a3 . . . ak+4  , d =  yn y′n y′′n y′′′n fn . . . fn+k  by solving equation (8) with k = 15 and j = 0(1)(k+4) ( where c is a 20-by-20 matrix), using gaussian elimination method, ak+4′s are obtained and substitution into equation (2) is made to give a continuous linear multistep method(lmm) of the form: y(x) = α0yn+hα1y′n+h 2α2y′′n +h 3α3y′′′n +h 4 k∑ j=0 β j fn+ j (9) where, ξ = x−xn α0 = 1 (10) α1 = ξ (11) α2 = 1 2 ξ2 (12) α3 = 1 6 ξ3 (13) 309 atabo and adee / j. nig. soc. phys. sci. 3 (2021) 308–333 310 β0 = 1 24 ξ4− 1195757 ξ5 43243200 h + 13215487 ξ6 1009008000 h2 − 35118025721 ξ7 7628100480000 h3 + 2065639 ξ8 1676505600 h4 − 277382447 ξ9 1086375628800 h5 + 2271089 ξ10 54867456000 h6 − 54576553 ξ11 10346434560000 h7 + 4783 ξ12 9053130240 h8 − 324509 ξ13 7846046208000 h9 + 109 ξ14 43589145600 h10 − 26921 ξ15 235381386240000 h11 + ξ16 261534873600 h12 − 47 ξ17 533531142144000 h13 + ξ18 800296713216000 h14 − ξ19 121645100408832000 h15 (14) β1 = 1 8 ξ5 h − 835397 ξ6 8648640 h2 + 60461593 ξ7 1412611200 h3 − 146723651 ξ8 11176704000 h4 + 8548607 ξ9 2874009600 h5 − 28162523 ξ10 54867456000 h6 + 1475953 ξ11 21555072000 h7 − 7346057 ξ12 1034643456000 h8 + 5973601 ξ13 10461394944000 h9 − 24599 ξ14 697426329600 h10 + 223 ξ15 135862272000 h11 − 2329 ξ16 41845579776000 h12 + 71 ξ17 54721142784000 h13 − ξ18 53801459712000 h14 + ξ19 8109673360588800 h15 (15) β2 = − 7 ξ5 16 h + 1015577 ξ6 2471040 h2 − 20848547 ξ7 100900800 h3 + 1546697837 ξ8 22353408000 h4 − 149466307 ξ9 8941363200 h5 + 664545493 ξ10 219469824000 h6 − 289311887 ξ11 689762304000 h7 + 5795249 ξ12 129330432000 h8 − 1379251 ξ13 373621248000 h9 + 40549 ξ14 174356582400 h10 − 5231 ξ15 475517952000 h11 + 989 ξ16 2615348736000 h12 − 61 ξ17 6840142848000 h13 + 59 ξ18 457312407552000 h14 − ξ19 1158524765798400 h15 (16) β3 = 91 ξ5 72 h − 1075637 ξ6 855360 h2 + 281523559 ξ7 419126400 h3 − 23857083361 ξ8 100590336000 h4 + 32534861339 ξ9 543187814400 h5 − 617568593 ξ10 54867456000 h6 + 207888451 ξ11 129330432000 h7 − 5522731 ξ12 31352832000 h8 + 66465877 ξ13 4483454976000 h9 − 1989607 ξ14 2092278988800 h10 + 390953 ξ15 8559323136000 h11 − 66571 ξ16 41845579776000 h12 + 6229 ξ17 164163428352000 h13 − 13 ξ18 23451918336000 h14 + ξ19 267351869030400 h15 (17) β4 = − 91 ξ5 32 h + 1105667 ξ6 380160 h2 − 18107311 ξ7 11289600 h3 + 3269363347 ξ8 5588352000 h4 − 6129294103 ξ9 40236134400 h5 + 3229334839 ξ10 109734912000 h6 − 2970738131 ξ11 689762304000 h7 + 31183253 ξ12 64665216000 h8 − 15450997 ξ13 373621248000 h9 + 235087 ξ14 87178291200 h10 − 62483 ξ15 475517952000 h11 + 12121 ξ16 2615348736000 h12 − 139 ξ17 1243662336000 h13 + 29 ξ18 17588938752000 h14 − ξ19 89117289676800 h15 (18) β5 = 1001 ξ5 200 h − 224737 ξ6 43200 h2 + 103207793 ξ7 35280000 h3 − 221496517 ξ8 203212800 h4 + 176798693 ξ9 609638400 h5 − 3138934651 ξ10 54867456000 h6 + 230071601 ξ11 26943840000 h7 − 201455717 ξ12 206928691200 h8 + 126849403 ξ13 1494484992000 h9 − 3917983 ξ14 697426329600 h10 + 1319249 ξ15 4755179520000 h11 − 1507 ξ16 152165744640 h12 + 1201 ξ17 4974649344000 h13 − 23 ξ18 6395977728000 h14 + ξ19 40507858944000 h15 (19) 310 atabo and adee / j. nig. soc. phys. sci. 3 (2021) 308–333 311 β6 = − 1001 ξ5 144 h + 1135697 ξ6 155520 h2 − 158534377 ξ7 38102400 h3 + 28776650641 ξ8 18289152000 h4 − 84007238759 ξ9 197522841600 h5 + 2079145531 ξ10 24385536000 h6 − 26754892001 ξ11 2069286912000 h7 + 193584983 ξ12 129330432000 h8 − 21199931 ξ13 160123392000 h9 + 4645801 ξ14 523069747200 h10 − 1900739 ξ15 4279661568000 h11 + 1271 ξ16 79252992000 h12 − 737 ξ17 1865493504000 h13 + 19 ξ18 3197988864000 h14 − ξ19 24304715366400 h15 (20) β7 = 429 ξ5 56 h − 1144277 ξ6 141120 h2 + 15356659 ξ7 3292800 h3 − 603869969 ξ8 338688000 h4 + 297351533 ξ9 609638400 h5 − 1811075923 ξ10 18289152000 h6 + 218600713 ξ11 14370048000 h7 − 4306835671 ξ12 2414168064000 h8 + 557256047 ξ13 3487131648000 h9 − 2522719 ξ14 232475443200 h10 + 174023 ξ15 317011968000 h11 − 25411 ξ16 1268047872000 h12 + 827 ξ17 1658216448000 h13 − 113 ξ18 14923948032000 h14 + ξ19 18903667507200 h15 (21) β8 = − 429 ξ5 64 h + 143839 ξ6 20160 h2 − 108870653 ξ7 26342400 h3 + 33726391 ξ8 21168000 h4 − 178809037 ξ9 406425600 h5 + 825125941 ξ10 9144576000 h6 − 460176289 ξ11 32845824000 h7 + 2237897 ξ12 1347192000 h8 − 131166143 ξ13 871782912000 h9 + 75043 ξ14 7264857600 h10 − 83719 ξ15 158505984000 h11 + 193 ξ16 9906624000 h12 − 29 ξ17 59222016000 h13 + ξ18 133249536000 h14 − ξ19 18903667507200 h15 (22) β9 = 1001 ξ5 216 h − 128413 ξ6 25920 h2 + 109950473 ξ7 38102400 h3 − 3413334089 ξ8 3048192000 h4 + 15383980811 ξ9 49380710400 h5 − 3534853769 ξ10 54867456000 h6 + 652662223 ξ11 64665216000 h7 − 1249849571 ξ12 1034643456000 h8 + 495342143 ξ13 4483454976000 h9 − 1780879 ξ14 232475443200 h10 + 3386083 ξ15 8559323136000 h11 − 56057 ξ16 3804143616000 h12 + 5581 ξ17 14923948032000 h13 − 37 ξ18 6395977728000 h14 + ξ19 24304715366400 h15 (23) β10 = − 1001 ξ5 400 h + 1159721 ξ6 432000 h2 − 13852667 ξ7 8820000 h3 + 249027661 ξ8 406425600 h4 − 1254757531 ξ9 7315660800 h5 + 1567631503 ξ10 43893964800 h6 − 19442163553 ξ11 3448811520000 h7 + 17594483 ξ12 25866086400 h8 − 23443657 ξ13 373621248000 h9 + 765371 ξ14 174356582400 h10 − 543937 ξ15 2377589760000 h11 + 409 ξ16 47551795200 h12 − 137 ξ17 621831168000 h13 + 11 ξ18 3197988864000 h14 − ξ19 40507858944000 h15 (24) 311 atabo and adee / j. nig. soc. phys. sci. 3 (2021) 308–333 312 β11 = 91 ξ5 88 h − 105727 ξ6 95040 h2 + 10139863 ξ7 15523200 h3 − 2860697731 ξ8 11176704000 h4 + 160975931 ξ9 2235340800 h5 − 827528203 ξ10 54867456000 h6 + 103295831 ξ11 43110144000 h7 − 301248457 ξ12 1034643456000 h8 + 40445413 ξ13 1494484992000 h9 − 1331119 ξ14 697426329600 h10 + 1049441 ξ15 10461394944000 h11 − 159209 ξ16 41845579776000 h12 + 5381 ξ17 54721142784000 h13 − 109 ξ18 70355755008000 h14 + ξ19 89117289676800 h15 (25) β12 = − 91 ξ5 288 h + 1165727 ξ6 3421440 h2 − 672835853 ξ7 3353011200 h3 + 3969230807 ξ8 50295168000 h4 − 24226741543 ξ9 1086375628800 h5 + 515061979 ξ10 109734912000 h6 − 1551629011 ξ11 2069286912000 h7 + 1976179 ξ12 21555072000 h8 − 9616213 ξ13 1120863744000 h9 + 14491 ξ14 23775897600 h10 − 138169 ξ15 4279661568000 h11 + 3229 ξ16 2615348736000 h12 − 1321 ξ17 41040857088000 h13 + ξ18 1954326528000 h14 − ξ19 267351869030400 h15 (26) β13 = 7 ξ5 104 h − 89849 ξ6 1235520 h2 + 8665381 ξ7 201801600 h3 − 189340169 ξ8 11176704000 h4 + 96662087 ξ9 20118067200 h5 − 55723553 ξ10 54867456000 h6 + 1757123 ξ11 10777536000 h7 − 20736923 ξ12 1034643456000 h8 + 402847 ξ13 213497856000 h9 − 94109 ξ14 697426329600 h10 + 6847 ξ15 951035904000 h11 − 11611 ξ16 41845579776000 h12 + 5189 ξ17 711374856192000 h13 − 107 ξ18 914624815104000 h14 + ξ19 1158524765798400 h15 (27) β14 = − ξ5 112 h + 1170017 ξ6 121080960 h2 − 8077807 ξ7 1412611200 h3 + 50565173 ξ8 22353408000 h4 − 17264887 ξ9 26824089600 h5 + 29969773 ξ10 219469824000 h6 − 15185081 ξ11 689762304000 h7 + 2461967 ξ12 905313024000 h8 − 673219 ξ13 2615348736000 h9 + 3229 ξ14 174356582400 h10 − 473 ξ15 475517952000 h11 + 101 ξ16 2615348736000 h12 − ξ17 977163264000 h13 + 53 ξ18 3201186852864000 h14 − ξ19 8109673360588800 h15 (28) β15 = ξ5 1800 h − 1171733 ξ6 1945944000 h2 + 30946717 ξ7 86682960000 h3 − 406841 ξ8 2874009600 h4 + 21939781 ξ9 543187814400 h5 − 3209 ξ10 373248000 h6 + 899683 ξ11 646652160000 h7 − 3247 ξ12 18811699200 h8 + 515261 ξ13 31384184832000 h9 − 71 ξ14 59779399680 h10 + 2747 ξ15 42796615680000 h11 − ξ16 398529331200 h12 + ξ17 14923948032000 h13 − ξ18 914624815104000 h14 + ξ19 121645100408832000 h15 (29) substituting equation (10) – (29) into equation (9), interpolating and collocating at xn+ j, j = 0, 1; 0, 2; 0, 3; ..., 0, k¡ and xn+ j, j = 0(1)k gives the solution to equation (1) and is given explicitly below: 312 atabo and adee / j. nig. soc. phys. sci. 3 (2021) 308–333 313 yn+1 = yn+hy ′ n+ 1 2 h2y′′n + 1 6 h3y′′′n + 27312614539002931 1161157776629760000 h4 fn+ 215021456509297 3547982095257600 h4 fn+1 − 2405950254864953 13516122267648000 h4 fn+2+ 228535928736331 465478700544000 h4 fn+3− 43830916431996773 40548366802944000 h4 fn+4 + 1983716565322187 1055947052160000 h4 fn+5− 188870621369290423 72987060245299200 h4 fn+6+ 16765512493838707 5913303492096000 h4 fn+7 − 77901096585757121 31537618624512000 h4 fn+8+ 38820512227495919 22808456326656000 h4 fn+9− 26549037149067547 28963119144960000 h4 fn+10 + 63857023621987 168951528345600 h4 fn+11− 42069828441973351 364935301226496000 h4 fn+12+ 31069509811453 1267136462592000 h4 fn+13 − 307336703482477 94612855873536000 h4 fn+14+ 14651877357209 72572361039360000 h4 fn+15 (30) yn+2 = yn+2 hy ′ n+2 h 2y′′n + 4 3 h3y′′′n + 776157610312243 3118343638410000 h4 fn+ 12110365177663 11549420883000 h4 fn+1 − 320666666657 123743795175 h4 fn+2+ 313723318722181 44547766263000 h4 fn+3− 7257112564561 471404934000 h4 fn+4 + 659444520473071 24748759035000 h4 fn+5− 815371875214873 22273883131500 h4 fn+6+ 30824521197751 769961392200 h4 fn+7 − 804553592927429 23098841766000 h4 fn+8+ 1068040546872449 44547766263000 h4 fn+9− 26608028437433 2062396586250 h4 fn+10 + 537006152783 101015343000 h4 fn+11− 2625137049893 1619918773200 h4 fn+12+ 568425874883 1649917269000 h4 fn+13 − 790392890503 17324131324500 h4 fn+14+ 4419708470941 1559171819205000 h4 fn+15 (31) yn+3 = yn+3 hy ′ n+ 9 2 h2y′′n + 9 2 h3y′′′n + 602530652428169 648922193920000 h4 fn+ 19434314988567 4055763712000 h4 fn+1 − 189942041947311 18540634112000 h4 fn+2+ 63328896091 2228441600 h4 fn+3− 1152715866154893 18540634112000 h4 fn+4 + 311770415766357 2896974080000 h4 fn+5− 2741097040974839 18540634112000 h4 fn+6+ 1311392773004949 8111527424000 h4 fn+7 − 3650761245166821 25956887756800 h4 fn+8+ 56087920654093 579394816000 h4 fn+9− 4828796962028649 92703170560000 h4 fn+10 + 12434976965421 579394816000 h4 fn+11− 121294082904151 18540634112000 h4 fn+12+ 14650769187 10534451200 h4 fn+13 − 23901979612131 129784438784000 h4 fn+14+ 464062820041 40557637120000 h4 fn+15 (32) yn+4 = yn+4 hy ′ n+8 h 2y′′n + 32 3 h3y′′′n + 451207826917664 194896477400625 h4 fn+ 57491991503264 4331032831125 h4 fn+1 − 5241628601168 206239658625 h4 fn+2+ 58358490710752 795495826125 h4 fn+3− 1525291110688 9518753475 h4 fn+4 + 286110321363872 1031198293125 h4 fn+5− 303253476312368 795495826125 h4 fn+6+ 601876165585312 1443677610375 h4 fn+7 − 174546287711456 481225870125 h4 fn+8+ 278040058459552 1113694156575 h4 fn+9− 59370049162288 441942125625 h4 fn+10 + 11415845225504 206239658625 h4 fn+11− 93955520375296 5568470782875 h4 fn+12+ 317044076768 88388425125 h4 fn+13 − 137148361136 288735522075 h4 fn+14+ 5751617237536 194896477400625 h4 fn+15 (33) 313 atabo and adee / j. nig. soc. phys. sci. 3 (2021) 308–333 314 yn+5 = yn+5 hy ′ n+ 25 2 h2y′′n + 125 6 h3y′′′n + 95264979269192125 20436376868683776 h4 fn + 168412149190625 5913303492096 h4 fn+1− 16289893064271875 324386934423552 h4 fn+2+ 2509888032209375 16587968237568 h4 fn+3 − 35503567768034375 108128978141184 h4 fn+4+ 80665948618375 141777506304 h4 fn+5− 2282582909693346875 2919482409811968 h4 fn+6 + 13484181814896875 15768809312256 h4 fn+7− 563136843495371875 756902846988288 h4 fn+8+ 46722464671271875 91233825306624 h4 fn+9 − 29797815959624875 108128978141184 h4 fn+10+ 2302123472646875 20274183401472 h4 fn+11− 101052862907884375 2919482409811968 h4 fn+12 + 3825296084375 519850856448 h4 fn+13− 2212679175659375 2270708540964864 h4 fn+14+ 7029876481375 116115777662976 h4 fn+15 (34) yn+6 = yn+6 hy ′ n+18 h 2y′′n +36 h 3y′′′n + 1302374156467 158428270000 h4 fn+ 166010903703 3168565400 h4 fn+1 − 98189047503 1131630500 h4 fn+2+ 615303609259 2263261000 h4 fn+3− 2646014872329 4526522000 h4 fn+4 + 11487075212241 11316305000 h4 fn+5− 1103785049 791350 h4 fn+6+ 24170242181949 15842827000 h4 fn+7 − 42060841828767 31685654000 h4 fn+8+ 2068042917719 2263261000 h4 fn+9− 2782160441883 5658152500 h4 fn+10 + 91711309509 452652200 h4 fn+11− 279567458927 4526522000 h4 fn+12 + 29717015319 2263261000 h4 fn+13− 3443405418 1980353375 h4 fn+14+ 8557581379 79214135000 h4 fn+15 (35) yn+7 = yn+7 hy ′ n+ 49 2 h2y′′n + 343 6 h3y′′′n + 493224413054950241 37238296043520000 h4 fn + 102260630103289 1175451264000 h4 fn+1− 7568023997612911 55167845990400 h4 fn+2+ 206970176772057433 465478700544000 h4 fn+3 − 783475292841925019 h4 fn+4 827517689856000 h4 fn+4+ 8886369763068371 5387484960000 h4 fn+5 − 16869125122720872113 7447659208704000 h4 fn+6+ 29871704176307 12055910400 h4 fn+7− 198233731133162081 91946409984000 h4 fn+8 + 43176101394439517 29092418784000 h4 fn+9− 3304470142638045703 4137588449280000 h4 fn+10+ 5673457014546587 17239951872000 h4 fn+11 − 149427441213124277 1489531841740800 h4 fn+12+ 275759583050483 12929963904000 h4 fn+13− 59927250442381 21218402304000 h4 fn+14 + 290057854303 1652978340000 h4 fn+15 (36) yn+8 = yn+8 hy ′ n+32 h 2y′′n + 256 3 h3y′′′n + 3895558043809792 194896477400625 h4 fn + 193850692468736 1443677610375 h4 fn+1− 125956008558592 618718975875 h4 fn+2+ 108065421131776 159099165225 h4 fn+3 − 295522157750272 206239658625 h4 fn+4+ 7751983865397248 3093594879375 h4 fn+5− 19137382555205632 5568470782875 h4 fn+6 + 1809910071427072 481225870125 h4 fn+7− 6607665740288 2019129525 h4 fn+8+ 12544381621239808 5568470782875 h4 fn+9 − 25512731459584 21044863125 h4 fn+10+ 309076971618304 618718975875 h4 fn+11− 847973274318848 5568470782875 h4 fn+12 + 1335378968576 41247931725 h4 fn+13− 18568423555072 4331032831125 h4 fn+14+ 51915343659008 194896477400625 h4 fn+15 (37) 314 atabo and adee / j. nig. soc. phys. sci. 3 (2021) 308–333 315 yn+9 = yn+9 hy ′ n+ 81 2 h2y′′n + 243 2 h3y′′′n + 18625133626628643 648922193920000 h4 fn + 1591587721573497 8111527424000 h4 fn+1− 5340901247722377 18540634112000 h4 fn+2+ 1141699342205313 1158789632000 h4 fn+3 − 1091696657937021 529732403200 h4 fn+4+ 10481199823834029 2896974080000 h4 fn+5− 91850956406711577 18540634112000 h4 fn+6 + 44009977111307523 8111527424000 h4 fn+7− 612567442391345667 129784438784000 h4 fn+8+ 5266011862773 1620684800 h4 fn+9 − 162099315257981103 92703170560000 h4 fn+10+ 119277892927593 165541376000 h4 fn+11− 4072438241603397 18540634112000 h4 fn+12 + 211374969363 4526522000 h4 fn+13− 14592591850059 2359717068800 h4 fn+14+ 15583153071177 40557637120000 h4 fn+15 (38) yn+10 = yn+10 hy ′ n+50 h 2y′′n + 500 3 h3y′′′n + 197774498840375 4989349821456 h4 fn + 76168854565625 277186101192 h4 fn+1− 1296060059375 3299834538 h4 fn+2+ 44460274821875 32398375464 h4 fn+3 − 20508142934375 7199638992 h4 fn+4+ 66257006246125 13199338152 h4 fn+5− 1222568114959375 178191065052 h4 fn+6 + 694980202403125 92395367064 h4 fn+7− 402852494434375 61596911376 h4 fn+8+ 1604910232878125 356382130104 h4 fn+9 − 83883071750 34613649 h4 fn+10+ 13181028190625 13199338152 h4 fn+11− 216981310746875 712764260208 h4 fn+12 + 2562778803125 39598014456 h4 fn+13− 395952809375 46197683532 h4 fn+14+ 1328461659125 2494674910728 h4 fn+15 (39) yn+11 = yn+11 hy ′ n+ 121 2 h2y′′n + 1331 6 h3y′′′n + 61603729660715825561 1161157776629760000 h4 fn + 19997710920159577 53757304473600 h4 fn+1− 1916068546902768599 3686215163904000 h4 fn+2 + 39142139354400203 21158122752000 h4 fn+3− 4686113275972795271 1228738387968000 h4 fn+4 + 3884821540842611237 575971119360000 h4 fn+5− 8722148090842645789 947883899289600 h4 fn+6 + 1809824212823561759 179191014912000 h4 fn+7− 75477320193022950763 8601168715776000 h4 fn+8 + 6267565444696350157 1036748014848000 h4 fn+9− 2854635393056124029 877670277120000 h4 fn+10 + 1187685734092033 886109414400 h4 fn+11− 13555618279103961511 33175936475136000 h4 fn+12 + 68072706364399 783634176000 h4 fn+13− 22834016763745847 1984885088256000 h4 fn+14 + 51871823638164079 72572361039360000 h4 fn+15 (40) 315 atabo and adee / j. nig. soc. phys. sci. 3 (2021) 308–333 316 yn+12 = yn+12 hy ′ n+72 h 2y′′n +288 h 3y′′′n + 62288947936 900160625 h4 fn+ 970010779104 1980353375 h4 fn+1 − 37981121616 56581525 h4 fn+2+ 686872050784 282907625 h4 fn+3− 1407334462848 282907625 h4 fn+4 + 12486203584416 1414538125 h4 fn+5− 3400616961584 282907625 h4 fn+6+ 5232260492064 396070675 h4 fn+7 − 22708918277856 1980353375 h4 fn+8+ 2236670113376 282907625 h4 fn+9− 6013168148016 1414538125 h4 fn+10 + 495777441312 282907625 h4 fn+11− 2324971232 4352425 h4 fn+12+ 32128912224 282907625 h4 fn+13 − 29783960496 1980353375 h4 fn+14+ 841151264 900160625 h4 fn+15 (41) yn+13 = yn+13 hy ′ n+ 169 2 h2y′′n + 2197 6 h3y′′′n + 86779718953850369369 982518118686720000 h4 fn + 430007608906153603 682304249088000 h4 fn+1− 883011992382639353 1039701712896000 h4 fn+2 + 1093285607607322649 350899328102400 h4 fn+3− 19802369214482134397 3119105138688000 h4 fn+4 + 3671517181341961757 324906785280000 h4 fn+5− 431204822493929017859 28071946248192000 h4 fn+6 + 7688899400227183177 454869499392000 h4 fn+7− 7111838608315340341 485194132684800 h4 fn+8 + 4437011288736058799 438624160128000 h4 fn+9− 84741976626999074461 15595525693440000 h4 fn+10 + 145708237149829601 64981357056000 h4 fn+11− 1743000544456694861 2551995113472000 h4 fn+12 + 5662729165476331 38988814233600 h4 fn+13− 139985003179042333 7277911990272000 h4 fn+14 + 73385695962908641 61407382417920000 h4 fn+15 (42) yn+14 = yn+14 hy ′ n+98 h 2y′′n + 1372 3 h3y′′′n + 1006280506843003 9091380870000 h4 fn + 26776412350933 33671781000 h4 fn+1− 53335962667519 50507671500 h4 fn+2+ 3566435834783353 909138087000 h4 fn+3 − 107133265333577 13468712400 h4 fn+4+ 7171100242899187 505076715000 h4 fn+5− 2189555205249664 113642260875 h4 fn+6 + 238278143310809 11223927000 h4 fn+7− 1238370377602421 67343562000 h4 fn+8+ 2310124038552169 181827617400 h4 fn+9 − 573731688450277 84179452500 h4 fn+10+ 284604385256219 101015343000 h4 fn+11− 1556780681533049 1818276174000 h4 fn+12 + 558564964069 3061071000 h4 fn+13− 121946014477 5050767150 h4 fn+14+ 6819244114711 4545690435000 h4 fn+15 (43) 316 atabo and adee / j. nig. soc. phys. sci. 3 (2021) 308–333 317 yn+15 = yn+15 hy ′ n+ 225 2 h2y′′n + 1125 2 h3y′′′n + 141760259867125 1038275510272 h4 fn + 16008433396875 16223054848 h4 fn+1− 191857931184375 148325072896 h4 fn+2+ 6435135446875 1324331008 h4 fn+3 − 1454924082628125 148325072896 h4 fn+4+ 1271375605125 72424352 h4 fn+5− 3527747563259375 148325072896 h4 fn+6 + 212840309896875 8111527424 h4 fn+7− 23572142846465625 1038275510272 h4 fn+8+ 1137431715625 72424352 h4 fn+9 − 178243328212875 21189296128 h4 fn+10+ 32313630965625 9270317056 h4 fn+11− 156469002934375 148325072896 h4 fn+12 + 525121903125 2317579264 h4 fn+13− 30805101646875 1038275510272 h4 fn+14+ 60144050875 32446109696 h4 fn+15 (44) thus, equation (30) – (44) is the new method named (ns4o15m). therefore, we take the first derivative of equation (9) and evaluate at xn+ j, j = 0(1)k to have: y′n+1 = y ′ n+hy ′′ n + 1 2 h2y′′′n + 2646946687904237 32011868528640000 h3 fn + 9331210633373 35568742809600 h3 fn+1− 525466618744679 711374856192000 h3 fn+2+ 11892772256669 5884534656000 h3 fn+3 − 50067223151677 11291664384000 h3 fn+4+ 6838723295804269 889218570240000 h3 fn+5− 13547754274847623 1280474741145600 h3 fn+6 + 6130273402717 529296768000 h3 fn+7− 2391130153285807 237124952064000 h3 fn+8+ 11115802946972267 1600593426432000 h3 fn+9 − 13298258493701081 3556874280960000 h3 fn+10+ 195767508553 127031224320 h3 fn+11− 3008598640512121 6402373705728000 h3 fn+12 + 5923791011249 59281238016000 h3 fn+13− 855970682491 64670441472000 h3 fn+14+ 822811814369 1000370891520000 h3 fn+15 (45) y′n+2 = y ′ n+2 hy ′′ n +2 h 2y′′′n + 6414418122631 15630795180000 h3 fn+ 176725029073 86837751000 h3 fn+1 − 653662511 141775920 h3 fn+2+ 9816667434881 781539759000 h3 fn+3− 866478322811 31577364000 h3 fn+4 + 404117238667 8513505000 h3 fn+5− 101892006563273 1563079518000 h3 fn+6+ 412585117387 5789183400 h3 fn+7 − 7177626750949 115783668000 h3 fn+8+ 680466388363 15949791000 h3 fn+9− 19933118316169 868377510000 h3 fn+10 + 821247749563 86837751000 h3 fn+11− 1802310446791 625231807200 h3 fn+12+ 53212387759 86837751000 h3 fn+13 − 1565842507 19297278000 h3 fn+14+ 19699416731 3907698795000 h3 fn+15 (46) y′n+3 = y ′ n+3 hy ′′ n + 9 2 h2y′′′n + 690720887139 697016320000 h3 fn+ 32059975167 5544448000 h3 fn+1 − 981108537597 88711168000 h3 fn+2+ 16959465003 536166400 h3 fn+3− 67574744254179 975822848000 h3 fn+4 + 746266958199 6223360000 h3 fn+5− 160777027856499 975822848000 h3 fn+6+ 2585730381813 14350336000 h3 fn+7 − 30594531344121 195164569600 h3 fn+8+ 6580669587693 60988928000 h3 fn+9− 40468694141943 697016320000 h3 fn+10 + 5836036102983 243955712000 h3 fn+11− 1016547199431 139403264000 h3 fn+12+ 18909103959 12197785600 h3 fn+13 − 200320234209 975822848000 h3 fn+14+ 15557090403 1219778560000 h3 fn+15 (47) 317 atabo and adee / j. nig. soc. phys. sci. 3 (2021) 308–333 318 y′n+4 = y ′ n+4 hy ′′ n +8 h 2y′′′n + 891141341312 488462349375 h3 fn+ 5951251456 516891375 h3 fn+1 − 395392552 20138625 h3 fn+2+ 5846193073216 97692469875 h3 fn+3− 443144332 3408075 h3 fn+4 + 12223798950016 54273594375 h3 fn+5− 30236186795768 97692469875 h3 fn+6+ 1224829520192 3618239625 h3 fn+7 − 1065684738736 3618239625 h3 fn+8+ 6657405184 32837805 h3 fn+9− 657887719304 6030399375 h3 fn+10 + 9958015424 221524875 h3 fn+11− 1338659514532 97692469875 h3 fn+12+ 31620742784 10854718875 h3 fn+13 − 837478216 2170943775 h3 fn+14+ 900559936 37574026875 h3 fn+15 (48) y′n+5 = y ′ n+5 hy ′′ n + 25 2 h2y′′′n + 149078112485525 51218989645824 h3 fn+ 27357607800625 1422749712384 fn+1 − 57524077045625 1896999616512 h3 fn+2+ 9768839123125 100037089152 h3 fn+3− 108220654263125 517363531776 h3 fn+4 + 516319144825 1421328384 h3 fn+5− 3653176292400625 7316998520832 h3 fn+6+ 32375202664375 59281238016 h3 fn+7 − 901457665911875 1896999616512 h3 fn+8+ 4188606865466875 12804747411456 h3 fn+9− 1001792026947025 5690998849536 h3 fn+10 + 252938010625 3487131648 h3 fn+11− 14708026245625 665181683712 h3 fn+12+ 955425548125 203249958912 h3 fn+13 − 3542687313125 5690998849536 h3 fn+14+ 1934553625 50018544576 h3 fn+15 (49) y′n+6 = y ′ n+6 hy ′′ n +18 h 2y′′′n + 10124252319 2382380000 h3 fn+ 689167593 23823800 h3 fn+1 − 10275078321 238238000 h3 fn+2+ 2468567643 17017000 h3 fn+3− 145912136193 476476000 h3 fn+4 + 318410446077 595595000 h3 fn+5− 34946139 47600 h3 fn+6+ 95665826601 119119000 h3 fn+7 − 47568246489 68068000 h3 fn+8+ 57304332759 119119000 h3 fn+9− 6293482551 24310000 h3 fn+10 + 2541447099 23823800 h3 fn+11− 15494777187 476476000 h3 fn+12+ 6920469 1001000 h3 fn+13 − 218119509 238238000 h3 fn+14+ 33879999 595595000 h3 fn+15 (50) y′n+7 = y ′ n+7 hy ′′ n + 49 2 h2y′′′n + 346932898878727 59391221760000 h3 fn + 36848690199839 907365888000 h3 fn+1− 168555897445747 2903570841600 h3 fn+2+ 6601074966881017 32665171968000 h3 fn+3 − 555890398779737 1319804928000 h3 fn+4+ 3356052363748651 4536829440000 h3 fn+5− 132440457899414987 130660687872000 h3 fn+6 + 625723050379 564019200 h3 fn+7− 4671757797600943 4839284736000 h3 fn+8+ 5427096160617551 8166292992000 h3 fn+9 − 25961087688408481 72589271040000 h3 fn+10+ 534888408252611 3629463552000 h3 fn+11− 234803718886513 5226427514880 h3 fn+12 + 787863604577 82487808000 h3 fn+13− 18363232338017 14517854208000 h3 fn+14+ 755030443391 9607403520000 h3 fn+15 (51) 318 atabo and adee / j. nig. soc. phys. sci. 3 (2021) 308–333 319 y′n+8 = y ′ n+8 hy ′′ n +32 h 2y′′′n + 16988198848 2210236875 h3 fn+ 53560565248 986792625 h3 fn+1 − 815067367168 10854718875 h3 fn+2+ 1050018913792 3907698795 h3 fn+3− 859150752128 1550674125 h3 fn+4 + 53114085282304 54273594375 h3 fn+5− 130629987954944 97692469875 h3 fn+6+ 757517948416 516891375 h3 fn+7 − 2150503744 1686825 h3 fn+8+ 85740073246208 97692469875 h3 fn+9− 25634459993344 54273594375 h3 fn+10 + 301807773184 1550674125 h3 fn+11− 5796336118912 97692469875 h3 fn+12+ 27384514048 2170943775 h3 fn+13 − 18132713216 10854718875 h3 fn+14+ 50697789952 488462349375 h3 fn+15 (52) y′n+9 = y ′ n+9 hy ′′ n + 81 2 h2y′′′n + 47735374876317 4879114240000 h3 fn + 1312259291757 18765824000 h3 fn+1− 10036420749 106496000 h3 fn+2+ 10518533830593 30494464000 h3 fn+3 − 137554269574383 195164569600 h3 fn+4+ 1526337505582101 1219778560000 h3 fn+5− 1663567405083603 975822848000 h3 fn+6 + 57085662261447 30494464000 h3 fn+7− 1587493105268517 975822848000 h3 fn+8+ 7800788673 6963200 h3 fn+9 − 2941612400903409 4879114240000 h3 fn+10+ 473504266647 1905904000 h3 fn+11− 73906005035817 975822848000 h3 fn+12 + 3928134597579 243955712000 h3 fn+13− 416165679981 195164569600 h3 fn+14+ 57389067 433160000 h3 fn+15 (53) y′n+10 = y ′ n+10 hy ′′ n +50 h 2y′′′n + 43351425625 3572753184 h3 fn+ 60830018125 694702008 h3 fn+1 − 12345544375 106877232 h3 fn+2+ 384753416875 893188296 h3 fn+3− 47592424375 54486432 h3 fn+4 + 22085668025 14177592 h3 fn+5− 26465861018125 12504636144 h3 fn+6+ 539270756875 231567336 h3 fn+7 − 1871692800625 926269344 h3 fn+8+ 8709301139375 6252318072 h3 fn+9− 1040166025 1388016 h3 fn+10 + 214532249375 694702008 h3 fn+11− 336344910625 3572753184 h3 fn+12+ 421341875 21051576 h3 fn+13 − 283286875 106877232 h3 fn+14+ 1029672275 6252318072 h3 fn+15 (54) y′n+11 = y ′ n+11 hy ′′ n + 121 2 h2y′′′n + 42887155137009007 2910169866240000 h3 fn + 12377607151943 115482931200 h3 fn+1− 183316518915541 1319804928000 h3 fn+2+ 76566704845774583 145508493312000 h3 fn+3 − 1399043605047649 1319804928000 h3 fn+4+ 4262048618111119 2245501440000 h3 fn+5− 3522716671290113 1369491701760 h3 fn+6 + 15283394277950909 5389203456000 h3 fn+7− 52954438598926117 21556813824000 h3 fn+8+ 8814989815349839 5196731904000 h3 fn+9 − 294676003269851291 323352207360000 h3 fn+10+ 1907340759533 5076172800 h3 fn+11− 66668404334979491 582033973248000 h3 fn+12 + 98429973965983 4041902592000 h3 fn+13− 69521484528617 21556813824000 h3 fn+14+ 11214181843469 55964805120000 h3 fn+15 (55) 319 atabo and adee / j. nig. soc. phys. sci. 3 (2021) 308–333 320 y′n+12 = y ′ n+12 hy ′′ n +72 h 2y′′′n + 1309780608 74449375 h3 fn+ 1917572544 14889875 h3 fn+1 − 489571704 2977975 h3 fn+2+ 9399103872 14889875 h3 fn+3− 18828258012 14889875 h3 fn+4+ 24164791488 10635625 h3 fn+5 − 6534821544 2127125 h3 fn+6+ 10105364736 2977975 h3 fn+7− 2570011632 875875 h3 fn+8 + 30227141184 14889875 h3 fn+9− 81008278056 74449375 h3 fn+10+ 6696008064 14889875 h3 fn+11 − 4481292 32725 h3 fn+12+ 61932096 2127125 h3 fn+13− 5219352 1353625 h3 fn+14+ 17836032 74449375 h3 fn+15 (56) y′n+13 = y ′ n+13 hy ′′ n + 169 2 h2y′′′n + 50976734501772521 2462451425280000 h3 fn + 231607232521561 1520031744000 h3 fn+1− 10507261454113931 54721142784000 h3 fn+2+ 327984650433739 439723468800 h3 fn+3 − 7396667135771797 4974649344000 h3 fn+4+ 183307899755534437 68401428480000 h3 fn+5− 254414562335962769 70355755008000 h3 fn+6 + 570182964741793 142502976000 h3 fn+7− 1800041137343969 521153740800 h3 fn+8+ 294815320789342379 123122571264000 h3 fn+9 − 140128717971047 109486080000 h3 fn+10+ 454258655878469 855017856000 h3 fn+11− 79225841723025829 492490285056000 h3 fn+12 + 2682311572421 78173061120 h3 fn+13− 248596418804357 54721142784000 h3 fn+14+ 21720378795557 76951607040000 h3 fn+15 (57) y′n+14 = y ′ n+14 hy ′′ n +98 h 2y′′′n + 7676045327671 318995820000 h3 fn+ 28667764727 161109000 h3 fn+1 − 23816981209 107406000 h3 fn+2+ 13878519596837 15949791000 h3 fn+3− 489732836761 283551840 h3 fn+4 + 27659304286957 8860995000 h3 fn+5− 134110052796167 31899582000 h3 fn+6+ 2752397206979 590733000 h3 fn+7 − 9488406575333 2362932000 h3 fn+8+ 8897984343731 3189958200 h3 fn+9− 26342918970319 17721990000 h3 fn+10 + 122149393583 196911000 h3 fn+11− 696997533883 3752892000 h3 fn+12+ 71589567763 1772199000 h3 fn+13 − 3740727473 708879600 h3 fn+14+ 26185092311 79748955000 h3 fn+15 (58) y′n+15 = y ′ n+15 hy ′′ n + 225 2 h2y′′′n + 216067472925 7806582784 h3 fn+ 100262210625 487911424 h3 fn+1 − 179965535625 709689344 h3 fn+2+ 1959327448125 1951645696 h3 fn+3− 2213858930625 1115226112 h3 fn+4 + 1754723396925 487911424 h3 fn+5− 37748275074375 7806582784 h3 fn+6+ 1496243626875 278806528 h3 fn+7 − 36049237025625 7806582784 h3 fn+8+ 1568033338125 487911424 h3 fn+9− 13327761065625 7806582784 h3 fn+10 + 199792130625 278806528 h3 fn+11− 1651019848125 7806582784 h3 fn+12+ 23381533125 487911424 h3 fn+13 − 2568718125 459210752 h3 fn+14+ 752826975 1951645696 h3 fn+15 (59) similarly, the second derivative of equation (9) is found and then evaluated at xn+ j, j = 0(1)k to give: 320 atabo and adee / j. nig. soc. phys. sci. 3 (2021) 308–333 321 y′′n+1 = y ′′ n +hy ′′′ n + 3206819325906617 16005934264320000 h2 fn + 111956703448001 133382785536000 h2 fn+1− 780902869927541 355687428096000 h2 fn+2+ 9502268460740177 1600593426432000 h2 fn+3 − 1975509756311887 152437469184000 h2 fn+4+ 9965996481056899 444609285120000 h2 fn+5− 98538747853228853 3201186852864000 h2 fn+6 + 5985224981757391 177843714048000 h2 fn+7− 3470876036880139 118562476032000 h2 fn+8+ 251938134205319 12504636144000 h2 fn+9 − 57837526128808793 5335311421440000 h2 fn+10+ 113475037891729 25406244864000 h2 fn+11− 4358164674336769 3201186852864000 h2 fn+12 + 77205249783577 266765571072000 h2 fn+13− 13631425781543 355687428096000 h2 fn+14+ 19055094997949 8002967132160000 h2 fn+15 (60) y′′n+2 = y ′′ n +2 hy ′′′ n + 7100763759181 15630795180000 h2 fn + 79686372907 28945917000 h2 fn+1− 198060940481 37216179000 h2 fn+2+ 11595391593263 781539759000 h2 fn+3 − 3762629808143 115783668000 h2 fn+4+ 73285372020391 1302566265000 h2 fn+5− 120811992265001 1563079518000 h2 fn+6 + 815470210927 9648639000 h2 fn+7− 25538181924601 347351004000 h2 fn+8+ 807085379107 15949791000 h2 fn+9 − 492563604418 18091198125 h2 fn+10+ 2922359996183 260513253000 h2 fn+11− 10689180138419 3126159036000 h2 fn+12 + 21039751529 28945917000 h2 fn+13− 4559016221 47366046000 h2 fn+14+ 23367229601 3907698795000 h2 fn+15 (61) y′′n+3 = y ′′ n +3 hy ′′′ n + 1724745170011 2439557120000 h2 fn+ 1156246932303 243955712000 h2 fn+1 − 3669218612307 487911424000 h2 fn+2+ 15741501473 670208000 h2 fn+3− 25023934662201 487911424000 h2 fn+4 + 15482577601863 174254080000 h2 fn+5− 59583820501187 487911424000 h2 fn+6+ 16294185200871 121977856000 h2 fn+7 − 56712794491323 487911424000 h2 fn+8+ 19519814663081 243955712000 h2 fn+9− 15006216511971 348508160000 h2 fn+10 + 8453922669 476476000 h2 fn+11− 376990411261 69701632000 h2 fn+12+ 25501122657 22177792000 h2 fn+13 − 74295242361 487911424000 h2 fn+14+ 5770019551 609889280000 h2 fn+15 (62) y′′n+4 = y ′′ n +4 hy ′′′ n + 468835748086 488462349375 h2 fn+ 31276065416 4652022375 h2 fn+1 − 237066268 24613875 h2 fn+2+ 3223897961912 97692469875 h2 fn+3− 25089869444 357847875 h2 fn+4 + 2201175147736 18091198125 h2 fn+5− 16340498219092 97692469875 h2 fn+6+ 1986215722936 10854718875 h2 fn+7 − 21338011906 134008875 h2 fn+8+ 1529765976712 13956067125 h2 fn+9− 9604572197884 162820783125 h2 fn+10 + 4188081368 172297125 h2 fn+11− 723896144432 97692469875 h2 fn+12+ 51299745704 32564156625 h2 fn+13 − 754844036 3618239625 h2 fn+14+ 6331431944 488462349375 h2 fn+15 (63) 321 atabo and adee / j. nig. soc. phys. sci. 3 (2021) 308–333 322 y′′n+5 = y ′′ n +5 hy ′′′ n + 31054788444685 25609494822912 h2 fn + 6194003019125 711374856192 h2 fn+1− 100287550861375 8536498274304 h2 fn+2+ 545635775073875 12804747411456 h2 fn+3 − 22793304846625 258681765888 h2 fn+4+ 164744717905 1065996288 h2 fn+5− 776841970197875 3658499260416 h2 fn+6 + 110166567234875 474249904128 h2 fn+7− 575215088287625 2845499424768 h2 fn+8+ 890981273520125 6402373705728 h2 fn+9 − 213109965786455 2845499424768 h2 fn+10+ 131726502560375 4268249137152 h2 fn+11− 3129099276125 332590841856 h2 fn+12 + 3176113625 1587890304 h2 fn+13− 2261231861125 8536498274304 h2 fn+14+ 210742341295 12804747411456 h2 fn+15 (64) y′′n+6 = y ′′ n +6 hy ′′′ n + 3491237903 2382380000 h2 fn+ 1273496769 119119000 h2 fn+1 − 3302866863 238238000 h2 fn+2+ 888545417 17017000 h2 fn+3− 50509176723 476476000 h2 fn+4+ 112104615069 595595000 h2 fn+5 − 30620993 119000 h2 fn+6+ 33546751263 119119000 h2 fn+7− 116779083507 476476000 h2 fn+8 + 20098920059 119119000 h2 fn+9− 15452545149 170170000 h2 fn+10+ 4457401749 119119000 h2 fn+11 − 5435403101 476476000 h2 fn+12+ 41270877 17017000 h2 fn+13− 4782384 14889875 h2 fn+14+ 11885633 595595000 h2 fn+15 (65) y′′n+7 = y ′′ n +7 hy ′′′ n + 561270290962031 326651719680000 h2 fn+ 138008915043671 10888390656000 h2 fn+1 − 115995161213911 7258927104000 h2 fn+2+ 1009684768894753 16332585984000 h2 fn+3− 2698346531507767 21776781312000 h2 fn+4 + 4028977853553919 18147317760000 h2 fn+5− 19697678682126767 65330343936000 h2 fn+6+ 2100668420753 6345216000 h2 fn+7 − 696945974650877 2419642368000 h2 fn+8+ 6477310161851851 32665171968000 h2 fn+9− 11619838717354259 108883906560000 h2 fn+10 + 79805888878681 1814731776000 h2 fn+11− 875850118609039 65330343936000 h2 fn+12+ 2387301668339 837568512000 h2 fn+13 − 2740021526389 7258927104000 h2 fn+14+ 3740727473 159497910000 h2 fn+15 (66) y′′n+8 = y ′′ n +8 hy ′′′ n + 962791153472 488462349375 h2 fn+ 159116760448 10854718875 h2 fn+1 − 589264474048 32564156625 h2 fn+2+ 6977664005248 97692469875 h2 fn+3− 219901261024 1550674125 h2 fn+4 − 33760546806464 97692469875 h2 fn+6+ 1379590216064 3618239625 h2 fn+7− 25115068768 75907125 h2 fn+8 + 22258735804544 97692469875 h2 fn+9− 6655353722816 54273594375 h2 fn+10+ 235080913792 4652022375 h2 fn+11 − 1504988356384 97692469875 h2 fn+12+ 35552129152 10854718875 h2 fn+13− 14124842944 32564156625 h2 fn+14 + 13164251008 488462349375 h2 fn+15 (67) 322 atabo and adee / j. nig. soc. phys. sci. 3 (2021) 308–333 323 y′′n+9 = y ′′ n +9 hy ′′′ n + 5425276143393 2439557120000 h2 fn+ 507508999443 30494464000 h2 fn+1− 1408763359683 69701632000 h2 fn+2 + 1797053294583 22177792000 h2 fn+3− 77925040957971 487911424000 h2 fn+4+ 176608903026609 609889280000 h2 fn+5 − 190121626816461 487911424000 h2 fn+6+ 105298772316291 243955712000 h2 fn+7− 181947002167377 487911424000 h2 fn+8 + 15689842209 60928000 h2 fn+9− 337995172999251 2439557120000 h2 fn+10+ 13927989639699 243955712000 h2 fn+11 − 8492014219401 487911424000 h2 fn+12+ 451361269269 121977856000 h2 fn+13− 18392351427 37531648000 h2 fn+14 + 37139996961 1219778560000 h2 fn+15 (68) y′′n+10 = y ′′ n +10 hy ′′′ n + 61940195105 25009272288 h2 fn+ 38819735375 2084106024 h2 fn+1 − 7755391625 347351004 h2 fn+2+ 11564784875 127598328 h2 fn+3− 1480662560375 8336424096 h2 fn+4 + 32091122155 99243144 h2 fn+5− 5423884076375 12504636144 h2 fn+6+ 334822805875 694702008 h2 fn+7 − 384255384875 926269344 h2 fn+8+ 1800509413375 6252318072 h2 fn+9− 321370705 2082024 h2 fn+10 + 44211042625 694702008 h2 fn+11− 69319839125 3572753184 h2 fn+12+ 8597247125 2084106024 h2 fn+13 − 759050375 1389404016 h2 fn+14+ 212230705 6252318072 h2 fn+15 (69) y′′n+11 = y ′′ n +11 hy ′′′ n + 3971658023273897 1455084933120000 h2 fn + 2266813707059 109983744000 h2 fn+1− 48389901241627 1979707392000 h2 fn+2+ 1823189496513649 18188561664000 h2 fn+3 − 100348334174341 513257472000 h2 fn+4+ 86610066741761629 242514155520000 h2 fn+5− 139054076635086833 291016986624000 h2 fn+6 + 53121794226743 99800064000 h2 fn+7− 14769476593075457 32335220736000 h2 fn+8+ 6621237757011361 20786927616000 h2 fn+9 − 9127089742965557 53892034560000 h2 fn+10+ 9366906277091 133249536000 h2 fn+11− 6230159494371769 291016986624000 h2 fn+12 + 8175398913517 1796401152000 h2 fn+13− 58463252748769 97005662208000 h2 fn+14+ 13621603946467 363771233280000 h2 fn+15 (70) y′′n+12 = y ′′ n +12 hy ′′′ n + 222031178 74449375 h2 fn+ 336427848 14889875 h2 fn+1 − 395464572 14889875 h2 fn+2+ 1635581336 14889875 h2 fn+3− 3177823608 14889875 h2 fn+4 + 4157834328 10635625 h2 fn+5− 1110232588 2127125 h2 fn+6+ 8675883864 14889875 h2 fn+7 − 7426541862 14889875 h2 fn+8+ 5198849816 14889875 h2 fn+9− 13721678676 74449375 h2 fn+10 + 1158088968 14889875 h2 fn+11− 3815012 163625 h2 fn+12+ 10574568 2127125 h2 fn+13 − 9806508 14889875 h2 fn+14+ 3046952 74449375 h2 fn+15 (71) 323 atabo and adee / j. nig. soc. phys. sci. 3 (2021) 308–333 324 y′′n+13 = y ′′ n +13 hy ′′′ n + 3983186767192217 1231225712640000 h2 fn + 504351825845881 20520428544000 h2 fn+1− 261504344043007 9120190464000 h2 fn+2+ 2100826446847817 17588938752000 h2 fn+3 − 18984767728152889 82081714176000 h2 fn+4+ 1210240131830197 2850059520000 h2 fn+5− 19906199411300267 35177877504000 h2 fn+6 + 8656591000332301 13680285696000 h2 fn+7− 1642958954761241 3040063488000 h2 fn+8+ 23360194872458951 61561285632000 h2 fn+9 − 11668896830043479 58629795840000 h2 fn+10+ 388626555599243 4560095232000 h2 fn+11− 5998735795932529 246245142528000 h2 fn+12 + 8030528674181 1465744896000 h2 fn+13− 594992073871 829108224000 h2 fn+14+ 27410337567119 615612856320000 h2 fn+15 (72) y′′n+14 = y ′′ n +14 hy ′′′ n + 1112609593123 318995820000 h2 fn+ 47075937433 1772199000 h2 fn+1 − 327517459171 10633194000 h2 fn+2+ 2059176926639 15949791000 h2 fn+3− 1767851783719 7088796000 h2 fn+4 + 12198755857183 26582985000 h2 fn+5− 4871439361817 7974895500 h2 fn+6+ 404310610109 590733000 h2 fn+7 − 4139166288943 7088796000 h2 fn+8+ 6562564086043 15949791000 h2 fn+9− 3810288983977 17721990000 h2 fn+10 + 496658084279 5316597000 h2 fn+11− 1636772074721 63799164000 h2 fn+12+ 1113970361 161109000 h2 fn+13 − 3740727473 5316597000 h2 fn+14+ 3740727473 79748955000 h2 fn+15 (73) y′′n+15 = y ′′ n +15 hy ′′′ n + 14605252055 3903291392 h2 fn+ 55679491125 1951645696 h2 fn+1 − 127956427875 3903291392 h2 fn+2+ 134732597875 975822848 h2 fn+3− 147877626375 557613056 h2 fn+4 + 951630836355 1951645696 h2 fn+5− 2519631003875 3903291392 h2 fn+6+ 175914493875 243955712 h2 fn+7 − 2381638669875 3903291392 h2 fn+8+ 834744780875 1951645696 h2 fn+9− 856960688385 3903291392 h2 fn+10 + 13247717625 139403264 h2 fn+11− 93624917875 3903291392 h2 fn+12+ 14479300125 1951645696 h2 fn+13 + 1501410375 3903291392 h2 fn+14+ 50188465 487911424 h2 fn+15 (74) also, we take the third derivative of equation (9) and evaluate at xn+ j, j = 0(1)k to give: y′′′n+1 = y ′′′ n + 25221445 98402304 h fn+ 105145058757073 62768369664000 h fn+1 − 20999287611259 5706215424000 h fn+2+ 612744541065337 62768369664000 h fn+3− 189568380436867 8966909952000 h fn+4 + 2285168598349733 62768369664000 h fn+5− 3129453071993581 62768369664000 h fn+6+ 1138313909617631 20922789888000 h fn+7 − 988788576755233 20922789888000 h fn+8+ 679781959848881 20922789888000 h fn+9− 1096355235402331 62768369664000 h fn+10 + 64486158419069 8966909952000 h fn+11− 137515713789319 62768369664000 h fn+12+ 29219384284087 62768369664000 h fn+13 − 3867689367599 62768369664000 h fn+14+ 2639651053 689762304000 h fn+15 (75) 324 atabo and adee / j. nig. soc. phys. sci. 3 (2021) 308–333 325 y′′′n+2 = y ′′′ n + 631693279 2501928000 h fn+ 15268129873 7662154500 h fn+1− 302093934503 122594472000 h fn+2 + 31510558777 3831077250 h fn+3− 2248853768341 122594472000 h fn+4+ 245046332159 7662154500 h fn+5 − 1802000069137 40864824000 h fn+6+ 30858062633 638512875 h fn+7− 1720640706481 40864824000 h fn+8 + 222285459167 7662154500 h fn+9− 1915228492981 122594472000 h fn+10+ 24675139849 3831077250 h fn+11 − 240803930183 122594472000 h fn+12+ 3200718769 7662154500 h fn+13− 6783778481 122594472000 h fn+14 + 3012 875875 h fn+15 (76) y′′′n+3 = y ′′′ n + 1674820979 6623232000 h fn+ 56902877331 28700672000 h fn+1 − 60273315579 28700672000 h fn+2+ 794168588449 86102016000 h fn+3− 550323797343 28700672000 h fn+4 + 135645163833 4100096000 h fn+5− 1303414182247 28700672000 h fn+6+ 203566719249 4100096000 h fn+7 − 1239549466761 28700672000 h fn+8+ 2559362190283 86102016000 h fn+9− 459042547569 28700672000 h fn+10 + 189136929441 28700672000 h fn+11− 3529763983 1757184000 h fn+12+ 12256162053 28700672000 h fn+13 − 1622998893 28700672000 h fn+14+ 9166839 2609152000 h fn+15 (77) y′′′n+4 = y ′′′ n + 1328507 5255250 h fn+ 3800835068 1915538625 h fn+1− 579872212 273648375 h fn+2 + 18445487108 1915538625 h fn+3− 70153091887 3831077250 h fn+4+ 62404505692 1915538625 h fn+5 − 86024128376 1915538625 h fn+6+ 31404619756 638512875 h fn+7− 54690363709 1277025750 h fn+8 + 2690307652 91216125 h fn+9− 30412523756 1915538625 h fn+10+ 12534712108 1915538625 h fn+11 − 7643539093 3831077250 + 812610068 1915538625 h fn+13 − 107625136 1915538625 h fn+14+ 53500 15324309 h fn+15 (78) y′′′n+5 = y ′′′ n + 887775845 3511517184 h fn+ 10946710975 5518098432 h fn+1 − 81687827825 38626689024 h fn+2+ 4819615460525 502146957312 h fn+3− 8971075692025 502146957312 h fn+4 + 16743731110985 502146957312 h fn+5− 1081685073325 23911759872 h fn+6+ 8274717147275 167382319104 h fn+7 − 7198523492725 167382319104 h fn+8+ 14864967700175 502146957312 h fn+9− 7999362395575 502146957312 h fn+10 + 3296252627975 502146957312 h fn+11− 1004846299475 502146957312 h fn+12+ 30518628325 71735279616 h fn+13 − 28291221275 502146957312 h fn+14+ 803745 229605376 h fn+15 (79) 325 atabo and adee / j. nig. soc. phys. sci. 3 (2021) 308–333 326 y′′′n+6 = y ′′′ n + 340115 1345344 h fn+ 6950787 3503500 h fn+1− 118634421 56056000 h fn+2 + 2404277 250250 h fn+3− 1004179551 56056000 h fn+4+ 118528077 3503500 h fn+5 − 7491091147 168168000 h fn+6+ 43106373 875875 h fn+7− 2403399321 56056000 h fn+8 + 310373351 10510500 h fn+9− 127297257 8008000 h fn+10+ 11476851 1751750 h fn+11 − 10179391 5096000 h fn+12+ 135273 318500 h fn+13− 450453 8008000 h fn+14+ 1312 375375 h fn+15 (80) y′′′n+7 = y ′′′ n + 148080429 585728000 h fn+ 2541298850209 1280987136000 h fn+1 − 2709977083289 1280987136000 h fn+2+ 1118294324987 116453376000 h fn+3− 22922743552933 1280987136000 h fn+4 + 43245378870197 1280987136000 h fn+5− 56382862132093 1280987136000 h fn+6+ 21283573174031 426995712000 h fn+7 − 18372161728753 426995712000 h fn+8+ 12638994737537 426995712000 h fn+9− 20401231357003 1280987136000 h fn+10 + 8406116820059 1280987136000 h fn+11− 232954185989 116453376000 h fn+12+ 544780276327 1280987136000 h fn+13 − 72145233503 1280987136000 h fn+14+ 4482518383 1280987136000 h fn+15 (81) y′′′n+8 = y ′′′ n + 69181108 273648375 h fn+ 3800295904 1915538625 h fn+1− 368501776 174139875 h fn+2 + 18400755872 1915538625 h fn+3− 4900116016 273648375 h fn+4+ 64738663904 1915538625 h fn+5 − 28171120816 638512875 h fn+6+ 32195047904 638512875 h fn+7− 27104604508 638512875 h fn+8 + 56498981024 1915538625 h fn+9− 30436015696 1915538625 h fn+10+ 1792450592 273648375 h fn+11 − 3825896576 1915538625 h fn+12+ 813518752 1915538625 h fn+13− 107747312 1915538625 h fn+14+ 1312 375375 h fn+15 (82) y′′′n+9 = y ′′′ n + 148080429 585728000 h fn+ 56938145427 28700672000 h fn+1 − 60718067451 28700672000 h fn+2+ 275617326987 28700672000 h fn+3− 513628536159 28700672000 h fn+4 + 969126008463 28700672000 h fn+5− 1264262164839 28700672000 h fn+6+ 1442792036727 28700672000 h fn+7 − 1200258046089 28700672000 h fn+8+ 861743554713 28700672000 h fn+9− 458089367409 28700672000 h fn+10 + 188548120737 28700672000 h fn+11− 57454723701 28700672000 h fn+12+ 12212476677 28700672000 h fn+13 − 1617125229 28700672000 h fn+14+ 803745 229605376 h fn+15 (83) 326 atabo and adee / j. nig. soc. phys. sci. 3 (2021) 308–333 327 y′′′n+10 = y ′′′ n + 340115 1345344 h fn+ 121610725 61297236 h fn+1− 2075551475 980755776 h fn+2 + 294433325 30648618 h fn+3− 17564076025 980755776 h fn+4+ 295984445 8756748 h fn+5 − 43273790575 980755776 h fn+6+ 36779675 729729 h fn+7− 13743878725 326918592 h fn+8 + 627611225 20432412 h fn+9− 15176096665 980755776 h fn+10+ 18169175 2786238 h fn+11 − 39878275 20015424 h fn+12+ 25987525 61297236 h fn+13− 55091525 980755776 h fn+14+ 53500 15324309 h fn+15 (84) y′′′n+11 = y ′′′ n + 887775845 3511517184 h fn+ 11319967290643 5706215424000 h fn+1 − 132626474849 62705664000 h fn+2+ 54784127878667 5706215424000 h fn+3− 102073339405279 5706215424000 h fn+4 + 192568525522703 5706215424000 h fn+5− 83717504794957 1902071808000 h fn+6+ 95525302183421 1902071808000 h fn+7 − 79485015923203 1902071808000 h fn+8+ 24771909147359 815173632000 h fn+9− 83924967291121 5706215424000 h fn+10 + 39756270033953 5706215424000 h fn+11− 11548005399029 5706215424000 h fn+12+ 2443388534917 5706215424000 h fn+13 − 322980342509 5706215424000 h fn+14+ 9166839 2609152000 h fn+15 (85) y′′′n+12 = y ′′′ n + 1328507 5255250 h fn+ 248268 125125 h fn+1− 12972 6125 h fn+2 + 25263044 2627625 h fn+3− 31416549 1751750 h fn+4+ 29661108 875875 h fn+5− 5536136 125125 h fn+6 + 44305452 875875 h fn+7− 74018277 1751750 h fn+8+ 81177812 2627625 h fn+9− 13324692 875875 h fn+10 + 6858276 875875 h fn+11− 8502661 5255250 h fn+12+ 51204 125125 h fn+13− 48192 875875 h fn+14+ 3012 875875 h fn+15 (86) y′′′n+13 = y ′′′ n + 1674820979 6623232000 h fn+ 9574328794069 4828336128000 h fn+1− 10180906367021 4828336128000 h fn+2 + 6600937081531 689762304000 h fn+3− 85872829106857 4828336128000 h fn+4+ 161714860824569 4828336128000 h fn+5 − 210181855059553 4828336128000 h fn+6+ 79669188209819 1609445376000 h fn+7− 65878891642213 1609445376000 h fn+8 + 47601806972213 1609445376000 h fn+9− 9733223634433 689762304000 h fn+10+ 33795680175959 4828336128000 h fn+11 − 2990410218211 4828336128000 h fn+12+ 3733892296147 4828336128000 h fn+13− 44874067901 689762304000 h fn+14 + 2639651053 689762304000 h fn+15 (87) 327 atabo and adee / j. nig. soc. phys. sci. 3 (2021) 308–333 328 y′′′n+14 = y ′′′ n + 631693279 2501928000 h fn+ 311056753 156370500 h fn+1− 5395044599 2501928000 h fn+2 + 765940609 78185250 h fn+3− 46375653541 2501928000 h fn+4+ 5525678207 156370500 h fn+5 − 39205297537 833976000 h fn+6+ 712193069 13030875 h fn+7− 39205297537 833976000 h fn+8 + 5525678207 156370500 h fn+9− 46375653541 2501928000 h fn+10 + 765940609 78185250 h fn+11− 5395044599 2501928000 h fn+12+ 311056753 156370500 h fn+13+ 631693279 2501928000 h fn+14 (88) y′′′n+15 = y ′′′ n + 25221445 98402304 h fn+ 442589775 229605376 h fn+1− 388226775 229605376 h fn+2+ 1746295975 229605376 h fn+3 − 372104925 32800768 h fn+4+ 4103141115 229605376 h fn+5− 10001664025 688816128 h fn+6+ 1698012675 229605376 h fn+7 + 1698012675 229605376 h fn+8− 10001664025 688816128 h fn+9+ 4103141115 229605376 h fn+10− 372104925 32800768 h fn+11 + 1746295975 229605376 h fn+12− 388226775 229605376 h fn+13+ 442589775 229605376 h fn+14+ 25221445 98402304 h fn+15 (89) 3. analysis of the basic properties theorem 3.1 (source: lambert [21]). no zero-stable linear multistep method of step number k can have order exceeding k+1 when k is odd, or k+2 when k is even. remark: for a proof of this theorem, the reader is referred to [18]. 3.1. order of the block method we define the linear operator associated with the new method, (ns4o15m) as: l(y(x); h) = a0ym−a1ym−1−ha2y ′ m−1−h 2 b0y ′′ m−1−h 3 b1y ′′′ m−1−h 4(c0 fm+c1 fm−1) (90) where, ym =  yn+1 yn+2 . . . yn+k  , ym−1 =  yn−k+1 yn−k+2 . . . yn  , y′m−1 =  y′n−k+1 y′n−k+2 . . . y′n  , y′′m−1 =  y′′n−k+1 y′′n−k+2 . . . y′′n  , y′′′m−1 =  y′′′n−k+1 y′′′n−k+2 . . . y′′′n  , fm =  fn+1 fn+2 . . . fn+k  , fm−1 =  fn−k+1 fn−k+2 . . . fn  taylor series expansion of equation (90) gives: l(y(x); h) = e0y(x) + he1y ′(x) + h22 e2y ′′(x) + . . . + h( p)p e py ( p)(x) + h( p+1)p+1 e p+1y ( p+1)(x) + . . . (91) the new method, (ns4o15m) and its linear operator equation (90) are said to be of order p if and only if e0 = e1 = e2 = . . . = e p = e p+1 = e p+2 = 0, e p+3 = 0, e p+4 , 0 in equation (91). thus, the new method, (ns4o15m) is of a uniform order 16 with the error constants: 328 atabo and adee / j. nig. soc. phys. sci. 3 (2021) 308–333 329 e p+4 =− 97656794693207524009663299584000 ,− 43313735102631663116607152000 ,− 109088471551231038275510272000 ,− 27085200034999469114875 ,− 4328588356506257785286426165248 ,− 25150957551253485232000 , 320104985832634119860424556544000 ,− 2890186588161181190772125 ,− 3664651642057711038275510272000 ,− 650885653187513304932857216 , − 31281013654043197 47637242118144000 ,− 242877078 282907625 ,− 575350492228736471 524009663299584000 ,− 514075874669 372979728000 ,− 14142409239375 8306204082176  t hence, our new derived method (ns4o15m) is inline with theorem 3.1. 3.2. consistency a numerical method, according to lambert [20], [21], is said to be consistent if its order, that is, p ≥ 1. therefore, the new method (ns4o15m), whose order is 16, is certainly consistent. 3.3. zero stability the new method (ns4o15m) is said to be zero stable if no roots of the first characteristic polynomial, |rs| > 1,∀s = 1, 2, . . . , n and that one of its roots is simple. that is, ρ(r) = ∣∣∣∣∣∣(ra0 − a1) ∣∣∣∣∣∣ (92) where, a0 =  1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1  , a1 =  0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1  figure 1. absolute stability region of ns4o15m(16) so that equation (92) is evaluated and solved for r to give: r = 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0. thus, the new method (ns4o15m) is zero stable. 3.4. convergency according to lambert [21], a numerical method is convergent if it is both zero stable and consistent. hence, the new method (ns4o15m) is certainly convergent inline with (3.2) and (3.3) 3.5. absolute stability definition 3.5.1 (source: lambert [20]). the lmm (ns4o15m) is said to be a-stable if the region of absolute stability includes the entire left half of the z-plane (that is z ∈ (−∞, 0)). therefore, the absolute stability region for the new method (ns4o15m) is plotted in the spirit lambert [20], [21] and okuonghae and ikhile [22] using matlab. hence, the stability polynomial is given by: r(t; h) = 141760259867125 1038275510272 t14(h)4 + 1125 2 t14(h)3 + 225 2 t14(h)2 + 15 t14(h) + (t − 1) t14 (93) where, h = λh since a-stability is a severe property that is desired by all numerical methods, by definition (3.5.1) the new method (ns4o15m) is a-stable. therefore, it is certain that dahlquist barrier no longer posses restrictions on the use of step sizes in the method. also, it is clear from figure 1 that the new method has small unstable region and a wider region of absolute stability which proved its convergence with approximate solutions having little or no deviation from the true solution as a result of small change in input data lambert [20]. hence, the new method is numerically stable and it is investigated in section four below. 329 atabo and adee / j. nig. soc. phys. sci. 3 (2021) 308–333 330 4. numerical examples the new method (ns4o15m) employed taylor series approach to incorporate all the initial values for the solution of (1). the following numerical examples are used to test the accuracy of the new method and comparison are made with some selected numerical methods in recent literature. all numerical iterations are carried out on matlab software environment. problem 4.1 (source: jena et al. [6]). y(iv) = ez ( z4 + 14 z3 + 49 z2 + 32 z − 12 ) , h = 0.1, y (0) = 0, y′ (0) = 0, y′′ (0) = 2, y′′′ (0) = −6, exact solution: y (z) = z2 (1 − z)2 ez, the numerical results for problem 4.1 is as shown in table 1 problem 4.2 (source: ukpebor et al. [13]). consider the special fourth order below u(iv) = x, u (0) = 0, u′ (0) = 1, u′′ (0) = 0, u′′′ (0) = 0, h = 0.1 exact solution: u (0) = x 5 120 + x the numerical results for problem 4.2 is as shown in table 2 problem 4.3 (source: ukpebor et al. [13]). consider the linear differential equation of fourth order: u(iv) + u′′ = 0, u (0) = 0, u′ (0) = −1.172−50π , u ′′ (0) = 1144−100π , u ′′′ (0) = 1.2144−100 π , h = 0.01, exact solution: u (x) = 1−x−cos(x)−1.2 sin(x)144−100 π the numerical results for problem 4.3 is as shown in table 3 problem 4.4. to confirm the application of the new method ns4o15m(16), we solve a physical problem from ship dynamics. as stated by familua and omole [17], when a sinusoidal wave of frequency ω passes long a ship or offshore structure, the resultant fluid actions vary with time t. therefore, consider the fourth-order problem as: y(iv) + 3y′′ + y(2 + ε cos(ωt)) = 0, t > 0 is subjected to the following initial conditions: y(0) = 1, y′(0) = 0, y′′(0) = 0, y′′′(0) = 0, h = 1320 where ε = 0 , for the existence of the theoretical solution: y(t) = 2 cos(t) − cos(t √ (2)). problem 4.5 (source: allogmany et al., [4]). we shall consider a nonlinear initial value problem of the form: yiv = (y′)2 − yy′′′ − 4x2 + ex(1 − 4x + x2), 0 ≤ x ≤ 1 y(0) = 1, y′(0) = 1, y′′(0) = 3, y′′′(0) = 1 exact solution: y(x) = x2 + ex we also compare the results for this problem that has been solved by [19]. the results are shown in table 5. to further illustrate the table 1. comparison of absolute maximum error for problem 4.1 when h=0.1 x error in error in error in ode45 jena et al. [6] ns4o15m 0.1 1.0326 e-07 1.5370 e-14 1.7347 e-18 0.2 1.6792 e-07 8.2021 e-14 1.6653 e-16 0.3 2.5340 e-07 3.6666 e-13 4.3715 e-16 0.4 3.6477 e-07 6.3424 e-13 8.3267 e-16 0.5 5.0816 e-07 6.7024 e-13 5.2736 e-16 0.6 6.9093 e-07 5.2608 e-13 3.7748 e-15 0.7 9.2191 e-07 3.3906 e-13 4.6629 e-15 0.8 1.2116 e-06 1.9011 e-13 5.1209 e-15 0.9 1.5727 e-06 9.6152 e-14 1.3854 e-14 1.0 4.1649 e-06 4.4983 e-14 1.9791 e-14 figure 2. efficiency curves for test problem 4.1 accuracy of (ns4o15m) with comparison to other recent existing methods such as jena et al. [6], ukpebor et al. [13] and familua et al. [17], we have also presented the efficiency curves for problem 4.1, 4.3 and 4.4 and are given below: 5. results and discussions this research paper has considered five special fourth order odes problems in recent literature. problems 4.1 and 4.2 are solved using a step size, h=0.1, problem 4.3 has been solved using a step-size, h=0.01 while problem 4.4 and 4.5 have been solved using a step330 atabo and adee / j. nig. soc. phys. sci. 3 (2021) 308–333 331 table 2. comparison of absolute maximum error for problem 4.2 when h=0.1 x error in error in error in omar et al. [8] ukpebor et al. [13] ns4o15m 0.1 1.002087 e-12 0.00 e-00 0.00 e+00 0.2 0.000000 e+00 0.00 e-00 0.00 e+00 0.3 0.000000 e+00 0.00 e-00 0.00 e+00 0.4 0.000000 e+00 0.00 e-00 0.00 e+00 0.5 1.002087 e-12 0.00 e-00 0.00 e+00 0.6 2.755907 e-12 0.00 e-00 0.00 e+00 0.7 3.507306 e-12 0.00 e-00 0.00 e+00 0.8 3.507306 e-12 2.00 e-18 0.00 e+00 0.9 4.175569 e-12 2.00 e-18 1.11 e-16 1.0 4.175569 e-12 1.00 e-17 0.00 e+00 table 3. comparison of absolute maximum error for problem 4.3 when h=0.01 x error in error in error in adesanya et al. [10] ukpebor et al. [13] ns4o15m 0.01 8.5052 e-19 8.0000 e-20 5.4210 e-20 0.02 1.3010 e-18 9.1500 e-19 5.4210 e-20 0.03 4.7704 e-18 3.4150 e-18 2.7105 e-19 0.04 1.7347 e-17 8.2220 e-18 1.0842 e-19 0.05 4.3368 e-17 1.5965 e-17 3.2526 e-19 0.06 9.5409 e-17 2.7404 e-17 3.2526 e-19 0.07 1.8127 e-16 3.4329 e-17 3.2526 e-19 0.08 3.1571 e-16 3.2551 e-17 4.3368 e-19 0.09 5.1868 e-16 6.5927 e-17 2.1684 e-19 0.10 8.0491 e-16 1.1919 e-16 6.5052 e-19 size of h = 1320 jena et al. [6] has solved problem 4.1 using 9-step block method with uniform order 12 in comparison with ode45 and awoyemi et al. [11]. similarly, ukpebor et al. [13] derived a 7-step block formula for solving fourth order odes with a uniform order 4 and results compared with duromola [9]. however, results for our test problems are shown in tables 1, 2 and 3 respectively. also, familua et al. [17] formulated a five point mono hybrid block method for the solution of nth order ordinary differential equations and results compared in table 4 with direct block and predictor-corrector block methods and also compared with our new method. results from tables 4, 5 and figure 4, showed that the new method is considerably efficient at solving application problems as ship dynamics and nonlinear ivps. it is certain from the tables that the new derived method (ns4o15m) gave better accuracy than jena et al. [6], ode45 omar et al. [8], adesanya et al. [10], awoyemi et al. [11], ukpebor et al. [13] and familua et al. [17] respectively. generally, the new method, (ns4o15m) showed improved approximations as h → 0 on some of the test examples considered. further analysis using the efficiency table 4. comparison of absolute maximum error for problem 4.4 when h = 1 320 (application problem arising from ship dynamics) x error in block error in pcerror in method method, p=7, in [17] method, p=7, [17] ns4o15m(16) 0.003125 6.685763 e-13 5.685763 e-10 0.000000e+00 0.006250 1.458489 e-11 1.767654 e-10 1.110223e-16 0.009375 1.082968 e-10 5.909878 e-09 0.000000e+00 0.001250 3.917803 e-10 5.767654 e-09 2.220446e-16 0.015625 1.025145 e-09 1.100202 e-08 7.771561e-16 0.018750 2.217319 e-09 6.898767 e-08 2.886580e-15 0.021875 4.226068 e-09 4.636354 e-08 8.548717e-15 0.025000 7.358019 e-09 5.787654 e-07 2.187139e-14 0.028125 1.196868 e-08 2.245763 e-07 4.951595e-14 0.031250 1.846249 e-08 2.846249 e-07 1.035838e-13 table 5. comparison of absolute maximum error for problem 4.5 when h = 1 320 x ihb8 [19] i2pbdo6 [4] ns4o15m(16) 0.003125 4.261834 e-14 0.0000000 e+00 2.220446 e-16 0.006250 7.314820 e-13 3.9099252 e-16 0.000000 e+00 0.009375 1.625321 e-12 7.2441945 e-16 2.220446 e-16 0.001250 6.568972 e-12 9.9936929 e-16 0.000000 e+00 0.015625 2.849147 e-12 1.2947886 e-15 0.000000 e+00 0.018750 4.231457 e-12 0.0000000 e+00 2.220446 e-16 0.021875 1.246852 e-11 1.5163355 e-15 6.661338 e-16 0.025000 5.146142 e-11 2.8632323 e-15 1.332268 e-15 0.028125 3.765423 e-11 4.5622733 e-15 2.886580 e-15 figure 3. efficiency curves for test problem 4.3 curves in figures 2, 3 and 4 showed that the new method has small scale errors unlike other existing methods compared. 331 atabo and adee / j. nig. soc. phys. sci. 3 (2021) 308–333 332 figure 4. efficiency curves for test problem 4.4 6. conclusion in this research paper, a new block method that approximates 15-step simultaneously for the solution of (1) has been formulated. the new method is derived through interpolation and collocation techniques using power series as a basis function. the numerical properties of the method have been investigated and proven to be zero stable, consistent and a-stable. test results showed that the new method is capable of solving general fourth order odes including physical problems from ship dynamics and nonlinear ivps with better accuracy than other existing methods considered in this research paper. therefore, the new method 15-step with uniform higher order 16, proved better accuracy over 7-step with uniform order 4 and 9-step with uniform higher order 12, among others. hence, the new method (ns4o15m) should be considered as a viable alternative for solving general fourth order odes. acknowledgements the authors wish to gracefully thank all editors and peer reviewers for their constructive comments at making this research paper a success. more grease to your elbows! references [1] r. conti, d. graffi & g. sansone, theory of nonlinear ordinary differential equations, proc. internat. sympos. non-linear vibrations (1961). 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(1973) [22] r. i. okhuonghae & m. n. o. ikhile, “a family of highly stable second derivative block methods for stiff ivps in odes”, numerical analysis and applications. pleiads publishing ltd. 7 (2012) 57. https://doi.org/10.1134/s1995423914010066. 333 j. nig. soc. phys. sci. 3 (2021) 159–164 journal of the nigerian society of physical sciences a new multi-step method for solving delay differential equations using lagrange interpolation v. j. shaalinia, s. e. fadugbab,∗ adepartment of mathematics, bishop heber college, trichy, india bdepartment of mathematics, ekiti state university, ado ekiti, nigeria abstract this paper presents 2-step p-th order ( p = 2, 3, 4) multi-step methods that are based on the combination of both polynomial and exponential functions for the solution of delay differential equations (ddes). furthermore, the delay argument is approximated using the lagrange interpolation. the local truncation errors and stability polynomials for each order are derived. the local grid search algorithm (lgsa) is used to determine the stability regions of the method. moreover, applicability and suitability of the method have been demonstrated by some numerical examples of ddes with constant delay, time dependent and state dependent delays. the numerical results are compared with the theoretical solution as well as the existing rational multi-step method2 (rmm2). doi:10.46481/jnsps.2021.247 keywords: multi-step method, delay differential equations, interpolating function, lagrange interpolation, stability polynomial, stability region article history : received: 08 june 2021 received in revised form: 12 july 2021 accepted for publication: 24 july 2021 published: 29 august 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: t. latunde 1. introduction delay differential equations (ddes) are differential equations in which the derivative of the unknown function depends not only at its present time but also at the previous times. in ordinary differential equations (odes), a simple initial condition is given. but to specify ddes, additional information is needed. because the derivative depends on the solution at the previous times, an initial history function which gives information about the solution in the past needs to be specified. a general form of the first order dde is y′ (t) = f (t, y (t) , y (t −τ)) , t > t0 ∗corresponding author tel. no: +2348067032044 email address: sunday.fadugba@eksu.edu.ng (s. e. fadugba ) y (t) = ϑ (t) , t ≤ t0 (1) where ϑ(t) is the initial function and τ is the delay term. the function ϑ(t) is also known as the ‘history function’, as it gives information about the solution in the past. if the delay term τ is a constant, then it is called constant delay. if it is function of time t, then it is called time dependent delay. if it is a function of time t and y(t), then it is called state dependent delay. these equations arise in population dynamics, control systems, chemical kinetic, and in several areas of science and engineering [1, 2, 3]. recently there has been a growing interest in obtaining the numerical solutions of ddes. rostann et al. [4] implemented adomian decomposition method for the solution of system of ddes. two and three point one-step block method for solving ddes was developed by [5]. block method for solving pantograph type functional ddes was described by 159 shaalini & fadugba / j. nig. soc. phys. sci. 3 (2021) 159–164 160 [6]. an exact/approximate solution of ddes by using the combination of laplace and the variational iteration method were obtained by [7]. rk method based on harmonic mean for solving ddes with constant lags was proposed by [8]. several numerical methods have been constructed for solving stiff ddes, see [9, 10, 11]. several multi-step techniques using varieties of interpolating polynomials and functions have been developed to solve odes such as [12, 13, 14, 15, 16, 17], just to mention a few. in this paper, we present the 2-step p-th order ( p = 2, 3, 4) multi-step method for solving ddes. this method has been referred here as epmm (2, p), (p = 2, 3, 4). the organization of this paper is as follows: in section two, the derivation of epmm (2, p), (p = 2, 3, 4) is given. in section three, the stability analysis of epmm (2, p), (p = 2, 3, 4) has been presented. in section four, numerical illustrations of ddes are provided. moreover, the numerical results are compared with the existing rational multi-step method2 (rmm2) to demonstrate the efficiency and suitability of the method. 2. derivation of epmm (2, p) for 2-step p-th order epmm, let us assume an approximation to the analytical solution y (tn+2) of (1) given by yn+2 = a0e 2h + 1 + p∑ j=1 b jh j, (2) where a0, b j, ( j=1,2,. . . p) are parameters that may contain the approximation of y (tn) and higher derivatives of y (tn). with epmm in (2) we associate the difference operator l defined by l[y (t) ; h]epm m = y (t + 2h) − (1 + p∑ j=0 b jh j)  − a0e2h(3) where y(t) is an arbitrary, continuous and differentiable function. expanding y(t + 2h) as taylor series and collecting terms in (3), l[y (t) ; h]epm m = c0h 0 +c1h 1 +· · ·+c ph p +c p+1h p+1 +. . .(4) where ci, i = 0, 1, . . . , p, p + 1 are the coefficients that need to be determined. for 2-step second order epmm, we take p = 2 and expand y(t + 2h) via taylor series, (3) becomes l [ y (t) ; h ] epm m(2,2) = −1 + y (t) + h ( −b1 + 2y ′ (t) ) +h2 ( 2y ′′ (t) − b2 ) + h3 ( 4 3 y ′′′ (t) ) + a0e 2h + o(h4) (5) using e2h ≈ 1 + 2h + 2h2 in (2), we get l [ y (t) ; h ] epm m(2,2) = −1 − a0 + y (t) + h ( −b1 − 2a0 + 2y ′ (t) ) +h2 ( 2y ′′ (t) − 2a0 − b2 ) + h3 ( 4 3 y ′′′ (t) ) + a0e 2h + o(h4)(6) comparing (5) and (6), we have c0 = −(1 + a0) + y (t) , c1 = −b1 − 2a0 + 2y ′ (t) , c2 = 2y ′′ (t) − 2a0 − b2, c3 = 4 3 y ′′′ (t) (7) for second order epmm, we put c0 = c1 = c2 = 0 in (7) and get the following solutions: a0 = y (t)−1, b1 = 2(y ′ (t)−y(t)+1), b2 = 2(y ′′ (t)−y (t)+1)(8) if we write yn = y(tn) and y (m) n = y (m)(tn) for m = 1, 2,. . . , then (8) becomes a = yn, b1 = 2 ( yn ′ − yn + 1 ) , b2 = 2(yn ′′ − yn + 1) (9) taking p = 2 and e2h ≈ 1 + 2h + 2h2 in (2), we get yn+2 = (a0 + 1) + h (2a0 + b1) + h 2(2a0 + b2) (10) substituting (9) into (10), we have yn+2 = yn + 2hyn ′ + 2h2yn ′′ (11) the local truncation error of epmm (2, 2) is given by, lt e epm m(2,2) = h 3 ( 4 3 yn ′′′ ) + o(h4) taking p = 3 in (2) and on simplification, we get the formula for epmm (2, 3) yn+2 = yn + 2hyn ′ + 2h2yn ′′ + 4 3 h3yn ′′′ (12) the local truncation error of epmm (2, 3) is given by, lt e epm m(2,3) = h 4 ( 2 3 yn (4) ) + o(h5) taking p = 4 in (2) and on simplification, we get the formula for epmm (2, 4) yn+2 = y (t)+2hy ′ (t)+2h2y ′′ (t)+ 4 3 h3y ′′′ (t)+ 2 3 h4y(4) (t)(13) the local truncation error of epmm (2, 4) is given by, lt e epm m(2,4) = h 5 ( 4 15 yn (5) ) + o(h6) 3. stability analysis of epmm in this section, we derive the stability polynomials of epmm (2, p), (p = 2, 3, 4) and their corresponding stability regions were obtained. we consider a commonly used linear test equation with a constant delay τ = mh where m is a positive integer, y ′ (t) = λy (t) + µy(t −τ), t > t0 160 shaalini & fadugba / j. nig. soc. phys. sci. 3 (2021) 159–164 161 y (t) = φ(t), t ≤ t0 (14) where λ,µ ∈ c, τ > 0 andφ is continuous. using (11) in (14), we get yn+2 = yn+2h (λyn + µy (tn −τ))+2h 2 (λy′n + µy′ (tn −τ))(15) y (tn − mh) = y (tn−m) = s1∑ l=−r1 ll (ci)yn−m+l with ll (ci) = s1∏ j=−r1 ci − j1 l − j1 , j1 , l and r1, s1 > 0 taking y (tn −τ) = s1∑ l=−r1 ll(c)yn−m+l and y ′ (tn −τ) = λ s1∑ l=−r1 ll (c) yn−m+l + µ s1∑ l=−r1 ll (c) yn−2m+l (16) then (15) becomes yn+2 = yn + 2h λyn + µ s1∑ l=−r1 ll (c) yn−m+l  +2h2  λ ( λyn + µ ∑s1 l=−r1 ll (c) yn−m+l ) +µλ ∑s1 l=−r1 ll (c) yn−m+l +µ ∑s1 l=−r1 ll (c) yn−2m+l  yn+2 = yn + 2λhyn + 2λ 2h 2 yn + s1∑ l=−r1 ll (c) yn−m+l ( 2µh + 4h2µλ ) + s1∑ l=−r1 ll (c) yn−2m+l ( 2h2µ2 ) yn+2 = yn ( 1 + 2λh + 2(λh)2 ) + s1∑ l=−r1 ll (c) yn−m+l (µh (2 + 4λh)) + s1∑ l=−r1 ll (c) yn−2m+l ( 2(µh)2 ) let α = λh and β = µh then the above equation becomes yn+2 = yn ( 1 + 2α + 2α2 ) + (β (2 + 4α)) s1∑ l=−r1 ll (c) yn−m+l +2β2 s1∑ l=−r1 ll (c) yn−2m+l to obtain the stability polynomial, the delay term is approximated using three points lagrange interpolation. by putting n − m + l = 0 and n − 2m + l = 0 and by taking l = -1, 0, 1, the stability polynomial will be in the standard form. the recurrence is stable if the zeros of ζi of the stability polynomial s (α,β : ζ) = ζn+2 − ( 1 + α + 2α2 ) ζn −β (2 + 4α) ( l−1 (c) + l0 (c) ζ + l1 (c) ζ 2 ) −2β2 ( l−1 (c) + l0 (c) ζ + l1 (c) ζ 2 ) satisfies the root condition|ζi| ≤ 1. from this, the stability polynomial for the method grmm (2, 2) with τ = 1 is given as s (α,β : ζ) = ζn+2 − ( 1 + α + 2α2 ) ζn − ( 2β + 2β2 + 4αβ ) similarly, by considering suitable number of points in lagrange interpolation according to the order of the method, we can obtain the corresponding stability polynomials of epmm (2, p). when p = 3, the stability polynomial for epmm (2, 3) is given as s (α,β : ζ) = ζn+2 − ( 1 + α + 2α2 + 4 3 α3 ) ζn − ( 2β + 2β2 + 4 3 β3 + 4αβ + 4α2β + 4αβ2 ) when p = 4, the stability polynomial for epmm (2, 4) is given as s (α,β : ζ) = ζn+2 − ( 1 + α + 2α2 + 4 3 α3 + 2 3 α4 ) ζn − ( 2β + 2β2 + 4 3 β3 + 2 3 β4 + 2αβ + 4α2β + 4αβ2 + 8 3 αβ3 + 8 3 α3β + 4α2β2 ) the stability regions of epmm (2, 2), epmm (2, 3) and epmm (2, 4) are given in figures 1 -3. in a similar manner, we can obtain the stability polynomials and their corresponding regions of epmm with r-step and of any order p. 4. numerical examples example 1: (stiff linear system with multiple delays) y1 ′ (t) = − 1 2 y1 (t) − 1 2 y2 (t − 1) + f1 (t) , y2 ′ (t) = −y2 (t) − 1 2 y1 ( t − 1 2 ) + f2(t), 0 ≤ t ≤ 1 with initial conditions y1 (t) = e −t/2, −1 2 ≤ t ≤ 0, y2 (t) = e −t, −1 ≤ t ≤ 0 161 shaalini & fadugba / j. nig. soc. phys. sci. 3 (2021) 159–164 162 table 1. comparison of absolute error results in epmm and rmm2 for example 1 time (t) y epmm (2, 2) rmm2 (2, 2) epmm (2, 3) rmm2 (2,3) epmm (2, 4) rmm2(2, 4) 0.2 y1 1.52e-06 7.54e-07 3.80e-09 1.26e-09 7.60e-12 9.06e-07 y2 3.26e-04 3.31e-04 3.26e-04 3.26e-04 3.15e-04 3.32e-04 0.4 y1 2.75e-06 1.36e-06 6.88e-09 2.28e-09 1.38e-11 1.64e-06 y2 5.62e-04 5.71e-04 5.62e-04 5.62e-04 5.43e-04 5.72e-04 0.6 y1 3.73e-06 1.85e-06 9.33e-09 3.10e-09 1.87e-11 2.22e-06 y2 7.27e-04 7.38e-04 7.27e-04 7.27e-04 7.04e-04 7.40e-04 0.8 y1 4.50e-06 2.23e-06 1.13e-08 3.73e-09 2.25e-11 2.68e-06 y2 8.36e-04 8.48e-04 8.36e-04 8.36e-04 8.12e-04 8.51e-04 1.0 y1 5.09e-06 2.53e-06 1.27e-08 4.22e-09 2.55e-11 3.04e-06 y2 9.03e-04 9.16e-04 9.03e-04 9.03e-04 8.78e-04 9.18e-04 table 2. comparison of absolute error results in epmm and rmm2 for example 2 time (t) epmm (2, 2) rmm2 (2, 2) epmm (2, 3) rmm2 (2, 3) epmm (2, 4) rmm2 (2, 4) 1.1 1.82e-06 4.04e-06 1.31e-06 1.46e-06 1.35e-06 9.58e-06 1.2 9.26e-07 2.51e-06 1.25e-06 1.20e-06 2.75e-06 8.03e-06 1.3 1.50e-06 3.23e-08 2.22e-06 4.13e-06 2.26e-07 8.41e-06 1.4 3.50e-06 5.92e-06 2.70e-06 3.29e-06 2.99e-06 1.19e-05 1.5 4.95e-05 1.96e-06 3.13e-06 6.16e-06 4.39e-06 6.83e-06 table 3. comparison of absolute error results in epmm and rmm2 for example 3 time(t) epmm (2, 2) rmm2 (2, 2) epmm (2, 3) rmm2 (2, 3) epmm (2, 4) rmm2 (2, 4) 0.2 1.33e-05 1.35e-05 1.97e-09 2.36e-07 4.92e-09 4.87e-07 0.4 2.60e-05 2.80e-05 2.46e-08 5.62e-07 1.76e-09 5.46e-06 0.6 3.78e-05 4.48e-05 5.67e-08 7.45e-07 3.03e-09 3.58e-05 0.8 4.79e-05 6.56e-05 9.93e-08 8.72e-07 2.45e-09 2.32e-04 1.0 5.63e-05 9.31e-05 2.85e-06 2.85e-06 3.24e-09 3.24e-04 figure 1. stability region of 2-step second order epmm where f1 (t) = 1 2 e−(t−1) and f2 (t) = 1 2 e−(t−1/2)/2 the exact solution is given by y1 (t) = e −t/2, y2 (t) = e −t figure 2. stability region of 2-step third order epmm example 2: (time-dependent delay) y ′ (t) = t − 1 t y(ln (t) − 1)y(t), 1 ≤ t ≤ 3 2 with initial condition y (t) = 1, 0 ≤ t ≤ 1 162 shaalini & fadugba / j. nig. soc. phys. sci. 3 (2021) 159–164 163 figure 3. stability region of 2-step fourth order epmm figure 4. comparison of error graph of y1 and y2 in example 1 figure 5. comparison of absolute error graph of y in example 2 and the exact solution is given by y (t) = exp(t − ln (t) − 1), 1 ≤ t ≤ 3 2 example 3: (state-dependent delay) y ′ (t) = cos(t)y ( y (t)−2 ) , t ≥ 0 figure 6. comparison of absolute error graph of y in example 3 with initial condition, y (t) = 1, t ≤ 0 and the exact solution is given by y (t) = sin (t) + 1, 0 ≤ t ≤ 1 by taking the step-size h = 0.01in the above examples, the absolute errors by using epmm and rmm2 are given in tables 1 – 3 and their corresponding error graphs are shown in figures 4 – 6. 5. conclusion in this paper, the new multi-step method of r-step and p-th order that are based on interpolating functions which consists of both polynomial and exponential function is presented for solving ddes. the local truncation errors have been determined. the stability polynomials of epmm (2, p) where p = 2, 3, 4 are derived and their corresponding stability regions are obtained and shown in figures 1–3. the delay argument is approximated using lagrange interpolation. numerical examples of ddes with constant delay, time dependent delay and state dependent delays have been considered to demonstrate the efficiency of the proposed method. the comparative absolute error analyses of epmm (2,p) in the context of rmm2 (2,p) for examples 1, 2 and 3 were shown in tables 1, 2 and 3, respectively. from the figures 4 – 6, it is evident that the newly proposed method gives results with good accuracy than the existing rmm2. hence, it is concluded that the proposed epmm (2, p) is suitable for solving ddes. acknowledgments we thank the referees and the editor for their contributions on the improvement of this paper. 163 shaalini & fadugba / j. nig. soc. phys. sci. 3 (2021) 159–164 164 references [1] y. kuang, delay differential equations with applications in population dynamics, academic press, boston, san diego, new york. 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[17] s. e. fadugba, s.n. ogunyebi & b.o. falodun, “an examination of a second order numerical method for solving initial value problems”, journal of the nigerian society of physical sciences 2 (2020) 120. 164 j. nig. soc. phys. sci. 4 (2022) 88–98 journal of the nigerian society of physical sciences optimal control of the coronavirus pandemic with impacts of implemented control measures t. t. yusuf, a. abidemi∗, a. s. afolabi, e. j. dansu department of mathematical sciences, federal university of technology, akure, nigeria abstract this paper considers the current global issue of containing the coronavirus pandemic as an optimal control problem. the goal is to determine the most advantageous levels of effectiveness of the various control and preventive measures that should be attained in order to effectively reduce the burden of the disease within a relatively short time. thus, the problem’s objective functional is constructed such that it minimizes the prevalence as well as the cost of implementing four control measures (namely personal protection, contact tracing and testing control for asymptotically infected individuals, detection control for symptomatic infected individuals and management control for hospitalized individuals) subject to a model for the disease transmission dynamics which incorporates four time-dependent control variables. the optimality system of the model is derived based on pontryagin’s maximum principle. thereafter, the resulting optimality system is solved numerically using the runge-kutta fourth order scheme with forward-backward sweep approach to evaluate the effect of combining at least any three of the control functions with personal protection always in use on the disease dynamics. findings from our results show that the new cases and the prevalence of the disease can be remarkably reduced in a cost effective way by implementing any of the control strategies analysed in this work. however, the results of efficiency analysis suggest that a strategy which combines all the four preventive and control measures is most effective in reducing the disease burden in the population. doi:10.46481/jnsps.2022.414 keywords: coronavirus, epidemiological model, objective functional, optimality system, pontryagin’s maximum principle, disease prevalence article history : received: 18 september 2021 received in revised form: 13 december 2021 accepted for publication: 06 february 2022 published: 28 february 2022 c©2022 journal of the nigerian society of physical sciences. all rights reserved. communicated by: t. latunde 1. introduction the coronavirus disease (covid-19), which originated from china towards the end of the year 2019, has the severe acute respiratory syndrome coronavirus-2 (sars-cov-2) as its causative agent. just like similar infections caused by coronaviruses, namely the severe acute respiratory syndrome (sars) and the middle east respiratory syndrome (mers), the tendency of rapid ∗corresponding author tel. no: +2348036412201 email address: aabidemi@futa.edu.ng (a. abidemi ) spread of covid-19 calls for concern [1]. coronaviruses are said to have the most common genes of the ribonucleic acid viruses, though with possibility of genetic mutations. covid-19 is transmitted from humans to humans by inhaling the respiratory droplets from humans infected by covid19 during coughing and sneezing and through direct contact with contaminated surfaces [2, 3]. the incubation period for covid-19 ranges between 0 and 14 days, although a longer incubation period of 25 days has also been reported in literature. the mortality rate of the disease is lower than the recovery rate, but this ratio varies from one country to another as well as re88 yusuf et al. / j. nig. soc. phys. sci. 4 (2022) 88–98 89 gion to region. the primary symptoms of covid-19 include body aches, headache, consistent dry coughing, severe chest pain, high fever, and complication in the respiratory system [3]. presently, there are no certified potent drugs for the treatment of the disease. though, there are ongoing efforts in this direction. recently, wu et al. undertook a review of therapies to prevent the contagion, control the spread and support patients infected with the disease through the use of antiviral, and supporting agents [4]. also, galluccio et al. proposed a multifaceted management approach in handling the complexity presented by the global covid-19 pandemic [5]. they encourage knowledge sharing, proper timing and setup of treatments as possible ways out of the situation. in a related study, becker investigated the usefulness of two cheap oral drugs -chloroquine and hydroxychloroquinein the treatment of the disease based on the fact that these drugs had hitherto been helpful in managing autoimmune illnesses [6]. in another study conducted by galvin et al., they found control and preventive interventions like the use of masks, sanitizers, social distancing and others to be quite helpful in curbing the spread of covid-19 in taiwan [7]. moreover, the foregoing is equally true in other parts of the world since these measures limit contacts and the possibility of transmission among people as demonstrated in the studies by shah et al. and tsay et al. [8, 9]. similarly, ngonghala et al. developed a model to evaluate the impact of non-pharmaceutical approaches on covid-19, namely social distancing, contact tracing, quarantine, isolation and the use of face mask [10]. these non pharmaceutical measures were found to be reliably effective based on the analysis of the model simulated using data from the united states of america. in addition, another closely related study with similar findings was conducted by okuonghae and omame, though their focus was on lagos, which is the epicenter of covid-19 in nigeria [2]. globally, there were 263,563,622 confirmed cases of the coronavirus disease and 5,232,562 deaths due to the disease as at 3rd december 2021 [11]. moreover, on the nigerian scene, the first reported case of covid-19 was on the 27th february 2020 due to an italian visitor [12, 13, 14]. however, as of 5th december 2021, 3,580,510 covid-19 samples had been tested in nigeria, there had been 214,567 confirmed cases, 207,427 discharged cases, 2,980 cases of death; and 4,160 active cases [14]. according to the nigeria centre for disease control (ncdc), from 27th february, 2020 up to 31st july, 2020, all the thirty-six states of the federal republic of nigeria and the federal capital territory (fct) had reported cases of covid-19 [13]. thus, it was instructive for ohia [15] to express the need for urgent attention as it concerns contextualizing the peculiarity of africa at large and nigeria in particular as it relates to the management of covid-19. this was necessary considering the deplorable state of the nation’s health system and the dearth of infrastructure for emergency care for victims of the pandemic. however, having put up a successful fight against polio and ebola in the past, ebenso and otu were of the opinion that nigeria had some leverage in confronting this covid-19 outbreak despite the below average capacity of the health system [16]. nevertheless, osseni opines that in spite of the fragile health care system in sub-saharan africa, covid-19 presents a huge opportunity for the region to demonstrate that they are equal to the task of combating the pandemic [17]. it is worthy of note that nigeria is a major player in the region with a large human population, thus all eyes are on the country to see how it responds to this disease of global concern. in recent years, optimal control theory has been applied to help inform policies and strategies which could be used in the control of infectious diseases. it is often used to determine the optimal levels of effectiveness of the different control measures that should be deplored per unit time in order to promptly contain an epidemic. for instance, ijalana and yusuf used the optimal control theory approach on a model describing the transmission dynamics of hepatitis b virus in highly endemic areas and recommended how the various control measures should be combined in order to effectively contain the disease in the area within a decade [18]. similarly, yusuf and olayinka used the same approach to propose strategies for containing the meningitis disease recurrence in the meningitis belt of africa [19]. in the same vein, obsu and balcha applied the theory on a model for covid-19 spread and they found that the prevention of infection, intensive care for the hospitalized and sanitizing of infected sites are the most effective strategies against the disease’s spread [20]. equally, yousefpour et al. worked on a model for covid-19 with a multi-objective genetic algorithm for rates of contact and transition to quarantine. they came up with optimal policies for curtailing the disease while they also evaluated the economic impact of the disease [21]. abioye et al. [22] emphasised the possibility of the eradication of covid-19 in nigeria if three control strategies, namely enforcement of preventive measures; prompt testing and treatment; and prevention of repeat infections, are duly sustained. using the prophet package, abioye et al. [23] was able to predict a reliable trend in the evolution of covid-19 infections in nigeria for the period considered by the study. diagne et al. [24] formulated and analyzed a covid-19 transmission model which highlights the significance of effective contact rate, vaccine efficacy, vaccination rate, and proportion of symptomatic exposed individuals in the spread of the disease. moreover, awareness programs and timely hospitalization were shown as very crucial in the containment of covid-19 in nigeria and other countries of the world by musa et al. [25]. also, peter et al. [26] investigated the behaviour of covid-19 infections based on fractional differentiation using the atangana-baleanu fractional derivative operator to establish existence. the iterative laplace transform method was employed to carry out numerical simulations. the most important factors in the model were identified to be the force of infection and the progression rate of individuals from the exposed class to the infectious class. using real data from pakistan, peter et al. [27] presented a novel model in order to understand the covid-19 outbreak. the work highlighted that public health policies leading to properly managed transmission rates were necessary for effective management of the disease. most of the works on mathematical analysis of covid-19 transmission dynamics do not consider the effect of combining most or all available control measures in an optimal way. that 89 yusuf et al. / j. nig. soc. phys. sci. 4 (2022) 88–98 90 is, controlling the pandemic in a way that the prevalence, incidence and cost of implementing the adopted control strategies are minimised while the efficiency of the proposed strategies is quantitatively evaluated. the foregoing is critical to the implementation of any proposed control strategy, since it guarantees measurable achievement on fund invested. consequently, this paper tries to fill this research gap as much as possible. in this work, we investigate optimal control strategies for containing the covid-19 pandemic with a focus on nigeria. the goal is to determine how to optimally combine the different disease control and preventive measures in order to contain the disease within the shortest possible time in a cost effective way. to this end, we formulate an optimal control problem involving a non-autonomous system of ordinary differential equations incorporating four time-dependent control variables representing preventive (precautionary) measures, contact tracing and testing control for asymptomatically infected individuals, detection control for symptomatic infected individuals, and management control for hospitalized individuals to describe the transmission dynamics of covid-19 in a population. pontryagin’s maximum principle is used to conduct the analysis of the formulated optimal control problem to determine the necessary conditions for the optimal strategies for controlling the disease spread in the community. the paper outline is as follows: section 2 describes the non-autonomous deterministic model for the spread of the disease; section 3 contains the formulation of the optimal control problem, proof of existence of optimal control, its characterization, and the derivation of the optimality system; estimation of model parameters and initial conditions is considered in section 4; section 5 has the numerical simulations and efficiency analysis; conclusion is made in section 6. 2. mathematical model we consider an improved version of the covid-19 transmission dynamics by yusuf et al.[28]. this work is different from the former because it considers a non-autonomous deterministic model for covid-19 transmission dynamics with time-dependent control measures. in addition, the goal of this paper is to solve an optimal control problem subject to the proposed model dynamics. thus, the results will not just be about the state variables, rather it will include the adjoint and the control variables. so, the solution determines optimal level of each of the control variables that should be applied at any point in time in order to minimize the incidence and prevalence of the disease as well as the cost of implementing these control and preventive measures. conventionally, the model subdivides the total population under consideration, denoted as n(t), into five mutually exclusive compartments, namely: susceptible s (t), asymptomatically infected e(t), symptomatically infected i(t), hospitalized h(t), and recovered r(t) compartments. thus, n(t) = s (t) + e(t) + i(t) + h(t) + r(t). the susceptible compartment contains individuals who can be infected with covid-19 disease if sufficiently exposed to the infection without taking adequate protective measures; the asymptomatically infected compartment is made up of individuals who have been infected with the disease, though do not exhibit the symptoms of the infection but can infect others; the infected compartment is made up individuals who have been infected and show symptoms of the infection; the hospitalized compartment comprises of individuals who have been confirmed to be infected with disease based on standard tests and are placed under isolation in designated isolation centres; the recovered compartments contains individuals who have recovered from the infection and now possess temporary immunity against re-infection. furthermore, four time-dependent control functions, ui(t) (i = 1, . . . , 4), are incorporated into the model to mitigate the spread of the disease in the population, where u1(t) represents the fraction of the whole population that adopt human personal protective intervention such as wearing of face mask, regular hand washing cum sanitizing and observing social distancing, u2(t) accounts for the successful effort of the contact tracing and testing for asymptomatically infected individuals, u3(t) denotes the successful detection effort for symptomatic infectious individuals, and lastly, the medical care successful effort (management control) for hospitalized individuals in designated isolation centers is represented by u4(t). therefore, the non-autonomous model system of ordinary differential equations governing the disease spread is as given in (1) as ds dt = λ − (1 − u1(t))(β1 e + β2 h + β3 i)s −µs + ωr, de dt = (1 − u1(t))(β1 e + β2 h + β3 i)s − u2(t)e −ρe −µe, di dt = ρe − u3(t)i −γi − (µ + δ2)i, dh dt = u2(t)e + u3(t)i − u4(t)h − (µ + δ1)h, dr dt = γi + u4(t)h −µr −ωr, (1) with given initial conditions at time t = 0 while the model parameters are as defined in table 1. note that lim t→∞ n(t) ≤ λ µ . however, under the dynamics described by (1), the region ω defined by ω = { (s, e, i, h, r) ∈ r5+|s + e + i + h + r ≤ λ µ } is positively invariant. 3. optimal control problem formulation in this work, the objective functional is designed such that it is quadratic in control terms to follow the standard in literature [18, 19, 29, 30, 31, 32, 33]. thus, our objective functional is defined as z = min u1,u2,u3,u4 ∫ t f 0 (w1 e + w2 i + w3 h + w4 2 u21 + w5 2 u22 + w6 2 u23 + w7 2 u24 ) dt (2) 90 yusuf et al. / j. nig. soc. phys. sci. 4 (2022) 88–98 91 table 1: description of model parameters parameter description λ recruitment rate into the s class µ natural death rate δ1 covid-19 induced death rate for hospitalized individuals δ2 covid-19 induced death rate for symptomatically infected individuals not hospitalized β1 transmission rate of the disease based on unprotected contact with individuals in the e class β2 transmission rate of the disease based on unprotected contact with individuals in the h class β3 transmission rate of the disease based on unprotected contact with individuals in the i class ρ progression rate from the e class into the i class γ recovery rate of individuals in the i class ω temporary immunity waning rate for individuals in the class r subject to the dynamical system (1) with appropriate states initial conditions and t f is the final time while the control set u is lebesgue measurable and it is defined as u = {(u1(t), u2(t), u3(t), u4(t))|0 ≤ ui ≤ ui max < 1, i = 1, . . . , 4, t ∈ [0, t f ]}. (3) in addition, the weight constants wi > 0 (i = 1, 2, . . . , 7) that appear in (2) are the relative weights, and they ensure that each term in the integrand does not dominate while also balancing terms therein. nevertheless, it is important to note that w4, w5, w6, w7 are relative measures of the cost or effort required to implement each of the associated controls while w1, w2, w3 are relative measures of the importance of reducing the associated classes on the spread of covid-19 in the country under consideration. the upper bounds (u1 max, u2 max, u3 max, u4 max) for each of the controls will depend on the budget allocated for the execution of each of the control measures and the practicability of achieving the set bounds. for this study, we shall hypothetically set u1 max = 0.7, u2 max = 0.5, u3 max = 0.5, and u4 max = 0.9 in our simulations with the assumptions that it is reasonably practicable to achieve the set bounds subject to the availability of needed resources. we wish to determine the optimal combination of different levels of effectiveness of the controls u1, u2, u3 and u4 that will be adequate to minimize the cost of implementing the preventive measures, the cost of contact tracing of the exposed individuals, the cost of medical testing of infected individuals, and the cost associated with hospitalizing cum treating hospitalized individuals as well as reducing the prevalence of the disease within a set time frame. 3.1. existence of an optimal control strategy for the problem now, we examine the sufficient conditions for the existence of a solution to the optimal control problem. theorem 3.1. there exists an optimal control set (u∗1, u ∗ 2, u ∗ 3, u ∗ 4) with corresponding solution (s ∗, e∗, i∗, h∗, r∗) to the model system (1) that minimize z over u. proof. the existence of the optimal control is guaranteed by the compactness of the control and the state space and the convexity in the problem based on theorem 3.1 and its corresponding corollary in fleming and rishel[34]. the following non-trivial requirements from fleming and rishel’s theorem are stated and verified below: (1) the set of all solutions to the model equations (1) and its associated initial conditions together with the corresponding control functions in u is non empty. (2) the state system can be written as a linear function of the control variables with coefficients dependent on time and state variables. (3) the integrand l in (2) is convex on u and additionally satisfies l(t, s, e, i, h, r, u1, u2, u3, u4) ≥ c1||(u1, u2, u3, u4)|| α − c2, where c1, c2 > 0 and α > 1. we refer to theorem 3.1 by picard-lindelof in [35]. based on this theorem, if the solutions to the state equations are a priori bounded and if the state equations are continuous and lipschitz in the state variables, then there is a unique solution corresponding to every admissible control set in u. using the fact that for all (s, e, i, h, r) ∈ ω, all the model states are bounded below and above, then the solutions to the state equations are bounded. in addition, it is direct to show the boundedness of the partial derivatives with respect to the state variables in the system and this shows that the system is lipschitz with respect to the state variables. hence, the proof of the first condition is established. equally, direct observation of the state equations (1) shows that they are linearly dependent on the controls u1, u2, u3 and u4. thus, the second condition is also satisfied. in order to establish the third condition, it is observed that the integrand l in our objective functional is convex since it is quadratic in the controls. thus, it suffices to show that there exists a bound on l to satisfy the remaining condition. this is 91 yusuf et al. / j. nig. soc. phys. sci. 4 (2022) 88–98 92 shown as below: l = w1 e + w2 i + w3 h + 1 2 ( w4u 2 1 + w5u 2 2 +w6u 2 3 + w7u 2 4 ) ≥ 1 2 ( w4u 2 1 + w5u 2 2 + w6u 2 3 + w7u 2 4 ) since wi > 0, i = 1, . . . , 7 ≥ 1 2 ( w4u 2 1 + w5u 2 2 + w6u 2 3 + w7u 2 4 ) − w4 since 0 ≤ u1 < 1, w4u 2 1 − w4 ≤ 0 ≥ min ( 1 2 w4, 1 2 w5, 1 2 w6, 1 2 w7 ) ( u21 + u 2 2 + u 2 3 + u 2 4 ) − w4 ≥ k||(u1, u2, u3, u4)|| 2 − w4 where k = min ( 1 2 w1, 1 2 w2, 1 2 w3, 1 2 w4 ) . (4) the foregoing shows that the integrand l is bounded below. hence, there exists a unique solution to the optimality system for small time intervals based on the opposite time orientations of the state and adjoint equations. in addition, the uniqueness of the solution of the optimality system guarantees the uniqueness of the optimal control if it exists. 3.2. characterization of the optimal controls we proceed to characterize the optimal controls u∗1, u ∗ 2, u ∗ 3, u ∗ 4 which gives the optimal levels for the various control measures and the corresponding states (s ∗, e∗, i∗, h∗, r∗). the necessary conditions for the optimal controls are obtained using the optimality conditions of the pontryagin’s maximum principle [36] while the terminal conditions on the adjoint variables are obtained based on its transversality conditions [37]. theorem 3.2. (necessary conditions) let (u∗1, u ∗ 2, u ∗ 3, u ∗ 4) ∈ u be an optimal control with the corresponding states s ∗, e∗, i∗, h∗ and r∗. then, there exist the adjoint variables λi for i = 1, . . . , 5 which satisfy dλ1 dt = λ1µ + (λ1 −λ2)(1 − u1)(β1 e + β2 h + β3 i), dλ2 dt = −w1 + λ2µ + (λ1 −λ2)(1 − u1)β1s + (λ2 −λ3)ρ + (λ2 −λ4)u2, dλ3 dt = −w2 + (λ1 −λ2)(1 − u1)β3s + (λ3 −λ4)u3 + (λ3 −λ5)γ + λ3(µ + δ2), dλ4 dt = −w3 + (λ1 −λ2)(1 − u1)β2s + (λ4 −λ5)u4 + λ4(µ + δ1), dλ5 dt = λ5µ + (λ5 −λ1)ω, (5) and the transversality conditions λi(t f ) = 0, f or i = 1, . . . , 5 (6) with the optimal controls defined as follows u∗1 = min { max ( 0, s (β1 e + β2 h + β3 i)(λ2 −λ1) w4 ) , u1 max} , u∗2 = min { max ( 0, e(λ2 −λ4) w5 ) , u2 max } , u∗3 = min { max ( 0, i(λ3 −λ4) w6 ) , u3 max } , u∗4 = min { max ( 0, h(λ4 −λ5) w7 ) , u4 max } . (7) proof. using pontryagin’s maximum principle, we obtain (5) from dλ1 dt = − ∂h ∂s , dλ2 dt = − ∂h ∂e , dλ3 dt = − ∂h ∂i , dλ4 dt = − ∂h ∂h , dλ5 dt = − ∂h ∂r , (8) where the hamiltonian h is given by h = w1 e + w2 i + w3 h + w4 2 u21 + w5 2 u22 + w6 2 u23 + w7 2 u24 + λ1(λ − (1 − u1)(β1 e + β2 h + β3 i)s −µs + ωr) + λ2((1 − u1)(β1 e + β2 h + β3 i)s − u2 e −ρe −µe) + λ3(ρe − u3 i −γi − (µ + δ2)i) + λ4(u2 e + u3 i − u4 h − (µ + δ1)q) + λ5(γi + u4 h −µr −ωr). (9) the transversality conditions have the form (6), since all the states are free at the terminal time. also, the hamiltonian is maximized with respect to the controls at the optimal control u∗ = (u∗1, u ∗ 2, u ∗ 3, u ∗ 4), thus we differentiate h with respect to u1, u2, u3, and u4 on u respectively to obtain ∂h ∂u1 = w4u1 − (λ2 −λ1)(β1 e + β2 h + β3 i)s = 0 at u1 = u ∗ 1, ∂h ∂u2 = w5u2 − (λ2 −λ4)e = 0 at u2 = u ∗ 2, ∂h ∂u3 = w6u3 − (λ3 −λ4)i = 0 at u3 = u ∗ 3, ∂h ∂u4 = w7u4 − (λ4 −λ5)h = 0 at u4 = u ∗ 4. (10) hence, solving for u∗1, u ∗ 2, u ∗ 3, and u ∗ 4 on the interior of u 92 yusuf et al. / j. nig. soc. phys. sci. 4 (2022) 88–98 93 gives u∗1 = s (β1 e + β2 h + β3 i)(λ2 −λ1) w4 , u∗2 = e(λ2 −λ4) w5 , u∗3 = i(λ3 −λ4) w6 , u∗4 = h(λ4 −λ5) w7 . (11) we can now impose the bounds 0 ≤ u1 ≤ u1 max, 0 ≤ u2 ≤ u1 max, 0 ≤ u3 ≤ u3 max, and 0 ≤ u4 ≤ u4 max on the controls to get u∗1 = min { max ( 0, s (β1 e + β2 h + β2 i)(λ2 −λ1) w4 ) , u1 max} , u∗2 = min { max ( 0, e(λ2 −λ4) w5 ) , u2 max } , u∗3 = min { max ( 0, i(λ3 −λ4) w6 ) , u3 max } , u∗4 = min { max ( 0, h(λ4 −λ5) w7 ) , u4 max } . (12) this ends the proof. 4. estimation of the model parameters and initial conditions 4.1. estimation of parameters the average new recruitment per unit time (λ) into the susceptible population was estimated using the annual increase in nigeria’s population from year 2000-2015. nigeria’s average annual population increase is estimated to be 3.922 million per year [38]. thus, the daily new recruitment will be λ = 3.922365 = 0.0107 humans per day. in order to estimate the daily death rate of nigerians, we used the average life expectancy of nigerians which is estimated as 12 (53 + 56) = 54.5 years and the concept that an individual losses the reciprocal of its life expectancy every day. this implies that µ = 1365×54.5 = 0.00005 per day [39]. using the half-life concept, we estimate the disease progression rate (ρ) into the symptomatic infected compartment based on the fact that about a fifth of asymptomatic infected individuals become symptomatic infected while it takes a maximum of 14 days for this transition to take place. so, ρ = − 114 ln(0.2) = 0.155 per day [40]. we estimated the natural recovery rate (γ) based on the fact that about 80% of infected individuals recover without treatment while an average of about a quarter infected individuals recover on daily basis [40, 41]. thus, γ = 0.8 × 0.25 = 0.2 per day. the disease-induced death rate (δ1) is estimated from the nigeria covid-19 epidemiological data of 79 consecutive days [40]. so, we estimate it as δ1 = 1 79 × ln 506 2 = 0.07 per day. in addition, we assumed that those covid-19 infected individuals who were not hospitalized are twice likely to die due to the infection. hence, δ2 = 2 × 0.07 = 0.14 per day. similarly, the disease transmission rate due to the exposed individuals (β1) is estimated using the nigeria covid-19 prevalence epidemiological data of 79 consecutive days [40]. so, we estimated 6β1 = 1 79 × ln 19808 190 = 0.0588. thus, β1 = 0.0098 per day. also, we assume that the hospitalized infected individuals are twice likely to transmit the disease when compared to asymptomatic infected individuals while the undetected symptomatically infected individuals are three times likely to transmit the disease when compared to asymptomatic infected individuals. hence, we estimate β2 = 2 ×β1 = 0.0196 per day and β3 = 3 ×β1 = 0.0294 per day. the disease waning rate ω is taken conservatively to be 0.005 per day, since the outbreak is unprecedented in human history. it is worth mentioning that this term is incorporated based on recent reports of cases of re-infection by the disease after recovery in some countries. we assumed that it will take about six months before an earlier infected person could be reinfected. 4.2. estimation of initial conditions we adopted the nigerian coronavirus surveillance data for the 17th may, 2020. we assumed the total number of people tested for the virus as at the date as initial cohort for the disease spread through which the entire country might eventually become susceptible. so, we take the initial conditions for each of the compartments as s (0) = 28193, e(0) = 1394, i(0) = 4183, h(0) = 4183, r(0) = 1594 with model parameter values as estimated in the preceding section. 5. numerical simulations and efficiency analysis 5.1. numerical simulations the resulting optimality system consisting the dynamical system (1) and its associated initial conditions, the adjoint system (5), and the transversality conditions (6) coupled with control characterizations given in (7) is solved numerically using a fourth order iterative runge-kutta scheme implemented in matlab. this method solves the state equation with an initial guess for u1, u2, u3 and u4 forward in time; after which it solves the adjoint equations backward in time while the controls are continuously updated based on the control characterization in (7). the computational procedure is done iteratively until the results converge. see lenhart and workman[42] for details. in order to investigate the impact of combined efforts of the controls ui (i = 1, . . . , 4) on the dynamics of covid-19 pandemic in the population, the weight constants wi (i = 1, 2, . . . , 7) that appear in the objective functional expressed by (2) are taken as wi = 10 with the assumption that the available resources can only accommodate the maximum levels of the controls as defined by u1 max = 0.7, u2 max = u3 max = 0.5 and u4 max = 0.9. keeping in mind that precautionary measures such as wearing of face masks, observing social distancing when in public, regular personal hygiene (hand washing and sanitizing) play crucial roles in the fight against covid-19 right from the onset of 93 yusuf et al. / j. nig. soc. phys. sci. 4 (2022) 88–98 94 the pandemic outbreak, the effect of different combinations of personal protection (precautionary) measure with at least any two other control interventions on the transmission dynamics of covid-19 in the community are investigated. thus, four different control strategies are considered, and are defined as follows: strategy 1: optimal use of personal protection, contact tracing and testing control for asymptomatically infected individuals, and detection control for symptomatic infected individuals (u1(t) , 0, u2(t) , 0, u3(t) , 0, u4(t) = 0), strategy 2: optimal use of personal protection, contact tracing and testing control for asymptomatically infected individuals, and management control for hospitalized individuals (u1(t) , 0, u2(t) , 0, u4(t) , 0, u3(t) = 0), strategy 3: optimal use of personal protection, detection control for symptomatic infected individuals, and management control for hospitalized individuals (u1(t) , 0, u3(t) , 0, u4(t) , 0, u2(t) = 0) and strategy 4: optimal use of personal protection, contact tracing and testing control for asymptomatically infected individuals, detection control for symptomatic infected individuals, and management control for hospitalized individuals (u1(t) , 0, u2(t) , 0, u3(t) , 0, u4(t) , 0). 5.1.1. strategy 1–optimal use of personal protection, contact tracing and testing control for asymptomatically infected individuals, and detection control for symptomatic infected individuals (u1(t) , 0, u2(t) , 0, u3(t) , 0, u4(t) = 0) the effect of the control intervention that considers the simultaneous use of optimal personal protection (u1(t)), optimal contact tracing and testing control for asymptomatic infected individuals (u2(t)) and optimal detection control for symptomatic infectious humans (u4(t)) on the community spread of covid19 is evaluated as shown in figure 1. it is seen in figure 1a that the size of covid-19 prevalence decreases more rapidly with control intervention than when no control is applied in the population. also, with application of this control strategy, the number of new cases is constantly managed near the initial number at the onset of the disease outbreak after about 50 days (counting from the commencement of the intervention) till the final time of the intervention (as shown in figure 1b). figure 1c reveals that personal protection (u1) is at maximum for about 190 days before undergoing stepwise fluctuation between the lower and upper bounds in the last intervention period, contact tracing and testing control for asymptomatic infected individuals (u2(t)) is held consistently close to the lower bound over the simulation period, while the detection control for symptomatic infectious humans (u3(t)) is at minimum level for about 175 days (counting from the beginning of the intervention period) before increasing and decreasing stepwisely between the lower and upper bounds in the remaining intervention period. 0 50 100 150 200 250 300 −2 0 2 4 6 8 10 12 time (days) e + i + h (′ 0 0 0 ) u1 6= 0, u2 6= 0, u3 6= 0, u4 = 0 u1 = u2 = u3 = u4 = 0 (a) disease prevalence with and without control strategy 1 0 50 100 150 200 250 300 0 10 20 30 40 50 time (days) n e w c a se s (′ 0 0 0 ) u1 6= 0, u2 6= 0, u3 6= 0, u4 = 0 u1 = u2 = u3 = u4 = 0 (b) covid-19 incidence with and without strategy 1 0 50 100 150 200 250 300 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 time (days) c o n tr o l p ro fil e s u1 u2 u3 u4 (c) profiles of optimal controls u1 , u2 , u3 with u4 = 0 figure 1: simulations depicting the optimal use u1, u2 and u3 only 5.1.2. strategy 2–optimal use of personal protection, contact tracing and testing control for asymptomatically infected individuals, and management control for hospitalized individuals (u1(t) , 0, u2(t) , 0, u4(t) , 0, u3(t) = 0) figure 2 illustrates how the implementation of the control intervention that combines optimal use of personal protection (u1(t)), contact tracing and testing control for asymptomatic infected individuals (u2(t)) and management control for hospitalized individuals (u4(t)) affects the dynamics of transmission 94 yusuf et al. / j. nig. soc. phys. sci. 4 (2022) 88–98 95 0 50 100 150 200 250 300 −2 0 2 4 6 8 10 12 time (days) e + i + h (′ 0 0 0 ) u1 6= 0, u2 6= 0, u4 6= 0, u3 = 0 u1 = u2 = u3 = u4 = 0 (a) disease prevalence with and without control strategy 2 0 50 100 150 200 250 300 0 10 20 30 40 50 time (days) n e w c a se s (′ 0 0 0 ) u1 6= 0, u2 6= 0, u4 6= 0, u3 = 0 u1 = u2 = u3 = u4 = 0 (b) covid-19 incidence with and without strategy 2 0 50 100 150 200 250 300 0 0.2 0.4 0.6 0.8 1 time (days) c o n tr o l p ro fil e s u1 u2 u3 u4 (c) profiles of optimal controls u1 , u2 , u4 with u3 = 0 figure 2: simulations depicting the optimal use of u1, u2 and u4 only and spread of covid-19 in the population. it is salient to state that with the implementation of this control strategy, the disease prevalence is sharply decreased to zero within a very short period as shown in figure 2a. further, the use of this intervention strategy helps to constantly hold the number of new cases at the initial value over the simulation time (see figure 2b). all the three optimal controls (u1, u2 and u4) fluctuate between the maximum and minimum levels in a stepwise manner over the simulation period as revealed by the control profiles in figure 2c. 5.1.3. strategy 3–optimal use of personal protection, detection control for symptomatic infected individuals, and management control for hospitalized individuals (u1(t) , 0, u3(t) , 0, u4(t) , 0, u2(t) = 0) 0 50 100 150 200 250 300 0 2 4 6 8 10 12 time (days) e + i + h (′ 0 0 0 ) u1 6= 0, u3 6= 0, u4 6= 0, u2 = 0 u1 = u2 = u3 = u4 = 0 (a) disease prevalence with and without control strategy 3 0 50 100 150 200 250 300 0 10 20 30 40 50 time (days) n e w c a se s (′ 0 0 0 ) u1 6= 0, u3 6= 0, u4 6= 0, u2 = 0 u1 = u2 = u3 = u4 = 0 (b) covid-19 incidence with and without strategy 3 0 50 100 150 200 250 300 0 0.2 0.4 0.6 0.8 1 time (days) c o n tr o l p ro fil e s u1 u2 u3 u4 (c) profiles of optimal controls u1 , u3 , u4 with u2 = 0 figure 3: simulations depicting the optimal use of u1, u3 and u4 only in figure 3, the effect of personal protection control (u1(t)) combined with the efforts of detection control for symptomatic infected individuals (u3(t)) and management control for hospitalized individuals (u4(t)) on the dynamics of community spread of covid-19 is demonstrated. figure 3a depicts that the size of covid-19 prevalence with control intervention diminishes 95 yusuf et al. / j. nig. soc. phys. sci. 4 (2022) 88–98 96 more rapidly than when no control is put in place. as shown in figure 3b, the number of new cases of covid-19 sharply increased till the end of the disease outbreak, but the number of new cases is successfully managed near the initial value throughout the intervention period. the control profiles in figure 3c suggest that personal protection (u1(t) is at maximum coverage for about 280 days, counting from the commencement of intervention, before dropping gradually to the lower bound at the final time of control implementation, while the detection control for symptomatic infected individuals (u3(t)) and the management control for hospitalized humans are at maximum coverage for the first 180 and 205 days, respectively, before decreasing in a stepwise fluctuation manner to the lower bound at the final time. 5.1.4. strategy 4–optimal use of personal protection, contact tracing and testing control for asymptomatically infected individuals, detection control for symptomatic infected individuals, and management control for hospitalized individuals (u1(t) , 0, u2(t) , 0, u3(t) , 0, u4(t) , 0) figure 4 illustrates the transmission dynamics of covid19 with implementation of control strategy that combines personal protection (u1(t)), contact tracing and testing control for asymptomatic infected individuals (u2(t)), detection control for symptomatic infected individuals (u3(t)) and management control for hospitalized individuals (u4(t)) and without any intervention. as shown in figure 4a, the disease prevalence with control strategy diminishes more rapidly than when no control is in use. it is also observed from figure 4b that the number of new cases of covid-19 is consistently managed close to zero throughout the control implementation period, whereas a rapid increase in the number of new cases is observed when no control strategy is implemented. to obtain this optimal solution, figure 4c shows that the four controls are maximally held at the upper bounds (70% for u1, 50% for u2, 50% for u3 and 90% for u4) and minimally held at lower bounds (0% for all the controls) in a stepwise manner over the control intervention period. 5.2. efficiency analysis based on the time series plots in figs. 1 to 4, it is observed that the implementation of each of the four strategies has similar effect on the dynamics of covid-19 prevalence and new cases. thus, it is pertinent to compare the efficacy of each of the four control strategies in reducing the spread of covid-19 in the population. therefore, we need to find out the most effective control strategy among the four strategies analysed in this work. in order to identify the most effective strategy among the proposed ones, efficiency analysis is carried out on the four strategies under consideration. taking a cue from previous studies [3, 43], efficiency index, denoted as ei, is defined as ei = ( 1 − ws wo ) × 100. (13) in the formula given by (13), ws and w0 are the respective cumulative numbers of infected humans with and without control strategy, respectively, which are determined from ∫ t f 0 (e(t)+ 0 50 100 150 200 250 300 −2 0 2 4 6 8 10 12 time (days) e + i + h (′ 0 0 0 ) u1 6= 0, u2 6= 0, u3 6= 0, u4 6= 0 u1 = u2 = u3 = u4 = 0 (a) disease prevalence with and without control strategy 4 0 50 100 150 200 250 300 0 10 20 30 40 50 time (days) n e w c a se s (′ 0 0 0 ) u1 6= 0, u2 6= 0, u3 6= 0, u4 6= 0 u1 = u2 = u3 = u4 = 0 (b) covid-19 incidence with and without strategy 4 0 50 100 150 200 250 300 0 0.2 0.4 0.6 0.8 1 time (days) c o n tr o l p ro fil e s u1 u2 u3 u4 (c) profiles of optimal controls u1 , u2 , u3 , u4 figure 4: simulations depicting the optimal use of u1, u2, u3 and u4 i(t) + h(t))dt, where t f = 300 days. the strategy with highest efficiency index value is the best strategy [3, 43]. the value wo is obtained in matlab as wo = ∫ t f 0 (e(t) + i(t) + h(t))dt = 550.55 per thousands. the values of ws and the corresponding efficiency index for the four control strategies ranked from lowest to the highest efficiency index are presented in table 2. a look at table 2 shows that strategy 4 has the highest ei value, followed by strategy 2, strategy 3, and lastly strategy 1. this result of control strategies ranking in terms of their ei values suggest that strategy 4 is the most effective strategy, 96 yusuf et al. / j. nig. soc. phys. sci. 4 (2022) 88–98 97 table 2: efficiency indices for the four intervention strategies strategy ws(′000) ei(%) strategy 1 245.068 55.48 strategy 3 233.543 57.58 strategy 2 198.493 63.94 strategy 4 175.288 68.16 followed by strategy 2, then strategy 3, while strategy 1 is the least effective strategy in decreasing the covid-19 prevalence and the number of new cases in the community. 6. conclusion in this paper, a mathematical analysis of the novel coronavirus 2019 (covid-19) pandemic transmission dynamics is carried out to gain insights into the transmission and effective control of the disease using precautionary and control measures. a non-linear mathematical model is formulated by stratifying the total human population into susceptible, exposed (infected but not yet infectious), symptomatic infectious, hospitalized and recovered classes. the model incorporates four timedependent control variables that account for personal protection, contact tracing and testing control for asymptomatic infected humans, detection control for symptomatic infectious humans, and management control for hospitalized humans to examine the impacts of their various combination strategies in minimizing covid-19 burden in the community. in particular, the developed model is parametrized to mimic the reported cases of covid-19 in nigeria. theoretical analysis of the model is carried out using pontryagin’s maximum principle. the associated optimality system is simulated to examine the effect of the four different proposed strategies on the dynamics of the disease spread. the four strategies are: strategy 1 (optimal use of personal protection, contact tracing and testing control of asymptomatically infected individuals, and detection control of symptomatic infected individuals), strategy 2 (optimal use of personal protection, contact tracing and testing control of asymptomatically infected individuals, and management control for hospitalized individuals), strategy 3 (optimal use of personal protection, detection control of symptomatic infected individuals, and management control for hospitalized individuals) and strategy 4 (optimal use of personal protection, contact tracing and testing control of asymptomatically infected individuals, detection control of symptomatic infected individuals, and management control for hospitalized individuals). the results reveal that both the prevalence and incidence of the disease can be reduced significantly by implementing any of the four proposed control strategies. efficiency analysis is carried out to determine the most effective strategy to reduce the community spread of covid-19 among the four strategies. the results obtained from the analysis further suggests that simultaneous implementation of the optimal use of personal protection, contact tracing and testing control of asymptomatically infected individuals, detection control of symptomatic infected individuals, and management control for hospitalized individuals is the most effective strategy. the epidemiological insight from the result is that to effectively control the dynamics of transmission and spread of covid-19 in nigeria (as well as in other countries affected by the pandemic), the implementation of these control and preventive measures should be sustained continuously for an appreciable length of time and relaxed gradually as new cases of the disease become rare and the prevalence rapidly wanes out in the population. acknowledgments the authors are grateful to the editors and the anonymous 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[43] s. a. carvalho, s. o. da silva & i. da cunha charret, “mathematical modeling of dengue epidemic: control methods and vaccination strategies”, theory in biosciences, 138 (2019) 223. 98 j. nig. soc. phys. sci. 4 (2022) 117–122 journal of the nigerian society of physical sciences modeling and forecasting the third wave of covid-19 incidence rate in nigeria using vector autoregressive model approach gabriel o. odekina, adedayo f. adedotun∗, ogbu f. imaga department of mathematics, covenant university ota, ogun, nigeria abstract modeling the onset of a pandemic is important for forming inferences and putting measures in place. in this study, we used the vector autoregressive model to model and forecast the number of confirmed covid-19 cases and deaths in nigeria, taking into account the relationship that exists between both multivariate variables. before using the vector autoregressive model, a co-integration test was performed. an autocorrelation test and a heteroscedasticity test were also performed, and it was discovered that there is no autocorrelation at lags 3 and 4, as well as no heteroscedasticity. according to the findings of the study, the number of covid-19 cases and deaths is on the rise. to forecast the number of cases and deaths, a vector autoregressive model with lag 4 was used. the projection likewise shows a steady increase in the number of deaths over time, but a minor drop in the number of confirmed covid-19 cases. doi:10.46481/jnsps.2021.431 keywords: covid-19, vector autoregressive model, akaike information criterion, final prediction error, hannan quinn information criterion article history : received: 07 october 2021 received in revised form: 10 february 2022 accepted for publication: 10 february 2022 published: 28 february 2022 c©2022 journal of the nigerian society of physical sciences. all rights reserved. communicated by: t. latunde 1. introduction coronavirus disease has now been declared a global pandemic. the first case in nigeria was discovered on february 27, 2020, and was confirmed at the lagos state university teaching hospital’s virology laboratory. in late december 2019, the coronavirus disease known as a form of severe acute respiratory syndrome was initially discovered in the city of wuhan, china and it was acknowledged as a pandemic by [1] on the 11th of march 2020 after infecting over 118,000 people globally. this virus has been the main worry of doctors of medicine, community health experts, and researchers of all fields. much international public/ community health inventiveness is being executed ∗corresponding author tel. no: +2348055711272 email address: adedayo.adedotun@covenantuniversity.edu.ng (adedayo f. adedotun ) and swift research of the biology of the virus and pathogenesis of the virus are being conducted in research institutes all around the globe. the virus which has spread at an exponential rate all over the world has negatively affected the healthcare system in many countries. the covid-19 pandemic is one of the worst pandemics mankind has ever been confronted after the spanish flu pandemic in 1918, which caused the deaths of about 50 million people at a time the world’s population was around 2 billion. economic and social interruption triggered by the pandemic is overwhelming. disease prevention and control are eager for disease prediction guidance. effective models for short-term forecasting have a pivotal role to develop strategic planning methods in the public health system. under the guidance of the prediction model, we know the severity and the trends of 117 odekina et al. / j. nig. soc. phys. sci. 4 (2022) 117–122 118 the pandemic under different strategies [2]. according to [3], those who would suffer in the medium-term as a result of actions made to prevent the spread of covid-19 are the influence of the virus on the socioeconomic determinants of health, as well as its consequences on the next generation. in nigeria, the first known and confirmed case of covid-19 was documented on the 27th of february 2020 in lagos state according to [2]. after the index case on the 27th of february, the number of confirmed cases has been on the rise with the earliest reported death case on 22nd march 2020. due to the speedy escalation in the number of cases in nigeria, the federal government had to enforce total lockdown in lagos, abuja, and ogun state. some states which were not included in the total lockdown by the federal government also had lockdown enforced by the state government to curtail the rise of the deadly virus. with the outbreak of covid-19, a lot of studies have been carried out in various science disciplines to either reduce the spread or control the increasing trend of the disease. therefore, to manage and comprehend the epidemic, various approaches of estimation, modeling, and forecasting have been introduced. based on five deep learning approaches, [4] conducted a relative analysis to forecast the new number of covid-19reported cases and retrieved instances. long short-term memory (l stm), gated recurrent units (grus), basic recurrent neural network (rnn), variational auto-encoder (vae) algorithms, and bidirectional l stm (bil stm) algorithms were utilized for the global prediction of covid-19 cases with a small amount of data. their research is based on daily verified instances and the number of cases retrieved from six countries: china, spain, italy, australia, the united states, and france. when the performance of each model was tested, it was discovered that vae had a higher predicting precision than the other models. forecasting the coronavirus (covid-19) cases and deaths, [5] proposed the approach of statistical time series to model and forecast the short period behavior of covid-19. they assumed a trend that is multiplicative which aims to capture the persistence of the two variables predicted (number of cases and mortality rate) as well as their uncertainty. the anticipated time series model showed an excellent level of precision and ambiguity as additional data were collected. in a study by [6], the effect of total lockdown on covid-19 prevalence rate and death rate in china was investigated, and it was concluded that lockdown is effective in lowering the incidence rate and mortality rate. the widespread increase in covid-19 and death as a result of corona-virus infection has been predicted by [7]. they also looked at a time series model that was used to forecast the number of confirmed and recovered coronavirus cases. the error distributions were carefully designed as a two-member scale combination of classical (tp-smn) models, with the best match carefully selected. the chosen model was used to forecast the global number of diseases and fatalities caused by covid19. the study [8] looked at epidemic data and statistics with a focus on covid-19 and found that lockdown measures in italy, spain, and china, as well as the closure of firms in hubei that provided non-essential services, were beneficial approaches. in the study carried out on the coronavirus (covid-19) in spain and italy by [9], two simple mathematical epidemiological models were applied where it was observed that the loglinear regression yielded an improved result and basic estimate of the everyday incidence for both countries. [10] studied the gender-based covid-19 prevalence rate and death rate in nigeria. in the study, a wilcoxon signed-rank test was adopted to examine disparity in the sex distributions of the daily prevalence. in the work of [11], the autoregressive integrated moving average was adopted to forecast the covid-19 incidence rate in india where an increasing tendency in the number of coronavirus cases was observed. the vector autoregressive model and co-integrated vector autoregressive models are time series models used for multivariate time series data set. a lot of research has been carried out through the adoption of this model which takes into account the linear dependence that exists among the variables. for example, [12] adopted the co-integrated vector autoregressive model in modeling wind speed along with some selected meteorological variables. using the (var) model, a time-series analysis was proposed to investigate the impact of environmental pollution on mortality in nigeria. the data set passes the stationarity test, indicating that the data is steady and that the var model would fit well. furthermore, environmental pollution has a considerable impact on mortality in nigeria, according to the study [13]. a study conducted in the united state by [14] used a var model to predict the covid-19 prevalence rate in the u.s. the result of the research stated that the situation of the pandemic will get shoddier if there is no active control. vector autoregressive integrated moving average analysis (varima) to establish the relationship between the number of deaths due to covid-19 and the number of new cases of covid19 in the country. the aicc was used to select the best model after it fulfilled all the assumptions [15]. modeling the outbreak of a pandemic is pertinent for inferencemaking and implementation of policies. in this paper, we adopted the vector autoregressive model in modeling and forecasting the number of covid-19 cases and deaths in nigeria. in the next section (section 2), we describe the data we used in the analyses and the var model and analysis plan for the research. the results section (section 3) provides the prediction results by var modeling and an internal validation/evaluation of the model. section 4 discusses the model performance, further improvement, and comparison with other models. 2. material and methods 2.1. data source the data used for this study is daily data on the number of covid-19 cases and death obtained from https://raw.githubusercontent. com/owid/covid-19-data/ master/public/data/ owid-covid-data.xlsx the methodology used for this paper is given below; 2.2. vector autoregressive model consider a k-dimensional vector autoregressive model of order 2 given below, where at ∼ n (0, ∑ ), φ1 and φ2 are k×k 118 odekina et al. / j. nig. soc. phys. sci. 4 (2022) 117–122 119 matrices of unknown coefficients. let st (t) denote the price of the risk free asset at time t and the model is given as follows yt = φ1yt−1 + φ2yt−2 + at (1) subtracting yt−1 from both sides of equation (1), and adding φ2yt−1 to the right-hand side of equation (1), we have ∆yt = φyt−1 + γ∆yt−1 + at, t = 1, 2, ..t (2) where φ = (φ1 + φ2 − ik×k). equation (2) is referred to as the vector error correction model, otherwise called the co-integrated vector autoregressive model. suppose we have a multivariate variable consisting of variables p and q, according to [16], equation (2) can be written in matrix form as;[ ∆p ∆q ] = [ p11 p12 p21 p22 ] [ ∆pt−1 ∆qt−1 ] + [ β11 β12 β21 β22 ] + [ a1t a2t ] in the formulation of the vector error correction model in equation (2), there are three cases of interest to be considered which are; 1. rank(φ) = 0 implies yt is not cointegrated and the vector error correction model in equation (2) reduces to a vector autoregressive model in (1) 2. rank(φ) = k, then yt contains no unit root. that is, yt is stationary and i(0), where k is the total number of variables. 3. 0 <rank(φ) = m < k, then there is at least one stationary linear combination of the variables which is the co-integrating relation. in this case,(φ) = αβ1, where α =vector of adjustment coefficients and β =co-integrating vector. for equation (2) to hold, φ should be of a reduced rank [17]. 2.3. augmented dickey fuller test (adf) the augmented dickey fuller test statistics were used to test the stationarity of the data. considering the hypothesis; h0:∅1 = 1 vsh0:∅1 < 1, the adf test statistic is given as; adf = β̂− 1 s td(̂β) (3) 2.4. akaike information criterion the aic for selecting the underlying vector autoregressive (p) is given as aic = ln (∣∣∣∣∣ ∼ωi ∣∣∣∣∣) + 2p2in (4) where n is the size of the sample, p is the parameter numbers, ∼ ω = 1 n n∑ n=i+1 ∧ ait( ∧ ait) ′ (5) at is the error term 2.5. final prediction error (fpe) the fpe for selecting the underlying vector autoregressive (p) is given as f pe = det  1n n∑ 1 e ( t, θ̂n ) ( e ( t, θ̂n ))t  ( 1 + d/n 1 − d/n ) (6) where n is the number of values in the estimation data set, e(t) is the n by 1 vector of prediction errors, θn represents the estimated parameters, d is the number of estimated parameters. 2.6. hannan-quinn information criterion (hqic) the hqic for selecting the underlying vector autoregressive (p) is given as hqic = −2lmax + 2kln(ln (n) ) (7) where lmaxthe log-likelihood, k is is the number of parameters, and n is the number of observations 2.7. lagrange multiplier statistic the autoregressive conditional heteroscedastic (arch) lagrange multiplier (lm) is used to test the hypotheses of homoscedasticity. this involves regressing the squared residuals on the conditional mean equation which may be an autoregressive, or moving average model. for example, considering an arma (1,1) process, rt = φrt−1 + φet−1 + et, e2t = β0 + β1e 2 t−1 + β2e 2 t−2 + ... + βqe 2 t−q (8) h0 : β1 = β2 = β3 = ...βq = 0vs h1 : βi > 0 lme = t r 2 (9) where t= sample size, r2= r squared 2.8. residual autocorrelation the box-pierce statistic which was proposed by [18] will be used to test the autocorrelation in the residuals. h0: no autocorrelation up to order k vs h1: there is autocorrelation up to order k. the statistic for the test is given as q = n k∑ j=1 r2j (10) where n = sample size, r = autocorrelation at lag j 119 odekina et al. / j. nig. soc. phys. sci. 4 (2022) 117–122 120 2.9. normality test the jarque-berra test was used to decide if the error correction model is gaussian distributed. the test is used to measure the discrepancy in skewness and kurtosis of a variable compared to those of the gaussian distributions. h0 : the variable is distributed normally vs h1: the variable is not distributed normally. j b = m − p 6 [ s 2 + (l − 3)2 4 ] , (11) where m= number of observations, p=number of estimated parameters, s= skewness, l=kurtosis. the condition is to reject the null hypotheses if the p-value ≤ level of significance. 3. data analysis and interpretation 3.1. time plot the foremost step during the study of the data is to generate the time plot of the variables. figure 1 is a time plot of the amount of covid-19 confirmed cases in nigeria and the number of covid-19 related deaths in nigeria. the graph shows an increasing inclination (trend) in the number of confirmed covid19 cases and deaths in the country with the first wave occurring early in the year 2020, the second wave in the late year 2020, and the third wave which is the greatest occurring in the early year 2021. there seems to be no decrease in the pandemic figure 1: time plot of covid-19 reported cases and deaths 3.2. descriptive statistics table 1 gives a descriptive summary of statistics of the number of confirmed covid-19 cases and covid-19 deaths. it is observed that there is a high level of variation in the data obtained. the difference between the three measures of central tendencies (mean, median, and mode) in the number of confirmed covid19 cases shows a departure from normality, the same with the number of covid-19 deaths. this departure from normality can be ignored considering the reasonably large sample size. table 1: adf descriptive statistics number of cases number of deaths mean 88164.28 1294.141 median 66523.00 1180.00 standard deviation 64682.12 706.9598 variance 4183776688 499792.141 range 185570 2246 3.3. stationarity test from table 2, it is seen that the null hypothesis of nonstationarity could not be rejected for the number of confirmed covid-19 cases as our p-value is greater than our level of significance 0.05, hence the need for differencing. h0: the series is non-stationary vs h1: the series is stationary table 2: adf stationarity test test statistic 1% critical value 5% critical value z(t) 1.371 -3.430 -2.860 mackinnon approximate p-value for z(t) = 0.9970 from table 3, it is seen that the null hypothesis of non-stationarity was rejected after taking the third difference as our p-value is less than our level of significance 0.05. the series is stationary after the first differencing. h0: the series is non-stationary vs h1: the series is stationary table 3: adf stationarity test test statistic 1% critical value 5% critical value z(t) -6.469 -3.430 -2.860 mackinnon approximate p-value for z(t) = 0.0000 from table 4, it is seen that the null hypothesis of non-stationarity was rejected for the number of covid-19 deaths as our p-value of 0.0002 is less than our level of significance 0.05 which means the series is stationary. h0: the series is non-stationary vs h1: the series is stationary mackinnon approximate p-value for z (t) = 0.0002 3.4. lag selection and model estimation a lag of order 4 was selected based on the following information criterion used where it is observed that the minimum occurs on the fourth lag. using equation 2 on the covid-19, the model for the number of deaths and number of cases are estimated and presented in equations 12 and 13 respectively with the model summary presented in tables 6 and 7. from table 6, it is evident that for every increase in the number of deaths, there is a 0.0012504, 0.0024392 and 0.0011357 upsurge in the number of reported 120 odekina et al. / j. nig. soc. phys. sci. 4 (2022) 117–122 121 table 4: adf stationarity test test statistic 1% critical value 5% critical value z(t) -4.559 -3.430 -2.860 table 5: akaike information criterion lag order aic fpe hqic 1 19.3886 902429 30.7735 2 19.1106 683387 19.4081 3 18.9981 610694 19.143 4 18.9129* 560827* 18.9713* covid-19 cases at lag 2, 3, and 4 respectively with a -0.0008647 decrease in the number of covid-19 cases at lag 1 with their respective confidence intervals. table 7 also gives the vector autoregressive model for the number of covid-19 cases and it is also evident that for every increase in the number of cases, there is a 2.924165 and 1.410195 upsurge in the number of deaths at lag 1 and lag 3 respectively with a −4.285465 and −0.280125 decrease at lag 3 and lag 4 respectively. table 8 gives the normality test of the disturbances as not being normally distributed. however, with practically huge sample sizes, the contravention of the gaussian hypothesis ought not to cause any setback [19] number of deaths = −0.0008647yt−1 + 0.0024392yt−2 + 0.0011357yt−3 + 0.0012504yt−4 (12) number of cases = 2.924165yt−1 − 4.285465yt−2 + 1.410195yt−3 − 0.280125yt−4 (13) 3.5. lagrange multiplier test for autoregressive conditional heteroscedasticity (arch) and autocorrelation table 9 reports the heteroscedasticity test of the number of confirmed covid-19 cases and deaths in nigeria. the test result shows that the null hypothesis of no arch effect was not rejected. that is, there is no arch effect. table 10 gives the box-pierce statistic for the autocorrelation test of the 4 lags which shows no autocorrelation is depicted across the four lags. h0 : noarche f f ectsv sh1 : arch ( p) disturbance h0 : noautocorrelationatlagorder 3.6. forecast precision figure 2 shows a forecast of the number of covid-19 cases and deaths for the next one year (365 days). from the graph, it can be seen that there is a sharp rise in the covid-19 mortality rate alongside a slight decrease in the number of confirmed cases with 95% confidence interval. this shows the future expectation of the current pandemic table 6: vector autoregressive model death coefficient p− value standard error 95% confidence interval number of cases l1 -.0008647 0.393 .0010122 −.0028487 .0011192 l2 0.0024392 0.033 .0011437 .0001977 .0046808 l3 0.0011357 0.334 .0011767 −.0011705 .003442 l4 0.0012504 0.258 .001105 −.0009153 .0034161 constant 1.888731 0.000 table 7: vector autoregressive model number of cases coefficient p− value standard error 95% confidence interval number of deaths l1 2.924165 0.131 1.935567 −0.8694764 6.717806 l2 -4.285465 0.141 2.913769 −9.996347 1.425417 l3 1.410195 0.630 2.930209 −4.332909 7.153298 l4 -0.280125 0.988 1.91525 −3.781833 3.3725808 constant 20.04022 0.457 table 8: jarque-berra test variables chi-square df prob>chi2 number of cases 5102.057 2 0.00000 number of deaths 1381.003 2 0.00000 all 6483.060 4 0.00000 4. conclusion the surge of covid-19 has crippled the health care system in nigeria and other parts of the world. the need to model and study the incidence rate of covid-9 cannot be overemphasized as it is pertinent for concrete decision-making. there has been a sharp rise in the figures of covid-19 cases and death as depicted by the time plot. the time plot of the number of cases and deaths shows an “s” shape which indicates the increase in the pandemic. a vector autoregressive model of lag 4 was adopted which was used to make a forecast on the number of cases and death. moreover, an autocorrelation test and a test of heteroscedasticity were carried out where it was observed that there exists no autocorrelation at lag 3 and lag 4 and there exists no heteroscedasticity. a jarque-berra test of normality of the disturbances was done on the vector autoregressive model 121 odekina et al. / j. nig. soc. phys. sci. 4 (2022) 117–122 122 figure 2: forecast of the number of covid-19 reported cases table 9: testing for arch effect lags (p) chi2 prob> chi2 4 1.035 0.3089 table 10: box-pierce statistic test of autocorrelation lag chi2 prob>chi2 1 12.2810 0.31327 2 42.5130 0.08523 3 86.0806 0.42145 4 53.4515 0.21436 which indicates a departure from normality. however, this result can be ignored for a reasonably large sample size of at least 30 according to [19]. the forecast also reveals an upward trend in the number of deaths with a slight decrease in the number of infections. this is an indication that in the future, the spread may be reduced however, there will be a high mortality rate resulting from this pandemic. though the cause of death can also be attributed to some other factors such as age or other underlying ailments the patient may have which could have been triggered by covid-19. this can be an area of further research. concerning the above findings, the following recommendations have been made; (1) the government needs to put further measures in place to curtail the spread of the virus and aim towards flattening the curve, (2) an awareness should be created for individuals to be properly enlightened and get vaccinated, (3) the government should also render proper health care to infected individuals to reduce the mortality rate. references [1] ”world health organization.” [2] ”nigeria center for disease control,” (2020). 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[22] m. jonsson & r. sircar, ”optimal investment problems and volatility homogenization approximations”, modern methods in scientific computing and applications 75 (2002) 255. 122 j. nig. soc. phys. sci. 2 (2020) 36–50 journal of the nigerian society of physical sciences original research a modified forced randomized response model a. t. adeniran∗, a. a. sodipo, c. g. udomboso department of statistics, university of ibadan, ibadan, nigeria. abstract this paper proposed a new randomized response model (rrm) to estimate proportion of people characterizing a sensitive variable (s ) under study. simple random sampling with replacement and stratified simple random sampling scheme were adopted. maximum likelihood and bayesian estimation procedures of the proposed model were developed and compared. the sampling distribution (expectation and variance) of the proposed estimator under the two sampling techniques, efficiency comparison of the proposed model with some existing models, and numerical illustration of all the compared models were also explored. the study found that the proposed model outperformed other existing rrms in terms of efficiency and it proved to be more protective in designing survey for sensitive related issues. keywords: randomized response, proportion, sensitive variable, maximum likelihood estimator, sampling distribution. article history : received: 25 december 2019 received in revised form: 15 february 2020 accepted for publication: 17 february 2020 published: 28 february 2020 c©2020 journal of the nigerian society of physical sciences. all rights reserved. communicated by: t. latunde 1. introduction surveys usually collect responses to a large number of items from each sample unit and many institutions use empirical evidence from surveys to make their policies. thus, surveys play a prominent role in society, hence collecting and interpreting survey data correctly is essential. one of the most obvious problems in census or sample surveys is the inability of the researcher to collect responses on some or all of the items for a sampled unit or when some responses are deleted because they fail to satisfy edit constraints [1]. in the vernacular of sample survey, this is the problem of non-response. non response can arise for a variety of reasons, one of which is the sensitive nature of the survey question. socially sensitive questions such as shoplifting, rash driving, tax evasion, felony, false declaration of assets, expenditure on addiction of various form (drug abuse), kidnapping, occult affiliation, psychosocial status, cheating in national examination; ∗corresponding author tel. no: +2348035529045 email address: at.adeniran@mail.ui.edu.ng (a. t. adeniran) or health habits and sexual orientation related questions such as hiv infection, induced abortion, masturbation, ravish, homosexuality, and other illegal, unethical, prohibited attitudes or practices that receive disapproval by the society are thought to be threatening to respondents [2-3]. when sensitive topics are studied, respondents often react in ways that negatively affect the validity of data by giving socially desirable answers to avoid social embarrassment and to project a positive self-image [4-5]. the randomized response model (rrm) also known as randomized response technique (rrt) or randomized response distribution/design (rrd) is a survey method specifically developed to improve the participation and precision of answers to sensitive questions because the frequency of socially undesirable, disgraceful, embarrassing, incriminating, ignominious, highly stigmatizing variable has been found usually underestimated in surveys. in the year 1965, warner explained that the reluctance of the respondents to elicit sensitive or probably harmful information would diminish when respondents are convinced that their anonymity be guaranteed, when incriminating answers could be covered 36 adeniran et al. / j. nig. soc. phys. sci. 2 (2020) 36–50 37 even from the interviewer, the need to present oneself in a positive way would decrease and honest answering would increase [6]. following this assumption, [7] did the pioneering work on a rrt which required the interviewee to give a “yes”or “no”answer either to the sensitive question or to its negative depending on the outcome of a randomizing device not revealed to anybody including the interviewer. after [7] original rrm, different authors developed various rr models including unrelated questioning technique (uqt), forced randomized response model (frrm), stratification of new and existing models in randomized response (rr) surveys, calibration techniques and so on. according to [8-9] ”frrm and uqt with known population prevalence of the innocuous attribute are the best, a major setback is that it is not easy to come up with unrelated question with known prevalence and variance near zero, hence, uqt seems harder to be adopted.” to circumvent this problem, this study proposed a modified forced randomized response model (mfrrm) with a definite no response to the unrelated question. the structure of the paper is as follows: in section 2, literatures that are related to the study are reviewed to identify all existing works, their contributions and limitations. section 3 introduces the newly proposed rrm namely modified forced randomized response model. in the same section, maximum likelihood (ml) and bayesian estimator (be) of the proposed rrm are explored, comprehensive treatment of statistical properties of the proposed estimator such as: unbiasedness, efficiency comparison with existing models, and cost efficiency of the proposed design are as well carried out. section 4 presents analysis of the simulated data sets, discussion of results, and the concluding remarks. tables and charts showing the results are also presented in this section. 2. existing models this section lucidly renders a vivid review of related previous studies, starting with warner original model, via intermediate models to the current advanced stage of robust quantitative rrm. 2.1. warner randomized response model in warner’s (1965) original design, respondents are provided with a randomizer (say, spinner) and instructed to answer one of two statements: (i) i belong to the sensitive group (selected with probability p). (ii) i do not belong to the sensitive group (selected with probability 1 − p). respondents then in turn answer “true”or “not true”or “yes”or “no”according to their status on sensitive question without revealing to the interviewer which statement was selected by the randomizer. the probability tree diagram below illustrates the randomization procedure of [7] original randomized response model. from figure 1 above, the probability of yes and no answers are λ = pπs + (1 − p)(1 − πs) and 1 − λ = p(1 − πs) + (1 − p)πs, respectively. suppose no and n − no denote the total number of “yes”and “no”answers in the sample of n respondents, following the maximum likelihood principle, the likelihood function to maximize is p(λ, n) = ( n no ) [pπs+(1−p)(1−πs)] no [ p(1−πs)+(1−p)πs] n−no.(1) setting the derivative (w.r.t. πs) of the natural logarithm of equation (1) to zero, warner unbiased estimator of πs is π̂s = no n − (1 − p) 2p − 1 = λ̂− (1 − p) 2p − 1 , p , 1 2 (2) with variance v ar(π̂s) = πs(1 −πs) n + p(1 − p) n(2p − 1)2 (3) where, n = number of observed respondents’ (sample size), p = probability of selecting (and answering) sensitive question, and πs = proportion of yes answers to the sensitive question. subsequently, both in theory and application, several other authors have suggested various alternatives rr models including [10-17] among many others. recently, [18] used [7] rrm to measure corruption among public bureaucrats in bolivia, brazil, and chile. 2.2. forced randomized response model (frrm) the forced response method otherwise referred to as uqt with known πu was originated by [12] and later simplified by [19] with the assumption that πu is a predetermined parameter. that is, πu exists or must be known either from independent ad-hoc studies or statistical abstracts. this assumption does not always hold. even though πu is available in any register or statistical abstracts, cost of retrieving from records may be expensive. if πu is known beforehand, solving for πs in [11] model, and replacing λ with its sample estimator ( λ̂ = no n ) gives π̂s = 1 p [ λ̂− (1 − p)πu ] = 1 p [ no n − (1 − p)πu ] (4) with variance v ar(π̂s) = πs(1 −πs) n + (1 − p) np [πs(1 −πu) + πu(1 −πs) + (1 − p)πu(1 −πu) p ] . (5) estimating πs in this manner is termed “forced randomized response model or technique (frrm or frrt)”. among applied researchers, frrm is the most famous design. illegal 37 adeniran et al. / j. nig. soc. phys. sci. 2 (2020) 36–50 38 centering origin first statement ∈ s yes πs < s no 1 −πs p second statement < s yes 1 −πs ∈ s no πs 1 − p figure 1: probability tree diagram of the warner original rrm waste disposal [9], prevalence of civilian cooperation with militant groups in southeastern nigeria [20], vote choice regarding a mississippi abortion referendum [21], use of performance enhancing drugs [22], xenophobia and anti-semitism in germany [23], illegal poaching among south african farmers [24], and violation of regulatory laws by commercial firms [25] are few evidences among numerous practical application of the forcedrr-uqt. 2.3. mangat improved two step procedure alternative form of forced response model was developed by [15] as an optimization of one of his earlier designs. mangat’s procedure requires all respondents that have sensitive attribute (s ) to answer truthfully without use of randomizer. all respondents who do not have the sensitive attribute are required to use the randomizer to choose which from warner’s statements. this means that all no-answers are true negatives, and that only the yes answers are contaminated. [15] estimator of πs is π̂s = 1 p ( no n − 1 + p ) = 1 p (λ̂− 1 + p) (6) with variance v ar(π̂s) = πs(1 −πs) n + (1 − p)(1 −πs) np . (7) even though mangat’s estimator is most efficient when compared with the earlier similar estimators or models, his design is unrealistic as respondents with the sensitive attribute will be more inclined to lie, making the population estimates less valid, and this causes the results and inferences of his procedure to be less trustworthy [8], [26]. recently, [27] modified and applied [15] rrm to estimate two sensitive attributes simultaneously. 2.4. stratified warner’s randomized response model [16] presented a stratified rrt of [7] original model using an optimal allocation which is more efficient and cost effective than using a proportional allocation of [2] technique. the maximum likelihood estimate of πs using [16] rrm is π̂s = l∑ h=1 whπ̂sh = l∑ h=1 wh  nohnh − (1 − ph)2ph − 1  = l∑ h=1 wh [ λ̂h − (1 − ph) 2 ph − 1 ] (8) and the minimal variance of π̂s is given by v ar(π̂s) = 1 n  l∑ h=1 wh { πsh(1 −πsh) + ph(1 − ph) (2ph − 1)2 } 1 2  2 (9) where, wh = nh n = stratum weight, λ̂h = noh nh = proportion of yes-answer in a stratum h, ph = probability that a respondent in the sample stratum h has a sensitive question (s) card, and π̂sh = λ̂h−(1−ph ) 2 ph−1 = proportion of respondents with the sensitive trait in a stratum h, for h = 1, 2, · · · , l. 2.5. mixed stratified randomized response model [17] proposed mixed stratified rrm by taking two independent samples of sizes n1 and n2. also, two randomization devices r1 and r2 were used in each sample to estimate human immune virus (hiv) seroprevalence rate in kaduna state, nigeria. following the same maximum likelihood estimation procedure, an unbiased mixed stratified seroprevalence rates estimator is given by π̂s = l∑ h=1 whπ̂sh = l∑ h=1 wh [ nh1 nh π̂h1 + nh2 nh π̂h2 ] (10) with variance given by v ar(π̂s) = l∑ h=1 w 2h  nh1 n2h { (1 −πsh)( ph1πsh + 1 − ph1) ph1 } 38 adeniran et al. / j. nig. soc. phys. sci. 2 (2020) 36–50 39 + nh2 n2h { (1 −πsh)(ph2πsh + 1 − ph2) ph2 } (11) where, π̂sh1 = λ̂h1−(1−ph1 ) ph1 , π̂sh2 = λ̂h2−(1−ph2 ) ph2 and π̂sh = nh1 nh π̂h1 + nh2 nh π̂h2. if ph1 = ph2 = ph and nh1 = nh2 = noh, the above estimator reduces to π̂s = l∑ h=1 whπ̂sh = l∑ h=1 wh 2noh nh [ λ̂h − (1 − ph) ph ] (12) with variance v ar(π̂s) = 1 n  l∑ h=1 wh {πsh(1 −πsh) (13) + (1 − ph)(1 −πsh) ph } 1 2  2 (14) where, π̂sh = 2λ̂h [ λ̂h−(1−ph ) ph ] for h = 1, 2, · · · , l. 3. methodology 3.1. randomization procedure a simple random sampling with replacement (srswr) of n respondents was selected from the population. an individual respondent in the sample of size n was instructed to use the randomization device (r) which consists of a sensitive question (s) card selected with probability p ( p , 1 is a pre-assigned value set by the researcher) and unrelated question (u) card selected with probability (1 − p). the paired unrelated or innocuous question has a definite “no”response, for example (i) sare you a member of the insurgence group? (ii) uis this month of february (when research is conducted in the month of september)? in an applied research, the phrase inside bracket of the unrelated question must be excluded. after the researcher explained the procedures, randomization procedure then followed. each subject was given a box1 (containing both cards) to select at random a card and tick “yes”or “no”(based on his or her true status, i.e., under the assumption that respondents answer truthfully) and return the ticked card through an opener in the box2. respondents were not placed in close proximity and instructed not to let anyone (the investigator inclusive) see the card he or she had drawn so as to maintain the integrity of the randomized response technique, and keep the individual respondent’s privacy or anonymity protected. the probability tree diagram below illustrates the proposed modified-frrm: 3.2. estimation procedure: classical approach let xi = 0, if ith respondent says no1, if ith respondent says yes. (15) elementary probability theory can then be used to get an unbiased estimate of the prevalence (π̂s) of sensitive issues in the population. the procedure followed thus: from figure 2, the probability of a yes and no response are λ = pπs and 1 − λ = p(1 − πs) + (1 − p) = 1 − pπs, respectively. therefore, the likelihood function of πs is l(πs) = ( n no ) [ pπs] no [p(1 −πs) + (1 − p)] n−no. (16) where no is the total number of yes answers in the sample of size n respondents. to estimate πs, ( n no ) in (16) does not contain πs, as a result, the study considers it to be constant. hence, the function to maximize reduces to l(πs) = [ pπs] no [ p(1 −πs) + (1 − p)] n−no. (17) since logarithm is a monotone function, π̂s that maximizes the likelihood function also maximizes its log-likelihood function. therefore, to facilitate computation, the natural logarithm of (17) yield the log-likelihood function as l(πs) = no ln[ pπs] + (n − no) ln[p(1 −πs) + (1 − p)]. (18) differentiating (18) with respect to πs, the differential coefficient is d dπs l(πs) = no p pπs + −(n − no) p p(1 −πs) + (1 − p) . (19) equating (19) to zero and simplifying the resulting algebraic equation, the maximum likelihood estimator of πs which shall be arbitrarily denoted in this study as πspropo is π̂spropo = no np , p , 1. (20) remarks 1 (i) putting p = 1 in either (16) or (20), the proposed mfrrm reduces to the conventional method of direct questioning and the proposed rrt estimator also becomes the traditional measure of proportion. the value of p = 1 is absolute lack of protection, hence, p = 1 is impracticable in any randomized response survey. when 0 < p < 12 or 1 2 < p < 1, the respondent gives partially useful but not accurate information to which class (sensitive or not) he or she belongs. (ii) like [11], [15] and [36], the proposed estimator π̂spropo is not admissible since the range of π̂spropo is not a subspace of (0, 1). in fact, if non > p, that is, observed proportion of yes is greater than the pre-assigned randomization parameter, the proposed estimator produce estimate whose value is greater than unity. 39 adeniran et al. / j. nig. soc. phys. sci. 2 (2020) 36–50 40 origin sensitive question ∈ s yes πs < s no 1 −πsp unrelated question u no1 1 − p figure 2: probability tree diagram of the proposed mfrr model 3.2.1. sampling distribution of the proposed estimator (π̂spropo) theorem 3.1 (unbiasedness of π̂spropo). the proposed estimator is an unbiased estimator of population proportion of the sensitive attribute under survey. proof 3.1. π̂spropo is said to be unbiased of the population parameter πs if and only if e(π̂spropo) = πs. hence, we take expectation of (20) as follows e(π̂spropo) = e [ no np ] = 1 np n∑ i=1 [e(xi)] = 1 np npπs = πs, (21) which proves the theorem. theorem 3.2 (variance of π̂spropo). the variance of the proposed estimator is v ar(π̂spropo) = πs(1 −πs) n + (1 − p)πs np . (22) proof 3.2. this follows from taking the variance of (20) and considering section 5.5 of [28], v ar(π̂spropo) = v ar [ no np ] = 1 (np)2 n∑ i=1 v ar(xi) = npπs [ p(1 −πs) + (1 − p) ] (np)2 = πs(1 −πs) n + (1 − p)πs np . the sample estimator of v ar(π̂spropo) is obtained by substituting (20) in the preceding equation to have v̂ ar(π̂spropo) = π̂s(1 − π̂s) n + (1 − p)π̂s np . (23) remark 2: equation (23) consists of two parts. the first is the variance of the population proportion of the sensitive attribute if all the respondents are willing to imbibe direct questioning approach. the second part is injected due to use of randomizer. putting πu = 0 in equation (4) and (5), the results coincident with the proposed estimator in (20) and (22), respectively. 3.2.2. stratified sampling strategy for the proposed mfrrm if in the proposed model, the population (p) is partitioned (using a suitable or appropriate stratification factor) into l strata, and a sample nh(h = 1, 2, · · · , l) is selected by simple random sampling with replacement in each stratum such that n = l∑ h=1 nh. to get the full benefit from stratification, the study assume that nh (the number of units in each stratum) is known. following the same randomization procedure of the proposed (srswr) model, a respondent belonging to the sample in different strata will perform different randomization devices, each having different preassigned probabilities ph. under the assumption that these respondents “yes”or “no”-reports are made truthfully and ph , 1 is set by the researcher. the probability of a ”yes” and ”no” response from stratum h are pr(xih = 1) = λh = phπsh and pr(xih = 0) = 1−λh = ph(1−πsh) + (1− ph) for h = 1, 2, · · · , l respectively. suppose noh report “yes”and (nh − noh) report “no”to the sensitive question in stratum h, the likelihood function of the sample in stratum h is l(πsh) = [ phπsh] noh [ ph(1 −πsh) + (1 − ph)] (nh−noh ) (24) for computational convenience, the natural logarithm of the likelihood function is l(πsh) = nohln[phπsh]+(nh−noh) ln[ph(1−πsh)+(1−ph)](25) to obtain the maximum likelihood estimator of πsh, the study differentiate (25) with respect to πsh and equating the derivative to zero to get 0 = noh ph phπsh − (nh − noh) ph ph(1 −πsh) + (1 − ph) (26) solving the resulting equation (26) for πsh, the unbiased estimators in terms of the responses of the respondent in stratum h is given by π̂sh = noh nh ph = λ̂h ph (27) where λ̂h = proportion of ”yes”-answers in stratum h and π̂sh is an unbiased estimate for πsh. variance of π̂sh: the variance of π̂sh is obtained by taking vari40 adeniran et al. / j. nig. soc. phys. sci. 2 (2020) 36–50 41 ance of (27) as follows: v ar(π̂sh) = v ar [ noh nh ph ] = 1 (nh ph)2 v ar(noh) = nh phπsh[ ph(1 −πsh) + (1 − ph)] (nh ph)2 = phπsh(1 −πsh) nh ph + πsh(1 − ph) nh ph . hence, v ar(π̂sh) = πsh(1 −πsh) nh + πsh(1 − ph) nh ph . (28) since selections in different strata are made independently, the maximum likelihood estimate of πs is easily shown to be π̂stpropo = l∑ h=1 whπ̂sh = l∑ h=1 wh λ̂h ph = 1 n l∑ h=1 nh noh nh ph (29) where n and nh denote the number of subjects in the whole population and in the stratum h, respectively. wh (stratum weight) = nh n , for h = 1, 2, · · · , l so that l∑ h=1 wh = 1. 3.2.3. sampling distribution of the proposed stratified estimator π̂st propo theorem 3.3 (unbiasedness of π̂stpropo). the proposed stratified estimator is an unbiased estimator of population proportion of the sensitive attribute under study. proof 3.3. π̂stpropo is said to be unbiased of the population parameter πs if e(π̂stpropo) = πs. hence, we take expectation of (29) and provided that π̂sh is unbiased for πsh, e ( π̂stpropo ) = e  l∑ i=1 whπ̂sh  = l∑ h=1 wh e(π̂sh) = l∑ h=1 whπsh = πs.(30) theorem 3.4 (variance of π̂stpropo). the variance of π̂stpropo is v ar(π̂stpropo) = 1 n  l∑ h=1 wh { πsh(1 −πsh) + πsh(1 − ph) ph } 1 2  2 .(31) proof 3.4. this follows from taking the variance of (29) and from corollary 1 in section 5.9 of [28], since each unbiased estimator π̂sh has its own variance, the variance of π̂stpropo is v ar ( π̂stpropo ) = v ar  l∑ h=1 whπ̂sh  = l∑ h=1 w 2h v ar(π̂sh). (32) substituting (28) into (32), we have v ar(π̂stpropo) = l∑ h=1 w 2h nh [ πsh(1 −πsh) + πsh(1 − ph) ph ] .(33) recall, simple random sampling scheme asserts that v ar(ȳ) = (1 − f ) s 2y n . similarly, v ar(π̂sh) = (1 − fh) s 2πsh nh = ( 1 − nh n ) s 2πsh nh . (34) [28] established that when sampling with replacement, the sampling fraction is ignorable as n −→ n relative to nh, nh n −→ 0 and 1 − fh = 1 − nh n −→ 1. therefore, equation (34) reduces to v ar(π̂sh) = s 2πsh nh , (35) making s πsh subject of relation from equation (35) and substituting equation (28) into the resulting expression produces s πsh = [ πsh(1 −πsh) + (1 − ph)πsh ph ] 1 2 . (36) however, by optimum allocation the sample sizes are defined to minimize variance with a given cost. for fixed cost, by the cauchy-schwarz inequality the sample size nh to minimize v ar(π̂st propo) is given by nh = nwhs πsh l∑ h=1 whs πsh . (37) the optimal allocation of n to n1, n2, · · · , nl−1, nl to derive the minimum variance of π̂s subject to n = l∑ h=1 nh is obtained by substituting (36) into (37) to give nh = nwh [ πsh(1 −πsh) + πsh (1−ph ) ph ] 1 2 l∑ h=1 wh [ πsh(1 −πsh) + πsh (1−ph ) ph ] 1 2 . (38) the minimal variance of π̂stpropo is then obtained by substituting (38) into (33), so that v ar(π̂stpropo) = l∑ h=1 w 2h { πsh(1 −πsh) + πsh(1 − ph) ph } ÷  nwh { πsh(1 −πsh) + πsh (1−ph ) ph } 1 2 l∑ h=1 { πsh(1 −πsh) + πsh (1−ph ) ph } 1 2  (39) v ar(π̂stpropo) = l∑ h=1 w 2h { πsh(1 −πsh) + πsh(1 − ph) ph } × l∑ h=1 { πsh(1 −πsh) + πsh (1−ph ) ph } 1 2 nwh { πsh(1 −πsh) + πsh (1−ph ) ph } 1 2 . (40) further simplification of (40) yields v ar(π̂stpropo) = 1 n  l∑ h=1 wh { πsh(1 −πsh) + πsh(1 − ph) ph } 1 2  2 . the unbiased minimal variance of π̂stpropo follows on replacing n by (n − 1) in equation (31) to get 41 adeniran et al. / j. nig. soc. phys. sci. 2 (2020) 36–50 42 v̂ ar(π̂stpropo) = 1 n − 1  l∑ h=1 wh {πsh(1 −πsh) + πsh(1 − ph) ph } 1 2  2 . (41) although, if n is large, the difference between equation (31) and (41) is negligible. if ph = p for h = 1, 2, · · · , l, (31) reduces to v ar(π̂stpropo) = 1 n  l∑ h=1 wh {πsh(1 −πsh) + πsh(1 − p) p } 1 2  2 (42) to facilitate computation of v ar(π̂stpropo), let {πsh(1 −πsh) + πsh (1−ph ) ph } 1 2 be replaced by φsh in equation (31) so that v ar(π̂stpropo) = 1 n  l∑ h=1 whφsh  2 . (43) 3.3. efficiency comparison the efficiency of the proposed model with warner’s original model, boruch’s frrm and mangat’s improved two step procedure is judged by mean square error (mse) criterion. it should be noted that since the existing and the proposed estimators are unbiased, then the criterion for judging the performance of the proposed estimator is now limited to variance comparison. 3.3.1. efficiency comparison with warner’s (1965) original rrm in this study, we arbitrarily denote variance under warner design as v ar(π̂sw) and the proposed model as v ar(π̂spropo). recall from equations (3) and (22) v ar(π̂s warner) = πs(1 −πs) n + p(1 − p) n(2p − 1)2 and v ar(π̂s propo) = πs(1 −πs) n + (1 − p)πs np the proposed modified frrm is more efficient than warner’s rrm if v ar(π̂s warner) − v ar(π̂s propo) ≥ 0. that is,{ πs(1 −πs) n + p(1 − p) n(2p − 1)2 } − { πs(1 −πs) n + (1 − p)πs np } ≥ 0 this implies that p2 − (2p − 1)2πs ≥ 0 ∀p ∈ [0, 1] (44) the inequality (44) always holds. therefore, the proposed model is more efficient than [7] original model. 3.3.2. efficiency comparison with boruch (1971) original frrm recall from equation (5), the boruch’s variance estimator of πs which is subjectively denoted here as v ar(π̂s.boruch) is v ar(π̂s.boruch) = πs(1 −πs) n + (1 − p) np [πs(1 −πu) + πu(1 −πs) + (1 − p)πu(1 −πu) p ] . (45) the proposed mfrrm is more efficient than boruch’s frrm if v ar(π̂s.boruch) − v ar(π̂spropo) ≥ 0. that is, πs(1 −πs) n + (1 − p) np [πs(1 −πu) + πu(1 −πs) + (1 − p)πu(1 −πu) p ] − { πs(1 −πs) n + (1 − p)πs np } (46) which implies πs(1 −πu) + πu(1 −πs) + (1 − p)πu(1 −πu) p −πs and further simplification gives πs ≤ 1 −πu(1 − p) 2 p . (47) if the condition in (47) is satisfied then the proposed mffrm is more efficient than [12] original frrm. 3.3.3. efficiency comparison with mangat’s (1994) improved two-step procedure from equation (7), the variance of πs under mangat design is v ar(π̂s) = πs(1 −πs) n + (1 − p)(1 −πs) np . the proposed mfrrm is more efficient than mangat’s improved two-step procedure if v ar(π̂s mangat) − v ar(π̂s propo) ≥ 0. that is,{ πs(1 −πs) n + (1 − p)(1 −πs) np } − { πs(1 −πs) n + (1 − p)πs np } ≥ 0. after simplification, 1 − 2πs ≥ 0, =⇒ πs ≤ 1 2 (48) the inequality (48) holds ∀πs ≤ 1 2 . hence, the proposed mfrrm is more efficient than mangat’s improved two step procedure if and only if πs ≤ 1 2 . 42 adeniran et al. / j. nig. soc. phys. sci. 2 (2020) 36–50 43 3.3.4. efficiency comparison with kim and warde stratified model recall, variance of πs under kim and warde literally represented as v ar(π̂skw) and proposed stratified mfrrm are v ar(π̂skw) = 1 n  l∑ h=1 wh { πsh(1 −πsh) + ph(1 − ph) (2ph − 1)2 } 1 2  2 and v ar(π̂stpropo) = 1 n  l∑ h=1 wh { πsh(1 −πsh) + πsh(1 − ph) ph } 1 2  2 , respectively. the proposed stratified mfrrm is more efficient than the [16] stratified model if the relative efficiency (re) = v ar(π̂skw ) v ar(π̂stpropo ) ≥ 1. that is, v ar(π̂skw) − v ar(π̂stpropo) ≥ 0. using this condition, 1 n  l∑ h=1 wh { πsh(1 −πsh) + ph(1 − ph) (2ph − 1)2 } 1 2  2 − 1 n  l∑ h=1 wh { πsh(1 −πsh) + πsh(1 − ph) ph } 1 2  2 ≥ 0. (49) the above inequality is true if for each stratum h, h = 1, 2, · · · , l we have[ πsh(1 −πsh) + ph(1 − ph) (2ph − 1)2 ] 1 2 − [ πsh(1 −πsh) + πsh(1 − ph) ph ] 1 2 ≥ 0 which implies ph(1 − ph) (2ph − 1)2 − πsh(1 − ph) ph ≥ 0 multiplying the preceding inequality throughout by ph (2ph−1) 2 1−ph gives p2h−(2ph−1) 2πsh ≥ 0 ∀ ph ∈ (0, 1) and ∀ πsh ∈ (0, 1)(50) the lhs of (50) is always non-negative, hence the proposed stratified mfrrm is more efficient than [16] stratified randomized response model. 3.3.5. efficiency comparison with usman and oshungade mixed stratified rrm [17] invented mixed stratified rrm for hiv sero-prevalence survey with v ar(π̂s) as v ar(π̂s) = 1 n  l∑ h=1 wh { πsh(1 −πsh) + (1 − ph)(1 −πsh) ph } 1 2  2 . the proposed stratified modified frrm is more efficient than the [17] mixed stratified rrm if the relative efficiency (re) = v ar(π̂s usman ) v ar(π̂stpropo ) ≥ 1. that is, v ar(π̂s usman)−v ar(π̂stpropo) ≥ 0. therefore, 1 n  l∑ h=1 wh { πsh(1 −πsh) + (1 − ph)(1 −πsh) ph } 1 2  2 − 1 n  l∑ h=1 wh { πsh(1 −πsh) + πsh(1 − ph) ph } 1 2  2 ≥ 0(51) the inequality (51) holds if for each stratum h (h = 1, 2, · · · , l) we have { πsh(1 −πsh) + (1 − ph)(1 −πsh) ph } 1 2 − { πsh(1 −πsh) + πsh(1 − ph) ph } 1 2 ≥ 0 (52) which reduces to (1 − ph)(1 −πsh) ph − πsh(1 − ph) ph ≥ 0. (53) multiplying the above inequality (53) through by ph1−ph yields 1 − 2πsh ≥ 0. (54) the inequality (54) holds ∀ πsh ≤ 1 2 . that is, the proposed model is more efficient than the [17] mixed stratified rr model if πsh ≤ 1 2 for h = 1, 2, · · · , l. 3.3.6. cost and efficiency of stratification it is essential to think about more than two or three strata cases in terms of efficiency. [28] showed that the variance for the mean of a stratified random sample decreases as the number of strata increases. so, this section explore the behaviour of v ar(π̂stpropo) as the number of strata increases. suppose l strata of equal sizes are created such that wh = 1 l , substituting wh = 1 l in equation (31) produces v ar(π̂stpropo) = 1 nl2  l∑ h=1 { πsh(1 −πsh) + πsh(1 − ph) ph } 1 2  2 .(55) let f (l) = 1l2 [ l∑ h=1 { πsh(1 −πsh) + πsh (1−ph ) ph } 1 2 ]2 where l is a positive integer, we want to show that f (l) − f (l + 1) ≥ 0 for l(πsh, ph) = { πsh(1 −πsh) + πsh (1−ph ) ph } 1 2 . f (l) − f (l + 1) = 1 l2  l∑ h=1 l(π̂sh, ph)  2 − 1 (l + 1)2 l+1∑ h=1 l(π̂sh, ph)  2 =   1l l∑ h=1 l(π̂sh, ph)   2 −   1l + 1 l+1∑ h=1 l(π̂sh, ph)   2 43 adeniran et al. / j. nig. soc. phys. sci. 2 (2020) 36–50 44 =  1l  l∑ h=1 l(π̂sh, ph)  + 1l + 1 l+1∑ h=1 l(π̂sh, ph)  × 1l  l∑ h=1 l(π̂sh, ph)  − 1l + 1 l+1∑ h=1 l(π̂sh, ph)   . as the number of strata increases, it may be possible to divide a heterogeneous population into sub-populations, each of which is more homogeneous. so we may get 1l  l∑ h=1 l(π̂sh, ph)  − 1l + 1 l+1∑ h=1 l(π̂sh, ph)   ≥ 0. (56) by this assumption, f (l) is a monotone decreasing function of l. therefore, the variance of the proposed estimator gets smaller as the number of strata increases. 3.4. bayesian estimation approach in making generalization, conclusion or prediction about unknown population parameter(s) the trend is to distinguish between classical method of estimating the population parameter(s) whereby inferences are based strictly on information obtained from a random sample and a bayesian one which utilizes prior subjective knowledge about the probability distribution in conjunction with the information provided by the sample data. [29] and [30] estimated warner’s rrm using bayesian approach with binomial likelihood and beta prior as π̂bw = (a+no ) (a+b+n) − (1 − p) (2p − 1) (57) with posterior variance defined as v ar(π̂bw) = nπs(1 −πs) (a + b + n)2 + np(1 − p) (2p − 1)2(a + b + n)2 . (58) similarly, the model for the proposed sensitive random variable x conditional on unknown parameter λ popularly called likelihood function is the density function f (x|λ) = l(λ) given as l(λ) = f (no|λ) = ( n no ) λno (1 −λ)n−no = ( n no ) ( pπs) no [ p(1 −πs) + (1 − p)] n−no. (59) the parameter λ = pπs is considered a random variable which has a distribution g(λ) called the prior distribution otherwise known as prior predictive which describes uncertainty about the parameter before data are observed. a more convenient family of densities for a proportion λ is the beta with kernel proportional to g(λ) ∝ λa−1(1 −λ)b−1, 0 < λ < 1. hence, g(λ) = 1 β(a, b) λa−1(1 −λ)b−1 = 1 β(a, b) (pπs) a−1[p(1 −πs) + (1 − p)] b−1, 0 < λ < 1 (60) where the hyper-parameters a and b are chosen to reflect the user’s prior beliefs about λ. our goal is to start with this prior information and update it using the data to make the best possible estimator of λ. multiplying this beta prior with the likelihood function gives the joint density function h(no,λ) as h(no,λ) = f (no|λ)g(λ) = ( n no ) β(a, b) ( pπs) a+no−1[ p(1 −πs) + (1 − p)] b+n−no−1.(61) the marginal distribution (m(no)) can be obtained by integrating out parameter λ from the joint distribution as m(no) = ∫ � h(no,λ)dλ = ∫ � f (no|λ)g(λ)dλ = ∫ 1 0 ( n no ) β(a, b) ( pπs) a+no−1[ p(1 −πs) + (1 − p)] b+n−no−1d( pπs) = ∫ 1 0 ( n no ) β(a, b) ( pπs) a+no−1(1 − pπs) b+n−no−1d( pπs). therefore, m(no) = ( n no ) β(a, b) β(a + no, b + n − no). (62) combining this beta prior with the likelihood function, one can find f (λ|no), called the posterior distribution/density for λ given x = no as f (λ|no) = f (no|λ)g(λ)∫ � f (no|λ)g(λ)dλ = h(no,λ) m(no) = ( n no ) β(a, b) ( pπs) a+no−1[ p(1 −πs) + (1 − p)] b+n−no−1 × β(a, b)( n no ) β(a + no, b + n − no) which gives f (λ|no) = ( pπs)a+no−1(1 − pπs)b+n−no−1 β(a + no, b + n − no) = (λ)a+no−1(1 −λ)b+n−no−1 β(a + no, b + n − no) . (63) like g(λ), f (λ|no) also follows beta-distribution with updated parameters a + no and b + n − no. this is an example of a conjugate analysis where the prior and posterior densities have the same functional form. the prior g(λ) is said to be a conjugate with respect to f (no|λ) [31, 32]. the bayes estimator of h(λ) under the squared error loss is a ratio of integrals ĥλ = ∫ � h(λ)π(λ|no)dλ = ∫ � h(λ) f (no|λ)g(λ)dλ∫ � f (no|λ)g(λ)dλ (64) 44 adeniran et al. / j. nig. soc. phys. sci. 2 (2020) 36–50 45 following from (64) above, the posterior mean λ̂ is λ̂ = ∫ 1 0 λ f (λ|no)dλ = ∫ 1 0 λ λ(a+no )−1(1 −λ)(b+n−no )−1 β [(a + no)(b + n − no)] dλ p̂πs = 1 β[(a + no)(b + n − no)] ∫ 1 0 λ(a+no +1)−1(1 −λ)(b+n−no )−1dλ = 1 β[(a + no)(b + n − no)] β[(a + no + 1)(b + n − no)] = γ(a + b + n) γ(a + no)γ(b + n − no) × γ(a + no + 1)γ(b + n − no) γ(a + b + n + 1) by definition, γ(α) = (α− 1)!. so, the up equation yields π̂s = a + no p(a + b + n) . (65) the variance of bayesian estimator of the proposed rrm follows by taking the variance of (65) as v ar(π̂s) = v ar [ a + no p(a + b + n) ] = 1 [ p(a + b + n)]2 npπs[ p(1 −πs) + (1 − p)]. (66) after some algebraic operations, equation (66) gives v̂ ar(π̂s) = nπs(1 −πs) (a + b + n)2 + n(1 − p)πs p(a + b + n)2 . (67) for large n, the influence of a and b in equations (65) and (67) become negligible because the weight na+b+n → 1. consequently, the posterior mean and variance are getting closer to classical (maximum likelihood) estimators in equations (20) and (23), respectively. 4. results and discussion this section presents analysis of the simulated data of different sample sizes that are binomially distributed (with parameters (n, p)) to validate the developed model. [33] established that the best average value of p across rr studies is 0.7. however, this study considered values of p within the range [0.6, 0.8] at equal step of 0.1. the proposed model was estimated and variance comparison was done to examine its performance with some other existing models using [34](r-software version 3.6.0) and ”learnbayes” package in r-software authored by [35]. the tables and figures showing the results are presented below and the results discussion follows: 4.1. discussion in terms of minimum variance, column (2) and column (6) of tables 4, 5 and 6 show that the proposed rr model is more efficient and precise than [7] original rrm. in a situation where πu , 0 and πs in the neighbourhood of 0.6 and below, columns 4 and 6 of tables 4, 5 and 6 reveal that the proposed mfrrm is more efficient than [12] original frrm. in addition, table 7 depicts that if πu = 0, [12] and the proposed model produce same results not only for v ar(π̂s) but also π̂s across different sample sizes regardless of pre-assigned randomization parameter ( p). mangat’s procedure is more efficient but less effective when compare with the earlier similar estimators (models). the proposed model is more protective than [15] rrm. moreover, the results demonstrate that it is more efficient given the following condition: when πs assume values ≤ 1 2 . this condition is already established in equation (48). a more detailed insight of the results is given in columns 5 and 6 of tables 4, 5 and 6 that showed conditional efficiency of the two designs. when πs > 1 2 , mangat is more efficient (see table 6). like [11] model, especially if the sensitive variable under study is highly rampant, the proposed estimator has tendency to produce estimate whose value is greater than unity. tables 1, 2 and 3 confirmed this possibility. second column of table 9 also show that kim and warde warner’s stratified rrm has the same property of producing proportion greater than unity. furthermore, bayesian estimation of both warner’s and the proposed mfrrm provide relatively more precise estimators than their classical (maximum likelihood) estimators, conditional on the sample size (n) and total of yes-responses (no). but in large samples, bayesian estimator asymptotically approaching the classical estimators (refer to columns 3 and 4, 7 and 8 of table 1 through 3; columns 2 and 3, 6 and 7 of table 4 through 6 for details). to update the user’s prior knowledge about πs, the study considers posterior of immediate sample size as a new prior in the subsequent sampling to generate a new posterior which produce figure 3. figure 3 above reveals that the posterior density compromises between the initial prior beliefs and the information in the data. again, using information from table 8, the numerical illustration shows that the proposed stratified mfrrm is more efficient than kim and warde warner’s stratified rrm as v ar(π̂stpropo) = 0.0007313622 < v ar(π̂skw) = 0.002960187 (compare column 4 of tables 10 and 14). in addition, the proposed model is less efficient when compared with mixed stratified rrm of [17] (see column 4 of tables 12 and 14). both warner’s stratified and mixed stratified rrms have tendency to produce negative estimate of π̂sh which has a seeming distance from reality. the possibility of negative stratum proportion is well known from their theoretical results particularly when λ̂h + ph < 1 (see first row, second column of table 11). theoretically and empirically, the proposed stratified rrm can not produce negative π̂sh (refer to equation (27) and table 13 for confirmation). 5. concluding remarks this study proposed a modified version of frrm for designing surveys that will elicit response to sensitive issues from sample units. although extant models such as [7] original rrm, [12] original frrm, and [15] frrm among others have focused on improving response in rr surveys exists in the literature. however, designs of some of these models are not 45 adeniran et al. / j. nig. soc. phys. sci. 2 (2020) 36–50 46 table 1: estimate of πs for the proposed model and some existing models when p = 0.6, πu = 0.75, a = 10.2 and b = 27.4 n no warner rrm boruch mangat proposed mfrrm π̂mle π̂bayesian π̂mle π̂mle π̂mle π̂bayesian 200 118 0.950000 0.6978114 0.4833333 0.3166667 0.9833333 0.8992705 400 244 1.050000 0.9044790 0.5166667 0.3500000 1.0166667 0.9681597 600 362 1.016667 0.9187578 0.5055556 0.3388889 1.0055556 0.9729193 800 476 0.975000 0.9023400 0.4916667 0.3250000 0.9916667 0.9674467 1000 591 0.955000 0.8970702 0.4850000 0.3183333 0.9850000 0.9656901 10000 5949 0.974500 0.9684387 0.4915000 0.3248333 0.9915000 0.9894796 100000 60303 1.015150 1.0145265 0.5050500 0.3383833 1.0050500 1.0048422 1000000 600418 1.002090 1.0020281 0.5006967 0.3340300 1.0006967 1.0006760 table 2: estimate of πs for the proposed model and some existing models when p = 0.7, πu = 0.75, a = 10.2 and b = 27.4 n no warner rrm boruch mangat proposed mfrrm π̂mle π̂bayesian π̂mle π̂mle π̂mle π̂bayesian 200 135 0.9375000 0.7777778 0.6428571 0.5357143 0.9642857 0.8730159 400 275 0.9687500 0.8793419 0.6607143 0.5535714 0.9821429 0.9310525 600 416 0.9833333 0.9211104 0.6690476 0.5619048 0.9904762 0.9549202 800 557 0.9906250 0.9429322 0.6732143 0.5660714 0.9946429 0.9673898 1000 699 0.9975000 0.9587510 0.6771429 0.5700000 0.9985714 0.9764291 10000 6958 0.9895000 0.9855244 0.6725714 0.5654286 0.9940000 0.9917282 100000 70104 1.0026000 1.0021962 0.6800571 0.5729143 1.0014857 1.0012550 1000000 700007 1.0000175 0.9999772 0.6785814 0.5714386 1.0000100 0.9999870 table 3: estimate of πs for the proposed model and some existing models when p = 0.8, πu = 0.75, a = 10.2 and b = 27.4 n no warner rrm boruch mangat proposed mfrrm π̂mle π̂bayesian π̂mle π̂mle π̂mle π̂bayesian 200 151 0.9250000 0.7974186 0.7562500 0.6937500 0.9437500 0.8480640 400 308 0.9500000 0.8785801 0.7750000 0.7125000 0.9625000 0.9089351 600 470 0.9722222 0.9218946 0.7916667 0.7291667 0.9791667 0.9414210 800 624 0.9666667 0.9286055 0.7875000 0.7250000 0.9750000 0.9464542 1000 785 0.9750000 0.9439733 0.7937500 0.7312500 0.9812500 0.9579800 10000 7977 0.9961667 0.9928801 0.8096250 0.7471250 0.9971250 0.9946601 100000 80061 1.0010167 1.0006851 0.8132625 0.7507625 1.0007625 1.0005138 1000000 800137 1.0002283 1.0001952 0.8126712 0.7501713 1.0001712 1.0001464 table 4: performance comparison of the proposed model with some existing models when p = 0.7, πs = 0.1, πu = 0.75, a = 10.2 and b = 27.4 n warner rrm boruch mangat proposed mfrrm v (π̂mle) v (π̂bayesian) v (π̂mle) v (π̂mle) v (π̂mle) v (π̂bayesian) 200 0.0070125000 4.968668e-03 2.122194e-03 2.378571e-03 6.642857e-04 4.706760e-04 400 0.0035062500 2.929599e-03 1.061097e-03 1.189286e-03 3.321429e-04 2.775174e-04 600 0.0023375000 2.069939e-03 7.073980e-04 7.928571e-04 2.214286e-04 1.960828e-04 800 0.0017531250 1.599262e-03 5.305485e-04 5.946429e-04 1.660714e-04 1.514961e-04 1000 0.0014025000 1.302696e-03 4.244388e-04 4.757143e-04 1.328571e-04 1.234028e-04 10000 0.0001402500 1.392012e-04 4.244388e-05 4.757143e-05 1.328571e-05 1.318637e-05 100000 0.0000140250 1.401446e-05 4.244388e-06 4.757143e-06 1.328571e-06 1.327573e-06 1000000 0.0000014025 1.402395e-06 4.244388e-07 4.757143e-07 1.328571e-07 1.328472e-07 effective and their estimators are not efficient. based on the simulated results and other evidence from variance comparison, the proposed mfrrm outperformed some of these models. in terms of efficiency, the proposed model is preferred to [7] rrm as it portents or exhibits minimal variance. in comparison with [12] frrm, the proposed mffrrm is more efficient and also more flexible as it does not requires prior knowledge of πu unlike boruch’s original frrm. to some extent, the proposed model is more efficient than [15] rrm. its practicability trascends mangat’s as it is wholly stochastic in procedure thus 46 adeniran et al. / j. nig. soc. phys. sci. 2 (2020) 36–50 47 table 5: performance comparison of the proposed model with some existing models when p = 0.7, πs = 0.4, πu = 0.75, a = 10.2 and b = 27.4 n warner rrm boruch mangat proposed mfrrm v (π̂mle) v (π̂bayesian) v (π̂mle) v (π̂mle) v (π̂mle) v (π̂bayesian) 200 0.0077625000 5.500077e-03 2.550765e-03 2.485714e-03 2.057143e-03 1.457577e-03 400 0.0038812500 3.242926e-03 1.275383e-03 1.242857e-03 1.028571e-03 8.594088e-04 600 0.0025875000 2.291323e-03 8.502551e-04 8.285714e-04 6.857143e-04 6.072242e-04 800 0.0019406250 1.770306e-03 6.376913e-04 6.214286e-04 5.142857e-04 4.691493e-04 1000 0.0015525000 1.442021e-03 5.101531e-04 4.971429e-04 4.114286e-04 3.821506e-04 10000 0.0001552500 1.540891e-04 5.101531e-05 4.971429e-05 4.114286e-05 4.083520e-05 100000 0.0000155250 1.551333e-05 5.101531e-06 4.971429e-06 4.114286e-06 4.111194e-06 1000000 0.0000015525 1.552383e-06 5.101531e-07 4.971429e-07 4.114286e-07 4.113976e-07 table 6: performance comparison of the proposed model with some existing models when p = 0.7, πs = 0.6, πu = 0.75, a = 10.2 and b = 27.4 n warner rrm boruch mangat proposed mfrrm v (π̂mle) v (π̂bayesian) v (π̂mle) v (π̂mle) v (π̂mle) v (π̂bayesian) 200 0.0077625000 5.500077e-03 2.501020e-03 2.057143e-03 2.485714e-03 1.761239e-03 400 0.0038812500 3.242926e-03 1.250510e-03 1.028571e-03 1.242857e-03 1.038452e-03 600 0.0025875000 2.291323e-03 8.336735e-04 6.857143e-04 8.285714e-04 7.337293e-04 800 0.0019406250 1.770306e-03 6.252551e-04 5.142857e-04 6.214286e-04 5.668888e-04 1000 0.0015525000 1.442021e-03 5.002041e-04 4.114286e-04 4.971429e-04 4.617653e-04 10000 0.0001552500 1.540891e-04 5.002041e-05 4.114286e-05 4.971429e-05 4.934253e-05 100000 0.0000155250 1.551333e-05 5.002041e-06 4.114286e-06 4.971429e-06 4.967692e-06 1000000 0.0000015525 1.552383e-06 5.002041e-07 4.114286e-07 4.971429e-07 4.971055e-07 table 7: comparison of original frrm and the proposed mfrrm when p = 0.7 and πu = 0 boruch original frrd proposed mfrrd n no π̂mle v (π̂mle) π̂mle v (π̂mle) 200 135 0.9642857 2.514286e-03 0.9642857 2.514286e-03 400 275 0.9821429 1.257143e-03 0.9821429 1.257143e-03 600 416 0.9904762 8.380952e-04 0.9904762 8.380952e-04 800 557 0.9946429 6.285714e-04 0.9946429 6.285714e-04 1000 699 0.9985714 5.028571e-04 0.9985714 5.028571e-04 10000 6958 0.9940000 5.028571e-05 0.9940000 5.028571e-05 100000 70104 1.0014857 5.028571e-06 1.0014857 5.028571e-06 1000000 700007 1.0000100 5.028571e-07 1.0000100 5.028571e-07 table 8: samples and strata sizes strata nh nh ph noh wh λ̂h 1 876 69 0.4 27 0.08981852 0.3913043 2 2412 118 0.6 51 0.24730852 0.4322034 3 3012 279 0.7 115 0.30882805 0.4121864 4 3453 288 0.8 102 0.35404491 0.3541667 total 9753 754 1.00 table 9: computational procedure of kim and warde warner’s stratified rrm strata π̂sh whπ̂sh φ̂sh whφsh 1 1.0434783 0.09372367 2.440211 0.2191762 2 0.1610169 0.03982086 2.476911 0.6125613 3 0.2804659 0.08661575 1.230571 0.3800348 4 0.2569444 0.09096987 0.797100 0.2822092 total 0.3111302 1.493982 47 adeniran et al. / j. nig. soc. phys. sci. 2 (2020) 36–50 48 0.0 0.2 0.4 0.6 0.8 1.0 0 2 4 6 8 1 0 1 2 p d e n si ty prior likelihood posterior 0.0 0.2 0.4 0.6 0.8 1.0 0 5 1 0 1 5 p d e n si ty prior likelihood posterior figure 3: the prior, the likelihood and the posterior for n = 200 and n = 400 table 10: summary of kim and warde warner’s stratified rrm n n π̂s v ar(π̂s) 95% ci 9753 754 0.3111302 0.002960187 0.2044932, 0.4177671 table 11: computational procedure of mixed stratified rrm strata π̂sh whπ̂sh φ̂sh whφsh 1 -0.40831758 -0.03667448 1.2399337 0.1113690 2 0.04639471 0.01147381 0.8246085 0.2039327 3 0.13211914 0.04080210 0.6975762 0.2154311 4 0.13650174 0.04832774 0.5777054 0.2045337 total 0.06392917 0.7352665 table 12: summary of mixed stratified rrm n n π̂s v ar(π̂s) 95% ci 9753 754 0.06392917 0.0007169984 0.01144755, 0.1164108 table 13: computational procedure of the proposed stratified mfrrm strata π̂sh whπ̂sh φ̂sh whφsh 1 0.9782609 0.08786594 1.2201057 0.1095881 2 0.7203390 0.17814597 0.8256372 0.2041871 3 0.5888377 0.18184960 0.7031834 0.2171628 4 0.4427083 0.15673863 0.5978250 0.2116569 total 0.6046001 0.7425948 table 14: summary of proposed stratified mfrrm n n π̂stpropo v ar(π̂stpropo) 95% ci 9753 754 0.6046001 0.0007313622 0.5515954, 0.6576048 avoiding any posiible identification of individual status. the motivation for adopting existing rrt in sensitive related survey is protection of respondents’ privacy, and for the fact that the proposed mfrrm is more efficient, flexible and practical over some of the leading rrt, the study suggest its adoption. the proposed model argues that majority of people have a wrong intuition of the concepts of probabilities. this ignorance was 48 adeniran et al. / j. nig. soc. phys. sci. 2 (2020) 36–50 49 used to the advantage of the survey under consideration by making the subjective privacy protection larger than the true statistics privacy protection. though, the proposed design is still in accordance with the principle of rrt, since the identity of the respondent is not 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[38] h. zawar & k. khushnoor, ”on estimation of sensitive mean using scrambled data”, world applied sciences journal, 23 (2013) 1201. 50 j. nig. soc. phys. sci. 3 (2021) 423–428 journal of the nigerian society of physical sciences assessment of radiation shielding properties of polymer-lead (ii) oxide composites m. a. salawua,∗, j. a. gbolahana, a. b. alabia adepartment of physics, university of ilorin, ilorin, nigeria abstract long term exposure to very high levels of radiations from medical diagnostic centres, industries, nuclear research establishments and nuclear weapon development have resulted in health effects such as cancer and acute radiation syndrome, hence the need for proper radiation shielding. this paper investigated epoxy-lead (ii) oxide (pbo) composite as radiation shielding. the composites were prepared by dispersion of microsized pbo particles into polymeric materials using effective melt-mixing method and cast in a 4 cm by 6 cm rectangular aluminium mold with a thickness of 5 mm and was allowed to set over night at room temperature. the gamma ray attenuation ability of the composites were studied using gamma ray transmission or attenuation coefficient determination for the gamma ray energy. three gamma ray sources ba-133, cs-137 and co-60 were employed. the density, linear attenuation coefficient, half value layer (hvl), relaxation length and heaviness of the samples were determined. the measured values of linear attenuation coefficient increased with increasing filler concentration in all the samples at all gamma ray energies. it was also noticed that 40 % and 50 % filler samples attenuates more relative to the other samples under study. the maximum linear attenuation attained was found at energy of 662 kev. the composites have been found to possessed medical gamma-ray attenuation characteristics among the sample materials over a certain photon energy range (0.08 mev–1.332 mev) and found useful as a biological radiation shielding against gamma rays. doi:10.46481/jnsps.2021.249 keywords: pbo, pani, hvl, epoxy, attenuation article history : received: 12 june 2021 received in revised form: 17 august 2021 accepted for publication: 18 august 2021 published: 29 november 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: e. etim 1. introduction electromagnetic radiation like gamma beams and x-beams are risky to human wellbeing. individuals get exposed to gamma and x-radiations from various kinds of sources including industries, medical diagnostic centres, nuclear research establishments, nuclear reactors, research involving radio-isotopes and nuclear weapon development facilities. ∗corresponding author tel. no: +2348030788553 email address: salawu.ma@unilorin.edu.ng, abideen2004@gmail.com (m. a. salawu ) to shield work force and delicate electronic gear from such ionizing radiation, protection is important. radiation protection is the act of shielding individuals and the environment from unsafe impact of ionizing radiation. radiation exposure is one of the significant concerns when setting up thermal energy stations, the utilization of solid (high action) poly aniline (pani) discover application in strong erosion safe materials, electrical apparatuses, electrolytes for batteries and electromagnetic protecting materials. lead is the most well-known protecting material used to 423 m. a. salawu et al. / j. nig. soc. phys. sci. 3 (2021) 423–428 424 shield against gamma beam and x-beams. be that as it may, the presence of a quickly changing attractive field in these radiations prompts whirlpool current in a conductor. thus, the utilization of standard metallic lead gets unsatisfactory. accordingly, it is attractive that the protecting material ought to be a non-conductor. this necessity is very much fulfilled by utilizing different mixes of metal powders in a polymer grid. composites are materials that contain solid burden conveying material (known as reinforcement) imbedded in a more vulnerable material known as matrix [1]. polymeric composites containing different measures of inorganic added substances have been widely read by a few specialists to contemplate the protecting properties toward gamma and x-radiation [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. the added substances might be any of the fitting high-electron thickness materials, for example, lead oxides, lead silicate, lead sulfide, and so forth, scattered in a plastic matrix or materials containing lead acrylate. lead material is dense and has radiation that can be employed against different forms of highenergy applications of radiation such as gamma rays, x-rays, and other types of nuclear radiation. in this study, the gamma ray attenuation ability of the fabricated composites was studied by means of gamma ray transmission or attenuation. coefficient determination for the gamma ray energy. this was performed by using three gamma ray sources which are ba-133, cs-137 and co-60. the sample preparation was based upon the dispersion of micro-sized particles into polymeric materials by an accurate and effective melt-mixing method which was becoming increasingly important so that a uniform mixture could be produced. 2. materials and methods epoxy resin (bisphenol-a-diglycidyl ether poly), hardener (isophoromediamine) and filler (lead (ii) oxide powder) of commercial grade were used with the modification of the method reported by [24]. the sample preparation was based upon the dispersion of particles into polymeric materials by melt-mixing methods. micron-sized pbo was added into the epoxy resin before the hardener was mixed into it. the ratio of epoxy resin to hardener used was 2:1. the mixing of pbo powder in epoxy resin was stirred thoroughly using a two-roll rheomixer device and a rotor speed of 60 revolution per minutes (rpm) for 10 minutes to ensure fairly uniform dispersion of the powder in epoxy matrix. the well-mixed mixture was then cast in a 4 cm by 6 cm rectangular aluminium mold with a thickness of 5 mm and was allowed to set over night at room temperature. the density, linear attenuation coefficient, half value layer (hvl), relaxation length and heaviness of the samples were determined. the apparent density of prepared samples was measured using: ρ = m v , (1) where m and v are the mass and volume of each sample, respectively. the measured densities were compared with the theoretical values, (with an assumption that the samples were free from voids) which are determined using: ρc = 100[ e ρ f − e ρe ] (2) where e is the weight % of epoxy, ρ f is the density of filler, and ρe is the density of epoxy. gamma ray transmission through the prepared samples was studied using beam of well collimated point source from co60, cs-137, ba-133. the initial intensity (io) of the generated gamma-rays was determined. the distance between the source and the detector was set to be 6.6 cm. then the transmitted gamma-ray beam (i) was measured with the sample placed in front of the detector. for each composite sample, the measurements were performed three times. the linear attenuation coefficient for each sample was then calculated using: µ = ln i0i x (3) where x is the thickness of the sample, i0 is initial intensity of the gamma ray, i is the transmitted intensity with shielding. figure 1 shows the schematic diagram of the set up. figure 1: schematic setup of experiment. the mass attenuation coefficient was determined using: µm = µ ρc (4) where µ is linear attenuation and ρc is theoretical density. the half value layer (hvl), relaxation length and % of heaviness were respectively determined using equations (5), (6) and (7), respectively. hv l = 0.693 µ , (5) relaxation length λ = 1 µ (6) 424 m. a. salawu et al. / j. nig. soc. phys. sci. 3 (2021) 423–428 425 where µ is the linear attenuation of the absorber. with reference to lead, the % of heaviness of other conventional shielding materials along with epoxy + pbo composites was evaluated using equation (7). % of heaviness = density of material density of lead × 100 (7) 3. results and discussion figure 2: density against the percentage weight of the sample the density of the samples increases as the percentage weight of the samples increases. the densities ranges from 0.55 g/cm3 to 1.1 g/cm3 from 0 to 50 percentage weight. table 1 and figure 3 illustrate the relationship between linear attenuation coefficient and weight percent of the filler of the composites for ba-133, cs-137 and co-60 gamma sources. the measured values of linear attenuation coefficient increased with increasing filler concentration in all the samples at all gamma ray energies and is in good agreement with the reported works of [24, 25]. the linear attenuation coefficient ranges between 1.0773 to 2.9833 for ba-133, 4.9945 to 7.2121 for cs-137 and 0.3942 to 0.9486 for co-60 gamma sources. figure 4 shows the linear attenuation coefficient against energy for the samples under study. it was noticed that 40 % and 50 % filler samples attenuate more than the rest of the samples. the maximum attenuation attained was found to be at table 1: linear attenuation coefficient (µ) of the samples for each gamma source. s/n filler % in sample linear attenuation (µ) ba-133 cs-137 co-60 1 0 1.0773 4.9945 0.3942 2 10 2.0877 5.1310 0.5467 3 20 2.1472 5.2417 0.6125 4 30 2.2455 5.4956 0.7190 5 40 2.8388 7.1113 0.8981 6 50 2.9833 7.2121 0.9486 figure 3: linear attenuation coefficient against % filler in the sample figure 4: linear attenuation coefficient (cm−1) against energy (kev) energy 662 kev as shown in figure 4. the shielding materials attenuates single energy more than the double range energies sources. linear attenuation decreased at 80 kev to around 400 kev and optimum at 662 kev and decreases down as the energy increases to 1332 kev. this is due to the fact that linear attenuation was pronounced for single energy sources and less for bi-energy source. at 662 kev energy, linear attenuation coefficient values was noticed to have increased due to cs-137 being single energy source and having low activity of 0.25 µci. figure 5: half value layer against % weight 425 m. a. salawu et al. / j. nig. soc. phys. sci. 3 (2021) 423–428 426 figure 5 shows the half value layer against % weight for the samples under study. the half value layers (hvls) for the samples are seen to have small values for cs-137 and ba-133 gamma radiations relative to co-60. also the significant reduction in values of hvls for cs-137 may be due to it being single energy source. the hvls also decreases with increasing filler content. therefore, it requires a thickness of 0.731 cm, 0.096 cm, 0.23 cm of 50 % filler to reduce the radiation intensity coming from gamma sources co-60, cs-137, and ba-133 to 50 % respectively. figure 6: half value layer against energy (kev). figure 6 illustrates the relationship between hvl and energy. it was found that the hvls of the composite samples behaves differently at different energies of the gamma radiation. the hvl decreased significantly at 662 kev and has optimum value at 1200 kev for all the samples under study. the figure 7: relaxation length against the energy (kev) for different weight percent filler in the samples. relaxation length (λ) is calculated from linear attenuation coefficient (µ) of all the samples at an energy range of 80 kev–1332 kev and it changes with photon energy as shown in figure 7. the relaxation length (λ) of any particular radiation represents the average distance between two successive interactions. the less the relaxation length of a material at a particular energy, the better the shielding properties it possesses. mathematically, the relaxation length (λ) is equivalent to the reciprocal of linear attenuation coefficient (µ). it is observed that at 662 kev energy, the relaxation length is relatively low relative to other energy sources. this indicates that the composites will be a good shielding material at 662 kev. the low energy photons loses their energy in a short distance while high energy photons need a longer distance to lose their energy. figure 8 shows figure 8: relaxation length against % weight of the samples the relaxation length against % weight of the sample. the relaxation length decreased with increased in filler content of the samples. this is as a result of increase in densities of the samples thereby causing the distances between the particles in the samples to decrease. figure 9: relaxation length against % weight of the samples figure 9 shows the % of heaviness of each of the % weight composition of epoxy pbo. the results obtained proved that the polymer composites considered exhibits excellent lightness relative to conventional radiation shielding materials such as lead, barite, steal, concrete lead glass, whose % heaviness ranges between 54.85 and 100. at the same time, their performance as radiation shielding materials are appreciable particularly at higher concentrations and for low energy gamma ray sources. figure 10 shows the scanning electron microscope (sem) micrographs of pbo epoxy filled composites for (a) pure epoxy (b) 10 % filler (c) 20 % filler (d) 30 % filler (e) 40 % filler and (f) 50 % filler. the micrographs showed uniform distributions 426 m. a. salawu et al. / j. nig. soc. phys. sci. 3 (2021) 423–428 427 (a) (b) (c) (d) (e) (f) figure 10: scanning electron microscope (sem) micrographs of fabricated samples of the filler within the matrix (figure 10a to 10f). the bright regions represent the filler particles (pbo) dispersed in the dark epoxy matrix. the fillers were seen to be quite uniformly dispersed in the composites, although with minor agglomerations. 4. conclusion the measured values of linear attenuation coefficient increased with increasing filler concentration in all the samples at all gamma ray energies. the 40 % and 50 % filler samples attenuate more 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[25] vasiliy cherkasov, valeriy avdonin, yuriy yurkin & dmitrii suntsov, “prediction of radiation shielding properties of self adhesive elastic coating”, materials physics and mechanics 42 (2019) 825. 428 j. nig. soc. phys. sci. 2 (2020) 55–60 journal of the nigerian society of physical sciences original research energy distribution of an ion cloud in a quadrupole penning trap b. m. dyavappa∗ department of physics, government college for women, kolar, india. abstract ions are confined in penning trap by the combination of electric and magnetic fields, as the electric field confines ions in the axial direction through an electric potential minimum and the magnetic field applied along the axis of the trap confines the ions in the radial direction. in the high temperature limit coulomb interaction of ions can be neglected and the total energy is due to the electrostatic potential energy of the charge of ions and kinetic energy due to thermal energy. however, in the low temperature limit the trapping potential created by the dc voltage applied between the end cap and ring electrodes is cancelled by coulomb interaction of ions and the total energy is mainly kinetic energy of ions. the probability density of energy distribution of ions along axial direction, in radial plane and the total probability density of energy distribution due to resulting motion of both axial and radial motion of ions under high temperature and low temperature limits in a quadrupole penning trap are presented here. these results reveal the energy properties of ion cloud and are useful to carry out accurate measurement experiments on single stored particle, antiparticles with energy related parameters, under high temperature and low temperature limits in a quadrupole penning trap. keywords: quadrupole penning trap, probability of energy distribution function, probability density of energy distribution function article history : received: 25 january 2020 received in revised form: 09 march 2020 accepted for publication: 20 march 2020 published: 14 may 2020 c©2020 journal of the nigerian society of physical sciences. all rights reserved. communicated by: b. j. falaye 1. introduction penning traps are used for high precision mass spectrometry, time of flight detection of ion cyclotron and magnetron resonances and for cooling of antiprotons, spectral intensity distribution of fermions [1], quantum information processing [2] etc. a thorough knowledge of different properties of stored ions like stability parameters, energy distribution, potential depths, space charge effects, anharmonicities etc. in the trap are essential. the penning trap uses a time-independent, spatially homogeneous uniform magnetic field along the symmetry axis of the trap. the end-cap electrodes are biased with a positive dc voltage for the confinement of positive ions or negative dc voltage for the confinement of negative ions with respect to the ∗corresponding author tel. no: +919483113600 email address: dyavappabm@gmail.com (b. m. dyavappa) ring electrode; hence both electric and magnetic fields are superposed to trap ions, as the resultant of quadrupolar potential in a small region of space (see figure 1). the quadrupole penning trap is made with three-electrodes; two are end-cap electrodes and a ring electrode. the equation of ring electrode is r2 r20 − z2 z20 = +1 and those of two similar end-cap electrodes is r2 r20 − z2 z20 = −1, where r0 is the inner radius of the ring electrode in the radial plane and z0 is half of the vertical distance between the two end-cap electrodes designed such that r0 = √ 2 z0 . the trap potential is [3, 4] vt (r, z) = v0 r20 + 2z 2 0 ( 2z2 − r2 ) (1) the potential well due to the electric field in the axial direction 55 dyavappa / j. nig. soc. phys. sci. 2 (2020) 55–60 56 figure 1: quadrupole penning trap with cloud of ions confined in trap space. and radial plane is given by [5] vt (r, z) = 1 4π�0r [ qe − ( r λd )] + md2ω2z 4q (2) where λd = ( �0kbt/nq2 ) is debye length, r is the radial coordinate, n is density of particles, t is temperature and q is the charge of ion. the plasma parameter ( nd = 4πnλ3d/3 ) gives the average number of particles found in the debye sphere of plasma. penning traps are used in high precision mass spectrometry [6, 7] for cooling antiprotons for measuring their gravitational acceleration [8]. ions are confined first in a catching trap and then transferred to high precision trap to carry out measurements. time of flight technique is used to detect ion cyclotron and magnetron resonances. a thorough knowledge of different properties of ions including energy distribution along axial direction, in radial plane and the total energy distribution is necessary to understand these processes. 2. theory we assume that the ion cloud is in thermal equilibrium through coulomb interaction between ions. the axial component of angular momentum of the ions is conserved as the penning trap is symmetric along its axis in the absence of external torques. this causes the rotation of the ion cloud as a whole [9] and the rotation in presence of magnetic field provides radial confining force to balance the electrostatic force of expansion. therefore the rotation in magnetic field is equivalent to neutralization by opposite charge of ions and the distribution of magnetically confined ions in thermal equilibrium without rotation can be treated as ions confined and neutralized by a cylinder of opposite charge. the axial, pure cyclotron, reduced cyclotron and magnetron frequencies are given respectively by [10, 11, 12, 13] fz = 1 2π √ 4ev0 md2 , fc = eb 2πm , f ′ c = fc + √ f 2c − 2 f 2z 2 , fm = fc− √ f 2c − 2 f 2z 2 , ∴ fm < fφ < fc − fm. the probability of energy distribution function in thermal-equilibrium is [14] dp ( er,φ,z,pr,pφ,pz ) = 2π (πkbt ) 3 2 e 1 2 exp [ − ( e−ωφpφ kb t )] · drdφdzdpr dpφdpz (3) where ωφ is the rotational frequency of ion cloud as a whole determined by total angular momentum and energy determined by the temperature of the ion cloud. energy of single ion is [14] e = 12m [ p2r + ( pφ r − qbr 2c )2 + p2z ] + q [ vt (r, z) + vq (r, z) ] (4) the momentums in radial plane, azimuthal and axial directions are [14] pr = mvr = m dr dt , pφ = mr 2 dφ dt + qb 2c r2, = mrvφ − ωc 2 mr2, pz = m dz dt = mvz  (5) coulomb interaction potential of ions is [14] vq (r, z) = ∫ ∞ 0 nq ( r ′ , z ′ ) |r − r′ | r ′ dr ′ dφ ′ dz ′ (6) 3. energy distribution of an ion cloud at the high temperature limit the penning trap is designed with its dimension to be 20 mm and magnetic field applied is 6 t , when the dc voltage applied is 3000 v for an ion cloud density of 1014 m−3 with debye length 2.4 × 10−3 m , the energy and temperature of ions at high temperature limit considered are 10 ev , 116045.05 k respectively. in high temperature limit [14] qvq (r, z) � kbt � qvt (r, z) + m 2 ωφ ( ωc −ωφ ) r2 (7) if we neglect coulomb interaction vq (r, z) then the probability of energy distribution function is [14] dp ( er,φ,z,pr,pφ,pz ) = a exp [ − ( ez +e ′ r kb t )] · drdφdzdpr dpφdpz (8) the energy along z-direction is [14] ez = 1 2m p2z + kzz 2 (9) the energy in the radial plane is [14] er = 1 2m ( p2r + p2φ r2 ) + ( mω2c 8 + qv0 d2 ) r2 + ωc 2 pφ (10) 56 dyavappa / j. nig. soc. phys. sci. 2 (2020) 55–60 57 e ′ r = er −ωφpφ = 12m ( p2r + p2φ r2 ) + ( mω2c 8 + qv0 d2 ) r2 +( ωc 2 −ωφ ) pφ (11) the energy distribution in z-direction is [14] dp ( ez,pz ) = bex p [ − ( ez kbt )] dzdpz (12) the probability density of energy distribution in z-direction is [14] ρz (ez) = 1 kbt exp ( − ez kbt ) (13) the average energy in z-direction is [14] 〈ez〉 = ∫ ∞ 0 ezρz (ez) dez = kbt (14) the energy distribution in the radial direction is [14] dp ( er,φ,pr,pφ ) = cex p ( − e ′ r kbt ) drdφdpr dpφ (15) dp ( e ′ r ) = ( 1 kbt )2 exp ( − e ′ r kbt ) e ′ r de ′ r (16) the probability density of energy distribution in the high temperature limit with ωφ � ωc 2 is [14] ρr ( e ′ r ) = ( 1 kbt )2 exp ( − e ′ r kbt ) e ′ r (17) ρr (er ) = ( 1 kbt )2 exp ( − er kb t ) er{ ∵ er ≈ e ′ r } (18) the average energy in the radial plane is [14] 〈er〉 = ∫ ∞ 0 erρr (er ) der = 2kbt (19) the total probability of energy distribution is [14] dp (e) = ρz (ez) ρr (er ) dezder = ( 1 kbt )3 exp ( − e kb t ) er dezder (20) the total probability density of energy distribution is [14] ρ ( e kbt ) = 1 2 ( 1 kbt )3 exp ( − e kbt ) e2 (21) the total average energy is [14] 〈e〉 = ∫ ∞ 0 eρ (e) de = 〈ez〉 + 〈er〉 = kbt + 2kbt = 3kbt (22) the total energy is [14] e = ez + er (23) the probability density of energy distribution in axial direction increases sharply up to 24.3559 × 1016 at e = 1.3375kbt , decreases abruptly and remains almost a constant for high temperature limit as shown in figure 2(a). the probability density of energy distribution in the radial plane increases sharply up to 34.4544 × 1016 at e = 2.11487kbt , decreases abruptly and remains almost a constant for the high temperature limit as shown in the figure 2(b). the total probability density of energy distribution increases sharply up to 42.5566×1016 at e = 3.0728kbt , decreases abruptly and remains almost a constant for the high temperature limit as shown in the figure 2(c). the axial, radial and total probability densities of energy distribution increase sharply up to 24.3559×1016, 34.4544×1016, 42.5566×1016 at e = 1.3375kbt , 2.1149kbt , 3.0728 kbt respectively, decrease abruptly and remain almost constant as k ′ r r 2/kbt ≈ 5.4 × 102 and kbt/qvq (r, z) ≈ (λd/r) 2 ≈ 6 for the high temperature limit as shown in figure 2(d). the axial probability density of energy distribution is less than the radial probability density of energy distribution, which in turn less than the total probability density of energy distribution. this analysis helps to design trap that consists of appropriate structure trap electrodes to minimize anharmonicities to improve storage time. 4. energy distribution of an ion cloud at the low temperature limit in a penning trap of dimension 10 mm , magnetic field 2 t , when the dc voltage applied is 10 v for an ion cloud density 2× 1014 m−3 and debye length 10−5 m , the energy and temperature of ions at low temperature limit considered are 3.6 × 10−4 ev , 4.2 k respectively. in the low temperature limit [14] qvq (r, z) � kbt (24) qvt (r, z) + qvq (r, z) + m 2 ωφ ( ωc −ωφ ) r2 = 0 (25) if we neglect coulomb interaction potential vq (r, z) then probability of energy distribution distribution function is [14] dp ( epr,pφ,pz ) = a′ exp [ − ( ez +e ′ r kb t )] · dpr d ( pφ r ) dpz (26) the energy in z-direction is [14] ez = 1 2m p2z (27) the energy in the radial plane is [14] er = 1 2m p2r + p2φr2  + ( mω2c 8 ) r2 + ωc 2 pφ (28) e ′ r = er −ωφpφ = 1 2m ( p2r + p2φ r2 ) +( mω2c 8 ) r2 + ( ωc 2 −ωφ ) pφ (29) 57 dyavappa / j. nig. soc. phys. sci. 2 (2020) 55–60 58 table 1: the values of axial, radial and total probability density of energy distribution in the high temperature limit. e kb t e− e kb t t (k) kbt ( 10−19 j ) ρz ( ez kb t ) ( 1016 ) ρr ( er kb t ) ( 1016 ) ρ ( e kb t ) ( 1016 ) 0 1 ∞ ∞ 0 0 0 1 0.367879 115942 15.999996 22.9924 22.9924 11.4962 2 0.135335 57971 7.999998 16.9169 33.8338 33.8338 3 0.049787 38647.3 5.3333274 9.33507 28.0052 42.0078 4 0.018316 28985.5 3.999999 4.579 18.316 36.632 5 0.006738 23188.4 3.1999992 2.1056 10.528 26.32 6 0.00247875 19323.7 2.6666706 0.92953 5.57718 16.73154 7 0.00091188 16563.1 2.2857078 0.39895 2.79265 9.774275 8 0.00033546 14492.75 1.9999995 0.16773 1.34184 5.36736 9 0.00012341 12882.4 1.7777712 0.06942 0.62478 2.8115 10 0.0000454 11594.2 1.5999996 0.028375 0.28375 1.41875 table 2: the values of axial, radial and total probability density of energy distribution in the low temperature limit. e kb t e− e kb t t (k) kbt ( 10−23 j ) ρz ( ez kb t ) ( 1020 ) ρr ( er kbt ) ( 1020 ) ρ ( e kb t ) ( 1020 ) 0 1 ∞ ∞ 0 0 0 1 0.367879 4.2 5.796 35.8098 63.471187 71.6195545 2 0.135335 2.1 2.898 18.63038 46.699448 74.521537 3 0.049787 1.4 1.932 8.39408 25.7696687 50.3644996 4 0.018316 1.05 1.449 5.166848 12.640442 28.5264222 5 0.006738 0.84 1.1592 1.4666034 5.812693 14.666034 6 0.00247875 0.7 0.966 0.591024 2.565994 7.0922868 7 0.00091188 0.6 0.828 0.234846 1.101304 3.2878457 8 0.00033546 0.525 0.7245 0.09235968 0.4630228 1.4777549 9 0.00012341 0.4667 0.644 0.01325787 0.1916304 0.6486954 10 0.0000454 0.42 0.5796 0.013975 0.07832988 0.2795 the probability of energy distribution in z-direction is [14] dp ( epz ) = b′ exp [ − ( ez kbt )] dpz (30) the probability density of energy distribution in z-direction is [14] ρz ( ez kbt ) = √ 1 πkbt [ exp ( − ez kbt )] 1√ ez (31) the average energy in z-direction is [14] 〈ez〉 = ∫ ∞ 0 ezρz (ez) dez = 1 2 kbt (32) the probability of energy distribution in the radial plane is [14] dp ( epr,pφ ) = c′ exp ( − e ′ r kbt ) dpr dpφ (33) dp ( e ′ r kbt ) = ( 1 kbt ) exp ( − e ′ r kbt ) de ′ r (34) the probability density of energy distribution in radial plane in the low temperature limit with ωφ � ωc/2 is [14] ρr ( e ′ r kbt ) = ( 1 kbt ) exp ( − e ′ r kbt ) (35) ρr ( er kbt ) = ( 1 kb t ) exp ( − er kb t ) { ∵ er ≈ e ′ r } (36) the average energy in radial plane is [14] 〈er〉 = ∫ ∞ 0 erρr (er ) der = kbt (37) the total probability density of energy distribution is [14] ρ (e) = ( 4 π ) 1 2 ( 1 kbt ) 3 2 exp ( − e kbt ) √ e (38) the total average energy is [14] 〈e〉 = 〈ez〉 + 〈er〉 = 1 2 kbt + kbt = 3 2 kbt (39) the probability density of energy distribution in axial direction increases sharply up to 36.6166 × 1020 at e = 1.1466kb, decreases abruptly and remains almost a constant for the low temperature limit as shown in figure 3(a). the total probability density of energy distribution in the radial plane increases sharply up to 63.59 × 1020 at e = 0.99kbt , decreases abruptly and remains almost a constant for the low temperature limit as shown in the figure 3(b). the total probability density of energy distribution increases sharply up to 83.39 × 1020 at e = 1.674kbt , decreases abruptly and remains almost a constant 58 dyavappa / j. nig. soc. phys. sci. 2 (2020) 55–60 59 figure 2: (a) the probability density of energy distribution in axial direction in the high temperature limit; (b) the probability density of energy distribution in the radial plane in the high temperature limit; (c) the total probability density of energy distribution in the high temperature limit; (d) the axial, radial and total probability densities of energy distribution in the high temperature limit. figure 3: (a) the probability density of energy distribution in axial direction in the low temperature limit; (b) the probability density of energy distribution in the radial plane in the low temperature limit; (c) the total probability density of energy distribution in the low temperature limit; (d) the axial, radial and total probability density of energy distribution in the low temperature limit. 59 dyavappa / j. nig. soc. phys. sci. 2 (2020) 55–60 60 for the low temperature limit as shown in the figure 3(c). the axial, radial and total probability densities of energy distribution increase sharply up to 36.6166×1020, 63.59×1020, 83.39× 1020 at e = 1.1466kbt , 0.99kbt , 1.674kbt respectively, decrease abruptly, remain almost constant as λd/r � 1, λd/z � 1 for the low temperature limit as shown in the figure 3(d). the axial probability density of energy distribution is less than the radial probability density of energy distribution, which in turn less than the total probability density of energy distribution. the total probability densities of energy distributions increase sharply up to 42.5566 × 1016 , 83.39 × 1020 at e = 3.0728kbt, 1.674kbt decrease abruptly and remain almost constants under high and low temperature limits respectively as shown in the figure 4. 0 1 2 3 4 5 6 7 8 9 10 0 10 20 30 40 50 60 70 80 0 1 2 3 4 5 6 7 8 9 10 0 10 20 30 40 50 60 70 80 ρ(1016) : high temperature limit ρ(1020) : low temperature limit ρ ( e / k b t ) e / k b t figure 4: the total probability density of energy distribution when fφ � fc/2 in the high temperature and low temperature limits. 5. conclusion the probability density of energy distribution of ions along axial direction and in radial plane together results total probability density of energy distribution under high temperature limit and similarly at low temperature limit measured in a quadrupole penning trap. the total probability density of energy distribution in the low temperature limit is much sharper and greater than that at the high temperature limit. these results reveal the energy properties of an ion cloud and are useful to design and carry out experiments on stored single particle, antiparticles with energy related parameters, under high temperature and low temperature limits in quadrupole penning trap. acknowledgments we thank the referees for the positive enlightening comments and suggestions, which have greatly helped us in making improvements to this paper. the author also acknowledges dr. satyajith k. t., assistant professor of physics, amritha vishwa vidhya peetam, coimbatur, tamil nadu, india, for important suggestion on energy distribution function. references [1] s. k. ghosh “spectral intensity distribution of trapped fermions”, pramana journal of physics 85 (2015) 616. [2] dong et al. “a review of silicon microfabricated ion traps for quantum information processing, micro and nano systems”, letters (2015). [3] f. g. major, v. n. gheorghe & g. werth, “charged particle traps, physics and techniques of charged particle confinement”, springer (2005). [4] pradip k.ghosh ion traps, clarendon press, oxford (1995) 72. [5] u. shumacher “basics of plasma physics”, lect.notes phys. 670 (2005) 20. [6] c. gerz, d. wilsdorf & g. werth, “a high precision penning trap mass spectrometer”, nuclear instruments and methods in physics research section b: beam interactions with materials and atoms. z.phys. d 47 (1990) 453. [7] s. becker, g. bollen, f. kern, h. j. kluge, r. b. moore, g. savard, l. schweikhard, h. stolzenberg & isolde collaboration, “mass measurements of very high accuracy by time-of-flight ion cyclotron resonance of ions injected into a penning trap”, int. j. mass spectrom. ion processes 99 (1990) 53. [8] n. beverini, b. e. bonner & j. h. billen, “a measurement of the gravitational acceleration of the antiprotons”, los alamos national laboratory document la-ur-86-260 (1986). [9] t. m. o’neil & c. f. driscoll, “transport to thermal equilibrium of a pure electron plasma”, the physics of fluids 22 (1979) 266. [10] d. datar, b. m. dyavappa, b. l. mahesh, k. t. satyajith & s. ananthamurthy, “energy distribution of electrons under axial motion in a quadrupole penning trap”, can. j. phys 94 (2016) 1249. [11] b. m. dyavappa, “velocity distribution of electrons along the symmetry axis of quadrupole penning trap”, discovery 56 (2020) 141. [12] b. m. dyavappa, d. d. prakash & s. ananthamurthy “dependence of the confinement time of an electron plasma on the magnetic field in a quadrupole penning trap”, epj techniques and instrumentation 4 (2017) doi: 10.1140/epjti/s40485-017-0039-4. [13] b. m. dyavappa spectroscopy of non-neutral plasmas in ion traps ph.d. thesis submitted to bangalore university (2017). [14] g. z. li & g. werth, “energy distribution of ions in penning trap”, int. j. mass spectro. ion processes 121 (1992) 75. 60 j. nig. soc. phys. sci. 2 (2020) 106–114 journal of the nigerian society of physical sciences original research comparison of the solution of the van der pol equation using the modified adomian decomposition method and truncated taylor series method j. n. ndama,∗, o. adedireb adepartment of mathematics, university of jos, nigeria bfederal college of forestry, jos, nigeria abstract in this paper, we compare the solution of the van der pol equation obtained by using truncated taylor series method and modified adomian decomposition method with the solution obtained by the poincare-lindstedt (p-l) method. the approximating 4-component modified adomian decomposition method behaves more like approximate p-l analytic method than tenth-order taylor series. also, with addition of just one term, the approximating 5-component modified adomian decomposition method produces more convergent solution to that of p-l analytic method than the twenty second-order taylor series approximation as the independent variable t representing time progressively increases. a general comparison of the two solutions revealed that the absolute errors generated by the approximating polynomial from taylor series are greater than the ones generated from modified adomian decomposition method. it was further revealed that very few components of the modified adomian decomposition could yield a series of about 3 times the order of the one obtained by using the taylor series method. hence, we recommend inclusion of the modified adomian decomposition method in modern mathematical tools. keywords: van der pol oscillator, modified adomian decomposition method, taylor series method article history : received: 01 january 2020 received in revised form: 04 march 2020 accepted for publication: 09 march 2020 published: 14 may 2020 c©2020 journal of the nigerian society of physical sciences. all rights reserved. communicated by: t. latunde 1. introduction the van der pol equation, otherwise called the van der pol oscillator is a model which describes the behaviour of electrical circuits [1-19]. it was formulated by an electrical engineer and physicist, balthasar van der pol, and no exact solution has been obtained for the second order differential equation since then [12]. however, asymptotic and numerical solutions have been obtained. one important method of obtaining approximate analytic solutions to differential equations is the taylor series ∗corresponding author tel. no: 07031950138 email address: ndamj@unijos.edu.ng (j. n. ndam) method. however, arguments have been advanced against it because the method has the problem of producing tedious computational work in determining the coefficient an which can be observed in deriving recurrence relation, particularly when the product of two or more infinite series are involved [17]. recently, another form of series solution procedure for solving both linear and nonlinear differential equations has been developed. this procedure, which generates series solutions to differential equations that converge very rapidly, is called adomian decomposition method, named after the author [18]. the adomian decomposition method has been used to obtain approximate analytic solutions to a wide range of differential equations, including linear and nonlinear equations, and has 106 j. n. ndam & o. adedire / j. nig. soc. phys. sci. 2 (2020) 106–114 107 been shown to converge more rapidly with proofs of convergence than other forms of series solutions ([1, 4, 13, 17, 19]). there have been various modifications to this procedure which have significantly improved the convergence of the method ([13, 5, 8, 9, 14, 18]). wazwaz [17] compared the solutions of a linear and nonlinear differential equations using the taylor series and the adomian decomposition methods and concluded that the solutions obtained by the decomposition method converged much more rapidly than the taylor series procedure. however, consideration of wazwaz [17] of the two series methods was in the sense of ease of getting adomian decomposition series compared with that of taylor series but did not consider error analysis of the series methods as his focus was on infinite series solution. in this work, we attempt approximate solutions of the van der pol equation using few orders of taylor series and few components of the modified adomian decomposition procedures with emphasis on finite series solution. the aim is to compare the solution of the van der pol equation using the modified adomian decomposition method (madm) and the truncated taylor series method (ttsm) with approximate poincare-lindstedt (p-l) technique of pertubation method. the specific objectives are to investigate the behaviour of the errors generated from both methods for parameter values of van der pol equation and to graphically display them. these objectives of the study will be achieved by comparing the solution of the van der pol equation from finite orders of ttsm and finite components of madm with that of approximate p-l analytic method. the choice of p-l analytic method for comparison in this study is based on its suitability and accuracy for the solution of nonlinear differential equations ([6], [11], [16]). the remaining parts of this paper are organised as follows: the second section considers the taylor series solution, while section 3 is dedicated to the modified adomian decomposition procedure. results and discussion comes up in section 4. finally, conclusion will be the subject of section 5. 2. taylor series method the van der pol oscillator is given by u′′(t) + µ(u2(t) − 1)u′(t) + u(t) = 0 (1) with the initial conditions u(0) = 1, u′(0) = 0, where µ � 1 or µ � 1. the taylor series solution about t = 0 is obtained as u(t) = u(0)+tu′(0)+ t2 2! u′′(0)+ t3 3! u′′′(0)+· · ·+ tn n! u(n)(0)(2) equation (1) can be written as u′′(t) = µ(1 − u2(t))u′(t) − u(t) u′′′(t) = −2µu(t)u′2(t) + µ(1 − u2(t))u′′(t) − u′(t) uiv(t) = −2µu′3 − 6µu(t)u′(t)u′′(t) − 8µu(t)u′(t)u′′′(t) +µ(1 − u(t)2)uiv(t) − u′′′(t) ... computing the derivatives as far as possible and evaluating at t = 0, we obtain u(0) = 1 u′(0) = 0 u(2)(0) = −1 u(3)(0) = 0 u(4)(0) = 1 u(5)(0) = −6µ u(6)(0) = −1 u(7)(0) = 66µ u(8)(0) = −252µ2 + 1 u(9)(0) = −612µ u(10)(0) = 11052µ2 − 1 u(11)(0) = 5532µ− 31752µ3 u(12)(0) = −341316µ2 + 1 u(13)(0) = −49818µ + 3714552µ3 u(14)(0) = 6259644µ2 − 8845200µ4 − 1 u(15)(0) = 448398µ− 274369248µ3 u(16)(0) = 2211091344µ4 − 241794792µ2 + 1 u(17)(0) = −4604457312µ5 + 15983461728µ3 − 4035384µ ... using the above values, tenth-order taylor series yields the following approximating polynomial: u(t) ≈ 1 − t2 2! + t4 4! − 6µ t5 5! + t6 6! + 66µ t7 7! −(252µ2 + 1) t8 8! − 612µ t9 9! + (11052µ2 − 1) t10 10! (3) analogously, twenty second-order taylor series yields the following approximating polynomial: u(t) ≈ 1 − t2 2! + t4 4! − 6µ t5 5! + t6 6! + 66µ t7 7! −(252µ2 + 1) t8 8! − 612µ t9 9! + (11052µ2 − 1) t10 10! +(5532µ− 31752µ3) t11 11! − (341316µ2 − 1) t12 12! −(49818µ− 3714552µ3) t13 13! + ( 6259644µ2 − 8845200µ4 −1) t14 14! + (448398µ− 274369248µ3) t15 15! 107 j. n. ndam & o. adedire / j. nig. soc. phys. sci. 2 (2020) 106–114 108 +(2211091344µ4 − 241794792µ2 + 1) t16 16! +(−4604457312µ5 + 15983461728µ3 − 4035384µ) t17 17! +(−325537770960µ4 + −325537770960µ2 − 1) t18 18! +(2136491544096µ5 − 924215233680µ3 +36320664µ) t19 19! + (−4018487578560µ6 +37614999137616µ4 − 154525796280µ2 + 1) t20 20! +(−557011268077248µ5 + 48491799847920µ3 −326886030µ) t21 21! + (3140584492618944µ6 −3803681636636400µ4 + 3873523538520µ2 − 1) t22 22! (4) 3. adomian decomposition method in this section, we follow the approach used by el-kalla [4] in which he considered kth order nonlinear ordinary differential equation whose nonlinear term has adomian polynomial representation. this approach is similar to that of wazwaz [17]. the adomian decomposition procedure seeks a solution of the form u(t) = ∞∑ n=0 un(t) (5) equation (1) can be expressed in operator form as ltu = µ(1 − u 2)u′ − u (6) where lt = d2 dt2 , and hence the solution of (1) can be expressed as u(t) = u(0) + l−1t { µ(u′ − u2u′) − u } (7) where l−1t = ∫ ∫ (.)dtdt is the two-fold integration operator. initial condition of eq. (1) given by u(0) = 1 is denoted by u0 = 1 so that from eq. (5), the components that make up the series becomes u0 = 1 (8) un+1(t) = ∫ t 0 ∫ t 0 { µ((un(s)) ′ − an(s)) − un(s) } d sd s (9) where an = 1 n! dn dλn n  ∞∑ i=0 uiλ i   λ=0 = u2n(un) ′ n = 0, 1, 2, · · · and λ is a parameter, are called the adomian polynomials. however, we shall use the modified procedure for generating the adomian polynomials as discussed below: suppose the nonlinear term in (1) is denoted by n(u), then we decompose it as a0 = n(u0) a1 = n(u0 + u1) − a0 a2 = n(u0 + u1 + u2) − (a0 + a1) a3 = n(u0 + u1 + u2 + u3) − (a0 + a1 + a2) ... thus we obtain the relation for generating the adomian polynomials as an = n  n∑ i=0 ui  − n−1∑ i=0 ai (10) from the initial value problem (1), n(u) = u2u′ and hence by using (10), the adomian polynomials are obtained as a0 = u 2 0u ′ 0 a1 = (u0 + u1) 2(u′0 + u ′ 1) − u 2 0u ′ 0 = u20u ′ 1 + 2u0u1u ′ 0 + 2u0u1u ′ 1 + u 2 1u ′ 0 + u 2 1u ′ 1 a2 = (u0 + u1 + u2) 2(u′0 + u ′ 1 + u ′ 2) − (a0 + a1) = u20u ′ 2 + 2u0u1u ′ 2 + 2u0u2u ′ 0 + 2u0u2u ′ 1 +2u0u2u ′ 2 + 2u1u2u ′ 0 + 2u1u2u ′ 1 + 2u1u2u ′ 2 +u22u ′ 1 + u 2 2u ′ 2 + u 2 1u ′ 2 + u 2 2u ′ 0 accordingly, we obtain u0 = 1 u1 = − t2 2 u2 = µ 168 t7 − µ 20 t5 + t4 4! u3 = − µ4 312947712 t22+ µ4 11289600 t20− µ4 1299600 t18− µ3 12870144 t19 + µ3 399840 t17− µ 240 ( 5µ 32256 − µ3 1600 ) t16− 617µ3 21168000 t15+ 3µ2 125440 t14 − µ 156 ( − 31µ2 1400 + 1 3456 ) t13− 41µ2 120960 t12− µ 110 ( − 5 576 + µ2 40 ) t11 + 19µ2 8400 t10 − 13µ 9072 t9 − µ2 160 t8 + µ 140 t7 − t6 6! ... the series solution for 4-components adomian decomposition then becomes u(t) = u0 + u1 + u2 + u3 (11) which gives u(t) = 1 − t2 2 + µ 168 t7 − µ 20 t5 + t4 4! − µ4 312947712 t22 + µ4 11289600 t20 108 j. n. ndam & o. adedire / j. nig. soc. phys. sci. 2 (2020) 106–114 109 − µ4 1299600 t18 − µ3 12870144 t19 + µ3 399840 t17 − µ 240 ( 5µ 32256 − µ3 1600 ) t16 − 617µ3 21168000 t15 + 3µ2 125440 t14 − µ 156 ( − 31µ2 1400 + 1 3456 ) t13 − 41µ2 120960 t12 − µ 110 ( − 5 576 + µ2 40 ) t11 + 19µ2 8400 t10 − 13µ 9072 t9 − µ2 160 t8 + µ 140 t7 − t6 6! (12) and the series solution for 5-component adomian decomposition then gives u(t) = u0 + u1 + u2 + u3 + u4 (13) we simplify (13) and choose µ to be arbitrarily small because approximate p-l method to be used for comparison can work effectively on differential equations of the form u′′ + w20u = µf(t, u, u ′), 0 < µ << 1 whose leading order is oscillatory with frequency w0 [10]. so eq.(13) using µ = 2 x 10−2 gives u(t) = 1 − 1/2 t2 + 0.0002619047619 t7 −0.001000000000 t5 − t6 720 + 1/24 t4 +1.329776164 × 10−50 t67 − 1.139867761 × 10−48 t65 +5.077535944 × 10−47 t64 + 4.357672234 × 10−47 t63 −4.592882729 × 10−45 t62 + 8.593423967 × 10−44 t61 −8.269818866 × 10−42 t59 − 1.845062816 × 10−13 t23 +1.866090683 × 10−43 t60 + 8.309132396 × 10−41 t58 +3.568516858 × 10−40 t57 − 8.714821000 × 10−39 t56 +4.811105327 × 10−38 t55 + 4.002845402 × 10−37 t54 −5.886109478 × 10−36 t53 + 1.424302351 × 10−35 t52 +2.902430667 × 10−34 t51 − 2.595201219 × 10−33 t50 −1.043753849 × 10−33 t49 + 1.407479522 × 10−31 t48 −7.032084376 × 10−31 t47 − 3.022579830 × 10−30 t46 +4.519343938 × 10−29 t45 − 8.093847241 × 10−29 t44 −1.418636597 × 10−27 t43 + 8.881156719 × 10−27 t42 +1.617446587 × 10−26 t41 − 3.583414553 × 10−25 t40 +7.270414170 × 10−25 t39 + 8.689959175 × 10−24 t38 −4.931206414 × 10−23 t37 − 1.020281668 × 10−22 t36 +1.652954579 × 10−21 t35 − 1.931801068 × 10−21 t34 −3.707212443 × 10−20 t33 + 1.482428707 × 10−19 t32 +5.109844495 × 10−19 t31 − 4.724699345 × 10−18 t30 +2.377613320 × 10−19 t29 + 1.016506228 × 10−16 t28 −2.556365546 × 10−16 t27 − 1.543943729 × 10−15 t26 +8.696083800 × 10−15 t25 + 1.257108256 × 10−14 t24 +1.611259824 × 10−13 t22 − 9.420685884 × 10−12 t20 +0.00002230158730 t8 + 2.799612144 × 10−12 t21 −3.001809900 × 10−11 t19 + 0.0000000002238474420 t18 +0.0000000001499617141 t17 − 0.000000003494172630 t16 +0.000000003368584091 t15 + 0.00000003773049937 t14 −0.0000001349519000 t13 − 0.0000002741969964 t12 +0.000002674211961 t11 + 0.000001218253968 t10 −0.00003373015873 t9 (14) 4. results and discussion here, comparison of results from the approximating polynomials (3), (4), (12) and (14) with approximate p-l method for the solution of (1) and its initial conditions are shown in tables 1, 2, 3 and 4. the choice of approximate p-l method is because exact solution has not been obtained for the van der pol equation. details of the approximate method used here can be obtained from ([10], [15]) and references contained therein. it should be noted that the initial conditions used in this research suggest that the periodic behaviour exhibits the amplitude 1 as against the amplitude 2 used in [15]. absolute errors (abs errors) are also obtained and are tabulated as follows: figure 1: graphical solutions using p-l method and approximating polynomials (3) and (12) of ttsm and 4-component madm. while the graphs of solutions of equation (1) and its initial conditions using approximate p-l analytic method and approximating polynomials (3), (4), (12) and (14) are shown in figures 109 j. n. ndam & o. adedire / j. nig. soc. phys. sci. 2 (2020) 106–114 110 table 1: solutions and errors from (3) and (12) of ttsm series and 4-component madm t poincare-lindstedt ttsm madm abs error ttsm abs error madm 0 1.000000000000 1.000000000000 1.000000000000 0.000000000000 0.000000000000 0.3 0.955854836508 0.955334115584 0.955334114058 0.000520720924 0.000520722450 0.6 0.828944246584 0.825264812016 0.825264451699 0.003679434568 0.003679794885 0.9 0.631240082296 0.621131026775 0.621122751091 0.010109055521 0.010117331205 1.2 0.378578337374 0.360630621478 0.360560141843 0.017947715896 0.018018195531 1.5 0.0906245246636 0.066327550293 0.065998964725 0.024296974371 0.024625559939 1.8 -0.20868615269 -0.23659702897 -0.23752492072 0.027910876280 0.028838768030 2.1 -0.491935529476 -0.52423546969 -0.52542008925 0.032299940214 0.033484559774 2.4 -0.731188366817 -0.78110829991 -0.77773900461 0.049919933093 0.046550637793 2.7 -0.902481373797 -1.01309133371 -0.98448885633 0.110609959913 0.082007482533 3.0 -0.989925613354 -1.26963992857 -1.15176986217 0.279714315216 0.161844248816 table 2: solutions and errors from (4) and (12) of ttsm and 4-component madm t poincare-lindstedt ttsm madm abs error ttsm abs error madm 0 1.000000000000 1.000000000000 1.000000000000 0.000000000000 0.000000000000 0.3 0.955854836508 0.955334115588 0.955334114058 0.000520720920 0.000520722450 0.6 0.828944246584 0.825264821264 0.825264451699 0.003679425320 0.003679794885 0.9 0.631240082296 0.621131785279 0.621122751091 0.010108297017 0.010117331205 1.2 0.378578337374 0.360647539325 0.360560141843 0.017930798049 0.018018195531 1.5 0.0906245246636 0.066512092157 0.065998964725 0.024112432507 0.024625559939 1.8 -0.20868615269 -0.23531787096 -0.23752492072 0.026631718270 0.028838768030 2.1 -0.491935529476 -0.51774914669 -0.52542008925 0.025813617214 0.033484559774 2.4 -0.731188366817 -0.75481924354 -0.77773900461 0.023630876723 0.046550637793 2.7 -0.902481373797 -0.9216069535 -0.98448885633 0.019125579703 0.082007482533 3.0 -0.989925613354 -0.97208752941 -1.15176986217 0.017838083944 0.161844248816 table 3: solutions and errors from (3) and (14) of ttsm and 5-component madm t poincare-lindstedt ttsm madm abs error ttsm abs error madm 0 1.000000000000 1.000000000000 1.000000000000 0.000000000000 0.000000000000 0.3 0.955854836508 0.955334115584 0.95533411559 0.000520720924 0.000520720918 0.6 0.828944246584 0.825264812016 0.825264822641 0.003679434568 0.003679423943 0.9 0.631240082296 0.621131026775 0.621131859045 0.010109055521 0.010108223251 1.2 0.378578337374 0.360630621478 0.360648788531 0.017947715896 0.017929548843 1.5 0.0906245246636 0.066327550293 0.066523502482 0.024296974371 0.024101022182 1.8 -0.20868615269 -0.23659702897 -0.23524709918 0.027910876280 0.026560946490 2.1 -0.491935529476 -0.52423546969 -0.51742153967 0.032299940214 0.025486010194 2.4 -0.731188366817 -0.78110829991 -0.75375297048 0.049919933093 0.022564603663 2.7 -0.902481373797 -1.01309133371 -0.92099274209 0.110609959913 0.018511368293 3.0 -0.989925613354 -1.26963992857 -0.99998289494 0.279714315216 0.010057281586 110 j. n. ndam & o. adedire / j. nig. soc. phys. sci. 2 (2020) 106–114 111 table 4: solutions and errors from (4) and (14) of ttsm and 5-component madm t poincare-lindstedt ttsm madm abs error ttsm abs error madm 0 1.000000000000 1.000000000000 1.000000000000 0.000000000000 0.000000000000 0.3 0.955854836508 0.955334115588 0.95533411559 0.000520720920 0.000520720918 0.6 0.828944246584 0.825264821264 0.825264822641 0.003679425320 0.003679423943 0.9 0.631240082296 0.621131785279 0.621131859045 0.010108297017 0.010108223251 1.2 0.378578337374 0.360647539325 0.360648788531 0.017930798049 0.017929548843 1.5 0.0906245246636 0.066512092157 0.066523502482 0.024112432507 0.024101022182 1.8 -0.20868615269 -0.23531787096 -0.23524709918 0.026631718270 0.026560946490 2.1 -0.491935529476 -0.51774914669 -0.51742153967 0.025813617214 0.025486010194 2.4 -0.731188366817 -0.75481924354 -0.75375297048 0.023630876723 0.022564603663 2.7 -0.902481373797 -0.9216069535 -0.92099274209 0.019125579703 0.018511368293 3.0 -0.989925613354 -0.97208752941 -0.99998289494 0.017838083944 0.010057281586 1, 2, 3 and 4 for µ = 2 x 10−2, the graphs of absolute errors are also indicated in figures 5, 6, 7 and 8. figure 2: graphical solutions using p-l method and approximating polynomials (4) and (12) of ttsm and 4-component madm. figure 3: graphical solutions using p-l method and approximating polynomials (3) and (14) of ttsm and 5-component madm. 111 j. n. ndam & o. adedire / j. nig. soc. phys. sci. 2 (2020) 106–114 112 figure 4: graphical solutions using p-l method and approximating polynomials (4) and (14) of ttsm and 5-component madm. figure 5: plot of absolute errors from approximating polynomials (3) and (12) of ttsm and 4-component madm. figure 6: plot of absolute errors from approximating polynomials (4) and (12) of ttsm and 4-component madm. figure 7: plot of absolute errors from approximating polynomials (3) and (14) of ttsm and 5-component madm. 112 j. n. ndam & o. adedire / j. nig. soc. phys. sci. 2 (2020) 106–114 113 figure 8: plot of absolute errors from approximating polynomials (4) and (14) of ttsm and 5-component madm. equation (12) of approximating 4-component madm behave more like approximate p-l method than approximating polynomial (3) of ttsm as shown in figure 1 and table 1. on the other hand, many more terms are added to approximating ttsm (3) to obtain ttsm (4). solutions from approximating polynomial (4) of ttsm with twenty-two order behave more closely as that of approximate p-l method than equation (12) of approximating 4-component madm series and are indicated in figure 2 and table 2. however, results from 5-component madm (14) shown in tables 3 and 4 are similar to the results obtained from approximate p-l method than those got from ttsm of equations (3) and (4) as shown in figures 3 and 4. the absolute errors generated by the approximating polynomial (3) are greater than the ones generated by (12) as shown in figure 5. absolute errors obtained from ttsm (4) are generally lower than the ones generated from (12) of madm due to many numbers of terms added to ttsm (3) as shown in figure 6. finally, figures 7 and 8 indicate that (14) shows smaller absolute errors compared with (3) and (4). at t=3 from table 1, equation (3) gives u(3) ≈ −1.27 and (12) gives u(3) ≈ −1.15. observe that with several terms added to (3) to obtain (4), the solution obtained in table 4 at t = 3 for (4) is not as close to that of approximate p-l method as (14) which is obtained from (12) by addition of just one component. the theoretical implication of this is that few components from madm obtained with ease can produce solution that is more convergent to that of pl method than for tediuos computational work of many terms added to ttsm. 5. conclusion in this paper, approximate series solutions of the van der pol equation have been obtained using the truncated taylor series method and the modified adomian decomposition method. the two series solutions for van der pol equation obtained as approximating polynomials (3), (4), (12) and (14) agree. from the approximating polynomials of the two series solutions, madm produces more convergent results to the p-l methods than the ttsm, as only very few components of the madm yielded a series much longer than order twenty of taylor series. hence, we recommend inclusion of the madm in modern mathematical tools. references [1] e. a. az-zobi, k. al-khaled & a. darweesh, “numeric-analytic solutions for nonlinear oscillators via the modified multi-stage decomposition method.”, mdpi journal of mathematics 7 (2019) 1. doi: 10.3390/math7060550 [2] j. biazar and y. shafiof, “a simple algorithm for calculating adomian polynomials. int. j. contemp. math. sciences, 975 982. 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[16] d. viswannath, “the construction of lindstedt-poincare technique as an algorithm for computing periodic orbits, siam journal of applied mathematics 43 (2001) 478. doi: 10.1137/s0036144500375292 [17] a. m. wazwaz, “a comparison between adomian decomposition method and taylor series method in the series solutions”, appl. math. comput 97 (1998) 37. doi: 10.1016/s0096-3003(97)10127-8 113 j. n. ndam & o. adedire / j. nig. soc. phys. sci. 2 (2020) 106–114 114 [18] a. m. wazwaz “a new algoritm for calculating adomian polynomials for nonlinear operators”, appl. math. comput 111 (2000) 53. doi: 10.1016/s0096-3003(99)00063-6 [19] a. m. wazwaz & a. gorguis, “an analytic study of fisher’s equation by using adomian decomposition method”, appl. math. comput 154 (2004) 609. doi: 10.1016/s0096-3003(03)00738-0 114 j. nig. soc. phys. sci. 4 (2022) 34–48 journal of the nigerian society of physical sciences numerical solution of stiff and oscillatory problems using third derivative trigonometrically fitted block method m. kidaa, s. adamub,∗, o. o. adurojac, t. p. pantuvod adepartment of mathematics, modibbo adama university yola, adamawa, nigeria bdepartment of mathematics, nigerian army university biu, borno, nigeria cdepartment of mathematics, osun state college of education ilesha, osun, nigeria dmathematics and statistics department, federal university wukari, taraba, nigeria abstract this paper considered the formulation of continuous third derivative trigonometrically fitted block method for the solution of stiff and oscillatory problems. the development of the technique involved the interpolation and collocation of the approximate solution which is the combination of polynomial and trigonometric functions. solving for the unknown parameters and substituting the results into the approximate solution yielded a continuous linear multistep method, which is evaluated at some selected grid points where two cases were considered at equal intervals to give the discrete schemes which are implemented in block form. the blocks are convergent and stable. numerical experiments show that the methods compete favorably with existing method and efficient for the solution of stiff and oscillatory problems. doi:10.46481/jnsps.2022.271 keywords: third derivative, trigonometrically fitted block method, oscillatory problems, stability, convergence article history : received: 22 june 2021 received in revised form: 27 january 2022 accepted for publication: 28 january 2022 published: 28 february 2022 c©2022 journal of the nigerian society of physical sciences. all rights reserved. communicated by: t. latunde 1. introduction in science and engineering, mathematical models are developed to help in the studying of physical phenomena, these models often yield equations that contain some derivatives of an unknown function of one or several variables, such equations are called differential equations (des) [1, 2]. interestingly, systems described by differential equations are so complex, or the systems that they describe are so large, that a purely analytical solution to the equations is not tractable, hence the need for numerical approximation [3]. ∗corresponding author tel. no: +2348069405058 email address: malgwisa@gmail.com (s. adamu ) block method is formulated in terms of lmm, it preserves the traditional advantage of one step methods, of being selfstarting and permitting easy change of step length [1, 4]. the advantages of block methods over predictor-corrector methods lies in the fact that they are less expensive in terms of number of functions evaluation, it is capable of giving evaluations at different grid points, without overlapping as done in the predictorcorrector method and it generates simultaneous solutions at all grid points [1, 4, 5]. despite the success recorded by linear multistep method (lmm) in the numerical solution of initial value problems, most of the approaches failed when the problems are stiff. adoption of trigonometrically fitted approximate solution as basis functions have been effective in handling this setback, but their de34 kida et al. / j. nig. soc. phys. sci. 4 (2022) 34–48 35 velopments are tedious. in this paper, we consider numerical procedures for approximating the solution of stiff and oscillatory problems of the form y′ = f (x, y, ), y (x0) = η0. (1) these type of differential equations are known to be highly oscillatory and some problems have special properties such as discontinuity and stiffness, hence it is quite difficult to get their numerical solutions accurately [6, 2]. they are numerically unstable, unless the step size is taken to be extremely small [7]. a stiff system is one involving rapidly changing components together with slowly changing ones, stiff systems are sometimes referred to as systems with large lipstchitz constant [7, 8]. the system of (1) is stiff if the eigenvalues of the matrix have negative real parts at every time x and varies greatly in magnitude [9, 10]. oscillation is the repetitive variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states [7]. this type of problem arises in different fields of science and engineering, which includes quantum mechanics, celestial mechanics, molecular dynamics, quantum chemistry, astrophysics, electronics and semi-discretizations of wave equation [5, 11]. numerous numerical methods have been derived for approximating the solutions of (1), some of which are linear multistep methods (lmms) [12, 13, 4, 14], block method [5, 15, 1, 16, 17, 18, 19, 20], trigonometrically fitted methods [21, 22, 23, 6, 24]. other methods for the solution of stiff and oscillatory problems includes [2, 9, 11]. modeled problems in engineering and sciences leads to complex problems such as stiff and oscillatory problems. it was observed that lmm fails to converge at the state of oscillatory as a result has poor stability properties [7]. second derivative formulas with trigonometric basis functions seems to solve such problems very well and has a better and stability properties [1, 17]. [25, 26] develop an algorithm for the solution of stiff initial value problems. furthermore, [27, 28] considers an implicit r−point block backward differentiation formula (bbdf) and block method that generates two values simultaneously respectively for solving stiff ivps of odes. the r−point block method simultaneously produces r−new values at the time discretization points. the total number of steps to complete the integration by the methods are reduced but not sufficiently and the computation time for the method can still be reduced. [29] developed a block integrator for the solution of stiff and oscillatory first order ivps of odes by means of collocation and interpolation of the combination of power series and exponential function to generate a continuous implicit linear multistep method. despite the success achieved by the above methods, it increases the dimension of the resulting systems of first order by the order of the differential equation, hence it wastes both the computer and human effort. polynomial approximate solution has been reported not to be efficient in handling stiff oscillatory problems, trigonometrically fitted methods handle oscillatory problems effectively, but the application of higher derivatives methods for the solution of first order oscillatory initial value problems has not given much attention. in this paper continuous trigonometrically fitted third derivative method (ctftdm) is constructed which provides a discrete method for direct solution of first order initial value problems. the coefficients of the tftdm are functions of the frequency and the step-size, hence the solutions that will be provided by the methods will be highly accurate if (1) has periodic solutions with the unknown frequencies. this new method considers the application of higher derivative and does not waste both the computer and human effort which is efficient in handling stiff oscillatory problems. trigonometrically fitted methods are powerful tool in handling stiff/oscillatory problems, order 2 methods developed by combining polynomial and trigonometric approximate solution and considering higher derivatives, conveniently solved stiff/oscillatory problems and it perform better than other methods of the same order. 1.1. preliminaries definition 1.1. stiff [8]: the system ẏ = ay + φ(x) where a is an m × m matrix and φ(x) an m dimensional vector, is said to be stiff if reλi < 0, i = 1, 2, . . . , m and maxi=1...m |reλi| mini=1,...,m |reλi| ≥ 0 where λi, i = 1, 2, . . . , m are the eigenvalues of a. the ratio s = maxi=1...m |reλi| mini=1,...,m |reλi| ≥ 0 is called the stiffness ratio. theorem 1.2. consistency [30]: a block method is said to be consistent if it has order p ≥ 1. theorem 1.3. zero-stable [30]: a block method is said to be zero stable if as h → 0, the roots r j, j = 1(1)k of the first characteristic polynomials ρ(r) = 0 that is ρ(r) = det [∑ a(0)rk−1 ] = 0 satisfying |r| ≤ 1, must be simple. theorem 1.4. convergent [31]: consistent and zero stability are sufficient conditions for a block method to be convergent. 2. methodology we considered the approximate solution y (x) = 2∑ n=0 αn x n + 2∑ n=1 βn sin nωx + 2∑ n=1 cn cos nωx (2) 35 kida et al. / j. nig. soc. phys. sci. 4 (2022) 34–48 36 where ω = 2nπ t , t is the period of the oscillation, α′s, β′s and c′s are parameters to be determined. interpolating (2) at point xn, collocating the first derivative of (2) at point xn, xn+u, xn+v, collocating the second derivative of (2) at point xn+u, xn+v, and collocating the third derivative of (2) at point xn+v for u and v, u < v to obtain the system xa = u (3) where a = [ α1 β1 β2 β3 c1 c2 d2 ]t u = [ yn fn fn+u fn+v gn+u gn+v ln+v ]t x =  1 xn x2n sin ωxn sin 2ωxn cos ωxn cos 2ωxn 0 1 2xn ω cos ωxn+v 2ω cos 2ωxn −ω sin ωxn −2ω sin 2ωxn 0 1 2xn+u ω cos ωxn+u 2ω cos 2ωxn+u −ω sin ωxn+u −2ω sin 2ωxn+u 0 1 2xn+v ω cos ωωxn+v 2ω cos 2ωxn+v −ω sin ωxn+v −2ω sin 2ωxn+v 0 0 2 −ω2 sin ωxn+u −4ω2 sin 2ωxn+u −ω2 cos ωxn+u −4ω2 cos 2ωxn+u 0 0 2 −ω2 sin ωxn+v −4ω2 sin 2ωxn+v −ω2 cos ωxn+v −4ω2 cos 2ωxn+v 0 0 0 −ω3 cos ωxn+v −8ω3 cos 2ωxn+v ω3 sin ωxn+v 8ω3 sin 2ωxn+v  (3) gives system of nonlinear equations which is then solved using newton raphson’s method. now considering two cases of third derivative trigonometrically fitted methods at equal intervals as specified below case i: u = 12 , v = 1 case ii: u = 1, v = 2 2.1. block method for case i considering u = 12 , v = 1 yn+t = α0 (t) yn + β0 (t) fn + β1 (t) fn+ 12 + β2 (t) fn+1 + γn+1 (t) gn+ 12 (4) +γn+2 (t) gn+2 + [ ln+2 (t) (ζn+2) ] evaluating (4) at t = 12 yields yn+ 12 = α01yn + β01 fn + β11 fn+ 12 + β21 fn+1 + γ11gn+ 12 (5) +γ21gn+1 + ζ11ln+1 where α01 = 1 β01 = 1 4  8 cos hω + 60 cos 2hω + 104 cos 12 hω− 94 cos 3 2 hω− 10 cos 5 2 hω + 30h 2ω2 +6h2ω2 cos hω− 33h2ω2 cos 12 hω− 3h 2ω2 cos 32 hω + 8hω sin hω +8hω sin 2hω− 56hω sin 12 hω + 6hω sin 3 2 hω− 2hω sin 5 2 hω− 68  ω [ 12 sin hω− 22 sin 2hω− 42 sin 12 hω + 27 sin 3 2 hω + 5 sin 5 2 hω + 26hω +4hω cos hω + 2hω cos 2hω− 32hω cos 12 hω + hω cos 3 2 hω− hω cos 5 2 hω ] β11 = 1 4  164 cos hω− 44 cos 2hω− 4 cos 3hω + 20 cos 12 hω− 42 cos 3 2 hω +22 cos 52 hω + 18h 2ω2 + 6h2ω2 cos 2hω− 30h2ω2 cos 12 hω +9h2ω2 cos 32 hω− 3h 2ω2 cos 52 hω + 32hω sin hω− 40hω sin 2hω +28hω sin 12 hω + 6hω sin 3 2 hω + 10hω sin 5 2 hω− 116  ω [ 12 sin hω− 22 sin 2hω− 42 sin 12 hω + 27 sin 3 2 hω + 5 sin 5 2 hω + 26hω +4hω cos hω + 2hω cos 2hω− 32hω cos 12 hω + hω cos 3 2 hω− hω cos 5 2 hω ] β21 = 1 4  −172 cos hω− 16 cos 2hω + 4 cos 3hω− 124 cos 12 hω + 136 cos 3 2 hω −12 cos 52 hω + 4h 2ω2 + 2h2ω2 cos hω− 2h2ω2 cos 2hω− h2ω2 cos 12 hω −4h2ω2 cos 32 hω + h 2ω2 cos 52 hω− 16hω sin hω− 12hω sin 2hω −56hω sin 12 hω + 42hω sin 3 2 hω + 2hω sin 5 2 hω + 184  ω [ 12 sin hω− 22 sin 2hω− 42 sin 12 hω + 27 sin 3 2 hω + 5 sin 5 2 hω + 26hω +4hω cos hω + 2hω cos 2hω− 32hω cos 12 hω + hω cos 3 2 hω− hω cos 5 2 hω ] 36 kida et al. / j. nig. soc. phys. sci. 4 (2022) 34–48 37 γ11 = 1 4  54 sin hω + 28 sin 2hω + 6 sin 3hω + 12 sin 12 hω− 66 sin 3 2 hω− 14 sin 5 2 hω −138hω + 6h2ω2 sin 2hω + 18h2ω2 sin 12 hω− 9h 2ω2 sin 32 hω −3h2ω2 sin 52 hω− 6hω cos hω + 18hω cos 2hω− 2hω cos 3hω +156hω cos 12 hω− 22hω cos 3 2 hω− 6hω cos 5 2 hω  ω2 [ 12 sin hω− 22 sin 2hω− 42 sin 12 hω + 27 sin 3 2 hω + 5 sin 5 2 hω + 26hω +4hω cos hω + 2hω cos 2hω− 32hω cos 12 hω + hω cos 3 2 hω− hω cos 5 2 hω ] γ21 = 1 8  −108 sin hω− 56 sin 2hω− 12 sin 3hω− 24 sin 12 hω + 132 sin 3 2 hω +24hω + 12h2ω2 sin hω + 30h2ω2 sin 2hω + 6h2ω2 sin 12 hω− 45h 2ω2 sin 32 hω −3h2ω2 sin 52 hω + 192hω cos hω + 40hω cos 2hω− 84hω cos 1 2 hω +28 sin 52 hω− 186hω cos 3 2 hω + 14hω cos 5 2 hω  ω2 [ 12 sin hω− 22 sin 2hω− 42 sin 12 hω + 27 sin 3 2 hω + 5 sin 5 2 hω + 26hω +4hω cos hω + 2hω cos 2hω− 32hω cos 12 hω + hω cos 3 2 hω− hω cos 5 2 hω ] ζ11 = 1 8  −156 cos hω− 32 cos 2hω− 4 cos 3hω− 144 cos 12 hω + 136 cos 3 2 hω +2h2ω2 + 16h2ω2 cos hω + 14h2ω2 cos 2hω− 2h2ω2 cos 12 hω −h2ω2 cos 52 hω− 144hω sin hω− 24hω sin 2hω + 36hω sin 1 2 hω +8 cos 52 hω− 29h 2ω2 cos 32 hω + 126hω sin 3 2 hω− 6hω sin 5 2 hω + 192  ω3 [ 12 sin hω− 22 sin 2hω− 42 sin 12 hω + 27 sin 3 2 hω + 5 sin 5 2 hω + 26hω +4hω cos hω + 2hω cos 2hω− 32hω cos 12 hω + hω cos 3 2 hω− hω cos 5 2 hω ] evaluating (4) at t = 1 yn+1 = α02yn + β02 fn + β12 fn+ 12 + β22 fn+1 + γ12gn+ 12 (6) +γ22gn+1 + ζ12ln+1 where α02 = 1 β02 = − 1 2  32 cos hω− 30 cos 2hω− 34 cos 12 hω + 29 cos 3 2 hω + 5 cos 5 2 hω− 20h 2ω2 +18h2ω2 cos 12 hω + 2h 2ω2 cos 32 hω− 16hω sin hω− 4hω sin 2hω +70hω sin 12 hω− 9hω sin 3 2 hω + hω sin 5 2 hω− 2  ω  12 sin hω− 22 sin 2hω− 42 sin 12 hω + 27 sin 3 2 hω + 5 sin 5 2 hω + 26hω +4hω cos hω + 2hω cos 2hω− 32hω cos 12 hω + hω cos 3 2 hω −hω cos 52 hω  β12 =  65 cos hω− 20 cos 2hω− cos 3hω + 2 cos 12 hω− 9 cos 3 2 hω +7 cos 52 hω + 6h 2ω2 + 4h2ω2 cos hω + 2h2ω2 cos 2hω +3h2ω2 cos 32 hω− h 2ω2 cos 52 hω + 8hω sin hω− 16hω sin 2hω −14h2ω2 cos 12 hω + 28hω sin 1 2 hω + 4hω sin 5 2 hω− 44  ω  12 sin hω− 22 sin 2hω− 42 sin 12 hω + 27 sin 3 2 hω + 5 sin 5 2 hω + 26hω +4hω cos hω + 2hω cos 2hω− 32hω cos 12 hω + hω cos 3 2 hω −hω cos 52 hω  β22 = − 1 2  98 cos hω− 10 cos 2hω− 2 cos 3hω + 38 cos 12 hω− 47 cos 3 2 hω +9 cos 52 hω− 20h 2ω2 + 18h2ω2 cos 12 hω + 2h 2ω2 cos 32 hω +8hω sin hω + 16hω sin 2hω + 70hω sin 12 hω− 45hω sin 3 2 hω −3hω sin 52 hω− 86  ω  12 sin hω− 22 sin 2hω− 42 sin 12 hω + 27 sin 3 2 hω + 5 sin 5 2 hω +26hω + 4hω cos hω + 2hω cos 2hω− 32hω cos 12 hω +hω cos 32 hω− hω cos 5 2 hω  γ12 = 1 2  39 sin hω− 24 sin 2hω + 3 sin 3hω− 12 sin 12 hω− 6 sin 3 2 hω +6 sin 52 hω− 26hω + 8h 2ω2 sin hω + 2h2ω2 sin 2hω + 4h2ω2 sin 12 hω −6h2ω2 sin 32 hω− 2h 2ω2 sin 52 hω + 17hω cos hω + 10hω cos 2hω −hω cos 3hω + 8hω cos 12 hω− 8hω cos 5 2 hω  ω2  12 sin hω− 22 sin 2hω− 42 sin 12 hω + 27 sin 3 2 hω + 5 sin 5 2 hω +4hω cos hω + 2hω cos 2hω− 32hω cos 12 hω + hω cos 3 2 hω +26hω− hω cos 52 hω  37 kida et al. / j. nig. soc. phys. sci. 4 (2022) 34–48 38 γ22 = 1 2  −39 sin hω + 24 sin 2hω− 3 sin 3hω + 12 sin 12 hω + 6 sin 3 2 hω −6 sin 52 hω− 16hω + 4h 2ω2 sin hω + 8h2ω2 sin 2hω −4h2ω2 sin 12 hω− 12h 2ω2 sin 32 hω + 16hω cos hω + 28hω cos 1 2 hω −38hω cos 32 hω + 10hω cos 5 2 hω  ω2  12 sin hω− 22 sin 2hω− 42 sin 12 hω + 27 sin 3 2 hω + 5 sin 5 2 hω +26hω + 4hω cos hω + 2hω cos 2hω− 32hω cos 12 hω + hω cos 3 2 hω −hω cos 52 hω  ζ12 = 1 2  −63 cos hω + 10 cos 2hω− cos 3hω− 24 cos 12 hω + 28 cos 3 2 hω −4 cos 52 hω + 8h 2ω2 + 4h2ω2 cos hω + 4h2ω2 cos 2hω −8h2ω2 cos 12 hω− 8h 2ω2 cos 32 hω− 24hω sin hω− 4hω sin 2hω −24hω sin 12 hω + 36hω sin 3 2 hω− 4hω sin 5 2 hω + 54  ω3  12 sin hω− 22 sin 2hω− 42 sin 12 hω + 27 sin 3 2 hω + 5 sin 5 2 hω +26hω + 4hω cos hω + 2hω cos 2hω− 32hω cos 12 hω +hω cos 32 hω− hω cos 5 2 hω  writing (5) and (6) in discrete block method[ 1 0 0 1 ] [ yn+ 12 yn+1 ] = [ 0 1 0 1 ] [ yn−1 yn ] + [ 0 β01 0 β02 ] [ fn−1 fn ] (7) + [ β11 β21 β12 β22 ] [ fn+ 12 fn+1 ] + [ γ11 γ12 γ21 γ22 ] [ gn+ 12 gn+1 ] + [ 0 ζ11 0 ζ21 ] [ ln−1 ln+1 ] 2.2. analysis of the stability properties for case i 2.2.1. order and error constant evaluating each row of (5) and (6) in taylor series about xn gives l [ y (x) ; h ] = yn+ 12 − yn − hy ′ n −β01 fn −β11 fn+ 12 −β21 fn+1 −γ11gn+ 12 −γ21gn+1 −γ11ln+1 = 0 therefore, the block methods are of order p = (2, 2)t with the following error constants k = − 1 96  −1872 cos hω− 384 cos 2hω− 48 cos 3hω− 1728 cos 12 hω + 1632 cos 3 2 hω +96 cos 52 hω + 516h 2ω2 + 50h4ω4 + 852h2ω2 cos hω + 540h2ω2 cos 2hω +12h2ω2 cos 3hω + 16h4ω4 cos hω− 10h4ω4 cos 2hω + 24h3ω3 sin hω +212h3ω3 sin 2hω− 588h2ω2 cos 12 hω− 1338h 2ω2 cos 32 hω + 6h 2ω2 cos 52 hω −38h4ω4 cos 12 hω− 23h 4ω4 cos 32 hω + 5h 4ω4 cos 52 hω− 216h 3ω3 sin 12 hω −180h3ω3 sin 32 hω− 28h 3ω3 sin 52 hω− 2376hω sin hω− 624hω sin 2hω −72hω sin 3hω + 288hω sin 12 hω + 2304hω sin 3 2 hω + 96hω sin 5 2 hω + 2304  ω3 [ 12 sin hω− 22 sin 2hω− 42 sin 12 hω + 27 sin 3 2 hω + 5 sin 5 2 hω + 26hω +4hω cos hω + 2hω cos 2hω− 32hω cos 12 hω + hω cos 3 2 hω− hω cos 5 2 hω ] k = 1 24  756 cos hω− 120 cos 2hω + 12 cos 3hω + 288 cos 12 hω− 336 cos 3 2 hω + 48 cos 5 2 hω −132h2ω2 − 34h4ω4 + 51h2ω2 cos hω− 108h2ω2 cos 2hω− 3h2ω2 cos 3hω +4h4ω4 cos hω + 2h4ω4 cos 2hω− 24h3ω3 sin hω− 52h3ω3 sin 2hω −66h2ω2 cos 12 hω + 297h 2ω2 cos 32 hω− 39h 2ω2 cos 52 hω + 22h 4ω4 cos 12 hω +7h4ω4 cos 32 hω− h 4ω4 cos 52 hω + 192h 3ω3 sin 12 hω + 18h 3ω3 sin 32 hω +2h3ω3 sin 52 hω + 522hω sin hω− 96hω sin 2hω + 18hω sin 3hω + 216hω sin 1 2 hω −468hω sin 32 hω + 84hω sin 5 2 hω− 648  ω3 [ 12 sin hω− 22 sin 2hω− 42 sin 12 hω + 27 sin 3 2 hω + 5 sin 5 2 hω + 26hω +4hω cos hω + 2hω cos 2hω− 32hω cos 12 hω + hω cos 3 2 hω− hω cos 5 2 hω ] 2.2.2. zero stability of the block − ρ(λ) = det [ λa(1) − a(0) ] = 0 (8) 38 kida et al. / j. nig. soc. phys. sci. 4 (2022) 34–48 39 where a(0), a(1) are from (7) λ [ 1 0 0 1 ] − [ 0 1 0 1 ] = det [ λ −1 0 λ− 1 ] ,λ2 −λ = λ (λ− 1) = 0 thus, solving for λ λ (λ− 1) = 0 which implies that λ1 = 0,λ2 = 1. hence, using theorem 1, theorem 2, theorem 3, the block method (7) is zero stable and also consistent as its order p = [2, 2]t > 1, thus it is convergent. 2.2.3. linear stability applying the test equation y(k) = λ(k)yn to yield yn+1 = µ (z) ym, µ (z) is the amplification equation given by µ (z) = − ( a(1) − zβ(1) − zγ(1) − zζ(1) )−1 ( a(0) + z2β(0) + z3γ(0) ) (9) where a(0) = [ 0 1 0 1 ] , β(0) = [ 0 β01 0 β02 ] , γ(0) = [ 0 ζ11 0 ζ21 ] , a(1) = [ 1 0 0 1 ] , β(1) = [ β11 β21 β12 β22 ] and γ(1) = [ γ11 γ12 γ21 γ22 ] the matrix µ (z) has eigenvalues (0, 0, ...,ξk) where ξk is called the stability function which is the rational function with coefficients. µ (z) =  72z − 234z2h + 9zh3 − 102z2h2 + 27z2h4 + 126zh −60h3ω4 − 64h4ω4 + 210zh2ω2 + 210zh3ω2 − 108zh4ω4 −136zh5ω4 + 180z2h3ω2 + 150z2h4ω2 + 36z2h5ω4 − 24z2h6ω4 [ 72z − 180z2h + 42z3h2 + 45z4h3 − 60h3ω4 + 210zh2ω2 −112zh4ω4 + 180z2h3ω2 − 135z3h4ω2 + 42z2h5ω4 ] , 0 2.2.4. region of absolute stability the region of absolute stability of the block method (7) is shown in figure 1. figure 1. region of absolute stability for case i 2.3. block method for case ii we considered u = 1, v = 2 yn+t = α0 (t) yn + β0 (t) fn + β1 (t) fn+1 + β2 (t) fn+2 + γn+1 (t) gn+1 (10) +γn+2 (t) gn+2 + ζn+2 (t) ln+2 evaluating (4) at t = 1 yields yn+1 = α01yn + [ β01 fn + β11 fn+1 + β21 fn+2 ] + [ γ11gn+1 + γ21gn+2 ] + ζ11ln+2 (11) 39 kida et al. / j. nig. soc. phys. sci. 4 (2022) 34–48 40 where α01 = 1 β01 = 1 2  −52 cos hω− 4 cos 2hω + 47 cos 3hω− 30 cos 4hω +5 cos 5hω− 60h2ω2 + 66h2ω2 cos hω− 12h2ω2 cos 2hω +6h2ω2 cos 3hω + 56hω sin hω− 8hω sin 2hω− 6hω sin 3hω −8hω sin 4hω + 2hω sin 5hω + 34  ω  42 sin hω− 12 sin 2hω− 27 sin 3hω + 22 sin 4hω −5 sin 5hω− 52hω + 64hω cos hω− 8hω cos 2hω −2hω cos 3hω− 4hω cos 4hω + 2hω cos 5hω  β11 = 1 2  −10 cos hω− 82 cos 2hω + 21 cos 3hω + 22 cos 4hω −11 cos 5hω + 2 cos 6hω− 36h2ω2 + 60h2ω2 cos hω −18h2ω2 cos 3hω− 12h2ω2 cos 4hω + 6h2ω2 cos 5hω −28hω sin hω− 32hω sin 2hω− 6hω sin 3hω + 40hω sin 4hω −10hω sin 5hω + 58  ω  42 sin hω− 12 sin 2hω− 27 sin 3hω + 22 sin 4hω −5 sin 5hω− 52hω + 64hω cos hω− 8hω cos 2hω −2hω cos 3hω− 4hω cos 4hω + 2hω cos 5hω  β21 =  31 cos hω + 43 cos 2hω− 34 cos 3hω + 4 cos 4hω + 3 cos 5hω −cos 6hω− 4h2ω2 + h2ω2 cos hω− 2h2ω2 cos 2hω +4h2ω2 cos 3hω + 2h2ω2 cos 4hω− h2ω2 cos 5hω +28hω sin hω + 8hω sin 2hω− 21hω sin 3hω +6hω sin 4hω− hω sin 5hω− 46  ω  42 sin hω− 12 sin 2hω− 27 sin 3hω + 22 sin 4hω −5 sin 5hω− 52hω + 64hω cos hω− 8hω cos 2hω −2hω cos 3hω− 4hω cos 4hω + 2hω cos 5hω  γ11 = − 1 2  6 sin hω + 27 sin 2hω− 33 sin 3hω + 14 sin 4hω− 7 sin 5hω +3 sin 6hω− 138hω + 36h2ω2 sin hω− 18h2ω2 sin 3hω +12h2ω2 sin 4hω− 6h2ω2 sin 5hω + 156hω cos hω −6hω cos 2hω− 22hω cos 3hω + 18hω cos 4hω −6hω cos 5hω− 2hω cos 6hω  ω2  42 sin hω− 12 sin 2hω− 27 sin 3hω + 22 sin 4hω −5 sin 5hω− 52hω + 64hω cos hω− 8hω cos 2hω −2hω cos 3hω− 4hω cos 4hω + 2hω cos 5hω  γ21 = − 1 2  −6 sin hω− 27 sin 2hω + 33 sin 3hω− 14 sin 4hω + 7 sin 5hω −3 sin 6hω + 12hω + 6h2ω2 sin hω + 12h2ω2 sin 2hω −45h2ω2 sin 3hω + 30h2ω2 sin 4hω− 3h2ω2 sin 5hω− 42hω cos hω +96hω cos 2hω− 93hω cos 3hω + 20hω cos 4hω + 7hω cos 5hω  ω2  42 sin hω− 12 sin 2hω− 27 sin 3hω + 22 sin 4hω −5 sin 5hω− 52hω + 64hω cos hω− 8hω cos 2hω −2hω cos 3hω− 4hω cos 4hω + 2hω cos 5hω  ζ11 = 1 2  36 cos hω + 39 cos 2hω− 34 cos 3hω + 8 cos 4hω− 2 cos 5hω + cos 6hω− 2h2ω2 + 2h2ω2 cos hω− 16h2ω2 cos 2hω +29h2ω2 cos 3hω− 14h2ω2 cos 4hω + h2ω2 cos 5hω −18hω sin hω + 72hω sin 2hω− 63hω sin 3hω + 12hω sin 4hω +3hω sin 5hω− 48  ω3  42 sin hω− 12 sin 2hω− 27 sin 3hω + 22 sin 4hω −5 sin 5hω− 52hω + 64hω cos hω− 8hω cos 2hω −2hω cos 3hω− 4hω cos 4hω + 2hω cos 5hω  evaluating (4) at t = 2 yields yn+2 = α02yn + [ β02 fn + β12 fn+1 + β22 fn+2 ] + [ γ12gn+1 + γ22gn+2 ] + ζ12ln+2 (12) 40 kida et al. / j. nig. soc. phys. sci. 4 (2022) 34–48 41 where α01 = 1 β02 = 1 2  −34 cos hω + 32 cos 2hω + 29 cos 3hω− 30 cos 4hω +5 cos 5hω− 80h2ω2 + 72h2ω2 cos hω + 8h2ω2 cos 3hω +140hω sin hω− 32hω sin 2hω− 18hω sin 3hω −8hω sin 4hω + 2hω sin 5hω− 2  ω  42 sin hω− 12 sin 2hω− 27 sin 3hω + 22 sin 4hω −5 sin 5hω− 52hω + 64hω cos hω− 8hω cos 2hω −2hω cos 3hω− 4hω cos 4hω + 2hω cos 5hω  β12 =  −2 cos hω− 65 cos 2hω + 9 cos 3hω + 20 cos 4hω −7 cos 5hω + cos 6hω− 24h2ω2 + 56h2ω2 cos hω −16h2ω2 cos 2hω− 12h2ω2 cos 3hω− 8h2ω2 cos 4hω +4h2ω2 cos 5hω− 56hω sin hω− 16hω sin 2hω +32hω sin 4hω− 8hω sin 5hω + 44  ω  42 sin hω− 12 sin 2hω− 27 sin 3hω + 22 sin 4hω −5 sin 5hω− 52hω + 64hω cos hω− 8hω cos 2hω −2hω cos 3hω− 4hω cos 4hω + 2hω cos 5hω  β22 = 1 2  38 cos hω + 98 cos 2hω− 47 cos 3hω− 10 cos 4hω +9 cos 5hω− 2 cos 6hω− 80h2ω2 + 72h2ω2 cos hω +8h2ω2 cos 3hω + 140hω sin hω + 16hω sin 2hω −90hω sin 3hω + 32hω sin 4hω− 6hω sin 5hω− 86  ω  42 sin hω− 12 sin 2hω− 27 sin 3hω + 22 sin 4hω −5 sin 5hω− 52hω + 64hω cos hω− 8hω cos 2hω −2hω cos 3hω− 4hω cos 4hω + 2hω cos 5hω  γ12 = − 1 2  −12 sin hω + 39 sin 2hω− 6 sin 3hω− 24 sin 4hω + 6 sin 5hω +3 sin 6hω− 52hω + 16h2ω2 sin hω + 32h2ω2 sin 2hω −24h2ω2 sin 3hω + 8h2ω2 sin 4hω− 8h2ω2 sin 5hω +16hω cos hω + 34hω cos 2hω + 20hω cos 4hω −16hω cos 5hω− 2hω cos 6hω  ω2  42 sin hω− 12 sin 2hω− 27 sin 3hω + 22 sin 4hω −5 sin 5hω− 52hω + 64hω cos hω− 8hω cos 2hω −2hω cos 3hω− 4hω cos 4hω + 2hω cos 5hω  γ22 = 1 2  −12 sin hω + 39 sin 2hω− 6 sin 3hω− 24 sin 4hω +6 sin 5hω + 3 sin 6hω + 32hω + 16h2ω2 sin hω −16h2ω2 sin 2hω + 48h2ω2 sin 3hω− 32h2ω2 sin 4hω −56hω cos hω− 32hω cos 2hω + 76hω cos 3hω− 20hω cos 5hω  ω2  42 sin hω− 12 sin 2hω− 27 sin 3hω + 22 sin 4hω −5 sin 5hω− 52hω + 64hω cos hω− 8hω cos 2hω −2hω cos 3hω− 4hω cos 4hω + 2hω cos 5hω  ζ12 = 1 2  24 cos hω + 63 cos 2hω− 28 cos 3hω− 10 cos 4hω +4 cos 5hω + cos 6hω− 32h2ω2 + 32h2ω2 cos hω −16h2ω2 cos 2hω + 32h2ω2 cos 3hω− 16h2ω2 cos 4hω +48hω sin hω + 48hω sin 2hω− 72hω sin 3hω + 8hω sin 4hω +8hω sin 5hω− 54  ω3  42 sin hω− 12 sin 2hω− 27 sin 3hω + 22 sin 4hω −5 sin 5hω− 52hω + 64hω cos hω− 8hω cos 2hω −2hω cos 3hω− 4hω cos 4hω + 2hω cos 5hω  writing (11) and (12) in discrete block method[ 1 0 0 1 ] [ yn+1 yn+2 ] = [ 0 1 0 1 ] [ yn−1 yn ] + [ 0 β01 0 β02 ] [ fn−1 fn ] (13) + [ β11 β21 β12 β22 ] [ fn+1 fn+2 ] + [ γ11 γ12 γ21 γ22 ] [ gn+1 gn+2 ] + [ 0 ζ11 0 ζ21 ] [ ln−1 ln+1 ] 41 kida et al. / j. nig. soc. phys. sci. 4 (2022) 34–48 42 2.4. analysis of the stability properties for case ii 2.4.1. order and error constant evaluating each row of (11) and (12) in taylor series about xn gives l [ y (x) ; h ] = yn+1 − yn −β01 fn −β11 fn+1 −β21 fn+2 −γ11gn+1 −γ21gn+2 − ζ11ln+2 = 0 therefore, the methods are of order p = (2, 2)t with the following error constants k = − 1 12  216 cos hω + 234 cos 2hω− 204 cos 3hω + 48 cos 4hω −12 cos 5hω + 6 cos 6hω− 258h2ω2 − 100h4ω4 + 294h2ω2 cos hω −426h2ω2 cos 2hω + 669h2ω2 cos 3hω− 270h2ω2 cos 4hω +76h4ω4 cos hω− 3h2ω2 cos 5hω− 6h2ω2 cos 6hω− 32h4ω4 cos 2hω +46h4ω4 cos 3hω + 20h4ω4 cos 4hω− 10h4ω4 cos 5hω + 216h3ω3 sin hω −24h3ω3 sin 2hω + 180h3ω3 sin 3hω− 212h3ω3 sin 4hω + 28h3ω3 sin 5hω −72hω sin hω + 594hω sin 2hω− 576hω sin 3hω + 156hω sin 4hω −24hω sin 5hω + 18hω sin 6hω− 288  ω3  42 sin hω− 12 sin 2hω− 27 sin 3hω + 22 sin 4hω −5 sin 5hω− 52hω + 64hω cos hω− 8hω cos 2hω− 2hω cos 3hω −4hω cos 4hω + 2hω cos 5hω  k = − 1 6  72 cos hω + 189 cos 2hω− 84 cos 3hω− 30 cos 4hω + 12 cos 5hω +3 cos 6hω− 132h2ω2 − 136h4ω4 − 66h2ω2 cos hω + 51h2ω2 cos 2hω +297h2ω2 cos 3hω− 108h2ω2 cos 4hω + 88h4ω4 cos hω− 39h2ω2 cos 5hω −3h2ω2 cos 6hω + 16h4ω4 cos 2hω + 28h4ω4 cos 3hω + 8h4ω4 cos 4hω −4h4ω4 cos 5hω + 384h3ω3 sin hω− 48h3ω3 sin 2hω + 36h3ω3 sin 3hω −104h3ω3 sin 4hω + 4h3ω3 sin 5hω + 108hω sin hω + 261hω sin 2hω −234hω sin 3hω− 48hω sin 4hω + 42hω sin 5hω + 9hω sin 6hω− 162  ω3  42 sin hω− 12 sin 2hω− 27 sin 3hω + 22 sin 4hω −5 sin 5hω− 52hω + 64hω cos hω− 8hω cos 2hω −2hω cos 3hω− 4hω cos 4hω + 2hω cos 5hω  2.4.2. zero stability of the block − ρ(λ) = det [ λa(1) − a(0) ] = 0 (14) where a(0), a(1) are from (13) λ [ 1 0 0 1 ] − [ 0 1 0 1 ] = det [ λ −1 0 λ− 1 ] ,λ2 −λ = λ (λ− 1) = 0 thus, solving for λ λ (λ− 1) = 0 which implies that λ1 = 0,λ2 = 1 hence, using theorem 1, theorem 2, theorem 3, the block method (13) is zero stable and also consistent as its order p = [2, 2]t > 1, thus it is convergent. 2.4.3. linear stability applying the test equation y(k) = λ(k)yn to yield yn+1 = µ (z) ym, µ (z) is the amplification equation given by µ (z) = − ( a(1) − zβ(1) − z2γ(1) − z3ζ(1) )−1 ( a(0) + zβ(0) + z2γ(0) ) (15) where a(0) = [ 0 1 0 1 ] , β(0) = [ 0 β01 0 β02 ] , γ(0) = [ 0 ζ11 0 ζ21 ] , a(1) = [ 1 0 0 1 ] , β(1) = [ β11 β21 β12 β22 ] and γ(1) = [ γ11 γ12 γ21 γ22 ] the matrix µ (z) has eigenvalues (0, 0, ...,ξk) where ξk is called the stability function which is the rational function with coefficients. µ (z) = [ 1364h5z4ω4 − 2816h4z2ω4 − 180h3z5 + 280h3z4ω2 +960h3ω4 + 270h2z3 − 420h2z2ω2 − 246hz4 + 369z2 ]  −288h5z4ω4 + 288h5z3ω4 + 560h4z2ω4 − 1260h3z4ω2 +360h3z3ω2 − 960h3ω4 + 150h2z5 + 2100h2z2ω2 +162hz4 − 342hz3 − 225z2  , 0 42 kida et al. / j. nig. soc. phys. sci. 4 (2022) 34–48 43 table 1. result for example i h 3bebdf maxe 3bbdf maxe error case i time error case ii time 10−2 1.68449e(−001) 6.62694e(−002) 3.6891e(−002) 0.178s 2.9440e(−002) 0.178s 10−3 5.14997e(−002) 7.44768e(−002) 3.6837e(−003) 0.178s 8.3499e(−003) 0.178s 10−4 6.95725e(−003) 8.45376e(−003) 2.6596e(−004) 0.179s 3.8627e(−004) 0.179s 10−5 7.16727e(−004) 8.53717e(−004) 2.6585e(−005) 0.179s 1.0017e(−005) 0.179s 2.4.4. region of absolute stability the region of absolute stability of the block method (13) is shown in figure 2 figure 2. region of absolute stability for case ii 3. numerical experiments evaluating the performance of the new block method on some challenging stiff and oscillatory problems which appeared in literature and compared the results with solution from some methods. the following notations were used in table: h step size maxe maximum error 3bbdf 3-point block backward differentiation formula 3be bdf 3-point block extended backward differentiation formula example i: consider a linear oscillatory problem in the interval 0 ≤ x ≤ 10( y′1 = 9y1 + 24y2 + 5 cos x − 1 3 sin x y′2 = −24y1 − 5y2 − 9 cos x + 1 3 sin x ) , ( y1 (0) y2 (0) ) = ( 4 3 2 3 ) the exact solution is given as ( y1 (x) y2 (x) ) = ( 4e−x − 3e−1000x −2e−x + 3e−1000x ) source: [25] table 1 shows that the new methods performed accurately and approximate better than the result of [25]. for instance, at h = 10−5, the error in [25] are 7.16727e(−004) and 8.53717e(−004) respectively, while the error in case 1 and case 2 are 2.6585e(−05) and 1.0017e(−05) respectively. the comparison of error in figure 3 clearly shows that, the new methods converge better than the existing method. 43 kida et al. / j. nig. soc. phys. sci. 4 (2022) 34–48 44 figure 3. error for example i table 2. results for example ii h 3bebdf maxe 3bbdf maxe case i time case ii time 10−2 8.35359e(−002) 1.08041e(−002) 1.7729e(−05) 0.3511s 1.7729e(−05) 0.3511s 10−3 9.10218e(−003) 1.08962e(−002) 1.0051e(−04) 0.3511s 2.1635e(−06) 0.3511s 10−4 9.18073e(−004) 1.09230e(−003) 5.0175e(−05) 0.3511s 2.0646e(−07) 0.3511s 10−5 9.18862e(−005) 1.09257e(−004) 4.1886e(−06) 0.3510s 7.7533e(−08) 0.3510s 10−6 9.18939e(−006) 1.09259e(−005) 4.3258e(−07) 0.3510s 7.4327e(−09) 0.3510s example ii: consider the linear stiff system in the interval 0 ≤ x ≤ 10( y′1 y′2 ) = ( −100y1 + 9.901y2 0.1y1 − y2 ) , ( y1 (0) y2 (0) ) = ( 1 10 ) the eigenvalues of the jacobian matrix of the system are λ1= −0.99 and λ2 = −100.01. the exact solution is given as( y1 (x) y2 (x) ) = ( exp (−0.99x) 10 exp (−0.99x) ) source: [25] table 2 shows that the new methods performed accurately and approximate better than the result of [25]. at h = 10−6, the error of [25] are 9.18939e(−006) and 1.09259e(−005) , while the error in case 1 and case 2 are 4.3258e(−07) and 7.4327e(−09) respectively. this clearly shows that the new methods performed better than of [25].the comparison of error in figure 4 clearly shows that, the new methods converge better than the existing method. example iii: consider a linear oscillatory problem in the interval 0 ≤ x ≤ 10( y′1 y′2 ) = ( −2y1 + y2 + 2 sin x 998y1 − 999y2 + 999 (cos x − sin x) ) , ( y1 (0) y2 (0) ) = ( 2 3 ) the exact solution is given as ( y1 (x) y2 (x) ) = ( 2 exp (−x) + sin x 2 exp (−x) + cos x ) source: [26] table 3 shows that the new methods performed accurately and approximate better than the result of [26]. from the result obtained for example iii as shown in the table 3, it is evident that the class of third derivative trigonometrically fitted method 44 kida et al. / j. nig. soc. phys. sci. 4 (2022) 34–48 45 figure 4. error for example ii table 3. result for example iii h maxe in [26] case i time case ii time 10−2 8.4000e (−003) 5.4393e(−04) 0.21322s 2.6947e(−04) 0.21322s 10−4 1.6621e (−004) 2.6947e(−05) 0.21322s 1.0276e(−06) 0.21322s 10−6 2.7506e (−006) 1.3195e(−08) 0.21321s 6.9756e(−08) 0.21321s figure 5. error for example iii 45 kida et al. / j. nig. soc. phys. sci. 4 (2022) 34–48 46 table 4. result for example iv s teps ohbm case i case ii 20 2.943 × 10−5 1.311 × 10−5 1.122 × 10−5 40 1.250×10−6 1.406×10−8 1.465×10−8 80 2.721×10−13 2.545×10−15 1.843×10−15 160 4.262×10−15 2.733×10−18 1.033×10−18 320 7.707×10−18 8.371×10−20 9.572×10−20 640 3.000×10−21 1.721×10−21 1.953×10−21 computational time 0.03125s 0.02034s 0.02034s performed better than those of [26]. the comparison of error in figure 5 clearly shows that, the new methods converge better than the existing method. example iv: consider the following linear systems y′1(x) = −21y1 + 19y2 − 20y3 y′2(x) = 19y1 − 21y2 + 20y3 y′3(x) = 40y1 − 40y2 − 40y3  ,  y1 (0) y2 (0) y3 (0)  =  1 0 −1  the exact solution is given as  y1 (x) y2 (x) y3 (x)  =  1 2 ( e−2x + e−40x (cos (40x) + sin (40x)) ) 1 2 ( e−2x − e−40x (cos (40x) + sin (40x)) ) 1 2 e −40x (cos (40x) + sin (40x))  source: [2] we compared the results of case i and case ii along side ohbm by comparing the maximum relative errors over the three components y1(x), y2(x) and y3(x). as shown in table 4, the new methods in case i and case ii proved to be superior in terms of accuracy and computation time. figure 6. error for example iv table 5. result for example v h [24] error case i error case ii 2π/300 2.58313×10−9 2.03334×10−9 2.03034×10−9 2π/600 1.07326×10−10 2.08320×10−12 2.34020×10−12 2π/1200 3.48507×10−12 5.02513×10−12 5.01013×10−12 46 kida et al. / j. nig. soc. phys. sci. 4 (2022) 34–48 47 example v: consider the following problem( y′′(x) = −100y(x) + 99 sin(x) ) , y1 (0) = 1, y ′ (0) = 11, x ∈ [0, 2π] the exact solution is given as y (x) = cos(10x) + sin(10x) + sin(x) source: [24] table 6. result for example v h [24] error case i error case 2π/300 2.58313×10−9 2.03334×10−9 2.03034×10−9 2π/600 1.07326×10−10 2.08320×10−12 2.34020×10−12 2π/1200 3.48507×10−12 5.02513×10−12 5.01013×10−12 results of case i and case ii was compared with [24] as shown in table 5. the new methods compute favorably with [24] and has better consistency. figure 7. error for example v 4. conclusion a class of third derivative continuous block methods for the solution of stiff and oscillatory problems is constructed using collocation and interpolation technique. the approximate solution adopted for the development of the methods comprises of a combination of polynomial and trigonometric functions. a consistent, zero stable and convergent block methods with continuous coefficients are developed and implemented by writing a code using matlab 8.5. the methods are presented in continuous and discrete form. the accuracy of the block methods were tested on some stiff and oscillatory ivp to generate results and compared the results with the results of some existing methods. the results of the new methods compete favorably with the existing methods with better accuracy, consistency and computational time. thus the methods are consistent, convergent and zero stable. hence the methods derived are suitable for solving stiff and oscillatory problems and computationally reliable. references [1] a. o. adesanya, m. r. odekunle & a. o. adeyeye, ”continuous block hybrid 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[31] r. abdelrahim & z. omar, ”a four-step implicit block method with three generalized off-step points for solving fourth order initial value problem directly”, journal of king saud university science 29 (2017) 401. 48 j. nig. soc. phys. sci. 3 (2021) 360–384 journal of the nigerian society of physical sciences computational studies on emodin (c15h10o5) from methanol extract of pteridium acquilinum leaves m. e. khana, e. e. etimb,, v. j. anyamc, a. abeld, i. g. osigbemhee, c. t agberf adepartment of chemistry, federal university lokoja, kogi state, nigeria bdepartment of chemical science, federal university wukari taraba state nigeria cdepartment of chemistry university of agriculture makurdi, benue state nigeria ddepartment of pure and applied chemistry, adamawa state university, mubi, adamawa state nigeria edepartment of industrial chemistry, federal university lokoja, kogi state, nigeria fdepartment of chemistry, benue state university, makurdi, benue state, nigeria abstract this research isolated, characterized, and studied the computational and frequency calculations of emodin, extracted from the leaf extract of pteridium acquilinum using methanol. vacuum liquid and tin layer chromatographic techniques were used for the purification of the molecule. the (vlc purified), fraction was analyzed by nuclear magnetic resonance (nmr) and the chemical structure of the compound isolated (anthraquinone), was confirmed by 1h & 13c-nmr analyses as emodin (c15h10o5) computational and frequency studies were done on the isolated molecule. optimized geometry, ir frequencies, bond distances (r) and angles (a), dipole moments and other parameters have been computationally determined for the isolated molecule from quantum chemical calculations using the gaussian 09 retinue programs. experimentally determined and computationally measured ir frequencies agreed perfectly with each other. computational studies have been used to predict unobserved chemical phenomena like design of new drugs and materials such as the positions of constituent atoms in relationship to their relative and absolute energies, electronic charge densities, dipoles, higher multiple moments, vibrational frequencies, relativity or other spectroscopic quantities and cross sections for collision with other molecules. this is the first time this anthraquinone, [emodin], with most of the parameters examined is reported from p. aquilinum. doi:10.46481/jnsps.2021.301 keywords: pteridium aquiilinum, isolated & characterized, chromatography, emodin, optimized geometry, computational and frequency studies. article history : received: 14 july 2021 received in revised form: 14 october 2021 accepted for publication: 01 november 2021 published: 29 november 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: b. j. falaye 360 khan et al. / j. nig. soc. phys. sci. 3 (2021) 360–384 361 1. introduction the phenomenon of screening plants for the presence of phyto-molecules of medicinal importance is common place. natural products have pharmaceutical / pharmacological activity that are useful in treating diseases and are the starting points for drugs discovery from which synthetic drug analogues can be prepared with improved efficacy, potency, safety and purity [1], when isolated, synthetic strategies and tactics are used by organic chemists providing challenging mechanisms that permit the biologically active product to the target site. a review reported that 577 plant species have been used traditionally due to the secondary metabolites in them, [2 & 3]. africa is blessed with its natural pharmacy of variety of plants. chemists take the pain to analyse these plants‘ secondary metabolites for the benefit of mankind as they are precursors for modern drugs, [4]. based on this concept and theory, the optimal geometry of the eagle fern was calculated. pteridium aquilinum, brake or common bracken, also known as ”eagle fern,” is a species of fern occurring in temperate and subtropical regions in both hemispheres. the extreme lightness of its spores has led to its global distribution, [5]. the large, roughly triangular fronts of the fern are herb-like rhizomes produced singly, arising upwards from an underground, and grow to 1–3 m (3–10 ft) tall; the main stem, or stipe is up to 1 cm (0.4 in) diameter at the base. it is an adaptable plant, which readily colonies disturbed areas. it is a vascular plant that reproduces via spores and has neither seeds nor flowers. it differs from moss by being vascular and it‘s of the class polypodiopsidae. they are the best house purifying plants with their evergreen leaves that help rid the home of harmful toxins and improve humanity by helping to restore moisture to air naturally and also combat winter dryness by raising indoor humidity. they are used as cooked vegetable in baffousam (cameroon) and consume with vernonia amygdalina, delile and triumfetta rhomboidae. when soaked in wood ash for 24 – 36 hours, free tannic acids are removed and the crosiers consumed and sold as “warabi” or “zenmai in japan. rhizomes are consumed in france, madagascar, and the canary island and also used as starch and confections. leaves are used as straw and bedding for cattle, also to filter oil and palm wine. in cote d‘ivoire, powdered crosiers are applied to old wounds and also as enema to overcome sterility in women. the rhizomes are mixed with zingiber officinal in juice form and taken as aphrodisiac and with others to calm mental disability. in china, water soaked leaves are used as pesticides. the ash used in europe for glass and soap production, [6]. studies carried on pteridium acquilinumin included, potential and historical uses for braken (l) kuhn, in organic agriculture where the braken were considered a serious weed species, due to toxic constituents and negativity on agriculture and conservation, [7]. the resistance of p. acquilinum (l)kuhn, to insect attack by trichoplusia ni (hubn) where dried braken leaf meals and extracts of the leaf was incorporated into an artificial diet for trichoplusia ni larvae and studied, [8]. isolation and characterization of the bio assay active molecule(s) from the exemail address: emmaetim@gmail.com (e. e. etim ) tract of the leaves of pteridium acquilinum using the aqueous and methanolic leaf extracts was carefully examined and the extracts used in boosting some female rats hormones, [2]. the plant has many chemical compounds such as emodin, quercetin, shikimic acid, prunasin, ptaquiloside and a ‘bleeding factor’ of other known and unknown structures [9] almost every health challenge in the world has a solution in natural products. thus, the discovery of pharmaceutical drugs remains one of the preeminent tasks in biomedical and related research areas. advances in science and technology coupled with the development of quantum chemistry, new computational models and software including user-friendly interfaces have reduced the barriers to the application of computational tools in the discovery and structure elucidation of natural products. consequently, the use of computational chemistry software as a tool to discover and determine the structure of natural products has become more common in recent years. there are several reports of recent studies where computational chemistry is applied to facilitate the discovery and structure elucidation of various natural products with the view to giving insights about the isolated compounds, [10, 11, 12, 13 & 14]. computational studies are based on quantum mechanics and basic physical constants, with involvement of approximations, but tractable; thus, help find entirely new chemical objects. they are used to find a starting point for a laboratory synthesis or to assist in understanding experimental data such as position and source of spectroscopic peaks and to predict the possibility of entirely unknown molecules or to explore reaction mechanisms not readily studied via experiments. this work is thus, designed to carry out the characterization and identification of bioactive compounds from p. aquilinum leaf extracts and the quantum chemical calculations are employed to further scrutinised and give a better insight about the isolated molecule for the enhancement of human life. 2. materials and methods 2.1. sample collection and authentication leaves of p. acquilinum that were still green but matured, were collected in and around michika l. g. a, adamawa state on 28th july 2014. authentication of the plant species was by [15], in the state ministry of forestry, mubi north lga, adamawa state and a specimen of the plant was kept in their herbarium. the forestry herbarium index number (fhi) is 1030. 2.2. preparation of the plant sample leaves of the plant were properly washed with running tap water to avoid dust and other unwanted materials that most have accumulated on the leaves from their natural habitat. then, dust free leaves were kept to dry under shade in the chemistry laboratory of adamawa state university mubi. these dried leaves were pulverized by using mortar and pestle. finally, fine powder was obtained from the pulverized leaves by sieving through the kitchen strainer and used for extraction. 361 khan et al. / j. nig. soc. phys. sci. 3 (2021) 360–384 362 2.3. successive extractions using microwave assisted extraction (mae) 250g of the powdered sample was poured into a glass (2.5 l) and 500cm3 of normal-hexane was added to it. the bottle was then put into the microwave set at defrost every 3 minutes and removed and cooled. this was repeated 10 times. after filtration, the residues were further extracted similarly using ethyl acetate followed by methanol, [16], and the yields thus: n-hexane, 10g, ethyl acetate, 15g and methanol, 28g respectively. 2.4. vacuum liquid chromatography (vlc) 15g of the methanol extract was dissolved and mixed with celite and left to dry. the dried mixture was then loaded on the vlc that had already been packed with silica gel. it was then rinsed 20 times with 20 cm3 each of n-hexane and ethyl acetate. ethyl acetate methanol gradient was used to elute the column and 60 fractions collected. 2.5. thin layer chromatography tlc was carried out on all the fractions using a solvent gradient system of 9:1 v/v chloroform in methanol. fractions 1219 were combined and allowed to dry. the dried sample was then dissolved in methanol but yellow crystals were left undissolved. this was carefully washed. a yellow component was thus purified, spotted on the tlc plate and was labelled (f1), [17] 2.6. sephadex column dissolved components of fractions 12-19, were put on a column loaded with sephadex and eluted with solvent gradient system of 1:3:3 v/v/v methanol, chloroform and ethyl acetate. seven (7) fractions of 2 cm3 each were collected and spotted on the tlc plate labelled (a) and (b) 2.7. analysis with nuclear magnetic resonance (nmr) machine fraction (a), rf: 0.65, of the vlc, (with a yellow colour) that was purified with the sephadex column was sent for nmr analyses. the nmr spectra were run at sipbs, university, strathclyde, glasgow, united kingdom on jeol-la-400 mhz ft-nmr spectrophotometer [2]. 2.8. quantum chemical calculations current advances in theoretical and computational thermodynamics have made it easier to study systems, molecular interactions, reactions and predict parameters which would have been experimentally impossible or very difficult to study. the gaussian 09 retinue of programs was used for all the quantum chemical calculations reported here. the molecule was optimized at the m06-2x level of theory with the 6 31g (d,p) 6 31+g* basis set. the m06-2x functional is a high-nonlocal functionality with double amount of nonlocal exchange (2x). the optimized structure was stable with real frequencies as shown from the frequency calculations [18, 19, 20, 22-26]. 3. results and discussion detailed computational and frequency studies of the molecule was done and all thermodynamic parameters investigated. this could help to properly identify and justly place the molecule in its chemical context and use it adequately in bioactivity studies and for some bioassays in researches. these investigations and detail analyses were carried out with the aid of chromatography (vlc, tlc and sephadex) and other available spectroscopic techniques like nuclear magnetic resonance (nmr) (1h & 13c-nmr). the computational and frequency studies were carried out using gaussian 09 retinue programs thus; levels of theory set: hf/6-311g, charge = 0, multiplicity = 1, stoichiometry c15h10o5, framework group c1[x(c15h10o5)], deg. of freedom: 84, full point group : c1, largest abelian subgroup :c1. largest concise abelian subgroup: c1, van-derwaal spheres, ir & raman spectra, bond distances (r) and angles (a), dipole moments; field –independent basis (debye), rotational constants, homo-lumo structures, molecular orbital levels and the band gaps of: 0.33925 were investigated to bring the structural definition of the molecule, as ‘emodin’ qualitative tlc, results of p. aquilinum leaf extract (table 1), performed on the methanol crude extracts are presented and discussed. the two spots detected on the tlc plate of the sephadex column fraction; yellow colour, implied that there are two different components in the extract. retention factor (rf) = distance travelled by the solution/ distance travelled by the solvent front; rf (fa) = 2.2/3.7 = 0.59 and rf (fb) = 2.5/3.7 = 0.68. table 1. result of qualitative tlc of leaf methanol extract of p. aquilinum extract no. of spots r f quantity leaf 1 2.2/3.7 = 0.59 0.5mg leaf 1 2.5/3.7=0.68 (small) 3.1. optimized geometry figure 1, portrays the optimized geometry of p. aquilinum isolated from the methanol extract. optimized geometry of p. aquilinum, (figure 1), obtained at the m062x/6-31g (d,p) level of theory and the van-der-waals sphere (figure 2), are representations of emodin illustrating where a surface might reside for the molecule based on the hard cutoffs of van-der-waals radii for the individual atoms making up the molecule. [12] 3.2. ftir spectra data for isolated emodin the ftir spectrum displayed c-oh stretching hydroxyl groups at 3500cm−1, c-h asymmetric stretching in -ch3 at 2925 cm−1, -ch2stretching frequency at 2900 cm−1, -c=o asymmetric stretching in carbonyl at 1750, and c-h bending in ch3 at 1400 cm−1. ftir analysis of isolated emodin is presented in table 2 and it also attested to the ir spectrum of the computationally obtained levels at the m062x/6-31g (d, p) of the molecule.. 362 khan et al. / j. nig. soc. phys. sci. 3 (2021) 360–384 363 table 2. the ftir spectral data and interpretation of isolated methanol extract frequency range cm -1 vibrational mode remarks 3500 c –oh stretching hydroxyl 2925 c-h-asymmetric stretching -ch3 2900 c-h stretching frequencies -ch21750 c=o stretching carbonyl 1400 c-h bending -ch2figure 1. models of optimized geometry of p. aquilinum obtained at the m06 2x /6-31g (d,p) level of theory all the major peaks obtained experimentally are in consonance with those obtained computationally at the m062x/631g (d, p) level. this further validates both the experimental and computational results. the frequency from 8cm−1 to 3902cm−1 and the corresponding intensity for the ir spectrum obtained at the m062x/6-31g (d, p) level are supporting information. regarding microwave (or rotational) spectroscopy, this molecule is active with a total dipole moment of 3.7801 debye obtained at the m062x/6-31g (d, p) level, table 7. also, its microwave spectrum has been measured. as an asymmetric top molecule with three different moments of inertia corresponding to the three principal axes, this molecule is expected and it has three different rotational constants. at the m062x/6-31g (d,p) level, the rotational constants obtained for the molecule figure 2. van-der-waals sphere for p. aquilinum obtained at the m062x/6-31g (d, p) level of optimized geometry figure 3. ir spectrum are 0.7462698, 0.2467690 and 0.1856618 ghz corresponding to the a, b and c rotational constants respectively, table 8. figure 3 portrays the ir spectrum of emodin obtained at the m062x/6-31g (d, p) level of theory, 1hnmr and 13cnmr spectra interpretation for methanol extract. 1h nmr of the sample (table 3) contains; 4 sets of aromatic peaks (7.53, 7.23, 7.12 and 6.62), two sets of phenolic peaks (12.12 and 12.06 ) and one set of methyl groups (2.38, attached to aromatic ring) protons. 13c nmr chemical shift values (table 4) revealed the presence of; 12 aromatic peaks (166.2, 164.9, 161.8, 148.7, 135.7, 133.4, 124.6, 121.0, 113.9, 109.5, 108.5 and 108.4), 2 carbonyls 363 khan et al. / j. nig. soc. phys. sci. 3 (2021) 360–384 364 figure 4. bonf distances and angles figure 5. homo lumo diagram figure 6. raman spectra (190.7 and 182.2) and methyl (22.0, benzylic) carbons. this inferred that there were two aromatic nuclei that were joined through two carbonyl carbons. thus, an anthraquinone. it can be suggested again that one of the aromatic nuclei of this anthraquinone contains a methyl and a hydroxyl substituent and the other contains two, hydroxyl substituents, one at position 8 and the other at 6. using 2d nmr spectra correlation (cosy), heteronuclear single quantum correlation (hsqc) and heteronuclear multiple bonds correlation (hmbc), emodin was identified as (1, 6, 8-trihydroxy-3-methylemodin), which is an anthraquinone. both 1d and 2d nmr spectra and also a comparison of the structure of the isolated molecule with literature values of same compound isolated from other plants like rumex japonica, further confirmation was alluded to [21, 12 &13]. table 3. 1h nmr (400 mhz, dmso solvent) position chemical shift (δ) ppm (j in hz) multiplicity 1 2 7.20 1h, m 3 4 7.53 (1.57) 1h, d 5 6.62 (2.40) 1h, d 6 7 7.14 (2.39) 1h, d 8 9 10 11 12 13 14 3-ch3 2.43 3h, s 1-oh 2.06 1h, s 8-oh 12.12 1h, s 6-oh 11.30 1h,s table 4. 13c nmr result (dmso) position chemical shift (δ) type of c 1 161.8 c 2 124.6 ch 3 148.7 c 4 121.0 ch 5 108.4 ch 6 166.2 c 7 109.5 ch 8 164.9 c 9 190.7 c 10 182.2 c 11 113.9 c 12 133.4 c 13 108.5 c 14 135.7 c 3-ch3 22.0 ch3 1h and 13c (nmr) of the spectra were recorded on nmr machine: jeol-la-400 mhz ft-nmr spectrophotometer, at sipbs, university, strathclyde, glasgow, united kingdom, using deuterated solvents as indicated by tables 3 and 4. 3.3. bond distances and bond angles figure 4 is the optimized geometry of emodin showing the atomic numbers. these numbers help in determining the 364 khan et al. / j. nig. soc. phys. sci. 3 (2021) 360–384 365 distance between two atoms (say atoms 3 and 5) and the angles between atoms. table 6 of the supporting information contains the complete bonds distances (in angstrom) and bond angles (in degrees), of emodin isolated from the methanol extract. 3.4. homolumo diagram here electron flow, shows whether there is constructive overlap / bonding interaction, or not, between the orbital of the homo and those of the lumo. it also indicates that the orbitals are either occupied or unoccupied, table 9 and figure 5. when the overlap is favourable, electron movement is symmetry allowed. the computational calculations have validated the theory of this interaction. 3.5. raman spectra this provided good information about the structure of emodin. in this write up, the phase and the polymorphy, crystalline and molecular interactions of the atoms were computed and they tallied with the theory. (see figure 6 and table 10). table 5: ir values (frequencies: cm−1 and their intensities) frequency, cm−1 ir intensity 40.5889 1.7505 58.1541 0.1401 72.4885 0.1147 123.9918 3.1669 152.1424 2.1859 180.5744 0.4699 185.0588 1.6663 255.505 3.307 256.5763 0.0736 277.7802 4.6633 294.7906 0.2492 297.6356 2.5751 351.015 0.4827 366.8926 1.6488 382.5217 190.7134 400.1286 9.5094 420.2978 0.9179 464.6792 34.443 477.8178 16.8814 505.9458 1.3444 520.2946 9.4524 557.7997 4.895 605.0876 14.3639 606.812 13.8679 613.2264 0.8377 652.8296 4.129 665.3268 23.4006 686.8013 28.6752 701.3117 10.2161 717.2705 0.6281 737.3912 2.1851 744.8034 9.2651 772.1549 283.7709 786.5867 5.1204 806.2587 84.9909 844.6811 86.8029 845.4904 24.5256 949.8981 67.5742 continued on next page 365 khan et al. / j. nig. soc. phys. sci. 3 (2021) 360–384 366 table 5 – continued from previous page frequency, cm−1 ir intensity 980.7929 33.8411 991.5636 66.5423 1008.3375 5.9754 1019.5568 2.2067 1063.7093 1.3686 1087.4272 13.171 1110.4729 52.0946 1138.1731 32.1339 1196.3708 5.8936 1205.8194 206.5051 1230.0187 7.0492 1253.0616 123.2033 1291.437 224.8896 1294.8107 424.9948 1311.5505 92.2745 1348.5262 102.8252 1393.5713 71.2756 1400.046 119.8228 1440.4723 349.6365 1449.158 118.6796 1486.7681 127.3453 1504.6975 67.6164 1515.7165 664.1199 1563.6894 1.0644 1577.6142 7.1853 1599.795 7.4221 1627.5681 97.4458 1640.3534 10.8432 1642.8372 63.6547 1660.0202 41.9661 1725.1233 26.9024 1743.1457 155.4465 1779.4404 42.6281 1800.8258 161.4526 1805.9447 688.232 1858.3537 0.0332 3168.3115 25.498 3223.758 22.4323 3253.8947 25.0516 3375.7378 3.6376 3377.4741 1.6247 3378.858 5.0178 3411.1628 0.2215 3907.6476 236.8743 3914.4907 145.2159 4084.1357 117.2641 structural illustrations of bond distance and bond angles table 6: numerical values of bond distances and angles r(1-2) 1.405 continued on next page 366 khan et al. / j. nig. soc. phys. sci. 3 (2021) 360–384 367 table 6 – continued from previous page r(1-8) 1.387 r(1-19) 1.349 r(2-3) 1.457 r(2-4) 1.405 r(3-7) 1.460 r(3-20) 1.251 r(4-5) 1.487 r(4-9) 1.373 r(5-6) 1.482 r(5-10) 1.221 r(6-7) 1.410 r(6-11) 1.370 r(7-12) 1.395 r(8-18) 1.374 r(8-21) 1.066 r(9-18) 1.395 r(9-22) 1.069 r(11-14) 1.405 r(11-15) 1.069 r(12-13) 1.396 r(12-16) 1.352 r(13-14) 1.375 r(13-17) 1.069 r(14-27) 1.505 r(16-25) 0.954 r(18-23) 1.364 r(19-24) 0.954 r(23-26) 0.946 r(27-28) 1.082 r(27-29) 1.082 r(27-30) 1.079 r(20-24) 1.828 r(20-25) 1.834 a(2-1-8) 120.5 a(2-1-19) 123.3 a(1-2-3) 121.1 a(1-2-4) 118.2 a(8-1-19) 116.2 a(1-8-18) 119.8 a(1-8-21) 119.5 a(1-19-24) 113.6 a(3-2-4) 120.7 a(2-3-7) 119.6 a(2-3-20) 120.2 a(2-4-5) 120.3 a(2-4-9) 121.4 a(7-3-20) 120.1 a(3-7-6) 120.8 a(3-7-12) 121.1 a(3-20-24) 105.4 a(3-20-25) 105.5 a(5-4-9) 118.3 a(4-5-6) 118.4 continued on next page 367 khan et al. / j. nig. soc. phys. sci. 3 (2021) 360–384 368 table 6 – continued from previous page a(4-5-10) 120.4 a(4-9-18) 119.1 a(4-9-22) 119.4 a(6-5-10) 121.2 a(5-6-7) 120.1 a(5-6-11) 118.8 a(7-6-11) 121.1 a(6-7-12) 118.1 a(6-11-14) 120.5 a(6-11-15) 118.9 a(7-12-13) 120.2 a(7-12-16) 123.9 a(18-8-21) 120.7 a(8-18-9) 121.1 a(8-18-23) 117.2 a(18-9-22) 121.5 a(9-18-23) 121.7 a(14-11-15) 120.6 a(11-14-13) 118.7 a(11-14-27) 119.9 a(13-12-16) 115.9 a(12-13-14) 121.3 a(12-13-17) 117.1 a(12-16-25) 113.5 a(14-13-17) 121.6 a(13-14-27) 121.4 a(14-27-28) 110.8 a(14-27-29) 110.8 a(14-27-30) 111.4 a(16-25-20) 135.9 a(18-23-26) 115.2 a(19-24-20) 136.3 a(28-27-29) 107.5 a(28-27-30) 108.1 a(29-27-30) 108.1 a(24-20-25) 149.1 w1(a) 40.6 w2(a) 58.2 w3(a) 72.5 w4(a) 124.0 w5(a) 152.1 w6(a) 180.6 w7(a) 185.1 w8(a) 255.5 w9(a) 256.6 w10(a) 277.8 w11(a) 294.8 w12(a) 297.6 w13(a) 351.0 w14(a) 366.9 w15(a) 382.5 w16(a) 400.1 w17(a) 420.3 continued on next page 368 khan et al. / j. nig. soc. phys. sci. 3 (2021) 360–384 369 table 6 – continued from previous page w18(a) 464.7 w19(a) 477.8 w20(a) 505.9 w21(a) 520.3 w22(a) 557.8 w23(a) 605.1 w24(a) 606.8 w25(a) 613.2 w26(a) 652.8 w27(a) 665.3 w28(a) 686.8 w29(a) 701.3 w30(a) 717.3 w31(a) 737.4 w32(a) 744.8 w33(a) 772.2 w34(a) 786.6 w35(a) 806.3 w36(a) 844.7 w37(a) 845.5 w38(a) 949.9 w39(a) 980.8 w40(a) 991.6 w41(a) 1008.3 w42(a) 1019.6 w43(a) 1063.7 w44(a) 1087.4 w45(a) 1110.5 w46(a) 1138.2 w47(a) 1196.4 w48(a) 1205.8 w49(a) 1230.0 w50(a) 1253.1 w51(a) 1291.4 w52(a) 1294.8 w53(a) 1311.6 w54(a) 1348.5 w55(a) 1393.6 w56(a) 1400.0 w57(a) 1440.5 w58(a) 1449.2 w59(a) 1486.8 w60(a) 1504.7 w61(a) 1515.7 w62(a) 1563.7 w63(a) 1577.6 w64(a) 1599.8 w65(a) 1627.6 w66(a) 1640.4 w67(a) 1642.8 w68(a) 1660.0 w69(a) 1725.1 w70(a) 1743.1 continued on next page 369 khan et al. / j. nig. soc. phys. sci. 3 (2021) 360–384 370 table 6 – continued from previous page w71(a) 1779.4 w72(a) 1800.8 w73(a) 1805.9 w74(a) 1858.4 w75(a) 3168.3 w76(a) 3223.8 w77(a) 3253.9 w78(a) 3375.7 w79(a) 3377.5 w80(a) 3378.9 w81(a) 3411.2 w82(a) 3907.6 w83(a) 3914.5 w84(a) 4084.1 table 7. dipole moment (field –independent basis, debye) x 1.9607 y -3.2319 z -0.0025 total 3.7801 table 8. rot. constants for emodin rotational constants ghz a 0.7462698 b 0.2467690 c 0.1856618 table 9: numerical indicators of occupied and unoccupied molecular orbitals 1 -20.6106 occupied a 2 -20.59061 occupied a 3 -20.58999 occupied a 4 -20.58463 occupied a 5 -20.58063 occupied a 6 -11.39494 occupied a 7 -11.38389 occupied a 8 -11.35976 occupied a 9 -11.3571 occupied a 10 -11.35506 occupied a continued on next page 370 khan et al. / j. nig. soc. phys. sci. 3 (2021) 360–384 371 table 9 – continued from previous page 11 -11.30275 occupied a 12 -11.29346 occupied a 13 -11.29192 occupied a 14 -11.28269 occupied a 15 -11.28221 occupied a 16 -11.2699 occupied a 17 -11.2639 occupied a 18 -11.26076 occupied a 19 -11.25788 occupied a 20 -11.24752 occupied a 21 -1.45133 occupied a 22 -1.443 occupied a 23 -1.43897 occupied a 24 -1.43163 occupied a 25 -1.4124 occupied a 26 -1.21648 occupied a 27 -1.19863 occupied a 28 -1.12143 occupied a 29 -1.07739 occupied a 30 -1.06499 occupied a 31 -1.05973 occupied a 32 -0.9952 occupied a 33 -0.9526 occupied a 34 -0.92195 occupied a 35 -0.89005 occupied a 36 -0.86663 occupied a 37 -0.83926 occupied a continued on next page 371 khan et al. / j. nig. soc. phys. sci. 3 (2021) 360–384 372 table 9 – continued from previous page 38 -0.79888 occupied a 39 -0.77404 occupied a 40 -0.76007 occupied a 41 -0.73987 occupied a 42 -0.72586 occupied a 43 -0.69958 occupied a 44 -0.69417 occupied a 45 -0.67067 occupied a 46 -0.66231 occupied a 47 -0.65235 occupied a 48 -0.64739 occupied a 49 -0.63103 occupied a 50 -0.62251 occupied a 51 -0.61638 occupied a 52 -0.60289 occupied a 53 -0.60155 occupied a 54 -0.59301 occupied a 55 -0.58622 occupied a 56 -0.58475 occupied a 57 -0.56457 occupied a 58 -0.56087 occupied a 59 -0.55885 occupied a 60 -0.54681 occupied a 61 -0.54236 occupied a 62 -0.53297 occupied a 63 -0.49741 occupied a 64 -0.48094 occupied a continued on next page 372 khan et al. / j. nig. soc. phys. sci. 3 (2021) 360–384 373 table 9 – continued from previous page 65 -0.46984 occupied a 66 -0.44326 occupied a 67 -0.37943 occupied a 68 -0.36226 occupied a 69 -0.35055 occupied a 70 -0.33995 occupied a 71 -0.0007 unoccupied a 72 0.06955 unoccupied a 73 0.10587 unoccupied a 74 0.13378 unoccupied a 75 0.13583 unoccupied a 76 0.14836 unoccupied a 77 0.18701 unoccupied a 78 0.2009 unoccupied a 79 0.2011 unoccupied a 80 0.2037 unoccupied a 81 0.21134 unoccupied a 82 0.22203 unoccupied a 83 0.22495 unoccupied a 84 0.23454 unoccupied a 85 0.23529 unoccupied a 86 0.25213 unoccupied a 87 0.32229 unoccupied a 88 0.34319 unoccupied a 89 0.34611 unoccupied a 90 0.35786 unoccupied a 91 0.37219 unoccupied a continued on next page 373 khan et al. / j. nig. soc. phys. sci. 3 (2021) 360–384 374 table 9 – continued from previous page 92 0.3831 unoccupied a 93 0.4018 unoccupied a 94 0.40497 unoccupied a 95 0.41765 unoccupied a 96 0.43383 unoccupied a 97 0.44363 unoccupied a 98 0.44626 unoccupied a 99 0.45481 unoccupied a 100 0.46011 unoccupied a 101 0.48707 unoccupied a 102 0.49981 unoccupied a 103 0.50292 unoccupied a 104 0.50877 unoccupied a 105 0.51194 unoccupied a 106 0.52321 unoccupied a 107 0.52819 unoccupied a 108 0.53979 unoccupied a 109 0.54525 unoccupied a 110 0.55944 unoccupied a 111 0.56865 unoccupied a 112 0.57009 unoccupied a 113 0.58434 unoccupied a 114 0.58662 unoccupied a 115 0.59683 unoccupied a 116 0.60621 unoccupied a 117 0.60814 unoccupied a 118 0.61375 unoccupied a continued on next page 374 khan et al. / j. nig. soc. phys. sci. 3 (2021) 360–384 375 table 9 – continued from previous page 119 0.61988 unoccupied a 120 0.62327 unoccupied a 121 0.62597 unoccupied a 122 0.62776 unoccupied a 123 0.63742 unoccupied a 124 0.65303 unoccupied a 125 0.65887 unoccupied a 126 0.66405 unoccupied a 127 0.67701 unoccupied a 128 0.68162 unoccupied a 129 0.68987 unoccupied a 130 0.69117 unoccupied a 131 0.69598 unoccupied a 132 0.7286 unoccupied a 133 0.73459 unoccupied a 134 0.75046 unoccupied a 135 0.75306 unoccupied a 136 0.76128 unoccupied a 137 0.77724 unoccupied a 138 0.77807 unoccupied a 139 0.77996 unoccupied a 140 0.80091 unoccupied a 141 0.80389 unoccupied a 142 0.81138 unoccupied a 143 0.81625 unoccupied a 144 0.82077 unoccupied a 145 0.83575 unoccupied a continued on next page 375 khan et al. / j. nig. soc. phys. sci. 3 (2021) 360–384 376 table 9 – continued from previous page 146 0.84813 unoccupied a 147 0.86274 unoccupied a 148 0.88017 unoccupied a 149 0.88642 unoccupied a 150 0.88885 unoccupied a 151 0.89606 unoccupied a 152 0.92417 unoccupied a 153 0.93098 unoccupied a 154 0.9795 unoccupied a 155 0.98929 unoccupied a 156 1.01128 unoccupied a 157 1.01814 unoccupied a 158 1.02265 unoccupied a 159 1.02918 unoccupied a 160 1.05372 unoccupied a 161 1.06017 unoccupied a 162 1.06199 unoccupied a 163 1.06346 unoccupied a 164 1.07991 unoccupied a 165 1.08652 unoccupied a 166 1.09138 unoccupied a 167 1.10578 unoccupied a 168 1.11793 unoccupied a 169 1.1187 unoccupied a 170 1.13433 unoccupied a 171 1.14179 unoccupied a 172 1.14713 unoccupied a continued on next page 376 khan et al. / j. nig. soc. phys. sci. 3 (2021) 360–384 377 table 9 – continued from previous page 173 1.16523 unoccupied a 174 1.17615 unoccupied a 175 1.1908 unoccupied a 176 1.19371 unoccupied a 177 1.20229 unoccupied a 178 1.21612 unoccupied a 179 1.25361 unoccupied a 180 1.26206 unoccupied a 181 1.27792 unoccupied a 182 1.29634 unoccupied a 183 1.34272 unoccupied a 184 1.36669 unoccupied a 185 1.37455 unoccupied a 186 1.38165 unoccupied a 187 1.39717 unoccupied a 188 1.45107 unoccupied a 189 1.48546 unoccupied a 190 1.49397 unoccupied a 191 1.50947 unoccupied a 192 1.51616 unoccupied a 193 1.54722 unoccupied a 194 1.55972 unoccupied a 195 1.62076 unoccupied a 196 1.92958 unoccupied a 197 1.9693 unoccupied a 198 1.99045 unoccupied a 199 2.01527 unoccupied a continued on next page 377 khan et al. / j. nig. soc. phys. sci. 3 (2021) 360–384 378 table 9 – continued from previous page 200 2.04241 unoccupied a 201 2.47872 unoccupied a 202 2.55557 unoccupied a 203 2.57245 unoccupied a 204 2.58411 unoccupied a 205 2.61213 unoccupied a 206 2.66724 unoccupied a 207 2.67833 unoccupied a 208 2.7261 unoccupied a 209 2.75184 unoccupied a 210 2.76667 unoccupied a 211 2.83591 unoccupied a 212 2.87279 unoccupied a 213 2.87898 unoccupied a 214 2.89883 unoccupied a 215 2.92545 unoccupied a 216 2.94814 unoccupied a 217 2.96326 unoccupied a 218 2.98075 unoccupied a 219 3.03619 unoccupied a 220 3.05634 unoccupied a 221 3.09158 unoccupied a 222 3.09641 unoccupied a 223 3.10212 unoccupied a 224 3.10332 unoccupied a 225 3.10964 unoccupied a 226 3.17293 unoccupied a continued on next page 378 khan et al. / j. nig. soc. phys. sci. 3 (2021) 360–384 379 table 9 – continued from previous page 227 3.23325 unoccupied a 228 3.248 unoccupied a 229 3.29189 unoccupied a 230 3.30673 unoccupied a 231 3.32245 unoccupied a 232 3.34172 unoccupied a 233 3.35642 unoccupied a 234 3.35848 unoccupied a 235 3.37538 unoccupied a 236 3.37812 unoccupied a 237 3.38723 unoccupied a 238 3.41087 unoccupied a 239 3.46033 unoccupied a 240 3.48637 unoccupied a 241 3.502 unoccupied a 242 3.53895 unoccupied a 243 3.56422 unoccupied a 244 3.57188 unoccupied a 245 3.59923 unoccupied a 246 3.61431 unoccupied a 247 3.62728 unoccupied a 248 3.65106 unoccupied a 249 3.67652 unoccupied a 250 3.70142 unoccupied a 251 3.71733 unoccupied a 252 3.74714 unoccupied a 253 3.76852 unoccupied a continued on next page 379 khan et al. / j. nig. soc. phys. sci. 3 (2021) 360–384 380 table 9 – continued from previous page 254 3.89765 unoccupied a 255 3.98257 unoccupied a 256 5.32337 unoccupied a 257 5.35191 unoccupied a 258 5.35493 unoccupied a 259 5.38283 unoccupied a 260 5.40342 unoccupied a 261 5.44518 unoccupied a 262 5.49454 unoccupied a 263 5.50494 unoccupied a 264 5.54169 unoccupied a 265 5.5829 unoccupied a 266 5.60217 unoccupied a 267 5.66417 unoccupied a 268 5.70088 unoccupied a 269 5.73321 unoccupied a 270 5.76144 unoccupied a 271 24.19134 unoccupied a 272 24.27042 unoccupied a 273 24.41848 unoccupied a 274 24.49212 unoccupied a 275 24.49889 unoccupied a 276 24.51616 unoccupied a 277 24.5837 unoccupied a 278 24.63317 unoccupied a 279 24.7082 unoccupied a 280 24.80808 unoccupied a continued on next page 380 khan et al. / j. nig. soc. phys. sci. 3 (2021) 360–384 381 table 9 – continued from previous page 281 24.81693 unoccupied a 282 24.83965 unoccupied a 283 24.88602 unoccupied a 284 25.00561 unoccupied a 285 25.02082 unoccupied a 286 51.55558 unoccupied a 287 51.58631 unoccupied a 288 51.59188 unoccupied a 289 51.59641 unoccupied a 290 51.60474 unoccupied a band gap: 0.33925 a.u table 10: frequencies cm−1 and raman activities frequency, cm−1 raman activity 40.5889 0.0308 58.1541 0.3493 72.4885 0.0255 123.9918 0.0304 152.1424 0.1427 180.5744 1.7709 185.0588 0.3809 255.505 1.9977 256.5763 1.6381 277.7802 0.0179 294.7906 0.6864 297.6356 0.4601 351.015 6.5213 366.8926 1.5406 382.5217 3.5322 400.1286 2.3964 420.2978 1.913 464.6792 0.0432 477.8178 3.0011 505.9458 35.2839 520.2946 2.4177 557.7997 3.7289 605.0876 23.6777 606.812 1.0858 613.2264 1.5184 652.8296 0.7743 665.3268 0.1479 686.8013 1.8993 continued on next page 381 khan et al. / j. nig. soc. phys. sci. 3 (2021) 360–384 382 table 10 – continued from previous page frequency, cm−1 raman activity 701.3117 0.4288 717.2705 0.0411 737.3912 0.1857 744.8034 0.5408 772.1549 0.8902 786.5867 6.7614 806.2587 0.5934 844.6811 0.2227 845.4904 1.5859 949.8981 0.3197 980.7929 0.8345 991.5636 4.1317 1008.3375 1.311 1019.5568 54.1302 1063.7093 0.7049 1087.4272 6.8712 1110.4729 1.8682 1138.1731 3.5729 1196.3708 0.5286 1205.8194 11.7034 1230.0187 26.9241 1253.0616 11.4807 1291.437 9.0978 1294.8107 18.6095 1311.5505 10.9375 1348.5262 55.9278 1393.5713 7.765 1400.046 11.0092 1440.4723 195.2705 1449.158 77.3445 1486.7681 104.1229 1504.6975 44.8716 1515.7165 31.9762 1563.6894 54.3211 1577.6142 15.8238 1599.795 32.9398 1627.5681 30.8732 1640.3534 21.3528 1642.8372 13.7481 1660.0202 9.3907 1725.1233 85.1739 1743.1457 150.3192 1779.4404 106.9382 1800.8258 151.5551 1805.9447 13.6314 1858.3537 221.7421 3168.3115 226.1542 3223.758 93.6794 3253.8947 74.8114 3375.7378 45.7586 3377.4741 59.3447 3378.858 113.9246 continued on next page 382 khan et al. / j. nig. soc. phys. sci. 3 (2021) 360–384 383 table 10 – continued from previous page frequency, cm−1 raman activity 3411.1628 138.4204 3907.6476 30.7164 3914.4907 128.1222 4084.1357 128.3823 4. conclusion this article is the first to have done complete computational and frequency studies on the isolated anthraquinone, “emodin” from p. aquilinum. optimized geometry, ir frequencies, bond distances (r) and angles (a), dipole moments and other parameters have been computationally determined for the isolated molecule from quantum chemical calculations using the gaussian 09 retinue programs. experimentally determined and computationally measured ir frequencies agreed perfectly with each other. this can be used to predict unobserved chemical phenomena like design of new drugs and materials such as the positions of constituent atoms, in relationship to their relative and absolute energies, electronic charge densities, dipoles, higher multiple moments, vibrational frequencies, relativity or other spectroscopic quantities and cross sections for collision with other molecules. acknowledgments our gratitude to prof j. o. igoli and sipbs, university of strathclyde, glasgow, uk for their help in the analyses and the indian institute of science, bangalore for facilitating the computational calculations. references [1] s. samala. & s. veeresham “enhanced bioavailability of glimepirida in the presence of boswellic acids in streptozotocin induced diabetic rat model”, natural products chemistry and research 1 (2013) 166. 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[26] . e. etim, o. e. godwin & s. a. olagboye “protonation in heteronuclear diatomic molecules: same molecule, different proton affinities”, communication in physical sciences 6 (2020) 835. 384 j. nig. soc. phys. sci. 3 (2021) 469–476 journal of the nigerian society of physical sciences effects of repeated frying on physical properties of cooking oil obtained from wurukum market in makurdi metropolis, benue state, nigeria. t. daniela,1,∗, f. eriba-idokoa, j. o. tsora, s. t. kungur1, e. o. enokelaa, f. gbaoruna, e. c. hembac, a. a. mcasule1, n. s. akiiga1, p. o. ushie1 adepartment of physics, benue state university, makurdi bdepartment of physics, college of education, katsina-ala, benue state cdepartment of physics, federal college of education, pankshin, plateau state ddepartment of physics, federal university of agriculture, makurdi, benue state edepartment of physics, faculty of physical sciences, cross river university of technology, calabar abstract the viscosity, density and specific gravity of different brands of cooking oil samples locally sourced for in makurdi have been measured with respect to change in temperature. the viscosity of the different brands of cooking oil was measured with the instrumentality of brookfield viscometer. the density and specific gravity were evaluated using the mass of the sampled oil obtained with the help of the density bottle. the result showed a pattern of rapid decrease in viscosity with increase in temperature for the oil samples, while density decrease is observed to be almost linear with increase in temperature for all samples. amongst the sampled cooking oils, palm kernel showed the least viscosity of 8.6 pascal-second when measured at 45.200c. this illustrates that palm kernel oil has a relatively low viscous nature at 45.200c as compared to other samples used in this work. doi:10.46481/jnsps.2021.298 keywords: cooking oil, frying, brookfield viscometer, viscosity and temperature article history : received: 30 june 2021 received in revised form: 04 september 2021 accepted for publication: 14 september 2021 published: 29 november 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: edward anand emile 1. introduction cooking oils are vital ingredients for preparing most of everyday food in many parts of the world and in particular, they are known to be rich sources of beneficial dietary nutrients that contribute to human health [1-2]. palm trees, olive plants, ∗corresponding author tel. no: +234(0)8167598988 email address: terver.daniel@yahoo.co.uk, tdaniel@bsum.edu.ng (p. o. ushie) cocoa-nut trees, groundnut and soybean plants are few examples of plants whose seeds and fruits can be processed in order to extract various kinds of cooking oils that are mostly consumed on daily basis in this part of the world. these plants are widely cultivated in different parts of the country of which benue state being the food basket of the nation [3-4] produces these on a high commercial scale [3]. the production of cooking oils can be done by pressing or crushing the plant seeds or fruits before extracting the oil. these processes are either local or modernized depending on the equipment used and their 469 daniel et al. / j. nig. soc. phys. sci. 3 (2021) 469–476 470 availability. some are further refined by application of heat to remove unwanted component(s), while others are marketed without further refining. out of the total production of cooking oils, 75 % is used for direct cooking of day-to-day meals both for frying and non-fried meals. the remaining 25 % is being used in industries as local sources for production of other industrial commodities such as biscuits, detergents etc. [5-6]. food frying involves immersing the food to be fried (such as sliced pieces of yam, plantain, potatoes, fish, beans cake etc.) completely in a bath of hot oil under the supply of heat (though not necessarily with a defined quantity of heat) for a particular period of time until the immersed food turns yellowish in nature. the hot cooking oil usually processes the food during the frying process by dehydrating the food moisture content leading to both chemical and physical changes in the food thereby reducing its mass and making it porous and crispy. frying usually affects the characteristics taste, texture and the color of the food [7]. food frying can be used for food preservation and food processing. the quantity of heat supplied affects both the chemical and the physical properties of the cooking oil. for instance, extreme supply of temperature to cooking oil is a factor that lead to increase in acid value in oils [8]. high acid value indicates high free fatty acid, which in turn, causes oil to become rancid [9] and ingestion of acids especially transfatty acids appears to increase blood cholesterol, in particular low-density lipoproteins to high-density [6]. high levels of low lipoproteins in the blood have been linked to arteriosclerosis, a disease of the heart that can cause stroke, heart attack [10-11] and other serious health problems as a result of cholesterol deposits that form plaques on the inner surfaces of the arteries obstructing blood flow [8]. this causes a disease called arteriosclerosis. due to increasing awareness on the health implications of bad cholesterols in diet, most people now prefer to purchase vegetable oils with low density lipoproteins [12]. it has been demonstrated that free fatty acids, change of color, smoke point, iodine values, total polar materials, interfacial tension, density, cloud point, pour point, flash point, boiling range, freezing point, refractive index, foaming properties and viscosity are indicators used to determine the quality of cooking oils [13-16]. this is to say that: frying of food may bring out the aroma, enhance the flavor and texture combination which is very desirable for consumption, making fried foods one of the most popular food products [17]. conversely, when cooking oils are stored for a long time, excessive recycling or exposed to excessive heat, oil degradation occurs [1819], and by-products are produced [20]. elevated temperatures, especially temperature above the smoking point of the oil, will cause significant change in its quality due to the chemical and physical reactions [21]. some by-products of oil degradation have adverse effects on human health [7, 20, 22]. previous studies have revealed the effect and health risks associated with eating fried food [7-8, 23-25]. also, other several researchers have focused on effect of temperature on physical properties of cooking oil at high temperature up to and above their smoking points [9-10, 17, 24, 26], without taking into account however, how repeated use of the same cooking oil for deep frying affects these properties. our present study focused on the effect of repeated frying on some physical properties of cooking oil purchased from local markets in makurdi, benue state. uniquely in this work, we maintain a minimal temperature (temperature well below the smoking point temperature of each cooking oil) [10, 17, 22] in the frying process, using the same cooking oil repeatedly for frying of yam and potatoes for several days while keeping track of the changes in some of the physical properties of each cooking oil on a daily basis. additionally, the motivation for this work was further honed by the limited availability of data on the physical properties of cooking oils in north central, nigeria. also owing to the fact that benue state is classified as the food basket of the nation in nigeria, information on the physical properties of oils with respect to changes in temperature will further call for high quality breeds of seedlings for farmers that could mitigate these short changes in the long run. 2. physical properties of cooking oil the physical properties of cooking oils are free fatty acids, change of color, smoke point, iodine values, total polar materials, interfacial tension, density, cloud point, pour point, flash point, boiling range, freezing point, refractive index, foaming properties and viscosity [13-16]. however, the physical properties of cooking oil measured in this work included the viscosity, density and specific gravity based on availability of instrumentation and other requirements. 2.1. viscosity viscosity is an important physical property of a fluid system including cooking oil being a function of shear rate, temperature, pressure, moisture content, concentration and flow rate. it differs from fluid to fluid because they are made of different particles with different forces of attraction between them. examples of these fluids can be engine oil, honey, cooking oils and water. it is important to note that the viscous forces are called into play as soon as fluid flow starts. if the external force causing the flow are constant, the rate of flow becomes constant and a steady state is attained with the resisting viscous forces equal to the applied force. the viscous force stops the flow when the applied force is removed. viscosity can be expressed in two distinct forms known as dynamic viscosity (η) and the kinematic viscosity (ν). they are mathematically expressed respectively as according to [27] as, ηoil = τ γ &νoil = ηoil ρoil (1) where τ is the shear stress, γ is the shear rate and ρoil is the oil density. the viscosities of the oil samples used in this work are measured using the brookfield viscometer. 2.2. density the density of cooking oil influences its absorption as it affects the drainage rate after frying and also the mass transfer rate during the cooling stage of frying. the density of a substance is the property that describes the amount of matter that is 470 daniel et al. / j. nig. soc. phys. sci. 3 (2021) 469–476 471 packed in a small space. it is measured throughout industry to gain insight into the purity, concentration, quality and behavior of substance. the density of cooking oil is expressed mathematically as [13, 27] ρoil = moil νoil (2) where ρoil is the oil density in g/cm3. moil is the mass of the oil in gram and νoil is the oils volume in cm3. 2.2.1. specific gravity specific gravity is an important parameter in cooking oil because of its correlation with cetane number, heating value and storage time. the greater the specific gravity of oil, the higher its energy content. the specific gravity of cooking oil was obtained using the equation 3 [13]; specific gravity = ρoil+bottle −ρbottle ρwater+bottle −ρbottle = ρoil ρwater (3) 3. materials and method 3.1. geology of area of study makurdi metropolis doubles as the state capital of benue state in nigeria and also serves as the headquarters of the makurdi local government area with the geographical coordinates 7.73o n and 8.53o e [28-29]. in 2016, makurdi and its environs had an estimated population of about 365,000 persons [30]. the town is divided by the river benue into the north and south banks, which are connected by two bridges of about 1 km long each. due to the fact that makurdi is located in the valley of river benue [29, 31], it experiences a temperature fluctuation of between 21−37o c in the year [30, 32]. within the makurdi metropolis, there are five major markets namely: wurukum, wadata, makurdi modern, international, north bank and the high-level markets which are the major trade centers for soybeans, groundnut, shea nut, millet, sorghum, rice, maize, yams, tomatoes, water melon, and other seedlings at various quantities etc. the markets and other features of makurdi town are as shown in figure 1. benue state is predominantly an agricultural area specializing in both cash and subsistence crops [30]. the vegetation is mainly savanna [32] with the southern part of the state characterized by forests, thus, fertile for growing palm trees, consequently, ranking benue state among producers of palm oil in nigeria. however, the oil samples used in this work were purchased from wurukum market due to the proximity to the university research laboratory. 3.2. sample collection the cooking oil samples used in this work include groundnut oil, palm oil, coconut oil, olive oil, palm kernel oil and soybeans oil. they were purchased from wurukum market a major market in makurdi metropolis that is situated few meters away from the benue state university, nigeria. while figure 1. map of makurdi metropolis, ministry of lands and survey, benue state showing wurukum market [33]. the soybeans oil was purchased at the seraph company along makurdi gboko road, makurdi, benue state. all the oil samples were moved to the research laboratory in the department of physics, benue university makurdi, nigeria in plastic bottles, where they were properly labelled and stored at a room temperature for two days before experiment. 3.3. measurement of viscosity the viscosities of the different brands of cooking oils were measured using the brookfield viscometer. brookfield viscometers employs the well-known principle of rotational viscometry; which measures viscosity by sensing the torque required to rotate a spindle at constant speed while immersed in the sample fluid. the torque is proportional to the viscous drag on the immersed spindle, and thus to the viscosity of the fluid. the continuous rotation of the spindle allows uninterrupted measurements to be made over long periods of time depending on the fluid properties. the viscosities of each of the six (6) samples of the cooking oil (at room temperature) without frying, were measured using the brookfield viscometer at room temperature. the measurements for each oil sample were performed and recorded as shown in table 1. 5. 00 ml of groundnut oil was poured in a frying pan and heated up to a temperature of 120o c . a tuber of yam and six (6) tubers of potatoes were sliced, poured into the frying oil and allowed to fry well enough for human consumption. the frying process for all the six (6) brands of oil was carried out almost concurrently. during the frying process, the temperature of the frying oil was maintained at a temperature below the smoking point of the oil. at the end of each frying, the oil was allowed to cool before taking the viscosity measurement at a constant temperature every day for five (5) consecutive days. viscosity for each oil sample was measured and plotted as shown in figure 2. again, while varying the temperature of each oil sample each consecutive day for five days, their viscosity was measured and recorded as shown in table 2. the same variety of yam and 471 daniel et al. / j. nig. soc. phys. sci. 3 (2021) 469–476 472 figure 2. viscous change measured at constant temperature per day after recycled usage of cooking oils for frying of yams and potatoes potatoes with the same cooking oils were repeatedly used in all the five days in the frying process. 3.4. determination of oil density to obtain the densities of the oil samples, we use the density bottle to measure the mass and volume of each oil sample before evaluating their corresponding densities using equation 2. to do that, we first of all measure the mass of the dry and empty density bottle using the electronic weighing balance and its volume was evaluated using equation 4 as expressed v olume = mwater − mbottle ρwater −ρair (4) the result of equation 4 is substituted for the volume of oil in equation 2 and the results for the densities of the cooking oils are recorded as shown in table 3. 3.5. experimental results in this section, the results that were gotten in the various investigations of the physical properties considered in this study are duly presented and discussed as shown below 4. discussions 4.1. viscosity the results for the viscosity of soybean oil, groundnut oil, olive oil, coconut oil, palm kernel oil and palm oil measured at room temperature without frying are presented in table 1 and plotted as in figure 3. while table 2 presents the results of viscosity of the sampled cooking oils measured each day after frying of sliced yams and potatoes repetitively for five consecutive days at varying temperature. from table 1, the viscosities ranged from 61 mpa.s to 111 mpa.s with palm oil having the highest viscosity of 110.96 mpa.s and the lowest value is reported in palm kernel oil as 61.12 mpa.s. this is clearly depicted in figure 3. the result for soybeans at 24o c. is 96.82 figure 3. viscosity of cooking oil samples without frying and measured at room temperature figure 4. viscosity versus temperature of cooking oil samples after repeated frying of yams and potatoes for five successive days 472 daniel et al. / j. nig. soc. phys. sci. 3 (2021) 469–476 473 table 1. viscosity (pas−1) of cooking oil sample without (that is, at room temperature) frying. oil sample viscosity (pas−1) temperature (0c) soya beans 96.82 23.90 groundnut 104.27 23.10 olive 68.62 22.70 coconut 64.26 24.10 palm kernel 61.12 23.60 palm oil 110.96 23.20 table 2. viscosity (pas−1) of cooking oils used for frying yams and potatoes for five consecutive days. viscosities of oil samples (pas−1) days temperature (0c) soya beans groundnut olive coconut palm kernel palm oil day 1 24.50 60.00 52.61 65.67 50.91 65.27 70.21 day 2 32.30 48.30 41.34 50.26 34.32 46.32 50.57 day 3 37.30 31.52 28.21 38.92 27.21 35.78 38.26 day 4 41.20 33.20 21.25 23.24 19.46 18.45 28.92 day 5 45.20 14.31 11.21 12.36 10.52 8.6 18.61 table 3. densities, ρ(g/cm3), of the cooking oil at different temperature obtained after repeatedly frying yams and potatoes for five days. density of oil samples (g/cm3) temperature (0c) soya beans groundnut olive coconut palm kernel palm oil 20.0 0.900 0.900 0.910 0.870 0.880 0.920 40.0 0.890 0.860 0.880 0.860 0.860 0.890 60.0 0.880 0.850 0.860 0.860 0.850 0.870 80.0 0.840 0.830 0.840 0.850 0.840 0.850 100.0 0.810 0.820 0.840 0.820 0.830 0.820 table 4. specific gravities of the cooking oil at different temperature obtained after repeatedly frying yams and potatoes for five days consecutively. specific gravity of oil samples temperature (0c) soya beans groundnut olive coconut palm kernel palm oil 20.0 0.903 0.903 0.913 0.873 0.883 0.883 40.0 0.893 0.832 0.883 0.863 0.853 0.873 60.0 0.883 0.842 0.863 0.853 0.842 0.873 80.0 0.842 0.853 0.842 0.863 0.832 0.893 100.0 0.812 0.822 0.842 0.822 0.832 0.822 473 daniel et al. / j. nig. soc. phys. sci. 3 (2021) 469–476 474 figure 5. density of different brands of cooking oil against temperature obtained after repeated frying of yams and potatoes for five successive days. figure 6. specific gravity of different samples of cooking against temperature for five successive days obtained after repeated frying of yams and potatoes mpa.s. this value is far higher when compared to [26] and [34] which are 60 mpa.s at 20o c and 57.1 mpa.s at 22o c respectively. while the result of [34] is a theoretical prediction that was modelled mathematically using the andrade equation, [2]6 is contrarily an experimental result measured with the use of mcr301 rheometer a device that employ the principle of rotational viscometery which is similar to the brookfield viscometer used in this work. however, the large disparity of about 36.8 mpa.s in this result and [26] is not only due to the temperature difference of 4o c but largely because, the viscometers were operated at different shear rate, which corresponds to the rate of spindle rotation in oil solution per minute (rpm). the brookfield viscometer was operated at a shear rate of 60 rpm while the rheometer as in [26] was operated at 750 rpm. whereas, higher shear rate corresponds to higher spindle speed in oil solution [35] and thus, lower viscosity measurement as observed in this result. the results in figure 2 shows that, viscosity decrease as the number of days of recycled use of the same oil to fry sliced yams and potatoes increases continually for five successive days. the viscosity of palm oil for instance decreased from 70.21 mpa.s in day 1 to 50.57 mpa.s in day 2 to 38.26 mpa.s in day 3 to 28.92 mpa.s in day 4 to 18.61 mpa.s in day 5. since the viscosity of a solution is dependent on the intermolecular forces and the interaction within its molecules, the continuous decrease in viscosity after each repeated frying is because, continuous application of heat to the oil each day for five days results in continuous breakdown of the oil layers. this breakdown weakens the intermolecular forces and the bond energies of oils molecules that restrict molecular rotational motion of the viscometer spindle within the oil solution, and therefore, lower viscosity is recorded as the number of days of repeated frying increase. the results plotted in figure 4 reveal a steady linear decrease in viscosity as the temperature of the cooking oils increases. a similar trend was observed for olive oil, palm oil and soybean in [27], and in coconut oil and soybean as in [14]. we observed that palm oil has the highest viscosity for all the five-oil follow by olive oil and groundnut oil while the oil with least viscosity is palm kernel. 4.2. density and specific gravity the density and specific gravity of cooking oil varied significantly with change in temperature as shown in tables 3 and 4 respectively. a general trend of decrease in these two physical properties of cooking oils were observed in figures 5 and 6 as temperature increases from 20o c to 100o c. this decrease in the density of the cooking oil with increased temperature was also observed [13, 27, 34] and same trend being applicable for specific gravity [10, 13, 36]. the decrease in density and specific gravity of the oils due to increase in temperature lead to breakdown of the oils structure. this causes the mass of the oils to reduce as frying progresses even though the volume or the quantity of this oil that was used in frying at the initial place may still be the same 474 daniel et al. / j. nig. soc. phys. sci. 3 (2021) 469–476 475 5. conclusion this work has shown that the thermal (heat) properties of different oils determine their superiority over others when subjected to the same amount of weather conditions. based on the results obtained from research, temperature had a significant effect on all three physical properties (viscosity, specific gravity and density) of oils measured. oil type was shown to have a significant effect on viscosity, density and specific gravity. a general trend for all the properties decreased with increased temperature as can be seen from figures 4 to 6. the effect of temperature on the viscosity for all the vegetable oils show a linear relationship for all temperatures used in this work. although [34] reported non-linear decrease in viscosity at very high temperature above 100o c. this is an indication that temperature had significant effect on viscosity of vegetable oils. oil that has been used repeatedly for so long at high temperatures are not advisable for human consumption because its qualities such as the viscosity, density, specific gravity etc have been reduced. density decreased linearly with increased temperature whereas specific gravity decreases almost linearly in comparison with the other physical properties measured in this work. 6. acknowledgment we wish to appreciate the head of the department of physics, professor frederick gbaorun for allowing this experiment to be conducted in the laboratory of the department. the authors also wished to thank all the laboratory staff of the department of physics for their assistance during measurement and data analysis as well as interpretations and discussions of the results obtained. lastly, the authors are grateful to god for a good life that all of them are enjoying and the wisdom to conduct this research. references [1] n. morgan, “food consumption: world vegetable oil consumption expands and diversifies”, food rev. 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[36] m.s. sarjadi, t.c. ling, m.s. khan, “analysis and comparison of olive cooking oil and palm cooking oil properties as biodiesel feedstock”, j. phys. conf. ser. 1358 (2019) 1. available at https://doi.org/10.1088/17426596/1358/1/012007. 476 j. nig. soc. phys. sci. 4 (2022) 130–137 journal of the nigerian society of physical sciences analysis of hydromagnetic double exothermic chemical reactive flow with convective cooling through a porous medium under bimolecular kinetics f. o. akinpelu, r. a. oderinu, a. d. ohaegbue∗ department of mathematics, ladoke akintola university of technology, ogbomoso, nigeria abstract in this study, the analytical solution of steady hydromagnetic double exothermic combustible reaction fluid flow in a porous medium with convective cooling wall is presented. the viscous heating reactive liquid is totaling exothermic without consumption of material. the combustion reaction of the fluid takes place in a poiseuille device, and it is been propelled by pressure gradient and pre-exponential bimolecular kinetics. the device is exposed to convective cooling to keep the reactive hydromagnetic fluid from distortion. the weighted residual method (wrm) is analytically used to get the numerical values for the dimensionless nonlinear governing equations. the solution to temperature and velocity distribution is carried out and the result is graphically depicted. the nusselt number and skin friction coefficient is also showed for some significant parameters engrained in the flow and the solution obtained is compared with numerical method. as obtained in the study, the second exothermic reaction term increases the combustion process; hence the term will assist in reducing toxic discharge from the engines that pollute the environment. the frank-kamenetskii parameter contributes highly to system thermo-fluid destruction; as such it must be monitored. doi:10.46481/jnsps.2022.525 keywords: exothermic reaction, convective cooling, bimolecular kinetics, porous medium article history : received: 19 december 2021 received in revised form: 06 february 2022 accepted for publication: 19 february 2022 published: 28 february 2022 c©2022 journal of the nigerian society of physical sciences. all rights reserved. communicated by: j. ndam 1. introduction the idea of laminar flow of fluid particles along a medium continues to receive tremendously attention because of its application to fluid mechanics particularly in industries. its significant has been influenced by many engineering analyst, biological science, atomic control, aerodynamics of plasma and magnetohydrodynamic (mhd) system, ellahiand afzal [1]. the flow of exothermic reactive fluid is useful in industrial and engi∗corresponding author tel. no: +2348061198971 email address: ohaegbueanthony@gmail.com (a. d. ohaegbue ) neering system and is frequently accompanied with heat transfer which has long been called combustion, makinde [2]. some convective fluids flows are caused by the generating or absorbing of heat which results to a fluid chemical reaction. heat source encourages the distribution of heat which then changes the amount of particles deposition within the arrangements such as electric clip, nuclear reactor, semiconductor etc. generation or the absorption of heat has been assumed to depend on temperature. the effect of heat production or absorption of nonhomogenous state in a micro polar fluid with variable conductivity was considered by chen [3]. the results show the dependent generation of heat at the surface is less to temperature 130 akinpelu et al. / j. nig. soc. phys. sci. 4 (2022) 130–137 131 dependent generation. the authors [4, 5] studied the impact of non-uniform heat supply on hydromagnetic micro polar fluid with radiative in permeable media. many studies have been carried out on exothermic chemical reactive fluid flows and its applicability in energy storage systems, atomic reactor plan, geophysics, polymer expulsion and mhd reactive fluid flow. consequent to its applications, gbadeyan and hassan [6] worked on the reaction of couette variable viscosity fluid flow for arrhenius case. transfer of exothermic heat flow through a pipe was investigated by marcello et al. [7] under different chemical kinetics, adomian decomposition was used to solve the problem. the comprehensive survey on generation of heat by hydromagnetic fluid flow considering all the basic physical properties was reported in [8-9]. among the numerous suggested models is a single step exothermic reaction while some are under convective cooling. hazarika et al. [10] worked on the relevance of variable properties on mhd flow through a perpendicular plate. they declared that the velocity outline rises and the parameter of heat conductivity decreases. dulal & hiranmoy [11] analyzed the effect of heat on variable heat conductivity and mhd of non-darcy mixed convective flow of concentration over an extending sheet. the impact of mhd on heat distribution over an extented walls fixed in porous medium with heat source, variable viscousity and viscous dissipation was discussed by hunegnaw & naikoti [12]. hassan and maritz [13] investigated the chemical reaction on hydromagnetic flow in stationary parallel convective cooling walls. the result shows that a rise in magnetic intensity and porous medium values result to a fall in the velocity profiles and a rise in the values of porous medium, activation energy and magnetic force with convective cooling terms decreases the temperature of the fluid. dulal [14] examined temperature and mass transfer in a fixed flow of viscous fluid on a distending perpendicular sheet, taking buoyancy and heat radiation into consideration. hazarika and utpal [15] investigated the effect of energy conductivity and variable properties on hydromagnetic flow across a perpendicular plate. they observed that when the heat conductivity value is reduced, the velocity distribution rises. adesanya and folade [16] investigated the flow of hydromagnetic material through a porous media at the third grade level. bala and suneetha [17] investigated the effect of homogeneous and heterogeneous chemical reactive stagnation flow on the mhd of a micropolar fluid through a permeability shrinking/stretching surface plate fixed in a porous wall. salawu [18] reported on the heat transfer on third grade combustive echemical reactive flow and absorption of heat with convective cooling in a porous medium. seth et al. [19] provided a computational model for the mhd natural convection casson fluid flow with nth order of chemical species and newtonian heating in porous channel. their research concentrated on the steady-state one-step exothermic reactions of hydromagnetic flows. however, analysis of steady hydromagnetic double exothermic combustible reaction fluid flow through a porous channel with convective cooling which takes place in a poiseuille device is yet to be studied. double exothermic reaction deals with an intermediary reaction. the second exothermic reaction term increases the combustion process. makinde et al. [20] reported on the heat stability of double steps combustive chemical reaction of a slab. they considered the diffusion reactant by assuming a chemical kinetics of variable factor of the unsteady and steady state. salawu and okedoye [21] provided a solution to the unsteady double step exothermic reaction along a conduit but they never considered the case when it’s under convective cooling. they studied second law of thermodynamics double step hydromagnetic exothermically combustible chemical reactive fluid flow considering viscous and gravity with heat absorbing along a channel. in the study, they observed that magnetic force decreases the flow rate and undesirable alterations in the fluid viscosity caused by an increase in temperature. labelo et al. [22] works on two-step cylindrical stockpiles combustible reaction. hence, the purpose of this research is to examine the steady hydromagnetic double reaction of pre-exponential combustible reactive fluid flow in porous channel with convective cooling. however, the important of double combustion reaction processes assist in improving the hydrodynamics lubricants that can leads to engine efficiency. thus, this study is significant to pollution control, enhancing of industrial machine output and improving lubricant viscosity. an analytical weighted residual technique is employed to provide solutions to the dimensionless viscous dissipative exothermic reaction flow equations. important of some properties of temperature and velocity field with heat gradient and skin friction are discussed. the present study will help in improving the performance of industrial engines and in reducing toxic discharge to the environment due to incomplete combustion. this study is motivated by previous report on the important and application on two step exothermic reaction to the chemical industries. introduction and background to the study is presented in the first section. the problem formulation equations are described in the second section. the collocation weighted residual method of solution is presented and utilized in the third section. the results and discussion is presented in fourth section. the final remark is stated in the last section. 2. the flow’s mathematical formulation consider viscous heating in a reactive fluid of double step combustible reaction between stationary parallel walls. the combustion reaction of the fluid takes place in a poiseuille device, and it is been propelled by pressure gradient and preexponential bimolecular kinetics. the flow is assumed to be in the direction of x with y − axis normal to it as illustrated in figure 1. the device is exposed to convective cooling to keep the reactive hydromagnetic fluid from distortion. according to chinyoka and makinde [23], and ignoring time-dependent and the fluid reactive viscose consumption. the velocity and temperature balance equation governing the flow are as follows: − d p d x + µ d2u dy2 −σb2ou − µ k1 u = 0 (1) 131 akinpelu et al. / j. nig. soc. phys. sci. 4 (2022) 130–137 132 figure 1: flow coordinates schematic α d2t dy2 + µ ( du dy )2 + σb2ou 2 + µ k1 u2 + q1c1 a1  kt vl m e− e1rt + q2c2 a2  ktvl m e− e2rt = 0 (2) the imposing boundary conditions are: y = a, u = 0, −k dtdy = h ( t − tw ) y = 0, u = 0, k dtdy = h ( t − tw ) (3) where b0, x , y, p,σ,u,µ,k1,α,q1, c1, a1, e1, kb, t , l ,v,r, q2, c2, a2, e2, tw, m, are magnetic field, dimensional coordinate flow alongside the surface, dimensional coordinate flow along the normal surface, modify pressure, electrical conductivity, axial fluid velocity, fluid viscosity, darcy porosity, thermal conductivity, first step heat reaction, concentration species of step one reactant, constant rate of step one reactant, energy activation of step one, boltzmann’s constant, dimensional fluid temperature, plank’s number, vibration frequency, general gas constant, step two heat reaction, concentration species of step two reactant, constant rate of step two reactant, energy activation of step two, temperature at the wall, computational index (m=0.5) respectively. introducing the following dimensionless quantities: p = µupa , u = uu , θ = e1 ( t−tw ) rt 2w , x = ax, ha2 = σb 2 o a 2 µ , g = −d pd x , y = y a , br = µu2 e1 αrt 2w , λ = q1c1 a1 a2 e1 αrt 2w ( kb tw vl )m e 1 ε , bi = ahk , ha 2 = σb2o u 2 a2 µ , k = a 2 k1 , γ = q2c2 a2 q1c1 a1 e− 1 ε e 1 ε e rθ (1+εθ) (4) substituting the dimensionless quantities of equation (4) into equation (1) to (2) gives: d2u dy2 − ( ha2 + k ) u + g = 0 (5) d2θ dy2 + br (k + ha2) u2 + ( du dy )2 + λ [ (1 + εθ)m ( e θ (1+εθ) + γe rθ (1+εθ) )] = 0 (6) alongside with boundaries conditions u(0) = 0, dθ dy = −biθ, u(1) = 0, dθ dy = biθ (7) where γ, λ, g, br, ha2, r, ε and bi are two-step exothermic reaction, frank-kamenetskii, pressure gradient, brinkman numbers, hartmann numbers, activation energy ratio, activation energy and biot number parameter respectively. 3. numerical setup the analytical method of solution used is weighted residual method (wrm). the goal of weighted residual method [24, 25], is to look for an estimated result in the form of polynomial considering the given differential equation. φ [u (v)] = f in the domain d,bµ [u] = γµon∂d (8) whereφ [u] is the non-linear differential operator relating of the dependent variables u and f the function of a known point, bµ [u]is the estimated number of boundary conditions with domain d and ∂d the boundary. the solution to the problem of the above estimated boundary isoften carried out by taking an estimate of the solutionu (v). the expression u (v) ≈ w (v, a1, a2, ...at) (9) relies on the number of parameters a1, a2, ....atand the arbitrary value aljof the boundary conditions are satisfied. in practice w (v, a1, a2, ...at) take the form w (v, a) = ϕo (v) + t∑ j=1 a jϕ j (10) where the functionϕ j(v) are given to satisfy the boundary conditions qµ (ϕo) = γµ, qµ ( ϕ j ) = 0, j = 1, 2. 3. ...t (11) for arbitrary values of al s, w (v, a1, a2, a3, ...at) fulfills the boundary condition (8). when equation (9 is substituted for equation (8), the differential equation’s residual becomes φ (v, a) = l (w (v, ai)) − f (v) (12) where a = (a1, a2, ....at).this yields the measure whereby the function w(v, a) satisfies the differential equation’s degree, as the number t of the function ’j is enhanced in subsequent estimations. the aim is to minimize the residual φ (v, a) to zero on an average basis throughout the domain. which is∫ c φ (v, a) w jdv = 0, j = 1, 2, ...t (13) 132 akinpelu et al. / j. nig. soc. phys. sci. 4 (2022) 130–137 133 where the unknown number of constants a jin w is the same as the value of weight functions w j.the outcome consist an algebraic equations of set t for all the unidentifiedconstantsa j. so, functions of the weighted are taken as dirac delta. that is,w j (v) = δ ( v − v j ) such that the chosen error at the nodev j is zero. integrating equation (13) with w j (v) = δ ( v − v j ) result in φ ( v, a j ) = 0. employing wrm to equations (5), (6) and (7), assuming that the parameters or polynomial with undefined coefficients will be found later, this function is referred to as the trial function. u (v) = 10∑ j=0 a jv j,θ (v) = 10∑ j=0 b jv j (14) introducing the boundary condition (7) on (14) which is the trial functions and then substitute the trial function on the equation (5) and (6) to get the residual of the momentum and energy equation as: ur = 90y8a10 + 72y7a9 + 56y6a8 + 42y5a7 + 30y4a6 + 20y3a5 + 12y2a4+ 6ya3 + 2a2 − ( ha2 + k )  y10a10 + y9a9 + y8a8 + y7a7 + y6a6 + y5a5+ y4a4 + y3a3 + y2a2 + ya1 + a0  + g (15) θr = 90y8b10 + 72y7b9 + 56y6b8 + 42y5b7 + 30y4b6 + 20y3b5 + 12y2b4 + 6yb3 + 2b2+ br  ( 10y9a10 + 9y8a9 + 8y7a8 + 7y6a7 + 6y5a6 + 5y4a5 + 4y3a4 + 3y2a3+ 2ya2 + a1 )2 +( ha2 + k )( y10a10 + y9a9 + y8a8 + y7a7 + y6a6 + y5a5 + y4a4 + y3a3+ y2a2 + ya1 + a0 )2  + λ  ( 1 + ε ( y10b10 + y9b9 + y8b8 + y7b7 + y6b6 + y5b5 + y4b4 + y3b3+ y2b2 + yb1 + bo ))m  e y10 b10 +y 9 b9 +y 8 b8 +y 7 b7 +y 6 b6 +y 5 b5 +y 4 b4 +y 3 b3 +y 2 b2 +yb1 +bo (1+ε(y10 b10 +y9 b9 +y8 b8 +y7 b7 +y6 b6 +y5 b5 +y4 b4 +y3 b3 +y2 b2 +yb1 +bo)) + γe r(y10 b10 +y9 b9 +y8 b8 +y7 b7 +y6 b6 +y5 b5 +y4 b4 +y3 b3 +y2 b2 +yb1 +bo) (1+ε(y10 b10 +y9 b9 +y8 b8 +y7 b7 +y6 b6 +y5 b5 +y4 b4 +y3 b3 +y2 b2 +yb1 +bo))   (16) the residual for the set of collocation points within the domain at regular interval is reduced to zero when m = 0.5, br = 1,ε = 1,γ = 1,λ = 0.5, g = 1, ha = 1, k = 1. that is nn = (q−p)n m where n = 1, 2, 3, .., m − 1andp = 0, q = 1, m = 9. the unknown constant coefficients are gotten with the aid of maple software. so, the dimensionless equations are as follow: u(y) = −0.000003521116925y10 + 0.00001760557764y9 − 0.0001970055654y8+ 0.0006823888193y7 − 0.005555067018y6 + 0.01435078358y5− 0.08333329281y4 + 0.1435095221y3 − 0.49999999994y2 + 0.4305285858y (17) θ(y) = 0.000317754724y10 − 0.001588772547y9 + 0.007075226439y8− 0.01876827406y7 + 0.05145794359y6 − 0.09535771165y5+ 0.1277972545y4 − 0.1153837668y3 + 0.2601529854y2− 0.2157026397y + 0.2157026397 (18) the process of wrm scheme is used for difference value of. k, λ,ha2,g, γ, ε and br. the physical quantities that are of interest to engineering are the heat gradient (nu)and skin friction ( c f ) respectively as [23, 27, 28]. nu = − dθ dy and c f = du dy (19) 133 akinpelu et al. / j. nig. soc. phys. sci. 4 (2022) 130–137 134 4. results and discussion the numerical result in table 1 depicts the influence of many physical factors on skin friction and nusselt numbers. a rise in values of the term g and λ accelerate the temperature gradient and skin friction at the wall since temperature at the boundary layer increases. also, an increase in values of ha and k decreases the effect flow at the wall surface. table 2 displays the relationship between the existing related work and the current results. thus, the current results are consistent with the previous results. table 3 demonstrates the comparison results of the exact and the existing work. the outcome shows that they are in good agreement.figure 2 demonstrates the profile of velocity over pressure gradient(g). an increase in(g) term enhances the fluid flow in the channel and heat within the system rises which cause the breaking of the force bonding in the fluid. hence, pressure gradient aids changes in formation of fluid in the system. figure 3 shows effect of fluid velocity on hartmann number (ha). the lorentz force present in (ha) increases the damping magnetic strength which then accelerate the fluid resistance and by that decelerate the motion of the fluid on convective cooling wall. thus, the velocity profile is affected by a very small reading scale system which helps in fluid viscosity of many industries. figure 4 demonstrates porosity parameter (k) which generates resistance force in the fluid as shown in the plot. increase in porosity (k) values lead to an increase of the resistance in the fluid. this is due to the saturated of the medium with pores that decreases the free flow of the fluid. the term enhances viscosity fluid as a result of decrease within the heat source terms and so reduces the velocity flow. figure 5 displayed the result of frank-kamenetskii parameter (λ). a rise in (λ) correspond to an increase in the rate of viscous heating in a reaction which enhanced the chemically reaction rate and consequently increase the distribution of the temperature within the system. this is due to enhancement in the momentum viscosity coupling that results in rational rises in the flow temperature. the parameter (λ) can cause a blow up if not well monitored. figure 6 shows the strength of activation energy (ε) on heat transfer in an exothermic chemical reactive system. the least energy which exists in a chemical system with potential reactants is known as activation energy. as the parameter term (ε) rises the level of heat transfer decreases. this is because of the presence of an appreciable number of molecules with translational energy equal to or greater than the activation energy. thus, this help the industries to inspect the spark of activation energy required to break the bond in a chemical reaction. figure 7 reveals that temperature falls while the porosity (k) values rises because the resistance to fluid motion offered by porous medium causes the fluid to heat up, so the temperature decreases within the boundary layer. the behavior is as a result of the wall of the plates that provides a supporting resistance to the fluid flow. figure 8 presents the influence of second step term (γ) on the temperature field. it reveals that the double step exothermic combustive reaction increases heat transfer in the system. thus, this leads to increase in the combustion process that in turn encourages complete burn of hydrocarbon in the engines. hence, second step exothermic reaction figure 2: profile of velocity against (g) figure 3: profile of velocity against(ha) figure 4: profile of heat against(k) reduces the release of carbon-monoxide that pollutes our environment as a result of unburned hydrocarbon. 134 akinpelu et al. / j. nig. soc. phys. sci. 4 (2022) 130–137 135 table 1: comparison of c f and nu at various values of g, ha2, k and λ parameter values weighted residual method 4th order of r-k absolute error c f nu c f nu c f nu g 1.0 0.430529 0.290443 0.430529 0.290443 7.0 × 10−8 1.4 × 10−8 2.0 0.861057 0.344304 0.861057 0.344304 1.7 × 10−9 1.2 × 10−7 3.0 1.291586 0.434728 1.291586 0.434729 3.0 × 10−9 7.3 × 10−7 4.0 1.722114 0.564353 1.722114 0.564357 1.8 × 10−9 4.0 × 10−6 ha2 1.0 0.430529 0.290443 0.430529 0.290443 8.0 × 10−10 1.4 × 10−8 3.0 0.290544 0.283715 0.290544 0.283717 5.1 × 10−8 1.1 × 10−6 5.0 0.193735 0.278940 0.193737 0.278951 2.4 × 10−6 1.0 × 10−5 7.0 0.141156 0.276443 0.141181 0.276472 2.5 × 10−5 2.9 × 10−5 k 1.0 0.430529 0.290443 0.430529 0.290443 2.8 × 10−8 1.4 × 10−8 1.5 0.416618 0.289792 0.416618 0.289792 1.4 × 10−9 2.1 × 10−8 2.0 0.403769 0.289188 0.403769 0.289187 1.9 × 10−9 3.3 × 10−8 2.5 0.391863 0.288624 0.391863 0.288624 2.6 × 10−9 5.0 × 10−8 λ 1.0 0.430529 0.388391 0.430529 0.388391 8.0 × 10−10 1.7 × 10−8 3.0 0.430529 0.526687 0.430529 0.526686 2.0 × 10−10 3.7 × 10−7 5.0 0.430529 0.599584 0.430529 0.599524 1.0 × 10−10 6.0 × 10−5 7.0 0.430529 0.703732 0.430529 0.707863 1.0 × 10−10 4.1 × 10−3 table 2: comparison of temperature profile for g = λ = 1, ε = ha = br = 1 y makinde and beg [28] hassan and maritz [13] present study 0.00 0.3544502181 0.3581076494 0.3582159913 0.25 0.3323243479 0.3358417798 0.3358879051 0.50 0.2660663845 0.2691056991 0.2690827375 0.75 0.1556934861 0.1577379743 0.1578252719 1.00 0 0.0002178316 0.0000182633 table 3: comparison of the exact results with wrm for u(y) when g = k = ha = 1 y exact solution for u(y) weighted residual method for u(y) 0.0 0 0 0.2 0.0671247049 0.0671247051 0.4 0.0993879030 0.0993879032 0.6 0.0993879029 0.0993879032 0.8 0.0671247053 0.0671247050 1.0 0.0000000001 0 135 akinpelu et al. / j. nig. soc. phys. sci. 4 (2022) 130–137 136 figure 5: profile of heat against(λ) figure 6: profile of heat against (ε) figure 7: profile of heat against (k) 5. conclusion the analysis of steady hydromagnetic double exothermic chemically reactive flow with convective cooling through a porous medium under bimolecular kinetics was mathematically formulated. the weighted residual collocation integration method were used to solve the dimensionless viscous flow equations and the computational result obtained were used to compare with the fourth order of runge-schemes coupled by shooting figure 8: profile of heat against (γ) techniques, and found to have very low absolute errors. as obtained in the study, the second exothermic reaction term increases the combustion of hydrocarbon due to rise in the heat distribution within the system. hence the term will assist in reducing toxic discharge from the engines that pollute the environment. also, terms that increase heat in the system must be guarded to prevent solution explosion. this work is significant in improving combustion process in the engines with many industrial applications. hence, this study is limited to newtonian fluid and could be extended to a non-newtonian flow in an annular cylindrical pipe. references [1] r. ellahi & s. afzal, “effects of variable viscosity in a third grade fluid with porous medium:an analytic solution”, communications in nonlinear science and numerical simulation 14 (2009) 2056. 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[28] o. d. makinde & o. a. beg, “on inherent irreversibility in a reactive hydromagnetic channel flow.” j. them. sci. 19 (2010) 72. 137 j. nig. soc. phys. sci. 1 (2019) 12–19 journal of the nigerian society of physical sciences original research application of the exponentiated log-logistic weibull distribution to censored data adeniyi f. fagbamigbea,∗, gomolemo k. baseleb, boikanyo makubateb, broderick o. oluyedec adepartment of epidemiology and medical statistics, college of medicine, faculty of public health, university of ibadan, nigeria bdepartment of mathematics and statistical sciences, botswana international university of science and technology, palapye, bw cdepartment of mathematical sciences, georgia southern university, statesboro, ga, 30460, usa abstract in a recent paper, a new model called the exponentiated log-logistic weibull (ellogw) distribution with applications to reliability, survival analysis and income data was proposed. in this study, we applied the recently developed ellogw model to a wide range of censored data. we found that the ellogw distribution is a very competitive model for describing censored observations in life-time reliability problems such as survival analysis. this work shows that in certain cases, the ellogw distribution performs better than other parametric model such as the log-logistic weibull, exponentiated log-logistic exponential, log-logistic exponential distributions and the non-nested gamma-dagum (gd). keywords: generalized distribution, exponentiated log-logistic distribution, weibull distribution, survival analysis. article history : received: 31-03-2019 received in revised form: 23-04-2019 accepted for publication: 23-04-2019 published: 26-04-2019 c©2019 journal of the nigerian society of physical sciences. all rights reserved. communicated by: b. j. falaye 1. introduction oluyede et al. had recently proposed a new model called the exponentiated log-logistic weibull (ellogw) distribution with applications to reliability, survival analysis and income data [1]. in this paper, we are primarily concerned with the application of the new model (ellogw) to censored data. the details of the model and all its associated properties including the hazard function, reverse hazard function, quantile function, moments, conditional moments, mean deviations, bonferroni and lorenz curves, entropy and order statistics have been reported earlier [1]. censored data arise when a study comes to an end before every subject been studied experience the event of interest [2, ∗corresponding author tel. no: +2348061348165 email address: fadeniyi@cartafrica.org (adeniyi f. fagbamigbe ) 3]. in this study we applied data collected in situations where an experimentalist involved n samples in a life-testing experiment. the samples were kept under observation until “failure” occur. the samples could be electrical or mechanical components, system, individuals, or perhaps computer chips in a reliability experiment. it could also be patients, subjects or individuals that are recruited for drug or clinical trials. in most cases, such experiments are terminated before all the samples could have “failed”. the same could be said of drug or clinical trials wherein the recruited subjects might have been lost to follow up, dropped out of the study, withdrew from the study or the study may come to an end due to unforeseen circumstances or economic conditions or exhausted time frame. also, samples could break accidentally, malfunction in an industrial experiment and thus referred to as “failure”. generally, the data obtained under these or similar scenarios are 12 fagbamigbe et al. / j. nig. soc. phys. sci. 1 (2019) 12–19 13 called censored data. literature is replete on application of statistical distribution to different types of data [1, 4, 5, 6]. a brief review of the model, the different types of censored data and the application of the model to different types of censored data are discussed in the subsequent sections. 2. the ellogw distribution in this section, the generalized or exponentiated log-logistic weibull (ellogw) distribution is presented. the generalized or exponentiated log-logistic weibull (ellogw) (see oluyede et al. [1] for details) distribution function is given by gellogw (x) = [ 1 − ( 1 + ( x s )c)−1 exp(−αxβ) ]δ , (1) for s, c, α, β, δ, and x ≥ 0. the corresponding probability density function (pdf) is given by gellogw (x) = δ [ 1 − ( 1 + ( x s )c)−1 exp(−αxβ) ]δ−1 e−αx β × [ 1 + ( x s )c]−1{ αβxβ−1 + cxc−1 (sc + xc) } , (2) for s, c, α, β, δ > 0, and x ≥ 0. the corresponding hazard function is given by hellogw (x) = δ [ 1 − ( 1 + ( x s )c)−1 e−αx β ]δ−1 × e−αx β [ 1 + ( x s )c]−1{ αβxβ−1 + cxc−1 (sc + xc) } × 1 − {1 − (1 + ( x s )c)−1 exp(−αxβ) }δ−1 . 3. censoring in the following section we construct log-likelihood functions of the ellogw distribution to deal with type i right censoring and type ii doubly censored observations. 3.1. type i right censoring this is the most common form of incomplete data often encountered in survival analysis. type i censoring arises when the study is conducted over a specified time period that can terminate before all the units have failed. each individual has a fixed censoring time ci, which would be the time between the date of entry and the end of the study, so that the complete failure time of an individual will be known only if it is less than or equal to the censoring time ti ≤ ci; otherwise, only a lower bound of the individual lifetime is available ti > ci. the data for this design are conveniently indicated by pairs of random variables (ti, εi), i = 1, ..., n. consider a sample of size n of independent positive random variables t1, ..., tn, such that ti is associated with an indicator variable εi=0 if ti is a censoring time. let θ=(s,c,α,β,δ)t , then the likelihood function, l(θ), of a type i right censored sample (t1,ε1), ..., (tn,εn) from the ellogw distribution with pdf gellogw (.) and survival function s ellogw (.) can be written as l(θ) = n∏ i=1 gellogw (ti) �i s ellogw (ti) 1−�i, (3) where s ellogw (ti)= 1 gellogw (ti). the log-likelihood function, l(θ), based on data, from the ellogw distribution is, l(θ) = n∑ i=1 { �i ln [ δ ( 1 − ( 1 + ( ti s )c)−1 e−αti β )δ−1 × e−αti β ( 1 + ( ti s )c)−1( αβti β−1 + ctic−1 sc + tic )] (4) + (1 − �i) ln [ 1 − ( 1 − ( 1 + ( ti s )c)−1 e−αti β )δ]} . elements of the score vector are given by( ∂l(θ) ∂s , ∂l(θ) ∂c , ∂l(θ) ∂α , ∂l(θ) ∂β , ∂l(θ) ∂δ )t . the mles θ̂ = (ŝ, ĉ, α̂, β̂, δ̂)t are obtained from the numerical maximization of equation (4). let θ = (s, c,α,β,δ)t be the parameter vector and θ̂ = (ŝ, ĉ, α̂, β̂, δ̂)t be the maximum likelihood estimate of θ = (s, c,α,β,δ)t . under the usual regularity conditions and that the parameters are in the interior of the parameter space, but not on the boundary, [12] we have: √ n(θ̂ − θ) d −→ n5(0, i−1(θ)), where i(θ) is the expected fisher information matrix. the asymptotic behavior is still valid if i(θ) is replaced by the observed information matrix evaluated at θ̂, that is j(θ̂). the multivariate normal distribution n5(0, j(θ̂)−1), where the mean vector 0 = (0, 0, 0, 0, 0)t , can be used to construct confidence intervals and confidence regions for the individual model parameters and for the survival and hazard functions. 3.2. type ii censoring for type ii censoring, the data consists of the rth smallest lifetimes x(1) ≤ x(2) ≤ ... ≤ x(r) out of a random sample of n lifetimes x1, x2, ..., xn from the ellogw distribution. assuming x1 ≤ x2 ≤ ... ≤ xn are independent and identically distributed and have a continuous distribution with pdf gellogw (.) and survival function s ellogw (.), it follows that the joint pdf of x(1), ..., x(r) is l(θ) = n! (n − r)! [ r∏ i=1 gellogw (x(i)) ][ s ellogw (x(r)) ]n−r ,(5) 13 fagbamigbe et al. / j. nig. soc. phys. sci. 1 (2019) 12–19 14 table 1: times to infection data estimates statistics model s c α β δ −2 log l aic aicc bic w∗ a∗ ks pvalue s s ellogw 2.1674 0.8092 0.0090 1.7083 4.1106 98.0372 108.0372 114.7039 111.5775 0.0225 0.1556 0.999 0.0196 (2.5612) (0.7057) (0.0450) (1.4104) (2.9263) llogw 20.4723 1.5446 0.0071 1.7720 1 98.2032 106.2032 112.8698 109.0354 0.0244 0.1655 0.9988 0.0213 (0.0309) (1.0376) (0.0193) (0.8868) elloge 0.0192 0.3982 0.1266 1 28.8554 98.4424 106.4424 113.1091 109.2746 0.0242 0.1697 0.9922 0.0208 (0.0991) (0.4326) (0.0633) (0.0002) lloge 12.5296 2.5301 0.0295 1 1 99.4833 105.4833 112.1500 107.6075 0.0283 0.1963 0.9981 0.0193 (7.2980) (1.0222) (0.0482) λ β δ α θ gd 9.0005 1.9151 0.4786 3.9043 0.0192 98.7984 108.7984 115.4651 112.3386 0.0274 0.1932 0.9676 0.0254 (13.2476) (2.5511) (0.4362) (3.4708) (0.0658) where s ellogw (x) = 1 − gellogw (x). the log-likelihood function, l(θ), based on data from the ellogw distribution is, l(θ) = ln [ n! (n − r)! ] + r∑ i=1 { ln δ + (δ− 1) × ln [ 1 − ( 1 + ( x(i) s )c)−1 e−αx β (i) ] −αxβ(i) (6) − ln [ 1 + ( x(i) s )c] + ln { αβxβ−1(i) + cxc−1(i) sc + xc(i) } +(n − r) ln [ 1 − ( 1 − ( 1 + ( x(r1 ) s )c) e−αx β (r1 ) )δ]} . elements of the score vector and mle’s are similar to those in the type i right censoring scheme. 3.3. type ii double censoring type ii double censoring is a censoring procedure wherein a fixed number of observations is missing at the two ends of a n sample size unlike in type i censoring, where the number of censored observations is totally random and the study time is predetermined. the data consist of the remaining ordered observations tr+1,...,tm when the r smallest observations and the n−m largest observations are out of a sample of size n from the ellogw distribution. the likelihood function, l(θ), of the type ii doubly censored sample t(r+1),...,t(m) from the ellogw distribution with pdf gellogw (.), cdf gellogw (.) and survival function s ellogw (.) is given by l(θ) = n! r!(n − m)! {gellogw (t(r+1))} r ×{s ellogw (t(m))} n−m m∏ i=r+1 gellogw (t(i)). (7) the log-likelihood function, l(θ), for a type ii doubly censored sample t(r+1),...t(m) from the ellogw distribution is given by l(θ) = ln ( n! r!(n − m)! ) +r ln [( 1 − ( 1 + ( t(r+1) s )c)−1 e−αt(r+1) β )δ] +(n − m) ln [ 1 − ( 1 − ( 1 + ( t(m) s )c)−1 e−αt(m) β )δ] + m∑ i=r+1 ln { δ ( 1 − ( 1 + ( t(i) s )c)−1 e−αt(i) β )δ−1 (8) ×e−αt(i) β ( 1 + ( t(i) s )c)−1( αβt(i) β−1 + ct(i)c−1 sc + t(i)c )} . elements of the score vector and their mle’s are similar to those in the type i right censoring scheme. 4. applications in this section, we give some applications to real life data. maximum likelihood estimates of the model parameters under type i right and type ii double censored data are obtained and comparisons with the log-logistic weibull (llogw), exponentiated log-logistic exponential (elloge) and log-logistic exponential (lloge) distributions as well as the non-nested gamma-dagum (gd) [11, 13] distribution are presented. the mles of the parameters (standard error in parenthesis), -2log-likelihood statistic (−2 ln(l)), akaike information criterion (aic = 2p − 2 ln(l)), bayesian information criterion (bic = p ln(n) − 2 ln(l)), and corrected akaike information criterion ( aicc = aic + 2 p(p+1)n−p−1 ) , where l = l(θ̂) is the value of the likelihood function evaluated at the parameter estimates, n is the number of observations, and p is the number of estimated parameters are presented. the goodness-of-fit statistics: kolmogorov-smirnov p-value (ks p-value), cramer-von mises (w∗) and anderson-darling (a∗) were also presented in the tables. these statistics can be used to verify which distribution fits better to the data. in general, the smaller the values of w∗ and a∗, the better the fit. the ellogw distribution is fitted to the data sets and these fits are compared to the fits using exponentiated log-logistic exponential (elloge), log-logistic weibull (llogw) and log-logistic exponential (lloge) distributions. 14 fagbamigbe et al. / j. nig. soc. phys. sci. 1 (2019) 12–19 15 figure 1: fitted densities, probability plots for times to infection data table 2: maintenance data: type i right censoring estimates statistics model s c α β δ −2 log l aic aicc bic w∗ a∗ ks p − value s s ellogw 0.4681 1.2222 0.0023 3.9238 3.2765 115.3248 125.3248 127.2603 133.3794 0.0330 0.2262 0.8368 0.0360 (0.4880) (0.2612) (0.0033) (0.8702) (2.9931) llogw 1.7033 1.4847 0.0049 3.4126 1 118.9152 126.9152 128.8507 133.3589 0.0744 0.4677 0.5637 0.0569 (0.5094) (0.3395) (0.0301) (3.6607) elloge 0.0169 0.6744 0.4229 1 27.9948 120.4324 128.4324 130.3678 134.8760 0.0576 0.3904 0.7290 0.0594 (0.0207) (0.2535) (0.1848) (0.0003) lloge 1.9005 1.8526 0.1235 1 1 124.1565 130.1565 132.0920 134.9893 0.0902 0.5814 0.6966 0.0641 (1.1114) (0.3484) (0.2375) λ β δ α θ gd 1.7891 2.6292 0.4189 3.1506 0.0332 121.6979 131.6979 133.6333 139.7525 0.0686 0.4541 0.6322 0.0724 (3.2287) (6.4012) (0.7477) (8.2541) (0.1681) table 3: maintenance data: type ii double censoring estimates statistics model s c α β δ −2 log l aic aicc bic w∗ a∗ ks p − value s s ellogw 0.5157 1.3934 0.0003 4.9765 5.0079 97.5309 107.5309 109.9309 114.7009 0.0450 0.3378 0.5716 0.0535 (0.1059) (0.2409) (0.0003) (0.7204) (0.0146) llogw 21.0083 1.6085 0.1990 1.6298 1 105.1903 113.1903 115.5903 118.9262 0.1017 0.6910 0.3093 0.0976 (0.0046) (2.4384) (0.0849) (0.2486) elloge 0.0221 0.9696 0.3636 1 98.3757 102.7356 110.7356 113.1356 116.4715 0.0777 0.5469 0.7063 0.0845 (0.0337) (0.4440) (0.2369) (0.00007) lloge 5.1884 25.1414 0.3791 1 1 106.8501 112.8501 115.2501 117.1521 0.0782 0.5817 0.1414 0.1575 (0.3930) (26.2939) (0.0784) λ β δ α θ gd 1.7696 2.9249 0.5022 3.0936 0.0341 104.1335 114.1335 116.5335 121.3034 0.0853 0.5947 0.5902 0.0895 (3.0503) (3.8793) (0.4692) (4.9479) (0.0966) 4.1. type i right censoring: times to infection data the following example utilizes the times to infection (in months) of kidney dialysis patients with a surgically placed catheter. this data is given in table 1.2 of klein and moeschberger [7]. the infection times are given in table a.1 and the censored observations are given in table a.2 in the appendix. the initial values for the ellogw distribution in r code are s = 2, c = 0.9, α = 0.01, β = 1.9 and δ = 4. the mles of the parameters of the ellogw distribution and its related submodels, aic, aicc, bic, w∗, a∗, ks p-value and ss are given in table 1, along with standard errors in parentheses. plots of the fitted densities and the histogram, observed probability vs predicted probability ([6]) are given in figure 1. the likelihood ratio (lr) test statistic of the hypothesis h0: llogw against ha: ellogw and h0: elloge against ha: ellogw are 0.166 (p-value = 0.6837) and 0.4052 (p-value 15 fagbamigbe et al. / j. nig. soc. phys. sci. 1 (2019) 12–19 16 fitted pdf x f( x) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ellogw llogw elloge lloge gd 0.0 0.2 0.4 0.6 0.8 1.0 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 the graph of observed vs expected probability observed probability e xp e ct e d p ro b a b ili ty ellogw(ss=0.0360) llogw(ss=0.0569) elloge(ss=0.0594) lloge(ss=0.0641) gd(ss=0.07479690) figure 2: fitted densities, probability plots for type i right censoring maintenance data fitted pdf x f( x) 0 1 2 3 4 5 6 0.0 0.1 0.2 0.3 0.4 0.5 ellogw llogw elloge lloge gd 0.0 0.2 0.4 0.6 0.8 1.0 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 the graph of observed vs expected probability observed probability e xp e ct e d p ro b a b ili ty ellogw(ss=0.0546) llogw(ss=0.0976) elloge(ss=0.0845) lloge(ss=0.1575) gd(ss=0.0895) figure 3: fitted densities, probability plots for type ii double censoring maintenance data = 0.5244). we can conclude that there are no significant differences between fits of the llogw and ellogw distributions, as well as between fits of the elloge and ellogw distributions. the values of the goodness-of-fit statistics a∗ and w∗ shows that the ellogw distribution is the better fit for the time to infection data. the kolmogorov-smirnov p-value shows that the ellogw distribution fits the data better than the other models. also, the value of ss from the probability plot in figure 1 is the smallest for ellogw model. also, figure 1 showed that the new model ellogw provided better fit for the density and probability plots than the llogw, ellogw, lloge and gd models. 16 fagbamigbe et al. / j. nig. soc. phys. sci. 1 (2019) 12–19 17 4.2. type i right censoring: maintenance data in the following example, we consider a maintenance data set. the set of data is the maintenance data with 46 observations reported on active repair times (hours) for an airborne communication transceiver. the data was discussed by alven [8], chhikara and folks [9], and dimitrakopoulou et al. [10]. the maintenance data set is given in table a.3 in the appendix. the data given in table a.4 has eight observations removed to illustrate type i right censoring (see appendix for the data). initial values for the ellogw model in r code are s = 0.3, c = 1, α = 0.002, β = 3 and δ = 3. the mles of the parameters of the ellogw distribution and its related submodels, aic, aicc, bic, w*, a*, ks p-value and ss are given in table 2. plots of the fitted densities and the histogram, observed probability vs predicted probability are given in figure 2. the lr test statistic of the hypothesis h0: llogw against ha: ellogw and h0: elloge against ha: ellogw are 3.5904 (p-value = 0.0581) and 5.1076 (p-value = 0.0238). we can conclude that there is a significant difference between the fits of the elloge and ellogw distributions. there is also a significant difference between fits of the llogw and ellogw distributions at the 10 % level of significance. the values of the goodness-of-fit statistics a∗ and w∗ shows that the ellogw distribution is by far the better fit for the maintenance data. the ks p-value shows that the ellogw distribution fits the maintenance data better than the other models. figure 2 showed that the new model ellogw provided better fit for the density and probability plots than the llogw, ellogw, lloge and gd models. 4.3. type ii double censoring: maintenance data in the following example we consider the maintenance data set with 15 observations removed to illustrate type ii double censoring. the data is given in table a.5 in the appendix. initial values for the ellogw model are s = 0.4, c = 1.4, α = 0.001, β = 5 and δ = 5. the mles of the parameters of the ellogw distribution and its related sub models, aic, aicc, bic, w*, a*, ks p-value and ss are given in table 3, along with standard errors in parentheses. plots of the fitted densities and the histogram, observed probability vs predicted probability are given in figure 3. the lr test statistic of the hypothesis h0: llogw against ha: ellogw and h0: elloge against ha: ellogw are 7.6594 (p-value = 0.0056) and 5.2047 (p-value = 0.0225). we can conclude that there is a significant difference between the fits of the llogw and the ellogw distributions as well as between the fits of the elloge and ellogw distributions. the values of the goodness-of-fit statistics a∗ and w∗ shows that the ellogw distribution is by far the better fit for the maintenance data. the ks p-value for the ellogw distribution is greater than that of the llogw and lloge models. figure 3 showed that the new model ellogw provided better fit for the density and probability plots than the llogw, ellogw, lloge and gd models. 5. concluding remarks we have presented the ellogw distribution under different censoring mechanisms. applications of the model, under type i right censoring and type ii double censoring, to real data are presented to illustrate its usefulness and applicability. the findings suggest that the ellogw distribution performed better than all other distributions considered. acknowledgment we thank the referees for the positive enlightening comments and suggestions, which have greatly helped us in making improvements to this paper. references [1] b. o. oluyede, g. basele, s. huang & b. makubate, “a new class of generalized log-logistic weibull distribution: theory, properties and applications”, journal of probability and statistical sciences, 14 (2016) 171. 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[13] b. o. oluyede, s. huang & m. pararai, “a new class of generalized dagum distribution with applications to income and lifetime data”, journal of statistical and econometric methods 3 (2014) 125. 17 fagbamigbe et al. / j. nig. soc. phys. sci. 1 (2019) 12–19 18 appendix a. elements of the score vector: type i right censoring elements of the score vector for the type i right censoring scheme are given by, ∂`(θ) ∂s = n∑ i=1 { �i [ δ(δ− 1) ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ−2 (−1) ( 1 + ( ti s ))−3( ti s )c−1( cti s2 ) e−αt β i ( αβtβ−1i + ctc−1i sc + tci ) + [ δ ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ−1 e−αt β i ][( 1 + ( ti s )c)−2( ti s )c−1( cti s2 )( αβtβ−1i + ctc−1i sc + tci ) − ( 1 + ( ti s )c)−1 c2(sti)c−1 (sc + tci ) 2 ]] × [ δ ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ−1 e−αt β i ( 1 + ( ti s ))−1( αβtβ−1i + ctc−1i sc + tci )]−1 +(1 − �i) [ δ ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ−1 e−αt β i ( 1 + ( ti s )c)−2( ti s )c−1( cti s2 )][ 1 − ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ]−1} , ∂`(θ) ∂c = n∑ i=1 { �i [ δ(δ− 1) ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ−2 (−1) ( 1 + ( ti s ))−3( ti s )c ln ( ti s ) e−αt β i ( αβtβ−1i + ctc−1i sc + tci ) + [ δ ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ−1 e−αt β i ][ (−1) ( 1 + ( ti s )c)−1( ti s )c ln ( ti s )( αβtβ−1i + ctc−1i sc + tci ) − ( 1 + ( ti s )c)−1 × t2(c−1)i ln(ti)(s c + tci ) − ct c−1 i (s c ln s + tci ln ti) (sc + tci ) 2 ]][ δ ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ−1 e−αt β i ( 1 + ( ti s ))−1( αβtβ−1i + ctc−1i sc + tci )]−1 +(1 − �i) [ δ ( 1 − ( 1 + ( ti s )c)−2 e−αt β i )δ−1 e−αt β i ( 1 + ( ti s )c)−2( ti s )c ln ( ti s )][ 1 − ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ]−1} , ∂`(θ) ∂α = n∑ i=1 { �i [[ δ(δ− 1) ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ−2( 1 + ( ti s ))−1 tβi e −2αtβi −δ ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ−1( 1 + ( ti s ))−1 tβi e −αtβi ] × ( 1 + ( ti s ))−1( αβtβ−1i + ctc−1i sc + tci ) + δ ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ−1 e−αt β i ( 1 + ( ti s )c)−1 βtβ−1i ] × [ δ ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ−1 e−αt β i ( 1 + ( ti s ))−1( αβtβ−1i + ctc−1i sc + tci )]−1 + (1 − �i) [ δ ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ−1 ×e−αt β i ( 1 + ( ti s )c)−1 (−tβi ) ][ 1 − ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ]−1} , ∂`(θ) ∂β = n∑ i=1 { �i [[ δ(δ− 1) ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ−2( 1 + ( ti s ))−1 αtβi e −2αtβi ln(ti) −δ ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ−1( 1 + ( ti s ))−1 ×αtβi e −αtβi ln(ti) ]( 1 + ( ti s ))−1( αβtβ−1i + ctc−1i sc + tci ) + δ ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ−1 e−αt β i ( 1 + ( ti s )c)−1 (αtβ−1i + αβt β−1 i ln(ti)) ] × [ δ ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ−1 e−αt β i ( 1 + ( ti s ))−1( αβtβ−1i + ctc−1i sc + tci )]−1 + (1 − �i) [ δ ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ−1 ×e−αt β i ( 1 + ( ti s )c)−1 (−αtβi ) ln(ti) ][ 1 − ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ]−1} and ∂`(θ) ∂δ = n∑ i=1 { �i [ δ ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ−1 e−αt β i ln ( 1 − ( 1 + ( ti s )c)−1 e−αt β i ) + ( 1 − ( 1 + ( ti s )c)−1 e−αt β i ] × [ δ ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ−1 e−αt β i ( 1 + ( ti s ))−1( αβtβ−1i + ctc−1i sc + tci )]−1 + (1 − �i) [( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ × ln ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )][ 1 − ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ]−1} . 18 fagbamigbe et al. / j. nig. soc. phys. sci. 1 (2019) 12–19 19 tables with complete and censored data the infection times are given in table a.1 and the censored observations are given in table a.2. the complete and censored maintenance data are given in tables a.3, a.4 and a.5, respectively. table a.1: infection times 1.5 3.5 4.5 5.5 8.5 8.5 9.5 10.5 11.5 15.5 16.5 18.5 23.5 26.5 table a.2: censored infection times 2.5 2.5 3.5 3.5 3.5 4.5 4.5 5.5 6.5 6.5 7.5 7.5 7.5 7.5 8.5 9.5 10.5 11.5 12.5 12.5 13.5 14.5 14.5 21.5 21.5 22.5 22.5 25.5 27.5 table a.3: complete maintenance data 0.2 0.3 0.5 0.5 0.5 0.5 0.6 0.6 0.7 0.7 0.7 0.8 0.8 1.0 1.0 1.0 1.0 1.1 1.3 1.5 1.5 1.5 1.5 2.0 2.0 2.2 2.5 2.7 3.0 3.0 3.3 3.3 4.0 4.0 4.5 4.7 5.0 5.4 5.4 7.0 7.5 8.8 9.0 10.3 22.0 24.5 table a.4: type i right censored maintenance data 0.2 0.3 0.5 0.5 0.5 0.5 0.6 0.6 0.7 0.7 0.7 0.8 0.8 1.0 1.0 1.0 1.1 1.3 1.5 1.5 1.5 1.5 2.0 2.0 2.2 2.5 2.7 3.0 3.0 3.3 3.3 4.0 4.0 4.5 4.7 5.0 5.4 5.4 table a.5: maintenance data ii 0.6 0.7 0.7 0.7 0.8 0.8 1.0 1.0 1.0 1.1 1.3 1.5 1.5 1.5 1.5 2.0 2.0 2.2 2.5 2.7 3.0 3.0 3.3 3.3 4.0 4.0 4.5 4.7 5.0 5.4 5.4 19 j. nig. soc. phys. sci. 4 (2022) 174–182 journal of the nigerian society of physical sciences modified szmidt and kacprzyk’s intuitionistic fuzzy distances and their applications in decision-making p. a. ejegwaa,∗, i. c. onyekeb, b. t. terhemena, m. p. onojaa, a. ogijia, c. u. opeha adepartment of mathematics, university of agriculture, p.m.b. 2373, makurdi, nigeria bdepartment of computer science, university of agriculture, p.m.b. 2373, makurdi, nigeria abstract intuitionistic fuzzy models are significant in resolving decision-making. distance measures under intuitionistic fuzzy environment are reliable techniques deployed to express the application of ifss. some approaches of estimating distances between ifss have been explored by szmidt and kacprzyk, where the complete parameters of ifss are considered. albeit, the distance operators lack reliability because of certain setbacks. in this paper, we modified szmidt and kacprzyk’s distance operators between ifss to enhance reliability in terms of applications. some theorems are given to substantiate the validity of the modified intuitionistic fuzzy distance operators. futhermore, decision-making cases of pattern recognition and disease identification are discussed using the szmidt and kacprzyk’s distances and their improved versions where information are represented in intuitionistic fuzzy pairs. from the study, it is observed that the modified szmidt and kacprzyk’s distance operators between ifss yield better results compare to the szmidt and kacprzyk’s distance operators between ifss. doi:10.46481/jnsps.2022.530 keywords: decision-making, distance measure, intuitionistic fuzzy set, pattern recognition, medical diagnosis article history : received: 12 december 2021 received in revised form: 21 april 2022 accepted for publication: 24 april 2022 published: 29 may 2022 c©2022 journal of the nigerian society of physical sciences. all rights reserved. communicated by: t. latunde 1. introduction decision-making is a critical task enmeshed with vagueness. with the introduction of fuzzy sets by zadeh [1], the problem of vagueness has be considerably tackled to enhance the solution of many decision-making problems including medical diagnosis, career determination, pattern recognition among others. fuzzy set theory although relevant has a setback in the ∗corresponding author tel. no: +2347062583323 email addresses: ejegwa.augustine@uam.edu.ng (p. a. ejegwa ), onyeke.idoko@uam.edu.ng (i. c. onyeke), terhementarfabenjamin@gmail.com (b. t. terhemen), onojaexcel1234@gmail.com (m. p. onoja), andrewogiji@yahoo.com (a. ogiji), christophernathaniel360@gmail.com (c. u. opeh) sense that it considers only the membership degree υ (md) of the case under consideration. because of this setback, atanassov [2] proposed a generalized fuzzy set called intuitionistic fuzzy set (ifs). ifs is described by membership degree υ, nonmembership degree ν and intuitionistic fuzzy index $ with the property that their sum is unity. ifss have been applied in sundry cases [3, 4, 5]. chen et al. [6] discussed fuzzy queries process based on intuitionistic fuzzy social networks, de et al. [7] applied ifss to medical decision via composite relation, ejegwa and onasanya [8] improved the intuitionistic fuzzy composite relation in [7] with application to medical diagnosis, and liu and chen [9] presented a group decision-making based on heronian aggregation operators of ifss. several information measures have been studied to 174 ejegwa et al. / j. nig. soc. phys. sci. 4 (2022) 174–182 175 enhance the application of ifss in real-life problems. ejegwa [10] proposed a new correlation coefficient of ifss and applied the measure to solve multi-criteria decision-making (mcdm) problems. many correlation coefficients of ifss have been proposed and applied to several decision-making problems [11, 12, 13, 14, 15]. medical diagnostic problems were solved based on intuitionistic fuzzy correlation coefficient as seen in [16, 17, 18]. garg [19] presented a correlation coefficient under intuitionistic multiplicative environment with application in decisionmaking process, and a new correlation coefficient of ifss based on the connection number of set pair analysis was studied with application [20]. a robust technique of computing the correlation coefficients of complex ifss and their applications in decision-making were discussed in [21]. topsis method based on correlation coefficient under intuitionistic fuzzy soft sets was discussed and with application [22]. the idea of aggregation operators has been applied in many cases of decision-making [23, 24, 25, 26], and fuzzy soft set-valued maps has been introduced and applied to homotopy [27]. similarly, the concepts of similarity and distance measures under intuitionistic fuzzy context have been discussed as reliable information measures. boran and akay [28] proposed a biparametric similarity measure and applied the measure to pattern recognition. a similarity measure of ifss based on transformation technique has been studied and applied to pattern recognition [29]. in [30, 31], some similarity measures of ifss were introduced and applied to medical diagnostic reasoning. in addition, some similarity measures based on dice and jaccard approaches have been studied on expected intervals of trapezoidal neutrosophic fuzzy numbers with application to mcdm [32]. burillo and bustince [33] initiated the concept of distances for ifss and interval-valued fuzzy sets. szmidt and kacprzyk [34] modified the distances in [33], and showed that all the three parameters describing ifss should be taken into account while calculating distances between ifss. hatzimichailidis et al. [35] introduced a novel distance measure between ifss with application to cases of pattern recognition. wang and xin [36] proposed a novel distance measure between ifss and its weighted version with application to the solution of pattern recognition problem, davvaz and sadrabadi [37] revised some existing distance measures and applied them to medical diagnostic process, and other applications of distance measures between ifss have been studied [38, 39, 40, 41, 42]. among the distances between ifss studied in literature, distances in [34] are prominent for their reliable interpretations. albeit, these distances show some limitations which needed to be strengthened to enhance reliable outputs. although szmidt and kacprzyk [34] modified the intuitionistic fuzzy distance measures in [33] with better rating, they do not take account the number of the considered parameters, but just added the hesitation margins to the methods introduced in [33]. this setback adversely influence the performance rating of szmidt and kacprzyk’s distances [34]. the motivation for this work is to propose modified szmidt and kacprzyk’s distances which have better performance indexes compare to szmidt and kacprzyk’s distances taking into account the number of the considered parameters to forestall inaccurate outputs. specifically, the objectives of this paper are to (i) revisit szmidt and kacprzyk’s distances between ifss, (ii) propose modified versions of the distances between ifss in [34], (iii) apply the modified distances between ifss to determine pattern recognition and disease diagnosis, and (iv) present comparison analyses of the modified distances with the szmidt and kacprzyk’s distances in intuitionistic fuzzy domain. the paper is outlined as follow; section 2 presents the concept of ifss and discusses the szmidt and kacprzyk’s distances between ifss [34], section 3 introduces the modified szmidt and kacprzyk’s distances between ifss, section 4 discusses the applications of szmidt and kacprzyk’s distances and their modified versions in pattern recognition and disease diagnosis, and section 5 concludes the paper with recommendations for future studies. 2. preliminaries this section presents the concept of ifss and discusses the szmidt and kacprzyk’s distances between ifss. 2.1. intuitionistic fuzzy sets numerous works on ifss have been carried out [2, 43, 3]. here, some basic concepts of ifss are presented. let us assume that y is a non-empty set throughout this paper. definition 2.1. [43] an intuitionistic fuzzy set c of y is defined by c = {〈y,υc (y),νc (y)〉 : y ∈ y}, where the functions υc, νc : y → [0, 1] are the membership and non-membership degrees of y ∈ y , and 0 ≤ υc (y) + νc (y) ≤ 1. for a ifs c in y , $c (y) ∈ [0, 1] = 1 −υc (y) −νc (y) is the intuitionistic fuzzy index or hesitation margin of c. definition 2.2. [3] suppose c and d are ifss in y , then for all y ∈ y we have (i) c = d iff υc (y) = υd(y), νc (y) = νd(y). (ii) c ⊆ d iff υc (y) ≤ υd(y), νc (y) ≥ νd(y). (iii) c = {〈y,νc (y),υc (y)〉 : y ∈ y}. (iv) c ∪ d = {〈y, max(υc (y),υd(y)), min(νc (y),νd(y))〉 : y ∈ y}. (v) c ∩ d = {〈y, min(υc (y),υd(y)), max(νc (y),νd(y))〉 : y ∈ y}. definition 2.3. [3] intuitionistic fuzzy pair (ifp) is characterized by the form 〈c, d〉 such that c + d ≤ 1 where c, d ∈ [0, 1]. ifp evaluate the ifs for which the components (c and d) are interpreted as membership and non-membership degrees. 2.2. distances between intuitionistic fuzzy sets distance measure is a soft computing tool use in the applications of ifss. the definition of distance measure between ifss is a follows. definition 2.4. [34] if c and d are ifss of y , then the distance between c and d denoted by φ(c, d) is a function φ: i fs × i fs → [0, 1] which satisfies (i) 0 ≤ φ(c, d) ≤ 1 175 ejegwa et al. / j. nig. soc. phys. sci. 4 (2022) 174–182 176 (ii) φ(c, d) = 0 iff c = d (iii) φ(c, d) = φ(d, c) (iv) φ(c, e) ≤ φ(c, d) + φ(d, e), where e is also an ifs of y . when φ(c, d) reaches 0, it shows that c and d are more close or related. again, if φ(c, d) reaches 1 then c and d are not related or close. for any two ifss c and d in y = {y1, · · · , yn}, we present the following distances between them. 2.2.1. burillo and bustince’s distances between ifss by extending the distances between fuzzy sets as presented in [44], burillo and bustince [33] proposed the following distances under intuitionistic fuzzy environment: φ1(c, d) = 1 2 σ n i=1 ( |υc (yi) −υd(yi)| + |νc (yi) −νd(yi)| ) (1) φ2(c, d) = (1 2 σ n i=1 ( (υc (yi)−υd(yi)) 2 +(νc (yi)−νd(yi)) 2))12 (2) φ3(c, d) = 1 2n σ n i=1 ( |υc (yi)−υd(yi)|+ |νc (yi)−νd(yi)| ) (3) φ4(c, d) = ( 1 2n σ n i=1 ( (υc (yi)−υd(yi)) 2 +(νc (yi)−νd(yi)) 2))12 (4) the denominator in eqs. (1-4) depicts the number of parameters of ifss considered similar to the approach in [44] where the denominator is unity because fuzzy set considers only membership function. the limitation of these approaches [33] is that the hesitation margin is not considered in the computations. 2.2.2. szmidt and kacprzyk’s distances between ifss because of the limitation in [33], szmidt and kacprzyk [34] proposed the some distances by incorporating hesitation margin of the considered ifss. for simplicity sake, let υc (yi) = υc , νc (yi) = νc , $c (yi) = $c , υd(yi) = υd, νd(yi) = νd, $d(yi) = $d. the distances are as follow: φ̂1(c, d) = 1 2 σ n i=1 ( |υc −υd| + |νc −νd| + |$c −$d| ) (5) φ̂2(c, d) = (1 2 σ n i=1 ( (υc−υd) 2 +(νc−νd) 2 +($c−$d) 2))12 (6) φ̂3(c, d) = 1 2n σ n i=1 ( |υc −υd| + |νc −νd| + |$c −$d| ) (7) φ̂4(c, d) = ( 1 2n σ n i=1 ( (υc−υd) 2 +(νc−νd) 2 +($c−$d) 2))12 (8) though the distances in [34] captured the three parameters of ifss, the denominators in each equations do not depict the number of parameters of the considered ifss. these omissions will lead to information loss and thus, affect the interpretations. 3. modified szmidt and kacprzyk’s distances between ifss to enhance reliable outputs and avoid information loss, we modified the szmidt and kacprzyk’s distances between ifss [34] using the same hypotheses in subsubsection 2.2.2 as follow: φ̃∗(c, d) = (1 3 σ n i=1 ( |υc−υd| r +|νc−νd| r +|$c−$d| r))1r (9) φ̃∗∗(c, d) = ( 1 3n σ n i=1 ( |υc−υd| r +|νc−νd| r +|$c−$d| r))1r (10) for r ≤ 2. if r = 1, we get φ̃1(c, d) = 1 3 σ n i=1 ( |υc −υd| + |νc −νd| + |$c −$d| ) (11) φ̃2(c, d) = 1 3n σ n i=1 ( |υc −υd|+ |νc −νd|+ |$c −$d| ) (12) if r = 2, we get φ̃3(c, d) = (1 3 σ n i=1 ( (υc−υd) 2 +(νc−νd) 2 +($c−$d) 2))12 (13) φ̃4(c, d) = ( 1 3n σ n i=1 ( (υc−υd) 2 +(νc−νd) 2 +($c−$d) 2))12 (14) proposition 3.1. suppose c and d are ifss of y = {y1, . . . , yn}, then we have (i) φ̃1(c, d) = nφ̃2(c, d) (ii) φ̃3(c, d) = √ nφ̃4(c, d). proof. given that φ̃2(c, d) = 1 3n σni=1 ( |υc − υd| + |νc − νd| + |$c −$d| ) . then it implies that φ̃2(c, d) = 1 3 σni=1 ( |υc −υd| + |νc −νd| + |$c −$d| ) n = φ̃1(c, d) n . hence (i) holds. similarly, φ̃4(c, d) = ( 1 3n σ n i=1 ( (υc −υd) 2 + (νc −νd) 2 + ($c −$d) 2))12 = (1 3 σni=1 ( (υc −υd)2 + (νc −νd)2 + ($c −$d)2 ))12 √ n = φ̃3(c, d) √ n , and so (ii) holds. proposition 3.2. if c and d are ifss in y , then the following hold: (i) φ̃∗(c, d) = φ̃∗(d, c) (ii) φ̃∗(c, d) = φ̃∗(c, d). 176 ejegwa et al. / j. nig. soc. phys. sci. 4 (2022) 174–182 177 proof. for the proof of (i), we have φ̃∗(c, d) = (1 3 σ n i=1 ( |υc −υd| + |νc −νd| + |$c −$d| ))1 r = (1 3 σ n i=1 ( |υd −υc| + |νd −νc| + |$d −$c| ))1 r = φ̃∗(d, c). hence (i) holds. the proof of (ii) is similar. similarly, we have the following proposition. proposition 3.3. suppose c and d are ifss in y , then the following hold: (i) φ̃∗∗(c, d) = φ̃∗∗(d, c) (ii) φ̃∗∗(c, d) = φ̃∗∗(c, d). theorem 3.4. suppose c, d and e are ifss in y , then the function φ̃∗(c, d) satisfies (i) 0 ≤ φ̃∗(c, d) ≤ 1 (ii) φ̃∗(c, d) = 0 iff c = d (iii) φ̃∗(c, d) = φ̃∗(d, c). proof. the proof of (i) is straightforward. recall that φ̃∗(c, d) = (1 3 σ n i=1 ( |υc −υd| r + |νc −νd| r + |$c −$d| r))1r . to proof (ii), suppose that φ̃∗(c, d) = 0. then |υc −υd| r = 0, |νc −νd| r = 0, and |$c −$d| r = 0, and thus, υc = υd, νc = νd and $c = $d, hence c = d. the converse is easy to see, so omitted. the proof of (iii) is the same as (i) of proposition 3.2. from theorem 3.4, we have the following result. theorem 3.5. if c, d and e are ifss in y , then the function φ̃∗∗(c, d) satisfies (i) 0 ≤ φ̃∗∗(c, d) ≤ 1 (ii) φ̃∗∗(c, d) = 0 iff c = d (iii) φ̃∗∗(c, d) = φ̃∗∗(d, c). theorem 3.6. suppose c, d and e are ifss in y and c ⊆ d ⊆ e. then we have (i) φ̃∗(c, e) ≥ φ̃∗(c, d), (ii) φ̃∗(c, e) ≥ φ̃∗(d, e), (iii) φ̃∗(c, e) ≥ max[φ̃∗(c, d), φ̃∗(d, e)]. proof. if c ⊆ d ⊆ e, then |υc −υe|r ≥ |υc −υd|r , |νc −νe|r ≥ |νc −νd| r and |$c −$e|r ≥ |$c −$d|r . thus |υc −υe| r + |νc −νe| r + |$c −$e| r ≥ |υc −υd| r + |νc −νd| r + |$c −$d| r. so, φ̃∗(c, e) ≥ φ̃∗(c, d), which proves (i). by the same logic we see that φ̃∗(c, e) ≥ φ̃∗(d, e), so (ii) holds. combining (i) and (ii), it follows that φ̃∗(c, e) ≥ max[φ̃∗(c, d), φ̃∗(d, e)], which proves (iii). by consequence, we have theorem 3.7. theorem 3.7. suppose c, d and e are ifss in y and c ⊆ d ⊆ e. then we have (i) φ̃∗∗(c, e) ≥ φ̃∗∗(c, d), (ii) φ̃∗∗(c, e) ≥ φ̃∗∗(d, e), (iii) φ̃∗∗(c, e) ≥ max[φ̃∗∗(c, d), φ̃∗∗(d, e)]. 4. experimental examples in this section, we apply both the szmidt and kacprzyk’s distances and their modified versions to problems of pattern recognition and medical diagnosis to determine which of the approaches are better in terms of performance indexes. 4.1. case i pattern recognition is the process of identifying patterns by using machine learning method. the idea of pattern recognition is important because of its application potential in diverse areas. assume there are three patterns p1, p2, p3 denoted with ifps in y = {y1, y2, y3}. if there is an unknown pattern q denoted with ifp in the same feature space y . the intuitionistic fuzzy representations of these patterns are in table 1. table 1: intuitionistic fuzzy representations of patterns feature space ifps y1 y2 y3 υp1 νp1 $p1 0.1000 0.1000 0.8000 0.5000 0.1000 0.4000 0.1000 0.9000 0.0000 υp2 νp2 $p2 0.5000 0.5000 0.0000 0.7000 0.3000 0.0000 0.0000 0.8000 0.2000 υp3 νp3 $p3 0.7000 0.2000 0.1000 0.1000 0.8000 0.1000 0.4000 0.4000 0.2000 υq νq $q 0.4000 0.4000 0.2000 0.6000 0.2000 0.2000 0.0000 0.8000 0.2000 take skd as szmidt and kacprzyk’s distance and msk as modified szmidt and kacprzyk’s distance. the task is to determine which of the patterns can the sample q be associated with. table 2 contains the distances using the szmidt and kacprzyk’s distances while table 3 contains the distances based on the modified szmidt and kacprzyk’s distances. from tables 2 and 3, the sample q can be classified with pattern p2 since the distances between them are the smallest. tables 2 and 3 can be represented by the following figures to show the superiority of the modified szmidt and kacprzyk’s distances over the szmidt and kacprzyk’s distances. 177 ejegwa et al. / j. nig. soc. phys. sci. 4 (2022) 174–182 178 table 2: results using szmidt and kacprzyk’s distances distances (p1, q) (p2, q) (p3, q) φ1 1.0000 0.4000 1.3000 φ2 0.3333 0.1333 0.4333 φ3 0.5745 0.2449 0.7348 φ4 0.3317 0.1414 0.4243 table 3: results using modified szmidt and kacprzyk’s distances distances (p1, q) (p2, q) (p3, q) φ̃1 0.6667 0.2667 0.8667 φ̃2 0.2222 0.0889 0.2889 φ̃3 0.4690 0.2000 0.6000 φ̃4 0.2708 0.1155 0.3464 table 4: intuitionistic fuzzy representations of diagnostic process clinical manifestations ifss s1 s2 s3 s4 s5 υv νv $v 0.4000 0.0000 0.6000 0.3000 0.5000 0.2000 0.1000 0.7000 0.2000 0.4000 0.3000 0.3000 0.1000 0.7000 0.2000 υm νm $m 0.7000 0.0000 0.3000 0.2000 0.6000 0.2000 0.0000 0.9000 0.1000 0.7000 0.0000 0.3000 0.1000 0.8000 0.1000 υt νt $t 0.3000 0.3000 0.4000 0.6000 0.1000 0.3000 0.2000 0.7000 0.1000 0.2000 0.6000 0.2000 0.1000 0.9000 0.0000 υs νs $s 0.1000 0.7000 0.2000 0.2000 0.4000 0.4000 0.8000 0.0000 0.2000 0.2000 0.7000 0.1000 0.2000 0.7000 0.1000 υc νc $c 0.1000 0.8000 0.1000 0.0000 0.8000 0.2000 0.2000 0.8000 0.0000 0.2000 0.8000 0.0000 0.8000 0.1000 0.1000 υp νp $p 0.6000 0.1000 0.3000 0.5000 0.4000 0.1000 0.3000 0.4000 0.3000 0.7000 0.2000 0.1000 0.3000 0.4000 0.3000 table 5: results using szmidt and kacprzyk’s distances distances (v, p) (m, p) (t, p) (s, p) (c, p) φ1 1.4000 1.5000 1.9000 2.2000 2.7000 φ2 0.5568 0.6557 0.7810 0.9644 1.1091 φ3 0.2800 0.3000 0.3800 0.4400 0.5400 φ4 0.2490 0.2933 0.3493 0.4313 0.4960 4.2. case ii diagnosis of diseases is challenging due to embedded fuzziness in the processes. here, we present a scenario of mathematical approach of diagnosing a patient medical status using the szmidt and kacprzyk’s distance and the modified szmidt and table 6: results using modified szmidt and kacprzyk’s distances distances (v, p) (m, p) (t, p) (s, p) (c, p) φ1 0.9333 1.0000 1.2667 1.4667 1.8000 φ2 0.4546 0.5354 0.6377 0.7874 0.8200 φ3 0.1867 0.2000 0.2533 0.2933 0.3600 φ4 0.2033 0.2394 0.2852 0.3521 0.4050 kacprzyk’s distance, where the symptoms or clinical manifestations of the diseases are represented as ifps using hypothetical case. suppose we have a set of diseases namely; viral fever (v), malaria (m), typhoid fever (t), stomach ulcer (s) and chest 178 ejegwa et al. / j. nig. soc. phys. sci. 4 (2022) 174–182 179 figure 1: distances using eqs. (5) and (11) (p1 q) (p2 q) (p3 q) 0.2 0.4 0.6 0.8 1 1.2 1.4 classifications d is ta nc es skd mskd figure 2: distances using eqs. (6) and (13) (p1 q) (p2 q) (p3 q) 0.2 0.4 0.6 classifications d is ta nc es skd mskd problem (c) represented by ifps, and a set of symptoms s = {s1, s2, s3, s4, s5} where s1 = temperature, s2 = headache, s3 = stomach pain, s4 = cough, s5 = chest pain. these symptoms are the clinical manifestations of the mentioned diseases. assume a patient p manifests symptoms as mentioned above, represented by ifps. table 4 contains intuitionistic fuzzy information of the diseases and patient p with respect to the symptoms. now, we find which of the diseases has the smallest distance with the patient with respect to the symptoms by deploying the szmidt and kacprzyk’s distances and their modifications. tafigure 3: distances using eqs. (7) and (12) (p1 q) (p2 q) (p3 q) 0.1 0.2 0.3 0.4 classifications d is ta nc es skd mskd figure 4: distances using eqs. (8) and (14) (p1 q) (p2 q) (p3 q) 0.1 0.2 0.3 0.4 classifications d is ta nc es skd mskd bles 5 and 6 contain the results. from tables 5 and 6, it is inferred that the patient is suffering from viral fever since the distance between the patient and viral fever is the smallest. tables 5 and 6 can be represented by the following figures to show the supremacy of the modified szmidt and kacprzyk’s distances over the szmidt and kacprzyk’s distances. from figs. 1–8, it is observed that the modified szmidt and kacprzyk’s distances outperformed the szmidt and kacprzyk’s distances in terms of accuracy because while the modified szmidt 179 ejegwa et al. / j. nig. soc. phys. sci. 4 (2022) 174–182 180 figure 5: distances using eqs. (5) and (11) (v p) (mp) (t p) (s p) (cp) 1 1.5 2 2.5 classifications d is ta nc es skd mskd figure 6: distances using eqs. (6) and (13) (v p) (mp) (t p) (s p) (cp) 0.4 0.6 0.8 1 classifications d is ta nc es skd mskd and kacprzyk’s distances take cognizance of the average of the differences among the three parameters of ifss, the szmidt and kacprzyk’s distances do not consider the average of the differences. since distance measure lies between 0 and 1, we conclude that φ1, φ2, φ̃1 and φ̃2 are not reliable distances. 5. conclusion in this paper, we have studied the szmidt and kacprzyk’s distances between ifss and noticed a setback with the distance figure 7: distances using eqs. (7) and (12) (v p) (mp) (t p) (s p) (cp) 0.2 0.3 0.4 0.5 classifications d is ta nc es skd mskd figure 8: distances using eqs. (8) and (14) (v p) (mp) (t p) (s p) (cp) 0.2 0.3 0.4 0.5 classifications d is ta nc es skd mskd measures. because of this setback, modifications of the szmidt and kacprzyk’s distances between ifss were proposed to enhance accuracy of measure. it was verified mathematically that the modified szmidt and kacprzyk’s distances between ifss satisfied the conditions for distance measure. both the szmidt and kacprzyk’s distances and their modified versions were applied to determine pattern recognition and diagnostic medical reasoning where information were represented as ifps. from the work, it is observed that the modified szmidt and kacprzyk’s distances outperformed the szmidt and kacprzyk’s distances 180 ejegwa et al. / j. nig. soc. phys. sci. 4 (2022) 174–182 181 in terms of accuracy because while the modified szmidt and kacprzyk’s distances take account of the number of the considered parameters, the szmidt and kacprzyk’s distances just added the hesitation margins to the methods introduced in [33]. in nutshell, the contributions of the paper includes; (i) modifications of the distance measures in [34] for better performance rating (ii) characterizations of the novel distance measures (iii) comparative analysis to show the edge of the novel approaches overs the approaches in [34] (iv) applications of the distances in pattern recognition and disease diagnosis. the modified szmidt and kacprzyk’s distances could be extended to topsis method as viable information measures to tackle multi-attributes decision-making problems. these novel distance measures of ifss can be studied in other variants of fuzzy sets with minimal modifications. acknowledgements we thank the referees for the positive enlightening comments and suggestions, which have greatly helped us in making improvements to this paper. references [1] l. a. zadeh, “fuzzy sets”, information and control 8 (1965) 338-353. 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[44] p. diamond & p. kloeden, metric spaces of fuzzy sets theory and applications. singapore: word scientific, 1994. 182 j. nig. soc. phys. sci. 3 (2021) 385–394 journal of the nigerian society of physical sciences sentiment analysis using various machine learning and deep learning techniques v. umarania,∗, a. juliana, j. deepab adepartment of computer science and engineering, saveetha engineering college, chennai, tamil nadu, india bdepartment of information technology, veltech institute of technology, chennai, tamil nadu, india abstract sentiment analysis has gained a lot of attention from researchers in the last year because it has been widely applied to a variety of application domains such as business, government, education, sports, tourism, biomedicine, and telecommunication services. sentiment analysis is an automated computational method for studying or evaluating sentiments, feelings, and emotions expressed as comments, feedbacks, or critiques. the sentiment analysis process is automated by using machine learning techniques, which analyses text patterns faster. the supervised machine learning technique is the most used mechanism for sentiment analysis. the proposed work discusses the flow of sentiment analysis process and investigates the common supervised machine learning techniques such as multinomial naive bayes, bernoulli naive bayes, logistic regression, support vector machine, random forest, k-nearest neighbor, decision tree, and deep learning techniques such as long short-term memory and convolution neural network. the work examines such learning methods using standard data set and the experimental results of sentiment analysis demonstrate the performance of various classifiers taken in terms of the precision, recall, f1-score, roc-curve, accuracy, running time and k-fold cross validation and helps in appreciating the novelty of the several deep learning techniques and also giving the user an overview of choosing the right technique for their application. doi:10.46481/jnsps.2021.308 keywords: convolution neural network, long short term memory, sentiment analysis, supervised machine learning, deep learning article history : received: 21 july 2021 received in revised form: 04 october 2021 accepted for publication: 05 october 2021 published: 29 november 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: w. a. yahya 1. introduction in the present days, social media is a popular technology that uses micro blogging platforms to connect millions of people. people can freely express their thoughts, ideas, and views as short messages called tweets on many micro blogging platforms in social networks (like twitter) and business websites or web forums [1]. researchers gather these unstructured tweets and use a variety of methods to extract information from them. ∗corresponding author tel. no: +918610352527 email address: umaranibharathy@gmail.com (v. umarani ) this analysis of tweets or opinions provides predictions or measures in a variety of application domains such as business, government, education, sports, tourism, biomedicine and telecommunication services [2]. sentiment analysis or opinion mining is the study of opinions and prediction. sentiment analysis (sa) is one of the text mining approach that use natural language processing for binary text classification. sentiment analysis can be performed in four levels based on the scope of text. they are document-level, sentence-level, aspect-level, and word level sentiment analysis [6]. in document level sa, overall opinion of the document 386 umarani et al. / j. nig. soc. phys. sci. 3 (2021) 385–394 387 about single entity is grouped into positive or negative. in sentence level sa, the opinion expressed in a sentence is classified as either positive or negative. in aspect level sa, opinions about entities are grouped based on specific entity elements. at word level sa, opinions about entities are grouped based on a specific word. in the proposed work, word level sentiment analysis model is developed for restaurant review data set based on machine-learning and deep learning algorithms to classify sentiments as positive or negative automatically. the sections in this paper are organized as follows: section 2 examines the various sentiment analysis models that have been discussed in the literature. section 3 discusses the various machine learning techniques. section 4 analyzes the performance of machine learning techniques used in sentiment analysis and finally discusses the conclusion and future work that need to be carried out in sentiment analysis. 2. related work the researchers create several sentiment analysis models based on data collected from social networks, business websites, and dataset providers. examples of these analysis models built from data include amazon reviews, tweets from twitter micro blogging sites, yelp travel recommendations, and movie reviews from imdb, kaggle, and others [3]. this section discusses some existing work as well as the benefits of their proposed model. endsuy [4] used twitter datasets to conduct exploratory data analysis on the us presidential election 2020. they compare the sentiment of location-based tweets to that of on-theground public opinion. they collect features such as latitude, longitude, city, country, continent, and state code using open cage api and scispacy ner. they use two datasets from kaggle about donald trump and joe biden, both dated november 18, 2020. for lexicon-based feature extraction, they use a valence aware dictionary for sentiment reasoning (vadar), and for classification, they use logistic regression machine learning approaches. bibi et al. [5] created a cooperative binary-clustering framework for sentiment analysis on indigenous data sets using twitter. they use majority voting to partition the data and combine single linkage, complete linkage, and average linkage approaches. based on the confusion matrix, they divide the cluster into positive and negative. for feature selection, they use unigram, tf-idf, and word polarity mechanisms. according to this analysis, the cooperative clustering approach outperforms the other individual partitioning techniques (75 %). cekik et al. [6] use a filter-based feature selection method called proportional rough feature selector (prfs) for feature selection and test it with various classifiers such as svm,dt,knn, and naive bayes. prfs uses rough set theory to determine whether documents belong to a specific class or not. it improves classifier performance at a 95% confidence level. peng et al. [7] proposed a sentiment model adversarial learning method. it is made up of three parts: a generator, a discriminator, and a sentiment classifier. to obtain efficient semantic and sentiment information, the generator uses a multi-head attention mechanism. the discriminator measures up the similarity of sentiment polarity in the generative vector and global vector produced by the generator and classifier, respectively. it uses gradients to update the weights in the generator, resulting in high-quality word embeddings. word vectors with opposing sentiment contexts will be classified as fake vectors in this context. it avoids the issue of similar context words with polar opposite sentiments. to improve the performance of aspect-based sentiment analysis, tan et al. [8] devised an aligning aspect embedding method (aae). using the cosine measure metric, the aae method discovered the relationship between aspect-categories and aspectterms. the aae method effectively solves the misalignment problem in aspect-based sentiment analysis and improves sentiment analysis performance. jain et al. [9] uses the apirori algorithm to minimize the feature set for sentiment analysis and developed a feature selection approach based on association rule mining. for experiments, they used supervised classification methods such as naive bayes, support vector machine, random forest, and logistic regression. kalaivani et al. [10] developed a hybrid feature selection method for sentiment analysis. for feature extraction, they use the unigram and bigram models, as well as the tfidf weighting technique. they use information gain to shrink the subset of features. to select optimal features, this method employs a genetic algorithm. then, using various classifiers such as naive bayes and logistic regression, support vector machine (svm), this hybrid model is put into practice. based on this analysis, the svm classifier outperformed the other classifiers. using statistical feature selection methods, ghosh et al. [11] proposed an ensemble feature selection technique to improve the performance of the sentiment analysis process. to find the best feature set, they use information gain, the gini index, and the chi-square method. they utilized five distinct classifiers: multinomial nave bayes, knn, maximum entropy, decision tree, and support vector machine. rodrigues et al. [12] formed a pattern-based method for extracting aspects and analyzing sentiment. pattern analysis is used in this case to extract the explicit aspect syntactic pattern from product sentiments. it extracts the bigram features and uses senti-wordnet to determine the sentence’s sentiment polarity. according to this analysis, the multi node clustering approach outperforms the single node clustering approach. for tweet sentiment classification, jianqiang et al. [13] created a glove-dcnn (global vectors for word representation – deep convolution neural network). they presented a method of word embedding that combines unigram and bigram feature vectors. a subset of sentiment features is formed by combining the twitter specific features vector and word sentiment polarity features. this feature set is used to train and predict sentiment classification labels in a deep convolution neural network. it resolves the data sparseness issue. imran et al. [14] use tweets from twitter and the sentiment 140 dataset. they utilize the lstm model to estimate sentiment polarity and emotion. for sentiment analysis, li et al. 387 umarani et al. / j. nig. soc. phys. sci. 3 (2021) 385–394 388 [15] developed lexicon integrated cnn family models. they implemented a sentiment padding approach to ensure that input data sizes are consistent and that the percentage of sentiment information in each row is increased. during neural network learning, the gradient vanishing problem between the input layer and the first hidden layer is solved using the sentiment padding method. the detailed literature survey on machine learning methods, illustrates that the suggested methods help in analyzing the text patterns faster and they can be used to automate the sentiment analysis process [16, 17]. deep learning handles large volume of data and it uses artificial neural network to analyze the text pattern faster and it can solve the misalignment problems by extracting local features from a sentiment. as a result, deep learning algorithm takes word embedding as input and provides high accuracy on these kinds of tasks. in the proposed work, the most widely used machine learning algorithms have been deployed to analyze the performance of deep learning models lstm and cnn. 3. methodology sentiment analysis process is carried out by several steps. first step is to collect data and perform preprocessing. in preprocessing step, data set is structured by normalization, stop word removal, stemming, and tokenization process [18]. next step in sentiment analysis is to extract and select most relevant text features from the opinion. feature extraction and feature selection methods are used for this purpose. these methods are used to reduce the number of input variables, avoid over fitting, decrease computational complexity or training time, and improve model accuracy [19]. vectorization and word embedding methods are used for feature extraction [20]. finally, machine learning techniques are used to classify or categories text as positive, negative, or neutral based on sentiment polarity of opinions. machine learning techniques classify the sentiments based on training and test data set [21]. in this section, the most widely used machine learning classifiers namely naı̈ve bayes, logistic regression, random forest, linear svc (support vector classifier) , k-nearest neighbor and decision tree and deep learning technique such as convolution neural network and long short term memory have been investigated the basics of these methods are discussed in detail. theoretical background naive bayes it is an intuitive approach and has good ability to work with small data with lower computation time for training and prediction [22]. it uses bayes theorem for finding the probability of the event. naive bayes classifier is based on equation (1). p (y/x1, x2, . . . , x2) = p (x1, x2, . . . , xn/y ) × p (y ) p(x1, x2, . . . , xn) (1) here p (y/x1, x2, . . . , x2) is called the posterior probability of an output class y given input features x1, x2, . . . , xn. figure 1: sentiment analysis process ¶ (x1,x2, . . . ,xn/y) is the likelihood of input features x1, x2, . . . , xn given their class y. p(y) is the prior probability of class y. p(x1, x2, . . . , xn) is the marginal probability. the common distributions are the normal (gaussian), multinomial, and bernoulli distributions used in naı̈ve bayes classifier. logistic regression (lr) lr is a most widely used binary classifier which uses logistic function to predict the probability of observation output value (yi) with input xi [23]. if p(yi/xi) is greater than 0.5 then predict class 1 otherwise considered it as 0. the logistic function is defined by equation (2). p(yi/x) = 1 1 + e−(1 +2 x) (2) where 1+2 are learning parameters, x is the training data, y is observation output value and e is the euler’s number support vector classifier (svc) svc classifies the data by finding hyper plane that maximize the margin between the predicted classes in the training data [24]. support vector classifier is represented by equation (3). f (x) =0 + ∑ j∈s ( j × k(x j, x j)) (3) 388 umarani et al. / j. nig. soc. phys. sci. 3 (2021) 385–394 389 where k is the kernel function which is used to compare the similarity of (x j, x j) observations and j is the learning parameter, s is the set of support vector observations. svc uses linear kernel or radial basis function kernel to create a hyper plane decision boundary. decision tree classifier decision tree classification is a non-parametric supervised method suitable for both classification and regression problem. decision tree contain set of decision nodes, each node specifies decision rule which create split to decrease the impurity recursively until all leaf nodes are belonging to specific class. decision tree classifier uses various split quality measures like gini impurity or entropy or mae (mean absolute error) to decrease the impurity at a node. by default, it uses gini impurity which is calculated by equation (4). g (n) = 1 − c∑ i−1 p2i (4) where g(n) is the giniimpurity at node n and pi specifies the proportion of observed class c at node n. random forest random forest is a kind of ensemble learning method in which many decision trees are trained and each tree carries bootstrapped sample of observations called out of bag observations. for every observation, random forest learning algorithm calculates the overall score by comparing the observation’s true value with the prediction from a subset of trees not trained using that observation. this overall score is taken as a measure of random forest performance. k-nearest neighbor knn is simple and most widely used classifier in supervised machine learning [24,25]. knn algorithm first identifies the k closest neighbors based on the distance metrics and kneighbors predict their classes based on k observations. most widely used distance metrics as euclidean or manhattan distance or minkowski distance. they are defined by equation (5), (6) and (7). deuclid = √√ n∑ i=1 (xi − yi) 2 (5) dmanhat = n∑ i=1 |(xi − yi) | (6) dminkowski = ( n∑ i=1 |xi − yi | p) 1 p (7) where xi and yi are the observations and p is the hyper parameter convolution neural network it is a type of feed forward neural network called convnet (convolution network) which uses hierarchical structure for fast feature extraction and classification [26,27]. most important layers form a cnn is shown in figure 2. 1. convolution layer it extracts the features and creates feature maps. convolution is a mathematical operation which is a measure of two overlapping function. it uses filter or kernels to detect important feature in the input. the number of output features (no) is calculated by equation (??) based on number of input features (ni) and convolution properties like kernel size k , padding size p and stride size s. no = ⌊ ni + 2 p − k s ⌋ + 1 (8) here, kernel size specifies the size of filter, stride specifies the rate at kernel increases and padding adjust the size of input according to the requirement of input matrix. when kernel detects the important features then it is stored in a feature map. zero padding adds many zeros to fit the input matrix with kernel size. if zero padding is used in convolution layer called wide convolution otherwise it is called narrow convolution. other variants are zero padding called valid padding which keeps only valid parts and ignore other parts. 2. pooling layer it progressively reduces size of input representation and the number of weights needed. hence it reduces overall number of input features and computation in the network as well as controls the over fitting. pooling layer reduce the dimensionality of feature map without reducing important information. it uses max pooling, average pooling or sum pooling technique to reduce dimensionality. max pooling takes the maximum value from the feature map. average pooling finds the average value from the feature map and sum pooling finds the sum of all values from the feature map. the new values form a matrix called pooled feature map. the pooling process is also called sub sampling or down sampling. 3. flattening layer – it convert the multidimensional data to single dimensional data. flattening layer is connected to fully connected layer for classification. 4. fully connected layer – in fully connected layer, neurons have full connections to all activations in neurons of previous layer. fully connected layers classify the feature which is obtained from the convolution and pooling layers into various classes based on the training set. it uses soft max activation function in output layer for text classification [33].soft max function [24] is defined by equation 9. it takes input vector x j of n real numbers and normalize the output into a range [0,1]. so f tmax ( x j ) = ex j∑n k=1 e x j (8) 389 umarani et al. / j. nig. soc. phys. sci. 3 (2021) 385–394 390 figure 2: layers in convolution neural network figure 3: hidden state transformation in lstm cell long short term memory lstm is a type of feedback or recurrent neural network designed to solve various sequential and time series problems [28,29]. lstm takes current observation and previous observation as input and store information in a gated cell or memory cell. input data is propagated to lstm layers to make a prediction. lstm layer contain set of recurrently connected blocks, each one carries memory cell and three gates namely input, output and forget gate. each memory cell has input weight, output weight and hidden state are used to process the input data. lstm control the data flow passes through cell with the help of gates. input gate describe how much of newly calculated state for present input, forget gate describe how much of previous state passes through it and output gate describe how much of internal state passes to next layer. these gates help to adjust the lstm hidden state and every time, cell state c is calculated based on hidden state g, previous cell state, forget gate and input gate. finally hidden state at time t is calculated by cell state with output gate. the transformations applied for hidden state is shown in figure 3. the calculations done by the hidden state transformation are described by the following equations i = ((wi × ht−1) + (ui × xt)) (9) f = (( w f × ht−1 ) + ( u f × xt )) (10) o = ((wo × ht−1) + (uo × xt)) (11) g = tanh (( wg × ht−1 ) + ( ug × xt )) (12) ct = ((ct−1 ◦ f ) + (g ∗ i)) (13) ht = tanh (ct) ◦o (14) here i,f,o,g, ctand ht refer input gate , forget gate ,output gate and hidden state at time t-1 ,cell state at time t and hidden state at time t. here is the sigmoid function which helps to adjust the output of gates between value 0 and 1. w, u are the weight matrix and transition matrix which helps to reduce the number of parameters learned by lstm. xt refer input at time t. the present input xt and hidden state ht−1 are used to calculate internal hidden state g. finally hidden state ht is calculated from ct and output gate value o. 4. results and discussion restaurant reviews from kaggle were taken to examine the various supervised learning techniques for assessing the sentiment analysis process. this data set contains restaurant review text containing 1000 text reviews. here anaconda python platform is used to evaluate and pre-process the restaurant reviews. 70% of the reviews are used for training, while 30% are used to test the supervised learning technique. nltk is used for pre-processing, and keras, tensor flow (backend) is used to create lstm (rnn with memory) and cnn neural network models[30]. experiments are carried out by google collaboratory which provide python development environment and run code in google cloud. the figure 4 shows the five rows of initial configuration of restaurant data set which contain 1000 reviews and its opinion classification positive or negative. the statistical summary of restaurant data set is shown in figure 5. evaluation parameters precision, recall, f1-score, accuracy, auc score, and training time were used to assess the classifier’s performance [31]. they are calculated by precision = t p t p + f p (15) 390 umarani et al. / j. nig. soc. phys. sci. 3 (2021) 385–394 391 figure 4: first five rows of restaurant review data set figure 5: statistical summary of dataset recall = t p t p + f n (16) f1 score = 2 × (recall × precision) recall + precision (17) accuracy = (t p + t n) (t p + f p + f n + t n) (18) here true positive (tp) refers to restaurant reviews that were initially classified as positive and are also expected to be positive. false positive (fp) reviews are those that are initially identified as negative but are predicted to be positive. true negative (tn) refers to restaurant reviews that were initially categorized as negative and that the classifier predicted to be negative. false negative (fn applies to restaurant reviews that are initially positive but are expected to be negative. auc score specifies the area under roc curve from prediction score. result analysis of machine learning technique for sentiment analysis initially, pre-processing is carried out in sentiment reviews by removing non-character data such as digits and symbols, as well as punctuation and converts the sentence into lowercase. after preprocessing, cleaned text reviews are converted to numerical data that contain sentiment tokens and sentiment score called feature vector. the feature vector is formed tfidf vectorizer. the tf-idf stands for term-frequency times inverse document-frequency and it assign weight to each word based on how often it appears in the review text. the tf-idf refer term-frequency times inverse document-frequency which assign weight to each word based on the frequency of that word appeared in review text. after extracting features with a vectorizer, six machine learning classifiers are used for sentiment figure 6a: roc curve of multinomial naı̈ve bayes figure 6b: roc curve of bernouli naı̈ve bayes analysis: naive bayes, logistic regression, random forest, linear svc (support vector classifier), k-nearest neighbor and decision tree. the performance of classifier is assessed by precision, recall, f1-score, accuracy, auc score, roc curve and training time. the classification report of classifier is shown in table 1. from this table, the highest auc score obtained for naı̈ve bayes classifier is 0.7642. so naive bayes model provide better prediction compared to other machine learning classifier for this restaurant data set. the table also shows that time taken for naive bayes model is low compared to other machine learning algorithms. roc curve of machine learning classifier is depicted in figures 6a to 6f. result analysis of deep learning techniques in sentiment analysis sentiment analysis is carried out by deep learning techniques cnn and lstm. after pre-processing the review text are converted into tokens. the result of tokenizer is shown in figure 7. following tokenization, the sentiment text is passed to word2vec model which converts word to vectors. the total number of words obtained for training data is 5118 words total, with a vocabulary size of 1839 and maximum sentence length is 18, while the total number of words obtained for test data is 391 umarani et al. / j. nig. soc. phys. sci. 3 (2021) 385–394 392 table 1: performance analysis of machine learning classifiers for sentiment analysis classifier class classification report accuracy auc score time to train (seconds) precision recall f1-score support train test naive bayes multinomial 0 1 0.74 0.79 0.78 0.75 0.76 0.77 256 244 0.76 0.7633 0.7642 0.009278 naive bayes bernouli 0 1 0.72 0.79 0.79 0.73 0.76 0.76 256 244 0.76 0.756 0.758 0.020505 logistic regression 0 1 0.67 0.83 0.87 0.61 0.75 0.70 256 244 0.73 0.73 0.736 0.06633 random forest 0 1 0.65 0.89 0.93 0.54 0.76 0.67 256 244 0.72 0.7233 0.732 0.57832 linear svc 0 1 0.69 0.80 0.82 0.67 0.75 0.73 256 244 0.74 0.74 0.743 0.014712 knn 0 1 0.70 0.74 0.72 0.71 0.71 0.72 256 244 0.72 0.716 0.716 0.079166 decision tree 0 1 0.65 0.77 0.80 0.61 0.72 0.68 256 244 0.70 0.70 0.704 0.1707437 figure 6c: roc curve of logistic regression figure 6d: roc curve of decision trees 574 words, with a vocabulary size of 415 and maximum sentence length of review text is 15. figure 6e: roc curve of knn figure 6f: roc curve of random forest the cnn model passes the input data to series of layers. it 392 umarani et al. / j. nig. soc. phys. sci. 3 (2021) 385–394 393 result of tokenizer figure 7a: roc curve of cnn figure 7b: roc curve of lstm uses convolution layer to extract features from input data, max pooling layer to reduce the dimensionality of trainable parameters and a sigmoid activation function in output dense layer. similarly, the input data in the lstm model is passed to embedding layer, spatial dropout, lstm, and output layer. the spatial dropout rate is taken as 0.2 for avoid over fitting. for compilation, both the cnn and lstm model use the adam optimizer. the classification report of cnn and lstm classifier is shown in table 2. the roc curve of cnn and lstm is shown in figure 7a and 7b. finally k-fold cross validation (k=10) is carried out to evaluate the machine learning and deep learning algorithm with ranfigure 8: accuracy score of machine learning and deep learning algorithms dom seed = 20. the results of accuracy score for each algorithm obtained is shown in figure 8. this box plot shows the spread of accuracy score across each cross validation fold of these classifiers. from this box plot, the mean value of accuracy score obtained for bernoulli naive ayes (bnb) is 0.7714, multinomial naı̈ve bayes (mnb) is 0.767, logistic regression (lr) is 0.762, random forest (rf) is 0.738, linear support vector machine (lsvc) is 0.748 k nearest neighbor (knn) is 0.752 and decision tree (dt) is 0.72. the mean score value of accuracy score for lstm is 0.823 and cnn is 0.828. hence deep learning algorithm (lstm and cnn) provide high prediction compared to machine learning classifier algorithms. according to this experimental study, machine learning classifiers such as naive bayes, logistic regression, svc, and knn are faster to train than deep learning models such as cnn and lstm. the results of the experiment show that lstm and cnn take longer to train but have higher accuracy in both training data (98%) and test data (84%) for this restaurant dataset. 393 umarani et al. / j. nig. soc. phys. sci. 3 (2021) 385–394 394 table 2: performance analysis of deep learning techniques for sentiment analysis classifier class classification report accuracy auc score time to train (seconds) precision recall f1-score support train test cnn 0 1 0.83 0.76 0.79 0.80 0.81 0.78 256 244 0.98 0.845 0.84 7.3 lstm 0 1 0.80 0.73 0.67 0.76 0.73 0.74 256 244 0.96 0.77 0.748 9.08 5. conclusion sentiment analysis on restaurant data set is carried out to classify the opinion as positive or negative. initially preprocessing is carried out to reduce the features and speed up the classification task. the classification task is performed by machine learning methods (multinomial naive bayes, bernoulli naive bayes, logistic regression, random forest, svc, knn, decision tree) and deep learning methods (lstm and cnn). machine learning methods use bag of model approach (tfidvectorizer) to convert text into vector form and deep learning methods use word embedding method for converting text into vector form. the classifier performance is evaluated by various metrics such as precision, recall, f1 score, auc score and time taken for training. the accuracy score of each classifier is tested by k-fold cross validation technique. from these findings of the experiments, neural network-based learning has a higher training accuracy and a longer running time compared with machine learning classifier. for future work, improve the performance 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[30] k. hirota & f. masahiro, “efficient attention mechanism by softmax function with trained coefficient”, ieice technical report 339 (2021) 52. 395 j. nig. soc. phys. sci. 3 (2021) 191–200 journal of the nigerian society of physical sciences goodness of fit test of an autocorrelated time series cubic smoothing spline model samuel olorunfemi adamsa,∗, davies abiodun obaromib, alumbugu auta irinewsc adepartment of statistics, university of abuja, abuja, nigeria bdepartment of statistics, confluence university of science and technology, osara, kogi state, nigeria cdepartment of mathematics and statistics, federal polytechnic, nasarawa, nasarawa state, nigeria abstract we investigated the finite properties as well as the goodness of fit test for the cubic smoothing spline selection methods like the generalized maximum likelihood (gml), generalized cross-validation (gcv) and mallow cp criterion (mcp) estimators for time-series observation when there is the presence of autocorrelation in the error term of the model. the monte-carlo study considered 1,000 replication with six sample sizes: 30; 60; 120; 240; 480 and 960, four degree of autocorrelations; 0.1; 0.3; 0.5; and 0.9 and three smoothing parameters; λgml= 0.07271685, λgcv= 0.005146929, λmcp= 0.7095105. the cubic smoothing spline selection methods were also applied to a real-life dataset. the predictive mean square error, r-square and adjusted r-square criteria for assessing finite properties and goodness of fit among competing models discovered that the performance of the estimators is affected by changes in the sample sizes and autocorrelation levels of the simulated and real-life data set. the study concluded that the generalized cross-validation estimator provides a better fit for autocorrelated time series observation. it is recommended that the gcv works well at the four autocorrelation levels and provides the best fit for time-series observations at all sample sizes considered. this study can be applied to; non-parametric regression, non-parametric forecasting, spatial, survival and econometric observations. doi:10.46481/jnsps.2021.265 keywords: cubic spline, goodness-of-fit test, generalized maximum likelihood (gml), generalized cross-validation (gcv), mallow cp criterion (mcp) article history : received: 18 june 2021 received in revised form: 21 july 2021 accepted for publication: 26 july 2021 published: 29 august 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: t. latunde 1. introduction a cubic spline is the most widely recognized example of the smoothing spline regression model. it’s anything but a piecewise cubic function that interpolates a bunch of observation focuses and ensures smoothness of the observations [1]. it is piecewise third-degree polynomials that go through a core of ∗corresponding author tel. no: +23480xxxx572 email address: samuel.adams@uniabuja.edu.ng (samuel olorunfemi adams ) interests. it has a nonstop first and second subordinate with the request for (d −1) coherence, where d is the polynomial degree [2]. the model with shortened force premise work b(t) changes the factors ti and fit a model utilizing these changed factors, which adds non-linearity to the model and empowers the splines to fit smoother and adaptable non-straight cubic measures. it is assumed that the variables (ti, yi)and (ti+1, yi+1) are connected by a cubic polynomial s it = ait3 + bit2 + cit + di that is valid for ti ≤ t ≤ tt+1 for i = 1, 2, . . . , n − 1[3]. the interpolation function is derived by firstly finding the coefficients ai, bi, ci, di, for each of the cubic functions. for n points, there are n − 1 191 adams et al. / j. nig. soc. phys. sci. 3 (2021) 191–200 192 cubic functions to find, and each cubic function requires four coefficients. therefore we have a total of 4(n − 1) unknowns, which implies that 4(n − 1) independent equation coefficients are required. firstly, cubic functions must intersect the observation on the left and the right: s i (ti) = yi, i = 1, 2, · · · , n − 1, s i (ti+1) = yi+1, i = 1, 2, · · · , n − 1 (1) equation (1) produces 2(n − 1) conditions. then, we need each cubic function to join as easily with its neighbours as could be expected, so we compel the splines to have consistent first and second subsidiaries at the observations i = 1, 2, . . . , n − 1: s ′ i (ti+1) = s ′ i+1 (ti+1) , i = 1, 2, · · · , n − 2, s ′′ i (ti+1) = s ′′ i+1 (ti+1) , i = 1, 2, · · · , n − 2 (2) besides, s i(t) is figured by choosing to fit the additional conditions being performed. a typical arrangement of definite imperatives accepts that the subsequent subsidiaries are zero at the endpoints; this implies that the bend is a ”straight line” at the endpoints, written as; s ′′ 1 (t1) = 0, s ′′ n−1 (tn) = 0 (3) there exist a few studies on the goodness-of-fit test for nonlinear regression models in the literature; these current studies can be grouped as a penalized smoothing, polynomial regression model and soothing spline test statistics, double coordinations regression and nonparametric regression models. [4] proposed another test estimation for testing uprightness of assault of a mth demand polynomial backslide model. the test measurement is; ∫ 10 [ µ (n) λ(t) ]2dt, where µ(n)λ is the zthrequest subsidiary of a zth request smoothing spline estimator for the regression model µ and λ is its related smoothing parameter. the huge example qualities of the test measurement are gotten from both the invalid and elective speculation. [5] portrays a goodness-of-fit technique for testing the parametric capacity for the regression model and the change in a parametric nonlinear regression model. [6] proposed a likelihood and restricted likelihood extent tests for decency of-attack of a nonlinear regression using first-request taylor assessment gauge around the maximum likelihood estimator of the relapse boundary to harsh the invalid and elective theory is shown nonparametrically using penalized splines. [7] applied bootstrap techniques that are computationally productive to assess the achievement of goodness-of-fit measurement and see that for the most part, the power and type one error of the goodness-of-fit measurements rely upon the model being scrutinized. [8] considered a smoothing-based test estimation and surmised its invalid scattering using a bootstrap philosophy to propose a goodness-offit test for examining parametric covariance capacities against general nonparametric choices for both irregularly noticed longitudinal perceptions and thickly noticed useful perception. [9] offered a goodness-of-fit test for nonparametric regression models with straight smoother structure by noticing factual dependence between the assessed error terms and the covariates using the hilbert-schmidt independence criterion (hsic). the bootstrap is used to acquire p-values and show the fitting type one error and power of the test execution through monte-carlo data reenactment. it is clear from the existing literature that the goodness-of-fit of smoothing spline for time series observations have not been investigated so far. this paper presents a goodness-of-fit test for time series observations using three classical cubic spline nonparametric regression functions. in section two, the cubic smoothing spline was discussed, smoothing spline selection parameters like generalized cross-validation, generalized maximum likelihood, mallow’s c.p. criterion and performance evaluation criteria were also addressed in this section. the simulation result is given in section three, while section four presented the real-life dataset result. finally, a discussion of findings and conclusion are presented in section five. 2. materials and methods 2.1. cubic smoothing spline the spline smoothing model is written as; yi = f (ti) + εi (4) where; yi is the response/dependent variable, f is an unknown smoothing function, ti is the independent/predictor variable and εi is zero mean autocorrelated stationary process [10]. the general cubic spline function is given as; f (t) = at3 + bt2 + ct + d + ε (5) where; a, b, c, and d is real number coefficients and a , 0, t is the independent variable, ε is the error term and d. f. is k−d−1 (k is number of knots and d is the degree of the cubic spline) the cubic smoothing spline estimate f̂ of the function, f is defined to be the minimizer (over the class of twice differentiable function) of; s ( f ) = n∑ i−1 ( yi − f̂ (ti) )2 + λ ∫ b a ( f̂ ′′ (t) )2 dt (6) where; 1. λ > 0 is a smoothing parameter, 2. the initial part of the equation is the residual sum of the square for the goodness of fit for the observation. 3. the subsequent term is a roughness penalty, which is enormous when the incorporated second derivative of a regression function f ′′ (t) is likewise huge 4. if λ approaches 0, then f (t) simply interpolates the observations. 5. if λ is very large, then f (t) will be chosen so that f ′′ (t) is wherever 0, which suggests a by and large direct leastsquares fit the perceptions. if f (t) values are fixed at f (t1) , . . . . , f (t2) the roughness∫ b a ( f̂ ′′ (t) )2 dt is minimized by a natural cubic spline, this solution is written as a basic function as; f (t) = β0 + β1 f1 (t) + . . . + βn+3 fn+3 (t) (7) 192 adams et al. / j. nig. soc. phys. sci. 3 (2021) 191–200 193 2.2. selection of the smoothing parameter the smoothing parameter in cubic spline smoothing is to control the smoothness of the fitted curve, to estimate the optimal value of the smoothing parameter λ, three smoothing parameter selection criteria are considered and compared in this study: generalized cross-validation (gcv), generalized maximum likelihood (gml) and mallow cp (mcp). the generalized cross-validation (gcv) selection method was suggested by [11, 12] as a substitution for cross-validation (cv), which is the most famous technique for selecting the intricacy of statistical models. the essential standard of crossvalidation is to leave the information that brings up each in turn and choose the estimation of λ under which the rest of the information best predicts the missing focuses [13, 14]. to be exact, let g−1 λ be the smoothing spline determined from all the information sets aside from (ti, yi) utilizing the worth λ for the smoothing boundary. the cross-approval decision of λ is then the estimation of λ, which limits the cross-validation score, be; cv (λ) = 1 n ∑ {yi − ĝ(ti)} 2 (8) equation (8) is similar to the criterion for model estimation in regression, generally [15]. define a matrix a (λ) by; ai j (λ) = n −1g ( ti, t j ) (9) cv (λ) = 1 n n∑ i=1 {yi − ĝ(ti)} 2 {1 − aii(λ)} 2 (10) [12, 16, 17] also suggest the use of a related criterion, called generalized cross-validation, obtained from (10) by replacing aii(λ) by its average value, n−1tra(λ), this gives the score. gcv (λ) = n−1rs s (λ)( 1 − n−1tra(λ) )2 (11) where; rss (λ) is the residual sum of squares, ∑ {yi − ĝ(ti)} 2. in their study [12] likewise give hypothetical contentions to show that generalized cross-validation ought to, asymptotically, pick an ideal estimation of λ in the sense of minimizing the average squared error at the design points. the predicted published practical examples bear out a good performance in [18]. the summed-up cross-validation technique is notable for its optimal properties [19]. if there exists an n x n, the impact matrix, with the property f̂n, λ (t1) f̂n, λ (t2) . . . f̂n, λ (tn)  = s (λ) y (12) and w0 (λ) = ∑n k=1 ( ak jy j − yk )2 (1 − akk) 2 (13) generalized cross-validation is the adjusted form of crossvalidation, a traditional technique for estimating the smoothing parameter. the gcv score is constructed by comparison to the cv score obtained from the ordinary residuals by dividing them by 1 − s (λ))ii. the accepted format of gcv is to replace the notation 1 − (s (λ)) in cross-validation with the mean score 1 − n−1 trace s (λ)thus, by adding the squared residual and notation {1 − n−1 trace s (λ)} 2 , by the already known ordinary cross-validation, the gcv smoothing method is written mathematically as; gcv (λ) = 1 n ∑n k=1 {y − fk (x1)} 2{ 1 − n−1trace (s λ) }2 (14) gcv (λ) = n−1‖(i − s λ) y‖2[ n−1trace (i − s λ) ]2 (15) where; n is the dataset (xi, yi), λ refers to the smoothing parameters and s (λ) is the ith diagonal member of a smoother matrix generalized maximum likelihood (gml) selection method; [20] proposed the gml technique for correlated data with one smoothing parameter. in a bivariate model, two smoothing parameters should be assessed simultaneously along with the covariance boundaries. following a comparative determination, gml is given as; gml (λ) = yi w (i − s (λ))[ det+w (i − s (λ) ) ] 1 n−m (16) det+ (i − s (λ)) is the product of the n− m nonzero eigenvalues of (i −− s (λ)) . λ is smoothing parameter, w is the correlation structure, s (λ) is the diagonal element of the smoother matrix, n is n1 + n2 pairs of measurements/observations and m are the number of zero eigenvalues. mallow’s c.p. criterion (mcp) selection method was developed by [21] to estimate the fit of a regression model dependent on ordinary least square. it is applied to a model choice situation where explanatory variables can predict a few results and locate the best model associated with subset independent variables. the more modest the estimation of the cp, the generally exact it is, the cp is written numerically as; mcp (λ) = ‖(s λ − i)y‖2 tr (i − s λ) (17) where; n is the measurements or observations, λ is the smoothing parameters and s (λ) is the ith diagonal member of the smoother matrix. the assumption underlying the application of the generalized cross-validation (gcv), generalized maximum likelihood (gml) and mallow’s cp criterion (mcp), the observations must be well represented by the model. 2.3. simulation study in this section, a simulation study is performed to assess the performance of the three cubic smoothing spline estimator, namely 193 adams et al. / j. nig. soc. phys. sci. 3 (2021) 191–200 194 generalized cross-validation (gcv), generalized maximum likelihood (gml) and mallow’s cp criterion (mcp) when autocorrelation is present in the error term. before the results were computed, datasets for the different simulation combinations are generated using codes written in the [22]. the data generation procedure, with accompanying explanation is presented in table 1. in the data generation process, n = 30, 60, 120, 240, 480 and 960, nrepl.= 1000 and ρ= 0.1, 0.3, 0.5 and 0.9, λgml = 0.07271685, λgcv = 0.005146929, λmcp = 0.7095105. in this study, a nonparametric smoothing function was used to generate the data under different conditions. the function is given as; y (xi) = 5s in ( π xi ) + εxi (18) where; xi = i−0.5 n ,π= 180 0, εx ∼ n ( 0, ρw−1 ) , a first-order autoregressive process with mean 0, standard deviation is 0.8 and autocorrelation levels ρ = 0.1, 0.3, 0.5 and 0.9 with 95% confidence interval. 2.4. performance evaluation criteria a comparative analysis was performed made to test the performance and goodness-of-fit test of the three cubic spline estimation methods (i.e. generalized crossed validation (gcv), generalized maximum likelihood (gml) and mallow cp criterion (mcp) in the presence of autocorrelation error. the predictive mean squared prediction error of a smoothing or curve fitting procedure according to [22, 23, 24] is the normal worth of the square distinction between the fitted value suggested by the predictive function f̂ (xi) and the value of the observed function f (xi). it is utilized to assess the performance and nature of explanatory variables or smoothing techniques like crossvalidation, generalized cross-validation, generalized maximum likelihood and so forth. the predictive mean square error (pmse) is written mathematically as; pms e (λ) = e  n∑ i=1 ( f (xi) − f̂ (xi) )2 (19) pms e (λ) = 1 n n∑ i = 1 ( f (xi) − f̂ (xi) )2 (20) pms e (λ) = n∑ i=1 ( e [ f̂ (xi) ] − f (xi) )2 = n∑ i=1 var [ f̂ (xi) ] (21) the predictive mean square error is usually grouped into two parts; the first part is the sum of square biases of the fitted observations, while the second is the total of variances of the fitted observations. where; f (xi) is the observed value and f̂ (xi) is the fitted/predicted/estimated value. based on each estimate of the parameter, the methods were ranked according to their performance at the criterion. the evaluation of methods was concluded at two levels using individual measures and the totality based on the standard. for the first level, the ranks were added for each technique and the whole method. then the methods of estimation were ranked by this total. the smoothing procedure with the least capacity was adjudged the most preferred method and the one with the largest sum the least preferred. these ranks were added together over all the criteria to know how each estimator performs in each parameter in the model. the best estimator in terms of the model was identified by further adding all the ranks over the model’s parameters. an estimator is ranked as the best if it has a minimum sum of levels. here the groups’ total was used, which will give identical results in terms of ranks if the mean levels had been used. but the consequences might be different if the median of the groups had been used. the disadvantage of the median is that if further work were to be done on these ranks, the mathematical procedure would be at least slightly more complex than with the mean. the goodness of fit of the smoothing methods explains how well the methods fit the simulated and real-life data. it also summarizes the differences between the observed value and estimated/predicted values. the adjusted r-square was used to determine the best-fit smoothing methods. it is written mathematically as; ad justed r−s quare = ( 1 − (1 − rsquare) × (n − 1) n − p ) (22) where; n = number of observations and p = number of parameters 3. simulation result table 2 presents the summary fit result of the cubic spline regression model and the model performance criteria, namely; the predictive mean square error (pmse), multiple and adjusted r-square based on six small sizes (t = 30, 60, 120, 240, 480 and 960) and autocorrelation level (ρ = 0.1). it was revealed from the result that all the coefficients of the smoothing methods’ parameters were significant at (p-value <0.001, <0.01 and < 0.05). the adjusted r-square result indicated that gcv had the highest values at all sample size levels (t = 30, 60, 120, 240, 480 and 960) and ρ=0.1 with adjusted r-squares of; 0.9963, 0.9979, 0.9971, 0.9992, 0.9986 and 0.9804 respectively. it can be inferred from the result above that; the gcv smoothing method provides the best fit to the time-series observations at time series size of (t = 30, 60, 120, 240, 480 and 960) and ρ=0.1. tables 3, 4 and 5 show the predictive mean square error (pmse), r-square and adjusted r-square simulation results of gcv, gml and mcp for autocorrelation levels 0.3, 0.5 and 0.9 for sample sizes; 30, 60, 120, 240, 480 and 960. the result indicated that the adjusted r-square value for gcv was greater than gml and mcp’s value; this is an indication that the cubic smoothing spline chosen by generalized cross-validation possesses the best fit model. figures 1 to 6 clearly show the comparisons of the behaviours of the cubic smoothing spline selected by gcv, gml and mcp for sample sizes 30, 60, 120, 240, 480 and 960, respectively. it 194 adams et al. / j. nig. soc. phys. sci. 3 (2021) 191–200 195 table 1. data generation process with explanations steps explanation step 1: obtain n, nrepl. and ρ the sample size of the simulated dataset, number of replication of n and autocorrelation levels respectively step 2: decide on xi, and y i, read the simulated sample data (xi, yi) for i = 1 − t and each i′s step 3: produce, λt, and f (λ) determine the pre-selected smoothing parameters λ1 , . . . ,λt, calculate the respective set of smoothing spline estimates f (λ) = { f̂λ1 , . . . , f̂λt } step 4: fit the values of f (xi) and f̂ (xi) for the given λ, σ and t use the data in 1 above to fit a curve and the estimate ahead by linear extension f (xi) and f̂ (xi) step 5: compute, gcv, gml and mcp compute values of the coefficients of gcv, gml and mcp step 6: generate the pmse values obtain the predictive mean square error pms e ( f̂ λ) =∑t i = 1 [( f ( x) − f̂ (xi) ))2] for these points sum up all the pmses to get the corresponding gcv, gml and mcp scores for the given values of n, λ and ρ. table 2. simulation result for gcv, gml and mcp for autocorrelation level = 0.1 cubic smoothing methods sample sizes predictive mean square error criterion r-square adj. rsquare β̂0 β̂1 β̂2 β̂3 gcv 30 -0.0965 11.3553 -33.4818 22.4204 0.9998 0.9963 60 -0.1753 11.4502 -33.3385 22.2447 0.9996 0.9979 120 -0.1767 11.7285 -34.2144 22.8765 0.9977 0.9971 240 -0.2000 11.9830 -34.7553 23.1753 0.9995 0.9992 480 -0.1854 11.8463 -34.4414 22.9781 0.9988 0.9986 960 -0.1893 11.8804 -34.4933 22.9920 0.9804 0.9804 gml 30 -0.1278 11.5373 -33.8893 22.6035 0.9824 0.9804 60 -0.1080 10.9222 -32.1454 21.4069 0.9827 0.9818 120 -0.2242 12.0912 -34.7044 22.9920 0.9769 0.9763 240 -0.1968 12.0617 -34.9399 23.2581 0.9804 0.9802 480 -0.1815 11.9975 -34.9133 23.3067 0.9770 0.9768 960 -0.2157 12.0444 -34.8217 23.2068 0.9784 0.9784 mcp 30 -0.0965 11.3553 -33.4818 22.4204 0.9698 0.9663 60 -0.1753 11.4502 -33.3385 22.2447 0.9696 0.9679 120 -0.1953 11.7876 -34.1166 22.6815 0.9778 0.9772 240 -0.2092 12.1147 -35.0511 23.3478 0.9837 0.9835 480 -0.1913 11.9395 -34.6317 23.0761 0.9798 0.9796 960 -0.2103 12.0782 -34.9856 23.3324 0.9799 0.9798 195 adams et al. / j. nig. soc. phys. sci. 3 (2021) 191–200 196 table 3. simulation result for gcv, gml and mcp for autocorrelation level = 0.3 cubic smoothing methods sample sizes predictive mean square error criterion r-square adj. rsquare β̂0 β̂1 β̂2 β̂3 gcv 30 -0.2800 13.376 -37.869 24.757 0.9106 0.9003 60 -0.3475 13.005 -36.993 24.732 0.9201 0.9137 120 -0.1212 10.645 -31.490 21.112 0.8699 0.8665 240 -0.1015 11.572 -34.618 23.474 0.9037 0.9025 480 -0.2467 12.431 -36.011 24.109 0.9304 0.9297 960 -0.1395 11.605 -34.211 22.962 0.8938 0.8935 gml 30 -0.0353 8.3035 -25.345 16.764 0.8526 0.8356 60 -0.0564 11.046 -32.398 21.367 0.8986 0.8932 120 -0.2749 13.134 -37.862 25.307 0.9151 0.9129 240 -0.1426 11.686 -34.316 22.989 0.8941 0.8927 480 -0.1887 12.008 -34.743 23.089 0.9021 0.9015 960 -0.2065 12.082 -35.038 23.379 0.8978 0.8975 mcp 30 -0.2806 12.211 -33.454 21.491 0.8358 0.8168 60 -0.1306 8.8825 -27.929 19.017 0.8569 0.8492 120 -0.1361 11.428 -33.283 22.081 0.9021 0.8996 240 -0.1511 11.424 -33.114 22.003 0.8881 0.8867 480 -0.2311 12.348 -35.816 23.998 0.8911 0.8904 960 -0.2375 12.676 -36.672 24.519 0.8995 0.8992 table 4. simulation result for gcv, gml and mcp for autocorrelation level = 0.5 cubic smoothing methods sample sizes predictive mean square error criterion r-square adj. rsquare β0 β1 β2 β3 gcv 30 -0.0847 11.0575 -33.6550 22.9325 0.8242 0.7923 60 -0.1924 10.4646 -30.4378 20.3870 0.8479 0.8291 120 -0.2296 11.5794 -32.9893 21.6783 0.8481 0.8416 240 -0.2663 12.9193 -36.9713 24.5493 0.8999 0.8974 480 -0.1601 11.8929 -34.5237 22.9573 0.8733 0.8718 960 -0.2469 12.5111 -36.1165 24.1212 0.8705 0.8698 gml 30 -0.1644 9.7970 -32.003 22.1713 0.8335 0.8143 60 -0.0666 11.0730 -34.2401 23.5602 0.7834 0.7718 120 -0.2531 13.5851 -40.1524 27.3208 0.7438 0.7372 240 -0.1301 11.9951 -35.3193 25.6299 0.8072 0.8048 480 -0.1119 11.5210 -34.3366 23.1544 0.7587 0.7571 960 -0.0844 11.3560 -33.9543 22.9082 0.7652 0.7645 mcp 30 -0.3708 12.9624 -35.0598 22.4159 0.7934 0.7696 60 -0.3290 13.8516 -38.0849 24.5585 0.7509 0.7376 120 -0.1718 12.078 -34.8942 23.1811 0.7514 0.7450 240 -0.1913 11.556 -33.2619 22.1224 0.7155 0.7119 480 -0.2059 12.1036 -35.2078 23.5514 0.7514 0.7499 960 -0.1775 12.3371 -35.9588 24.0466 0.7738 0.7731 196 adams et al. / j. nig. soc. phys. sci. 3 (2021) 191–200 197 figure 1. cubic smoothing spline and fitted curve using gcv, gml and mcp, n=30 figure 2. cubic smoothing spline and fitted curve using gcv, gml and mcp, n = 60 figure 3. cubic smoothing spline and fitted curve using gcv, gml and mcp, n = 120 was observed that generalized cross-validation provided the best fitted/estimated value when compared to the generalized maximum likelihood (gml) and mallow’s cp criterion (mcp). 4. application to real-life data in this section, the performance of the cubic smoothing spline selection methods on the real-life dataset of the federal government capital expenditure (in billion nairas) in nigeria between 1981-2019 sourced from [25] is presented as our example. this series has 39 datasets of the expenditure, the cubic spline was fitted for the mean function (i.e. f ∈ w) and a first-order autoregressive process ar (1) for the disturbance. the generalized cross-validation (gcv) was used in the example because it was found to perform better than the other cubic smoothing spline competing models in the simulation study presented in section three. this cubic smoothing spline curve presented in figure 7 showed that the observed data in our curve is very 197 adams et al. / j. nig. soc. phys. sci. 3 (2021) 191–200 198 figure 4. cubic smoothing spline and fitted curve using gcv, gml and mcp, n = 240 figure 5. cubic smoothing spline and fitted curve using gcv, gml and mcp, n = 480 figure 6. cubic smoothing spline and fitted curve using gcv, gml and mcp, n = 960 close to the estimated data. this provided great insight on the cubic smoothing spline selection method whose model produces the best fit for the time-series observations used as an example to validate generalized cross-validation (gcv) cubic spline selection method as the preferred model for time series observation. 5. discussion and conclusion this paper presents the goodness-of-fit test for time series observations using three cubic spline nonparametric regression functions. a simulation study and real-life dataset on the total federal government capital expenditure (in billion nairas) between 1981-2019 in nigeria were used to demonstrate how the three classical cubic smoothing spline selection methods perform when a time series dataset possesses autocorrelation in its error term. 198 adams et al. / j. nig. soc. phys. sci. 3 (2021) 191–200 199 table 5. simulation result for gcv, gml and mcp for autocorrelation level = 0.9 cubic smoothing methods sample sizes predictive mean square error criterion r-square adj. rsquare β0 β1 β2 β3 gcv 30 -0.2659 10.7596 -33.8282 23.4447 0.4702 0.4575 60 -0.2095 9.4790 -27.5371 18.5292 0.3126 0.4198 120 -0.2638 11.3711 -31.8619 20.6751 0.4751 0.6665 240 -0.3234 13.7239 -38.8914 25.7508 0.5659 0.4267 480 -0.1290 11.8463 -34.4156 22.8385 0.5244 0.5226 960 -0.2833 12.9439 -37.2473 24.9099 0.5186 0.4717 gml 30 -0.4094 8.4909 -30.8536 22.0745 0.642 0.6007 60 -0.0169 10.5741 -34.3813 24.2411 0.5504 0.5263 120 -0.5117 11.8106 -31.1566 19.9740 0.3301 0.3128 240 -0.0496 10.5045 -31.9189 21.8216 0.4964 0.4900 480 -0.3971 13.7294 -39.3648 26.5502 0.4985 0.4954 960 -0.0487 11.0139 -33.4744 22.7946 0.4958 0.4942 mcp 30 0.2659 10.7596 -33.8282 23.4447 0.4702 0.4091 60 -0.2095 9.4790 -27.5371 18.5292 0.3126 0.2757 120 -0.2638 11.3711 -31.8619 20.6751 0.4751 0.4615 240 -0.3234 13.7239 -38.8914 25.7508 0.5659 0.5604 480 -0.1290 11.8463 -34.4156 22.8385 0.5244 0.5214 960 -0.2834 12.9439 -37.2473 24.9099 0.5186 0.5171 figure 7. smoothing curve of nigeria federal government capital expenditure (in billion nairas) between 1981-2019 (green line) and estimates (red line) with smoothing parameters chosen by gcv. in the general structure of the simulated result, it was observed that an increase in the sample size and changes in the level of disturbances from autocorrelation affect the performance of the three cubic smoothing spline methods see (tables 1-4 and figures 1–6). the adjusted r-square result indicated that the gcv had the highest values of 0.9992 at n = 240 and ρ = 0.1, closely followed by the gml and mcp. it was discovered that the generalized cross-validation (gcv) smoothing method provides the best model fit and proved to be more efficient than the other smoothing methods for the simulated time-series observations with autocorrelation levels (ρ = 0.1, 0.3, 0.5 and 0.9) in the error term and for sample sizes (n = 30, 60, 120, 240, 480 and 960). the smoothing curve of the real-life data set of nigeria federal government capital expenditure (in billion nairas) between 1981-2019 validated the efficiency of the generalized cross-validation smoothing method as its’ model provided the best fit model without any defection and shortcoming under cubic spline functional form when compared with the competing smoothing methods. our findings also revealed that the gcv smoothing spline estimator out-performed the other competing selection methods for time series observation disturbed with four autocorrelation levels (ρ = 0.1, 0.3, 0.5 and 0.9). the gcv is a smoothing spline method fitted without any defection and shortcoming under cubic spline functional form with the highest adjusted rsquare of 0.9992, 0.9297, 0.8974, at ρ = 0.1, 0.3, 0.5, for n = 240 and 480 respectively. this finding is corroborated by [13, 26, 27, 28 ] whose findings found that gcv was fairly better when compared to gml for n = 64. that gcv was distinctly unrivalled for n = 128, while for n= 32, gcv was better for more modest σ2 and the examination 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[28] a. r. devi, i .n. budiantara & v. vita-ratnasari, “ unbiased risk and cross-validation method for selecting optimal knots in multivariable nonparametric regression spline truncated (case study: the unemployment rate in central java, indonesia, 2015)”, aip conference proceedings, (2018). 200 j. nig. soc. phys. sci. 4 (2022) 165–173 journal of the nigerian society of physical sciences green information and communication technologies implementation in textile industry using multicriteria method j. a. adebisia,∗, o. m. babatundeb a university of namibia, namibia b tshwane university of technology, pretoria, south africa abstract it has been recognized that green information communication technology (ict) is fast becoming popular in every sector. it’s fundamental relevance to textile industry requires urgent attention than ever. over the years selections of technologies to drive the textile industry has been a debate. there is a need for an efficient technology selection method to realize this goal. selection of green ict alternatives is a multicriteria decision making problem which has been sparsely explored in open literature. this study presents green ict adoption in the textile industry using a fuzzy-topsis multi-criteria approach for the most preferred ict alternative in a textile industry. criteria for green ict selection were identified by administering interview with selected textile and ict industry experts at the managerial cadre of organizations and academics. criteria considered were implementation cost (ic), operating and maintenance cost (omc), environmental impact (ei), improved system performance and use (ispu), supply chain management (scm) and employment opportunities (eo). results shows that the most preferred ict alternative is power management with overall coefficient of 0.60 while the least preferred is software optimization with coefficient of 0.23. this work will allow clean industrial process in the textile industry and also promote sustainable cities and communities through responsible consumption and production as highlighted by sustainable development goals (sdg) 11 and 12. doi:10.46481/jnsps.2022.518 keywords: green ict, textile, topsis, multicriteria, technology. article history : received: 16 december 2021 received in revised form: 26 march 2022 accepted for publication: 30 march 2022 published: 29 may 2022 c©2022 journal of the nigerian society of physical sciences. all rights reserved. communicated by: w. a. yahya 1. introduction now commercialized and computerized, the textile industry dates back to the early civilization of men when conventional techniques were adopted to establish the industry. the textile industry is one of the most important sectors of the economy. although the various technological breakthroughs around the world has led to the globalization of the textile industry, the requirements for sustainability in the industry involves ∗corresponding author tel. no: +234-8034496354 email address: adebisi_tunji@yahoo.com (j. a. adebisi ) constant monitoring of the emerging trends in the market with respect to standards and regulations, raw materials, customer tastes and technical aspects of production [1]. in order to accomplish some of the targets under the 12th sustainable development goal, the textile industry is now more conscious of the issues of sustainability. as a result, the subject of sustainability in the textile and clothing industry is receiving special attention especially among the producers. some of the strategies adopted are directed at securing the environment, protecting the people, adoption of renewable energy, energy efficiency and conservation, and environmentally friendly 165 adebisi & babatunde / j. nig. soc. phys. sci. 4 (2022) 165–173 166 communication. as such, for sustainability purpose, the textile industry requires extensive and intensive application of various types of state-of-theart enabling technologies. the adoption of information and communication technologies in the textile industry is one of the main technologies that can drive the required growth, development and expansion in the textile industry. the adoption of ict in the textile industry would drive the changes required to enhance the sustainability of textile and clothing manufacturing and trade. apart from making transactions and the production processes easier, the adoption of ict in the textile industry would make response to emerging fashion trends prompt; ensure effectiveness and efficiency during design, processing and mass production; and expansion of market through the adoption of e-commerce that ensures ease of transactions and prompt feedback from customers [2]. to stay relevant and sustain growth in the competitive market, business organizations increasingly invest in way to improve services delivery. in this quest, the adoption of information and communication technologies is currently driving the dynamic of many markets to satisfy the increased thirsts of the consumers. based on this, many business processes across almost all sectors depend on ict for sustainability. furthermore, with the mode by which data and information is being managed in the emerging markets and business, ict has a great impact on sustenance of modern [3, 4, 5]. green ict/green computing deals with the aspect of computer resource relation process towards reducing power consumption. it concentrates on central processing unit (cpu) efficiency, recycling and proper disposal of e-waste. the term “green ict” has been associated with manufacturing, usage, design and disposal of computer subsystems (printers, servers, hard disks, memory, monitors, network cards and so on) in a more efficient and effective manner with little or no negative impact on the environment [6]. to attain sustainability in the textile industry, the production and supply chain must be green as much as possible. these include the production, transportation, energy and most especially the communication departments of the industry. however, the adoption of green ict technologies comes with various technical, costs and environmental implications that must be considered before they are implemented; hence, making the selection of the most appropriate green ict in a textile industry is a multi-criteria problem. based on aforementioned, this study presents a framework for the selection of the most appropriate green ict adoption in the textile industry using a fuzzytopsis multi-criteria approach. 2. literature karabasevic et al., discussed the influence of ict in modern society due to increase societal demands [7]. the authors emphasized on the fact that; there is an aggressive search of new ideas and approaches to production especially in the textile industry. the business climate is more competitive, the market is dynamic, data exchange rate is high, systems integration is enormous and business support can barely survive without the technology support. they further emphasized on the need for green technology and its establishment in the textile industry. however, the work proposed bipolar fuzzy miltimoora method. today, professionals are fast considering green it both in academics and industry. istrat and lalić conducted a research with the objective to analyze various transactions in the textile industry [8]. most of the transactions used were extracted using various database techniques to find the patterns of buyer’s attitude towards improving decision-making. with this work, data mining techniques enhanced the decision-making process. close to 2000 transactions records were used from the cloth making industry as a knowledge base. interesting rules and various measures including customized offers were proposed in this work. in [9] the possible use of automation tools in the manufacture of garment has been proposed. according to [10], supply selection is simply the process of identifying, contracting and evaluating suppliers through the purchaser. in another study [11]; various criteria for supplier evaluation and their importance were checked towards selecting and retaining competent suppliers. some of the criteria identified in this are not based on specific organisation type or industry rather generalized. a separate research proposed multiple criteria decision analysis for supplier selection with focus on factors that can affect selection decision among others. [12]. to the best of our knowledge, the application of diverse green ict alternatives in the textile industry is rare and emerging. in [13] various applications of decision-making methods in the textile industry were described. this paper presents a fantastic approach to select suitable information communication technology approach (green ict) to enhance the supply chain activities in the textile industry. the advantages of green ict such as power management, energy efficiency and conservation, virtualization and cloud computing, materials recycling, software optimization, user friendliness to mention few. this approach will provide more interesting result in the textile industry. a subdivision of possible selection criteria in the textile industry under service, product, cost and supplier performances as been presented [14]. in [15], researchers gave a proposal on voting process as a method for making selection on textile related activities through one of the stages in the supply chain; the approach encouraged ranking as preferred over comparison for determining a potential supplier. fuzzy ahp method was utilized by [16] for textile supplier’s evaluation in india. from the same researcher, another review was carried out on various issues related to suitable supplier selections and their relationships with buyer. no doubt issues relating to supplier are dynamic. organizations tend to invest in green ict to support decision making process for the best result in the business especially; to survive competition. an interesting study on general multicriteria decision making system and its implementation has been described in [13]. in addition, is an original approach that represents customers aspect in the 166 adebisi & babatunde / j. nig. soc. phys. sci. 4 (2022) 165–173 167 fashion business, so this work addresses research in green ict specifically for textile industry. in [17] a theoretical model with dynamic application of technology related to skills and the impact of technology in textile industry was considered. this work is only from the skills perspective rather than holistic and no alternatives were proposed. however, the paper used common hypothesis in economics and embedded information systems to complement new skills required in the textile industry. it further reveals the room for continuous on-the-job training and higher level of education. in another research by [18, 19], various surveys on skilled and unskilled workers in textile industry was carried out to complement the work of [17]. the work demonstrates how new technologies can drive workers in the textile industry for better performances rather than sending deskilled workers out of job due to required technology skills. the authors of [20] studied strategies related to textile industry from smart perspective through nanotechnology and electronic telecommunication. the study explored the high potential of smart materials in the field of textile. they used innovative technology applications different from the conventional textile approach to cover various market segments. micro systems, material engineering and production technology were the focus of the work. this led to flexible economical fabrication, production compliance textile-based information system with more user applications. this work is relevant to next generation of fabrics and fibre coupled with items they produce. albeit this work serves as a leverage for the work under study however, the current research work considers various experts in the field of technology especially the green ict which brings out more potential of unpopular application of technology to textile, therefore presents further implementation of scientific significance. from the literature review, it is important to note that enhancing the supply chain process in the textile industry is tightly coupled with criteria decision challenges, which green ict can handle through various alternatives. [21, 22, 23, 24] carried out various analysis and algorithm on energy systems, distribution, performance analysis among other alternative power sources that could enhance going green and eventually saving costs of service production in textile industry. hence this study presents a multi-criteria framework for the ranking of green ict alternatives based on techno-economic and environmental criteria. albeit the framework is developed for the textile production industry; other industrial and production related sector can adopt and modify the framework proposed in this study for energy consumption reduction and going green. 3. green ict alternatives in textile industy this section discusses the various green ict alternatives that could be considered for production, services and process enhancement in the textile industry. these include power management, virtualization, cloud computing, telecommuting, software optimization and user friendliness. the considered alternatives are: • power management (r1) • virtualization (r2) • cloud computing (r2) • telecommuting (r4) • software optimization (r5) • user friendliness (r6) 3.1. power management one of the drivers of green computing is the reduction of power consumption while using latest technology as tools in business enhancement. this no doubt can help various automated systems deployed across high end computers, servers, workstations to mention are few in within textile industry operation [25]. the efficient handling of various power related systems and aspects of computer hardware through the use of advanced configuration and power interface (acpi) to temporarily preserve power on terminals that are idle for long period is fast becoming important. acpi is an advanced power management tool developed by intel-microsoft, it is useful for power consumption control facilitated by various technologies such as “speedstep” and “powernow” for under-volting depending on the workload adjustments [6]. workers in the textile industry making use of computers need to be aware of power management tools available to minimize cost along economic factors in achieving sustainable development. 3.2. virtualization virtualization as part of sustainability goals in green ict reduces hardware requirements by ratio of 10:1 or even better [4]. operational expenses are cut by 50% with resource virtualization on each server. most textile industry can take advantage of this technology to cut cost of computing resources as part of the alternatives explored in this work. it is affordable, robust with high uptime. virtualization results in far more efficient use of resources, including energy. one of the main purposes of virtualization is to make a single computer hardware function as if it is multiple places. most of the computing resources that can be virtualized include but not limited to the hard disk, memory, network accessories among others. these components can be isolated into different parts and making them accessible by individual as a virtual infrastructure without necessarily making a new purchase for different purpose. application of virtualization in textile can be replicated in the various supply chain activities in among sole distributors in the use of various enterprise resource planning solutions with heavy traffic and user requirements. this whole idea also helps in energy consumption reduction in data centers. 3.3. user friendliness this is also another alternative; most ict solution users look out for this a lot. while making choice of software solutions, user instructiveness and friendliness is one of the vital factors that drives software acceptability even before testing 167 adebisi & babatunde / j. nig. soc. phys. sci. 4 (2022) 165–173 168 by clients. this proposal as part of the green ict alternatives in textile is very important. textile industry is fashion driven with various styles colour combinations and attractiveness. so also, the solutions users expect a colour interface solution, easy to use, straight and direct while satisfying the business need. graphical user interface and interoperability is also very important with this alternative. various graphical packages involved in aforementioned, consume heavy resources including power hence need for green ict consideration in fulfilling this aspect of textile industrial production processes. 3.4. cloud computing this is a scalable style of computing with multiple access information technology. it does not depend on immediate physical resource to function. with the idea of sustainable development (environment, economic and social). cloud computing offers greater flexibility to various business model. textile industry being a dynamic one; no doubt needs to update software solutions used in various stages of production. most machines can now record production activities electronically, inventory is being managed using computer systems. with the proposal of cloud computing as an alternative; energy cost will be reduced and data in cloud are more secured with less worry about updates. most of the maintenance activities are shifted to the cloud services providers. the solutions offered through cloud computing are pay as you use thereby eliminating unnecessary overhead associated with incorrect resource demand in organisation’s planning [26]. 3.5. telecommuting the importance of telecommunication-related technology in textile industry cannot be overemphasized. it ranges from phone calls, fleet management, email services, online meetings among others. most devices used in the aforementioned are related to green ict, with lesser power consumption and large memory support. working from anywhere, rendering remote services especially with the advent of the pandemic is becoming more popular in every sector including textile. customers and wholesalers can nowbook textile products remotely, organizations can have internal and external meetings through telecommuting resources hence a green ict alternative that cannot be avoided. this technology has taken green ict even to document management, e-auditing, electronic signatories and approvals. thus, more space for teleworking with aggressive reduction of negative environmental impact. it enables video solutions and online collaborations in business environment like textile. [27]. 3.6. software optimization textile industry is associated with various software solution ranging from erp, customer relationship management (crm) among other supply chain solutions. optimizing these solutions for optimum performance to drive positive services delivery is very important. with this green ict factor, no doubt; serious of unique and efficient initiatives will play a vital role in the textile industry. software architectures are becoming smart through green ict and this alternative will address series of conventional software architectural challenges in design, analysis, synthesis and evaluation. 4. methods to rank various alternatives based on conflicting criteria and metrics, decision makers have developed many robust multi-criteria-decision-making (mcdm) techniques [28]. although these techniques typically most times produce same results, there are times when the techniques produces conflicting results [14, 29, 30]. mcdm methods allow experts to select the most preferred alternative from finite number of feasible alternatives. one of the most robust mcdm methods that has been extensively used by expert is the topsis method; this is because of its rationality, simplicity, and comprehensibility. also attributed to topsis is its ability to quantify and estimate the relative performance of the alternatives in an easy mathematical expression and robust computational efficacy [13]. while it is acknowledged that topsis generates quality output, professionals usually have preference for giving their opinions using linguistics terms; this usually makes evaluating uncertainties easier as compared to crisp value. because of its robust nature, fuzzy topsis method was extended by chen to capture uncertainty and the fuzzy nature of some multicriteria decision problems [31]; and this has been deployed by researchers in various fields [11, 32, 33, 34, 35]. for this study, the efficacy of fuzzy-topsis is deployed to evaluate and rank the most relevant green ict strategy applicable to a textile factory. the ranks of the alternatives are based on the relative closeness index from the fuzzy negative ideal solution (fnis) and fuzzy positive ideal solution (fpis). the steps followed in the implementation of the fuzzy-topsis include: step 1: develop a decision matrix in this step, the literature is explored to identify relevant green ict alternatives and the relevant criteria. these alternatives were rated by 5 experts using a ten-point fuzzy scale. the data was collected with the aid of a well-designed questionnaire. step 2: develop the normalized decision matrix using the following expression which is based on negative and positive ideal solutions: r̃i j =  ai jc∗j , bi j c∗j , ci j c∗j  and c∗j = maxi {ci j} (beneficial) (1) or r̃i j =  a−j ci j , a−j bi j , a−j ai j  and a−j = mini {ai j} (non − beneficial)(2) step 3: develop the weighted normalized decision matrix using the following expression: ṽ = ( ṽi j ) ; ṽi j = r̃i j × w j (3) 168 adebisi & babatunde / j. nig. soc. phys. sci. 4 (2022) 165–173 169 step 4: calculate the fuzzy positive-ideal solution (fpis) and fuzzy negative-ideal solution (fnis) using the mathematical expression: fpis(a∗) = ( ṽ∗1, ṽ ∗ 2, . . . ṽ ∗ n ) , where ṽ∗j = maxi { vi j3 } ; (4) fnis(a−) = ( ṽ−1 , ṽ − 2 , . . . ṽ − n ) , where ṽ−j = maxi { vi j1 } ; (5) step 5: compute the distance between each alternative and fpis and fnis using the expression: d∗i = n∑ j=1 d ( ṽi j, ṽ ∗ j ) , (6a) d−i = n∑ j=1 d ( ṽi j, ṽ − j ) . (6b) but d(x̃, ỹ) := √ 1 3 [ (a1 − a2) 2 + (b1 − b2) 2 + (c1 − c2) 2 ] . (7) step 6: evaluate the closeness coefficient and rank the alternatives cci = d−i d−i + d ∗ i (8) 4.1. illustrative example green computing and technology have gained popularity in recent times however, in future services arms of original equipment manufacturers (oems) will consider incorporating green ict into their service offerings with good focus on optimization, integration and customer centric offers. telecommunication service providers will also play more role in the future of green ict through recycling programs and sustainable innovations [28, 36, 37]. recently, some government agencies took radical actions resurrect textile industry so as to regulate their importation especially in western africa. one of the major issues that lead to importation is the economic recession that affected many nations in past few years. considering all forms of textile materials being blessed by some african countries as a case study, the adoption of green ict in the journey will provide better results considering the various criteria identified in this work [38]. the textile industry in taiwan also faced with some challenges such as low labour and manufacturing cost. this work will create increased awareness on how green ict alternative/multicriteria framework will encourage competition, reduce environmental hazards in the midst of harsh economy and global recession. in [38] an illustrative overview of a typical textile industry was provided. the challenges and possible solution reveal from this work is leveraged upon to support the green ict perspectives and impact in the textile industry. the authors mentioned a textile industry in western africa as the largest employer of labour, and a major player in the manufacturing sector of the economy in the past. unfortunately, it has seized to be a major contributor to the foreign exchange due to inadequate power supply, inconsistent government policies smuggling of foreign materials among others. a critical assessment of this cases study and its challenges proposed solutions along political will for sustainable development. the solution can be best implemented using the green ict, green ict drives policy, the use of modern software solutions for tracking of importation, automation of supply chain activities with technology that consume less power will elevate and resuscitate the sector. most western african countries used to be the most thriving textiles industry in africa. the discovering of other economic resources such as oil no doubt impacted its sustainability among other challenges [39]. with the advent of green ict, most of these challenges can be solved and application of technology to this industry can possibly make it another oil for most nations whom have suffered similar setback in the textile industry. the future is green, and green ict is here to stay. the rapid increase of technology development also reveals huge opportunities power management, decreasing pollution effects in the environment as identified in table 1. 5. results to establish the viability of the proposed framework, the proposed model is adapted to a textile industry. six green ict alternatives that will enhance the textile industry operations were evaluated based on six criteria. the green ict alternatives considered in this regard include power management (r1), virtualization (r2), cloud computing (r3), telecommuting (r4), software optimization (r5), user friendliness (r6). for ease, the alternatives are denoted by the set q = r1,r2,r3,r4,r5,r6, respectively. using the six criteria c = w1,w2,w3,w4,w5,w6 described in table 1, the alternatives were ranked in order of preference. in this illustrative example, a total of five experts (g ∈ g) from the industry and the academics with a sound knowledge on the issue of various textile production and value chain related to computing and ict were contacted for their knowledge and expertise on the subject. some of the experts have a doctorate degree while the minimum educational qualification of the experts is masters in the field of textile, processing, production, supply changing management and ict with not less than 10 to 20 years of experience respectively. they all acknowledged that they are aware of the numerous impact green ict could make in the textile industry. they also agreed that various criteria should be considered in selecting green ict technologies with respect to textile industry. table 2 presents the details of some of the questions and their responses. the opinions of the experts were collected with the aid of a well-structured questionnaire that points toward the importance of the criteria and the relationship between the alternatives and the criteria that was selected. the experts were assigned equal weights during the analysis of their responses. the linguistic responses obtained from the experts are presented in table 3 based on the triangular fuzzy number [1, 2, 26, 40]. the fuzzy 169 adebisi & babatunde / j. nig. soc. phys. sci. 4 (2022) 165–173 170 table 1: criteria considered in the study factor criteria description remark environmental environmental impact (w1) use of ict to drive policies that reduces negative impact on environment and general human living conditions. beneficial operating and maintenance (w2) smooth and continuous operation, services and textile equipment. non-beneficial economic implementation cost (w3) cost of deployment and implementation of green ict tools in line with global practices and standards. non-beneficial improved system performance and use (w4) improvement of services and general performance ranging from training of staff, acquisition of additional knowledge and skills in the area of green ict beneficial technical supply chain management (w5) simple, friendly, and interactive ict solutions that are usable by citizen of different categories and specializations towards climate change adaptation. beneficial employment(w6) opportunities ict solutions that are flexible with easy upgrade features in case of change. modifiable technical components as climate change evolves beneficial table 2: excerpt of expert response questions g1 g2 g3 g4 g5 have you heard of “green ict”? yes yes yes yes yes do you feel the green ict can impact textile industry positively? yes yes yes yes yes if yes, do you agree that different criteria should be considered in selecting green ict technologies with respect to textile industry processes? yes yes yes yes yes highest qualification ph.d. m.sc. m.sc. ph.d. ph.d. years of experience 16-20 16-20 16-20 11-15 16-20 organisation academics industry industry academics industry decision matrix of each expert’s opinions are combined using the elementary operators of the fuzzy triangular numbers such that the overall judgments and preference rating are extracted for the various green ict alternatives (table 4). the same procedure is followed to obtain the weights of the criteria from the preferences provided by the experts before conversion to crisp values. the aggregate results of the weights of the criteria are presented in table 5. the weighted normalized decision matrix and the fuzzy pis and fuzzy nis are given in tables 6 and 7, respectively. 6. discussion the closeness coefficient of the various green ict alternatives is obtained as 0.60, 0.43, 0.47, 0.45, 0.23 and 0.27 for r1, r2, r3, r4, r5 and r6, respectively (figure 1). the result shows that power management is the most effective green ict alternative for the textile industry. this is not far-fetched from the fact that textile industry is greeted with large and industrial machineries which are now being integrated with information technology tools through data centers, server management and end to end technology driven processes that are used in the production and manufacturing processes hence power plays a vital 170 adebisi & babatunde / j. nig. soc. phys. sci. 4 (2022) 165–173 171 table 3: alternative rating using linguistic terms g1 g2 w1 w2 w3 w4 w5 w6 w1 w2 w3 w4 w5 w6 r1 h eh vvh vh a aa r1 vl l aa vh l vh r2 a eh eh eh h ba r2 vl l a eh ba a r3 vh vh vh vvh aa aa r3 vl l a aa h a r4 aa eh vvh vvh vh vvh r4 vl l a vvh a a r5 a vh eh eh vh h r5 vl eh a eh vh vh r6 l vvh a vvh vvh vh r6 vl eh a vvh vvh a g3 g4 w1 w2 w3 w4 w5 w6 w1 w2 w3 w4 w5 w6 r1 vl l a vh l h r1 vl l a vh l h r2 l l l h l h r2 l l l h l h r3 vl vl a vh l h r3 vl vl a vh l h r4 vl l a vh l h r4 vl l a vh l h r5 vl a a vh vl h r5 vl a a vh vl h r6 vl l a h h h r6 vl l a h h h g5 w1 w2 w3 w4 w5 w6 r1 a h eh a aa h r2 a aa vh a aa l r3 a aa h a eh l r4 aa a h a eh l r5 aa a a a vh l r6 l l a a h l table 4: combined responses of the experts w1 w2 w3 w4 w5 w6 r1 (1,3.6,8) (1,4.8,10) (1,5.2,10) (47.4,9) (1,3.6,7) (5,7,9) r2 (1,3.6,6) (1,4.6,10) (2,5.8,10) (4,7.8,10) (2,4.6,8) (2,5.2,8) r3 (1,3.8,9) (1,3.8,9) (4,6,9) (4,6.6,9) (2,5.8,10) (2,5.4,8) r4 (1,3.6,7) (1,4.4,10) (4,6.2,10) (4,7.8,10) (2,5.8,10) (2,6.2,10) r5 (1,3.4,7) (4,6.6,10) (4,6,10) (4,8.2,10) (1,5.6,9) (2,6.4,9) r6 (1,2.4,4) (2,5.6,10) (4,5,6) (47.4,10) (6,7.8,10) (2,6,9) table 5: consolidated weight allocated to criteria criteria weight of the selection index combined weight g1 g2 g3 g4 g5 w1 emi vmi emi vvmi mi (6,8.8,10)) w2 emi i vvmi emi vvmi (5, 8.8, 10) w3 vvmi i i mi vvmi (5, 7.4, 10) w4 vvmi vmi vmi i vmi (5, 8, 10) w5 emi i i i vvmi (5, 7.4, 10) w6 mi mi i i mi (7, 6.6, 8) role and results shows, it is the most preferred alternative, followed by cloud computing, telecommuting and virtualization respectively; these closeness in the results shows how tightly coupled the technology drivers of the production output in textile industry are. most processes are now reachable by very wide audience due to popular and increased rate of cloud service migration; down the line are some previous on premise technologies as shown in the result (telecommuting and virtualization). user friendliness and software optimization are the least preferred alternatives based on the output of this research; although also important but compared to other options the effect of these criteria in line with the textile industries operations and service delivery could be taken care of at a relatively moderate level in the supply chain process without creating much negative impact in the entire service delivery. overall, the most preferred green ict alternative is making a choice of green ict 171 adebisi & babatunde / j. nig. soc. phys. sci. 4 (2022) 165–173 172 table 6: weighted normalized decision matrix w1 w2 w3 w4 w5 w6 r1 (0.7,3.5,8.9) (0.5,1.8,10,0) (0.5,1.4,10.0 ) (2.0,5.9,9.0) (0.5,2.7,7.0) (2.5,3.5,7.2 ) r2 (0.7,3.5,6.7) (0.5, 1.9, 10.0) (0.5,1.3,5.0) (2.0,6.2,10.0 ) (1.0,3.4,8.0) (1.0,2.6,6.4 ) r3 (0.7,3.7,10.0 ) (0.5, 2.3,10) (0.5,1.2,2.5) (2.0,5.3,9.0) (1.0,4.3,10) (1.0,2.7,6.4 ) r4 (0.7,3.5,7.8) (0.5,2.0,10.0) (0.5,1.2,2.5) (2.0,6.2,10.0 ) (1.0,4.3,10.0 ) (1.0,3.1,8.0 ) r5 (0.7,3.3,7.8) (0.5,1.3,2.5) (0.5,1.2,2.5) (2.0,6.6,10.0 ) (0.5,4.1,9.0) (1.0,3.2,7.2 ) r6 (0.7,2.3,4.4) (0.5,1.6,5.0) (0.8,1.5,2.5) (2.0,5.9,10.0 ) (3.0,5.8,10) (1.0,3.0,7.2 ) a* (0.7,3.7,10.0 ) (0.5, 2.3,10.0) (0.8,1.5,10.0 ) (2.0,6.6,10.0 ) (3.0,5.8,10) (2.5,3.5,8.0 ) a(0.7,2.3,4.4) (0.5,1.3,2.5) (0.5,1.2,2.5) (2.0,5.3,9.0) (0.5,2.7,7.0) (1.0,2.6,6.4 ) table 7: fuzzy pis and fuzzy nis distance from fpis w1 w2 w3 w4 w5 w6 di* r1 0.7 0.3 0.2 0.7 2.9 3.9 8.56 r2 1.9 0.2 2.9 0.2 2.1 4.4 11.75 r3 0 0 4.3 0.9 1.4 4.4 11.09 r4 1.3 0.2 4.3 0.2 1.4 4.8 12.25 r5 1.3 4.4 4.3 0 1.8 4.6 16.41 r6 3.3 2.9 4.3 0.4 0 4.6 15.5 distance from fnis r1 w1 w2 w3 w4 w5 w 6 dir2 2.7 4.3 4.3 0.4 0 1.1 24.76 r3 1.5 4.3 1.4 0.8 0.8 0 33.08 r4 3.3 4.4 0 0 2 0.1 33.26 r5 2 4.3 0 0.8 2 1 35.27 r6 2 0 0 0.9 1.4 0.6 43.57 r7 0 1.4 0.3 0.7 2.9 0.5 40.28 related services that makes provision for effective power management in achieving set goals and objectives. figure 1: closeness coefficient 7. conclusion the application of green ict using a multicriteria decision making method is a highly important task especially in textile industry. the great potential positive impact in the textile industry will not only boost performance in the entire supply chain trend but also create diverse opportunities to tackle various challenges at different stages resulting to improved customer satisfaction. this research provided a method for selecting the best technology that will enhance textile industry activities using green ict with the use of real case study as an example. three major novel contribution has been achieved with this research namely, application of green ict to textile industry, confident selection of the best technology that will encourage transparent and profit-oriented service delivery, incorporation of customers perspectives into service delivery through the use of ict techniques and lastly, this research is capable of being a decision-making reference tool in the selection of it tools especially for startups in the textile industry. the outcome of this study is expected to save cost, reduce energy consumption as well as emission in the society. furthermore, climate change, environmental and health impact will be drastically reduced. it is anticipated that decision makers will adopt and implement the proposed framework in developing a green textile production. although the advantage of the fuzzy topsis approach are numerous which include simplicity, rationality, comprehensibility, good computational efficiency and ability to measure the relative performance for each alternative in a simple mathematical form. the limitation of this method include high labor input, large amount of initial data, and limited nature of the assessment scale. future study will adopt/develop robust multi criteria methods to test the efficacy of the proposed framework and possibly use other mcdm methods to benchmark the present fuzzy-topsis approach developed as well as comparison of the result obtained from the study. references [1] g. kokeza and s. uros̆ević, “strategic directions of 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[40] c.-y. chen, c.-y. liu, c.-c. kuo, and c.-f. yang, “web-based remote control of a building’s electrical power, green power generation and environmental system using a distributive microcontroller”, micromachines 8 (2017) 241. 173 j. nig. soc. phys. sci. 3 (2021) 292–297 journal of the nigerian society of physical sciences application of computational chemistry in chemical reactivity: a review c. w. chidieberea,∗, c. e. durua, jp. c. mbagwub adepartment of chemistry, imo state university, surface chemistry and environmental technology (scent) research unit, department of chemistry, imo state university, owerri, imo state, nigeria bdepartment of physics, faculty of physical sciences, imo state university, owerri, nigeria. abstract molecular orbitals are vital to giving reasons several chemical reactions occur. although, fukui and coworkers were able to propose a postulate which shows that highest occupied molecular orbital (homo) and lowest unoccupied molecular orbital (lumo) is incredibly important in predicting chemical reactions. it should be kept in mind that this postulate could be a rigorous one therefore it requires an awfully serious attention in order to be understood. however, there has been an excellent breakthrough since the introduction of computational chemistry which is mostly used when a mathematical method is fully well built that it is automated for effectuation and intrinsically can predict chemical reactivity. at the cause of this review, we’ve reported on how homo and lumo molecular orbitals may be employed in predicting a chemical change by the utilization of an automatic data processing (adp) system through the utilization of quantum physics approximations. doi:10.46481/jnsps.2021.347 keywords: homo-lumo, computational methods, and chemical reactivity article history : received: 14 august 2021 received in revised form: 30 september 2021 accepted for publication: 01 october 2021 published: 29 november 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: e. a. emile 1. introduction frontier molecular orbital theory was postulated in 1952 by kenichi fukui where he described highest molecular orbital (homo) and lowest unoccupied molecular orbital (lumo) interaction from molecular orbital theory [1]. in 20th century, electronic structure of a molecule was described using molecular orbital theory from quantum physics [2], where electrons are treated as moving under the control of the atomic nucleus within the bulk molecule but aren’t treated as a personal attraction between atoms in line with molecular orbital theory. ∗corresponding author tel. no: +234 email address: oraclewizzy@gmail.com (c. w. chidiebere ) the reason of the spacing of electrons was given by quantum physics. however, quantum physics also explains the energetic behaviour of electrons as molecular orbital which house two or more atoms [3, 4]. there’s little question, that molecular orbital theory has changed how chemical bonding is viewed through the approximation of the states of bonded electrons i.e., viewing the molecular orbital as linear combinations of atomic orbital (lcao) [5]. these approximations are made by applying the density functional theory (dft) model or hartreefork (hf) models to the time independent differential equation. these approximations are the fundamental theories in computational chemistry [6]. they’ll be modeled and automatic in a very automatic data processing system to help easy calculation. however, computational chemistry may be utilized in different 292 c. w. chidiebere et al. / j. nig. soc. phys. sci. 3 (2021) 292–297 293 number of ways but the foremost important one is to model a molecular system by synthesizing the molecule in a laboratory [7]. confine in mind that computational models don’t seem to be always perfect but it can rule out about 90% of possible molecules that don’t seem to be good for his or her intended use [8], however, computational chemistry makes it easy to understanding a controversy fine. in the study of chemical reactivity, both state and open system (involving radicals, cations, and anions) calculations are performed to establish a result [9]. these methods are frequently unreliable, especially when diffuse functions must be included in the basis set. as a result, it is useful to have a technique that can extract all of the information required directly from the findings of the molecular system’s basis state calculation [10]. thus, the knowledge of frontier molecular orbitals has imparted on the understanding of chemical reactions and is accustomed to distinguish between the parts of orbital that have opposite signs. 2. highest occupied and lowest unoccupied molecular orbital (homo-lumo) highest occupied molecular orbital (homo) is that outermost orbital where the very best energy orbital is contained, the orbital acts as an electron donor while lowest unoccupied molecular orbital (lumo) is that orbital with the lowest unoccupied energy level which acts as the electron receiver. according to the frontier molecular orbital theory, the formation of a transition state is because of a synergy between the frontier orbitals of the reactants [11]. the stability of a molecule is determined by homo-lumo gap which is known as the dissimilarity in energy between the homo and lumo of the molecule [12]; thus, a large homo-lumo gap implies high stability for the molecule in chemical reactions while a tiny low one implies low stability. thus, the homo-lumo gap may be defined mathematically as [13]: ∆e = elu mo − ehomo (1) where ∆e is the energy gap, elu mo is the energy of lumo and ehomo is the energy of homo. keep in mind that when the homo-lumo gap is large, the molecule is said to be hard and a soft molecule has a small energy gap [14]. thus, chemical hardness is the opposition to change in electron cloud density or electron distribution of a chemical system and chemical softness is the inverse of chemical hardness [15]. chemical hardness and chemical softness can be shown mathematically as η = i − ea 2 (2) σ = 1 η (3) where η is known as chemical hardness, σ is known as chemical softness. ionization potential relates to energy of the homo while the energy of the lumo relates to the electron affinity thus the homo and lumo energies can be applied in the calculation of the ionization potential (i) and electron affinity ea. from koopmans theorem [16] i = −eh (4) ea = −el (5) 3. density functional theory (dft) density functional theory also referred chemical reactivity theory is a variational method that’s currently the foremost promising approach which is employed to determining the electronic structure of matter [17]. the n-particle differential equation yields density functional theory, which is totally stated in terms of the density distribution of the base state and the single particle wave function. it’s been a helpful tool which is employed in analysis, prediction and elucidation of the results of chemical reactions. following the work of parr et al [18], advantageous number of ideas has been derived from the resolution of the density of molecular systems using dft. its requisite to own recourse on the kohn-sham theory by calculating the molecular density, the system’s energy and energies of the orbital especially; those connected to the frontier orbitals. the kohn-sham method creates a precise connection between the density and state energy of a non-interacting fermion system and the real many bodies system given by the schrodinger wave equation [19]. −~ 2m ∂2 ∂x2 = i~ ∂ψ ∂t (6) the basis of dft has confirmed the existence of a universal density functional, and the system’s properties are acquired by completing calculations using the functionals [20]. because they’re approximate functionals rather than universal functionals, some of them are good at predicting certain qualities while others are good at predicting others. density functionals that are excellent at defining the features of each molecule system with a specific functional group can occasionally be discovered. for the study of molecular systems, it’s quite advantageous to employ different density functionals for a separate functional group that you merely want to require in [21]. dft reduces the time it takes to compute the bottom state parameters of systems of synergizing particles to the same precision as the results of single-particle hartree-type equations, which is why it’s so beneficial for systems with many electrons. the goal of dft calculations is to show how the electron density is related to the bottom state. by using the results of the kohn-sham equations, the definition of electron density is given [22]. ρ(r) = ∑ i ψi(r)ψ ∗ i (r) (7) 293 c. w. chidiebere et al. / j. nig. soc. phys. sci. 3 (2021) 292–297 294 figure 1: aromatic compounds with the highest occupied molecular orbital (homo) and the lowest unoccupied molecular orbital (lumo) were discovered using dft. 4. hatree-fock equation the hartree-fork fork technique is an approximation approach used in computational chemistry to derive the wave function with the energy of a quantum body system in an immobile mmobile condition [23]. hartree-fork theory is a fundamental notion in electronic structure theory; it is the cornerstone of molecular orbital theory, which states that each electron’s mobility may be described by a single-particle particle function orbital that is independent of the other electrons’immediate motion [24].the hartree-fork resolves the set of spin orbitals which reduce the energy and devote the best single determinant. the equation is given as [25]. h(x1)χi(x1) + h ∑ j,i [∫ d x2 ∣∣∣x j(x−1)∣∣∣2 r−112 ] χi(x −1) − ∑ j,i [ d x2χ ∗ j (x 2)χi(x 2)r−112 x j(x 1) = �iχi(x1) ] , (8) where �i is the energy eigenvalue associated with orbital (χi). hence the hartree-fork equations can be resolved mathematically i.e., exact hartree-fork, or can be solved in the spaced spanned by a set of fundamental functions (hartree-fork-roothan equations) where their solutions are dependent on the orbitals [26]. reducing hartree-fork equation to hartree-fork-roothan equation in matrix form, we have:∑ ν fµνcνi = �i ∑ ν s µνcνi (9) 5. frontier molecular orbital calculation and application of computer system in computational chemistry calculation of molecular orbital energy may be done using ab initio methods; the tactic applies electron and also the nuclei synergy as its fundamental component. it attempts to calculate the electronic differential equation by assigning the locations of the nuclei and therefore the electrons surrounding it. hence, electronic energy of a molecule may be calculated by treating the general electronic wave function as being a product of molecular orbitals which is predicated on slater determinant of molecular orbitals so as to satisfy the pauli exclusive principle [27]. so, we will write a molecular wave function as: ψ = φ1φ2φ3φ4 · · ·φn (10) where each molecular orbital is written in form of a linear combination of atomic orbitals that is, ψi = ∑ i cikφ (11) the molecular orbitals will be calculated by using the hartreefock method [28]. the hamiltonian operator has terms for the 294 c. w. chidiebere et al. / j. nig. soc. phys. sci. 3 (2021) 292–297 295 potential and mechanical energy of the electrons; it involves the interactions between the nuclei and the electrons, and also the electron-electron repulsion. note the molecular orbitals are conceived by making an initial supposition at the coefficients [29]. hence h − es = 0 (12) h contains the hamiltonian matrix elements and s is known as the overlap matrix elements. solving this determinant will give the answer of molecular orbital energy e, which is substituted into the overall equations to allow new values of the coefficients (ci ). this whole process is repeated until the coefficients don’t change from one cycle to the other. the orbitals can then be described as self-consistent, and this process is thought as a selfconsistent field procedure [30] where each molecular orbital is created from a mixture of atomic orbitals. when one function is employed for every filled atomic orbital for instance, using, an orbital for an atom, this basis set is termed a minimal basis [31]. using more functions than this may improve the calculation i.e., it’ll tend to extend the pliability and quality of the general wave function. from the variational principle we all know that a stronger wave function will provides a lower energy [32]. the larger the idea set the higher the calculation, if a large number of functions i.e., a really large basis set is employed to point out the atomic orbitals; we approach the hartree-fock limiting energy. however, a very cheap energy is achieved by the hartree-fock method [33]. frontier molecular orbital theory is an exemplification of molecular orbital theory which elucidates homo/lumo synergies [34]. during 1952, kenichi fukui put forth a paper within the journal of chemical physics which he titled “a molecular theory of reactivity in aromatic hydrocarbons”. his work examined the frontier orbitals and additionally the consequences of the homo and also the lumo on reaction mechanism specifically which is the reason why it’s called frontier molecular orbital theory (fmot) [35]. fukui noticed that an honest approximation for reactivity can be established by observing the homo-lumo. this was founded on principal point observance of molecular orbital theory of two molecules which shows that the occupied orbital of unlike molecules opposes one another, thus, the negative charges of a molecule attract positive charges of another molecule as such, the occupied orbital of another molecule and the unoccupied orbital of the opposite synergize with one another per se causing attraction [36]. though this theory could also be a rigorous narrative of chemical phenomena, impediment within the mathematical aspect may well be too difficult that it’ll not be viable to resolve a controversy accurately. since the introduction of computational chemistry, most rigorous theories can now be treated with little difficulty. computational chemistry is employed in a very number of various ways, one in all which is to model a molecular system before synthesizing the molecule within the laboratory [37]. however computational models don’t seem to be always perfect but it can rule out about 90% of possible molecules that don’t seem to be good for its intended use [38]. however, it is very helpful because to synthesizing only one (1) compound could take months of labour and raw materials and furthermore generate waste matter. computational chemistry helps in comprehending an issue wholly. there are some properties of a molecule that may be gotten computationally more easily than by experimental means [39]. there also are deep views into molecular bonding, which might be gotten from the inference of computations that can’t be obtained from any scientific method. thus, many experimental chemists are now making use of computational modeling to induce additional understanding of the compounds being observed within the laboratory [40]. going back to history, paul cruzan, mario molina, and sherwood rowland who are computational chemists were awarded the laurels in chemistry in 1995 for forming mathematical models that applied the thermodynamic and chemical laws to elucidate how ozone were formed and decomposed within the atmosphere [42]. although, computational chemistry wasn’t usually imagined as its own separate field of study until 1998, when walter kohn and john pople were awarded the nobel prize in chemistry for their exhibition of density functional theory and disclosure of computational methods using quantum chemistry respectively [43]. computational chemistry is not the same as engineering science; however professionals from both fields frequently collaborate. computer scientists spend the majority of their work assembling and testing computer algorithms, data visualization skills, and software and hardware solutions [44].computational chemists work alongside laboratory scientists and theoretical scientists to put all of this modeling, simulation, data analysis, and visualization expertise to good use in their study. computational chemists can employ supercomputers and computing clusters, which are high-performance computing systems, to solve issues and run simulations that require a large quantity of data [45]. electronic structure approaches, molecular dynamics simulations, quantitative structure-activity connections, and complete statistical analysis are among the tools used by computational chemists. to combine chemical theory and modeling with actual observations, computational chemists use statistics, mathematical algorithms, and massive databases. some computational chemists create physical step models and simulations, and then utilize statistics and data analysis approaches to extract useful data [46]. progress in computer visualization allows computational chemists to highlight complex analyses in a more comprehensive format, which they may use to explain experiments and current materials, as well as validate the results. computational chemistry advances our understanding of how the world works, aids manufacturers in developing more efficient and productive operations, allows for the characterization of new chemicals and materials, and aids other researchers in extracting relevant knowledge from large amounts of data. 295 c. w. chidiebere et al. / j. nig. soc. phys. sci. 3 (2021) 292–297 296 figure 2: shows the aid of computer system in elucidation of chemical behaviour quantum physics and thermodynamics are used in computational chemistry to learn about the character and fundamental properties of atoms, molecules, and chemical reactions [47]. recent researches showed that quantum physics (qm) and molecular mechanics are the foremost used models in computation. most computation software’s utilized by researchers include: • integrated packages • ab initio and dft software • semi empirical software • graphics package • special purpose program • molecular mechanics/molecular dynamics/monte carlo software software packages differ in cost, functionality, efficiency, automation, robustness and ease to use. these concerns make a difference in determining what computational projects is successful and the way much work which will be involved [48]. 6. conclusion the quest for understanding chemical reaction mechanism has really caused many researchers to go into in-depth study to source out path of reactions and the cause of the reactions. homo-lumo molecular orbitals have changed the perspective at which chemists view chemical reactions with the help of quantum mechanics which has given the exact explanation on the behaviour of electrons. apart from this, scientists went further to develop models 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[48] h. l. peña, d. a. boateng, s. mcpherson, & k. m. tibbetts, “using computational chemistry to design pump-probe schemes for measuring radical cation dynamics”, physical chemistry chemical physics 76 (2021) 205. 297 j. nig. soc. phys. sci. 3 (2021) 414–422 journal of the nigerian society of physical sciences waste glass: an excellent adsorbent for crystal violet dye, pb2+ and cd2+ heavy metal ions decontamination from wastewater k. o. sodeinde∗, s. o. olusanya, d. u. momodu, v. f. enogheghase, o. s. lawal materials and nanoresearch unit, department of industrial chemistry, federal university oye-ekiti, ekiti state, nigeria. abstract the suitability of waste glass as an eco-friendly adsorbent for the removal of crystal violet (cv) dye, pb2+ and cd2+ heavy metal ions in waste water samples was investigated in batch mode. waste glass sample was pulverized and characterized by sem/edx, xrd, bet and ftir. effects of variation in temperature, ph, contact time and recyclability of the adsorbent were studied. ftir spectra revealed major peaks around 491.53 and 3444.12 cm−1 corresponding to the bending vibrations of si-o-si and -oh groups respectively. sem/edx analysis showed a dense, coarse, porous morphology with predominantly silica component. the effective surface area and size of the adsorbent were 557.912 m2/g and 2.099 nm respectively. increase in temperature, dosage, contact time resulted in increase in adsorption efficiency. optimum adsorption efficiency of 94%, 97.5% and 89.1% was attained for pb2+, cd2+ ions and cv dye respectively at 70◦c. adsorption process followed more accurately pseudo-first order model and isotherm fitted perfectly into freundlich model indicating a multilayer adsorption mechanism for cv dye and the heavy metals. 89.87% reduction in chemical oxygen demand (cod) level of wastewater was reported upon treatment with waste glass adsorbent affirming its efficiency for dye and heavy metal pollutants removal. doi:10.46481/jnsps.2021.261 keywords: waste glass, crystal violet dye, heavy metals, adsorption, wastewater article history : received: 18 june 2021 received in revised form: 26 september 2021 accepted for publication: 27 september 2021 published: 29 november 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: e. etim 1. introduction glass is one of the earliest man-made materials. it is an inorganic, non-crystalline material obtained by melting, homogenization, cooling and finishing of components such as sand, sodium oxide, limestone, etc. glass has found useful applications in television screens, glass bottles, doors, car windscreens, packaging materials, certain industrial and scientific equipment, etc, [1]. increasing global population and industrialization have led to a sharp rise in the use of glass products in ∗corresponding author tel. no: +2348147773137 email address: kehinde.sodeinde@fuoye.edu.ng (k. o. sodeinde ) household and industrial devices resulting in the generation of huge volume of waste glass. according to the united nations estimates, the annual global amount of disposed solid waste is said to be 200 million tons, comprising 7% glass [2]. generally, glass is relatively stable, non-biodegradable, resistant to attack and outdoor temperature [3]. recycling is a useful method of reducing pollution load associated with waste glass. however, poor financial and technological capacity for proper waste glass recycling remain a major problem particularly in developing countries. although dyes have numerous applications in food, cosmet414 k. o. sodeinde et al. / j. nig. soc. phys. sci. 3 (2021) 414–422 415 ics, pharmaceutical and textile industries, improper discharge of effluents containing dyes into water bodies negatively impact on their aesthetic value; threaten the survival of aquatic species by impairing photosynthesis, etc, [3, 4]. bioaccumulation of dyes and heavy metals especially lead and cadmium in living cells can lead to impairment of haemoglobin synthesis, neurological disorder, kidney damage, carcinogenicity, mutagenicity, etc. [5]. in particular, exposure of pregnant women to elevated levels of lead may result in miscarriage, stillbirth, premature birth, deformities, etc, [6]. several methods such as adsorption, ion exchange, photo-degradation, physical precipitation, etc, have been reported for the removal of dyes and heavy metals in the environment [7, 8, 9, 10, 11]. of these methods, adsorption remains the most attractive based on cost, ease of operation, analysis time, etc [12]. different natural and synthetic materials including clay, activated carbon, termite mounds, agricultural wastes (sawdust, brans, banana peels), etc. have been investigated as economic adsorbents for decontamination of dyes and heavy metals [13, 14, 15, 16]. for instance, ahmad reported the removal of crystal violet (cv) dye by adsorption using coniferous pinus bark powder [17]. the amount of cv dye uptake increased with increase in initial ph, dye concentration and contact time but decreased with adsorbent dosage increase. equilibrium was attained in 2 h. the isotherm fitted most into langmuir model indicating a monolayer adsorption mechanism while the kinetics followed perfectly a pseudo second order rate equation. also, fly ash and rice husk adsorbents were investigated for the removal of copper, lead, cadmium, nickel and iron heavy metals [18]. the rice husk adsorbent showed better affinity towards iron, nickel and lead while fly ash was more effective for the removal of copper and cadmium [18]. crystal violet (cv) is categorized under the triarylmethane dyes. it is equally referred to as hexamethyl pararosaniline chloride, gentian violet, etc [9]. it possesses antibacterial, antifungal and antihelminthic properties [7]. it is widely used as a histological stain, for paints and textile applications [19]. cv dye is carcinogenic, non-biodegradable, persistent in the environment due to its poor metabolism by microorganisms [11]. access to clean water and environmental sanitation remains a key objective of the sustainable development goals (sdgs) of the united nations (un). utilisation of waste glass as a cheap and effective adsorbent for the decontamination of toxicants in polluted water bodies would promote cleaner environment, minimise water pollution and improve quality of life especially in rural communities of developing nations. thus, the focus of the work was to utilise waste glass from the local environment for the decontamination of crystal violet (cv) dye and toxic heavy metal ions (pb2+ and cd2+ ions) in aqueous medium. effects of variations in ph, temperature, adsorbent dosage, pollutant concentration, recyclability on the adsorption efficiency, kinetics, adsorption isotherm and practical application in the treatment of waste water are herein presented. 2. methodology 2.1. materials/reagents all the chemicals used were of analytical reagent grade or the highest purity available. 2.2. sample collection and treatment 2.0 kg of transparent soda-lime wasteglass bottle samples were collected around oye town, ekiti state, nigeria. the samples were cleaned and ground into fine powder by dry ball milling machine operated at ambient temperature for 8 h. the pulverised wasteglass (particle size 75 µm) was carefully kept in a glass vial and stored at room temperature (30 ± 2◦c) till further use. 2.3. scanning electron microscopy/energydispersive x-ray (sem/edx) analysis morphological features and surface characteristics of glass sample were obtained from scanning electron microscopy/energy dispersive x-ray (sem/edx) using a jeol usa model: jsm7900f. a thin layer of waste glass sample granule was placed on aluminium specimen holder by double-sided tape. the specimen holder was loaded in a polaron sc 7610 sputter coater and coated with gold to a thickness of about 30nm to prevent charging. the specimen holder was transferred to xl-20 series. sem/edx analysis was conducted at an accelerating voltage of15-20 kv. 2.4. fourier transform infra red (ftir) spectroscopic analysis ftir spectra of the waste glass samples were run as kbr pellets on perkin elmer spectrum two tm spectrometer in the frequency range 4000 − 400 cm−1. 2.5. x-ray diffraction studies x-ray diffraction patterns of powdered waste glass sample was obtained using empyrean xrd diffractometer at 40 ma and 45 kv with cu kα (1.5418 å) radiation at an angular incidence of 10 − 75◦. 2.6. brunauer-emmett-teller (bet) surface area analysis the specific surface area (bet) of powdered waste glass sample was measured with a quantachrome surface analyzer using n2 as the standard adsorbate. prior to analysis, the sample was degassed at 523 k for 3 h. the pore size and pore volume were estimated using barrett-joyner-halenda (bhj) theory. 2.7. batch adsorption experiment batch adsorption method as reported by ayanda et al. [20] was adopted with slight modification. briefly, a range of powdered waste glass (0.1 g 1.0 g) was contacted with 40 ml of 5 mg/l cv dye and stirred at 300 rpm at 40◦c for 30 min for the effect of waste glass adsorbent dosage. to investigate the influence of ph on the adsorption process, the ph of 40 ml of 5 ppm cv dye solution containing 0.5 g waste glass adsorbent was adjusted to ph range 2-10 using either 0.1 m hcl or 0.1 m naoh. the effect of time was studied between 20-60 min 415 k. o. sodeinde et al. / j. nig. soc. phys. sci. 3 (2021) 414–422 416 while the effect of temperature was determined at a temperature range of 30-70 oc. on the effect of initial dye concentration, 40 ml of 1, 5, 10, 15 and 20 ppm cv dye were used along with 0.5 g powdered waste glass adsorbent at 40◦c for 30 min. the resulting mixture after each experiment was filtered and absorbance of the aliquot of the filtrate was determined using t-60 uv-visible spectrophotometer at 592 nm. for the pb2+ and cd2+ heavy metal ions determination, the same procedure was repeated. the pb2+ and cd2+ levels were analysed using atomic absorption spectrophotometer (aas buck scientific model 211 vgp) with the respective hollow cathode lamps. the adsorption efficiency experiments were carried out for three repeated cycles to study the recyclability and stability of the waste glass sorbent. after each cycle, the adsorbent was filtered and oven-dried at 70◦c before reuse for the next cycle. for practical application of the adsorption efficiency of the waste glass sorbent, effluent from local textile (adire) industry was collected in a glass bottle and analysed before and after treatment (0.5 g waste glass contacted with 40 ml effluent at 300 rpm, ph 6, 40 oc for 30 min) for the levels of pb2+, cd2+ ions and cv dye. the chemical oxygen demand (cod) was equally determined before and after treatment using standard method [21]. duplicate analysis was conducted for all blank solutions, standards and samples. the percentage cv dye and heavy metal ions (pb2+ and cd2+) adsorption efficiency was obtained using: % adsorption efficiency = c0 − ce/c0 × 100, (1) where c0 is the initial concentration of analyte (cv dye/heavy metal) and ce is the equilibrium concentration of analyte. the equilibrium adsorption capacity (mg of analyte per g waste glass) was calculated using: qe = c0 − ce/w × v, (2) where c0 and ce (mg/l) are the initial and equilibrium concentrations of the cv dye/heavy metal solution. v (ml) is the volume of the solution, w (g) is the mass of waste glass used and qe (mg/g) is the amount adsorbed. 2.8. adsorption isotherm in this study, the adsorption isotherms described by langmuir and freundlich were used. the langmuir isotherm is based on the theoretical principle that the adsorption sites are of equal energy and the coverage of adsorbate molecules on a solid surface occurs only in a monolayer. the linearised form of the langmuir model is given by 1 qe = 1 qm kl × 1 ce + 1 qm , (3) where ce is the equilibrium concentration of pollutant analyte solution (mg/l), qe is the concentration of pollutant adsorbed per unit mass of the waste glass (mg/g), qm is the langmuir constant denoting adsorption capacity (mg/g), and kl is langmuir constant denoting energy of adsorption (l/mg). a linear plot of 1/qe against 1/ce is the underlying basis of this equation with kl and qm determined from the slope and intercept respectively. the linear form of the freundlich equation is written as: log qe = log kf + 1/n log ce, (4) where ce is the equilibrium concentration of pollutant analyte solution (mg/l), qe is the concentration of pollutant adsorbed per unit mass of the waste glass (mg/g) while n is the number of layers and kf is the freundlich constant. kf and n values are deduced from the respective intercept and slope of the linear plot of log qe against log ce. 2.9. kinetic models studies for detailed studies of the adsorption kinetics of pb2+, cd2+ ions and cv dye on the waste glass sample, the lagergren pseudo-first-order model [22] and pseudo-second-order kinetic model [23] were used. they are given, respectively, as: ln(qe − qt) = ln qe − k1t, (5) t/qt = 1/(k2 · q 2 e ) + t/qe, (6) where qt and qe are the amount of pollutant adsorbed (mg/g) at time t and equilibrium, respectively. k1 (min−1) is the rate constant of the pseudo-first-order kinetic model and k2 (g · mg−1· min−1) is the rate constant of the pseudo-second-order kinetic model. 3. result and discussion 3.1. sem/edx and bet surface area analyses of waste glass powder surface morphology and chemical composition of an adsorbent play a major role in ascertaining the efficiency and effectiveness of the adsorption process. the sem image (figure 1a) of the porous waste glass showed a dense, coarse, porous surface morphology. edx analysis (figure 1b) revealed the % composition of the waste glass in the following order: si>mg>c>o>ca>fe>s>na>k. these elements are present in form of oxides (e.g. sio2, mgo, fe2o3, etc.), carbonates (calcium carbonate, sodium carbonate, potassium carbonate, etc), silicates; with silica as the predominant component. the bet specific surface area of the waste glass was about 557.912 m2/g while the barrett, joyner, halenda (bjh) surface area was measured to be about 663.9 m2/g. the bjh pore volume was calculated to be about 0.3264 cc/g with an average pore size of 2.099 nm (see figure 2).the dense, coarse, porous nature of the glass surface coupled with interaction with various chemical components facilitated adsorption of pollutant analytes onto its surface under prevailing conditions. 3.2. ftir spectra of waste glass the result of the ftir analysis of the porous waste glass particles is shown in figure 3. the peaks at 412.31, 419.22, 431.95, 444.07, 451.09, 455.22, 473.95, 482.59 and 491.53 cm−1 revealed bending vibrations of si–o–si, while the peaks at 528.01, 519.90, 504.54, 511.85 cm−1 showed the stretching vibration of o–si–o. the peaks observed at 605.00, 645.43, 416 k. o. sodeinde et al. / j. nig. soc. phys. sci. 3 (2021) 414–422 417 (a) (b) figure 1: (a) sem image (b) edx analysis of waste glass powder figure 2: bet surface area plot of waste glass 644 and 694.53 cm−1 correspond to stretching vibrations of si– o–al and m–x [3]. the peaks at 3444.12, 3563.40, 3605.19 and 3838.03 cm−1 indicate the –oh group from the silanol (si– oh) or adsorbed water molecule trapped within the silicate structure of the waste glass particles [24]. in general, the various peaks in the spectra confirmed the presence of silicates, carbon, metal oxides, and metal halides in the waste glass particles [3]. these functional groups played crucial role in the overall adsorption of analytes at the sorbent surface through electrostatic interaction, precipitation, etc. 3.3. xrd pattern of waste glass xrd pattern of the waste glass powder is shown in figure 4. the prominent peak at 2θ = 27.1◦ corresponds to the silicasilica bond. smaller peaks at 2θ = 22.5◦, 29.5◦, 35.2◦, 37.3◦, 40.4◦, 44.3◦, 51.5◦, 60.2◦, and 68.3◦ are due to the presence of other mineral components including sodium oxide, calcium oxide, etc in the waste glass [3]. this result agreed with other previous studies on waste glass [25]. figure 3: ftir spectra of waste glass powder figure 4: xrd pattern of waste glass powder. 3.4. result of adsorption studies 3.4.1. effect of temperature the effect of temperature on the adsorption efficiency is presented in figure 5a. increasing the temperature of the medium from 30 − 70◦c resulted in increase in the % adsorption efficiency of analytes (cv dye, pb2+ and cd2+ metal ions). maximum % adsorption efficiency was attained at 70◦c in all the 417 k. o. sodeinde et al. / j. nig. soc. phys. sci. 3 (2021) 414–422 418 analytes. this might be due to the increase in the average kinetic energy, collision frequency and higher diffusion rate of the analytes to the porous glass powder surface at higher temperature. aniagor and menkiti [7] reported similar observation at elevated temperature using lignified bamboo isolate for the removal of crystal violet dye effluent. 3.4.2. effect of ph the degree of acidity or alkalinity of the medium plays crucial role in the overall adsorption process by facilitating precipitation, co-precipitation and sorption processes through electrostatic interaction (attraction or repulsion) of charges between the adsorbent and adsorbate at the surface. for this study, the effect of ph was studied between ph 2-10 at 40◦c, 0.5 g adsorbent dosage for 30 min. in the case of cv dye, a reduction in % adsorption efficiency was observed as the ph progressed from 2-10 (figure 5b). the silicate glass surface (adsorbent) acquired a negative surface charge density in aqueous medium largely due to the dissociation of the silanol (si–oh) group. the observed decrease in % adsorption efficiency might be due to an increase in electrostatic repulsion between the negatively charged –oh group of the silanol (in the adsorbent) and the –oh group of the aqueous medium of equal ionic size and charge at higher ph. these negatively charged –oh ions thus occupy position far away as possible from the adsorbent surface to minimize electrostatic repulsion. consequently, less negatively charged species are available at the adsorbent surface for adsorption (via electrostatic attraction with the cationic cv dye) and hence the reduction in % adsorption efficiency. unlike the case of cv dye, increasing the ph of the medium resulted in gradual increase in the adsorption efficiency of the heavy metals (pb2+ and cd2+). at higher ph (alkaline medium), precipitation of pb2+ and cd2+ ions readily occur on the glass (adsorbent) surface thus favouring higher adsorption efficiency. this implied that the energy barrier required for precipitation is much less than electrostatic repulsion between the –oh group of the silanol (adsorbent) and the –oh group of the aqueous alkaline medium. the observed variation in ph of this work is in contrast to the earlier studies on cv dye removal using treated ginger waste [26]. 3.4.3. effect of adsorbate concentration adsorption experiments were carried out at a concentration range of 1-20 ppm with an adsorbent dosage of 0.5 g of waste glass adsorbent at 30◦c for 30 min (figure 6a). the result showed a gradual increase in the % adsorption efficiency as the adsorbate concentration increased for the heavy metal ions and the cv dye. this might be due to large surface area (with sufficient active sites), retardation of resistance towards adsorbate uptake, which increase diffusion to the surface and relatively small adsorbent pore size for effective trapping of adsorbate ions [9]. at 15 ppm, % maximum adsorption efficiency (equilibrium) was attained for the pollutant (adsorbate) molecules signaling the maximum saturation at the adsorbent surface. above 15 ppm, the adsorbent surface became supersaturated with the pollutant molecules hence the slight reduction in the adsorption efficiency. 3.4.4. effect of adsorbent dosage the study of the adsorbent dosage effect was important in order to ascertain the minimum possible dosage amount with the maximum adsorption efficiency. from figure 6b, it was observed that the adsorption efficiency increased progressively as a function of the waste glass dosage (0.1-1.0 g). this is due to the fact that more porous active sites became more readily accessible to the adsorbate ions (pb2+, cd2+ ions and cv dye) for surface adsorption. maximum % adsorption efficiency of 75%, 94% and 97.5% was recorded for cv dye, pb2+ and cd2+ ions, respectively. 3.4.5. regeneration efficiency of waste glass adsorbent the regeneration efficiency is a measure of recyclability and stability of an adsorbent per number of consecutive adsorption cycle. for this study, the waste glass adsorbent was investigated for three consecutive runs for the removal of cv dye, pb2+ and cd2+ ions (figure 7). the result showed a slow, steady reduction in regeneration efficiency of the adsorbent over the three runs in the following order: cd2+ >cv dye > pb2+. % regeneration efficiency for the first repeated cycle is 78%, 72% and 61% for cd2+, cv dye and pb2+ ions respectively. this reduced to 57%, 50% and 45% for cd2+, cv dye and pb2+, respectively after the third cycle. the relative stability and efficiency of the waste glass adsorbent might be due to its relatively large pore size, specific surface area and pore volume which facilitated easy adsorption and separation of adsorbates at the porous glass surface. 3.5. adsorption isotherm and kinetic models figure 8 shows the plots obtained for the langmuir and freundlich adsorption isotherm models for the pollutants; cv dye, pb2+ and cd2+ ions. the slopes, intercepts and the corresponding constants of the models are presented in table 1. in all the cases studied, the regression (r2) values are much relatively higher for the freundlich isotherm model than the langmuir isotherm model indicating that the freundlich model is more favoured by the adsorption process. this implied that the uptake of cv dye and the heavy metal analytes occurred on a heterogeneous surface by multilayer adsorption and there is a non-uniform distribution of the heat of adsorption over the porous waste glass surface. based on the foregoing, the separation factor (rl); a useful non-dimensional parameter in isotherm studies was determined. rl described the suitability of the porous waste glass powder and its affinity towards cv dye, pb (ii) and cd (ii) ions. rl value was calculated using: rl = 1/(1 + klc0), (7) where c0 is the initial concentration and kl signifies the langmuir constant. four probabilities exist for the value of rl : rl > 1.0, rl = 1, 0 < rl < 1, and rl = 0, indicating unfavorable, linear, suitable, and irreversible degrees, respectively. from table 1, the calculated rl values are 0.19, 0.41 and 0.33 for cv dye, pb (ii) and cd (ii) metal ions, respectively, indicating the suitability of the adsorption process using porous waste 418 k. o. sodeinde et al. / j. nig. soc. phys. sci. 3 (2021) 414–422 419 (a) (b) figure 5: effect of: (a) temperature on the adsorption efficiency of waste glass (stirring speed = 300 rpm, contact time = 30 min, dosage = 0.5 g, initial concentration = 5 ppm each of cv dye at 592 nm, pb2+ and cd2+), (b) ph on the adsorption efficiency of waste glass (temperature = 30◦c, stirring speed = 300 rpm, contact time = 30 min, dosage = 0.5 g, initial concentration = 5 ppm each of cv dye at 592 nm, pb2+ and cd2+). (a) (b) figure 6: (c) effect of initial concentration on the adsorption efficiency of waste glass (temperature = 30◦c, stirring speed = 300 rpm, contact time = 30 min, dosage = 0.5g). (d): effect of dosage on the adsorption efficiency of waste glass (temperature = 30◦c, stirring speed = 300 rpm, contact time = 30 min, concentration = 5 ppm). table 1: adsorption isotherms and separation factor parameters adsorbate langmuir isotherm model constants freundlich isotherm model constants separation factor (rl) cv dye kl = 11.57 qm = 12.34 mg/g r2 = 0.895 kf = 12.08 n = 3.66 r2 = 0.991 0.19 pb2+ kl = 1.50 qm = 166.7 mg/g r2 = 0.972 kf = 89.54 n = 1.78 r2 = 0.985 0.41 cd2+ kl = 1.88 qm = 66.67 mg/g r2 = 0.936 kf = 422.67 n = 0.65 r2 = 0.952 0.33 glass powder. we have also studied the kinetics of adsorption of the three pollutants onto the surface of the waste glass using the pseudo-first and second order kinetic models (figure 9). the pseudo-first and -second rate constants derived from fig419 k. o. sodeinde et al. / j. nig. soc. phys. sci. 3 (2021) 414–422 420 figure 7: regeneration efficiency of waste glass. (a) (b) figure 8: (a) langmuir and (b) freundlich adsorption isotherms ure 9 are summarized in table 2. the data in table 2 suggested that the rate of adsorption of the three pollutants is faster with the first order kinetic and that the adsorption processes conform more to the first order kinetics than the second order kinetics (from the regression values) implying a favourable physisorption adsorption mechanism of the pollutants. (a) (b) figure 9: pseudo-first (a) and pseudo-second (b) order kinetic plots. table 2: pseudo first order and second order kinetic model parameters. parameter cv dye cd (ii) pb (ii) k1(min−1) 0.0065 0.0491 0.0166 r2 0.9066 0.8688 0.9711 k2(g · mg−1min−1) 0.0007 0.0471 0.0036 r2 0.8999 0.8294 0.9525 3.6. treatment of textile effluent practical application of the waste glass powder adsorbent was investigated in the treatment of textile dye effluent. the cod, cv dye, pb2+ and cd2+ ions levels of the effluent were determined before and after treatment of the wastewater. the cod is a measure of the amount of oxygen required for chemical oxidation of organic pollutants in an effluent (e.g. textile wastewater). the result showed a significant reduction in the cod value from 694.26 ± 0.21 mg/l to 70.34 ± 0.08 mg/l after treatment representing 89.87% decrease in cod level thus affirming its pollutant removal capacity. the result of this work is comparable to previous studies on treatment of textile effluents using different adsorbents [20, 27]. exposure to high levels of cadmium from untreated effluent discharge had been reported as a major cause of “itai-itai” disease; an ailment characterised by kidney malfunctioning, osteomalacia, marked decalcification, etc [28]. upon treatment of the textile effluent 420 k. o. sodeinde et al. / j. nig. soc. phys. sci. 3 (2021) 414–422 421 with waste glass powder adsorbent, % adsorption efficiency of 81.69%, 82.2% and 89.5% was achieved for cv dye, pb (ii) and cd (ii) ions, respectively (figure 10). pb2+(0.004 ± 0.19 mg/l) and cd2+(0.001 ± 0.14 mg/l) concentrations obtained after treatment are relatively much lower than the 0.01 mg/l and 0.003 mg/l world health organisation recommended values for pb (ii) and cd (ii) metal ions respectively in drinking water [6]. figure 10: adsorption efficiency of waste glass for the removal of cv dye at 592 nm, pb2+ and cd2+ ions in a textile dye wastewater (effluent volume= 40 ml, stirring speed = 300 rpm, temperature = 30◦c, contact time = 30 min, dosage = 0.5 g). 4. conclusion waste glass from local environment was pulverized, characterized and successfully applied for the removal of pb2+, cd2+ ions and cv dye in waste water. structural crystallinity, surface morphology, functional groups and surface area analyses of the porous waste glass showed predominantly silica crystalline phases with highly active –oh group on the dense, coarse, porous glass adsorbent. the ph of the medium strongly influenced the adsorption process via electrostatic interaction and precipitation. the adsorption isotherm and kinetics of the process favoured freundlich isotherm and pseudo-first order models, respectively. the waste glass adsorbent displayed remarkable efficiency and effectiveness in the practical treatment of textile effluent and can thus be recommended for the removal of other environmental pollutants. references [1] a. macfarlane & g. martin, “a world of glass”, science 305 (2004) 1407. 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the adsorbents surface features were profiled using scanning electron microscopic (sem) and ph point of zero charge (phpzc) analyses. the prospects of the adsorbents for the removal of trimethoprim from water were verified. the adsorption processes were performed under different experimental conditions. the adsorption isotherm, the kinetics, and the thermodynamics were studied; and the data fitting output revealed that both chemisorptions and physisorption occurred. surface and pore diffusion played active role in the adsorption of tmp by the adsorbents. the optimum conditions for adsorption of tmp by the adsorbents were ph at slightly acidic to neutral medium and temperature at room temperature. the fitting isotherm models were: langmuir (r2 = 0.993) for the zinc-chloride-activatedcarbon, temkin (r2 = 0.962) for the phosphoric-acid-activated-carbon, and the kinetics: pseudo-second order (r2 = 0.997) for both. the maximum monolayer adsorption capacities of the adsorbents for tmp was 4.115 and 6.495 mg/g, respectively. the thermodynamic parameters determined suggested feasibility, spontaneity, and endothermicity of the adsorption processes. the results reveal that the adsorbents were good prospects for the removal of tmp from water. doi:10.46481/jnsps.2021.320 keywords: carbonization, activation, adsorption, daniellia-oliveri sawdust, activated carbon, trimethoprim. article history : received: 28 july 2021 received in revised form: 15 september 2021 accepted for publication: 16 september 2021 published: 29 november 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: e. etim 1. introduction necessities for procurement of means of livelihood, maintenance of quality life and attainment of luxurious status had prompted man to recreate the environment. advancements in various frontiers of human endeavors had created lots of pollutants and contaminants. sources of pollutants and contaminants in the environment include: agricultural practices, medicine/pharmaceutical indus∗corresponding author tel. no: +2348035065456 email address: giwa1010@gmail.com (a. a. giwa ) tries, automobile/mechanical works, transportation, construction, manufacturing, tourism, research, productions and so on [1]. the propensity of these hazardous substances into the lithosphere and hydrosphere was high as a result of their proximity to man. most of these anthropogenic toxic materials upset the delicate balance of the ecosystem. the resultant of this upsetting was untoward environmental episodes. the advancement of man in various facets has led to new products, generation of new by-products and new wastes. some of new wastes had been identified as emerging contaminants of concerns (eccs). the eccs were of different categories: 344 s. a. adesokan et al. / j. nig. soc. phys. sci. 3 (2021) 344–353 345 nano-materials [1] pesticides, herbicides, personal care products and pharmaceuticals (pcpps) [2]. the pharmaceuticals are designed to fight specific ailment and/or microorganisms that cause diseases. when administered through involuntary exposure, the pharmaceuticals had been appraised to induce an array of health and ecological complications [3]. some of the identified ecological complications include and not limited to: hermaphroditic feature in fish; regeneration impairment in injured toads; and drug resistance in microbe populations [4, 5, 6]. the environmental levels of pharmaceuticals are generally low to cause acute health effect on human. however considering chronic exposure, precautionary principle approach should be adopted to forestall incidence of health complications. the involuntary presence of pharmaceuticals should be prevented in public resources like drinking water and air to protect the vulnerable groups like children and populations with compromised immune system from involuntary exposure. pharmaceuticals were of different categories classified by their functions. one of the main categories with high dispensability was antibiotics e.g. trimethoprim (tmp) [7, 8, 9]. high dispensability of antibiotics was responsible for their high levels in the environment. presence of antibiotics in the environment was highly undesirable as they induce drug resistance in microorganisms and this is a major factor of pandemic [10, 11]. tmp is an antibacterial and anti-malarial drug. it has reported half-lives in the range 5 100 days. it is insignificantly photodegraded, which suggests a relative persistence [12] in the environment. tmp is mutagenic, teratogenic, embryotoxic and folate antagonism [13]. hence tmp should be removed from water resources. the researchers are working to develop technologies [14] and strategies [15, 16, 17, 18] to remove various forms of pollutants and contaminants, including pharmaceuticals from water. recently the most sustainable technology so far developed was adsorption process [19, 20, 21, 22, 23, 24, 25]. the trending strategy was the use of agro-wastes as adsorbents for the removal of pollutants and contaminants. in nigeria, wood waste like sawdust remains one of the high volumes, posing great challenge to manage. in 2010, nigeria generated over 1,000,000 m3 (353,146.667 tonnes) [26] of sawdust and most managed by open air combustion. this combustion method led to generation of cox, nox, sox: radiation forcing substances [26]. conversion of the sawdust to highly demanded product like adsorbent is both economically and environmentally sustainable strategy. in this work, daniellia-oliveri sawdust, a high volume agroprocessing waste in nigeria and not yet reported for any usage, was processed and used to remove from contaminated water. 2. materials and methods 2.1. materials some of the materials used include daniellia–oliveri sawdust (collected from a local factory, washed, dried and segregated), analytical grade zncl2, h3 po4, (purchased from local chemical dealer) tmp (primary standard, >98.5% purity) (supplied bond chemical industries limited, awe, oyo) and hcl. the equipment made used of were furnace (carbolite aaf 1100), ultraviolet-visible (uv) (b-uv 1800pc) spectrophotometer, ph meter (jenway 3520), and scanning electron microscope (sem) (aspex 3020) and oven (carbolite). 2.2. tmp the structural formula and the properties of trimethoprim are given in figure 1 [27] and table 1 [28] below. figure 1: trimethoprim 2.3. preparation of adsorbents the daniellia—oliveri sawdust was washed with plenty of distilled water to remove surface impurities and then sundry. the sample was dried in an oven at 70 ◦c for 72 hours; then ground and stored as raw daniellia sawdust (rds). the method was described elsewhere [29], 3g of rds was mixed with 3 ml of 1 m h3po4 and 3 ml of 1 m zncl2 each in separate crucibles. these were subjected to the furnace at 800 ◦c (carbonization and activation in self-generated atmosphere) for 5 minutes. the adsorbents were stored in air tight container as h3po4 activated carbon (pacb) and zncl2 activated carbon (zac) respectively. 2.4. characterization of adsorbents sem and phpzc analyses were carried out using methods [30] and [31] respectively to study the surface morphology and ph of point of zero charge of the adsorbents. 2.5. batch adsorption studies the triplicate optimized batch adsorption processes were carried out. the adsorption bottles containing adsorbent-adsorbate mixtures were shake on a mechanical shaker operating at 160 rpm. the effects of contact times (0 240 min), adsorbent dosages (0.05 0.5 g), phs (2, 5, 8), initial adsorbate concentrations (15 50 mg/l), and temperatures (29 60 ◦c) were determined. after the processes reached equilibrium, the mixtures were filtered using filter papers and the filtrates analyzed at 282 345 s. a. adesokan et al. / j. nig. soc. phys. sci. 3 (2021) 344–353 346 table 1: properties of trimethoprim common names: trimethoprim, trimethoprime, trimethoprimum, trimetoprima, trimpex iupac names 5-[(3,45-trimethoxyphenyl)methyl]-2,4-pyrimidinediamine molecular formula c14 h18 n4o3 molecular weight 290.32 g/mol un number 2811 max λ 282 nm solubility 400 mg/l (25◦c) toxicity vomiting, headache, mental depression, confusion, bone marrow depression pka 7.12 description antibacterial, antimalaria interaction sites π−π, n −π, -nh2, ≡ n: nm with the uv spectrometer to determine the absorbances of trimethoprim in solutions after adsorption. the absorbances were converted to corresponding concentrations using standard curve equation. each experiment was performed in triplicate. equations (1) and (2) were used to determine amount of adsorbate adsorbed per unit mass of adsorbent and percentage adsorbate removal respectively: qe = (ci − ce)v m (1) % adsorbate removal = (ci − ce)v ci × 100 (2) where qe, ci, ce, m and v are the amount of trimethoprim adsorbed by the adsorbent at equilibrium (mg/g); the initial trimethoprim concentration (mg/l); the trimethoprim concentration at equilibrium (mg/l); the mass of the adsorbent (g); and the volume of the solution (l) respectively. 2.6. adsorption isotherms table 2 contains selected isotherm models used to process experimental data and then described the adsorption processes. the langmuir model: a plot of ce/qe versus ce gives a straight line with intercept: 1 /k · q0; and slope: 1/ q0. langmuir isotherm equilibrium parameter, rl: rl = 1 1+klci [39]. ci = highest initial adsorbate concentration (mg/l); rl>1 = unfavourable; rl=1=linear; favorable 0<rl<1=favourable and rl = 0 = irreversible. the freundlich model: a log-log plot should yield an intercept of log kf and a slope of 1/n. n defines isotherm shape; 1/n = 0 = irreversible; (0<1/n<1) = favourable; unfavourable (1/n>1); n = 1=linear; n > 1 = physical process and n < 1= chemical process. the temkin model: kt =temkin isotherm equilibrium binding constant (l/g); bt = table 2: isotherm equations model linearised formulae reference halsey harkin-jura langmuir lnqe= 1 n ln k− 1 n lnce 1 q2 = b a − 1 a logce ce qe = 1k·q0 + ce q0 [32] [33] [34] freundlich ln qe = ln kf + 1 n ln ce [35] temkin qe = b ln kt + b ln ce [36] redlich-peterson log ceqe = log kr + βrt log ce [37] dubinin– radushkevich lnqe = ln(qs ) − (kadε2) [38] temkin isotherm constant; r= universal gas constant (8.314 j/mol/k); t= temperature at 298 k; b = constant related to heat of sorption (j/mol). the redlich-peterson isotherm: ce = equilibrium concentration; qe = adsorption capacity of the adsorbent; β = desorption constant and kr = r-p isotherm constant (g/l). the dubinin-radushkevich model: ε = rt ln[1 + 1ce ] ; e = [ 1√ 2bdr ]; kad = bdr; e<8 kj/mol (physical adsorption); e = 20–40 kj/mol (chemical adsorption) [38]; e = 8–16 kj/mol (ion-exchange) [40]; qe = amount of adsorbate on the adsorbent at equilibrium(mg/g); qs = theoretical isotherm saturation capacity (mg/g); kad = dubinin-radushkevich isotherm constant (mol2/kj2). 346 s. a. adesokan et al. / j. nig. soc. phys. sci. 3 (2021) 344–353 347 2.7. the kinetics samples were taken from adsorption bottles at time intervals and the amounts of adsorbates were measured. the amount of adsorbed adsorbate at time t, qt (mg/g), was calculated using equation 3 and kinetic parameters were calculated using kinetic equations presented in table 3. qt = (c0 − ct) v m (3) the pseudo-first order: qe and qt are the amounts of the dye adsorbed (mg/g) at equilibrium and at time t (min), respectively, and k1 is the rate constant adsorption (min−1). a plot of log(qe − qt) versus t gives a slope of ( − k1 2.303 ) and intercept of logqe . the pseudo-second order: h = k2 . q2e ; qe is the amount of the solute adsorbed at equilibrium per unit mass of adsorbent (mg/g) , qt is the amount of solute adsorbed (mg/g) at any given time, t (min) and k2 is the rate constant for pseudosecond-order adsorption (g/mg/min) and h, known as the initial sorption rate. the values of qe and k2 can be obtained from the slope and intercept of the plot of t/qt versus t respectively. if the sorption follows pseudo-second order, h, is described as the initial rate constant as t approaches zero. the elovich: α is the initial adsorption rate (mg/g. min), β is the desorption constant (g/mg) and qt is the amount of solute adsorbed (mg/g) at any given time, t (min). a plot of qt versus ln t gives a straight line with intercept 1 β ln(αβ) and slope ( 1 β ) . intra-particle diffusion: qt (mg/g) = quantity of adsorbate adsorbed at time, t and kid (mg/gh0.5) = intra-particle diffusion constant. a graph of qt versus t0.5 gives a straight line with slope, kid , and intercept, x. x defines the thickness of the boundary layer. the boyd kinetic: b = π 2 di r2 ; b is a constant; di (m 2/s) is diffusion coefficient; r is radius of adsorbent particle; f is the fraction of adsorbate adsorbed at given time, t, and bt is a function of f. if graph of [–0.4977 – ln (1 − f)] vs time, t, is linear and intercept = 0, then the determining step of the adsorption process is intraparticle diffusion; but if intercept , 0, the determining step may be film diffusion. 2.8. thermodynamic study experiments were performed at 29, 40, 50, 60 and 70 ◦c to establish the effect of temperature on the adsorption capacities of zac and pacb for the trimethoprim. the thermodynamic table 3: kinetic equations kinetic equation reference pseudo-first order log(qe − qt) = logqe − k1 2.303 .t [41] pseudosecond order t qt = 1h + t qe [42] elovich qt = 1 β ln(αβ) + ( 1 β ) lnt [43, 44] intra-particle diffusion qt = kid t0.5 + x [45] boyd −0.4977−ln(1 − f) =bt [46] parameters of the adsorption were determined using the equations (4) and (5). ∆g0 = −2.303rt log k (4) log ( cae ce ) = ∆s 0 2.303r − ∆h0 2.303rt (5) k = ( cae ce ) ; cae = (ci – ce) (mg/l) = amount of trimethoprim adsorbed on adsorbent at equilibrium; ce (mg/l) = amount of trimethoprim in solution at equilibrium; r = gas constant (8.314 j/mol/k); t = temperature (k). the enthalpy change (∆h), the entropy change (∆h) and change in satndard free energy (∆g) were calculated. the spontaneity of the processes was thereby determined. 3. results and discussion 3.1. morphology of zac and pacb figure 2 showed characteristic surface morphology of the adsorbents. carbonization and activation processes impacted the adsorbents’ surfaces with characteristic morphology and chemical profile. using gwyddion 2.23-1 application for sem image processing, the average pore diameters respectively of zac and pacb were approximately 30 and 55 nm. high temperature of carbonization removed volatile compounds in the matrix of raw sawdust thereby created pores. high temperature could also break down cellulosic contents of the precursor sawdust and aided aromatization of carbon chains. high temperature chemical activation effected dehydration and decarboxylation of sawdust. it could be deduced from the changes in the surface morphology after adsorption that pores were active in adsorbing trimethoprim molecules. 3.2. determination of equilibrium time and effect of contact time on trimethoprim adsorption the amount of tmp adsorbed per unit mass (q) of each of zac and pacb increased rapidly within first 40 min. the q thereafter increased slowly and insignificantly over next two hours for both systems. after 60 min, almost all the available adsorption sites had been occupied for both systems. the q at time t (qt) for the tmp-zac system was 1.123 mg/g within first 60 min while it was 0.272 mg/g for the 180 min afterwards. the qe for the tmp-pacb system was 0.996 mg/g in first 60 min while 0.204 mg/g for the 180 min afterwards (figure 3a). these increments for the next 180 min after first 60 min were economically insignificant. the economic time of adsorption for both systems was 60 min. 3.3. effect of adsorbent dose on trimethiprim (tmp) adsorption the increase in mass of each of zac and pacb led to decreased amount of tmp adsorbed per unit mass during adsorption processes. the increased mass of the adsorbents provided more sites of adsorption for constant amount of tmp molecules. zac and pacb adsorbed equal amounts of the 347 s. a. adesokan et al. / j. nig. soc. phys. sci. 3 (2021) 344–353 348 (a) (b) (c) (d) figure 2: (a) pacb (before adsorption); (b) pacb (after adsorption); (c) zac (before adsorption); (d) zac (after adsorption) at magnification × 500. tmp per unit mass at 0.15g each (at the point of intersection) (figure 3b). 0.2, 0.3 and 0.4 g of zac adsorbed 68 %, 82 % and 87 %; and pacb adsorbed 65 %, 76% and 75% of tmp respectively at equilibrium time. the equilibrium dose was reached at 0.3 g for both adsorbents. the economic dose for zac was 0.3 g while 0.2 for pacb. 3.4. effect of ph on trimethoprim adsorption the phpzc is the ph of solution at which electrical charge of a solid surface in the solution is neutral. the pka of trimethoprim and the phpzc of the adsorbents are parameters whose electrical charge changes with ph of solution. the maximum attraction between the adsorbate and the adsorbents would take place when electrical charges on them are opposite. tmp is a basic compound with pka 7.12. at ph above 7.12, tmp was neutral while at ph below 7.12 tmp was positively charged (the hypothesized interactions were presented in equations 6 to 10). at ph below 7.12, the oxygen and nitrogen atoms on tmp donated lone pairs of electrons to the protons in solution and tmp became positively charged. at ph above 7.12, the lone pairs of electrons on tmp oxygen and nitrogen atoms repelled lone pairs of electrons on hydroxyl groups in solution and tmp remained neutral. at about ph 7, tmp was at the threshold of changing from positively charged (+) to neutral (0), zac (phpzc 7.6) positively charged (+) while the pacb (phpzc 6.6) negatively charged (-). at ph below 6, the adsorbents and trimethoprim were all positively charged and the repulsive effect de-enhanced adsorption. at ph above 8, lone pairs of electrons on neutral tmp repelled negatively charged adsorbents’ surface and chemisorption was de-enhanced. the maximum interactions occurred at slightly acidic to neutral ph (ph 6) where electrical charges on the adsorbents and trimethoprim were opposite (figure 3c). at ph 2.5, zac adsorbed 1.262 mg/g of tmp; at ph 5, 1.266 mg/g and at ph 8, 1.25 mg/g. and for pacb, at ph 2.5, 1.164 mg/g of tmp was adsorbed, at ph 5, 1.215 mg/g and at ph 8, 1.138 mg/g. tmp (pka) ph>7.12 −→ tmp(o) (6) meö ≡ n:(t mp) + oh− → meö (7) ≡ n:(t mp) ↔ oh− (neutralt mp) tmp (pka) ph<7.12 −→ tmp(+) (8) meo ≡ n:(tmp) + 2h+ → meoh+ (9) ≡ nh+ ( positively charged tmp ) 3.5. effect of temperature on trimethoprim adsorption tmp is a biphenyl compound with functional group branches. the bulky structure of tmp may constitute stearic hindrance to 348 s. a. adesokan et al. / j. nig. soc. phys. sci. 3 (2021) 344–353 349 pore accessibility. at elevated temperatures, tmp molecules acquired more kinetic energy which led to frequent knocking with the adsorbents’ pore openings. the impact of the knocking might ‘mend and tend’ the molecules or wear the pore openings and eventually forced the molecules into the pores. the higher the temperature, the higher was the kinetic energy and the contact between molecules and pores. at equilibrium, 1.6 mg/g (64 %) and 1.904 mg/g (76 %) of tmp were adsorbed by zac at 29 ◦c (room temperature) and 60 ◦c respectively. also 1.196 mg/g (72 %) and 1.326 mg/g (80 %) adsorbed respectively at 29 ◦c and 60 ◦c by the pacb (figure 3d). the enhanced adsorption of tmp on zac and pacb as a result of increase in temperature was economically unfavourable. from economic perspective, room temperature would be chosen considering costs of heating and cooling the system after adsorption before discharge, or environmental impact of discharged warm/hot wastewater. it would be less expensive to subject the tmp treated water to secondary adsorption process. 3.6. effect of initial tmp concentrations the qe (mg/g) of the adsorption of tmp onto zac and pacb increased as the initial tmp concentrations increased. this was due to the more tmp molecules in the solutions which were in contact with constant active adsorption sites on the adsorbents. for the tmp-zac system, at 15, 20, 25, 30, 35, 40 and 50 mg/l, 82.6 %, 83.5 %, 82.8 %, 81 %, 79.5 % 75 % and 70 % of tmp were adsorbed respectively. and for the tmppacb system, at 15, 20, 25, 30, 35, 40 and 50 mg/l, 75 %, 77.9 %, 81.7 %, 79.9 %, 78.8 %, 80 % and 75 % were adsorbed respectively (figure 3e). a unit mass of zac had highest removal efficiency for tmp at 20 mg/l of solution while pacb at 25 mg/l of solution. 3.7. adsorption isotherms for tmp adsorption the isotherms applied to describe the adsorption of tmp onto zac and pacb were langmuir, freudlich, temkin, redlichpeterson, dubinin-radushkevich, harkin-jura and halsey. the isotherms that best fit were langmuir (r2=0.993) and temkin (r2=0.962) for tmp-zac adsorption and tmp-pacb respectively (table 4). these suggested that tmp-zac adsorption system was monolayer and tmp-pacb multilayer. the langmuir isotherm equilibrium parameter, rl values: 0.108 and 0.206 for the tmp-zac and the tmp-pacb meant that both processes were favorable. the adsorption affinity (kl) for zactmp process was higher, which implied that it was more favorable compare to pacb-tmp process. the qmax for zac was 4.115 mg/g and pacb 6.495 mg/g (table 6). the multilayer coverage exhibited by the tmp-pacb process explained its higher qmax. also wider pores exhibited by pacb (figure 2) might be more accessible to tmp molecules than for zac. comparison of qmax of some adsorbents reported in literatures indicated that zac and pacb were better adsorbents for tmp adsorption from aqueous solution. the 1/n as determined from the freundlich isotherm for the tmp-zac and the tmp-pacb processes were 0.472 g/l and 0.636 g/l respectively. these further confirmed the processes to be favorable. both processes were physical because n greater than 1 for table 4: isotherm parameters for tmp adsorption onto the zac and the pacb isotherm zac pacb langmuir q0 (mg/g) 4.115 6.495 kl (l/mg) 0.165 0.077 rl 0.108 0.206 r2 0.993 0.913 freundlich kf (mg/g) 0.849 0.653 n (l/g) 2.120 1.572 1/n 0.472 0.636 r2 0.959 0.955 temkin kt (mg/l) 1.306 1.673 bt (kj/mol/g.l) 2.531 0.664 β (l/g) 0.992 1.500 r2 0.988 0.962 redlich-peterson kr (g/l) 1.178 1.528 βr (g/mg.k) 0.0017 0.0013 r2 0.966 0.879 dubinin-radushkevich qs (mg/g) 2.821 3.320 kad (jl/mol.mg) 2×10−6 1×10−8 e (kjl/mol.mg) 0.50 2.5 r2 0.966 0.921 harkin-jura qe (mg/g) 1.287 1.220 a (mg3/g3) 1.783 1.745 r2 0.840 0.908 halsey n (g/l) -2.105 -1.572 k (mg/l) 1.415 0.668 r2 0.959 0.957 349 s. a. adesokan et al. / j. nig. soc. phys. sci. 3 (2021) 344–353 350 (a) (b) (c) (d) (e) figure 3: effect of (a) contact time, (b) adsorbent dose, (c) solution ph, (d) temperature and (e) initial adsorbate concentration on the adsorption of tmp onto zac and pacb each (table 4). adsorption intensity (n) was higher for zactmp process which in agreement with explanation given under langmuir isotherm. 1/n, a function of surface heterogeneity, was higher for pacb-tmp system. positive values of bonding energy (bt ) obtained from temkin isotherm for both processes indicated endothermic processes. the adsorption energy, e (kj/mol), derived from dubinin-radushkevich isotherm for the tmp-zac and the tmp-pacb processes were 0.50 and 2.5 respectively. these values described both processes as physical. halsey isotherm described a heterogeneous adsorbent surface and multilayer adsorption system. relatively close values of adsorption coefficients of both halsey and temkin isotherms further established pacb-tmp system as multilayer adsorption and pacb had heterogeneous surface.as could be deduced from the coefficient values of the isotherm and kinetic (section 3.9 below) models, the adsorption of tmp by zac and pacb involved both chemisorption and physisorption. chemisorption involves reaction among functional groups on both the adsorbents (zac and pacb) and the adsorbates (tmp). hydroxyl group would like form hydrogen bond with adsorbents’ n and 350 s. a. adesokan et al. / j. nig. soc. phys. sci. 3 (2021) 344–353 351 (a) (b) figure 4: van’t hoff plot of log k vs 1/t adsorption of tmp onto (a) zac and (b) pacb . table 5: kinetic parameters for tmp adsorption onto the zac and the pacb kinetic zac pacb qe, ex p (mg/g) pseudo-first order 1.395 1.200 qe (mg/g) 0.670 0.581 k1 (min−1) 0.014 0.014 r2 0.964 0.920 pseudo-second order qe (mg/g) 1.425 1.230 h (min−1) 0.107 0.097 k2 0.053 0.064 r2 0.997 0.997 elovich a 0.535 0.445 β (min.g/mg) 4.40 5.05 r2 0.939 0.969 intraparticle diffusion x (mg/g) 0.594 0.518 kid (mg/g.min1/2) 0.061 0.053 r2 0.877 0.888 boyd x 1.475 1.617 b (min.g/mg) -0.004 0.003 r2 0.211 0.017 table 6: comparison of maximum monolayer adsorption capacities of different adsorbents for tmp adsorbent adsorbent capacity (mg/g) reference gac 0.51 [47] pac 0.51 [47] clay 47 [48] bhb 0.508 [49] lzb 0.294 [49] zac 4.115 this work pacb 6.495 this work gac – granular activated carbon pac – powdered activated carbon sediment bhb; sediment lzb o bearing functional groups, and vice versa: nh2 − tmp − nh2+ho − zac/pacb→h2n − tmp (10) − hn· · ·ho − zac/pacb multilayer adsorption of tmp onto pacb was as a result of reaction between electron withdrawing and electron donating functional groups on already adsorbed tmp and free adsorbate molecules in solution: h2n − t mp − hn· · ·ho − pacb+h2n − tmp→ (11) →tmp−h2n· · ·h2n − tmp−h2n· · ·oh − pacb π-π and n-π are electron density centers capable of facilitating electrostatic interactions between tmp molecules and the adsorbents through resonance of conjugated double bonds and aromaticity of benzene rings: tmp − n−−c =n+h2+o −c6h6−pacb/zac→ (12) tmp−n+· · ·o−c6h6−pacb/zac 3.8. kinetic studies of adsorption of tmp onto zac and pacb the data of adsorption of tmp on zac and pacb were tested with0 five different kinetic models: pseudo-first order, 351 s. a. adesokan et al. / j. nig. soc. phys. sci. 3 (2021) 344–353 352 table 7: thermodynamic parameters of the adsorption of tmp onto the zac and pacb ∆g0 (kj/mol/k) ∆s0 (j/mol/k) ∆h0 (kj/mol/k) 302 313 323 333 zac -1.444 -2.39 -2.64 -3.22 +54.60 +14.99 pacb -2.342 -2.74 -2.90 -3.76 +42.16 +10.46 pseudo-second order, elovich, intraparticle and boyd. both tmp-zac and the tmp-pacb processes data fitted pseudosecond order equation with r2 = 0.997 (table 5). the rate of adsorption in both processes depended on the states of both the adsorbate and the adsorbents. the initial rate constants for zac and pacb, h (mg/g/min), were 0.107 and 0.097 respectively, implied that rate at which tmp adsorbed on to the zac was higher initially. however, overall rate constant of pacbtmp process was higher. high adsorption coefficient (r2) for elovich model suggested a level of chemisorptions for both systems. higher elovich r2 for pacb-tmp process also indicated considerable multilayer coverage compare to zac-tmp process. however, earlier it was explained from the isotherm data fitting that adsorption of tmp by the adsorbents were both chemisorptions and physisorption. high coefficient values for pseudo first order kinetic and relatively high r2 for intraparticle diffusion model were pointers to physical adsorption. surface and pore diffusions played important role in the adsorption of tmp by zac and pacb. pacb having wider average pore diameter than zac could easily accommodate bulky tmp molecules and hence higher qmax. diffusion of tmp molecules at the surface of the adsorbents occurred rapidly while diffusion down the adsorbents’ pores took place steadily until equilibrium is attained. 3.9. thermodynamic study of the tmp adsorption the functions of temperature, the standard enthalpy (∆h0), the standard entropy (∆s0) and the standard gibb’s free energy (∆g0) of the adsorption processes were investigated for spontaneity and feasibility. the van’t hoff plot of graph of log k versus 1/t gave the slope and the intercept as ∆h0 and ∆s0 respectively (table 7, figure 4). the negative values for ∆g0 at all investigated temperatures indicated that the adsorption of tmp onto zac and pacb were spontaneous. the positive value of ∆s0 showed high degree of randomness of the processes while positive value of ∆h0 indicated endothermic processes. 4. conclusion this study verified the prospects of the processed daniella— oliveri sawdust as adsorbents for adsorption of tmp from aqueous solution. the economic time of adsorption of tmp by both adsorbents was 60 minutes while economic doses of adsorbents were 0.3 and 0.2 g for zac and pacb respectively at 29 ◦c. the qmax of both the zac and the pacb was obtained at slightly acidic-neutral ph. the zac data fitted best into the langmuir isotherm and the pacb, the temkin while data for both fitted best the pseudo-second order kinetic. multilayer adsorption and average pore diameter were suspected to be responsible for higher 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[49] l. jia & z. hua “factors influencing adsorption and desorption of trimethoprim on marine sediments: mechanisms and kinetics”, environ sci pollut res 24 (2017) 21929. doi 10.1007/s11356-017-9693-y. 353 j. nig. soc. phys. sci. 3 (2021) 298–307 journal of the nigerian society of physical sciences optimized breast cancer classification using feature selection and outliers detection a. b. yusufa, r. m. dimab,∗, s. k. ainac adepartment of information and communication technology, usmanu danfodiyo university, sokoto state bdepartment of computer science, federal university dutsinma, katsina state cdepartment of computer science, federal university gashua, yobe state abstract breast cancer is the second most commonly diagnosed cancer in women throughout the world. it is on the rise, especially in developing countries, where majority of the cases are discovered late. breast cancer develops when cancerous tumors form on the surface of the breast cells. the absence of accurate prognostic models to assist physicians recognize symptoms early makes it difficult to develop a treatment plan that would help patients live longer. however, machine learning techniques have recently been used to improve the accuracy and speed of breast cancer diagnosis. if the accuracy is flawless, the model will be more efficient, and the solution to breast cancer diagnosis will be better. nevertheless, the primary difficulty for systems developed to detect breast cancer using machine-learning models is attaining the greatest classification accuracy and picking the most predictive feature useful for increasing accuracy. as a result, breast cancer prognosis remains a difficulty in today’s society. this research seeks to address a flaw in an existing technique that is unable to enhance classification of continuous-valued data, particularly its accuracy and the selection of optimal features for breast cancer prediction. in order to address these issues, this study examines the impact of outliers and feature reduction on the wisconsin diagnostic breast cancer dataset, which was tested using seven different machine learning algorithms. the results show that logistic regression, random forest, and adaboost classifiers achieved the greatest accuracy of 99.12%, on removal of outliers from the dataset. also, this filtered dataset with feature selection, on the other hand, has the greatest accuracy of 100% and 99.12% with random forest and gradient boost classifiers, respectively. when compared to other state-of-the-art approaches, the two suggested strategies outperformed the unfiltered data in terms of accuracy. the suggested architecture might be a useful tool for radiologists to reduce the number of false negatives and positives. as a result, the efficiency of breast cancer diagnosis analysis will be increased. doi:10.46481/jnsps.2021.331 keywords: breast cancer, machine learning, accuracy, feature selection and outliers article history : received: 05 august 2021 received in revised form: 10 october 2021 accepted for publication: 01 november 2021 published: 29 november 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: t. latunde 1. introduction cancer is a group of disorders characterized by the growth of abnormal cells that have the ability to infiltrate or spread ∗corresponding author tel. no: email address: rocinta976@gmail.com (r. m. dima ) throughout the body[1]. breast cancer is the world’s second most common disease and public health concern, especially among women, with high mortality and substantial morbidity[2]. breast cancer is on the rise in emerging countries, and a fiveyear study indicated that it is the most common malignancy[3]. according to the american cancer society, 93,600 new instances of breast cancer are diagnosed each year in africa, with 298 yusuf et al. / j. nig. soc. phys. sci. 3 (2021) 298–307 299 around 50,000 fatalities. breast cancer was diagnosed in 2.3 million women, resulting in 685,000 deaths in 2020, according to the world health organization (who, 2021). it is a disease characterized by aberrant cell proliferation in the breast[4], which is caused largely by dna mutations. breast cancer tumors are classified as either malignant or benign [5]. this classification is applied in the analysis of breast tumors, lumps, or any other abnormal development in the breast tissue. cancer that is classified as benign is typically not lifethreatening and has a greater chance of survival, whereas cancer that is classified as malignant is life-threatening [6]. a malignant tumor can develop fast, infiltrating the lymph system and encroaching on other healthy tissues in the surrounding area, causing disastrous effects; on the other hand, a benign tumor cannot grow beyond a specific size and remains confined inside its bulk. early cancer identification guarantees successful therapy and enhances the likelihood of survival [7]. scientists have attempted to pinpoint the specific cause of breast cancer since there are only a few risk factors that promote a woman’s chances of developing the disease. breast cancer risk factors include age, genetic risk, family history, obesity, gene variation, smoking and alcohol consumption. due to the small size of the cancer cell as seen from the outside, it is nearly hard to identify breast cancer in its early stages. mammography, ultrasound [8], dynamic mri[9], and elastography are the only ways to detect cancer at an early stage[10]. in many cases, clinicians would be required to read a large amount of imaging data, which would compromise accuracy. this method is extremely time-consuming and, in some cases, it incorrectly diagnoses the cancer. medical professionals continue to make this sort of diagnosis in order to see which one has the most impact. in recent years, however, machine learning (ml) [11]–[14], deep learning [15], [16], and bio-inspired computing [17] approaches have been employed in a variety of medical diagnoses. machine learning in the detection of breast cancer has been the subject of several studies [18]–[21]. several studies use different datasets collected from the university of california-irvine (uci) repository for clinical prediction of this disease. among these are wisconsin breast cancer dataset (wbcd), wisconsin diagnostic breast cancer (wdbc) and wisconsin prognostic breast cancer (wpbc) dataset to mention few. regardless of the nature of the dataset, the focus of the research is always aimed at enhancing accuracy of the prediction in order to correctly diagnose the cancer. however, despite the popularity of ml algorithms modalities proven on different breast cancer dataset, it still cannot offer accurate and consistent outcome in diagnosis unless improved with some data mining techniques [22]. among the popular dataset used is wdbc with continuous-valued data problem [23] hence finding the linearity among the features pose a difficulty [24], [25] when applied to ml algorithms thus leading to poor accuracy when applied on some algorithms. recently, in the research work carried out on wdbc dataset, study and analysis of ml algorithms were reported along with the approaches used in improving the performance of ml algorithms [25]–[27]. in their work, [25] proposed a method using clustering and noise removal on wdbc dataset before it was applied on some ml algorithms. in the research, expectation maximization (em) was used for data clustering, classification and regression trees (cart) automatically generated the fuzzy rules from the data hence causing removal of noise while the principal component analysis (pca) as a dimensionality reduction technique was used to overcome the multicollinearity issue in the data. the proposed technique was evaluated with wdbc and mammographic mass datasets; then its effectiveness was demonstrated. when compared to pca-support vector machine (pca-svm,), pca-k nearest neighbour (pcaknn) and decision tree (dt), em-pca-cart-fuzzy rulebased had the greatest accuracy of 93.2%, whereas pca-svm, pca-knn and dt had an accuracy of 86.7%, 82.3% and 92.9%, respectively. in order to improve classification accuracy of breast cancer disease, [27] applied preprocessing step on wdbc dataset. here, features were selected using gain ratio and modeled with six algorithms using 10-fold cross-validation method. the accuracy of these algorithms was: svm-linear (98.07%), knn at k=3 (97.36%), naı̈ve bayes (95.08%), j48 (98.07%), multilayer perception (98.41%) and random forest (98.77%). the results demonstrated a considerable improvement above the stateof-the-art since rf performed the best. similarly, [26] focused on integrating ml algorithms with different feature selection methods and compared their performances to identify the most suitable approach.. the selected features were correlation based feature selection (cfs), recursive feature elimination (rfe), linear discriminant analysis (lda) and pca. the ml algorithms tested were svm (using radial basis kernel), neural networks (nn) and naı̈ve bayes (nb) carried out on wdbc dataset. it was observed that svm-lda combination and nn-lda combination obtained the best performance in terms of accuracy (98.82%). in the comparative study carried out by [18] to determine how to improve classification algorithm on the wdbc dataset, the investigation was conducted on different level of cross validations and percentage of splitting the training dataset. the nb, j48, random forest (rf), smo, multilayer perceptron algorithms when trained with 85.5% of the dataset at 10-fold cross validation, the evaluation result showed the nb as 97.28%, j48 as 94.27%, rf as 95.56%, smo as 96.13% and multilayer perceptron as 96.13% accuracy. nb having the highest accuracy was further investigated with 5, 10 and 15 cross validation at 66.6% and 85.5% splitting. the result showed improvement when trained with 85.5% trained set resulting into 99% accuracy. it can be justified that such improvement exists as a result of overfitting of the training set. in a similar investigation conducted on three distinct datasets: wbdc, wdbc and coimbra by [23], the study proposed a fuzzy technique in improving the ml algorithms. in order to resolve the limitation of an existing method, where id3 algorithm was unable to classify the continuous-valued data and increase the classification accuracy of the decision tree, fuzzydbd method an automatic fuzzy database was used to design the fuzzy database for fuzzification of data in the fid3 algorithm. it was used to generate a predefined fuzzy database 299 yusuf et al. / j. nig. soc. phys. sci. 3 (2021) 298–307 300 before the generation of the fuzzy rule base. the fuzzified dataset was applied to fuzzy-id3 algorithm. the accuracy of fuzzy-id3 applied to fuzzified dataset was 94.362% when compared with non-fuzzy wdbc dataset applied to id3 (91.059%), svm (86.1%), c4.5(92.97%), nb(91.81%), rf(91.66%) and knn(92.57%). the study conducted by [28] in order to improve accuracy of ml algorithms, investigated the best features suitable for wdbc dataset. light gradient boosting model (lgbm), catboost, and extreme gradient boosting (xgb) were applied as the feature selection approaches tested on naive bayes algorithm. the findings revealed best accuracy with lgbm (97%), followed by catboost (96%) and xgb (96%). again, [29] experimented with feature selection techniques of correlation based feature selection(cfs), univariate selection (selectkbest), and recursive feature elimination (rfe). the rf was applied on these feature selection methods and evaluated on wdbc dataset. it was shown that when 5 features were picked, the rf model had the best accuracy with a cfs of 95.32%, selectkbest 94.15% and rfe 94.15%. as reviewed from the literature on the state-of-the-art approaches employed, the problem of multi linearity in the wdbc dataset still exists because to the best of our knowledge, none of the existing work has investigated the presence of outliers on the wdbc dataset. in addition, the research investigates the extent to which the removal of outliers, when combined with feature selection method can improve the accuracy of other weak algorithms. hence, this study analyzes improvement of ml algorithms for detecting disease based on outlier detection and feature selection method using pearson correlation based feature selection. the goal of this research was to create an optimal model that would fill a knowledge gap. in this study, the characteristics of the wisconsin diagnostic breast cancer (wdbc) dataset were examined in depth for the presence or absence of outliers. when outliers were discovered, some of the instances were dropped. the filtered dataset was further refined as the classification system’s inputs using correlation feature selection (cfs). the pearson correlation technique was used to find the appropriate continuous features and their associated weight (importance) in order to discover variables that are relevant for prediction. this method assists in the resolution of overfitting and underfitting difficulties in ml. the accuracy was used to evaluate the performance of the unfiltered (conventional technique), filtered (outliers approach), and outliers correlation feature selection (ocfs) datasets. the findings were assessed and compared using seven classifiers: logistic regression (lr), k-nearest neighbor (knn), support vector machines (svm), decision tree (dt), random forest (rf), gradient boost (gb), and adaboost (ab). 2. materials and method the research architecture for predicting the presence of breast cancer disease is shown in figure 1. the approach includes acquiring a breast cancer disease dataset and preprocessing it to remove missing values and outliers. also, using the already processed dataset, an algorithm to discover strongly correlated features was applied, and the results were engaged in ml techniques to predict whether a patient had benign or malignant tumors. finally, the outcome was compared using a performance score based on the confusion matrix. figure 1 depicts the process of recommended techniques for implementing ml algorithms. 2.1. data description the data for this study is acquired from the uci repository. this dataset, identified as the wdbc dataset, has 569 cases that are either benign or malignant. in these situations, 357 cases (62.74%) are benign and 212 cases (37.26%) are malignant. the distribution of the number of benign and malignant classes in the dataset is displayed in figure 2. the dataset contains 33 attributes: class attribute labels (diagnosis: b= benign, m= malignant), id, and 31 real value attributes. these attributes are derived from a digitized image of a biopsy procedure for a breast mass and are used to describe the characteristics of the cell nuclei in the image. the wdbc dataset is a computation of ten real-valued features of cell nucleus: radius, texture, perimeter area smoothness, compactness, concavity, concave points, and symmetry fractal dimension. each of these qualities was estimated of their respective mean, standard error, and worst values, resulting in a total of 30 attributes. the attributes of the wdbc dataset, as well as their datatypes, are listed in table 1. the unique id numbers of the instances and the accompanying class label (diagnosis: m=malignant, b=benign) are stored in the first two columns of the dataset, respectively. columns 332 contain 30 real-value features derived from digitized images of cell nuclei which can be used to create a model to predict whether a tumor is benign (i.e., cancer-free) or malignant (i.e., cancerous). 2.2. data pre-processing purification and modification of the dataset are required before applying ml algorithms to the dataset, it is a necessary step to pre-process the data. performance and accuracy of the predictive model are not only affected by the algorithms used but also by the quality of the dataset and pre-processing. the phases of pre-processing used in this investigation are as follows: 2.2.1. missing values checking the dataset contains 569 instances of 33 variables. however, it was discovered that the variable id had no effect on the dataset description or on disease prediction because it merely keeps a serial record of the instances. as a result, the dataset’s id feature was removed. additionally, while conducting additional preprocessing operations on the dataset, it was discovered that the last feature, unnamed:32, had the value null for all occurrences. this might be a mistake in the data collection process, because of this the feature was also removed from the dataset. 300 yusuf et al. / j. nig. soc. phys. sci. 3 (2021) 298–307 301 figure 1. proposed architecture table 1. attributes and their description on wdbc dataset s/n attributes datatypes s/n attributes datatypes 1 id numeric 18 compactness se numeric 2 diagnosis nominal 19 concavity se numeric 3 radius mean numeric 20 concave points se numeric 4 texture mean numeric 21 symmetry se numeric 5 perimeter mean numeric 22 fractal dimension se numeric 6 area mean numeric 23 radius worst numeric 7 smoothness mean numeric 24 texture worst numeric 8 compactness mean numeric 25 perimeter worst numeric 9 concavity mean numeric 26 area worst numeric 10 concave points mean numeric 27 smoothness worst numeric 11 symmetry mean numeric 28 compactness worst numeric 12 fractal dimension mean numeric 29 concavity worst numeric 13 radius se numeric 30 concave points worst numeric 14 texture se numeric 31 symmetry worst numeric 15 perimeter se numeric 32 fractal dimension worst numeric 16 area se numeric 33 unnamed:32 numeric 17 smoothness se numeric 1 2.2.2. encoding data the performance of machine models depends on various aspects. one element that influences performance of the models are the methods used to analyze data and feed it to the model. as such, vital step in encoding data is turning data into categorical variables understood by ml models. encoding data elevates model quality and helps in feature engineering. the class label ”diagnosis” was expressed as strings of (b= benign, m= malignant). this category characteristic must be converted to restricted numbers. this is done to transform data into a format that ml algorithms can understand. label encoding was used to encode the diagnostic occurrences in this study, and the result was (m=1, b =0). 2.2.3. outliers checking an outlier is a statistic or observation that deviates from a distribution’s overall pattern. if few data are significantly different or not in range of main trend then those are termed outliers. there skewness results, affecting the mean and standard deviation of the distribution. as shown in figure 3, this study detects the existence of outliers in the dataset. as a result, outliers were identified and eliminated from their respective features. 2.2.4. data transformation data must be normalized or standardized before ml algorithms can be applied. the data is standardized to have a mean of 0 (µ) and a standard deviation ( ∑ ) of 1. equation 1 gives the 301 yusuf et al. / j. nig. soc. phys. sci. 3 (2021) 298–307 302 figure 2. dataset’s class level showing malignant and benign in wdbc dataset figure 3. plots of data points to show presence of outliers conversion formula: x = x − µ σ (1) 2.3. dimension reduction dimension reduction is delineated as the mapping of data to a lower dimensional space by removing irrelevant variance in data, resulting in the detection of a subspace in which the data exist [30]. feature extraction and feature selection are two types of dimensional reduction [30]. the process of identifying and discarding irrelevant, less relevant or duplicated features of dimensions in a dataset is known as feature extraction. creating strong learning models, feature selection may be used to detect and remove as much unnecessary and redundant information as feasible. as a result, feature selection not only decreases computational and processing costs, but also improves the model created from the chosen data [31], [32]. on healthcare data, the feature selection method has been used in a number of previous studies [27]–[29], [33]. previous research that are partly relevant to this study and deal with the datasets utilized here; nevertheless, in most cases, the performance of such systems was not as predicted. one of the reasons for some systems’ poor performance is their inability to recognize the most important and highly correlated features. the goal of this research is to devise a method for identifying the best set of features and then investigate which algorithms work best with those features. filters, wrappers, and embedding techniques are the three types of algorithms that may be used to select features[34]. the correlation feature selection method is used in this study to identify the best predictive features. correlation feature selection is a technique that uses the filter approach. the relationship between the independent and dependent variables is determined using a mathematical function. the features are chosen based on the values of their correlation coefficients. the most predictive feature with the class variable is considered to be highly associated, and it is included in the final feature set. pearson’s correlation: consider a dataset d having feature set f f = {x1, x2, x3. . . ,xn} (2) and classes c with values c, where x, c are treated as random variables, pearson’s linear correlation coefficient is defined as r = ∑n i=1 (xi − x) 2 (ci − c)√[∑n i=1 (xi − x) 2 ] [ ∑n i=1 (ci − c) 2 ] (3) where xi and ci the ith value of x and c respectively. percentage of (r) = ±1 if x and y are linearly dependent and zero if they are completely uncorrelated. the values of the correlation coefficients between independent features and the dependent class variable in the wdbc data are shown in figure 4. a filter approach was used to identify the most predictive characteristics. the correlation between all features is calculated and displayed in this article. the correlation criterion utilized is 0.6, and features having a correlation of less than 0.6 are removed from the training dataset. while the other qualities that have a higher threshold are chosen. the following figure 5 depicted based on the highly correlated 10 features with predicted attribute (diagnosis). 2.4. data splitting the goal of dividing the data is to avoid overfitting the model during model testing on the testing dataset. the dataset for this study was split into two parts: training data (80%) and test data (20%). 2.5. trained model based on the seven classifiers, two intelligent systems were built. different forms of ml algorithms have previously been the subject of numerous studies. five of the most prevalent approaches (lr, knn, svm, dt, and rf) were chosen, as well as two infrequent techniques (ab and gb). when combined with feature selection approaches, several prior research have demonstrated that the projected accuracy of lr, knn, svm, dt[25], [26] and rf[27] algorithms was fairly high. furthermore, to the best of our knowledge, no research in this field have shown that ab and gb can perform very well with a high degree of accuracy. these methods were investigated in this study using hyperparameter tuning to improve the proposed model’s efficiency. the following are the algorithms that will be discussed: 302 yusuf et al. / j. nig. soc. phys. sci. 3 (2021) 298–307 303 figure 4. correlation coefficient values between independent features and dependent class variable figure 5. plot of highly correlated features 2.5.1. linear regression lr is a simple supervised learning method for projecting the relationship between explanatory variables and dependent variables by fitting a linear equation to experience data. lr can be mathematically modelled as shown in equation 4 [35]: y = β0 + β1 x1 + e (4) where y is the response variable, β0 and β1 are the model coefficients, and e is the model coefficients error. the intercept and slope are represented by these unknown constant values, which are learnt during the training phase. equation 5 is used to predict after the model has been fitted in the training phase: y = β0 + β1 x (5) where y is the predicted value based on x, and the error is calculated as shown in equation 6 as follows: ei = yi + ŷ1 (6) in this study, the solver and the maximum number of iterations were the two lr parameters used. the solver is the algorithm that is utilized to solve the optimization issue. the algorithm options include “newton-cg”, “sag”, “saga”, “liblinear” and “lbfgs”. the number of iterations required for the solvers to converge is specified between 100 and 10000. 2.5.2. k-nearest neighbor knn is a supervised classifier that learns from data samples that have been labeled. it is a lazy method since it makes 303 yusuf et al. / j. nig. soc. phys. sci. 3 (2021) 298–307 304 no generalizations about the sample data points and all calculations are put on hold until the classification is completed. knn works by determining the k closest neighbors given n training vectors. knn converts the training data set into a multi-dimensional feature space and divides it into various areas based on the training dataset’s classifications. the concept of number of neighbors is fundamental to this method. the number of neighbors specifies how many neighbors should be checked when an item is classified. the parameter range in this study was randomly searched between 1 and 30, with the case being assigned to the most frequent class among its k nearest neighbors, as determined by a distance role. 2.5.3. support vector machine svm are a classification approach that involves projecting input data points into n-dimensional vector space and determining the optimal hyper-plane that maximizes the difference between the two classes. the choice of parameters such as kernel, c, and gamma has a significant impact on svm performance. kernels are a function that converts a low-dimensional space into a high-dimensional one, making categorization simple. the non-linearity is controlled by the kernel. “rbf”, “poly” or “sigmoid” are all possible kernel coefficients. the kernel coefficients were adjusted in this investigation. 2.5.4. decision tree algorithm dt is a strong predictive learning tool used to solve classification and regression problems. it uses a tree-based top-down progression method. it employs a tiered splitting method to divide data into two or more groups at each layer, ensuring that data in each group is comparable. every inner node in a dt’s tie ups to a test attribute, every branch to a test result, and each leaf node to a different class. before applying ‘splitting’, the tree develops from the root node by selecting a ‘best feature’ or ‘best attribute’ from the set of accessible attributes using entropy and information gain measures. the most useful information is provided by the ‘best attribute’. information gain is the pace at which the entropy of attributes increases or decreases, and entropy reflects how homogenous the dataset is. the key parameters that are optimized are the maximum depth of the tree, the number of features to check when looking for the optimum split, the lowest number of samples required to divide an internal node, and the criterion used for splitting. 2.5.5. random forest rf is an ensemble of several separate randomized decision trees that work together. bootstrap sampling of the data is used to create the trees. based on the set of predictor values entered, each individual tree in the random forest casts a unit vote, and the class with the most votes becomes the model’s prediction for categorizing an input vector[36]. the recurrent division of a binary tree into comparable nodes is used to create rf. by inheritance, the parent node impacts the similarity of the child node. 2.5.6. gradient boosting gb is a strategy for enhancing ideas that have poor learning or predictability. the goal of gb is to combine numerous concepts with a weak predictive component and a clever algorithm to create a decision tree with a considerably greater connectivity. if there aren’t many ideas in common across the data components, this notion is especially effective with large datasets. search engine rankings are one of the most popular uses of this technology. search engine rankings must filter a large number of possible queries, some of which may or may not be related, into a limited number of rankable words. 2.5.7. adaboost ab is a boosting method that uses weight modification to solve classification problems without requiring any prior information of the learner’s learning. the goal of adaboost is to enhance classification performance by combining different weak learners or classifiers. a basic collection of training examples is used to train each weak learner. each sample has a weight, which is adjusted iteratively across all samples. the robustness of the weak learner is represented by this weight. the adaboost algorithm consists of the following main steps: (i) sampling, which involves selecting some samples from the training set while iterating. (ii) the sample data is used to train different classifiers, and the error rates for each classifier are computed. (iii) the last stage is the combination of all trained models. 3. performance evaluation metric the metric accuracy was calculated using the 2 x 2 confusion matrix to test the validity of the prediction models, as shown in table 2. the accuracy determines the proportion or possibility of a total number of correct predictions [37]. as seen in equations 7, the following formula is used to quantitatively represent this measurement. where tp, tn, fp, and fn stand for true positive (number of positive data correctly labeled by the classifier), true negative (number of negative data correctly labeled by the classifier), false positive (number of negative data incorrectly labeled as positive), and false negative (number of positive data incorrectly labeled as negative) respectively. table 2. illustration of confusion matrix table actual values positive negative predicted values positive tp fp negative fn tn accuracy = t p + t n (t p + f p + t n + f n) (7) 304 yusuf et al. / j. nig. soc. phys. sci. 3 (2021) 298–307 305 table 3. comparison table between the accuracy of the proposed models and existing techniques authors dataset techniques accuracy (%) [25] wdbc dt 92.9 pca-svm 86.7 pca-knn 82.3 em-pca-cart-fuzzy rule-based 93.2 [27] wdbc rf 98.7 [26] wdbc svm 96.47 cfs-svm 96.47 rfe-svm 96.47 lda-svm 98.82 [23] wdbc svm 61.96 rf 89.37 knn 92.77 fuzzy-id3 94.53 [29] wdbc cfs-rf 95.32 ufs-rf 94.15 rfe-rf 94.15 proposed work with outliers wdbc lr 99.1 knn 96.5 svm 95.6 dt 96.5 rf 99.1 gb 98.3 ab 99.1 proposed work with pearson correlation feature selection wdbc ocfs-lr 96.5 ocfs-knn 96.5 ocfs-svm 95.6 ocfs-dt 94.7 ocfs-rf 100 ocfs-gb 99.1 ocfs-ab 98.3 4. result and discussion the two proposed approaches were subjected to various ml techniques. in order to create the performance statistic, a 2 x 2 confusion matrix was created, which allowed all of the algorithms to be compared. the suggested models were evaluated using the performance indicator “accuracy”. 4.1. comparison between different machine learning algorithms based on accuracy the most essential metrics for evaluating ml algorithms is accuracy. seven classifiers were applied to the wdbc features and processed by resolving the problem of missing values and scaling their instances, as previously indicated, this is called conventional approach. second, the outliers instances were detected in five input features and were dropped from their respective features this is termed outliers approach. finally, the remaining instances of the outliers technique were subjected to pearson correlation feature selection, which resulted in the selection of 10 features, which is known as the ocfs approach. the accuracy of several types of classifiers was carried out on each of these three phases and the result is plotted in figure 6. figure 6. comparison between different machine learning algorithms based on accuracy as demonstrated in figure 6, the conventional approach performs the poorest of all the strategies owing to the presence of outliers, which increases data variability and reduces statistical power. knn, rf, and gb all have similar accuracy 305 yusuf et al. / j. nig. soc. phys. sci. 3 (2021) 298–307 306 (96.5%); however, they perform poorly when compared in other approaches. furthermore, the conventional approach has an exceptional high accuracy of 98.2% in the svm classifier and the least accuracy of 93.9% in the dt. the outlier approach produces considerably higher results for all its classifiers, with three classifiers having maximum accuracy of 99.1% for lr, rf, and ab, respectively. lastly, when the approach of outlier is used with the outcomes of pearson correlation technique (10 features), the accuracies for all predictive classifiers improve the most compared to their foils in other approaches. this is feasible because feature selection allows for the removal of noise from data while also picking the most valuable features; this strategy, when paired with the outlier approach, yields the greatest results. figure 6 shows that ocfs approach rf has the best accuracy (100%); this is because rf added additional randomness to the model when developing the trees. instead of working with the suggested features, it also performs additional searching for the most important feature while splitting a node, it searches for the best feature among a random subset of features. 4.2. comparison table between the accuracy of the proposed models and existing techniques for this research, wdbc datasets were used. the architecture of our suggested system is depicted in figure 1. table 3 shows the outliers and ocfs results, as well as previous work on the wdbc dataset. as a result, distinct results based on the wdbc dataset have been published in the literature. it was reported that after changing the number of selected features by implementing selection algorithms like principal component analysis(pca), correlation feature selection(cfs), recursive feature selection(rfe), linear discriminant analysis(lda), univariate feature selection(ufs), fuzzy rule based and expectation maximization (em)-pca-cart-fuzzy rule-based, there were significant improvements in the accuracy. so, this work compared the accuracy of our two approaches. when it came to removing outliers, the lr model had the greatest accuracy (99.1%), while the svm model had the lowest accuracy score (95.6%). when the ocfs is used, it results in some significant modifications. the rf model achieved the best accuracy (100%), whereas the svm model fared the poorest. table 3 shows a comparison of our findings to current models and datasets. the table’s “techniques” column contains information on the methodologies that were utilized in the previous study, as well as our own methodology and the findings that were published. the table depicts the overall performance of the algorithms in our study in comparison to other comparable works. the greatest outcome of previous rf findings is 94.15% [29], and our ocfs performance has improved to 100%. 5. conclusion the research primarily focuses on improving ml models in order to improve accuracy in forecasting breast cancer disease outcomes. the results show that outlier detection and ocfs methods, in combination with various classification algorithms, might provide useful tools for inference in this area. more study in this area is needed to improve the classification systems’ performance on diverse feature selection approaches so that they can predict on more variables. acknowledgments the authors will like to appreciate the handling editor and the anonymous referees for their contributions to the success of this research. references [1] m. r. mohebian, h. r. marateb, m. mansourian, m. a. mañanas & f. mokarian, “a hybrid computer-aided-diagnosis system for prediction of breast cancer recurrence (hpbcr) using optimized ensemble learning,” computational and structural biotechnology journal 15 (2017) 75. 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[37] s. v. stehman, “selecting and interpreting measures of thematic classification accuracy,” remote sensing of environment 62 (1997) 77. 307 j. nig. soc. phys. sci. 4 (2022) 193–204 journal of the nigerian society of physical sciences mathematical model of in-host dynamics of snakebite envenoming s. a. abdullahia,c, a. g. habibb, n. hussainic,∗ adepartment of mathematics, modibbo adama university yola, adamawa state, nigeria binfectious and tropical diseases unit, department of medicine, bayero univesrity kano, nigeria cdepartment of mathematical sciences, bayero university kano, p.m.b. 3011, kano, nigeria abstract in this paper, we develop an in-host mathematical model of snakebite envenoming that includes tissue, red blood and platelet cells of humans as specific targets of different kinds of toxins in the snake venom. the model is use to study some harmful effects of cytotoxic and hemotoxic snake venom on their target cells under the influence of snake antivenom. the model has two equilibrium points, namely, trivial and venom free. it has been shown that both the equilibrium points are globally asymptotically stable and numerical simulations illustrate the global asymptotic stability of the venom free equilibrium point. furthermore, simulations reveal the importance of administering antivenom to avert the possible damage from venom toxins on the target cells. it is also shown through simulation that administering the required dose of antivenom can lead to the elimination of venom toxins within one week. therefore, we recommend the administration of an adequate dose of antivenom therapy as it helps in deactivating venom toxins faster and consequently enhances the recovery time. doi:10.46481/jnsps.2022.548 keywords: snakebite, in-host model, venom, antivenom, stability analysis article history : received: 21 december 2021 received in revised form: 17 february 2022 accepted for publication: 25 february 2022 published: 29 may 2022 c©2022 journal of the nigerian society of physical sciences. all rights reserved. communicated by: t. latunde 1. introduction snakebite envenoming is one of the most devastating neglected tropical diseases. it continues to impose a major public health and socio-economic burden in low and middle income countries around the world [1, 2, 3, 4, 5]. this neglected tropical disease, which is considered a serious public health issue in tropical and subtropical countries worldwide, is caused by venomous snake bites [6, 7, 8]. globally, more than five million people are bitten by snakes every year [3, 4, 9]. it has been estimated that 1.8 2.7 million envenomations and 81 000 ∗corresponding author tel. no: +2348037594137 email address: nhussaini.mth@buk.edu.ng (n. hussaini) 138 000 deaths occur annually, with an additional 400 000 amputations and other severe health consequences, such as infection, tetanus, scarring, contractures, and psychological sequelae [3, 4, 6, 7, 8, 10, 11]. the majority of snakebites occur in africa and southeast asia. snakebites are most common among people living in rural, resource-poor settings. furthermore, agricultural workers, women, and children are the groups most at risk of being bitten by snakes [6, 7, 8, 9, 12, 13]. according to the world health organization (who) [14], there are more than 3000 species of snakes in the world, of which 600 are venomous and more than 200 are medically important. venomous snakes cause substantial morbidity and mortality in humans. when a venomous snake strikes, it may likely inject venom into its vic193 abdullahi et al. / j. nig. soc. phys. sci. 4 (2022) 193–204 194 tim, as some bites may be dry, which is a bite that will not lead to the release of venom into its target [15]. these venomous snakes administer bites either as a method of hunting their prey, or as a means of protection against their predators [3]. according to young and zahn in [16] the rate of venom discharge and total volume of venom ejected by venomous snakes is much greater during defensive strikes than during predatory strikes. a defensive strike can have 10 times as much venom volume expelled at 8.5 times the flow rate. snake species accountable for the majority of envenoming in the human population are categorized within the families of elapidae (cobra, king cobra, krait, etc.) and viperidae (saw-scaled viper, pit viper, russell’s viper). further, species belonging to the families atractaspididae and colubridae can also induce significant envenomation [17, 18, 19, 20, 21, 22, 23]. snake venom is the poisonous, naturally yellow fluid stored in the adapted salivary glands of venomous snakes. snake venom is mainly categorized into three namely: cytotoxins, neurotoxins and hemotoxins, which target body cells, nervous system and cardiovascular system respectively [14, 24]. according to león et al. in [25] when a snake bites and injects venom into its victim, two concurrent processes occur within the organism: (a) the development of toxic effects and (b) stimulation of immune responses in order to neutralize venom proteins. if the capability of snake venom to cause local and systemic mutilation surpasses the ability of the animal to assimilate and respond to the violence, an envenomation is established. according to some studies, the majority of snakebites occur on the hands, arms, or legs [26, 27]. some signs and symptoms of snakebite may include: redness, swelling, and severe pain at the bite site, which may take up to an hour to appear. further, vomiting, blurred vision, tingling of the limbs, and sweating may likely occur [26, 28]. the severity of snakebite depends on some factors, such as the kind of snake, the part of the body bitten, the quantity of venom injected, and the general health of the person bitten. the severity of snakebite is higher in children than in adults, due to their smaller size [9, 13, 29]. snakebites can be avoided by wearing protective footwear, avoiding snake-infested areas, and not handling snakes [7, 13, 28, 30]. the use of snake antivenom remains the most effective method of averting death from bites. nevertheless, antivenoms often have side effects [9, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48]. some researches on snake venom focused on the use of venom toxins to control infections such as protozoal infections, cancers, and so on (see for instance [49, 50, 51, 52, 53] and references therein). numerous population-based mathematical models have been developed to study the transmission dynamics and control of neglected tropical diseases and other diseases (see, for instance, [54, 55, 56, 57, 58, 59, 60] and references therein). also, several in-host mathematical models for diseases such as hiv, hbv, hcv, chagas disease, etc. have been successfully developed and were used to study the transmission dynamics and possible control of the pathogens inside the victim’s body (for instance, see [61, 62, 63, 64, 65, 66] and references therein). however, the literature on the use of mathematical modeling techniques to gain an in-depth understanding of the dynamics of snake venom inside an envenomed person is scarce. although not a mathematical model, kini and evans in [67] proposed a hypothetical model and explained how venom phospholipase a2 enzymes exhibit specific pharmacological effects such as presynaptic neurotoxicity, cardiotoxicity, myotoxicity, anticoagulant and platelet effects. the research revealed how toxins and enzymes contained within the venom are drawn to specific tissues or cells due to their affinity for specific proteins. in 1994, stagg [68] developed a mathematical model and studied the effect of cobra venom and its anti-venom inside the envenomed body (in-vivo). the work primarily described the neurotoxic type of snake venom and the cell that it targets. furthermore, tanos et al. [69] developed a semi-mechanistic model of the clotting cascade to describe the effect of procoagulant toxin from taipan venom and the effect of anti-venom in controlling venom-induced consumptive coagulapathy. their work focuses on how the hemotoxic venom affects or disrupts blood clotting factors. we are motivated by the fact that in june 2017, who added snakebite envenoming to its priority list of neglected tropical diseases and has set a target to reduce its mortality and disability by 50% before the year 2030 [70]. also, to the best of our knowledge, not much work has been done in terms of mathematical modeling to gain more insight into the dynamics and control of toxic effects of snake venom inside the host body. thus, the aim of this study is to develop a novel inhost mathematical model that will describe the dynamics and control of cytotoxic snake venom that was not considered by the aforementioned studies. moreover, the study will also investigate the impact and control of hemotoxic snake venom on red blood cells and platelets. additionally, it is important to note that providing effective control of snakebite envenoming in a population may depend on a comprehensive understanding of the effects and dynamics of various kinds of toxins inside snake venom on their target cells (in-vivo). therefore, this paper is committed to study the effects of interactions between snake venom (cytotoxic and hemotoxic) and their associated target cells with and without antivenom therapy. the paper is organized as follows: the model is formulated in section 2, analyzed in section 3. results and discussion are reported in section 4 and conclusion is provided in section 5. 2. model formulation a mathematical model that consists of nine in-host compartments is proposed in order to investigate the effects of snake venom and its target cells inside the human victim at a given time. the in-host compartments are: quantity of healthy tissue cells denoted by t (t), quantity of healthy red blood cells represented by r(t), quantity of healthy platelets denoted by p(t), quantity of snake venom with cytotoxic and hemotoxic components expressed as v (t), quantity of antivenom to be administered denoted by a(t). we assumed that the quantity of antivenom to be administered is sufficient to neutralize the circulating venom within the victim’s bloodstream. we also included damaged tissue cells (necrotic cells) caused by the cytotoxic component of venom toxins (necrotoxins), denoted by n(t), damaged red blood cells (hemolyzed cells) caused by the hemotoxic component of venom toxins, denoted by h(t), damaged 194 abdullahi et al. / j. nig. soc. phys. sci. 4 (2022) 193–204 195 platelets caused by the hemotoxic component of venom toxins (platelets toxins), denoted by d(t), and venom-antivenom complexes expressed as c(t). the quantity of healthy tissue cells is replenished at a logistic growth rate given by γt t ( 1 − tkt ) with t ≤ kt . it is decreased as it undergoes necrosis by direct action of the venom’s cytotoxic components (necrotoxin) that begin at the bite site at a constant destruction rate that follows the law of mass action given by (β1t v ). the damaged cells caused by venom-induced necrosis are known as necrotic cells. the healthy tissue cells further decreased due to the elimination of the dead cells at a rate µt . thus, the equation governing the dynamics of healthy tissue cells is given by dt dt = γt t ( 1 − t kt ) −β1t v −µt t. (1) the quantity of healthy red blood cells is replenished at a logistic growth rate denoted by γrr ( 1 − rkr ) (r ≤ kr). after 120 days it is reduced by natural elimination from the host body at a rate µr. it further decreased because of the damage it undergoes as a result of the deleterious effects of the hemotoxic component of the venom as it circulates in the victim’s blood stream. the action of the hemotoxic component of the venom also follows the law of mass action given by (β2rv ). the damaged red blood cells due to venom-induced hemolysis are termed hemolyzed cells. thus, we have dr dt = γrr ( 1 − r kr ) −β2rv −µrr. (2) the quantity of healthy platelets is replenished at a logistic growth rate of γp p ( 1 − pkp ) (p ≤ kp). it is reduced by natural elimination of the dead platelet cells at a rate of µp. it decreases further because the hemotoxic component of the venom (platelet toxins) activates and damages platelets at a rate of β3. this effect may cause serious internal bleeding. thus, the following equation is obtained dp dt = γp p ( 1 − p kp ) −β3 pv −µp p. (3) it is assumed that an initial quantity of venom injected by an offending snake is in circulation in the host body. therefore, the quantity of venom decreases because of its reaction with the various target cells, i.e. tissue, red blood, and platelet cells at the rates β1,β2 and β3 respectively. it further diminishes due to its neutralization by antivenom that is describe by the law of mass action (β4 av ). this reaction produces venom-antivenom complexes that are not harmful to the body. the amount of venom is also eliminated at a rate µv . therefor, the following equation is formed dv dt = −β1t v −β2rv −β3 pv −β4 av −µv v. (4) when the required dose of snake antivenom is administered, its action is to neutralize the circulating venom toxins inside the victim’s body. the quantity of antivenom inside the host body is decreased because of the reaction between venom and antivenom that follows the law of mass action given by (β4 av ). the unused antivenom is then removed at a rate of µa. thus, we have da dt = −β4 av −µa a. (5) the quantity of necrotic cells is formed because of the reaction between the cytotoxic component of the venom (necrotoxin) and the tissue cells at the rate β1 and is eliminated at a rate µn . therefore, we obtain dn dt = β1t v −µn n. (6) the quantity of hemolyzed cells is formed due to the reaction between the hemotoxic component of the venom and the red blood cells at the rate β2 and is eliminated at a rate µh . thus, we have dh dt = β2rv −µh h. (7) the formation of the quantity of damaged platelets occurs as a result of reaction between the platelet toxins and the healthy platelet cells at the rate β3. it decays at a rate of µd. therefore, we have dd dt = β3 pv −µd d. (8) the quantity of venom-antivenom complexes is produced because of the neutralization of venom by antivenom and is eliminated from the body at a rate µc . so that dc dt = β4 av −µcc. (9) based on the aforementioned description of the model, the inhost dynamics of interaction between cytotoxic and hemotoxic venom, their target cells, and antivenom is described by the following system of first order nonlinear ordinary differential equations. the schematic diagram is depicted in figure 1, and the corresponding variables and parameters are tabulated in tables 1 and 2, respectively: dt dt = γt t ( 1 − t kt ) −β1t v −µt t, dr dt = γrr ( 1 − r kr ) −β2rv −µrr, dp dt = γp p ( 1 − p kp ) −β3 pv −µp p, dv dt = −β1t v −β2rv −β3 pv −β4 av −µv v, da dt = −β4 av −µa a, dn dt = β1t v −µn n, dh dt = β2rv −µh h, dd dt = β3 pv −µd d, dc dt = β4 av −µcc, (10) 195 abdullahi et al. / j. nig. soc. phys. sci. 4 (2022) 193–204 196 figure 1. schematic diagram of the model. note that, in the schematic diagram of the model, the dotted lines indicate the reactions between venom toxins and their target cells (which produce damaged cells) and the neutralization of venom toxins by antivenom (which leads to the formation of venom-antivenom complexes). the solid arrows represent the self-replenishment of the healthy cells, the formation of damaged cells and venom-antivenom complexes. it also indicates the natural elimination of damaged or dead cells from the host body. table 1. description of state variables of the model. variables description t(t) quantity of healthy tissue cells at time t r(t) quantity of healthy red blood cells at time t p(t) quantity of healthy platelets at time t v(t) quantity of venom in the victim blood stream at time t a(t) quantity of antivenom in the victim blood stream at time t n(t) quantity of damaged tissue (necrotic cells) at time t h(t) quantity of damaged red blood cells (hemolyzed cells) at time t d(t) quantity of damaged platelets at time t c(t) quantity of venom-antivenom complexes at time t with the initial conditions t (0) > 0, r(0) > 0, p(0) > 0, v (0) = v0 ≥ 0, a(0) = a0 ≥ 0, n(0) ≥ 0, h(0) ≥ 0, d(0) ≥ 0, c(0) ≥ 0. (11) 3. model analysis 3.1. basic properties of the model this section presents some essential properties of the in-host model. it is vital to show that the state variables of the model table 2. description of parameter of the model. parameter description γi(i = t, r, p) replenishment rates of tissue cells, red blood cells and platelets respectively ki(i = t, r, p) maximum quantities of tissue cells, red blood cells and platelets respectively µi(i = t, r, p) elimination rates of tissue, red blood, platelet cells respectively µi(i = n, h, d) elimination rates of damaged tissue, red blood, and platelet cells respectively µi(i = v, a, c) elimination rates of venom, antivenom and venom-antivenom complexes respectively β1 rates at which cytotoxic component of the venom (necrotoxins) destroys tissue cells β2 rate at which hemotoxic component of the venom (hemotoxins) destroys red blood cells β3 rate at which hemotoxic component of the venom (platelet toxins) damages platelets β4 naturalization rate of venom by antivenom are non-negative for all t > 0. if this result is established, then it is sufficient to say that the model (10) is mathematically and biologically reasonable. it is crucial to assume that the model parameters given in table 2 are positive. for mathematical convenience let µ1 = min{µt ,µn}, µ2 = min{µr,µh} and µ3 = min{µp,µd}. further, let v0 denotes the amount of venom that the offending snake injected into its victim at the initial time. thus, we assume that quantity of venom inside the host at any given time is presented as v (t) ≤ v0. (12) similarly, let a0 be the initial dose of antivenom administered into the victim’s bloodstream, which we assumed was sufficient to neutralize the circulating venom for simplicity. thus, it is assumed that the quantity of antivenom inside the host at any given time is given a(t) ≤ a0. (13) theorem 3.1. let the initial data of the model (10) be nonnegative, then the model solutions given by (t, r, p, v, a, n, h, d, c) are all non-negative for every t > 0. proof. let t1 = sup{t > 0 : t > 0, r > 0, p > 0, v > 0, a > 0, n > 0, h > 0, d > 0, c > 0, }. thus, t1 > 0. taking the first equation of the model (10) we have, dt dt = γt t ( 1 − t kt ) −β1t v −µt t. (14) since, t ≤ kt we have dt dt ≥−(β1t v + µt t ) . (15) 196 abdullahi et al. / j. nig. soc. phys. sci. 4 (2022) 193–204 197 (a) (b) (c) figure 2. graph showing the lethal effects of russell’s viper venom with and without antivenom on (a) tissue cells (b) red blood cells (c) platelets. in figures (a), (b) and (c), the red dotted lines indicate the quantities of damaged tissue cells, red blood cells and platelets without the influence of antivenom. while the blue solid lines represent quantities of damaged tissue cells, red blood cells and platelets under the influence of antivenom. now integrating both sides from t=0 to t = t1 we obtain,∫ t1 0 ( 1 t ) dt ≥− ∫ t1 0 (µ) dt + ∫ t1 0 (β1v (τ)) dτ  t (t) ≥ t (0) exp − µt1 + ∫ t1 0 β1v (τ)dτ   > 0. thus, t ( t1 ) > 0 for all t 1 > 0, hence, t (t) > 0 for t > 0. following similar technique, it can be established that the remaining eight variables r, p, v, a, n, h, d, c are all positive for t > 0. consequently , the solutions of the system (10) remain positive for all t > 0. lemma 3.1. the closed set ω = {(t, r, p, v, a, n, h, d, c) ∈ r9+ : p1 ≤ γt kt µt , p2 ≤ γr kr µr , p3 ≤ γp kp µp , c ≤ β4 a0 v0 µc }, is positively invariant. proof. let p1 represents the total quantity of healthy and damaged tissue cells (necrotic). thus, p1 = t + n and dp1 dt = dt dt + dn dt . now, adding the first and sixth equations of system (10) we get dp1 dt = γt t ( 1 − t kt ) −µ1 p1. (16) it follows that dp1 dt ≤ γt kt −µ1 p1. applying comparison theorem [73] it can be shown that p1(t) ≤ p1(0)e −µ1 t + γt kt µ1 [ 1 − e−µ1 t ] . 197 abdullahi et al. / j. nig. soc. phys. sci. 4 (2022) 193–204 198 (a) (b) (c) figure 3. simulations showing the effects of varying the initial amount of russell’s viper venom injected on (a) tissue cells (b) red blood cells (c) platelets. the red dashed, black solid, green dashed, and blue dashed lines denote the quantities of damaged tissue, red blood and platelet cells when 0mg, 50mg, 100mg and 150mg of venom are injected, respectively. thus, lim sup t→∞ p1(t) ≤ γt kt µ1 . similarly, let p2 denotes the total quantity of healthy and damaged red blood cells (hemolyzed). thus, p2 = r + h and dp2 dt = dr dt + dh dt . now, adding the second and seventh equations of system (10) we obtain dp2 dt = γrr ( 1 − r kr ) −µ2 p2. (17) it follows that dp2 dt ≤ γr kr −µ2 p2. applying comparison theorem [73] it can be shown that p2(t) ≤ p2(0)e −µ2 t + γr kr µ2 [ 1 − e−µ2 t ] . thus, lim sup t→∞ p2(t) ≤ γr kr µ2 . in the same manner let p3 denotes the total quantity of healthy and damaged platelets. thus, p3 = p + d and dp3dt = dp dt + dd dt . now, adding the third and eighth equations of system (10) we get dp3 dt = γp p ( 1 − p kp ) −µ3 p3. (18) it follows that dp3 dt ≤ γp kp −µ3 p3. applying comparison theorem [73] it can be shown that p3(t) ≤ p3(0)e −µ3 t + γp kp µ3 [ 1 − e−µ3 t ] . thus, lim sup t→∞ p3(t) ≤ γp kp µ3 . finally, taking the ninth equation of 198 abdullahi et al. / j. nig. soc. phys. sci. 4 (2022) 193–204 199 (a) (b) (c) (d) figure 4. simulations showing the effects of varying the initial dose of snake antivenom in averting damaged (a) tissue cells (b) red blood cells (c) platelets and (d) deactivating venom inside the host body. in figures 4a, 4b and 4c, the red dashed, black solid, green dashed, and blue dashed lines represent the quantities of necrotic, hemolyzed and damaged platelet cells when 0ml, 100ml, 150ml and 250ml of antivenom are administered, respectively. while in figure 4d, the black dashed, blue dashed and red solid lines represent the quantities of venom inside the host body when 50ml, 100ml and 250ml of antivenom are administered, respectively. system (10) and using v (t) ≤ v0, a(t) ≤ a0 we have dc dt ≤ β4 a0v0 −µcc. (19) consequently, using the comparison theorem in conjunction with the integrating factor technique of solving first order linear differential equations, we have c(t) ≤ c(0)e−µc t + β4 a0v0 µc [ 1 − e−µc t ] . thus, lim sup t→∞ c(t) ≤ β4 a0 v0 µc . hence, ω is positively invariant, meaning that, all the solutions starting in ω stay in ω for t > 0. thus, it is established that ω is an attracting set. since, the region ω is positively invariant, it suffice to investigate the dynamics of the model equations given by (10) in ω, where the usual existence, uniqueness, continuation of solutions hold. 4. results and discussion 4.1. equilibrium points of the model to obtain the equilibrium points of the model equations given by system (10) the right hand side of each equation in system (10), is set to zero and solves for the associated state variables. the model equations have two non-negative equilibrium points, viz; the trivial and the venom free equilibrium points, denoted 199 abdullahi et al. / j. nig. soc. phys. sci. 4 (2022) 193–204 200 by ε0 and ε1 correspondingly. the results obtained are presented in the following subsections. 4.1.1. trivial equilibrium point the trivial equilibrium point is presented as follows: ε0 = ( t 0, r0, p0, v 0, a0, n0, h0, d0, c0 ) = (0, 0, 0, 0, 0, 0, 0, 0, 0) . it is important to note that this equilibrium point is biologically not feasible. however, its qualitative analysis is of interest. 4.1.2. venom free equilibrium point the venom free equilibrium point of the model is given by: ε1 = (t ∗, r∗, p∗, v∗, a∗, n∗, h∗, d∗, c∗) = ( kt (γt −µt ) γt , kr (γr −µr) γr , kp (γp −µp) γp , 0, 0, 0, 0, 0, 0 ) . (20) this equilibrium point is biologically and mathematical feasible. it describes the healthy state of tissue, red blood cells and platelets. observe that if γt = µt ,γr = µr and γp = µp then the equilibrium point ε1 reduces to ε0. therefore, for the equilibrium point ε1 to be positive we must have γt > µt ,γr > µr and γp > µp. 4.1.3. global stability of equilibrium points of the model here we shall use lyapunov function theory in conjunction with la-salle’s invariance principle to establish the global asymptotic stability of the trivial and venom free equilibrium points. 4.1.4. global stability of trivial equilibrium point theorem 4.1. the trivial equilibrium point (ε0) of model (10), is globally asymptotically stable in ω. proof. the stability analysis of (ε0) is simplified by defining a perturbation w = x −ε0, x = (t, r, p, v, a, n, h, d, c). (21) therefore, model (10) is expressed as: dw dt = b(w)w, (22) where w = w ti (i = 1, ..., 9) and b(w) is a matrix of coefficients of the system (10) with the variables wi(i = 1, ..., 9), this is presented as follows: where, z11 = (γt−β1 w4−µt )kt−2 γt w1 kt , z22 = (γr−β2 w4−µr )kr−2 γr r kr , z33 = (γp−β3 w4−µp )kp−2 γp p kp , z44 = −β1w1 − β2w2 − β3w3 − β4w5 −µv. observe that εw = (0, 0, 0, 0, 0, 0, 0, 0, 0) , is the only equilibrium point of equation (22). furthermore, we consider the following lypunov function as applied in the following studies [74, 75, 76] and references therein. v (w) = 〈y, w〉, y = (1, 1, 1, 1, 1, 1, 1, 1, 1) > 0. (24) v′(w) = 〈y, b(w)w〉, = z11w1 + z22w2 + z33w3 − (w1β1 + w2β2 + w3β3 + w5β4 + µv ) w4 − (w4β4 + µa) w5 −µn w6 −µh w7 −µdw8 −µc w9, = ( γt ( 1 − w1 kt ) − γt w1 kt −β1w4 −µt ) w1 + ( γr ( 1 − w2 kr ) − γrw2 kr −β2w4 −µr ) w2 + ( γp ( 1 − w3 kp ) − γpw3 kp −β3w4 −µp ) w3 − (β1w1 + β2w2 + β3w3 + β4w5 + µv ) w4 − (β4w4 + µa) w5 −µn w6 −µh w7 −µdw8 −µdw9. (25) consider the following conditions: w1 ≤ kt , w2 ≤ kr, w3 ≤ kp. (26) therefore, applying the conditions given in (26) in equation (25) we obtain v′(w) ≤− ( γt w1 kt + β1w4 + µt ) w1− ( γrw2 kr + β2w4 + µr ) w2 − ( γpw3 kp + β3w4 + µp ) w3 − (β1w1 + β2w2 + β3w3 + β4w5 + µv ) w4 − (β4w4 + µa) w5 −µn w6 −µh w7 −µdw8 −µc w9. (27) thus, v′(w) ≤ 0 with v′(w) = 0, if and only if wi = 0, (i = 1, 2, ..., 9). hence, v (w) is lypunov function in ω. further, since, ω is an invariant and attracting set as established in lemma (3.1) it follows from la-salle’s invariance principle in [77] that the maximum invariant set contained in {v/v′(w) = 0} is εw . consequently, the transformed equilibrium point εw, is globally asymptotically stable in ω. hence, the trivial equilibrium, ε0, is also globally asymptotically stable in ω. 4.1.5. global stability of venom free equilibrium point theorem 4.2. the venom free equilibrium point (ε1) of model (10), is globally asymptotically stable in ω. proof. consider the following linear lyapunov function given by: w (t) = a1v (t) + a2 n (t) + a3 h (t) + a4 d (t) + a5c (t) ,(28) where ai, i = 1, ...5 are positive constants to be chosen later. the time derivative of the lyapunov function along the solutions of model (10) is given as; ẇ (t) = a1v̇ (t) + a2 ṅ (t) + a3 ḣ (t) + a4 ḋ (t) + a5ċ (t) , = a1 [ −β1t v −β2rv −β3 av −µv v ] + a2 [ β1t v −µn n ] + a3 [ β2rv −µh h ] + a4 [ β3 pv −µd d ] + a5 [ β4 av −µcc ] . (29) 200 abdullahi et al. / j. nig. soc. phys. sci. 4 (2022) 193–204 201 b (w) =  z11 0 0 −β1w1 0 0 0 0 0 0 z22 0 −β2w2 0 0 0 0 0 0 0 z33 −β3w3 0 0 0 0 0 −β1w4 −β2w4 −β3w4 z44 −β4w4 0 0 0 0 0 0 0 −β4w5 −β4w4 −µa 0 0 0 0 β1w4 0 0 β1w1 0 −µn 0 0 0 0 β2w4 0 β2w2 0 0 −µh 0 0 0 0 β3w4 β3w3 0 0 0 −µd 0 0 0 0 β4w5 β4w4 0 0 0 −µc  , (23) ẇ = (a1 − a2) β1t v + (a1 − a3) β2rv + (a1 − a4) β3 pv + (a1 − a5) β4 av − a1µv v − a2µn n − a3µh h − a4µd d − a5µcc. (30) choosing ai = 1 µv µnµhµdµc for (i = 1, ...5) and applying it in equation (30) we obtain ẇ = − [( 1 µnµhµdµc ) v + ( 1 µvµhµdµc ) n + ( 1 µvµnµdµc ) h ] − [( 1 µvµnµhµc ) d + ( 1 µvµnµhµd ) c ] . (31) consequently, using equation (31), it follows that ẇ < 0 unconditionally, since all the model parameters are positive in ω. for ẇ = 0 we must have v = n = h = d = c = 0. following the claim that ω is invariant and attracting set, it follows that the largest possible invariant set in ω is the singleton ε1. according to la-salle’s invariance principle [77], ε1 is globally attractive. therefore, the venom free equilibrium, ε1, is globally asymptotically stable. 4.2. numerical simulation in this section, some numerical simulations were performed using the values of the model parameters in table 3. according to bawaskar in [79], the deadly dose of russell’s viper venom is v0 = 150mg and the initial dose of antivenom needed to neutralize the venom is a0 = 250ml. the numerical simulation results are shown in figures 2, 3, 4, 5 and 6. 4.2.1. effects of venom toxins on target cells with and without antivenom figure 2a shows the deleterious effects of toxins inside the russell’s viper venom on the tissue cells. it can be observed that the quantity of damaged tissue cells is on the increase when antivenom is not administered (red dot line). on the other hand, a negligible quantity of damaged tissue cells is noticed when antivenom is administered (blue line). also, in figures 2b and 2c table 3. values of parameter used in numerical simulation. parameter values reference γt 0.009 week −1 estimated γr 1 17.143 week −1 [71, 72] γp 1 1.429 week −1 [78] ki(i = t, r, p) 80000, 10000, 5000 estimated µt 1 3 week −1 estimated µr 1 17.143 week −1 [71, 72] µp 1 1.429 week −1 [78] µi(i = v, a, n) 0.1, 0.013, 0.001 estimated µi(i = h, d, c) 0.0012, 0.00013, 0.0002 estimated βi(i = 1, 2) 7.1 × 10−11, 5.1 × 10−06 estimated βi(i = 3, 4) 3.7 × 10−04, 0.87 estimated we notice a significant amount of damaged red blood cells and platelets in the absence of antivenom to neutralize the circulating venom. however, an insignificant number of damaged red blood cells and platelets are observed when snake antivenom is administered. 4.2.2. effects of varying initial quantity of venom on the target cells the dynamical effect of increasing the initial dose of venom on the target cells is depicted in figure 3. the outcome reveals that the amounts of injured tissue, red blood cells, and platelet cells increase as the initial dose of venom increases. this result suggests that the magnitude of damage to be inflicted by venom toxins on the target cells depends on the initial quantity of venom injected by the offending snake. therefore, determining the amount of venom injected by the offending snake is crucial in the management of snakebite patients. 4.2.3. effects of varying initial dose of antivenom on the damaged cells and the venom the simulations in figures 4a, 4b and 4c demonstrate the effect of varying antivenom doses in preventing damaged cells as a result of venom toxins. it is obvious that the application of the re201 abdullahi et al. / j. nig. soc. phys. sci. 4 (2022) 193–204 202 figure 5. simulation showing the effect of using adequate dose of antivenom in decreasing the quantity of venom inside the host. quired dose of antivenom is more effective in terms of averting damaged cells. also, the dynamics of varying the initial dose of antivenom in deactivating venom toxins inside the host body is depicted in figure 4d. it is evident that when a lower-thanrequired amount of antivenom is administered, it takes longer time to neutralize the quantity of venom inside the host. on the other hand, when the required amount of antivenom is applied, it takes less time to neutralize the venom toxins. this finding supports the need for administering an adequate dose of antivenom in order to enhance the time of neutralizing the lethal dose of venom within the host. 4.2.4. effect of adequate dose of antivenom in decreasing the quantity of venom inside the host body according to the simulation result shown in figure 5, application of adequate dose of antivenom leads to the elimination of a deadly dose of russell viper venom within one week. this result is in line with the recovery time for snakebite envenoming in treatment and research hospital, katungo, gombe state, as reported in [80]. this finding supports the use of the recommended dose of antivenom in the treatment of snakebites envenoming, since it will not only prevent harm to the cells but also speed up recovery time. it is worth mentioning that the epidemiological implication of the numerical results provided in figures 2, 3, 4 and 5 is that whenever snakebite envenomation is confirmed, the required doses of antivenom therapy should be administered as soon as possible to avoid the potential damage that venom toxins will inflict on target cells. as a result, envenomation-related deaths and disabilities will be avoided. 4.2.5. numerical illustration of global asymptotic stability of the venom free equilibrium point the results in figures 6a, 6b and 6c illustrate the convergence of solutions to the venom free equilibrium point which uphold the result of global asymptotic stability of venom free equilibrium point presented in theorem 4.2. (a) (b) (c) figure 6. simulation results showing total quantity of damaged (a) tissue cells (b) red blood cells (c) platelets. in all the cases different initial conditions were used to illustrate the result of global asymptotic stability of the venom free equilibrium point. 5. conclusions in this work, we developed and analyzed an in-host model of snakebite envenoming. the model was used to study some toxic effects of snake venom on tissue, red blood, and platelet cells of humans under the influence of snake antivenom. the basic properties of the model in terms of positvity and boundedness of solutions were explored. the qualitative study of the model reveals that the model has two non-negative equilibrium points, namely, trivial and venomfree. the equilibrium points were shown to be globally asymptotically stable and numerical simulations were used to illustrate the result of the global asymptotic stability of the venom free equilibrium point as established in theorem 4.2. furthermore, numerical simulations also revealed the importance of antivenom therapy in preventing the deleterious effects of snake venom on target cells. it is also shown through simulation that the application of required dose of antivenom can lead to the elimination of venom toxins within one week. the public health implication of the simulation results in figures 2, 3, 4 and 5 in terms of managing snakebite envenomation is that determining the quantity of venom injected by the offending snake is crucial. in addition, the required dose of antivenom should be administered as soon as possible so that envenomation-related deaths and disabilities can be prevented. some of the numerical findings in this study provide some useful insights into the management and control of snakebite envenoming patients, such as determining the initial quantity of venom injected by the offending snake and applying the required dose of antivenom to neutralize the venom toxins. this work is limited to the effect of cytotoxic and hemotoxic venoms on the tissue, red blood, and platelet 202 abdullahi et al. / j. nig. soc. phys. sci. 4 (2022) 193–204 203 cells of humans under the influence of snake antivenom. in the future, we intend to use delay differential equations to model the impact of early and late treatment. future studies may also be directed towards modeling the effects of other types of snake venom toxins, such as neurotoxins and anticoagulant toxins, on their target cells. acknowledgments the authors are grateful to the anonymous reviewers and the handling editor for their constructive comments, which have undoubtedly improved the quality and clarity of the manuscript. we would also like to thank dr. abubakar s. balla and dr. agom d. ibrahim of snakebite treatment and research hospital kaltungo, gombe state, for providing us with useful information on the management of snakebite envenoming patients. references [1] j. m. gutierrez, r. d. g. theakston & d. a. warrell, “confronting the neglected problem of snake bite envenoming: the need for a global partnership”, plos med 3 (2006) 6. 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[80] a. g. habib, m. lamorde, m. m. dalhat, z. g. habib & a. kuznik, “cost-effectiveness of antivenoms for snakebite envenoming in nigeria”, plos negl trop dis 9 (2015) 1, doi.org/10.1371/journal.pntd.0003381 204 j. nig. soc. phys. sci. 4 (2022) 146–156 journal of the nigerian society of physical sciences modified gradient flow method for solving one-dimensional optimal control problem governed by linear equality constraint olusegun olotua, charles aladesayeb, kazeem adebowale dawodua,∗ adepartment of mathematical sciences,the federal university of technology akure, nigeria bdept. of mathematics, school of science, college of education, ikere-ekiti, ekiti state, nigeria abstract this study presents a computational technique developed for solving linearly constraint optimal control problems using the gradient flow method. this proposed method, called the modified gradient flow method (mgfm), is based on the continuous gradient flow reformulation of constrained optimization problem with three-level implicit time discretization scheme. the three-level splitting parameters for the discretization of the gradient flow equations are such that the sum of the parameters equal to one (θ1 + θ2 + θ3 = 1). the linear and quadratic convergence of the scheme were analyzed and were shown to have first order scheme when each parameter exist in the domain [0, 1] and second order when the third parameter equal to one. numerical experiments were carried out and the results showed that the approach is very effective for handling this class of constrained optimal control problems. it also compared favorably with the analytical solutions and performed better than the existing schemes in terms of convergence and accuracy. doi:10.46481/jnsps.2022.589 keywords: optimal control, gradient flow, three-level splitting parameters, discretization scheme, linear and quadratic convergence article history : received: 15 january 2022 received in revised form: 23 february 2022 accepted for publication: 23 february 2022 published: 28 february 2022 c©2022 journal of the nigerian society of physical sciences. all rights reserved. communicated by: j. ndam 1. introduction many authors, in literature, have attempted the class of linearly constrained optimal control problems using the classical analytical techniques of the hamilton-jacobi-bellman (hjb) or "pontryagin maximum principle" by pontryagin et al. in [14]; though this approach may proof difficult in some cases when inclusive of non-differentiable delay terms. the numerical approach has also been fully handled by several authors like betts ∗corresponding author tel. no: +2348032541380 email address: dawodukazeem@futa.edu.ng (kazeem adebowale dawodu) in [4], olotu and dawodu in [11, 12], adekunle [1] and akeremale [2], using the direct numerical methods. this method deploys the "first discretized then optimize" technique which is known for its amenability to well-structured discretization schemes and optimization solvers after the unconstrained formulation of the control problem using any of the penalty function methods or method of multipliers. the direct method of nonlinear programming formulation tends to eradicate the problem of ill-conditioning associated with control problems with delay dynamical systems. however, this research paper seeks for a better approach in the 146 olotu et al. / j. nig. soc. phys. sci. 4 (2022) 146–156 147 formulation of the control operators in the optimization of the discretized optimal control problem. the gradient flow method was first applied to nonlinear programming problems by evtushenko [5] and later improved by evtushenko and zhadan [6]; while the application to unconstrained problems was by behrman [3] and the unified approach to nonlinear optimization problem was by wang et al. [15]. the unified gradient flow algorithm exhibits linear convergence for θ ∈ [0, 1) and quadratic convergence at θ = 1. as an improvement over [15], a modified gradient flow method mgfm was proposed using the gradient formula for lagrange multiplier formulation. the development of this method requires a continuous gradient flow approach based on a three level implicit time discretization scheme with splitting parameter θ1, θ2 and θ3. the convergence of the solution of the modified discretized gradient flow equation to a local minimum of the original problem, either linearly or quadratically depending on the choice of θ, will as well be demonstrated. however, the technique can also be adapted to handle inequality constraint problems by introducing slack variables. three test problems were solved and their numerical results presented in comparison with existing methods. this is to demonstrate the effectiveness of the method in solving constrained nonlinear programming problems with either continuous variables or a mixture of continuous and discrete variables. the structure of this paper is as follows: section 3 is dedicated to the idea of gradient flow approach for unconstrained optimization while section 4 presents the modified gradient flow algorithm for constrained optimization in section 2 using the 3level splitting parameters and as well as the convergence analysis. the algorithm was experimented on two numerical examples and comparison made with the conjugate gradient method (cgm) in section 4. 2. statement of problem the general form of the one-dimensional optimal control problem is given as minimize j(x, u) = ∫ t t0 f(t, x(t), u(t))dt, (1) subject to ẋ(t) = f (t, x(t), u(t)), t ∈ [t0, t ], (2) x(t0) = x0, (3) where x(t) ∈ c1[t0, t ] ∈ r, u(t) ∈ l2[t0, t ] ∈ r, f : [t0, t ] × r × r → r and f : [t0, t ] × r × r → r. thus, x(t) is the state variable which describes the state of the system at any point in time and u(t) is the control variable in the optimization problem. however, for linear-quadratic problem, the objective functional is written as f(t, x(t), u(t)) = ( px2(t) + qu2(t)) while the linear differential constraint is written as f (t, x(t), u(t)) = ax(t) + bu(t), where p, q , a , b ∈ r and p, q > 0. 3. research methodology this paper considered both the theoretical and numerical analyses of the problem in the development of the algorithm for solving eqns. (1) to (3). the discrete form of the quadratic objective functional and the linear differential constraint are constructed using the trapezoidal and euler method respectively. the recurrence relation for both the objective functional and constraint were developed to help construct their respective matrix operators. furthermore, the symmetry and positive definite properties of the matrix operators were analyzed to guarantee their consistency and invertibilty in the developed mgfm. 3.1. preamble let f : x → r be a functional such that the solution x(t) is a minimizer of f . the gradient flow method, with an initial point x(0) ∈ x, seeks to find a minimizer of f by the curve x(t) defined by the differential equation d x(t) dt = −∇ f (x(t)), x(0) = x0, (4) where ∇ f is called the gradient of f and the solution is the called the integral curve such that at each instance proceeds in the direction of the steepest descent of f . in an unconstrained optimization problem, the equilibrium points of the gradient flow are the zeros of ∇ f , which are exactly the minimizers of f . the gradient flow solves the problem min f (x), x(0) = x0 such that in every trajectory x(t) of the gradient flow, then f (x(t)) → p∗ for p∗ is the minimizer of f . for clarity, the derivative x′(t) is descretized using the forward euler step which yields xk+1 := xk − h∇ f (xk) (5) or the backward discretization scheme which yields xk+1 := xk − h∇ f (xk+1). (6) the backward euler method can be applied for the numerical integration of gradient flow differential equation, also known as the proximal minimization method expressed thus: xk+1 := proxh, f (xk). however, the proximal gradient flow algorithm can be interpreted as gradient flows, d x(t) dt = −∇ f (x(t)) −∇g(x(t)). (7) where g is differentiable. using the forward-backward integration of the gradient flow, eqn. (7) can be expressed as xk+1 = xk − h∇ f (xk) − h∇g(xk+1), xk+1 := (i + h∇g)−1(i − h∇ f )xk, 147 olotu et al. / j. nig. soc. phys. sci. 4 (2022) 146–156 148 where the forward-backward splitting method above numerically integrates the gradient flow differential equations. the time-stepping discretization of eqns. (5) and (6) is expressed as follows: xk+1 := xk − h [ (1 − θ)∇ f (xk) + θ∇ f (xk+1) ] , (8) where θ ∈ [0, 1] is a parameter. when θ = 0, the above discretization is the explicit euler’s scheme and on the other hand, the implicit backward euler when θ = 1. 3.2. discretization of the objective function the discretization of the objective function in eqn. (1) above, the trapezoidal rule will be used given as;∫ t t0 f (x)d x = h 2 n−1∑ i=0 [ f (xi) + f (xi+1)], (9) expanding eqn. (9) above, yields; j(x, u) ≈ h 2 p n−1∑ i=0 (x2i + x 2 i+1) + h 2 q n−1∑ i=0 (u2i + u 2 i+1), (10) where for ti = t0 + ih then x(ti) ≈ x(t0 + ih) = xi, u(ti) ≈ u(t0 + ih) = ui for i = 1, 2, ..., n and n = (t − t0)/h. rearranging eqn. (10) into matrix form yields; minimize j(z) = 1 2 zt mz + c, (11) where z = (x1, x2, ...xn, u0, u1, ..., un )t ∈ r2n+1, c = 12 x 2 0hp ∈ r and m ∈ r(2n+1)×(2n+1) is a (2n +1)×(2n +1) matrix operator with diagonal entries mi j as expressed below: [mi j] =  2hp, 1 ≤ i ≤ n − 1, j = i, hp, i = n, j = i, hq, i = n + 1, 2n + 1, j = i, 2hq, n + 2 ≤ i ≤ 2n, j = i, 0, elsewhere . (12) 3.3. discretization of the constraint function the differential constraint ẋ(t) = f (t, x(t), u(t)) = ax(t) + bu(t) was discretized by slotting it into the euler’s scheme xi+1 = xi + ẋh with ti = t0 + ih, to arrive at xi+1 = cxi + dui, (13) where c = 1 + ha, d = hb and i = 0, 1, 2, ..., n − 1. and upon expansion and re-arrangement of eqn. (13) into matrix form yields gz = k express as g(z) = gz − k = 0, (14) where z is of dimension (2n + 1) × 1 and g is of dimension n × (2n + 1) as shown below; g =  1 0 · · · 0 −d 0 · · · · · · 0 −c 1 ... ... 0 −d 0 ... ... 0 ... ... 0 ... ... ... ... 0 ... ... −c 1 0 · · · 0 −d 0  and k =  cx0 0 ... 0  , with the entries defined as follows: gi j = 1 for 1 ≤ i ≤ n and j = i; gi j = −d for 1 ≤ i ≤ n and j = n + i; gi j = −c for 2 ≤ i ≤ n and j = i − 1 while gi j = 0 otherwise. hence, eqns. (11) and (14) give the quadratic representation for the constrained minimization problem as minimize j(z) = 1 2 zt mz + c, subject to gz − k = 0. (15) 3.4. lagrangian formulation of the constraint problem the unconstrained form of the optimal control problem can be obtained using the lagrangian function on eqn. (15) as stated below: l(z,λ) = 1 2 zt mz + c + λt (gz − k), (16) l(z,λ) = j(z) + λt g(z), (17) where λ = (λ1,λ2, ...,λn )t ∈ rn is the lagrangian multiplier. the pair of points (z∗,λ∗) is said to be a stationary pair points of eqn. (17) if the following first order necessary optimality conditions hold: lz (z∗,λ∗) = jz (z ∗) + gtz (z ∗)λ∗ = 0, (18) lλ(z∗,λ∗) = g(z ∗) = 0, (19) where lz (z,λ) = ( ∂l(z,λ) ∂x1 , · · · , ∂l(z,λ) ∂un )t , lλ(z,λ) = ( ∂l(z,λ) ∂λ1 , · · · , ∂l(z,λ) ∂λn )t , jz (z) = ( ∂j(z) ∂x1 , · · · , ∂j(z) ∂un )t , denote the gradient vectors of l and j with the dimensions (2n + 1)×1, n ×1 and (2n + 1)×1 respectively. similarly, we 148 olotu et al. / j. nig. soc. phys. sci. 4 (2022) 146–156 149 have the jacobian of g with dimension n ×(2n + 1) represented below: gz (z) =  ∂g1 (z) ∂x1 ∂g1 (z) ∂x2 . . . ∂g1 (z) ∂un ∂g2 (z) ∂x1 ∂g2 (z) ∂x2 . . . ∂g2 (z) ∂un ... ... ... ... ∂gn (z) ∂x1 ∂gn (z) ∂x2 . . . ∂gn (z) ∂un  . (20) in order to fulfill the above optimality conditions, the continuous gradient flow reformulation with the given initial condition z0 is derived as follows; dz dt = −lz (z(t),λ(z(t))), z(0) = z0, (21) dz dt = ϕ(z). (22) substituting eqn. (18) into (21) yields, dz dt = −[jz (z(t)) + g t z (z(t))λ 0(z(t))], z(0) = z0 (23) such that the choice of λ0(z) is to satisfy lz (z,λ 0(z)) = τgtz (z)g(z), (24) in the least square sense, where τ > 0 is a parameter. substituting eqn. (18) into (24) yields, jz (z) + λ 0(z)gtz (z) = τg t z (z)g(z). (25) obtaining the least squares solution of λ0(z) in en. (25) yields, λ0(z)gtz (z) = τg t z (z)g(z) − jz (z), (26) λ0(z) = [gtz (z)] † {τgtz (z)g(z) − jz (z)}, (27) where the superscript ‘†’ refers to the pseudo inverse of gtz (z) in layton [9]. theorem 3.1. the constraint equations for the discretized optimal control problem g(z) satisfies the linearly independent constraint qualification (licq) for any feasible point z∗. the licq states that the gradient of the active inequality constraints and the gradient of the equality constraints are linearly independent at z∗. eyiu licq holds if g1z (z), g2z (z), ..., gnz (z) are linearly independent for any feasible point z∗. thus α1g1z (z) + α2g2z (z) + ... + αn gnz (z) = 0 (28) implies that αi = 0. hence, the dimension of the row space of g must be n. applying the row-reduction process to g in eqn. (3.3), we have ḡ∗ = [i|v ] written as ḡ∗ =  1 0 · · · 0 −d 0 · · · · · · 0 0 1 ... ... −cd −d ... ... ... ... ... ... 0 ... ... ... ... ... 0 · · · 0 1 −cn−1d · · · −cd −d 0  ,(29) where i ∈ rn×n and v ∈ rn×(n+1). the entries g∗i j of ḡ∗ can be defined as follows: [g∗i j] =  1, 1 ≤ i ≤ n, j = i; −ckd, k + 1 ≤ i ≤ n, j = n + i − 2; for k = 1, 2, ..., n − 1, 0, elsewhere. (30) thus, rank of ḡ∗ equals rank of g and αi = 0 are the parameters that make g1z (z), g2z (z), ..., gnz (z) linearly independent for any feasible point z∗ and g(z) satisfies licq. from theorem (3.1), we have for any feasible point z∗ that ḡ∗ ∈ rn×(2n+1) is re-structured into a n×n non-singular (invertible) gram matrix g∗ given as g∗ = gz (z)g t z (z). (31) multiplying both sides of eqn. (25) by gz (z) arrives at gz (z)(jz (z) + λ 0(z)gtz (z)) = τgz (z)g t z (z)g(z). (32) therefore, with g∗(z) = gz (z)gtz (z), eqn. (32) becomes gz (z)jz (z) + gz (z)λ 0(z)gtz (z) = τg∗g(z) and simplified to get g∗λ 0(z) = τg∗g(z) − gz (z)jz (z). (33) pre-multiplying eqn. (33) by g−1∗ yields g−1∗ g∗λ 0(z) = g−1∗ τg∗g(z) − g −1 ∗ gz (z)jz (z), (34) and solving for λ0(z) to obtain λ0(z) = τg(z) − g−1∗ gz (z)jz (z), (35) where eqns. (27) and (35) are equivalent. consequently, eqn. (27) can be express as λ0(z) = (gtz (z)) †[τgtz (z)gz (z) − jz (z)], = τgz (z) − g −1 ∗ jz (z)gz (z), where g−1∗ = [g t z (z)gz (z)] −1, hence the pseudo-inverse of [gtz (z)] is [gtz (z)] † = g−1∗ gz (z). however, eqn. (35) is true when z is sufficiently close to an equilibrium point z∗. the choice of λ0(z) has the advantage of reducing the complexity in the evaluation of the gradient of λ0 with respect to z. thus substituting λ0(z) in eqn. (35) above into eqn. (23) yields; ż = ϕ(z) = −(jz (z) + g t z (z)[τg(z) − g−1(z)gz (z)jz (z)]). (36) in testing the convergence of the gradient flow formulation expressed in eqns. (23) and (24), we then considered the licq theorem below. 149 olotu et al. / j. nig. soc. phys. sci. 4 (2022) 146–156 150 theorem 3.2. suppose that (z∗,λ∗) is a stationary point, then 1. licq holds at z∗. 2. for any z̄ satisfying gz (z∗)z̄ = 0, then, z̄t lzz (z∗,λ∗)z̄ > 0 where z̄ = z − z∗. it indicates that the pair (z(t),λ0(z(t))) tends to (z∗,λ∗) as t → ∞ provided the starting point z0 is sufficiently close to point z∗. proof : we start by deploying the poincare-lyapunov stability theorem stated in haddad et al. [8] and morris [10] thus: theorem 3.3 (liapunov stability). let x∗ be an equilibrium point for x ′ = j(x). let l : ω → r be a differentiable function defined on an open set ω containing x∗. suppose further that 1. l(x∗) = ω and l(x) > 0 if x , x∗; 2. l̇ ≤ 0 in ω−x∗; then x∗ is stable. furthermore, if l also satisfies 3. l̇ < 0 in ω − x∗, then x∗ is asymptotically stable, it then requires we show that z∗ is an asymptotically stable point for ż = ϕ(z) if ϕ(z) is continuously differentiable and the original of the linearized system ẏ = ϕz (z ∗)y, (37) where y := z − z∗, is exponentially stable (i.e. all eigenvalues of ϕz (z∗) are strictly negative). it was however noted that at z∗, (z∗,λ0(z∗)) satisfies eqns. (18) and (19) because the right hand side of eqn. (24) is identically zero; implying that λ∗ = λ0(z∗). applying taylor’s expansion to the rhs of eqn. (36), about the point z∗, gives the following linearized equation dz dt = ϕ(z) + ϕz (z)y. (38) comparing eqn. (38) with the rhs of eqn. (23) yields dz dt = ϕ(z∗) + ϕz (z ∗)(z − z∗). (39) recall that l(z,λ) = j(z) + λt g(z), (40) lz (z,λ) = jz (z) + λ t gz (z), (41) lzz (z,λ) = jzz (z) + g t zz (z)λ, (42) then from eqn. (38), dz dt = ϕ(z∗) + ϕz (z ∗)y, [where y = z − z∗]. (43) suppose ϕ̇(z∗) = lz (z,λ) which is equal to eqn. (41); then differentiating eqn. (41) arrives at ϕz (z ∗) = [jzz (z ∗) + λt gzz (z ∗) + gz (z ∗)λz ]y, = [lzz (z ∗,λ∗)) + gz (z ∗)λz ]y. (44) therefore, eqn. (39) becomes dz dt = −[lz (z ∗,λ∗) + [lzz (z ∗,λ∗) + gtz (z ∗)λ0z (z ∗)](z − z∗)], (45) where ϕ(z∗) = lz (z∗,λ∗) and ϕz (z∗) = lzz (z∗,λ∗) + gtz (z ∗)λ0z (z ∗). considering the unknown coefficient λ0z (z ∗) and setting lz (z∗,λ∗) = 0 yields, dz dt = −[lzz (z ∗,λ∗) + gtz (z ∗)λ0z (z ∗)](z − z∗)]. (46) we seek to differentiate both sides of eqn. (32) with respect to z and evaluating at z∗. firstly, we differentiate the lhs of eqn. (32) as follows: gz (z ∗) d dz [jz (z) + λ 0(z)gtz (z)] + [jz (z) + λ 0(z)gtz (z)] d dz gz (z) = gz (z ∗)[jzz (z ∗) + λ0(z ∗) d dz gtz (z ∗) + gtz (z) d dz λ0(z)] + [jz (z ∗)gzz (z ∗)λ0(z)gtz (z ∗)gzz (z ∗) (47) and arrives at [jzz (z ∗)gz (z ∗) + λ0gtzz (z ∗)]gz (z ∗) +[jz (z ∗) + λ0z (z ∗)gtz (z ∗)]gzz (z ∗) + λ0z (z ∗)gtz (z ∗)gz (z ∗). (48) secondly, we differentiate the rhs of eqn. (32) as follows; τ d dz [ gtz (z)gz (z)g(z) ] = τ [ gtz (z) d dz gz (z)g(z) +gz (z)g(z) d dz gtz (z) ] , = τ[gtz (z ∗)gz (z ∗)]gz (z ∗) +τ[gtz (z ∗)gzz (z ∗)]g(z∗) + τ[gtzz (z ∗)]gz (z ∗)g(z∗). (49) equating eqns. (48) and (49), we have [jzz (z ∗)gz (z ∗) + λ0gtzz (z ∗)]gz (z ∗) + [jz (z ∗) + λ0z (z ∗)gtz (z ∗)]gzz (z ∗) + λ0z (z ∗)gtz (z ∗)gz (z ∗) (50) = τ[gtz (z ∗)gz (z ∗)]gz (z ∗) + τ[gtz (z ∗)gzz (z ∗)]g(z∗) + τ[gtzz (z ∗)]gz (z ∗)g(z∗). (51) 150 olotu et al. / j. nig. soc. phys. sci. 4 (2022) 146–156 151 applying eqns. (18) and (19) on eqn. (51), we have gz (z ∗)[lzz (z ∗,λ∗) + gtz (z ∗)λ0z (z ∗)] = τgz (z ∗)gtz (z ∗)gz (z ∗), (52) where lzz (z∗,λ∗) = jzz (z∗)gz (z∗) +λ0gtzz (z ∗) and lz (z∗,λ) = jz (z∗) + λ0z (z ∗)gtz (z ∗)). eqn. (52) becomes gz (z ∗)lzz (z ∗,λ∗) + g(z∗)λ0z (z ∗) = τg(z∗)gz (z ∗), (53) where g(z∗) = gtz (z ∗)gz (z∗). making λ0z (z ∗) the subject of eqn. (53) yields λ0z (z ∗) = −g−1(z∗)gz (z ∗)lzz (z ∗,λ∗) + τgz (z ∗). (54) recall that y = z − z∗, (55) ⇒ dy dt = dz dt − dz∗ dt , (56) and dy dt = dz dt . (57) substituting eqn. (54) into eqn. (46) and considering eqn. (57) yields dy dt = −[lzz (z ∗,λ∗) + gtz (z ∗)(−g−1(z∗)gz (z ∗)lzz (z ∗,λ∗) +τgz (z ∗))]y, (58) dy dt = −[lzz (z ∗,λ∗) + +gtz (z ∗)gz(z ∗)lzz (z ∗,λ∗)g−1(z∗) −τgz (z ∗)gtz (z ∗)]y, (59) dy dt = [−lzz (z ∗,λ∗)[i − gtz (z)g −1(z)gz (z)] −τgtz (z)gz (z)]y. (60) since z∗ is zero at equilibrium point, then dy dt = −[(i − gtz (z)g −1(z)gz (z))lzz (z ∗,λ∗) + τgtz (z)gz (z)]y. (61) let s (z) = i − gtz (z)g −1(z)gz (z), (62) v (z) = gz (z)g t z (z), (63) where i is the n × n identity matrix. therefore, eqn. (61) becomes dy dt = −[s (z∗)lzz (z ∗,λ∗) + τv (z∗)]y. (64) from above, eqns. (62) and (63) are orthogonal where eqn. (62) is an operator which projects a vector to the null space of gz (z), because gz (z)s (z) = 0. multiplying eqn. (64) from the left by gz (z∗) yields gz (z ∗) dy dt = −[τgz (z ∗)v (z∗)]y. (65) eqn. (65) is equivalent to dy dt = −[τv (z∗)]y. (66) hence, since τ > 0, it remains to show that v (z∗) is positive definite. recall that v (z∗) = gtz (z ∗)gz (z∗) and gz (z∗) = g; where g whose dimension is n × (2n + 1) is given by eqn. (3.3). therefore, q = ggt with dimension n × n is given by q =  1 + d2 −c 0 · · · 0 −c α −c 0 ... 0 ... ... ... 0 ... ... ... ... −c 0 · · · 0 −c α  , (67) where α = 1 + c2 + d2. theorem 3.4 (sylvester’s criterion). a real, symmetric matrix is positive definite if and only if all the principal minors are positive definite. gilbert [7]. considering the implementation of theorem 3.4 on the matrix q, we seek to establish that the matrix is positive definite. firstly, q is real since for every qi, j ∈ q, qi, j ∈ r (i.e. all the diagonal entries in eqn. (67) are real). secondly, q = gtz (z ∗)gz (z∗) is symmetric since (q)t = [gtz (z ∗)gz (z∗)]t = gtz (z ∗)gz (z∗) = q. however, we also seek to establish that the principal minors qi, ∀i = 1, 2, 3, ...n, of q are all positive definite. when i = 1, 2, 3, and 4, q1, q2, q3 and q4 are 1, 1 + c2, 1 + c2 + c4 and 1 respectively; and are all positive ∀c, d > 0. we then assumed, by mathematical induction, that it is true for qk when i = k and qk+1 when i = k + 1. and by cholesky, it was shown that qk+1 is positive definite and that all the principal minors of q are positive definite [see appendix]. we, therefore, concluded that the matrix q = ggt is positive definite. 3.5. construction of the modified gradient flow method (mgfm) we now considered the discretization of the formulation of eqns. (23) and (24) using three level implicit time discretization scheme with splitting parameters θ1 and θ2 where θ3 = (1−θ1 − θ2). let 0 = t0 < t1 < t2 < t3 < ... < tk = t be a sequence time points for the time t ≥ t0, hk = tk+1 − tk and δzk+1 = zk+2 − zk being the sequence of solution distance between two successive 151 olotu et al. / j. nig. soc. phys. sci. 4 (2022) 146–156 152 equilibrium points. we discretize eqn. (23) by the following implicit time stepping algorithm; zk+2 − zk+1 hk+1 = −[θ3 lz (zk+2,λ 0 k+2)+ θ2 lz (zk+1,λ 0 k+1) + θ1 lz (zk,λ 0 k )]. (68) simplified to zk+2 − zk+1 = −hk+1[θ3 lz (zk+2,λ 0 k+2)+ θ2 lz (zk+1,λ 0 k+1) + θ1 lz (zk,λ 0 k )]. (69) employing taylor’s expansion about (zk+2,λ0k+2) arrives at zk+1 = zk + hkzk + o||hk|| 2, (70) lz (zk+2,λ 0 k+2) = lz (zk,λ 0 k ) + h(zk,λ 0 k )δzk+1 + o||δzk+1 || 3, (71) lz (zk+1,λ 0 k+1) = lz (zk,λ 0 k ) + h(zk,λ 0 k )δzk + o||δzk || 3, (72) where, δzk = zk+1 − zk and h(zk,λk) = lzz (zk,λ0k ) + lzλ0k ((zk,λk))λz (zk) is the hessian of l(z,λ). substituting eqns. (70), (71) and (72) into eqn. (69) yields zk+2 = zk + hkz ′ k + o||hk|| 2 −hk+1[θ3(lz (zk,λ 0 k ) +h(zk,λ 0 k )δzk+1 + o||δzk+1 || 3) + θ2(lz (zk,λ 0 k ) + h(zk,λ 0 k )δzk +o||δzk || 2) + θ1 lz (zk,λ 0 k )]. (73) neglecting the higher order terms in eqn. (73) and solving the resultant equation for zk+2 yields zk+2 = zk + hkzk − hk+1θ3 lzk −hk+1θ3 hzkzk+2 + hk+1θ3 hzkzk −hk+1θ2 lzk + +hk+1hkθ2 hzk lzk −hk+1θ1 lzk. (74) for equal step size, hk+1 = hk with θ1 = 0 we obtain zk+2 expressed below zk+2 = zk + hkzk − hkθ3 lzk − hkθ3 hzkzk+2 +hkθ3 hzkzk − hkθ2 lzk +h2kθ2 hzk lzk. (75) in collecting like terms and factorizing eqn. (75), it becomes (i + hkθ3 hzk)zk+2 = (i + hkθ3 hzk )zk − (θ2 + θ3)hk lzk + h2kθ2 hzk lzk, = (i + hkθ3 hzk )zk + (−hk i + h2kθ2 hzk)lzk. (76) making zk+2 the subject of formula in eqn. (76) yields zk+2 = (i + hkθ3 hzk) −1[(i + hkθ3 hzk)zk +(−hk i + h 2 kθ2 hzk)lzk], = zk − hk(i + hkθ3 hzk) −1 ×[i − hkθ2 hzk]lzk, for any θ2,θ3 ∈ [0, 1] and θ2 + θ3 = 1 by the time-stepping algorithm for convex functions. also, from eqn. (76), we obtained (i + hkθ3 hzk )(zk+2 − zk) = −hk(i + hkθ2 hzk)lzk. (77) given that δk+1 = zk+2 − zk and θ2 = 0, eqn. (77) becomes (i + hkθ3 hzk )δzk+1 = −hk lzk. (78) thus, for any initial guess z0, eqn. (77) defines a series converging to the solution of the optimal control problem stated in eqns. (1) to (3). hence, the modified gradient flow (mgf) algorithm for the discretized optimal control problem (ocp) is stated as follows: mgf algorithm for discretized ocp step 1: input the operator g ∈ rn×(2n+1), z0 ∈ r2n+1, parameters θ2,θ3 ∈ [0, 1], sequence of time step-size {hk} and tolerance (σ > 0) sufficiently small. set k = 0. step 2: compute λ0zk using eqn. (54). step 3: solve for δzk+1 in the system using the cgm (i + hkθ3 hzk )δzk+1 = −hk lzk (where δzk+1 = zk+2 − zk) step 4: update: zk+2 = zk + δzk+1 . step 5: stop the process if ||hzk || ≤ σ, otherwise, set k = k + 1 and repeat step 3. 3.6. error and convergence analysis the convergence analyses of the mgfm requires us to shown the rate and type of convergence of the algorithm for varying values of parameters θ1,θ2 and θ3. we shall show that the algorithm is super-linearly convergent for some θ2 and θ3 and quadratically convergent if θ1 = θ2 = 0 and θ3 = 1. theorem 3.5. let {zk} be the sequence defined in eqn. (77) and (zå,λå) be the solution of eqns. (1) and (2). if the initial guess z0 is sufficiently close to z, then we have 1. zk converges linearly to zå if θ1,θ2,θ3 ∈ [0, 1] and hk > 0 is sufficiently small; 2. zk converges to zå quadratically if θ2 = 0, θ3 = 1 and hk →∞. we shall prove the two cases separately. 152 olotu et al. / j. nig. soc. phys. sci. 4 (2022) 146–156 153 • case 1. θ1,θ2,θ3 ∈ [0, 1]. from eqn. (77) we have: (i + hkθ3 hzk )(zk+2 − zk) = −hk(i − hkθ2 hzk )lzk . hence, zk+2 = zk − hk[(i − hkθ2 hzk )lzk + θ3 hzk (zk+2 − zk)]. let (z∗,λ∗) be the minimum points of eqns. (16) and (17) and ek+1 = zk+1 − z∗. subtracting z∗ from both sides of the above equation and noting that we have ek+1 = ek − hk[(i − hkθ2 hzk )(lzk − lz∗k ) +θ3 hzk (ek+1 − ek)]. (79) thus, by the mean value theorem, ek+1 = ek − hk[(i − hkθ2 hzk )h ζk zk ek +θ3 hzk (ek+1 − ek)], (80) where ζk denotes a point in the line segment connecting zk and z∗. taking the norm of both sides, we have ||ek+1|| ≤ γ(zk,λ 0 k,ζk,θ1,θ2,θ3, hk)||ek||, (81) where γ(zk,λ 0 k,ζk,θ1,θ2,θ3, hk) = ||{i − hk(i + θ3 hzk ) −1(i − θ2 hzk )h ζk zk }||. (82) from eqn. (81), we see that if γ(zk,λ0k,ζk,θ1,θ2,θ3, hk) < 1, then ek converges to zero linearly. to show this, we need to prove that hzk is positive definite. but by theorems and , hzk is positive definite. with these, when zk is sufficiently close to z∗, from eqn. (82), it is seen that if θ1,θ2,θ3 ∈ [0, 1] and hk sufficiently small, we have γ(zk,λ 0 k,ζk,θ1,θ2,θ3, hk) ≤ 1 − hk(1 − hkθ2λkmin)λkmin 1 + hkθ3λkmax  < 1, (83) where λkmin and λ k max denote the minimum and maximum eigenvalues of hzk , respectively. therefore, eqn. (81) implies that limk→∞ ek = 0 is linear. this proves that zk converges linearly to zå if θ1,θ2,θ3 ∈ [0, 1] and hk > 0 is sufficiently small. • case 2. θ3 = 1. from eqn. (77), when θ3 = 1 and hk = h, we have zk+2 − zk h = −(lzk + hzkδzk+1 ). when h →∞, the above equation becomes hzkδzk+1 + lzk = 0. (84) eqn. (84) coincides with the equation resulting from the application of the newton’s method to lz = 0. hence, zk converges quadratically to z∗ if z0 is sufficiently close to z∗. figure 1. state behaviors in various algorithms figure 2. control behaviors in various algorithms 4. examples and numerical results the numerical examples and respective results are presented below based on the construction of modified gradient flow method (mgfm) for discretized optimal control problems. the results show the robustness and efficiency of the scheme (mgfm) as compared with existing schemes. example 1 considering the optimal control problem presented by [1] minimize i(x, u) = ∫ 1 0 (x2(t) + u2(t))dt, (85) subject to ẋ = 2x(t) + 5u(t), x(0) = 1. (86) the analytical solution is given as( x λ ) = ( 1.0000 −0.5908 ) e−5.3852t, (87) and the control variable is given as u(t) = − 1.4770e−5.3852t. the solution of eqns. (85) and (86) were obtained, using h = 0.1 and hk = 20, by the modified gradient algorithm for discretized optimal control problems and the results were compared with the existing algorithms as shown in table 1 below. however, the step-size h = 0.1 was chosen for the purpose of illustration. meanwhile smaller choices of step-size could be used for better refinement. 153 olotu et al. / j. nig. soc. phys. sci. 4 (2022) 146–156 154 algorithm objective functional value analytical method 0.2953894 adekunle [1] 0.3033885 akeremale [2] 0.2955932 mgfm 0.2955879 table 1: comparison of objective functional values of existing methods for example 1 results: from table 1 above, it was observed that the objective functional value (0.2955879) for the developed mgfm agreed favourably with the value (0.2953894) obtained from the analytical method and better than the values obtained by adekunle [1] and akeremale [2] from conjugate gradient method (cgm) and extended conjugate gradient method (ecgm) respectively. table 2a: errors in the state variables for example 1 |x(t) − x̂(t)| (×10−3) t adekunle [1] akeremale [2] mgfm 0.0 0 0 0 0.1 3.2204 0.5528 0.5525 0.2 3.8300 0.3680 0.3677 0.3 3.4841 0.3414 0.3410 0.4 2.9424 0.4447 0.4442 0.5 2.5506 0.6968 0.6962 0.6 2.4892 1.1667 1.1650 0.7 2.9118 1.9867 1.9858 0.8 4.0417 3.4014 3.4005 0.9 6.2660 5.8292 5.8279 1.0 10.266 9.9981 9.9789 |u(t) − û(t)| (×10−4) t adekunle [1] akeremale [2] mgfm 0.0 38.796 38.419 38.4202 0.1 18.131 0.6668 0.6555 0.2 7.9765 0.2654 0.2562 0.3 3.1973 0.0149 0.0755 0.4 1.1303 0.1891 0.1949 0.5 4.3839 0.4248 0.4294 0.6 4.8076 0.7726 0.7761 0.7 1.0193 1.3417 1.3442 0.8 2.0619 2.3045 2.3062 0.9 3.8245 3.9497 3.9505 1.0 6.7705 6.4026 6.4027 table 2 above shows the errors in the state and control variables for example 1 while the errors generated with the mgfm were minimal compared with the errors generated from adekunle [1] and akeremale [2]. figures 1 and 2 below clearly show the algorithm objective functional value analytical method 0.5647 adekunle [1] 0.5746 akeremale [2] 0.5649 mgfm 0.5648 comparisons of the behaviors of the state and control trajectories to the various algorithms respectively. it was observed that the trajectories for both the state and control variables overlap due to closeness in results. example 2 considering the optimal control problem presented by [1] minimizej(x, u) = ∫ 1 0 (x2(t) + u2(t))dt, (88) subject to ẋ = 1.705x(t) + 3.021u(t), (89) x(0) = 1. (90) table 3: comparison of objective functional values of existing methods for example 2 table 4a: errors in the state variables for example 1 |x(t) − x̂(t)| (×10−4) t adekunle [1] akeremale [2] mgfm 0.0 10.000 10.0000 1.0000e 0.1 3.2204 5.528 5.5253 0.2 2.0509 4.347 3.9394 0.3 3.3242 2.604 2.3946 0.4 3.2376 2.435 2.2688 0.5 2.9852 2.646 2.4997 0.6 2.6832 3.2433 3.0992 0.7 2.3921 4.2860 4.1303 0.8 2.1365 5.8933 5.7138 0.9 1.9144 8.2567 8.0424 1.0 1.7012 10.716 11.403 table 4b: errors in the control variables for example 1 results: from table 3 above, it was observed that the objective functional value, 0.5648, for mgfm agreed favourably with the value (0.5647) obtained from the analytical method; and better than the values obtained by adekunle [1] and akeremale [2] from conjugate gradient method (cgm) and extended conjugate gradient method (ecgm) respectively as expressed in table 3 above. results: table 4 above shows the errors in the state and control variables for example 2 where the errors generated from the use of mgfm were minimal compared to the errors generated from same examples in adekunle [1] and akeremale [2]. figures 3 and 4 below clearly show the comparisons of the behaviors of the state and control trajectories to the various algorithms 154 olotu et al. / j. nig. soc. phys. sci. 4 (2022) 146–156 155 |u(t) − û(t)|(×10−4) t adekunle [1] akeremale [2] mgfm 0.0 0.2928 0.3404 0.2893 0.1 1.8337 0.5220 0.3179 0.2 1.0622 1.8094 2.4019 0.3 5.9872 1.1203 2.3334 0.4 3.3442 0.4000 0.4941 0.5 1.9971 0.7234 0.7965 0.6 1.5137 1.1255 1.1823 0.7 1.6411 1.6579 1.7019 0.8 2.2516 2.3872 2.4210 0.9 3.3101 3.4037 3.4311 1.0 2.9285 3.4045 2.8932 figure 3. state behaviors in various algorithms figure 4. control behaviors in various algorithms respectively. it was observed that the trajectories for both the state and control variables overlap due to closeness in results. 5. conclusion this paper developed a new scheme for the solution of a quadratic optimal control problem with linear equality constraint. the method is a gradient flow formulation that requires discretization techniques and a splitting parameter approach for the gradient of the equation whose solution converges to a local minimum of the original problem; either linearly or quadratically. two linear equality constrained optimal control problems were considered and their results compared with the analytical and two known existing schemes. thus, it was observed that the new scheme compared favorably with the analytic solutions and performed better than the conjugate gradient method used in [2] in terms of convergence and accuracy. it is therefore obvious that this developed algorithm, due to its robustness and efficiency, would provide a new platform for solving optimal control problems with or without box constraints, equality or inequality constraints or with delay or non-delay dynamical systems. acknowledgements the authors are grateful to the editors and the anonymous reviewers for their constructive comments and suggestions. references [1] a. i. adekunle, “algorithm for a class of discretized optimal control problems", m.tech. thesis, federal university of technology, akure, nigeria (2011) (unpublished). [2] o. c. akeremale, “optimization of quadratic constrained optimal control problems using augumented lagrangian method", m.tech. thesis, federal university of technology, akure, nigeria (2012) (unpublished) [3] w. behrman, “an efficient gradient flow method for unconstrained optimization", phd thesis, stanford university (1998) (unpublished). [4] j. t. betts, “practical methods for optimal control problem using non linear programming", siam, philadelphia, (2001). [5] y. evtushenko, “generalized lagrange multipliers technique for nonlinear programming", jota, 21 (1977) 121. [6] y. g. evtushenko & v. g. zhadan, “stable barrier projection and barrier newton methods", nonlinear programming optimization methods and software, 3 (1994) 237. [7] g. t. gilbert, “positive definite matrices and sylvester’s criterion", the american mathematical monthly, taylor & francis, 98 (1991) 44. [8] w. m. haddad, s. g. nersesov & v.s. chellaboina, “lyapunov function proof of poincare’s theorem", international journal of systems science, 35 (2004) 287. [9] j. b. layton, “efficient direct computatioion of the pseudo-inverse and its gradient", international journal for numerical methods in engineering, 40 (1997) 4211. [10] w. h. morris, s. stephen & l. d. robert, “differential equations, dynamical systems, and an introduction to chaos", elsevier academic press, usa (2004) 194. [11] o. olotu & k. a. dawodu, “quasi-newton embedded augmented lagrangian algorithm for discretized optimal proportional control problems", journal of mathematical theory and modeling, 3 (2013a) 67. [12] o. olotu & k. a. dawodu, “on the discretized algorithm for optimal proportional control problems constrained by delay differential equations", journal of mathematical theory and modeling, 3 (2013b) 157. [13] o. olotu & s. a. olorunsola, “an algorithm for a discretized constrained, continuous quadratic control problem", journal of applied sciences, 8 (2006) 6249. [14] l. s. pontryagin, v. g. boltyanskii, r. v. gamkrelidze & e. f. mishchenko, “the mathematical theory of optimal processes", interscience publishers, london (1962). [15] s. wang, x. q. yang k. l. teo “a unified gradient flow approach to constrained nonlinear optimization problems", computational optimization and applications, 25 (2003) 251. 155 olotu et al. / j. nig. soc. phys. sci. 4 (2022) 146–156 156 appendix let qi represent the leading principal minor of q, ∀i = 1, 2, 3, ..., n of the matrix q = ggt ; we then considered the determinants of the principal minors q1, q2, q3, ... as follows: for i = 1, q1 = 1 + d2 > 0, ∀d. for i = 2, q2 = [ 1 + d2 −c −c 1 + c2 + d2 ] , (91) such that det (q2) = 1 + 2d2 + c2d2 + d4 > 0 for all c, d. for i = 3, q3 =  1 + d2 −c 0 −c 1 + c2 + d2 −c 0 −c 1 + c2 + d2  , (92) such that det (q3) = 1 > 0 ∀c. therefore, |q3| is positive. for i = 4, q4 =  1 + c2 −c 0 −d −c 1 + c2 −c 0 0 −c 1 0 −d 0 0 2d2  , (93) such that det. (q4) = d2 > 0. ∀c, d. therefore, det (q4) = d2 > 0, ∀d, i.e. |q4| is positive. hence, by mathematical induction, since it is true for values of qi, i = 1, 2, 3, 4, then we assume that it is true for i = k, i.e. qk, while we shall prove that it is true for i = k + 1, i.e. qk+1. set σ2 = 1 + c2 > 0, c2 > 0, 1 > 0, 2d2 > 0 and d2 > 0 where 1 + c2, c2, 1, 2d2 and d2 are principal diagonals of q. thus, qk+1 = [ qk βk+1 βtk+1 σ 2 ] , (94) where βk+1 = [ q1,k+1 q2,k+1 . . . qk,k+1 ]t . (95) by cholesky, qk+1 is said to be positive definite if there exists a lower triangular matrix li, j such that qk+1 = llt . that is,[ qk βk+1 βtk+1 c 2 ] = [ hk 0 ltk+1,1 lk+1,k+1 ] [ htk l t k+1,1 0 lk+1,k+1 ] , where hk is a lower triangular matrix with positive diagonal entries. however, eqn. (5) becomes [ qk βk+1 βtk+1 c 2 ] = [ hk htk hk lk+1,1 ltk+1,1 h t k l t k+1,1 lk+1,1 + l 2 k+1,k+1 ] , (96) such that qk = hk htk , β t k+1 = l t k+1,1 h t k , βk+1 = hk lk+1,1 and σ2 = ltk+1,1 lk+1,1 + l 2 k+1 lk+1. thus, since the diagonal entries of hk are greater than zero, it is non-singular and therefore the linear system of the equation has a unique solution given by lk+1,1 = h−1k βk+1 and a positive value for lk+1,k+1 can be obtained if σ2 − ltk+1 lk+1 > 0. hence, 0 < |qk| = det [ qk βk+1 βtk+1 σ 2 ] , = qk[σ 2 i −βtk+1(qk) −1βk+1]. substituting the values of βk+1 and qk, we have q = qk[σ 2 i − (hk lk+1,1) t (hk h t k ) −1 hk lk+1,1], = qkσ 2 − ltk+1,1 lk+1,1, where (hk h t k ) −1 = q−1k , qk = [σ 2 − ltk+1,1 lk+1,1]. since detqk > 0, it follows that σ2 − ltk+1,1 lk+1,1 > 0. hence, lk+1,k+1 = √ σ2 − ltk+1,1 lk+1,1. (97) thus li > 0 arises from the fact that g is positive definite (licq). hence, the pair (z(t),λ0(z(t))) tends to (z∗,λ∗) as t → ∞ provided the starting point z0 is sufficiently close to point z∗. 156 j. nig. soc. phys. sci. 2 (2020) 91–105 journal of the nigerian society of physical sciences original research optimal power point on the i-v curve of a photovoltaic solar system (modelling and analysis) baba alfaa, yakubu adamua,∗, daniel alberto pena perezb adepartment of physics, ibrahim babangida university, lapai niger state nigeria. bav, paseo oriente 950, cd industrial, 36541, lapem irapuato guanajuato méxico abstract motivated by recent interest in improving the performance of pv cells, we explored the optimal power point in photovoltaic (pv) cells by using three different topologies to compare its function and efficacy. firstly, we investigate the consequence of connecting a pv directly to the load. secondly, the efficacy of an electronic device that generates a pulse width modulation (pwm) to control a boost converter connected to the pv panel and the load and finally the maximum power point tracking (mppt) method by using the algorithm perturb and observe (p&o), with the implementation of the dc-dc converter between the pv panel and the load. in doing so, a mathematical model of the pv cell was employed and using matlab simulink, the behaviour of the voltage and current signals acquired were analysed, this helps in understanding the performance of the solar cell under different meteorological circumstances, and its effect on the power generated by the pv cell. finally, based on the performance simulations on the three methods implemented, the results were tested for the response time of the mppt under different load conditions in order to ascertain it performance. keywords: photovoltaic cell, perturb & observe (p&o), mppt, load. article history : received: 16 february 2020 received in revised form: 05 may 2020 accepted for publication: 06 may 2020 published: 14 may 2020 c©2020 journal of the nigerian society of physical sciences. all rights reserved. communicated by: b. j. falaye 1. introduction in the last two decades, the field of photovoltaic has made a profound impact in it widespread use, this represents a good progress from a stand-alone pv to a utility interactive system. despite the remarkable growth of this technology, it’s often characterized with low efficiency, which are mostly due to bad solar orientation or adaptation which makes it difficult to extract maximum output power from the pv panel [1]. it is a proven fact that pv panels have a non-linear characteristic (i = f (v )) with a single point where the power generated is maximum (mpp). this maximum power point strongly ∗corresponding author tel. no: 08034511406 email address: yaakubu@ibbu.edu.ng (yakubu adamu) depends on the intensity of solar radiation and the temperature of the location, which changes during the day [2]. such nonlinearity in the characteristic of a pv panel will give rise to a perfect non-coupling between photovoltaic generator (gpv) and the load, thus, creating a problem of transferring the maximum power from the photovoltaic generator (gpv) to the load [3]. therefore, it becomes difficult under these scenarios to extract the maximum output power that could be available at any given weather condition. to get around these problems and extract the maximum power available at the gpv terminals and transfer it to the load at any weather condition all times, an mppt scheme that responds to the weather changes and obtains the highest possible energy at each moment is necessary. though there have been many research approaches in the liter91 b. alfa et al. / j. nig. soc. phys. sci. 2 (2020) 91–103 92 ature that have been tested and implemented, ranging from the simplest method like disrupt & observer (d & o), estimate, perturb and perturb (ep &p), modified perturb and observe (mp&o). etc. to more complex methods, none have reached the desired potential [4]. in this paper, we explored different topology to achieve the optimal power point on a pv cell using the maximum power point tracking (mppt) method which is controlled and implemented by the perturb and observe (p&o) algorithm. this method is intended to track the point of maximum energy generated by the pv panel according to the weather variations. therefore, to show the effectiveness of the method compare with others, a matlab simulink model was used for the numerical analysis of different values of irradiation and temperature and for the analysis of the relationship of module parameters with characteristics curves of pv module. 2. pv model formulation and equations of the pv module in the model formulation, we consider a single-diode model for the pv module, which consists of a few pv cells basically a p-n diode, connected both in series and parallel to obtain the desired voltage. the circuit model is designed to show agreement between a current source and diode such that it can be used to calculate the power supplied by the gpv under all irradiation and temperature conditions (figure 1) [5]. for simplicity, isc (in figure 1) is the current of the source otherwise known as the cell photocurrent isc while rsh and rs are, respectively, the intrinsic shunt and series resistances of the solar cell. 2.1. equations of the pv module the mathematical equations that described the relationship between the cell terminal current (a) and voltage (v) is given by [6], such relationship is based on i-v curve characteristics of the solar cell and pv module, and a generalization from shockley ideal diode equation lead to the following equations: id = io [ exp ( qvd nnskbt ) − 1 ] , (1) where id is the current through the diode, io is the value of the saturation current, and it depends on the temperature, q is the absolute value of the electron’s charge, n represent the quality factor of the diode that is used as adjustment parameter, which ranges between 1 to 5 (n = 1 for the ideal diode model), ns is the number of cells that are connected in series, t is the p-n junction temperature that is generally assumed to be close to the temperature of the cell [7]. now, based on the circuit for single-diode model (figure 1), the current flowing through the parallel resistor ip, is given by: ip = v + rs i rsh . (2) then, analysis of the total current in figure 1 will provide equation (3) i = iph − id − ip, (3) figure 1: equivalent circuit for single-diode model of solar cell [8]. where, iph is the current of the source, id the current of the diode, ip current in the parallel resistor and i represents the current through the series resistors. substituting the equation (1) and (2) into equation (3), and considering that vd = v + irs , this leads to a generalization (based on shockley ideal diode equation) given as: i = i ph − io [ exp ( qv + irs nnskbt ) − 1 ] − v + rs i rsh (4) here, equation (4) is used for the numerical methods to find the electrical current value. 2.2. equations for open-circuit voltage and short-circuit current in the model, there are two critical parameters to measure, the open-circuit voltage and the short circuit current. while, the short-circuit current is affected by the solar radiation, the temperature condition affects the open-circuit voltage [8]. therefore, from theory of semiconductors and photovoltaic, pv cells can be characterized base on the operational point, such as the short-circuit point where i = isc and v = 0 , open-circuit point where v = v oc and i = 0 , and the maximum power point where v = vmpp and i = impp, these conditions can enhance the achievement of maximum electrical power[9]. therefore, considering equation (4), it is possible to determine the isc, using the condition that v = 0 and i = i sc this leads to the equation (5): is c = i ph − io [ exp ( q(rs isc) nnskbt ) − 1 ] − rs isc rsh , (5) where, is c is the short-circuit current in the model (figure 1). thus, to obtain the equation for open circuit point voc we consider the condition that i0c = 0. ioc = 0 = i ph − io [ exp ( q(voc) nnskbt ) − 1 ] − voc rsh (6) from the conditions that v = vmpp and i = impp, (which permit that the maximum electrical power to be achieved), it can be deduce that: impp = i ph−io [ exp ( q(vmpp + rs impp) nnskbt ) − 1 ] − vmpp + rs impp rsh (7) equation (7) suggests that a pv cell may have a hybrid behaviour, which may depend on the values of open-circuit point and short current point, where every region is characterized by 92 b. alfa et al. / j. nig. soc. phys. sci. 2 (2020) 91–103 93 an asymptote in their respective axis [9]. therefore, it is possible to use the tangent of the relationship of i−v curve, to evaluate the transition from the current source to the voltage source controller regions. in that respect, we considered the analysis from equation (4), which lead to the mathematical expression as: di dv = −qio nnskbt ( 1 + di dv ∗ rs ) ∗ ex p [ −q(v + rs i nnskbt ] − 1 rsh ( 1 + di dv ∗ rs ) (8) where the didv is the derivative of the current with respect to the voltage that is evaluated at the maximum power point, hence, from ohm’s law for power, we see that at any point on the i−v curve p = iv . the derivative didv (equation (8)) must be equal to zero to be able evaluate the maximum point values. therefore; dp dv = v di dv + i = 0 (9) furthermore, if equation (9) is the maximum power value and equated to zero, then the equation can be written as dp dv = − impp vmpp (10) where impp is the current (a), and vmpp the voltage (v), both at the maximum power point. also, from the equation (8), it we can be deducted that equation 11 − impp vmpp = −qio nnskbt ( 1 − impp vmpp ∗ rs ) ∗ ex p [ q(vmpp + rs impp nnskbt ] − 1 rsh ( 1 − impp vmpp ∗ rs ) (11) represents the mathematical equation for the maximum power point voltage and current, which can be solved numerically. the photodiode current or photocurrent iph can be determined using the mathematical expression for open circuit voltage in equation (6), taking into account (to neglect) the term ‘-1’ of the shockley equation (equation 1), this is because, in silicon devices the dark saturation current is lower than the exponential term value [10], thus the equation is expressed as: iph = voc rsh + io [ exp ( qvoc nnskbt ) ] (12) then, by inserting equation (12) into equation (5), to obtain is c, which is expressed is c = voc − rs is c rsh + io [ exp ( qvoc nnskbt ) ] (13) and solving equation (13) to obtain the value of io as: io = ( is c − voc − rs is c rsh ) ∗ ex p [ − ( qvoc nnskbt ) ] (14) similarly, inserting equation (14) into equation (12), to obtain: iph = is c ( rsh − rs rsh ) (15) equation (15) represents the value of the photodiode current iph [a], which is a light generated current that depends on three factors, shunt resistance (rsh), series resistance (rs) and shortcircuit current (is c ) and as well depend linearly on solar irradiance. 2.3. reverse saturation current the module reverse saturation current irs exhibit a certain behaviour when the diode is operating in the reverse bias region, thus the equation of such behaviour is represented as: irs = is c ex p (nnskbt ) − 1 (16) this (equation 16) depends on short circuit current, open circuit voltage and influenced by the changed in temperature [10]. such temperature dependence of the diode saturation current is also given as; itrs = irs ( to tc )3 ∗ ex p [ qeg kn ( 1 t0 − 1 tc ) ] , (17) where itrs is the temperature dependent reverse saturation current and eg is the band gap energy. the photocurrent generated in the solar cell generally depend on temperature and solar irradiance as such the value of the saturation current varies linearly and proportional to the solar radiation. thus, equation (18) describes the saturation effect as: iph = [ is c + ks c ( tc − to ∗ g gs )] , (18) where ks c is short circuit current temperature coefficient and gs is the solar irradiance at standard test conditions, g is the operating radiation condition, tc is the temperature at standard conditions (k) and to is the operating cell temperature (k). the equations (6), (5), (11) are needed to solve the parameters iph, io, rs and rs h. therefore, to find the numerical values of the all the equations ( voc, is c , vmpp and impp) a newtonraphson technique is used to get the quantitative values. also, a maximum power point equation (pmpp = vmpp ∗ impp) was utilized and the detailed simulink model of all the parameters were analysed. the value of module short-circuits current is c is taken from the field experimental procedures. 3. implementation and analysis (environmental and electrical effects) the mathematical model used depend on factors that varies according to environmental conditions such as temperature and irradiance and those factors that also depend on electrical parameters. to analyse the effects of these factors, matlab simulink with solarex msx-60 specification values were used (see appendix table a.1). these values are based on the commercial polycrystalline silicon cells from solarex with a parallel string of 36 solar cells [11]. this is because polycrystalline cells are the most used in the market and so accommodate changes in a better way [12]. furthermore, the algorithm of p & o is implemented to evaluate current and voltage points 93 b. alfa et al. / j. nig. soc. phys. sci. 2 (2020) 91–103 94 figure 2: (a) i-v curve under variable radiation. appendix c.2 (b). p-v curve under variable radiation. appendix c.2. continuously under a changing weather or electrical variations and the maximum power point (mpp) was found at every moment in time. 3.1. effect of solar variation the variational effects of solar radiation (1000-200 w/m2) on voltage, current and power were observed at a temperature of 25oc (table 1). the maximum power reached under different irradiance values from 1000-200 w/m2 was 59.59 w at 1000 w/m2, as such, the voltage changes from 20.32 v to 18.2 v representing a change of 9.10% while current changes from 3.83 a to 0.86 a representing a drop of 23.88% of the maximum energy possible, this happened at the 20.00% of the total irradiance. the i-v curve of the changes in the irradiance from 200w/m2 to 1000w/m2 at a constant temperature 25oc (figure 2a) shows that the value of the mppt current increases at higher irradiance values and decreases at lower irradiance. similarly, the p-v curve (figure 2b), describes the maximum power achieved based on the values for irradiation. here, the value of the power increases with increasing values of irradiance and voltage. thus, the maximum power achieved on the changing irradiance for this model was at 59.59w at 1000 w/m2, this is akin to the findings in [13]. 3.2. effects of temperature variation it is a well-known fact that the pv cells are susceptible under the influence of temperature change. table 2 is indicative of the behaviour of pv cells under a changing temperature, an indication that those changes are not linear. to represent the effect of temperature variation at a constant irradiance (1000 w/m2), the numerical calculations (figures 3a & 3b. ) were plotted to observe the behaviour. firstly, the value of the current changes very slightly while the value of the voltage has a significant change in the different values of the temperatures, from 0oc to 40oc. (figures 2b, 3a, 3b). figure 3: (a) i-v curve at constant radiation (b) p-v curve at constant radiation on the other hand, the value of the power varies accordingly as the temperature increases, thereby making the voltage to drop while the current increases. hence, the best performance (efficiency) occurred at low temperatures conditions with high irradiance. the energy production drops when the panel reaches high temperature as such the pv panel works most effciently in cold temperatures. hence, it can be deduced that, even under cold and sunny environment, this model of the pv cells will perform to it maximum. 3.3. effects of electrical parameters electrical parameters in a pv system such as resistors (shunt or series) can affect the performance of the solar cell considerably, such parameters tends to reduce the efficiency of the solar cell by dissipating power in the resistance. the most common parasitic resistances are the shunt and series resistors (figure 1), both are not strictly constant with respect to temperature changes [14]. 3.4. effects of variation of shunt resistor the presence of the shunt resistance rsh can reduce the power output of the pv cell, this power loss arises from the imperfections on the surface of the solar cell (or in bulk) and by leakage current across the edge of the cell. the efficiency of the cell decreases when the value of the shunt resistor decreases [14]. therefore, it is possible to approximate the value of rsh to the slope of is c . in figure 4a & 4b (i-v curve of the pv from 5 ω to 500 ω), the small resistance values show that the curve is not a perfect curve, but as the values of the resistance increases higher than 100 ω, a proper i-v curve was observed. it noteworthy that the level of the output power remains low when rsh values are lower than 100ω, as such the efficiency can be affected for an incorrect value of rsh, hence, the importance of calculating rsh correctly. table 3 shows the performance of the power output, in this case, the resistance values from 5-500 ohms were used. the iv curve (figure 5), which is almost linear for some parameters 94 b. alfa et al. / j. nig. soc. phys. sci. 2 (2020) 91–103 95 table 1: analysis of voltage, current and power under solar radiation changes at 25oc. irradiance (w/m2) voc (v) is c (a) mppt voltage (v) mppt current (a) mppt power (w) 1000 20.32 3.83 16.93 3.52 59.59 800 20.03 3.06 16.65 2.80 46.60 600 19.66 2.29 16.3 2.09 34.06 400 19.20 1.53 15.79 1.38 21.79 200 18.2 0.76 14.91 0.67 9.98 table 2: analysis of voltage, current and power under temperature changes at 1000 w/m2. temperature oc voc (v) is c (a) mppt voltage (v) mppt current (a) mppt power (w) 0 22.76 3.63 19.44 3.38 65.70 10 21.93 3.70 18.60 3.43 63.79 20 21.12 3.76 17.76 3.47 61.62 30 20.32 3.83 16.93 3.52 59.59 40 19.49 3.89 16.10 3.56 57.31 table 3: analysis of voltage, current and power under shunt resistance changes at 1000 w/m2 and 25 oc.. shunt resistance (ω) voc (v) isc (a) mppt voltage (v) mppt current (a) mppt power (w) 5 17.31 3.83 9.57 1.91 18.27 10 19.42 3.83 15.11 2.25 33.99 50 20.20 3.83 16.73 3.27 54.70 100 20.27 3.83 16.85 3.41 57.45 500 20.33 3.83 16.93 3.53 59.76 table 4: analysis of voltage, current and power under shunt resistance changes at 1000 w/m2 and 25 oc.. irradiation vs shunt resistance 1000 w/m2 800 w/m2 600 w/m2 400 w/m2 200 w/m2 10 40.81w 40.25 w 39.7 w 39.15 w 38.61w 20 20.79 w 20.51 w 20.23 w 19.96 w 19.68 w 30 13.95 w 13.76 w 13.57 w 13.39 w 13.2 w 40 10.49 w 10.35 w 10.21 w 10.07 w 9.93 w 50 8.41 w 8.29 w 8.18 w 8.07 w 7.96 w 100 4.22 w 4.16 w 4.10 w 4.05 w 3.99 w 200 2.15 w 2.08 w 2.05 w 2.03 w 2.00 w 300 1.40 w 1.38 w 1.36 w 1.34 w 1.32 w indicates that the efficiency of the power is reduced at low shunt resistances while at high values of shunt resistances the efficient improves. it is noteworthy that the current remains at the same value in all the cases, hence the value of the maximum power point increases with increase in the shunt resistance. 3.5. power dissipation of shunt resistance table 4 shows the variation of the shunt resistance against different levels of solar radiation. this represents a reduction of the efficiency due to the power loses inside the pv cell in the form of heat. similarly, figure 6, shows that the temperature affects the voltage value to a greater extent than the current, in this case as temperature increases, the power dissipated by the resistor reduces, this is because the voltage generated by the pv cell was less. on the case, where the temperature decreases the voltage increases, thus, making the power consumed by the resistor to increase. table 5 shows the effect of temperature on the shunt resistance. firstly, the variation of the resistance affects the value of the power consumed under different temperature, as the temperature increases the voltage value also decreases, this generates less power delivered by the pv cells. it was noted that the performance of the pv cell was better in low temperatures and high shunt resistance values. although the power consumed by the shunt resistance may be the result of the quality of the materials used in its manufacture, as an imperfection in the manufacture of the cell can create the scenario [15]. it is important not to exceed the allowed specified value as this could lead to low performance by the cell. 3.6. effects of variation of series resistor in solar cells, series resistance variation is generated due to; the flow of current through the emitter and base of the solar cell; the resistance between the metal contact and the silicon and the resistance from the connections [15]. thus, series resistor in high values can reduce the short circuit current. fur95 b. alfa et al. / j. nig. soc. phys. sci. 2 (2020) 91–103 96 table 5: power dissipate in shunt resistance under temperature changes. appendix c.7 temperature vs shunt resistance 0oc 10oc 20oc 40oc 50oc 10 ω 45.22 w 44.55 w 43.23 w 40.81 w 36.64 w 20 ω 23.02 w 22.68 w 22.02 w 20.79 w 18.69 w 30 ω 15.44 w 15.21 w 14.77 w 13.95 w 12.54 w 40 ω 11.61 w 11.44 w 11.11 w 10.49 w 9.43 w 50 ω 9.30 w 9.17 w 8.90 w 8.41 w 7.56 w 100ω 4.66 w 4.60 w 4.46 w 4.22 w 3.79 w 200ω 2.33 w 2.30 w 2.23 w 2.11 w 1.90 w 300ω 1.55 w 1.52 w 1.48 w 1.40 w 1.26 w figure 4: (a) i-v curve for rsh variable at stc (b)p-v curve for rsh variable at stc. figure 5: power dissipated of shunt resistance thermore, the maximum percentage that can affect the variation of this values is 1.19% and less of 0.04% in nominal operating cell temperature and the values over 1.0 ω can be considered as high-power losses. moreover, the age of the panel tends to increase the value of rs hence it decreases the value of the is c [15] figure 6: power dissipated of shunt resistance figure 7(a) corresponds to the change in the i-v curve when the value of the series resistor is varied, in this case values were varied between 0.001 ω to 0.180 ω and the slight change of the voltage was observed, while the current remains at the same value. figure 7: (a) i-v curve for rs variable at stc (b) p-v curve for rs variable at stc. 96 b. alfa et al. / j. nig. soc. phys. sci. 2 (2020) 91–103 97 in the case for the power values (figure 7b), the difference in the value of voltage remains small, but can affect the value of the power in the same proportion. table 6 is a comparison of the values of the different parameters when the series resistor changes. the power decreases in small values of the series resistor, but when the resistance is increased the power also increased. the isc remains the same value, only the value of the voltage changes as rs varies. in these parameters, the i-v and p-v curve have its characteristic shape from the behaviour of the power and voltage. 3.7. power dissipation of series resistor figures 8a represents the behaviour of the power with respect to the series resistor value. the value of the irradiance changes when the power is absorbed by the resistance. in this case, the power absorbed tends to reduce irradiance values, this is because the series resistor is small, and the shunt resistance is relatively large, this allows more current to flow through series resistor path. figure 8: (a) power dissipated of series resistor at 25?oc (b) power dissipated of series resistor at 1000w/m2. table 7 shows the variation of the power regarding with respect to changing irradiation and series resistance. the power the power consumed between 0.0 ω and 0.4 ω were less despite the high value of irradiance variation. for the values above 0.4 ω the power consumed by the resistance increases in different proportions according to the resistance value, this represents the power lost, which can potentially affect the efficiency of the pv panel. figure 8b shows the responses of the series resistor under different temperature changes; the power losses increase logarithmically at 0.7 ω. the numerical values of the temperature variation on the series resistance (table 8), between 0.0 ω and 0.06 ω exhibit a low power dissipation. as previously noted, the effect of the radiation and the temperature changes can affect the energy generation by a pv cell and the value of the power consumed by the series resistor can vary depending on weather conditions [16]. if these changing values are compared for all the figures the best recommendation range for less power consumption will be values under 0.5 ω. also, the worst case is at 1 ω where the power consumption is signifficant enough for any temperature changes. for this reason, it is important to control this value adequately because it can affect the performances of the pv cell by consuming energy that is delivered to the load. for the power lost based on the material properties, it is necessary to use conductors with less resistance to improve the flow of the electrons, and thus reduce the power consumed by the pv panel [16]. 4. implementation and analysis (topological effects) the implementation of the topologies was conducted in stages, firstly, a pv system without a converter, this is to compare the power output of the pv module. secondly, a pv system driven by pwm controller, this is to control the power output of the pv systems, and finally, the implementation with the mppt using the p&o algorithm, to compare the performance of the different topologies. the methods were modelled and simulated by matlab-simulink and the analysis of the behaviour of the signal (at some specific points) and the change in the variation of the temperature and irradiance conditions were investigated. furthermore, an inquiry of the response time under different loads were also analysed. 4.1. pv system without a boost converter to determine the values of the pv output under the temperature and irradiance changes with a constant load of 60 ω, the pv module was connected without a dc-dc converter (figure 9): the obtained values (table 9) from the pv module (figure 10) and the signal produced by this model is shown in figure 10, the analysis of the performance shows that changes in the values occurred after the first milliseconds of the test. this topology generates a maximum power of 6.97 w of 60 w available. therefore, it can be considered that this topology has low efficacy at the beginning of implementation with 11.61% of the total energy that can be produced. the result of this topology indicates that maximum power cannot be extracted when a pv panel is connected directly to the load. apart from the low efficiency of this topology, it also reduces the time-life of the storage system due to the fluctuations in the charger. 4.2. pulse width modulation controller (pwm) method the pwm topology (figure 11) was used to investigate the performance of the pwm under different weather conditions, in this case the pv was connected to a boost converter with a constant load resistance of 60 ω and a duty cycle value at d = 0.6478. the signals produced by the input of the pv panel and the output of the dc-dc converter when compare with the behaviour of the pv before and after the pwm was connected to the boost converter shows a low signals values at the beginning (table 10), except for the current, which was due to the 97 b. alfa et al. / j. nig. soc. phys. sci. 2 (2020) 91–103 98 table 6: under series resistance changes at 1000w/m2 and 25oc temperature vs shunt resistance 0oc 10oc 20oc 30oc 40oc 10 ω 45.22 w 44.55 w 43.23 w 40.81 w 36.64 w 20 ω 23.02 w 22.68 w 22.02 w 20.79 w 18.69 w 30 ω 15.44 w 15.21 w 14.77 w 13.95 w 12.54 w 40 ω 11.61 w 11.44 w 11.11 w 10.49 w 9.43 w 50 ω 9.30 w 9.17 w 8.90 w 8.41 w 7.56 w 100 ω 4.66 w 4.60 w 4.46 w 4.22 w 3.79 w 200 ω 2.33 w 2.30 w 2.23 w 2.11 w 1.90 w 300 ω 1.55 w 1.52 w 1.48 w 1.40 w 1.26 w table 7: power dissipate in series resistance under irradiance changes irrad. vs shunt resistance 1000 w/m2 800 w/m2 600 w/m2 400 w/m2 200 w/m2 0.1 0.26 w 0.43 w 0.65 w 0.92 w 1.26 w 0.2 0.40 w 0.10 w 0.23 w 0.43 w 0.72 w 0.3 0.06 w 0.01 w 0.01 w 0.06w 0.24 w 0.4 0.57 w 0.38 w 0.16 w 0.02 w 0.02 w 0.5 1.54 w 1.15 w 0.64 w 0.21 w 0.06w 0.6 2.91 w 2.20 w 1.31 w 0.53 w 0.07 w 0.7 4.54 w 3.40 w 2.06 w 0.90 w 0.16 w 0.8 6.33 w 4.68 w 2.84 w 1.27 w 0.26 w 0.9 8.28 w 5.99 w 3.60 w 1.61 w 0.36 w 1 10.25 w 7.29 w 4.31 w 1.93 w 0.45 w table 8: power dissipate in series resistance under temperature changes temp. vs shunt resistance 0oc 10oc 20oc 30oc 40oc 0.1 0.63 w 0.55 w 0.48 w 0.40 w 0.32 w 0.2 0.98 w 0.84 w 0.69 w 0.54 w 0.41 w 0.3 1.03 w 0.82 w 0.63 w 0.45 w 0.30 w 0.4 0.77 w 0.54 w 0.35 w 0.20 w 0.10 w 0.5 0.32 w 0.15 w 0.05 w 0.005 w 0.004w 0.6 0.002 w 0.02 w 0.11 w 0.25 w 0.39 w 0.7 0.52 w 0.93 w 1.35 w 1.72 w 1.94 w 0.8 3.44 w 4.43 w 5.27 w 5.76 w 5.78 w 0.9 11.51 w 13.35 w 14.49 w 14.66 w 13.73 w 1 30.31 w 32.70 w 33.59 w 32.23 w 28.78w table 9: parameter values obtained at the beginning of the test time (s) 0.000 0.005 0.010 0.015 0.020 0.025 0.030 power (w) 0.00 0.63 2.46 5.06 6.68 6.96 6.97 current (a) 0.00 0.10 0.20 0.29 0.33 0.34 0.34 voltage (v) 0.00 6.30 12.30 17.95 20.26 20.49 20.51 98 b. alfa et al. / j. nig. soc. phys. sci. 2 (2020) 91–103 99 figure 9: pv module without a boost converter [17]. figure 10: behaviour of pv without a boost . fixed irradiance value during simulation. the power increases up to 49.04 w and remains around at this value until the next change of irradiance. the output of the boost converter reduces the value of the current to increase the voltage. the power output that delivers this method increases if compared with the topology earlier mentioned. the variations (figure 12) were observed from 40oc to 35oc at 800 w/m2. these values remain the same with a less modification, since the temperature changes within time lag were quite negligible as such the parameters have less effect than the irradiance changes in the power. table 11 and (12) shows parameter values under temperature change and under irradiance change. the voltage and current input responded differently under the irradiance changes (table 12). the most significant change occurred in the power output, due to the abrupt change in the current, which is generated by the irradiance that passes from 1000 w/m2 to 800w/m2 as the voltage decreases. these values were found to affect the power output directly, as the converter maintains voltage value. the power output (level) was minimally affected by the changes in the current, which resulted in slow response time and stable power level, which is important for the storage system. 4.3. mppt control system in the mppt (topology) control system (figure ??), the boost converter was implemented and controlled by the algorithm called perturb and observes (p & o), which controls the duty cycle to achieve the maximum point. in this topology, the power output under different climatological conditions were evaluated and the behaviour of the signals from the pv and the boost converter were analysed. in figure 14, the power increases to a maximum level point (highest power output value) and after some fluctuations became stable, the voltage input generated by the pv panel work well on the maximum value as the output voltage increases to 60 v. this power point was achieved as a result of the boost converter implementation. apparently, as the voltage increases, it becomes evident that the pv current decreases in a steady manner without fluctuations. also, figure 15 shows the variation of the signals due to the temperature changes, in this case, the temperature decreases from 40oc to 30oc with a sudden increase in the input power from 56 w to 58 w. the output power signal reacts slowly to these changes as the mppt tracks the maximum power based on the input signal. the change in the irradiance is an essential factor to consider on pv panels with the mppt implementation. this factor (figure 16) shows that the output and input signals of the model could change under the influence of irradiance. the power input drop abruptly because the irradiance varies from 1000 w/m2 to 800 w/m2, this difference is due to the change in the pv current, that is the current is most affected by irradiance change while the voltage has a very slight change. concerning the output signals, the voltage remains at the highest possible value, and the mppt tract the power output at the highest point available under these conditions. the change in the output does not occur spontaneously it takes a time lag to achieve the maximum possible level under such weather conditions. also, the signal has no sudden changes due to the irradiance variation. the implementation of the boost converter controlled by an mppt algorithm offers the best way to extract the maximum 99 b. alfa et al. / j. nig. soc. phys. sci. 2 (2020) 91–103 100 figure 11: pwm controller topology [17]. table 10: parameter vales generated at the beginning of the test time (s) 0.00 0.005 0.010 0.015 0.020 0.025 0.030 pv power (w) 0.00 31.54 44.72 47.72 48.49 48.82 49.04 pv voltage (v) 0.00 8.62 12.22 12.94 13.25 13.34 13.4 pv current (a) 3.66 3.66 3.66 3.66 3.66 3.66 3.66 output power (w) 0.00 8.78 24.60 34.89 40.82 43.35 44.64 output current (a) 0.00 0.38 0.64 0.76 0.82 0.85 0.86 output voltage (v) 0.00 23.12 38.45 45.92 49.79 51.00 51.91 figure 12: variation of voltage, current and power etc generated at the start of the test power available in the system, and the implemented algorithm works to find the optimal point available despite the temperature and the irradiance changes. on the results of this model implemented, this topology (mppt) is the most efficient way to achieve the maximum performance from pv panels because it reaches a high efficiency of the power output. moreover, this system can reduce the time required to fully charge the storage system due to the voltage optimal point level. the best performance of this system is in temperatures under 40 celsius degrees, as we can see in the figure 4b, the reduction of the temperature increased the power output. 4.4. comparison of model topologies every topology has a different efficiency in the power that it delivered. figure 16 shows a comparison of the three-topology 100 b. alfa et al. / j. nig. soc. phys. sci. 2 (2020) 91–103 101 table 11: parameter values under temperature time (s) 3.32 3.33 3.34 3.35 pv power (w) 32.60 32.60 32.43 32.43 pv voltage (v) 10.94 10.94 10.92 10.92 pv current (a) 2.98 2.98 2.97 2.97 output power (w) 30.35 30.36 30.35 30.35 output current (a) 0.71 0.71 0.71 0.71 output voltage (v) 42.76 42.77 42.76 42.74 table 12: parameter values under irradiance change time (s) 2.00 2.05 2.10 2.15 2.20 2.25 2.30 pv power (w) 49.37 33.62 33.11 32.78 32.75 32.69 32.63 pv voltage (v) 13.49 11.32 11.15 11.04 10.99 10.97 10.95 pv current (a) 3.66 2.97 2.97 2.97 2.98 2.98 2.98 output power (w) 46.68 38.87 34.69 32.27 31.29 30.60 30.48 output current (a) 0.88 0.80 0.76 0.73 0.72 0.71 0.71 output voltage (v) 53.05 48.59 45.65 44.21 43.47 43.11 42.93 figure 13: mppt simulation model [17] figure 14: signals variation for the mppt in the pv system used under the same weather conditions at the same time. also, table 13 is shown the numerical values of the comparison at the time of one second. the power delivery for the mppt is the highest, it is 10.64% higher than the pwm power output and 86.17% greater than the pv without control. these results show that the output energy performance of mmpt under different technologies is more efficient. 101 b. alfa et al. / j. nig. soc. phys. sci. 2 (2020) 91–103 102 figure 15: implementation of the mppt in pv system with variable temperature figure 16: mppt implementation on the pv system under irradiance change the implementation of the boost converter by a pwm controller tend to improve the efficacy of the energy delivered by the pv panel. in the mppt topology, power of 44.64 w was realized under the same conditions without sudden fluctuations and more stable signals. however, the implementation of the boost converter controller by an mppt with a p&o algorithm is the most efficient of the three topologies used to control the system, reaching 51.20 w, that is the energy that the system can deliver under the climate conditions at one second. 4.5. 4.5 mppt under different loads the response time of the mppt under three different loads (20 ω, 60 ω and 100 ω) were examined, in order to understand the behaviour of the power output when an unexpected change of temperature or irradiance occurred that may affect its value. therefore, the behaviour of the input and output power under di?erent loads were investigated, and the results (table 14), shows that, as the input power changes suddenly within 3.33 sec, the signal stabilizes rapidly. while, the power output signal takes a time of 3.2 sec. to reach a stable value after a change in the input signal, this corresponds to the time that the mppt takes to find the maximum point under a perturbation in the input signal, this process consists of sending the optimal duty cycle to the boost converter that calculate the algorithm and instruct the switching between on and off to control the inductance in reaching the optimal power of delivery. in the case of 60 ω load, the temperature changes at a time of 3.33s (seesupplementary material figure 17). the figure 18 represents the input and output power signals under a load of 20 ω, this shows an abrupt temperature change to mppt response. the mppt tracks the power input and stabilises the power output after 3.36 s to remain constant until a new variation. also, the variation of the load at 60 ω is in the figure 19 of the supplementary material, shows the maximum value for the load in the converter and after the perturbation in the input signal the power output responds in a fraction of time, and after 3.4s it reaches a constant value close to the power input. these behaviour in the time lapse is consistence with the values in table 6. similarly, in the figure 20 (supplementary material), under the effect of 100 ω load, the power input response time was slower and not close to the maximum power point, hence, the efficiency of the mppt at this point was reduced (table 14). conclusion from the foregoing analysis, three different topologies has been used to achieve an optimal power point on the i-v curve 102 b. alfa et al. / j. nig. soc. phys. sci. 2 (2020) 91–103 103 table 13: power comparison of topologies at 1000 w/m2 and 25oc topology pv pwm mppt voltage (v) 20.50 51.31 55.66 current (a) 0.34 0.87 0.92 power (w) 6.97 44.64 51.20 table 14: response time of the mppt under different loads time (s) 3.32 3.34 3.36 3.38 3.4 3.42 3.44 input 20 ω 54.10 w 54.80 w 54.72 w 54.40 w 54.37 w 54.36 w 54.33 w output 20 ω 53.12 w 53.23 w 53.31 w 53.46 w 53.57 w 53.60 w 53.62 w input 40 ω 55.50 w 56.10 w 55.94 w 55.80 w 55.77 w 55.75 w 55.74 w output 40 ω 54.70 w 54.83 w 54.96 w 55.00 w 55.10 w 55.12 w 55.15 w input 60 ω 55.00 w 55.70 w 55.54 w 55.37 w 55.35 w 55.33 w 55.32 w output 60 ω 53.82 w 53.85 w 53.90 w 53.93 w 53.90 w 54.00 w 54.04 w from pv system from the less to the most efficient way of connection. firstly, the pv panel was connected directly to the load without a control boost converter and the results shows a reduced lifetime storage for the system, which cannot be said to be efficient. secondly, the system was connected to boost converter and controlled by pwm controller, in this case, the efficiency increases but it does not reach the maximum power point, because the pwm has a ?x duty cycle to generate a pwm pulse. this method is useful for small applications where it is not necessary to get the maximum power available because the energy needed by the load is reached without a boost, and enough for the use required. finally, the mppt technique was implemented using the algorithm p&o, this delivered more power because it tracks the optimal point to get the maximum energy available in every moment, this method is recommended for a large system where it 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[17] p. giroux, g. sybille, c. osorio & s. chandrachood, “detailed model of a 100 kw grid connected pv array matlab and simulink mathworks”, united kingdom, 2018. 103 b. alfa et al. / j. nig. soc. phys. sci. 2 (2020) 91–103 104 table 15: solarex msx-60 specifications (1kw/m2, 25oc) [11] characteristics specifications maximum power point (pmpp) 60 w maximum voltage vmpp 17.1 v maximum current impp 3.5 a short-circuit current is c 3.8 a open circuit voltage voc 21.1 v temperature coefficient of open-circuit voltage kv -(80± 10)mv/oc temperature coefficient of short-circuit current kl (0.065±0.1)%/oc approximate effect of temperature on power -(0.5±0.015)%/oc nominal operating cell temperature (noct) 47±2 oc figure 17: simulation of temperature variations figure 18: mppt at load of 20 ω 104 bbj020 typewriter supplementary materials b. alfa et al. / j. nig. soc. phys. sci. 2 (2020) 91–103 105 figure 19: mppt at load of 60 ω figure 20: mppt at load of 60 ω 105 j. nig. soc. phys. sci. 4 (2022) 620 journal of the nigerian society of physical sciences a study of the relationship between southward bz > −10 nt and storm time disturbance index during solar cycle 23 t. w. davida,b,∗, b. j. adekoyaa, c. m. michaelb, s. a. adekoyaa, o. a. adenugaa, s. o. kareemc, h. t. oladunjoyea, a. e. ajetunmobia, o. t. williamsa, d. t. ogundelea adepartment of physics, olabisi onabanjo university, ago-iwoye, nigeria bdepartment of physics and astronomy, university of leicester, leicester, uk cdepartment of physics, mountain top university, prayer city, nigeria abstract magnetic reconnection can be used for studying the geoeffective processes in the coupled sun–solar wind – magnetosphere dynamics leading to geomagnetic disturbance. in this study, 1-hour resolution solar wind plasma parameters from omniweb were used to investigate the relationship between moderate southward interplanetary magnetic field, imf-bz (i.e., bz > −10 nt) and geomagnetic storm time disturbance, dst , during the ascending, maximum and descending phases of solar cycle 23. occurrences of different classes of geomagnetic storms during moderate southward bz are reported. the occurrence of weak and moderate geomagnetic storms is more predominant during maximum solar activity than intense and super intense storms. it was found that 10.11 % (181) of all the classes of the storm were intense, and 0.17 % (3) were super intense storms. furthermore, it was found that 4 (2.2 %) out of the 181 intense storms were caused by southward bz > −10 nt which were associated with the complex structure due to the high-speed solar wind stream and corotating interacting region. in such a complex structure and bz > −10 nt, we observed that an intense geomagnetic storm rarely occurs and if it does, would be predominant around solar maximum. it was found that longduration (∆t > 6 hrs) of southward bz (i.e., −10 nt < bz ≤−3.6 nt ) can also lead to an intense geomagnetic storm during the solar maximum and descending phase (moderate solar activity) of a solar cycle. the complex structure of intense geomagnetic storms associated with the bz > −10 nt is rare and possesses a special configuration of magnetic field and solar wind parameters structures which are cir manifestations. doi:10.46481/jnsps.2022.620 keywords: magnetic reconnection, dst , intense geomagnetic storm, moderate southward imfbz, solar maximum, solar minimum, solar cycle. article history : received: 28 january 2022 received in revised form: 15 september 2022 accepted for publication: 16 september 2022 published: 11 november 2022 c©2022 journal of the nigerian society of physical sciences. all rights reserved. communicated by: s. j. adebiyi 1. introduction the earth’s near-space environment is not surrounded by vacuum, but by a highly dynamic and coupled system of plasmas and magnetic fields, whose complex interplay with ∗corresponding author tel. no: +2348055268531 email address: david.testimony@yahoo.com; david.timothy@oouagoiwoye.edu.ng (t. w. david ) sunspot number, coronal mass ejections (cmes) and solar flares eruption could cause a time-varying condition that constitutes the subject of space weather [1-3]. the solar wind, consisting mainly of protons and electrons, originates from the sun and streams radially into space, pervading the solar system. the sun’s magnetic field flows with these particles. the magnetic field of the earth poses an obstacle to this oncoming flow of plasma from the sun and thereby carves out the magnetosphere. the solar wind interacts with the earth’s magnetic 1 t. w. david et al. / j. nig. soc. phys. sci. 4 (2022) 620 2 field and reconfigures its dipolar shape by compressing it on the sunward side while stretching it to a tail-like shape on the night side. during an anti-parallel orientation of the interplanetary magnetic field (imf) to the geomagnetic field lines, solar wind energy, momentum and mass could be transferred into the earth’s magnetosphere, thereby influencing its dynamics and geomagnetic storms. landmark studies [3-9] have shown that bz, which is the north-south direction component of the imf, is a good index that plays a vital role in the reconnection between the solar and terrestrial magnetic fields, leading to additional energy in the magnetospheric flux tube and the geoeffectiveness of geomagnetic storms. a southward bz, fluctuating in the corotating interacting regions (cirs), is a prerequisite for magnetic reconnection leading to the development of geomagnetic storms [4] whereas, a northward bz rarely leads to storms except during extremely high solar wind speed [5, 10]. on the other hand, the storm time disturbance index, dst, has been extensively used to measure the degree of moderate magnetic perturbation experienced by the dipolar field of the terrestrial planet [11]. the dst index represents the magnetic depletion resulting from the westward drift of the ring current formed by ions and electron energy which would be detectable by an appropriate ground-based instrument [10]. the interplanetary origin of intense geomagnetic storms has been widely studied [3,12-21]. many have explained the interrelation between the southward interplanetary magnetic field and the depletion in the d st index leading to intense magnetic storms. they recognised that the orientation of the interplanetary magnetic field played a great part in the strength of the main phase of geomagnetic storms. gonzalez et al. [22, 23] were among the leading research work on the relationship between the interplanetary magnetic field and the resulting ring current system. meanwhile, gonzalez and tsurutani [24] and gonzalez et al. [15] had drawn an inference that bz and the geomagnetic storm’s development are interrelated, where they observed that the magnitude of the southward bz leading to an intense storm is ≤−10 nt and over a period exceeding 3 hours. this intense nature of bz appears to originate from the different interplanetary and solar wind structures and the interaction between them. also, it was established that the poynting flux from the interplanetary medium is a driver of the main phase of storm development in the magnetosphere and determines the strength and magnitude of the storm [3, 15, 21]. several studies [25-28] while studying the ring current, tried to quantify the energy deposited in the magnetosphere by the stream of charged particle/solar wind, coupled with the solar magnetic field it drags. they also looked at the magnetospheric dissipation rate of this energy. the stored magnetosheath energy (i.e., pressure gradient force and magnetic tension force) realigns the magnetic field, the energy along the field lines causes the westward ring current of the plasma under the influence of gradient and curvature drift [29]. the main phase of geomagnetic storms results from ring current development, in which energisation depends on storm time disturbance index, dst , solar wind speed, ram pressure and bz component of the imf [3, 26, 30]. the drivers of events with large geomagnetic storms are described to be of complex interplay [14]. the complexity includes several steps at the main phase of the storm, the phase of the solar cycle, the occurrence of shock at the sheath, the speed of the interplanetary coronal mass ejection (icme), etc. borovsky and denton [16] explained that while coronal mass ejection-driven storms are inimical to the electrodynamics system on earth, storms driven by the corotating interacting region (cir) affect assets in the space region. many of the cirs have transient properties that are similar to that of icmes and contain small icme-like transient, that is, the solar wind properties feature many icme signatures, such as smooth/coherent field rotation and enhanced fields, but that is significantly reduced in duration (a few hours on average) than typical large-scale icmes [31, 32, 33]. entrainment of such transients, when they have southward fields, may enhance the geoeffectiveness of a cir. the interaction between such transients may enhance the geoeffectiveness of a cir, especially, when they have moderate southward fields [32]. adekoya and chukwuma [3] reported that these types of storms may be originated from small explosions that are more intense than the cmes-caused storms. and were caused by the interactions between a shock driven by the symmetric transient disturbances and the corotating stream [34]. gonzalez et al. [15] added bs structures (southward magnetic field) to the list of causative elements of events with large geomagnetic storms. they pointed out that the long duration of the enhanced bs would offer support for a magnetic cloud to produce a large magnetic storm. kumar et al. [35] observed that the magnitude and duration of the southward bz are fundamental in the occurrence and development of the geomagnetic storm. in all, none of these investigations has been able to categorically report or linked the geoeffectiveness of an intense magnetic storm to a lower magnitude of southward bz. similarly, studies exist on the interplay of solar wind parameters and geomagnetic activity during the solar cycle 24. for example, rathore [36] in a case study of a geomagnetic storm during the solar cycle 24 shows that imf parameters including the bz component are linked to geomagnetic storm given the increase in the value of bz at the storm commencement. this is relatable to a previous observation of a high bz value by rawat et al. [37] during several intense geomagnetic storms in solar cycles 23 and 24. however, the bz cannot be isolated as the main driver following rathore [36], since the value of other solar wind parameters such as solar wind speed and plasma temperature were equally large during the same period of the storm commencement. pokharia et al. [38] suggest that combining solar wind speed and bz is a better approach to depicting the production of geomagnetic storms than considering both parameters separately. the foregoing shows bz is a relevant interplanetary condition for geomagnetic storm commencement, which is in tandem with the requirements suggested in gonzalez and tsurutani [24] for intense geomagnetic activity. this work looks at the statistical analysis of the occurrence 2 t. w. david et al. / j. nig. soc. phys. sci. 4 (2022) 620 3 of geomagnetic storms associated with the imf-bz as well as the geoeffectiveness of bz > −10 nt. in comparison to previous works, a complete solar cycle will be investigated in this study, which will allow every phase of the solar cycle to be investigated. previous studies [22, 24, 39] had argued that an intense storm can only occur when the bz component of the imf stays at least three hours in the southward direction with a threshold of bz ≤ −10 nt. attention is restricted in the present study to events during a southward turning of bz such that bz > −10 nt. the term “moderate southward bz ” in this work refers to the range of bz component of the imf such that −10 nt < bz ≤ −5 nt. solar cycle 23 has been chosen not for any special reason, but it is worth mentioning that it is the longest solar cycle in history. figure 1 shows the time series of the sunspot number enclosing the period of solar cycle 23, where the peak (solar maximum) shows a double hump between the year 2000 2003 and a solar minimum in 2008 ending the cycle. the black line in figure 1 is the daily total while the red line is the yearly smoothed sunspot numbers. the minimum and maximum sunspot numbers during the cycle are respectively about 5 and 180. figure 1: the time series sunspot number enclosing the period of solar cycle 23. 2. data source and method the period under investigation is the solar cycle 23, with a time interval from august 1996 to december 2008. for this period, the solar wind plasma and geomagnetic parameters data employed in this study consist of hourly values of the bz component of the interplanetary magnetic field (bz, nt), the storm time disturbance index (dst index, nt), plasma flow speed (vsw, km/s), proton density (ρ, n/cm3) and corresponding plasma temperature (t, k). these interplanetary and hourly geomagnetic data are obtained from the space physics data facility (spdf) omniweb website (http://omniweb.gsfc.nasa.gov/). this research is focused on a statistical analysis of the different classes of geomagnetic storms indicated by the storm time disturbance index as a result of moderate southward bz at the nose of the earth’s magnetopause. the geomagnetic storms classifications are: weak (−30 nt ≤ dst > −50 nt), moderate (−50 nt ≥ dst > −100), intense (−250 nt < dst ≤ −100 nt), and super intense (dst < −250 nt), while (dst ≥ −30 nt) is regarded as period of quiet activity [39, 40, 41]. in this study, the period at which the omni data set of solar wind plasma parameters at the bow shock region of the earth records −10nt < bz ≤ −5 nt is taken to be a moderate southward bz event. there were 1790 geomagnetic events during the solar cycle 23, and the frequency of the hourly values is shown in table 1. 3. result and discussion 3.1. hourly analysis of events table 2 indicates the hourly distribution (in ut) of geomagnetic storms as indicated by the dst index in table 1. the data in table 2 shows the frequency of the hourly values of the 1790 events during the solar cycle 23 and its occurrence, which is the frequency of a particular class at a particular hour divided by the total frequency for that hour (see equation (1)). it is clearly shown that weak geomagnetic storms dominate every hour throughout the period under investigation, while the super intense class is a rare phenomenon. figure 2 shows the bar chart of the temporal distribution of events with weak, moderate, intense and super intense geomagnetic storms caused by moderate southward bz during the period under investigation. the hourly occurrence of the different classes of storms is calculated as the percentage ratio of the number of events in each class to the total events caused by moderate bz at the particular hour (see table 2). dstoccurrence = total of each storm category total storm event ×100%.(1) the blue, green, yellow, and red colours represent the percentage occurrence of the weak, moderate, intense and super intense geomagnetic storms, respectively. as more field lines are opened due to dayside re-connection, the open field lines stretch in an anti-sun ward direction and the solar wind energy is transported into the magnetosphere, where it is stored in the magnetotail [42]. the eventual release of the energy as a result of reconnection in the neutral sheet gives rise to varying degrees of storm categories. it could be seen from figure 2 that apart from the super intense geomagnetic storms that rarely occur, all other classes of geomagnetic storms are general phenomena at all times. furthermore, figure 2 indicates that though moderate geomagnetic storms have no distinct peak, it attains a maximum occurrence of about 49 % during the moderate southward bz conditions. however, when the class of geomagnetic storms is above moderate, the occurrence peak table 1: frequency of different range of classes of dst corresponding to moderate southward bz 165 during the solar cycle 23. dst weak moderate intense super intense frequency 878 728 181 3 3 t. w. david et al. / j. nig. soc. phys. sci. 4 (2022) 620 4 table 2: frequency of hourly peak values and percentage occurrence for different classes of geomagnetic storms during solar cycle 23. ut frequency of dst dst occurrence (%) weak moderate intense super intense weak moderate intense super intense 0 42 28 7 0 54.55 36.36 9.09 0.00 1 39 28 7 0 52.7 37.84 9.46 0.00 2 29 34 7 0 41.43 48.57 10 0.00 3 30 31 7 0 44.12 45.59 10.29 0.00 4 32 30 7 0 46.38 43.48 10.14 0.00 5 34 28 8 0 48.57 40 11.43 0.00 6 33 30 13 0 43.42 39.47 17.11 0.00 7 34 33 5 0 47.22 45.83 6.94 0.00 8 48 34 5 0 55.17 39.08 5.75 0.00 9 31 24 5 0 51.67 40 8.33 0.00 10 37 37 5 0 46.84 46.84 6.33 0.00 11 29 29 6 0 45.31 45.31 9.38 0.00 12 45 37 8 0 50 41.11 8.89 0.00 13 45 28 8 0 55.56 34.57 9.88 0.00 14 45 34 8 0 51.72 39.08 9.2 0.00 15 33 26 3 0 53.23 41.94 4.84 0.00 16 38 30 4 0 52.78 41.67 5.56 0.00 17 33 31 4 0 48.53 45.59 5.88 0.00 18 36 38 7 0 44.44 46.91 8.64 0.00 19 39 27 14 0 48.75 33.75 17.5 0.00 20 33 31 14 0 42.31 39.74 17.95 0.00 21 28 24 13 2 41.79 35.82 19.4 2.99 22 47 30 9 0 54.65 34.88 10.47 0.00 23 38 26 7 1 52.78 36.11 9.72 1.39 total 878 728 181 3 figure 2: the hourly occurrence values for different classes of geomagnetic storms associated with southward moderate bz during solar cycle 23. reduces below 20 %. a clear observation of figure 2 shows a higher occurrence of intense storms during the night sector with respect to the universal time. around 19 − 23 ut and 00 − 06 ut, the occurrence of an intense storm is higher in comparison to during the period 08 − 18 ut. less than 0.2 % of the events are super intense storms, and about two-thirds are driven by midnight mechanisms. 3.2. yearly analysis of events table 3 indicates how the total number of 1790 events indicated by the d st index in table 1, is distributed across the years in solar cycle 23. as stated earlier, frequency is the number of events for the different classes of geomagnetic storms (measured by the dst index) according to the southward bz due to the magnetic reconnection. the occurrence on the other hand is the frequency of a particular class in a particular year divided by the total frequency for that year. it can be seen that the maximum frequency of all the classes of geomagnetic storms is during solar maximum (see figure 1). in all, the percentage occurrence of intense and super intense storms peaked during solar maximum, while the occurrence of weak geomagnetic storms was dominant during solar minimum. this is partly because the sunspot number being at maximum, increases solar activity level and provides an opportunity for geoeffectiveness. the occurrence of moderate geomagnetic storms was higher at the declining phase of the solar cycle, which occurs at solar moderate periods, meanwhile, the lowest occurrence rate was at solar minimum. this result was congruent with the report 4 t. w. david et al. / j. nig. soc. phys. sci. 4 (2022) 620 5 of echer et al. [10] who reported the highest rate of moderate storm occurrence in the declining phase of the solar cycle 23. the moderate storms were dominantly driven by cirs and high-speed streams (hsss), but with variable contributions from icmes, their shocks (sheaths), and combined occurrence within the solar cycle [10, 21, 43]. whereas, the weak class of geomagnetic storm’s highest occurrence was found at the solar minimum, lowest during the solar maximum periods, the ascending and descending phases of the solar cycle shows non-linear variations. 3.3. analysis of the southward bz > −10 nt and solar wind conditions leading to an intense geomagnetic storm the intense geomagnetic storms (dst ≤ −100 nt) are caused by the intense southwardly directed imf-bz of magnitude > 10 nt, with a duration greater than 3 hours [22, 24, 39]. however, not all intense storms are associated with the intense nature of bz [3, 10]. that is, there are intense storms that are likely caused by moderate southward bz, (−10 nt < bz ≤ −5 nt) whose complex solar wind and interplanetary interplay may be different from other storms [44, 45]. kumar et al. [35] observed that the magnitude and duration of the southward bz are fundamental in the occurrence and development of geomagnetic storms. therefore, onward in this section, the interplay of the intense storms associated with the moderate southward b z (−10 nt < bz ≤ −5 nt) turning for long-duration of three consecutive hours and solar wind conditions during the solar cycle 23 is presented. this event is a rare occurrence and only occurred during the solar maximum and descending phase (moderate solar activity) of the solar cycle 23. its properties and geoeffectiveness as related to the moderate southward bz are explained below. figure 3 shows the interplanetary and solar wind plasma characteristics during the geomagnetic storm event of 25 − 27 september 2001, representing the moderate southward bz event during the solar maximum phase of the solar cycle 23. the region within the two red vertical lines presents the structure of the solar wind characteristics associated with the moderate southward bz . the duration of the moderate southward bz turning is represented with ∆t, that is, the changes in time at the first and second vertical lines. looking at figure 3, the peak value of southward bz was -3.6 nt which is contrary to the earlier suggested magnitude of the moderate southward bz [19, 22]. from the view of the solar wind plasma structure associated with this southward bz , the intense signature of dst = −102 nt corresponds with the respective high-scale variations of the plasma density, ρ is 39.2 n/cm3, plasma temperature, t is 706094 k, plasma flow speed, vsw is 675 km/s above the typical value of 450 km/s and plasma beta, β, is 4.88. choi et al. [7] referred to such an event with a duration ≥ 3 hrs as long-weak bz and has characteristics of hsss/cirs driven intense storms. the event properties emanated from coronal holes and are associated with stream interaction regions [5] and are identified as complex ejecta [3, 45]. in a similar configuration, the solar wind properties during figure 3: interplanetary and solar wind plasma parameters of a moderate southward bz turning caused intense storm during the solar maximum phase of the solar cycle 23. the shaded region marked the corresponding interplanetary and geomagnetic profiles associated with the moderate southward bz. the september 25 − 27, 2001 geomagnetic storm event. the intense geomagnetic storm of 3 − 5 april 2004 represent the moderate southward bz event during the descending phase of the solar cycle 23 is presented in figure 4. looking at this figure, the intense storm (dst ≥ -100 nt) originated from the long-duration (∆ t > 7 hours) moderate southward bz turning of magnitude -7.9 nt. the corresponding solar wind parameters suggest that there is a high-scale variation of flow speed with a peak magnitude of 504 km/s below the typical value of 450 km/s. the plasma flow speed increases from 423 km/s around 1300 ut on april 3 to a peak magnitude of 504 km/s around 0000 ut on april 4 corresponding to the periods of the moderate southward orientation of the bz. contrary to the high plasma temperature of the events during the solar maximum phase of the solar cycle, the plasma temperature during the descending phase was reduced (the peak plasma temperature at the confined period of the moderate southward bz was 121,155 k) below the typical of 400,000 k [41]. corresponding to the moderate southward directed bz are high proton density, ρ with a peak value of 24.5 n/cm3 and high plasma beta, β, with a peak value of 3.46. the characteristic signature of this intense storm is depression in the magnitude of the dst resulting from the westward ring current associated with the moderate southward directed bz system encircling the earth. following the sudden storm commencement of the storm, the dst decreases to a minimum value of -117 nt at exactly 0000 ut on april 4. during the window of the southward directed bz, the characteristic of this complex ejecta-driven storm during the solar 5 t. w. david et al. / j. nig. soc. phys. sci. 4 (2022) 620 6 table 3: frequency of yearly peak values and percentage occurrence for different classes of dst during solar cycle 23. year frequency of dst dst occurrence (%) weak moderate intense super intense weak moderate intense super intense 1996 13 3 1 0 76.47 17.65 5.88 0.00 1997 52 66 5 0 42.28 53.66 4.07 0.00 1998 120 61 22 0 59.11 30.05 10.84 0.00 1999 100 77 11 0 53.19 40.96 5.85 0.00 2000 173 105 26 1 56.72 34.43 8.52 0.33 2001 118 119 57 2 39.86 40.2 19.26 0.68 2002 50 97 38 0 27.03 52.43 20.54 0.00 2003 136 103 2 0 56.43 42.74 0.83 0.00 2004 20 25 10 0 36.36 45.45 18.18 0.00 2004 40 30 2 0 55.56 41.67 2.78 0.00 2005 26 30 7 0 41.27 47.62 11.11 0.00 2006 19 11 0 0 63.33 36.67 0.0 0.00 2007 11 1 0 0 91.67 8.33 0.0 0.00 2008 878 728 180 3 figure 4: same as figure 3, but during the descending phase of the solar cycle 23. the april 3 − 5, 2004 geomagnetic storm event. maximum and descending phase are presented in table 4. in table 4, the peak value of the solar wind parameters and magnetic index associated with the moderate southward directed bz were highlighted. four (4) intense geomagnetic storm events were identified to be related to the bz > −10 nt, three (3) were during the solar maximum and one during the descending phase (around moderate solar activity periods) of the solar cycle 23. this rare occurrence event was not found during the ascending phase of the solar cycle 23, which indicates the possibility of not having such an event during the ascending phase of a solar cycle. the occurrence rate of this moderate southward bz -driven storm is very rare. the only condition for an increase in the occurrence of the moderate southward bz is when the geomagnetic storm classification is moderate (i.e. −100 nt < dst ≤ −50 nt) and during solar minimum [5, 32]. from table 4, one can see that the peak response of southward bz during the storm of 25 27 sept. 2001 was -3.6 nt, which is above the typical value for moderate southward bz of (−10nt < bz ≤ −5 nt). aside from this low-weak bz, the parameters show highscale variables compared to other storms. choi et al. [7] have suggested that this weakly event of southward bz should be considered significant from the viewpoint of their geoeffectiveness. therefore, the occurrence rate may be too low to be considered, but the solar wind characteristics and geoeffectiveness of the storm compared to the other storms with a higher magnitude of southward bz should be classified as moderate southward bz-driven intense storm. the sudden increase and high flow speed and proton density of the plasma indicate that the geoeffectiveness of stream interacting region/high-speed streams can be explained by solar wind properties. therefore, either low-weak southward bz or moderate southward bz-related intense storms should be considered as geoeffective significant events that can affect space-based and ground-based technology as much as those of the intense southward bz signature. 4. summary and conclusions we have carried out a statistical analysis of the occurrence of geomagnetic storms associated with the southward interplanetary magnetic field in the range of bz ¿ -10 nt, using the complete solar cycle 23 data. the complex structure of intense geomagnetic storms associated with the bz > -10 nt is rare 6 t. w. david et al. / j. nig. soc. phys. sci. 4 (2022) 620 7 table 4: the characteristics of the solar wind parameters associated with the southward directed bz ( > −10 nt) with long-duration of intense geomagnetic storm during solar cycle 23. the peak response of the parameter during the southward bz turning was highlighted. s/n year doy bz (nt) ∆t (hr) t (k) vsw (km/s) ρ (n/cm3) β dst (nt) 1 2001 269 -3.6 > 8 706094 675 39.2 4.88 -104 2 2002 233 -9.2 > 12 35238 495 7 0.88 -106 3 2002 325 9.4 > 6 478599 728 54.8 1.26 -128 4 2004 95 -7.9 > 7 121155 504 24.5 3.46 -117 and possesses a special configuration of magnetic field and solar wind parameters structures which are hsss/cir manifestations. it was found that 10.11 % (181) of all the classes of the storm were intense, 0.17 % (3) were super intense storms, 40.67 % (728) were moderate geomagnetic storms while 49.05 % (878) were weak storms. furthermore, it was found that only 4 (2.2 %) out of the 181 intense storms were caused by the southward bz > −10 nt which were associated with the complex structure due to the high-speed solar wind stream and corotating interacting region. the results are further summarised as follows: • the complex structure and bz > −10 nt event of the intense geomagnetic storm is a rare occurrence and only occurred during the solar maximum and descending phase (moderate solar activity) of the solar cycle 23. • sometimes the southward turning of bz > −5 nt with a complex configuration of solar wind characteristics may lead to an intense geomagnetic storm which may occur during solar maximum. therefore, the geoeffectiveness of an intense storm associated with the southward bz > −10 nt is controlled by the solar wind characteristics (i.e., high plasma flow speed, high proton density and complex structure of plasma beta). • the yearly analysis shows a predominance in the weak and moderate geomagnetic storms resulting from -10 nt ¡ bz ≤−5 nt. • the complex structure associated with the moderate geomagnetic storms is more predominance during solar minimum. • the peak occurrence for all classes of geomagnetic storms, associated with moderate southward bz, is observed during solar maximum. • in contrast to previous studies that a 3-hour intense southward bz is a pre-requisite for an intense geomagnetic storm, a 3-hour southward bz > −10 nt with a complex structure can lead to an intense geomagnetic storm at the phases of the solar cycle. • about 75 % of the southward bz > −10 nt leads to intense geomagnetic storms occurring at the maximum phase of the solar cycle. acknowledgements the authors appreciate the staff of space physics data facility (spdf) omniweb (website: http://omniweb.gsfc.nasa.gov/) for free accessibility of data. we thank the olabisi onabanjo university, ago-iwoye, nigeria, for creating the enabling environment for this research. the university of leicester, united kingdom, is deeply appreciated for making uol spectre available for the analysis. references [1] c. g. falthammar, “magnetosphere ionosphere interactions near-earth manifestations of the plasma universe”, ieee transactions on plasma science 14 (1986) 616. 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[45] a. ojeda-gonzalez, v. klausner, o. mendes, m.o. domingues & a. prestes, “characterization of the complex ejecta measured in situ on 19 22 march 2001 by six different methods”, solar physics 292 (2017) 160. 8 j. nig. soc. phys. sci. 1 (2019) 122–130 journal of the nigerian society of physical sciences original research software process ontology: a case study of software organisations software process sub domains r. o. oveh∗, o. efevberha-ogodo, f. a. egbokhare department of mathematics and computer science, western delta university, oghara, delta state, nigeria abstract in a domain like software process that is intensively knowledge driven, transforming intellectual knowledge by formal representation is an invaluable requirement. an improved use of this knowledge could lead to maximum payoff in software organisations which is key. the purpose of formal representation is to help organisations achieve success by modelling successful organisations. in this paper, software process knowledge from successful organisations was harvested and formally modelled using ontology. domain specific knowledge base ontology was produced for core software process subdomain, with its resulting software process ontology produced. keywords: software process, software process ontology, ontology, knowledge, formal representation, knowledge representation article history : received: 06 june 2019 received in revised form: 09 september 2019 accepted for publication: 12 september 2019 published: 17 december 2019 c©2019 journal of the nigerian society of physical sciences. all rights reserved. communicated by: t. latunde 1. introduction software process is a knowledge driven and knowledge intensive process that involves several other sub-processes. software process can be defined as the set of related activities that are used in developing software. knowledge in software engineering (se) is diverse and organizations have problems capturing, retrieving, and reusing it. an improved use of this knowledge is the basic motivation and driver for knowledge management (km) in se [1-23]. harvesting, representing and reusing knowledge within a domain leads to maximum payoff, which is desirable in most organisations [23]. knowledge management (km) is defined as an effort to capture critical knowledge and share it within an organization [3, 17]. it capitalizes on the collective organiza∗corresponding author tel. no: +2347036142579 email address: omo_rich@yahoo.com (r.o.oveh) tional memory to improve decision making, enhance productivity, and promote innovation [18, 19]. it is also the process of transforming information and intellectual assets into persisting value. km connects people with the knowledge that they need to take action, when they need it [20]. knowledge management involves the identification and analysis of available and required knowledge [21] and helps an organization to gain insight and understanding from its own experience. specific knowledge management activities focus on acquiring, storing and utilizing knowledge for problem solving, dynamic leaning, strategic planning and decision making. this prevents intellectual assets from decay, adds to a firm’s intelligence and provides increased flexibility [22]. se comprises several interrelated subdomains such as requirements, design, coding, testing, project management, and configuration management. there are several software process models which describe the sequence of activities carried out in developing software. these software process models are a stan122 oveh et al. / j. nig. soc. phys. sci. 1 (2019) 122–130 123 dard way of planning and organizing a software process. the major phases are requirement gathering, design and coding, implementation and maintenance. it has been identified that there are few works in literature that aim at developing ontologies covering wide portions of the se domain, such as [4-6]. a lot of se domain ontologies model se subdomains [711]. ref. [12] described these subdomain ontologies as weak or not interrelated, and are often applied in isolation. thus he made an attempt to provide an integrated solution for better dealing with km-related problems in se by means of a software engineering ontology network (seon). it was designed with mechanisms for easing the development and integration of se domain ontologies, covering the main technical software engineering subdomains (i.e requirements, design, coding and testing). however, he only represented a small portion of software engineering ontology. [7], identified that the combination of ontologies of all se subdomains would result in an ontology of the complete se domain. he further stated that the reality is that this goal is extremely laborious, not only due to its size, but also due to the numerous problems related to ontology integration and merging, such as overlapping concepts, diverse foundational theories, and different representation and description levels, among others. he concluded that despite the challenges involved, an ontological representation covering a large extension of the se domain remains a desired solution. this paper represents a software process ontology covering major se subdomains (i.e. requirement gathering, design and coding, implementation and maintenance). 2. related literature ontologies have been widely recognized as a key enabling technology for km. they are used for establishing a common conceptualization of the domain of interest to support knowledge representation, integration, storage, search and communication [2]. a domain ontology identifies the key concepts, objects and entities that exist in some knowledge domain or area of interest and the relationships between them [15, 16]. ontologies play a significant role for knowledge sharing and as knowledge models in instructional science, technology-enhanced learning, knowledge management and training [15, 14, 13]. ontologies consist of instances, properties and classes, where instances represent specific project data, properties represent binary relations held among software engineering concepts/instances, and classes represent the software engineering concepts interpreted as sets that contain specific project data [25]. [7], did an extensive review of se ontologies, where he classified them into generic and specific ontology. generic se ontologies, have the ambitious goal of modeling the complete se body of knowledge; while specific se ontologies, attempting to conceptualize only part (a subdomain) of this discipline. the management of knowledge and experience are key means by which systematic software development and process improvement occur. within the domain of software engineering (se), quality continues to remain an issue of concern. knowledge management (km) gives organizations the opportunity to appreciate the challenges and complexities inherent in software development [24]. successful organisations continuously improve their processes. like organisational standard process definition, systematic process improvement is more effective and efficient if it is done guided by process quality models and standards. the purpose of most standards is to help software organisations achieve excellence by following the processes and activities adopted by the most successful organisations [26]. 3. methodology two complementary methods were used for data collection. they are: case study and interview methods. for the case study, four (4) software development organizations was studied. for reason of privacy and confidentially, the organizations studied are not referenced by their names. table 1 shows the details of the organisations. from table 4, the domain concept, the data value/property and the instances are specified. for example the entity domain expert in number 9 has the property: name that can take a data value string, data property domain that can take a data value string and a data property years of experience that can take a data property integer. it also has instances of the class as: business rules and directory of experts. figure 6 shows the domain concepts and their instances 4. result four main subdomains were identified as knowledge entities in a typical software development process irrespective of the life cycle model adopted: requirements definition, design & coding, implementation and maintenance. these are core human centric activities performed by developers that create opportunities for sharing tacit knowledge during the software development process. this research used both inductive and deductive analysis by first identifying keywords related to software development process and then grouping the keywords into categories related to requirements definition, coding, implementation and maintenance. for each process activity, a union of the useful themes obtained each case study was used to determine the useful knowledge constructs for that activity. that is, for each process activity (pa), knowledge harvested (kh) for the software process is given as kh(pa)=case 1(pa) ⋃ case 2 (pa)⋃ case 3 (pa) ⋃ case 4 (pa) (1) 5. software process ontology there is no globally accepted methodology for ontology construction [28] but the development is usually an iterative process. a 5-step iterative process from [29, 30] was adopted for the ontology construction: 123 oveh et al. / j. nig. soc. phys. sci. 1 (2019) 122–130 124 table 1: details of the organisations case study years of existence software area 1 15 human resources systems, payroll systems, and portal for schools. 2 22 banking solutions, human resource management systems, process control systems, payroll systems, security systems, materials management systems 3 10 security solutions, and web development 4 12 human resource management systems, third party online payment solutions, payroll and other financial systems figure 1: requirement definition sub-domain ontology visualization • step 1: identify the key concepts of the domain the first step in the ontology building process is to identify the concepts and map them into the relevant knowledge entities. to provide clarity and efficiency in the formal representation of software process knowledge, the concepts from the harvested knowledge were categorized into the four main knowledge entities: 1. knowledge-creation 2. knowledge-transfer 3. knowledge-sharing 4. knowledge-documentation. each concept (table 2) connotes a software process activity that describes a task, function, action, strategy, or reasoning process. a concept is a collection of objects. it is the fundamental element of a domain and usually represents a group or class whose members share common properties [30] • step 2: organisation of the concepts into a hierarchy the purpose of this categorization is to establish a systematic relationship between the knowledge entities and the specific software development process activities. competency questions was used here in creating the ontology. competency questions as defined by some methodologies for ontology engineering describe what kind of knowledge the resulting ontology is supposed to answer. according to [27] one of the ways to determine the scope of the ontology is to sketch a list of questions that a knowledge based on the ontology should be able to answer. the following competency questions were used: 1. ethnographic study in requirements gathering involves the study of? 2. what are the processes for gathering requirements from stakeholders and end-users? 3. what are the processes of software development? 4. what are the physiological processes for blocks resolution during coding? 124 oveh et al. / j. nig. soc. phys. sci. 1 (2019) 122–130 125 figure 2: design and coding sub-domain ontology visualization figure 3: implementation sub-domain ontology visualization figure 4: maintenance sub-domain ontology visualization 5. what are the appropriate approach to software design? 6. what are the people and processes involved in user tasks? 7. what are the tools used in ethnographic study of stakeholders and end-users? 8. what are the stages involved in implementation? 9. what does a deployed system need for maintenance? 10. who can business rules be obtained from in a domain? 11. how can low bus factor issues be handled in coding? 12. how can code ownership issues be resolved in coding? 13. how can knowledge be shared in software process? 14. how can knowledge be transferred in software process? 125 oveh et al. / j. nig. soc. phys. sci. 1 (2019) 122–130 126 figure 5: software process ontology visualization • step 3: determine the properties of each class every objects have both data property and object property. the object property shows the relationship between classes or instances, and the data property shows the relationship between instances and the data value. it provides a logical relationship between objects. tables 3 and 4 present sample object and data property defined for some objects in the software process ontology. • step 4: add logical expressions axioms represent assertions formulated in a logical form that together comprise the core knowledge that the ontology describes in its domain of application. they are used to model sentences that are always true. they provide a powerful way to add logical expressions to ontology and they are used to verify the consistency of the ontology. axioms are usually added by default in protege. • step 5: create the ontology protégé 5.0 was then used to build the software process knowledge ontology, with its subdomains.as shown in figure 1-5 6. discussion the requirements definition (figure 1) is the first stage in software development process. it is a very critical phase. a major finding from the organizations studied is that ethnography study was used to obtain the software requirements. it involves studying the organization’s culture to understand the key elements and observable patterns of behaviour. this usually requires interaction between stakeholders from the software de126 oveh et al. / j. nig. soc. phys. sci. 1 (2019) 122–130 127 figure 6: software process ontology showing concept, instances and their relationship velopment organization and those from the user organization in order to gain a better understanding of the problem. it was observed from the study that quality time is dedicated to identifying and interacting with the end users during requirements elicitation. these series of interactions provide opportunity for the company to distinguish itself and to learn about how the users’ tasks are performed. users are asked to discuss their routine tasks with the requirements elicitation team and they are informed about the services to be provided by the proposed system. this helps to prevent user resistance to the new software system and provides opportunity for users to discuss their daily routines with the developers in an informal way. during the process, apart from using informal interviews to gather data, participant observation, questionnaires, documents review and other suitable requirements elicitation techniques are also used when necessary. sometimes, the user’s expressions and reactions send useful signals regarding what they expect in the system. one of the organizations studied adopted the agile software method were the initial requirements are quickly developed into the first build which is installed for the users and additional features are reported as feedbacks for the next build. users e-mail addresses and phone numbers are collected. a repository is created to store user complains and any additional features requested. figure 2 shows the approach to design and coding. for design it should be broken down into mathematical process and made modular. approaches to coding should include: pairwise coding, avoiding code ownership, code review, and encourage knowledge retention. bus factor issue should be resolved by creating clean code and documenting codes. blocks which are dead ends during coding should be resolved by going to: senior and experienced programmers, problem domain, back to definition and design, physical objects like games, and interactive blogs like stack overflow or stack exchange. implementation in figure 3 should adopt phased change over approach instead of a holistic approach. it should be done incrementally. maintenance in figure 4 requires the system to be first deployed, and further requirements obtained from stakeholders and end users based on their usage. figure 5 is the various subdomain put together to form the software process ontology. 7. conclusion software process knowledge is a knowledge driven process with sub-processes. this knowledge is latent and could be lost if not formally harvested and documented. an improved use of this knowledge could lead to maximum payoff in software organisations. this is the heart of knowledge management, which focuses on knowledge capturing and sharing. this paper presents a generic software process domain ontology, covering the main technical software engineering subdomains of requirements definition, design & coding, implementation and maintenance. acknowledgments we thank the referees for the positive enlightening comments and 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[29] a. f. sawsaa, ontological engineering approach of developing 128 oveh et al. / j. nig. soc. phys. sci. 1 (2019) 122–130 129 table 3: object property s/n object property domain range 1 identifies ethnographic study stakeholders, users tasks 2 todetermine users tasks who is, what is, how is 3 require maintenance deployed system 4 toobtain domain experts business rules 5 through implementation incremental software process, phased changeover 6 using design modular design, mathematical process 7 with modular design data flow diagram 8 bygoingto block resolution senior and experienced programmers, problem domain, physical objects, interactive blogs, definition and design, user requirements 9 obtainedfrom deployed system users comments, stakeholders 10 requirementelicitationthrough stakeholders and endusers observation, interview, document review, brainstorming, questionnaire 11 transfersknowledgethrough coding code review, pairwise coding table 4: data property s/n domain concept 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name(string), organisation (string), portfolio (string), domain (string) observation, interview 11 block resolution definition and design (string), problem domain (string), experience programmers (string), interactive blogs (string), physical objects (string) document review, brainstorming, senior and experienced programmers, problem domain, physical objects, interactive blogs, definition and design ontology of information science, anchor academic publishing. https://books.google.com.ng/books?isbn=3954894483 (2015) [30] o. corcho and a. gomez-perez, “a roadmap to ontology specification languages”, in rose dieng and olivier corby (eds.), knowledge 129 oveh et al. / j. nig. soc. phys. sci. 1 (2019) 122–130 130 engineering and knowledge management. methods, models and tools, springer, berlin, (2000) 80. 130 j. nig. soc. phys. sci. 3 (2021) 59–65 journal of the nigerian society of physical sciences application of remote sensing and geoinformatics techniques in erosion mapping and groundwater management in the river amba watershed, central nigeria nuhu degree umara,∗, aliyu itari abdullahib adepartment of geology, federal university of lafia, nasarawa state, nigeria bdepartment of geology, university of nigeria, nsukka, enugu state, nigeria abstract this research integrated an easy-to-handle remote sensing data and geoinformatics techniques for erosion mapping and groundwater management in the river amba watershed, central nigeria. it is aimed at: (a) the determination of the erosion prone areas, and (b) the estimation of the groundwater potential contamination risk under current and future anthropogenic activities. rainfall intensity was evaluated from monthly rainfall data (2001 2011) from station located within the river amba watershed. digital elevation model (dem) for the terrain was created using 3d analyst tool (surfer 14) and was used to determine the flow direction and lineament features in each raster cells. remote sensing data (aerial photographs and landsat imagery) were used to develop land use map, while geological mapping was used to determine the local geology of the watershed area. the contributions of the various factors to the erosion hazardous areas are: elevation 31.49 %, land use 21 %, slope 14 %, geology 12.52 %, rainfall intensity 10.5 % and flow accumulation 10.5 %. the combined influences of these factors to erosion susceptibility as either: very high, high, moderate, low and very low withthe south-western part characterized as high while other parts of the study area moderate to very low erosion vulnerability. the groundwater level is shallow (4.0 –28.5 m) and discharges through the amba river and many springs. these springs along with boreholes and wells supply drinking water to lafia and environs. doi:10.46481/jnsps.2021.63 keywords: erosion, river amba, watershed, landsat imagery, gis article history : received: 17 march 2020 received in revised form: 18 march 2021 accepted for publication: 03 april 2021 published: 29 may 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: o. j. abimbola 1. introduction erosion is the removal and transport of soil, rock debris, or weathered materials by the action of water, wind, air and human activities from one location to another on the earth’s crust. this process is selective in a way as it removes mainly fine particles that contain relatively high proportion of organic matter. this ∗corresponding author tel. no: +2347069247822 email address: aliyu.itari@gmail.com (aliyu itari abdullahi ) portion of the soil that is always removed is very rich in fertility and therefore supports plants growth. soil erosion therefore, remains the main mechanism of soil degradation which threatens the global sustainability of the food production systems [1]. in the amba watershed, soil erosion has often been exacerbated by crude agricultural practices, and particularly nonimplementation of appropriate soil conservation measures such as crop rotation, run-off control and contour farming. studies showed that soil degradation leads to the loss of basic soil properties relevant to the farming system and/or an increase of the 59 nuhu & aliyu / j. nig. soc. phys. sci. 3 (2021) 59–65 60 production costs [2]. significant increase in erosion has been recorded all around the world in the last three decades especially in the tropics. these events which are quickly transformed into runoff, due to the high capacity of transport, can be characterized as the most significant weather-related hazards in many parts of the tropics all around the world, causing considerable economic and human losses [3]. the main causes of erosion are climate change, changes in land use and other anthropogenic interventions. the most common anthropogenic interventions are urban growth, the partial or total cover of torrent banks, watercourse alignment, improperly dimensioned bridges, deforestation and the consequent erosion, the construction of roads or other structures across the watercourse, subsidence observed in flat regions due to anthropogenic interventions such as over-pumping, and finally, the change or deviation of the watercourse [4]. due to rapid urbanization and economic growth in lafia in the past two decades, large percentage of cultivated land and forest land have been built up. areas where deforestation, farming or urbanization leads to the formation of direct runoff will be more prone to erosion than areas where land use changes such as afforestation minimizes direct runoff [5]. one of the major contributing factors of erosion is rainfall: the degree of impact is dependent on its intensity, distribution, duration, frequency and kinetic energy. erosion induced by rainfall causes many problems such as decreased in agricultural productivity due to the loss of arable land, increased landslide activity and ecosystem disturbance [6]. the lack of detailed climatic and hydrogeological data which are expensive and time consuming is a hindrance to the use of detail modelling approach. thus, in erosion studies, precipitation records and flow data have been widely used [7]. thus, the effective use of geographic information systems (gis) management tool is essential to delineate the flood prone areas [8]. furthermore, the synthesis of available data and the mapping of the relationships between groundwater hazard and the elements at risk require the use of tools such as gis. integrating gis and remote sensing techniques can provide the base for analyzing quantitatively environmental process with an appropriate degree of accuracy. whereas research has been conducted into the geology [9, 10], mineral resources [11], hydrogeology [9, 12] and aquifers characteristics and groundwater quality [10], little or no attention has been channeled towards assessing the vulnerability of lafia formation to erosion. the aim of this work is to use an integrated and easy to handle gis tool that incorporates geoinformatics techniques for erosion and groundwater management in tropics. the proposed water management tool has two components: (a) the determination of the erosion hazardous areas, and (b) the estimation of groundwater flow and the potential contamination risk under current and future anthropogenic pressures and activities. 2. location and accessibility of the study area river amba catchment is located in the southwest of nasarawa state, central nigeria. it lies between latitude 8◦ 24 ′ and 8◦ 36 ′ and longitude 8◦ 26 ′ and 8◦36 ′ (fig. 1) and drains a surface area of about 800 km2 (fig. 2) with the monitoring station is located at the outlet (coordinates: 8◦ 29 ′ 36.4” n and 8◦ 30 ′ 31.4” e). river amba is the major river that drained the area and is characterized by the guinean savannah vegetation. the original vegetation has been tampered with due to human activities such as farming, bush burning and grazing, which has given rise to a secondary forest [13, 10]. the area is characterized by two major distinct seasons (wet and dry), the former lasts from march to october, while the later lasts from november to february. the annual average rainfall ranges between 1000 and 1500 mm while the mean annual humidity is 70 % with relative humidity of 60 to 80 % [14]. temperatures range from 33 – 36 ◦c with an annual average temperature of 28.5 ◦c. an annual average sunshine hour of 6.7 per day is also experienced. the elevations of this area range from 110 to 243 m above sea level. the geological bedrock is sandstone, and it is overlaid by deep and highly loose soils (oxisols, ultisols, and alfisols), with the oxisols being the dominant soil class in the catchment. these soils are enriched in iron oxides and laterite. the landscape is generally characterized by gentle slopes (3–6 %), whereas steeper slopes (5–9 %) are found near the drainage channels. accessibility is by the trunk ‘a’ lafia – akwanga and further improved by minor routes and footpaths linking villages, farm lands and streams (fig. 3). the major stream flows in northeast southwest direction which corresponds to the strike direction of most structures generally found in the area indicating structurally controlled stream. figure 1. topographic map of the study area. 2.1. geology and land use geologically, the study area is underlain by the laa formation (fig. 4) also called the ‘laa sandstone’ [15]. it lies within central benue trough, nigeria, and characterized by ferruginized sandstones, red loose sands, flaggy mudstones, clays and claystones. hydrogeologically, the lafia formation comprises 60 nuhu & aliyu / j. nig. soc. phys. sci. 3 (2021) 59–65 61 figure 2. river amba watershed map. figure 3. accessibility map of the study area. mainly fine-to-coarse grain sandstones, which are highly porous and permeable [13] and is the most prolific in the central benue trough. land use including farming, forests, wetlands, and urban areas cover less than 15 % of the total catchment surface area. the riparian areas (< 10 m wide) found along the river catchment network are narrow. they are further affected by cattle trampling, which prevents them from providing effective traps to stop sediment emanating from upper parts of the catchment. the hydrogeology of the lafia formation and the central benue trough has been studied by [9, 15] who made a brief survey of the water resources of the area in the light of its importance to the development of the region. an evaluation of some hydrogeological characteristics of the aquifers of lafia formations [10], where he reported the aquifer to be about 150 m thick, highly prolific having dominantly moderate aquifer protective capacity. four types of aquifers were identified [9, 15]. these are; the awe formation, the keana / ezeaku formation, the awgu formation and the lafia formation (being the youngest). in obi village, the lafia sandstone is sub-artesian and quite near the surface which can be tapped for domestic and industrial purposes [16]. 3. materials and methods 3.1. estimating the erosion hazardous areas in the amba watershed based on factors that form and influence erosion; slope, land use, rainfall intensity, geology and flow accumulation the study area was divided into five regions characterized by different degrees of erosion susceptibility (very high, high, moderate, low and very low). relative weights were assigned to each factor based on its influence in triggering erosion according to kourgialas & karatzas [4]. thematic maps are produced for each parameter in a gis environment. a combination of these thematic maps and the selection of the weights yield the final map of erosion prone areas. the original data include the topographic map (fig. 1) and the monthly rainfall data (table 1) from station located in the surrounding area. the digital elevation model (dem) for the terrain was created by surfer 14. remote sensing data (aerial photographs and landsat imagery) were used to determine the land use, while geological mapping was used to determine the geology of the watershed area. from the dem, the flow direction in each raster cell was determined. this was followed by the identification of the water accumulation points. the flow concentration map, which indicates the number of cells that hydrologically contribute to each raster cell was developed by using the flow direction map combined with a suitable algorithm (flow accumulation arc hydro). output cells with a high flow accumulation (pixels) are areas of concentrated flow [4]. the rainfall intensity was determined from the meteorological data (2001-2011) of area surrounding the amba watershed (table 1). the rainfall intensity map was created by using the modified fourier index (mfi) methodology [17]: mf i = 12∑ 1 p2 p (1) where ∑12 1 is the 12-month summation, p 2 is the average monthly rainfall and p is the average annual rainfall. the mfi indicator expresses the sum of the average monthly rainfall intensity at a station. 3.2. estimating the groundwater direction and groundwater contamination risk in the amba watershed the static water levels (swl) of over two hundred wells and boreholes were measured using dipper (the dipper-t model) while the topographic elevation and coordinates (latitude and longitude) of each was established using geographical positioning system (e-trek 20 garmin model). hydraulic head for 61 nuhu & aliyu / j. nig. soc. phys. sci. 3 (2021) 59–65 62 figure 4. geologic map of the amba watershed [10]. each well was estimated from the swl and topographic elevation above mean sea level measured for such well. these measurements were used to construct hydraulic head map. this map describes the groundwater flow direction in the study area. groundwater contamination risk was estimated in the extended area by combining information of the equipotential contours map created in a gis environment with lineament map of the area (fig. 5). the primary data for the creation of the above maps include: a) land uses based on aerial photographs, b) hydrogeological layers maps, c) soil maps, and d) groundwater level data from wells in the study area. figure 5. lineaments density map of the amba watershed. 4. results and discussion 4.1. erosion prone areas the six factors viz; flow accumulation (fig. 6), slope (fig. 7), elevation (fig. 8), rainfall intensity (table 1), geology (fig. 4), and land use (fig. 9), were used to estimate the erosion susceptible areas and create the corresponding maps. these factors were assigned numeric values, except geology and land use which were expressed in descriptive form [4]. in the case of the numeric-valued factors, five classes of susceptibility were identified while in the case of the non-numeric-valued factors, classification depends mainly on the influence of the factor on the generation of erosion process. for instance, for the geology factor, a loamy soil indicates very low erosion vulnerability while in the case of the land use factor, limited land cover (low land cover) indicates a very high erosion susceptibility. it is almost impossible to estimate erosion susceptibility of an area by considering the influence of a single factor, therefore the integration of all related factors is necessary in order to obtain the overall erosion vulnerability map as all factors have varying degrees of influence on the vulnerable areas. a weighting approach, where a different weight is assigned to each factor, was applied while the factor weights were determined according to kourgialas & karatzas [4]. based on this methodology, the effects of each factor on all other factors are depicted in fig. 10. a solid line between two factors indicates that one factor has a main effect on the other pointed by the arrow, that is, a change of the first factor has a direct effect on the other (main avenue). a dashed line between two factors indicates that one factor has a secondary effect on the factor pointed by the arrow, that is, a change of the first factor has an indirect effect on the other (minor avenue). for example, flow accumulation has a main effect on land use and a secondary effect on slope. in order to quantify the two different 62 nuhu & aliyu / j. nig. soc. phys. sci. 3 (2021) 59–65 63 table 1. mean monthly rainfall data (mm) from 2001 – 2011 (source: nimet, lafia). year 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 mean std dev jan 0 0 0 0 0 0 0 0 0 0 0 0.0 0 feb 0 0 26 0 0 0 0 0 0 0 9.3 3.2 7.68 mar 0.4 0 0 0 58 40 21 14 0 0 0 12 19.03 apr 118 55 109 148 43 1.9 89 92 128 75 28 81 42.91 may 205 143 170 159 162 225 164 181 190 116 198 176 29.01 jun 231 48 128 181 106 166 255 229 324 125 222 183 74.98 jul 278 298 262 268 242 304 244 188 230 382 74 252 73.35 aug 312 337 266 283 356 251 233 241 193 230 275 271 46.67 sep 216 162 319 183 154 248 274 109 146 312 228 214 66.40 oct 54 139 105 84 169 84 0 77 376 177 227 136 160.99 nov 0 12 24 0 0 0 0 0 8.8 20 0 5.9 8.61 dec 0 0 0 0 0 0 0 0 0 0 0 0.0 0 total 1415 1192 1406 1305 1129 1319 1279 1136 1595 1438 1261 1316 529.63 figure 6. flow direction and accumulation map of the study area (note: colour bands show demarcation of watersheds). types of effects, one (1) point is assigned to a main and half a point ( 12 ) to a secondary effects [17]. then, the rate for a factor is computed as the summation of the points corresponding to the effects emanating from the factor. based on the above weighting approach, the contribution of each factor to the erosion vulnerable areas, expressed as a percentage, is for the elevation: 31.49 %, land use: 21 %, slope: 14 %, geology: 12.52 %, rainfall intensity: 10.5 %, and flow accumulation: 10.5 %. the resulting map of the erosion vulnerability areas includes the combination of the above six variables that are related directly to any erosion event that occurs in the watershed. specifically, the six maps that were developed after the classification method were combined using a weighted linear combination approach in a gis environment. accordingly, each factor is multiplied by its percentage weight and the summation of all factors yields the final hazardous area map [18]. s = ∑ wi xi (2) figure 7. slope classification map of the study area. figure 8. elevation map of the study area. 63 nuhu & aliyu / j. nig. soc. phys. sci. 3 (2021) 59–65 64 figure 9. land use map of the study area. where, s is the final hazardous areas map, wi is the weight of factor i (percentage) and xi is the rate of the factor i. the factors (maps) were combined according to equation (2) and the final flood hazardous areas map was produced (fig. 11). according to this figure, the study area of amba watershed can be classified with respect to erosion hazard from high to very low. based on the results, the south-eastern parts of the watershed can be characterized as high flood prone areas. figure 10. a schematic depiction of the interaction between factors that influence the flood hazard [4]. 4.2. groundwater risk under an intensive agriculture due to recent agricultural activities in the study area, particularly fertilizers and pesticides application and the hypothetical scenario of charging pollutants and whether these may create environmental problems in the extended area, the map of hydraulic heads (equipotential lines) for the entire watershed area was created (fig. 4) based on the groundwater level data from over two hundred wells. it describes the flow direction, perpendicular to the equipotential lines. the direction of groundwater flow is northwest, through this flow direction groundwater encountered various soil types of varying porosities resulting in many springs particularly at contact points. from the study, it was observed that a) based on the groundwater level data the aquifer in the amba watershed is shallow (4.0–28.5 m) [10],b)the study area receives significant amounts of rainfall, fact that lead to a rapid recharging of groundwater resources, and c) the groundwater flow in the study area discharges through the amba river and many springs. these springs supplies drinking water to lafia settlements and environs. based on the above, the highly significant groundwater contamination risk in the study area from any excessive use of agrochemicals through intensification of agricultural activity is obvious. any deposition of contaminants such as nitrates in this area may have the final recipient of the amba river and adjoining springs. figure 11. erosion vulnerability zone map of the study area. 5. conclusion this study specifically evaluated the erosion prone areas and estimated the groundwater contamination risk in the amba watershed by utilizing a combined remote sensing and geoinformatics approach. thus, the following conclusions based on the results of this study: 1. the erosion hazard phenomena in the study area is characterized as high in south-western part and moderate to very low in other parts of the study area. 2. the loamy soil of the amba watershed is characterized by high clay content and shallowness. all these features contribute to a high groundwater contamination risk due to increased fertilizers and pesticides application. regarding the possible contamination in surrounding areas, 64 nuhu & aliyu / j. nig. soc. phys. sci. 3 (2021) 59–65 65 especially the spring, the groundwater hazard can be classified as very high. the estimation of erosion and the groundwater hazardous areas are fundamental components of a water management strategy. this study could become a useful tool for better management plan for the amba watershed and others which are at risk of potential erosion and the high risk of groundwater contamination. acknowledgments we thank the referees for the positive enlightening comments and suggestions, which have greatly helped us in making improvements to this paper. the authors are grateful to dr. solomon o. onwuka of the department of geology, university of nigeria, nsukka, for his assistance at 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[18] r. p. c. morgan, soil erosion and conservation, blackwell publishing ltd, oxford uk, 2005. 65 j. nig. soc. phys. sci. 2 (2020) 170–179 journal of the nigerian society of physical sciences original research on the investment strategy, effect of inflation and impact of hedging on pension wealth during accumulation and distribution phases. b. o. osua,∗, k. n. c. njokub, b. i. oruha adepartment of mathematics, michael okpara university of agriculture, umudike, nigeria. bdepartment of mathematics, imo state university, owerri, nigeria. abstract this paper studies the various results obtained in literature on the investment strategy, the effect of inflation and impact of hedging on the pension wealth generation. the explicit solution of the constant relative risk aversion (crra) and constant absolute risk aversion (cara) utility functions are obtained, both in the accumulation and distribution phase, using legendre transform, dual theory, and change of variable techniques. it is established herein that the elastic parameter (β) is not equal to one (β , 1), based on the assumption of our model. theorems are constructed and proved on the various wealth investment strategies. observations and significant results are made and obtained, respectively in the comparison of our various utility functions and some previous results in literature. sensitivity analysis and simulations on the various utility functions and optimal strategies during the accumulation and distribution phase are presented; when the existence of a elastic parameter that is not equal to one, when there is existence of modifying factors, when there is need for diversification of investment, when there is no significant effect of the choice of risk aversion strategy on investment returns during inflation period, when there is hedging ability of inflation-indexed bond and inflation-linked stock and when there is insignificant effect of the orthogonal relationship between stock and time and nonpayment of pension benefits on the satisfaction of the investors. keywords: hedging, constant relative risk aversion (crra), constant absolute risk aversion (cara), defined contribution (dc), constant elasticity of variance (cev), optimal strategy. article history : received: 16 march 2020 received in revised form: 29 april 2020 accepted for publication: 29 april 2020 published: 01 august 2020 c©2020 journal of the nigerian society of physical sciences. all rights reserved. communicated by: t. latunde 1. introduction one of the major challenges that old people are faced with is economic insecurity. by economic security we mean; having access to information, good health condition, good education, and many more, and security as it concerns work. particularly, with the work related economic security, it is only but natural that at retirement age, they would become economically vul∗corresponding author tel. no: +2348032628251 email address: osu.bright@mouau.edu.ng (b. o. osu ) nerable. this is true because at old age, they are mentally, economically and physically inactive. thus, in a bid to providing this required economic security, the national pension scheme was established [1]. pension scheme, an economic security scheme that takes care of people at retirement was introduced in 1951, by colonial british administration, through an instrument called “pension ordinance” (the nation, may, 2015). this ordinance had retroactive effect from 1946, and was only applicable to those british officials that was posted to nigeria (the nation, may, 2015) and as such, attracted some legislation, such as; pension 170 osu et al. / j. nig. soc. phys. sci. 2 (2020) 170–179 171 law for military, and many more. roughly speaking, pension can be said to be a specific pull fund into which a sum of money is paid during an employee’s active years, and that is used in turn to service his/her periodic payments after retirement. according to the encyclopedia britannica, pension is simply “a series of periodic” money payments made to a person(s) who retires from employment due to age barrier, challenges due to disability, or the completion of an agreed period of service. these payments continue as long as the beneficiary is alive, and even sometimes, extends to a next-of-kin. pension consists of lump sum payment made to an employee upon his/her disengagement from active service. payments are mostly made installmentally, every month. pension scheme could be contributory (i.e., defined contribution plan), where a fixed amount of money is invested through the pension wealth, or noncontributory (i.e., defined benefit plan), where a fixed amount of money is paid to a person, regularly. 2. literature review recent publications in economic journals and other reputable mathematics and science journals have brought to light, variety of methods of optimizing investment strategies and returns. for instance, some researchers have made various contributions in this direction, particularly, in dc pension plan. in [2] a work on stochastic life styling optimal dynamic asset allocation for defined contribution pension plans was done. in their work, various properties and characteristics of the optimal asset allocation strategy, both with and without the presence of non-hedge-able salary risk were discussed. the significance of alternative optimal strategy by pension providers was established. in [3], a defined contribution (dc) pension plan investment problem during the accumulation phase under the multi-period mean-variance criterion was investigated. in [4] the optimal investment strategies for a dc pension fund under the hull-white interest rate model. under this model, the pension fund manager can invest capital in the bank account, stock index, and real estates was analysed. more so, [5] studied optimal pension management in a stochastic framework, they came out with a significant result. in order to deal with optimal investment strategy, the need for maximization of the expected utility of the terminal wealth became necessary. example, the constant relative risk aversion (crra) utility function, and (or) the constant absolute risk aversion (cara) utility function were used to maximize the terminal wealth. in [5, 6] and [7, 8], crra was used to maximize terminal wealth. however, [9] used the crra and the cara to maximize terminal wealth, and this triggered our research. in [4] an optimal investment strategy for a dc pension fund with a stochastic salary, under the affine interest rate model was constructed. in this work, they introduced the notion of ” relative pension wealth”, that is, where a pension plan member only considers his/her post-retirement benefit, the ratio of the pension wealth to his terminal salary. [9] considered a minimum guarantee while obtaining their optimal policy, under inflationary market. this minimum guarantee served as an amortization fund to make up for lost fund due to inflation. in 2009, [10] obtained an optimal investment strategy for annuity contracts, under the constant elasticity of variance (cev) model. in his work made an assertion without a proof that the elastic parameter is not equal to 1. the work did not consider in its pension wealth model the fact that the next-of-kin of the dead contributors should be paid some benefits. neither did it consider categories of contributors, since in every employment period, a certain group of people of the same cadre is employed, therefore, in real life situation, the right thing should be multiple and categorized contributors. it did not also indicate that contributors will not willingly withdraw from the scheme. sequel to the above, [11, 12] came up with a modification. however, these modifications are not void of mistakes, omission and limitations, maybe due to environmental and financial factors. hence, we further modify of this works, by considering different categories of contributors, with some other additional assumptions made. our task in this work is to compare the various results obtained in literature, and categorically state the optimal policy in the various trading period. 3. preliminaries we start with a complete and frictionless financial market that is continuously open over the fixed time interval [ 0, t ], for t > 0 , representing the retirement time of any plan member. we assume that the market is composed of the risk-free asset (cash), and risky asset (stock). let (ω, f, p) be a complete probability space, where ω is a real space and p is a probability measure, {ws (s) , wt (t)} are two standard non-orthogonal brownian motions, {fs (s) , ft (t)} are right continuous filtrations whose information are generated by the two standard brownian motions {ws (s) , wt (t)} , whose sources of uncertainties are respectively to the stock market and time evolution. 3.1. materials and methods assume we represent u = us as the strategy and we define the utility attained by the contributor from a given state y at time t as ∅u (t, r, y) = eu [ u (y (t)) : r (t) , y (t) = y ] (1) where t is the time, r is the short interest rate andy is the wealth. our interest here is to find the optimal value function ∅ (t, r, y) = s upu∅u (t, r, y) (2) and the optimal strategy u∗ = u∗s such that ∅u∗ (t, r, y) = ∅ (t, r, y) (3) 3.2. legendre transformation the legendre transform and dual theory help to transform the nonlinear partial differential equation that is formed due to 1, to a linear partial differential equation. 171 osu et al. / j. nig. soc. phys. sci. 2 (2020) 170–179 172 theorem 1 [13] : let f : rn → r be a convex function for z > 0, then the legendre transform is defined as; l (z) = max y { f (y) = zy} , (4) where l (z) is the legendre dual of f (y). since f (y) is convex, from theorem 1 we define the legendre transform ∅̂ (t, r, z) = sup {∅ (t, r, y) − zy} (5) 0 < y < ∞, 0 < t < t, where ∅̂ is the dual of ∅ and z > 0 is the dual variable of x. the value of ywhere this optimum is attained is denoted by h (t, r, z) , so that h (t, r, z) = in f { y : ∅ (t, r, y) ≥ zy + ∅̂ (t, r, z) } (6) 0¡t¡t, the function h and φ̂ are closely related and can be referred to as the dual of φ . these functions are related as follows ∅̂ (t, r, z) = ∅ (t, r, y) − zh (7) where, h (t, r, z) = y,∅y = z, h = −∅̂z. (8) at terminal time t , we denote û = s up {u(y) − zy : 0 < y < t} and ∅ (z) = s up { u (y) ≥ zy + u (z) } . as a result ∅ (z) = u−1 (z) , (9) where φ is the inverse of the marginal utility uand note that ∅ (t, r, y) = u (y). at terminal time t , we can define h (t, r, y) = infy>0 { y : u (y) ≥ zy + φ̂ (t, r, z) } and φ̂ (t, r, y) = s upy>0 {u (y) − zy} so that h (t, r, z) = u−1 (z). 4. the model this section introduces the financial market and proposes the optimization problems in the pension distribution phase. (i.e., pension payment/disbursement period). 4.1. the financial market here, we consider a financial market that consists of a riskfree asset (i.e., cash in the bank) and a risky asset (stock). let the risk-free asset ct , say, at any positive time, t, evolve thus; dct = rctdt (10) where r represents constant rate of interest. next, we denote the price of the risky asset (stock) at any positive time r, by s t, as in [11, 14, 15] thus; ds t = µs tdt + ks β+1 t dwt (11) where µ(µ > r) represents the instantaneous rate of return on stock, β (β≤ 0) is the elastic constant parameter, k is a constant, ks βt represents the instantaneous volatility. let {wt; t ≥ 0} denote a standard brownian motion, defined on a probability space, (ω, f, p) where f = {ft} is an augmented filtration generated by the brownian motion. 5. model assumption consistent with the nigerian pension reform act of 2004 [1], we make the following assumptions: 1. the pension scheme accumulates wealth. 2. there are different categories of contributions. 3. the contributors will not willingly withdraw from the scheme. 4. payments are made to the retirees. 5. an accumulated amount is paid to the next-of-kin of the dead contributors, at the instance of death by any contributor(s). 6. a certain amount is retained from the payment made to the families of dead contributors, by the pension managers (i.e., management fee). 5.1. model formation (i.e., the optimization program) the fund accruing from the contributors can be invested in both bank and stock. particularly, the fund to be invested by the fund manager is the surplus, which is the fund that is available after each period of routine disbursements. that is, let the contribution process be: dy = (1 + θi) ci+1dt (12) and the payment process d j = bi+1dt + (ai+1 −η) dws (13) then the surplus dp = dy − d j = (1 + θi) ci+1dt − (14)[ bi+1dt + (ai+1 −η) dws ] = (ci+1 + θici+1 − bi+1) dt − (ai+1 −η) dws therefore, our task here is to construct an optimal investment strategy for the assets for the remaining periods after retirement, to enable us maximize the expected utility at each retirement period. without loss of generality, the pension wealth is denoted by at any time 0 < t < t < t + n and it evolves stochastically, thus (see [12]): dy (t) = usy (t) ds t ds t + (15) (1 − us) y (t) dct dct + (ci+1 + θici+1) dt − (ai+1 −η) dws i = 0, 1, . . . , n − 1 and θi = 0,θ1 = 1,θ2 = 2, . . . ,θi (an integer) = staff loading, where; ai+1 > 0 represents various amount that 172 osu et al. / j. nig. soc. phys. sci. 2 (2020) 170–179 173 is paid to the next-of-kin of the dead contributors, bi+1 > 0 represents various amount paid to retired contributors, ci+1 > 0 represents various amount contributed, η is the service charge deducted from the ai+1. however, relevant to the provisions of the nigerian pension reform act of 2004, on the eligibility condition for signing up on the pension scheme, by both government and private sectors, we have; dy (t) = usy (t) ds t ds t + (16) (1 − us) y (t) dct dct + (ci+1 + θici+1 − bi+1) dt − (ai+1 −η) dws i = 4, 5, . . . , n − 1, and θ4 = 4,θ5 = 5,θ6 = 6, . . . ,θn = n (a positive integer)= staff loading. assuming, bi+1 = ci+1 + rdci+1; r < rd , and rd represents discounted interest, then dy (t) = usy (t) ds t ds t + (1 − us) y (t) dct dct + (17) (θi − rd ) ci+1dt − (ai+1 −η) dws, i = 4, 5, . . . , n − 1, and θ4 = 4,θ5 = 5,θ6 = 6, . . . ,θn = n,θi > 0 (integer)= staff loading. taking into 12, 13 and 18, one obtains the wealth process dy (t) usy (t) [ µdt + ks βdwt ] + (1 − us) y (t) rdt + (θi − rd ) ci+1dt − (ai+1 −η) dws us y (t) µdt + usy (t) ks βdwt + y (t) rdt + us y (t) rdt + (θi − rd ) ci+1dt − (ai+1 −η) dws (us y (t) µ + y (t) r − usy (t) r + (θi − rd ) ci+1) dt +us y (t) ks βdwt − (ai+1 −η) dws (18) i = 4, 5, . . . , n − 1, and θ4 = 4,θ5 = 5,θ6 = 6, . . . ,θn = n,θi > 0 (integer)= staff loading. based on the wealth process in 19, the pension manager seeks a strategy, u∗t , which maximizes the utility function, such that u∗t = max e (u (y (t ))) ,∀u(t). where u(•) is an increasing concave utility function, which satisfies the inada conditions; u′(+∞) = 0,, and u′(0) + ∞, (cf. [10]). 5.2. equations for utility functions and optimal policies for crra utility function in the pension distribution phase under noninflationary market h (t, r, z) = z 1 p−1 e d r 1−p −r+ 1 q(2−p) − u2s k2 (1−p) + u2s (2−p) j 2k2 (1−p) e[t−t] −ert [ 2k (θi − rd ) ci+1 + αγus (ai+1 −η) ] [t − t] (19) and u∗s = αγ (ai+1 −η) 2x (t) k − uz ( p − 1) z 2p−3 p−1 ×e d r 1−p −r+ 1 q(2−p) − u2s k2 (1−p) + u2s (2−p) j 2k2 (1−p) e[t−t] (20) where as in [16], c1 = 1 2 (ai+1 −η) 2 + 38 (αγ) 2 (ai+1 −η) 2, as required. for cara utility function in the distribution phase under noninflationary market g∗ (t.s.z) = q−1er(t−t) ln z + q−1er(t−t) (∫ w (t ) dt− ∫ w (t) dt + e− (αγ)2 k(ai−1−η) 2 q ∫ s(t)dt ) −q−1er(t−t) u2l 2k2 [∫ e− (αγ)2 k(ai−1−η) 2 q ∫ s(t)dtdt − ∫ e− (αγ)2 k(ai−1−η) 2 q ∫ s(t )dt dt ] +e−rt [∫ ert 2k (θi − rd ) ci+1 + αγul (ai+1 −η) 2k dt −ert 2k (θi − rd ) ci+1 + αγul (ai+1 −η) 2k dt ] (21) and u∗s = αγ (ai+1 −η) 2x (t) ks β + ule (t−t) αγ 2qx (t) s β−1 d ds × ( e−(αγ) 2 k(an+1−η)q ∫ s (t)dt ∫ e−(αγ) 2 k(an+1−η)q ∫ s (t )dt dt ) (22) where q > 0, µ = ul + r (see [17]). for crra utility function in the accumulation phase under inflationary market u∗s = σip σss −  −λ1σ s s −λ2σ i sθi + σ i sσ i p + σ i s ( θi − σip 2 ) σss  × 1 p − 1 e { a±(a2−2k2 h)1/2 k−12 t } e { a±(a2−2k2 h)1/2 k−12 t }  e { a±(a2−2k2 h)1/2 k−12 t } e { a±(a2−2k2 h)1/2 k−12 t } − ρ∗z 1 p−1 k ( 1 − e−ρ∗(t−t) )  . (23) u∗b = σip σi − σipσ i s σssσi +  [( σis )2 θi −σ i sσiσ i p −σ i sλ1σ s s ] (σss) 2 − ( σis )2 λ2θi + (σis) 2 σip 2 + σssσ i sσ i p 2 (σss) 2  1 p − 1 + e { a±(a2−2k2 h) 1 2 k−12 t } e { a±(a2−2k2 h) 1 2 k−12 t }  e { a±(a2−2k2 h) 1 2 k−12 t } e { a±(a2−2k2 h) 1 2 k−12 t } − ρ∗z 1 p−1 k ( 1 − e−ρ∗(t−t) )  + θi p − 1 e { a±(a2−2k2 h) 1 2 k−12 t } e { a±(a2−2k2 h) 1 2 k−12 t }  e { a±(a2−2k2 h) 1 2 k−12 t } e { a±(a2−2k2 h) 1 2 k−12 t } − ρ∗z 1 p−1 σi k ( 1 − e−ρ∗(t−t) )  , ρ∗ = 1 2 ρ5 + ρ1, σ i p = σ s p (24) 173 osu et al. / j. nig. soc. phys. sci. 2 (2020) 170–179 174 where ρ1, ρ2, . . .ρ5 are as in [16]. h = ρ5 2 (1 − p) + ρ1 1 − p − 1 2 ρ5 − 1 2 ρ1 + 2 ( 2θ2i 1 2 ( σip )2 − θiσ i p + ρ2 + ρ4 ) 1 − p − (2 − p) ( 2θ2i 1 2 ( σip )2 − θiσ i p + ρ2 + ρ4 ) (1 − p)2 n (t) = 2b [t − t ] k1 , d1 = 4b 2k1 , d2 = 0, (25) 6. sensitivity analysis for crra utility function in the pension distribution phase under noninflationary market; setting αγ = 0 (that is, saying that stock and time have orthogonal relationship), and ai+1−η = 0 (that is, no money is paid to the next-of-kin of the dead contributors) in equations ?? and 20 yield, respectively; h (t, s, z) = z 1 p−1 e d r 1−p −r+ 1 q(2−p) − u2s k2 (1−p) + u2s (2−p) j 2k2 (1−p) e[t−t] −ert [2k (θi − rd ) ci+1] [t − t] , (26) u∗s = −uz (p − 1) z 2p−3 p−1 e −d r 1−p −r+ 1 q(2−p) − u2s k2 (1−p) + u2s (2−p) j 2k2 (1−p) e[t−t] (27) and h (t, s, z) = z 1 p−1 e d r 1−p −r+ 1 q(2−p) − u2s k2 (1−p) + u2s (2−p) j 2k2 (1−p) e[t−t] −e−rt [2k (θi − rd ) ci+1] [t − t] , (28) u∗s = −us ( p − 1) z 2p−3 p−1 e −d r 1−p −r+ 1 q(2−p) − u2s k2 (1−p) + u2s (2−p) j 2k2 (1−p) e[t−t] (29) for cara utility function in the pension distribution phase under noninflationary market, setting αγ = 0 (that is, saying that stock and time have orthogonal relationship), and ai+1 − η = 0 (that is, no money is paid to the next-of-kin of the dead contributors) in equations 21 and 22 yield, respectively; h (t.s.z) = q−1 [ er(t−t) ( ln z + ∫ w (t ) dt−∫ w (t) dt + u2l 2k2 [∫ (dt − dt ) ] + table 1. parameters and their respective values . name of parameters symbol used values constant rate of interest r 0.02 expected stock returns µ 0.10 instantaneous stock returns us = ul 0.07 stock volatility k 0.55 risk aversion q = p 0.50 rate of contribution rd 0.075 management fee ηi 0.025 e−rt [∫ ert 2k (θi − rd ) ci+1 2k dt −ert 2k (θi − rd ) ci+1 2k dt ] (30) u∗l = ule r(t−t) (31) and h (t.s.z) = −q−1 [ er(t−t) ( ln z + ∫ w (t ) dt − ∫ w (t) dt + u2l 2k2 [∫ (dt − dt ) ] +e−rt [∫ ert 2k (θi − rd ) ci+1 2k dt −ert 2k (θi − rd ) ci+1 2k dt ] , (32) u∗s = ule −iωt − αγ 2y (t) s β−2 q−1 f ′ (s ) , (33) where= f ′ (s ) = 1q d ds (∫ dt ) . for cara utility function in the pension accumulation phase under inflationary market; setting σip = σ i s = θi = 0 , makes the optimal strategies in equations 23 and 24 would be of the form of the [4]. what this means is that in the absence of inflation vis-a-vis risk premium due to inflation, we would have a model of the form of [18]. 7. numerical simulation a numerical example of the proposed model was given to demonstrate the dynamic behavior of a dc pension fund and optimal investment strategy. nigeria-national pension fund administration (nnpfa) real data was used to illustrate the efficiency of the model. the parameters used are summarized in the table below, for t = 30. t = 0, 5, 15, 20, 25, 30, with i = 4, 5, 6, . . . , n − 1 and θ4 = 4,θ5 = 5,θ6 = 6,θ7 = 7, . . . ,θn = n. assume for perfect correlation αγ = 1, ai+1 = 274, 464.68 × 30 = 8233940.4 as at 30th june (for april, 2016), 2019, x (t) = 0.07 × 8233940.4 + 82333940.4 = 576375.83 + 8233940.4 = 8810316.2 , β = 0, s (t) = 7657564.6, x (t0) = 8233940.4 , which is initial pension wealth for 30 contributors, from january 2015 to april 2016 of imsu academic staff of 2014 set (trust fund pfa returns of june, 2019). for crra utility function in the pension distribution phase under noninflationary market, below are various three-dimensional 174 osu et al. / j. nig. soc. phys. sci. 2 (2020) 170–179 175 table 2. the numerical values for the optimal policy and the dual variable for the time period from 0 to 29th year, for the crra model. u 1.57 1.52 1.51 1.50 1.49 1.39 0.38 0.38 0.34 0.32 0.46 0.42 0.41 0.34 0.32 z 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 t 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 u 1.46 0.42 0.41 0.34 0.32 1.41 0.40 1.39 0.39 0.38 0.38 0.34 0.32 0.46 0.42 z 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 t 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 table 3. values for optimal policy, dual variable and continuous time for crra model, when there is orthogonal relationship between time and stock. u 0.47 0.042 0.041 0.040 0.039 0.038 0.038 0.34 0.32 0.46 0.42 0.41 0.34 0.32 0.46 z 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 t 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 u 0.42 0.41 0.34 0.32 0.41 1.40 0.39 0.39 0.38 0.38 0.34 0.32 0.46 0.42 0.42 z 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 t 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 figure 1. the graph of dual variable and instantaneous time against optimal policy for crra mode. tables (that is, tables 1, 2 and 3 and their plots respectively in figures 1, 2 and 3). table 4 above are various three-dimensional used in the plot of figure 3. for cara utility function in the distribution phase under noninflationary market, table 6 below are various threedimensional used in the plot of figure 6. table 7 above are various three-dimensional used in the plot of figure 7. table 8 above are various three-dimensional used in the plot of figure 8. 8. discussion for crra utility function in the distribution phase under noninflationary market; result shows that if we set αγ = 0 and ai+1−η = 0 , then in equations 24 and 27, the orthogonal relation between stock and time has the same effect on contributors’ satisfaction. figure 2. the graph of dual variable and instantaneous time against optimal policy for crra model, when there is orthogonal relationship between time and stock. in equations 26 and 28, we have the same negative effect (i.e., a decline in stock investment). considering the dual relationship between pension wealth, it follows that the optimal strategy will appreciate frictionless, which is unrealistic. there exists a modifying factor which depends on time, only, just like in [13] and this modifying factor controls investment decision of the ppm. from proposition 4.4.1 in [14], result shows that the elastic parameter, β is not equal to 1 (the elastic parameter β , 1 ). for cara utility function in the distribution phase under noninflationary market, results show that if we set αγ = 0 and ai+1 −η = 0 then in equation 30 and 32, the orthogonal relation between stock and time has the same effect on contributor’s satisfaction. however, both have significant effect. in equations 31 and 33, investments made in stock slightly reduce, vis-a-vis the stock returns. though, more money is invested in cash account than stock when there is orthogonal relationship between stock and time. here, there are minimal stock returns at each trading 175 osu et al. / j. nig. soc. phys. sci. 2 (2020) 170–179 176 table 4. values for optimal policy, dual variable and continuous time for crra model, when no payment was made to the next of kin of the dead contributors. u 0.047 0.042 0.041 0.040 0.039 0.039 0.038 0.38 0.34 0.32 0.46 0.42 0.41 0.34 0.32 z 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 t 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 u 0.46 0.42 0.41 0.34 0.32 1.41 0.40 0.39 0.39 0.38 0.38 0.34 0.46 0.32 0.46 z 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 t 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 table 5. values for optimal policy, dual variable and continuous time for crra model, for fixed values ranging from 1-3. t z u t z u t z u 0 1 0.01126233265 0 2 0.1801973224 0 3 0.9122489445 1 1 0.01163316951 1 2 0.1861307122 1 3 0.9422867306 5 1 0.01324271030 5 2 0.2118833648 5 3 1.072659535 10 1 0.01557131915 10 2 0.2491411064 10 3 1.261276851 15 1 0.01830939246 15 2 0.2929502793 15 3 1.483060789 20 1 0.02152893078 20 2 0.3444628925 20 3 1.743843393 25 1 0.02531459532 25 2 0.4050335251 25 3 2.050482221 30 1 0.02976593415 30 2 0.4762549464 30 3 2.411040666 35 1 0.03500000000 35 2 0.5600000000 35 3 2.835000000 table 6. values for optimal policy, dual variable and continuous time for cara model. u 1.57 1.52 1.51 1.50 1.49 1.39 0.38 0.38 0.34 0.32 0.46 0.42 0.41 0.34 0.32 z 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 t 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 u 1.46 0.42 0.41 0.34 0.32 1.41 0.40 1.39 0.39 0.38 0.38 0.34 0.32 0.46 0.42 z 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 t 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 table 7. values for optimal policy, dual variable and continuous time for cara model, when there is orthogonal relationship between time and stock. u 0.47 0.042 0.041 0.040 0.039 0.038 0.038 0.34 0.32 0.46 0.42 0.41 0.34 0.32 0.46 z 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 t 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 u 0.42 0.41 0.34 0.32 0.41 1.40 0.39 0.39 0.38 0.38 0.34 0.32 0.46 0.42 0.42 z 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 t 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 table 8. values for optimal policy, dual variable and continuous time for cara model, when there no payment was made to the next of kin of the dead contributors. u 0.047 0.042 0.041 0.040 0.039 0.039 0.038 0.38 0.34 0.32 0.46 0.42 0.41 0.34 0.32 z 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 t 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 u 0.46 0.42 0.41 0.34 0.32 1.41 0.40 0.39 0.39 0.38 0.38 0.34 0.46 0.32 0.46 z 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 t 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 176 osu et al. / j. nig. soc. phys. sci. 2 (2020) 170–179 177 figure 3. the graph of dual variable and instantaneous time against optimal policy for crra model, when there is no payment made to the next of kin of the dead contributors. figure 4. two dimensional view of crra model, for fixed values of the dual variable. period. this suggests that diversification of investments should be advised in order to enhance investment returns. there exists a modifying factor, which depends on both time and the coefficient of the utility function, just like in [10], and this modifying factor controls investment decision of the ppm. particularly, it reflects the investor’s decision to hedge the volatility risk. for crra utility function in the accumulation phase under inflationary market, the effect of inflation on pension wealth investment; the crra utility function has little or no effect on figure 5. the three dimensional view of the crra model, using maple of table 5. figure 6. the graph of dual variable and instantaneous time against optimal policy for cara model. the investment strategy. recall from [4], the coefficients d1, d2 degenerates to 4b2k1 and zero, in the absence of the coefficient of the crra (i.e., as p → 0 ), however, in this work, even in the presence of the coefficient of crra, the coefficients d1, d2 are already degenerate. for the effect of hedging, using inflationindexed bond and inflation-linked stock, the inflation-indexed bond and inflation-linked stock served as a hedging mechanism against inflation. in the absence of the coefficient of the crra (i.e., as → 0 ), the coefficients d1, d2 , still retains its value (i.e., will never degenerate further than this). there exists also a modifying factor, which depends on time, only, just like in [12], for crra utility function, and this modifying factor controls investment decision of the ppm. there exists a modifying factor which depends on both time and the coefficient 177 osu et al. / j. nig. soc. phys. sci. 2 (2020) 170–179 178 figure 7. the graph of dual variable and instantaneous time against optimal policy for cara model, when there is orthogonal relationship between time and stock. figure 8. the graph of dual variable and instantaneous time against optimal policy for cara model, when no payment was made to the next of kin of the dead contributors. of the cara utility function as in [10], for the cara utility function. this modifying factor controls investment decision of the ppm. particularly, it reflects the investor’s decision to hedge the volatility risk. there exists an elastic parameter that takes values other than unity. more so, the inflation-linked bond and inflation-indexed stock served as hedging mechanism in the optimization of pension wealth. lastly, comparing the utilities in the noninflationary market, during decummulation phase, we conclude that the policy, u∗s (t) due to crra utility function is the optimal policy, while in that of the inflationary market, during accumulation phase, the policies, u∗s (t), and u ∗ b(t) due to crra utility function is not the best fit, based on the assumptions of our models .based on the result above, there exists some optimal policies that can permit energetic retirees to invest in risky assets. 9. conclusion for the noninflationary market, we studied and modified the optimal investment strategy for annuity contract under the constant elasticity of variance by [10], and proved that the elastic parameter takes values other than unity. we constructed an optimal investment strategy in the pension benefit distribution phase, for both crra and cara utility functions, and the associated utility functions (see equations ??, 20, 21, and 21). we tested for sensitivity of some parameters in both the investment strategies and the utility functions (see sensitivity analysis). we simulated and compared the expected utilities of the accumulation phase (see figures 1 through 8 above using tables 1 through 8 as appropriate). based on these comparisons, we conclude that the investment strategy with the crra utility function provides the optimal strategy, though, the investment returns is still low, and this suggests that diversification of investment should be advised to enhance returns. more so, there exists an investment strategy that permits investments into risky assets, even after retirement. we conclude that both for crra and cara utility, the orthogonal relationship between stock and time have the same effect on contributor’s satisfaction. there exists a modifying factor which depends on both time and the coefficient of crra utility function, and also depends on time, only just like in [10], and this modifying factor controls investment decision of the pension plan member. particularly, it reflects the investor’s decision to hedge the volatility risk. for inflationary market, we constructed an optimal investment strategy in the pension accumulation phase, for the crra utility function and the associated expected utility functions (see equations 23 and 24). we also tested for sensitivity of parameters in both the investment strategies and the utility functions. we conclude that the crra utility approach has little or no effect on the investment strategy, and this depicts the effect of inflation on optimal investment strategy. more so, the inflation-linked bond and the inflation-indexed stock serve as hedging mechanism in the optimization of pension wealth. as a result of uncertainties of the markets, the substitute investment and the encouragement of higher investment yield, for the noninflationary markets, we recommend diversified investment of pension wealth into assets like government bond and even corporate (private) bond. for the inflationary market (in order to cushion the effect of inflation on pension wealth generation in a dc pension scheme) we recommend the investigation of the effect of extra stochastic contributions. acknowledgments we thank the referees for the positive enlightening comments and suggestions, which have greatly helped us in making improvements to this paper. references [1] nigerian pension reform act of (2004). 178 osu et al. / j. nig. soc. phys. sci. 2 (2020) 170–179 179 [2] a. j. g. cairns, d. blake & k. dowd “stochastic 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[18] p. k. mwanakatwe, l song & e hagenimana “management strategies for a defined contribution pension fund under the hull-white interest rate model”, advances in intelligent systems research (amms) 153 (2017) 239. 179 j. nig. soc. phys. sci. 2 (2020) 141–148 journal of the nigerian society of physical sciences original research perturbed collocation method for solving singular multi-order fractional differential equations of lane-emden type o. a. uwaherena,∗, a. f. adebisib, o. a. taiwoa adepartment of mathematics, university of ilorin, ilorin, nigeria bdepartment of mathematics, osun state university, oshogbo, nigeria abstract in this work, a general class of multi-order fractional differential equations of lane-emden type is considered. here, an assumed approximate solution is substituted into a slightly perturbed form of the general class and the resulting equation is collocated at equally spaced interior points to give a system of linear algebraic equations which are then solved by suitable computer software; maple 18. keywords: perturbed collocation method, legendre polynomial, singular multi-order differential equations, lane-emden equations article history : received: 17 march 2020 received in revised form: 15 may 2020 accepted for publication: 16 may 2020 published: 01 august 2020 c©2020 journal of the nigerian society of physical sciences. all rights reserved. communicated by: t. latunde 1. introduction perturbation collocation method is a technique of numerical approximation. it has been described as a very useful tool for solving differential and integro-differential equations of different kinds [1, 2, 3]. fractional order differential equation is a special kind of differential equations with fractional order derivative say α, which is a non-integer. many phenomena in science and engineering resulting in fractional order differential equations have been successfully represented using mathematical models in the field of mechanics, elasticity, science, education to mention but a few. most often, the equations arising from the mathematical modelling of fractional order differential equations are difficult to solve analytically. this is because many of them do not have solutions in closed form ∗corresponding author tel. no: +23480xxxx572 email address: uwaheren.ao@unilorin.edu.ng (o. a. uwaheren ) and hence seeking an approximate solution by numerical treatment becomes helpful. to achieve this, a lot of numerical methods have been advanced. [4, 5] proposed variational iteration method (vim) and used the method to solve multi-order fractional integro-differential equations and singular ivps of laneemden type respectively. their results showed that the method is an accurate one that yields the exact solution within a few iterations. their study, however, noted that in some cases, the method may require more calculations which will add some difficulties. [6] solved nonlinear multi-order fractional differential equations using legendre pseudo-spectral method (lpsm). the results reveal that the method is effective as a small number of shifted legendre polynomials were needed to obtain a satisfactory result. [7] presented a numerical solution method for solving fractional differential equations using bernstein polynomials. other methods used by many researchers are chebyshev wavelet method [8], chebyshev polynomial method [9, 10] collocation method [11], pade approximate method [12], 141 uwaheren et al. / j. nig. soc. phys. sci. 2 (2020) 141–148 142 etc. [12] described the lane-emden differential equation as a singular initial value problem relating to second-order differential equations which have been used to model several phenomena in mathematical physics and astrophysics. according to [13], lane-emden type of differential equation is used to describe a variety of phenomena in physics and astrophysics, including isothermal gas spheres and thermionic currents and determining their numerical solutions is very challenging because of the singularity behaviours at the point of origin. the study also solved some fractional order differential equations of lane-emden type using the collocation method and admitted that a collocation method is a suitable tool for solving a class of lane-emden type of differential equations. [14] worked on the numerical solution of lane-emden type equation using multilayer perceptron neural network method. the study aimed at producing an optimal solution of laneemden equation with less computation using multilayer perceptron artificial neural network technique. the results obtained show that the technique can develop into an effective approach for solving lane-emden type problems with less computational time and memory space. in this study, we are concerned with multi-order fractional differential equations of lane-emden type and their solutions by perturbation collocation method. to transform the problems to a system of linear algebraic equations, the collocation method algorithms as given by [13] was implemented. other researchers that have solved different problems of laneemden type using different methods include [15, 16, 17 ]. the general form of the singular multi-order fractional differential equation is given as n∑ i=0 piy (i)(x) + dα (y(x)) + k xα−β dβ (y(x)) + k xα−2 (y(x)) = f (x) (1) subject to conditions y(0) = a, y′(0) = b (2) where a, b are constants, pi, (i = 0, 1, 2, ..n) are arbitrary constants to be determined, n is the highest integer order derivative, α and β are fractional order derivatives, and α > β. 2. definition of relevant terms 2.1. assumed approximate solution an assumed approximate solution also called trial solution refers to a predetermined polynomial taken to represent the solution of the problem been solved. 2.2. maple 18 is a mathematical software that is capable of performing mathematical computation, for example, solving an equation symbolically or numerically, or construct maple objects etc depending on the command. 2.3. fractional differential equation a differential equation is called a fractional differential equation if it contains at least one fractional order derivatives, dα of the unknown function y(x). the general form of a fractional differential equation is given as dαy(x) = f (x, y(x)) (3) subject to the conditions: dαy(0) = ωk, k = 0, 1, ..., n where dα is the fractional order derivative in the caputo sense and α is a non-integer value, n = dαe, called the ceiling α, α > 0 is the highest order of the equation. the operator dα is defined as dα∗ f (x) = 1 γ(m −α) ∫ x 0 (x − t)m−α−1 dm dtm f (t)dt (4) in the caputo sense for function f (x) with (m−1) absolute continuous derivatives and riemann-liouville defined it as c d β x f (x) = 1 γ(n −β) dn d xn ∫ x c (x − t)β−1 f (t)dt (5) the advantage of the caputo’s fractional derivative is that one can specify the initial conditions of the fractional differential equation in classical form. i.e. d(k)y(k) = bk, k = 0, 1, 2, · · · , m − 1 (6) the generalized factorial form of non-integer order derivatives in euler’s gamma function f (x) = xm is given as dα d xα xm = γ(m + 1) γ(m − n + 1) xn−m (7) equation (7) is a property of fractional derivatives given in euler gamma functions which makes the definition more suitable and compatible for application. 2.4. multi-order fractional differential equations of lane-emden type a fractional multi-order differential equation of lane-emden type is of the form equation (1). however, when the highest order derivative of the equation is fractional, then equation (1) becomes: dα (y(t)) + k tα−β dβ (y(t)) + k tα−2 (y(t)) = f (t) (8) subject to conditions y(0) = a, y′(0) = b (9) where a, b, α and β are as described earlier. 142 uwaheren et al. / j. nig. soc. phys. sci. 2 (2020) 141–148 143 3. methodology consider a general class of multi-order fractional differential equation of lane-emden type given in equation (1) together with the conditions in equation (2). in order to solve equations (1) and (2), an assumed approximate solution of the form: yn (t) = n∑ j=0 a j l j(t) (10) is taken. where l j(t) is legendre polynomial and n is the degree of the assumed approximant. thus equation(10) is substituted into a slightly perturbed equation (1) to give n∑ i=0 pi  n∑ j=0 a j l ( j) j (t)  + dα  n∑ j=0 a j l j(t)  + k tα−β dβ  n∑ j=0 a j l j(t)  + ktα−2  n∑ j=0 a j l j(t)  = f (t) + hn(t) (11) where hn(t) = n∑ (r=1) τr t(n−r−1)(t) (12) is called the perturbation term, n is as stated earlier and t (t) is chebyshev polynomial. putting equation (12) into (11), gives n∑ i=0 pi  n∑ j=0 a j l ( j) j (t)  + dα  n∑ j=0 a j l j(t)  + k tα−β dβ  n∑ j=0 a j l j(t)  + ktα−2  n∑ j=0 a j l j(t)  = f (t) + n∑ (r=1) τr t(n−r−1)(t) (13) subject to conditions n∑ k=1 y(k)n (0) = φk; k = 1, 2, ...n (14) where τr (r = 1(1)n) are the free tau parameters to be determined, t(n−r−1)(t) are the chebyshev polynomials and a j are the unknown constants also to be determined. applying equations (7) on (13) accordingly gives n∑ i=0 pi  n∑ j=0 a j l ( j) j (t)  + ktα−β n∑ j=0 a j γ( j + 1) γ( j −α + 1) t j−α + k tα−2 n∑ j=0 γ( j + 1) γ( j −α + 1) t j−α + n∑ j=0 a j γ( j + 1) γ( j −β + 1) t j−β = f (t) + n∑ (r=1) τr t(n−r−1)(t) (15) equation (15) is further simplified and then collocated at equally spaced interior points, t = te on [a,b]; te = a + (b−a)i (n) ; i = 1,2,...,n to obtain a system of linear algebraic equations, including those obtained from the use of initial conditions. the system of linear algebraic equations are solved using gaussian elimination method (using a computer package; maple 18) to obtain the unknown constants. the constants obtained are then substituted back into the assumed approximate solution to give the required approximate solution. 4. numerical examples example 1 consider the multi-order fractional differential equation dα (y(x)) + k x(α−β) dβ (y(x)) + k x(α−2) (y(x)) = f (x) (16) where = [ 6x ( γ(4 −β) + kγ(4 −α) γ(4 −α)γ(4 −β) + x2 6 ) −2 ( γ(3 −β) + kγ(3 −α) γ(3 −α)γ(3 −β) + x2 2 )] x2−α (17) for k = 1, α = 32 and β = 1 2 subject to the conditions y(0) = y′(0) = 0 0 ≤ x ≤ 1 the exact solution is y(x) = x3 − x2 equation (16) is perturbed as d 3 2 (y(x)) + 1 x( 3 2 − 1 2 ) d 1 2 (y(x)) + 1 x( 3 2 −2) (y(x)) = f (x) + n∑ r=1 τr t(n−r−1)(x) (18) taking an approximate solution for n = 4 y4 = 4∑ j=0 y(x) = 4∑ j=0 a j l j(x) = a0 + a1 (2 x − 1) + a2 ( 6 x2 − 6 x + 1 ) + a3 ( 20 x3 − 30 x2 + 12 x − 1 ) + a4 ( 70 x4 − 140 x3 + 90 x2 − 20 x2 + 1 ) (19) substituting equations (17) and (19) into (18), we have dα  4∑ j=0 y(x)  + 1x(α−β) dβ  4∑ j=0 y(x)  + 1x(α−2)  4∑ j=0 y(x)  = 143 uwaheren et al. / j. nig. soc. phys. sci. 2 (2020) 141–148 144[ 6x( γ(4 −β) + γ(4 −α) γ(4 −α)γ(4 −β) + x2 6 − 2( γ(3 −β) + γ(3 −α) γ(3 −α)γ(3 −β) + x2 2 ) ] x2−α + n∑ r=1 τr t(n−r−1)(x) (20) further simplifications with α = 32 and β = 1 2 gives dα(a0 +a1(2x−1)+a2(6x 2 −6x+1)+a3(20x 3 −30x2 +12x−1) + a4(70x 4 − 140x3 + 90x2 − 20x + 1)) + 1 x dα(a0 + a1(2x − 1) + a2(6x 2 −6x + 1) + a3(20x 3 −30x2 + 12x−1) + a4(70x 4 −140x3 + 90x2 − 20x + 1)) + 1 x −1 2 (a0 + a1(2x − 1) + a2(6x 2 − 6x + 1) + a3(20x 3 −30x2 + 12x−1) + a4(70x 4 −140x3 + 90x2−20x + 1)) − ( 6 x ( 28 15 √ π + 1/6 x2 ) − 20 3 √ π − x2 ) √ x −τ1 ( 8 x2 − 8 x + 1 ) −τ2 (2 x − 1) = 0 (21) applying the operator dα on equation (21) and after some simplifications gives −1.7724538509 x9/2 + 1.7724538509 x7/2− 56 x5/2 5 + 20 x3/2 3 +124.0717696 x11/2a4+35.44907702 x 9/2a3+10.63472311 x 7/2a2 +3.544907702 x5/2a1+1.7724538509 x 3/2a0−3.544907702 τ2 x 2 −14.17963081 x3τ1−248.1435391 x 9/2a4−53.17361553 x 7/2a3 +1311.520847 x7/2a4−10.63472311 x 5/2a2+245.2694462 x 5/2a3 −1603.449077 x5/2a4−1.7724538509 x 3/2a1+41.77245385 x 3/2a2 −201.7724539 x3/2a3 +601.7724539 x 3/2a4 +6 √ xa1−18 √ xa2 + 36 √ xa3 −60 √ xa4 + 1.7724538509 τ2 x + 14.17963081 x 2τ1 − 1.7724538509 xτ1 = 0 (22) equation (22) is collocated at n − 2 equal spaced points in the interval [0, 1] and the remaining two equations are obtained from the use of the conditions attached. this gives 7 system of linear equations. solving the system of equations, we get y4(x) = −1.48141262×10 −10 + 3.2×10−9 x−1.000000039x2 + 1.00000004x3 − 5.469888349 × 10−9 x4 (23) when the same example was solved α = 52 , β = 3 2 following the same methodology,the approximate solution obtained is y4(x) = 5.00 × 10 −11 − 3 × 10−10 x − 1.397902893 x2 + 1.664907851 x3 − 0.1566880882 x4 (24) example 2 consider the multi-order fractional differential equation dα (y(x)) + k x(α−β) dβ (y(x)) + k x(α−2) (y(x)) = [ 2 ( γ(3 −β) + kγ(3 −α) γ(3 −α)γ(3 −β) + x2 2 ) − 6x ( γ(4 −β) + kγ(4 −α) γ(4 −α)γ(4 −β) + x2 6 )] x2−α (25) for k = 1, α = 32 and β = 1 2 subject to conditions y(0) = y′(0) = 0 0 < x ≤ 1 with exact solution y(x) = −x3 + x2 taking an assumed approximate solution for n = 4, and computing the α = 32 and β = 1 2 -derivatives of equation (25) and other necessary simplifications and substitution, gives 35.44907702 x9/2a3+10.63472311 x 7/2a2+3.544907702 x 5/2a1 + 1.7724538509 x3/2a0−3.544907702 τ2 x 2 −14.17963081 x3τ1 −248.1435391 x9/2a4−53.17361553 x 7/2a3+1311.520847 x 7/2a4 −10.63472311 x5/2a2+245.2694462 x 5/2a3−1603.449077 x 5/2a4 −1.7724538509 x3/2a1+41.77245385 x 3/2a2−201.7721531 x 3/2a3 +1.7724538509 x9/2+124.0717696 x11/2a4+601.7724539 x 3/2a4 + 6 √ xa1 −18 √ xa2 + 36 √ xa3 −60 √ xa4 + 1.7724538509 τ2 x + 14.17963081 x2τ1 − 1.7724538509 xτ1 − 1.7724538509 x 7/2 − 20 x3/2 + 51 x2/3 = 0 (26) equation (26) is collocated at n − 2 equal spaced points in the interval [0, 1] and the remaining two equations are obtained from the use of the conditions attached. this gives 7 system of linear equations. solving the system of equations, we get y4(x) = 3.77824 × 10 −12 − 1.× 10−10 x + 1.000000109x2 − 0.9999984717x3 − 0.1415135523e − 5x4 (27) when the same example was solved α = 52 , β = 3 2 following the same method,the approximate solution obtained is y4(x) = 6.2 × 10 −9 − 1.449037318 x2 − 8.450067824 x3 + 5.265645764 x4 (28) example 3 consider the multi-order fractional differential equation dα (y(x)) + 2 x dβ (y(x)) + xy(x) = x2 + x3 + x4 + 96 √ x √ π + 3 √ π √ x ( 6 + 1 √ x ) (29) subject to conditions y(0) = y′(0) = 0 0 < x ≤ 1 with exact solution y(x) = x + x2 + x3 taking an assumed approximate solution for n = 4, and computing the α = 52 and β = 3 2 -derivatives of equation (29) and other necessary simplifications and substitution, gives 70 x5a4 + 20 x 4a3 − 140 x 4a4 + 6 x 3a2 − 30 x 3a3 + 1490 x 3a4 + 2 x2a1 − 6 x 2a2 + 252 x 2a3 − 1700 x 2a4 + xa0 − xa1 + 37 xa2 144 uwaheren et al. / j. nig. soc. phys. sci. 2 (2020) 141–148 145 −181 xa3 +541 xa4 +4 a1−12 a2 +24 a3−40 a4 = x 4 +8 x3τ1 + x 3 − 8 x2τ1 + 2 x 2τ2 + 12 x 2 + xτ1 − xτ2 + xτ3 + x (30) equation (30) is collocated at n − 2 equal spaced points in the interval [0, 1] and the remaining two equations are obtained from the use of the conditions attached. this gives 8 system of linear equations. solving the system of equations, we get y4(x) = 3.2749743×10 (−10) + 1.6×10(−8) x + 1.109848402x2 + 1.000005386x3 − 0.2049775180e − 5x4 (31) when the same example was solved α = 32 , β = 1 2 following the same method,the approximate solution obtained is y4(x) = −4.9638971×10 (−10) + 2.6×10(−8) x + 1.48521236x2 + 1.000023102x3 − 0.346404728e − 5x4 (32) example 4 consider the multi-order fractional differential equation dα (y(x)) + 8 x dβ (y(x)) + xy(x) = x5 − x4 + 44x2 −30x(33) subject to conditions y(0) = y′(0) = 0 0 < x ≤ 1 with exact solution y(x) = x4 − x3 taking an assumed approximate solution for n = 4, and computing the α = 52 and β = 3 2 -derivatives of equation (33) and other necessary simplifications and substitution, gives − 140 x5a4 + 6 x 4a2 − 30 x 4a3 + 370 x 4a4 + 2 x 3a1 − 6 x 3a2 + 72 x3a3 + 1800 x 3a4 + x 2a0 − x 2a1 + 13 x 2a2 + 419 x 2a3 − 3179 x2a4 + 2 xa1 + 9 xa2 + 70 x 6a4 + 20 x 5a3 − 468 xa3 + 142 xa4 + 16 a1 − 48 a2 + 96 a3 − 160 a4 = x6 −x5 + 8 x3τ1 + 44 x 3 −8 x2τ1 + 2 x 2τ2 −30 x 2 + xτ1 −xτ2 (34) equation (34) is collocated at n − 2 equal spaced points in the interval [0, 1] and the remaining two equations are obtained from the use of the conditions attached. this gives 8 system of linear equations. solving the system of equations, we obtained y4(x) = −8.× 10 (−11) + 1.× 10(−9) x − 2.857225303x2 + 4.713873487x3 − .9046244956x4 (35) when the same example was solved α = 32 , β = 1 2 following the same method,the approximate solution obtained is y4(x) = −1.×10 (−10) x−1.000000001x3+1.000000000x4(36) 5. conclusion in this study, the proposed method was used to solve multiorder fractional differential equations successfully. four examples were solved and the results of the numerical solutions presented in tables 1, 2, 3 and 4. when α = 32 and β = 1 2 the proposed method produced accurate approximate solution curve figure 1. graphical representation of error in table 1 figure 2. graphical representation of error in table 2 figure 3. graphical representation of error in table 3 which is exactly over the curve of the exact solution for examples 1, and 2. but, the results are not as satisfactory at α = 52 and β = 32 as seen in figures 1 and 2. however, for examples 3 and 4, there is a change of results. here, the results are better at α = 52 and β = 3 2 than at α = 3 2 and β = 1 2 as shown in figure 3 and 4. this is likely because k in these examples is higher than 1 which is 2 and 8 in examples 3 and 4 respectively. the value 145 uwaheren et al. / j. nig. soc. phys. sci. 2 (2020) 141–148 146 table 1. error of results for example 1 x α = 52 α = 3 2 β = 32 β = 1 2 exact appx error appx error 0.0 0.00000 0.00000 5.0000e-11 -0.000000 5.9232e-13 0.1 0.00900 -0.01232 3.3298e-03 0.008999 2.6668e-10 0.2 0.03200 -0.04284 1.0848e-02 0.031999 9.6994e-10 0.3 0.06300 -0.08213 1.9128e-02 0.062999 1.9604e-09 0.4 0.09600 -0.12112 2.5122e-02 0.095999 3.0781e-09 0.5 0.12500 -0.15115 2.6155e-02 0.124999 4.1532e-09 0.6 0.14400 -0.16393 1.9932e-02 0.144000 5.0056e-09 0.7 0.14700 -0.15153 4.5298e-03 0.146999 5.4456e-09 0.8 0.12800 -0.10640 2.1596e-02 0.127999 5.2733e-09 0.9 0.08100 -0.02139 5.9613e-02 0.080999 4.2787e-09 1.0 0.00000 0.11032 1.1032e-01 0.000000 2.2421e-09 table 2. error of results for example 2 x α = 52 α = 3 2 β = 32 β = 1 2 exact appx error appx error 0.0 0.000000 0.000000 6.2000e-10 0.000000 3.7782e-12 0.1 0.009000 -0.022414 3.1414e-02 0.0090000 2.4706e-09 0.2 0.032000 -0.117137 1.4914e-01 0.0320000 1.4306e-08 0.3 0.063000 -0.315914 3.7891e-01 0.0630000 3.9585e-08 0.4 0.096000 -0.637850 7.3385e-01 0.0960000 7.8988e-08 0.5 0.125000 -1.089415 1.2144e+00 0.125000 1.2980e-07 0.6 0.144000 -1.664440 1.8084e+00 0.1440001 1.8590e-07 0.7 0.147000 -2.344120 2.4911e+00 0.1470002 2.3778e-07 0.8 0.128000 -3.097010 3.2250e+00 0.1280002 2.7253e-07 0.9 0.081000 -3.879029 3.9600e+00 0.0810002 2.7386e-07 1.0 0.000000 -4.633459 4.6335e+00 0.0000002 2.2207e-07 table 3. error of results for example 3 x α = 32 α = 5 2 β = 12 β = 3 2 exact appx error appx error 0.0 0.0000000000 -0.0000000005 4.9639e-10 0.0000000000 3.4707e-11 0.1 0.0110000000 0.0158521485 4.8521e-03 0.0110000001 8.6920e-11 0.2 0.0480000000 0.0674086784 1.9409e-02 0.0480000002 2.1491e-10 0.3 0.1170000000 0.1606697155 4.3670e-02 0.1170000004 3.8136e-10 0.4 0.2240000000 0.3016353775 7.7635e-02 0.2240000006 5.5642e-10 0.5 0.3750000000 0.4963057740 1.2131e-01 0.3750000007 7.1780e-10 0.6 0.5760000000 0.7506810062 1.7468e-01 0.5760000009 8.5069e-10 0.7 0.8330000000 1.0707611670 2.3776e-01 0.8330000009 9.4779e-10 0.8 1.1520000000 1.4625463410 3.1055e-01 1.1520000010 1.0093e-09 0.9 1.5390000000 1.9320366030 3.9304e-01 1.5390000010 1.0431e-09 1.0 2.0000000000 2.4852320250 4.8523e-01 2.0000000010 1.0642e-09 146 uwaheren et al. / j. nig. soc. phys. sci. 2 (2020) 141–148 147 table 4. error of results for example 4 x α = 32 α = 5 2 β = 12 β = 3 2 exact appx error appx error 0.0 0.0000000000 -0.0000000001 8.0000e-11 0.0000000000 0.0000e+00 0.1 -0.0009000000 -0.0239488420 2.3049e-02 -0.0009000000 1.1000e-11 0.2 -0.0064000000 -0.0780254233 7.1625e-02 -0.0064000000 2.8000e-11 0.3 -0.0189000000 -0.1372031514 1.1830e-01 -0.0189000001 5.7000e-11 0.4 -0.0384000000 -0.1786265321 1.4023e-01 -0.0384000001 1.0400e-10 0.5 -0.0625000000 -0.1816111705 1.1911e-01 -0.0625000002 1.7500e-10 0.6 -0.0864000000 -0.1276437696 4.1244e-02 -0.0864000003 2.7600e-10 0.7 -0.1029000000 -0.0003821324 1.0252e-01 -0.1029000004 4.1300e-10 0.8 -0.1024000000 0.2143448386 3.1674e-01 -0.1024000006 5.9200e-10 0.9 -0.0729000000 0.5285371464 6.0144e-01 -0.0729000008 8.1900e-10 1.0 0.0000000000 0.9520236894 9.5202e-01 -0.0000000010 1.1000e-09 figure 4. graphical representation of error in table 4 of the exact solution and the source functions count. from numerical results, it can be seen that the proposed method is an accurate estimate for the class of differential equations considered. acknowledgments the authors wish to sincerely appreciate mr akorede m. t, for his immense contributions in the typesetting of this work. we are also grateful to the referees and editor for their invaluable comments and 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[15] k. parand, m. dehgham, a. rezaeia & s. ghaderi. “an applica147 uwaheren et al. / j. nig. soc. phys. sci. 2 (2020) 141–148 148 tion algorithm for the solution of the nonlinear lane-emden type equations arising in astrophysics using hermite functions collocation method”, computer physics communications 181 (2010) 1096. doi.org/10.1016/j.cpc.2010.02.018 [16] w. al-hayani, l. alzubaidy & a. entesar. “solutions of singular ivps of lane-emden type by homotopy analysis method with genetic algorithm”, applied mathematics and information sciences 11 (2019) 1. doi.org/10.18576/amis/paper 148 j. nig. soc. phys. sci. 1 (2019) 20–29 journal of the nigerian society of physical sciences original research an epidemic model of zoonotic visceral leishmaniasis with time delay l. adamu, n. hussaini∗ department of mathematical sciences, bayero university, kano, nigeria. abstract this paper presents a mathematical model with time delay for the transmission dynamics of zoonotic visceral leishmaniasis (zvl which is caused by protozoan parasite leishmania infantum and transmitted by female sandflies). qualitative analysis of the ode version of the model reveals that the disease-free equilibrium of the model is globally asymptotically stable when the basic reproduction number, r0, is less than unity. using time delay as a bifurcation parameter in the delay-differential version of the model, it has been shown that the incubation period has significant effect on the stability of the equilibria and we derived the condition for hopf bifurcation to occur. keywords: zvl, stability, hopf bifurcation, time delay article history : received: 01 april 2019 received in revised form: 05 may2019 accepted for publication: 06 may 2019 published: 09 may 2019 c©2019 journal of the nigerian society of physical sciences. all rights reserved. communicated by: b. j. falaye 1. introduction leishmaniasis is a disease caused by a group of protozoan parasite called leishmania. it is transmitted to humans by the bite of infected female phlebotomine sandfly that fed previously on an infected reservoir or infected human. more than 20 leishmania species are known to be transmitted to humans and over 90 sandfly species are known to transmit leishmania parasites [1]. there are basically three forms of the diseases namely: visceral leishmaniasis (vl), cutaneous leishmaniasis (cl), and mucocutaneous leishmaniasis. vl also known as kala azar has two major forms which differ in their characteristics transmission, namely (i) zoonotic visceral leishmaniasis (zvl) and (ii) anthroponotic visceral leishmaniasis (avl). zvl, infects ∗corresponding author tel. no: +2348037594137 email address: nhussaini.mth@buk.edu.ng (n. hussaini ) mostly children and immunosuppressed individuals, is transmitted from animal reservoir to female phlebotomine sandflies and then to humans [2]. zvl posed a serious public health threat due to the fact that 200,000–400,000 incidences are estimated yearly [3, 4]. as noted in [5], zvl is endemic in africa, europe (particularly the mediterranean region) and asia (particularly the indian subcontinent). a complication of visceral leishmaniasis is known as post kala-azar dermal leishmaniasis (pkdl) usually appear 6 months to 1 or more years after treatment. pkdl is self-healing [6, 7, 8]. there is no licensed vaccine against any form of leishmaniasis including the zvl (although a number of candidate vaccines are at various stages of development and clinical trials) [912]. on the other hand, vaccine against zvl for reservoir exists [13, 14]. furthermore, drugs like miltefosine, paromomycin and liposomal amphotericin b can be used to cure zvl patients [15, 16]. evidence has shown that when a susceptible human, sand20 l. adamu & n. hussaini / j. nig. soc. phys. sci. 1 (2019) 20–29 21 fly or reservoir is infected, there is an incubation period during which the infectious agent develops. the incubation periods for humans, sandflies and reservoir range from 2 to 6 months, 8 to 6 days, and 3 to 7 years, respectively [6, 17, 18]. furthermore, to capture the effect of the incubation period on the transmission dynamics of zvl it is important to incorporate delay caused by the incubation period in the zvl model. mathematical models have been developed to study the dynamics of zoonotic visceral leishmaniasis (for instance, see refs. [12, 19-23] and some reference therein). the aforementioned models do not, however, incorporate an important aspect of the disease, that is the incubation period. a few mathematical modeling studies, such as those by [11, 24, 25] incorporated time delay in the transmission dynamics of the vector. since the incubation period in the reservoir is the longest in comparison to the time the parasite takes to develop in humans or sandflies. thus, it is instructive to study the effect of zvl latency period in the reservoir population on the transmission dynamics of the disease. the paper is organized as follows. the model is formulated in section 2. the ode version of the model is introduced and analyzed in section 3. the analysis of the model with time delay is carried out in section 4. the paper is concluded in section 5. 2. model formulation three different populations, at time t, are considered, namely: human host population denoted by nh (t), vector population denoted by nv (t), and reservoir host population denoted by nr(t). the total human host population is compartmentalize into four subpopulation namely: susceptible humans (s h (t)), zvl infected individuals (ih (t)), individuals who developed pkdl (ph (t)), and individuals who recovered from zvl (rh (t)), so that: nh (t) = s h (t) + ih (t) + ph (t) + rh (t). similarly, the total population of vector (nv (t)) is sub-divided into susceptible vector (s v (t)), and zvl-infected vector (ih (t)), so that: nv (t) = s v (t) + iv (t). furthermore, the reservoir population is categorized into susceptible reservoir (s r(t)), and zvl-infected reservoir (ir(t)), such that: nr(t) = s r(t) + ir(t). the delay-differential model for the transmission dynamics of zoonotic visceral leishmaniasis in human, sandfly and reservoir populations is given by the following deterministic autonomous system of non-linear delay differential equations (a schematic diagram of the model is depicted in fig 1, the state variables and the parameters of the model are described in table 1 and 2, respectively): s ′h = πh −λh s h iv −µh s h, i′h = λh s h iv − (ζh + µh + δh )ih, p′h = (1 − f1)ζh ih − (β + γ + µh )ph, r′h = f1ζh ih + (γ + β)ph −µh rh, s ′v = πv −λv s v ir −µv s v, i′v = λv s v ir −µv iv, s ′r = πr −λrs r(t −τ)iv (t −τ) −µrs r, i′r = λrs r(t −τ)iv (t −τ) −µr ir, (1) with n′h = πh −µh nh, n′v = πv −µh nv, n′r = πr −µr nr, where τ ≥ 0 represents the incubation time that the reservoir needed to become infectious. the susceptible human population (s h (t)) is recruited at a constant rate πh and diminished as a result of infection from an infected sandfly (iv ) at per capita rate (λh ) so that the incidence of new infection is given by λh s h iv . the population is decreased by natural death µh (assumed to be the same in all compartments of humans). here we neglect the incubation period in humans, since in zvl humans are not contributing to the disease transmission unlike in avl [26] zvl symptomatic individuals ih are generated at the rate λh and decreased by recovery at a rate ζh ;. a fraction f1 of individuals recovered and the remaining fraction (1 − f1) developed pkdl. the population of individuals with pkdl ph is decreased by treatment at a rate γ or natural recovery at rate β. the population of individuals recovered from zvl is generated at the rates f1ζh , β and γ. the population of susceptible female sandflies is generated at a constant rate πv and diminished following contact with infected reservoir ir at per capita infection rate λv so that the incidence of new infection is given by the mass action term λv s v ir. we neglect the incubation period in sandflies since it is very short. furthermore, the population is decreased by natural death at a rate µv (assumed to be the same in both the compartments of sandflies). the latency period in the reservoir is very long, as such we, therefore introduced time delay τ to capture this period. at time t, susceptible reservoirs (that are recruited into the population at constant rate πr) get zvl infection as a result of contact with sandflies that have been infected at time t − τ (assumed reservoirs survived the incubation period τ) when bitten by an infectious sandfly at a rate λr. thus, the population of infected reservoirs is generated at the rate λrs r(t − τ)iv (t − τ) and diminished by natural death (at a rate µr). we assume that all the parameters of the model to be non-negative. for the human, sandfly, and reservoir the corresponding population sizes are asymptotically constant i.e, nh (t) → πh µh as t → ∞, nv (t) → πv µv , nr(t) → πr µr as t → ∞. without loss of generality we assume that nh (t) = πh µh , nv (t) = πv µv , nr(t) = πr µr ∀t ≥ 0. the dynamics of system (1) are qualitatively equivalent to 21 l. adamu & n. hussaini / j. nig. soc. phys. sci. 1 (2019) 20–29 22 the dynamics of the following system given by s ′h = πh −λh s h iv −µh s h, i′h = λh s h iv − (ζh + µh )ih, p′h = (1 − f1)ζh ih − (β + γ + µh )ph, i′v = λv ( πv µv − iv ) ir −µv iv, i′r = λr ( πr µr − ir(t −τ) ) iv (t −τ) −µr ir. (2) the initial conditions for the above system take the form: s h (θ) = φ1(θ), ih (θ) = φ2(θ), ph (θ) = φ3(θ), iv (θ) = φ4(θ), ir(θ) = φ5(θ), φi ≥ 0, i = 1, 2, 3, 4, 5, θ ∈ [−τ, 0], φi(0) > 0 where φ = (φi(θ)) ∈ c+ × c+. here, c denote the banach space c = c([−τ, 0],r) of continuous function mapping the interval [−τ, 0] into r. the non-negativity of the cone is defined as c+ = c([−τ, 0],r+). some of the main assumptions made in the formulation of the model are as follows; i. homogeneous mixing among the vector-hosts populations (that is, a sandfly has an equal chance of biting any human or reservoir host); ii. pkdl infected individuals can recover naturally or due to treatment [7]; iii. rate of pkdl relapse to zvl is assumed negligible (hence not considered)[27]; iv. pkdl is not life-threatening (no pkdl-induced death is assumed)[12]; v. zvl-infected individuals recover successfully or develops pkdl [7]; vi. treated infected reservoirs do not usually get cured but develop an immune response that prevents them from becoming infectious [28, 29]; vii. no direct transmission within reservoirs and also within sandflies [2]. viii. death rate for humans due to zvl-infection is assumed to be negligible (hence not accounted for)[18] 3. analysis of the model without delay we consider a special case of the delay-differential model where there is no incubation period (i.e τ = 0) . the ode version of the model is given by s ′h = πh −λh s h iv −µh s h, i′h = λh s h iv − (ζh + µh )ih, p′h = (1 − f1)ζh ih − (β + γ + µh )ph, i′v = λv ( πv µv − iv ) ir −µv iv, i′r = λr ( πr µr − ir ) iv −µr ir. (3) 3.1. invariant region: it can be shown that the biologically-feasible region: ω = { (s h, ih, ph, iv, ir) ∈ r5+ : s h, ih, ph, iv, ir ≥ 0, nh ≤ πh µh , nv ≤ πv µv , nr ≤ πr µr } , is positively-invariant (see for instance [5, 30]) i.e all solutions starting in ω remain in ω ∀t ≥ 0. thus it is sufficient to consider the dynamics of system (3) in ω. 3.2. stability of disease-free equilibrium (dfe) the model (3) has a dfe obtained by setting the right-hand sides of the equations in the model to zero, given by e0 = (s o h, i o h, p o h i o v, i o r) = ( πh µh , 0, 0, 0, 0 ) . (4) it follows that the associated basic reproduction number of the model (2) denoted by r0 is defined as: r0 = √ λrπrλv πv µr2µv 2 (5) 3.3. epidemiological interpretation of r0 we can epidemiologically interpret the terms in the expression for the threshold quantity r0 as follows. the term gives the number of secondary infections that one infectious host can generate only through a vector and reservoir transmission. susceptible sandflies get zvl-infection following effective contact with an infected reservoir (ir ), the number of susceptible sandfly πv µv generated by infected reservoir near dfe is the product of infection rate of infected reservoir λr πv µv and the average life expectancy of the infected reservoir ( 1 µr ). thus the average number of new sandfly infections generated by an infected reservoir is given by λr µr πv µv . similarly, the expression λv µv πr µr account for the number of new reservoir infections caused by an infected sandfly over the expected infection period. the humans contribute nothing to the zvl transmission. theorem 3.1. if r0 ≤ 1, then the disease-free equilibrium e0 of (3) is globally asymptotically stable in ω proof. consider the following liapunov function l = λrπr ih + λrπr ph + µvµr iv + λv πv ir l′ = λrπr [ λh s h iv − (ζh + µh + ] +λrπr [ (1 − f1)ζh ih − (β + γ + µh )ph ] +µvµr [ λv ( πv µv − iv ) ir −µv iv ] +λv πv [ λr ( πr µr − ir ) iv −µr ir ] l′ = λrλrπrs h iv − (µhλrπr + λrπr f1ζh )ih 22 l. adamu & n. hussaini / j. nig. soc. phys. sci. 1 (2019) 20–29 23 μh μh sh ih sv iv ir sr rh ph πh μh μh μv μv μr μr πr πv δhf1ih (1-f1)δhih λh λv λr (β+γ)ph figure 1: schematic diagram of the model (1). solid arrows indicate transitions and dashed arrow indicates interaction. expressions next to arrows show the per capita flow rate between compartments. table 1: description of the state variables of the model. variable interpretation nh total population of humans s h population of susceptible humans with risk of zvl infection ih population of zvl-infected humans with symptoms of zvl ph zvl-infected humans with pkdl rh zvl-recovered individuals nv total population of sandflies s v population of susceptible sandflies iv population of zvl-infected sandflies nr total population of reservoir s r population of susceptible reservoir with risk of zvl-infection ir zvl-infected reservoir with symptoms of zvl + ( µvµrλvλrπv µv − πvλvµr ) ir −(µvµrλvλr)ir iv +µ2vµr  πvλvλrπr µ2vµ 2 r − 1  iv ≤ µ2µr  πvλvλrπr µ2vµ 2 r − 1  iv l′ = µ2µr(r 2 0 − 1)iv ≤ 0 if r0 ≤ 1. thus if r0 ≤ 1 the derivative l′ = 0 if and only if and iv = 0. furthermore the case r0 = 1 the derivative l′ = 0, consequently the largest compact invariant set in {(s h, ih, ph, iv, ir) ∈ 23 l. adamu & n. hussaini / j. nig. soc. phys. sci. 1 (2019) 20–29 24 table 2: description of the state variables of the model. parameter interpretation πh ,πv , πr recruitment rate for humans,vector and reservoir λh rates of direct transmission in humans λv rates of direct transmission in sandflies λr rates of direct transmission in reservoir µh,µv,µr natural death rates of humans, sandflies and reservoir , respectively ζh recovery rate of a human from zvl-infection f1 fraction of those who recovered from zvl τ incubation period of the disease β pkdl natural recovery rate γ pkdl treatment rate ω : l′ = 0} when r0 ≤ 1, is the singleton set {e0}, therefore by lasalle-invariance principle [31], every solution that starts in ω approaches e0 as t →∞. 3.4. existence of an endemic equilibrium and its stability in order to obtain an endemic equilibrium where at least one of the infected component is non zero, we let e∗ = (s ∗h, i ∗ h, p ∗ h, i ∗ v, i ∗ r) be an arbitrary endemic equilibrium of the model (3), solving (3) at equilibrium, the non-trivial equilibrium is given by: s ∗h = λrµv πh (λv πr + µvµr) g1 , i∗h = πhλhµ 2 vµ 2 r(r0 − 1) (ζh + µh + δh )g1 , p∗h = λh πhµ 2 vµ 2 r(r0 − 1)ζh (1 − f1) g2g1 , i∗v = µ2vµ 2 r(r0 − 1) (µvµr + πrλv )λrµv , i∗r = µ2vµ 2 r(r0 − 1) (πvλr + µvµr)λvµr , where, g1 = µ 2 vµ 2 rλh (r0 − 1) + λrµvµh (λv πr + µvµr), g2 = (ζh + µh )(β + γ + µh ). lemma 3.2. the endemic equilibrium e∗ of model (3) is locally asymptotically stable if r0 ≥ 1. proof. we linearize the system (3) at an endemic equilibrium to obtain the corresponding characteristic equation as follows: det  k1 −λ 0 0 −λh s ∗h 0 −λh i∗v k2 −λ 0 λh s ∗ h 0 0 (1 − f1)ζh k3 −λ 0 0 0 0 0 k4 −λ λv ( πv µv − i∗v ) 0 0 0 λr( πr µr − i∗r) k5 −λ  = 0, (6) where, k1 = −λh i∗v − µh, k2 = −(ζh + µh ), k3 = −(β + γ + µh ), k4 = −(λv i∗r + µv ), k5 = −(λr i ∗ r + µr). then we have the following characteristics equation (k1 −λ) (k2 −λ) (k3 −λ) (λ 2 + (λv i ∗ r + λr i ∗ v + µv + µr)λ + λvµr i ∗ r + µvλr i ∗ v + λvλr i ∗ v i ∗ r) = 0, (7) which has negative roots λ1 = −(ζh + µh ) < 0, and λ2 = −(β + γ + µh ) < 0. furthermore, λ3 = −(λh i∗v + µh ) < 0 if r0 > 1 (i.e., i∗v > 0). other roots are given by the following equation λ2 + (λv i ∗ r + λr i ∗ v + µv + µr)λ + λvµr i ∗ r + µvλr i ∗ v + λvλr i ∗ v i ∗ r = 0. (8) according to descarte’s rule of sign changes, if r0 ≥ 1 (i.e., i∗v ≥ 0, and i ∗ r ≥ 0), then the above equation has no sign changes in its coefficients and therefore there will be no roots whose real part is positive. this shows that the the endemic equilibrium is stable. furthermore, if r0 ≤ 1, then the above equation has sign changes in its coefficients. therefore one of the roots has its real part positive and the endemic equilibrium is unstable. 4. analysis of the model with delay next we analyze the dynamics of system (2) where the time delay is non-zero (τ , 0). theorem 4.1. the disease-free equilibrium e0 of model (2) is locally asymptotically stable if r0 < 1 and unstable if r0 > 1. proof. we linearize the system (2) with delay around the diseasefree equilibrium, then the characteristics equation correspond24 l. adamu & n. hussaini / j. nig. soc. phys. sci. 1 (2019) 20–29 25 ing to the jacobian matrix is given by: det  λ + µh 0 0 λh πh µh 0 0 λ + p1 0 −λh πh µh 0 0 − (1 − f ) ζh λ + p2 0 0 0 0 0 λ + µv − λv πv µv 0 0 0 −λr πr e −λτ µr λ + µr  = 0, (9) where, p1 = ζh + µh, p2 = β + γ + µh. thus, (λ + µh )(λ + ζhµh )(λ + β + γ + µh )(−µv µrλ 2 − hλ−µv 2µr 2 + λv πvλrπre −λτ) = 0, (10) where h = µvµr2 + µv 2µr. the equation (10) has the following roots with negative real part, namely: λ = −µh , λ = −(µh + ζh ), λ = −(β + γ + µh ) and the remaining roots of f(λ,τ) = −µvµrλ 2 − (µvµr 2 + µv 2µr)λ −µv 2µr 2 + λv πvλrπre −λτ = λ2 + (µv + µr)λ + µvµr(1 − r0e −λτ) = 0. (11) for τ = 0 we obtain the same characteristics polynomial as in ode case (without delay) i.e, f(λ, 0) = λ2 + (µv + µr)λ + µvµr(1 − r0) = 0. (12) again, it follows from descarte’s rule of signs changes that equation (12) has roots with negative real part whenever r0 < 1 and thus, the disease free equilibrium is stable. if r0 > 1 then equation (12) has at most one of its root whose real part is positive and the disease free equilibrium is unstable. for τ , 0, suppose r0 > 1, we show that equation (11) has a positive roots and the disease free equilibrium is unstable. to see this, we rearrange the equation (11) in form: λ2 + (µv + µr)λ = µvµr(r0e −λτ − 1). (13) suppose λ ∈ r, let h(λ) be the left-hand side of (13) and g(λ) be the right-hand side. then h(0) = 0 and limλ→∞ h(λ) = ∞. moreso, the function g(λ) is decreasing function of λ and g(0) = µvµr(r0 − 1) > 0 as such the two functions must intersect for some λ∗ > 0. as such equation (13) has a positive real solution and the disease-free equilibrium is unstable. now for r0 < 1 we claim equation (13) does not have a non-negative real roots. in this case for λ ≥ 0, h(λ) is increasing and g(λ) is still decreasing function of λ but g(0) = µvµr(r0 − 1) < 0. thus if equation (13) has roots whose real parts is non-negative, there must be complex conjugate roots which cross the imaginary axis. also by rouché’s theorem, if instability occur for a particular value of delay τ, a characteristic root of equation (13) intersect the imaginary axis. consequently, equation (13) must have a pair of purely imaginary solutions for some τ > 0. assume that λ = iω, and without loss of generality assume that ω > 0 is a root of (13) that is ω satisfies the following equation: −ω2 + (µv + µr)iω + µvµr − λv πvλrπr µvµr (cos(ωτ) − i sin(ωτ)) = 0. separate the real and the imaginary part and obtain the following: −ω2 + µvµr = λv πv λr πr µv µr cos(ωτ), (14) (µv + µr)ω = − λv πv λr πr µv µr sin(ωτ). (15) square both sides of each equation above and add the squared equations to obtain the following forth order equations in ω : ω4 + (µv + µr)ω 2 + µ2vµ 2 r − λ2v π 2 vλ 2 rπ 2 r µ2vµ 2 r = 0. (16) let σ = ω2 so that equation (16) can be reduced to quadratic form as follows: σ2 + (µv + µr)σ + µ 2 vµ 2 r − λ2v π 2 vλ 2 rπ 2 r µ2vµ 2 r = 0. (17) we denote the coefficient as: a11 = (µ 2 v + µ 2 r) > 0, a12 = µ 2 vµ 2 r − λ2v π 2 vλ 2 rπ 2 r µ2vµ 2 r = ( µvµr − λv πvλrπr µvµr ) ( µvµr + λv πvλrπr µvµr ) = µvµr 1 − λv πvλrπr µ2vµ 2 r  (µvµr + λv πvλrπr µvµr ) = µvµr(1 − r0) ( µvµr + λv πvλrπr µvµr ) > 0, i f r0 < 1. then equation (17) can be written as: σ2 + a11σ + a12 = 0. (18) it follows that a12 is positive whenever r0 < 1. thus, the two roots of equation (18) have positive product which means that they are complex or they are real but they have the same sign. moreso, they have negative sum which implies they are either real and negative or complex conjugate with negative real parts. hence, equation (18) does not have positive real roots which leads to the conclusion that there is no ω such that iω is a solution of equation (18). therefore, it follows from kuang’s theorem [ref. [32], pp 83] that all the eigenvalues of the characteristics equation (13) have negative real parts for all delay values τ ≥ 0. this implies that the disease free equilibrium e′ is locally asymptotically stable if r0 < 1. 25 l. adamu & n. hussaini / j. nig. soc. phys. sci. 1 (2019) 20–29 26 4.1. hopf bifurcation analysis let r0 > 1 (i.e, the endemic equilibrium e∗ exists) and τ be bifurcation parameter. the characteristics equation is obtained in form of transcendental equation after linearizing system (2) at an endemic equilibrium e∗ and obtained the roots (−λh i∗v − µh ), −(ζh + µh ), −(β + γ + µh ) and the roots of equation (19) can be written in the form: λ2 + b2λ + b3 = e −λτ (g1λ + g2) (19) where, b2 = µr + µv + λv i ∗ r, b3 = µrλv i ∗ v + µvµr, g1 = −λr i ∗ v, g2 = −µvλr i ∗ v + λvλrπv πr µvµr − λvλrπv i∗r µv − λvλrπr i∗v µr now from system (3) we have λv πv µv = λv i ∗ v + µv i∗v i∗r and λrπr µr = λr i ∗ r + µr i∗r i∗v (20) now, g2 = −µvλr i ∗ v + λv πv µv λrπr µr − λv πv µv λr i ∗ r − λrπr µr λv i ∗ v substituting (20) in g2 we obtained: g2 = −µvλr i ∗ v + (λv i ∗ v + µv i∗v i∗r )(λr i ∗ r + µr i∗r i∗v ) − (λv i ∗ v + µv i∗v i∗r )λr i ∗ r − (λr i ∗ r + µr i∗r i∗v )λv i ∗ v = −µvλr i ∗ v + λvλr i ∗ v i ∗ r + λvµr i ∗ r + λrµv i ∗ v + µvµr −λvλr i ∗ v i ∗ r −λrµv i ∗ v −λvλr i ∗ v i ∗ r −λvµr i ∗ r = −µvλr i ∗ v + µvµr −λvλr i ∗ v i ∗ r. therefore, g2 = µvµr −λr i ∗ v (λv i ∗ v + µv ). for τ = 0, it follow from lemma (3.2) above that the endemic equilibrium is locally asymptotically stable. moreover we claim that for any τ > 0, equation (19) does not have positive real solution. to see this, we have b2 > 0, b3 > 0. we move the positive terms from the right-hand side to left-hand side. the rewritten equation (19) takes the form: λ2 + b2λ + b̃3 = e −λτ ( g1λ + g̃2 ) , (21) where b̃3 = b3 − e−λτ (µvµr) > 0, ∀λ ≥ 0, ∀τ > 0. on the other hand g1 < 0, and g̃2 = −λr i∗v (λv i ∗ v + µv ) < 0. consequently, for all λ ≥ 0 the left-hand side in equation (21) is positive while the right hand side is negative and the two cannot be equal for all λ ≥ 0. to obtain the complex conjugate solutions with positive real parts. let λ = iω with ω > 0 be the root of equation (19). substituting λ = iω into equation (19), the following equation is obtained: −ω2 + b2iω + b3 = (cos(ωτ) − i sin(ωτ))(g1iω + g2). (22) separating the real and imaginary parts, we obtain that b3 −ω 2 = g2 cos(ωτ) + g1ω sin(ωτ), (23) b2ω = g1ω cos(ωτ) − g2 sin(ωτ), (24) we obtained a polynomial equation by eliminating the trigonometric term, this is done by squaring both side of each equation above and adding the resulting equation we have: ω4 + (b22 − 2b3 − g 2 1)ω 2 + b23 − g2 2 = 0. (25) notice that this a forth degree polynomial and the delay, τ, has been eliminated. let ω2 = σ ∈ r the equation (25) become a quadratic in σ: σ2 + ασ + κ = 0, (26) where, α = b22 − 2b3 − g 2 1, κ = b23 − g 2 2. (27) we establish the conditions for endemic equilibrium e∗ to be locally stable, that is equation 25 cannot have a purely imaginary solutions. lemma 4.2. the following results follows from polynomial equation (26) (i). if κ < 0 or κ ≥ 0 and α < 0, then equation (26) has only one positive root. (ii). if κ > 0 and α > 0 or ∆ = α2 − 4κ < 0, then equation (26) has no positive root. (iii). if κ > 0, α < 0 and ∆ = α2 − 4κ ≥ 0, then equation (26) has at least one positive root. suppose that condition (ii) of lemma (4.2) is satisfied then endemic equilibrium is stable, that is equation (26) does not have a positive real solution. conditon (ii) of lemma 4.2 shows that for τ > 0 there is no positive σ such that iσ is an eigenvalue of the characteristics equation (19). therefore, it follows from kuang [32] any root λ of (19) satisfies the relation reλ < 0. theorem 4.3. assume that condition (ii) of lemma(4.2) is satisfied and r0 > 1, then the endemic equilibrium e∗ of system (2) is absolutely stable, that is e∗ is asymptotically stable for all values of delay τ ≥ 0 . remarks. theorem 4.3 indicates that for all values of delay, the endemic equilibrium e∗ of system (2) is asymptotically stable if the parameters satisfy conditions (ii) of lemma(4.2), the stability of the endemic equilibrium will depend on the 26 l. adamu & n. hussaini / j. nig. soc. phys. sci. 1 (2019) 20–29 27 value of the delay if the conditions in theorem 4.3 are not satisfied. as the delay varies the endemic equilibrium can lose stability which can lead to oscillations. for instance equation (26) has at least one positive root say σ0 if κ < 0 denoted by ω = √ σ0. using time delay (i.e incubation time) τ as the bifurcation parameter, we drive the condition for hopf bifurcation to occur. we can think of the roots of (21), λ as a continuous function in terms of the delay parameter λ(τ). theorem 4.4. suppose that r0 > 1 and if condition (i) of lemma (4.2) is satisfied, then the endemic equilibrium e∗ of the delay model (2) is asymptotically stable when τ ∈ [0,τ0), and unstable when τ > τ0 provided that ω0 is the largest positive simple root of equation (25), furthermore the delay model (2) undergoes a hopf bifurcation at an endemic equilibrium e∗ when τ = τ0 where, τ0 = 1 ω0 arcot  g2(b3 −ω20) + g1b2ω20 g1(ω30 − b3ω0) + g2b2ω0  proof. let λ(τ) = η(τ) + iσ(τ) be the eigenvalue of equation (21) such that for some initial value of the bifurcation τ0 we have η(τ0) = 0, and ω(τ0) = ω0 (without loss of generality we may assume ω0 > 0 ) from equation (23) and (24) we obtained a sequence of positive values of τn corresponding to any positive root ω given by: τn = 1 ω0 arcot  g2(b3 −ω20) + g1b2ω20 g1(ω30 − b3ω0) + g2b2ω0  + 2nπ ω0 , n = 0, 1, 2, ... it follows from lemma (3.2) that for τ = 0, e∗ is asymptotically stable, by kuang’s theorem [ref[32], p.83], the endemic equilibrium e∗ is asymptotically stable for τ ∈ [0,τ0) and unstable for all τ > τ0. now we show the transversal condition dreλ(τ) dτ |τ=τo> 0 holds. when τ passes the critical value τ0 (i.e τ > τ0) by continuity, the real part of λ(τ) becomes positive and the steady state become unstable. moreover, a hopf bifurcation occur when τ passes through a critical value τ0 (see ref. [33]) to establish hopf bifurcation at τ = τ0 we need to show that dre(λ(τ)) dτ > 0. differentiating equation (21) with respect to τ we have (2λ + b2) dλ dτ =[−τe−λτ(g1λ + g2) + g1e −λτ] dλ dτ −λe−λτ(g1λ + g2). this gives( dλ dτ )−1 = 2λ + b2 + τe−λτ(g1λ + g2) − g1e−λτ −λe−λτ(g1λ + g2) = 2λ + b2 −λe−λτ(g1λ + g2) + g1 λ(g1λ + g2) − τ λ = λ2 − b3 + e−λτ(g1λ + g2) −λ2(e−λτ(g1λ + g2)) + g1 λ(g1λ + g2) − τ λ = λ2 − b3 λ2(λ2 + b2λ + b3) − g2 λ2(g1λ + g2) − τ λ thus, sign {( d(reλ) dτ )} = sign { re ( dλ dτ )−1} λ=iω0 = sign { re [ λ2 − b3 −λ2(z(λ)) ] λ=iω0 + re [ −g2 λ2(g1λ + g2) ] λ=iω0 } = sign { re  −ω20 − b3 ω20(b3 −ω 2 0 + b2ω0i)  + re  g2 ω20(g2 + g1ω0i) } = sign { ω40 + g22 − b23 ω20((b3 −ω 2 0) 2 + b22ω 2 0) } = sign { 2ω20 + (b22 − 2b3 − g21) (b2 −ω20) 2 + b22ω 2 0 } , where z(λ) = λ2 + b2λ + b3. since f (σ) = σ2 + ασ + κ, thus, d f (σ) dσ = 2σ + α = 2σ + (b22 − 2b3 − g 2 1) since ω0 is the simple largest positive root, we have d f (σ) dσ ∣∣∣∣∣ σ=ω20 > 0. hence dreλ dτ ∣∣∣∣∣ ω=ω0,τ=τ0 = d f (ω20 ) dσ (b2 −ω20) 2 + b22ω 2 0 > 0, remarks if an endemic equilibrium exists and the parameters κ < 0 or κ ≥ 0 and α < 0 is satisfied, an increase in the length of the time delay of zvl disease transmission below the critical value of the time delay τ0 will cause an endemic equilibrium be stable. as the time delay progresses beyond the critical delay τ0, a locally asymptotically stable endemic equilibrium will lose its stability. 27 l. adamu & n. hussaini / j. nig. soc. phys. sci. 1 (2019) 20–29 28 5. numerical simulation in this section, numerical simulations were performed for the model (2). the following parameter values λh = 0.5,β = 0.016,λh = 0.5,µh = 0.0067,µv = 0.087,µr = 0.1,γ = 0.033,δh = 0.011, f1 = 0.64, πh = 0.05,ζh = 0.036 are used. we observed that the number of infected sandflies, infected reservoir and infected humans decay faster for higher values of the delay (incubation period) as shown in figures 2, 3, and 4. therefore, the delay has impact on the transmission dynamics of the disease. 0 10 20 30 40 50 0 0.5 1 1.5 time n u m b e r o f in fe c te d h u m a n s tau=1 tau=2 tau=3 tau=4 tau=6 figure 2: numerical simulation of model (2) with parameter values λv = 0.250,λr = 0.20, πv = 0.0026, πr = 0.022,. 0 10 20 30 40 50 0 0.2 0.4 0.6 0.8 1 time n u m b e r in fe c te d s a n d fl ie s tau=1 tau=2 tau=3 tau=4 tau=5 figure 3: numerical simulation of model (2) with parameter values λv = 0.50,λr = 0.1, πv = 0.0016, πr = 0.052. 0 10 20 30 40 50 0 0.2 0.4 0.6 0.8 1 time n u m b e r o f in fe c te d r e se rv o ir tau=0.01 tau=1 tau=3 tau=5 tau=8 figure 4: numerical simulation of model (2) with parameter values λv = 0.250,λr = 0.20, πv = 0.005, πr = 0.000523,. 6. conclusions this paper presents a new deterministic delay-deferential model for assessing the effect of time delay on the dynamics of zvl by looking at the stability of the equilibria. the time delay accounts for the incubation period of reservoirs needed to become infectious. the main theoretical and epidemiological findings of the study are summarized as follows: the ode version of the model has a dfe (zvl-free equilibrium) which is locally-asymptotically stable whenever a certain threshold quantity (r0) is less than unity. if r0 < 1 the disease-free equilibrium is globally asymptotically stable. also a unique endemic equilibrium exists and is locally asymptotically stable in the interior of the feasible region. base on the delay-deferential model we determined the criteria for hopf-bifurcation to occur using time delay as the bifurcation parameter, when time delay is small the positive equilibrium is locally asymptotically stable, while instability can occur by hopf-bifurcation as the delay increases. hopf-bifurcation has helped us in finding the region of instability in a neighborhood of endemic equilibrium where the population will undergo regular oscillations. acknowledgments we thank the referees for the positive enlightening comments and suggestions, which have greatly helped us in making improvements to this paper. references [1] world health organization (2016), fact sheet http://www.who.int/ mediacentre/factsheets/fs375/en/. 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[33] b. d. hassard, n. d. k. azarinoff & y. h. wan, theory and applications of hopf bifurcation, cambridge university, cambridge, (1981). 29 j. nig. soc. phys. sci. 4 (2022) 251–264 journal of the nigerian society of physical sciences jackknife kibria-lukman m-estimator: simulation and application segun l. jegedea,∗, adewale f. lukmanb,c, kayode ayinded, kehinde a. odeniyie adepartment of mathematical sciences, kent state university bdepartment of mathematics, university of medical sciences, nigeria cinstitute of mathematics, eotvos lorand university, hungary ddepartment of statistics, federal university of technology, akure, nigeria edepartment of agricultural economics and agribusiness management, osun state university, nigeria abstract the ordinary least square (ols) method is very efficient in estimating the regression parameters in a linear regression model under classical assumptions. if the model contains outliers, the performance of the ols estimator becomes imprecise. multicollinearity is another issue that can reduce the performance of the ols estimator. this study proposed the robust jackknife kibria-lukman (rjkl) estimator based on the m-estimator to deal with multicollinearity and outliers. we examine the superiority of the estimator over existing estimators using theoretical proofs and monte carlo simulations. we put the estimator to the test once more using real-world data. we observed that the estimator performs better than the existing estimators. doi:10.46481/jnsps.2022.664 keywords: jackknife kibria-lukman, m-estimator, monte carlo simulation, multicollinearity, outliers, robust article history : received: 17 february 2022 received in revised form: 21 march 2022 accepted for publication: 24 march 2022 published: 29 may 2022 c©2022 journal of the nigerian society of physical sciences. all rights reserved. communicated by: j. ndam 1. introduction the regression model is commonly used in many disciplines to analyze data. the model’s analysis is a form of predictive modelling technique which statistically examines the relationship between two different sets of variables. the first variable is called the dependent, also known as the target variable. the second variable is called the independent, also known as the predictors because they are usually more than one. the model is frequently used for forecasting. mathematically, the ∗corresponding author tel. no: email address: sjegede@kent.edu (segun l. jegede ) general regression model includes; an n × 1 vector of observations referred to as dependent variable labelled y, a known full column rank of n × p standardize and centered independent variables labelled x, a p × 1 vector of unknown parameters labelled β and an n × 1 vector of disturbances labelled ε. ε is assumed to be normally distributed with e (ε) = 0 and dispersion matrix cov (ε) = σ2 i. the model is mathematically written as y = xβ + ε (1) the ordinary least square (ols) method is very efficient in estimating the regression parameters in a linear regression model under classical assumptions. the gauss markov theorem establishes this fact. the theorem stated the ols estimator has the best linear unbiased estimator (blue) with minimum vari251 jegede et al. / j. nig. soc. phys. sci. 4 (2022) 251–264 252 ance in the class of all unbiased linear estimators. however, if the dataset for the regression analysis contains outliers, the performance of the ols estimator becomes imprecise [1–3]. the ols estimator of β is given by β̂ = s −1 x ′ y (2) where s = x ′ x. when outliers are present, robust regression is used, which gives better results than the ols method[4–7]. the m-estimation approach is the most common robust regression method which is used to handle outlier in the y-direction [8]. it is a generalization to maximum likelihood estimation in context of location models. the means that it is nearly as efficient as the ols. the approach involves minimizing residual function rather than minimizing the number of squared errors as the objective. generally, the approach considers the likelihood function of β and l (β,σ) = n∏ i=1 1 σ f ( yi − xi ′ β σ ) = 1 σn n∏ i=1 f ( yi − xi ′ β σ ) , (3) and by replacing the ols criterion with a robust criterion, mestimator of β is β̂m = minβ n∑ i=1 ρ ( yi − xi ′ β σ̂m ) (4) the purpose of this study is to propose an estimator that solves the problem of multicollinearity and outlier in linear regression model. we investigate the superiority of the estimator through theoretical comparison, simulations and practical application. 2. existing shrinkage estimators a popular shrinkage estimator is the ridge estimator (re), developed by [9] and expressed as: β̂ k = (s + ki) −1 x ′ y = w(k)β̂ (5) where w(k) = (s + ki)−1s and k > 0. however, re can be sensitive to outliers in the y-direction. silvapulle [10] combined the advantage of the ridge estimator and the m-estimator to form the ridge m-estimator (rme), expressed as follows: β̂ m k = w(k)β̂ m, (6) kibria and lukman [11] recently proposed an estimator called the kibria-lukman (kl) estimator, the estimator is expressed as: β̂ k l = (s + ki) −1 (s − ki) β̂ = m(k)β̂ (7) where m (k) = (s + ki)−1 (s − ki) = ( i − 2k(s + ki)−1 ) . grafting the m-estimator into the kl estimator produced the robust kl estimator as follows: β̂ m k l = m(k)β̂ m, (8) the use of jackknife is also a popular approach for reducing the biasness in bias estimator proposed for handling multicollinearity [12–14]. ugwuowo et al. [15] proposed a jackknife kl (jkl) estimator under the proposition that the use of jackknife procedure reduces the bias of the ridge estimator [16]. the jkl estimator is expressed as: β̂ jk l = ( i − 2k(s + ki)−1 )2 ( i + 2k(s + ki)−1 ) β̂ = (m(k))2 n(k)β̂ (9) where n (k) = (s + ki)−1 (s + 3ki) = ( i + 2k(s + ki)−1 ) . 2.1. a new robust estimator using the same approach as [10,17,3], we combined the jkl estimator in (9) and the m-estimator to form the robust jackknife kibria-lukman (rjkl) estimator. it is obvious that the presence of outliers in the y-direction will reduced the efficiency of the jkl estimator. thus, we defined the rjkl estimator as follows β̂ rjkl = (m(k)) 2 n(k)β̂ m (10) where k > 0. the canonical form of model (1) is written as y = zα + ε, (11) where z = xt, α = t′β and t is the ortogonal matrix whose columns contains the eigenvectors of x′x. then z ′ z = t ′ x ′ xt = λ = diag ( λ1, λ2, . . . , λp ) , (12) where λ1, λ2, . . . , λp > 0 are the ordered eigenvalues of x′x. let α̂m be an m-defined by the solution of the m-estimating equations ∑ ϕ(ei/s)zi = 0 where ei = yi − ziα̂m, s is an estimator of scale for the errors and ϕ(.) is some suitably chosen function [18]. thus, the estimators presented in (2-9) can be written in canonical form as follows: α̂ = λ−1z ′ y (13) α̂k = (λ+ki) −1z ′ y = w∗(k)α̂ (14) α̂m k = w ∗(k)α̂m (15) α̂k l = ( i − 2k(λ + ki)−1 ) α̂ = m∗(k)α̂ (16) α̂m k l = m ∗(k)α̂m (17) α̂jk l = ( i − 2k(λ + ki)−1 )2 ( i + 2k(λ + ki)−1 ) α̂ = (m∗(k))2 n∗(k)α̂ (18) α̂rjk l = (m ∗ (k))2 n∗ (k) α̂m (19) where w∗(k) = λ (λ+ki)−1, m∗(k) = ( i − 2k(λ + ki)−1 ) , n ∗(k) = ( i + 2k(λ + ki)−1 ) for k>0. the organization of this article is as follows. the theoretical comparison among estimators is given in section 2.2. robust choice of the biasing parameters are discussed in section 3, a simulation study conducted to evaluate the performance of the proposed estimator in section 4 and a real life data was analyzed in section 6 to illustrate the finding of the paper. section 7 ends with some concluding remarks. 252 jegede et al. / j. nig. soc. phys. sci. 4 (2022) 251–264 253 2.2. superiority of the rjkl estimator the bias of an estimator β̃ is expressed in equation (20), its mean squared error matrix (msem) in equation (21) and its scalar mean squared error (mse) in equation (22). bias ( β̃ ) = e ( β̃ ) −β (20) ms e m ( β̃ ) = e ( β̃−β ) ( β̃−β )′ = d ( β̃ ) + bias ( β̃ ) bias ( β̃ )′ (21) ms e ( β̃ ) = e ( β̃−β ) ( β̃−β )′ = tr ( d ( β̃ ) + bias ( β̃ ) bias ( β̃ )′) (22) where d ( β̃ ) is the variance matrix of β̃, e ( β̃ ) is the expectation of β̃ and tr (a) is the trace of a matrix, a. also, for α̃ = tβ̃, ms e m (α̃) = t′ms e m ( β̃ ) t and ms e (α̃) = ms e ( β̃ ) . additionally, for two estimators β̃1 and β̃2, β̃1 is said to be superior to β̃2 with respect to the msem criterion, if and only if, ms e m ( β̃1 ) −ms e m ( β̃2 ) ≥ 0. if ms e m ( β̃1 ) −ms e m ( β̃2 ) ≥ 0 then ms e ( β̃1 ) − ms e ( β̃2 ) ≥ 0 for j = 1, 2. the converse is not true. we also made use of the following lemmas for the theoretical comparison: lemma 2.1 [19] for some vector, α and a positive definite matrix, a (that is, a > 0); a −αα ′ ≥ 0 if and only if αa−1α ′ ≤ 1. lemma 2.2 [20] let β̃1 and β̃2 be two competing estimators of β. suppose that the difference between the covariance of the two estimators, d = cov ( β̃1 ) − cov ( β̃2 ) > 0, then ∆ (̂ β1, β̂2 ) = ms e m ( β̃1 ) −ms e m ( β̃2 ) ≥ 0 if and only if d ′ 2(d + d1d ′ 1) −1 d1 ≤ 1. ms e m ( β̃ j ) and d j denote the mean squared error matrix and bias vector of β̃ j respectively for j = 1, 2. we used the mse to prove the superiority of the rjkl. the mean squared error of the ols estimator and the m-estimator is expressed in equation (23) and (24), respectively. ms e ( α̂ ) = σ2 p∑ i=1 1 λi (23) ms e ( α̂m ) = p∑ i=1 ωii (24) where ωii = cov ( α̂m ) . the mean squared error matrix and the scalar mean squared error of the jkl is defined as follows: ms e m ( α̂jk l ) = σ̂2 ( i − (2k(λ + ki)−1) 2 )2 λ −1 ( i − 2k(λ + ki)−1 )2 + bias ( α̂jk l )′ bias ( α̂jk l ) (25) ms e ( α̂jk l ) = σ̂2 p∑ i=1 ( (λi + k) 2 − 4k2 )2 (λi − k) 2 λi(λi + k) 6 + p∑ i=1 ( (λi − k) 2 (λi + 3k) − (λi + k) 3 )2 α2i (λi + k) 6 (26) where bias ( α̂jk l ) = [( i − 2k(λ + ki)−1 )2 ( i + 2k(λ + ki)−1 ) − i ] α. the corresponding mean square error matrix and the scalar mean square error for the rjkl estimator is: ms e m ( α̂rjk l ) = ( i − (2k(λ + ki)−1) 2 )2 ω ( i + 2k(λ + ki)−1 )2 + bias ( α̂jk l )′ bias ( α̂jk l ) (27) ms e ( α̂rjk l ) = p∑ i=1 ωii ( (λi + k) 2 − 4k2 )2 (λi − k) 2 (λi + k) 6 + p∑ i=1 ( (λi − k) 2 (λi + 3k) − (λi + k) 3 )2 α2i (λi + k) 6 (28) we only consider the jkl and the robust estimators in the theoretical comparison and we impose the following conditions to present the main theorems: 1. ϕ is skew-symmetric and non-decreasing; 2. the errors are symmetric; 3. o is finite. 2.2.1. superiority of the rjkl estimator over the jkl estimator we state a theorem for rjkl to be superior to the jkl estimator: theorem 2.1: the proposed estimator, α̂rjk l is superior to the jackknife kibria lukman estimator, α̂jkl if and only if then o < σ2λ−1 for every k > 0. proof: the difference between the mse of the rjkl and the jkl estimator from (28) and (26) is ∆1 = ms e ( α̂rjk l ) − ms e ( α̂jkl ) = p∑ i=1 ωii ( (λi + k) 2 − 4k2 )2 (λi − k) 2 (λi + k) 6 − σ̂2 ( (λi + k) 2 − 4k2 )2 (λi − k) 2 λi(λi + k) 6 = p∑ i=1 ( ωii − σ2 λi )  ( (λi + k) 2 − 4k2 )2 (λi − k) 2 (λi + k) 6  (29) for ∆1 < 0 will imply that ∑p i=1 ωii < ∑p i=1 σ 2/λi, k > 0. 253 jegede et al. / j. nig. soc. phys. sci. 4 (2022) 251–264 254 2.2.2. superiority of the rjkl estimator over the robust kl estimator the scalar mean square error of the kl estimator is ms e(α̂m kl) = p∑ i=1 (λi − k) 2 (λi + k) 2 ωii + p∑ i=1 4α2i k 2 (λi + k) 2 (30) theorem 2.2: the proposed estimator, α̂rjk l is superior to the robust kibria lukman estimator, α̂m kl if and only if αω−1 [ 2 ( i − 2k(λ + ki)−1 )−1(( i − 2k(λ + ki)−1 ) ( i + 2k(λ + ki)−1 ) + 1 )−1 − i ] α ′ < 1 for k > 0. proof: the difference between the mse of the rjkl and the kl estimator from (28) and (30) is ∆2 = ms e ( α̂rjkl ) − ms e ( α̂mk l ) = p∑ i=1 ωii(λi − k) 2 (λi + k) 2  ( (λi + k) 2 − 4k2 )2 (λi + k) 4 − 1  + p∑ i=1 α2i (λi + k) 2  ( (λi − k) 2 (λi + 3k) − (λi + k) 3 )2 (λi + k) 4 − 4k2  (31) consequently, ω [ 2 ( i − 2k(λ + ki)−1 )−1(( i − 2k(λ + ki)−1 ) ( i + 2k(λ + ki)−1 ) + 1 )−1 − i ]−1 is positive definite, provided that ( (λi + k) 2 − 4k2 )2 > (λi + k) 4 and ( (λi − k) 2 (λi + 3k) − (λi + k) 3 )2 > 4k2 (λi + k) 4 . 2.2.3. superiority of the rjkl estimator over the ridge m-estimator the scalar mean square error of the ridge m-estimator is ms e(α̂m (k)) = p∑ i=1 λi 2 (λi + k) 2 ωii + p∑ i=1 k2α2i (λi + k) 2 (32) theorem 2.3: the proposed estimator, α̂rjk l is superior to the ridge m-estimator, α̂m k if and only if αω−1 [(( i − 2k(λ + ki)−1 )2 ( i + 2k(λ + ki)−1 ) − i )2 − (( i − k(λ + ki)−1 ) − i )2] [( i − k(λ + ki)−1 )2 − ( i − (2k(λ + ki)−1) 2 )2( i + 2k(λ + ki)−1 )2]−1 α ′ < 1 for k > 0. proof: the difference between the mse of the rjkl and the ridge m-estimator from (28) and (32) is ∆3 = ms e ( α̂rjkl ) − ms e ( α̂m k ) = p∑ i=1 ωii (λi + k) 2  ( (λi + k) 2 − 4k2 )2 (λi − k) 2 (λi + k) 4 −λi 2  + p∑ i=1 α2i (λi + k) 2  ( (λi − k) 2 (λi + 3k) − (λi + k) 3 )2 (λi + k) 4 − k2  (33) consequently, ω [(( i − 2k(λ + ki)−1 )2 ( i + 2k(λ + ki)−1 ) − i )2 − (( i − k(λ + ki)−1 ) − i )2]−1 [( i − k(λ + ki)−1 )2 − ( i − (2k(λ + ki)−1) 2 )2( i + 2k(λ + ki)−1 )2] is positive definite, provided that ( (λi + k) 2 − 4k2 )2 (λi − k) 2 > λi 2 (λi + k) 4 and ( (λi − k) 2 (λi + 3k) − (λi + k) 3 )2 > k2 (λi + k) 4 . 254 jegede et al. / j. nig. soc. phys. sci. 4 (2022) 251–264 255 2.2.4. superiority of the rjkl estimator over the m-estimator we state a theorem for rjkl to be superior to the m-estimator: theorem 2.4: the proposed estimator, α̂rjk l is superior to the m-estimator, α̂m if and only if αω−1 ( i − ( i − ( 2k(λ + ki)−1 )2)2( i + 2k(λ + ki)−1 )2)−1 (( i − 2k(λ + ki)−1 )2 ( i + 2k(λ + ki)−1 ) − i )2 α ′ < 1 for k > 0. proof: the difference between the mse of the rjkl and the m estimator from (28) and (24) is ∆4 = ms e ( α̂rjk l ) − ms e ( α̂m ) = p∑ i=1 ωii  ( (λi + k) 2 − 4k2 )2 (λi − k) 2 (λi + k) 6 − 1  + p∑ i=1 ( (λi − k) 2 (λi + 3k) − (λi + k) 3 )2 α2i (λi + k) 6 (34) consequently, ω ( i − ( i − ( 2k(λ + ki)−1 )2)2( i + 2k(λ + ki)−1 )2) (( i − 2k(λ + ki)−1 )2 ( i + 2k(λ + ki)−1 ) − i )2−1 is positive definite, provided that ( (λi + k) 2 − 4k2 )2 (λi − k) 2 > (λi + k) 6. notice that the right hand expression after the addition sign is positive, this follows from bias squared. 3. robust choice of the biasing parameter it is customary to use the optimization procedure to obtain the biasing parameter of an estimator [3]. this is done by minimizing equation (28), which is rewritten in (35), with respect to k. f (k) = p∑ i=1 ωii ( (λi + k) 2 − 4k2 )2 (λi − k) 2 (λi + k) 6 + p∑ i=1 ( (λi − k) 2 (λi + 3k) − (λi + k) 3 )2 α2i (λi + k) 6 (35) this can be obtained by setting ∂ f (k) ∂k = 0 (36) proceeding with (36) will yield a rather complex estimation for k. thus, we propose to use the robust version of the biasing parameter used for the jackknife kl estimator [15]. the biasing parameter used for the jackknife kl estimator is presented in (37). k̂ = √ pσ̂2∑p i=1 α̂ 2 i (37) this parameter (37) is the squared root of harmonic mean of the biasing parameter used for ridge estimator. the robust equivalence of the parameter, k̂ is k̂m = √ pâ2∑p i=1 α̂ 2 mi (38) where â2 is given by huber [8] as â2 = s2(n − p)−1 ∑p i=1 (ϕ(ei/s)) 2 ( ∑p i=1 (1/n)ϕ ′(ei/s)) 2 (39) with the assumption that α̂m ∼ n(α,a2λ−1). and, this holds since n 1 2 ( α̂2m −α ) λn ( 0, a2λ−1 ) , where a2 = s2o e(ϕ 2(ε/so)) (e(ϕ′(ε/so))) 2 , (40) with the scale estimate so. 4. monte carlo simulation study we adopt the monte carlo simulation design [21, 22] to observe the superiority of the robust jackknife kl estimator. the design was also recently adopted in related studies [23–28]. r programming language was used for the simulation. the following equation is used to generate the predictors: xi j = (1 −ρ 2) 1/2 zi j + ρzi,p+1, i = 1, 2, . . . , n, j = 1, 2, . . . , p (41) where ρ2 denotes the correlation between independent variables and zi j are pseudo-random numbers from the standard normal distribution. the coefficients β1,β2, . . . , βp are selected as the normalized eigenvectors corresponding to the largest eigenvalue of x ′ x such that β ′ β = 1. this is a common restriction in simulation studies of this kind [3, 25–26, 29–30]. the dependent variable determined by yi = β0 + β1 xi1 + β2 xi2 + · · · + βp xip + εi, i = 1, 2, . . . , n (42) where the ε′i s are independently generated from n(0, σ 2). we consider number of independent variables, p = 3 and p = 7. we introduced 10%, 20% and 30% outlier into each sample size considered in the simulation study [31]. the other specifications considered in the simulation design is as follows: 1. ρ = 0.70, 0.80, 0.90, 0.99 2. σ = 1, 5, 10 255 jegede et al. / j. nig. soc. phys. sci. 4 (2022) 251–264 256 table 1: estimated mses of the estimators when there are no outliers and p=3 n σ̂ ρ α̂ α̂m α̂k α̂m k α̂k l α̂m k l α̂jk l α̂rjk l 50 1 0.7 0.0906 0.0945 0.0817 0.0856 0.0743 0.0781 0.0640 0.0675 0.8 0.1201 0.1250 0.1033 0.1082 0.0886 0.0935 0.0663 0.0708 0.9 0.2145 0.2232 0.1633 0.1720 0.1201 0.1286 0.0618 0.0689 0.99 1.9546 2.0313 0.5968 0.6575 0.0671 0.0920 0.0373 0.0400 5 0.7 2.2644 2.3616 1.7243 1.8253 1.2825 1.3828 0.7235 0.8078 0.8 3.0013 3.1259 2.1305 2.2606 1.4436 1.5713 0.6654 0.7631 0.9 5.3623 5.5804 3.2613 3.4870 1.7660 1.9747 0.5260 0.6413 0.99 48.8650 50.7823 13.0782 14.5443 2.0620 2.4665 5.4359 5.1779 10 0.7 9.0574 9.4463 6.6716 7.0819 4.7377 5.1484 2.3424 2.6843 0.8 12.0052 12.5036 8.2686 8.7944 5.3753 5.8912 2.2291 2.6120 0.9 21.4491 22.3218 12.7356 13.6432 6.6773 7.5085 1.8716 2.3134 0.99 195.4598 203.1294 52.0364 57.8922 8.4338 10.0135 23.8768 22.6897 100 1 0.7 0.0470 0.0492 0.0443 0.0464 0.0418 0.0438 0.0376 0.0395 0.8 0.0637 0.0666 0.0583 0.0611 0.0533 0.0560 0.0445 0.0470 0.9 0.1159 0.1213 0.0985 0.1034 0.0826 0.0871 0.0562 0.0600 0.99 1.0714 1.1209 0.4493 0.4816 0.1042 0.1212 0.0088 0.0096 5 0.7 1.1756 1.2303 0.9732 1.0242 0.7955 0.8426 0.5246 0.5635 0.8 1.5919 1.6658 1.2475 1.3150 0.9527 1.0137 0.5344 0.5815 0.9 2.8982 3.0318 2.0106 2.1281 1.3059 1.4063 0.5010 0.5642 0.99 26.7861 28.0236 9.3727 10.1409 1.7215 2.0322 0.7853 0.7860 10 0.7 4.7023 4.9212 3.7496 3.9546 2.9304 3.1197 1.7457 1.8980 0.8 6.3674 6.6630 4.8106 5.0824 3.5146 3.7596 1.7988 1.9809 0.9 11.5929 12.1274 7.7879 8.2604 4.8593 5.2603 1.7529 1.9918 0.99 107.1445 112.0945 37.1706 40.2385 6.9026 8.1198 4.0429 3.9826 200 1 0.7 0.0229 0.0240 0.0222 0.0233 0.0216 0.0226 0.0205 0.0215 0.8 0.0310 0.0325 0.0297 0.0311 0.0284 0.0298 0.0260 0.0273 0.9 0.0566 0.0593 0.0520 0.0545 0.0475 0.0499 0.0394 0.0415 0.99 0.5249 0.5498 0.2979 0.3151 0.1365 0.1474 0.0149 0.0181 5 0.7 0.5723 0.6000 0.5105 0.5363 0.4536 0.4775 0.3558 0.3763 0.8 0.7761 0.8133 0.6660 0.6999 0.5659 0.5968 0.3999 0.4251 0.9 1.4158 1.4833 1.1108 1.1697 0.8468 0.8978 0.4615 0.4986 0.99 13.1226 13.7447 5.9327 6.3356 1.8403 2.0648 0.1855 0.2230 10 0.7 2.2894 2.4000 1.9666 2.0687 1.6748 1.7687 1.1970 1.2755 0.8 3.1043 3.2531 2.5609 2.6958 2.0798 2.2015 1.3328 1.4300 0.9 5.6634 5.9333 4.2718 4.5066 3.1064 3.3082 1.5428 1.6854 0.99 52.4905 54.9787 23.3878 24.9955 7.1815 8.0669 0.9502 1.0706 250 1 0.7 0.0185 0.0196 0.0180 0.0191 0.0176 0.0186 0.0168 0.0178 0.8 0.0250 0.0265 0.0241 0.0256 0.0232 0.0246 0.0216 0.0229 0.9 0.0456 0.0484 0.0424 0.0450 0.0393 0.0418 0.0337 0.0358 0.99 0.4227 0.4494 0.2553 0.2738 0.1308 0.1426 0.0200 0.0236 5 0.7 0.4614 0.4892 0.4180 0.4438 0.3775 0.4014 0.3062 0.3266 0.8 0.6248 0.6630 0.5462 0.5810 0.4738 0.5055 0.3503 0.3760 0.9 1.1391 1.2097 0.9160 0.9777 0.7197 0.7730 0.4200 0.4584 0.99 10.5672 11.2344 5.0044 5.4380 1.6925 1.9306 0.1565 0.1963 10 0.7 1.8457 1.9567 1.6107 1.7131 1.3959 1.4901 1.0355 1.1138 0.8 2.4993 2.6521 2.0988 2.2374 1.7399 1.8648 1.1655 1.2645 0.9 4.5565 4.8390 3.5136 3.7599 2.6261 2.8375 1.3832 1.5302 0.99 42.2689 44.9375 19.6719 21.4040 6.5464 7.4852 0.7302 0.8829 3. n = 50, 100, 200, 250 the biasing parameter, k̂ was used for the ridge, kl and jackknife kl estimator while the robust version of the biasing parameter, k̂m is used for the robust version of the estimators, including the robust jackknife kl. the simulation was done 2000 times by generating new pseudo-random numbers and the esti256 jegede et al. / j. nig. soc. phys. sci. 4 (2022) 251–264 257 table 2: estimated mses of the estimators when there is 10% outlier and p=3 n σ̂ ρ α̂ α̂m α̂k α̂m k α̂k l α̂m k l α̂jk l α̂rjk l 50 1 0.7 13.6963 0.1479 10.0387 0.1313 7.0596 0.1169 3.3265 0.0949 0.8 19.9972 0.1966 13.7140 0.1671 8.8318 0.1409 3.4809 0.1002 0.9 39.1477 0.3526 23.1621 0.2666 12.0224 0.1943 3.1734 0.0968 0.99 384.1922 3.2231 101.8056 1.0736 15.5236 0.1687 43.3645 0.0674 5 0.7 137.1126 3.6961 99.5904 2.8692 69.2274 2.1761 31.7675 1.2357 0.8 184.4161 4.9145 125.4914 3.5931 80.0792 2.5253 31.1814 1.2177 0.9 334.6314 8.8142 196.9320 5.6335 101.7029 3.2920 27.0002 1.0943 0.99 3091.902 80.5657 819.9766 24.6852 126.6444 4.3096 359.7422 6.3585 10 0.7 515.965 14.7841 374.3237 11.2446 259.8384 8.3034 119.0229 4.3970 0.8 689.075 19.6578 468.3979 14.1016 298.5428 9.6738 116.2113 4.4187 0.9 1239.973 35.2564 729.1771 22.1921 376.3083 12.7275 100.0509 4.1134 0.99 11372.96 322.2612 3017.1655 98.4042 465.9051 17.3591 1325.6418 27.3046 100 1 0.7 7.6998 0.0710 6.0621 0.0665 4.6620 0.0622 2.6673 0.0549 0.8 11.3464 0.0960 8.4537 0.0872 6.0668 0.0790 2.9758 0.0645 0.9 22.3993 0.1742 14.8158 0.1462 9.0488 0.1209 3.1156 0.0796 0.99 221.3106 1.6064 75.0130 0.6830 13.3166 0.1701 9.6412 0.0139 5 0.7 79.8287 1.7757 62.0647 1.4746 46.9970 1.2094 25.9263 0.8017 0.8 109.6894 2.3985 80.9090 1.8927 57.3702 1.4589 27.4833 0.8379 0.9 202.3623 4.3550 133.0403 3.0711 80.7291 2.0451 27.6453 0.8410 0.99 1888.093 40.1531 641.9309 14.9983 116.0079 3.2139 84.8115 1.0515 10 0.7 300.1889 7.1028 233.0442 5.7389 176.1599 4.5550 96.8446 2.8014 0.8 409.2441 9.5939 301.4447 7.3756 213.4082 5.5098 102.0062 2.9647 0.9 749.2885 17.4197 492.0093 12.0162 298.1691 7.7904 102.0787 3.0693 0.99 6946.875 160.6115 2359.3943 59.6581 427.1866 12.8245 315.4987 5.0330 200 1 0.7 4.4751 0.0355 3.7892 0.0344 3.1754 0.0333 2.1944 0.0314 0.8 6.5295 0.0480 5.3021 0.0457 4.2303 0.0434 2.6193 0.0394 0.9 12.9185 0.0872 9.5866 0.0793 6.8417 0.0719 3.2967 0.0584 0.99 129.3138 0.8062 56.8508 0.4555 17.1766 0.2086 2.6167 0.0263 5 0.7 46.0261 0.8880 38.4685 0.7913 31.7480 0.7021 21.1678 0.5494 0.8 62.9938 1.1988 50.6022 1.0295 39.8739 0.8757 24.0539 0.6218 0.9 116.2244 2.1794 85.6010 1.7212 60.5908 1.3247 28.8401 0.7453 0.99 1090.0300 20.1500 479.2827 9.6267 145.7462 3.4039 23.5782 0.4220 10 0.7 175.6885 3.5519 146.6835 3.0737 120.9005 2.6397 80.3387 1.9222 0.8 238.7105 4.7953 191.5810 3.9980 150.8015 3.2894 90.7374 2.1769 0.9 436.7494 8.7175 321.4645 6.6959 227.3712 4.9901 108.0523 2.6457 0.99 4062.8734 80.5996 1786.2549 38.1570 543.0836 13.4103 87.9686 1.8838 250 1 0.7 4.0422 0.0296 3.4584 0.0288 2.9318 0.0280 2.0741 0.0266 0.8 5.8499 0.0403 4.8094 0.0387 3.8919 0.0371 2.4794 0.0341 0.9 11.4636 0.0738 8.6490 0.0682 6.2987 0.0627 3.1569 0.0527 0.99 114.3099 0.6868 52.1028 0.4135 16.7411 0.2115 2.3191 0.0344 5 0.7 38.3098 0.7405 32.3421 0.6688 26.9940 0.6018 18.4195 0.4841 0.8 52.5104 1.0077 42.6718 0.8802 34.0728 0.7629 21.1009 0.5630 0.9 96.9599 1.8450 72.5000 1.4935 52.2625 1.1835 25.7424 0.7081 0.99 910.4138 17.1656 413.6027 8.6770 132.6110 3.3776 19.6279 0.4017 10 0.7 145.3148 2.9619 122.5263 2.5978 102.1118 2.2638 69.4115 1.6977 0.8 197.7114 4.0306 160.4884 3.4168 127.9780 2.8640 79.0100 1.9680 0.9 362.2294 7.3800 270.6038 5.8033 194.8606 4.4479 95.7777 2.4934 0.99 3377.0204 68.6613 1533.5258 34.3510 491.4155 13.2504 72.9195 1.7006 mated mse calculated as: mse ( α̂ ) = 1 2000 2000∑ j=1 ( α̂i j −αi )′ ( α̂i j −αi ) (43) as the standard deviation, σ and the degree of multicollinearity, ρ increases, the mean square error, mses of the estimators, α̂, α̂m, α̂k, α̂m k, α̂k l, α̂m k l, α̂jk l and α̂rjk l also increases. the mean square error, mses of the estimators, 257 jegede et al. / j. nig. soc. phys. sci. 4 (2022) 251–264 258 table 3: estimated mses of the estimators when there are 20% outliers and p=3 n σ̂ ρ α̂ α̂m α̂k α̂m k α̂k l α̂m k l α̂jk l α̂rjk l 50 1 0.7 27.4909 0.2521 20.2574 0.2197 14.3415 0.1908 6.8243 0.1449 0.8 40.3804 0.3364 27.9507 0.2809 18.2475 0.2316 7.4111 0.1545 0.9 79.5740 0.6035 47.8373 0.4495 25.5539 0.3206 7.1857 0.1496 0.99 790.303 5.5425 218.992 1.9246 35.8040 0.3346 86.3257 0.1074 5 0.7 272.653 6.3015 199.545 4.9528 140.154 3.8028 65.9791 2.1712 0.8 368.888 8.4073 253.726 6.2640 164.505 4.4998 66.6474 2.2272 0.9 672.473 15.0834 402.222 9.9382 213.812 6.0549 60.7675 2.1345 0.99 6245.01 138.519 1727.07 45.6565 287.645 8.3804 699.754 7.5586 10 0.7 1025.56 25.2056 749.801 19.5587 526.109 14.7806 247.809 8.1339 0.8 1377.33 33.6286 946.375 24.7504 613.041 17.5204 248.908 8.4380 0.9 2490.26 60.3325 1488.33 39.3541 790.911 23.7011 226.165 8.2671 0.99 22949.5 554.071 6349.96 182.212 1065.63 33.6397 2586.37 32.0533 100 1 0.7 15.1429 0.1200 11.8102 0.1110 8.9723 0.1026 4.9686 0.0879 0.8 22.3024 0.1624 16.4695 0.1456 11.6798 0.1300 5.5478 0.1027 0.9 43.9747 0.2953 28.8485 0.2439 17.4039 0.1979 5.7482 0.1243 0.99 433.363 2.7235 145.132 1.1828 24.3159 0.3201 20.1608 0.0265 5 0.7 156.829 2.9992 121.501 2.5008 91.5606 2.0585 49.7936 1.3663 0.8 215.022 4.0580 157.956 3.2314 111.340 2.5171 52.3434 1.4738 0.9 396.164 7.3797 259.142 5.3047 155.911 3.6269 51.5900 1.5822 0.99 3695.54 68.0707 1242.22 27.1764 212.261 6.7520 172.944 1.4096 10 0.7 593.815 11.9965 459.802 9.8262 346.290 7.9204 188.139 5.0142 0.8 808.576 16.2319 593.809 12.7085 418.459 9.7046 196.747 5.4576 0.9 1479.48 29.5183 967.986 20.9228 582.716 14.0706 193.291 5.9866 0.99 13717.4 272.281 4619.94 108.333 794.266 26.9217 641.662 6.2819 200 1 0.7 8.5590 0.0546 7.1615 0.0526 5.9191 0.0507 3.9641 0.0472 0.8 12.4987 0.0740 10.0274 0.0700 7.8863 0.0661 4.7286 0.0589 0.9 24.7452 0.1350 18.1275 0.1214 12.7229 0.1086 5.8933 0.0856 0.99 247.230 1.2522 106.201 0.6986 30.1372 0.3139 4.8016 0.0399 5 0.7 88.0770 1.3649 73.1962 1.2110 59.9960 1.0693 39.3377 0.8269 0.8 120.611 1.8505 96.2486 1.5846 75.2182 1.3437 44.4431 0.9475 0.9 222.578 3.3753 162.506 2.6714 113.635 2.0624 52.3102 1.1721 0.99 2086.82 31.2969 897.195 15.4657 255.893 5.8296 41.8475 0.7606 10 0.7 336.691 5.4597 279.756 4.7377 229.235 4.0805 150.121 2.9871 0.8 457.457 7.4019 365.064 6.2033 285.290 5.1345 168.490 3.4419 0.9 836.317 13.5011 610.777 10.4839 427.256 7.9212 196.828 4.3292 0.99 7773.24 125.185 3346.15 61.5152 957.602 23.0985 155.941 3.2643 250 1 0.7 7.6189 0.0448 6.4429 0.0434 5.3895 0.0421 3.7021 0.0396 0.8 11.067 0.0609 8.9921 0.0581 7.1774 0.0554 4.4397 0.0502 0.9 21.7733 0.1115 16.2260 0.1019 11.6384 0.0927 5.6469 0.0760 0.99 218.570 1.0377 97.8022 0.6138 30.2036 0.3058 4.1657 0.0496 5 0.7 72.4863 1.1186 60.8732 1.0056 50.4973 0.9004 33.9836 0.7165 0.8 99.6846 1.5225 80.5435 1.3247 63.8719 1.1432 38.9391 0.8365 0.9 184.767 2.7871 137.222 2.2539 98.0540 1.7852 47.2803 1.0714 0.99 1741.86 25.9362 780.620 13.4438 242.517 5.5131 34.7890 0.7911 10 0.7 274.650 4.4745 230.508 3.9307 191.072 3.4313 128.323 2.5835 0.8 374.584 6.0901 302.465 5.1794 239.672 4.3583 145.838 3.0239 0.9 687.970 11.1482 510.550 8.8300 364.476 6.8313 175.387 3.9233 0.99 6427.93 103.743 2876.86 53.4205 891.498 21.7854 128.508 3.2871 α̂, α̂m, α̂k, α̂m k, α̂k l, α̂m k l, α̂jk l and α̂rjk l decrease as the sample size increases. as expected, the mses increases as the outliers increases. this is what births the purpose of our study, to propose a better estimator which handles outliers jointly with multicollinearity by yielding a minimum mse. as we know it, the performance of the ols is poor in the presence of outliers and multicollinearity. this can be observed in table (1-8) and when there is no outlier (table 1 & 5), the jackknife 258 jegede et al. / j. nig. soc. phys. sci. 4 (2022) 251–264 259 table 4: estimated mses of the estimators when there are 30% outliers and p=3 n σ̂ ρ α̂ α̂m α̂k α̂m k α̂k l α̂m k l α̂jk l α̂rjk l 50 1 0.7 42.8128 1.1147 31.6134 0.9605 22.4115 0.8201 10.6056 0.5859 0.8 63.1076 1.5114 43.7884 1.2524 28.6259 1.0210 11.5133 0.6538 0.9 124.554 2.8334 75.1313 2.1200 40.1568 1.5214 10.9966 0.7209 0.99 1235.20 26.6215 344.127 11.0545 51.1102 2.7550 114.977 0.3498 5 0.7 425.609 22.2058 313.429 18.2857 221.616 14.8223 104.952 9.4545 0.8 577.241 30.1395 399.724 23.7643 260.936 18.2928 105.598 10.4202 0.9 1054.87 53.3573 635.995 37.9961 340.290 25.6981 94.3308 11.0380 0.99 9841.98 506.709 2750.33 201.981 405.678 48.7122 923.202 12.2529 10 0.7 1603.19 88.0314 1180.05 72.1510 834.108 58.1816 395.727 36.7551 0.8 2157.41 115.778 1493.33 90.7417 974.720 69.3911 395.734 39.1248 0.9 3908.96 210.583 2356.57 149.327 1261.65 100.545 351.986 43.0182 0.99 36170.2 1987.74 10123.2 789.441 1503.91 189.681 3415.795 50.4719 100 1 0.7 21.9449 0.2312 16.9722 0.2108 12.7585 0.1917 6.8909 0.1579 0.8 32.4753 0.3138 23.7871 0.2769 16.6932 0.2427 7.7505 0.1834 0.9 64.4363 0.5716 41.9240 0.4641 25.0097 0.3685 8.1158 0.2194 0.99 638.769 5.2860 211.730 2.3982 35.7890 0.7310 30.2949 0.0588 5 0.7 229.929 5.7788 176.924 4.8626 132.203 4.0421 70.5340 2.7288 0.8 315.839 7.8429 230.364 6.3371 160.890 5.0219 74.1414 3.0440 0.9 582.709 14.2855 378.129 10.5470 224.968 7.4702 72.9364 3.5126 0.99 5437.17 132.0971 1804.28 57.3248 306.655 16.8488 262.016 2.5864 10 0.7 870.874 23.1149 669.761 19.2563 500.176 15.8246 266.649 10.4209 0.8 1187.82 31.3710 865.995 25.1117 604.560 19.6900 278.519 11.6933 0.9 2175.94 57.1419 1411.80 41.8709 839.947 29.4090 272.570 13.6617 0.99 20172.5 528.373 6699.34 228.894 1142.26 67.2673 975.172 11.0599 200 1 0.7 12.3718 0.1028 10.2729 0.0983 8.4133 0.0940 5.5140 0.0860 0.8 18.2202 0.1403 14.5162 0.1312 11.3207 0.1225 6.6581 0.1065 0.9 36.2226 0.2572 26.3754 0.2276 18.3733 0.2001 8.3854 0.1516 0.99 362.358 2.3898 154.744 1.3440 43.7583 0.6179 7.0546 0.0898 5 0.7 127.504 2.5702 105.398 2.2789 85.8416 2.0101 55.4493 1.5485 0.8 175.938 3.5056 139.749 3.0109 108.592 2.5619 63.3166 1.8203 0.9 326.571 6.4269 237.550 5.1467 165.332 4.0331 75.3718 2.3770 0.99 3073.00 59.7253 1318.54 31.2958 377.572 13.1203 61.8303 1.9138 10 0.7 485.060 10.2807 400.873 8.9939 326.389 7.8142 210.598 5.8199 0.8 664.064 14.0223 527.384 11.8961 409.708 9.9837 238.676 6.8910 0.9 1221.52 25.7073 888.342 20.3805 618.079 15.7932 281.502 9.1218 0.99 11394.4 238.904 4885.92 124.855 1397.64 52.2367 229.319 7.8085 250 1 0.7 10.0629 0.0740 8.4145 0.0713 6.9507 0.0687 4.6546 0.0639 0.8 14.6414 0.1007 11.7454 0.0952 9.2400 0.0899 5.5572 0.0800 0.9 28.8334 0.1844 21.1552 0.1661 14.8922 0.1487 6.9958 0.1176 0.99 288.820 1.7173 125.332 1.0015 37.0075 0.4891 6.3407 0.0793 5 0.7 95.2388 1.8496 79.1862 1.6547 64.9614 1.4739 42.7579 1.1599 0.8 130.944 2.5165 104.553 2.1822 81.7941 1.8767 48.5731 1.3643 0.9 242.623 4.6092 177.371 3.7318 124.329 2.9620 57.9075 1.7933 0.99 2288.42 42.9196 991.234 22.8923 291.752 9.9008 51.2254 1.5034 10 0.7 361.592 7.3981 300.594 6.5134 246.530 5.6998 162.094 4.3143 0.8 493.526 10.0661 394.058 8.5973 308.269 7.2706 182.987 5.1045 0.9 907.090 18.4365 663.258 14.7336 465.033 11.5268 216.678 6.8007 0.99 8483.85 171.678 3677.92 91.2278 1084.86 39.3404 190.115 6.1180 kl estimator performs better than other non-robust estimator as argued by ugwuowo et al. [15]. the proposed estimator in this study, α̂rjkl performs much better than the existing estimators considered in this study. that is, the robust jackknife kibria lukman estimator has the smallest mean square error. 259 jegede et al. / j. nig. soc. phys. sci. 4 (2022) 251–264 260 table 5: estimated mses of the estimators when there are no outliers and p=7 n σ̂ ρ α̂ α̂m α̂k α̂m k α̂k l α̂m k l α̂jk l α̂rjk l 50 1 0.7 0.2697 0.2861 0.2208 0.2391 0.1776 0.1971 0.1106 0.1302 0.8 0.3781 0.4014 0.2898 0.3163 0.2144 0.2426 0.1084 0.1346 0.9 0.7106 0.7545 0.4639 0.5153 0.2738 0.3261 0.0779 0.1134 0.99 6.7379 7.1589 1.7016 2.0728 0.2011 0.2786 0.4034 0.2812 5 0.7 6.7424 7.1523 4.6574 5.1544 3.0057 3.5329 1.1284 1.5434 0.8 9.4532 10.0345 6.0780 6.7871 3.5309 4.2619 1.0342 1.5256 0.9 17.7638 18.8637 9.7972 11.1263 4.4276 5.6791 0.7544 1.3060 0.99 168.448 178.973 40.1434 49.2488 5.8421 7.1385 16.9292 11.8110 10 0.7 26.9695 28.6090 18.3182 20.3240 11.5532 13.6742 4.1293 5.7498 0.8 37.8129 40.1378 23.9652 26.8178 13.6418 16.5678 3.8478 5.7570 0.9 71.0552 75.4550 38.7965 44.1234 17.2632 22.2467 2.8872 5.0310 0.99 673.791 715.892 160.257 196.655 23.5174 28.6142 68.9781 48.1329 100 1 0.7 0.1398 0.1480 0.1250 0.1330 0.1112 0.1190 0.0868 0.0942 0.8 0.1966 0.2081 0.1687 0.1799 0.1431 0.1540 0.1000 0.1099 0.9 0.3704 0.3920 0.2855 0.3062 0.2123 0.2318 0.1072 0.1228 0.99 3.5231 3.7282 1.2434 1.3929 0.1913 0.2582 0.0273 0.0273 5 0.7 3.4947 3.6994 2.7012 2.9052 2.0214 2.2188 1.0551 1.2194 0.8 4.9144 5.2019 3.5908 3.8762 2.4962 2.7675 1.0860 1.2920 0.9 9.2599 9.8007 5.9888 6.5146 3.4971 3.9695 0.9865 1.2604 0.99 88.0776 93.2049 28.4307 32.0951 3.8710 5.2275 1.7745 1.5159 10 0.7 13.9790 14.7976 10.5672 11.3906 7.6909 8.4885 3.7718 4.4183 0.8 19.6575 20.8078 14.0815 15.2312 9.5407 10.6323 3.9301 4.7328 0.9 37.0396 39.2029 23.6031 25.7142 13.4985 15.3861 3.6637 4.7223 0.99 352.311 372.820 113.347 127.995 15.4177 20.7938 7.4482 6.3335 200 1 0.7 0.0647 0.0683 0.0614 0.0649 0.0582 0.0616 0.0522 0.0554 0.8 0.0909 0.0959 0.0844 0.0892 0.0782 0.0829 0.0667 0.0710 0.9 0.1710 0.1805 0.1498 0.1587 0.1300 0.1383 0.0955 0.1027 0.99 1.6255 1.7153 0.8128 0.8796 0.2906 0.3334 0.0165 0.0237 5 0.7 1.6178 1.7071 1.3832 1.4686 1.1688 1.2499 0.8092 0.8805 0.8 2.2718 2.3972 1.8662 1.9850 1.5036 1.6150 0.9280 1.0217 0.9 4.2758 4.5118 3.2115 3.4303 2.3108 2.5089 1.0747 1.2205 0.99 40.6370 42.8824 17.8112 19.4537 4.8583 5.8012 0.2790 0.3684 10 0.7 6.4711 6.8284 5.3885 5.7326 4.4155 4.7435 2.8485 3.1339 0.8 9.0871 9.5887 7.2798 7.7584 5.6921 6.1419 3.2789 3.6510 0.9 17.1032 18.0474 12.5815 13.4614 8.8149 9.6115 3.8625 4.4339 0.99 162.548 171.530 70.8341 77.4027 19.1209 22.8692 1.1435 1.4841 250 1 0.7 0.0507 0.0535 0.0486 0.0513 0.0466 0.0492 0.0427 0.0452 0.8 0.0712 0.0752 0.0671 0.0710 0.0632 0.0668 0.0557 0.0590 0.9 0.1341 0.1417 0.1205 0.1275 0.1076 0.1141 0.0843 0.0898 0.99 1.2751 1.3466 0.6942 0.7454 0.2948 0.3277 0.0261 0.0336 5 0.7 1.2667 1.3375 1.1070 1.1734 0.9588 1.0211 0.7022 0.7559 0.8 1.7806 1.8802 1.5009 1.5932 1.2466 1.3317 0.8261 0.8965 0.9 3.3536 3.5415 2.6044 2.7738 1.9560 2.1063 1.0100 1.1204 0.99 31.8780 33.6647 14.9915 16.2498 4.7865 5.5181 0.2975 0.3886 10 0.7 5.0670 5.3499 4.3067 4.5737 3.6140 3.8643 2.4612 2.6752 0.8 7.1226 7.5208 5.8442 6.2154 4.7030 5.0456 2.8969 3.1764 0.9 13.4145 14.1658 10.1797 10.8600 7.4268 8.0308 3.5881 4.0222 0.99 127.512 134.659 59.5520 64.5848 18.7782 21.6873 1.1794 1.5301 5. real-life application we adopted the hussein data for the data analysis yi = β0 +β1 xi1 + β2 xi2 +β3 xi3 +εi, i = 1, 2, . . . , 31(44) hussein and zari [32] were the first to originally adopt the hussein data and was recently used to test the performance of a proposed two parameter estimator in the presence of outlier [3]. the data contains 31 observations and three independent vari260 jegede et al. / j. nig. soc. phys. sci. 4 (2022) 251–264 261 table 6: estimated mses of the estimators when there is 10% outlier and p=7 n σ̂ ρ α̂ α̂m α̂k α̂m k α̂k l α̂m k l α̂jk l α̂rjk l 50 1 0.7 60.8029 0.4791 40.8878 0.3938 25.4231 0.3180 8.8012 0.1997 0.8 99.2107 0.6718 62.2013 0.5205 34.8383 0.3907 9.5212 0.2056 0.9 216.044 1.2625 116.569 0.8524 50.8361 0.5308 8.2968 0.1780 0.99 2333.39 11.9803 545.305 3.6804 84.8065 0.5415 260.208 0.3678 5 0.7 422.505 11.9761 283.381 8.7943 175.556 6.1779 60.2840 2.8467 0.8 611.046 16.7925 382.392 11.6306 213.564 7.5504 57.9704 2.9104 0.9 1187.80 31.5545 640.163 19.2652 278.541 10.3922 45.1488 2.6870 0.99 11631.7 299.420 2716.67 89.2637 420.106 13.4739 1286.53 13.2986 10 0.7 1510.69 47.9040 1013.53 34.8725 628.169 24.2323 215.967 10.9305 0.8 2136.76 67.1685 1337.73 46.1804 747.614 29.6979 203.205 11.2630 0.9 4054.03 126.217 2185.96 76.6677 951.854 41.0706 154.244 10.5271 0.99 38783.8 1197.65 9062.67 356.691 1393.78 53.9159 4259.15 53.9040 100 1 0.7 33.3801 0.2180 24.8173 0.1937 17.6889 0.1711 8.2997 0.1315 0.8 54.6386 0.3065 38.4207 0.2612 25.3986 0.2199 9.9254 0.1510 0.9 119.500 0.5773 74.5251 0.4430 41.2638 0.3280 10.4418 0.1645 0.99 1282.08 5.4894 397.063 2.0845 50.4080 0.4107 34.2417 0.0428 5 0.7 239.390 5.4501 177.582 4.2958 126.232 3.2974 58.9316 1.8394 0.8 344.928 7.6607 242.290 5.7469 159.995 4.1420 62.5281 1.9869 0.9 668.299 14.4296 416.904 9.7203 231.095 6.0453 58.9708 2.0327 0.99 6539.09 137.201 2032.17 49.2646 262.700 8.9174 174.470 1.9351 10 0.7 873.665 21.8004 647.966 16.9514 460.494 12.7991 214.904 6.8875 0.8 1236.35 30.6427 868.354 22.7161 573.360 16.1320 224.127 7.5045 0.9 2347.62 57.7179 1464.59 38.5455 812.026 23.6971 207.553 7.8010 0.99 22517.5 548.797 7002.33 196.679 908.997 35.5307 602.934 7.9639 200 1 0.7 26.5675 0.1008 21.7406 0.0952 17.4507 0.0898 10.7326 0.0796 0.8 43.6640 0.1415 34.3492 0.1307 26.2661 0.1203 14.3672 0.1011 0.9 94.7433 0.2663 68.3657 0.2314 46.6883 0.1991 19.2553 0.1433 0.99 988.387 2.5302 417.755 1.3002 104.913 0.4966 6.8410 0.0385 5 0.7 138.754 2.5188 113.288 2.1673 90.6932 1.8448 55.4578 1.2999 0.8 200.219 3.5364 157.283 2.9357 120.070 2.3959 65.4601 1.5274 0.9 387.860 6.6543 279.733 5.1010 190.928 3.7714 78.7039 1.8875 0.99 3780.44 63.2284 1600.02 29.7487 403.397 9.6212 26.2144 0.6796 10 0.7 499.716 10.0752 407.821 8.5153 326.312 7.1013 199.297 4.7765 0.8 706.782 14.1453 555.039 11.5504 423.557 9.2456 230.727 5.6423 0.9 1341.19 26.6170 967.189 20.1369 660.061 14.6469 272.047 7.0741 0.99 12846.0 252.909 5439.26 118.575 1372.75 38.1181 89.1150 2.7126 250 1 0.7 23.7443 0.0798 19.9966 0.0761 16.6024 0.0726 11.0348 0.0658 0.8 39.3865 0.1121 32.0454 0.1051 25.5322 0.0983 15.3790 0.0855 0.9 86.3019 0.2113 65.0884 0.1882 47.1256 0.1665 22.3962 0.1279 0.99 926.147 2.0085 432.778 1.1122 136.614 0.4907 8.4626 0.0533 5 0.7 125.902 1.9932 105.646 1.7495 87.3520 1.5231 57.5402 1.1285 0.8 183.632 2.8026 149.001 2.3814 118.342 1.9971 70.8069 1.3553 0.9 359.783 5.2793 270.848 4.1691 195.660 3.1997 92.5738 1.7509 0.99 3548.57 50.1888 1656.48 25.1650 522.139 9.2833 32.6286 0.7798 10 0.7 445.692 7.9727 373.720 6.8642 308.752 5.8468 203.024 4.1248 0.8 633.414 11.2102 513.634 9.3527 407.647 7.6792 243.524 4.9686 0.9 1206.99 21.1167 908.186 16.4252 655.669 12.3754 309.832 6.5003 0.99 11598.9 200.749 5412.89 100.221 1705.35 36.7024 106.558 3.0723 ables. details description can be found in [32,3]. the data contains multicollinearity with its variance inflation factor, vif>10 and about 19.4% outliers in the y-direction at observations 12, 14, 15, 16, 30 and 31. hence, it is sufficient to use the data in this study. the output of the analysis is presented in table 9. the rjkl estimator has the smallest mean square value. thus, the rjkl estimator performed better. the percentage mse 261 jegede et al. / j. nig. soc. phys. sci. 4 (2022) 251–264 262 table 7: estimated mses of the estimators when there are 20% outliers and p=7 n σ̂ ρ α̂ α̂m α̂k α̂m k α̂k l α̂m k l α̂jk l α̂rjk l 50 1 0.7 119.914 1.0249 80.5354 0.8315 50.0589 0.6608 17.4943 0.3986 0.8 196.356 1.4329 123.079 1.1009 69.0594 0.8176 19.2267 0.4192 0.9 428.611 2.9971 231.566 2.0723 101.677 1.3390 17.2449 0.5015 0.99 4636.04 29.4453 1092.23 10.7202 177.533 2.2999 545.556 0.5486 5 0.7 818.077 25.1608 547.263 19.0514 338.257 13.9238 116.863 7.0046 0.8 1184.59 35.2298 739.444 25.3346 412.225 17.3117 113.255 7.4746 0.9 2304.40 66.2106 1238.94 42.5563 539.034 24.8324 90.0158 7.5425 0.99 22578.0 629.137 5270.26 209.884 866.359 36.1990 2712.14 16.1937 10 0.7 2921.54 100.502 1955.11 75.7647 1209.05 55.0819 418.169 27.4365 0.8 4139.97 140.781 2585.75 100.868 1442.75 68.6166 397.063 29.4010 0.9 7866.74 264.728 4232.82 169.731 1843.98 98.7288 308.658 29.8633 0.99 75384.1 2517.78 17619.9 839.856 2884.57 145.069 8964.29 65.3513 100 1 0.7 64.3033 0.3636 47.2958 0.3187 33.2345 0.2772 15.0659 0.2057 0.8 105.357 0.5111 73.2423 0.4290 47.6494 0.3549 17.9011 0.2338 0.9 230.299 0.9633 141.697 0.7288 76.7823 0.5297 18.3979 0.2537 0.99 2467.38 9.1679 743.891 3.6115 89.5646 0.7829 73.7920 0.0692 5 0.7 451.313 9.0874 331.732 7.2409 232.916 5.6325 105.360 3.2388 0.8 652.082 12.7723 453.225 9.7249 294.798 7.1444 110.705 3.5839 0.9 1265.65 24.0712 778.849 16.5974 422.200 10.6691 101.254 3.8743 0.99 12401.6 229.079 3743.17 87.2753 451.618 17.9354 373.088 2.3922 10 0.7 1645.11 36.3492 1209.82 28.7364 850.020 22.1430 385.168 12.4717 0.8 2330.35 51.0881 1620.83 38.6357 1055.34 28.1486 397.402 13.8775 0.9 4431.27 96.2822 2729.89 66.0659 1482.56 42.2002 357.451 15.1433 0.99 42575.8 916.292 12885.5 348.721 1564.79 71.5612 1266.77 9.7050 200 1 0.7 51.8951 0.1592 42.1616 0.1490 33.5487 0.1391 20.2075 0.1207 0.8 85.2974 0.2235 66.6005 0.2040 50.4516 0.1855 26.9671 0.1516 0.9 185.026 0.4206 132.354 0.3597 89.3118 0.3039 35.7030 0.2093 0.99 1931.06 3.9945 801.767 2.0537 192.314 0.7878 13.5335 0.0631 5 0.7 265.637 3.9787 215.353 3.4238 170.926 2.9148 102.374 2.0546 0.8 384.132 5.5850 299.383 4.6468 226.286 3.8029 120.372 2.4415 0.9 745.649 10.5089 532.507 8.1126 358.563 6.0534 142.701 3.1030 0.99 7282.62 99.7940 3017.36 48.3717 720.906 16.6721 51.9463 1.2305 10 0.7 956.170 15.9145 774.997 13.5292 614.952 11.3592 368.095 7.7612 0.8 1354.06 22.3397 1055.12 18.3830 797.330 14.8522 423.951 9.2666 0.9 2571.73 42.0342 1836.38 32.1730 1236.35 23.7566 491.943 11.9066 0.99 24647.7 399.164 10211.6 193.054 2439.54 66.2922 174.788 4.9018 250 1 0.7 49.7937 0.1376 42.0037 0.1305 34.9404 0.1235 23.3224 0.1103 0.8 82.8689 0.1937 67.6386 0.1800 54.0985 0.1669 32.8819 0.1424 0.9 181.992 0.3651 138.031 0.3218 100.670 0.2813 48.7098 0.2099 0.99 1954.84 3.4733 927.931 1.9613 303.128 0.8999 18.7292 0.1124 5 0.7 266.658 3.4385 224.513 3.0327 186.357 2.6544 123.821 1.9913 0.8 389.032 4.8396 317.098 4.1467 253.220 3.5115 153.400 2.4393 0.9 761.738 9.1229 577.253 7.3240 420.579 5.7381 203.068 3.3072 0.99 7506.45 86.7781 3562.79 46.1198 1163.49 19.0945 71.3877 2.0149 10 0.7 948.340 13.7537 798.193 11.9834 662.298 10.3470 439.725 7.5317 0.8 1349.17 19.3579 1099.43 16.4008 877.710 13.7119 531.431 9.2588 0.9 2572.53 36.4902 1949.31 29.0336 1420.11 22.5058 685.605 12.6739 0.99 24732.5 347.098 11741.3 184.017 3836.20 75.8862 234.766 7.9755 change of ols estimator to the proposed is about 73%. the jackknife kl estimator performs better than the ridge and the kl estimator[15]. however, when there is both multicollinearity and outlier, the proposed rjkl study dominates other estimators. the estimator with the closest performance is the robust kibria-lukman estimator with about 8% difference. the application result agrees with the theoretical proofs and the simulation. all the estimators produced the same regression sign. 262 jegede et al. / j. nig. soc. phys. sci. 4 (2022) 251–264 263 table 8: estimated mses of the estimators when there are 30% outliers and p=7 n σ̂ ρ α̂ α̂m α̂k α̂m k α̂k l α̂m k l α̂jk l α̂rjk l 50 1 0.7 176.029 25.4476 117.630 20.5623 72.4841 16.2500 24.5439 9.6180 0.8 289.091 40.7835 180.230 31.6595 100.078 23.8026 26.7521 12.4852 0.9 632.508 87.9825 339.479 62.5223 146.664 41.7969 23.2801 16.3585 0.99 6842.66 927.927 1587.39 379.980 250.968 88.9143 789.588 5.9982 5 0.7 1198.50 249.099 799.409 198.493 491.401 154.303 165.777 88.1495 0.8 1738.52 356.258 1082.10 272.434 599.513 201.112 159.759 101.484 0.9 3389.34 676.860 1816.30 472.164 782.782 308.093 124.102 114.791 0.99 33298.5 6511.84 7712.37 2575.63 1228.32 565.161 3889.30 51.3169 10 0.7 4302.62 939.749 2870.54 747.659 1765.13 580.130 596.058 330.089 0.8 6101.25 1325.89 3799.03 1012.04 2106.00 745.378 562.091 374.207 0.9 11603.0 2464.02 6221.37 1714.74 2683.96 1115.34 426.552 412.784 0.99 111319.1 23366.9 25810.1 9187.00 4102.53 1989.79 12934.1 190.624 100 1 0.7 87.9633 0.7618 63.8612 0.6566 44.1094 0.5603 19.2165 0.3977 0.8 144.012 1.0742 98.6666 0.8876 62.8889 0.7205 22.5087 0.4541 0.9 314.994 2.0255 190.263 1.5195 100.081 1.0925 22.3053 0.5107 0.99 3387.24 19.2659 983.693 8.0896 111.755 2.0612 121.875 0.1558 5 0.7 628.585 19.0303 456.143 15.4409 314.869 12.2794 136.945 7.4362 0.8 907.147 26.8310 621.375 20.9103 395.976 15.8243 141.763 8.5224 0.9 1759.23 50.5780 1062.86 36.0667 559.477 24.2995 125.289 9.8913 0.99 17235.4 481.069 5012.56 198.963 576.233 49.5735 624.349 4.4673 10 0.7 2290.08 76.1193 1661.75 61.5544 1146.97 48.7586 498.577 29.2766 0.8 3245.87 107.318 2223.17 83.3977 1416.52 62.8993 506.765 33.6416 0.9 6171.17 202.300 3727.83 143.967 1961.68 96.7536 438.654 39.2026 0.99 59262.7 1924.14 17227.7 795.448 1975.93 198.076 2146.44 17.9635 200 1 0.7 73.1493 0.2996 58.8538 0.2771 46.2797 0.2555 27.1012 0.2157 0.8 120.479 0.4212 93.0287 0.3793 69.4843 0.3398 35.8860 0.2685 0.9 261.752 0.7935 184.570 0.6684 122.085 0.5546 46.3870 0.3661 0.99 2734.52 7.5420 1099.80 3.9736 242.901 1.6104 18.6298 0.1589 5 0.7 375.503 7.4839 301.842 6.4787 237.110 5.5537 138.592 3.9787 0.8 543.927 10.5212 419.764 8.8351 313.348 7.3109 161.745 4.8223 0.9 1057.87 19.8201 745.870 15.5525 493.402 11.8510 187.766 6.4108 0.99 10343.8 188.360 4164.78 96.1097 924.537 36.8939 71.6864 3.4590 10 0.7 1349.72 29.9343 1084.69 25.7389 851.829 21.8968 497.604 15.4259 0.8 1912.79 42.0830 1475.82 35.1269 1101.40 28.8672 568.271 18.7578 0.9 3636.66 79.2775 2563.59 61.9265 1695.46 46.9327 645.146 25.1003 0.99 34893.8 753.406 14047.6 383.982 3119.88 147.128 242.676 13.7813 250 1 0.7 77.3227 0.2738 65.2093 0.2573 54.2173 0.2414 36.1045 0.2114 0.8 128.843 0.3857 105.175 0.3550 84.1090 0.3256 51.0087 0.2711 0.9 283.208 0.7277 214.894 0.6342 156.734 0.5474 75.5029 0.3965 0.99 3040.22 6.9228 1442.93 4.0329 467.134 1.9628 26.1510 0.2998 5 0.7 408.258 6.8396 343.645 6.0789 285.092 5.3668 188.916 4.1066 0.8 597.276 9.6350 486.802 8.3489 388.581 7.1625 234.658 5.1307 0.9 1172.97 18.1740 888.968 14.8722 647.371 11.9286 310.650 7.2851 0.99 11587.6 172.879 5493.24 97.5796 1773.18 45.1740 98.5925 6.3387 10 0.7 1439.39 27.3570 1210.94 24.1521 1004.01 21.1663 664.426 15.9385 0.8 2051.01 38.5383 1670.75 33.1929 1332.80 28.2843 803.719 19.9655 0.9 3915.69 72.6924 2965.88 59.2099 2158.22 47.2351 1033.85 28.5155 0.99 37674.9 691.468 17842.1 389.808 5746.51 180.118 318.295 25.2059 though, the intercept value of ols estimator is the highest. we noticed a sharp reduction in the intercept for other estimators, especially the jackknife kl and the proposed. 6. conclusion we introduced a new robust estimator for the linear regression model in this study and named it the robust jackknife kibria 263 jegede et al. / j. nig. soc. phys. sci. 4 (2022) 251–264 264 table 9: regression coefficients and mses of estimator adopting the hussein data. coefficients α̂ α̂m α̂k α̂m k α̂k l α̂m k l α̂jk l α̂rjk l β0 208.8853 173.3445 178.6308 155.4823 148.3763 137.6202 92.9456 103.4135 β1 0.6130 0.9976 0.8627 1.1450 1.1124 1.2924 1.5698 1.5747 β2 1.2563 1.1153 1.1574 1.0569 1.0584 0.9985 0.8772 0.8867 β3 -1.2213 -1.1159 -1.2635 -1.1408 -1.3058 -1.1658 -1.3832 -1.2135 mses 1850.4816 864.4366 1353.9507 695.6977 935.0263 545.3211 785.1599 500.0968 lukman (rjkl) estimator. the rjkl estimator was proposed to handle outlier and multicollinearity together. the estimator was formed by grafting the m-estimator into the jackknife kibria lukman (jkl) estimator. we presented theorems that state the necessary conditions for the new estimator to perform better than the jkl estimator and other existing robust estimators discussed. we observed a good performance in the simulation study of the new estimator and the real-life data analysis. both results supports the efficiency of the new estimator. references [1] k. ayinde, a. f. lukman & o. arowolo, “robust regression diagnostics of influential observations in linear regression model”, open journal of statistics 05 (2015) 273. 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[32] y. hussein & m. zari, “generalized two stage ridge regression estimator tr for multicollinearity and autocorrelated errors”, canadian journal on science and engineering mathematics 3 (2012) 79. 264 j. nig. soc. phys. sci. 4 (2022) 711 journal of the nigerian society of physical sciences estimation of critical and thermophysical properties of saturated cyclic alkanes by group contributions c. otobrise∗, g. a. orotomah department of chemistry, delta state university, p.m.b. 1, abraka, nigeria. abstract group contribution methods (gcms) of marrero & gani (m & g), constantinou & gani (c & g) and lydersen (lyd) were employed in the prediction of some critical and thermophysical properties(tpps), namely; critical temperature (tc ), critical pressure (pc ), critical volume (vc ), boiling temperature (tb ) and melting temperature (tm ), for various cycloalkanes. the predicted properties were compared with available experimental data. experimental data for tm were unfortunately very scanty in the open literature; no comparison was done to appraise any of the methods. for tc, lyd, c & g and m & g gave average relative deviation (ard) values of 0.02%, 1.57% and 14.64%, respectively. in the case of vc, ard values for lyd, c & g and m & g are 2.6 %, 23.97 %, -4.53 %, respectively. predicted pc values using the methods gave ard of -31.55 %, -1.49 %, -277.98 %, respectively. c & g and m & g recorded ard values of 0.63 % and 10.30 % for tb, respectively. doi:10.46481/jnsps.2022.711 keywords: group contributions, critical properties, thermophysical properties, cycloalkanes. article history : received: 16 march 2022 received in revised form: 17 july 2022 accepted for publication: 27 july 2022 published: 20 august 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: emmanuel etim 1. introduction cycloalkanes have numerous applications. they are widely used in the pharmaceutical industry as organic solvent for the production of drugs. they are largely employed in the petroleum industry for the production of fuels. in modern chemical processes hundreds of thousands of components are used. chemical processing units are designed on the basis of a set of physical and thermodynamic properties of compounds [1]. it is not feasible to measure them as the need arises due thermal instability in most cases, estimation/prediction methods are generally employed in such situations. different esti∗corresponding author tel. no: +2348038930023 email address: otobrisec@delsu.edu.ng ( c. otobrise ) mation methods have been developed over the years to provide data that cannot be sourced readily in open literature [2]. such models are validated by comparing available experimental data with predicted values [3]. predictive methods can replace measurements/experiments if they provide sufficiently good estimations. more chemists now employ computational models to elucidate details of compounds observed in the laboratory [4]. the estimated properties cannot be as precise as experimental measurements, but for many purposes the quality of the estimated properties is sufficient [5]. estimation methods depending on the required input data have been divided into quantity-property-property-relationship (qppr) or quantity-structure-property-relationship (qspr) [6]. qppr methods are input data intensive. qspr methods 1 c. otobrise & g. a. orotomah / j. nig. soc. phys. sci. 4 (2022) 711 2 need only knowledge of the chemical structure of a compound to predict the estimated property [6, 7]. gcms are presented as empirical qspr approaches. they have been found to be very suitable and easy to employ for the prediction of a large number of pure components properties [8], as well as physicochemical properties of thousands of compounds [1]. a lot of gcms have been designed already with varying predictive capabilities. they have been used to predict critical properties [9, 10], state equation parameters [11, 12], acentric factor [2, 9], activity coefficients [13], normal boiling temperature [1, 14], liquid density [10, 15, 16], and flash temperatures [17]. the purpose of this work is to predict with good accuracy, some critical and tpps of saturated cyclic hydrocarbons. such data are significant in the design of process plants. 2. numerical methods the gcms of lyd, c & g and m & g were employed in the determination of tc, pc, vc, tb and tm, for c3 − c13 cycloalkanes. this range of cycloalkanes with or without side chains are 49 common and experimental data can be readily sourced for a good number of them. the rest are large, thermally labile compounds with paucity of experimental data to compare with the results of predictive models. 2.1. the method of lydersen the lyd estimates tc, pc, and vc. the method is the prototype for many new models. it employs structural combinations and the relations are shown in equations (1), (2), (3) below: tc = tb 0.567 + σ∆t − (σ∆t ) 2 (1) pc = m (0.34 + σ∆p) 2 (2) vc = 40 + σ∆v (3) m represents the molar mass of the compound, ∆t , ∆p, ∆v are the group contributions for different groups of atoms in the molecule. tc and tc are in kelvin (k), pc is measured in atmospheres (atm), and vc is in cubic centimeters per gram mole (cm3/gm). contributions of atoms and molecules as stipulated by lyd are accessible [18]. 2.2. the method of constantinou & gani this method makes use of first and second order level group contributions. the second level involves groups that permit a better description of proximity effects and differentiation among isomers. property estimation in this model takes the form of equation (4) below: f (x) = σi nici + wς j m j d j (4) ci in equation (4) is the contribution of the first order group type-i which occurs ni times and d j is the contribution of the second order group typej with m j occurrence in a compound. for tc, f (x) = exp(tc/tc0). hence, for tc equation (4) becomes exp(tc/tc0) = ( σi nitc1i + σ j m jtc2 j ) (5) this can also be expressed as: tc = tc0 ln ( σi nitc1i + σ j m jtc2 j ) (6) tc0 is a constant (universal adjustable parameter) with the value 181.128 k, tc1i represents group contribution of the first order group of type-i, tc2 j represents group contribution of the second order group of typej. for pc f (x) = (pc − pci) −0.5 −pc2. hence, pc takes the form (pc − pci) −0.5 − pc2 = σi ni pc1i + σ j m j pc2 j (7) pc1 and pc2 are universal constants (adjustable parameters) with values 1.3705 bar and 0.100220 82 bar−0.5, respectively, pc1i represents group contribution of the first order of type-i, pc2 j represents group contribution of the second order of typej. for vc, f (x) = vc − vc0. therefore, we can write: vc − vc0 = σi nivc1i + σ j m jvc2 j (8) vc0 is a universal constant (adjustable parameter) with a value of -0.004350 m3/kmol, vc1i represents group contribution of the first order of type-i, vc2 j represents group contribution of the second order of typej. for tm, f (x) = exp (tm/tm0). equation (4) will take the form of equation (9) for tm exp (tm/tm0) = σi nitc1i + σ j m jtc2 j (9) this can be written as: tm = tm0 ln ( σi nitm1i + σ j m jtm2 j ) (10) tm0 is a constant (adjustable parameter) with value 102.425 k. tm1i represents group contribution of first order of type-i, tm2 j represents group contribution of the second order of typej. for tb, f (x) = exp (tb/tb0). this can be expressed as: exp (tb/tb0) = σi nitb1i + σ j m jtb2 j (11) tb = tb0 ln ( σi nitb1i + σ j m jbm2 j ) (12) tb0 is a constant (adjustable parameter) with value 204.359 k. tb1i represents group contribution of first order of type-i, tb2 j represents group contribution of the second order of typej. a table of the contributions for various atoms or groups as proposed by c & g can be readily sourced [2]. 2.3. the method of marrero & gani the above method considers the molecular structure of a compound to be collection of three types of groups. the property estimation method takes the form of equation (13). f (x) = σi nici + ως j m j d j + zσkok ek (13) 2 c. otobrise & g. a. orotomah / j. nig. soc. phys. sci. 4 (2022) 711 3 table 1: critical properties predicted by the gcms cycloalkanes nc tc/k pc/bar vc/cm3/gmol lyd c & g m & g lyd c & g m & g lyd c & g m & g cyclopropane 3 397.04 379.91 320.46 66.14 55.75 611.42 173.5 162.92 176.79 cyclobutane 4 463.25 459.93 386.98 64.48 49.85 346.54 218 217.82 233.07 methylene cyclobutane 5 499.7 498.61 442.24 59.65 42.78 268 254.5 171.26 268.19 spiropentane 5 511.81 480.44 388.3 60.97 58.35 331.72 249 217.79 249.08 cyclopentane 5 514.52 510.83 438.58 59.89 43.85 223.94 262.5 273.58 289.35 cyclohexane 6 553.59 558.22 480.74 54.71 37.74 157.34 307 331.85 345.63 methyl cyclopentane 6 535.76 533.33 469.91 50.94 38.51 160.16 319 228.48 340.64 cycloheptane 7 602.57 578.95 516.38 49.77 38.4 117.19 351.5 378.92 401.91 1,1-dimethyl cyclopentane 7 559.89 555.95 504.53 46.35 35.06 122.51 359 232.8 385.8 cis-1,2 dimethyl cyclopentane 7 565.19 553.33 512.09 43.78 34.89 113.27 375.5 182.98 388.18 trans-1,2 dimethyl cyclopentane 7 553.57 562.24 512.09 43.78 34.82 113.27 375.5 182.98 388.18 trans-1,3 dimethyl 7 553.36 553.33 497.49 43.78 34.11 120.77 375.5 183.38 391.93 cyclopentane ethyl cyclopentane 7 569.54 563.89 508.34 44.22 34.15 120.1 374 284.24 395.21 methyl cyclohexane 7 571.35 575.78 507.13 46.64 33.48 118.96 363.5 286.74 396.92 cyclooctane 8 642.25 603.14 547.26 45.36 30.69 91.12 396 441.73 458.19 1,1-dimethyl cyclohexane 8 598.11 593.9 536.92 42.51 30.69 94.67 403.5 291.06 442.08 cis-1,2-dimethyl cyclohexane 8 601.16 599.02 543.51 40.35 30.5 88.49 420 241.24 444.46 trans-1,2-dimethyl cyclohexane 8 592.19 599.02 543.51 40.35 30.5 88.49 420 241.24 444.46 cis-1,3-dimethyl cyclohexane 8 586.97 591.78 530.82 40.35 29.92 93.52 420 241.64 448.21 trans-1,3 dimethylcyclohexane 8 592.94 591.78 530.82 40.35 29.92 93.52 420 241.64 448.21 cis-1,4-dimethyl cyclohexane 8 593.69 591.78 530.82 40.35 29.92 93.52 420 241.64 448.21 trans-1,4-dimethyl cyclohexane 8 585.47 591.78 530.82 40.35 29.92 93.52 420 241.64 448.21 ethyl cyclohexane 8 602.71 600.36 540.24 40.72 29.95 93.07 418.5 342.5 451.49 propyl cyclopentane 8 0.00 590.03 540.24 38.82 30.52 93.07 429 340 451.49 cis-octahydro-1h9 688.01 640.73 517.9 43.59 42.46 89.62 413.5 386.67 433.93 indene trans-octahydro-1h9 677.04 640.73 517.9 43.59 42.46 89.62 413.5 386.67 433.93 indene butyl cyclopentane 9 0.00 612.86 568.27 34.47 27.46 74.62 484 395.76 507.77 propyl cyclohexane 9 624.61 622.01 568.27 35.98 26.98 74.62 473.5 398.26 507.77 table 2: predicted critical properties continued cycloalkanes nc tc/k pc/bar vc/cm3/gmol lyd c & g m & g lyd c & g m & g lyd c & g m & g 1,1-bicyclopentyl 10 0.00 642.13 571.77 37.01 29.62 72.19 488 446.3 533.19 cisdecahydronaphthalene 10 718.14 650.8 548.59 39.93 36.64 87.42 458 444.93 490.21 transdecahydronaphthalene 10 705.81 650.8 548.59 39.93 36.64 87.42 458 444.93 490.21 butyl cyclohexane 10 646.31 641.34 593.26 32.15 24.96 61.47 482.5 454.02 564.05 1,4-diethyl cyclohexane 10 0.00 634.53 587.52 31.92 24.43 62.49 530 353.16 557.35 1,1-dimethylethyl cyclohexane 10 642.12 629.19 581.52 39.98 24.64 63.03 514.5 296.3 346 decahydro-111 0.00 667.78 567.33 32.74 25.76 60.29 544.5 406.2 584.48 methylnaphthalene decahydro-2methylnaphthalene 11 0.00 667.78 567.33 32.74 25.76 60.29 544.5 406.2 584.48 ethyl octahydro-1hindene 11 0.00 660.72 590.75 31.53 26.21 60.67 555 697.39 582.77 decahydro dimethyl naphthalene 12 0.00 677.58 615.92 29.3 23.38 49.44 601 361.1 632.02 ethyl decahydronaphthalene 12 0.00 682.97 613.54 29.49 23.4 51.17 599.5 461.97 639.05 1, ethyl decahydronaphthalene 12 0.00 682.97 613.54 29.49 23.4 51.17 599.5 461.97 639.05 2, ethyl decahydronaphthalene 12 0.00 682.97 613.54 29.49 23.4 51.17 599.5 461.96 639.05 octahydro (1methylethyl)-1hindene 12 0.00 670.9 608.29 34.43 23.76 51.92 606 358.6 632.35 decahydro (1methylethyl) naphthalene 13 0.00 692 629.48 31.92 21.35 44.52 650.5 416.86 688.63 decahydro-1-(1methylethyl) naphthalene 13 0.00 692 629.48 31.92 21.35 44.52 650.5 416.86 688.63 decahydro-1-propyl naphthalene 13 0.00 682.97 634.28 26.75 21.37 43.94 654.5 517.73 695.33 2 methyl-1,1bicyclohexyl 13 0.00 696.99 633.58 28.56 21.37 43.72 633.5 517.73 697.04 1,1-methylene biscyclohexane 13 0.00 701.84 638.98 28.73 19.61 43.37 632 618.59 702.03 heptyl cyclohexane 13 701.58 689.38 655.24 24.2 26.98 38.68 647.5 621.3 732.89 in equation (13), ci is the contribution of the first order group of type-i that occurs ni times, d j is the contribution of the second order group of typej that occurs m j times, the ek is the contribution of the third order group of type-k that has ok occurrence in a compound. in the first level of estimation, the constants ω and z are assigned zero values because only first order groups are employed. in the second order level, the constants ω and z are assigned unit and zero values, respectively, because only 3 c. otobrise & g. a. orotomah / j. nig. soc. phys. sci. 4 (2022) 711 4 table 3: thermophysical properties predicted by the gcms cycloalkanes nc tm/k tb/k c & g m & g c & g m & g cyclopropane 3 145.79 -41.54 240.37 169.55 cyclobutane 4 133.96 0.88 285.65 233.57 methylene cyclobutane 5 155.49 61.81 322.07 279.07 spiropentane 5 184.01 4.59 291.06 228.28 cyclopentane 5 170.92 33.78 320.75 283.23 cyclohexane 6 200.95 60.67 356.78 323.81 methyl cyclopentane 6 168.87 54.36 343.67 308.2 cycloheptane 7 218.76 83.4 391.93 358.11 1,1-dimethyl cyclopentane 7 198.67 130.45 363.88 334.1 cis-1,2 dimethyl cyclopentane 7 175.27 92.54 378.87 346.04 trans-1,2 dimethyl cyclopentane 7 175.27 92.54 378.87 346.04 trans-1,3 dimethyl cyclopentane 7 166.79 72.41 364.27 330.65 ethyl cyclopentane 7 185.63 67.43 376.04 344.97 methyl cyclohexane 7 199.43 78.01 376.16 344.81 cyclooctane 8 204.96 103.09 408.47 387.83 1,1-dimethyl cyclohexane 8 222.33 145.01 393.53 366.97 cis-1,2-dimethyl cyclohexane 8 204.22 111.12 406.55 377.3 trans-1,2-dimethyl cyclohexane 8 204.22 111.12 406.55 377.3 cis-1,3-dimethyl cyclohexane 8 197.89 93.52 393.87 363.99 trans-1,3 dimethyl-cyclohexane 8 197.89 93.52 393.87 363.99 cis-1,4-dimethyl cyclohexane 8 197.89 93.52 393.87 363.99 trans-1,4-dimethyl cyclohexane 8 197.89 93.52 393.87 363.99 ethyl cyclohexane 8 212.12 89.21 404.08 376.37 propyl cyclopentane 8 190.05 89.21 403.98 376.37 cis-octahydro-1h-indene 9 254.65 87.62 408.76 352.06 trans-octahydro-1h-indene 9 254.65 87.62 408.76 352.06 butyl cyclopentane 9 212.66 108.19 428.55 403.89 propyl cyclohexane 9 223.42 108.19 428.65 403.89 table 4: predicted thermophysical properties continued cycloalkanes nc tm/k tb/k c & g m & g c & g m & g 1,1-bicyclopentyl 10 230.3 34.78 448.35 409.52 cis-decahydronaphthalene 10 269.01 106.79 432.89 382.54 trans-decahydronaphthalene 10 269.01 106.79 432.89 382.54 butyl cyclohexane 10 233.59 125 450.57 428.37 1,4-diethyl cyclohexane 10 222.2 113.11 442.47 418.87 1,1-dimethylethyl cyclohexane 10 210.17 155.69 433.76 411.37 decahydro-1-methylnaphthalene 11 246.85 55.22 461.11 424.02 decahydro-2-methylnaphthalene 11 246.85 55.22 461.11 424.02 ethyl octahydro-1h-indene 11 238.38 42.8 461.02 424.14 decahydro dimethyl naphthalene 12 245.88 93.21 468.97 447.27 ethyl decahydronaphthalene 12 255.02 68.22 479.95 446.6 1, ethyl decahydronaphthalene 12 255.02 68.22 479.95 446.6 2, ethyl decahydronaphthalene 12 255.02 68.22 479.95 446.6 octahydro (1-methylethyl)-1h-indene 12 237.33 50.41 472.88 437.85 decahydro (1-methylethyl) naphthalene 13 254.12 74.65 490.79 459.03 decahydro-1-(1-methylethyl) naphthalene 13 254.12 74.65 490.79 459.03 decahydro-1-propyl naphthalene 13 262.59 89.89 497.21 466.99 2 methyl-1,1-bicyclohexyl 13 262.59 99.02 497.21 466.9 1,1-methylene biscyclohexane 13 270.4 103.71 503.43 474.68 heptyl cyclohexane 13 259.15 166.29 505 488.9 4 c. otobrise & g. a. orotomah / j. nig. soc. phys. sci. 4 (2022) 711 5 table 5: experimental data for the various properties used in the comparison [19]. cycloalkanes nc tc/k pc/bar vc/cm3/gmol tb/k (m) g/mol cyclopropane 3 397.91 55.77 162.8 240.37 42.081 cyclobutane 4 459.93 49.85 210 285.66 56.107 methylene cyclobutane 5 na na na na 68.118 spiropentane 5 499.74 52.13 236.5 312.19 68.118 cyclopentane 5 511.76 45.02 258.3 322.4 70.134 cyclohexane 6 553.54 40.75 307.9 353.87 84.161 methyl cyclopentane 6 532.79 37.85 318.9 344.96 84.162 cycloheptane 7 604.32 38.4 359 391.94 98.188 1,1-dimethyl cyclopentane 7 547 34.45 360 361 98.188 cis-1,2 dimethyl cyclopentane 7 565.15 34.45 370 372.68 98.188 trans-1,2 dimethyl cyclopentane 7 553.15 34.45 360 365.02 98.188 trans-1,3 dimethyl cyclopentane 7 553 34.45 360 364.88 98.188 ethyl cyclopentane 7 569.52 33.98 374.5 376.62 98.189 methyl cyclohexane 7 572.19 34.71 368 374.08 98.186 cyclooctane 8 647.2 35.5 410 424.3 112.21 1,1-dimethyl cyclohexane 8 591.15 29.38 450 392.7 112.22 cis-1,2-dimethyl cyclohexane 8 606.15 29.38 460 402.94 112.21 trans-1,2-dimethyl cyclohexane 8 596.15 29.38 460 396.58 112.21 cis-1,3-dimethyl cyclohexane 8 591.15 29.38 450 393.24 112.21 trans-1,3 dimethylcyclohexane 8 598 29.38 460 397.61 112.21 cis-1,4-dimethyl cyclohexane 8 598.15 29.38 460 397.47 112.21 trans-1,4-dimethyl cyclohexane 8 590.15 29.38 450 392.51 112.21 ethyl cyclohexane 8 609.15 30.4 450 404.95 112.21 propyl cyclopentane 8 na na na na 112.21 cis-octahydro-1hindene 9 na na na na 124.22 na = not available. the first and the second order groups are involved while the third level both ω and z are set to unity values. the left hand side of equation (13) is a simple function f (x) of the target property “x”. for tc, f (x) = exp (tc/tc0). hence, exp (tc/tc0) = ( σi nitc1i + σ j m jtc2 j + σkoktc3k ) (14) tc = tc0 ln ( σi nitc1i + σ j m jtc2 j + σkoktc3k ) . (15) tc0 is a constant (adjustable parameter) with value 231.239 k. tc1i represents group contribution of first order of type-i, tc2 j represents group contribution of the second order of typej, tc3k represents group contribution of third order of type-k. for pc, f (x) = (pc − pc1) −0.5 − pc2. therefore (pc − pc1) −0.5 −pc2 = ( σi ni pc1i + σ j m j pc2 j + σkok pc3k ) (16) pc = 1/ ( σi ni pc1i + σ j m j pc2 j + σkok pc3k + pc2 ) +pc1(17) figure 1: plot of experimental tc versus tcs obtained from the gcms. pc1 and pc2 are both universal constants (adjustable parameters) with values 5.9827 bar and 0.108998 bar-0.5, respectively. pc1i represents group contribution of first order of type-i, pc2 j represents group contribution of second order of typej and pc3k 5 c. otobrise & g. a. orotomah / j. nig. soc. phys. sci. 4 (2022) 711 6 table 6: experimental properties used in the comparison continued [19]. cycloalkanes nc tc/k pc/bar vc/cm3/gmol tb/k (m) g/mol trans-octahydro-1hindene 9 na na na na 124.22 butyl cyclopentane 9 625.05 27.64 480.5 429.76 126.24 propyl cyclohexane 9 639.15 28.87 477 429.9 126.24 1,1-bicyclopentyl 10 na na na na 138.25 cisdecahydronaphthalene 10 702.25 32.42 480 468.97 138.25 transdecahydronaphthalene 10 687.05 28.37 480 460.46 138.25 butyl cyclohexane 10 667 25.7 574 454.13 140.27 1,4-diethyl cyclohexane 10 na na na na 140.27 1,1-dimethylethyl cyclohexane 10 na na na na 152.28 decahydro-1methylnaphthalene 11 na na na na 152.28 decahydro-2methylnaphthalene 11 na na na na 152.28 ethyl octahydro-1hindene 11 na na na na 152.28 decahydro dimethyl naphthalene 12 na na na na 166.3 ethyl decahydronaphthalene 12 na na na na 166.3 1, ethyl decahydronaphthalene 12 na na na na 166.3 2, ethyl decahydronaphthalene 12 na na na na 166.3 octahydro (1methylethyl)-1hindene decahydro (1-methylethyl) 12 na na na na 204.35 naphthalene decahydro1-(1-methylethyl) 13 na na na na 208.38 naphthalene 13 na na na na 208.38 decahydro-1-propyl naphthalene 13 na na na na 208.38 2 methyl-1,1bicyclohexyl 13 na na na na 180.33 1,1-methylene biscyclohexane 13 na na na na 180.33 heptyl cyclohexane 13 na na na na 182.35 na = not available. figure 2: plot of experimental pc versus pcs obtained from the gcms. figure 3: plot of experimental vc versus vcs obtained from the gcms. 6 c. otobrise & g. a. orotomah / j. nig. soc. phys. sci. 4 (2022) 711 7 table 7: deviations of predicted properties from experimental data cycloalkanes nc tc (% dev.) pc (% dev.) vc (% dev.) tb (% dev.) lyd c & g m & g lyd c & g m & g lyd c & g m & g c & g m & g cyclopropane 3 0.22 4.52 24.17 0.04 -6.57 -0.07 -8.59 0.00 29.4 cyclobutane 4 0.00 18.85 -18.59 0.00 -996 3 -3.81 -3.72 0.00 6 18.2 methylene cyclobutane 5 -0.72 -29.35 3 595. -1 -10.99 -3 spiropentane 5 3.86 28.7 -7 -5.29 7.91 -5.32 6.77 26.8 cyclopentane 5 -2.41 0.18 16.69 -16.96 11.93 2.6 -536 3 -1.63 -5.92 0.51 12.1 8 cyclohexane 6 -0.54 -0.85 15.14 -33.03 7.39 -397.4 4 0.29 -7.78 -12.02 -0.82 5 8.50 methyl cyclopentane 6 -0.01 -0.1 13.38 -34.58 -1.74 286. 3 1 -0.03 28.35 -6.82 12.25 0.37 10.6 cycloheptane 7 0.56 0.29 4.2 17.03 -34.58 0.00 323. 2 1 2.09 -5.55 0.00 8.63 6 1,1-dimethyl cyclopentane 7 -1.64 8.42 -29.61 -1.77 5 205. 1 0.28 35.33 -7.17 11.95 -0.88 7.45 cis-1,2 dimethyl 7 -2.36 2.09 10.36 -34.54 -1.28 255. 7 6 -1.49 50.55 -4.91 -1.66 7.15 cyclopentane trans-1,2 dimethyl 7 -0.01 -1.64 8.02 -27.08 -1.07 -228.2 8 -4.31 49.17 -7.83 -3.79 5.2 trans-1,3 cyclopentane dimethyl 7 -0.08 -0.06 11.16 -27.08 0.99 1 228. 8 -4.31 49.06 -8.87 0.17 9.38 cyclopentane ethyl cyclopentane 7 0.06 0.00 0.99 12.03 -27.08 -0.5 1 250. 5 0.13 24.1 -5.53 0.15 8.4 methyl cyclohexane 7 0.15 -0.63 12.83 -30.14 3.54 253. 8 4 1.22 22.08 -7.86 -0.56 7.83 cyclooctane 8 0.77 6.88 18.26 -34.37 13.55 5 242. 7 3.41 -7.74 3.73 8.6 1,1-dimethyl cyclohexane 8 -0.47 10.1 -27.77 -4.46 156. 2 6 10.33 35.32 11.75 1.76 -0.21 6.55 cis-1,2-dimethyl 8 0.18 0.82 1.18 11.53 -44.69 -3.81 222. 8 2 8.70 47.56 3.38 -0.9 6.36 cyclohexane trans-1,2-dimethyl 8 0.66 -0.48 9.69 -37.34 -3.81 201. 4 2 8.70 47.56 3.38 -2.51 4.86 cyclohexane cis-1,3-dimethyl 8 0.71 -0.11 11.37 -37.34 -1.84 -201 2 6.67 46.3 0.4 -0.16 7.44 cyclohexane trans-1,3 dimethyl8 0.85 1.04 12.66 -37.34 -1.84 -218 3 8.70 47.47 2.56 0.94 8.45 cyclohexane cis-1,4-dimethyl 8 0.75 1.06 12.68 -37.34 -1.84 -218.1 3 8.70 47.47 2.56 0.91 8.42 cyclohexane trans-1,4-dimethyl 8 0.79 -0.28 11.18 -37.34 -1.84 -218.1 3 6.67 46.3 0.4 -0.35 7.27 cyclohexane ethyl cyclohexane 8 1.06 1.44 12.76 -37.34 1.48 1 218. 3 7.00 23.89 -0.33 0.21 7.06 propyl cyclopentane 8 -39.95 206. 1 -1 cis-octahydro-1h-indene 9 -6 trans-octahydro-1h-indene 9 butyl cyclopentane 9 1.95 9.99 0.65 -0.73 17.64 -5.68 0.28 6.02 propyl cyclohexane 9 2.28 2.68 12.47 -24.71 6.55 -169 6 0.73 16.51 -6.45 0.29 6.05 1,1-bicyclopentyl 10 -24.63 4 158. -4 cis-decahydronaphthalene 10 7.33 28.01 -7 4.58 7.31 -2.13 7.69 18.4 trans-decahydronaphthalene 10 -2.26 5.28 25.24 -23.16 -13.02 -169 6 4.58 7.31 -2.13 5.99 16.9 3 butyl cyclohexane 10 2.73 3.10 3.85 12.43 -25.1 40.75 2.88 29.15 208. 4 1 15.94 20.9 1.73 0.78 5.67 2 figure 4: plot of experimental tb versus tbs obtained from the gcms. represents group contribution of third order of type-k. for vc, f (x) = vc − vc0. consequently, vc − vc0 = ( σi nivc1i + σ j m jvc2 j + σkokvc3k ) (18) vc = 1/ ( σi nivc1i + σ j m jvc2 j + σkokvc3k ) + vc0 (19) vc0 is a universal constant (adjustable parameter) with a value 7.95 cm3/mol, vc1i represents group contribution of first order of type-i, vc2 j represents group contribution of second order of typej and vc3k represents group contribution of third order of type-k. for tm, f (x) = exp (tm/tm0). therefore, exp (tm/tm0) = ( σi nitm1i + σ j m jtm2 j + σkoktm3k ) (20) tm = tm0 ln ( σi nitm1i + σ j m jtm2 j + σkoktm3k ) . (21) tm0 is a constant (adjustable parameter) with value 147.450 k. tm1i represents group contribution of first order of type-i, tm2 j represents group contribution of the second order of typej, tm3k represents group contribution of third order of type-k. for tb, f (x) = exp (tb/tb0). hence, exp (tb/tb0) = ( σi nitb1i + σ j m jtb2 j + σkoktb3k ) (22) tb = tb0 ln ( σi nitb1i + σ j m jtb2 j + σkoktb3k ) . (23) tb0 is a constant (adjustable parameter) with value 222.543 k. tb1i represents group contribution of first order of type-i, tb2 j represents group contribution of the second order of typej, tb3k represents group contribution of third order of type-k. a table of the contributions for various atoms or groups as proposed by m & g is readily available [1]. 3. results and discussion three gcms were employed in the prediction of the critical properties. tpps were estimated by c & g and m & g. tables 1 and 2 show the predicted tc, pc and vc values. tables 3 and 4 contain predicted tb and tm for the cycloalkanes. experimental data for the various tpps used in the comparison are presented on tables 5 and 6. lyd required molecular weight (m) and 7 c. otobrise & g. a. orotomah / j. nig. soc. phys. sci. 4 (2022) 711 8 t b as input parameters, they are also shown on tables 5 and 6 where experimental tb was not found in the literature for a particular cycloalkane, tb predicted by the method of c & g was utilized. deviations of the predicted properties from available experimental data were calculated using the equations below: deviation% = exp (data) − pred (data) exp (data) × 100. average relative deviation = σdeviation n table 7 shows the deviations of predicted critical properties and tpps from available experimental data. the experimental data were obtained from the handbook of chemical compound data for process safety [19]. lyd provided better results for tc prediction with average relative deviation of 0.02 %. m & g with ard of 14.64 % under predicted tc for all the compounds. c & g with ard of 1.57 %, like lydersen’s method, gave results comparable with experimental data. figure 1 is a comparative plot of experimental tc and the tcs obtained from the gcms. as the number of carbon atoms increased, the predicted pc decreased for the three methods. this trend is typical of organic compounds. lyd and m & g predicted pc values that were similar. the latter with ard of −1.49 % gave a better prediction of pc. c & g with an ard of −31.55 % under predicted pc as shown on figure 2. c & g under predicted vc for most of the compounds as can be seen on figure 3. as the number of carbon atoms increased the accuracy of the predicted vc by lyd declined. lyd and m & g proved more efficient in this particular critical property as they gave values close to available experimental data with ard of 2.6 % and −4.35 %, respectively. ard for c & g was 23.97 %. the method of c & g with ard of 0.63 % proved a more reliable method for the prediction of tb. it was able to differentiate between structural isomers having different boiling points in most cases. the method of m & g slightly under predicted tb with ard of 10.30 % as shown on figure 4. 4. conclusion three gcms were employed in the prediction of some critical and tpps, namely; tc, pc, vc, tb and tm, for cycloalkanes. the predicted properties were compared with available experimental data. lyd provided better results for tc prediction with average relative deviation of 0.02 %. c & g predicted pc values that were similar to values obtained experimentally. with ard of -1.49 %, it gave a better prediction of pc. lyd proved more efficient in the prediction of vc with ard of 2.6 %. c & g with ard of 0.63 % proved a more reliable method for the prediction of tb. references [1] j. marrero & r. gani, “group-contribution based estimation of pure component properties”, fluid phase equilibria 183 (2001) 183. 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[19] c.l. yaws, handbook of chemical compound data for process safety, united states of america, gulf publishing company, houston, texas (1997). 8 j. nig. soc. phys. sci. 4 (2022) 713 journal of the nigerian society of physical sciences combating the multicollinearity in bell regression model: simulation and application g. a. shewaa,∗, f. i. ugwuowob adepartment of mathematical sciences, taraba state university, jalingo/taraba, nigeria bdepartment of statistics, university of nigeria, nsukka/enugu, nigeria abstract poisson regression model has been popularly used to model count data. however, over-dispersion is a threat to the performance of the poisson regression model. the bell regression model (brm) is an alternative means of modelling count data with over-dispersion. conventionally, the parameters in brm is popularly estimated using the method of maximum likelihood (mml). multicollinearity posed challenge on the efficiency of mml. in this study, we developed a new estimator to overcome the problem of multicollinearity. the theoretical, simulation and application results were in favor of this new method. doi:10.46481/jnsps.2022.713 keywords: bell regression, liu, multicollinearity, poisson regression, ridge article history : received: 17 march 2022 received in revised form: 07 june 2022 accepted for publication: 08 june 2022 published: 15 august 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: t. latunde 1. introduction regression modeling is crucial in describing the outcome (response) variable of interest as a function of predictor variable(s). the outcome variable usually assumed to follow a normal distribution in the linear regression model. however, in practice this may not hold. the generalized linear model (glm) is employed when the response variable fails to follow a normal distribution. glm is the generalization of the linear regression model that allow the linear model to be related to the response variable via a link function. examples include the poisson regression, negative binomial, the bell regression, the beta regression models and others. it becomes inappropriate ∗corresponding author tel. no: email address: gladysshewa@yahoo.com (g. a. shewa ) to adopt the linear regression model when the response variable is a count data. example of count data includes the number of thunderstorms occurrences, the number of accidents, the number of insurance claims, the number of species in a habitat among others. the poisson regression model is popularly employed to model the count data. however, the major drawback of the model is that the model restricts the variance to be equal to the mean and when the variance exceeds the mean [1]. the bell regression model was introduced as an alternative to the poisson regression model to model count data with over-dispersion [1]. the properties of the bell regression model were discussed in detail by [1]. the parameters of the model is determined using the method of maximum likelihood (mml), but the efficiency of mml suffers drawback in the presence of multicollinearity. the ridge and the liu estimators were proposed for the 1 shewa & ugwuowo / j. nig. soc. phys. sci. 4 (2022) 713 2 parameter estimation of the bell regression model with multicollinearity [2,3]. the main objective given in this study is to propose a new estimator to account for multicollinearity in the bell regression model, derive its property. we illustrate the proposed estimator using a real life data and compare with the popular poisson regression model. the rest of this article is organized as follows. in section 2, we introduce the bell regression model and the parameter estimation. also, we discuss the new method of estimation and its property. the simulation and the practical illustration are in section 3 and 4, respectively. the concluding remark is given in the last section. 2. existing estimators in bell regression model assuming the probability distribution of the response variable yi is as follows: f (y) = θye1−e θ by y! , y = 0, 1, 2, . . . , (1) where θ > 0 and by = (1/e) ∑ ∞ d=0(d y/d!) denotes the bell numbers [1,4,5]. the bell distribution in (1) has the following properties: e (y) = θeθ, (2) v ar (y) = θ (1 + θ) eθ, (3) the model is expressed as a function of the mean response. assume there exists a function, ϕ = θeθ and θ = wo (ϕ) , where wo represent the lambert function [1]. therefore, equation (1) can be re-parameterize as follows: f (y) = e1−e wo (ϕ) wo (ϕ) y by y! , y = 0, 1, 2, . . . , (4) where ϕ > 0 is the mean response. consequently, e (y) = ϕ, (5) v ar (y) = ϕ (1 + wo (ϕ)) , (6) the probability mass function (pmf) in equation (4) is an example of the one-parameter exponential family. the bell distribution is fit for modelling over-dispersed data because v ar (y) > e (y) . assume yi follows a bell distribution with mean ϕi, yi ∼ bell (wo (ϕi)) , such that g(ϕi) = ηi = x t i β, i = 1, 2, . . . n, (7) where β = ( β1,β2, . . . ,βp )t is the vector of the regression parameters, ηi is the linear predictor, and xti = ( xi1, xi2, . . . , xip ) denotes the p-known predictors. the bell regression model (brm) can be modeled by assuming that ϕi = ex t i βee xti β and ln(ϕi) = xti βe xti β as yi ∼ bell (ln (ϕi)) . the log-likelihood function becomes l (β,ϕi) = n∑ i=1 yilog ( ex t i βee xti β ) + n∑ i=1 ( 1 − ee xti βee xti β ) + logby − log  n∏ i=1 yi!  (8) thus, the method of maximum likelihood (mml) is obtained by equating the first derivative of equation (8) to zero. the first derivative of equation (8) cannot be solved analytically since it is nonlinear in β. so, β is obtained iteratively using the fisherscoring algorithm [6] defined as follows: β(r+1) = β(r) + i−1β(r)s (β(r)) (9) where i−1 (β) = ( −e ( ∂2l (β,ϕi) /∂β∂βt ))−1 . consequently, the mml of β is defined as β̂m ml = h −1 xt ŵû (10) where h = xt ŵ x and û = log ( ϕ̂i ) + yi−ϕ̂i√ var(ϕ̂i ) , and ŵ = diag [ (∂ϕi/∂ηi) 2/v (yi) ] . the asymptotic covariance matrix is given by: cov(̂βm ml) = ( xt ŵ x )−1 . (11) multicollinearity threatens the efficiency of the mml. multicollinearity occurs when the predictors are correlated and makes the mml estimate unstable. also, multicollinearity inflates the covariance matrix of mml [7-9]. [10] developed the ridge estimator for the linear regression model while [2] developed the bell ridge regression model and defined it as follows: β̂k−b = (h + ki) −1 xt ŵû, (12) where the tuning parameter k>0. the bias, variance and matrix mean squared error (msem) of bell ridge estimator are shown as follows: bias (̂ βk−b ) = − kq (b + ki)−1γ (13) variance (̂ βk−b ) =q (h + ki)−1h(h + ki)−1 qt , (14) msem (̂ βk−b ) =q (h + ki)−1h(h + ki)−1 qt + k2(h + ki)−2γγt , (15) where q denotes the eigenvectors of the xt ŵx matrix, and γ = qt β. [11] developed the liu estimator for linear regression model while [3] developed the bell liu estimator and defined as follows: β̂d−b = (h + i) −1(h + di)̂βm ml. (16) 2 shewa & ugwuowo / j. nig. soc. phys. sci. 4 (2022) 713 3 where the tuning parameter d > 0. the bias, variance and matrix mean squared error (msem) of bell liu estimator are as follows: bias (̂ βd−b ) = −(1 − d)q (h + i)−1γ (17) variance (̂ βd−b ) = q (h + i)−1 (h + di) h−1 (h + di) (h + i)−1 qt , (18) msem (̂ βd−b ) = q (h + i)−1 (h + di) h−1 (h + di) (h + i)−1 qt + (1 − d)2(h + i)−2ββt , (19) where q denotes the eigenvectors of the xt ŵx matrix. let qt xt ŵ xq = e = diag ( e1, ..., ep ) , e1 ≥ e2 ≥ ... ≥ ep, e is the matrix of eigenvalues of xt ŵ xand q is the matrix whose columns are the eigenvectors of xt ŵ x.thus, the canonical model can be expressed in terms of m = xq,γ = qt βand mt ŵ m = e. consequently, the mml in equation (10) can be re-written as γ̂m ml = e −1 mt ŵû, (20) cov(̂γm ml) = e −1. (21) thus, the scalar mean squared error (smse) is as follows: s ms e(̂γm ml) = p∑ j=1 e−1j , (22) the ridge estimator in canonical form is as follows: γ̂k−b = (e + ki) −1 mt ŵû, (23) the msem and smse of the ridge estimator in canonical form is calculated as msem (̂ γk−b ) = q (e + ki)−1 e(e + ki)−1 qt + k2(e + ki)−2γγt , (24) s ms e (̂ γk−b ) = p∑ j=1 e j( e j + k )2 +k2 p∑ j=1 γ2j( e j + k )2 (25) the liu estimator and its msem and smse in canonical form are given by: γ̂d−b = (e + i) −1(e + di)̂γm ml. (26) msem (̂ γd−b ) = q (e + i)−1 (e + di) e−1 (e + di) (e + i)−1 qt + (1 − d)2(e + i)−2γγt , (27) s ms e (̂ γd−b ) = p∑ j=1 ( e j + d )2 e j ( e j + 1 )2 + (1 − d)2 p∑ j=1 γ2j( e j + 1 )2 (28) 2.1. the proposed estimator the kibria-lukman estimator for the linear regression model is defined as follows: γ̂k l = ( xt x + ki )−1 (xt x − ki)̂γm ml. (29) hence, the kibria-lukman estimator for the bell regression model will be as follows: γ̂k l = (e + ki) −1(e − ki)̂γm ml. (30) msem (̂ γkl−b ) = q (e + ki)−1 (e − ki) e−1 (e − ki) (e + ki)−1 qt + (2k)2(e + ki)−2γγt , (31) s ms e (̂ γkl−b ) = p∑ j=1 ( e j − k )2 e j ( e j + k )2 +(2k)2 p∑ j=1 γ2j( e j + k )2 (32) also, [12] developed the modified ridge estimator for the linear regression model which is given by: γ̂mrt = ( xt x + k(1 + d)i )−1 xt y (33) the proposed estimator in this study is motivated by replacing γ̂m ml in equation (30) with γ̂mrt . hence, the modified kibrialukman estimator can be defined as follows: γ̂kd−b = (x t x − ki) ( xt x + ki )−1( xt x + k(1 + d)i )−1 xt ŵû, k = 0. (34) the proposed can be written in canonical form as follows: γ̂kd−b = (e − ki) (e + ki) −1(e + k(1 + d)i)−1 mt ŵû, k = 0. (35) the statistical properties of γ̂kd−b are as follows: bias (̂ γkd−b ) = −k ((3e + de) + k(1 + d)) (e + ki)−1(e + k(1 + d)i)−1γ (36) v ar (̂ γkd−b ) = (e − ki)2 (e + ki) −2 (e + k(1 + d)i)−2 (37) ms e m (̂ γkd−b ) = (e + ki)−2(e − ki)2 (e + k(1 + d)i) −2 + k2((3 + d) e + k(1 + d))2 (e + ki) −2 (e + k(1 + d)i)−2γt γ (38) s ms e (̂ γkd−b ) = p∑ j=1 ( λ j − k )2 ( λ j + k )2( λ j + k(1 + d) )2 + k2 p∑ j=1 ( 3λ j + λ jd + k(1 + d) )2 γ2j( λ j + k )2( λ j + k(1 + d) )2 (39) 3 shewa & ugwuowo / j. nig. soc. phys. sci. 4 (2022) 713 4 2.2. theoretical comparison based on msem and mse lemma 2.1. given a positive definite (p.d) matrix m*, and θ∗ be some vector, then m∗ − θ∗θ∗t ≥ 0 if and only if θ∗ ′ m∗−1θ∗ ≤ 1[13]. lemma 2.2. θ̂1 = c1y and θ̂2 = c2y are two estimators of θ with covariance matrix cov(̂θ1) and cov (̂ θ2 ) , respectively. suppose that cov(̂θ1) > cov (̂ θ2 ) , and the bias fi = (ci x − i)θ, i = 1, 2 then, msem (̂ θ1 ) −msem (̂ θ2 ) >0 if and only if f ′t2 [ φv + f1 f t1 1 ]−1 f2 < 1 where ms e m ( θ̂i ) = cov ( θ̂i ) + fi f ti [14]. theorem 1. under the bell regression model, if k>0 and d>0, then the proposed estimator γ̂kd−b is preferred to γ̂k−b if and only if, γt b [ (e + ki)−1 e(e + ki)−1 + k2(e + ki)−2γγt − (e + ki)−2(e − ki)2(e + k(1 + d)i)−2 ]−1 bt γ < 1 where b = −k ((3e + de) + k(1 + d)) (e + ki)−1(e + k(1 + d)i)−1. proof: we show that the difference in the bias is positive definite. bias(̂γk−b) − bias ( γ̂kd−b ) = − k(e + ki)−1γ+k ((3e + de) + k(1 + d)) (e + ki)−1(e + k(1 + d)i)−1γ = −kdiag { 1 e j + k − 3e j + de j + k (1 + d) (e j + k)(e j + k (1 + d)) } p j=1 γ (40) −k [( e j + k (1 + d) ) − ( 3e j + de j + k (1 + d) )] γ = k(2e j + e jd)γ > 0. consequently, bias (̂γk−b)-bias ( γ̂kd−b ) > 0. we show that the variance difference of the two estimators is positive definite (pd). var(̂γk−b) − var ( γ̂kd−b ) = (e + ki)−2 e(e + ki)−2 − (e + ki)−2(e − ki)2(e + k(1 + d)i)−2 = diag { e j (e j + k)2 − (e j − k)2 (e j + k)2(e j + k(1 + d))2 } p j=1 (41) (e + ki)−2 e(e + ki)−2-(e + ki)−2(e − ki)2(e + k(1 + d)i)−2 is pd since e j ( e j + k(1 + d) )2 − ( e j − k )2 > 0 for k,d>0. consequently, the proposed estimator is preferred. theorem 2. under the bell regression model, if k>0 and d>0, then the proposed estimator γ̂kd−b is preferred to γ̂d−b if and only if γt b [ (e + i)−2 e−1(e + di)2 + (1 − d)2(e + i)−2γγt − (e + ki)−2(e − ki)2(e + k(1 + d)i)−2 ]−1 bt γ < 1 where b = −k ((3e + de) + k(1 + d)) (e + ki)−1(e + k(1 + d)i)−1. proof: we show that the difference in the bias is positive definite. bias(̂γk−b)-bias ( γ̂kd−b ) = −(1 + d)(e + i)−1γ+k ((3e + de) + k(1 + d)) (e + ki)−1(e + k(1 + d)i)−1γ = −kdiag { 1 + d e j + 1 − 3e j + de j + k (1 + d) (e j + k)(e j + k (1 + d)) } p j=1 γ (42) − (1 + d) (e + i)−1γ+k ((3e + de) + k (1 + d)) (e + ki)−1(e + k (1 + d) i)−1γ = k (1 + d) [ k + e jd + dk − 1 ] +e j(k+dk−3−d) > 0 for k,d>0 consequently, bias (̂γd−b)-bias ( γ̂kd−b ) > 0. we show that the variance difference of the two estimators is positive definite (pd). var(̂γk−b)-var ( γ̂kd−b ) =(e + i)−2(e + di)2-(e + ki)−2(e − ki)2(e + k(1 + d)i)−2 = diag  ( e j + d )2 e j ( e j + 1 )2 − (e j − k)2(e j + k)2(e j + k(1 + d))2  p j=1 (43) (e + ki)−2 e(e + ki)−2-(e + ki)−2(e − ki)2(e + k(1 + d)i)−2 is pd since ( e j + d )2( e j + k (1 + d) )2 (e j +k)2 > e j ( e j + 1 )2 ( e j − k )2 > 0 for k,d>0. 4 shewa & ugwuowo / j. nig. soc. phys. sci. 4 (2022) 713 5 consequently, the proposed estimator is preferred. theorem 3. under the bell regression model, if k>0 and d>0, then the proposed estimator γ̂kd−b is preferred to γ̂kl−b if and only if, γt b [ (e + ki)−1 (e − ki) e−1 (e − ki) (e + ki)−1 + (2k)2(e + ki)−2γγt − (e + ki)−2(e − ki)2(e + k(1 + d)i)−2 ]−1 bt γ < 1 where b = −k ((3e + de) + k(1 + d)) (e + ki)−1(e + k(1 + d)i)−1. proof: we show that the difference in the bias is positive definite. bias(̂γkl−b) − bias ( γ̂kd−b ) = −2k(e + i)−1γ+k ((3e + de) + k(1 + d)) (e + ki)−1(e + k(1 + d)i)−1γ = −kdiag { 2 e j + k − 3e j + de j + k (1 + d) (e j + k)(e j + k (1 + d)) } p j=1 γ (44) −2k(e + ki)−1γ+k ((3e + de) + k (1 + d)) (e + ki)−1(e + k (1 + d) i)−1γ > 0 for k,d>0 consequently, bias (̂γkl−b)-bias ( γ̂kd−b ) > 0. we show that the variance difference of the two estimators is positive definite (pd). var(̂γk−b)-var ( γ̂kd−b ) =(e − ki)2 e−1(e + ki)−2-(e + ki)−2(e − ki)2(e + k(1 + d)i)−2 = diag  ( e j − k )2 e j ( e j + k )2 − (e j − k)2(e j + k)2(e j + k(1 + d))2  p j=1 (45) (e − ki)2 e−1(e + ki)−2-(e + ki)−2(e − ki)2(e + k(1 + d)i)−2 is pd since ( e j − k )2( e j + k (1 + d) )2 (e j + k)2 > e j ( e j + k )2 ( e j − k )2 > 0 for k,d>0. consequently, the proposed estimator is preferred. 2.3. estimation of shrinkage parameters k and d the shrinkage parameter for the proposed is obtained by differentiating its mean squared error. we adopted the maple software for the simplification and simplest form of the calculated shrinkage parameter. hence, the ridge parameter k is defined as k̂ =  p∏ j=1 ( d + 1 + √ 2d2 + 6d + 4 ) e j 1 + d  1/p (46) d̂ = min  γ̂2j1 e j + γ̂2j  [15] (47) where e j is the eigenvalue of m ′ ŵ m. k̂ in (46) produced optimum performance for the proposed and kibria-lukman estimator. also, d in (47) is adopted for the proposed and the liu estimator. the shrinkage parameter for the ridge estimator is defined as: k̂ = 1 max(̂γ2j ) (48) 3. simulation study in this study, we simulate using r software with the help of bellreg-package [1,16]. the predictors are generated in accordance to [7,8,9,17,18,19,20,21,22,23,24,25]: xi j = √ (1 −ρ2)mi j + ρmi( j+1), i = 1, . . . , n; j = 1, . . . p, (49) where mi j are independent standard normal pseudo-random numbers and ρ2 denotes the correlation between the explanatory variables such that ρ = 0.8, 0.9, 0.99 and 0.999. it is assumed that yi ∼ bell (wo (µi)) , such that log(µi) = ηi = β1 xi1 + β2 xi2 + · · · + βp xip (50) the sample sizes n are 50, 100, 200 and 500 while p is taken to be 3, 8 and 12. we choose the true regression parameters β such that ∑p i=1 β̂ 2 i = 1 [26]. 5 shewa & ugwuowo / j. nig. soc. phys. sci. 4 (2022) 713 6 table 1. simulated result in terms of mse when p= 4 coef 50 100 200 500 0.8 γ̂mml 5.1368 1.9656 1.1792 1.0398 γ̂k−b 4.3841 1.8570 1.1404 1.0386 γ̂d−b 3.3761 1.9492 1.1699 1.0392 γ̂kl−b 1.9286 1.5390 1.1393 1.0234 γ̂kd−b 1.4073 1.3149 1.2070 1.0183 0.9 γ̂mml 5.6531 1.9926 1.3388 1.1347 γ̂k−b 4.8511 1.9512 1.3192 1.1335 γ̂d−b 4.7792 1.9629 1.3342 1.1338 γ̂kl−b 3.8790 1.7835 1.3128 1.1234 γ̂kd−b 2.5191 1.5851 1.3093 1.0191 0.99 γ̂mml 6.8980 2.9614 1.8329 1.2415 γ̂k−b 5.5356 2.2753 1.3614 1.2360 γ̂d−b 5.2118 2.4122 1.4868 1.2081 γ̂kl−b 4.2356 2.1345 1.3545 1.2076 γ̂kd−b 3.4242 1.6709 1.4037 1.2001 0.999 γ̂mml 15.4196 8.3391 8.2790 3.4418 γ̂k−b 6.1161 3.7683 7.2700 2.0326 γ̂d−b 5.6982 2.9212 4.2932 1.7886 γ̂kl−b 4.8923 2.7634 2.4512 1.5531 γ̂kd−b 3.4986 2.0303 1.6574 1.2747 the simulation study was conducted by adopting the rstudio programming language. the experiment was replicated 1000 times and the mean squared error (mse) was employed to evaluate the performance of the estimators’. ms e(β∗) = 1 1000 1000∑ j=1 (β∗i j −βi) t (β∗i j −βi) (51) where β∗i j is the estimator and βi is the parameter. the estimator with the minimum mse is preferred to the estimator with maximum mse. the simulation result is presented in tables 1-3. the following observations were made from tables 1-3. the mean squared errors for each of the estimators were computed at different specifications. the method of maximum likelihood has the least performance in this study. this is in line with the literature as expected. we observed that its performance drops as the level of multicollinearity changes. the new estimator produces a better performance in terms of minimum mean squared error than the ridge and the liu estimator. the other noticeable trend from tables 1-3 are as follows: 1. the mse rises as the level of multicollinearity rises, keeping other factors constant. 2. also, mse rises as the predictor variable increases keeping other factors constant. 3. increasing the sample size n results in a decrease in the mse for all the estimators’ keeping other factors constant 4. these results agreed with the theoretical section. table 2. simulated result in terms of mse when p= 8 coef 50 100 200 500 0.8 γ̂mml 5.5490 5.2430 4.9503 1.4723 γ̂k−b 5.5467 4.9460 4.1475 1.4700 γ̂d−b 5.5375 4.9272 3.1725 1.4710 γ̂kl−b 4.7675 4.3211 3.2415 1.3980 γ̂kd−b 3.8533 3.1229 1.2796 1.3476 0.9 γ̂mml 6.1409 6.0730 5.2045 2.0049 γ̂k−b 6.1372 6.0708 4.8290 2.0023 γ̂d−b 6.1169 6.0617 4.1508 2.0024 γ̂kl−b 5.9890 5.3678 4.0234 1.9087 γ̂kd−b 4.3141 4.4619 3.4109 1.7746 0.99 γ̂mml 7.3489 7.7949 6.8973 3.0801 γ̂k−b 7.3374 6.7782 5.8251 3.0555 γ̂d−b 7.2683 6.7231 5.0666 3.0412 γ̂kl−b 6.9764 6.0123 5.1034 2.9807 γ̂kd−b 6.1818 5.5338 4.0534 2.4082 0.999 γ̂mml 17.7282 10.7036 10.0083 5.9225 γ̂k−b 7.5111 6.9166 5.5853 4.7222 γ̂d−b 8.0643 7.0671 6.7115 4.1167 γ̂kl−b 8.0423 7.0654 6.0980 4.0088 γ̂kd−b 7.0007 6.9086 4.4547 2.5738 table 3. simulated result in terms of mse when p= 12 coef 50 100 200 500 0.8 γ̂mml 8.3479 6.6716 5.6898 2.5792 γ̂k−b 8.3446 5.6696 5.4483 2.1417 γ̂d−b 8.3357 5.6696 4.6284 2.2176 γ̂kl−b 6.7845 5.3222 3.9908 2.2214 γ̂kd−b 5.9400 4.1370 3.8202 1.8786 0.9 γ̂mml 9.7388 7.6736 6.0451 3.0170 γ̂k−b 9.7362 6.8409 5.0426 2.6473 γ̂d−b 9.7256 5.8649 5.0422 2.8627 γ̂kl−b 8.7656 5.0119 4.8765 2.9080 γ̂kd−b 7.8950 4.4621 4.2318 2.5720 0.99 γ̂mml 15.5725 9.5476 5.6815 4.7363 γ̂k−b 10.6565 7.4973 5.6736 4.6573 γ̂d−b 10.6124 7.3872 5.1511 4.6358 γ̂kl−b 9.6787 7.0577 5.0335 4.4631 γ̂kd−b 9.2586 6.7495 4.9946 2.9945 0.999 γ̂mml 31.7659 21.8870 11.1505 8.6753 γ̂k−b 16.2341 10.0161 9.0282 6.1041 γ̂d−b 12.1141 9.0713 7.0907 5.7521 γ̂kl−b 11.0886 8.0989 6.5656 4.8989 γ̂kd−b 11.1061 7.0443 5.0506 3.2270 6 shewa & ugwuowo / j. nig. soc. phys. sci. 4 (2022) 713 7 table 4. bell regression estimates coef. γ̂m ml γ̂k−b γ̂d−b γ̂kl−b γ̂kd−b intercept -0.5423 -0.1484 -0.3006 0.0375 0.0322 x1 0.5992 0.3396 -0.0034 0.3286 -0.0459 x2 0.1630 0.1667 0.0119 0.1605 0.1685 x3 -0.0117 -0.0146 0.0023 -0.0161 -0.0136 k/d 2.78554 0.1108 1.1333 18.0445 (0.1108) mse 1.7447 0.3914 0.5327 0.1493 0.0181 4. application in this section, we adopt the aircraft data to evaluate the performance of the existing estimators and the proposed. this dataset is originally assumed to follow the poisson regression model [8, 9, 27]), among others. there is one response variable and three predictors (see [8, 9] for the details). poisson distribution fits well to the outcome variable [8, 9, 27]). the model suffers from multicollinearity because the condition number is 219.3654 [8, 9]. however, the variance of the number of locations with damage on the aircraft is more than twice the mean (2.0569). with this, it is evident that the data exhibit overdispersion. we fit the bell regression model as alternative to account for the over-dispersion in the data. table 4 provides the regression estimates and the mean squared error of each of the adopted estimators in this study. it is obvious from the result in table 4, the proposed estimator produced the lowest mse and dominates the mml, the ridge and the liu estimators. the mml has the highest mean squared error. thus, not recommended when there is multicollinearity. 5. conclusion the poisson regression is often employed to model count data. however, it is certain that the poisson regression model gives poor fit for count data with over-dispersion. recently, the bell regression model was introduced as alternative to the poisson regression model for the purpose of accounting for overdispersion in count data modelling. the conventional method of maximum likelihood (mml) is employed to estimate the regression parameters. the estimator flop when the regressors are correlated. the ridge and the liu estimator were developed to combat correlated regressors in bell regression model. in this study, we developed a new method of parameter estimation in bell regression model to compete with the existing ones. we compared the performance of the new method with some methods. the new method dominates the existing methods by giving a minimum mean squared error. acknowledgments the authors are very grateful to anonymous referees for their careful and diligent reading of the paper and helpful suggestions. this research has not received any sponsorship. references [1] f. castellares, s. l. p. ferrari & a. j. lemonte, “on the bell distribution and its associated regression model for count data”, applied mathematical modelling 56 (2017) 172, https://doi.org/10.1016/j.apm.2017.12.014 [2] m. amin, m. n. akram, & a. majid, “on the estimation of bell regression model using ridge estimator”, communications in statistics simulation and computation 2021 (2021), https://doi.org/10.1080/03610918.2020.1870694 [3] a. majid, m. amin, & m. n. akram, “on the liu estimation of bell regression model in the 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[26] b. m. g. kibria & a. f. lukman, “a new ridge – type estimator for the linear regression model: simulations and applications”, scientific 2020 (2020) 9758378, https://doi.org/10.1155/2020/9758378 [27] r. h. myers, d. c. montgomery, g. g. vining & t. j. robinson, generalized linear models: with applications in engineering and the sciences, wiley, new york (2012) 791 8 j. nig. soc. phys. sci. 4 (2022) 746 journal of the nigerian society of physical sciences field strength variability mapping of nigeria m. m. tankoa,b,∗, m. s. limanb,c, w. l. lumbib, u. s. aliyuc, m. u. sarkib anumerical weather prediction, nigerian meteorological agency, abuja, nigeria bdepartment of physics, nasarawa state university, keffi, nigeria cdepartment of physics, federal university of lafia, nigeria abstract the analysis of the field strength variability using the meteorological parameters (temperature, pressure and relative humidity) retrieved from the archive of national aeronautical and space administration (nasa) was conducted for 61 locations between 2016 and 2020 to check for the spatial and seasonal variation. the result shows a seasonal variation with higher field strength variability during the dry season and lower field strength variability during the wet season. the spatial variation shows a significant difference between stations in the drier locations up north and those in the coastal areas. this could be attributed to the moisture contents of the atmosphere. further analysis of the inter-tropical discontinuity (itd) position during the study period has confirmed this assertion where we discovered the northward movement of the itd brings along with it more moisture and consequently a weak field strength. the variation in the high grounds is not manifesting because of the fact that pressure has little influence on the field strength variability. a careful study of the pressure contour explains that. the mean field strength is found to be between 2.8 db and 17.9 db. this implies that the output of the receiving antenna should be less than 2.8 db and can be as high as 17.9 db. doi:10.46481/jnsps.2022.746 keywords: field strength variability, tropospheric refractivity, radio wave, electromagnetic propagation, inter tropical discontinuity article history: received: 29 march 2022 received in revised form: 04 june 2022 accepted for publication: 13 june 2022 published: 15 august 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: o. j. abimbola 1. introduction atmosphere is a non-homogenous space having different layers exhibiting different physical behavior in terms of density, temperature, humidity and pressure. this erratic behavior has a significant impact on the propagation of radio as it travels from a transmitter to a receiver. in comparison to other communication channels, the notion of radio wave communication over time has been used to a higher extent. the wireless communication links are used for phone, data, and video services. ∗corresponding author tel. no: +234(0)8039608210 email address: mhtanko@gmail.com (m. m. tanko ) the terrestrial radio link and mobile telecommunication can be deployed using the point-to-point radio line-of-sight link [1]. these areas of application are being deployed for both civil and military missions around the world. an important system in the radio propagation is the transmission channel. this could be the atmosphere, optic fiber, or coaxial cable. the extent of signal distortion during the propagation within a chosen transmission channel determines the integrity of the information received. the variation in refractive index is a factor that determines the refraction in the lower part of the atmosphere. the ratio of the speed of propagation of a radio wave in a vacuum to its speed in a specific medium is known as the radio refractive index. the changes in this refrac1 tanko et al. / j. nig. soc. phys. sci. 4 (2022) 746 2 tive index determines the propagation of the radio wave in the troposphere [2]. radio wave propagation in the atmosphere is largely influenced by the variation in relative humidity, temperature, and atmospheric pressure. therefore, radio refractivity variation in the troposphere is a direct function of relative humidity, temperature, and pressure [3, 4]. however, it is worthy of note that large percentage of the values of refractivity can be determined by temperature in the wet season compared to dry season which is largely driven by relative humidity [5]. the radio wave propagation in the troposphere is broadly impacted by the properties of the earth and atmosphere [6]. the curvature of the earth and the condition of the atmosphere are responsible for the electromagnetic wave multi-directional refraction. the refractive index of the medium through which the electromagnetic wave travels determines the clearing of the microwave route. surface refractivity correlates negatively with the radio field strength, especially at very high frequencies [7]. a study by [8] concluded that radio refractivity is synchronous with the seasonal moisture in the atmosphere. higher field strength are recorded when the refractivity is low. this corresponds to a time when there is less humidity and moisture. for every unit change in surface refractivity (ns), a factor of 0.2 db change in field strength can be used in the frequency range 30−300 mhz [9]. for the characterization of radio channels, surface and elevated refractivity data are frequently required. surface refractivity, in particular, is highly useful for predicting propagation effects [10]. the condition of the atmosphere is very important for the proper planning of terrestrial radio links, navigation and remote sensing installations like radar. the field strength variability determines the sound level of a signal antenna. this will determine the height of the antenna and the distance between successive antennas. the basic principle of this research is snail’s law of refraction where the ratio of the angle of incidence to that of refraction is equal to unity. this paper for the first time analyzed field strength variability for 61 locations in nigeria compared to previous studies. our decision to analyze 61 locations is to achieve significant spread. the period of the study (2016 – 2020) was decided for convenience. 2. data and methodology satellite meteorological data comprising of temperature (oc), relative humidity (%), and pressure (hpa) were obtained from the us national aeronautics and space administration (nasa) at 2-meter height [11]. while the inter-tropical discontinuity (itd) position data was obtained from the nigerian meteorological agency (nimet) for the period of five years (20162020). these data are however obtained daily. the analysis of the surface radio refractivity depends on relative humidity, air temperature, and pressure. the atmospheric refractivity n (n-units) is analyzed using the equation from [12]: n = 77.6 t ( p + 4810 e t ) (1) where p is atmospheric pressure (hpa), t is absolute temperature (k), and e is the atmospheric water vapour pressure (hpa). the water vapour pressure can be determined from [2]: e = hes 100 (2) e = h × 6.1121ex p ( 17.50t t+240.97 ) 100 (3) where t is temperature in celsius (oc), and h is the relative humidity. the field strength variability (fsv) can be analyzed from the monthly maximum and minimum of surface refractivity obtained from equation 1. the fsv is determined from the monthly ranges of surface refractivity (ns ) as expressed in equation 4. the 0.2 db is a factor adopted for a frequencies between 30 − 300 mhz [9]: fs v = [nsmax − nsmin] × 0.2 (4) equations 1, 2, and 3 were used to compute the surface radio refractivity after which the equation 4 was used to compute the field strength variability: the surface refractivity was computed daily, this was to enable us obtain the monthly range of the surface radio refractivity which will subsequently be used to compute field strength variability using equation 4. the monthly mean and the study period mean where obtained for final plotting using surfer software. 2.1. study area the study was undertaken from the 61 stations within nigeria, as shown on figure 1. these stations were sampled for convenience from the five geo-climatic regions in the country using the definition of [13], these stations are: • sahel: gusau, sokoto, maiduguri, katsina, b/kebbi, damaturu, nguru and abadam. • sudan savannah: yelwa, maru, zaria, giade, dutse and potiskum and kano. • guinea savannah: abuja, bida, jalingo, kaiama, wase, ilorin, jos, lafia, lokoja, makurdi, minna, ibi, m/plateau, yola, kaduna, bauchi, gombe and shaki. • tropical rainforest: abeokuta, ado-ekiti, akure, ibadan, iseyin, ondo, oshogbo, benin, usieki, abakaliki, asaba, umuahia, owerri, awka, enugu, ogoja, obudu and ikom. • coastal: ijebu-ode, ikeja, kokamarine, calabar, eket, onne, port harcourt, uyo, yenagoa and warri. 3. results and discussion 3.1. field strength variability of nigeria the results from this study can be seen as displayed in table 1 as well as on figure 2 to figure 7. the field strength variability is found to be prevalent in the dry season and lower in the 2 tanko et al. / j. nig. soc. phys. sci. 4 (2022) 746 3 table 1: field strength variability (db) of the study area (2016-2020). region january february march april may june july august september october november december sahel 5.6 8.1 14.2 17.7 14.0 8.5 4.7 3.5 4.1 13.1 8.2 5.5 sudan 6.1 9.2 16.0 17.9 8.7 5.1 3.9 3.5 3.2 10.5 9.2 5.6 guinea 9.6 12.3 14.3 10.0 4.3 3.9 3.1 3.2 3.3 6.0 9.3 7.9 rainforest 12.4 12.9 4.8 4.0 3.6 4.1 3.2 3.2 3.1 3.6 5.6 8.6 coastal 10.0 9.5 3.3 3.3 3.3 3.6 3.1 2.8 2.6 3.2 3.5 7.4 figure 1: map of the study area. rainy season. a negative correlation between the field strength and the moisture content in the atmosphere has been established at -0.90, unlike the refractivity which correlated positively at 0.93. more moisture means less field strength variability hence the rise in the dry season. having higher field strength variability in the dry season is directly as a result of elevated surface temperature, low humidity and almost constant pressure. the field strength variability values in a particular location determines the output of the receiving antenna in that location. figure 2: field strength variability of nigeria. 3.2. seasonal variation a seasonal variation has been observed in this study of field strength variability. this work agrees with [6, 7], which all confirmed that field strength variability varies with season. figure 2 revealed elevated values in the early and late part of the year figure 3: field strength variability of sahel zone of nigeria. figure 4: field strength variability of sudan savannah of nigeria. except in stations within sudan and sahel areas where elevate values are also recorded in the april. the general pattern of field strength variability variation in the tropics is exhibiting higher values in the dry season and lower values in wet season. the field strength variability in the sahel stations in this study as shown in figure 3 is found to rise from january at 5.6 db to a peak of about 17.7 db in april. it further declined sharply to about 3.5 db in august. the elevated values were recorded in the dry season while the lower values in the rainy season. the curve did not completely follow the expected trend due to erratic nature of rain in the zone where the period is short with heavy rainfall. the output of antenna in this zone can be as high as 17.7 db but should not be lower than 3.5 db. the electric field strength in this zone followed the same pattern as the sahel zone. careful observation of figure 4 shows a sudden rise from january at about 6.1 db to a peak of about 17.9 db in april. the fall is as sudden as the rise to a dip of about 3.2 db in september accounting for the lowest value in the zone. the output of the antenna in this zone is at the same range as that of the sahel which is expected to be between 3.2 db and 17.9 db. this confirmed the similarities between the two zones. the field strength variability in the guinea savannah of 3 tanko et al. / j. nig. soc. phys. sci. 4 (2022) 746 4 figure 5: field strength variability of guinea savannah of nigeria. figure 6: field strength variability of tropical rainforest of nigeria. nigeria is displayed in figure 5. the maximum field strength is measured in march at about 14.3 db and the minimum in july at about 3.1 db. the curve showed a little deviation from those of the sahel and sudan showing nearly flat bottom from may at about 4.3 db to september at about 3.3 db. this clearly the months where maximum rain and atmospheric refraction is recorded in the zone. for planning of radio link and radar network in this zone, the output of the antenna can be as high as 14.3 db but should not be less than 3.1 db. figure 6 revealed the outcome of the study on the field strength studied in the tropical rainforest zone of nigeria. elevated values are recorded in the month of january and february at about 12.4 db and 12.9 db respectively. december, january and february are known to be dry months in this zone. a near consistent low values within the range of about 5 db is recorded between march and november confirming the length of wet season in the zone. from the curve we can suggest that antenna output in the zone should as high as 12.9 db and not lower than 3.1 db. the field strength of coastal stations in this study as shown in figure 7 is found to follow the same pattern as that of the tropical rainforest zone. the maximum value in this zone is recorded in january at about 10 db declining sharply to about 3.3 db in march. a near consistent low values are recorded from march to november at the range of about 2 to 3.5 db. this is in agreement with the established length of wet period in the zone. the output of the receiving antenna in this zone can be as high as 10 db but should not be less than 2.6 db when planning radar network and terrestrial radio link. figure 7: field strength variability of coastal areas of nigeria. 3.3. spatial variation of electric field strength variability the variation of field strength variability across the study stations within the geo-climatic zones is shown in figure 9 to figure 12. for convenience, we divided the years under study into: first dry quarters (december, january & february), second dry quarter (march, april & may), first rain quarter (june, july & august) and second rain quarter (september, october & november). the spatial variation of field strength variability in the first dry quarter (december, january & february) is revealed in figure 9. the approximate position of the itd during this quarter as revealed in figure 8 is around 8.3 o n with more than half of the country north of the imaginary line. during this period we expect a rise in the field strength variability due to reduced moisture and humidity in the atmosphere as showed in the humidity contour. the minimum value recorded in this quarter was in giade, a station in sudan savannah at about 4.0 db. while the maximum value was recorded in benin a station in the tropical rainforest zone at about 15.1 db. lower values are found to strangely dominate the sahel and sudan savannah zone during this quarter with most of the higher values in the lower part of the guinea savannah into the tropical rainforest. this could be attributed to high range of atmospheric radio refractivity in the tropical rainforest zone. figure 8: average itd position during the study period. showing in figure 10 is the field strength variability during the second dry quarter. this is in consistent with the natural behavior of field strength variability where higher values are recorded in the dry regions and lower values in the wet regions. 4 tanko et al. / j. nig. soc. phys. sci. 4 (2022) 746 5 figure 9: spatial variation of fsv of nigeria in first dry quarter (december, january & february). the nearness of the coastal and the tropical rainforest to the atlantic ocean led to higher atmospheric refraction which adversely affects the values of field strength variability in those regions. about half of the country accounts for the higher values of field strength variability especially in march and april. this could be attributed to lower refraction recorded during this periods all over the country. a maximum of about 21.5 db was recorded in potiskum, a station in the sahel zone while the minimum of about 2.6 db was recorded in eket, a station in the coastal zone. this agrees completely with the established trend of the field strength variability. higher temperature give rise to higher field strength variability while the reverse is the case with humidity as can be seen on the temperature and humidity contours. the itd position during this quarter is about 12.7 o n as can be seen in figure 8. this puts more than half of the country under the line. figure 10: spatial variation of fsv of nigeria in second dry quarter (march, april & may). the variation in the first rain quarter as depicted in figure 11 revealed elevated values at abadam an extreme north eastern flank of the sahel zone at about 16.0 db and lowest values at eket in the coastal zone at about 2.0 db. during this period, the large part of the country is dominated by lower values except the extreme north of the sahel where higher values were recorded. this confirms the facts that june, july and aug as wet months of the year. the approximate position of the itd during this quarter is around 19.7 o n putting the entire country under the influence of south westerly moisture laden wind. this consequently rise the atmospheric refractivity of the entire country. as the south westerly monsoon wind intensifies, it pushes the elevated values northward as can be seen clearly from the chart. the average minimum and maximum field strength variability during this period is about 2 db and 16 db respectively. figure 11: spatial variation of fsv of nigeria in first rain quarter (june, july & august). figure 12: spatial variation of fsv of nigeria in second rain quarter (september, october & november). the spatial variation of field strength variability of nigeria for the second rain quarter (september, october & november) is shown in figure 12. this is the period when rainfall dominate the tropical rainforest and coastal areas while receding in the sahel, sudan savannah, and guinea savannah. looking at the months of october and november it shows the dominance of values in most part of the country. the itd position in this quarter is approximately around 15 o n it is clear that the imaginary meteorological line is retreating southward due to the push 5 tanko et al. / j. nig. soc. phys. sci. 4 (2022) 746 6 from the north-easterly trade wind. the retreat will continue until the whole country in under the influence of north easterly trade wind. for every southward retreat of the itd, an increase in the field strength variability is recorded. the maximum field strength variability was recorded in abadam at about 16.8 db and the minimum in eket at about 2 db. 4. conclusion the field strength variability of nigeria studied from 2016 to 2020 has revealed both seasonal and spatial variation. the seasonal variation has revealed a dominance in the dry season and a decline in the wet season. this is believe to be due moisture variation in the atmosphere during these two seasons. the presence of moisture in the atmosphere is believed to dampen the field strength, hence the prevalence of higher values in the dry season. the stations in the sahel recorded higher field strength while stations in the coastal area recorded the least field strength. this further buttress the fact that field strength depend largely on the moisture content of the atmosphere. the value of field strength found during the study was between 17.9 db to 2.8 db. the implication of the field strength variability values in this study is that the output of a receiving antenna in nigeria may generally be not less than 2.8 db, but can be as high as 17.9 db. acknowledgments we thank the referees for the positive enlightening comments and suggestions, which have greatly helped us in making improvements to this paper. references [1] n.k. chaudhary, d.k. trivedi & r. gupta, “the impact of k-factor on wireless link in indian semi-desert terrain”, international journal of advanced networking and applications, 2 (2011) 776. [2] a. t. adediji & m. o. ajewole, “vertical profile of radio refractivity gradient in akure, south-west nigeria”, progress in electromagnetic research, c (2008) 157. [3] k. saha, s. raju & k. parameswaran, “neutral atmospheric refractivity on microwave propagation and its implication on gps based ranging system”, proceedings of the xxviiith general assembly, new delhi (2005). [4] m. s. tomar, “measurement and analysis of radio refractive index over patiala during monsoon season with respect to its diurnal and monthly characteristics”, mausam, 63 (2012) 334. [5] m. ajewole, a. adediji, j. ojo, s. falodun, k. adedayo, t. ewetumo, a. ashidi, s. adebusola, k. ogunjobi, a. oluleye, a. dada & o. ojo, “variation of microwave radio refractivity profiles with temperature over akure, nigeria”, global journal of computer science and technology: g interdisciplinary, 20 2020. [6] a. t. adediji, m. o. ajewole & s. e. falodun, “distribution of radio refractivity gradient and effective earth radius factor (k-factor) over akure, south western nigeria”, journal of atmospheric and solar terrestrial physics, 70 (2011) 2300. [7] m. m. tanko, a. akinbobola, s. ojo & m. s. liman, “assessment of tropospheric variation of radio refractivity and field strength variability over some selected stations in northern nigeria”, physical science international journal, 18 (2018) 1. [8] o. j. abimbola, e. bada, a. o. falaiye, y. m. sukam, m. s. otto & s. muhammad, “estimation of radio refractivity from satellite-derived meteorological data over a decade for west africa”, scientific african, 14 (2021) e01054. [9] a. t. adediji & m. o. ajewole, “microwave anomalous propagation (ap) measurement over akure south-western nigeria”, journal of atmospheric and terrestrial physics, 72 (2010) 550. [10] r. b. bean & e. j. dutton, “radio meteorology”, national bureau of standards. u.s. (1966) pp 20. [11] nasa, “power data” national aeronautical and space administration, may 5 (2019) https://power.larc.nasa.gov/data-access-viewer/ [12] e. k. smith & s. weintraub, “the constants in the equation for atmospheric refractive index at radio frequencies”, proceedings of the i.r.e, 4 (1953) 1035. [13] k. ogunjobi, d. ragatoa, a. b. nana, a. okhimamhe & j. eichie, “change comparison of heat wave aspects in climatic zones of nigeria”, environmental earth sciences, (2019). 6 j. nig. soc. phys. sci. 4 (2022) 281–286 journal of the nigerian society of physical sciences diagnostic reference levels (drls) and image quality evaluation for digital mammography in a nigerian facility a. e. anasthesiaa, u. ibrahimb, s. d. yusufb, d. z. josephc,∗, n. flaviousd, m. sidid, s. sheme, a. mundif, a. dareg, d. s. josephh, y. a. ningii aradiology unit, nyanya general hospital abuja, nigeria bdepartment of physics, nasarawa state university keffi, nigeria cdepartment of radiography, federal university, lafia, nigeria ddepartment of radiography, university of maiduguri edepartment of radiography, ahmadu bello university zaria fdepartment of radiography, bayero university kano, nigeria gdepartment of radiography and radiation sciences, baze university abuja hdepartment of radiology, federal medical center katsina iradiology department, abubakar tafawa balewa university bauchi abstract diagnostic reference levels (drls) for digital mammography and image quality evaluation are important optimization tools in medical imaging. high quality mammograms are essential to the successful early detection of breast cancer. the objective of the study is to establish drls for digital mammography and to assess image quality of the mammograms for optimization. drls were established using thermoluminescent dosimeter (tld) chips to estimate the mean glandular dose for both cranio-caudal and medio-lateral oblique projections. the tld chips were calibrated. the drls were set at the 75th percentile of the distribution of the median value of mean glandular dose. image quality was assessed using european commission guideline for mammographic image quality assessment. results for drls were 0.53 mgy for cranio-caudal and also 0.53 mgy for medio-lateral oblique. image quality evaluation showed criteria scores for cranio-caudal and medio-lateral oblique projections as 76 % and 61.2 % respectively. the mammograms scored the highest and lowest score of 100 % and 44 % on criteria 2 and criteria 6 (absence of skin fold) respectively for cranio-caudal projections while for the mediolateral oblique projections, criteria 1 (all breast tissue clearly shown) and criteria 5 (inframammary angle clearly demonstrated) have the highest and lowest score of 96 % and 8 % respectively. the study showed that the drls in this study was lower than the established values in other regions of nigeria and international established values. image quality was within acceptable level. drls for digital mammography and image quality evaluation are important optimization tool that should be adopted by every radiology department with mammography unit. doi:10.46481/jnsps.2022.734 keywords: diagnostic reference level, mean glandular dose, image quality evaluation, dose, thermoluminiscent dosimeters article history : received :25 march 2022 received in revised form: 05 may 2022 accepted for publication: 10 may 2022 published: 29 may 2022 c©2022 journal of the nigerian society of physical sciences. all rights reserved. communicated by: w. a. yahya ∗corresponding author tel. no: +2348130582721 email address: josephdlama@gmail.com (d. z. joseph ) 1. introduction medical uses of ionizing radiation are among the longest established applications of ionizing radiation [1]. the risk associ281 anasthesia et al. / j. nig. soc. phys. sci. 4 (2022) 281–286 282 ated with medical use of ionizing radiation varies significantly depending strongly on the radiological procedure [2]. for medical exposure, the optimization of radiological protection is best described as management of the radiation dose to the patient to be commensurate with the medical purpose [3]. diagnostic reference level is an important tool for optimization of protection. it is a quality assurance tool introduced in 1996 as a term for a form of investigation level used to identify situation where optimization of protection may be required in the medical exposure of the patient [4]. it is used in medical imaging to indicate whether in routine conditions, the radiation dose used for a specific radiological examination or the amount of radiopharmaceutical administered to a patient is unusually high or unusually low for that procedure [5]. compliance with diagnostic reference value does not necessarily indicate image quality is adequate or that the examination is performed using optimal amount of radiation [5]. mammography is a specialized non-invasive radiographic imaging of the breast tissue (soft tissue radiography) using low dose of x-ray. the breast is largely composed of two types of tissue, namely adipose (fatty) tissue and glandular/ fibroglandular tissue (including acinar and ductal epithelium and associated stroma) [6]. the goal of mammography is the early detection, characterization, and evaluation of findings suggestive of breast cancer and other breast diseases [7]. it is carried out on both symptomatic women with a known history or suspected abnormality of the breast and as a screening procedure in, asymptomatic women [8]. patient dose and image quality are the main concerns in mammography particularly in screening mammography where comparison with former films is often essential. the accuracy of diagnosis is very dependent on the image quality [8]. adequate quality dose is the current trend recommended by icrp in the optimization of protection for patient undergoing mammography procedure. it is the recommended dose needed to produce images that meet the clinical needs of the patient. [8]. the glandular tissue that makes up the breast is one of the radiosensitive tissues in the human body. it is therefore imperative] to use the required amount of radiation to produce images that are clinically acceptable. overexposure of the breast to radiation may increase radiation induced carcinogenesis while under exposure may hide anatomical and pathological details needed to make proper diagnosis. optimization process that encompasses both establishment of diagnostic reference level process and image quality evaluation should be encouraged and implemented in radiology facilities [8]. the role of the radiographer is central to the success of breast screening programme in producing high quality mammograms which are crucial for early diagnosis of breast cancer (euref, 2006a). the radiographic technique adopted by the radiographer has the greatest influence on the dose received by the glandular tissue of the breast and correct positioning techniques during mammography has been shown to improve breast cancer detection rates [9]. diagnostic reference level and image quality evaluation are key components in the optimization of protection for patient undergoing mammography examination (icrp, 2017) [10]. establishment of diagnostic reference level for mammography and image quality evaluation of mammographic images ensures that the required amount of radiation is used to produce clinically diagnostic images. diagnostic reference level for 4 mammography if consistently exceeded calls for audit of the procedure in order to determine the cause of the higher doses and recommendation implemented urgently (euref, 2006a) [9]. in situation where the diagnostic reference level is unusually low, image quality may be compromised and there might be increased likelihood of repetition arising from poor image quality. the glandular tissues that make up the breast are very radiosensitive thus must be protected from unnecessary exposure to radiation. diagnostic reference level (drl) for mammography has been established in many parts of the world as a necessary tool for optimization of protection of patients undergoing mammography examination. in nigeria, diagnostic reference level for mammography has been established in north –eastern nigeria but diagnostic reference level specifically for digital mammography as well as image quality evaluation has not been done in any of the federal capital territory hospital. the purpose of this study is to establish diagnostic reference level for digital mammography and to evaluate image quality in asokoro district hospital in abuja, nigeria the outcome of this study, will serve as a clinical audit of mammography procedure in the selected hospital in order to promote good practice and dose optimization. it will further enhance training of radiographers, radiologist and medical physicist involved in the production of mammographic images, interpretation of images and quality assurance procedure through comparison of results with international established work. the research will add to the pool of data for the establishment of national and regional diagnostic reference level for mammography procedure. it will also serve as a comparative guide for practitioners, researchers and regulators. 2. materials and methods 2.1. materials general electric senograph essential digital mammography machine manufactured june, 2011 with maximum kv p of 49, inherent filtration of 0.66 be and focal spot size of 0.1 − 0.3 mm was used for the research. this machine is for generation of x-ray with low energy between 18 − 35 kv p (figure 1). the thermoluminescent dosimeter chips are manufactured by rad pro international germany. they are mcp tld chips and measure 3.2 mm x3.3 mm xo.9 mm. density 2.5 g/cm3. the chips are for measurement of incident air kerma and estimation of mean glandular dose. the tld chips are made up of lithium fluoride which is near tissue equivalent. the tld chips were annealed and exposed for the purpose of generating reader calibration factors (rcf) for the reader and elemental correction coefficient (ecc) for each of the tld chip. all exposures for the calibration of the reader and dosimeter were done at the national institute of radiation protection research (nirpr), university of ibadan. tld chips was also set aside as control to record the background radiation. the control tld chips were kept away from every form of irradiation. 282 anasthesia et al. / j. nig. soc. phys. sci. 4 (2022) 281–286 283 2.2. data collection data was obtained using prospective cross sectional approach involving fifty (50) women who came for both screening and diagnostic mammography examinations within the period of the research june 2019 to december 2019. the women were between the ages of 4064 and consented to the research. .mediolateral and cranio-caudal projections were obtained. positioning for both projections was done by a radiographer in mammography. the machine uses automatic exposure control (aec); therefore provided the exposure factors used automatically nam− ely kv p, mas, anode/filter combination according to the breast granularity and thickness. the above parameters were recorded on a work sheet for the patients and projections. the exposed tlds were labeled ptcc1-50 for cranio-caudal projections and ptmlo1-50 for medio-lateral oblique projections for proper identification and kept in black nylon away from radiation. 2.3. estimation of doses the tld chips were read with a reader harshaw model 3500. the values obtained were subtracted from the control tld value to give the value of the incident air kerma used for each patient and view. to estimate the mean glandular dose, the conversion factors from the works of dance [10] and dance et al [12, 13], were used to calculate the mean glandular dose (mgd) for both cranio-caudal and medio-lateral oblique projections of the breast. mgd = kgcs, (1) where k is the incident air kerma obtained at the upper quadrant of the breast without back scatter; g = k to mgd conversion factor on the assumption that the entire breast has a glandularity 50 %, c is the conversion factors for difference in breast composition other than 50 % glandularity, s = conversion factor for different x-ray spectrum which can be due to different anode/filter combination, for example, mo/mo, mo/rh, rh/rh. the total mean glandular dose for both cranio-caudal and medio-lateral oblique projections was calculated using the statistical package for social science version 21.0 and the formula is given as x = ∑ x n , (2) where n = number of patients, x = mean glandular dose for each view, x = mean glandular dose for cranio-caudal and mediolateral oblique projections respectively. 2.4. establishment of diagnostic reference level the drl was set at the 75th percentile (third quartile) distribution of the median glandular dose for both cranio-caudal and medio-lateral oblique projection. 2.5. image quality evaluation image quality evaluation was carried out by the researcher and two specialist radiographers in mammography using the ec criteria for image quality evaluation for mammography. absolute grading score of yes (1) or no (0) were used for the criteria. scores were converted to percentages for each of the patient for cranio-caudal and medio-lateral projections. number of patients mammogram that met each criterion were also assessed and converted to percentages. 3. results the data from table 2 show that the mean values for incident air kerma and direct digital mammography reading for both cranio-caudal and medio-lateral oblique projections are 2.05 mgy, 1.99 mgy and 0.46 mgy, 0.46 mgy, respectively. using the data obtained in table 2, the mean glandular doses for each both cranio-caudal and medio-lateral oblique projection were calculated using equation 1 and the total mean glandular dose for cranio-caudal and medio-lateral oblique projections were calculated using equation 2. the total mean glandular dose for cranio-caudal projection was 0.46±0.07 mgy while the total mean glandular dose for medio-lateral oblique projections was 0.46 ± 0.11 mgy. the median value for both craniocaudal and medio –lateral oblique projections was 0.47 mgy. the local diagnostic reference level for tld was set at the 75th percentile of the distribution of the mean glandular dose. statistical package for social science was used to obtain the 75th (3rd quartile) percentile values of the mean glandular dose. the tld value was found to be 0.53 mgy for cranio-caudal oblique and also 0.53 mgy for medio-lateral oblique projections, while that of ddmr is 1.46 mgy and 1.46 mgy. table 3 and 4 image quality was assessed using european guidelines for image quality assessment. the scores obtained were converted to percentages. the mammograms scored the highest and lowest score of 100 % and 44 % on criteria 2 (as much as possible of the lateral aspect of the breast is shown) and criteria 6 (absence of skin fold) respectively for cranio-caudal projections while for the mediolateral oblique projections, criteria 1 (all breast tissue clearly shown) and criteria 5 (inframammary angle clearly demonstrated) have the highest and lowest score of 96 % and 8 % respectively. the total average score of the mammogram for cranio-caudal and medio-lateral oblique projections are 76 and 61.2 %, respectively. 4. discussion the research has established a facility based diagnostic reference level for digital mammography and image quality evaluation in a selected hospital under federal capital territory administration. based on the findings, the total mean glandular dose using thermoluminescent dosimeters for cranio-caudal and medio-lateral oblique view of the breast are 0.46±0.07 mgy and 0.46±0.11 mgy respectively, which is within the recommended value (2.5 mgy and not more than 3 mgy) given by euref [8]. the result of the current study is in line with studies done 283 anasthesia et al. / j. nig. soc. phys. sci. 4 (2022) 281–286 284 table 1. demographic distribution of patients. variable number of patients percentage (%) age 40-45 years 28 56.0 45-55 years 18 36.0 56-64 years 4 8.0 body mass 51-70 kg 7 14.0 71-90 kg 27 54.0 91-110 kg 14 28.0 111-130 kg 2 4.0 table 2. mean distribution of incident air kerma (iak) and direct digital mammography reading (ddmr) for cc and mlo. cbt(mm) (mgy) mean cc mlo cc mlo tld iak 57.34 56.36 2.05 1.99 ddmr 57.34 56.36 0.46 0.46 table 3. local diagnostic reference level for the hospital. cc(mgy) mlo(mgy) tld 0.52 0.52 ddmr 1.46 1.46 by joseph et al. [15], joshua et al. [16] and ogundare et al. [17] who estimated mean glandular doses for cranio-caudal and medio-lateral oblique projections using tld chips. findings from the comparison of dicom mean glandular dose values and tld mean glandular dose values show significant difference in the two methods of data collection. the dicom mean glandular dose values were found to be 1.30 and 1.36 mgy for cranio-caudal and medio-lateral oblique view respectively. while the tld mean glandular dose values were 0.46 mgy for both cranio-caudal and medio-lateral projections respectively, which difference in conversion factors used by different vendors of mammography units may be responsible. the value obtained for dicom mean glandular dose is similar to the work by researchers who used the dose values on the dicom viewer to obtain the mean glandular dose. diagnostic reference level obtained by tld calculations are 0.52 and 0.52 mgy for cranio-caudal and medio-lateral oblique view respectively. while the 75th percentile of distribution of the mean glandular dose obtained from the dicom viewer of the digital machine are 1.46 and 1.46 mgy for cranio-caudial and medio-lateral oblique. the findings reveal that the drl (diagnostic reference level) obtained from this study using tld chips is slightly lower when compared to work by joseph et al. [15]. this might be due to combination of screen film mammography and digital mammography unit and the output of the machines. digital units are known to use lower dose when compared with screen film unit. the value obtained is however within the recommended range for both tld and dicom values. the image quality was evaluated using european guideline figure 1. ge senograph essential digital mammography unit. figure 2. tld chips. for image quality assessment for both cranio-caudal and mediolateral projections of the mammograms. for the cranio-caudal projection, 42 (84 %) of the mammograms have the medial border of the breast shown. 50 (100 %) have much as possible lateral aspect of the breast shown. 40 (80 %) of the mammogram have the pectoral muscle shadow shown at the posterior 284 anasthesia et al. / j. nig. soc. phys. sci. 4 (2022) 281–286 285 table 4. mammogram image quality evaluation for cranio-caudal view (cc). criteria for assessment no. of percentage mammograms (%) the medial border of the breast is shown 42 84 as much as possible of lateral aspect of the breast is shown 50 100 if possible, the pectoral muscle shadow is shown on the posterior edge of the breast 40 80 the nipple is in profile 26 52 symmetrical images of both breast 46 92 absence of skin fold 22 44 table 5. mammogram image quality evaluation for medio-lateral oblique view (mlo). criteria for assessment no. of percentage mammograms (%) all breast tissue clearly shown 48 96 pectoral muscle to nipple level 40 80 symmetrical images of both breast 46 92 nipple in profile 18 36 inframammary angle clearly demonstrated 4 8 table 6. comparison of dose values with other established works . tld mgd tld drl (mgy) (mgy) researcher cc mlo cc mlo current study 0.46 0.46 0.53 0.53 joseph et al. [15] 0.31 0.69 0.63 1.04 joshua et al. [16] 0.25 0.51 ogundare et al. [17] 0.33 1.43 edge of the breast. 26 (52 %) of the mammograms have the nipple in profile, the inability to achieve a higher percentage of this criteria might be due to patients with retracted nipple and poor positioning techniques. 46 (92 %) of the mammograms have symmetrical images of both breast while 22 (44 %) of the mammogram have no skin. the mean glandular doses and diagnostic reference levels obtained from this study, are in line with the works done by joseph et al. [15] that established drls for mammography in north-eastern nigeria. the mean glandular doses obtained from this study is also in tandem with the findings of joseph et al. [15, 16] and ogundare et al. [17]. the absence of skin fold has the lowest percentage achieved criteria. this could be attributed to inability of the radiographer to properly spread the breast tissue before obtaining the projections, thus the need for improvement on positioning technique which has the greatest influence on the dose received by the glandular tissue and it is also central in producing high quality mammograms. for the medio-lateral projections, 48 (96 %) of the mammograms have all breast tissue shown. 40 (80 %) of the mammograms have the pectoral muscle to the nipple level. 46 (92 %) of the mammograms has symmetrical images of both breast. 18 (36 %) of the mammograms has nipple in profile while only 4 (8%) of the mammograms has the infra-mammary angle clearly demonstrated. inability to clearly demonstrate the inframmary angle is the criteria with the least score and improper positioning of the breast support plate might be responsible. overall average score for both cranio-caudal and mediolateral oblique mammograms are 76 and 61.2 % respectively. continuous training and education is highly recommended for radiographers in order to meet up with international best practice. 5. conclusion local diagnostic reference level (ldrls) obtained in the current study 0.52 mgy for cranio-caudal and medio-lateral oblique were lower than the established diagnostic reference level. image quality was within acceptable percentage. however, ldrls is an important optimization tool that ensures that unnecessary high or low doses are not used during mammography examination. optimization process that encompasses both establishment of ldrls and image quality evaluation is encouraged in our various radio-diagnostic facilities with mammography unit. radiographic techniques which play essential roles in the production of high-quality mammograms are important in the detection of early breast lesion thus the need to prioritize continuous education and training among personnel. acknowledgments we thank the referees for the positive enlightening comments and suggestions, which have greatly helped us in making improvements to this paper. the authors wish to acknowledge the support of the ethical committee of health and human services secretariat of the federal capital territory administration who consented to the use of one of their mammography unit for this research 285 anasthesia et al. / j. nig. soc. phys. sci. 4 (2022) 281–286 286 references [1] international atomic energy agency (iaea), “radiation protection and safety in medical uses of ionizing radiation”, iaeasafety standard series no ssg -46, vienna (2018). [2] international commission on radiological protection (icrp), “diagnostic reference level in medical imaging, review and additional advice”, icrp supporting guidance 2 ann. icrp, 31 (2001). [3] international commission on radiological protection (icrp), “the 2007 recommendations of international commission on radiological protection” icrp publication 103.ann.icrp37 (2007). [4] international commission on radiological protection (icrp), “radiological protection and safety in medicine”, icrp publication 73.ann. icrp, 26 (1996). [5] international commission on radiological protection (icrp), “diagnostic reference level in medical imaging”, icrp publication 135.ann. icrp, 46 (2017). [6] institute of physical science in medicine (ipsm), “the commissioning and routine testing of mammographic x-ray systems”, (2nd ed.). report no, 59 (1994). [7] american college of radiology (acr), “mammography quality control manual”, american college of radiology committee on quality assurance in mammography. reston, va. (1998). [8] a. s. whitley, c. sloane, g. hoodley, a. d. moore & c. w. aslop, “clark positioning in radiography”, (12th ed.). new york, (2009) 436. [9] s. h. taplin, c. m. rutter, c. finder, m. t. mandelson, f. houn & e. white, “screening mammography clinical image quality and the risk of interval breast cancer”, american journal of roentgenology, 178 (2002) 797. [10] d. r. dance, “monte carlo calculation of conversion factors for the estimation of mean glandular breast dose”, phys. med. biol.,35 (1990) 1211. [11] d. r. dance, c. skinner, k. c. young, j. r. beckett & c. j. kotre, “additional factors for the estimation of mean glandular breast dose using the uk mammography dosimetry protocol”, phys. med. biol., 45 (2000) 3225. [12] d. r. dance, k. c. young & r. e. van engen, “further factors for the estimation of mean glandular dose using the united kingdom, european and iaea breast dosimetry protocols”, phys. med. biol., 56 (2009) 4372. [13] d. r. dance, k. c. young & r. e. van engen, “estimation of mean glandular dose for breast tomosynthesis using the united kingdom, european and iaea breast dosimetry protocols”, phys. med. biol., 54 (2011) 453. [14] euref, “european guidelines for quality assurance in breast cancer screening and diagnosis”, (4th ed.) official publications of the european commission, luxembourg (2006a). [15] d. z. joseph, c. c. nzotta, j. d. skam, m. s. umar & d. y. musa, “diagnostic reference level for mammography examination in northern nigeria”, african journal of medical & health sciences, 17(2018) 58. [16] j. i. joshua, c. c. nzotta, g. m. abubakar & f. b. nkubli, “assessment of mean glandular doses in some selected hospital in lagos state”, global scientific journal, 6 (2018). [17] f. o. ogundare, a. n. oditta, r. i. obed & f. a. balogun, “mean glandular dose for women undergoing breast screening in oyo state nigeria”, international journal of diagnostic imaging and radiation therapy, 15 (2009) 327. 286 j. nig. soc. phys. sci. 3 (2021) 334–339 journal of the nigerian society of physical sciences effect of substitutional point defect of gold (au) in indium (in) site of double halide perovskite (cs2insbcl6) i. magajia,∗, a. shuaibua, m. s. abubakara, m. isaha, a. hussainia adepartment of physics, faculty of science, kaduna state university, p.m.b, 2339, kaduna, nigeria abstract lead (pb) free (non-toxic) perovskite solar cells materials have attracted great interest in the commercialization of the photovoltaic devices. in this work, density functional theory (dft) and linear response time-dependent within density functional theory (tddft) are used to simulate and investigate the effect of gold (au) doped pb-free double halide perovskite a2bb′x6(a = cs; b = in, au; b′ = sb; x = cl) on the structural, electronic, and optical properties for perovskite solar cell application. on the structural properties, bond length and bulk modulus calculations show that the doped compound is more likely to resist deformation than the undoped compound. the calculated band structure for both materials (doped and undoped) reveals the presence of the valence band maximum (vbm) and the conduction band minimum (cbm) at around the same symmetry point which indicates a direct band gap nature (at γ− point). the band gap value for the initial compound (eg= 0.99 ev) agrees with published theoretical values. for the gold doped compound, the value of the band gap increased to a value of 1.25 ev. the result of the optical properties shows that the au-doped material has higher absorption coefficient, lower reflectivity and higher optical conductivity when compared with the initial, as such demonstrates better properties as a candidate for solar cell applications and in other optoelectronic devices. doi:10.46481/jnsps.2021.352 keywords: perovskite, photovoltaic solar cell, optical, structural article history : received: 17 august 2021 received in revised form: 05 october 2021 accepted for publication: 07 october 2021 published: 29 november 2021 ©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: e. etim 1. introduction traditionally, silicon has been the most widely used materials in most successful photovoltaic solar cells. however, high processing or manufacturing cost, unavailability of effective mass production technique, reflection, and recombination and absorption losses limit the use of silicon photovoltaic solar cells. perovskite materials which are generally photonic materials have shown great promising qualities to replace silicon as solar cell materials. the lead-based halide perovskite ∗corresponding author tel. no: email address: imagaji05@gmail.com (i. magaji ) are considered to be most promising material for solar cell absorbers [1]. however the environmental hazard and the toxicity nature associated with the lead-based halide perovskite still remain a major concern. recently, there has been tremendous efforts in developing new halide perovskite which will potentially turnaround the instability and toxicity problems associated with emerging lead-based perovskite solar cell devices. indium (in)-based double perovskite; cs2inbicl6 and cs2insbcl6 were proposed to be good return photovoltaic solar cell material, evidently due to the suitable direct band gaps foretold by spinorbit coupling with hybrid functional (soc+hse) calculation, minimal effective masses for electron-hole, and thermal stability towards decomposition [5]. the large dielectric con334 magaji et al. / j. nig. soc. phys. sci. 3 (2021) 334–339 335 stant [2], low cost fabrication [3], high optical absorption coefficient [4], low exciton binding energy, [5], efficient solar absorption when used as light absorbing layer [6], long carrier diffusion length [7], and swift increase in power conversion efficiency (pce) [8], are among the properties that makes the double halide and other similar perovskite solar cells very promising for commercialization. the major concern of the proposed indium (in) based double perovskite is the stability of indium (in) state specifically against oxidizing to in (iii) [1]. point defect is a promising way to control such instability particularly against reduction oxidation reaction (redox). in semiconductors, point defects play a fundamental role in enhancing material properties. substitutional point defect was employed by total substitution of indium with gold elements in the compound. generally, transition metals when used as dopant improves material conductivity. the surface plasmon resonance (spr) property of gold influences the used of it as the substitute in the compound. in this work, first principle dft and linear response tddft calculations were used to study the effect of substitutional point defect of transitional metal (gold) alloyed along indium (in) region of cs2insbcl6 for potential solar cell application. 2. computational details the first principle calculations were carried out within the frame work of dft and tddft by the use of plane-wave (pw) pseudopotential technique as implemented in quantum espresso suite. generalized gradient approximation (gga) was used as exchange-correlation functional formulated by perdew burke ernzerhof (pbe). the lead-free halide perovskite compound (cs2insbcl6) is a face-centered-cubic (fcc) which is a member of the cubic lattice structure in the space group fm-3m. it simple primitive cell contains 10 atoms. however, a super cell of 40 atoms was used throughout the course of this work. the structural parameters (lattice parameters and internal atomic coordinate) were optimized and fully relaxed using broyden-flectcher-goldfarb-shanno (bfgs) quasi-newton algorithm. 60 and 240 ry were set for the convergence test of the kinetic energy cut-off for wave function and charge densities respectively. the monkhorst-packk-point mesh sampling of brillouin zone integration was set as 6 x 6 x 6 and a denser k-point mesh of 12 x 12 x 12 for density of state (dos). 10−7 was set as the convergence threshold for self-consistent-field (scf) iteration. the output of the scf calculation provides the value of pressure p in equation 1 which was used to calculate the bulk modulus of the materials. b0 = −v d p dv = b1 + b2, b1 = −v 1 (p1− p2 ) (v1− v2 ) , b2 = −v 1 (p1− p3 ) (v1− v3 ) where b0 is the bulk modulus, v is the volume of the structure, and p is the pressure 3. result and discussion 3.1. structural properties the optimized crystal lattice parameters obtained are a = b = c = 11.342 å and α = β = γ = 900. the value obtained is in consistent with the experimental value 11.32 å [9]. the cesium cs atom site (1) in figure 1(a) was observed to have a bond with at least an atom site of each of the element existing in the material. it has ten (10) neighboring atoms sites with three (3) indium (in), 3 chlorine (cl), two (2) antimony (sb), and 2 cesium (cs) atoms. a total bond length of 30.594 å exists between the cs atom site (1) and the neighboring atoms. the alloying of au element in the in site of the compound reduces the total bond length to 30.588 å. the alloyed compound cs2ausbcl6 figure 1(b) having lesser total bond length when compared to the initial compound cs2insbcl6 is likely to be more stable. this is because the lesser the bond length the stronger the bond [10]. the strength, hardness, compressibility or flexibility of materials is often determined by the bulk modulus of the material. the bulk modulus (which is a function of pressure and volume) of the material was calculated (using equation 1). the result obtained shows that the bulk modulus of the material cs2insbcl6 calculated is 28.44 gpa. the substitution of in with au raises this property to 30.93 gpa. the ability of a material to resist deformation under applied stress is often defined by its bulk modulus [11]. greater value of the bulk modulus implies strong resistance to deformation and it indicates high compressive strength [12]. this means that the material (cs2ausbcl6) will offer more durability and resistance to deformation and as such it is suitable for solar cell application [11]. 3.2. electronic properties the band structure of the compounds cs2insbcl6 and cs2ausbcl6 were calculated using the projected augmented wave (paw) within dft-gga functionals. the band was calculated in the high symmetry directions of the brillion zone (x γ w γ l) for both compounds. it is observed that both materials demonstrate semiconducting behavior. the structures clearly reveals that both the valence band maximum (vbm) and the conduction band minimum (cbm) exist at γ-point symmetry. this indicates that the parent compound (cs2insbcl6) and the alloyed (cs2ausbcl6) are direct band gap materials. the band gap of the material (figure 2(b)) increased to 1.25 ev after substitution of in with au element (cs2ausbcl6) in the parent compound (figure 2(a) ) which has a band gap value of 0.99 ev that is in excellent agreement with the value of 0.98 ev obtained by [1] and most recently 1.02 ev obtained by [9]. the density of state (dos) shows the number of available states in the system, the number of the allowed electron (hole) states per volume at given energy. thus, it’s a tool for determining the energy distributions and carrier concentrations within semiconductor [13]. the plotted dos (figure 3(a) and 3(b)) clearly reveals the semiconducting behavior of the materials. it clearly shows the existence of energy range below and above the fermi level (0 ev). it shows that the higher peaks are situated at the valence band for cs2insbcl6 (figure 3(a)) and at conduction band for cs2ausbcl6 (figure 3(b)). this indicates the presence of many electrons at the particular energy state. 335 magaji et al. / j. nig. soc. phys. sci. 3 (2021) 334–339 336 figure 1. crystal structure figure 2. band structure figure 3. density of state (dos) 3.3. optical properties the study of optical properties involves the determining material response to an applied electromagnetic field particularly to visible light [14].the knowledge is essential in determination of the efficiency of photovoltaic materials. theoretically, an important function used in solid state physics to determine the linear response is the complex dielectric function [15]. the dielectric function ε (ω) = ε1 (ω) + iε2 (ω) (where ε1 (ω) is 336 magaji et al. / j. nig. soc. phys. sci. 3 (2021) 334–339 337 figure 4. dielectric functions the real part and iε2 (ω) is the imaginary part) of the materials were calculated within the linear response tddft. the optical transition mechanism is illustrated by the imaginary part of the dielectric function. a greater value for dielectric constant of material indicates that the material has comparatively low charge carrier recombination rate and as such the entire performance of the optoelectronics material will be improved [13]. the result obtained for cs2insbcl6 (figure 4(a)) shows that the value of the imaginary part (ε2) becomes zero at about 23.5 ev. this means that the material becomes transparent above 23.5 ev [14]. generally, the value of ε2 becomes nonzero immediately absorption starts. it can be seen that for ε2 the main peaks are located at 2 ev, 3 ev, 4 ev, and 5 ev (figure 4(a)). an important remark to make from the real part (ε1) plot is that ε1 decreases to negative values. the negative values of (ε1) indicates that at the particular energy level, there is mostly rejection of the incidence electromagnetic waves from the medium, as such the material exhibit metallic behavior [14]. as such it serves as protective means from the radiations in the energy region. comparing the result with the alloyed compound (cs2ausbcl6) figure 4(b) shows that the alloyed material will absorb radiations up to about 24 ev which is higher than the initial compound (13 ev). this is in line with the difference in the electronic band gap obtained from the band structure (figure 2(a) and (b)). also, it is observed that here’s a difference in the high positive peak value of ε1 with about 100 ev that exists between the initial (about 50 ev) and the alloyed compound (about 150 ev). this means that the alloyed compound is a good absorber of radiations. as such it is considered as potential good photovoltaic material for optoelectronics application. the absorption coefficient spectrum provides knowledge on the extent at which material absorbs photons. it indicates how far the light of a specific energy can penetrate into a material before being absorbed [14]. figure 5(a) shows the absorption coefficient graph which provides information about optimum solar energy conversion efficiency. it reveals the absorption spectra for cs2insbcl6and cs2ausbcl6. the absorption abruptly rises for energies higher than 0.5 ev which is in line with the band gap energy value. the absorption values are in the order of 10 4cm −1 from the infrared to visible and down to the ultraviolet range (1.24 ev to 1.7 ev to above 3.3 ev). several peaks were noticed to exist with cs2insbcl6 having its highest peak located just within the infrared energy region while the compound cs2ausbcl6 was noticed to have its utmost peaks located within the ultraviolet region (above 3.3 ev). this affirms the result obtained by the dft calculation of the electronic band structure (figure 2(a) & (b)) which reveals that the compound cs2ausbcl6 has a wider band gap than cs2insbcl6. this is evidently the reason for the difference in the energy range of the highest peaks. an important remark to make is that the material with higher peak in the high energy region will be a good absorber material and as such it is considered as a better candidate for photovoltaic applications. the electron energy loss function eels is a vital optical parameter used in explaining the energy loss of a fast electron traversing a material which is large at plasma frequency [14]. figure 5(b) represent the corresponding eels in function of photon energy of the materials. the prominent peaks were found to be at 28.02 ev for cs2insbcl6, 29.5 ev for cs2ausbcl6. the sharp maxima peak for eels indicates the existence of plasma resonance which appears at a specific incident light frequency which corresponds to the trailing edges in the reflection spectra often called plasma frequency (ωρ). the imaginary part of the dielectric function is zero at this point of energy which indicates rapid reduction in reflectance [15]. reflectivity is an important optical parameter used to describe 337 magaji et al. / j. nig. soc. phys. sci. 3 (2021) 334–339 338 figure 5. (a) absorption coefficient, (b) electron energy loss spectra, (c) reflectivity, (d) conductivity, (e) refractive index the ratio of reflected photon energy from material surface to the photon energy incident on the surface. figure 5(c) shows the reflectivity of the materials (cs2insbcl6 and cs2ausbcl6) as function of photon energy. the first edges were found to be at 90%, and 80% for cs2insbcl6, and cs2ausbcl6 respectively. these edges were noticed to be within the low energy level. 338 magaji et al. / j. nig. soc. phys. sci. 3 (2021) 334–339 339 the reflectivity begins to decrease as the energy increases from infrared region. it then increases to other peaks of 75, and 70 percent for cs2ausbcl6 and cs2insbcl6 respectively. these peaks were found to be located from visible to ultraviolet energy range. the decrease in the reflectivity in the high energy region means that the material is transmitting in the ultraviolet wavelength as a result of lower reflectance within the energy range. an important remark to make is that, the material with high reflectivity in the high energy region can be used as a good coating material to avoid solar heating [14]. conductivity is an important optical property that provides information on the electrical conductivity of material. the measure of ease at which heat or electric charge pass through a material is termed conductivity. figure 5(d) below shows the conductivity of the materials. it is observed that the material cs2insbcl6 has its own highest optical conductivity value of 400 ω−1cm−1. the conductivity rises to about 500 ω−1cm−1 for cs2ausbcl6. the increase in the photoconductivity of cs2ausbcl6 is due to high absorption of photons [14]. the material cs2ausbcl6 will consequently be a better candidate for photovoltaic applications. the refractive index of a material is a dimensionless quantity which describes how much light travels or refracted after entering a material [15]. figure 5(e) shows the refractive index n (ω) as a function of photon energy for the materials. the static refractive index n (0) was found to be 2.5 and 3.4 for cs2ausbcl6 and cs2insbcl6 respectively. from the graph, the materials possess a high refractive index at infrared region around (1.24 ev-1.7 ev) to visible region (1.7 ev-3.3 ev) and drastically decrease at higher energy from the visible to ultraviolet region (above 3.3 ev). furthermore, after 2.5 ev, and 6.0 ev the velocity of light is greater than the light celerity because n (ω) is less than one for cs2insbcl6, and cs2ausbcl6 respectively. the n (0) attain by cs2ausbcl6 (2.5) which is lower than the initial cs2insbcl6 (3.4) will be less dense as such, the light rays will bend more away from the normal line. the result obtained for cs2ausbcl6 (2.5) is in consistent with the result obtained for other lead-free halide cs2tii6 double perovskite of n (0) = 2.6 [18]. as such, cs2ausbcl6 can be considered as a good photovoltaic material. 4. conclusion in this research work, investigation was carried out on the effect of substitutional point defect in the structural, electronic and optical properties of a lead-free halide perovskites cs2insbcl6 for photovoltaic application within the framework of dft-gga (for the structural and electronic properties) and tddft (for optical properties) as implemented in quantum espresso (q.e-6.5). a substitutional point defect was performed in the indium site of the compound with substitution of indium with a gold element. the effect was investigated on both the structural, electronic and optical properties. the result of the structural investigation shows that the alloyed compound (cs2ausbcl6) has less total bond length when compared with the initial compound (cs2insbcl6) and such it makes more stable. the higher magnitude of bulk modulus attained by the alloyed compound means that it will offer more durability and resistance to deformation which makes it suitable for solar cell application. the electronic band structure (figure 2) shows the substitution widened the band gap from 0.99 ev to 1.25 ev. this increase in the band gap value of the defect compound indicates higher absorption of photons energy. the result of the optical properties shows that the defect material has higher absorption coefficient, lower reflectivity and higher optical conductivity when compared with the initial, this means that the defect compound is a good photovoltaic material and potentially better candidate for solar cell application. references [1] z. xiao, k. du, w. meng, j. wang, d. b. mitzi & y. yan, “intrinsic instability of cs2in (i) m (iii) x6 (m= bi, sb; 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[15] a. lawal, a. shaari, r. ahmed & n. jarkoni, “first-principles investigations of electron-hole inclusion effects on optoelectronic properties of bi2te3, a topological insulator for broadband photodetector.” physica b: condensed matter 520 (2017) 69. 339 j. nig. soc. phys. sci. 4 (2022) 49–53 journal of the nigerian society of physical sciences jensen-based new cryptographic scheme s. a. osikoya∗, e. o. adeyefa department of mathematics, faculty of science, federal university oye-ekiti, ekiti state, nigeria abstract the surgent availability of data and the demand for more data to implement several technological products which heavily depends on data have made the issue of cyber attacks, a global threat, of great concern. to ensure the protection of information, the heart of every nation, the development of formidable techniques which contribute to the advancement of present schemes is crucial to combat the challenge. in this research work, a mathematical framework, from the idea of cryptography, is developed using jensen polynomial. jensen polynomial is used for the encryption algorithm, and the laplace transform is used as a transformation tool to convert the plain text message to cipher text message. the decryption is a reversal which involves the use of structured mathematical techniques and the key for the message. doi:10.46481/jnsps.2022.325 keywords: jensen polynomial, laplace transform, cryptographic algorithm, information security,cryptographic scheme article history : received: 31 july 2021 received in revised form: 22 november 2021 accepted for publication: 23 november 2021 published: 28 february 2022 c©2022 journal of the nigerian society of physical sciences. all rights reserved. communicated by: j. ndam 1. introduction cryptography has always played major role in the security of information. there are many existing cryptographic schemes which are being adopted for internet security measure. this research is to contribute to the new research direction of developing cryptographic schemes using laplace transform as a transformation tool for encryption of text message from plain text to cipher text. in the existing literature, there are limitations imposed by the polynomials and functions adopted for the encryption. the integer coefficients, which play an important role in the encryption algorithm of the message, are restricted to the coefficients of the polynomials adopted for the schemes. this may open these schemes to brute force attack, or cryptanalytic attack, or both. ∗corresponding author tel. no: +234(0)8076344051 email address: samuelabidemi2@gmail.com (s. a. osikoya ) in [1], hiwarekar introduced the application of laplace transform for cryptographic scheme using hyperbolic function as the mathematical object for the transformation of the message into ciphertext using laplace transform. the concept was further extended in [2] adopting the ascii values and hyperbolic functions for the encryption of secret message using laplace transform for conversion into ciphertexts. adeyefa et al. introduced a new cryptographic scheme using chebyshev polynomials and laplace transform which further solidifies the idea introduced in [1]. the concept of cryptographic scheme developed with the aid of laplace transform was applied to network security in [4] and [5]. all of the schemes developed by the authors aforementioned are symmetric key cryptography. as an extension of laplace-based cryptography to public key cryptography, nagalakshmi et al. in [9] implemented the concept of laplace transform to elgamal scheme but this possesses an inherent computational problem at the decryption phase. the symmetric schemes developed in he existing literature pos49 osikoya, s. a. & adeyefa, e. o. / j. nig. soc. phys. sci. 4 (2022) 49–53 50 sess a unique problem. the functions and polynomials used by the authors are to generate integer coefficients for the laplace transformation which plays an integral part of the schemes. but such coefficients which are restricted to the coefficients of the hyperbolic functions, exponential functions or polynomials which are well-known can expose such schemes to cryptanalysis attacks. in this research, jensen polynomial which include an arithmetic function is adopted to overcome the problem inherent with other polynomials, exponential functions or hyperbolic functions used in the existing literature. the coefficients of jensen polynomial are generated through the arithmetic function, thus making the coefficients unrestricted by the polynomial. this non-restriction of integer coefficients makes the algorithm developed using jensen polynomials more robust and versatile. jensen polynomials permit the use of any arithmetic function thus making it more reasonable for practical usage compare to other functions and polynomials. the rest of the paper is summarized as follows. in section 2, the basic mathematical tools required to develop the cryptographic algorithm is discussed. section 3 introduces the crytographic algorithm with illustration, followed by an example to substantiate the algorithm. the paper is concluded in section 4 in which further research directions are suggested for the readers. 2. major result preliminaries in this section, the mathematical tools needed for the development of the algorithm are presented. 2.1. jensen polynomial [8] jensen polynomial of degree d and shift n of an arbitrary sequence {αk}∞k=1 of real numbers is the polynomial jd,nα = d∑ j=0 ( d j ) α(n + j)x j (1) 2.2. the laplace transform [7] let f (t) be a function for t ≥ 0. the laplace transform of f (t) denoted by l[ f (t)] or f(s) is defined by the equation: l[ f (t)] = f (s) = ∫ ∞ 0 e−st f (t) dt (2) where ∈< or c, supposing the integral converges. the inverse of the laplace transform in (1) is defined as: l−1[f (s)] = l−1 (∫ ∞ 0 e−st f (t) dt ) (3) 3. methodology the encryption function is the function in equation (1). for the purpose of this study, we shall slightly modify equation (1) to enable its adaptability for the encryption process of the text message. define f (x) = d∑ j=1 φ jα (n + j) x j (4) where φ j is the ascii value of each string in the message. the required sequence to be chosen must be such that satisfies the following properties. 1. the range of {αk}∞k=1 is the set of positive integers z + ⊂ r; 2. the sequence {αk}∞k=1 is a non-decreasing sequence. 3. the nth term of the sequence should be a large number of about 128-bits. the following subsections outline the steps involved in the cryptographic algorithm. 3.1. encryption algorithm let m be the message to be encrypted with length l. suppose φ1,φ2, . . . ,φl are the strings in the message. the encryption of the message is outline in the following steps. step i: obtain the ascii value of each of the alphabets( and symbols) in the text message. define an arbitrary sequence which satisfies the properties above. the degree of the polynomial is the length of the string to be encrypted. the shift n is is equivalent to the length of the message. step ii: obtain the laplace transform of the function obtained in step i. the numerators of the laplace transform are saved as resultant values, r j , for the message to be encrypted. step iii: obtain the cipher text by taking the modulo 52 of each of the resultant values, that is (r j + x)mod52, where x is a value known to emmanuel and samuel. the cipher presented using the alphabet system provided in table 1. example 3.1. the message “attack” is to be sent through a channel an eavesdropper has access to just like emmanuel and samuel who are to share the information. define the arbitrary sequence: α (n) = 22n − 1 α (n + j) = 22n+2 j − 1 the jensen polynomial is the equation f (x) = d∑ j=1 ( j d ) φ j ( 22n+2 j − 1 ) x j where φ j is the ascii value of each letter in the message. therefore, the ascii value of each letter in the message is, φ1 = 65,φ2 = 116,φ3 = 116,φ4 = 97,φ5 = 99,φ6 = 107. choosen = 13. then the encryption function is computed in the simplest form as: f (x) = 104689827450x + 1868310792020x2 (5) + 9964324124400x3 + 24996709661265x4 + 40819369180590x5 + 29411936042901x6 50 osikoya, s. a. & adeyefa, e. o. / j. nig. soc. phys. sci. 4 (2022) 49–53 51 table 1: the alphabet system for the encryption and decryption a b c d e f g h i j k l m 0 1 2 3 4 5 6 7 8 9 10 11 12 n 0 p q r s t u v w x y z 13 14 15 16 17 18 19 20 21 22 23 24 25 a b c d e f g h i j k l m 26 27 28 29 30 31 32 33 34 35 36 37 38 n 0 p q r s t u v w x y z 39 40 41 42 43 44 45 46 47 48 49 50 51 take the laplace transform of the function above. l[ f (x)] = 104689827450 s2 + 3736621544040 s3 (6) + 59785944746400 s 4 + 599921031870360 s5 + 4898324301670800 s6 + 21176593950888720 s7 the resultant values are obtained from the last equation are presented in table 2. next, the cipher text are obtable 2: the resultant values from the laplace transform r1 r2 r3 104689827450 3736621544040 59785944746400 r4 r15 r6 599921031870360 4898324301670800 21176593950888720 tained with the equation, φcj = ( r j + x ) mod52, where x = 17.φc1 = 43,φ c 2 = 25,φ c 3 = 45,φ c 4 = 9,φ c 5 = 17,φ c 6 = 49. using the english letters presented in the table 1, we obtain the cipher text “rztjrx” with the key: 2013265912, 71858106616, 1149729706661, 11536942920584, 94198544262900, 407242191363244. 3.2. decryption algorithm the following steps describe the encryption phase of the cryptographic scheme. step i: obtain the resultant value of each string in the cipher text received with the equation: ri = 52ki + ci − x step ii: expand equation (1) in terms of the φ′i s. step iii: the ascii values of the original message, that is the plain text, is computed using the equation: φi = ri coe f fi ∗ i! (7) where coe f fi ∗ i! is the coefficient of ri in the equation obtained in step ii. using the steps outlined in the decryption algorithm, the resultant values in table 2 are obtained. next, equation (1) is expanded in terms of the φi s to obtain: f (x) = 6∑ j=1 φ ( j 6 ) ( 226+2 j − 1 ) x j (8) = 1610612730φ1 + 16106127345φ2 + 85899345900φ3 + 257698037745φ4 + 412316860410φ6 + 274877906943φ6 the ascii values of the plain text is obtained using the equation (8). therefore,φ1 = 65; φ2 = 116; φ3 = 116; φ4 = 97; φ5 = 99; φ6 = 107. this implies the original message. the next example below is an improvement of the illustration given in the algorithm. the value of j chosen is 1024 bits. the value of n chosen is such that have only two primes as its prime factors. one of the primes is used in the arithmetic function while the other is used for the value of x. the message to be encrypted is ‘missile’. although in the next example, we shall not use a 1024 bits number because of space. but we shall demonstrate the idea proposed. example 3.2. the encryption phase first, the ascii value of each letter in the message is obtained as follows:m = 77, i = 105, s = 115, s = 115, i = 105, l = 108, e = 101. the encryption function in the illustration is used in this example. f (x) = d∑ j=1 φ ( j d ) ( 22n+2 j − 1 ) x j the large integer chosen is the number, q = n = x = 221 where n = 13, and x = 17. the jensen polynomial for the encryption is computed as: f (x) = 144686710245x + 2367600719715x2 (9) + 17287243362375x3 + 69148973461575x4 + 151526446200675x5 + 207807697648908x6 + 111050674405275x7 51 osikoya, s. a. & adeyefa, e. o. / j. nig. soc. phys. sci. 4 (2022) 49–53 52 the laplace transform of equation (7) is next obtained: l[ f (x)] = 144686710245 s2 + 4735201439430 s3 (10) + 103723460174250 s4 + 1659575363077800 s5 + 18183173544081000 s6 + 149621542307213760 s7 + 559695399002586000 s8 thus, the resultant values of the message are stored in a computer memory in table 3. table 3: the resultant values of the message. r1 r2 144686710245 4735201439430 r3 r4 103723460174250 1659575363077800 r5 r6 18183173544081000 149621542307213760 r7 559695399002586000 next, the cipher text values are computed using the equation: φ′i = (ri + x) mod52 where x = 17. the cipher text values and the respective keys are presented in table 4. the corresponding cipher text is ‘qlldrvp’. table 4: the cipher text value and the corresponding key value. φi cipher value key φ1 42 2782436735 φ2 11 91061566143 φ3 11 1994681926428 φ4 29 31914910828419 φ15 17 349676414309250 φ6 21 2877337352061803 φ7 41 10763373057742038 the decryption phase the resultant values of the original message are first obtained using the equation: ri = 52ki + γi − x where the γi is the respective cipher value of the cipher text. substituting the respective cipher value and key, the resultant values in table 3 are obtained. the equation (4) is expanded in form of the φi s as follows: f (x) = 1879048185φ1 x + 22548578283φ2 x 2 (11) + 150323855325φ3 x 3 + 601295421405φ4 x 4 + 1443109011435φ5 x 5 + 1924145348601φ6 x 6 + 1099511627775φ1 x 7 the plain text is obtained using equation (8) to get the φi.φ1 = 77,φ2 = 105,φ3 = 115,φ4 = 115,φ5 = 105,φ6 = 108,φ7 = 101. these are the ascii values of the original message. hence, the decryption of the cipher text is obtainable. 4. security analysis the plaintext is distinctively different from the ciphertext. the number of shifts cannot be calculated under this scheme as the same letter corresponds to different letter in the ciphertext thereby making the scheme secured against passive attack. the developed scheme is secured against chosen-plaintext attack (cpa) which is an active type of attacks against cryptographic scheme. it involves the attacker’s performance of encryption queries for plaintext of their choice and observation of resulting ciphertexts [10]. but the model developed involved the use of arithmetic function which may be kept secret between two parties thus protecting the transmission of the message. the knowledge of the arithmetic function does not guarantee the decryption of the message because the algorithm permits the addition of other keys in the steps thus making it harder for the attackers to break. thus, the scheme is protected against cpa. the length of message string is the same as the length of the ciphertext, thus making the issue of storage memory of little concern. 5. conclusion the mathematical modeling for cryptography using jensen polynomials is free from the integer coefficients constraint associated with other polynomials like chebyshev, hermite, legendre etc., or functions adopted for the encryption and decryption of text message. the possibility of using a suitable encryption through the jensen polynomial makes the scheme more robust and versatile. the coefficient of the polynomial can be made as large as possible, a property which constitutes the security of the scheme against brute force attack. the complexity aspect of the algorithm is in the management of the key; as a key cannot be shared between more than two parties. this implies that a user will have about ten different keys in order to communicate with ten different people. therefore, the concept introduced can be explored in different directions. first, the study can be used as the basis for public key cryptographic scheme, also known as asymmetric cryptography which overcome the issue of key management. this kind of extension will further help the implementation of the scheme for more practical use. also, the research can be deeply explored to handle image encryption which it is hoped to be the next research, and the introduction of the concept to asymmeric key cryptography. references [1] a. p. hiwarekar, “application of laplace transform for cryptographic scheme”, proceedings of world congress on engineering, 1 (2013) 95. [2] c. jayanthi & v. srinivas, “mathematical modeling for cryptography using laplace transform”,international journal of mathematical trends and technology, 2 (2019) 10. 52 osikoya, s. a. & adeyefa, e. o. / j. nig. soc. phys. sci. 4 (2022) 49–53 53 [3] e. o. adeyefa, l. s. akinola, & o. d. agbolade, “a new cryptographic scheme using chebyshev polynomials”, 2020 international conference in mathematics, computer engineering (icmcecs), (2020) 3. [4] a. mittal & r.gupta, “kamal transformation based cyrptography technique in network security involving ascii value”, international journal of innovative technology and exploring engineering, 8 (2019) 3448. [5] d. swati, a. archana and j. swati, “laplace transformation based cryptographic technique in network security”, international journal of computer applications, 7 (2016) 10. [6] m. saha, “application of laplace-mellin transform for cryptography”, rai journal of technology, research and innovation, 1 (2017) 12. [7] b. d. gupta, “mathematical physics”, ist ed., reprint, ansari road, new delhi: india, (1980). [8] m. griffin, k. ono, l. rolen & d. zagier, “jensen polynomials for the riemann zeta function and other sequences”, (2019). [9] g. nagalakshmi, s. a. chandra, & s. n. ravi, “an implementation of elgamal scheme for laplace transform cryptosystem”, indian journal of computer science and engineering (ijcse, 1 (2020) 48. [10] jean-philippe aumasson, “serious cryptography: a practical introduction to modern encryption”, (2018). 53 j. nig. soc. phys. sci. 3 (2022) 429–445 journal of the nigerian society of physical sciences quantum chemical studies on c4h4n2 isomeric molecular species e. e. etima,∗, m. e. khanb, o. e. godwina, g. o. ogofothab adepartment of chemical science, federal university wukari, nigeria bdepartment of chemistry, federal university lokoja, kogi state, nigeria abstract quantum chemical calculations have been carried out on c4h4n2 isomeric molecular species using the g4 method and compared with experimental values where available, probing parameters like thermochemistry, structural parameters (e.g. bond length, bond angles), rotational constants, vibrational spectroscopy and dipole moments. pyrimidine was found to be the most stable of all the isomers with ∆ f h0 =37.1 kcal/mol. a critical analysis showed high correlation and consistency between the computed and experimental values of all the parameters under study and therefore providing the needed rationale to validate the values provided for the isomers which do not have available experimental data. doi:10.46481/jnsps.2021.282 keywords: isomers, pyrimidine, experimental, computational article history : received: 30 june 2021 received in revised form: 16 august 2021 accepted for publication: 07 september 2021 published: 29 november 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: b. j. falaye 1. introduction quantum chemistry is an area of computational chemistry which could reproduce experimental chemical phenomena mathematically, this branch of study avails one the opportunity to understand the electronic structures and model the nature of interactions molecules undergo not just for stable molecules as usually provided by experimental procedures but also for the short-lived intermediates or unstable analogues [14]. it has been applied in different fields in chemistry where researchers have been able to make accurate predictions of future reactions [5, 6], physicochemical properties, docking, rate constants, protein calculations, calculations of potential energy surfaces, electronic structures of molecules and their isomers, molecules in the interstellar medium (ism) [7]. ∗corresponding author tel. no: email address: emmaetim@gmail.com (e. e. etim ) c4h4n2 has many isomeric species of wide spread relevance. pyrazines (c4h4n2) also known as 1,4-diazines are heterocyclic aromatic organic compounds commonly distributed in nature such as in bacteria, fungi, insects and plants e.g. potatoes, coffee, nuts. they are responsible for the nutty and roasty smell which is reminiscent of cocoa and coffee. these compounds are used to improve aroma/flavor in food and cosmetic industries [8 – 11]. nucleotides (such as cytosine, thymine), vitamin such as thiamine (i.e. vitamin b1), hiv drug zidovudine and synthetic compounds like barbitutrates all contain the isomer pyrimidine [12]. the isomer pyridazine finds applications as herbicides (such as pyridafol, pyridate, credazine), as drugs such as minaprine, cadralazine, cefozopran[13]. it has been cumbersome studying certain molecules whose isomers or themselves are unstable molecules such as the present case. this challenge may be overcome to a certain level by the application of computational approach which has been shown 429 etim et al. / j. nig. soc. phys. sci. 3 (2022) 429–445 430 table 1: standard enthalpy of formation of c4h4n2 isomeric species molecules ∆ f h0 (kcal/mol) 1,2-diisocyanoethane 92.4 1,3-butadiene-1,4-diimine 88.0 4-amino-2-butynenitrile 83.0 iminopyrrole 62.3 2-methylene-2h-imidazole 60.1 pyridazine 59.0 1,1-dicyanoethane 54.7 pyrazine 41.0 pyrimidine 37.1 to be a relative good substitute for experimental approach. thus, the present study aims at using computational methods to investigate the isomers of the c4h4n2 group by predicting the standard enthalpy of formation, bond length and angles, dipole moment, vibrational frequencies, rotational constants etc., and comparing their results with the available experimental values where available. 2. computational methods gaussian 09 suite of program was employed in performing all the quantum chemical calculations reported in this study. the effectiveness of the g4 composite method has been reported in many literatures [14-16] in addition to experience from our previous studies [1723]; as such the molecule under study was optimized using the g4 level of theory. 3. result and discussion the possible isomers of the c4h4n2 isomeric group include; 1,2-diisocyanoethane, 1,3-butadiene-1,4-diimine, 4-amino2-butynenitrile, iminopyrrole, 2-methylene-2h-imidazole, pyridazine, 1,1-dicyanoethane, pyrazine, pyrimidine. the results of the quantum chemical calculations carried out on the c4h4n2 isomeric species using the g4 level of theory are presented and discussed below under the different subheadings: 3.1. thermochemistry the focal point of thermochemistry is energy changes such as enthalpy of formation and many other energy related parameters. fig 1 shows the optimized geometries of the c4h4n2 isomeric molecular species while the standard enthalpies of formation (∆fho) computed for these isomeric species are presented in table 1. pyrimidine has the least enthalpy of formation of 37.1 kcal/mol which corresponds to the most stable of all the isomers, 1,2-diisocyanoethane is shown to be the least stable of all the isomers of the c4h4n2 isomeric group having the value of 92.4 kcal/mol as the heat of formation. the experimentally reported standard heat of formation of pyrimidine ranges from 46.1±0.5 to 47.75± 0.4 [24-26], the disparity between the computed and experimental values point towards the possibility of an error in the experimentally reported value as weisenburger and co-workers [27]. according to them, experimental measurement of heat of formation is always inaccurate and impractical. from previous studies, the g4 method has proven to be effective in predicting the enthalpy of formation that is in good accuracy with experimental results [14-23]. standard enthalpy of formation is an important parameter that can be applied for safe and scaling up of chemical processes involving thermal stability. it helps researchers in predicting the spontaneity of a reaction, know whether a reaction can be favourable or not and the reactants and products quantities [27]. 3.2. vibrational spectroscopy table 2 depicts the vibrational frequencies of pyrimidine (the most stable isomer of the c4h4n2 isomeric group) with the corresponding spectrum in figure 2. the vibrational frequencies and the corresponding spectra for other isomers of the c4h4n2 isomeric group are presented in the appendix (tables a1-a3 and figure a respectively). table 1 contains the calculated and experimental values of the vibrational frequencies of pyrimidine. the error between the values ranges between 0.3-4 cm−1. the computed values are in excellent agreement with the reported experimental values. thus, for the other isomers with no experimentally measured vibrational frequencies, the values computed at the g4 level (presented in the appendix) are believed to be accurate. the g4 composite method has also been reported to give accurate predictions for vibrational spectroscopic parameters for other molecular species with experimentally known values [14-23]. among other applications, the vibrational spectroscopy parameters are useful in the chemical examination of the interstellar medium especially for the astronomical observation of interstellar molecular species with no dipole moment [17,20]. 3.3. rotational constants rotational spectroscopy remains the most important spectroscopic technique employed in the astronomical observation of molecular species from different regions of the interstellar medium. the experimentally measured rotational constants (from the nist webbook) for pyrimidine and the values obtained at the g4 level are presented in table 3 below. as shown in the table, there is a good agreement between the experimental and the computed values of the rotational constant of pyrimidine. the table also contains the rotational constants calculated for other isomers of the c4h4n2 isomeric group at the g4 level of theory with no experimentally measured values. analysis of the difference showed errors of 0.0261635, 0.0330026 and 0.0156889 ghz for the a, b and c rotational constants of pyrimidine respectively. this level of accuracy suggests a good level of accuracy for the rotational constants obtained for other isomers at the g4 level with no experimentally measured values. 3.4. structural parameters the bond lengths and bond angles of pyrimidine are presented in table 4 while fig. 3 depicts the optimized geometry. as shown in the table, there is an excellent agreement 430 etim et al. / j. nig. soc. phys. sci. 3 (2022) 429–445 431 1,2-diisocyanoethane 1,3-butadiene-1,4-diimine 4-amino-2-butynenitrile iminopyrrole 2-methylene-2h-imidazole pyridazine 1,1-dicyanoethane pyrazine pyrimidine figure 1: optimized geometry of c4h4n2 isomeric groups figure 2: calculated ir frequencies of pyrimidine between the experimentally measured values and the computationally predicted values. for example, both the experimental (1.087å) and the computational (1.087å) values for rch bond length. for the other bond lengths reported for pyrimidine, the difference between the experimental and the computational values range 0.36-0.46 å while for the predicted bond angles, the difference between the experimental and the computational values range from 0.19-1.10 degrees. these findings suggest that 431 etim et al. / j. nig. soc. phys. sci. 3 (2022) 429–445 432 table 2: vibrational frequencies of pyrimidine calculated frequency (cm−1) experimentalfrequency (cm−1) error(cm−1) 353 347 1.69 411 398 2.9 632 621 1.58 693 679 1.45 744 719 3.36 835 804 3.59 988 960 2.74 1009 969 3.96 1011 980 2.97 1030 1033 -0.29 1083 1065 1.01 1095 1155 2.7 1165 1158 0.85 1227 1224 5.6 1260 1356 2.78 1395 1224 2.79 1439 1356 1.95 1496 1411 2.07 1610 1465 2.55 1612 1569 2.48 3154 1572 3.36 3157 3047 3.39 3165 3053 3.51 3207 3082 3.89 table 3: rotational constant of c4h4n2 isomers molecules rotation constants (ghz) a b c pyrimidine calculated 6.3010414 6.0983826 3.0990279 experimental 6.2748779 6.06538 3.083339 error 0.0261635 0.0330026 0.0156889 1,2diisocyanoethane calculated 7.4362588 2.5722569 2.0604923 1,3butadiene1,4-diimine calculated 23.1647044 1.3418215 1.2867453 4-amino-2butynenitrile calculated 23.0996150 1.3419073 1.2868109 iminopyrrole calculated 8.5048965 4.1234186 2.7770331 2-methylene2h-imidazole calculated 8.7736412 4.2378633 2.8575861 pyridazine calculated 6.4337599 5.9503379 3.0913068 pyrazine calculated 6.4337599 5.9503379 3.0913068 the bond lengths and bond angles predicted with the g4 method for the other isomers of the c4h4n2 isomeric group presented in the appendix (tables a4-a6) with no experimental values will have a good level of accuracy and can be used when required. 3.5. dipole moments dipole moment is useful in determining the polar nature of the chemical bond. it is also useful in astrophysics and related areas such as astrochemistry and astrobiology as the dipole of a molecule plays an important role in the astronomical observation of such molecule [2]. the dipole moments obtained at the g4 level for all the isomeric molecular species in this study 432 etim et al. / j. nig. soc. phys. sci. 3 (2022) 429–445 433 table 4: bond distances/ angles of pyrimidine and its isomers description calculated value (å) exp. value (å) error connectivity atom 1 atom 2 atom 3 rch 1.087 1.087 0 4 4 rch 1.079 1.083 0.36 3 3 rch 1.082 1.087 0.46 2 2 rcc 1.393 1.389 0.29 3 2 rcn 1.328 1.334 0.45 1 1 accc 117.8 116.5 1.10 4 3 2 ahcc 120.90 121.13 0.19 2 2 3 accn 121.20 122.37 0.97 3 4 2 table 5: dipole moment of c4h4n2 isomers molecule calculated dipole moment (debye) experimental dipole moment (debye) 1,2-diisocyanoethane 5.1506 1,3-butadiene-1,4diimine 3.6290 4-amino-2-butynenitrile 3.7238 iminopyrrole 3.2055 2-methylene-2himidazole 1.0758 pyridazine 4.5926 1,1-dicyanoethane 4.5904 pyrazine 0.0000 pyrimidine 2.4133 2.33 figure 3: optimized geometry of pyrimidine are presented in table 5. the experimental dipole moment of 2.33d [29] reported for pyrimidine is in good agreement with the value (2.41d) calculated at the g4 level. there are no experimentally reported dipole moments values for the other isomers of the pyrimidine isomeric group. however, the good agreement between the experinmentally measured and the computationally calculated values for pyrimidine suggest a good accuracy for the dipole moments predicted for those molecular species with no experimental values. 4. conclusion the gaussian g4 compound model has been applied in computing some quantum chemical properties for the c4h4n2 isomeric molecular species. spectroscopic parameters (rotational and vibrational), bond distances, bond angles and dipole moments have been calculated for all the isomeric molecular species considered in this study. the results show a good agreement between the values obtained with the g4 method and the available experimentally measured values. this good agreement suggests a good accuracy for those the computationally predicted values with no experimental values. thus, the predicted values at the g4 level of theory could serve as useful data where there are no experimental values. acknowledgments the authors will like to appreciate the handling editor and the anonymous reviewers for their advice to make this work a success. appendix figure a: ir spectra of c4h4n2 isomeric species. 433 etim et al. / j. nig. soc. phys. sci. 3 (2022) 429–445 434 table a1: ir frequencies and intensities of c4h4n2 isomers pyrazine 1,1-dicyanoethane pyridazine frequency (cm−1) ir intensities frequency (cm−1) ir intensities frequency (cm−1) ir intensities 350.2454 0 142.6498 10.3325 377.4241 8.4376 436.8441 20.8309 212.9721 0.0061 378.1465 0 609.3268 0 217.325 10.417 629.7531 0.1251 718.6392 0 232.5641 0.1562 678.7734 3.7111 784.4478 0 390.8231 0.6653 773.826 26.977 813.6882 16.8768 501.3777 0.5045 784.7741 0 954.5671 0 577.079 2.0394 955.9237 0 995.7849 0 591.9736 0.0401 988.5043 0.0273 1004.4743 0 789.326 0.9436 1015.5701 6.7412 1032.2801 41.3863 924.1966 0.5222 1025.6928 0 1043.5004 0 1024.5549 3.0293 1057.3562 1.8513 1091.2542 11.3623 1077.6373 1.9773 1085.4055 1.9376 1168.3175 3.0353 1137.7852 11.4915 1094.3822 10.9875 1234.2883 3.3451 1293.1894 0.2956 1172.7042 0 1257.5615 0 1325.8192 10.5516 1198.4094 0.0296 1377.4992 0 1412.7563 0.8374 1315.9121 2.7813 1443.5577 32.5811 1489.6743 7.1366 1438.4468 17.1702 1516.3516 1.6121 1495.8979 2.9216 1480.3479 1.0699 1581.3618 0 2369.8219 1.4472 1604.0217 4.1776 1617.6288 0 2375.5886 1.0261 1608.3885 6.5778 3154.1328 0 3040.9897 0.0176 3171.291 11.3104 3154.5805 6.8491 3064.7956 7.0787 3175.4201 0.6737 3170.0288 69.5698 3150.5629 6.2676 3191.228 18.3941 3176.6122 0 3154.8613 4.5128 3205.5438 8.2119 434 etim et al. / j. nig. soc. phys. sci. 3 (2022) 429–445 435 table a2: ir frequencies and intensities of c4h4n2 isomers 2-methylene-2himidazole iminopyrrole 4-amino-2-butynenitrile frequency (cm−1) ir intensities frequency (cm−1) ir intensities frequency (cm−1) ir intensities 231.2641 16.2028 222.6459 1.5177 102.9298 3.212 367.2831 5.0478 437.8852 15.624 136.5392 6.7159 545.8568 0 513.7051 0.5416 249.5854 0.2682 729.7105 0.9392 673.5182 13.329 273.9337 21.0114 757.4695 6.9218 709.63 1.5332 384.9431 30.924 778.714 0.0001 821.884 0.5162 500.8402 11.2464 893.0866 0.2762 839.2175 34.5306 575.9942 2.9544 912.9179 8.5062 878.6648 11.3524 582.7078 1.7895 915.8468 10.8072 926.5924 10.2062 699.5133 11.1841 954.3095 42.6853 967.2053 5.8961 889.2243 0.0003 961.4052 0.0001 969.8546 11.5774 895.8998 201.8104 979.8251 16.2158 993.4477 44.2831 1098.3395 26.2627 991.4666 31.7917 1058.7038 51.7342 1148.1344 4.2125 1202.2146 17.8701 1093.3592 5.4566 1183.4347 0.2291 1303.1681 6.2567 1280.4929 34.9662 1356.3555 33.0669 1346.2606 21.0282 1341.6019 75.8567 1383.0345 0.0121 1412.9732 17.8688 1353.3948 11.9765 1459.7535 5.1009 1496.1998 11.0291 1548.0035 19.6616 1669.8384 18.9874 1613.199 2.8089 1646.7243 11.3723 2258.8654 0.001 1714.8692 3.0856 1740.6104 20.5283 2396.7797 101.4735 3180.8768 0.0002 3175.7293 18.3444 3036.9215 13.0137 3196.1465 6.9809 3234.5262 3.2481 3070.8682 4.6604 3211.1464 15.5153 3266.179 1.1482 3496.6081 1.6966 3283.7487 0.0178 3434.2972 4.1245 3574.5662 3.4152 435 etim et al. / j. nig. soc. phys. sci. 3 (2022) 429–445 436 table a3: ir frequencies and intensities of c4h4n2 isomers 1,3-butadiene1,4-diimine 1,2-diisocyanoethane frequency (cm−1) ir intensities frequency (cm−1) ir intensities 83.6289 4.2328 78.7298 4.2564 127.629 1.9142 168.9156 0.5605 247.6388 0 192.4169 3.7141 413.9178 39.0839 262.623 0.1196 422.8897 0.0001 299.0842 0.3214 553.3442 90.5491 389.5249 0.2258 572.2602 0.0004 551.2205 13.3477 602.0439 17.4793 827.5204 7.2786 678.4837 0 859.6152 9.5185 879.0897 0.0004 1035.2344 3.9097 905.6999 99.6452 1041.9629 7.1071 1033.4248 0.0001 1091.3162 0.7493 1042.7608 612.5875 1268.8807 1.8639 1064.7913 0.0002 1304.1572 0.188 1144.81 40.0234 1384.277 13.328 1186.8856 0 1392.4053 22.9043 1291.0522 42.9681 1485.0234 17.2458 1485.9837 0 1486.765 0.1757 2123.9009 896.411 2224.3091 150.9267 2130.4724 0.1389 2226.309 166.4745 3174.8502 0.0003 3052.5648 3.5487 436 etim et al. / j. nig. soc. phys. sci. 3 (2022) 429–445 437 table a4: bond radius and angles of c4h4n2 isomers pyrazine 1,1-dicyanoethane pyridazine description cal. value description cal. value description cal. value r(1-2) 1.393 r(1-2) 1.154 r(1-2) 1.333 r(1-6) 1.334 r(1-3) 1.470 r(1-6) 1.332 r(1-7) 1.087 r(3-4) 1.470 r(2-3) 1.395 r(2-3) 1.334 r(3-6) 1.549 r(2-8) 1.086 r(2-8) 1.087 r(3-10) 1.098 r(3-4) 1.381 r(3-4) 1.334 r(4-5) 1.154 r(3-7) 1.084 r(4-5) 1.393 r(6-7) 1.092 r(4-5) 1.395 r(4-9) 1.087 r(6-8) 1.091 r(4-9) 1.084 r(5-6) 1.334 r(6-9) 1.092 r(5-6) 1.333 r(5-10) 1.087 a(2-1-3) 178.0 r(5-10) 1.086 a(2-1-6) 122.1 a(1-3-4) 110.7 a(2-1-6) 119.4 a(2-1-7) 120.8 a(1-3-6) 111.3 a(1-2-3) 123.8 a(1-2-3) 122.1 a(1-3-10) 107.3 a(1-2-8) 114.9 a(1-2-8) 120.8 a(4-3-6) 111.3 a(1-6-5) 119.4 a(6-1-7) 117.1 a(4-3-10) 107.3 a(3-2-8) 121.2 a(1-6-5) 115.9 a(3-4-5) 178.0 a(2-3-4) 116.8 a(3-2-8) 117.1 a(6-3-10) 108.9 a(2-3-7) 120.9 a(2-3-4) 115.9 a(3-6-7) 109.5 a(4-3-7) 122.3 a(3-4-5) 122.1 a(3-6-8) 110.5 a(3-4-5) 116.8 a(3-4-9) 117.1 a(3-6-9) 109.5 a(3-4-9) 122.3 a(5-4-9) 120.8 a(7-6-8) 109.0 a(5-4-9) 120.9 a(4-5-6) 122.1 a(7-6-9) 109.3 a(4-5-6) 123.8 a(4-5-10) 120.8 a(8-6-9) 109.0 a(4-5-10) 121.2 a(6-5-10) 117.1 r(1-2) 1.154 a(6-5-10) 114.9 r(1-2) 1.393 r(1-3) 1.470 r(1-2) 1.333 r(1-6) 1.334 r(3-4) 1.470 r(1-6) 1.332 r(1-7) 1.087 r(3-6) 1.549 r(2-3) 1.395 r(2-3) 1.334 r(3-10) 1.098 r(2-8) 1.086 r(2-8) 1.087 r(4-5) 1.154 r(3-4) 1.381 r(3-4) 1.334 r(6-7) 1.092 r(3-7) 1.084 r(4-5) 1.393 r(6-8) 1.091 r(4-5) 1.395 r(4-9) 1.087 r(6-9) 1.092 r(4-9) 1.084 r(5-6) 1.334 a(2-1-3) 178.0 r(5-6) 1.333 r(5-10) 1.087 a(1-3-4) 110.7 r(5-10) 1.086 a(2-1-6) 122.1 a(1-3-6) 111.3 a(2-1-6) 119.4 a(2-1-7) 120.8 a(1-3-10) 107.3 a(1-2-3) 123.8 a(1-2-3) 122.1 a(4-3-6) 111.3 a(1-2-8) 114.9 a(1-2-8) 120.8 a(4-3-10) 107.3 a(1-6-5) 119.4 a(6-1-7) 117.1 a(3-4-5) 178.0 a(3-2-8) 121.2 a(1-6-5) 115.9 a(6-3-10) 108.9 a(2-3-4) 116.8 a(3-2-8) 117.1 a(3-6-7) 109.5 a(2-3-7) 120.9 a(2-3-4) 115.9 a(3-6-8) 110.5 a(4-3-7) 122.3 a(3-4-5) 122.1 a(3-6-9) 109.5 a(3-4-5) 116.8 a(3-4-9) 117.1 a(7-6-8) 109.0 a(3-4-9) 122.3 a(5-4-9) 120.8 a(7-6-9) 109.3 a(5-4-9) 120.9 a(4-5-6) 122.1 a(8-6-9) 109.0 a(4-5-6) 123.8 a(4-5-10) 120.8 r(1-2) 1.154 a(4-5-10) 121.2 a(6-5-10) 117.1 r(1-3) 1.470 a(6-5-10) 114.9 437 etim et al. / j. nig. soc. phys. sci. 3 (2022) 429–445 438 table a5: bond radius and angles of c4h4n2 isomers 2-methylene2h-imidazole iminopyrrole 4-amino-2-butynenitrile description cal. value description cal. value description cal. value r(1-2) 1.413 r(1-2) 1.437 r(1-2) 1.466 r(1-3) 1.295 r(1-3) 1.289 r(1-5) 1.465 r(2-5) 1.414 r(2-8) 1.484 r(1-9) 1.097 r(2-8) 1.339 r(2-9) 1.270 r(1-10) 1.097 r(3-4) 1.473 r(3-4) 1.486 r(2-3) 1.209 r(3-7) 1.084 r(3-7) 1.087 r(3-4) 1.365 r(4-5) 1.295 r(4-6) 1.081 r(4-6) 1.162 r(4-6) 1.084 r(4-8) 1.341 r(5-7) 1.016 r(8-9) 1.082 r(5-8) 1.079 r(5-8) 1.016 r(8-10) 1.083 r(9-10) 1.025 a(2-1-5) 115.2 a(2-1-3) 103.6 a(2-1-3) 104.8 a(2-1-9) 108.9 a(1-2-5) 113.0 a(1-2-8) 109.2 a(2-1-10) 108.9 a(1-2-8) 123.6 a(1-2-9) 125.3 a(1-2-3) 177.1 a(1-3-4) 109.9 a(1-3-4) 114.1 a(5-1-9) 108.7 a(1-3-7) 122.8 a(1-3-7) 121.5 a(5-1-10) 108.7 a(5-2-8) 123.5 a(8-2-9) 125.5 a(1-5-7) 109.7 a(2-5-4) 103.5 a(2-8-4) 106.2 a(1-5-8) 109.7 a(2-8-9) 120.3 a(2-8-5) 124.1 a(9-1-10) 106.0 a(2-8-10) 120.2 a(2-9-10) 108.7 a(2-3-4) 179.7 a(4-3-7) 127.2 a(4-3-7) 124.4 a(3-4-6) 180.0 a(3-4-5) 110.0 a(3-4-6) 125.4 a(7-5-8) 106.2 a(3-4-6) 127.2 a(3-4-8) 105.9 w1(a) 102.9 a(5-4-6) 122.8 a(6-4-8) 128.7 w2(a) 136.5 a(9-8-10) 119.5 a(4-8-5) 129.8 w3(a) 249.6 w1(a) 231.3 w1(a) 222.6 w4(a) 273.9 w2(a) 367.3 w2(a) 437.9 w5(a) 384.9 w3(a) 545.9 w3(a) 513.7 w6(a) 500.8 w4(a) 729.7 w4(a) 673.5 w7(a) 576.0 w5(a) 757.5 w5(a) 709.6 w8(a) 582.7 w6(a) 778.7 w6(a) 821.9 w9(a) 699.5 w7(a) 893.1 w7(a) 839.2 w10(a) 889.2 w8(a) 912.9 w8(a) 878.7 w11(a) 895.9 w9(a) 915.8 w9(a) 926.6 w12(a) 1098.3 w10(a) 954.3 w10(a) 967.2 w13(a) 1148.1 w11(a) 961.4 w11(a) 969.9 w14(a) 1183.4 w12(a) 979.8 w12(a) 993.4 w15(a) 1356.4 w13(a) 991.5 w13(a) 1058.7 w16(a) 1383.0 w14(a) 1202.2 w14(a) 1093.4 w17(a) 1459.8 w15(a) 1303.2 w15(a) 1280.5 w18(a) 1669.8 w16(a) 1346.3 w16(a) 1341.6 w19(a) 2258.9 w17(a) 1413.0 w17(a) 1353.4 w20(a) 2396.8 w18(a) 1496.2 w18(a) 1548.0 w21(a) 3036.9 w19(a) 1613.2 w19(a) 1646.7 w22(a) 3070.9 w20(a) 1714.9 w20(a) 1740.6 w23(a) 3496.6 w21(a) 3180.9 w21(a) 3175.7 w24(a) 3574.6 w22(a) 3196.1 w22(a) 3234.5 r(1-2) 1.466 w23(a) 3211.1 w23(a) 3266.2 r(1-5) 1.465 w24(a) 3283.7 w24(a) 3434.3 r(1-9) 1.097 438 etim et al. / j. nig. soc. phys. sci. 3 (2022) 429–445 439 table a6: bond radius and angles of c4h4n2 isomers 1,3-butadiene1,4-diimine 1,2-diisocyanoethane description cal. value description cal. value r(1-2) 1.316 r(1-5) 1.173 r(1-5) 1.226 r(2-3) 1.538 r(2-3) 1.461 r(2-5) 1.416 r(2-9) 1.085 r(2-7) 1.094 r(3-4) 1.316 r(2-8) 1.095 r(3-10) 1.085 r(3-6) 1.416 r(4-6) 1.225 r(3-9) 1.095 r(5-7) 1.023 r(3-10) 1.094 r(6-8) 1.023 r(4-6) 1.173 a(2-1-5) 173.7 a(1-5-2) 179.0 a(1-2-3) 124.3 a(3-2-5) 112.1 a(1-2-9) 116.9 a(3-2-7) 109.7 a(1-5-7) 115.8 a(3-2-8) 108.4 a(3-2-9) 118.8 a(2-3-6) 112.1 a(2-3-4) 124.3 a(2-3-9) 108.4 a(2-3-10) 118.8 a(2-3-10) 109.7 a(4-3-10) 116.9 a(5-2-7) 109.2 a(3-4-6) 173.7 a(5-2-8) 109.2 a(4-6-8) 115.8 a(7-2-8) 108.1 w1(a) 83.6 a(6-3-9) 109.2 w2(a) 127.6 a(6-3-10) 109.2 w3(a) 247.6 a(3-6-4) 179.1 w4(a) 413.9 a(9-3-10) 108.1 w5(a) 422.9 w1(a) 78.7 w6(a) 553.3 w2(a) 168.9 w7(a) 572.3 w3(a) 192.4 w8(a) 602.0 w4(a) 262.6 w9(a) 678.5 w5(a) 299.1 w10(a) 879.1 w6(a) 389.5 w11(a) 905.7 w7(a) 551.2 w12(a) 1033.4 w8(a) 827.5 w13(a) 1042.8 w9(a) 859.6 w14(a) 1064.8 w10(a) 1035.2 w15(a) 1144.8 w11(a) 1042.0 w16(a) 1186.9 w12(a) 1091.3 w17(a) 1291.1 w13(a) 1268.9 w18(a) 1486.0 w14(a) 1304.2 w19(a) 2123.9 w15(a) 1384.3 w20(a) 2130.5 w16(a) 1392.4 w21(a) 3174.9 w17(a) 1485.0 w22(a) 3183.6 w18(a) 1486.8 w23(a) 3422.3 w19(a) 2224.3 w24(a) 3422.6 w20(a) 2226.3 r(1-2) 1.316 w21(a) 3052.6 r(1-5) 1.226 w22(a) 3056.6 r(2-3) 1.461 w23(a) 3096.1 r(2-9) 1.085 w24(a) 3107.6 r(3-4) 1.316 r(1-5) 1.173 439 etim et al. / j. nig. soc. phys. sci. 3 (2022) 429–445 440 pyrazine 440 etim et al. / j. nig. soc. phys. sci. 3 (2022) 429–445 441 1,1-dicyanoethane pyridazine 441 etim et al. / j. nig. soc. phys. sci. 3 (2022) 429–445 442 2-methylene-2h-imidazole iminopyrrole 442 etim et al. / j. nig. soc. phys. sci. 3 (2022) 429–445 443 4-amino-2-butynenitrile 1,3-butadiene-1,4-diimine 443 etim et al. / j. nig. soc. phys. 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[28] p. perrot, a to z of thermodynamics, oxford university press, (1998). isbn 0-19-856552-6 [29] p. c. mishra & r. d. tewari, r. d. “study of the para-directing effect of n-singlrt transition by isopotential mapping”, indian j. pure appl. phys. 25 (1987) 387. 445 j. nig. soc. phys. sci. 4 (2022) 310–317 journal of the nigerian society of physical sciences comparative analysis of the implication of periods before and during vaccination of covid-19 infection in some regional leading african countries abiola t. owolabia,∗, kayode ayindeb, taiwo j. adejumoa, wakeel a. kasalia, emmanuel t. adewuyia adepartment of statistics, ladoke akintola university of technology, ogbomoso, oyo state, nigeria bdepartment of statistics, federal university of technology, akure, nigeria abstract there has been a high expectation about the efficacy of coronavirus disease 2019 (covid-19) vaccines. this research investigates and compares the efficiency of covid-19 vaccines in five (5) african countries and evaluates the risk or preventive factors inherent in covid-19 spread. five different covid-19 leading african countries in their respective regions (nigeria, ethiopia, south africa, morocco, and cameroon) were considered in this study. population sampling proportional to size concept was used to draw data for two periods (before and during covid-19 vaccination). a sequential analysis approach was adopted, focusing on the estimates of some epidemiological metrics for the two distinct periods. nigeria (a wet region) has the lowest risk of covid-19 incidence during vaccination. the risk of being reported covid-19 positive in south africa (a high semi-arid region) is approximately 137 times the number in nigeria. this study suggests that while vaccination has successfully reduced the case fatality rate in most countries considered except ethiopia, infection and incidence rates increase during vaccination in all countries except nigeria. methods other than vaccination like wearing a face mask, washing hands, and avoiding large gatherings should be intensified to curtail incidence and infection rates. doi:10.46481/jnsps.2022.702 keywords: covid -19, vaccination, case fatality rate, incidence rate, infection rate, relative risk article history : received: 12 march 2022 received in revised form: 20 may 2022 accepted for publication: 20 may 2022 published: 29 may 2022 c©2022 journal of the nigerian society of physical sciences. all rights reserved. communicated by: t. latunde 1. introduction coronavirus 2019 (covid-19) started in wuhan, china, in december 2019 [1-3]. this virus was investigated to have an origin connected with the kingdom of the riboviria, the nidovirales order, orthocoronavirinae subfamily of the coronaviridae family, and corona-like species at 125mn size larger than in∗corresponding author tel. no: +2348062956731 email address: atowolabi@lautech.edu.ng (abiola t. owolabi ) fluenza, severe acute respiratory syndrome (sars), and middle east respiratory syndrome (mers) [4]. coronavirus is derived from ”corona,” which implies a ”halo” shape. under a two-dimensional transmission microscope, the virus exhibits a unique appearance because its surface is covered with a rodshaped protein tip. the incidence of 1981 influenza or pandemic smallpox earlier confronted the human race has not been successful as an epidermal viral pathogen of sars-cov-2, a member of the family of viruses that severely attack and cause severe diseases in many animals with backbones [5]. 310 owolabi et al. / j. nig. soc. phys. sci. 4 (2022) 310–317 311 the virulence power of sars cov-2 remained formidable and has spread exponentially worldwide, with negative effects on the healthcare system, economic, financial, commercial, and social development across the globe [6-8]. about 240,631,670 infected people with covid-19 and 4,899,169 death tolls were recorded worldwide as of 19th october 2021 [9]. following the first wave in china, the second wave of covid-19 hits iran, south korea, and italy. only about some moments later, the outbreak of covid-19 swiftly swept through the united states, africa, and more heavily into european countries [10]. because of many unknowns, a partial understanding of covid19 transmissibility has fueled the level of concern among the public health community, as most modeling suggests that the severity of the illness is more like influenza than sars [4, 11]. similar to this is the fold of research affirming semblance of certain flu-like symptomatic features that covid-19 shares with influenza [12], suggesting in this light the potentiality of mitigating factors on influenza being equally appropriate for quelling covid-19. the development efforts to curtail the spread from hotspot to hotspot fervently proved abortive. during the first surge on the regional level in lombardy, italy, the confirmed cases doubled from 41,035 to 86,498 in just one week (between 19th march and 27th march 2020), allotting italy as one country with a large outbreak of covid-19. in the wake of the u.k. variant emergence (b.1.1.7) that settled the origin lineage (d614g) immensely, italy encountered a covid19 sweep from 1st october 2020 as an indicator of a second surge [13]. from available data, incidence rate and mortality vary from country to country and from continent to continent. the variation may be due to factors such as dietary habits, climate, social activities, genetic differences, governance structures, the use of chloroquine (c.q.), and the anti-tuberculosis (t.b.) vaccination (bacillus calmette–guerin, b.c.g.) [14]. following the first confirmed case report on 14th february in egypt, the second in algeria on 25th february 2020, and the third in nigeria on 27th february 2020, there has been a continuous increment in covid-19 incidence in africa. one of the challenges faced in the fight against covid19 is the emergence of new sars-cov-2 variants in the latter part of 2020 and the early part of 2021. in july 2020, the eu2 variant (mutation s: 447n) was first observed in western europe and increased virus infectivity [15]. then b.1.1.7 developed firstly in the u.k. in september 2020 [16], b.1.351 in south africa in december 2020 [17], p.1 in brazil in january 2021 [18], and the ’indian’ variant b.1.617 reported firstly in maharashtra in january 2021 [19]. different countries adopted several preventive measures for the covid-19 outbreak. cooperate measures like lock-down, social distancing and mask use, and individual hygiene measures like frequent hand washing and avoiding touching eyes, nose, or mouth [20] were adopted by different countries. vaccines as part of the exit strategy to enable a return to previous working, schooling, and socializing patterns have been widely considered. numbers of different vaccines have been developed and analyzed in clinical trials. clinical trials on vaccine efficacy revealed that pfizerbiontech had 95% efficacy at preventing symptomatic covid19 infection in people without prior infection [21]. 94.1% efficacy was reported for moderna [22], 70.4% for oxford -astrazeneca [23], 66.5% for johnson & johnson [24], and 96.4% for novavax [25]. regarding the prevention of the disease, 100% efficacy was reported for pfizer, moderna, astrazeneca, and novavax, whereas 84% was observed for johnson & johnson [25]. by 24th september 2020, many vaccines (not less than 200) had been considered for preclinical development, although mixed with some concepts that have not previously been licensed for human vaccines, from which 43 had entered clinical trials. as of 19th october 2021, 6,545,309,084 vaccine doses have been administered worldwide (w.h.o, 2021). there has been a high expectation about the covid-19 vaccine’s efficacy, but this seems to vary from country to country, region to region, and then continent to continent. many nations have to depend on the development of vaccines when all efforts to cure and eradicate the covid-19 pandemic proved abortive. however, there are reports of side effects of the covid-19 vaccine, especially in africa, which has led to hesitation by some individuals to take the jab. in this light, this study aims at investigating the effectiveness of covid-19 vaccines, comparing case fatality, incidence, and infection rates caused by covid-19 in the presence of non-pharmaceutical intervention (pre-vaccination) and during the inception of pharmaceutical intervention (vaccination) in five regional leading african countries. 2. method 2.1. study location nigeria is located on the western coast of africa, bordered to the north by niger, to the east by chad and cameroon, to the south by the gulf of guinea of the atlantic ocean, and to the west by the benin republic. nigeria has a warm desert climate in the northeast, a tropical savannah climate in the southwest and middle belt, a warm semi-arid climate in some parts of the north, and the monsoon climate in the niger delta. nigeria is a major player in sub-saharan africa with its large human population [26]. nigeria is the most populous african country, with an estimated population of about 200 million people. ethiopia is another african country on the eastern coast, bounded by somalia to the east, south sudan and sudan to the west, kenya to the south, and djibouti to the northeast. due to its varied topography, its climate is tropical in the north-eastern lowlands and south-eastern lowlands, while it’s temperate and cool in the highlands. the country shares 1.47% of the world population, making it the second-largest african country. south africa is in the southern part of africa, surrounding lesotho in the east, bordered by zimbabwe, swaziland, namibia, and botswana. the country is a climate patchwork of warm coastal subtropics with humid highlands, hot deserts, and mediterranean weather in the southwest. south africa has about 60 million people, sitting 25th in the world population ranking. morocco is located in the northern part of africa, bordered by the mediterranean sea to the north, algeria to the east and southeast, western sahara to the south, and the atlantic ocean 311 owolabi et al. / j. nig. soc. phys. sci. 4 (2022) 310–317 312 to the west. the climate is humid temperate at a higher elevation, dry and hot in the south-western part, and the mediterranean on the coasts. morocco’s equivalence is 0.47% of the global population. cameroon is situated in the central part of africa. the central african republic borders its triangular structure to the east, the republic of the congo to the southeast, nigeria to the northeast, gabon and equatorial guinea to the south, and the atlantic ocean to the southwest. cameroon takes number 52 in the list of countries by population. 2.2. data extraction datasets on daily new cases and death cases of covid-19, including population size, cardiovascular death rate, diabetes prevalence, and people vaccinated for 5 (five) different covid19 leading african countries in their respective regions, were surveyed for 2 (two) periods. these were extracted from the official website of ”our world in data” [27]. the world annual averaged temperatures and precipitations data were sourced from the world bank data catalog [28]. 2.3. analysis of data conceptualizing the sampling strategy for relative efficiency under a super population model, the approximated 0.05% confirmed cases of the total population of each country in this study were sampled for ”before-vaccination period” and data for 5 (five) months after the intervention of covid-19 vaccines were surveyed. this study estimates the infection rate (irr) according to its definition by [29], incidence rate (i.r.), and relative risk (r.r.) for the two periods using equations 1, 2, and 3. irr = a b (1) where a is the number of people infected and b is the days of infection. ir = m n (2) where m is the number of persons who develop the disease over time and n is the number of persons initially without the disease who were followed for the defined period. rr = disease incidence in group 1 disease incidence in group 2 (3) where rr = 1 means that exposure does not affect the outcome, rr < 1 means that the risk of the outcome is decreased by the exposure, which is a ”protective factor,” and rr> 1 means that the risk of the outcome is increased by the exposure, which is a ”risk factor.” the case fatality rate (cfr) was obtained using the regression model approach [29]. the equation is given as: death cases = β0 + (β1 × confirmed cases) + εi (4) an aridity index (ai) is a numerical indicator of the degree of dryness of the climate at a given location as classified by de martonne [30], [31] and denoted i. for i < 5 is classified as hyper-arid, arid when 5 <i < 10, semi-arid when 10 <i < 20, mediterranean when 20 <i < 24, semi-wet when 24 <i < 28, wet when 28 <i < 35, very wet when 35 <i < 55 and extremely wet when i > 55. the index (i) is obtained as: i = p t + 10 (5) where p is the average annual precipitation (mm), t is the average yearly temperature (0c), and i is the de martonne aridity coefficient. in equation (5), evaporation is indirectly considered. the availability of the factors and the classification of this method which can define diverse climates made this method more widely used in iran [32]. 3. results this study examined the daily behavior of covid-19 in incidence, spread, and fatality rates in association with the level of vaccines efficacy when there was no vaccination and during vaccination in the aggregation of risks associated with the exposure to a climatic condition of five sub-saharan african countries having recorded high covid-19 cases in their respective regions. 3.1. descriptive statistics and estimated cases the descriptive statistics of covid – 19 outcomes on daily confirmed and death cases in the analysis explains the sampling technique and measures of the data surveyed in the period before vaccination as the sampling revolved from the day of the first reported covid-19 case till when the case reached 0.05% of the total population of every country under study. at the same time, the sampling during vaccination lasted for the first five months of vaccination, as presented in table 1. the sample for each country was taken in proportion to its size, as used by [33]. right before the infected people with covid-19 in cameroon summed up to about 15,000 on 7th august 2020 [9], the lowest and highest daily confirmed cases recorded were 0 and 2,324, respectively, while the lowest and highest ever recorded death cases were 0 and 64 respectively. the average within the pre-vaccination period is 119 and 3 for confirmed and death cases, respectively. within the first five months of vaccination, 22,479 cases were reported from amongst uninfected populations of 27,162,531 that made it to the vaccination period in cameroon. between 12th april 2021 and 12th september 2021, the average daily reported confirmed cases and death cases were 146 and 3, respectively. focusing on the realistic patterns of figures, south africa recorded a shock in the daily average confirmed cases of 5,289 and death cases of 122 during vaccination when infection was observed in larger people than in any other population on the list. according to de martonne’s global classification [30], cameroon falls within ”very wet region,” ethiopia within ”semiwet regions,” morocco within ”semi-arid regions,” nigeria within ”wet regions,” and south africa within ”semi-arid regions,” all measured by an index that characterizes climate. 312 owolabi et al. / j. nig. soc. phys. sci. 4 (2022) 310–317 313 table 1: descriptive statistics of the datasets countries period total population sample temp. (oc) prec. (mm) de martonne (i) confirmed/death cases min max mean std dev cameroon before 27,224,262 14,916 17.60 0/0 2324/64 119/3 279/9 46.72 during 27,162,531 22,479 474.78 0/0 6252/116 146/3 650/12 ethiopia before 117,876,226 59,648 17.23 0/0 1829/28 333/6 481/8 25.66 during 117,654,682 97,557 322.27 39/0 2149/47 633/12 585/10 morocco before 37,344,787 18,834 26.78 0/0 570/12 130/2 115/6 11.84 during 36,864,731 78,729 1138.53 35/0 2853/18 521/6 469/4 nigeria before 211,400,704 107,345 22.37 0/0 1867/31 332/5 331/5 30.95 during 21,124,0047 21,846 830.51 0/0 790/28 142/1 181/3 south africa before 60,041,996 30,967 24.53 0/0 1837/52 356/10 441/12 17.20 during 58,545,557 798,656 1613.07 0/0 26485/63 5289/122 6398/109 table 2: behavioral expression of rates of covid-19 infection with and without vaccines countries case fatality rate infection rate incidence rate before during before during before during cameroon 0.024 (r2= 0.958) 0.022(r2=0.982) 119.328 145.97 0.000562 0.0008276 ethiopia 0.016 (r2= 0.996) 0.020(r2=0.928) 333.23 633.49 0.000519 0.0008292 morocco 0.014(r2 = 0.806) 0.013(r2=0.977) 129.89 511.23 0.00051 0.0021356 nigeria 0.015 (r2=0.948) 0.009(r2=0.890) 332.33 141.86 0.000521 0.000103 south-africa 0.021 (r2=0.994) 0.019(r2=0.921) 355.94 5,186.09 0.000522 0.013642 3.2. health indicators the proportion of cameroon people who died of covid19 among 14,916 diagnosed with the disease within the defined time-interval for the study is 2.4% before vaccination (see table 2). within the first five (5) months of vaccination, 22,479 people diagnosed with the disease were sampled. the fatality rate dropped to 2.2%. before vaccination, it took 125 days for cameroon to reach 0.05% of the total population, making an approximate 119.328 infection rate. during the first five (5) months of vaccination, the infection rate rose to 145.97 within 154 days, with 22,479 individuals diagnosed with the disease. however, the incidence rate (the number of new cases in cameroon within the defined time-interval for this study before vaccination as a proportion of the 27,224,262 people at risk) is 0.562 per one-thousand people. this rate increases to 0.828 per one-thousand among 27,162,531 people at risk during the first five (5) months of vaccination. in ethiopia, the percentage of people who died of covid19 among 59,648 diagnosed with the disease within the defined time-interval for the study is 1.6% before vaccination. within the first five (5) months of vaccination, 97,557 people diagnosed with the disease were sampled. the fatality rate also increased to 2.0%. the infection rate that expresses the spread power among the population was estimated to be 333.23, with the infection taking 179 days to reach 0.05% of the total population of ethiopia before vaccination. in comparison, within the first five (5) months (154 days) of vaccination, the infection rate almost doubled to 633.49. the incidence rate for 117,876,226 people at risk was 0.519 per one-thousand people before vaccination. this rate among 117,654,682 people of ethiopia at risk during the first five (5) months of vaccination increased to 0.829. the proportion of morocco people who died of covid-19 among 18,834 diagnosed with the disease within the defined time-interval for the study is 1.4% before vaccination. within the first five (5) months of vaccination, 97,557 people diagnosed with the disease were sampled. the fatality rate dropped to 1.3%, leaving behind an indication that the vaccine has a significant mitigating influence on the case fatality rate. however, the covid-19 spread rate was estimated to be 129.89 within 145 days of the period before vaccination, and this rate increased to about 4-times during vaccination for 154 days of people receiving covid-19 vaccines. moreover, the incidence rate taking 37,344,787 of morocco people at risk before vaccination into consideration gives 0.51 per one-thousand people, and this rate during the vaccination period of five months accounted for 2.41 per one-thousand people of 36,864,731 in the population at risk, meaning that the incidence is five-times more of what it was before vaccination. this suggests that vaccination had less effect 313 owolabi et al. / j. nig. soc. phys. sci. 4 (2022) 310–317 314 on reducing the incidence rate. however, the tendency of these rates to behave differently in the period before and during vaccination can be explained by climate phenomenon, containment measures, the population in dominance, cardiovascular death rate, and diabetes prevalence in the country, even though the morocco vaccination campaign had been claimed as rapid and effective compared to other african countries. the nigerian people who died of covid-19 among 107,345 diagnosed with the disease within the defined time-interval for the study is 1.5% before vaccination. within the first five (5) months of vaccination, 21,846 people diagnosed with the disease were sampled, and the fatality rate was 0.9%. this indicates that vaccine has a mitigating influence on the case fatality rate. also, within 323 days after nigeria recorded its first case, the infection rate had reached about 0.05% of the total population, making 332.33 spread rate, similar to that of ethiopia before vaccination, while this reduced its spread drastically during the first five (5) months of vaccination. this implies covid-19 spread rate in nigeria is being affected positively by vaccines. a similar inflow is the incidence rate before vaccination taking 211,400,704 people at risk into consideration. the estimated 0.521 per one-thousand reduces among 211,240,047 people at risk during vaccination to 0.103 per one-thousand. these results make nigeria a benchmark for vaccine efficacy among the african countries considered in this study. the proportion of south africa people who died of covid19 among 30,967 diagnosed with the disease within the defined time-interval for the study is 1.5% before vaccination. within the first five (5) months of vaccination, 798,656 people diagnosed with the disease were sampled, and the fatality rate appeared to be 0.9%. this indicates that vaccine has a mitigating effect on the case fatality rate. before vaccination, the case dispersed across 0.05% of the total population within 87 days, making its spread the fastest compared to other countries in the study with a 355.94 infection rate. the spread becomes about 15-times faster during the first five (5) months of vaccination, estimating 5,186.09 as the value of infection rate. this implies covid-19 vaccines show no efficacy on the behavioral spread of the infection. however, the incidence rate taking 60,041,996 of south african people at risk into consideration gives 0.522 per one-thousand people, as this rate during the vaccination period of five months accounted for 13.642 per one-thousand people of 58,545,557 in the population at risk meaning that the incidence is by 26-times more of what it was before vaccination. 3.3. test of association the challenges of climatic and environmental changes and other events amidst the pandemic have been a dominating involution facing the social development and evolution of human civilization epidemics [3, 15, 34]; however, this shows dynamism in intensity from area to area. south africa, ethiopia, and nigeria, in the fresh spread of covid-19, reported a mean of 330 confirmed cases per day; this waved exponentially in south africa and mildly in ethiopia amidst vaccination. prevalent health issues among patients hospitalized are hypertension, diabetes, and other cardiovascular diseases [3, 35]. morocco has a cardiovascular death rate of 419.146 and a diabetes prevalence of 7.14, and there is a possibility of these rates depending on some influencing components of epidemics factors. spearmanrank correlation test was used to establish the relationship’s strength and direction of some of the features relative to the rates estimated in this study, as presented in table 3. combinative analysis of this trend from table 3 shows that only the case fatality rate is significantly correlated with the covid-19 incidence rate before vaccination (r=0.9; p < 0.05), while only the infection rate is significantly associated with the incidence rate during vaccination (r=0.9; p < 0.05). the finding affirms the undaunted input of the covid-19 vaccine in reducing the case fatality rate in africa. 3.4. test of significant differences between the periods due to the social upheaval of the pandemic, vaccine development was aimed to achieve direct certainty of vaccine efficacy in protecting humans against covid-19 & sars-cov2 infection so that manufacture of efficacious vaccines could reach its peak [36]. the most important efficacy endpoint of the vaccine against this severe disease is evaluated by comparing cases when it wasn’t within the population and when endorsed for use by citizens. some complex behavioral variables are constituted in the diversity of understanding efficacy in the use of vaccines. independent t-test conducted shows there is no statistically significant difference in the means of case fatality rate before vaccination and during the first five months of vaccination at (90% and 95% confidence levels, p > 0.1 and 0.05, respectively), while there are significant differences in the means of infection and incidence rates across the two periods (before and during vaccination) with (p < 0.1 and 0.05) as it can be demonstrated in the vaccine’s licensure of outcomes that include consequences attributable to the reduction in infection, transmission dynamics, lowered severity [37] and duration of infectivity [38] all defining vaccine efficacy. 3.5. association of exposure to aridity and covid – 19 outcome the control column in table 4 for ”before vaccination” expresses the difference in the total population and confirmed cases of the region (country) in concern when there was no intervention of actual covid-19 vaccines globally accepted, while control for ”during vaccination” demonstrates the population size that had not received vaccines during the first five (5) months of their vaccines distribution as cases-column describes the number of cases reported within these time intervals. for the period before vaccination, the assumption is made that every patient confirmed with covid-19 case and reported was under treatment with a specific vaccine, meaning that the population without infection is the control. while for the period ”during-vaccination,” it is explained through the control column, the population size that did not receive treatment (covid19 vaccine) during the first five (5) months of its distribution, through the difference of total people vaccinated and the total country population taken, as cases-column shows the number 314 owolabi et al. / j. nig. soc. phys. sci. 4 (2022) 310–317 315 table 3: spearman rank correlation of the rates across the periods of no-vaccination & vaccination vaccination period case fatality rate incidence rate infection rate case fatality rate before ρ = 0.90, (p=0.037) ρ = 0, absent during ρ = -0.1, (p=0.873) ρ = 0.3, (p=0.624) incidence rate before ρ = 0.90, (p=0.037) ρ = -0.1, (p=0.873) during ρ = -0.1, (p=0.873) ρ = 0.90, (p=0.037) infection rate before ρ = 0, absent ρ = -0.1, (p=0.873) during ρ = 0.3, (p=0.624) ρ = 0.90, (p=0.037) table 4: relative risks table for pairwise severity of the incidence in different aridity zones aridity period cases controls semi-arid (h) semiarid(l) semiwet wet very wet semi-arid (h) before 30,967 60,011,029 0.9778 0.9812 0.9845 1.0623 during 798,656 55,706,479 0.2129 0.0589 0.0073 0.2129 semi-arid(l) before 18,834 37,325,953 1.0227 1.0034 1.0069 1.0864 during 78,729 25,788,817 4.6962 0.2766 0.0343 0.2741 semi-wet before 59,648 117,816,578 1.0192 0.9966 1.0035 1.0828 during 97,557 115,549,695 16.981 3.6157 0.1239 0.9912 wet before 107,345 211,293,359 1.0157 0.9932 0.9965 1.079 during 21,846 208,850,314 137.0621 29.1855 8.0715 8.0007 very wet before 14,916 27,209,346 0.9413 0.9204 0.9235 0.9267 during 22,479 26,860,489 17.1314 3.6479 1.0088 0.125 of cases reported within this time interval. between 12th april 2021 and 12th september 2021, cameroon recorded 22,479 cases, while the total number of people vaccinated was 363,773 (1% of the total population). in the same vein, ethiopia progressed with 2,326,531 (2% of the total population) vaccinated people, and 97,557 cases were confirmed between 08th april 2021 and 08th september 2021. in morocco, the entire case confirmed was 78,729, and 11,555,970 (31% of the total population) people were vaccinated between 19th february 2021 and 19th july 2021. nigeria vaccinated 2,550,390 (1% of the total population) and recorded 21,846 confirmed cases between 15th march 2021 and 15th august 2021. the vaccination in south africa reached 4,335,517 (7% of the total population), while 798,656 cases were confirmed between 18th february 2021 and 18th july 2021. the odd ratios table for the regions in a pairwise manner is shown in table 4, indicating that the individual estimate ranges between 0.9 and 1.0; meaning that exposure to aridity (temperature and precipitation) either served as a protective factor or did not affect the outcomes during the spread of covid-19 among the 0.05% of the total population of each of the countries in the study during the period of no-vaccination. the wet region in de martonne’s aridity index [30] has the lowest risk of having covid-19 disease during vaccination. the part that falls under the wet region in this study is nigeria. the risk of being reported covid-19 positive for semiarid (high) region people (south-africa) is approximately 137 times the number of people in the wet region (nigeria). the risk of being reported covid-19 positive for the semi-arid (low) region (morocco) is approximately 29 times more for the people living in the wet area (nigeria). the risk of being reported covid-19 positive for the people living in the semi-wet region (ethiopia and cameroon) is approximately eight times more for those living in the wet region (nigeria). 4. discussions even though different countries adopted several preventive measures for the covid-19 outbreak; while corporate and clinical measures were adopted by various governments, south africa recorded a shock in the daily average confirmed cases of 5,289 and death cases of 122 during vaccination when infection was observed in larger people than in any other population on the list. this is closely followed by ethiopia, which recorded daily average confirmed cases of 633 and death cases of 12. however, nigeria of the five countries recorded the least daily average confirmed cases of 142 and 1 death case. the surge in south africa even during vaccination may be consistent with the report that one of the popular vaccines adopted by the south african government which is the oxford-astrazeneca covid19 vaccine in south africa, is not efficient against the 501y.v2 variant, which accounts for around 90% of cases in south africa [39, 40]. violation of social distancing guidelines in many parts of south africa may also be responsible for the spike in the infection rates. in the eastern cape province, 80% of all infections in the province resulted from burial ceremonies [41]. 315 owolabi et al. / j. nig. soc. phys. sci. 4 (2022) 310–317 316 this study suggests that the climatic classification of the countries considered relates to the incidence rate. the wet region in de martonne’s aridity index [26] has the lowest risk of having covid-19 disease during vaccination. the part that falls under the wet region in this study is nigeria. the risk of being reported covid-19 positive for semi-arid (high) region people (south africa) is approximately 137 times the number of people in the wet region (nigeria). the risk of being reported covid-19 positive for the semi-arid (low) region (morocco) is approximately 29 times the number of people living in the wet region (nigeria). the risk of being reported covid-19 positive for the people living in the semi-wet region (ethiopia and cameroon) is approximately eight times of those living in the wet region (nigeria). this agrees with the report of [36], who pontificated that transmission of viruses can be affected by many factors, including climate conditions such as temperature and humidity. although visual inspection of world maps describes that (covid-19) is less common in countries closer to the equator, where humidity & heat seem to be higher, the relationship between covid-19 and climatic conditions may be confounded by many factors. temperature and absolute humidity were reported as crucial weather indices associated with the spread of covid-19 [42, 43]. from this study, the ability of vaccines to curb the infection (spread) and incidence rates of the covid-19 pandemic is highly doubtful. although there is a scarcity of literature that has compared the effect of the vaccine on infection and incidence rates of covid-19, four (cameroon, ethiopia, morocco, and south africa) of the five countries considered in this study consistently show that infection and incidence rates increased (rapidly in some cases) during vaccination compared to what was obtainable before vaccination. only nigeria has reduced infection and incidence rates during vaccination. an increase in infection and incidence rates might be due to relaxing the preventive measures observed in all these countries before vaccination during the advent of vaccines. from the results of this study, the case fatality rate representing the proportion of cases that eventually die from the disease was mitigated through vaccination. four (cameroon, nigeria, morocco, and south africa) of the five countries considered in this study consistently show that the case fatality rate decreased during vaccination compared with the pre-vaccination period. analysis, however, shows no statistically significant difference in the means of case fatality rate before and during the first five months of vaccination. from this study, only ethiopia experienced an increase in case fatality rate during vaccination. this result is in tandem with [37], who suggest that vaccination against seasonal influenza (though nor covid-19 vaccine) might beneficially impact on incidence and severity of the novel coronavirus epidemic. also, according to miller [38] et al., bacillus calmette-guérin (bcg) vaccination correlates with covid-19 case fatality rates and offers protection against severe cases of sars-cov2. the differences in the behavior of vaccines among the population for the studied rates across the periods before-vaccination and during vaccination might have resulted from effects in stringency index by the government, diabetes prevalence, cardiovascular death rate, population-class in dominance & climatic condition as the study covered beyond a period [43, 44]. from spearman rank correlation, it was established for the five african countries that before introducing the covid-19 vaccines, the case fatality rate and incidence rate rose together as incidence rate and infection rate also show direct variation during the vaccination period. the reports of this research should be limited because the data collected for the five countries involved in this study are datasets for five months into the vaccination period. this can still be referred to as the early period into the vaccination period. although the climatic condition of the countries considered, which is one of the factors that could affect case fatality, incidence, and infection rates, was considered in this study; many other factors could influence the results like the level of strictness in preventive measures for the covid-19 from country to country. another factor is comorbidities that will affect the case fatality rate from covid-19. 5. conclusion from this study’s findings, vaccines effectively reduce the proportion of cases that eventually die from covid-19 in cameroon, nigeria, morocco, and south africa. however, vaccines seem not to affect the case fatality rate in ethiopia during the period of this study. it was observed that the vaccine is ineffective in curbing infection and incidence rates as there are increases in cameroon, ethiopia, morocco, and south africa. however, these two rates decrease in nigeria during vaccination compared with before vaccination. this implies that methods other than vaccination should be considered to curtail the incidence and infection (spread) rates. there is no yet observed ability of vaccines to produce a desired effect on the infection (spread) rate of covid-19 as the disease spread faster in south africa in this study than in any other country during the vaccination period. there should be more awareness on reporting every infected individual whenever observed. climatic condition is an important factor contributing to the risk of being infected. based on the categories 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[44] s. g. gupta, s. raghuwanshi & a. chanda, “effect of weather on covid-19 spread in the us: a prediction model for india in 2020”, science of the total environment 728 (2020). 317 j. nig. soc. phys. sci. 3 (2021) 282–286 journal of the nigerian society of physical sciences eigensolution to morse potential for scandium and nitrogen monoiodides c. a. onatea,b,∗, g. o. egharevbaa, d. t. bankolea adepartment of physical sciences, landmark university, omu-aran, nigeria. blandmark university sdg 4 (quality education) abstract the solutions for morse potential energy function under the influence of schrödinger equation are examined using supersymmetric approach. the energy equation obtained was used to generate eigenvalues forx1 ∑+state of scandium monoiodide (sci) andx3 ∑−state of nitrogen monoiodide (ni) respectively by imputing their respective spectroscopic parameters. the calculated results for the two molecules aligned excellently with the predicted/observed values. doi:10.46481/jnsps.2021.407 keywords: eigensolutions, wave equation, bound state, molecules article history : received: 13 september 2021 received in revised form: 27 september 2027 accepted for publication: 28 october 2021 published: 29 november 2021 c©2021 journal of the nigerian society of physical sciences. all rights reserved. communicated by: w. a. yahya 1. introduction the study of the interactions for atomic molecules can be carried out using some physical potential models. over the years, the energy eigenvalues of different molecules were obtained for different potential terms and reported. among the reported works are, the energy eigenvalues of four molecules for kratzer potential by bayrak et al. [1]. in ref. [2], ikhdair obtained energy eigenvalues of eight molecules for manningrosen potential. ikhdair and sever [3], obtained energy eigenvalues of six molecules for kratzer-type potential, falaye et al. [4], under the study of tietz-wei diatomic molecular potential function, obtained energy eigenvalues of ten molecules. in ref. [5], the energy eigenvalues of some molecules were obtained for pseudoharmonic potential. onate et al. [6], obtained energy eigenvalues of six molecules for hyperbolic-sinus ∗corresponding author tel. no: email address: oaclems14@physicist.net (c. a. onate ) potential model. recently, farout et al. [7], obtained exact momentum states of some molecules for improved deformed exponential-type potential. despite all the reports given above, it has not been easy to deduce a potential energy for various molecules, thus, a challenging issue subsist in the study. however, some potential energy function that can explain various diatomic molecules in good agreement with experimental data have been proposed based on the experimental constants such as the dissociation energy and equilibrium bond length, e.g. deng-fan, improved rosen-morse, morse, tietz-hua oscillator, improved generalized poschl-teller oscillator [8-13]. the energies of these potential functions were calculated for cesium molecule, sodium dimer, nitrogen dimer, hydrogen molecule and potassium. the result obtained for each molecule aligned with the experimental data. however, it is noted that only few potential functions were examined and their results compared with the experimental results. motivated by the interest in the morse potential function as one of the different potential en282 onate et al. / j. nig. soc. phys. sci. 3 (2021) 282–286 283 figure 1: the morse potential function for scandium monoiodide and nitrogen monoiodide ergy functions suggested to obtain information about diatomic molecular structures, this work aims to determine the energy eigenvalues of scadium monoiodide and nitrogen monoiodide for a morse potential energy function. the morse potential model is a molecular potential model that is used to describe the interaction between two atoms in the atomic domain, as well as interaction closed to the surface. several works have been reported on the morse potential with different methods. however, most of the reports given are on the morse potential of the form [17] v (r) = −2dee −α(r−re ) + de −2α(r−re ), (1) where deis the dissociation energy, reis the equilibrium bond length, ris the internuclear separation and αis a screening parameter. according to akanbi et al. [18], the morse potential given above has a minimum value at r = re and it is zero at r = ∞. the authors further emphasized that the barrier is assumed to be outside the influence of the morse oscillator. in refs. [19], another form of morse potential called shifted morse was studied. the form of morse potential is given as v (r) = (` + 1)2 − (2` + 3)e−x + e−2x. (2) in the present study, the interacting morse potential v (r) = de(1 − 2e −α(r−re ) + e−2α(r−re )) (3) will be considered. the form of morse potential model was studied by desai et al. [20] in one of their papers. however, the explicit detail analysis for the solutions of the morse potential as well as the energy equation were not given. in this study, the analysis of eigenvalues and eigenfunctions for the morse potential will be given in detail. figure 1 below, depicts the shape of morse potential function for scandium monoiodide (sci) and nitrogen monoiodide (ni). 2. bound state solution of the schrödinger equation using supersymmetric approach here, the eigenvalue and eigenfunction for the morse potential is obtained. the radial schrödinger equation with an interacting potential v (r)is given by[ − ~2 2µ d2 dr2 − en,` + v (r) ] rn,`(r) = 0, (4) where en,`is the non-relativistic energy of the system, ~is a reduced planck’s constant, µ is the reduced mass, rn,`(r)is the wave function. the solutions of eq. (2) can be obtained using different traditional methodologies. in this work, as said earlier, the supersymmetric approach will be used for the calculation. the supersymmetric approach is one of the approximate methods used to solve wave equations. the method depends on the proposition of superpotential. to use this method, first we plug eq. (3) into eq. (4) to have d2rn(r) dr2 = 2µ(de − en − 2dee−α(r−re ) + dee−2α(r−re )) ~2 rn(r) = 0. (5) to solve the equation above using supersymmetric and shape invariance approach [21-23], the next step is to write a ground state wave function r0(r) = ex p ( − ∫ u(r)dr ) , (6) where, u(r)is the superpotential function. invoking eq. (6) onto eq. (5) gives another relation satisfied by the superpotential functionu(r) : u 2(r) − du(r) dr = 2µ(de − en) ~2 + 2µ(dee−α(r−re ) − 2de) ~2 e−α(r−re ) (7) to validate the compatibility of the two sides of eq. (7) [24], we express the superpotential function as u(r) = ρ0 −ρ1e −αr. (8) the two terms ρ0 and ρ1 in eq. (8) are superpotential constants and their respective values will soon be determined. plugging eq. (8) into eq. (7) with some mathematical manipulations and simplifications leads to the following reasonable equations ρ20 = 2µde ~2 − 2µen ~2 , (9) ρ1 = √ 2µdee2αre ~2 , (10) ρ0 = 4µde eαre ~2 −αρ1 2ρ1 . (11) we consider the bound state solutions that demand the wave function rn(r)which satisfy the boundary conditions for rn(∞) = 283 onate et al. / j. nig. soc. phys. sci. 3 (2021) 282–286 284 0and rn(0)is limitary. these regularity conditions suggest that both ρ0and ρ1are greater than zero. having determined the two superpotential constants, the supersymmetric partner potentials u 2(r) ± du(r)dr can easily be constructed as follows v+(r) = u 2(r)+ du(r) dr = ρ20−ρ1(2a−α)e −αr +ρ21e −2αr,(12) v−(r) = u 2(r)− du(r) dr = ρ20−ρ1(2a+α)e −αr +ρ21e −2αr.(13) eq. (12) and eq. (13) are related by a simple relation v+(r, a0) = v−(r, a1) + r(a1), (14) where a0is an old set of parameters and a1is a new set of parameters uniquely determined from the old set of parameters. the termr(a1)is called a reminder term that do not dependent of the variable r. in the concept of shape invariance potential, a0 = ρ0 as ρ0 → ρ0 + αn. using the partner potential v−(r, a1), eq. (9), eq. (10) and eq. (11), the energy of the morse potential function can be obtain following e(−)n = ∑ k=1 r(ak) = r(a1) + r(a2) + r(a3) +...... + r(an−1) + r(an) = ( ρ1 −αa0 2a0 )2 + ( ρ1 −αa1 2a1 )2 − ( ρ1 −αa1 2a1 )2 + ( ρ1 −αa2 2a2 )2 − ( ρ1 −αa2 2a2 )2 + ( ρ1 −αa3 2a3 )2 − ( ρ1 −αa3 2a3 )2 + ( ρ1 −αa4 2a4 )2 + ....( ρ1 −αan−1 2an−1 )2 + ( ρ1 −αan 2an )2 = ( ρ1 −αa0 2a0 )2 + ( ρ1 −αan 2an )2 = ( 4µdeeαre −αa0 2a0 )2 + ( 4µdeeαre −αan 2an )2 (15) following the formalism of supersymmetric approach, the complete energy equation is obtain as en = de − α2~2 2µ  2µde eαre α2~2 − ( n + 12 ) √ 2µde e2αre α2~2√ 2µde e2αre α2~2  2 . (16) 2.1. wave function to obtain the wave function, we make a transformation of the form y = 1eαr and invoke it on eq. (5) to have d2rn(y) dy2 − 1 y drn(y) dy + 2µ[en−de +de eαre y(2−eαre y)] α2~2 y2 rn(y) = 0.(17) following the paper of tezcan and sever [25], the radial wave function for the morse potential becomes rn(y) = ny √ 2µ(en−de ) α2~2 e −y √ 2µde e2αre α2~2 l 2 √ 2µ(en−de ) α2~2 n ( 2 √ 2µ(en−de ) α2~2 y ) .(18) (a) (b) figure 2: variation of en (cm−1)against de (cm−1) and αrespectively 3. discussion the shape of morse potential for scandium monoiodide and nitrogen monoiodide is shown in figure 1. in figures 2 (a) and (b), the variation of energy against the dissociation energy and screening parameter respectively are shown. the energy and each of the dissociation energy and screening parameter respectively varies inversely with each other. in each case, the highest quantum state has the highest energy. the energies at various quantum state at deand α = 0respectively are zero, which is the point of convergence for the energies at different quantum states. the vibrational energies of the morse potential for various values of the screening parameter, quantum number and dissociation energy are presented in table 1. the energy of the system rises with an increase in the quantum number, screening parameter and dissociation energy respectively. for a unity value of the dissociation energy, the energy of the system has turning point as the quantum number and the screening param284 onate et al. / j. nig. soc. phys. sci. 3 (2021) 282–286 285 table 1: vibrational energies (incm−1) of morse potential model with µ = ~ = 1 and re = 0.5ȧ for three values ofde(cm1) and various values ofn and α(cm−1). n α en(de = 1) en(de = 5) en(de = 10) 0 1 2 3 0.05 0.25 0.55 0.75 0.05 0.25 0.55 0.75 0.05 0.25 0.55 0.75 0.05 0.25 0.55 0.75 0.0350428 0.0787444 0.1114909 0.1689642 0.3874722 0.5512045 0.3510962 0.8318137 1.1920249 0.4600176 1.1155416 1.6067385 0.1032535 0.2343583 0.3325977 0.4600176 1.1155416 1.6067385 0.8264137 2.2685666 3.3491997 0.9581778 2.9247499 4.3983404 0.1689642 0.3874722 0.5512045 0.6885710 1.7811110 2.5997725 0.9992312 3.4028193 5.2038744 0.8938379 4.1714581 6.6274424 0.2321749 0.5380861 0.7673113 0.8546244 2.3841805 3.5303065 0.8695486 4.2345720 6.7560492 0.2669981 4.8556664 8.2940444 table 2: comparison of the calculated energies (incm−1) forx1 ∑+ state of sci andx3 ∑ − state of ni with the predicted experimental rkr values of the morse potential function n s ci n i calculated [27] lte calculated [27] lte 0 138.4121 138.3 0.1121 301.6 301.1 0.5000 1 414.7976 413.9 0.4976 898.0135 896.6 1.4135 2 688.6233 687.7 0.9233 1486.4617 1482.3 3.1617 3 961.1402 959.9 1.2402 2062.5594 2058.8 3.7594 4 1232.1101 1230.4 1.7101 2629.9899 2625.9 4.0899 5 1501.6044 1499.3 2.3044 3187.9671 3183.6 4.3671 6 1769.1278 1766.5 2.6278 3736.8788 3731.9 4.9788 eter increases respectively. the observed data and the calculated energies for x1 ∑+state of sci and x3 ∑ −state of ni obtained using eq. (15) are given in table 2. the relative deviation lte (calculated result minus experimental result) of the calculated results from the experimental results are also given in table 2. the experimental values used for this work are taken from ref. [27]. for sci, de = 2.858ev, re = 2.6708å, ωe = 277.18cm−1while that of ni are de = 1.648ev, re = 1.9653å, ωe = 604.7cm−1. the screening parameter is calculated using α = πcωe √ 2µ de . (19) it is shown that the relative deviation becomes larger as the vibrational quantum state increases for the two molecules. however, the relative deviation forx3 ∑ −state of ni are higher compared to the relative deviation forx1 ∑+state of sci. this simply means that the sci are more fitted for the calculation compared to ni. the average absolute percentage deviation for each of the molecule is calculated using the formula σav = 100 n ∑ v ∣∣∣∣∣ ev − erkrerkr ∣∣∣∣∣ . (20) where erkr is the observed values, ev is the present results and is the number of observed data points. the formula seems to be the revised version of what is given in ref. [18]. the calculated results in this study are greater than the experimental values, thus, to avoid negative deviation, we revised the order of the both the experimental and calculated values. the average absolute percentage deviation for sci is 0.021% while that of ni is 0.032%. to determine the proximity of the present results to the predicted values, the percentage error for each of the calculated result is computed using the formula errp = ∑ v lt e erkr × 100. 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[26] c. a. onate, m. c. onyeaju, a. n. ikot, j. o. a. idiodi & j. o. ojonubah, “eigen solutions, shannon entropy and fisher information under the eckart manning rosen potential model”, journal of the korean physical society 70 (2017) 339 [27] r. r. reddy, y. n. ahammed, b. s. devi, p. a. azeem, k. r. gopal & t. v. r. rao, “potential energy curves, dissociation energies and franck condon factors for ni and sci molecules”, journal of quantum spectrosc andopy radiation transfer 74 (2002) 125 286 j. nig. soc. phys. sci. 4 (2022) 83–87 journal of the nigerian society of physical sciences on some pursuit differential game problem in a hilbert space jamilu adamua,∗, aminu sulaiman hallirub, bala ma’aji abdulhamidc adepartment of mathematics, federal university, gashua, nigeria. bdepartment of mathematical sciences, bayero university, kano, nigeria. cdepartment of mathematical sciences, abubakar tafawa balewa university, bauchi, nigeria. abstract we study pursuit differential game problem in which a countable number of pursuers chase one evader. the problem is formulated in a hilbert space l2 with pursuers’ motions described by nth order differential equations and that of the evader by mth order differential equation. the control functions of the pursuers and evader are subject to integral and geometric constraints respectively. duration of the game is denoted by the positive number θ. pursuit is said to be completed if there exist strategies u j of the pursuers p j such that for any admissible control v(·) of the evader e the inequality ‖y(θ) − x j(θ)‖ ≤ r j is satisfied for some j ∈ {1, 2, . . . }. in this paper, sufficient condition for completion of pursuit were obtained and also strategies of the pursuers that ensure completion of pursuit are constructed. doi:10.46481/jnsps.2022.379 keywords: differential game, pursuer, evader, geometric constraint, integral constraint, hilbert space. article history : received: 03 september 2021 received in revised form: 25 january 2022 accepted for publication: 25 january 2022 published: 28 february 2022 c©2022 journal of the nigerian society of physical sciences. all rights reserved. communicated by: t. latunde 1. introduction a considerable amount of literature on differential game problem in which finite number of pursuers chase one evader in the hilbert space l2, control function of players subjected to either geometric, integral or both constraints has been published. (see for example, [1][25] and some reference their in. in many studies of differential game problems, motions of the two players (i.e. pursuer and evader) are explicitly stated and are considered to be differential equations of the same order. for example, in the papers [3, 8, 10, 13, 20, 24], motion of each of the player is considered to obey first order differential ∗corresponding author tel. no: +2347033885836 email address: jamiluadamu88@gmail.com (jamilu adamu ) equation. in other studies such as [5, 6, 17, 22], players’ motions are described by second order differential equations. whereas in [1, 2, 4, 21], motion of the pursuers and evader are described by first and second order differential equations respectively. adamu et. al [1], studied pursuit-evasion differential game problem in a hilbert space l2, in which motions of pursuers and evader are described by first and second order differential equations respectively. control functions of both the pursuers and evader are subject to integral constrains. under certain condition the authors found the value of the game and constructed optimal strategies of the players. in the paper [2] by badakaya, a pursuit differential game problem with finite number of pursuers and one evader in the space l2 is considered. pursuers’ motions are described 83 adamu et al. / j. nig. soc. phys. sci. 4 (2022) 83–87 84 by a first order differential equations and that of the evader by a second order differential equation. control functions of the pursuers and evader are subject to integral and geometric constraints respectively. duration of the game is denoted by the positive number θ. theorems are stated and proved each of which provides a condition for completion of pursuit. consequently, strategies of the pursuers that ensure completion of pursuit are constructed. in the other hand, the paper [4], is about study of pursuit differential game problem in l-catch sense with finite number of pursuers and one evader in the space l2. in this paper, control functions of the pursuers and evader are subject to integral and geometric constraints respectively. sufficient condition for completion of pursuit were obtained. leong and ibragimov [12] studied simple motion pursuit differential game with m pursuers and one evader on a closed convex subset of the hilbert space l2 . control functions of the players are subjected to integral constraints. the total resource of the pursuers is assumed to be greater than that of the evader. strategy of pursuers were constructed sufficient to complete the pursuit from any initial position. in the paper [20], simple motion differential game with many players and geometric constraints on the control functions of the players was studied. by using lyapunov function method for an auxiliary problem, they obtained sufficient conditions to find the pursuit time in rn. the paper [21] deal with the study of a pursuit differential game problem for the so-called ”boy and crocodile” game in the space rn. boy’s motion is described by a first order differential equations and that of the crocodile by a second order differential equation. control functions of the pursuer and evader are subject to integral and geometric constraints respectively. they obtained sufficient conditions of completion of pursuit. vagin and petrov [25] studied a pursuit differential game problem with finite number pursuers and evader in the space rn. motions of the players are described by nth order differential equations. control functions of the players are subject to geometric constraints. sufficient condition for completion of pursuit was obtained. in the present paper, motivated by the above development, we proposed to study pursuit differential game problem with countable number of pursuers and one evader in the hilbert space l2. the motions of the pursuers and evader are described by nth and mth order differential equations respectively. control functions of the pursuers are subject to integral constrains. whereas, geometric constraint is imposed on the control function of the evader. we found the sufficient condition of completion of pursuit in l-catch sense. 2. differential game formulation let l2 = ξ = (ξ1,ξ2, . . . ) : ∞∑ j=1 ξ2j < ∞  , with inner product 〈·, ·〉 : l2 × l2 → r and norm ‖ · ‖ : l2 → [0, +∞), defined as follows: 〈ξ, y〉 = ∞∑ j=1 ξ jy j, ‖z‖ =  ∞∑ j=1 z2j  1/2 , where ξ, y, z ∈ l2. we consider a differential game described by the following initial value problem: p j : dn x j dtn = u j(t), e : d m y dtm = v(t), (1) subject to: x j(0) = x0j, d x j dt (0) = x 1 j, d2 x j dt2 (0) = x 2 j, . . . , dn−1 x j dtn−1 (0) = x n−1 j , j ∈ j, y(0) = y0, dydt (0) = y 1, d2 y dt2 (0) = y 2, . . . , dm−1 y dtm−1 (0) = y m−1 (2) where x j, x0j, x 1 j, x 2 j, . . . , x n−1 j , u j, y, y 0, y1, y2, . . . , ym−1, v ∈ l2, u j and v are controls parameter of the pursuer p j and evader e respectively. here and below n ≤ m for all n, m ∈ n, j = {1, 2, . . . } and ‖y0 − x j0‖ > r j, where r j ≥ 0. let c(0,θ; l2) be the space of functions h(t) = (h1(t), h2(t), · · · , h j(t), · · · ) ∈ l2, t ≥ 0, (3) such that the following conditions hold: (a) h j(t), 0 ≤ t ≤ θ, j = 1, 2, · · · , are absolutely continuous functions; (b) h(t), 0 ≤ t ≤ θ, is a continuous function in the norm of l2. definition 2.1. a function u j(t) = (u j1(t), u j2(t), . . . ) with borel measurable coordinates such that∫ θ 0 ||u j(t)|| 2dt ≤ β2j, (4) where β j is given positive number, is called an admissible control of the jth pursuer. definition 2.2. a function v(t) = (v1(t), v2(t), . . . ) with borel measurable coordinates such that ||v(t)|| ≤ γ, t ≥ 0 (5) where γ is given positive number, is called admissible control of the evader e. definition 2.3. a function u j(t, x j, y, v), u j : [0,θ] × l2 × l2 × l2 → l2, is called a strategy of the jth pursuer if, for any admissible control v(·) of the evader e, the system (1)-(2) has a unique solution (x j(·), y(·)), with x j(·), y(·) ∈ c(0,θ, l2). a strategy u j is admissible if each control involved in the formation of this strategy is admissible. 84 adamu et al. / j. nig. soc. phys. sci. 4 (2022) 83–87 85 definition 2.4. the system described by (1)-(2) in which the controls u j(·) and v(·) satisfy the inequalities (4) and (5) respectively is called game g1. definition 2.5. pursuit is said to be completed in l-catch sense in the game g1 if there exist strategies u j of the pursuers p j such that for any admissible control v(·) of the evader e the inequality ‖y(θ)− x j(θ)‖ ≤ r j is satisfied for some j ∈ {1, 2, . . . }. research problem: in the game g1, find sufficient condition for completion of pursuit. 3. main result once the players’ admissible controls u j(·) and v(·) are chosen, the solution to the equations of motion for the jth pursuer and evader in (1) with initial condition (2) are respectively given by x j(θ) = x j0 + ∫ θ 0 ∫ t1 0 ∫ t2 0 · · · ∫ tn−1 0 u j(t)dt dtn−1 . . . dt2 dt1, (6) and y(θ) = y0 + ∫ θ 0 ∫ t1 0 ∫ t2 0 · · · ∫ tm−1 0 v(t)dt dtm−1 . . . dt1, (7) where x j0 = x 0 j + θx 1 j + θ2 2! x2j + · · · + θn−1 (n − 1)! xn−1j (8) y0 = y 0 + θy1 + θ2 2! y2 + · · · + θm−1 (m − 1)! ym−1 (9) the expression with the multiple integrals in (6) and (7) can be reduced to that with single integral using the formula below (see [27])∫ θ 0 ∫ t1 0 ∫ t2 0 · · · ∫ tn−1 0 u(t) dtn−1 . . . dt2 dt1 = ∫ θ 0 (θ− t)n−1 (n − 1)! u(t)dt. (10) therefore, equation (6) and (7) become x j(θ) = x j0 + ∫ θ 0 (θ− t)n−1 (n − 1)! u j(t)dt, (11) y(θ) = y0 + ∫ θ 0 (θ− t)m−1 (m − 1)! v(t)dt, (12) alternatively, in place of the players’ dynamic equations (1), we can consider an equivalent differential game described by the following first order differential equations: p j : d x j dt = (θ−t)n−1 (n−1)! u j(t), x j(0) = x j0, e : dydt = (θ−t)m−1 (m−1)! v(t), y(0) = y0, (13) let π1 = ω0(β2j − γ 2ω1) + ||y0|| 2 − ∣∣∣∣∣∣x j0∣∣∣∣∣∣2 and π2 = ω0(β2j − θγ 2) + ||y0|| 2 − ∣∣∣∣∣∣x j0∣∣∣∣∣∣2, where ω0 = θ2n−1(2n−1)((n−1)!)2 and ω1 = ((n−1)!)2θ2(m−n)+1 (2(m−n)+1)((m−1)!)2 . we define the sets x :=  ⋃ j∈j { z ∈ l2 : 2〈y0 − x j0, z〉 ≤ π1 } , n < m.⋃ j∈j { z ∈ l2 : 2〈y0 − x j0, z〉 ≤ π2 } , n = m. (14) the following statement gives sufficient condition for completion of pursuit. theorem 3.1. . if y(θ) ∈ x then the pursuit can be completed in the game g1. proof. to prove this theorem, we first introduce dummy pursuer with state variable z whose motion obey the following initial value problem. dz dt = (θ− t)n−1 (n − 1)! ū(t), z(0) = z0 = x j0, (15) and that the control function ū(·) is such that(∫ θ 0 ∣∣∣∣∣∣ū(t)∣∣∣∣∣∣2 dt) 12 ≤ β̄ = β j + r j√ ω0 (16) clearly, β̄ > β j and (β̄−β j) √ ω0 = r j for all j ∈ j. for all j ∈ j and 0 ≤ t ≤ θ, we construct a dummy pursuer’s strategy as follows: if n = m, we set ū(t) = (2n − 1)(n − 1)! θ2n−1(θ− t)1−n (y0 − z0) + v(t). (17) if n < m, we set ū(t) = (2n − 1)(n − 1)! θ2n−1(θ− t)1−n (y0−z0)+ (n − 1)!v(t) (m − 1)!(θ− t)n−m .(18) for the case n = m, let the dummy pursuer z use the strategy (17), we show that this strategy is admissible and ensure the equality z(θ) = y(θ). indeed, z(θ) = z0 + ∫ θ 0 (θ− t)n−1 (n − 1)! × ( (2n − 1)(n − 1)! θ2n−1(θ− t)1−n (y0 − x j0) + v(t) ) dt = z0 + (2n − 1)(y0 − x j0) θ2n−1 ∫ θ 0 (θ− t)2n−2dt + ∫ θ 0 (θ− t)n−1 (n − 1)! v(t)dt = z0 + y0 − x j0 + ∫ θ 0 (θ− t)n−1 (n − 1)! v(t)dt = y0 + ∫ θ 0 (θ− t)m−1 (m − 1)! v(t)dt = y(θ). 85 adamu et al. / j. nig. soc. phys. sci. 4 (2022) 83–87 86 it is left to show that the strategy (17) is admissible. by assumption y(θ) ∈ x and if n = m, we have, 2〈y0 − x j0, y(θ)〉 ≤ ω0 ( β2j − θγ 2 ) + ||y0|| 2 − ||x j0|| 2 (19) in accordance with inequality (19) and using state equation (12) of the evader, we have: 2 〈 y0 − x j0, ∫ θ 0 (θ− t)m−1 (m − 1)! v(t)dt 〉 ≤ ω0 ( β2j − θγ 2 ) − ||y0 − x j0|| 2. (20) using the inequality (20), we obtained∫ θ 0 ||ū(t)||2dt = ∫ θ 0 ∣∣∣∣∣ ∣∣∣∣∣ (2n − 1)(n − 1)!θ2n−1(θ− t)1−n (y0 − z0) + v(t) ∣∣∣∣∣ ∣∣∣∣∣2 dt = ∫ θ 0 ∣∣∣∣∣ ∣∣∣∣∣ (2n − 1)(n − 1)!θ2n−1(θ− t)1−n (y0 − x j0) ∣∣∣∣∣ ∣∣∣∣∣2 dt + 2 ∫ θ 0 〈 (2n − 1)(n − 1)! θ2n−1(θ− t)1−n (y0 − x j0), v(t) 〉 dt + ∫ θ 0 ||v(t)||2 dt ≤ ||y0 − x j0||2 ω0 + 2 ω0 〈 y0 − x j0, ∫ θ 0 (θ− t)m−1 (m − 1)! v(t)dt 〉 + θγ2 ≤ ||y0 − x j0||2 ω0 + 1 ω0 ( ω0 ( β2j − θγ 2 ) − ||y0 − x j0|| 2 ) + θγ2 = β2j < β̄ 2 for the case n < m and if the dummy pursuer z uses the strategy (18), we have z(θ) = y(θ). indeed, clearly z(θ) = x j0 + (2n − 1)(y0 − x j0) θ2n−1 ∫ θ 0 (θ− t)2n−2dt + ∫ θ 0 (θ− t)m−1 (m − 1)! v(t)dt = x j0 + (2n − 1)(y0 − x j0) θ2n−1 θ2n−1 2n − 1 + ∫ θ 0 (θ− t)m−1 (m − 1)! v(t)dt = x j0 + y0 − x j0 + ∫ θ 0 (θ− t)m−1 (m − 1)! v(t)dt = y0 + ∫ θ 0 (θ− t)m−1 (m − 1)! v(t)dt = y(θ). the admissibility of the strategy (18) can be shown as follows. the inclusion y(θ) ∈ x, implies that, 2〈y0 − x j0, y(θ)〉 ≤ ω0 ( β2j −γ 2ω1 ) + ||y0|| 2 − ||x j0|| 2 (21) in accordance with inequality (21) and using state equation (12) of the evader , we have: 2 〈 y0 − x j0, ∫ θ 0 (θ− t)m−1 (m − 1)! v(t)dt 〉 ≤ ω0 ( β2j −γ 2ω1 ) − ||y0 − x j0|| 2. using the inequality (22), we obtained∫ θ 0 ||ū(t)||2dt = ∫ θ 0 ∣∣∣∣∣ ∣∣∣∣∣ (2n − 1)(n − 1)!θ2n−1(θ− t)1−n (y0 − x j0) + (n − 1)!v(t)(m − 1)!(θ− t)n−m ∣∣∣∣∣ ∣∣∣∣∣2 dt = ∫ θ 0 ∣∣∣∣∣ ∣∣∣∣∣ (2n − 1)(n − 1)!θ2n−1(θ− t)1−n (y0 − x j0) ∣∣∣∣∣ ∣∣∣∣∣2 dt + 2 ∫ θ 0 〈 (2n − 1)(n − 1)! θ2n−1(θ− t)1−n (y0 − x j0), (n − 1)!v(t) (m − 1)!(θ− t)n−m 〉 dt + ∫ θ 0 ∣∣∣∣∣ ∣∣∣∣∣ (n − 1)!v(t)(m − 1)!(θ− t)n−m ∣∣∣∣∣ ∣∣∣∣∣2 dt ≤ ||y0 − x j0||2 ω0 + 2 ω0 〈 y0 − x j0, ∫ θ 0 (θ− t)m−1 (m − 1)! v(t)dt 〉 + γ2ω1 ≤ ||y0 − x j0||2 ω0 + 1 ω0 ( ω0 ( β2j −γ 2ω1 ) − ||y0 − x j0|| 2 ) + γ2ω1 = β2j < β̄ 2 this shows that for each two cases we established that z(θ) = y(θ). with this and if the real pursuers use the strategies u j(t) = β j β̄ ū(t), 0 ≤ t ≤ θ, (22) we show that∣∣∣∣∣∣y(θ) − x j(θ)∣∣∣∣∣∣ ≤ r j (23) indeed, using equation (22) and cauchy-schwartz inequality, we get∣∣∣∣∣∣y(θ) − x j(θ)∣∣∣∣∣∣ = ∣∣∣∣∣∣z(θ) − x j(θ)∣∣∣∣∣∣ = ∣∣∣∣∣∣ ∣∣∣∣∣∣z0 + ∫ θ 0 (θ− t)n−1 (n − 1)! ū(t)dt − x j0 − ∫ θ 0 (θ− t)n−1 (n − 1)! u j(t)dt ∣∣∣∣∣∣ ∣∣∣∣∣∣ = ∣∣∣∣∣∣ ∣∣∣∣∣∣ ∫ θ 0 (θ− t)n−1 (n − 1)! ( ū(t) − β j β̄ ū(t) ) dt ∣∣∣∣∣∣ ∣∣∣∣∣∣ ≤ ∫ θ 0 ∣∣∣∣∣∣ ∣∣∣∣∣∣ (θ− t)n−1(n − 1)! ( 1 − β j β̄ ) ū(t) ∣∣∣∣∣∣ ∣∣∣∣∣∣ dt = ( β̄−β j β̄ ) ∫ θ 0 (θ− t)n−1 (n − 1)! ∣∣∣∣∣∣ū(t)∣∣∣∣∣∣ dt ≤ ( β̄−β j β̄ )  (∫ θ 0 (θ− t)2n−2 ((n − 1)!)2 dt ) 1 2 (∫ θ 0 ∣∣∣∣∣∣ū(t)∣∣∣∣∣∣2 dt) 12  ≤ ( β̄−β j β̄ ) ( θ2n−1 (2n − 1)((n − 1)!)2 ) 1 2 β̄ ≤ (β̄−β j) √ ω0 = r j. this complete the prove of the theorem. 86 adamu et al. / j. nig. soc. phys. sci. 4 (2022) 83–87 87 4. conclusion we studied pursuit differential game problem in which a countable number of pursuers chase one evader in the hilbert space l2. control function of the pursuers and evader are subject to integral and geometric constraints respectively. pursuers’ motions are described by nth order differential equations and that of the evader by mth order differential equation where n ≤ m, n, m ∈ n. in this piece of research the strategies of the pursuers are constructed and sufficient condition for completion of pursuit were obtained. the result of this paper is the generalization of some result in the literature. for example, the result of the paper [4] is a corrolary to this paper when we set n = 1 and m = 2 in the players’ dynamic equation (1)-(2). acknowledgements the authors would like to thank the referee for giving useful comments and suggestions for improvement of this paper. references [1] j. adamu, k. muangehoo, a. j. badakaya and j. rilwan, ”on pursuitevasion differential game problem in a hilbert space”, aims mathematics 5 (2020) 7467. 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[27] w. abdul-majid, linear and nonlinear integral equation methods and applications, 10-20, saints xavier university, chicago , 11,60655, usa, 2011. 87 j. nig. soc. phys. sci. 4 (2022) 769 journal of the nigerian society of physical sciences application of machine learning and resampling techniques to credit card fraud detection chinedu l. udeze, idongesit e. eteng∗, ayei e. ibor department of computer science, university of calabar, calabar, cross river state, nigeria abstract the application of machine learning algorithms to the detection of fraudulent credit card transactions is a challenging problem domain due to the high imbalance in the datasets and confidentiality of financial data. this implies that legitimate transactions make up a high majority of the datasets such that a weak model with 99% accuracy and faulty predictions may still be assessed as high-performing. to build optimal models, four techniques were used in this research to sample the datasets including the baseline train test split method, the class weighted hyperparameter approach, and the undersampling and oversampling techniques. three machine learning algorithms were implemented for the development of the models including the random forest, xgboost and tensorflow deep neural network (dnn). our observation is that the dnn is more efficient than the other 2 algorithms in modelling the under-sampled dataset while overall, the three algorithms had a better performance in the oversampling technique than in the undersampling technique. however, the random forest performed better than the other algorithms in the baseline approach. after comparing our results with some existing state-of-the-art works, we achieved an improved performance using real-world datasets. doi:10.46481/jnsps.2022.769 keywords: machine learning, fraud detection, random forest, resampling techniques, xgboost, tensorflow, deep neural network article history : received: 18 april 2022 received in revised form: 25 july 2022 accepted for publication: 01 august 2022 published: 15 august 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: t. latunde 1. introduction in the usa, fraudulent activities amount to a loss of more than 12 billion dollars by 2020 [1]. fraudulent records often make up about 0.2% of datasets making it a needle in the haystack task. credit card fraud detection is often carried out using large datasets and historical data. fraud often takes place when the card is lost, stolen or cloned by adversaries who perform third-party fraud by impersonating the identity of the original user. ∗corresponding author tel. no: +2348038722277 email address: ideteng@unical.edu.ng (idongesit e. eteng) fraud could happen when a point of sales (pos) is compromised. false detection could lead to unjustified blocking of credit cards without fraudulent transactions which could lead to customer complaints and loss of reputation whereas nonblocking of violated cards can lead to huge financial fraud. in general, fraud detection can be categorized as a case of binary classification like other similar problems such as spam filtering. such classification can be tackled using approaches like decision trees, support vector machines (svm), k-nearest neighbour, logistic regression, random forest, xgboost, neural networks and others. the major contributions of this research are: 1 udeze et al. / j. nig. soc. phys. sci. 4 (2022) 769 2 1. the comparative analysis of results from four dataset sampling techniques using some evaluation metrics. 2. use of machine learning (ml) and deep learning techniques to model fraud detection in real-world datasets using python libraries. 2. related works initially, rule-based management systems were used for creating fraud patterns, but this became too complex for manual analysis and machine learning (ml) techniques were adopted. ml techniques have also been applied to solve other complex problems [2-6]. although neural networks emerged first, they have the limitation of using a black-box model. consequently, random forests, which could provide a description for the reason why a transaction was fraudulent and also avoid over-fitting became more applicable. ml needs online algorithms to learn from streams with a continuous flow of information. sarno et al. [7] devised a method of obtaining fraudulent records from transaction datasets using a process mining technique. the records comprise variables that are used together with professional expertise to build a collection of association rules. association rule-based methods have the limitation of the inability to deal with the imbalance between negative and positive transactions. statistical models such as logistic regression, multiple discriminant analysis, regression analysis and others are applicable in financial data mining. ivo et al. [8] used a dataset provided by fraud detection and mitigation experts from feedzai comprising 5600 million transactions that have been anonymized as a security and privacy measure. the dataset contained only 0.05% fraudulent transactions. they successfully detected 80% of the data that had fraud with 70% accuracy by using the ibm proactive technology online (proton) as their open-source baseline engine. they classified credit card fraud as offline and online fraud. the limitation of their research is that they were not able to correctly establish the precision and recall of their solution, partly because the dataset was historical and anonymized. they recommended the use of learning techniques to create rules after analysis of historical data. ishan et al. [9] observed that random forest is more accurate for identifying non-fraudulent cases while feed-forward neural network does better in the detection of fraudulent transactions. in their research, they combined both algorithms in an ensemble machine learning method to achieve better accuracy in detection. ensemble learning involves the combination of different classifiers to improve the performance and predictive ability of the model. they identified the following challenges in the detection of credit card fraud: unavailability of datasets, dynamic behaviour of fraudsters, highly skewed datasets and parameters for evaluation. the dataset they used was made up of credit card transactions by europeans in september 2013 consisting of a total of 284, 807 transactions with 0.172% fraudulent records. to discover whether fraudulent records are outliers or clusters they applied unsupervised learning methods using k-means and other clustering techniques but found out that the fraudulent transactions were rather uniformly distributed in the clusters. the limitation of their work is that some of their classifiers did not attain optimal accuracy and the scope of their work was limited to a dataset with numerical values and not text. phuong et al. [10] used anomaly detection methods to perform credit card fraud detection using two data-driven approaches. the dataset consists of real data of european credit card users for e-commerce transactions. the dataset has some numerical values obtained from a principal component analysis (pca) transformation. in the training phase, they used 284000 transactions while in the testing phase they used 200 non-fraudulent transactions and 200 fraudulent transactions. with this, they realized a high detection accuracy and low false negatives and false positives. they used a one-class support vector machine (ocsvm) using the optimal kernel parameter selection and t2 control chart. artikis et al. [11] used the machine learning model of speedd which constructs the fraud pattern and contains a user interface for fraud analysts. the ml component of speedd can be used for the online development of fraud patterns permitting it to adjust to the dynamic nature of fraud varieties. they were able to develop a good prototype through the assistance of the feedzai company which specializes in ml fraud detection. the algorithm tries to learn the patterns of fraudulent transactions using inductive logic programming (ilp), a technique that uses the divide-and-conquer strategy. ilp makes use of logic programming to represent training sets and learnt rules in learning a logical theory called hypothesis which describes the pattern of the fraudulent records. they used the online learning of event definitions (oled) to resolve problems of velocity and volume in training sets. oled implements a heuristic for searching using statistical measures for learning with only a little portion of the records. the oled and ilp algorithms were developed with the scala language, using the clingo solver for reasoning. they used precision as the evaluation criteria for the rules. the limitation of their research is that the obtained precision, recall and runtime need further improvement. kang et al. [12] used a convolutional neural network based on the principle of the animal visual cortex to perform classification to determine when transactions are fraudulent. abakarim et al. [13] designed a live credit card fraud detection system that implements a deep neural network (dnn) technology that uses an auto-encoder for the classification of operations as legitimate or illegitimate. the dnn is made up of 6 hidden layers comprising 3 encoders and 3 decoders. the dnn is modelled after the biological structure of human neurons made up of multilevel concealed layers of processing units that communicate with each other. their model performs classification on a live feed of credit card transactions making real-time decisions. in the first phase, they developed an ml model through periodic offline training of historical data. this stage is used for the transformation of the credit card operations into features and labels that would aid the classifier. after this, the dataset is broken into training and test sets used for training and testing the model. 2 udeze et al. / j. nig. soc. phys. sci. 4 (2022) 769 3 the second stage involves predicting with the model on a live stream of transactions. to achieve this they used the symbolic math library of the open-source python tensorflow together with keras a neural network library, and an apache distributed streaming system kafka that works with the memsql. they compared their solution with the results of four other classifiers and recorded fair precision, accuracy and recall. the algorithms they used for comparison include linear svm, logistic regression, non-linear auto ann and nn-based classifier. the limitation of their solution is that although they achieved an improved recall and f1 score, the precision was relatively low. another weakness is that they implemented simple dnn which could have been improved with more hyper-parameter tuning. lucas et al. [14] proposed a system that creates historybased features using hidden markov models (hmm). the features are generated by computing the likelihood of a set of operations being legitimate. they measured the common traits between a transaction and previous fraudulent or normal operations recorded for the cardholders. most current feature engineering methods used for this purpose generate descriptive features of card-holder historical data. this approach could be limited because it does not take into account the history of the account holder. however hmm remediates these drawbacks as a generative probabilistic model for sequence modelling. hence, these authors used a system with eight hmm-based features that were used to study the similarity between historical data and eight distributions previously learned with an unsupervised algorithm to obtain a fitting model. the hmm models used four training sets containing genuine and compromised credit cards and terminals. they developed the program using python hmm learn. they used a dataset comprising 4.7 * 107 anonymized credit card transactions from 01/03/2015 to 31/05/2015 to measure the increase in the hmm detection. the dataset was split into a training set, validation set and testing set. they also trained a random forest algorithm to compare the accuracy of the system when hmm features are integrated. deshan et al. [15] analysed the dataset of european credit cardholders by applying sampling and resampling techniques to mitigate the effects of the unbalanced nature on the models. they applied some metrics in the evaluation of the performance of classification models on the unbalanced dataset. with five decision-tree-based algorithms, they were able to train the dataset, and determine the optimal model using cross-validation. with their research, they established a scheme for the analysis and prediction of credit card fraud. yuxin et al. [16] used a similar dataset to train machine learning models using logistic regression, gradient boosting tree model and 5-layer neural network with three hidden layers. the first hidden layer consists of 120 neurons, the second 60 neurons and the last 30 neurons. they realized that out of the three algorithms, logistic regression had the least performance. however, the tree model seems to involve overfitting and requires further regularization. 3. materials and methods 3.1. dataset the dataset used for this research comprises 30 features and was obtained from the kaggle website [17]. it was derived from credit card transactions of european cardholders generated over 2 days in 2013. it has been pre-processed using principal component analysis which transformed the columns v1 to v28 to numerical values, except ‘amount’ and ‘time’ to avoid exposure of sensitive data to privacy violations. the dataset contains 284, 807 records which comprise 284, 315 legitimate and 492 fraudulent transactions. 3.2. methodology the dataset only contains about 0.173% fraudulent transactions, hence, it is highly unbalanced. to train a more balanced sample of the dataset for optimal models that are not skewed towards the majority of the non-fraudulent transactions, the following four techniques were applied separately with different results in the confusion matrix: 1. the baseline approach: the dataset was first split into a training set and test set using a test size of 20% while the training set was further split into the training set and validation set using the same proportion with the train test split function. 2. the class weighted approach: in the random forest classifier, a ‘balanced’ class weight was used to enhance the model. for the xgboost classifier, the scale pos weight was computed by taking the square root of the ratio of the frequency of the legitimate to the frequency of the fraudulent transactions. for the tensorflow dnn, the reciprocal of the frequencies of the two classes were used in computing the class weight used for training the model. 3. oversampling method: using the adasyn from the imbalanced-learn (imblearn) library, oversampling techniques were applied such that the original data contained 181961 legitimate and 315 fraudulent transactions but after the oversampling, it contained 181961 legitimate and 181956 fraudulent records. 4. undersampling method: the randomundersampler from the imblearn library was also applied to the dataset and it reduced the samples to 315 legitimate and 315 fraudulent data. as shown in figure 1, first, the dataset was split into training and testing sets like in the baseline approach. if a resampling technique is to be applied, then a balanced dataset is created before the training and validation of the model. at this stage, the random forest, xgboost (extreme gradient boosting) and tensorflow dnn (deep neural network) algorithms were applied in training the model for each of the four ways of sampling the dataset. the confusion matrix for each of the models is obtained as the output of the python programs which could be used to compute the evaluation metrics. the program also generates the numerical value for the auprc (area under the precision-recall curve) for each model. 3 udeze et al. / j. nig. soc. phys. sci. 4 (2022) 769 4 figure 1. the architecture of the proposed system figure 2. undersampling and oversampling [18] 3.2.1. random forest random forest algorithms function like bagged decision trees with the major exception that every tree can only split on a subset of features m. in this case, the features m must be different for every classifier. in the random forest classifier, a ‘balanced’ class weight was used to enhance the model. in running the code, the algorithm performed better than the other algorithms in the baseline approach. in support of [9], while running the experiments, random forest was more accurate for identifying non-fraudulent cases but not fraudulent cases. it was also noticed that the algorithm prevents overfitting of data and is fast to train with test data. table 1. model of confusion matrix predictions from the model 1 0 actual class 1 true positive (tp) false negative (fn) 0 false positive (fp) true negative (tn) table 2. confusion matrix after baseline splitting fraudulent legitimate random forest fraudulent 77 21 legitimate 4 56860 xgboost actual class fraudulent 80 18 legitimate 6 56858 tensorflow dnn fraudulent 77 21 legitimate 14 56860 table 3. evaluation metrics after baseline splitting accuracy precision recall f1-score random forest 0.99956 0.95062 0.78571 0.86033 xgboost 0.99958 0.93023 0.81632 0.86956 tensorflow dnn 0.99944 0.84615 0.78571 0.81481 3.2.2. xgboost xgboost is an implementation of gradient boosted decision trees. for the xgboost classifier used in our experiment, the weight used was computed by taking the square root of the ratio of the frequency of the legitimate transactions to the frequency of the fraudulent transactions. overall, this algorithm performed better using the oversampling technique. this is supported by results reported in tables 2 to 9. 3.2.3. tensorflow dnn the tensorflow dnn was run with 200 epochs but implemented an early stopping that could prevent it from making all the iterations if a desirable result is achieved. it also plots the curves for the train, loss and validation loss together, and then it generates another plot of the training accuracy and validation accuracy using the number of epochs as the x-axis and also showing the auprc. 3.2.4. resampling techniques due to heavy imbalance in the dataset, undersampling and oversampling methods were applied in some of the machine learning models used in the system. undersampling involves reducing the number of majority class samples while oversampling is achieved by multiplying the number of minority class samples by repeating some records. the synthetic minority oversampling technique (smote) is a popular example of an oversampling technique, however, in this research, we have implemented the adasyn python library for oversampling. 4 udeze et al. / j. nig. soc. phys. sci. 4 (2022) 769 5 figure 3. train and validation accuracy curves using the baseline approach with the tensorflow dnn figure 4. bar chart showing area under precision-recall curve for the 4 sampling methods roweida et al. [18] took studies with two resampling techniques and some classifiers on a kaggle dataset used to predict customers that will make specific transactions in the future, using scikit-learn, numpy and pandas. they implemented a nonheuristic algorithm for oversampling to balance class distribution by a random iteration of minority class records. one of the limitations of such an approach is overfitting since it replicates similar records of the minority class. however, oversampling could be beneficial when the dataset size is small, or when a particular class is small making the dataset to be skewed to5 udeze et al. / j. nig. soc. phys. sci. 4 (2022) 769 6 table 4. class weighted model confusion matrix fraudulent legitimate random forest fraudulent 74 24 legitimate 3 56861 xgboost actual class fraudulent 82 16 legitimate 8 56856 tensorflow dnn fraudulent 88 10 legitimate 576 56288 table 5. class weighted model evaluation metrics accuracy precision recall f1-score random forest 0.99953 0.96104 0.75510 0.84571 xgboost 0.99958 0.91111 0.83673 0.87234 tensorflow dnn 0.98971 0.13253 0.89796 0.23097 table 6. random undersampling confusion matrix fraudulent legitimate random forest fraudulent 90 8 legitimate 1780 54707 xgboost actual class fraudulent 89 9 legitimate 2157 56856 tensorflow dnn fraudulent 87 11 legitimate 320 56544 table 7. random undersampling evaluation metrics accuracy precision recall f1-score random forest 0.96840 0.04813 0.91837 0.0884 xgboost 0.96335 0.03963 0.90816 0.07595 tensorflow dnn 0.99419 0.21376 0.88776 0.34456 table 8. adasyn oversampling confusion matrix fraudulent legitimate random forest fraudulent 77 21 legitimate 12 56852 xgboost actual class fraudulent 85 13 legitimate 27 56837 tensorflow dnn fraudulent 82 16 legitimate 70 56794 table 9. adasyn oversampling evaluation metrics accuracy precision recall f1-score random forest 0.99942 0.86517 0.78571 0.82353 xgboost 0.99930 0.75893 0.86735 0.80953 tensorflow dnn 0.99849 0.53947 0.83673 0.65600 wards the majority class instances. they also implemented a random undersampling algorithm, a non-heuristic method of balancing class spreading by reducing the number of records in the majority class. the drawback is that it can result in the loss of information through the elimination of valuable data. nevertheless, when the dataset size is large, the effect can be negligible. their results show that oversampling techniques give better performance with the classifier models when compared to the undersampling techniques. a depiction of undersampling and oversampling techniques is given in figure 2. 3.3. evaluation metrics the following evaluation metrics are used to assess the performance of our approach as well as other comparative approaches. these are standard evaluation metrics, which have been used in several other works [8, 19]. 1. confusion matrix: this is a table that shows the performance of a machine learning model. the rows contain the results from an actual class while the column represents the predictions from an algorithm. in table 1, 1 stands for fraudulent transaction while 0 stands for a legitimate transaction, such that tn is for legitimate transactions correctly detected, fp is for legitimate transactions classified as fraudulent, fn is for fraudulent transactions classified as legitimate and tp is for a correctly detected fraudulent transaction. 2. accuracy: this is the proportion of the right predictions (tp + tn) amidst the entire number of transactions examined. acc = t p + t n t p + t n + f p + f n 3. precision: this is the measure of the ability of the model to correctly predict a positive value. it is given as the proportion of true positives from the actual total number of transactions in the positive class. a precision of 80% implies that out of every 100 detected fraud, at least 80 of them are correctly inferred. precision = t p t p + f p 4. recall: this is also known as the sensitivity of the model and it is given as the proportion of true positives in the entire collection of transactions that were in the original positive class. a recall of 90% implies that of every 100 cases of fraud that traverses through the system, at least 90 are detected. recall = t p t p + f n 6 udeze et al. / j. nig. soc. phys. sci. 4 (2022) 769 7 table 10. auprc figures for the algorithms and sampling techniques baseline approach class weighted undersampling oversampling random forest 0.74889 0.72789 0.04467 0.68173 xgboost 0.76111 0.76392 0.03648 0.65952 tensorflow dnn 0.84296 0.72615 0.66570 0.67274 5. f1-score: this is the harmonic mean of the recall and precision f1 = 2 ∗ precision ∗ recall precision + recall 4. results and discussion 4.1. experimental testbed the computation was carried out on a windows 11 64-bit operating system with 8gb ram and 500gb ssd. python 3.9.7 was used on an anaconda navigator and jupyter environment. the major libraries used include tensorflow, sklearn, matplotlib, pandas, imblearn, xgboost, numpy, seaborn and math. the code samples from [20] were resourceful in optimizing our models. 4.2. results the results of experimentation are tabulated in tables 2 to 9 and discussed in this section. 4.3. discussion of results in the baseline splitting of datasets, the other two decisiontree-based algorithms achieved better performance than the deep learning model as shown in tables 2 to 3. for both the baseline dataset splitting and the class weighted approach (tables 2 to 5), the xgboost achieved a better performance than the other two algorithms while the dnn resulted in relatively low precision and f1 score. after the random undersampling, the performance of the three algorithms were reduced with a very poor precision by the random forest and xgboost models as tabulated in tables 6 and 7, however, although the dnn performance was reduced, it produced a better accuracy, precision and f1-score than the other two models. after the adasyn oversampling, the performance of the three algorithms was better than the undersampling results and the random forest model produced a better accuracy, precision and f1 score than the other two algorithms. this is illustrated in tables 8-9. a summary of these results is given in table 10. furthermore, the train and validation accuracy curves of the tensorflow dnn are given in figure 3. similarly, in figure 4, the comparative performance of the four sampling methods is depicted. 4.4. comparison of results our work was compared with the some existing work [9], [13], [15] and [16]. this comparison, which was based on the accuracy and fi scores of each model s tabulated in table 11. from table 11, it can be observed that our approach demonstrated promising performance in terms of accuracy and f1 in table 11. comparison of approaches approach accuracy f1 yuxin et al. [16] 0.9993 0.777 deshan et al. [15] 0.850 abakarim et al. [13] 0.9861 0.294 ishan et al. [9] 0.9995 our approach 0.99958 0.87234 comparison to other approaches using the same dataset. having used three algorithms and four sampling techniques, we arrived at the best performance using the xgboost with the classweighted approach. furthermore, the random forest model also achieved good performance with the oversampling technique while the tensorflow dnn also resulted in a fair accuracy using the undersampling technique. this also affirms that neural networks could outperform other models when the dataset size is small. overall, observation from the auprc figures of the three algorithms used shows that the tensorflow dnn had more stable values and obtained the highest auprc so far using the baseline approach. from this, we have shown that the use of machine learning algorithms and deep neural work can be applied to the problem of fraud detection with significant results. this is very useful since deep neural networks can learn from data by understanding the representations in the data through input-output mapping [20]. 5. conclusion in this research, we conducted a comparative analysis of the performance of three machine learning algorithms in the detection of credit card fraud. since the fraud cases constituted a minority group in the dataset, adasyn oversampling and random undersampling techniques were used for resampling to create uniformly distributed classes. asides from these two approaches, results were also obtained with the baseline train test split method and the class weighted method. evaluation metrics were defined for the machine learning and deep learning classifiers including random forest, xgboost and tensorflow dnn. each of these algorithms has special cases where they are best fitted from the results obtained. in the future, it will be expedient to create an ensemble of machine learning and deep learning algorithms to take advantage of the strengths of each of the models. it is also important to try some resampling techniques like smote, more machine learning algorithms and another dataset that is not entirely made up of numeric values. 7 udeze et al. / j. nig. soc. phys. sci. 4 (2022) 769 8 acknowledgments we thank the referees for the positive enlightening comments and suggestions, which have greatly helped us in making improvements to this paper. references [1] r. aitken, “u.s. card fraud losses could exceed 12b usd by 2020”, forbes, (2016), http://www.forbes.com/sites/rogeraitken/2016/10/26/uscard-fraud-losses-could-exceed-12bn-by-2020/ [2] v. umarani, a. julian & j. deepa, “sentiment analysis using various machine learning and deep learning techniques”, journal of the nigerian society of physical sciences (2021) 385. 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[19] a. e. ibor, o. b. okunoye, f. a. oladeji, andk. a. abdulsalam, “novel hybrid model for intrusion prediction on cyber-physical systems’ communication networks based on bio-inspired deep neural network structure”, journal of information security and applications 65 (2022). [20] g. zoto, “credit card fraud detection using ml and deep learning”, youtube, (2020), https://www.youtube.com/watch?v=yx1 idv0e50 8 j. nig. soc. phys. sci. 4 (2022) 315 journal of the nigerian society of physical sciences introducing article numbering to journal of the nigerian society of physical sciences starting from the august 2022 issue, journal of the nigerian society of physical sciences will use article number in place of the traditional method of continuous pagination through the volume. this step helps us to maintain a rapid, efficient production process by being able to define pagination as soon as a paper is accepted. for papers that use article numbers, the page number will start from 1 and the article number is used instead of the page range in the citation. e.g., c. a. oyelami, w. akande & t. o. kolawole ”a preliminary geotechnical assessment of residual tropical soils around osogbo metropolis as materials for road subgrade”, journal of the nigerian society of physical sciences 4 (2022) 417.(article number: 417. https://doi.org/10.46481/jnsps.2022.417). while journal volumes and issue numbers will remain in place, article numbering will now play the key role in identifying specific articles. with the introduction of article numbers, the article becomes citable as soon as the proof corrections have been incorporated, ensuring readers have access to the latest research faster. in light of this, the table of contents in the print issue will reflect the order of the online issue; print readers will need to refer to the article number, clearly visible at the top of every page, to identify articles of interest. the printed issue spine will no longer include a page range. doi:10.46481/jnsps.2022.315 1 fulafia mis typewriter nigerian society of physical sciences fulafia mis typewriter publisher’s note j. nig. soc. phys. sci. 4 (2022) 265–280 journal of the nigerian society of physical sciences approximate analytical solution of fractional lane-emden equation by mittag-leffler function method richard olu awonusika∗, oluwaseun akinlo mogbojuri department of mathematical sciences, adekunle ajasin university, p.m.b. 001, akungba akoko, ondo state, nigeria abstract the classical lane-emden differential equation, a nonlinear second-order differential equation, models the structure of an isothermal gas sphere in equilibrium under its own gravitation. in this paper, the mittag-leffler function expansion method is used to solve a class of fractional laneemden differential equation. in the proposed differential equation, the polytropic term f (y(x)) = ym(x) (where m = 0, 1, 2, . . . is the polytropic index; 0 < x ≤ 1) is replaced with a linear combination f (y(x)) = a0 + a1y(x) + a2y2(x) + · · · + amym(x) + · · · + an yn (x), 0 ≤ m ≤ n, n ∈ n0. explicit solutions of the fractional equation, when f (y) are elementary functions are presented. in particular, we consider the special cases of the trigonometric, hyperbolic and exponential functions. several examples are given to illustrate the method. comparison of the mittag-leffler function method with other methods indicates that the method gives accurate and reliable approximate solutions of the fractional lane-emden differential equation. doi:10.46481/jnsps.2022.687 keywords: fractional differential equation, lane-emden differential equation, mittag-leffler function, elementary functions, series solution, eigenfunction expansion method article history : received: 02 march 2022 received in revised form: 07 april 2022 accepted for publication: 10 april 2022 published: 29 may 2022 c©2022 journal of the nigerian society of physical sciences. all rights reserved. communicated by: o. j. oluwadare 1. introduction fractional calculus is a generalization and extension of classical calculus to non-integer orders. in recent times, fractional calculus has attracted the attention of researchers in several areas including mathematics, physics, biology, chemistry, engineering, economics and psychology ([1], [2], [3], [4], [5], [6], [7], [8], [9]). several definitions of fractional calculus have been formulated by researchers. the most common and widely used definitions are the caputo, grünwald-letnikov and ∗corresponding author tel. no: email address: richard.awonusika@aaua.edu.ng (richard olu awonusika) riemann-liouville derivatives ([5], [8], [10]). the grünwaldletnikov derivative is mostly limited to numerical algorithms. the riemann-liouville fractional derivative has certain limitations and so becomes unsuitable in modeling some real-life phenomena since it requires the definition of fractional order initial conditions which have no physical meaningful explanation yet ([8]). the main advantage of the caputo derivative is that it takes on the same form of initial conditions for the integer-order differential equations. recently, the lane-emden equation has been investigated by several researchers due to its significant applications in mathematical physics and astrophysics ([11]). the classical laneemden equation, first introduced in 1870 by lane and studied 265 awonusika et al. / j. nig. soc. phys. sci. 4 (2022) 265–280 266 in 1907 by emden is of the form: y′′(x) + 2 x y′(x) + f (y(x)) = 0 (1) with initial conditions y(0) = a, y′(0) = b. (2) equation (1) with various forms of f (y(x)) has been used to model several phenomena such as the theory of stellar structure, thermal explosions, the thermal behavior of a spherical cloud of gas, isothermal gas spheres and thermionic currents ([12], [13]). various methods have been presented by several authors to obtain the solution of the initial value problem (1) – (2). the adomian decomposition method was employed in [11] to investigate the initial value problem (1) – (2). solutions of the lane-emden equation have also been obtained via the series method (see e.g., [14], [15]). the series solutions obtained in [15] were compared with the results obtained using the homotopy perturbation method. a numerical algorithm was developed by vanani and aminataei [16] for solving the laneemden equation. ogunniran et al. in [17] investigated the linear stabilities of some explicit members of runge-kutta methods in integrating the lane-emden equation. however, the standard lane-emden differential equation is the one given precisely by the initial value problem ([14]) y′′(x) + 2 x y′(x) + ym(x) = 0, m ≥ 0; x > 0 y(0) = 1, y′(0) = 0, (3) where m is the polytropic index and y = y(x) is the polytrope. clearly, the equation (3) is linear when m = 0, 1 and as a result the analytical solutions of the corresponding equations are realisable in closed forms. by extension it is mentioned in [14] that a closed form solution is also possible for m = 5. for numerical solutions of second order ordinary differential equations, see [18] and [19]. as a result of the significant importance of fractional calculus in modeling real-life phenomena accurately, the fractional lane-emden equation has been formulated and studied by researchers in very recent times. by using the collocation method, a numerical solution of (3) was obtained in [20]. some other researchers have also sought numerical solutions to the fractional lane-emden equation (see, e.g., [21], [22]). approximate solutions based on orthonormal bernoulli’s polynomials method, homotopy-adomian decomposition method and the series expansion method were treated in [23], [24] and [25] respectively. other treatise on the fractional lane-emden equation can be found in [26], [27], [28], [29] and [30]. in [28, subsection 5.2.2], the authors considered the power series solutions of the fractional lane-emden equation with the polytropic term ym with the fractional derivative described in the caputo sense, while malik and mohammed [25] presented approximate solutions of fractional lane-emden equation using conformable homotopy–adomian decomposition method and conformable residual power series method. arafa et al. in [31] used the mittag-leffler function method to solve a simple fractional differential equation of the form dαy = ay2, 0 < α ≤ 1, (4) with the fractional derivative described in the caputo sense. in this paper, the mittag-leffler function expansion method is employed to solve a class of fractional lane-emden initial value problem whose polytropic term f (y(x)) = ym(x) (where m = 0, 1, 2, . . . is the polytropic index; 0 < x ≤ 1) is replaced with a finite series f (y(x)) = a0 + a1y(x) + a2y2(x) + · · · + amym(x) + · · · + an yn (x), 0 ≤ m ≤ n, n ∈ n0, with the fractional derivative described in the caputo sense. the analytical solutions of the corresponding fractional equations, when f (y) are elementary functions, are presented, from which several examples of fractional lane-emden equations are given. in particular, we consider the special cases where f (y) are trigonometric, hyperbolic and exponential functions. comparison of the mittag-leffler function method with other methods shows that the method is reliably capable of solving analytically, nonlinear fractional lane-emden differential equations, and by extension, one can apply the method to solve several nonlinear fractional lane-emden equations when the functions f (y) are given by other special functions. the motivation for the forms of the nonlinear function f (y) considered in this paper arises from the need to address those situations where the expansion coefficients a0, a1, . . . , an−1 are not identically zero. 2. caputo fractional derivative and its basic properties this section presents the definition and basic properties of caputo derivative needed. for further discussions on fractional calculus and fractional differential equations, see, e.g., [5], [28], [32], [33], [34], [35], [36], [37], [38], [39], [40]. definition 2.1. for α > 0, the caputo fractional derivative of order α is defined as follows (m ≥ 1) : c dα f (x) =jm−αdm f (x) =  1 γ(m−α) ∫ x 0 (x − ξ)m−α−1 f (m)(ξ) dξ, m − 1 < α < m, dm d xm f (x) α = m, (5) where j α f (x) := 1 γ(α) ∫ x 0 (x − ξ)α−1 f (ξ) dξ (6) is the riemann-liouville fractional integral of order α. here f (m)(x) = d m f (x) d xm . remark 2.1. the caputo derivative of the constant function c is zero, i.e c dαc = 0. 266 awonusika et al. / j. nig. soc. phys. sci. 4 (2022) 265–280 267 corollary 2.1. for m − 1 < α < m, we have c dαxβ =  0, β ∈ {0, 1, · · · , m − 1} , γ(ν+1) γ(β−α+1) x β−α β > m − 1 non existing otherwise. (7) the following properties of the caputo derivative hold ( [8], [10]). lemma 2.1. suppose m − 1 < α < m; n, m ∈ n; α ∈ r; λ,σ ∈ c. let f (x) and g(x) be such that c dα f (x) and c dαg(x) exist. we have that (a) lim α→n c dα f (x) = f (n)(x) (b) lim α→n−1 c dα f (x) = f (n−1)(x) − f (n−1)(0) (c) c dα(λ f (x) + σg(x)) = λ c dα f (x) + σ c dαg(x) (d) c dαdm f (x) = c dα+m f (x) , dm c dα f (x). 3. mittag-leffler function in this section we examine the definition and properties of the mittag-leffler function as eigenfunctions of fractional derivatives ([1]). the mittag-leffler function plays an important role in the theory of fractional calculus. just as the exponential function naturally comes out of the solutions of classical differential equations, the mittag-leffler function plays an analogous role in the solution of fractional differential equations. it is thus a generalization of the exponential function. definition 3.1. the mittag-leffler function is defined by: eα(z) = ∞∑ k=0 zk γ(αk + 1) , α > 0,α ∈ r, z ∈ c. (8) definition 3.2. the mittag-leffler function can also be represented in two arguments α and β as eα,β(z) = ∞∑ k=0 zk γ(αk + β) , α,β > 0,α,β ∈ r, z ∈ c; (9) with eα,1 = eα. in particular, the significance of the mittag-leffler function is in the fact that it serves as the eigenfunction of the caputo and riemann-liouville derivatives in fractional calculus ([1]). it is easy to see that ([8], [41]) e1,1(z) = e1(z) = e z, e1,2(z) = ez − 1 z , e2,1(z 2) = cosh z, z ∈ c. e2,1(−z 2) = cos z, e2,2(z 2) = sinh z z , e2,2(−z 2) = sin z z , z ∈ c. corollary 3.1. for α,β ∈ r, z ∈ c and m ∈ n, the following recurrence relations hold for the mittag-leffler function ([41]) : (a) eα,β(z) = zeα,α+β(z) + 1 γ(β) (b) eα,β(z) = βeα,β+1(z) + αz d dz eα,β(z) = βeα,β+1(z) (c) ( d dz )m [zβ−1 eα,β(zα)] = zβ−m−1 eα,β−m(zα). 4. fractional lane-emden equations and their approximate solutions in this section, we apply the mittag-leffler function discussed in section 3 to obtain approximate solutions to the fractional lane-emden initial value problem c dαy(x) + ω xα−β c dβy(x) + f (y(x)) = 0 y(0) = a, y′(0) = 0, a ∈ [0, 1], (10) with 0 < x ≤ 1, 0 < β ≤ 1, 1 < α ≤ 2, ω ∈ r. here c dµ denotes the caputo fractional differential operator of order µ, µ > 0. the resulting solution is given as the mittag-leffler function. the function f (y(x)) will be chosen in such a way that it can be expanded in a power series, in particular, the maclaurin series. towards this end, let the function f (y(x)) be given by the power series f (y(x)) = ∞∑ m=0 amy m(x), (11) where am, m = 0, 1, 2, 3, . . . , are the expansion coefficients (constants). for computation reasons, one is interested in the approximation fn (y(x)) = n∑ m=0 amy m(x), an , 0. (12) we proceed by giving the following result that will be needed in the sequel. proposition 4.1. for 1 < α ≤ 2, 0 < x ≤ 1, the function fn (y(x)) admits the power series expansion fαn (x) := fn (y(x)) = ∞∑ `=0 cα`,n x α`, n ∈ n0, (13) where the constants cα `,n are given by cα`,n = n∑ m=0 amb α,n `,m . (14) here the numbers bα,n `,m , m = 3, 4, 5, . . . are given by the finite series bα,n `,m = ∑̀ pm−1 =0 pm−1∑ pm−2 =0 · · · p2∑ p1 =0 aα`−pm−1,n a α pm−1−pm−2,n · · ·aαp1,n, (15) with the mittag-leffler expansion coefficients given by aα`,n = b`n γ(α` + 1) , b`n are constants. (16) in particular, the special cases bα,n `,m , m = 0, 1, 2, are given respectively by bα,n `,0 =  1, ` = 0, 0, ` = 1, 2, · · · , bα,n `,1 = a α `,n, b α,n `,2 = ∑̀ p1 =0 bp1n b `−p1 n γ(αp1 + 1)γ(α` −αp1 + 1) . (17) 267 awonusika et al. / j. nig. soc. phys. sci. 4 (2022) 265–280 268 proof. by definition, we use the mittag-leffler function y(x) = eα(bn x α) = ∞∑ k=0 bkn x αk γ(αk + 1) (18) in (12) to see that fαn (x) := fn (y(x)) = n∑ m=0 am  ∞∑ k=0 bkn x αk γ(αk + 1) m = n∑ m=0 ams α m,n (x), (19) where we have written (m ≥ 1) sα0,n = 1, s α m,n (x) := y m(x) =  ∞∑ k=0 bkn x αk γ(αk + 1) m . (20) let aαk,n be given by aαk,n = bkn γ(αk + 1) . (21) then clearly, sα1,n (x) = ∞∑ `=0 aα`,n x α` = ∞∑ `=0 bα,n `,1 x α`. (22) furthermore, the classical cauchy product tells us that sα2,n (x) =  ∞∑ k=0 aαk,n x αk 2 = ∞∑ m=0 aαm,n x αm ∞∑ n=0 aαn,n x αn = ∞∑ `=0 bα,n `,2 x α`, (23) where bα,n `,2 = ∑̀ p1 =0 aαp1,n a α `−p1,n . (24) also, in a similar way, we see that sα3,n (x) =  ∞∑ k=0 aαk,n x αk 3 =  ∞∑ k=0 aαk,n x αk 2 ∞∑ n=0 aαn,n x αn. (25) that is, one has sα3,n (x) = s α 2,n (x)s α 1,n (x) =  ∞∑ m=0 bα,nm,2 x αm   ∞∑ m=0 bα,nn,1 x αn  = ∞∑ `=0 bα,n `,3 x α`, (26) where bα,n `,3 = ∑̀ p2 =0 bα,np2,2b α,n `−p2,1 = ∑̀ p2 =0 p2∑ p1 =0 aαp1,n a α p2−p1,n aα`−p2,n. (27) similarly, by taking more products, we see that sα4,n (x) =  ∞∑ k=0 aαk,n x αk 4 =sα3,n (x)s α 1,n (x) =  ∞∑ m=0 bα,nm,3 x αm   ∞∑ n=0 bα,nn,1 x αn  = ∞∑ `=0 ∑̀ p3 =0 bα,np3,3b α,n `−p3,1 xα` = ∞∑ `=0 bα,n `,4 x α`, (28) where bα,n `,4 = ∑̀ p3 =0 p3∑ p2 =0 p2∑ p1 =0 aαp1,n a α p2−p1,n aαp3−p2,n a α `−p3,n . (29) continuing in this way, we obtain sαm,n (x) =s α m−1,n (x)s α 1,n (x) =  ∞∑ k=0 bα,nk,m−1 x αk   ∞∑ n=0 bα,nn,1 x αn  = ∞∑ `=0 ∑̀ pm−1 =0 bα,npm−1,m−1b α,n `−pm−1,1 xα` = ∞∑ `=0 bα,n `,m x α`, (30) where (m = 3, 4, 5, . . . ) bα,n `,m = ∑̀ pm−1 =0 pm−1∑ pm−2 =0 · · · p2∑ p1 =0 aα`−pm−1,n a α pm−1−pm−2,n · · ·aαp1,n. (31) upon inserting (30) – (31) into (19), one gets fαn (x) = n∑ m=0 am ∞∑ `=0 bα,n `,m x α` = ∞∑ `=0 cα`,n x α`, (32) where cα`,n = n∑ m=0 amb α,n `,m (33) and we obtain the result as required. with the proposition 4.1 in place we now present the main result of this paper. theorem 4.1. for 0 < x ≤ 1, 0 < β ≤ 1, 1 < α ≤ 2, ω ∈ r, n ∈ n0; the fractional initial value problem c dαy(x) + ω xα−β c dβy(x) + n∑ m=0 amy m(x) = 0 y(0) = a, y′(0) = 0, a ∈ [0, 1], (34) 268 awonusika et al. / j. nig. soc. phys. sci. 4 (2022) 265–280 269 admits a series solution given by the mittag-leffler function y(x) = ∞∑ `=0 b`n (α,β; ω) γ(α` + 1) xα`, (35) where b 0 n (α,β; ω) := b 0 n = a, (36) b `+1 n (α,β; ω) := b `+1 n = − γ(α` + 1)γ(α(` + 1) −β + 1)cα `,n γ(α(` + 1) −β + 1) + ωγ(α` + 1) , ` ≥ 0. (37) the coefficients cα `,n, ` ≥ 0, are as given in proposition 4.1. proof. the starting point is to substitute the mittag-leffler function (18) in the equation (34) to evaluate the caputo fractional derivatives c dαy(x) and xβ−α c dβy(x). towards this end, using corollary 2.1, we have that c dαy(x) = ∞∑ k=0 bkn γ(αk + 1) c dαxαk = ∞∑ k=1 bkn γ(αk + 1 −α) xαk−α = ∞∑ k=0 bk+1n γ(αk + 1) xαk (38) and similarly one sees that xβ−α c dβy(x) =xβ−α ∞∑ k=0 bkn γ(αk + 1) c dβxαk = ∞∑ k=0 bk+1n γ(αk + α + 1 −β) xαk. (39) substituting the series (38), (39) in (34) and then applying proposition 4.1 gives ∞∑ `=0 b`+1n γ(α` + 1) xα` + ∞∑ `=0 ωb`+1n γ(α(` + 1) −β + 1) xα` + ∞∑ `=0 cα`,n x α` = 0, (40) which implies that ∞∑ `=0  b`+1n γ(α` + 1) + ωb`+1n γ(α(` + 1) −β + 1) + cα`,n  xα` = 0. (41) comparing the coefficients of xα`, ` = 0, 1, 2, · · · on both sides of (41) gives b`+1n ( 1 γ(α` + 1) + ω γ(α(` + 1) −β + 1) ) = −cα`,n and as a result, b`+1n = − γ(α` + 1)γ(α(` + 1) −β + 1)cα `,n γ(α(` + 1) −β + 1) + ωγ(α` + 1) , (42) where the coefficients cα `,n are as given in proposition 4.1. remark 4.1. for computation purposes, in addition to the first two coefficients bα,n `,m , m = 1, 2, given in propostion 4.1, we present explicitly the coefficients bα,n `,m , 3 ≤ m ≤ 20: bα,n `,3 = ∑̀ p2 =0 p2∑ p1 =0 bp1n b p2−p1 n b `−p2 n γ(αp1 + 1)γ(αp2 −αp1 + 1)γ(α` −αp2 + 1) bα,n `,4 = ∑̀ p3 =0 p3∑ p2 =0 p2∑ p1 =0 bp1n b p2−p1 n b p3−p2 n b `−p3 n γ(αp1 + 1)γ(αp2 −αp1 + 1)γ(αp3 −αp2 + 1)γ(α` −αp3 + 1) ... bα,n `,10 = ∑̀ p9 =0 p9∑ p8 =0 · · · p2∑ p1 =0 b`−p9n b p9−p8 n · · ·b p1 n γ(α` −αp9 + 1)γ(αp9 −αp8 + 1) · · ·γ(αp1 + 1) ... bα,n `,20 = ∑̀ p19 =0 p19∑ p18 =0 · · · p2∑ p1 =0 b`−p19n b p19−p18 n · · ·b p1 n γ(α` −αp19 + 1)γ(αp19 −αp18 + 1) · · ·γ(αp1 + 1) . (43) one sees that the fractional differential equation under consideration now takes the interesting form c dαy(x) + ω xα−β c dβy(x) + a0 + a1y(x) + a2y 2(x) + · · · + an y n (x) = 0, (44) which clearly is a generalization of the fractional lane-emden equation c dαy(x) + ω xα−β c dβy(x) + ym(x) = 0, (45) in the sense that the new function f (y(x)) is a linear combination of the classical polytropic term ym(x), 0 ≤ m ≤ n. equation (45) is the one considered in [28, subsection 5.2.2] using the power series method. to make our calculations explicit and computationally interesting we consider those elementary functions whose expansion coefficients are explicitly known. such elementary functions to be examined with their explicit associated expansion coefficients am, m = 0, 1, 2, 3, . . . , are the trigonometric, hyperbolic and exponential functions. these functions are enumerated as follows. (a) f (y(x)) = sin y(x), a2m = 0, a2m+1 = (−1)m/(2m + 1)!; (b) f (y(x)) = cos y(x), a2m+1 = 0, a2m = (−1)m/(2m)!; (c) f (y(x)) = sinh y(x), a2m = 0, a2m+1 = 1/(2m + 1)!; 269 awonusika et al. / j. nig. soc. phys. sci. 4 (2022) 265–280 270 (d) f (y(x)) = cosh y(x), a2m+1 = 0, a2m = 1/(2m)!; (e) f (y(x)) = e±y(x), am = (±1)m/m!. for extensive discussions of models involving these functions as well as the physical interpretations of the solutions, see [12], [13], [42]. we proceed with the computation of the approximate solution of the problem (10) according to whether the functions f (y) are trigonometric, hyperbolic or exponential functions. 4.1. lane-emden equation involving trigonometric functions in this subsection, we treat the equation (10) as well as (34) for which the functions f (y(x)) are trigonometric functions, namely, the sine function and cosine function. the maclaurin series representations for these trigonometric functions are employed. 4.1.1. the special case f (y(x)) = sin y(x) in this case the fractional initial value problem (10) reduces to a special one c dαy(x) + ω xα−β c dβy(x) + sin y(x) = 0, y(0) = 1, y′(0) = 0. (46) in solving the initial value problem (46), we consider the series expansion of sin y(x), namely, sin y(x) = n∑ m=0 a2m+1y 2m+1(x), a2m+1 = (−1)m (2m + 1)! , 0 ≤ m ≤ n, n ∈ n0. (47) in this case the problem (34) becomes c dαy(x) + ω xα−β c dβy(x) + n∑ m=0 (−1)m y2m+1(x) (2m + 1)! = 0 y(0) = 1, y′(0) = 0. (48) by theorem 4.1, the analytical solution of the reduced fractional problem (48) is then given by the mittag-leffler function expansion y(x) = ∞∑ `=0 b`n (α,β; ω) γ(α` + 1) xα`, (49) where the associated expansion coefficients are given explicitly by b 0 n (α,β; ω) := b 0 n = 1, (50) b `+1 n (α,β; ω) := b `+1 n = − γ(α` + 1)γ(α(` + 1) −β + 1)cα `,n γ(α(` + 1) −β + 1) + ωγ(α` + 1) , ` ≥ 0. (51) the coefficients cα `,n, ` ≥ 0, appearing in (51) take the formulation cα`,n = n∑ m=0 (−1)m (2m + 1)! bα,n `,2m+1. (52) in order to see explicitly the values of the coefficients cα `,n , we consider the first values n ∈ n0 and this is presented as follows. (a) n = 0. in this case, equation (48) reduces to the initial value problem c dαy(x) + ω xα−β c dβy(x) + y(x) = 0, y(0) = 1, y′(0) = 0. (53) it follows from theorem 4.1 that the solution of the initial value problem (53) is given by the mittag-leffler expansion y(x) = ∞∑ `=0 b`0(α,β; ω) γ(α` + 1) xα`, (54) where the expansion coefficients b`0 admit the explicit formulation (` ≥ 0) b 0 0(α,β; ω) = 1, b `+1 0 (α,β; ω) = − γ(α` + 1)γ(α(` + 1) −β + 1)cα `,0 γ(α(` + 1) −β + 1) + ωγ(α` + 1) , (55) with the numbers cα `,0, ` ≥ 1, given by cα`,0 = b α,0 `,1 = a α `,0 = b`0 γ(α` + 1) . (56) for further explicit calculations we proceed by assigning special values to the parameters α,β and ω and this is done in the following way as examples. example 4.1. setting α = 2,β = 1,ω = 2, we see in this case that equation (53) reduces to the initial value problem y′′ + 2 x y′ + y = 0, y(0) = 1, y′(0) = 0. (57) it is seen from the series (54) that the solution of the initial value problem (57) takes the formulation y(x) = ∞∑ `=0 b`0(2, 1; 2) γ(2` + 1) x2`, (58) where the expansion coefficients b`0 = b ` 0(2, 1; 2) admit the explicit formulation (` = 0, 1, 2, . . . ) b 0 0(2, 1; 2) = 1, b `+1 0 (2, 1; 2) = − γ(2` + 1)γ(2` + 2)c2 `,0 γ(2` + 2) + 2γ(2` + 1) , (59) 270 awonusika et al. / j. nig. soc. phys. sci. 4 (2022) 265–280 271 and c2`,0 = b 2,0 `,1 = a 2 `,0 = b`0 γ(2` + 1) , ` = 0, 1, 2, . . . . (60) clearly, the first mittag-leffler expansion coefficients b`0(2, 1; 2), ` = 1, 2, 3, 4 yield the following values. b 1 0(2, 1; 2) = − c 2 0,0 · 1 3 = −b00 · 1 3 = − 1 3 b 2 0(2, 1; 2) = − c 2 1,0 · 6 5 = − b10 2! · 6 5 = 1 6 · 6 5 = 1 5 b 3 0(2, 1; 2) = − c 2 2,0 · 5! 7 = − b20 4! · 5! 7 = − 1 5! · 5! 7 = − 1 7 b 4 0(2, 1; 2) = − c 2 3,0 · 560 = − b30 6! · 560 = 1 7! · 560 = 1 9 . (61) substituting these coefficients into (58) gives the solution of the initial value problem (57): y(x) = 1 − 1 3! x2 + 1 5! x4 − 1 7! x6 + 1 9! x8 + · · · = sin x x . (62) example 4.2. indeed if we set α = 3/2,β = 1/2,ω = 2, then one sees that equation (53) becomes the initial value problem c d 3 2 y(x) + 2 x c d 1 2 y(x) + y(x) = 0, y(0) = 1, y′(0) = 0. (63) one clearly sees from the series (54) that the solution of the initial value problem (63) gives y(x) = ∞∑ `=0 b`0 ( 3 2 , 1 2 ; 2 ) γ ( 3` 2 + 1 ) x 3`2 , (64) where the expansion coefficients b`0 = b ` 0 ( 3 2 , 1 2 ; 2 ) admit the gamma function representation b 0 0 ( 3 2 , 1 2 ; 2 ) = 1, b `+1 0 ( 3 2 , 1 2 ; 2 ) = − γ( 32 ` + 1)γ( 3 2 ` + 2)c 3 2 `,0 γ( 32 ` + 2) + 2γ( 3 2 ` + 1) , ` ≥ 0, (65) and c 3 2 `,0 = b 3 2 ,0 `,1 = b 3 2 ,0 `,1 = b`0 γ ( 3 2 ` + 1 ), ` = 0, 1, 2, . . . . (66) the first mittag-leffler expansion coefficients b`0 ( 3 2 , 1 2 ; 2 ) , ` = 1, 2, 3, 4, 5, 6, are computed as follows: b 1 0 ( 3 2 , 1 2 ; 2 ) = − c 3 2 0,0 · 1 3 = −b00 · 1 3 = − 1 3 b 2 0 ( 3 2 , 1 2 ; 2 ) = − c 3 2 1,0 · γ( 52 ) · γ( 7 2 ) γ( 72 ) + 2γ( 5 2 ) = − b10 γ( 52 ) 5 √ π 12 = 5 27 b 3 0 ( 3 2 , 1 2 ; 2 ) = − c 3 2 2,0 · γ(4) · γ(5) γ(5) + 2γ(4) = − b20 γ(4) · (4) = − 10 81 b 4 0 ( 3 2 , 1 2 ; 2 ) = − c 3 2 3,0 · γ( 112 ) · γ( 13 2 ) γ( 132 ) + 2γ( 11 2 ) = − b30 γ( 112 ) · 693 √ π 32 = 22 243 b 5 0 ( 3 2 , 1 2 ; 2 ) = − c 3 2 4,0 · γ(7) · γ(8) γ(8) + 2γ(7) = − b40 γ(7) · (560) = − 154 2187 b 6 0 ( 3 2 , 1 2 ; 2 ) = − c 3 2 5,0 · γ ( 17 2 ) · γ( 192 ) γ( 192 ) + 2γ( 17 2 ) = − b50 γ ( 17 2 ) · 1640925√π 256 = 374 6561 . (67) upon substituting these coefficients into (64) yields the solution y(x) = 1 − 4x 3 2 9 √ π + 5x3 162 − 64x9/2 15309 √ π + 11x6 87480 − 512x15/2 57572775 √ π + 187x9 1190427840 + · · · . (68) (b) n = 1. indeed it is understood here that equation (48) reduces to the initial value problem c dαy(x) + ω xα−β c dβy(x) + y(x) − y3(x) 3! = 0, y(0) = 1, y′(0) = 0. (69) it is also seen here that the solution of the fractional problem (69) gives y(x) = ∞∑ `=0 b`1(α,β; ω) γ(α` + 1) xα`, (70) with the expansion coefficients b`1 = b ` 1(α,β; ω) (` ≥ 0) given by b 0 1(α,β; ω) = 1, b `+1 1 (α,β; ω) = − γ(α` + 1)γ(α(` + 1) −β + 1)cα `,1 γ(α(` + 1) −β + 1) + ωγ(α` + 1) . (71) here the coefficients cα `,1, ` ≥ 0, on the right of (71) are given by cα`,1 = b α,1 `,1 − 1 3! bα,1 `,3 , (72) 271 awonusika et al. / j. nig. soc. phys. sci. 4 (2022) 265–280 272 where the numbers bα,n `,m , m = 1, 3, are as illustrated in (16) and (27) respectively. example 4.3. in this case, another special lane-emden equation can be obtained by setting α = 2,β = 1, ω = 2; and as a result, equation (69) reduces to y′′ + 2 x y′ + y − y3 3! = 0, y(0) = 1, y′(0) = 0, (73) whose solution admits the series representation y(x) = ∞∑ `=0 b`1(2, 1; 2) γ(2` + 1) x2`. (74) from equations (71) and (72), we respectively have b 0 1(2, 1; 2) = 1, b `+1 1 (2, 1; 2) = − γ(2` + 1)γ(2` + 2)c2 `,1 γ(2` + 2) + 2γ(2` + 1) , ` ≥ 0; (75) and c2`,1 = b`1 γ(2` + 1) − 1 6 ∑̀ p2 =0 p2∑ p1 =0 bp11 b p2−p1 1 b `−p2 1 γ(2p1 + 1)γ(2p2 − 2 p1 + 1)γ(2` − 2p2 + 1)  . (76) it is straightforward to see that the first mittag-leffler expansion coefficients b`1(2, 1; 2), ` = 1, 2, 3, 4, 5, 6, are calculated as follows. b 1 1(2, 1; 2) = −c 2 0,1 · 1 3 = − 5b01 6 · 1 3 = − 5 6 · 1 3 = − 5 18 b 2 1(2, 1; 2) = −c 2 1,1 · 6 5 = − b11 3 · 6 5 = 5 54 · 6 5 = 1 9 b 3 1(2, 1; 2) = −c 2 2,1 · 5! 7 = − b2136 − b 1 1b 1 1 4  · 1207 = 1 7776 · 120 7 = 5 2268 b 4 1(2, 1; 2) = −c 2 3,1 · 560 = −  b311080 − b 1 1b 2 1 144  · 560 = − 53 244944 · 560 = − 265 2187 b 5 1(2, 1; 2) = −c 2 4,1 · 362880 11 = −  b4160480 − b 1 1b 3 1 4320 − b21b 2 1 3456  · 36288011 = 287516038 b 6 1(2, 1; 2) = −c 2 5,1 · 39916800 13 = −  b515443200 − b 1 1b 4 1 241920 − b21b 3 1 51840  · 3991680013 = 2905085293. (77) upon inserting the first coefficients (77) in (74) gives the solution y(x) =1 − 5 18 · 1 2! x2 + 1 9 · 1 4! x4 + 5 2268 · 1 6! x6 − 265 2187 · 1 8! x8 + 2875 16038 · 1 10! x10 + 29050 85293 · 1 12! x12 · · · =1 − 0.138889x2 + 0.00347222x4 + 0.0000030619x6 − 0.0000030052x8 + 0.0000000494x10 + 0.0000000007x12 + · · · (78) (c) n = 2. another approximate solution of the problem(46) can be obtained in this case in view of the reduced problem (48): c dαy(x) + ω xα−β c dβy(x) + y(x) − y3(x) 3! + y5(x) 5! = 0, y(0) = 1, y′(0) = 0. (79) example 4.4. similarly, in this case of n = 2, we specialise to the values α = 2,β = 1, ω = 2; to see that equation (79) becomes y′′ + 2 x y′ + y− y3 3! + y5 5! = 0, y(0) = 1, y′(0) = 0.(80) clearly, the solution of the problem (80) takes the series formula y(x) = ∞∑ `=0 b`2(2, 1; 2) γ(2` + 1) x2`, (81) with the coefficients described as follows: b 0 2(2, 1; 2) = 1, b `+1 2 (2, 1; 2) = − γ(2` + 1)γ(2` + 2)c2 `,2 γ(2` + 2) + 2γ(2` + 1) , c2`,2 = a1b 2,2 `,1 + a3b 2,2 `,3 + a5b 2,2 `,5 = b 2,2 `,1 − 1 3! b2,2 `,3 + 1 5! b2,2 `,5. (82) here, one clearly understands that b2,2 `,1 = b`2 γ(2` + 1) , b2,2 `,3 = ∑̀ p2 =0 p2∑ p1 =0 bp12 b p2−p1 2 b `−p2 2 γ(2p1 + 1)γ(2p2 − 2p1 + 1)γ(2` − 2 p2 + 1) b2,2 `,5 = ∑̀ p4 =0 p4∑ p3 =0 p3∑ p2 =0 p2∑ p1 =0 bp12 b p2−p1 2 b p3−p2 2 b p4−p3 2 b `−p4 2 b ` 2[γ(2` − 2p4 + 1)] −1 γ(2p1 + 1)γ(2p2 − 2p1 + 1)γ(2 p3 − 2p2 + 1)γ(2p4 − 2 p3 + 1) . (83) 272 awonusika et al. / j. nig. soc. phys. sci. 4 (2022) 265–280 273 as a result, the first mittag-leffler expansion coefficients b`3(2, 1; 2), ` = 1, 2, 3, 4, are computed as follows. b 1 2(2, 1; 2) = − c 2 0,2 · 1 3 = − 101 120 · 1 3 = − 101 360 b 2 2(2, 1; 2) = − c 2 1,2 · 6 5 = − 41b12 120 · 6 5 = 4141 36000 b 3 2(2, 1; 2) = − c 2 2,2 · 5! 7 = −  41b221440 − 19b 1 2b 1 2 480  · 5!7 = 49591 18144000 b 4 2(2, 1; 2) = − c 2 3,2 · (560) = −  41b3243200 − 19b 1 2b 2 2 2880  · 560 = − 16891139139968000 b 5 2(2, 1; 2) = − c 2 4,2 · 362880 11 = −  41b422419200 − 19b 1 2b 3 2 86400 − 19b22b 2 2 69120  · 36288011 = 93349921751 513216000000 . (84) upon substituting these coefficients into the series formula (81) gives the solution y(x) =1 − 101 360 · 1 2! x2 + 4141 36000 · 1 4! x4 + 49591 18144000 · 1 6! x6 − 16891139 139968000 · 1 8! x8 + 93349921751 513216000000 · 1 10! x10 + · · · =1 − 0.14027778x2 + 0.00479282x4 + 0.000003796x6 + 0.000002993x8 + 0.0000000501x10 + · · · (85) 4.1.2. the special case f (y(x)) = cos y(x) it is seen here that the fractional lane-emden equation under consideration is c dαy(x) + ω xα−β c dβy(x) + cos y(x) = 0, y(0) = 1, y′(0) = 0. (86) finding the approximate solution of the initial value problem (86) amounts to solving the problem c dαy(x) + ω xα−β c dβy(x) + n∑ m=0 (−1)m (2m)! y2m(x) = 0, y(0) = 1, y′(0) = 0, (87) where we have used the truncated power series cos y(x) = n∑ m=0 a2my 2m(x) = n∑ m=0 (−1)m (2m)! y2m(x), n ∈ n0. (88) theorem 4.1 tells us that the analytical solution of the initial value problem (87) admits the series solution y(x) = ∞∑ `=0 b`n (α,β; ω) γ(α` + 1) xα`, (89) where we have b 0 n (α,β; ω) := b 0 n = 1, (90) b `+1 n (α,β; ω) := b `+1 n = − γ(α` + 1)γ(α(` + 1) −β + 1)cα `,n γ(α(` + 1) −β + 1) + ωγ(α` + 1) , ` ≥ 0. (91) with cα`,n = n∑ m=0 (−1)m (2m)! bα,n `,2m. (92) we now proceed to the explicit computation of the first expansion coefficients b`n (α,β; ω). this is carried out by first considering the first values of n ∈ n0. (a) n = 0. in this case, equation (86) reduces to the initial value problem c dαy(x) + ω xα−β c dβy(x) + 1 = 0, y(0) = 1, y′(0) = 0. (93) one sees from (89) that the solution of the initial value problem (93) is given by y(x) = ∞∑ `=0 b`0(α,β; ω) γ(α` + 1) xα`, (94) where (` ≥ 0) b 0 0(α,β; ω) = 1, b `+1 0 (α,β; ω) = − γ(α` + 1)γ(α(` + 1) −β + 1)cα `,0 γ(α(` + 1) −β + 1) + ωγ(α` + 1) , (95) with the numbers cα `,0, ` ≥ 0, given by cα`,0 = a0b α,0 `,0 = b α,0 `,0 = δ`0. (96) example 4.5. with the special values α = 2,β = 1 and ω = 2, we obtain a special case of the standard lane-emden equation y′′ + 2 x y′ + 1 = 0, y(0) = 1, y′(0) = 0, (97) whose solution is of the form y(x) = ∞∑ `=0 b`0(2, 1; 2) γ(2` + 1) x2`. (98) here we have b 0 0(2, 1; 2) = 1, b `+1 0 (2, 1; 2) = − γ(2` + 1)γ(2` + 2)c2 `,0 γ(2` + 2) + 2γ(2` + 1) , ` ≥ 0, (99) 273 awonusika et al. / j. nig. soc. phys. sci. 4 (2022) 265–280 274 with c2 `,0 = δ`0. interestingly one has the formula b 1 0(2, 1; 2) =  − 1 3 , ` = 0, 0, ` = 1, 2, · · · , (100) and as a result we obtain the closed form solution y(x) = 1 − 1 6 x2. (101) (b) n = 1. in this case, equation (87) reduces to the initial value problem c dαy(x) + ω xα−β c dβy(x) + 1 − y2 2! = 0, y(0) = 1, y′(0) = 0. (102) example 4.6. considering the the special values α = 2,β = 1 and ω = 2; equation (102) becomes the initial value problem y′′ + 2 x y′ + 1 − y2 2! = 0, y(0) = 1, y′(0) = 0, (103) with the solution y(x) = ∞∑ `=0 b`1(2, 1; 2) γ(2` + 1) x2`. (104) it follows that b 0 1(2, 1; 2) = 1, b `+1 1 (2, 1; 2) = − γ(2` + 1)γ(2` + 2)c2 `,1 γ(2` + 2) + 2γ(2` + 1) , ` ≥ 0 (105) with c2`,1 =a0b 2,1 `,0 + a2b 2,1 `,2 =δ`0 − 1 2 ∑̀ p1 =0 bp11 b `−p1 1 γ(2p1 + 1)γ(2`− 2 p1 + 1) . consequently, the first mittag-leffler expansion coefficients b`1(2, 1; 2), ` = 1, 2, 3, 4, 5, are now computed as follows. b 1 1(2, 1; 2) = − c 2 0,1 · 1 3 = − b01 2 · 1 3 = − 1 6 b 2 1(2, 1; 2) = − c 2 1,1 · 6 5 = b11 2 · 6 5 = − 1 10 b 3 1(2, 1; 2) = − c 2 2,1 · 5! 7 = b2124 + b 1 1b 1 1 8  · 5!7 = − 184 b 4 1(2, 1; 2) = − c 2 3,1 · 560 =  b31720 + b 1 1b 2 1 48  · 560 = 527 b 5 1(2, 1; 2) = − c 2 4,1 · 362880 11 =  b4140320 + b 1 1b 3 1 1440 + b21b 2 1 1152  · 36288011 = 2960. (106) upon using the coefficients (106), one obtains the solution y(x) =1 − 1 6 · 1 2! x2 − 1 10 · 1 4! x4 − 1 84 · 1 6! x6 + 5 27 · 1 8! x8 + 29 60 · 1 10! x10 + · · · =1 − 0.0833333x2 − 0.00416667x4 − 0.00001653x6 + 0.00000459x8 + 0.00000013x10 + · · · (107) (c) n = 2. here, equation (87) becomes the initial value problem c dαy(x) + ω xα−β c dβy(x) + 1 − y2 2! + y4 4! = 0, y(0) = 1, y′(0) = 0. (108) example 4.7. consider the problem y′′+ 2 x y′+1− y2 2! + y4 4! = 0, y(0) = 1, y′(0) = 0.(109) the solution of (109) takes the form y(x) = ∞∑ `=0 b`2(2, 1; 2) γ(2` + 1) x2`, (110) where the coefficients b`2(2, 1; 2) are given by b 0 2(2, 1; 2) = 1, b `+1 2 (2, 1; 2) = − γ(2` + 1)γ(2` + 2)c2 `,2 γ(2` + 2) + 2γ(2` + 1) , (` ≥ 0) c2`,2 = a0b 2,2 `,0 + a2b 2,2 `,2 + a4b 2,2 `,4 = b 2,2 `,0 − 1 2! b2,2 `,2 + 1 4! b2,2 `,4, (111) with b2,2 `,0 = δ`0, b 2,2 `,2 = ∑̀ p1 =0 bp12 b `−p1 2 γ(2p1 + 1)γ(2`− 2 p1 + 1) b2,2 `,4 = ∑̀ p3 =0 p3∑ p2 =0 p2∑ p1 =0 bp12 b p2−p1 2 b p3−p2 2 b `−p3 2 γ(2p1 + 1)γ(2p2 − 2p1 + 1)γ(2 p3 − 2p2 + 1)γ(2`− 2p3 + 1) . (112) the first mittag-leffler expansion coefficients b`+12 (2, 1; 2), ` = 0, 1, 2, . . . are computed as follows. b 1 2(2, 1; 2) = −c 2 0,2 · 1 3 = − 1 − b022 + b 0 2 24  · 13 = −1372 b 2 2(2, 1; 2) = −c 2 1,2 · 6 5 = − −b122 + b 1 2 24  · 65 = − 1431440 274 awonusika et al. / j. nig. soc. phys. sci. 4 (2022) 265–280 275 b 3 2(2, 1; 2) = − c 2 2,2 · 5! 7 =  11b22288 + 11b 1 2b 1 2 96  · 1207 = − 143 145152 b 4 2(2, 1; 2) = − c 2 3,2 · 560 =  11b328640 + 11b 1 2b 2 2 576  · 560 = 26741 139968 b 5 2(2, 1; 2) = − c 2 4,2 · 362880 11 =  11b42483840 + 11b 1 2b 3 2 17280 + 11b22b 2 2 1152  · 36288011 = 6059911 14929920 . (113) upon substituting these coefficients in (110) yields the series solution y(x) =1 − 13 72 · 1 2! x2 − 143 1440 · 1 4! x4 − 143 145152 · 1 6! x6 + 26741 139968 · 1 8! x8 + 6059911 14929920 · 1 10! x10 + · · · =1 − 0.09027778x2 − 0.00413773x4 − 0.00000137x6 + 0.000004738x8 + 0.000000112x10 + · · · . (114) 4.2. lane-emden equation involving hyperbolic functions this subsection discusses the approximate solution of the initial value problems (10) and (34) for which the functions f (y(x)) are the hyperbolic sine function and the hyperbolic cosine function. we also make use of the maclaurin series representations for these hyperbolic functions. 4.2.1. the special case f (y(x)) = sinh y(x) we consider the fractional lane-emden initial value problem c dαy(x) + ω xα−β c dβy(x) + sinh y(x) = 0, y(0) = 1, y′(0) = 0. (115) in solving the initial value problem (115), we are interested in the series expansion of sinh y(x), namely (0 ≤ m ≤ n, n ∈ n0), sinh y(x) = n∑ m=0 a2m+1y 2m+1(x), a2m+1 = 1 (2m + 1)! , (116) so that finding the approximate solution of the problem (115) amounts to solving the corresponding problem c dαy(x) + ω xα−β c dβy(x) + n∑ m=0 1 (2m + 1)! y2m+1(x) = 0, y(0) = 1, y′(0) = 0. (117) we proceed to obtaining the solutions of the initial value problem (117) for the first values of n ∈ n. (a) n = 0. clearly, in this case a1 = 1 and as a result the resulting initial value problem coincides with the problem in (53). (b) n = 1. here, we have from (117) the problem c dαy(x) + ω xα−β c dβy(x) + y(x) + y3(x) 3! = 0, y(0) = 1, y′(0) = 0. (118) example 4.8. consider the initial value problem y′′ + 2 x y′ + y + y3 3! = 0, y(0) = 1, y′(0) = 0 (119) clearly the expansion coefficients appearing in the series solution of (119) are given by b 0 1(2, 1; 2) = 1, b `+1 1 (2, 1; 2) = − γ(2` + 1)γ(2` + 2)c2 `,1 γ(2` + 2) + 2γ(2` + 1) , ` ≥ 0. (120) with c2`,1 = a1b 2,1 `,1 + a3b 2,1 `,3 = b`1 γ(2` + 1) + 1 6 ∑̀ p2 =0 p2∑ p1 =0 bp11 b p2−p1 1 b `−p2 1 γ(2p1 + 1)γ(2p2 − 2 p1 + 1)γ(2` − 2p2 + 1)  . (121) the first mittag-leffler expansion coefficients b`1(2, 1; 2), ` = 0, 1, 2, 3, 4, 5, are presented as follows. b 1 1(2, 1; 2) = −c 2 0,1 · 1 3 = − 7b01 6 · 1 3 = − 7 18 b 2 1(2, 1; 2) = −c 2 1,1 · 6 5 = − 2b11 3 · 6 5 = 14 45 b 3 1(2, 1; 2) = −c 2 2,1 · 5! 7 = − b2118 − b 1 1b 1 1 24  · 1207 = −131324 b 4 1(2, 1; 2) = −c 2 3,1 · 560 = −  b31540 + b 1 1b 2 1 144  · 560 = 19462187 b 5 1(2, 1; 2) = −c 2 4,1 · 362880 11 = −  b4130240 + b 1 1b 3 1 4320 + b21b 2 1 3456  · 36288011 = −24821380190 (122) upon substituting the first coefficients (122) we obtain the 275 awonusika et al. / j. nig. soc. phys. sci. 4 (2022) 265–280 276 solution y(x) = ∞∑ `=0 b`1(2, 1; 2) γ(2` + 1) x2` =1 − 7 18 · 1 2! x2 + 14 45 · 1 4! x4 − 131 324 · 1 6! x6 + 1946 2187 · 1 8! x8 − 248213 80190 · 1 10! x10 + · · · =1 − 0.19444444x2 + 0.01296296x4 − 0.00056156x6 + 0.000022069x8 − 0.000000853x10 + · · · . (123) (c) n = 2. indeed it is easy to understand here that from (117) one obtains the initial value problem c dαy(x) + ω xα−β c dβy(x) + y(x) + y3(x) 3! + y5(x) 5! = 0, y(0) = 1, y′(0) = 0. (124) example 4.9. given the problem y′′+ 2 x y′+y+ y3 3! + y5 5! = 0, y(0) = 1, y′(0) = 0.(125) clearly, the expansion coefficients of the series solution of (125) are b 0 2(2, 1; 2) = 1, b `+1 2 (2, 1; 2) = − γ(2` + 1)γ(2` + 2)c2 `,2 γ(2` + 2) + 2γ(2` + 1) , c2`,2 = a1b 2,2 `,1 + a3b 2,2 `,3 + a5b 2,2 `,5 = b 2,2 `,1 + 1 3! b2,2 `,3 + 1 5! b2,2 `,5, (126) where b2,2 `,1 = b`2 γ(2` + 1) , b2,2 `,3 = ∑̀ p2 =0 p2∑ p1 =0 bp12 b p2−p1 2 b `−p2 2 γ(2p1 + 1)γ(2p2 − 2 p1 + 1)γ(2` − 2p2 + 1) b2,2 `,5 = ∑̀ p4 =0 p4∑ p3 =0 p3∑ p2 =0 p2∑ p1 =0 bp12 b p2−p1 2 b p3−p2 2 b p4−p3 2 b `−p4 2 [γ(2` − 2 p4 + 1)] −1 γ(2p1 + 1)γ(2p2 − 2 p1 + 1)γ(2p3 − 2p2 + 1)γ(2p4 − 2p3 + 1) . (127) the first expansion coefficients b`1(2, 1; 2), ` = 1, 2, 3, 4, 5 are computed as follows. b 1 2(2, 1; 2) = −c 2 0,2 · 1 3 = − 47b02 40 · 1 3 = − 47 120 b 2 2(2, 1; 2) = −c 2 1,2 · 6 5 = − 27b12 3 · 6 5 = 1269 4000 b 3 2(2, 1; 2) = −c 2 2,2 · 5! 7 = −  9b22160 − 7b 1 2b 1 2 160  · 1207 = − 282893 672000 b 4 2(2, 1; 2) = −c 2 3,2 · 560 = −  3b321600 + 7b 1 2b 2 2 960  · 560 = 4747 5000 b 5 2(2, 1; 2) = −c 2 4,2 · 362880 11 = −  3b4289600 + 7b 1 2b 3 2 28800 + 7b22b 2 2 23040  · 36288011 = −2379140611704000000 (128) upon inserting the first coefficients into the series solution one obtains y(x) = ∞∑ `=0 b`2(2, 1; 2) γ(2` + 1) x2` = 1− 47 120 · 1 2! x2 + 1269 4000 · 1 4! x4− 282893 672000 · 1 6! x6 + 4747 5000 · 1 8! x8 − 2379140611 704000000 · 1 10! x10 + · · · = 1 − 0.19583333x2 + 0.01321875x4 − 0.00058468x6 + 0.000023547x8 − 0.000000931x10 + · · · . (129) 4.2.2. the special case f (y(x)) = cosh y(x) we consider the fractional lane-emden equation c dαy(x) + ω xα−β c dβy(x) + cosh y(x) = 0, y(0) = 1, y′(0) = 0. (130) in order to solve the initial value problem (130), we consider the series expansion of cosh y(x): cosh y(x) = n∑ m=0 a2my 2m(x) = n∑ m=0 y2m(x) (2m)! , 0 ≤ m ≤ n, n ∈ n0, (131) and as a result we have the associated problem c dαy(x) + ω xα−β c dβy(x) + n∑ m=0 y2m(x) (2m)! = 0, y(0) = 1, y′(0) = 0. (132) for n = 0, we observe that the resulting initial value problem coincides with the problem in (93). 276 awonusika et al. / j. nig. soc. phys. sci. 4 (2022) 265–280 277 example 4.10. given the initial value problem y′′ + 2 x y′ + 1 + y2 2! = 0, y(0) = 1, y′(0) = 0. (133) it follows immediately that the solution to the problem (133) is given by y(x) =1 − 1 2 · 1 2! x2 + 3 10 · 1 4! x4 − 3 4 · 1 6! x6 + 7 3 · 1 8! x8 − 2877 220 · 1 10! x10 + · · · =1 − 0.25000000x2 + 0.01250000x4 − 0.00104167x6 + 0.00005787x8 − 0.00000360x10 + · · · (134) example 4.11. consider the problem y′′ + 2 x y′ + 1 + y2 2! + y4 4! = 0, y(0) = 1, y′(0) = 0. (135) one easily sees that the solution to the problem (135) gives y(x) =1 − 37 72 · 1 2! x2 + 481 1440 · 1 4! x4 − 126503 145152 · 1 6! x6 + 406445 139968 · 1 8! x8 − 256054097 14929920 · 1 10! x10 · · · =1 − 0.25694444x2 + 0.01391782x4 − 0.00121045x6 + 0.00007202x8 − 0.00000473x10 + · · · . (136) 4.3. the case of exponential functions f (y) = exp(±y) in this subsection, we compute the approximate solution of the fractional lane-emden problem (10) as well as the initial problem (34) for which the functions f (y(x)) are exponential functions. the maclaurin series expansion for these exponential functions are applied. 4.3.1. fractional isothermal gas sphere equation the fractional isothermal gas sphere equation is given by c dαy(x) + ω xα−β c dβy(x) + ey(x) = 0, y(0) = 0, y′(0) = 0. (137) also in this case, we look for the approximate solution of the problem (137) by solving the complementary equation c dαy(x) + ω xα−β c dβy(x) + n∑ m=0 ym(x) m! = 0, y(0) = 0, y′(0) = 0. (138) the problem y′′ + 2 x y′ + ey = 0, y(0) = 0, y′(0) = 0 (139) models the isothermal gas spheres where the temperature remains constant and the index m is infinite ([42]). example 4.12. given the problem y′′ + 2 x y′ + 1 = 0, y(0) = 0, y′(0) = 0. (140) it is straightforward to see by the initial conditions in (140) that the expansion coefficients appearing in the series solution of (140) are given by b 0 0(2, 1; 2) = 0, b `+1 0 (2, 1; 2) = − γ(2` + 1)γ(2` + 2)c2 `,0 γ(2` + 2) + 2γ(2` + 1) , ` ≥ 0, (141) with c2 `,0 = δ`0. thus one has the mittag-leffler expansion coefficient b 1 0(2, 1; 2) =  − 1 3 , ` = 0, 0, ` = 1, 2, · · · . (142) consequently we obtain the closed form solution y(x) = ∞∑ `=0 b`0(2, 1; 2) γ(2` + 1) x2` = − 1 6 x2. (143) example 4.13. n = 1: it is seen here that y′′ + 2 x y′ + 1 + y = 0, y(0) = 0, y′(0) = 0. (144) the coefficients of the series solution of (144) has the formulation b 0 1(2, 1; 2) = 0, b `+1 1 (2, 1; 2) = − γ(2` + 1)γ(2` + 2)c2 `,1 γ(2` + 2) + 2γ(2` + 1) , ` ≥ 0 (145) with c2`,1 = δ`0 + b`1 γ(2` + 1) . (146) further simplification gives the following first expansion coefficients b`1(2, 1; 2), ` = 1, 2, 3, 4, 5. b 1 1(2, 1; 2) = − c 2 0,1 · 1 3 = −(1 + b01) · 1 3 = − 1 3 b 2 1(2, 1; 2) = − c 2 1,1 · 6 5 = − b11 2 · 6 5 = 1 5 b 3 1(2, 1; 2) = − c 2 2,1 · 5! 7 = − b21 24 · 120 7 = − 1 7 b 4 1(2, 1; 2) = − c 2 3,1 · 560 = − b31 720 · 560 = 1 9 b 5 1(2, 1; 2) = − c 2 4,1 · 362880 11 = − b41 40320 · 362880 11 = − 1 11 (147) 277 awonusika et al. / j. nig. soc. phys. sci. 4 (2022) 265–280 278 upon substituting we obtain the solution y(x) = ∞∑ `=0 b`1(2, 1; 2) γ(2` + 1) x2` = − 1 3 · 1 2! x2 + 1 5 · 1 4! x4 − 1 7 · 1 6! x6 + 1 9 · 1 8! x8 − 1 11 · 1 10! x10 + · · · = − 1 3! x2 + 1 5! x4 − 1 7! x6 + 1 9! x8 − 1 11! x10 + · · · (148) example 4.14. n = 2 : consider the initial value problem y′′ + 2 x y′ + 1 + y + y2 2! = 0, y(0) = 0, y′(0) = 0. (149) the series solution of (149) has the following expansion coefficients. b 0 2(2, 1; 2) = 0, b `+1 2 (2, 1; 2) = − γ(2` + 1)γ(2` + 2)c2 `,2 γ(2` + 2) + 2γ(2` + 1) , ` ≥ 0, (150) where c2`,2 = δ`0 + b`2 γ(2` + 1) + 1 2 ∑̀ p1 =0 bp12 b `−p1 2 γ(2p1 + 1)γ(2`− 2 p1 + 1)  . (151) simplifying further gives the following first expansion coefficients b`2(2, 1; 2), ` = 0, 1, 2, . . . . b 1 2(2, 1; 2) = −c 2 0,2 · 1 3 = −1 · 1 3 = − 1 3 b 2 2(2, 1; 2) = −c 2 1,2 · 6 5 = − b12 2 · 6 5 = 1 5 b 3 2(2, 1; 2) = −c 2 2,2 · 5! 7 = − b2224 + b 1 2b 1 2 8  · 1207 = − 821 b 4 2(2, 1; 2) = −c 2 3,2 · 560 = −  b32720 + b 1 2b 2 2 48  · 560 = 2927 b 5 2(2, 1; 2) = −c 2 4,2 · 362880 11 = −  b4240320 + b 1 2b 3 2 1440 + b22b 2 2 576  · 36288011 = −403208173 . (152) as a result we obtain the solution y(x) = ∞∑ `=0 b`2(2, 1; 2) γ(2` + 1) x2` = − 1 3 · 1 2! x2 + 1 5 · 1 4! x4− 8 21 · 1 6! x6 + 29 27 · 1 8! x8− 40320 8173 · 1 10! x10 +· · · = − 1 3! x2 + 1 5! x4 − 8 3 · 7! x6 + 29 3 · 9! x8 − 40320 743 · 11! x10 + · · · (153) 4.3.2. richardson’s model of thermionic current here, we consider the initial value problem c dαy(x) + ω xα−β c dβy(x) + e−y(x) = 0, y(0) = 0, y′(0) = 0. (154) the problem (richardson’s model of thermionic current) y′′ + 2 x y′ + e−y = 0, y(0) = 0, y′(0) = 0 (155) models the density and electric force of an electron gas in the neighbourhood of a hot body in thermal equilibrium ( [13], [42]). clearly, for n = 0 with α = 2,β = 1 and ω = 2, (154) coincides with (140). example 4.15. consider the problem y′′ + 2 x y′ + 1 − y = 0, y(0) = 0, y′(0) = 0. (156) as in the preceding calculations we easily see that the initial value problem (156) has the solution y(x) = − 1 3 · 1 2! x2− 1 5 · 1 4! x4− 1 7 · 1 6! x6− 1 9 · 1 8! x8− 1 11 · 1 10! x10+· · · = − 1 3! x2 − 1 5! x4 − 1 7! x6 − 1 9! x8 − 1 11! x10 + · · · . (157) example 4.16. n = 2 : given the initial value problem y′′ + 2 x y′ + 1 − y + y2 2! = 0, y(0) = 0, y′(0) = 0. (158) it is also easy to show that the solution to (158) yields y(x) = − 1 3 · 1 2! x2 − 1 5 · 1 4! x4 − 8 21 · 1 6! x6 − 29 27 · 1 8! x8 − 40320 8173 · 1 10! x10 + · · · = − 1 3! x2 − 1 5! x4 − 8 3 · 7! x6 − 29 3 · 9! x8 − 40320 743 · 11! x10 + · · · . (159) 5. concluding remarks in this paper, we have used the mittag-leffler function expansion method to find approximate solutions of a class of fractional lane-emden equation whose nonlinear forms of f (y) are expressible as f (y(x)) = a0 + a1y(x) + a2y2(x) + · · ·+ amym(x) + · · · + an yn (x), 0 ≤ m ≤ n, n ∈ n0; the values of the expansion coefficients am, 0 ≤ m ≤ n, were explicitly provided. we considered the cases for which the functions f (y(x)) are trigonometric, hyperbolic and exponential functions. in all these cases, the lane-emden equations for the special values n = 0; α = 2,β = 1,ω = 2, coincide with the classical and standard ones ((57), (97), (140)) and consequently the corresponding solutions are given in closed forms ((62), (101), (143)). in the case where the functions f (y) are exponential functions, our results can be compared with those of [11, equations 278 awonusika et al. / j. nig. soc. phys. sci. 4 (2022) 265–280 279 (59) and (70)] (see also [25, equation (25)]). the other nonlinear forms of f (y) are the trigonometric and hyperbolic functions. our approximate solutions in the case of trigonometric functions are comparable with the solutions [11, equations (81) and (83) ]; and in the case of hyperbolic functions, see [11, equations (89) and (91)] (see also [22]). the method employed in this paper can be applied to other similar cases in applied sciences where the models are given as strongly nonlinear ordinary differential equations. by extension, the method, can therefore, be accurately and reliably used to compute approximate solutions of nonlinear fractional differential equations of lane-emden type where the nonlinear forms of f (y) involve several other special functions. acknowledgements the authors thank the anonymous referees for their careful reading of the manuscript and valuable comments for the improvement of the quality of the paper. references [1] r. garrappa, e. kaslik & m. popolizio, “evaluation of fractional integrals and derivatives of elementary functions: overview and tutorial”, mathematics 7, (2019) 407. 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[42] h. t. davis, introduction to nonlinear differential and integral equations, dover, new york, (1962). 280 j. nig. soc. phys. sci. 4 (2022) 64–74 journal of the nigerian society of physical sciences a review on transforming plastic wastes into fuel k. manickavelana, s. ahmeda, k. mithuna,∗, p. sathisha, r. rajasekaranb, n. sellappana a department of engineering, mechanical engineering section, university of technology and applied sciences salalah, salalah oman b department of engineering, chemical engineering section, university of technology and applied sciences salalah, salalah oman abstract the application of plastics in various sectors led to its increased production globally and this demand, in turn, caused an overflow of plastic waste in landfills, illegal dumping in the sea, and environmental pollution. to overcome this issue, several alternatives for managing plastic wastes have been developed and among them, reuse, recycling, and energy recovery methods are highly acknowledged methods. nonetheless, recycling methods come with certain disadvantages like mixing and segregation of wastes, high labour costs associated with segregation and processing, byproduct disposal, and its usage. researchers have shifted their focus to energy recovery systems because of these drawbacks. extensive research in this area led to the development of converting waste plastics into liquid fuel through the process called pyrolysis. the pyrolysis process can thermally degrade plastics in the absence of oxygenproducing oil and monomers. the temperature has the most impact on the pyrolysis process and depending on the types of plastic wastes, the pyrolysis temperature varies between 300 – 800 oc. the oil yield due to the variation in temperature varies between 45 – 95 wt.% and the calorific value of the oil has been observed to be in the range of 9679 – 11428.5 kcal/kg, which is similar to the other commercial fuels. also, the review indicates that it is possible to extract up to 84% of fuel from 1-kg plastic at 360 oc. as a result, following refining/blending with conventional fuels, pyrolysis oil can be utilised as an alternate source of energy and transportation fuel. apart from the temperature, the other influencing factors include, the reactor design and its size, pressure, heating rate, residence time and feedstock composition. the pyrolysis process was examined in terms of plastic types and primary process factors that impacted the end result, such as oil, gaseous, and char. temperatures, reactor types, residence duration, pressure, catalysts, and other critical factors were examined in this work. furthermore, the study examines technological problems and current advances. doi:10.46481/jnsps.2022.364 keywords: plastic wastes, pyrolysis, liquid fuel, energy recovery, recycling article history : received: 26 august 2021 received in revised form: 16 january 2022 accepted for publication: 5 february 2022 published: 28 february 2022 c©2022 journal of the nigerian society of physical sciences. all rights reserved. communicated by: w. a. yahya 1. introduction environmental problems and energy demand concerns have increased the global attention on recycling pathways. however, due to low recycling rates and the ever-expanding use of plastics has led to an exponential growth in the plastic waste generation. thus, necessitating the development of new waste re∗corresponding author tel. no: +968-96726334 email address: mv.kulkarni@sct.edu.om (k. mithun ) fining technologies [1]. it is estimated that in the sultanate of oman alone, the percentage of plastics in the municipal solid waste (msw) constitutes to be about 21% and by 2025 globally, the msw is expected to rise by 2600 million tonnes per year. plastics have high molecular weight ranging from thousands to millions. the excessive molecular size is the main cause for the non-biodegradable properties of plastics, making them persistent in soil environment for a long time. plastics, fundamentally, are differentiated depending on their composi64 m. kolandasami et al. / j. nig. soc. phys. sci. 4 (2022) 64–74 65 tion, and normally reported based on their proximate analysis. proximate analysis refers to a method for determining the chemical properties of the plastic compound based on four elements(1) moisture content, (2) fixed carbon, (3) volatile matter, and (4) ash content [2]. particularly, the ash content and volatile matter influence oil production. high ash content favours gas and char formation and on the other hand high volatile matter favours liquid oil production [3]. it has been observed that high carbon ratio in the plastics favours producing products with high calorific value. the plastics listed in table 1, have c ratio of 63.94%, 85.6%, 86.88%, 91.57% and 63.94% for high-density polyethylene (hdpe), low-density polyethylene (ldpe), polypropylene (pp), polystyrene (ps) and polyethylene terephthalate (pet), respectively. depending on the chemical structure, plastics have different thermal deposition temperatures. usually for plastics listed in table 1, the thermal decomposition starts at 350 oc. however, if the desired output is other than liquid, then the operating temperature should be above 500 ◦c , which helps in producing gas and char. research indicates that plastics such as pet and pvc produced little liquid yield and some plastics like pvc will never be used in the pyrolysis process as they produce harmful gases with harmful substances as by-products [4]. table 1: proximate analysis of various plastic types plastic type moisture (wt.%) fixed carbon (wt.%) volatile matter (wt.%) ash content (wt.%) reference hdpe 0.00 0.00 99.92 0.08 [5] 0.00 0.01 99.81 0.18 [6] ldpe 0.3 0.00 99.70 0.00 [7] pp 0.00 0.09 99.9 0.01 [6] ps – 0.1 99.9 – [8] pet 0.1 11.8 88.1 – [8] waste plastics 0.41 0.28 96.88 2.43 [9] as shown in table 2, plastics such as pet finds its use in household applications, in packaging industry, mineral water bottles, fruit juice containers and cans, etc. pet is also used in other applications such as printing sheets, magnetic tapes, x-ray films, etc. since, most applications use pet, its pyrolysis has been observed to yield a liquid of 23.1 wt.% and gas of 76.9% [10, 11]. acidic characteristics have also been observed in the pyrolysis oil that makes it difficult to directly blend with conventional fuels and/or use it directly in ic engines [12]. another major contributor to the waste plastics generation from table 2 is hdpe. hdpe due to its lost cost, high strength properties, ease of forming and break resistance properties is used in several applications. some applications are listed in table 2. in a work by ahmad et al. [6] a steel reactor was used to pyrolyze hdpe at temperatures ranging from 300 to 400◦c. nitrogen was employed as the fluidising medium. at 300 oc, 33.05 wt.% of solid residue was obtained and at 350◦c, a total of 80.88 wt.% liquid was collected. kumar and singh [13] used a semibatch reactor to conduct a thermal pyrolysis investigation of hdpe at a higher temperature of 400–550◦c. they noted that the thermal decomposition of waste hdpe can be enhanced to produce useful products by using appropriate catalysts. catalysts are observed to increase the chemical reaction and help in distributing hydrocarbons to obtain pyrolysis liquid that has properties nearly similar to conventional fuels such as diesel and petrol. the most typical catalysts that were employed in this process included zeolite, alumina, and zirconium oxide. also, the impact of different catalysts on pyrolysis of hdpe is being explored by a variety of researchers. however, the details about the use of the catalyst in the pyrolysis process will be discussed in detail in the sections to follow. low-density polyethylene (ldpe) plastics due to their excellent water resistance are used in a wide variety of applications. bagri and williams [14] and marcilla et al. [15] used ldpe as a raw material in the pyrolysis processes for obtaining the liquid fuel oil, a fixed-bed reactor at 500◦c with a heating rate of 10◦c/min and a batch reactor at 550◦c with a heating rate of 5◦c/min and liquid yields of 95 and 93.1 percent, were reported by them, respectively [11]. the raw materials derived from waste plastics will play a decisive role in the transformation of the petrochemical industry by 2030 and with technological development, one-third of the plastics entering the global market will be manufactured from recycled plastics [16]. however, recycling of plastics waste must meet several obstacles and problems, along with fiscal unfeasibility. there arises a slew of technological hurdles to overcome, when dealing with increasingly difficult to melt plastic items, such as multi-layer materials or polymers that include many toxic compounds, such as brominated flame retardants, raw material availability, raw material selection, pre-treatment, choice of reactor, etc., [17]. however, before discussing the technical challenges, it is important to discuss various plastic recycling methods. figure 1 lists the four main plastic recycling methods (primary, secondary, tertiary and quaternary). figure 1: various approaches for recycling of plastic wastes [19] 65 m. kolandasami et al. / j. nig. soc. phys. sci. 4 (2022) 64–74 66 table 2: municipal waste sources, major plastic types from msws and pyrolysis products [18] source types of solid waste major plastic types from msws pyrolysis products from waste plastics from msws residential food wastes, paper, cardboard, plastics, textiles, leather, garden wastes, wood, glass, metals, ashes, special wastes (e.g., bulky items, consumer electronics, white goods, batteries, oil, tyres), detergent bottles and household hazardous wastes. pe, pp, ps, pet, hdpe waxes, paraffins, olefins, benzoic acid, vinyl terephthalate industrial and automotive housekeeping wastes, packaging products, food wastes, construction hazardous wastes, ashes, plastics & polymers, special wastes. pur, pmma, pa6, waxes, paraffins, olefins, benzene, methane, ethylene, nh3, hcn, caprolactam construction & demolition wood, steel, concrete, dirt, plastic doors, etc. pur, pvc benzene, methane, ethylene, nh3, hcn, hcl (<300 ◦c). municipal services street sweepings; landscape and tree trimmings; general wastes from parks, beaches, and other recreational areas; sludge process, single use plastic bags (manufacturing, etc.) pe, pp waxes, paraffins, olefins. agriculture spoiled food wastes, agricultural wastes, hazardous wastes (e.g., pesticides), plastics used for storing pesticides, etc pe, hdpe waxes, paraffins, olefins pe=polyethylene, pp=polypropylene, ps=polystyrene, pa-6=polyamide 6, pmma=polymethyl methacrylate, pet=polyethylene terephthalate, pur=polyurethane resin, pvc=polyvinyl chloride, hcl=hydrochloride (a) primary recycling in this process, the uncontaminated single-use plastic after sorting out or recovered from plastic wastes having properties and characteristics close to virgin materials is recycled [20, 21]. this method needs a better sorting technique as plastics recovered from msw may not be suitable and useful for recycling [19]. to obtain better materials than the original products, many times clean scarp is added during the recycling process. often techniques such as injection moulding, blow moulding and other mechanical recycling techniques are used in primary recycling technique [21] and is also one of the simplest methods for recycling plastics. but the process also carries a certain disadvantage like it puts a limit on the number of cycles that waste plastics can be recycled [22]. (b) secondary recycling secondary recycling (sr) refers to the transformation of scrapped thermoplastics into products that are less demanding than compared to the original products. but looking at both primary and secondary techniques, these are wellestablished techniques and are widely applied in the transformation of plastic wastes into useful products [23]. as an example for sr, polyolefins are used in the preparation of flooring tiles [19]. sr is also referred to as mechanical recycling [22] and involves cutting, separation of contaminants, and segregation followed by processing, milling, washing, adding pigments and additives for obtaining the granulated form. these are further used for preparing useful products through techniques such as injection moulding, blow moulding, screw extrusion and so on. (c) tertiary recycling in this method, plastics are completely broken down into chemical component materials [24] that help in the production of raw materials used for making plastic components, and thus giving an opportunity for the recyclers to employ this method for recycling giving preference over the primary and secondary techniques [19]. for example, for tertiary recycling, glycolysis of pet into diols and dimethyl terephthalate can then be used to make virgin pet [20]. tertiary methods can be sub-classified into chemical and thermolysis methods. thermolysis is further classified into pyrolysis, gasification and hydrogenation as seen from figure 1. among these, pyrolysis process is an area where most of the research on recycling plastic wastes has taken 66 m. kolandasami et al. / j. nig. soc. phys. sci. 4 (2022) 64–74 67 place. (d) quaternary recycling in this process, the energy from waste plastics is recovered by burning or firing them. fuel obtained from scrapped tyres, generally known as tyre-derived fuel (tdf) is an example of the quaternary recycling method. 2. technical challenges pyrolysis oil as observed from various literature is made up of various hydrocarbons, many of which are partially oxygenated with the presence of water, ash, and charcoal. the composition of the oil is much influenced by the various process parameters and must overcome the challenges posed by these process parameters. the toxicity of oil and its unstableness in air possess furthermore challenges for having to be used directly into the engine. although, the main goal of the pyrolysis of plastic waste oil is to use it in engines directly, but the existing technology does not allow it to do so. the presence of charcoal, water and ash particles needs to be removed before blending it along with conventional fuels and or to replace the existing conventional fuels, raw material availability, viscosity, combustion behaviour, wax formation, the choice of reactors, etc., are other challenges that are discussed below. 2.1. raw material availability even with various approaches, attempts to popularise and make the plastic recycling methods economical have failed in the past due to the non-continuous / non-steady supply of consistent quality raw materials [25]. according to ragaert et al. [26], the economics of plastic recycling depends on the quantity of the raw material available at any given point in time. previously, china was the largest importer of plastic wastes, off late it has banned all imports and thus large volume of waste plastics needs to be treated locally [25]. but the ban by china has opened the door of opportunities in waste management sector such as, awareness of reuse and recycling wastes, preserving the ecosystem, employment generation, alternate fuels, alternate materials in construction sector, etc. however, to tap the opportunities, it is necessary to have a steady flow of consistent quality of raw materials, with bans and strict laws by several countries in place supply of raw materials may not be a problem in the future. 2.2. feedstock selection the plastic wastes collected from various sources are heterogeneous. the types of polymers used, vary for different applications. the plastic wastes, if segregated at the source of generation could solve most of the problems associated with raw material availability and selection. but for this, awareness among the public is essential. polyolefin occupies a major share of plastic items, owing to their chemistry and synergistic reasons, they are considered an ideal choice as a feed for pyrolysis process. also, most applications use pet and pvc plastics, which contributes largely towards waste plastics generation, yet they are not an ideal choice for the pyrolysis process. upon thermal degradation of pet and pvc, the latter has corroded the equipment due to the release of the chlorine containing compounds and renders the oil halogenated. alternatively, pet tends to decompose into phthalic acids deteriorating the oil quality [25]. 2.3. pre-treatment many plastics come with different forms and sizes that make it mandatory to uniformly size them before they are used as a raw material for various other steps. thus, this step adds an extra cost for the entire process [25]. 2.4. material feeding one of the most difficult tasks in plastic recycling is the material feeding job, as non-uniform size and shape of waste plastics pose a direct challenge during certain recycling processes. some recycling methods include screw feeders for processing but plastic wastes when in the form of sheet, wrap around the screw feeders and form a lump/melt of semi solid plastic and make it difficult to continue with the feeding [25]. 2.5. wax formation one of the principal results of the pyrolysis of plastics, particularly polyolefins, is wax. when wax is the primary result of plastic pyrolysis, the recovery system must be built specifically for waxes. when using standard condensing systems, the wax tends to condense on the condenser walls, making it harder to reclaim the material during the process. 2.6. choice of reactor although there are many types of reactors in use, only four important types of reactors have been reviewed and compared (refer table 3) in this section. 2.6.1. fluidized-bed reactor (fb reactor) in this type of reactor (figure 2a), due to the presence of solid particles and the fluidisation gas (flowing continuously) on the bed, makes the bed to be in a fluid-like state. the desired reaction occurs when the raw material is fed into the reactor [27]. the fb reactor are known for their ease of operation and less maintenance cost. they are considered the best for continuous operation and provide a degree of freedom when choosing different fluidisation agents and offers excellent heat and mass transfer capabilities. but when the operating conditions are poorly selected possibility of bed defluidization can occur and plastics tend to stick to the surface of the bed particles, thus making the operation a little complicated. even some times feeding can also be difficult due to particle size restriction [28]. 2.6.2. fluid catalytic cracking reactor (fcc reactor) a chemical process (figure 2b) used to produce gasoline and distillate fuels with the help of catalyst [29]. it is one of the widely used methods in conventional refinery [28]. the catalyst in the solid form is made fluid when the hot vapour and fluid are fed into the fcc. now, since the catalyst is in the liquid form 67 m. kolandasami et al. / j. nig. soc. phys. sci. 4 (2022) 64–74 68 it can easily move around between reactor and regenerator vessels in the fcc reactor. thus, the catalysts and heat are used to break down the larger gas molecules into molecules of petrol, butane, distillates etc., [29]. in this process, during the cracking process, carbon gets deposited on the catalyst and thus will be referred to as catalyst coke. the main disadvantage of carbon deposition is that the ability of the catalyst to crack the oil gets reduced. in the process, flue gas is produced as a cause of regeneration, passes through the environmental control equipment, and finally discharged into the atmosphere. 2.6.3. continuous stirred tank reactor (cstr) in this type of reactor, the product(s) of the reaction concomitantly moves out of the vessel when the reactants, solvents and reagents enter the reactor. thus making the reactor an important tool in the processing of the chemicals continuously and hence the name cstr. the advantages of this reactor are effectively mixing, easy accessibility to the interior of the reactor, the cost of reactor construction, and the capability to work under steady-state with uniform properties [30]. another characteristic feature of cstr is that the material’s composition inside the reactor is the same as the output composition, which is a function of the reaction rate and residence time. 2.6.4. screw reactor screw reactors contains a tubular reactor and a screw conveyor. this rector can be used to pyrolyze thermoset plastics, which are otherwise considered as a difficult to treat plastics. the reactor can effectively and efficiently remove chlorine in the case of mixed plastic waste. also, by varying the screw length and screw speed, it is possible to vary the residence time. excellent heat transfer and selective zone heating are the added features of screw reactors. 3. what is pyrolysis? the thermal degradation of long-chain polymer molecules into smaller and simpler ones is known as pyrolysis. in the absence of oxygen, the technique needs extremely high temperatures for a short duration. the three main products produced by pyrolysis are oil, gas, and char, which are important in industries such as manufacturing and refining. many researchers selected pyrolysis because it can create a large volume of liquid oil (up to 80 wt.%) at a modest temperature of roughly 500 ◦c [4]. furthermore, pyrolysis is extremely adaptable since process parameters may be tweaked to enhance product yield based on preferences. the liquid oil generated may be utilised in various applications, including furnaces, boilers, turbines, and diesel engines, without any further treatment or upgrade [5]. biomass pyrolysis oil has received a lot of interest as a more environmentally friendly fuel since it helps lower co2levels in the atmosphere. in contrast to recycling, the process handling is significantly easier and more flexible because it does not require a thorough sorting method, making it less labour intensive [6]. table 3: reactor comparison for plastic thermolysis. modified from [49] reactor type fixed bed vacuum pyrolysis cstr# screw type temperature control p s g p heat transfer p s g p particle size flexibility s g s s residence time flexibility g g g g process flexibility p s s s thermal mode operation s g s g catalytic mode operation s s na s value of obtained product g s g s scale-up flexibility p s g s economic feasibility s s g s # values from various sources; p = poor; s = satisfactory; g = good, na = not available in the literature pyrolysis produces a combination of molecules in the form of liquid or wax as the major products, depending on the process circumstances [10]. the liquid or wax that is produced can subsequently be utilised as a feedstock for refining chemicals or fuels [11, 12]. the primary pyrolysis products from various polymers are listed in table 2. some resins, such as ps, pa, and pmma, have high monomer yields, which is of particular importance. however, municipalities collected plastic garbage is frequently a combination of different polymers, resulting in a mixture of these goods. 3.1. types of pyrolysis pyrolysis is mainly categorised into three types viz., slow, intermediate, and fast as shown in figure 3. literature indicate that fast pyrolysis liquid oil obtained can be used in internal combustion engines [11]. slow pyrolysis includes the gradual heating of waste plastics at over 500◦c in the absence of oxygen. carbonisation is another name for slow pyrolysis. solid charcoal is a major result of slow pyrolysis [11], where fast pyrolysis involves a great and rapid heating in the absence of air, often in the temperature range of 450 – 650 ◦c produces products such as char, gas, and organic vapours. in most cases, 60–75 wt.% of the feedstock is converted to oil [31]. intermediate pyrolysis, also known as 68 m. kolandasami et al. / j. nig. soc. phys. sci. 4 (2022) 64–74 69 figure 2: common reactors for plastic pyrolysis [28] figure 3: pyrolysis types and their operating parameters and product yields [11] flash pyrolysis is a rapid heating process with a short residence time of less than 1 second. around 75% of the feedstock can be converted into liquid yield using the flash pyrolysis process. 3.2. process parameters influence on product yield in pyrolysis, parameters are crucial for optimising product yield and composition.temperature, reactor type, pressure, residence time, etc., are factors that impact the liquid oil output during pyrolysis. table 4 lists the comparison of various parameters with the different reactors.among the parameters, the temperature controls the cracking reaction; thus, treated as the most important operational parameter in pyrolysis.depending on the plastic type and its chemical structure, the degradation temperature varies.usually, the type of product required from pyrolysis is known to be influenced by temperature.for instance, if the char and gaseous product is desired, temperature more than 500 ◦c is preferred and in case liquid (oil) is desired then the temperature in the range of 300 – 500 ◦c is preferred for plastics such as pet, pp, ps, etc. [2]. besides temperature, reactors play an important role in improving the reaction efficiency and producing the desired end product and a few of the important reactors have already been discussed in the previous sections.pressure and residence time are temperature-dependent factors that influence the pyrolysis process, especially at lower temperatures.higher pressure increases the gaseous yield, yet most research is conducted at atmospheric pressure only.the effect of residence time becomes less apparent at higher temperatures.however, at temperature below 450 ◦c both pressure and residence time factors should be given much importance. catalysts are known to and have been tried to enhance the reaction efficiency and improve the hydrocarbon distribution in the pyrolysis process.the use of catalyst helps in obtaining the pyrolysis oil that has characteristics close to regular fuels such as diesel and petrol.the catalyst such as zeolite was used in the pyrolysis by miskolezi et al. [32]. it was proved that zeolite helps in reducing impurities and enhances the efficiency of liquid yield.a study on the use of bentonite clay in the form of pelletized catalyst for the pyrolysis of plastic wastes such as ps, pp, ldpe, and hdpe was taken up by supattra budsaereechai et al. [33]; observations revealed that the liquid oil produced, had a higher calorific value than compared with the experiments that used catalyst.maximum production of gas was reported in the case of pyrolysis of pp with an alumina-loaded catalyst by obali et al. [34]. a pyrolysis comparison study of pp and ps using thermally activated natural zeolite catalyst and acid (hcl) activated natural zeolite catalyst indicated more gas formation using the latter catalyst by syamsiro et al. [35]. a summary of the effect of process parameters is shown in table 4. 3.3. liquid fuel properties of plastic pyrolysis oil the fuel properties of waste ldpe, hdpe, pp and mixed plastics using the pyrolysis process are summarised in table 5.the experimental calorific value of waste plastics discussed in table 4 are close to the commercial standard values as depicted in table 5. however, work by onwudili et al. [40] noted that ps had a lower calorific value than polyolefin plastics because of the presence of an aromatic ring in the chemical structure, which had a lower combustion energy than aliphatic hydrocarbons.also, sharuddin et al. [2] in their research review paper noted that, because of the presence of benzoic acid in pet and chlorine compound in pvc, reduced the fuel quality, pet and pvc had the lowest calorific value.the viscosity measurement 69 m. kolandasami et al. / j. nig. soc. phys. sci. 4 (2022) 64–74 70 table 4: effect of process parameters on plastic waste pyrolysis process parameters findings effect of heating rate and operating temperature higher heating rates and operating temperature causes breaking of bond and thus helps in the production of smaller molecules [36]. rise in temperature causes easier conversion by reducing the aliphatic content [37]. effect of pressure coke is formed as a result of condensation of reactive fragments at lower pressure [36]. during thermal degradation of pe plastics, the reaction pressure was seen to have a significant impact on the distribution of degradation products, providing a mechanism to regulate product distribution [38]. residence time usually, residence time increases the conversion. secondary conversion of primary products happens with longer residence time thereby producing more tar, thermally stable products, and coke [37]. presence of gases such as n2, ar, o2 or h2. presence of gases have an influence over the kinetics, equilibrium, and mechanism. the gases also help in heat generation and product dilution [36, 37]. catalyst effect to fast-track the chemical reaction, catalysts are employed. the most widely utilized catalysts in the pyrolysis process are zeolite, silica (sio2), calcium oxide (cao), and alumina (al2o3). these catalysts help to shorten the reaction time and enhances the reaction rate [39]. of waste plastics was also made, as it is an important property, and is defined as the fluid resistance to flow [2, 40].the viscosity values determined at different temperatures can be noted from table 5. except for ldpe and mixed plastics from table 5 and ps [2], the viscosity of hdpe and pp were close to the viscosity of commercial standard diesel value from table 6. pour point refers to the temperature at which a fluid ceases to flow owing to an increase in viscosity and it is also an important quality specification for diesel fuels [41].generally, greater aromatic content and lesser wax (paraffin) is observed in fuels with lower pop [2, 41].hdpe, mixed plastics, pp from table 5 and ps [2] have lower pop than commercial diesel, which has a pop of 6 ◦c as shown in table 5.the values obtained indicate that pyrolysis oil is rich with aromatic content and possesses low calorific value compared to commercial diesel and petrol.one of the important properties to avoid or prevent fire accidents during storage of fuel is its flash point (fp).the fp is the lowest temperature at which a solvent may ignite a flammable combination in air near the liquid’s surface.the lower the flash point of a liquid solvent, the easier it is to ignite it [42]. from table 5 it is observed that waste plastics possess low fp temperature compared to commercial fuels and it is essential to exercise precaution while handling and storing the pyrolysis oil. 3.4. by-products of the plastic pyrolysis along with oil, as discussed previously, the pyrolysis process also produces char and gas as by-products. researchers have noted that process parameters such as heating rate, the presence of gases such as n2, ar, o2 or h2, pressure, residence time, and temperature play a greater role in the by-product generation. table 7 lists the by-products of the pyrolysis process. char as a by-product is formed due to (1) slow heating at a low temperature (2) longer residence time. this by-product has certain advantages like being used as an adsorbent in water treatment to remove heavy metals through an upgrading treatment [2], its low sulphur content makes it usable as a fuel as the calorific value of char is about 18.8 mj/kg, and can be used in laying of roads and building materials [51], char-based sensors and supercapacitors [52, 53, 54, 55]. the pyrolysis of pvc and pet compared to polyolefins and ps produced a large amount of gases. the composition of the gas mainly depends on the type of material used in the pyrolysis process. for instance, plastics such as hdpe, pet, pvc produced gases such as h2, ch4, c4h10, c2h6, c3h8, etc. interestingly, pet produced additional gases such as co2 and co, while hcl was produced by pvc. the gases produced using the pyrolysis process also has a good calorific value (pe and pp have a cv of 42 and 50 mj/kg, respectively) [44]. for maximising the gas production in the pyrolysis process, a longer residence time and higher temperatures are required. but these come with a price, that’s lower oil production [56]. 4. recent developments in plastic waste pyrolysis most of the recent works on waste plastic pyrolysis has taken place in the areas related to the use of renewable energy, biorefineries and bimetallic catalysts.the details of which have been discussed below. pyrolysis process can be treated as one of the sources of renewable energy, but the process has some drawbacks like it needs an external source of energy to heat the reactors and thus pollution associated with it has also to be taken care of. to deal with these issues, researchers have focussed their attention on the use of solar energy for heating the reactors.aklilu et al. [62] focussed on design and testing a solar-assisted pyrolysis process for the production of liquid fuel from hdpe plastics.the results of the study indicated that 1360 kwh/m2 of solar energy was required to produce 14.2 litres of liquid fuel from hdpe waste.further, they noted that the heating value of the produced liquid fuel was 41.8 mj/kg.chaouki ghenai et al. [63] designed a grid-tied solar pv power system to meet the energy demand 70 m. kolandasami et al. / j. nig. soc. phys. sci. 4 (2022) 64–74 71 table 5: characteristics of pyrolysis oil waste plastic type type of reactor characteristics of pyrolysis oil reference v (cst) ρ, g/cc fp, ◦c pop, ◦c cv, kcal/kg hdpe fixed bed pyrolysis batch reactor 1.980a 0.7477 a 14 <-15 9829.35 [43] astm d445 ip 131/57 astm d93 astm d97 bomb calorimeter 12/58 semi-batch 3.3 a 0.8006 c 10 18 10,573.71 [5] is:1448 p:25 is:1448 p:16 is:1448 p:20 is:1448 p:10 is:1448 p:6 pilot scale two stage reactor using batch system 2.319a 0.7991 c <10 24 10234.226 [44] astm d 445 astm d1298 astm d93 astm d 97 astm d240 horizontal steel 5.08 a 0.89 c 48 -5 9679.732 [2, 6] ip 71/87 ip160/87 ip 34/85 ip 15/67 ip 12/80 ldpe batch reactor 1.24a 0.773 c – – – [45] astm d445 astm d1298 – – – tube reactor 1.117a – 22 – 11089.87 [46] – – – – – mixed plastic bags (pe, pp) self-designed 0.8a 0.735 a 29 – 11428.5 [47] – – – – – mixed plastics self-designed 2.8b 0.76 a 35 – 10809.27 [48] – – – – – self-designed – 0.7365 c 22 <-20 11262 [37, 49] – – – – – polyolefin self-designed – 0.68 a 29 – – [50] – – – – – pp horizontal steel 4.09 a 0.86c 30 -9 9751.434 [2, 6] ip 71/87 ip160/87 ip 34/85 ip 15/67 ip 12/80 table 6: commercial fuel values fuel type v, cst ρ, g/cc fp, ◦c pp, ◦c cv, kcal/kg reference gasoline 1.17 0.780 42 – 10157.74 [43] diesel 1.9 – 4.1 0.807 52 6 10277.2 [43] viscosity – v; flash point – fp; ploud point – pop; density – ρ; calorific value – cv a@ 40◦c; b @ 25◦c; c @ 15◦c of the bfk type magnetic rotary stirred pyrolysis reactor.the main objective of their study was to employ some form of renewable energy for the thermal chemical conversion process.a solar powered mobile unit, capable of converting waste plastics into fuel, which is comparable with conventional fuels has been developed by a team of students at iit – madras.the best part of this technology is that being a mobile unit, can be driven directly to the place of waste collection and then process the waste.it is a total decentralised mechanism that needs no necessity to erect a processing plant and to transfer plastic wastes from the point of collection to the point of processing.the unit is capable of handling plastic bags, packaging material and other plastic items that are not normally collected for recycling by ragpickers can be used as raw materials.the team claims that waste plastic oil can be used as an alternative in generators, boilers, furnaces, pumps and diesel-powered vehicles [64].guozhan jiang et al. [65] performed a process simulation and a technoeconomic analysis of pyrolysis of mixed plastic 71 m. kolandasami et al. / j. nig. soc. phys. sci. 4 (2022) 64–74 72 table 7: by-products of plastic pyrolysis. waste plastic type pyrolysis temperature crude oil (wt. %) residue (wt. %) gas (wt. %) wax (wt. %) references pe 500 93 0 7 – [57] pp 95 0 5 – 300 69.82 1.34 28.84 – [6] mixed 370 – 420 90 5 5 – [58] hdpe 500 44.32 1.29 25.4 28.99 [5] 350 80.88 1.88 17.24 – [6] 440 74 17 9 – [59] mixed plastic 300-400 60-80 7-10 10-20 [49] 400 500 76.7 ± 1.2 16 ± 1.2 7.3 ± 0.5 [47, 60] 500 ∼ 55 ∼ 5 ∼ 40 – [9] 800 73 23.5 30.4 – [61] ldpe 600 51.07 – 19.35 – [45] 400 45 – 26 – [46] pet 500 23.1 – 76.9 – [11] waste (mpw) containing pvc that utilised molten solar salt as the heating medium.they observed that by adding calcium carbonate (caco3) to the salt mixture, a practical way to remove chlorine from pvc was possible.further, to melt the salt, the electricity generated from the non-condensable pyrolysis gas or concentrated solar power was used.a work on solar energy driven dielectric microwave oven was taken up by debadyoti ghosh et al. [66],the work involved use of sodium bentonite clay as a catalyst to endorse the conversion efficiency. hdpe and ldpe wastes were pyrolyzed in the temperature range of 450 ◦c to 500 ◦c under sub-atmospheric pressure inside the reactor chamber (vacuum chamber).due to the vacuum pyrolysis, the harmful gases produced was considered negligible. pyrolysis-based biorefineries have a high potential for converting waste into energy and other useful goods, which might aid in the development of circular economies.miandad et al. [67] worked on pyrolysis-based biorefineries that can convert plastic and biomass waste into energy and has the capability to convert them into valuable products.they used natural zeolite catalysts in a pilot scale reactor fabricated.plastics such as ps, pe, pp, and pet in mixed ratio was used in the study.work on co-pyrolysis of mahua seeds (mh) with ps and waste nitrile gloves (wng) was investigated by ranjeetkumar mishra and kaustubha mohanty [68].the ps and wng wastes were mixed with mh in varying proportions.the results of blending showed that waste plastics at 20 wt.% yielded a maximum liquid (44.18 ± 1.2 wt.% and 45.89 ± 1.4 wt.% for mh + wng and mh + ps respectively), which was higher than the thermal pyrolysis of individual mh (39.26 ± 1.2 wt.%).further, characterization of the obtained co-pyrolytic oil revealed that the oil could be used either directly or blended with the diesel after little upgradation. bimetallic catalysts are regarded as an important type of heterogeneous catalysts with exceptional catalytic characteristics compared to their separate metal components [69].because of the existence of two metals, they can develop new and characteristic features.ganjar fadillah et al. [70] employed bimetallic catalysts in the pyrolysis of plastic waste to biofuel.cia et al. [71] employed a carbon-based fe-ni bimetallic catalyst for rapid pyrolysis of plastic trash.in the procedure, a fixed bed reactor was employed, and the plastic-to-catalyst ratio was kept at 2:1 at a temperature of 500 ◦c.the researchers indicated that the fe-ni bimetallic catalyst was critical in the oxygen reduction process.because of the presence of an oxygen-containing functional group on the surface, the catalyst is also capable of avoiding corrosion on its surface.zhou et al. [72] employed a bimetallic ni-fe/zro2 catalyst in their research involving polystyrene waste.the findings demonstrated that bimetallic catalysts increased catalytic activity in waste breakdown. also due to the pandemic and post-pandemic stage, the use and disposal of ppes has increased and to effectively deal with the disposed ppes which are mainly derived out of plastics, cold plasma technology in pyrolysis system has been adopted.this technology is effective in dealing with infectious medical plastic wastes, along with the high temperature of pyrolysis.the technique involves use of electrons to break the chemical bonds in plastics compared to the conventional pyrolysis.however, the process comes with a disadvantage that it needs an advanced cooling system [73, 74]. 5. conclusion pyrolysis process has the capacity to transform most plastic wastes into various valuable products such as liquid oil, wax, and char over other thermal treatment methods. pyrolysis can be achieved via thermal or catalytic process; use of catalyst helps in attaining higher output of liquid oil for most polymers. the process’s long-term viability is unquestionable, given that the amount of plastic waste at any point in the year in each nation is in the millions of tons. waste management becomes more efficient using the pyrolysis process, which requires less landfill capacity, produces less pollution, and is also more cost72 m. kolandasami et al. / j. nig. soc. phys. sci. 4 (2022) 64–74 73 effective. furthermore, because of the presence of the pyrolysis process for decomposing plastic into useful energy fuel, the reliance on non-renewable energy sources such as fossil fuels may be minimized, therefore alleviating the growth in energy demand. among the various plastics, pe and pp plastics have the capacity to yield a greater amount of crude oil when subjected to pyrolysis at 500◦c as compared to other plastics. temperature below 500◦c leads to production of more gases and other residues. acknowledgement the authors would like to thank the research council oman for fully funding the work described in this publication through the trc reg no: bfp/rgp/ebr/20/519. a special thanks is also extended to the university of technology and applied sciences – salalah (utas). references [1] w. a. rasaq, m. golonka, m. scholz, and a. bialowiec, “opportunities and challenges of high-pressure fast pyrolysis of biomass: a review”, energies 14 (2021) 120, doi: 10.3390/en14175426. 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[74] m. m. harussani, s. m. sapuan, u. rashid, a. khalina, and r. a. ilyas, “pyrolysis of polypropylene plastic waste into carbonaceous char: priority of plastic waste management amidst covid-19 pandemic”, sci. total environ. 803 (2022) 149911, doi: 10.1016/j.scitotenv.2021.149911. 74 j. nig. soc. phys. sci. 4 (2022) 818 journal of the nigerian society of physical sciences time-fractional differential equations with an approximate solution lamees k. alzaki∗, hassan kamil jassim department of mathematics, faculty of education for pure science, university of thi-qar, nasiriyah, iraq. abstract this paper shows how to use the fractional sumudu homotopy perturbation technique (shp) with the caputo fractional operator (cf) to solve time fractional linear and nonlinear partial differential equations. the sumudu transform (st) and the homotopy perturbation technique (hp) are combined in this approach. in the caputo definition, the fractional derivative is defined. in general, the method is straightforward to execute and yields good results. there are some examples offered to demonstrate the technique’s validity and use. doi:10.46481/jnsps.2022.818 keywords: fractional differential equations, homotopy perturbation method, sumudu transform article history : received: 16 may 2022 received in revised form: 05 july 2022 accepted for publication: 26 july 2022 published: 18 august 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: o. j. oluwadare 1. introduction for characterizing nonlocal structures, fractional calculus has arisen as a new mathematical technique. due to their numerous applications in physics and engineering, fractional differential equations have attracted a lot of attention during the last decade [1-4]. many authors have investigated fractional calculus theory, and current study is centered on proving the important benefits of fractional calculus over classical calculus [5, 6]. in several instances, the fractional calculus gave the best outcomes than the conventional method. there is a significant literature dealing with the subject area of approximating solutions to fractional differential equations ∗corresponding author tel. no: +9647819374140 email addresses: lkalzaki@utq.edu.iq (lamees k. alzaki ), hassankamil@utq.edu.iq (hassan kamil jassim) using various approaches known as perturbation methods. the perturbation techniques have certain disadvantages; for example, the approximate solution requires a succession of smaller parameters, which is challenging because the majority of nonlinear problems do not have any small values. despite proper selections of smaller parameters might occasionally lead to an optimal solution, in most circumstances, bad choices have major consequences in the solutions. as a result, an analytical technique that does not need a smaller parameter in the equation representing the phenomena is preferred. he [7] was the first to propose the homotopy perturbation technique (hpm). many writers investigated the hpm to analyze linear and nonlinear equations encountered in numerous scientific and technical sectors. in general, higher performance of the fractional calculus is demonstrated by reduced error levels created during an estimating procedure. various approximation and methodologies, like 1 alzaki & jassim / j. nig. soc. phys. sci. 4 (2022) 818 2 the fractional adomian decomposition method (fadm) [8-10], fractional homotopy method (fhpm) [11, 12, 13], fractional function decomposition method [14, 15], fractional variational iteration method (fvim) [16-18], fractional reduce differential transform method (frdtm) [18, 19, 20, 21], fractional differential transform method [22, 23, 24], fractional laplace variational iteration method [25-32], fractional laplace homotopy perturbation method (flhpm) [33], fractional laplace decomposition method (fldm) [34, 35], fractional sumudu homotopy analysis method [36], fractional sumudu variational iteration method (fvim) [37, 38], fractional sumudu decomposition method (fsdm) [39-41], fractional natural decomposition method (fndm) [42, 43], fractional sumudu homotopy perturbation method (fshpm) [44, 45], energy balance method (ebm) [46], power series methods (psm) [47], have been used in latest years to analyze partial differential equations within caputo sense.moreover, many studies provide functional differential equations using various types of transforms, such as the mellin transform [48]. also, the lagrange interpolation is used to estimate the delay argument [49]. furthermore, the local truncation errors and stability polynomials are calculated [50-52]. our objective is to develop and use the sumudu homotopy perturbation methodology, which combines the sumudu transform with the homotopy perturbation method to solve nonlinear fractional differential equations. 2. preliminaries this section goes through some of the fractional calculus definitions and notation that will be utilized in this study [4, 45, 46, 47, 53, 54, 55]. definition 2.1. the riemann liouville fractional integral operator of order δ ≥ 0 of a function ϕ(µ) ∈ cϑ,ϑ ≥ −1 is given by the form iδϕ(µ) =  1 γ(δ) ∫ µ 0 (µ−τ)δ−1ϕ(τ)dτ, δ > 0,µ > 0 i0ϕ(µ) = ϕ(µ), δ = 0 (1) where γ(·) is the well-known gamma function. properties of the operator iδare as follows: for ϕ ∈ cϑ,ϑ ≥ −1,δ,γ ≥ 0, then 1. iδiγϕ(µ) = iδ+γϕ(µ) 2. iδiγϕ(µ) = iγiδϕ(µ) 3. iδµm = γ(m + 1) γ(δ + m + 1) µδ+m definition 2.2. in the caputo interpretation, the fractional derivative of ϕ(µ) is given as dδϕ(µ) = im−δdmϕ(µ) = 1 γ(m −δ) ∫ µ 0 (µ−τ)m−δ−1ϕ(m)(τ)dτ, (2) for m − 1 < α ≤ m, m ∈ n,µ > 0 and ϕ ∈ cm −1. the fundamental properties of the operator dδ are given as follows: 1. dδiδϕ(µ) = ϕ(µ) 2. dδiδϕ(µ) = ϕ(µ) − ∑n−1 k=0 ϕ (k)(0) µk k! definition 2.3. for δ > 0, the gamma function γ(·), is defined as follows: γ(δ) = ∫ ∞ 0 xδ−1e−xd x (3) definition 2.4. by considering eδ with δ > 0, the definition of the mittag–leffler function given as the following: eδ(z) = ∞∑ m=0 zδ γ(mδ + 1) (4) some special cases of the mittag-leffler function eδ(z) 1. e0(z) = 1 1−z , |z| < 1, 2. e1(z) = ez, 3. e2(z) = cosh √ z, z ∈ c, 4. e2(−z2) = cosz, z ∈ c definition 2.5. the sumudu transform is identified based on a collection of functions a = {ϕ(τ) : ∃m,ω1,ω2 > 0, with |ϕ(τ)| ≤ me |τ| ω j , ifτ ∈ (−1) j × [0,∞)} as determined by the formula s[ϕ(ω)] = g(ω) = ∫ ∞ 0 e−τϕ(ωτ)dτ, ω ∈ (−ω1,ω2) some properties of sumudu transform 1. s[k] = k, k constant 2. s [ τmδ γ(mδ+1) ] = ωmδ definition 2.6. the caputo fractional derivative for the sumudu transform is given as the following s [ dµδτ ϕ(µ,τ) ] = ω−µδs [ ϕ(µ,τ) ] − m−1∑ k=0 ω(−µδ+k)ϕ(k)(µ, 0), m − 1 < mδ < m (5) 3. fractional sumudu homotopy perturbation method (fshpm) consider this generic fractional nonlinear pdes: dmδτ ϕi(µ,τ) + r [ ϕ(µ,τ) ] + n [ ϕ(µ,τ) ] = ð(µ,τ), m − 1 < mδ ≤ m (6) with ϕ(µ, 0) = f (µ), (7) in which dmδτ ϕ(µ,τ) is the caputo fractional derivative of the function ϕ(µ,τ), r refers for the linear differential operator, 2 alzaki & jassim / j. nig. soc. phys. sci. 4 (2022) 818 3 while n refers for a nonlinear system differential operator, and ð(µ,τ) is the origin term. using the st to both sides of (6), we get s[dµδτ ϕ(µ,τ)] + s[r [ ϕ(µ,τ) ] ] + s[n [ ϕ(µ,τ) ] ] = s[ð(µ,τ)] (8) we obtain by utilizing the st’s property s[ϕ(µ,τ)] = −ωδ m−1∑ k=0 ϕ(k)(µ, 0) + ωδs [ ð(µ,τ) ] −ωδs [ r[ϕ(µ,τ)] + n[ϕ(µ,τ)] ] (9) operating the inverse sumudu transform on both sides of (9), we get ϕ(µ,τ) = s−1 [ ωδ m−1∑ k=0 ϕ(k)(µ, 0) ] + s−1 [ ωδs [ ð(µ,τ) ]] −s−1 [ ωδs [ r[ϕ(µ,τ)] + n[ϕ(µ,τ)] ]] (10) we implement the hpm: ϕ(µ,τ) = ∞∑ n=0 pnϕn(µ,τ), (11) thus the nonlinear term may be decomposed as follows: n[ϕ(µ,τ)] = ∞∑ n=0 pn hn(ϕ0,ϕ1, . . . ,ϕn), (12) where hn(ϕ0,ϕ1, . . . ,ϕn) = 1 n! ∂n ∂pn [ n [ ∞∑ i=0 piϕi(µ,τ) ]] p=0 when we substitute (11) and (12) into (10), we obtain ∞∑ n=0 pnϕn(µ,τ) = s −1 [ ωδ m−1∑ k=0 ϕ(k)(µ, 0) ] + s−1 [ ωδs [ ð(µ,τ) ]] −ps−1 ( ωδs [ r [ ∞∑ n=0 pnϕn(µ,τ) ] + ∞∑ n=0 pn hn ]) we get the following set of equations by comparing the terms with comparable powers of p: p0 : ϕ0(µ,τ) = s −1 [ ωδ m−1∑ k=0 ϕ(k)(µ, 0) ] +s−1 [ ωδs [ ð(µ,τ) ]] p1 : ϕ1(µ,τ) = −s −1 [ ωδs [ r[ϕ0(µ,τ)] + h0 ]] p2 : ϕ2(µ,τ) = −s −1 [ ωδs [ r[ϕ1(µ,τ)] + h1 ]] ... pn : ϕn(µ,τ) = −s −1 [ ωδs [ r[ϕn−1(µ,τ)] + hn−1 ]] , n ≥ 1 (13) consequently, we use truncated series to estimate the analytical result ϕ(µ,τ) ϕ(µ,τ) = lim p→1 ∞∑ n=0 pnϕn(µ,τ) (14) 4. applications this section will put the proposed method for solving time fractional partial differential equations into application. 4.1. example firstly, examine the time-fractional cauchy reaction–diffusion equation shown below dδτϕ(µ,τ) = ϕµµ(µ,τ) −ϕ(µ,τ), 0 < δ ≤ 1 (15) with ϕ(µ, 0) = e−µ + µ (16) for the (15), applying the sumudu transform st on both sides of it, we achieve s[ϕ(µ,τ)] = ϕ(µ, 0) + ωδs [ ϕµµ(µ,τ) −ϕ(µ,τ) ] (17) using the inverse st to (17), we obtain s[ϕ(µ,τ)] = e−µ + µ + s−1 ( ωδs [ ϕµµ(µ,τ) −ϕ(µ,τ) ]) (18) according to the hpm, and substituting ϕ(µ,τ) = ∞∑ n=0 pnϕn(µ,τ) in (18), we have ∞∑ n=0 pnϕn(µ,τ) = e −µ + µ +ps−1 ( ωδs [ ∂2 ∂µ2 [ ∞∑ n=0 pnϕn(µ,τ) ] − ∞∑ n=0 pnϕn(µ,τ) ]) (19) we get the following set of equations by comparing the terms with comparable powers of p: p0 : ϕ0(µ,τ) = e −µ + µ p1 : ϕ1(µ,τ) = s −1 ( ωδs [ ∂2ϕ0(µ,τ) ∂µ2 −ϕ0(µ,τ) ]) = s−1 ( ωδs[−µ] ) = − µτδ γ(δ + 1) , p2 : ϕ2(µ,τ) = s −1 ( ωδs [ ∂2ϕ1(µ,τ) ∂µ2 −ϕ1(µ,τ) ]) = s−1 ( ωδs [ − µτδ γ(δ + 1) ]) = − µτ2δ γ(2δ + 1) , 3 alzaki & jassim / j. nig. soc. phys. sci. 4 (2022) 818 4 p3 : ϕ3(µ,τ) = s −1 ( ωδs [ ∂2ϕ2(µ,τ) ∂µ2 −ϕ2(µ,τ) ]) = s−1 ( ωδs [ − µτ2δ γ(2δ + 1) ]) = − µτ3δ γ(3δ + 1) , p4 : ϕ4(µ,τ) = s −1 ( ωδs [ ∂2ϕ3(µ,τ) ∂µ2 −ϕ3(µ,τ) ]) = s−1 ( ωδs [ − µτ3δ γ(3δ + 1) ]) = − µτ4δ γ(4δ + 1) , ... pn : ϕn(µ,τ) = s −1 ( ωδs [ ∂2ϕn−1(µ,τ) ∂µ2 −ϕn−1(µ,τ) ]) = (−1)n µτnδ γ(nδ + 1) hence, the outcome of (15) is provided by ϕ(µ,τ) = lim p→1 ∞∑ n=0 pnϕn(µ,τ) = e−µ + µ [ 1 − τδ γ(δ + 1) + τ2δ γ(2δ + 1) + τ3δ γ(3δ + 1) . . . +(−1)n τnδ γ(nδ + 1) ] = e−µ + µeδ(τ δ) (20) for δ = 1, the (20) is close to the form ϕ(µ,τ) = e−µ + µe−τ, which is the precise solution of (15) for δ = 1. the outcome is same as with fsdjm [53]. 4.2. example we determine the most important time fractional cauchy reaction-diffusion equation: dδτϕ(µ,τ) = ϕµµ(µ,τ) − (1 + 4µ 2)ϕ(µ,τ), 0 < δ ≤ 1 (21) with ϕ(µ, 0) = eµ 2 (22) we obtain by utilizing the st’s property on both sides of (21), we get s[ϕ(µ,τ)] = ϕ(µ, 0) + ωδs [ ϕµµ(µ,τ) −(1 + 4µ2)ϕ(µ,τ) ] (23) using the inverse st to (23), we have s[ϕ(µ,τ)] = eµ2 + s−1 ( ωδs [ ϕµµ(µ,τ) −(1 + 4µ2)ϕ(µ,τ) ]) (24) according to the hpm, and substituting ϕ(µ,τ) = ∞∑ n=0 pnϕn(µ,τ) in (24), we have ∞∑ n=0 pnϕn(µ,τ) = e µ2 + ps−1 ( ωδs [ ∂2 ∂µ2 [ ∞∑ n=0 pnϕn(µ,τ) ] − (1 + 4µ2) ∞∑ n=0 pnϕn(µ,τ) ]) (25) we get the following set of equations by comparing the terms with comparable powers of p: p0 : ϕ0(µ,τ) = e µ2 p1 : ϕ1(µ,τ) = s −1 ( ωδs [ ∂2ϕ0(µ,τ) ∂µ2 −(1 + 4µ2)ϕ0(µ,τ) ]) = τδ γ(δ + 1) eµ 2 , p2 : ϕ2(µ,τ) = s −1 ( ωδs [ ∂2ϕ1(µ,τ) ∂µ2 − (1 + 4µ2)ϕ1(µ,τ) ]) = τ2δ γ(2δ + 1) eµ 2 , p3 : ϕ3(µ,τ) = s −1 ( ωδs [ ∂2ϕ2(µ,τ) ∂µ2 − (1 + 4µ2)ϕ2(µ,τ) ]) = τ3δ γ(3δ + 1) eµ 2 , p4 : ϕ4(µ,τ) = s −1 ( ωδs [ ∂2ϕ3(µ,τ) ∂µ2 − (1 + 4µ2)ϕ3(µ,τ) ]) = τ4δ γ(4δ + 1) eµ 2 , ... pn : ϕn(µ,τ) = s −1 ( ωδs [ ∂2ϕn−1(µ,τ) ∂µ2 − (1 + 4µ2)ϕn−1(µ,τ) ]) = τnδ γ(nδ + 1) eµ 2 hence, the outcome of (21) is provided by ϕ(µ,τ) = lim p→1 ∞∑ n=0 pnϕn(µ,τ) = eµ 2 [ 1 + τδ γ(δ + 1) + τ2δ γ(2δ + 1) + τ3δ γ(3δ + 1) · · · + 4 alzaki & jassim / j. nig. soc. phys. sci. 4 (2022) 818 5 τnδ γ(nδ + 1) ] = eµ 2 eδ(τ δ) (26) for δ = 1, the (26) is close to the form ϕ(µ,τ) = eµ 2 +τ, which is the precise solution of (21) for δ = 1. the outcome is same as with fsdjm [53]. 4.3. example consider the nonlinear fractional cauchy reaction–diffusion equation is given as the following dδτϕ(µ,τ) = ϕµµ(µ,τ) −ϕµ(µ,τ) + ϕ(µ,τ)ϕµµ(µ,τ) −ϕ2(µ,τ) + ϕ(µ,τ), 0 < δ ≤ 1 (27) w.r.t initial condition ϕ(µ, 0) = eµ (28) operating the st on both sides of (27), and employing st’s differential property, we obtain s[ϕ(µ,τ)] = ϕ(µ, 0) + ωδs [ ϕµµ(µ,τ) −ϕµ(µ,τ) +ϕ(µ,τ)ϕµµ(µ,τ) −ϕ 2(µ,τ) + ϕ(µ,τ) ] (29) we obtain by applying the inverse sumudu transform on both sides of (29) ϕ(µ,τ) = eµ + s−1 ( ωδs [ ϕµµ(µ,τ) −ϕµ(µ,τ) + ϕ(µ,τ)ϕµµ(µ,τ) −ϕ 2(µ,τ) + ϕ(µ,τ) ]) (30) according to the hpm, and substituting ϕ(µ,τ) = ∞∑ n=0 pnϕn(µ,τ) ϕϕµµ = ∞∑ n=0 pn hn ϕ2 = ∞∑ n=0 pngn in (30), we have ∞∑ n=0 pnϕn(µ,τ) = eµ + ps−1 ( ωδs [ ∂2 ∂µ2 [ ∞∑ n=0 pnϕn ] − ∂ ∂µ [ ∞∑ n=0 pnϕn ] + ∞∑ n=0 pn hn − ∞∑ n=0 pngn − ∞∑ n=0 pnϕn ]) (31) we get the following set of equations by comparing the terms with comparable powers of p: p0 : ϕ0(µ,τ) = e µ, p1 : ϕ1(µ,τ) = s −1 ( ωδs [ ∂2ϕ0 ∂µ2 − ∂ϕ0 ∂µ + h0 − g0 +ϕ0 ]) = τδ γ(δ + 1) eµ, p2 : ϕ2(µ,τ) = s −1 ( ωδs [ ∂2ϕ1 ∂µ2 − ∂ϕ1 ∂µ + h1 − g1 +ϕ1 ]) = τ2δ γ(2δ + 1) eµ, p3 : ϕ3(µ,τ) = s −1 ( ωδs [ ∂2ϕ2 ∂µ2 − ∂ϕ2 ∂µ + h2 − g2 +ϕ2 ]) = τ3δ γ(3δ + 1) eµ, p4 : ϕ4(µ,τ) = s −1 ( ωδs [ ∂2ϕ3 ∂µ2 − ∂ϕ3 ∂µ + h3 − g3 +ϕ3 ]) = τ4δ γ(4δ + 1) eµ, (32) ... pn : ϕn(µ,τ) = s −1 ( ωδs [ ∂2ϕn−1 ∂µ2 − ∂ϕn−1 ∂µ +hn−1 − gn−1 + ϕn−1 ]) = τnδ γ(nδ + 1) eµ hence, the outcome of (27) is provided by ϕ(µ,τ) = lim p→1 ∞∑ n=0 pnϕn(µ,τ) = eµ [ 1 + τδ γ(δ + 1) + τ2δ γ(2δ + 1) + · · · + τnδ γ(nδ + 1) ] = eµeδ(τ δ) (33) for δ = 1, the (33) is close to the form ϕ(µ,τ) = eµ+τ, which is the precise solution of (27) for δ = 1. the outcome is same as with fsdjm [53]. 4.4. example assume the coupled fractional burger’s equations shown below where 0 < δ ≤ 1, 0 < γ ≤ 1 dδτϕ(µ,τ) −ϕµµ − 2ϕϕµ + (ϕψ)µ = 0, dδτψ(µ,τ) −ψµµ − 2ψψµ + (ϕψ)µ = 0 (34) with ϕ(µ, 0) = eµ ψ(µ, 0) = eµ (35) 5 alzaki & jassim / j. nig. soc. phys. sci. 4 (2022) 818 6 operating the st on both sides of (34), and employing st’s differential property, we obtain s [ ϕ(µ,τ) ] = ϕ(µ, 0) + ωδs [ ϕµµ + 2ϕϕµ − (ϕψ)µ ] , s [ ψ(µ,τ) ] = ψ(µ, 0) + ωγs [ ψµµ + 2ψψµ − (ϕψ)µ ] (36) implementing with the inverse st to (36), we have ϕ(µ,τ) = eµ + s−1 ( ωδs [ ϕµµ + 2ϕϕµ − (ϕψ)µ ]) , ψ(µ,τ) = eµ + s−1 ( ωγs [ ψµµ + 2ψψµ − (ϕψ)µ ]) (37) assume that ϕ(µ,τ) = ∞∑ n=0 pnϕn(µ,τ) (38) ψ = ∞∑ n=0 pnψn (39) ϕϕµ = ∞∑ n=0 pn hn (40) ψψµ = ∞∑ n=0 pn kn (41) (ϕψ)µ = ∞∑ n=0 pngn (42) by applying the hpm, and substituting (38)-(42) in (37), we get ∞∑ n=0 pnϕn(µ,τ) = e µ + ps−1 ( ωδs [ ∂2 ∂µ2 [ ∞∑ n=0 pnϕn ] +2 ∞∑ n=0 pn hn − ∞∑ n=0 pngn ]) , ∞∑ n=0 pnψn(µ,τ) = e µ + ps−1 ( ωγs [ ∂2 ∂µ2 [ ∞∑ n=0 pnψn ] +2 ∞∑ n=0 pn kn − ∞∑ n=0 pngn ]) (43) we get the following set of equations by comparing the terms with comparable powers of p: p0 : ϕ0(µ,τ) = e µ, : ψ0(µ,τ) = e µ p1 : ϕ1(µ,τ) = s −1 ( ωδs [ ∂2ϕ0 ∂µ2 + 2h0 − g0 ]) : ψ1(µ,τ) = s −1 ( ωγs [ ∂2ψ0 ∂µ2 + 2k0 − g0 ]) p2 : ϕ2(µ,τ) = s −1 ( ωδs [ ∂2ϕ1 ∂µ2 + 2h1 − g1 ]) : ψ2(µ,τ) = s −1 ( ωγs [ ∂2ψ1 ∂µ2 + 2k1 − g1 ]) p3 : ϕ3(µ,τ) = s −1 ( ωδs [ ∂2ϕ2 ∂µ2 + 2h2 − g2 ]) : ψ3(µ,τ) = s −1 ( ωγs [ ∂2ψ2 ∂µ2 + 2k2 − g2 ]) ... then, we have p0 : ϕ0(µ,τ) = e µ, : ψ0(µ,τ) = e µ p1 : ϕ1(µ,τ) = s −1 ( ωδs [ eµ + 2e2µ − 2e2µ ]) : ψ1(µ,τ) = s −1 ( ωγs [ eµ + 2e2µ − 2e2µ ]) = s−1(ωδeµ) = τδ γ(δ + 1) eµ = s−1(ωγeµ) = τγ γ(γ + 1) eµ p2 : ϕ2(µ,τ) = s −1 ( ωδs [ τδ γ(δ + 1) eµ +2 τδ γ(δ + 1) e2µ − 2 τγ γ(γ + 1) e2µ ]) : ψ2(µ,τ) = s −1 ( ωγs [ τγ γ(γ + 1) eµ +2 τγ γ(γ + 1) e2µ − 2 τδ γ(δ + 1) e2µ ]) (44) = s−1 ( ω2δeµ + 2ω2δe2µ − 2ωδ+γe2µ ) = s−1 ( ω2γeµ + 2ω2γe2µ − 2ωδ+γe2µ ) = τ2δ γ(2δ + 1) eµ + 2 τ2δ γ(2δ + 1) e2µ − 2 τδ+γ γ(δ + γ + 1) e2µ = τ2γ γ(2γ + 1) eµ + 2 τ2γ γ(2γ + 1) e2µ − 2 τδ+γ γ(δ + γ + 1) e2µ ... thus, the outcome of (34) is given by ϕ(µ,τ) = eµ [ 1 − τδ γ(δ + 1) + τ2δ γ(2δ + 1) . . . ] +e2µ [ 2 τ2δ γ(2δ + 1) − 2 τδ+γ γ(δ + γ + 1) . . . ] ψ(µ,τ) = eµ [ 1 − τγ γ(γ + 1) + τ2γ γ(2γ + 1) . . . ] +e2µ [ 2 τ2γ γ(2γ + 1) − 2 τδ+γ γ(δ + γ + 1) . . . ] 6 alzaki & jassim / j. nig. soc. phys. sci. 4 (2022) 818 7 (45) setting δ = γ in (45), we obtain ϕ(µ,τ) = eµ [ 1 − τδ γ(δ + 1) + τ2δ γ(2δ + 1) . . . ] = eδ(−τ δ)eµ ψ(µ,τ) = eµ [ 1 − τγ γ(γ + 1) + τ2γ γ(2γ + 1) . . . ] = eγ(−τ γ)eµ (46) the (46) is close to the form ϕ(µ,τ) = ψ(µ,τ) = eµ−τ for δ = γ = 1 , which is the precise solution of (34) for δ = γ = 1. 5. conclusion the sumudu homotopy perturbation approach was effectively used in this work to discover approximate solutions to time-fractional partial differential equations. the analytical approach generates a convergence analysis that fast converges to the optimal solution. the simplicity and high precision of the analytical method are clearly illustrated, solving equations includes linear and nonlinear fractional pdes and a nonlinear system of fractional pdes. references [1] g. aguilar, y. martinez, t. jimenez, c. fraga, e. jimenez, o. peregrino, homotopy perturbation transform method for nonlinear differential equations involving to fractional operator with 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[55] h. eltayeb, a. kilicmai, a note on the sumudu transform and des, applied math. sci. 4 (2010) 1089. 8 j. nig. soc. phys. sci. 4 (2022) 787 journal of the nigerian society of physical sciences age prediction from sclera images using deep learning p. o. odiona, m. n. musab,∗, s. u. shuaibua acomputer science dept, nigerian defence academy kaduna, nigeria bcyber security dept, nigerian defence academy kaduna, nigeria abstract automatic age classification has drawn the interest of many scholars in the fields of machine learning and deep learning. in this study, we looked at the problem of estimating age groups using different biometric modalities of human beings. we looked at the problem of determining age groups in humans using various biometric modalities. specifically, we focused on the use of transfer learning for sclera age group classification. 2000 sclera images were collected from 250 individuals of various ages, and otsu thresholding was used to segment the images using morphological processes. experiment was conducted to determine how accurately the age group of a person can be classified from sclera images using pretrained cnn architectures. the segmented images were trained and tested on four different pre-trained models (vgg16, resnet50, mobilenetv2, efficientnet-b1), which were compared based on different performance metrics in which resnet-50 was shown to outperform the others, resulting in an accuracy, precision, recall and f1-score of 95% while vgg-16, efficientnetb1, and mobilenetv2 had 94%, 93%, and 91%, respectively. the findings from the study showed that there is an aging template in the sclera that can be utilized to classify age. doi:10.46481/jnsps.2022.787 keywords: sclera images, pre-trained cnn, age estimation, deep learning, segmentation, sbvpi dataset article history: received: 26 april 2022 received in revised form: 30 june 2022 accepted for publication: 15 july 2022 published: 15 august 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: t. latunde 1. introduction every human being possesses distinct biometric traits, which are employed in individual identification, recognition, verification, age estimation, and authentication due to their uniqueness. face recognition, ear recognition, voice recognition, iris recognition, retina scan, finger recognition, vein recognition, and deoxyribonucleic acid (dna) matching are just a few of the biometric technologies that have become a hot topic in academia. facial recognition is the most prominent area of human recognition since the human face reveals many features such as age, gender, race, emotion, expression, and identity [1]. ∗corresponding author tel. no: +234(0)8166046507 email address: muhammadmusa2502@nda.edu.ng (m. n. musa) age, as a human characteristic, plays a significant role in facilitating or limiting communication. it functions as a barrier, affecting both how we interact with one another and how we comprehend what others are saying, much as language, culture, beliefs, and experience do. however, as people age, their faces change and their skin thickens, as well as their colour and texture. the tissue composition begins to be more sub-cutaneous and the facial skeleton lines or wrinkles appear. the ability of a machine to recognize and interpret faces or facial traits such as age, emotion, or gender in real time gave rise to the concept of computer vision [2]. machine learning (ml) algorithms have been used to address issues in several domains, by analysing and understanding vast amounts of data [3-6]. machine learning has benefited 1 odion et al. / j. nig. soc. phys. sci. 4 (2022) 787 2 in the detection and identification of facial images, as well as the age estimation [7]. the method employs classic computer vision techniques that employ handcrafted features, as well as a step framework that include an aging learning pattern [2]. most facial recognition algorithms that use ml lack generalizability when applied to unseen photos [7]. factors that affect facial recognition and facial age estimation include race, beards, plastic surgery, skin tone, etc. this has led to age estimation research using facial features being extensively studied with the aim of finding out aging patterns and variations and how to best characterize an aging face for accurate age estimation [8]. the failings in using facial images for age estimation led to research in other areas, among which include eye regions like the iris, sclera, and retina, in which iris recognition is the predominant technology in the area [9]. the problem with using ml method in eye region recognition for age estimation occurs due to the eye region been so small and extracting the features manually using histogram of oriented gradients (hog), principal component analysis (pca) is undesirable as we might miss important features during the cause of building the model [10]. deep learning using convolutional neural networks (cnn) has recently piqued the interest of many computer vision researchers due to its superior ability to learn a series of nonlinear features directly from raw pixels [11]. this prompted researchers to concentrate their efforts on various eye region experiments using deep learning to observe template changes. images of the retina reveal considerable age-related changes. bruch’s membrane, which is present in the retina, is especially prone to aging [12]. high-resolution retinal images, on the other hand, are taken under limited imaging settings by professional specialists using sophisticated imaging devices. there is also evidence of studies into the use of the iris as a method of determining age [13]. infrared cameras are typically used to capture iris images from a close range. retina or iris images are not an obvious choice for age estimation because image gathering is becoming more difficult and expensive. this made researchers consider the sclera for age estimation because the sclera is the white visible portion of the eye image and thus it remains visible for various gaze directions [14]. it can also be captured with hand-held cameras or mobile cameras. even though the sclera is relatively stable over time, its colour changes with age and health [15, 16]. due to the fact that training deep learning models consumes a lot of resources and is computationally expensive [17], coupled with the inability to gather millions of eye images to train deep learning models from scratch, this motivates us to explore the use of some of the best pre-trained models that were state-of-the-art in the imagenet contest (vgg-16 [18], resnet50 [19], mobilenetv2 [20], efficientnet [21]) to predict human age using the sclera of the eye. to achieve this, we were able to acquire eye images from individuals between the ages of 5 to 30 years old, apply otsu thresholding to execute segmentation of the sclera from the acquired eye images, train, validate and interpret the result generated using different performance metrics. 2. literature review the first research conducted on age estimation was in 1994 by kwon and da vitoria lobo, in which the age was simply divided into numerous ranges [11]. many studies have focused on age estimation using facial images, with few studies focusing on eye regions [10]. there is evidence of research using the iris as a modality for age estimation [14]. some of the research that used iris for classification includes [22] who conducted an experiment using 596 iris images, where 300 consisted of young subjects and two hundred and 296 elderly subjects by employing a random forest algorithm to extract features and classify the images, resulting in a classification rate of 64.68%. [23] suggested an alternative method that relies on five geometric features extracted from the human iris using a total of 210 respondents, spanning from 18 to 73, and were classified as young (25), adult (25–60), and senior (> 60). the authors used segmentation to aid in efficiently detecting iris and pupil boundaries. they were able to extract 12 geometric features from the iris and pupil parameters, which were trained and tested on different classifiers. they achieved a 75% accuracy rate. again, [24] proposed a technique that used the iris structure to estimate a person’s age group. the image input was obtained from an iris database and was divided into three age groups. the iris boundaries were localized using a circular hough transform technique for image pre-processing and segmentation. they used five different classifiers (k nearest neighbour, fine gaussian support vector machine, decision tree, bagged ensemble, and linear discriminant) were used, with the bagged ensemble classifier outperforming the others because it reduces the problem of overfitting of training data and reduces the variance of the estimate. the suggested model attained an overall accuracy of 83.7% and outperformed prior state-of-the-art models. researchers have recently begun to use deep learning methods in the field of eye region biometrics for age prediction, as shown in the research by [25]. they used deep learning methods to test the iris for gender and age classification. they began by estimating gender using deep cnn models like alexnet and googlenet, which were trained on a real-time database of 213 people spanning 3-73 years. the features extracted from the human iris were fed into a multi-class svm using these cnn models. in comparison to googlenet, alexnet performed better, with an overall accuracy level of 95.34%. similarly, their trained model for age prediction indicates that the anticipated age was correct for virtually all of the subjects. they also indicate that gender classification performs better than age classification. in recent years, sclera images obtained in visible light have been used in biometric recognition systems [26, 27]. however, [27] used the vgg architecture to create a segnet and scleranet model for sclera segmentation and recognition, respectively. they conducted rigorous experiments and tested their findings using sclera blood vessel, periocular, and iris (sbvpi), a new public dataset that captures diverse gaze directions of the eye to comprehensively represent all parts of the sclera. the sbvpi collection contains images of 55 distinct people looking at four different directions, with the iris, sclera, 2 odion et al. / j. nig. soc. phys. sci. 4 (2022) 787 3 figure 1. proposed methodology table 1. age-group classes for the dataset age group age range number of subjects number of images kids 5 13 84 672 teens 14 20 85 680 adult 21 30 80 640 total 250 2000 blood vessels, eyelashes, and periocular segment. other models (u-net, refine-net) were compared to segnet, and it surpassed the preceding model in terms of processing speed and accuracy, resulting in highly competitive recognition performance. [14] recently published the first study on the analysis of sclera for age estimation, in which they employed a modified vgg-16 network on a custom dataset. they approached it as a regression problem, resulting in a mean absolute error (mae) of 0.06. this indicates that there is an aging template in the sclera and can be used for age prediction. hence, this research employed a custom dataset for use in age prediction using a classification technique. the newly captured images are then compared using some state-of-the-art pre-trained models. this research is novel, as to the best of our knowledge, there is no work that uses sclera to predict age using a classification technique. 3. methodology the proposed sclera-based age group classification method is shown in figure 1. 3.1. image acquisition this is the first step towards putting the proposed approach into action. this was done with an iphone 11 pro max set at the highest resolution and quality. the photos were taken in an unstructured setting. a total of 2000 photographs were taken of 250 people, ranging in age from 5 to 30, and were divided into three categories: children, teenagers, and adults, as in table 1. the image acquisition setup was inspired by [27]. in their work, they generated the first standardized dataset that consists of images captured in different gaze directions for sclera segmentation and recognition. each subject was asked to change gaze direction four times, that is, straight, upward, left, and right for both left and right eye as shown in figure 2. figure 2. image captured at different gaze direction figure 3. image segmentation using otsu thresholding algorithm 3.2. pre-processing several undesirable facial features, such as the eyebrows, a portion of the nose, the cheeks, and so on, are included in the captured image. as a result, the images were manually cropped and resized to reduce the amount of noise and focus just on the region of interest, which is the eye, while maintaining the aspect ratio. since 10 of the initial collected images were fuzzy and the region of interest was obscured, they were not used for segmentation. as a result, we now have a final image of 1990. the otsu thresholding technique was used to segment the images, as well as perform morphological operations. it is thought to be one of the simplest and most successful binary image segmentation methods [28]. it divides the image into two classes, white for the foreground and black for the background, based on its grayscale characteristics as seen in figures 3 and 4, and then we mask the region of interest. according to [29], the algorithm for the otsu method is outlined below: 1. compute histogram and probabilities of each intensity level 2. initialize wi and µi to zero (0) respectively 3. iterate over all possible threshold t = 0 . . ., max-intensity (a) update wi and µi (b) compute the between the class variance σ2b(t) 4. the final threshold is the maximum σ2b(t) to increase the size of the training dataset, image augmentation was performed, which involves applying geometric transformation techniques to the training images, such as shearing, contrasting, horizontally flipping, spinning, zooming, and blurring. the dataset is then downsized to 224 × 224 pixels and converted to array format, with the pixel intensities adjusted to the range [-1, 1]. this has proved to be a successful approach [30]. 3 odion et al. / j. nig. soc. phys. sci. 4 (2022) 787 4 table 2. comparison with different pre-trained cnn model with respect to performance metrics layer name output size 18-layer 34-layer 50-layer 101-layer 152-layer conv1 112 × 112 7 × 7, 64, stride 2 3 × 3, max pool, stride 2 conv2 x 56 × 56 [ 3 × 3, 64 3 × 3, 64 ] × 2 [ 3 × 3, 64 3 × 3, 64 ] × 3  1 × 1, 64 3 × 3, 64 1 × 1, 256 × 3  1 × 1, 64 3 × 3, 64 1 × 1, 256 × 3  1 × 1, 64 3 × 3, 64 1 × 1, 256 × 3 conv3 x 28 × 28 [ 3 × 3, 128 3 × 3, 128 ] × 2 [ 3 × 3, 128 3 × 3, 128 ] × 4  1 × 1, 128 3 × 3, 128 1 × 1, 512 × 4  1 × 1, 128 3 × 3, 128 1 × 1, 512 × 4  1 × 1, 128 3 × 3, 128 1 × 1, 512 × 8 conv4 x 14 × 14 [ 3 × 3, 256 3 × 3, 256 ] × 2 [ 3 × 3, 256 3 × 3, 256 ] × 6  1 × 1, 256 3 × 3, 256 1 × 1, 1024 × 6  1 × 1, 256 3 × 3, 256 1 × 1, 1024 × 23  1 × 1, 256 3 × 3, 256 1 × 1, 1024 × 36 conv5 x 7 × 7 [ 3 × 3, 512 3 × 3, 512 ] × 2 [ 3 × 3, 512 3 × 3, 512 ] × 3  1 × 1, 512 3 × 3, 512 1 × 1, 2048 × 3  1 × 1, 512 3 × 3, 512 1 × 1, 2048 × 3  1 × 1, 512 3 × 3, 512 1 × 1, 2048 × 3 1 × 1 average pool, 1000-d fc, softmax flops 1.8 × 109 3.6 × 109 3.8 × 109 7.6 × 109 11.3 × 109 figure 4. masking of rgb images using otsu thresholding figure 5. vgg architecture [18] table 3. comparison with different pre-trained cnn model with respect to performance metrics cnn model accuracy precision recall f1 score mobilenetv2 0.92 0.92 0.91 0.92 efficientnet-b1 0.93 0.93 0.93 0.93 resnet-50 0.95 0.95 0.95 0.95 vgg-16 0.94 0.94 0.94 0.94 3.3. age classification the goal of image classification is to determine the class of an input image using its attributes [31]. we used state-ofthe-art pre-trained cnn models for both feature extraction and classification. the pre-trained models were trained and tested on the dataset. we divided the dataset into 80% for training and set hyperparameter constants such as the initial learning rate of 0.00001, the number of training epochs to 50, and the batch size to 32. the model was evaluated on the remaining 20% of the dataset using various performance metrics such as accuracy, precision, recall, and f1 score. 3.4. vgg-16 model previous cnn architectures have proven successful in recent years, particularly in image recognition applications, attracting the attention of many academics. [18] presented a simple effective deep cnn architecture known as visual geometry group (vgg) in 2015, with a network of sixteen layers (vgg16) and nineteen layers (vgg19), which was far deeper than previous zfnet and alexnet models. it was created to demonstrate that the depth of a network is vital in getting better recognition or classification in cnns. the vgg replaced the 11*11 and 5*5 filters with a stack 3*3 filter layer, demonstrating that using smaller filters with fewer parameters reduces the computational cost of the network. the network also adds a set of 1*1 convolutions in between the convolutional layer as shown figure 5. vgg16 performed exceptionally well in image classification and localization. furthermore, it ranked second in the 2014-ilsvrc competition and has become well-known for its simplicity, enhanced depth, and homogenous topology. 3.5. resnet model residual network was developed by [19] to eliminate the vanishing gradient problem that previous networks experienced. it was declared the 2015-ilsvrc winner. the resnet architecture introduced the residual learning framework notion. it was created with a variety of layers, including 34, 50, 101, and 152 layers, as shown in table 2. despite the increased depth, it has low computational complexity when compared to earlier models. resnet established shortcut connections within layers to facilitate cross-layer interconnection, and these residual linkages tend to accelerate deep network convergence, aiding the network in avoiding gradient fading. the resnet architecture’s representational depth is thought to be advantageous for any 4 odion et al. / j. nig. soc. phys. sci. 4 (2022) 787 5 figure 6. comparison of model scaling [21] figure 7. evolution of separable convolution blocks [20] image recognition challenge. the authors demonstrated experimentally that the resnet with 50/101/152 layers has fewer errors in image classification problems and likewise gain an improvement of 28% on the well-known image recognition dataset name coco. 3.6. efficientnet the efficientnet model was developed by [21] to scale models uniformly in all dimensions of depth/width/resolution using a simple yet extremely effective compound scaling that can lead to greater performance. this method may scale up mobilenets and resnets by employing neural architecture search to construct a new baseline network and scale it up to obtain a family of models known as efficient-nets, which achieve significantly higher accuracy and efficiency than earlier convnets. as demonstrated in figure 6, efficientnet employs a compound scaling method that uniformly scales all three dimensions with a constant ratio. the efficientnet family ranges from b0 to b7. starting with the baseline efficientnet-b0, the compound scaling method is used to scale it up in two steps: the compound scaling approach begins with a grid search to determine the relationship between multiple scaling dimensions of the baseline network under a set resource limitation (e.g., 2x more flops). this finds the appropriate scaling coefficient for each dimension before scaling up the baseline network to the specified target model size or computational budget. when scaling up existing models, this compound scaling strategy reliably enhances model accuracy and efficiency such as mobilenet (+1.4% imagenet accuracy), and resnet (+0.7%), compared to conventional scaling methods. 3.7. mobilenetv2 mobilenetv2 network was developed by [20] based on depthwise separable convolution. mobilenetv2 is a tensorflow-based family of mobile computer vision models designed specifically to meet the requirements of low-resource efficient systems, maximize accuracy, and improve the state-of-the-art performance of mobile models on multiple tasks and benchmarks, as well as across a spectrum of model sizes. it is built on an inverted residual structure with quick connections between narrow bottleneck layers as shown in figure 7. as a source of non-linearity, the intermediate expansion layer filters features using lightweight depthwise convolutions. non-linearities were also reduced in the narrow layers to maintain representational power and increase performance. figure 7’s diagonally hatched texture denotes non-linear layers. the final (lightly colored) layer marks the start of the next block. when stacked, c and d are equivalent blocks. the network performance was measured on imagenet classification, coco object detection, voc image segmentation, the architecture improved the state of the art for wide range of performance points [20]. 3.8. performance metrics the practice of analysing how well a model works against real data is known as performance evaluation. standard performance metrics like as precision, recall, and the f1-score, which 5 odion et al. / j. nig. soc. phys. sci. 4 (2022) 787 6 figure 8. graphs of accuracy and loss score for the four pre-trained models are defined as follows [27], are used to assess the classification models’ performance: accuracy accuracy = t p + t n t p + t n + f p + f n × 100 (1) this is one of the popular metrics for assessing a classification model. it is the percentage of correct predictions in a given test data. precision it is about how precise or how often the prediction is correct. it is the ratio of true positives to the sum of true positives and false positives. it is calculated as precision = t p t p + f p (2) recall how often is the prediction correct when the actual value is positive? it is calculated mathematically as the ratio of true positives to the sum of true positives and false negatives as in equation 3. recall = t p t p + f n (3) f1-measure the f1-measure is also known as the f1 score. it is the harmonic mean of precision and recall. a harmonic mean is appropriate for situations where the average of rates (a ratio between two related quantities) is desired. it is calculated as f1 s core = 2 × precision × recall (precision × recall) (4) where, tp denotes the number of true positive pixels, fp stands for the number of false positive pixels, fn represents the number of false negative pixels and tn represents the number of true negative pixels. 6 odion et al. / j. nig. soc. phys. sci. 4 (2022) 787 7 figure 9. confusion matrix of different pre-train networks used 4. results and discussion the goal of the study was to see how accurate a pre-trained deep learning model could be in classifying a person’s age group from the sclera of the human eye. previous studies have demonstrated that the sclera ages and changes over time, and that it can be utilized to estimate age [11]. the dataset was trained and tested on four distinct pre-trained convolutional neural networks, including mobilenetv2, vgg-16, resnet-50, and efficientnet-b1, as shown in the results. table 3 compares the four different neural networks with respect to accuracy, precision, recall, and f1-score while figure 8 shows their accuracy and loss score across 50 epochs. the comparison of the four pre-trained models is shown in figures 8 and 9. the resnet-50 achieved the best accuracy of 95%, while vgg-16, efficientnet-b1, and mobilenetv2 had 94, 93, and 91%, respectively. it also shows that the adult class has a higher rate of misclassification, with that class accounting for the majority of the misclassification. this could be due to the large quantity of images for children aged 13 and 14, with 13 falling into the children’s class and 14 falling into the teen category. these are the boundaries for each class, meaning only months separate the two classes, which could result in misclassification. 5. conclusion in this study, we used a classification technique to evaluate an approach to age prediction. other methods for age prediction were carefully investigated. it was discovered that facial images for age prediction lack generalizability while retina and iris images for age prediction requires the images to be captured using a specialized cameras. our method demonstrated that sclera images captured through hand-held cameras can also provide a very high accuracy. about 2000 images were taken in an uncontrolled environment from 250 people. to solve the problem of inadequate data, pre-trained networks were used to 7 odion et al. / j. nig. soc. phys. sci. 4 (2022) 787 8 train the data. the classification process was evaluated with four different models: mobilenetv2, efficientnet-b1, resnet50, and vgg-16 resulting in an accuracy of 92%, 93%, 95% and 94% respectively. based on the findings, it can be stated that template aging of the sclera in the eye does exist over time, as evidenced by the findings of this study and also resnet-50 was most successful when applied for sclera classification. the study also suggests that with more data and better segmentation, greater accuracy can be attained, potentially rivalling other areas such as facial and iris age prediction. acknowledgments we thank the referees for the positive enlightening 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[31] l. h. thai, t. s. hai & n. t. thuy, “image classification using support vector machine and artificial neural network”, international journal of information technology and computer science 4 (2012) 32. 8 j. nig. soc. phys. sci. 4 (2022) 797 journal of the nigerian society of physical sciences a one-step block hybrid integrator for solving fifth order korteweg-de vries equations olumide o. olaiyaa, mark i. modebeia,b,∗, saheed a. belloc amathematics programme, national mathematical centre, abuja, nigeria bdepartment of mathematics, university of abuja, nigeria cnational water resources institute, kaduna, nigeria abstract fifth-order korteweg-de vries (kdv) equations, arise in modeling waves phenomena such as the propagation of shallow water waves over a flat surface, gravity-capillary waves and sound waves in plasmas. in this work, a one-step block hybrid linear multistep method was derived using the collocation technique, to solve fifth-order kdv models via the method of line (mol). the consistency, stability and convergence of the method were established. the efficiency of the method can be seen from comparison of the exact solutions of problems and other methods cited from literature. doi:10.46481/jnsps.2022.797 keywords: korteweg-de vries (kdv) equations, fifth-order pde, linear multistep, block method, convergence article history : received: 03 may 2022 received in revised form: 12 july 2022 accepted for publication: 25 july 2022 published: 19 august 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: t. latunde 1. introduction in this work, the numerical solution of fifth order pde of the form;{ a(x)ytt + b(x)yxxxxx = g(x, t, y, y2, yx, yt, yxx, yxxx), y(x, b) = g1(x), y(a, t) = g3(t) (1) defined on the domain ω = {(x, t) : a < x < b, c < t < d}, and the boundary ∂ω. where a, b, c, d are real numbers. formulation of models into pdes of the type in equation (1) arise in sciences, for example, fifth-order korteweg-de vries (kdv) type equations, models different wave phenomena which ∗corresponding author tel. no: +2348031576262 email address: gmarc.ify@gmail.com (mark i. modebei) includes the propagation of shallow water waves over a flat surface, gravity-capillary waves and sound wave propagation in plasmas see [1, 2, 3]. prominent examples includes the ”good” boussinesq equation, the kaup-kupershmidt equation and an extended kdv5 equation, [4]. these equations model phenomenon like laser optics, nonlinear dispersive waves in a wide range of applications, water waves, and plasma: generally, these are pdes with higher order spatial derivatives, [4, 5, 6]. some of the notable general (linear and nonlinear) fifth order kdv equations include though not limited to: the kawahara equation [4]; the lax equation [7]; the caudrey-dodd-gibbon (c-dg) equation [8]; the fifth-order kdv equation [9]; the sawadakotera (sk) equation [10]. they take the general form ayt + by 2yx + cyxyxx + dyyxxx + eyxxxxx = 0 (2) 1 olaiya et al. / j. nig. soc. phys. sci. 4 (2022) 797 2 with appropriate initial-boundary conditions, where a, b, c, d, e are constants. the form (2) is referred to as the generalized fifthorder korteweg-de vries (gfkdv), [11, 12]. these gfkdv equations models diverse important physical phenomena. this equation do not just show the movement of long waves in shallow water under gravity and in a onedimensional nonlinear lattice, but at the same time, it is a significant mathematical model for magneto-sound propagation in plasmas [13] and a chain of coupled nonlinear oscillators [14]. obtaining the exact solution of the gfkdv equation appears to be unreachable in most cases especially for the nonlinear case, save the special case of solitary waves in [15]. some methods which are semi numerical and complete numerical in literature that were derived for solving fifth order kdv types include, modified variational iteration technique [11], modified variational iteration algorithm-ii (mvia-ii) [12], he’s variational iteration method, [16], homotopy analysis method, [17], group analysis method [18], among others. in this work, we derive a 1-step block hybrid integrator (1sbhi), assembled in block form using collocation technique, for the solution of fifth-order korteweg-de vries (kdv) type equations using the mols approach, see [19, 20, 21]. the linear multistep method (lmm) derived using the collocation technique is conveniently used to approximate, with high accuracy, continuous initial-boundary value problems (ibvps) of both the ordinary and partial differential equations [20]. it has several advantages among which are the fact that they do not require any staring value to kick-start their implementation, they are self-starting [22, 23]. they also have the advantage of producing smaller global errors (at the end of the range of integration) than those produced by the step-by-step methods due to the presence of accumulated errors at each step in the step-by-step method, see [24, 25, 26, 27, 28]. the method of lines (mols) approach is a very important tool used for solving partial differential equations (pdes), where a pde is transformed into a system of ordinary differential equations (odes) whereby appropriate derivatives are replaced by finite difference approximations. in this work, we transform the pdes into a system of odes such that only one of the spatial derivatives is substituted with central difference approximations. the system that results is solved using 1shbi. in the case of this work, we discretize the space variable t with meshing ∆t = d − c m , tm = c + m∆t, m = 0, 1, . . . m and then define the vector y = [y1,1, y1,2, y2,1, . . . , yn,m−1, yn−1,m, . . . , yn−1,m−1] t and g = [g1,1, g1,2, g2,1, . . . , gn,m−1, gn−1,m, . . . , gn−1,m−1] t where ym ≈ y(x, tm). the central differences approximation formulae for the first and second derivatives are yt ≈ ym+1 − ym−1 (2∆t) ytt ≈ ym+1 − 2ym + ym−1 (∆t)2 the solution y(x, t), of equation (1), where (x, t) is in the rectangle [a, b]×[c, d], is obtained by solving the system of fifth-order odes in the spatial variable x. here the mesh of x is spaced is: πn = a = x0 < x1 < .. . < xn−1 < xn = b, with the step-size h given as h = b − a n , xn = a + nh, n = 0, 1, . . . , n then, the equation (1) has the semidiscretized form dy5n,m d x5 ≈− an,m bn,m + 1 bn,m ( ym+1 − 2ym + ym−1 (∆t)2 + gn,m ) (3) where gn,m = xn, tm, yn,m, y2n,m, dyn,md x , ym+1 − um−1(2∆t) , dy 2 n,m d x2 , dy3n,m d x3  and bn,m , 0 for n = 0, 1, . . . , n and m = 0, 1, . . . , m. the equivalent form for equation (3) is the following system of fifth order ode given by y(5) = f (x, y, y′, y′′, y′′′, y(4)), a < x < b (4) subject to the boundary conditions: y(a) = y0, y’(a) = y ′ 0, y”(a) = y ′′ 0 , y(b) = y1n,m, y ′(b) = y2n,m where f (x, y, y′, y′′, y′′′, y(4)) = ay + g and a is a k by k matrix of constants with (k = (n − 1)(m − 1)) which results from the semi-discretized system of equation (3), expressed in the form the equation (4) and hence solved using 1sbhi. here g is a vector of constants. note that j in y jn,m, for j = 1, 2, are only index notation for different yn,m. in what follows, the derivation of the approximate solution of equation (4) is discussed. it is noteworthy to see that this technique is easy to derive and implement for solving this class of problem defined in equation (1). 2. derivation of the method suppose the approximate solution yn(x) of equation (4) have the continuous form y(x) ≈ u(x) = 10∑ j=0 a j x j (5) where its fifth derivative is given by y(v)(x) ≈ u(v)(x) = 10∑ j=5 j( j−1)( j−2)( j−3)( j−4)a j x j−5(6) evaluating equation (5) at the points x = xn+v j , j = 0(1)4 where v0 = 0, v1 = 1 6 , v2 = 1 3 , v3 = 2 3 , v4 = 5 6 and equation (6) at the 2 olaiya et al. / j. nig. soc. phys. sci. 4 (2022) 797 3 points x = xn+v j , j = 0(1)5 where v5 = 1, the following system of equations are obtained. u(xn) = yn; u ′(xn) = y ′ n; u ′′(xn) = y ′′ n ; u ′′′(xn) = y ′′′ n ; u(iv)(xn) = y (iv) n ; u (v)(xn) = fn; u (v)(xn+v) = fn+v j, j = 1(1)5. (7) solving the above system simultaneously with the use of cas in mathematica 12.0, the coefficients a j, j = 0(1)10 are obtained (though not included here due to their cumbersome terms). they are in turn substituted into equation (5), to obtain the following continuous method: yn+vk = 4∑ j=0 αkj h iy( j)n + h 5 5∑ j=0 βkj fn+v j ∣∣∣∣∣ k=1(1)5 (8) where α j and β j are shown in the table 1. the first derivative of equation (8) is given as y′n+vk = 4∑ i=1 αih i−1y(i)n + h 4 5∑ j=0 βi fn+vi ∣∣∣∣∣ k=1(1)5 (9) here, α j and β j are shown in the table 2. the second derivative of equation (8) is given as y′′n+vk = 4∑ i=2 αih i−2y(i)n + h 3 5∑ j=0 βi fn+vi ∣∣∣∣∣ k=1(1)5 (10) where α j and β j are shown in the table 3 the third derivative of equation (8) is given as y′′′n+vk = 4∑ i=3 αih i−3y(i)n + h 2 5∑ j=0 βi fn+vi ∣∣∣∣∣ k=1(1)5 (11) here α j and β j are shown in the table 4. the fourth derivative of equation (8) is given as y(iv)n+vk = y (iv) n + h 5∑ j=0 βi fn+vi ∣∣∣∣∣ k=1(1)5 (12) where the coefficients α j and β j are shown in the table 5. 2.1. characteristics of the method the formula in equation (8) (and its derivatives in equations (9)-(12)) is a continuous schemes associated with the linear differential operator defined by l[z(x + vkh); h] = 4∑ i=0 αkj h jz( j)(x) − 5∑ j=0 h5βiz (5)(x + v jh) ∣∣∣∣∣ k=1(1)5 (13) expanding equation (13) in taylor series, the constants c′i s written as linear combination of the derivatives of z(x) up to p + 1 derivative as l[z(x); h] = c0z(x)+c1hz ′(x)+c2h 2z′′(x)+· · ·+c ph pz(p)(x)+o(h( p+1)) (14) figure 1. the region of stability where p is the order of equation (13) and consequently the formula (13). by definition, the lmm (8) is of order p if c0 = c1 = c2 = · · · = c p+4 = 0, and c p+5 , 0 in which l[z(x); h] = c p+5h p+5z(p+5)(x) + o(h(p+6)) (15) in this case, c p+5 is the principal error constant, see [29]. definition 2.1. a linear difference operator l[z(x); h] of order p is consistent if p > 1. thus, the order of the formula in equation (8) is p = 6 with the following error constants c p+5 = (−6.87857 × 10 −13,−2.36466 × 10−11, − 5.40035 × 10−10,−1.47456 × 10−9,−3.4408 × 10−9)t the order of the formulas in equations (9)-(12) is p = 6 and their error constants are obtained similarly. 2.2. zero-stability and convergence a numerical method in equations (8)-(12) is zero-stable provided as h → 0, the solutions is bounded. following [20], the block method in (8)-(12) may be rewritten in a matrix form as a0yµ = a1yµ−1 + h 5(fµ + fµ−1) (16) where where yµ = ( y 0µ, y 1 µ, . . . , y 5 µ )t , yµ−1 = ( y 0µ−1, y 1 µ−1, . . . , y 5 µ−1 )t 3 olaiya et al. / j. nig. soc. phys. sci. 4 (2022) 797 4 table 1. coefficients of α j, j = 0(1)4, and β j, j = 0(1)5 in equation (8) k α0 α1 α2 α3 α4 β0 β1 β2 β3 β4 β5 1 1 16 1 72 1 1296 1 31104 86197η1 58392η1 −30815η1 14415η1 −9112η1 1883η1 2 1 13 1 18 1 162 1 1944 7963η2 10528η2 −4365η2 1985η2 −1248η2 257η2 3 1 23 2 9 4 81 2 243 622η3 1392η3 −215η3 180η3 −112η3 23η3 4 1 56 25 72 125 1296 625 31104 6685η4 16600η4 −375η4 2375η4 −1368η4 275η4 5 1 1 12 1 6 1 24 401η5 1056η5 105η5 195η5 −96η5 19η5 η01 = 1 112870195200 ; η 0 2 = 1 440899200 ; η 0 3 = 2 3444525 ; η 0 4 = 625 4514807808 ; η 0 5 = 1 201600 table 2. coefficients of α j, j = 1(1)4, and β j, j = 0(1)5 in equation (9) k α1 α2 α3 α4 β0 β1 β2 β3 β4 β5 1 1 16 1 18 1 162 100795η 1 1 82832 1 1 −42175 1 1 19525 1 1 −12320 1 1 2543 1 1 2 1 13 1 18 1 162 2209η 1 2 3518η 1 2 −1300η 1 2 595η 1 2 −374η 1 2 77η 1 2 3 1 23 2 9 4 81 1316η 1 3 3328η 1 3 −140η 1 3 425η 1 3 −256η 1 3 52η 1 3 4 1 56 25 72 125 1296 7027η 1 4 19040η 1 4 2225η 1 4 3325η 1 4 −1712η 1 4 335η 1 4 5 1 1 12 1 6 35η 1 5 98η 1 5 25η 1 5 25η 1 5 −10η 1 5 2η 1 5 η11 = 1 4702924800 ; η 1 2 = 1 9185400 ; η 1 3 = 2 1148175 ; η 1 4 = 125 188116992 ; η 1 5 = 1 4200 ; table 3. coefficients of α j, j = 2, 3, 4, and β j, j = 0(1)5 in equation (10) k α2 α3 α4 β0 β1 β2 β3 β4 β5 1 1 16 1 72 40501η 2 1 42484η 2 1 −20455η 2 1 9335η 2 1 −5876η 2 1 0 2 1 13 1 18 1646η 2 2 3304η 2 2 −985η 2 2 470η 2 2 −296η 2 2 61η 2 2 3 1 13 1 9 467η 2 3 1328η 2 3 190η 2 3 205η 2 3 −112η 2 3 22η 2 3 4 1 56 25 72 497η 2 4 1444η 2 4 485η 2 4 395η 2 4 −164η 2 4 31η 2 4 5 1 1 12 222η 2 5 648η 2 5 315η 2 5 270η 2 5 −72η 2 5 17η 2 5 η21 = 1 87091200 ; η 2 2 = 1 680400 ; η 2 3 = 1 42525 ; η 2 4 = 125 3483648 ; η 2 5 = 1 8400 ; table 4. coefficients of α j, j = 3, 4, and β j, j = 0(1)5 in equation (11) k α3 α4 β0 β1 β2 β3 β4 β5 1 1 16 5561η 3 1 37504η 3 1 −16325η 3 1 7295η 3 1 −4576η 3 1 941η 3 1 2 1 13 1843η 3 2 5032η 3 2 −830η 3 2 515η 3 2 −328η 3 2 68η 3 2 3 1 23 254η 3 3 776η 3 3 395η 3 3 235η 3 3 −104η 3 3 19η 3 3 4 1 56 269η 3 4 800η 3 4 575η 3 4 475η 3 4 −128η 3 4 25η 3 4 5 1 1 239η34 696η 3 4 600η 3 4 555η 3 4 −24η 3 4 34η 3 4 η31 = 1 3628800 ; η 3 2 = 1 113400 ; η 3 3 = 2 14175 ; η 3 4 = 25 145152 ; η 3 5 = 1 4200 ; table 5. coefficients of α4 and β j, j = 0(1)5 in equation (12) . k α4 β0 β1 β2 β3 β4 β5 1 1 4991η41 12824η 4 1 −4365η 4 1 1885η 4 1 −1176η 4 1 241η 4 1 2 1 287η42 1248η 4 2 245η 4 2 45η 4 2 −32η 4 2 7η 4 2 3 1 43η43 112η 4 3 180η 4 3 155η 4 3 −48η 4 3 8η 4 3 4 1 43η44 120η 4 4 175η 4 4 225η 4 4 8η 4 4 5η 4 4 5 1 13η45 32η 4 5 55η 4 5 55η 4 5 32η 4 5 13η 4 5 η41 = 1 86400 ; η 4 2 = 1 5400 ; η 4 3 = 1 675 ; η 4 4 = 5 3456 ; η 4 5 = 1 200 ; y 0µ = (yn+ 16 , yn+ 13 , yn+ 23 , yn+ 56 , yn+1) t ... y 5µ = (y (v) n+ 16 , y(v) n+ 13 , y(v) n+ 23 , y(v) n+ 56 , y(v)n+1) t y 0µ−1 = (yn− 56 , yn− 23 , yn− 13 , yn− 16 , yn) t ... y 5µ−1 = (y (v) n− 56 , y(v) n− 23 , y(v) n− 13 , y(v) n− 16 , y(v)n ) t fµ = ( fn+ 16 , fn+ 13 , fn+ 23 , fn+ 56 , fn+1) t fµ−1 = ( fn− 56 , fn− 23 , fn−13 , fn− 16 , fn) t hence, as h → 0 the method in equation (16) becomes a0yµ = a1yµ−1 (17) where a0 is the identity matrix of order 25, a0 = i25, and a1 is 4 olaiya et al. / j. nig. soc. phys. sci. 4 (2022) 797 5 table 6. absolute error for problem 1 x = 1.0 x = 1.5 t 1shbi mvia-i 1shbi mvia-i 0.01 0.00000 0.00000 2.14572 × 10−18 0.00000 0.02 1.11257 × 10−18 8.88178 × 10−16 2.22141 × 10−18 8.88178 × 10−16 0.03 8.47894 × 10−17 1.19904 × 10−14 4.71125 × 10−17 1.95399 × 10−14 0.04 5.41381 × 10−17 8.79296 × 10−14 3.12412 × 10−17 1.43884 × 10−13 0.05 9.12414 × 10−16 4.19229 × 10−13 1.49787 × 10−17 6.90114 × 10−13 comparing absolute errors of 1shbi with the mvia-i in [11] for example 1. table 7. error for example 2 x yex yapprox error 0.12566 -0.07426755 -0.07426785 2.9720e-7 0.25132 0.05130957 0.05130487 4.7001e-6 0.31415 0.11392287 0.11391146 1.1407e-5 0.43982 0.23757397 0.23753066 4.3304e-5 0.50265 0.29812884 0.29805541 7.3430e-5 0.62831 0.41551891 0.41534181 1.7710e-4 0.75398 0.52644076 0.52607803 3.6273e-4 0.81681 0.57893577 0.57843927 4.9650e-4 0.94247 0.67698465 0.67611561 8.6903e-4 1.00530 0.72216309 0.72104522 1.1178e-3 0 ≤ x ≤ 2π, t = 1, h = 0.0628319. table 8. maximum error for problem 2 at t = 1 method n l1 l2 l∞ dgfem 10 0.11e-1 0.12e-1 0.17 20 0.39e-2 0.43e-2 0.61e-2 40 0.12e-3 0.14e-3 0.19e-3 80 0.36e-5 0.40e-5 0.57e-5 1shbi 10 1.52e-6 3.23e-6 1.17e-7 20 2.15e-8 3.24e-6 3.21e-8 40 5.18e-8 2.79e-9 5.27e-8 80 2.32e-10 4.71e-10 8.11e-10 dgfem is discontinuous galerkin finite element method developed in [32]. the p-norm in table 8 is given as lp = ||yi − y(xi)||p =  inf∑ i=1 |yi − y(xi)| p  1 p l∞ = ||yi − y(xi)||∞ = max 1≤i≤n {|yi − y(xi)|} a 25 × 25 matrix given by a11 a22 a33 a44 a55  with the a11, . . ., a55 being 5 × 5 matrices respectively, given yexact ynumerical error figure 2. the shaded regions shows the surface plots for the numerical solution, the analytic solution and the residue (error) for example 1. by aii =  0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0  , i = 1, . . . , 5 5 olaiya et al. / j. nig. soc. phys. sci. 4 (2022) 797 6 table 9. numerical results with t = 1 for example 3 x exact approximate error 0.1 1.999999999998e-10 1.999999999998e-10 0.000000000000000 0.2 1.999999999992e-10 1.999999999991e-10 1.292469707114105e-25 0.3 1.999999999982e-10 1.999999999981e-10 3.618915179919496e-25 0.4 1.999999999968e-10 1.999999999967e-10 6.720842476993354e-25 0.5 1.999999999950e-10 1.999999999949e-10 1.214921524687259e-24 0.6 1.999999999928e-10 1.999999999927e-10 1.757758801675183e-24 0.7 1.999999999902e-10 1.999999999901e-10 2.429843049374518e-24 0.8 1.999999999872e-10 1.999999999871e-10 2.998529720504725e-24 0.9 1.999999999838e-10 1.999999999837e-10 3.127776691216136e-24 1.0 1.999999999800e-10 1.999999999799e-10 2.403993655232236e-24 table 10. maximum error for example 3 at t = 1 method ga with rk mq with rk imq with rk 1shbi l2 1.9774e-21 5.9692e-12 1.9774e-21 5.96847e-24 l∞ 1.2705e-21 3.2896e-12 1.2705e-21 3.12778e-24 the matrices aii have the characteristic polynomial |a j j−λi5| = 0, for j = 1, . . . , 5, that is, λ4(λ − 1) = 0. the characteristic polynomial has the root λr = 0, for r = 1, . . . , 4 and λ5 = 1. the method is zero-stable, since the roots of the characteristic polynomial have one unity while others are zero (see [29]). for convergence, the following theorem apply. theorem 2.1. [30]. a linear multistep method is convergent provided it is consistent (having order p ≥ 1) and zero-stable. the derived method has p = 6, and equally zero-stable and hence it is convergent. 2.3. linear stability analysis the linear stability for any given h > 0 is concern with the behaviour of the underlined problem not just the numerical method. here, the stability properties for the intending numerical method is analyzed by considering the linear test equation for µ > 0 of the form y(v) = −µ5y (18) the test in equation (18) is appropriate since it has a bounded solution as µ → 0. here, µ is the frequency. set δ = µh and then applying to the test equation in (18), and after some manipulations, the following equation results zµ = m(δ)zµ−1, δ = µh (19) where m(δ) = a0 + δb0 (a1 −δb1) is the required stability matrix. the amplification matrix in equation (19) results on the dominant eigenvalues given as m(δdominant) = 6(−2)4/531/5 ( −1 − 4δ1/6 + 5δ1/3 − 5δ2/3 + 4δ5/6 −δ )1/5 ( 11 − 56δ1/6 + 285δ1/3 + 285δ2/3 − 56δ5/6 + 11δ )1/5 (20) having the stability region in the figure 1. these roots (containing the real and imaginary parts) are plotted and as shown in figure 1. here we study the boundedness of the solution under consideration through the eigenvalues of the stability matrix m(δ) for which these eigenvalues δ is such that |δ| < 1 for the derived method to be stable. if δ is real, then the absolute stability region is reduced to a real interval which is called an interval of stability, see [31]. figure 1 shows the stability region for the proposed method, with the stability interval (0, 8.7893). 2.4. implementation the system formed by the semi-discretization given by equation (4) is solved using the unified block scheme formed by equations (8)-(12) with 5n equations and 5n unknowns solve simultaneously, whose solution provides a set of approximate values of (1) using codes written in mathematica 12.0, enhanced by the feature nsolve[] for linear problems while nonlinear problems were solved by newton’s method enhanced by the feature findroot[]. following [20], the following algorithm summarizes the computational procedures. we begin by noting that the solution of the problem in equation (1) is sought in the subintervals dn = a = x0 < x1 < .. . < xn = b and dm = c = t0 < t1 < .. . < tm = d, where h = b−a n , k = d−cm are constant step-size. step 1: use the block method setting n = 0, m = 0, to obtain v1 on the rectangle [y0,m, y1,m](n,m)∈[a,b]×[c,d], similarly, for, n = 1 so that v2 is obtained on the rectangle [y1,m, y2,m](n,m)∈[a,b]×[c,d], and on the rectangle [y3,m, y4,m](n,m)∈[a,b]×[c,d] . . . [yn=n−1,m, yn=n,m](n,m)∈[a,b]×[c,d] for m = 0, 1, 2, . . . , m we thus obtain v3, v4, . . . , vd. step 2: solve the unified block given by the system v1 ⋃ v2 ⋃ , · · · , vd obtained in step 1. step 3: the solution of equation (1) is approximated by the solutions in step 2 as yn,m = [y(x1, t); y(x2, t) . . . y(xn, t)]t for t > 0, n = 1, 2, . . . n. 6 olaiya et al. / j. nig. soc. phys. sci. 4 (2022) 797 7 yexact ynumerical error figure 3. surface plots showing the numerical solution, the analytic solution and the residue for example 2. 3. numerical examples in this section, we test the efficiency of the derived method and thus compare only with the exact solution of the given problem extracted from literature. example 1.. consider the fifth order kdv equation, yt + yyx − yyxxx + yxxxxx = 0 y(x, 0) = ex, } (21) with exact solution y(x, t) = ex−t. after discretization of the ”t” variable, we obtain, ym+1 − ym−1 (2∆t) + ym dym d x − ym d3ym d x3 + d5ym d x5 = gm, x ∈ [0, 1] m = 1, . . . m − 1 (22) figure 4. surface plots for the numerical solution for example 4 where ∆t, tm, m = 0, 1, . . . , m, y, ym(x) ≈ y(x, tm), and gm = 0, expressed as y(v) = f (x, y, y′, y′′′) = ay + g (23) a is a k by k matrix. to show the efficiency of 1shb1, the above table 6 shows the absolute error for different values of t the values x = 1.0 and 1.5 respectively. it can be observe that the 1shbi outperformed mvia-i for the example considered. the figure 2 shows a good agreement of the numerical and exact solutions. example 2.. consider a time dependent biharmonic fifth order equation discussed in [32]. yt + yx + yxxxxx = 0, 0 ≤ x ≤ 2π, t ≥ 0 y(x, 0) = sin (x), y(0, t) = y(2π, t)  (24) the analytic solution for this problem is y(x, t) = sin(x − 2t). upon discretization of the time variable, we obtain, ym+1 − ym−1 (2∆t) + dym d x + d5ym d x5 = gm, 0 ≤ x ≤ 1, m = 1, . . . m − 1 (25) 7 olaiya et al. / j. nig. soc. phys. sci. 4 (2022) 797 8 where ∆t, tm, m = 0, 1, . . . , m, y, ym(x) ≈ y(x, tm), g and gm are expressed in the form y(v) = f (x, y, y′, y′′, y′′′, y(iv)) = ay + g (26) a is a k × k matrix (k = (n − 1)(m − 1)) and gm = 0. table 7 compares the errors obtained for example 2. figure 3 shows the representation of the numerical solution, analytical solution and residue (errors), for example 2. table 8, for different subinterval n, shows the maximum errors l∞ and the lp with p = 1, 2 − norm obtained, in comparison with the 1shbi and the method in [32]. this shows the superiority of the 1shbi. example 3.. consider the nonlinear fifth order kdv equation called lax’s cases of generalized kdv equation discussed [33]. yt + 45y2yx + 15yxyxx + 15yyxxx + yxxxxx = 0, 0 ≤ x ≤ 1, t ≥ 0 y(x, 0) = 2k2 sec h2(k(x − x0))  (27) the exact solution for this problem is y(x, t) = 2k2 sec h2(k(x − 16k4t − x0)) upon discretization of the time variable, we obtain, ym+1 − ym−1 (2∆t) + 45ymym dym d x + 15 dym d x d2ym d x2 + 15ym d3ym d x3 + d5ym d x5 = gm, 0 ≤ x ≤ 1, m = 1, . . . m − 1 (28) where ∆t, tm, m = 0, 1, . . . , m, y, ym(x) ≈ y(x, tm), g and gm are expressed in the form y(v) = f (x, y, y′, y′′, y′′′, y(iv)) = ay + g (29) a is as expressed in example 1 and gm = 0. in [33], the authors presented a numerical solution of equation (27) using a meshless method of lines. this method uses meshless, multiquadric (mq), inverse multiquadric (imq) and gaussian (ga) as radial basis function (rbf) for spatial derivatives and runge-kutta method as a time integrator. the constant step-size used for table 9 and table 10 is h = 0.1. figure 4 shows the graphical representation with surface plot for the numerical solution (shaded region), analytical solution (shaded region) and residue or error (shaded region), for example 3. the method presented in [33] for the numerical solution of example 4 uses a meshless method of lines. this method uses meshless, multiquadric (mq), inverse multiquadric (imq) and gaussian (ga) as radial basis function (rbf) for spatial derivatives and runge-kutta method as a time integrator. 4. conclusion in this work, a one step hybrid block integrator (ishbi) was derived consisting of continuous linear multistep methods. this was used to solve certain fifthorder pdes subject to appropriate initial-boundary conditions. to show the robustness of method derived, fifth-order kdv pdes were solved were transformed into a system of fifth-order odes using the method of lines. from the numerical experiments performed, it’s evident that the 1shbi performed well in terms of accuracy as compared to exact solutions of the problems considered in literature. acknowledgments we thank the referees for the positive enlightening comments and suggestions, which have greatly helped us in making improvements to this paper. references [1] m. kazeminia, s. soleimani-amiri & s.a. zahedi, “exact and numerical solutions for nonlinear higher order modified kdv equations by using variational iteration method”, adv. stud. theor. phys. 4 (2010) 437. 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[31] m. a. rufai & h. ramos, “numerical solution of second-order singular problems arising in astrophysics by combining a pair of one-step hybrid block nyström methods”, astrophys space sci. 365 (2020) 96, https://doi.org/10.1007/s10509-020-03811-8 [32] y. cheng & c shu, “a discontinuous galerkin finite element method for time dependent partial differential equations with higher order derivatives”, mathematics of computation 77 (2008) 699. [33] s. t. mohyud-din, e. negahdary & m. usman, “a meshless numerical solution of the family of generalized fifth-order korteweg-de vries equations”, mathematics faculty publications paper 10 (2012), htt p : //ecommons.udayton.edu/mth f acpub/10. 9 j. nig. soc. phys. sci. 4 (2022) 823 journal of the nigerian society of physical sciences mathematical modeling of waves in a porous micropolar fibrereinforced structure and liquid interface augustine igwebuike anyaa,∗, uko ofeb, aftab khanc agst-mathematics division, veritas university abuja, bwari-abuja, nigeria bdepartment of pure and applied physics, veritas university abuja, bwari-abuja, nigeria cdepartment of mathematics, comsats university islamabad, park road chak shahzad, 44000 islamabad, pakistan abstract the present investigation envisages on the mathematical modeling of waves propagating in a porous micropolar fibre-reinforced structure in a half-space and liquid interface. the harmonic method of wave analysis is utilized, such that, the reflection and transmission of waves in the media were modelled and it’s equations of motion analytically derived. it was deduced that incident longitudinal wave in the solid structure yielded four reflected waves given as; quasi–p wave (qld), quasi–sv wave, quasi–transverse microrotational (qtm) wave and a wave due to voids and one transmitted wave known as the quasi-longitudinal transmitted (qlt) wave. the phase velocity in the liquid medium is independent of angle of propagation as observed. the corresponding amplitude ratios of propagations for both reflected and transmitted waves are analytically derived by employing snell’s law. the model would prove to be of relevance in the understanding of modeling of the behavior of propagation phenomena of waves in micropolar fibre-reinforecd machination systems resulting in solid/liquid interfaces especially in earth sciences and in particular seismology, amongst others. doi:10.46481/jnsps.2022.823 keywords: micropolar, fibre reinforced, reflection/transmission, voids/porosity, liquid interface article history : received: 20 may 2022 received in revised form: 17 june 2022 accepted for publication: 21 june 2022 published: 20 august 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: t. latunde 1. introduction researchers in solid mechanics and in particular elastodynamics have always pursued for ways in trying to decipher the behaviors of certain disturbances in materials caused either naturally or artificially, especially in disciplines such as; engineering, structural designing, seismology, material sciences, and geophysics among others. although of particular interest in ∗corresponding author tel. no: +2347038933954 email address: anyaaugustineigwebuike@gmail.com, anyaa@veritas.edu.ng (augustine igwebuike anya) this study is the porous micopolar fibre reinforced materials. fibre reinforced materials play a collective host to the strength of materials used by construction engineers, physicists, material scientists etc., owing to their high tensile strength, low weight, and efficacy of fibre-reinforcement cum flexibility of usage [1]. fibre reinforced media tends to be similar to another important type of material known as the orthotropic material; which could be considered in the investigation of elastodynamic models. some of these materials possess micro-rotation and translation of local points as given by erigen [2], theory of micropolar elasticity. thus, in describing the propagation of waves 1 anya et al. / j. nig. soc. phys. sci. 4 (2022) 823 2 in such materials, a mathematical model is accompanied along with certain physical properties and parameters such as; voids in the material [3], pre-stress, as the case maybe. voids are pores in a material which are taken as a volume fraction fields in equations describing such media. mathematical models in terms of equations used for the study of reflection and transmission at a plane half-space of elastic media were initially introduced by knott [4] and subsequently modified by jeffreys [5], gutenberg [6], etc. in a similar vein, reflection and transmission coefficients in fluid-saturated porous media and boundary surfaces, have received great attention in literature [7-11]. extensive works on fiber reinforced media and reflection of waves in an elastic media with some other physical properties of rotation, gravity, thermal effects etc., were also conducted [12-18]. reflections of waves in a micropolar fibre reinforced medium with other physical properties of magneto-thermoelastic effects, rotation, etc., were also carried out [19-20]. furthermore, boundary surfaces of some materials are plane, grooved or entirely of different shapes in nature. grooved boundary surface could be visualized as a series of parallel furrows and ridges whose encounter in mechanical propagation of wave results to several effects especially across interfaces. interestingly, some authors had worked on this concept of corrugated boundaries and other related wave propagation phenomena [2129]. also, certain discussions on materials [30-33] could aid understanding of material characterizations. in spite of these contributions, the present study is also of particular interest, as it seeks to investigate the propagation of waves in plane boundary of a porous micropolar fibre reinforced solid structure and liquid interface. also the motivation of the study stems from the fact that classical theories have short comings in modeling solid structure interactions and its understanding of behaviors for any given impact on it; hence micropolar theory of elasticity by eringen [2], suggests the basic assumptions and subsequently, the consideration of 9x9 nonsymmetric material matrix of micropolar fibre-reinforcement. in studying this, we employed two concepts of wave propagation analyses called the normal mode analysis or harmonic analysis and snell’s law. due to the composition of the media, when p-wave is incident on the solid structure, four waves are reflected in the solid structure while only quasi-longitudinal transmitted (qlt) wave is transmitted in the liquid medium. and by taken continuity conditions at the interface, the reflection and transmitted wave’s propagation coefficients are derived. 2. mathematical formulation of the problem the constitutive relations for a micropolar fibre–reinforced elastic anisotropic solid with voids, considering some existing works [2-3], [34-35] follow as: σi j = ni jmn emn + s i jmnψmn + ζφδi j, (1) mi j = n jimn emn + s mn jiψmn, (2) we define the deformations and wryness tensors as: ei j = u j,i + ε jimφ∗m, ψmn = φ ∗ m,n , i = j = m = n = 1, 2, 3. (3) the balance laws in the absence of body forces are presented below as: σi j,i = ρü j, (4) mi j,i + ε jmnσmn = ρjφ̈ j ∗, (5) ξ1(φ,ii) −ωoφ−$φ̇− ζ(ui,i) = ρκφ̈. (6) where mi j, σi j, φ∗j, u j and φ are the couple stress tensor, stress tensor, microrotation vector, displacement vector and volume fraction field respectively. ρ is the density of the solid medium; ζrepresents the voids parameter; j is the microinertia, ni jmn, n jimn are constants of material characterization such that non symmetric properties of ni jmn, n jimn and s i jmn are observed. ε jim is the levi–civita tensor and δi j, is the kronecker–delta function. the given index after comma denotes partial derivative with respect to coordinate space and the superscript dot stipulates partial derivative with respect to time. we tried to consider the deformation in x1 x3–plane. this is such that the displacements; u1 , u3 , 0, while u2 = 0. thus, this implies that the micro-rotation; φ∗ = (0,φ∗2, 0). repeated indexes of einstein summation convention is used. considering the fact the tensors are not symmetric in micropolar solid, in the sense that, a 9x9 matrix of the solid material characterization is utilized, eqs. (4-6) in components form are given as: n1u1,11 + (n2 + n3)u3,13 + n6u1,33 +n∗1φ ∗ 2,3 + ζφ,1 = ρü1, (7) n4u3,11 + n2u1,13 + n5u3,33 + n4φ ∗ 2,1 + ζφ,3 = ρü3, (8) s 1φ ∗ 2,11 + s 2φ ∗ 2,33 + n ∗ 2φ ∗ 2 + χ1u1,3 + χ2u3,1 = ρjφ̈ ∗ 2, (9) ξ1(φ,ii) −ωoφ−$φ̇− ζ(ui,i) = ρκφ̈, (10) where n1 = (λ + 2α + β + 4µl − 2µt ), n2 = (λ + α), n3 = 2µt , n4 = 2µl, n5 = (λ + 2µt ), n6 = (2µl −µt ), n∗1 = (3µt − 2µl), n ∗ 2 = (3µt − 4µl), χ1 = n4 − (n3/2),χ2 = n4 − n3 3. analytical solution of the problem we consider a micropalar fibre–reinforced structure occupying the half–space x3 < 0and a liquid medium occupying the half–spacex3 > 0. thus, the two media constitutes an interface at the boundary. hence, an assumption is made for the displacements in the media as taken below: u1 = pe i{k (x j p j )−ωt}, (11) 2 anya et al. / j. nig. soc. phys. sci. 4 (2022) 823 3 u3 = qe i{k (x j p j )−ωt}, (12) φ∗2 = φ ∗ei{k (x j p j )−ωt}, (13) φ = φ0e i{k (x j p j )−ωt}, j = 1, 2. (14) where p, q, φ∗ and φ0 are amplitudes of the wave displacements respectively. c = ωk is the phase velocity of the wave, k is the wave number, and ω is the angular velocity of the wave. introducing eqs. (11-14) into eqs. (7-10), yields the following equations below: {k2 d1 − k2c2ρ}p + {k2(n2 + n3) p1 p3}q −ikn∗1 p3φ ∗ − ikζ p1φ0 = 0 (15) {n2k2 p1 p3}p + {k2 d2 − c2k2ρ}q in4k p1φ∗ − iζk p3φ0 = 0, (16) −{ikχ1 p3}p −{ikχ2 p1}q +{k2 d3 − n2 −ρjk2c2}φ∗ = 0, (17) {i ζk p1}p + {iζk p3}q +(ξ1k2 + ω0 −$ikc −ρκk2c2)φ0 = 0, (18) where d1 = n1 p21 + n6 p 2 2, d2 = n4 p 2 1 + n5 p 2 2, d3 = s 1 p 2 1 + s 2 p23,p1 = s inα and p2 = cosα. for non–trivial solution, eqs. (15-18) gives the quartic polynomial below: γ4 + e1γ 3 + e2γ 2 + e3γ + e4 = 0 . (19) here γ = k2. this means that the characteristic eq. (19) with complex coefficients e1,e2,e3,and e4 (see appendix) gives four complex roots. thus, the four waves propagate with complex phase velocities: c1, c2, c3 and c4 in the solid medium corresponding to the wave numbers k1, k2, k3and k4 respectively. hence, the two dimensional model in the x1 x3plane of the micropolar fibre–reinforced solid half space, have four waves; quasi–p wave (qld), quasi–sv wave, quasi–transverse microrotational (qtm) wave and wave due to voids travelling in the solid medium. following singh’s work [36], for the liquid medium, consider n1 = n2 = n5 = λ5,ρ = ρ5,φ∗2 = χ2 = χ1 = ζ = n4 = n3 = φ = 0 into eqs. (7-10), we obtain the equation below for a non–trivial solution: c∗4ρ2 − c∗2λ5ρ ( p21 + p 2 3 ) = 0 (20) the roots of the characteristics eq. (20) for the liquid medium can simply be represented asc∗ = ± √ λ5( p21 + p 2 3)/ρ5. this shows that one phase velocity; c5,corresponding to the wave numberk5is taken. thus, in the liquid medium, one of the roots of eq. (20) will be negligible; this entails that only quasi– longitudinal transmitted (qlt) wave can propagate. also observe that p21 + p 2 3 = 1. hence, the phase velocity in the liquid medium will be independent of angle of propagation. figure 1. schematic of the problem showing micropolar fibre–reinforced solid half-space with voids and liquid interface 4. reflection/transmission of waves and the geometry of the problem let us consider the propagation of p-wave (longitudinal wave) incident on a micopolar fibre-reinforced half-space in the x1 x3-plane with voids such that the boundary is plane in nature and it’s in an interface with a liquid medium. the geometry is demonstrated in figure 1 below. also any one of the four waves can be chosen as incident wave. figure 1 shows that when quasi–p wave(p0) is incident at micropolar fibre–reinforced anisotropic solid and liquid interface, there exist four reflected waves as quasi–p (p1) or qld, quasi–sv (p2) or qtd, quasi–tm (p3) and wave due to voids (p4)with their angles as α0,α1, α2,α3,α4 respectively. also the transmitted wave exists as; quasi–longitudinal transmitted (qlt); (p5)wave, with angleα5. 5. boundary conditions and results the following are the boundary conditions taken at the interface of the micropolar fibre reinforced solid with voids and liquid: 1. stresses at the common interface are continuous i.e. σα33 = σl33, σ α 13 = σ l 11,atx3 = 0. 2. the conditions due to microrotation and voids takes the form: (a) mα32 = 0, observe that m32 = 0 ⇒ φ ∗ 2,3 = 0, and (b) φα ,3 = 0, atx3 = 0, respectively. 3. normal displacements are continuous at the common interface: uα3 = u l 3, at x3 = 0. 3 anya et al. / j. nig. soc. phys. sci. 4 (2022) 823 4 we choose the displacement components, micro-rotation vectors and the volume fraction field as: uα1 = aα d α 1 e iµα, uα3 = f αaα dα1 e iµα, φ∗2 = ikαg αaα dα1 e iµα, φα = hαaα dα1 e iµα, ul1 = al d l 1e iµl, ul3 = i l al dl1e iµl.  , (21) where µα = kα(x1 pα1 + x3 p α 3 − cαt), α = 0 correspond to incident wave, α = 1, 2, 3, 4 corresponds to reflected waves in the solid medium andµ` = k`(x1 p`1 + x3 p ` 3 − c`t), where ` = 5 corresponds to quasi-longitudinal transmitted (qlt) wave in the inviscid liquid medium. also, the coupled relations fα, gαand hαare obtained from eqs (15-18) for the solid medium, i.e. fα = −fα1 /f α 2 , gα = ikα(χ2 pα1 f α + χ1 pα3 )/(k 2 α(d α 3 −ρjc 2 α) − n2), hα = −(kαζi(pα3 f α + pα1 ))/(k 2 α(ξ1 −ρκc 2 α) + ω0 −$ikc). where fα1 = p α 3{(d α 1 −ρc 2 α − n2( p α 1 ) 2)(k2α(d α 3 −ρjc 2 α) − n2) +χ1((pα3 ) 2 n∗1 − ( p α 1 ) 2 n4)}, fα2 = p α 1{(χ2(p α 3 ) 2 n∗1 − (p α 1 ) 2 n4) + ((pα3 ) 2(n2 + n3) −dα2 + ρc 2 α)(k 2 α(d α 3 −ρjc 2 α) − n2)}, dα1 = n1( p α 1 ) 2 + n6(pα3 ) 2, dα2 = n4(p α 1 ) 2 + n5( pα3 ) 2, and dα3 = s 1( p α 1 ) 2 + s 2( pα3 ) 2, similarly, the relation i` for the liquid medium assumes the form: i` = {λ`(( p`1) 2 − p`1 p ` 3) −ρ`c 2 ` }/λ`((p`3) 2 −p`1 p ` 3) −ρ`c 2 ` , ` = 5. using eq. (21) into the boundary conditions, a system of equation is obtained after using snell’s law i.e., the coefficients;ai j of the system are made possible by using snell’s law such that:k0 p (0) 1 = k1 p11 = k2 p 2 1 = k3 p 3 1 = k4 p 4 1 = k5 p 5 1 = k, and k0c0 = k1c1 = k2c2 = k3c3= k4c4 = k5c5 = ω. thus, the system obtained takes the form: ai jz j = bi, i = j = 1, 2, 3, 4, 5. (22) where a1r = kr d r 1{(n2 p r 1 + n5 p r 3 f r )}/k0d 0 1{(n2 p 0 1 + n5 p 0 3 f 0)}, a1` = −k`d ` 1λ`{( p ` 1 + p ` 3 i `)}/k0d 0 1{(n2 p 0 1 + n5 p 0 3 f 0)}, a2r = kr d r 1{(n4 f r pr1 + g r )}/k0d 0 1{(n4 f 0 p01 + g 0)}, a2` = −k`d ` 1λ`{(i ` p`1)}/k0d 0 1{(n4 f 0 p01 + g 0)}, a3r = k 2 r d r 1 p r 3g r/k20 d 0 1 p 0 3g 0, a3` = 0, a4r = kr p r 3 h r dr1/k0 p 0 3 h 0d01, a4` = 0 a5r = f r dr1/f 0d01, a5` = −i `d`1/f 0d01. hence, z j is the reflection and transmission coefficients, and the amplitude ratios of the reflected and transmitted waves arezi = |ai/a0| , bi = −1,i = j = 1, 2, 3, 4, 5, r = 1, 2, 3, 4, and` = 5. also observe that k0=k1, c0=c1and the speed of the waves; c0, c1, c2, c3, c4, and c5 depends upon the material parameters. the components of propagation and unit displacement vectors are as follows: p(0)1 = s inα0, p (0) 3 = cosα0, d (0) 1 = s inα0, d(0)3 = cosα0, p(1)1 = s inα1, p (1) 3 = −cosα1, d (1) 1 = s inα1, d(1)3 = −cosα1, p(2)1 = s inα2, p (2) 3 = −cosα2, d (2) 1 = cosα2, d(2)3 = s inα2, p(3)1 = s inα3, p (3) 3 = −cosα3, d (3) 1 = cosα3, d(3)3 = s inα3, p(4)1 = s inα3, p (4) 3 = −cosα4, d (4) 1 = cosα4, d(4)3 = s inα4, p(5)1 = s inα5, , p (5) 3 = cosα5, d (5) 1 = s inα5, d(5)3 = cosα5. (23) 6. conclusion this article dealt wholly on the formulation and the analytical solution of waves in a micropolar fibre-reinforced medium having pores and interfaced with liquid medium. four reflected waves namely; quasi–p or qld, quasi–sv or qtd, quasi–tm and wave due to voids traveling in the solid medium were found while quasi–longitudinal transmitted (qlt) was found traveling in the liquid medium due to the negligible viscosity of the liquid. the analytical derivation of the amplitude ratios of both reflected and transmitted waves respectively, were derived and presented. also, we observed that the phase velocity in the liquid medium is independent of the angle of propagation and the reverse is the case in the solid medium such that the phase velocity of propagation were found to be dependent on the angle of propagation. thus, it is worthy of note to state that the significance of the model should prove useful to new researchers, scientists, material scientists working to ascertain models that could be imperative in predicting some seismological analyses in some solid/liquid structures in terms of propagating phenomena. future works to this study-“mathematical modeling of waves in a porous micropolar fibre reinforced structure and liquid interface”, could incorporate rotating viscoelasticity of the solid/liquid media, grooved boundary conditions and non-local effects to the model. acknowledgments the authors are very grateful to the editors and anonymous referees for their careful reading of this manuscript and helpful suggestions. references [1] a. j. m. spencer, deformations of fibre-reinforced materials, oxford university press london, (1972). 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[36] b. singh, “reflection and transmission at the interface between a liquid and micropolar viscoelastic solid with stretch”, sadhana-academy proceeding in engineering sciences 25 (2000) 589. appendix e1 = (d1(d3 ( i2ζ2 p23 −ω 2ρξ1 ) − i2 n4 p21ξ1χ2 −d2( ( jω2ρ + n2 ) ξ1 + d3 ( iω$ + ω2κρ−ω0 ) )) +d2 ( d3 ( i2ζ2 p1 p3 −ω2ρξ1 ) − i2 p23ξ1χ1 (n1) ∗ ) +p1 p23(−d3(i 2ζ2 n3 p1 − n22 p1 (ω (i$ + ωκρ) −ω0) +n2(i2ζ2 p3 + p1(i2ζ2 + n3 (−ω (i$ + ωκρ) + ω0)))) +p1ξ1(n32 + n 2 2 ( jω2ρ + n3 ) + i2 n3 n4χ1+ n2  jω2ρn3 +i2 ( n4χ1+ χ2 (n1)∗ ) )))/d3 (d1 d2 − n2 (n2 + n3) p21 p23) ξ1, e2 = (−i2ω2ζ2ρd3 p1 p3 − i2ω2ζ2ρd3 p23 + i 2 jω2ζ2 ρn2 p21 p 2 3 + i 2ζ2 n22 p 2 1 p 2 3 − ijω 3$ρn22 p 2 1 p 2 3 −jω4κρ2 n22 p 2 1 p 2 3 − iω$n 3 2 p 2 1 p 2 3 −ω 2κρn32 p 2 1 p 2 3 +i2 jω2ζ2ρn3 p21 p 2 3 + i 2ζ2 n2 n3 p21 p 2 3 −ijω3$ρn2 n3 p21 p 2 3 − jω 4κρ2 n2 n3 p21 p 2 3 −iω$n22 n3 p 2 1 p 2 3 −ω 2κρn22 n3 p 2 1 p 2 3 +i2 jω2ζ2ρn2 p1 p33 + i 2ζ2 n22 p1 p 3 3 + ω 4ρ2 d3ξ1 −i3ω$n2 n4 p21 p 2 3χ1 − i 2ω2κρn2 n4 p21 p 2 3χ1 −i3ω$n3 n4 p21 p 2 3χ1 − i 2ω2κρn3 n4 p21 p 2 3χ1 +i4ζ2 n4 p1 p33χ1 − i 4ζ2 n4 p31 p3χ2 + i 2ω2ρn4 p21ξ1χ2 +jω2ρn22 p 2 1 p 2 3ω0 + n 3 2 p 2 1 p 2 3ω0 + jω 2ρn2 n3 p21 p 2 3ω0 +n22 n3 p 2 1 p 2 3ω0 + i 2 n2 n4 p21 p 2 3χ1ω0 +i2 n3 n4 p21 p 2 3χ1ω0 + d1(−i 2 jω2ζ2ρp23 − i 2ζ2 n2 p23 +jω4ρ2ξ1 + ω2ρn2ξ1 + i3ω$n4 p21χ2 +i2ω2κρn4 p21χ2 + ω 2ρd3 ( iω$ + ω2κρ−ω0 ) +d2 ( jω2ρ + n2 ) (ω (i$ + ωκρ) −ω0) −i2 n4 p21χ2ω0) − i 4ζ2 p43χ1 (n1) ∗ + i2ω2ρp23ξ1χ1 (n1) ∗ 5 anya et al. / j. nig. soc. phys. sci. 4 (2022) 823 6 +i4ζ2 p21 p 2 3χ2 (n1) ∗ − i3ω$n2 p21 p 2 3χ2 (n1)∗ − i2ω2κρn2 p21 p 2 3χ2 (n1) ∗ + i2 n2 p21 p 2 3χ2ω0 (n1) ∗ +d2(−i2ζ2 ( jω2ρ + n2 ) p1 p3 +jω4ρ2ξ1 + ω2ρn2ξ1 + ω2ρd3(ω (i$ + ωκρ) −ω0) +i3ω$p23χ1 (n1) ∗ + i2ω2κρp23χ1 (n1)∗ − i2 p23χ1ω0 (n1) ∗))/d3 ( d1 d2 − n2 ( n2+ n3 ) p21 p 2 3 ) ξ1, e3 = −ω2ρ(iω3$ρd3 + ω4κρ2 d3 − i2 jω2ζ2ρp1 p3− i2ζ2 n2 p1 p3 − i2 jω2ζ2ρp23 − i 2ζ2 n2 p23 +jω4ρ2ξ1 + ω2ρn2ξ1 + i3ω$n4 p21χ2 +i2ω2κρn4 p21χ2 + d2 ( jω2ρ + n2 ) ( iω$ + ω2κρ−ω0 ) +d1 ( jω2ρ + n2 ) (ω (i$ + ωκρ) −ω0) −ω2ρd3ω0 −i2 n4 p21χ2ω0 + i 3ω$p23χ1 (n1) ∗ + i2ω2κρp23χ1 (n1) ∗ −i2 p23χ1ω0 (n1) ∗)/d3 ( d1 d2− n2 (n2 + n3) p21 p 2 3 ) ξ1, e4 = ω4ρ2 ( jω2ρ + n2 )( ω ( i$+ ωκρ ) −ω0 ) /d3 ( d1 d2− n2 (n2 + n3) p21 p 2 3 ) ξ1. 6 j. nig. soc. phys. sci. 4 (2022) 105–116 journal of the nigerian society of physical sciences physicochemical characteristics and toxicity studies of crude oil, dispersant and crude oil-dispersant test media to marine organisms e. p. onokarea, l. o. odokumab, f. d. sikokic, b. m. nziwub, p. o. iniaghe d,∗, j. c. ossaie a world bank africa centre of excellence, centre for oil field chemicals research (ace-cefor), institute of petroleum studies, university of port harcourt, choba, rivers state b department of microbiology, university of port harcourt, nigeria c department of animal and environmental biology, university of port harcourt, nigeria d department of chemistry, federal university otuoke, nigeria. e department of chemistry, delta state university abraka, nigeria. abstract in this study, the physicochemical characteristics of crude oil, dispersant (ecobestr) and crude oil-dispersant testsystems,and their toxicities on representative marine organismswas assessed. the test media included mechanically dispersed crude oil-in-water(mdo) and its water accommodated fraction (waf), chemically dispersed crude oil-in-water (cdo) and its water accommodated fraction (cewaf), the dispersant (d), and a reference toxicant, sodium dodecyl sulphate (sds). these test media were used to carry out toxicity studies on tilapia guineensis, palaeomontesafricanus, and bacteria – heterotrophic bacteria and hydrocarbon utilizing bacteria. physicochemical characteristics of the test media were done using standard methods. the static with renewal bioassay option was employed for toxicity tests involving tilapia guineensis and palaeomontesafricanus, while the static without renewal option was used for microbial bioassays.marine organisms were exposed to the following concentrations: 100%, 50%, 25%, 12.5%, 6.25% and 0% of mdo, cdo, waf, cewaf and d, respectively. the 96 h lc50and toxicity factors were determined. results for physicochemical characteristics of the test media showed that the ph and dissolved oxygen levels were sufficient for sustaining aquatic habitation. pb metal was present in high amounts in d, but relatively low in cdo and cewaf. toxicity data showed that ecobestr was non-toxic to the test organisms relative to sds.the 96h lc50 of mdo, cdo, d, waf, cewaf and sds were 89.8, 225.6, 1891.8, 683.9, 528.1 and 1.37 for t. guineensis, 275.5, 137.7, 3800.9, 76.1, 168.3 and 9.99 for p. africanus; 1658.5, 944.1, 17221.9, 228641.0, 1036319.3 and 3.84 for heterotrophic bacteria, and 250.6, 9544.1, 77.2, 141.4, 12780.8 and 3.6 for hydrocarbon utilizing bacteria. sds exhibited the greatest toxicity, but the dispersant reduced its toxicity by several folds.however, with increased levels of some heavy metalsand polycyclic aromatic hydrocarbons in the test media water, there may be likelihood for bioaccumulation to occur in the tissues of marine organisms. doi:10.46481/jnsps.2022.427 keywords: ecobestr dispersant, tilapia guineensis, palaeomontesafricanus, crude oil, toxicity study article history : received: 04 october 2021 received in revised form: 23 november 2021 accepted for publication: 24 november 2021 published: 28 february 2022 c©2022 journal of the nigerian society of physical sciences. all rights reserved. communicated by: e. etim ∗corresponding author tel. no: +2347035191298 email address: po.iniaghe@gmail.com ( p. o. iniaghe ) 1. introduction marine pollution is defined as “the introduction by man, directly or indirectly, of substances or energy into the marine 105 onokare et al. / j. nig. soc. phys. sci. 4 (2022) 105–116 106 environment (including estuaries) resulting in such deleterious effects as harm to living resources, hazardsto human health, hindrance to marine activities including fishing, impairment of quality for use of seawater and reduction of amenities” [1]. oil spills are inevitable during oil exploration activities, and they are a significant threat to the marine environment. since oil is of a lower density than water, spilled oil floats on surface water. this can cause harm to aquatic organisms, primarily by limiting the amount of dissolved oxygen required for respiration. oil spill response techniques have been developed for remediating the ecological consequences of spilled oil in the marine environment [2]. mechanical methods (skimming) and chemical methods (dispersants and sinking agents) are common methods employed for such purposes. with several shortcomings associated with several cleanup methods, the use of dispersants have been seen as a useful alternative [3]. dispersants are made up of different chemical compositions, and are available in the market. they help to break up crude oil into oil-in-water emulsions, which allows for easy degradation by hydrocarbon utilizing bacteria indigenous to the environment [4]. they also help in diluting crude oil rapidly in the water column, thereby preventing a drift of the oil slick into ecologically relevant shorelines. the use of dispersants is still controversial because, although they help in dispersing the spilled oil quickly, they do not necessarily remove the oil [5]. dispersants could also induce high levels of toxic metals and polycyclic aromatic hydrocarbons in the water column, thereby creating harmful conditions for marine organisms. many of the earlier used dispersant formulations (prior to 1970) were inherently toxic to aquatic organisms and possessed bioaccumulation potentials (these were historically referred to as first and second generation dispersants, respectively). for instance, the aquatic toxicity of corexit dispersants to a range of marine organisms in the united states has been reported [6].the toxicity of these early dispersants led to the development of third-generation dispersants with the belief that they were nontoxic and biodegradable, as against their historical counterparts [7, 8]. the acute toxicity values of some of these third generation dispersants have been reported to range between 190 – 500 mg l−1, as against the earlier formulations, which had acute toxicity values between 20 – 50 mg l−1 [9]. however, the use of these third generation dispersants still needs to be monitored so that they do not cause harm to the environment. the toxicological effects resulting from dispersant use is not a straightforward task, as it requires at least five components: the dispersant, the oil being dispersed, the nature of the exposure, the age and species of the test organism(s). for instance, finasol osr52r, a third generation dispersant was reported to be toxic to sea urchin embryos, and toxicity was enhanced after the dispersant was added [10]. the same dispersant was also determined to be very toxic to juvenile sea bass, while finasol osr51r possessed toxic characteristics to sea urchin embryos [8]. however, nalco-d4106 was reported to significantly reduce the toxicity of crude oil to t. guineensis and d. trispinosa [10, 11]. ecobestr dispersant is a third generation dispersant that is about to be introduced into the nigeria oil industry for cleanup purposes. it is therefore necessary that its toxicity to organisms in the marine environment is evaluated. presently, there are no studies on the toxicities of this dispersant to tropical marine organisms. similarly, most studies do not take into account the properties of the habitat water used in their toxicity studies. the aim of this study was to evaluate the physicochemical characteristics of crude oil, dispersant and crude oil-dispersant test media using ecobestr dispersant, and to study the toxicity of its interaction with crude oil on marine organisms using laboratory-scale experimental approach. 2. research methodology 2.1. 2.1 source of crude oil, dispersant and habitat water samples of bonny light crude oil were obtained from an onshore operational facility situated in the niger delta. ecobestr oil dispersant was obtained from the national oil spill detection and response agency (nosdra), while habitat water was brackish water collected at intervals from off the shores of onneport. 2.2. collection of test organisms test organisms recommended by the department of petroleum resources (dpr) of nigeria were obtained from the african regional aquaculture centre/national institute of oceanography and marine research (arac/niomr) in buguma, rivers state, nigeria. they include fish – tilapia guineensis (tertiary consumer) and crustacean – palaemonetesafricanus (secondary consumer). bacteria (primary consumer) was however isolated from brackish water. these test organisms represented the three trophic levels in the food chain within a tropical marine ecosystem. the fish (body weight ranging between 4.80 g and 6.20 g)were caught with nets and transported to the laboratory in transparent polyethylene bags (air bags) containing habitat water and enough space for air. on reaching the laboratory, organisms were introduced into tanks (1 m × 3 m) containing habitat water for acclimatization. crustaceans were collected during the dry season (january) when the salinity of the river was high for easy acclimatization. they were transported to the laboratory in the early hours of the morning in transparent polyethylene bags (air bags) containing habitat water and enough space for air. heterotrophic bacteria (hb) and hydrocarbon utilizing bacteria (hub) were isolated from brackish water using the spread plate technique on nutrient agar [37]. incubation of cultures was 24 to 48 h for hb and 5-7 days for hub, all at 24 ± 2◦c. 2.3. acclimatization of test organisms 2.3.1. tilapia guineensis due to the difference in salinity between the habitat water and sea water, the test organisms were gradually exposed to increasing levels of salinity. they were first acclimatized in 100% habitat water for 72 h. the acclimatization water was then changed every 72 hwith 50:50 of habitat and sea water, followed by 30:70 of habitat and sea water and lastly, 100% sea water for 14 days [13]. 106 onokare et al. / j. nig. soc. phys. sci. 4 (2022) 105–116 107 2.3.2. palaemonetesafricanus the test organisms were introduced directly into the brackish water, since its habitat was within the salinity range of the brackish water, and they were acclimatized in the laboratory for 14 days at ambient temperature. 2.4. preparation of test media the test media was made up of five different exposure mixtures. these were the mechanically dispersed crude oil, chemically dispersed crude oil, dispersant, the water accommodated fractions of the mechanically dispersed and chemically dispersed crude oil, respectively. there was also a control (habitat water) set up without crude oil or dispersant, as well as a reference toxicant, sodium dodecyl sulphate (sds), which is a historically used toxic dispersant. 2.5. preparation of test media 2.5.1. crude oil preparation crude oil used for the study was bubbled using aerator for 24h at 25oc to mimic external weathering conditions in a tropical environment to reduce oil volume by 10 percent [13]. 2.5.2. mechanically dispersed crude oil mechanically dispersed crude oil (mdo) was prepared using a loading rate of 1g l−1 made up with habitat water [15]. 10 g of crude oil was diluted with 10 l of habitat water in a 15 l pyrex bottle, and agitated with a magnetic stirrer for 30 minutes at medium energy to mimic field conditions. five different concentrations were prepared (i.e. 100%, 50%, 25%, 12.5% and 6.25%, respectively). 2.5.3. chemically dispersed crude oil chemically dispersed crude oil (cdo) (containing 10 g of crude oil and 1g of dispersant) was obtained by adding each components respectively into 10 l of habitat water in a 15 l pyrexbottle and agitated for 30 minutes using a magnetic stirrer [14]. five different concentrations were also prepared. 2.5.4. dispersant each gram of dispersant used was mixed with 1 l of water in a glass beaker using a magnetic stirrer for 30 minutes [2]. five different concentrations were also prepared. 2.5.5. water accommodated fractions the water accommodated fractions were prepared according to [17]. the water accommodated fraction obtained from mechanically dispersed crude oil (i.e. waf) was prepared by adding 1 part of crude oil to 9 parts of habitatwater in a 15 l pyrex bottle with gentle stirring using a magnetic stirrer. mixing was done at ambient temperature for 20 h. after mixing, the oil and water phases were allowed to separate for 6 h. the water phase was then collected and immediately used for analysis. five different concentrations were also prepared. the same procedure described above was used for preparing the chemically enhanced water accommodated fraction (cewaf) from cdo. 2.6. physicochemical analysis of test media unstable parameters such as the ph, electrical conductivity (ec), temperature and total dissolved solids (tds) were measured in-situ. the ph was measured using a hand-held digital ph metre (phepr hanna, usa). the ec and tds were measured using a tds/conductivity metre. total solids (ts) and total suspended solids (tss) were carried out gravimetrically. dissolved oxygen (do), biochemical oxygen demand (bod5), chemical oxygen demand (cod), nitrate (no3−), and phosphate were determined using standard methods [18]. 2.7. heavy metals analysis water samples from the different test media were digested using nitric acid according to [19]. lead, chromium, vanadium, cadmium, arsenic, mercury, nickel, iron and zinc were quantified using atomic absorption spectrometer (perkin elmer 3110). the equipment was calibrated using analytical standards of the respective metals. these standards (1000 mgl−1) were diluted serially to get the working concentrations and subsequently, a calibration graph. 2.8. total petroleum hydrocarbons total petroleum hydrocarbon (tph), was determined accordingto astm d 7066, d3921. 100 ml of water sample was measured using a graduated cylinder into a separating funnel. 1ml of h2so4 was added, followed by 20 ml of tetrachloroethylene at 10 ml each. this was sealed, shaken vigorously for about 1-2minutes with periodic venting to release the inbuilt pressure, and allowed to stand for 10 mins for separation into organic and inorganic layers. the organic layer (i.e. the lower layer) was collected in a beaker. a filter paper was placed in a filter glass funnel and, approximately, 1 teaspoon of silica gel was added, and the solvent layer was drained through it. a cuvette was filled with the solvent layer and placed on an infracal 2 analyzer. “run” was selected, and the result was displayed in ppm result(mg/l) = c × v1 v2 where c = concentration obtained from the instrument, v 1 = volume of solvent used for extraction, and v 2 = volume of sample used for extraction. 2.9. polycyclic aromatic hydrocarbons polycyclic aromatic hydrocarbons (pahs) was extracted using the us epa method 3510 liquid-liquid extraction. 100 ml of water sample was collected into a separatory funnel. a known volume of n-hexane and methylene chloride (3:1) was added, and the sample was spiked with ortho-terphenyl. this was sealed and shaken vigorously for 1 to 2 minutes with periodic venting to release excess pressure, and the organic layer was allowed to separate from the water phase for a minimum of 5 minutes. the extract was filtered through a glass funnel with glass wool and anhydrous sodium sulphate. the volume of the sample extracted was recorded and the extract was transferred to a teflon-lined screw-cap vial ready for pah analysis. 107 onokare et al. / j. nig. soc. phys. sci. 4 (2022) 105–116 108 a method blank was similarly carried out. the extracts were quantified using gas chromatograph with a flame ionization detector (gc-fid) model 6890 (agilent instruments, usa). 2.10. microbiological analyses microbiological characteristics of the sea water and crude oil used for the acute toxicity test was done by adopting methods from [37] and [38]. 2.11. acute toxicity tests acute toxicity tests were conducted in two steps: first, a preliminary range finding test was conducted to evaluate the lowest concentration that would result in 100% mortality (lethal concentration) or 100% inhibition of bioluminescence (effective concentration) and the highest concentration that will cause 0% mortality or 0% inhibition of bioluminescence. second, the actual toxicity test, which involved exposure of the test organisms to varying concentrations of the toxicants, employing the least concentration that would cause mortality or inhibit bioluminescence, which was obtained from the preliminary range finding test as the highest concentration. two options were adopted for acute toxicity test. these were the static with renewal option and the static without renewal option. this test was conducted with reference to [20] and [21] guidelines in part iii e, section 4.3.2 of the ‘environmental guidelines and standards for petroleum industry in nigeria’ 2018. 2.11.1. acute toxicity test for t. guineensis the static renewal option was employed for t. guineensis. the habitat water was renewed every 48 h, while the organisms were fed with commercial feed of 5% body weight every 24 h. the acute toxicity test involved exposure of test organisms to five different concentrations (i.e. 100%, 50%, 25%, 12.5% and 6.25%) of mdo, cdo, d, waf, and cewaf. the method is described as follows: 10 healthy acclimatized fish were introduced into an aerated laboratory glass aquarium (18” x 10” x 18”) into each unit of the differentconcentrations. uniformed concentrations were used for all crude oil systems to allow for comparability of results following croserf standards [22]. the setup was done in triplicate at room temperature (22 – 27oc). mortality and impaired movement were the indices for scoring toxicity. dead organisms were removed and counted at 0, 24, 48, 72 and 96 hours. probit analysis [23] was used to analysze the number of mortalities recorded. this was carried out in order to establish the lc50 (median lethal concentration)of the reference toxicant [13]. 2.11.2. acute toxicity test for p.africanus the static renewal option was also employed for p. africanus: 10 healthy acclimatized organisms were introduced into 1000 ml glass jar of each unit of different concentrations of toxicants. control was sds for dispersant, and habitat water for crude oil. the set-up was done at room temperature (22-27o c). mortality was end point for toxicity. it was scored by organisms’ inability to swim and immediately organisms settled at the bottom of the jar. these organisms were counted at 0, 24, 48, 72 and 96 h. probit analysis [23] was used to analyze the number of mortality recorded. this was carried out in order to establish the lc50 (median lethal concentration). the toxicity factor was also calculated with respect to the lc50 of the reference toxicant. 2.11.3. acute toxicity test for bacteria for bacteria, the static nonrenewal method was employed. 100%, 50%, 25%, 12.5% and 6.25% of mdo, cdo, d, waf and cewaf were introduced into 1 l glass beakers. the habitat water without toxicant was used as control. the experiment was set up in triplicate at 22 – 25oc. aerobic plate count in nutrient agar treated with antifungal antibiotics was used to evaluate toxicity and was done at 0, 24, 48 and 96 hours. probit analysis was used to analyze the number of mortality recorded. 2.11.4. toxicity factor the toxicity factor (tf) for dispersant and synergistic or joint action factor (jaf) for crude oil-plus-dispersant was determined using the formula described by odiete (1999): t f = 96 h lc50 value o f dispersant 96 h lc50 value o f s ds jaf = 96 h lc50 o f dispersant − plus − crude − oil 96 h lc50 value o f cude oil 2.12. statistical analysis all analyses were performed in triplicates. data were expressed as mean±sd. significant differences among parameters in the different test media were analysed using analysis of variance (anova) using spss 22.0 version. data obtained from acute toxicity tests were also subjected to statistical analysis by probitmethod. 3. results and discussions 3.1. physicochemical characteristics of the crude oil test media the physicochemical characteristics of mdo, cdo, cewaf, waf, d and habitat water are presented in tables 16, respectively. the ph of all test media ranged from near-neutral to slightly alkaline. a gradual reduction in ph was observed with decreasing concentration of toxicant in cewaf and waf, while in mdo and cdo, the reduction was observed at toxicant concentration of 50%. a ph range of 7.9 8.0 has been previously reported in a similar study [16, 24]. a ph range of 6.09 – 8.45 is reportedly ideal for supporting aquatic life including fish [25]; while, waters with a ph value less than 6.0 may result in stunted, reduced or even absent fish population [26]. the ph of the studied test media will therefore support aquatic habitation. the temperature of the test media fluctuated between 29.0◦c and 31.3◦c. the temperature of surface waters usually range between 0 – 30◦c, but can get to as high as 40◦c, depending on the season and the prevailing environmental condition. temperature affects physical, chemical and biological 108 onokare et al. / j. nig. soc. phys. sci. 4 (2022) 105–116 109 processes in water bodies and, therefore, influences the concentration of many variables. the mean concentration of tds in the test media ranged as follows: 4365 – 4530 mg l−1in mdo; 4390 – 4435 mg l−1in cdo; 3915 – 4450mg l−1 in cewaf, 3900 – 3910mg l−1 in waf and from 4365 – 4410mg l−1 in d. there was no significant difference (p<0.05) in tds levels across the different test media. in terms of tds, the water in all test media, including the sea water, can still be classified as fresh water, since their concentrations were less than 5,000mg l−1 [27]. the do levels ranged from 8.07 – 11.1mg l−1 in mdo; 9.83 – 10.5mg l−1 in cdo; 10.3 – 10.9mg l−1 in cewaf; 10.1 – 10.2mg l−1 in waf; and from 10.5 – 13.1mg l−1 in d. a decreasing trend with decreasing concentration of toxicant in mdo and d was observed. comparatively, cewaf had slightly higher do levels than waf; while same trend was observed with mdo and cdo, but at toxicant concentration below 100%. the oxygen level in natural waters usually varies with salinity, temperature, turbulence, the photosynthetic activity of algae and plants, and atmospheric pressure [28]. in freshwaters, do at sea level normally ranges from 15 mg l−1 at 0◦c to 8mg l−1 at 25◦c, while concentrations in unpolluted waters are usually close to, but less than, 10mg l−1. the do levels in all test media indicated that the all media water were effectively aerated. the minimum acceptable do levels that can maintain fish population in aquatic environment is reported to range between 4 and 5mg l−1, while fish mortality occurs when do levels are less than 3mg l−1 [29, 30]. the range of do concentration that can support fisheries and aquatic lives is 5 – 9mg l−1 in the eu, 5 – 9.5mg l−1 in canada, and 4 – 6mg l−1 in russia [28]. therefore, the toxicant levels were not sufficient to cause any drastic reduction in oxygen levels and hence, could support fish growth and survival. biochemical oxygen demand (bod5) levels in the river water were relatively high, ranging from 6.23 – 8.25mg l−1 in mdo; 4.02 – 9.17mg l−1 in cdo; 4.03 – 9.41mg l−1 in cewaf; 4.01 – 9.45mg l−1 in waf; and from 4.02 – 6.95mg l−1 in d. high values were observed in the chemically enhanced media (i.e. cdo and cewaf, respectively). additionally, an increase in bod5 levels was obtained with increasing concentration of toxicants in the different test media. with respect to bod5 levels and aquatic pollution status of waters, bod concentrations < 1.0mg l−1 have been classified as being unpolluted; bod ≥ 2 ≤ 9mg l−1 has been classified as being moderately polluted, while bod > 10 mg l−1 has been classified as being heavily polluted [31, 32, 33]. similarly, the maximum acceptable limits set by [21] and [34] are 10mg l−1 and 5.0mg l−1, respectively. introduction of toxicants significantly altered the bod levels in all test media at toxicant concentrations greater than 250 mg l−1. for chemical oxygen demand (cod), the concentrations ranged from 31.8 – 124.1mg l−1 in mdo; 20.0 – 62.4mg l−1in cdo; 21.5 – 64.3 in cewaf; 20.0 – 64.2mg l−1in waf; and from 20.0 – 48.3mg l−1in d. water is considered relatively unpolluted when the cod levels is less than or equal to 20 mg l−1. in this study, the sea water could be said to be relatively unpolluted, with a cod concentration of 20.0 mg l−1. in the different test media and varying toxicant concentrations, as expected, cod levels were all greater those of the sea water, and an increasing concentration was observed with increasing toxicant concentration in all test media. this shows that there is a positive influence on the amount of oxidisable organic matter in water by the added toxicants. 3.2. heavy metals content in test media the concentration of heavy metals at different exposure concentrations in test media are also presented in tables 1-6 above. ni had the highest concentration with respect to other metals in all test media. pb was present in high amounts in d; and its concentration decreased with decreasing concentration of exposure test media. trace metals such as ni, v, cu, cd and pb are naturally found in crude oil, and, water contaminated with crude oil may exhibit higher concentrations of these metals. the concentration of cu was within the 2 mg l−1permissible limit for drinking water in all test media and at the different exposure concentrations. pb is also a normal constituent of crude oil. its maximum permissible limit of 0.01 mg l−1 in drinking water was exceeded at all exposure concentrations in d. this indicates that the dispersant itself is a toxin of ecotoxicological importance. in mdo and waf, exposure concentrations < 50% were within the permissible limit for pb.in cewaf, only exposure concentration at 100% exceeded the permissible limit, while in cdo, pb concentration was within the permissible limit in the different exposure concentrations. for cewaf and cdo, some form of positive synergy between crude oil and dispersant, which limited the bioavailability of pb in the test media, was observed. cd and ni, like pb and cu, are also natural constituents of crude oil. their permissible limits (0.003mg l−1 for cd and 0.1mg l−1 for ni) were significantly exceeded in all test media and at the different exposure concentrations. this also is indicative of some degree of toxicity due to cd and ni, to aquatic species. for cr, its permissible limit of 0.05mg l−1 was slightly exceeded even in sea water. with increasing toxicant concentrations, this limit was well exceeded in all test media, which implied that there will likely be some level of toxicity to aquatic species. fe levels were slightly elevated in the different test media compared to control medium. however, fe concentrations were within the who permissible limit of 0.3 mg l−1 in mdo, waf and d; but were slightly elevated in cdo and cewaf. zn, as and v were found in concentrations that were within their respective permissible limits in drinking water, hence, no toxic property is expected from these metals. 3.3. total petroleum hydrocarbons and pahs in test media the total concentration of tph in the different test media are shown in figures 1-3. the tph concentrations did not vary significantly in mdo and cdo, but there were significant variations in tph levels in waf and cewaf test media. a limit value for tph has been set at 10mg l−1 in many water contamination regulations, i.e., the maximum concentration of tph 109 onokare et al. / j. nig. soc. phys. sci. 4 (2022) 105–116 110 table 1: physicochemical characteristics of mdo test media parameters concentration of exposure medium (%) 100 50 25 12.5 6.25 ph 6.9±0.01a 7.00±0.02a 7.90±0.02a 6.90±0.02a 6.90±0.01a temp. (◦c) 29.9±1.03a 31.1±0.5a 31.3±0.1a 29.1±0.1a 29.1±0.1a ec (us/cm) 9060±3.1a 8770±2.5a 8860±2.1a 8810±3.0a 8750±2.1a tds (mg/l) 4530±2.3a 4385±3.1a 4450±3.0a 4430±2.5a 4415±1.9a tss (mg/l) 1.20±.0.5a 1.00±0.02a 1.05±0.1a 1.05±0.01a 1.05±0.01a salinity (psu) 0.50±0.03a 0.40±0.01a 0.40±0.01a 0.40±0.2a 0.40±0.2a do (mg/l) 11.1±1.1a 9.80±0.1a 10.6±0.2a 10.9±1.1a 10.8±0.2a bod (mg/l) 18.2±1.1a 13.0±2.1b 9.40±1.5b 4.70±1.1c 4.30±1.05c cod (mg/l) 124±2.5a 88.4±2.1b 64.3±1.8b 32.2±1.9c 29.7±1.5c pb (mg/l) 0.17±0.01a 0.06±0.002b nd nd nd cr (mg/l) 0.17±0.001a 0.13±0.001a 0.13±0.01a 0.13±0.01a 0.10±0.001a cd (mg/l) 0.13±0.01a 0.11±0.01a 0.09±0.001b 0.08±0.01b 0.06±0.01b ni (mg/l) 0.54±0.02a 0.75±0.15a 0.97±0.20b 1.19±0.25c 1.20±0.20c fe (mg/l) 0.25±0.001a 0.39±0.01a 0.35±0.002a 0.30±0.001a 0.39±0.02a cu (mg/l) 0.06±0.001a 0.16±0.02b 0.32±0.02c 0.45±0.02d 0.51±0.15d zn (mg/l) 0.04±0.001a 0.05±0.001a 0.04±0.015a 0.04±0.002a 0.03±0001a v (mg/l) 0.09±0.001a 0.09±0.001a 0.09±0.001a 0.09±0.01a 0.09±0.01a values are expressed as mean ± standard deviation. superscripts with same letters on the same row indicate no significant difference, while superscripts with different letters indicate significant difference at p < 0.05 table 2: physicochemical characteristics of cdo test media parameters concentration of exposure medium (%) 100 50 25 12.5 6.25 ph 6.82±0.20a 6.63±0.02a 6.86±0.01a 6.83±0.01a 6.81±0.02a temp. (◦c) 29.1±0.25a 30.5±0.05a 29.0±0.10a 31.0±0.25a 29.0±1.05a ec (us/cm) 8720±3.00a 8710±1.25a 8830±2.05a 8820±1.95a 8810±2.05a tds (mg/l) 4410±2.05a 3925±0.50a 4415±1.05a 4415±2.05a 4410±1.85a tss (mg/l) 1.05±0.50a 1.04±0.01a 1.06±0.01a 1.06±0.02a 1.06±0.01a salinity (psu) 0.43±0.10a 0.43±0.01a 0.44±0.01a 0.44±0.01a 0.44±0.02a do (mg/l) 10.3±0.15a 10.3±0.15a 10.1±1.50a 10.3±1.95a 10.2±0.50a bod (mg/l) 4.05±1.05a 4.03±1.00a 4.72±0.05a 4.05±0.01a 4.02±0.20a cod (mg/l) 21.9±0.95a 21.5±1.00a 32.1±1.05b 28.6±1.05b 22.9±1.05a pb (mg/l) nd nd nd nd nd cr (mg/l) 0.11±0.01a 0.11±0.02a 0.12±0.001a 0.12±0.02a 0.11±0.001a cd (mg/l) 0.12±0.02a 0.16±0.15a 0.18±0.01a 0.18±0.04a 0.16±0.01a ni (mg/l) 1.04±0.10a 2.19±0.02b 1.80±0.15b 1.20±0.15a 1.15±0.30a fe (mg/l) 0.09±0.001a 1.00±0.001b 0.47±0.20c 0.47±0.20c 0.41±0.02c cu (mg/l) 0.75±0.01a 0.30±0.03b 0.32±0.002b 0.31±0.10b 0.31±0.01b zn (mg/l) 0.07±0.02a 0.07±0.02a 0.05±0.001a 0.05±0.001a 0.05±0.001a v (mg/l) 0.10±0.03a 0.08±0.001a 0.24±0.01b 0.21±0.001b 0.20±0.01b 110 onokare et al. / j. nig. soc. phys. sci. 4 (2022) 105–116 111 table 3: physicochemical characteristics of waf test media parameters concentration of exposure medium (%) 100 50 25 12.5 6.25 ph 7.38±0.01a 7.33±0.10a 7.29±0.25a 7.02±0.15a 6.98±0.01a temp. (◦c) 29.5±1.05a 29.5±0.50a 29.3±1.05a 29.1±1.05a 29.1±0.50a ec (us/cm) 7820±2.00a 7810±3.05a 7990±2.95a 7970±3.50a 7970±2.50a tds (mg/l) 3915±1.95a 3910±2.95a 3905±2.50a 3900±2.80a 3900±1.85a tss (mg/l) 0.55±0.01a 0.55±0.01a 0.49±0.10a 0.42±0.01a 0.40±0.15a salinity (psu) 0.39±0.01a 0.39±0.02a 0.43±0.01a 0.43±0.01a 0.43±0.02a do (mg/l) 10.2±2.05a 10.2±0.10a 10.3±0.05a 10.2±0.05a 10.1±0.10a bod (mg/l) 9.45±2.05a 9.13±0.20a 7.33±0.10a 4.79±0.10b 4.01±1.50b cod (mg/l) 64.3±1.05a 62.9±0.25a 46.3±0.55b 33.5±0.10c 20.8±0.05d pb (mg/l) 0.88±0.01a 0.06±0.001b nd nd nd cr (mg/l) 0.13±0.01a 0.09±0.001a 0.07±0.001a 0.03±0.001b 0.008±0.001c cd (mg/l) 0.24±0.001a 0.19±0.02a 0.17±0.001a 0.12±0.01b 0.11±0.01b ni (mg/l) 0.45±0.01a 0.75±0.02a 0.97±0.03b 1.19±0.40c 0.54±0.002a fe (mg/l) 0.40±0.01a 0.40±0.001a 0.40±0.01a 0.30±0.01a 0.25±0.02a cu (mg/l) 0.50±0.02a 0.50±0.01a 0.50±0.01a 0.40±0.01a 0.06±0.002a zn (mg/l) 0.04±0.001a 0.04±0.001a 0.04±0.001a 0.04±0.001a 0.04±0.001a v (mg/l) 0.14±0.001a 0.09±0.001a 0.09±0.001a 0.09±0.001a 0.09±0.001a table 4: physicochemical characteristics of cewaf test media parameters concentration of exposure medium (%) 100 50 25 12.5 6.25 ph 7.98±0.15a 6.99±0.02a 6.93±0.15a 6.82±0.15a 6.63±0.02a temp. (◦c) 31.3±0.10a 29.1±0.05a 29.1±0.10a 29.1±1.05a 30.5±0.05a ec (us/cm) 8860±2.05a 8810±0.30a 8750±1.55a 8720±2.55a 8710±1.65a tds (mg/l) 4450±3.15a 4430±1.05a 4415±1.50a 4410±1.50a 3925±1.05a tss (mg/l) 1.05±0.01a 1.07±0.15a 1.06±0.10a 1.05±0.10a 1.04±0.01a salinity (psu) 0.44±0.01a 0.44±0.01a 0.43±0.01a 0.43±0.01a 0.43±0.01a do (mg/l) 10.6±1.55a 10.9±1.05a 10.8±0.20a 10.3±1.55a 10.3±0.01a bod (mg/l) 9.41±1.05a 4.70±0.15b 4.25±1.05b 4.06±0.25b 4.03±0.01b cod (mg/l) 64.3±2.05a 32.1±1.55b 29.7±0.20b 21.9±1.05c 21.5±0.50c pb (mg/l) nd nd nd nd nd cr (mg/l) 0.10±0.01a 0.10±0.001a 0.12±0.001a 0.12±0.01a 0.10±0.01a cd (mg/l) 0.16±0.01a 0.15±0.01a 0.18±0.02a 0.15±0.001a 0.15±0.001a ni (mg/l) 1.40±0.10a 0.97±0.15b 0.80±0.02b 0.80±0.02b 0.57±0.025c fe (mg/l) 0.43±0.20a 0.20±0.001b 0.13±0.001c 0.09±0.001c 0.08±0.01c cu (mg/l) 0.27±0.01a 0.16±0.01b 0.13±0.01b 0.13±0.001b 0.12±0.01b zn (mg/l) 0.04±0.001a 0.04±0.01a 0.04±0.01a 0.04±0.01a 0.04±0.015a v (mg/l) 0.23±0.10a 0.18±0.01a 0.17±0.02a 0.17±0.02a 0.17±0.001a 111 onokare et al. / j. nig. soc. phys. sci. 4 (2022) 105–116 112 table 5: physicochemical characteristics of dispersant test media parameters concentration of exposure medium (%) 100 50 25 12.5 6.25 ph 7.38±0.01a 7.33±0.10a 7.29±0.25a 7.02±0.15a 6.98±0.01a temp. (◦c) 29.5±1.05a 29.5±0.50a 29.3±1.05a 29.1±1.05a 29.1±0.50a ec (us/cm) 7820±2.00a 7810±3.05a 7990±2.95a 7970±3.50a 7970±2.50a tds (mg/l) 3915±1.95a 3910±2.95a 3905±2.50a 3900±2.80a 3900±1.85a tss (mg/l) 0.56±0.01a 0.55±0.01a 0.49±0.10a 0.42±0.01a 0.40±0.15a salinity (psu) 0.39±0.01a 0.39±0.02a 0.43±0.01a 0.43±0.01a 0.43±0.02a do (mg/l) 10.2±2.05a 10.3±0.10a 10.3±0.05a 10.2±0.05a 10.1±0.10a bod (mg/l) 9.45±2.05a 9.13±0.20a 7.33±0.10a 4.78±0.10b 4.01±1.50b cod (mg/l) 64.3±1.05a 62.9±0.25a 46.3±0.55b 33.5±0.10b 20.8±0.05c pb (mg/l) 0.85±0.25a 0.73±0.25a 0.54±0.01a 0.52±001a 0.08±0.01b cr (mg/l) 0.13±0.01a 0.08±0.01a 0.07±0.02a 0.07±0.01a 0.08±0.015a cd (mg/l) 0.20±0.001a 0.19±0.01a 0.16±0.02a 0.14±0.01a 0.13±0.01a ni (mg/l) 1.04±0.50a 0.99±0.01a 0.88±0.025a 0.80±0.02a nd fe (mg/l) nd nd nd nd nd cu (mg/l) 0.45±0.05a 0.40±0.01a 0.36±0.01a 0.40±0.01a 0.30±0.01a zn (mg/l) 0.05±0.01a 0.04±0.01a 0.04±0.001a 0.04±0.001a 0.03±0.001a v (mg/l) nd nd nd nd nd table 6: physicochemical characteristics of habitat water parameters concentration of brackish water (%) ph 7.23±0.01 temp. (◦c) 31.3±0.50 ec (us/cm) 8780±1.05 tds (mg/l) 4390±1.00 tss (mg/l) 1.04±0.01 salinity (psu) 0.43±0.01 do (mg/l) 9.20±0.02 bod (mg/l) 4.71±0.50 cod (mg/l) 20.0±0.50 pb (mg/l) nd cr (mg/l) 0.08±0.001 cd (mg/l) 0.20±0.01 ni (mg/l) 0.42±0.02 fe (mg/l) nd cu (mg/l) 0.12±0.001 zn (mg/l) 0.37±0.015 v (mg/l) 0.16±0.001 should be present in waters during the discharge of oilfield waters (özdemir, 2018). for drinking water, the limit is 0.5 mg l−1 [35] 2008) and 10 mg l−1 [21]. in this study, the tph level in sea water was 4.43mg l−1, which was within the dpr limit. in the test media, the tph concentrations ranged from 25.8 – 914.7 mg l−1 in cdo; 19.8 – 239.9mg l−1 in cewaf; 7.5 – 21.8 mg l−1 in waf; and from 4.30 – 11.1mg l−1 in d. the levels in d were generally within the limit for tph in discharged oilfield waters, but well above the drinking water limit. for all crude oil-based media, significant pollution was observed. results of pahs concentration in the different test media are presented in table 7. only two concentrations of the different test media (i.e., 100% and 50%) were reported in this study due to the significance of their results relative to the other concentrations. a decreasing concentration of pahs was observed in all test media with decreasing concentration of test media. when compared with the pah concentration of the crude oil (464.3 mg l−1), there was a significant drop in pahs concentration in the crude oil test media. a maximum pahs concentration of 55.1 mg l−1 was observed in cdo test media concentration of 100%, while the least concentration of 2.04mg l−1 was found in d test media concentration of 50%. all 16 us epa priority pahs were detected in waf test media, with ?16pahs concentrations of 21.3mg l−1and 12.5mg l−1 at toxicant concentrations of 100% and 50%, respectively. the least ?16pahs concentrations was detected in d, with a concentration of 2.04mg l−1.a value of 0.2 µgl−1 (0.0002 mg l−1) has been set as the maximum permissible limit for total pahs in drinking water, while that of benzo{a}pyrene is 0.1 µgl−1 (0.0001 mg l−1) [34]. the results of this study indicated that all test media, including the control, had ?16pahs that greatly exceeded the permissible limit by several orders of magnitude. this in112 onokare et al. / j. nig. soc. phys. sci. 4 (2022) 105–116 113 dicated a strong likelihood for significant bioaccumulation of these contaminants in aquatic organisms. for benzo{a}pyrene, which is regarded as the most carcinogenic of all pahs, it was detected in all test media except cewaf and the control. figure 1: comparing the concentration of tph in mdo and cdo figure 2: comparing the concentration of tph in waf and cewaf figure 3: concentration of tph in d 3.4. acute toxicity test result for test organisms the acute toxicity of the control toxicant and the different test media on the t. guineensis is presented as the 96h median lethal concentration (lc50) as shown in tables 8, 9 and 10, respectively. the lc50 of the different toxicant test media on t. guineensis followed the order: sds > mdo > cdo > cewaf > waf > d. following the description on 96h lc50, the reference toxicant was moderately toxic, with a value of 1.37. the dispersant solution exhibited the least toxicity, and was non-toxic to t. guineensis, having a value greater than 1,000. several studies have also shown that sds exhibited the greatest toxicity to marine organisms relative to the dispersant and the dispersant plus crude oil test system [11, 12]. the different exposure media ranged from being non-toxic to being practically non-toxic. the toxicity factors (i.e. the index for assessing toxicity) showed that sds was 1,380 times more toxic to t. guineensis that the dispersant, and 65.7 times more toxic to t. guineensis than mdo. acute toxicity tests on p. africanus showed that the chemically modified test media (i.e., cdo and cewaf) exhibited slightly greater toxicities to p. africanus than mdo and d; but lesser toxicities compared to the reference sds toxicant (table 9). the lc50 of the different toxicant test media followed the order: sds > cdo > cewaf> waf > mdo > d. the dispersant also exhibited the least toxicity to p. africanus. for the two tested organisms, which are representatives of the secondary and tertiary consumers in the marine food chain, ecobestr dispersant was shown to be lesstoxic to both test organisms. however, the lc50 test result suggested that p. africanus was more tolerant to sds relative to t. guineensis, as seen in the lc50 values for sds in both organisms. the toxicity of the different dispersants on the two types of microbial populations were determined using the 96hlc50 of the toxicants on the test organisms are presented in table 9. the toxicity of the toxicant test media followed the order: cewaf < d <cdo < mdo < coa < sds for heterotrophic bacteria, and cewaf < cdo <mdo < waf< d < sds for hydrocarbon-utilising bacteria. the toxicants generally expressed a lesser toxic effect on the hydrocarbon utilizing bacteria than the total heterotrophic bacteria when compared with their 96 h lc50 values. the toxicity factor of d to t. guineensis and p. africanus were 1380 and 380, which implied that sds was about 1380 and 380 times more toxic to t. guineensis and p. africanus, respectively. the synergistic action /joint action factor of crude oil-dispersant test systems to the tested organisms implied that the dispersant reduced the toxicity of crude oil. the toxicity of a given crude oil-dispersant mixture varies, and this depends on the proportion of addition of the mixture components [36]. 4. conclusion ecobestr dispersant and its product of interaction with crude oil in a marine environment was shown to be non-toxic to representative marine organisms in this study, based on the lc50 values obtained. p. africanus exhibited greater tolerance to the effect of crude oil-dispersant in the different test media relative to t. guineensis. however, with increased levels of some metals and polycyclic aromatic hydrocarbons present in 113 onokare et al. / j. nig. soc. phys. sci. 4 (2022) 105–116 114 table 7: concentration of pahs (mg l−1) in different test media cewaf waf cdo mdo d brackish water pahs concentration (%) of toxicant in test media 100 50 100 50 100 50 100 50 100 50 nap 0.08 0.0727 0.9 0.6 3.5 0.72 0.13 0.25 0.0069 0.036 0.027 ace 0.37 0.238 3.3 2.1 14.8 2.6 1.3 1.7 nd nd 0.037 acp 0.23 0.15 1.04 0.9 6.88 1.6 2.49 1.09 0.04 nd 0.009 flu 0.19 0.14 2.56 1.13 9.94 1.88 2.65 0.64 0.0042 0.035 0.031 phe 0.61 1.36 3.98 2.44 nd 2.35 nd 1.33 0.064 0.024 0.017 ant 0.72 nd 0.61 0.23 19.6 0.25 6.96 2.38 0.007 0.019 nd flo nd 0.25 0.59 0.35 nd 26.8 0.27 nd 0.01 0.0075 0.014 pyr nd 0.8 1.28 0.92 nd nd 12.6 5.74 0.0005 0.016 0.0076 chry 0.17 0.04 2.35 1.45 nd nd nd nd 0.043 0.03 0.021 baa 0.054 0.02 1.49 0.9 nd nd nd nd 0.076 0.024 0.067 bbf 0.22 nd 1.05 0.35 nd 0.55 0.11 0.23 0.16 nd 0.56 bkf 0.16 nd 0.62 0.25 nd 1.6 0.13 0.13 0.084 nd nd bap nd nd 0.21 0.1 0.3 0.67 0.18 0.078 0.41 nd nd indeno nd nd 0.37 0.25 nd 0.27 4.3 nd 1.8 nd nd dah 4.3 nd 0.46 0.3 nd 0.34 0.24 nd nd 1.6 nd bghi 0.048 0.33 0.38 0.2 nd 0.29 0.18 4.08 1.13 0.017 4.08 total 7.28 3.43 21.27 12.47 55.1 39.4 31.6 17.7 3.89 2.04 4.87 table 8: acute effect of the different test media on tilapia guineensis s/n toxicants 96 h lc50 (ppt) toxicity factor 1 (mdo) toxicity factor 2 (sds) inference 1. mdo 89.99 65.7 mechanically dispersed crude oil is 65.7 times less toxic than sds. 2. cdo 225.60 2.51 164.7 chemically dispersed crude oil is at least 2.5 times less toxic than mechanically dispersed crude and 164.7 times less toxic than sds. 3. d 1891.80 21.04 1380.8 dispersant is at least 21.04 times less toxic than mechanically dispersed crude oil and 1380.88 times less toxic the sds. 4. waf 683.99 7.60 684 waf is 7.6 less toxic than mechanically dispersed crude oil and 684 times less toxic than sds. 5. cewaf 528.02 5.87 385.4 cewaf is 5.86 times less toxic than mechanically dispersed crude oil and 385.41 times less toxic than sds. 6. sds 1.37 0.015 sds is 0.015 times more toxic than mechanically dispersed crude oil. 114 onokare et al. / j. nig. soc. phys. sci. 4 (2022) 105–116 115 table 9: acute effect of the different test media on palaemonetesafricanus s/n toxicants 96 h lc50 (ppt) toxicity factor 1 (mdo) toxicity factor 2 (sds) inference 1. mdo 275.5 27.8 mechanically dispersed crude oil is 27.8 less toxic than sds. 2. cdo 137.77 0.05 13.78 chemically dispersed crude oil is 0.05 times more toxic to crustacean than mechanically dispersed crude but 13.78 times less toxic than sds. 3. d 3800.92 13.80 380.24 dispersant is 13.80 times less toxic than mechanically dispersed crude oil and 380.24 times less toxic than sds. 4. waf 176.13 0.64 17.62 waf is 0.64 times less toxic than mechanically dispersed crude oil. 5. cewaf 168.29 0.61 16.83 cewaf is 0.61 more toxic than mechanically dispersed crude oil but 16.83 times less toxic than sds. 6. sds 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[38] c. b. chikere, c. a. ajuzieogu & m. c. miller, “characterisation of indigenous bacterial communities in crude oil-impacted sites at obagi town, onelga, rivers state, nigeria”, fine focus journal 2 (2015) 7. 116 j. nig. soc. phys. sci. 2 (2020) 69–76 journal of the nigerian society of physical sciences original research solving third order ordinary differential equations directly using hybrid numerical models j. o. kuboye, o. f. quadri∗, o. r. elusakin department of mathematics, federal university oye-ekiti, nigeria abstract in this work, numerical methods for solving third order initial value problems of ordinary differential equations are developed. multi-step collocation is used in deriving the methods, where power series approximate solution is employed as a basis function. gaussian elimination approach is considered in finding the unknown variables a j, j = 0(1)8 in interpolation and collocation equations, which are substituted into the power series to give the continuous implicit schemes. the discrete schemes and its derivatives are derived by evaluating the grid and non-grid points. these schemes are arranged in a matrix form to produce the block methods. the order of the developed methods are found to be six. the numerical results proved the efficiency of the methods over the existing methods. keywords: interpolation, collocation, block method, multistep method and third order initial value problem article history : received: 21 january 2020 received in revised form: 09 march 2020 accepted for publication: 16 march 2020 published: 14 may 2020 c©2020 journal of the nigerian society of physical sciences. all rights reserved. communicated by: t. latunde 1. introduction in time past, to solve third order ordinary differential equation, a reduction method is employed and this involves reducing the differentials to three systems of first order ordinary differential equation, and the numerical approach for first order ordinary differential equation is used to solve the resulting system. this technique has a lot of drawbacks like wastage of time as a result of many functions to be evaluated and the computational effort required. this is extensively discussed in ref. [1-5]. advancing in numerical methods, direct method for solving higher order initial value problems without going through the process of reduction was developed by fatunla [2], kuboye and omar ∗corresponding author tel. no: +2347068522865 email addresses: kubbysholly2007@yahoo.com (j. o. kuboye), qomotolaniq@gmail.com (o. f. quadri), opeyemielusakin21@gmail.com (o. r. elusakin) [4], olabode [6], olabode [7], omar and suleiman [8], sagir [9], sunday [10] amongst others. the accuracy of the methods developed by these researchers are not efficient in terms of error, as this can be improved by the introduction of off grid points within the interval of integration. hence this paper addresses the shortcoming of low accuracy found in the existing method mentioned above. 2. methodology 2.1. derivation of first block method (fbm) this section focuses on the derivation of first block method. a power series y(x) = k+4∑ j=0 a j x j (1) 69 j. o. kuboye et al. / j. nig. soc. phys. sci. 2 (2020) 69–76 70 is considered as an approximate solution to third order ordinary differential equations of the form y ′′′ = f (x, y(x), y ′ (x), y ′′ (x)), y(x0) = y0, y ′ (x0) = y ′ 0, y ′′ (x0) = y ′′ 0 . equation (1) is differentiated thrice to give: y ′′′ (x) = k+4∑ j=0 j( j − 1)( j − 2)a j x j−3 (2) where step number k=4. interpolating (1) at x = xn+i, i = 1, 2, 9 4 and collocating its highest derivative at x = xn+m, i = 0(1)4 and 94 , the resulting equations produce a system of non-linear equations of the form k+4∑ j=0 a j x j n+i = yn+i k+4∑ j=3 j( j − 1)( j − 2)a j x j−3 n+m = fn+m (3) solving for the unknowns constants a j, j = 0(1)8 in (3) using gaussian elimination approach and substituting into (1) to give a continuous implicit scheme of the form k−2∑ j=1 α j(t)yn+ j + α 9 4 (t)yn+ 94 = h 3 [ k∑ j=0 β j(t) fn+ j + γ 9 4 (t) fn+ 94 ] (4) where t = x−xn+k−1h α1 α2 α 9 4  =  3 5 7 5 4 5 −6 −11 −4 32 5 48 5 16 5   t0 t1 t2  (5)  β0 β1 β2 β 9 4 β3 β4  = e  t0 t1 t2 t4 t5 t6 t7 t8  (6) where e has been defined in appendix b. equation (4) is differentiated twice and the coefficients are as follow:  α ′ 1 α ′ 2 α ′ 9 4  =  5 87 −11 −8 48 5 32 5   t0 t1  (7)  β ′ 0 β ′ 1 β ′ 2 β ′ 9 4 β ′ 3 β ′ 4  = f  t0 t1 t3 t4 t5 t6 t7  (8) where f has been defined in appendix b. α ′′ 1 α ′′ 2 α ′′ 9 4  =  8 5 −8 32 5  [ t0 ] (9)  β ′′ 0 β ′′ 1 β ′′ 2 β ′′ 9 4 β ′′ 3 β ′′ 4  = g  t0 t2 t3 t4 t5 t6  (10) and g =  −573440 185794560 −1892352 185794560 430080 185794560 3153920 185794560 2580480 185794560 −585111 185794560 114688 5160960 516096 5160960 215040 5160960 −860160 5160960 −774144 5160960 262947 5160960 −573440 3440640 −3268608 3440640 −3440640 3440640 6021120 3440640 7741440 3440640 496249 3440640 114688 635040 688128 635040 860160 635040 −1146880 635040 −2064384 635040 487059 635040 −573440 15482880 −3956736 15482880 −7526400 15482880 1433600 15482880 16773120 15482880 4595169 15482880 573440 144506880 4644864 144506880 13332480 144506880 16343040 144506880 7741440 144506880 −837513 144506880  the discrete schemes and its derivatives which are generated from (4) are represented in a matrix to form a block  yn+1 yn+2 yn+ 94 yn+3 yn+4  =  1 1 1 1 1  [ yn ] +  1 2 9 4 3 4  [ hy ′ n ] +  1 2 2 81 32 9 2 8  h2y ′′ n +h3  0 0 0 0 422945360 0 0 0 0 26995670 0 0 0 0 4539264373400320 0 0 0 0 8170 0 0 0 0 60562835   fn−4 fn−3 fn−2 fn−1 fn  +h3  271 1680 −19 40 8896 19845 −1003 15120 63 11760 148 105 −64 21 8192 2835 −404 945 1 30 5137263 2621440 −144020511 36700160 943569 250880 −10281087 18350080 22497669 22497669 2349 560 −243 35 1728 245 −117 112 81 980 128 15 −416 35 262144 19845 −1408 945 8 49   fn+1 fn+2 fn+ 52 fn+3 fn+4  (11) 2.2. derivation of second block method(sbm) equation (1) is interpolated at x = xn+ j, j = 1, 2 and 5 2 . equation(2) is collocated at x = xn+ j, j = 0(1)4 and 5 2 . the 70 j. o. kuboye et al. / j. nig. soc. phys. sci. 2 (2020) 69–76 71 same step employed in deriving the first block method (fbm) are also used and this gives the second block of the form  yn+1 yn+2 yn+ 52 yn+3 yn+4  =  1 1 1 1 1  [ yn ] +  1 2 5 2 3 4  [ hy ′ n ] +  1 2 2 25 8 9 2 8  h2y ′′ n +h3  0 0 0 0 3794032 0 0 0 0 101210 0 0 0 0 116125147456 0 0 0 0 26192240 0 0 0 0 13663   fn−4 fn−3 fn−2 fn−1 fn  +h3  115 756 −901 3360 278 945 −283 2520 59 8640 1276 945 −12 7 256 135 −76 105 83 1890 1943125 774144 −460625 172032 75125 24192 −308125 258048 225625 3096576 81 20 −4131 1120 162 35 −99 56 243 2240 7808 945 −608 105 8192 945 −128 45 40 189   fn+1 fn+2 fn+ 52 fn+3 fn+4  (12) 3. analysis of the method 3.1. order of the method the method proposed by lambert [5] is applied in finding the order of the methods (11) and (12) where yand f-functions are expanded in taylor series and comparing the coefficients of h, this gives the methods to be of order [6, 6, 6, 6, 6]t 3.2. zero stability the continuous implicit scheme (4) is zero-stable if the roots rn = 1, 2, ..., n of the first characteristic polynomial defined by p(r) = det[a ′ − b ′ ] satisfy |rn|≤ 1 and |r|= 1 where a ′ =  1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1  and b ′ =  0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1  applying the condition above the methods are found to be zero stable. 3.3. convergence according to henrici [3], equation(4) converges if it is zerostable and consistent. therefore, since the order of the methods are greater than one and also satisfies other conditions of zero stability. hence, the new methods (11) and (12) converged. 4. application of the method the following problems are considered in order to examine the accuracy of the new block methods. problem 1: consider the initial value problem(i.v.p) y ′′′ = 3 sin x, y(0) = 1, y ′ (0) = 0, y ′′ (0) = −2, h = 0.1 theoretical solution y(x) = 3 cos x + x 2 2 − 2 problem 2: consider the i.v.p y ′′′ − y ′′ + y ′ − y = 0 y(0) = 1 y ′ (0) = 0 y ′′ (0) = −1 h = 0.01 with exact solution of y(x) = cos x problem 3: consider the initial value problem(i.v.p) y ′′′ = exp x, y(0) = 3, y ′ (0) = 1, y ′′ (0) = 5, h = 0.1 theoretical solution y(x) = 2 + 2x2 + exp x the following acronyms are used • es=exact solution • cs=computed solution • eifbm=error in first block method with s=5/2 • eisbm=error in second block method with s=9/4 • ei0(2009)=error in olabode(2009) • eio(2013)=error in olabode(2013) • eis(2014)=error in sagir(2014) • eisu(2018)=error in j.sunday (2018) 5. discussion of results in table 1, the results generated from the new developed block methods (eifbm and eisbm) are compared with eio(2009) and eio(2013). it is clearly shown that the accuracy in terms of error reveals the efficiency of the new derived methods over eio(2009,2013) inspite of the high step number k considered. furthermore, the results produced from the new numerical models outperform the results in eis(2014) and eisu(2018) as evident in table 2. additionally, the effectiveness of the new developed numerical methods is demonstrated in table 3 as it compared favorably with eio(2013) in terms of error. 71 j. o. kuboye et al. / j. nig. soc. phys. sci. 2 (2020) 69–76 72 table 1: comparing the new method with olabode(2009, 2013) for solving problem 1 x-value es cs eifbm eisbm eio(2009) eio(2013) 0.1 0.990012495834077020 0.990012495833478170 6.8911543e-13 5.9885430e-13 3.4077519e-11 1.65922e-10 0.2 0.960199733523725120 0.960199733519903840 4.4015902e-12 3.8212766e-12 1.2372514e-10 4.76275e-10 0.3 0.911009467376818090 0.911009467367234980 1.0999868e-11 9.5831121e-12 1.7681812e-10 6.23182e-10 0.4 0.843182982008655380 0.843182981990707400 2.0601632e-11 1.7947976e-11 4.0865533e-10 19.9134e-10 0.5 0.757747685671118280 0.757747685638492150 3.6853520e-11 3.2626124e-11 3.7111825e-10 3.28882e-10 0.6 0.656006844729034810 0.656006844668665210 6.7268413e-11 6.0369598e-11 7.0964790e-10 1.27096e-09 0.7 0.539526561853465480 0.539526561752478040 1.1150603e-10 1.0098744e-10 7.4653450e-10 4.84653e-09 0.8 0.410120128041496560 0.410120127886882570 1.6985002e-10 1.5461399e-10 1.9585035e-09 1.09585e-08 0.9 0.269829904811992980 0.269829904583073650 2.4948449e-10 2.2891933e-10 3.8880070e-09 2.0188e-08 1.0 0.120906917604419300 0.120906917269670430 3.6226498e-10 3.3474887e-10 6.3955807e-09 3.53956e-08 1.1 -0.034211635723267797 -0.034211636195096484 5.0769700e-10 4.7182869e-10 9.5232678e-09 5.66233e-08 1.2 -0.192926736569979610 -0.192926737210326740 6.8618927e-10 6.4034714e-10 1.3169979e-08 8.35700e-08 table 2: comparing the new method with sagir(2014)and j.sunday(2018) for solving problem 2 x-value es cs eifbm eisbm eis(2014) eisu(2018) 0.01 0.999950000416665260 0.999950000416665370 1.1102230e-16 0.0000000e+00 1.9990e-07 1.1102e-016 0.02 0.999800006666577760 0.999800006666578310 5.5511151e-16 5.5511151e-16 1.9560e-07 1.3323e-015 0.03 0.999550033748987540 0.999550033748996200 8.6597396e-15 8.7707619e-15 1.3651e-07 9.6589e-015 0.04 0.999200106660977920 0.999200106661042750 6.4837025e-14 6.4614980e-14 2.5210e-07 3.2974e-014 0.05 0.998750260394966280 0.998750260395229290 2.6301183e-14 2.6290081e-14 1.3039e-06 8.2379e-014 table 3: comparing the new method with olabode(2013) for solving problem 3 x-value es cs eifbm eisbm eio (2013) 0.1 3.125170918075647700 3.125170918077170500 1.5227819e-12 1.3447021e-12 9.24352e-10 0.2 3.301402758160169700 3.301402758169862000 9.6922470e-12 8.5487173e-12 18.3983e-10 0.3 3.529858807576003300 3.529858807600271000 2.4267699e-11 2.1475266e-11 24.2400e-10 0.4 3.811824697641270600 3.811824697686721800 4.5451198e-11 4.0219827e-11 53.5873e-10 0.5 4.148721270700128200 4.148721270778516200 7.8387963e-11 7.0274453e-11 7.00128e-10 0.6 4.542118800390509700 4.542118800522103200 1.3159340e-10 1.1942358e-10 3.90509e-10 0.7 4.993752707470477500 4.993752707675188400 2.0471091e-10 1.8746782e-10 6.52952e-09 0.8 5.505540928492468600 5.505540928790510200 2.9804159e-10 2.7454572e-10 2.15075e-08 0.9 6.079603111156950000 6.079603111576208400 4.1925841e-10 3.8885162e-10 3.88430e-08 1.0 6.718281828459045500 6.718281829040118500 5.8107297e-10 5.4199667e-10 6.15410e-08 72 j. o. kuboye et al. / j. nig. soc. phys. sci. 2 (2020) 69–76 73 6. conclusion the methods are derived via multistep collocation technique which is used to develop the block methods. the methods are zero stable, consistent, converged and of order six. the accuracy of the methods is better when compared with existing methods in terms of error as displayed in tables 1,2 and 3. hence fbm and sbm are reliable and efficient numerical models for solving third order initial value problems of ordinary differential equations. references [1] d. a. awoyemi, “class of continuous methods for general second order initial value problems in ordinary differential equations”, international journal of computer mathematics 72 (2009) 29. [2] s. o. fatunla, numerical methods for initial value problems in ordinary differential equations, academic press inc. harcourt brace jovanovich publishers, new york (1988) [3] p. henrici, “discrete variable methods in ordinary differential equations”, new york, ny; wiley 571.91 (1962) 419. [4] j. o. kuboye & z. omar, “numerical solution of third order ordinary differential equations using a seven-step block method”, international journal of mathematical analysis 9 (2015) 743. [5] j. d. lambert, “computational method in ordinary differential equation”, john wiley and sons, inc 571.91 (1973) 288. [6] b. t. olabode, “a six-step scheme for the solution of fourth order ordinary differential equation”, the pacific journal of science and technology 10 (2009) 143. [7] b. t. olabode, “block multistep method for third order ordinary differential equations”, futa journal of research in sciences 2 (2013) 194. [8] z. omar & m. suleiman, “solving higher order odes directly using parallel 2-point explicit block method”, matematika. pengintegrasian teknologi dalam sains matematik. universiti sains malaysia 21 (2005) 15. [9] a. m. sagir, “on the approximate solution of continuous coefficients for solving third order ordinary differential equations”, international journal of mathematical, computational, physical, electrical and computer engineering 8 (2018) 67. [10] j. sunday, “on the oscillation criteria and computation of third order oscillatory differential equations”, communications in mathematics and applications 9 (2018) 615. appendix a a0 = (xn + hs) × (2h + xn) h2(s − 1) yn+1 − (xn + hs) × (h + xn) h2(s − 2) yn+2 + 2h2 + 3hxn + xn2 h2(s2 − 3s + 2) yn+s − 2h2 + 3hxn + xn2 40320h5 s ( 3h6 s6 − 51h6 s5 +331h6 s4 − 1005h6 s3 + 1363h6 s2 − 425h6 s + 1228h5 sxn −425h5 xn + 1420h4 sxn2 − 135h4 xn2 + 616h3 sxn3 + 415h3 xn3 +116h2 sxn4 + 285h2 xn4 + 8hsxn5 + 65hxn5 + 5xn6 ) fn + 2h2 + 3hxn + xn2 10080h5(s − 1) ( 3h6 s6 − 45h6 s5 +223h6 s4 − 249h6 s3 − 1193h6 s2 + 1161h6 s − 1586h5 sxn +1161h5 xn + 258h4 sxn2 − 393h4 xn2 + 406h3 sxn3 + 9h3 xn3 +102h2 sxn4 + 183h2 xn4 + 8hsxn5 + 57hxn5 + 5xn6 ) fn+1 − 2h2 + 3hxn + xn2 6720h5(s − 2) ( 3h6 s6 − 39h6 s5 +143h6 s4 + 3h6 s3 − 277h6 s2 + 31h6 s − 228h5 sxn + 31h5 xn −92h4 sxn2 + 49h4 xn2 + 252h3 sxn3 − 89h3 xn3 + 88h2 sxn4 +109h2 xn4 + 8hsxn5 + 49hxn5 + 5xn6 ) fn+2 + 2h2 + 3hxn + xn2 336h5 s(s4 − 10s3 + 35s2 − 50s + 24) ( 13h6 s5 −h6 s6 − 57h6 s4 + 83h6 s3 + 27h6 s2 − 85h6 s − 85h5 xn −27h4 xn2 + 83h3 xn3 + 57h2 xn4 + 13hxn5 + xn6 ) fn+s + 2h2 + 3hxn + xn2 10080h5(s − 3) ( 3h6 s6 − 33h6 s5 +91h6 s4 + 3h6 s3 − 173h6 s2 + 49h6 s − 158h5 sxn + 49h5 xn −50h4 sxn2 + 15h4 xn2 + 154h3 sxn3 − 47h3 xn3 + 74h2 sxn4 +63h2 xn4 + 8hsxn5 + 41hxn5 + 5xn6 ) fn+3 − 2h2 + 3hxn + xn2 40320h5(s − 4) ( 3h6 s6 − 27h6 s5 +67h6 s4 + 3h6 s3 − 125h6 s2 + 39h6 s − 116h5 sxn + 39h5 xn −36 ∗ h4 sxn2 + 9h4 xn2 + 112h3 sxn3 − 33h3 xn3 + 60h2 sxn4 +45h2 xn4 + 8hsxn5 + 33hxn5 + 5xn6 ) fn+4 a1 = − 2h + 2xn + hs h2(s − 1) yn+1 + h + 2xn + hs (h2(s − 2) yn+2 − 3h + 2xn h2(s2 − 3s + 2) yn+s + 1 40320h5 s ( 9h7 s6 − 153h7 s5 + 993h7 s4 − 3015h7 s3 +4089h7 s2 + 1181h7 s − 850h7 + 6h6 s6 xn − 102h6 s5 xn +662h6 s4 xn − 2010h6 s3 xn + 2726h6 s2 xn + 12198h6 sxn −3090h6 xn + 20160h5 sxn2 + 14000h4 sxn3 + 6720h4 xn3 +4900h3 sxn4 + 7000h3 xn4 + 840h2 sxn5 + 2940h2 xn5 + 56hsxn6 +560hxn6 + 40xn7 ) fn − 1 10080h5(s − 1) ( 9h7 s6 − 135h7 s5 + 669h7 s4 − 747h7 s3 −3579h7 s2 + 311h7 s + 2322h7 + 6h6 s6 xn − 90h6 s5 xn +446h6 s4 xn − 498h6 s3 xn − 2386h6 s2 xn − 6162h6 sxn + 5394h6 xn +6720h4 sxn3 + 3640h3 sxn4 + 3360h3 xn4 + 756h2 sxn5 +2184h2 xn5 + 56hsxn6 + 504hxn6 + 40xn7 ) fn+1 + 1 6720h5(s − 2) ( 9h7 s6 − 117h7 s5 + 429h7 s4 + 9h7 s3 −831h7 s2 − 363h7 s + 62h7 + 6h6 s6 xn − 78h6 s5 xn +286h6 s4 xn + 6h6 s3 xn − 554h6 s2 xn − 1674h6 sxn + 382h6 xn +3360h4 sxn3 + 2660h3 sxn4 + 1680h3 xn4 + 672h2 sxn5 +1596h2 xn5 + 56hsxn6 + 448hxn6 + 40xn7 ) fn+2 − 1 336h5 s(s4 − 10s3 + 35s2 − 50s + 24) ( 39h7 s5 − 3h7 s6 73 j. o. kuboye et al. / j. nig. soc. phys. sci. 2 (2020) 69–76 74 −171h7 s4 + 249h7 s3 + 81h7 s2 − 255h7 s − 170h7 − 2h6 s6 xn +26h6 s5 xn − 114h6 s4 xn + 166h6 s3 xn + 54h6 s2 xn − 170h6 sxn −618h6 xn + 1344h4 xn3 + 1400h3 xn4 + 588h2 xn5 +112hxn6 + 8xn7 ) fn+s − 1 10080h5(s − 3) ( 9h7 s6 − 99h7 s5 + 273h7 s4 + 9h7 s3 −519h7 s2 − 169h7 s + 98h7 + 6h6 s6 xn − 66h6 s5 xn +182h6 s4 xn + 6h6 s3 xn − 346h6 s2 xn − 1050h6 sxn + 354h6 xn +2240h4 sxn3 + 1960h3 sxn4 + 1120h3 xn4 + 588h2 sxn5 +1176h2 xn5 + 56hsxn6 + 392hxn6 + 40xn7 ) fn+3 + 1 40320h5(s − 4) ( 9h7 s6 − 81h7 s5 + 201h7 s4 + 9h7 s3 −375h7 s2 − 115h7 s + 78h7 + 6h6 s6 xn − 54h6 s5 xn + 134h6 s4 xn +6h6 s3 xn − 250h6 s2 xn − 762h6 sxn + 270h6 xn + 1680h4 sxn3 +1540h3 sxn4 + 840h3 xn4 + 504h2 sxn5 + 924h2 xn5 +56hsxn6 + 336hxn6 + 40xn7 ) fn+4 a2 = 1 h2(s − 1) yn+1 − 1 h2(s − 2) yn+2 + 1 h2(s2 − 3s + 2) yn+s − 1 40320h5 s ( 3h6 s6 − 51h6 s5 + 331h6 s4 − 1005h6 s3 +1363h6 s2 + 6099h6 s − 1545h6 + 20160h5 sxn + 21000h4 sxn2 +10080h4 xn2 + 9800h3 sxn3 + 14000h3 xn3 + 2100h2 sxn4 +7350h2 xn4 + 168hsxn5 + 1680hxn5 + 140xn6 ) fn + 1 10080h5(s − 1) ( 3h6 s6 − 45h6 s5 + 223h6 s4 − 249h6 s3 −1193h6 s2 − 3081h6 s + 2697h6 + 10080h4 sxn2 + 7280h3 sxn3 +6720h3 xn3 + 1890h2 sxn4 + 5460h2 xn4 + 168hsxn5 + 1512hxn5 + 140xn6 ) fn+1 − 1 6720h5(s − 2) ( 3h6 s6 − 39h6 s5 + 143h6 s4 + 3h6 s3 −277h6 s2 − 837h6 s + 191h6 + 5040h4 sxn2 + 5320h3 sxn3 +3360h3 xn3 + 1680h2 sxn4 + 3990h2 xn4 + 168hsxn5 + 1344hxn5 + 140xn6 ) fn+2 + 1 336h5 s(s4 − 10s3 + 35s2 − 50s + 24) ( 13h6 s5 − h6 s6 − 57h6 s4 + 83h6 s3 +27h6 s2 − 85h6 s − 309h6 + 2016h4 xn2 + 2800h3 xn3 + 1470h2 xn4 + 336hxn5 + 28xn6 ) fn+s + 1 10080h5(s − 3) ( 3h6 s6 − 33h6 s5 + 91h6 s4 + 3h6 s3 −173h6 s2 − 525h6 s + 177h6 + 3360h4 sxn2 + 3920h3 sxn3 +2240h3 xn3 + 1470h2 sxn4 + 2940h2 xn4 + 168hsxn5 + 1176hxn5 + 140xn6 ) fn+3 − 1 40320h5(s − 4) ( 3h6 s6 − 27h6 s5 + 67h6 s4 + 3h6 s3 −125h6 s2 − 381h6 s + 135h6 + 2520h4 sxn2 + 3080h3 sxn3 +1680h3 xn3 + 1260h2 sxn4 + 2310h2 xn4 + 168hsxn5 + 1008hxn5 + 140xn6 ) fn+4 a3 = xn + hs 144h5 s (24h4 + 50h3 xn + 35h2 xn2 + 10hxn3 + xn4) fn 74 j. o. kuboye et al. / j. nig. soc. phys. sci. 2 (2020) 69–76 75 − xn(xn + hs) 36h5(s − 1) (24h3 + 26h2 xn + 9hxn2 + xn3) fn+1 + xn(xn + hs) 24h5(s − 2) (12h3 + 19h2 xn + 8hxn2 + xn3) fn+2 − xn 6h5 s(s4 − 10s3 + 35s2 − 50s + 24) (24h4 + 50h3 xn + 35h2 xn2 + 10hxn3 + xn4) fn+s − xn(xn + hs) 36h5(s − 3) (8h3 + 14h2 xn + 7hxn2 + xn3) fn+3 + xn(xn + hs) 144h5(s − 4) (6h3 + 11h2 xn + 6hxn2 + xn3) fn+4 a4 = − 1 576h5 s (50h4 s + 40hxn3 + 100h3 xn + 24h4 + 5xn4 + 105h2 xn2 + 4hsxn3 + 70h3 sxn + 30h2 sxn2) fn + 1 144h5(s − 1) (24h4 s + 36hxn3 + 48h3 xn + 5xn4 + 78h2 xn2 + 4hsxn3 + 52h3 sxn + 27h2 sxn2) fn+1 − 1 96h5(s − 2) (12h4 s + 32hxn3 + 24h3 xn + 5xn4 + 57h2 xn2 + 4hsxn3 + 38h3 sxn + 24h2 sxn2) fn+2 + 1 24h5 s(s4 − 10s3 + 35s2 − 50s + 24) (24h4 + 100h3 xn + 105h2 xn2 + 40hxn3 + 5xn4) fn+s + 1 144h5(s − 3) (8h4 s + 28hxn3 + 16h3 xn + 5xn4 + 42h2 xn2 + 4hsxn3 + 28h3 sxn + 21h2 sxn2)) fn+3 − 1 576h5(s − 4) (6h4 s + 24hxn3 + 12h3 xn + 5xn4 + 33h2 xn2 + 4hsxn3 + 22h3 sxn + 18h2 sxn2) fn+4 a5 = 1 1440h5 s (35h3 s + 60hxn2 + 105h2 xn + 50h3 + 10xn3 + 6hsxn2 + 30h2 sxn) fn − 1 360 ∗ h5 ∗ (s − 1) (26h3 s + 54hxn2 + 78h2 xn + 24h3 + 10xn3 + 6hsxn2 + 27h2 sxn) f n + 1 + 1 240h5(s − 2) (19h3 s + 48hxn2 + 57h2 xn + 12h3 + 10xn3 + 6hsxn2 + 24h2 sxn)) fn+2 − 1 12h5 s(s4 − 10s3 + 35s2 − 50s + 24) (10h3 + 21h2 xn + 12hxn2 + 2xn3) fn+s − 1 360h5(s − 3) (14h3 s + 42hxn2 + 42h2 xn + 8h3 + 10xn3 + 6hsxn2 + 21h2 sxn) fn+3 + 1 1440h5(s − 4) (11h3 s + 36hxn2 + 33h2 xn + 6h3 + 10xn3 + 6hsxn2 + 18h2 sxn) fn+4 a6 = − 1 2880h5 s (40hxn + 10h2 s + 35h2 + 10xn2 + 4hsxn) fn + 1 720h5(s − 1) (36hxn + 9h2 s + 26h2 + 10xn2 + 4hsxn) fn+1 − 1 480h5(s − 2) (32hxn + 8h2 s + 19h2 + 10xn2 + 4hsxn) fn+2 + 7h2 + 8hxn + 2xn2 24h5 s ∗ (s4 − 10s3 + 35s2 − 50s + 24 fn+s + 1 720h5(s − 3) (28hxn + 7h2 s + 14h2 + 10xn2 + 4hsxn) fn+3 − 1 2880h5(s − 4) (24hxn + 6h2 s + 11h2 + 10xn2 + 4hsxn) fn+4 a7 = 10h + 5xn + hs 5040h5 s fn − 9h + 5xn + hs 1260h5(s − 1) fn+1 + 8h + 5xn + hs 840h5(s − 2) fn+2 − 2h + xn 42h5 s(s4 − 10s3 + 35s2 − 50s + 24) fn+s − 7h + 5xn + hs 12605(s − 3) fn+3 + 6h + 5xn + hs 5040h5(s − 4) fn+4 a8 = − 1 8064h5 s(s4 − 10s3 + 35s2 − 50s + 24) (24 fn − 24 fn+s − 50s fn + 96s fn+1 75 j. o. kuboye et al. / j. nig. soc. phys. sci. 2 (2020) 69–76 76 −72s fn+2 + 32s fn+3 − 6s fn+4 + 35s 2 fn − 10s 3 fn − 104s 2 fn+1 + s 4 fn + 36s 3 fn+1 +114s2 fn+2 +11s2 fn+4 − 4s 4 fn+3 − 6s 3 fn+4 + s 4 fn+4 ) appendix b e =  20480 371589120 90112 371589120 28672 371589120 315392 371589120 430080 371589120 585111 371589120 665797 371589120 236334 371589120 4096 10321920 24576 10321920 14336 10321920 86016 10321920 129024 10321920 262947 10321920 346473 10321920 132678 10321920 −20480 6881280 −155648 6881280 −229376 6881280 602112 6881280 1290240 6881280 496249 6881280 2271083 6881280 1180914 6881280 4096 1270080 32768 1270080 57344 1270080 114688 1270080 344064 1270080 487059 1270080 351161 1270080 64806 1270080 −20480 30965760 −188416 30965760 −501760 30965760 143360 30965760 2795520 30965760 4595169 30965760 2242979 30965760 490434 30965760 20480 289013760 221184 289013760 888832 289013760 1634304 289013760 1290240 289013760 −837513 3870720 −693915 289013760 −200466 289013760  f =  −163840 371589120 −630784 371589120 172032 371589120 1576960 371589120 1720320 371589120 −1170222 371589120 −665797 371589120 32768 10321920 172032 10321920 86016 10321920 −430080 10321920 −516096 10321920 525894 10321920 346473 10321920 −163840 6881280 −1089536 6881280 −1376256 6881280 3010560 6881280 5160960 6881280 992498 6881280 2271083 6881280 32768 1270080 229376 1270080 344064 1270080 −573440 1270080 −3176256 1270080 974118 1270080 351161 1270080 −163840 30965760 −1318912 30965760 −3010560 30965760 716800 30965760 11182080 30965760 9190338 30965760 2242979 30965760 163840 289013760 1548288 289013760 752645332992 289013760 8171520 289013760 5160960 289013760 −1675026 289013760 −693915 289013760  76 j. nig. soc. phys. sci. 2 (2020) 51–54 journal of the nigerian society of physical sciences original research exponentially fitted chebyshev based algorithm as second order initial value solver e. o. adeyefaa,∗, o. s. esana amathematics department, faculty of science, federal university oye-ekiti, ekiti state, nigeria abstract in this research work, we focus on development of a numerical algorithm which is well suited as integrator of initial value problems of order two. exponential function is fitted into the chebyshev polynomials for the formulation of this new numerical integrator. the efficiency, ingenuity and computational reliability of any numerical integrator are determined by investigating the zero stability, consistency and convergence of the integrator. findings reveal that this algorithm is convergent. on comparison, the solutions obtained through the algorithm compare favourably well with the analytical solutions. keywords: chebyshev polynomial, convergence, ivps, numerical integrator, odes article history : received: 10 august 2019 received in revised form: 23 january 2020 accepted for publication: 19 february 2020 published: 05 march 2020 c©2020 journal of the nigerian society of physical sciences. all rights reserved. communicated by: b. j. falaye 1. introduction the search for numerical methods to accurately integrate ordinary differential equation is on the increase and of recent, much effort has been concentrated on solving initial and boundary value problems. variety of tools (polynomials) have been employed to develop numerical methods for integrating mathematical problems and modelling, an important perspective which cuts across all kind of mathematical problems. butcher [1], lambert [2, 3] and henrici [4] discussed extensively the approach of reducing higher order odes to a system of lower order, specifically, order one and then applying various methods available for solving the resulting system of rst order ivps. the increase in the number of equations resulting from this approach has been reported as its major setback (see [5, 6]). jator and li [7] formulated block methods which ∗corresponding author tel. no: +2348063307256 email address: adeyefa@gmail.com (o. s. esan) directly solve higher order ordinary differential equations without reducing the odes to system of first order equations. researchers such as [8-11] modelled series of algorithms which on their implementation, solve first order initial value problems. enoch and ibijola [12] developed a self-adjusting numerical integrator with an inbuilt switch for discontinuous initial value problems. the aim of developing new methods has always been to introduce a new approach with a target to reduce the error of approximation and thereby improve on the accuracy and efficiency of existing methods hence, this research paper. in this paper, we consider chebyshev polynomial owing to its elegant properties whereby exponential function shall be fitted into chebyshev polynomials to develop a direct integrator of second order ivps. 51 adeyefa & esan / j. nig. soc. phys. sci. 2 (2020) 51–54 52 2. formulation of exponentially fitted chebyshev method this section describes the formulation of an algorithm which directly integrate second order initial value problems. we set out by considering a function of the form f (x) = antn(x) + e x (1) where tn(x) is a chebyshev polynomial of first degree, ex is exponential function and an is a constant to be determined in the interval [a, b] , xn = a + nh, n = 0, 1, 2, . . . and h is the step size. we consider yn as numerical estimate to the theoretical value y(xn) and fn represents f (xn, yn) with assumption that the theoretical solution y(x) of a given ordinary differential equation can be locally represented in the interval [xn, xn+1] by the interpolating function described above. considering tn(x) in equation 1 for n = 0, 1, 2, . . ., we obtain f (x) = a0 + a1 x + a2(2x 2 − 1) + ex (2) where a0, a1 and a2 are constants and also real undetermined coefficient. interpolating equation 2 at point x, we obtain, for x = xn f (xn) = a0 + a1 xn + a2(2x 2 n − 1) + e xn ≡ yn (3) and for x = xn+1 f (xn+1) = a0 + a1(xn + h) + a2(2(xn + h)2 − 1) + exn +h ≡ yn+1 (4) where xn+1 = xn + h differentiating the interpolating function with respect to x and obtaining the first, second and third derivatives of the function, we have f ′ (xn) = a1 + 4a2 x + e x = fn (5) f ′′ (xn) = 4a2 + e x = f ′ n (6) f ′′′ (xn) = e x = f ′′ n (7) thus, equations 5, 6 and 7 become a1 + 4a2 x + e x = fn ≡ d1 (8) 4a2 + e x = f ′ n ≡ d2 (9) ex = f ′′ n ≡ d3 (10) subtracting equation 3 from equation 4 and simplifying yields y = a1(xn + h) + a2(2(xn + h)2 − 1) + exn +h − a1 xn − a2(2x2n − 1) − e xn (11) where y = yn+1 − yn, which gives yn+1 = yn + a1(xn + h) + a2(2(xn + h)2 − 1) + exn +h − a1 xn − a2(2x2n − 1) − e xn (12) solving equations 8 and 9, the values coefficients a’s are obtained and substituted in equation 2 to have yn+1 = yn + (d1 − (d2 − ex)x − ex)(xn + h) + 1 4 (d2 − x x)(2(xn + h)2 − 1) + exn +h − (d1 − (d2 − ex)x − ex)x − ex)xn − 1 4 (d2 − e x) (2x2n − 1) − e xn (13) equation 13 is therefore the new numerical integrator for integrating second order initial value problems. 3. analysis of the method the convergence of the scheme is investigated using lipschitz continuity theorem as discussed by fatunla [13, 14] and lambert [2]. consider f (x, y)− f (x, y∗) = ∂ f (x,y) ∂y (x, y ∗) and l = sup(x,y)∈dom ∂ f (x,y) ∂y where f (x, y) is defined for all points (x, y) in the region d = {(x, y)|xo ≤ x ≤ xn,−∞ < y < ∞}, xo and xn are finite and l is lipschitz constant such that for l ≤ 0 , | f (x, y) − f (x, y∗)| ≤ l|y − y∗| (14) f (x, y) − f (x, y∗) = ∂ f (x, y) ∂y (15) equation 14 can be satisfied if we choose l = sup ∂ f (x, y) ∂y (16) from equation 13 that is, the numerical scheme developed, we have yn+1 = yn + ( d1 − 4 ( 1 4 d2 − 1 4 e x ) x − ex ) (xn + h) + ( 1 4 d2 − 1 4 e x ) ( 2 (xn + h) 2 − 1 ) + exn +h − ( d1 − 4 ( 1 4 d2 − 1 4 e x ) x − ex ) xn −( 1 4 d2 − 1 4 e x ) ( 2x2n − 1 ) − exn by lambert [2] and fatunla [13] as mentioned above, f (x, y) − f (x, y∗) = ∂ f (x, y) ∂y (17) l = sup(x,y)∈dom ∂ f (x, y) ∂y by henrici [9], |φ (x, y∗; h) −φ (x, y; h)| ≤ l |y∗ − y| where x ∈ (a, b) , y ∈ (−∞,∞), a ≤ h ≤ ho; ho > 0. from equation 14, yn+1 = yn + ( d1 − 4 ( 1 4 d2 − 1 4 e x ) x − ex ) (xn + h) + ( 1 4 d2 − 1 4 e x ) ( 2 (xn + h) 2 − 1 ) + exn +h 52 adeyefa & esan / j. nig. soc. phys. sci. 2 (2020) 51–54 53 − ( d1 − 4 ( 1 4 d2 − 1 4 e x ) x − ex ) xn −( 1 4 d2 − 1 4 e x ) ( 2x2n − 1 ) − exn this is further simplified to have yn+1 = yn + h [(d2 − ex) x + ex − d1 + 1 4 (d2 − e x) (2h + 4x)] + ex+h − ex (18) recall that d1 = f(x, y) (19) d2 = f ′ (x, y) (20) substituting equations 19 and 20 into 18 and simplifying the resulting equation, we have yn+1 = yn + h [ 2(x + h2 ) ( f ′ (x, y) − ex ) + ex − f (x, y)] + ex+h − ex (21) we write the functional dependence yn+1 on the quantities xn, yn and h in the form φ (xn, yn; h) = [ ( f ′ (x, y) − ex ) ( 2x + h2 ) − f (x, y) + ex] (22) φ ( xn, y ∗ n; h ) = [ ( f ′ (x, y∗) − ex ) ( 2x + h2 ) − f (x, y∗) + ex] φ (xn, yn; h)− φ ( xn, y∗n; h ) = [ ( f ′ (x, y) − ex ) ( 2x + h2 ) − f (x, y) + ex] − [ ( f ′ (x, y∗) − ex ) ( 2x + h2 ) − f (x, y∗) + ex] (23) φ (xn, yn; h)− φ ( xn, y∗n; h ) = [[ ( f ′ (x, y∗) − ex ) ] − ( f ′ (x, y) − ex ) ] ( 2x + h2 ) +[ f (x, y∗) − f (x, y) ] (24) yn+1 = yn + hφ(xn, yn; h) applying equation 17, we have f(xn, y ∗ n) − f(xn, yn) = ∂f(xn, yn) ∂y (y∗n − yn) (25) f ′ (xn, y ∗ n) − f ′ (xn, yn) = ∂f ′ (xn, yn) ∂y (y∗n − yn) (26) where l1 = sup(x,y)∈dom ∂f(xn,yn ) ∂yn and l2 = sup(x,y)∈dom ∂f ′ (xn,yn ) ∂yn putting equations 25 and 26 into 24, we have φ (xn, yn; h)− φ ( xn, y∗n; h ) = [ ∂f ′ (xn,y) ∂y (y ∗ n − yn) (2x + h2 ] + ∂f(xn,y) ∂y (y ∗ n − yn) [f(x, y∗) − f(x, y)] (27) putting l1 and l2 into equation 27 and letting p = ( 2x + h2 ) , we obtain φ ( xn, y ∗ n; h ) −φ ( xn, y ∗ n; h ) = [ (l1 + pl2) ( y∗n − yn )] (28) if l1+pl2 = l, equation 28 becomes φ ( xn, y∗n; h ) −φ ( xn, y∗n; h ) = l ( y∗n − yn ) and ∣∣∣φ (xn, y∗n; h) −φ (xn, y∗n; h)∣∣∣ ≤ l ∣∣∣y∗n − yn∣∣∣. this shows clearly that the numerical scheme is convergent as it satisfies the lipschitz condition. 4. numerical experiments two test problems are considered here to implement the derived scheme. example 1: y ′′ = −6x y ′ − 4 x2 y, y(1) = 1, y ′ = 1; h = 0.132 . exact solution: y(x) = 53x − 2 3x4 . example 2: y ′′ = −y ′ , y(0) = 1, y ′ (0) = 1, h = 0.001. exact solution: y(x) = cosx. the graphical solutions of the examples above are shown below. figure 1: comparison of the exact and numerical solutions for example 1. 4.1. discussion of results we implement the constructed scheme on the two examples considered in this work. in figures 1 and 2, we compare the 53 adeyefa & esan / j. nig. soc. phys. sci. 2 (2020) 51–54 54 figure 2: comparison of the exact and numerical solutions for example 2. accuracy of the proposed method with the exact solution. from the graphs displayed, it is evident that the new scheme compares favourably with the exact solutions. 5. conclusion we have developed a scheme to solve second order initial value problems in ordinary differential equations (ode) using chebyshev polynomials with exponential function fitted into it. formulation of numerical integrator using the generated polynomials has been demonstrated. the scheme has been implemented using two test problems. on comparison, the solutions obtained through the numerical scheme recovers the analytical solutions. we therefore recommend the technique for numerical algorithm as we hope to extend the approach to solve boundary value problems. acknowledgments we thank the referees for the positive enlightening comments and suggestions, which have greatly helped us in making improvements to this paper. references [1] j. c. butcher “a modified multistep method for the numerical integration of ordinary differential equations”, journal of the acm, 12 (1965) 124. [2] j. d. lambert “computational methods in ordinary differential system”, john wiley, new york (1973). [3] j. d. lambert “numerical methods for ordinary differential systems”, john wiley, new york (1991). [4] p. henrici “discrete variable methods in ode”, new york, john wiley and sons (1962). [5] e. o. adeyefa “orthogonal-based hybrid block method for solving general second order initial value problems”, italian journal of pure and applied mathematics 37 (2017) 659. [6] s. n. jator & e. o. adeyefa “direct integration of fourth order initial and boundary value problems using nystrom type methods”, iaeng international journal of applied mathematics 49 (2019) 638. [7] s. n jator & j. li “a self-starting linear multistep method for a direct solution of the general second order initial value problem”, international journal of computational mathematics 86 (2009) 827. [8] e. a. ibijola “new schemes for numerical integration of special initial value problems in ordinary differential equations”, ph.d thesis, university of benin, nigeria (1997). [9] e. a. ibijola & p. kama “on the convergence, consistency and stability of a new one-step method for numerical integration of ode”, international journal of computational mathematics 73 (1999) 261. [10] o. o. enoch & a. a. olatunji: “a new self-adjusting numerical integrator for the numerical solutions of ordinary differential equations”, global journal gjsfr 12 (2012) 1022. [11] s. o. ayinde & e. a. ibijola “a new numerical method for solving first order differential equations”, american journal of applied mathematics and statistics 3 (2015) 156. [12] o. o. enoch & e. a. ibijola “a self-adjusting numerical integrator with an inbuilt switch for discontinuous initial value problems”, australian journal of basic and applied sciences 5 (2011) 1560. [13] s. o. fatunla “numerical methods for ivps, in odes”, academic press. usa (1988). [14] s. o. fatunla “a new algorithm for numerical solution of odes. computer and mathematics with applications”, 2 (1976) 247. 54 j. nig. soc. phys. sci. 4 (2022) 157–164 journal of the nigerian society of physical sciences a preliminary geotechnical assessment of residual tropical soils around osogbo metropolis as materials for road subgrade c. a. oyelamia,∗, w. akandeb, t. o. kolawoleb a department of geology, federal university, oye-ekiti, nigeria b department of geological sciences, osun state university, osogbo, osun-state. abstract failure of roads is common in most part of the tropical region of the world. majority of which can be attributed to poor selection of soil materials for road construction and climate. this study is aimed at investigating the properties of lateritic soil as material for road construction. following an initial visit to the study area, soil samples were taken and subjected to geotechnical analyses such as particle size analysis, natural moisture content, atterberg limits, compaction test, california bearing ratio (cbr), and direct shear test. all the tests were carried out according to the bs standard 1377 (1990). results revealed a well graded fairly coarse soil with natural moisture content varying from 6.2-29.4 %. atterberg limits shows a medium plasticity soil with medium compressibility and the soils classified within the a − 2, a − 6 and a − 7 of the aashto classification scheme. in terms of soil strength, the maximum dry density (mdd) and optimum moisture content varies from 1210-1520 kg/m3 and 13-24 %, respectively while california bearing ratio (cbr) value varied between 2.2745.32 % at 2.5 mm and 3.01 44.59 % at 5.0 mm. the shear strength of the samples ranged from 31.28-186.41 kn/m2. based on the results, the study concluded that the clayey nature of the soils was responsible for poor geotechnical properties which were generally below the specifications of the federal ministry of works and housing (fmwh), hence mostly not suitable in its present condition but could be improved to satisfy the conditions for use as sub-grade and sub-base soil in road construction. doi:10.46481/jnsps.2022.417 keywords: subgrade, subbase, geotechnics, cbr, soil strength, lateritic soils. article history : received: 20 september 2021 received in revised form: 14 december 2021 accepted for publication: 28 december 2021 published: 29 may 2022 c©2022 journal of the nigerian society of physical sciences. all rights reserved. communicated by: s. j. adebiyi 1. introduction all engineering structures such as roads, buildings, bridges, are founded on the ground. the majority of failed roads in the tropics can be attributed to geotechnical factors such as poor selection of sub-grade and sub-base materials [1, 2, 3] . the problem of premature failure of roads have been of great concern. most of the roads are placed on laterite (sub-grade) and ∗corresponding author tel. no: +2348033811411 email address: charles.oyelami@fuoye.edu.ng (c. a. oyelami ) laterite has also been used as part of their construction materials. it is therefore, important to study the properties of these residual soils on which the road is constructed. road failure is most time responsible for a greater percentage of road accidents in nigeria, this failure is often caused by poor selection sub-grade soils with invariably poor geotechnical properties such as high natural moisture and optimum moisture contents (nmc and omc), high plasticity index, low maximum dry density (mdd), poor bearing capacity, high compressibility and poor mineralogy [4, 5, 6, 7, 8] . for a material 157 oyelami et al. / j. nig. soc. phys. sci. 4 (2022) 157–164 158 to be used as either a base course or sub-base course depends on its strength in transmitting the axle-load applied on it [4, 5, 9]. roads are generally founded on natural soil (sub-grade) and are constructed with geological materials (rocks or soils), whose properties influence their durability and performances as transport medium. another factor that influences the performance of road especially in the south-western part of nigeria is the climatic factor, particularly temperature fluctuations, rainfall intensity and acid rain attacks; these have been adjudged to reduce the strength of the road base materials [10, 11, 12] . the laterite soil’s mineralogical composition determines geotechnical parameters such as specific gravity, shear strength, atterberg limits, and petrographic properties [13, 41] . the rapid cracking and damage to roads is worrying. the goal of this study is therefore to investigate the suitability of soils used as sub-grade or sub-base materials for the construction of new ring roads in and around osogbo, thus enhancing their durability. therefore, this current research work is focused on the preliminary assessment of the geotechnical properties of a proposed road, this is with the aim of establishing the background values which are intended to be followed up subsequently over a period of time. the paper presents the preliminary report of the geotechnical evaluation as a feasibility study for road construction. 2. location, accessibility and physiography the new ring road is located within osogbo metropolis. situated in the south-western part of osun state. geographically, it is bounded between latitudes n7◦44′0′′ n7◦48′0′′ and longitudes e4◦32′0′′ e4◦36′0′′. the area can be accessed through oke-baale road, from osogbo. it is less than 3 km away from osun state university osogbo campus and less than 1 km away from oja oba market. it is very close to ilesa garage. landform influences the formation and characteristics of tropical soils, especially lateritic soils [3, 14, 15, 16] . the physical settings of the study area and geology are shown in figure 1. the topography of the area can be described as undulating and the altitude of sampling points within the study area ranges from 333.0 346.2 m. the major river in the area is the osun river. the drainage pattern of the study area is dendritic. the area is well drained and a good number of streams exist as tributaries to the major rivers. the drainage map of the study area is shown in figure 2. 3. geology of the study area the study area (osogbo) is underlain by gneisses of the basement complex of nigeria which are precambrian in age [20, 21] . these gneisses are mostly weathered, but several figure 1: topographic map of osogbo (adapted from nigerian survey of iwo sheet, 1964) figure 2: drainage map of osogbo (adapted from nigerian survey of iwo sheet, 1964) road-cuts through these rocks still show features such as relict banding, quartz veins and quartz bands and including several pegmatites. although the pegmatites are wide spread, they are weathered [21, 22] . bodies of tonalite intruded into the gneisses, and the imprints from the gneisses probably formed relict bands on the tonalite. dolerite occurs as dykes within the tonalite, and locally causing colour changes within the tonalite near their contact. quartzites (massive and foliated) occur mainly as bands apparently sandwiched within the gneiss complex. the gneisses appear to be the oldest rocks, and it is on these that the metasediments (quartzites and possibly schist) were probably emplaced [20, 21, 22, 23] . the gneisses are formed from regional metamorphism of the pre-existing crustal rocks that were in place at the time. the sediments overlying the crustal rocks were metamorphosed in this process to produce the metasediments. tonalites are believed to represent diapiric emplacements of homogenized and mobilized basement [21, 22] . the pegmatites associated with the tonalite probably originated from the residual fluid phase of the tonalitic magma. some of the pegmatites present within the gneisses may be of replacement origin [20] . the dolerite, a basic igneous body intruded the pre-existing rocks in form of 158 oyelami et al. / j. nig. soc. phys. sci. 4 (2022) 157–164 159 dyke. figure 3 shows the geological map of osun state. figure 3: geologic map of osogbo (adapted after [40] ) 4. materials and methods the research involved collection of soil samples and laboratory analyses of the collected samples. six bulk disturbed samples were collected each at depths below 2m along the southern part of the new ring road, osogbo. proper labelling (obo 1 − 6) was done to avoid mix up and human error. samples collected were to the civil engineering department laboratory, osun state university, osogbo and soil laboratory of the federal polytechnic ado-ekiti for both index and engineering property analyses. index properties include natural moisture content, grain size analysis and atterberg limits while engineering properties test include compaction, california bearing ratio (cbr) and direct shear tests. all tests were carried out according to the bs 1377 [24] specification for soil tests. 5. results and discussions 5.1. mineralogical characteristics the summary of the analytical results of the mineralogy of soils from the study area is presented in table 1. it revealed the following minerals present in terms of abundance: kaolinite, goethite, quartz, muscovite, haematite and anatase. kaolinite is by far the most abundant mineral present in the soil with percentage as high as 94% in some of the sampled pits. this indicates the prevailing conditions within environment of decomposition with specific leaching and drainage conditions favouring kaolinite as the stable mineral [8, 25] . the lateritic nature of the soil samples is reflected in the percentage of kaolinite present, while the low percentages of haematite, quartz and goethite reflects the possible weak nature of the soil as previously stated [8] . generally speaking, the results of mineralogy reflect in particular the influence of underlying lithology, that is, parent rock of the soils derived. for instance, soils derived from pegmatite will have higher percentage of feldspar and mica which often decompose into clay particles like kaolinite as reflected in the present study. table 1: minerals present in the sampled soil. location no minerals present weight (%) obo1 anatase 7.79 hematite 2.99 kaolinite 88.64 quartz 2.58 obo2 anatase 1.64 hematite 4.46 kaolinite 72.93 muscovite 10.64 quartz 10.34 obo3 anatase 2.97 goethite 16.01 hematite 3.9 kaolinite 40.96 quartz 36.15 obo4 goethite 11.87 hematite 2.23 kaolinite 77.93 quartz 7.97 obo5 hematite 1.08 kaolinite 94.27 muscovite 2.25 quartz 2.40 obo6 hematite 0.82 kaolinite 89.4 muscovite 5.99 quartz 3.79 5.2. index properties of soil (grain size distribution and atterberg limits) results of the particle size distributions as presented in table 2 indicate that the soils are well graded with clay content ranging from 14%-62%. the percentage fine and coarse components of the soil samples from obo 1obo 6 range from 14-62% and 38-86%, respectively. the grain size distribution curves are presented in figure 4. according to the federal ministry of works and housing (fmwh) [26] specification, the fine content (clay + silt) for sub-grade, sub-base and base materials must not exceed 35%. based on this specification, obo 3 and obo 6 are the only fair soils for sub-grade, sub-base and base materials. high clay content in soil will be responsible for instability of road pavements in the area. nonetheless, soils with fines of less than 50% are expected to have stronger engineering properties, while those with fines of more than 50% are expected to result 159 oyelami et al. / j. nig. soc. phys. sci. 4 (2022) 157–164 160 in compaction problems in the field when used either as base course or as sub-base materials [27] . consequently, in terms of the amount of fines, obo 1 and obo 2a are not appropriate for use in road construction as sub-base or base course materials, because they may present problems. [28] and [29] noted that the sand particles contribute to the mechanical strength while the clay colloid content provides the plasticity or workability required. based on this specification, obo 2b-obo6 has higher mechanical strength than obo1 and obo2a. this variation is apparently due to the difference in mineralogical composition of the parent rocks from which the soils were derived. based on the american association of state highways and transportation officials (aashto) [30] soil classification, obo 3 (a-2-6(0)) and obo 6 (a-2-4(0)) are classified as good soils for sub-grade and sub-base, obo 2b and 5 are fair with gi of 4 and 5, respectively. the group index (gi) are useful in evaluating the quality of soil as highway sub-grade material. it gives an indication of the load carrying capacity of a subgrade soil within the aashto classification. the higher the gi number the lower the bearing capacity of the soil [12, 31] . figure 4: grain size distribution curves atterberg limits involves the determination of the liquid limit and plastic limit, and can be used to estimate the strength and settlement/compressibility characteristics of soils for road construction. the liquid limit and plastic limit ranges between 32.0%-47.2% and 19.7%25.9%, respectively (table 2). liquid limits of 40-60 % and above are typical of clay soils while values of 25-50% are typical of silty soils as illustrated in [24] . figure 5 shows the soil plasticity chart in the area under study. soils with liquid limits of ¡ 30 per cent are considered poor in plasticity and compressibility, those with liquid limits between 30 percent and 50 percent show medium plasticity, whereas those with liquid limits of > 50 per cent show high plasticity and compressibility. plasticity chart shows that obo 1-6 falls within medium plasticity. accordingly, [32] recommended a maximum liquid limit of 45% for base course materials with 15 for plasticity index. more recently, [26] , recommended 50%, 30% and 20% maximum for liquid limits, plastic limits and plasticity index respectively for use as sub-grade, sub-base and base materials. according to the new specification, all the samples were acceptable in terms of liquid and plastic limits. equally, all the soil samples except 2 (obo1 and 4) were found suitable based on pi. the average plasticity index which is the difference between the liquid limit and plastic limit was found to be 16.46. the plasticity index ranges between 10.1%-22.2%. according to [33] , laterite soils having a liquid limit above 30% and plasticity index above 12% are rated poor for use under bituminous surfacing. therefore, the samples can be rated poor for subgrade material. the average linear shrinkage for the samples is 3.16%. the linear shrinkage values for the soils ranged from 2.14%-4.29%. these values are within the limits of 8% maximum as specified by the [26]. figure 5: plasticity chart for soils in the study area. 5.3. engineering properties of soil (compaction, cbr and strength parameters) 5.3.1. natural moisture content (nmc) nmc is one of the most common parameters for soil. low moisture indicates that the soil is a dry one, whereas high moisture content indicates a wet soil. the study area has moisture content ranging from 6.2% -29.4% (table 2). some of these values are higher than the [26] recommended 5-15 percent range. samples with high nmc indicates poor response to ingress of water and weaker subgrade. only three samples, obo 3, 4 and 5 are suitable soils while the rest are not suitable for road construction [34, 12] . high nmc values have been found to negatively influence the shear strength of soils and are not generally suitable for road constructions [11] . 5.3.2. compaction test the higher the maximum dry density and the lower the optimum moisture content, the better the soil sample for construction work. maximum dry density and optimum moisture contents were derived from compaction tests conducted at 95% modified aashto (modified proctor), the maximum dry density (mdd) ranges from 1210 kg/m3 to 1520 kg/m3, while the optimum moisture content (omc) ranges from 13.0% to 24% (see table 3). according to [35] , samples classified as 160 oyelami et al. / j. nig. soc. phys. sci. 4 (2022) 157–164 161 table 2: index properties of sample soils sample no % gravel % sand % silt % clay % finer % coarse liquid limit % plastic limit % plasticity index linear shrinkage % fmwh (1997) aashto gi uscs remarks obo1 18 26 32 24 56 44 47.2 25 22.2 2.86 not suitable a-7-6 10 ml sandy silt obo2a 6 32 36 26 62 38 42 25.9 16.1 3.57 not suitable a-7-6 9 ml sandy silt obo2b 11 41 23 25 48 52 36.5 20.3 16.2 2.86 not suitable a-6 4 ml sandy silt obo3 66 20 2 12 14 86 33 22 11 3.57 suitable a-2-6 0 ml sandy silt with gravel obo4 12 40 24 26 48 52 41.1 19.7 21.4 2.86 suitable a-7-6 6 ml sandy silt obo5 18 36 18 28 46 54 40 21.8 18.2 2.14 suitable a-6 5 ml sandy silt obo6 51 17 4 28 32 68 32 21.9 10.1 4.29 suitable a-2-4 0 ml sandy silt with gravel range 666 1741 236 1228 1462 3886 32.047.2 19.725.9 10.122.2 2.144.29 average 26 30.29 19.86 24.14 43.71 56.29 38.83 22.37 16.46 3.16 table 3: engineering properties of sampled soils sample no moisture content (%) optimum moisture content % @ ma maximum dry density (kg/m3) @ ma cbr value (%) cbr rating (fmwh, 2010) cohesion (c) angle of shearing resistance (φ) shear strength (kn/m2) obo1 29.4 22.8 1370 4.01 poor (s2 subgrade) 30 7 31.28 obo2a 17.9 24 1260 3.01 poor (s2 subgrade) 50 5 50.87 obo2b 13 19.2 1460 30.06 good (s6 subbase/ subgrade) 60 5 60.87 obo3 6.2 15 1520 44.59 good (s6 subbase/ subgrade) 185 8 186.41 obo4 13 21.8 1210 42.59 good (s6 subbase/ subgrade) 98 11 99.94 obo5 11 18.2 1410 26.55 good(s5 subgrade) 80 10 81.76 obo6 17.8 13 1340 26.05 good(s5 subgrade) 120 8 121.41 range 6.2-29.4 13.024.0 1210-1520 3.0144.59 30-185 5-11 31.28186.41 average 15.47 19.14 1367.14 25.27 89 7.71 90.36 sub-base and sub-grade materials with high value of maximum dry density and low optimum humidity content are great. also, [36] specified optimum moisture content (omc) less than 18% and minimum 1700kg/m3 mdd for both sub-base and sub-grade materials. based on these specifications, most of 161 oyelami et al. / j. nig. soc. phys. sci. 4 (2022) 157–164 162 the soil are not suitable as sub-base and sub-grade material especially for a high load carrying flexible pavement. [37] , stated that for a good soil, the lower the optimum moisture content, the better its workability and that increase in dry density is an indicator of improvement. according to [38] , soils with mdd greater than 1610 kg/m3 are suitable for use as landfill barrier materials. none of the samples conform to the specification. 5.3.3. california bearing ratio (cbr) the cbr values vary from 3.01% to 44.59% (table 2). recommendation from the federal ministry of works and housing [36] for use as: sub-grade, sub-base and base materials are: approximately 10%, approximately 30% and approximately 80 %, respectively for unsoaked soil. this means that obo 1 and obo2a with values below 10 percent are excellent cbr-based sub-grade materials; obo 1, 2a, 5, and 6 having values below 30 percent are decent sub-base materials. all areas have less than 80 per cent of their unsoaked cbr values, which is the overall acceptable value for soils to be used as base materials [36] . by interpretation in terms of the strength of the soils, they are adequate for use as a sub-grade or sub-base road construction materials. obo 1 and obo 2a with poor cbr are not likely to provide a stable sub-grade material. the rating of soil according to cbr and their corresponding use has been classified according to [26] is presented in table 3. 5.3.4. shear strength characteristics the shear strength has been defined as the maximum resistance of a soil to shear stress before failure and it is an important mechanical property of soil which controls its stability under load. shear strength is dependent on the nature of the soil, inter-particle frictional resistance, force of attraction (cohesion) and fluid pressure within its pores [39] . the strength parameters from the tested soils (obo 1obo 6) shows that the cohesion varies between 30-185 kn/m2, angle of shearing resistance between 5 − 11◦ and the shear strength between 31.28-186.41 kn/m2 . from the result shown in table 2, the impact of percentage fine and percentage coarse affected the strength of the soils. for instance, obo 1 and obo 2a contain a higher percentage of fine and lower percentage of coarse while obo 2bobo 6 contain lower percentage of fine and higher percentage of coarse. the influence is reflected in the strength parameters as obo 1 and 2a have lower strength compared to obo 2bobo 6 with higher strength value. obo 3 has the lowest value of moisture content with 6.2 and it has the highest shear strength 186.41 kn/m2 . figures 6-8 illustrates the relationships cbr and pl, shear strength and liquid limits. a negative correlation exists between cbr against plastic limits and liquid limits while a fairly strong correlation exists between cbr and shear strength. 6. conclusions and recommendations the subgrade investigation of the south-east (se) section of the new ring road in osogbo metropolis has been undertaken. figure 6: relationship between cbr and pl figure 7: relationship between cbr and shear strength figure 8: relationship between cbr and liquid limits the result revealed a well-graded inorganic sandy silt soils, with fair compaction characteristics and good drainage. the study reveals that the soil samples are mostly wet soil based on the prevailing hydrological and climatic conditions of the study area and hence, would require an adequate drainage structure to maximise the potentials of the soil as subgrade material. atterberg limits reflected a subgrade with appropriate ll, pl and pi according to specifications. engineering properties of the soils revealed mostly cohesive soil with 162 oyelami et al. / j. nig. soc. phys. sci. 4 (2022) 157–164 163 reduced strength, whereas, some of the sampled soil possess a good strength parameter especially those with low moisture contents. cbr classified the samples between poor and good subgrade categories (s2-s6) with 3 samples meeting the cbr conditions for use as subbase and subgrade materials (s5 and s6). meanwhile, the weak nature of the soil have 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[41] a. n. amadi, c. j. eze, c. o. igwe, i. a. okunlola, & n. o. okoye, “architect’s and geologist’s view on the causes of building failures in nigeria”, modern aplied science, 6 (2012) 31. 164 j. nig. soc. phys. sci. 4 (2022) 832 journal of the nigerian society of physical sciences countermeasure to structured query language injection attack for web applications using hybrid logistic regression technique shehu magawata shagaria,∗, danlami gabia, nasiru muhammad dankoloa, noah ndakotsu ganab adepartment of computer science, kebbi state university of science and technology, aliero, nigeria bdepartment of cyber security science, federal university of technology, minna, nigeria abstract the new generation of security threats has been promoted by real-time applications, where several users develop new ways to communicate on the internet via web applications. structured query language injection attacks (sqlias) is one of the major threats to web application security. here, unauthorised users usually gain access to the database via web applications. despite the giant strides made in the detection and prevention of sqlias by several researchers, an ideal approach is still far from over as most existing techniques still require improvement, especially in the area of addressing the weak characterisation of input vectors which often leads to low prediction accuracy. to deal with this concern, this paper put forward a hybrid optimised logistic regression (lr) model with improved term frequency inverse document-frequency (itfidf-lr). to show the effectiveness of the proposed approach, attack datasets is used and evaluated using selected performance metrics, i.e., accuracy, recall, specificity and false positive rate. the experimental results via simulation when compared with the benchmarked techniques, achieved performance record of 0.99781 for accuracy, recall and f1-score as well as 0.99782, 0.99409 and 0.00591 for precision, specificity and false positive rate (fpr) respectively. this is an indication that the proposed approach is efficient and when deployed is capable of detecting sqlia on web applications. doi:10.46481/jnsps.2022.832 keywords: database management system, logistic regression, sql injection attack article history : received: 25 may 2022 received in revised form: 07 august 2022 accepted for publication: 08 august 2022 published: 01 october 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: j. ndam 1. introduction the internet, has witnessed tremendous growth over the past decade, making the web and various internet products of great research interest [1]. with this growth, the netcraft web server overviews as at january 2019 estimated that there are ∗corresponding author tel. no: +234 8036256995 email address: shagari1978@gmail.com (shehu magawata shagari) over 1.8 billion sites and over 4.39 billion internet users. with these increased number of websites comes with increased risks of security challenges confronting websites and other web applications. top on the list of security risks on web applications and sites is the structured query language injection attacks (sqlias) and are lethal and account for 51% of attacks [2]. the sqlias are attacks in which an attacker finds out the weakness of a web application and exploits the weakness by executing malicious statement through the internet to 1 shagari et al. / j. nig. soc. phys. sci. 4 (2022) 832 2 access private data from the database [3]. a database is a collection of interrelated and organised data [4]. the management of database is most significant to ensuring information are kept in secrecy while authorising only those that have permission to access resources within the database. to interact with the database, a query language such as structured query language (sql) is required. thus, database management system (dbms) is therefore software that provides a functionality of creating database, maintaining or updating the data in the database [5]. the common types of dbms are hierarchical, network, object-oriented and relational database models [6]. the unauthorised access of the database greatly endangers web applications as the database is a key component used by all web applications to store all the data required by the application [2]. sqlias occur when there is no validation of user inputs, cookies and input parameters, before they are passed to sql queries that will be executed on the database. there are three basic types of sqlias: union based sql injection attack, error based sql injection attack and blind sql injection attack [7]. union based sqlias are attacks where union statement is used. that is, two statements are joined by the attackers to get information from the database, on the other hand, error based sqlias are perpetrated by attackers through querying the server with a view to causing error and using the error message to determine vulnerabilities that can be exploited to attack the system. the blind sqlias are dubbed the hardest forms of sqlias as the attackers only extract data by querying the server. blind sqlias can either be boolean or time based [8-11]. when successful, these attacks enable attackers bypass the system authentication and gain control of the database and consequently private information [12, 13]. thus, enabling attacker to change user’s password, retrieve users, make illegal transaction, delete table or can damage the database as well as perpetrate several other illegalities. the widespread use of sqlias has led to development of several methods for the detection and prevention of such attacks. some of these methods can widely be seen e.g., in [1422]. although, these solutions have contributed immensely towards providing an understanding on how sqlias attacks do occur and addressed, however despites these giant leaps, an ideal solution is far from been achieved. with the negative impact of sqlias, several researchers in [23-29] have tried to address concerns arising from such attacks through provision of techniques to serve as potential solutions. despites the giant strides made in the detection and prevention of sqlias, the following limitations still exist: 1. weak characterisation of the input vector as witnessed by most machine learning based methods, leading to low accuracy and unsatisfactory recall rate. 2. weak representation of attributes as witnessed by text vectorisation-based algorithms which leads to inaccurate description of keyword weight and consequently low prediction accuracy. 3. the use of svm by some of the existing schemes reduces accuracy as they inherit the weaknesses of svm which lacks the ability to perform well when the data set has more noise and the number of features for each data point exceeds the number of training data sample therefore, in this paper, we proposed an optimised logistic regression model based based-improved term-frequencyinverse-document-frequency (itfidf-lr) to serve as a potential solution. simulation results show the proposed approach achieved performance record of 0.99781 for accuracy, recall and f1-score as well as 0.99782, 0.99409 and 0.00591 for precision, specificity and false positive rate (fpr) respectively. this is an indication that the proposed approach is efficient and when deployed is capable of detecting and preventing sqlia in dbms. the contribution of this paper is as follows: 1. determination of existing gaps through exploring existing techniques for countering sql injection attacks from literatures, 2. an optimised logistic regression model for prevention of sql injection attacks in dbms, 3. an improved itfidf-lr approach for the detection of sqlia in dbms the rest of this article is organised as follows. section 2 provides discussion on related work. discussion on idf, tfidf and logistic regression are provided in section 3. section 4 discusses the experimental model with proposed testbed and simulation. performance evaluation metric were presented in section 5. section 6 is the results and discussion section while section 7 concludes the paper. 2. related work with the negative impact of sqlias, several researchers have tried to address concerns arising from such attacks through provision of techniques to serve as an ideal. some of these researcher(s) and their techniques are discussed as follows: hassan et al. [23] proposed a deep neural network-based technique for the detection of sql injection vulnerability. the proposed method which is targeted at addressing the challenging effects associated with financial loss in business, application compromise and administrative exploit claimed to have outperformed existing methods in detections of sql injection attack with the accuracy of 98.04% over 1850 dataset records, however, improve performance can be recorded if a significantly available dataset is used for training the machine learning algorithms. yu et al. [24] proposed a novel technique for the detection of sql injection attacks, the technique encompasses tokenization, skin-gram model in word2vec and svm algorithm with eigenvectors, the proposed model was said to have obtained an effective means of detection of sql injection attack with an unspecified claimed good accuracy, 0.35%, and 1.9% was also achieved for fpr and fnr respectively, however, though the accuracy score was undisclosed, better performance can be attained, with regards to aforementioned scores 2 shagari et al. / j. nig. soc. phys. sci. 4 (2022) 832 3 in fpr and fnr it is of no doubt that performance can be improved upon. pan et al. [25] proposed a robust software modelling tool with the capability of detecting threats from sql injection attacks and cross-site scripting, the following machine learning models were used in the experimental analysis for sql injection attacks, naı̈ve bayes, random forest, and svm, the study claimed to have achieved an efficient and accurate method of detecting and checkmating threats from sql injection and cross-site scripting attacks with the following performance scores; 0.941, 1.00 and 0.933, for precision score and 0.800 each for recall, while f-score of 0.865 and 0.889 respectively for naı̈ve bayes, random forest and svm, nevertheless, a considerable significant sample dataset can aid in the effective detection of sql attack injection as the sample dataset used for this study is slightly above 200 sample point. krishnan et al. [26] performed research on sql injection detection based machine learning, the effectiveness of some selected machine learning algorithms was evaluated in the quest to establish the most robust learning model, the selected models that were explored are naı̈ve bayes, logistic regression, cnn, svm and passive-aggressive algorithms, with cnn proving the most optimal model for sqli attack detection after outperforming other models compared against with 97%, 0.92 and 0.96 for the performance matric accuracy, precision and recall respectively, however, the study further proposed research of other machine learning model and its optimisation for enhancing performance. farooq [27], an ensemble machine learning model was employed to detect sqli attack, gradient boosting machine, adaptive boosting, extended gradient boosting machine and light gradient boosting machine learning algorithms were analysed, the optimal result was obtained from light gradient boosting machine with the following performance scores; 0.9934 each for accuracy, precision, recall, and f1 score respectively, and 0.0093, 0.0146, 0.1208 and 0.007 for mae, mse, rmse and fpr respectively, however, with the tuning of model the performance rate for sqli attack can be improved. in order to address the challenge associated with machine learning-based detection for sqli attack, a novel system termed semantic query-featured ensemble learning model for sqli attack was proposed in [28]. it was claimed that the proposed model achieved the following optimal performance for accuracy, f-score, and auc of 98%, 0.989, 0.999 respectively, however, with an optimised model a more enhanced performance can be achieved when compared to contemporary techniques of model optimisation and dataset enhancement. a random forest-based sqli attack detection approach was introduced in [29] to address the challenge of detection efficiency for sqli attack detection, the performance score achieved in the experiment was 97% and 95% for precision and accuracy respectively, however, contemporary research has it that more effective means of detection of malware attack exist that can offer a robust performance result. despites the giant strides made in the detection and prevention of sqlias, the following limitations still exist. the existing schemes; 1. weak characterisation of the input vector as witnessed by most machine learning based methods, leading to low accuracy and unsatisfactory recall rate. 2. weak representation of attributes as witnessed by text vectorisation-based algorithms which leads to inaccurate description of keyword weight and consequently low prediction accuracy. 3. the use of svm by some of the existing schemes reduces accuracy as they inherit the weaknesses of svm which lacks the ability to perform well when the data set has more noise and the number of features for each data point exceeds the number of training data sample. 3. inverse document frequency (idf), term frequencyinverse document frequency (tf-idf) and logic regression models this section discusses three existing approaches in the proposed technique. these approaches were later improved as shown in section three (3) to achieve better results when compared to the benchmarked scheme. 3.1. inverse document frequency (idf) the inverse document frequency (idf), is the logarithm of the number of the queries in the corpus divided by the number of queries where a specific term appears [30]. inverse document frequency (idf), measures how important a term is. the idf attempts to weigh down the frequent terms and scale up the rare ones. it is given as: i df(t) = log ( total number o f documents number o f documents with term t in it ), (1) 3.2. term frequency-inverse document frequency (tf-idf) the tf-idf is a statistical measure used to evaluate how important a word is to a document in a collection or corpus. the importance increases proportionally to the number of times a word appears in the document but is offset by the frequency of the word in the corpus. it is often used as a central tool in scoring and ranking a document’s relevance given a user query. tf-idf can be successfully used for stop-words filtering intext summarisation and classification [31]. the tf-idf weight comprises of two elements: the first element is the normalised term frequency (tf), also referred to as the number of times a word appears in a query, divided by the total number of words in that query; the second term is the inverse document frequency (idf), which is the logarithm of the number of the queries in the corpus divided by the number of queries where the specific term appears [30]. term frequency, measures how frequently a term occurs in a document. since every document is different in length, it is possible that a term would appear much more times in long documents than shorter ones. thus, the term frequency is often divided by the document length also known as the total number of terms in the document as a way of normalisation: t f(t) = number o f times term t appears in a document t otal number o f terms in the document , (2) 3 shagari et al. / j. nig. soc. phys. sci. 4 (2022) 832 4 while computing tf, all terms are considered equally important but since certain terms, such as ”is”, ”of”, and ”that”, may appear a lot of times but have little importance. thus, the commonly used formular for the tf-idf is shown in equation 3 [32]: w(t, d) = t f(t, d) log ( n nt ) , (3) where, t f(t, d) is the term frequency of the keyword t appear in the text d,n represents the number of full textsnt represents the number of texts which have the word t. 3.3. logistic regression model logistic regression (lr) model is used to solve the classification problems. it is an extension of linear regression [3335], where the dependent variable is categorical based on the concept of probability. in logistic regression, dependent variable is a binary variable that contains data represent as 1 (yes, success, spam) or 0 (no, failure, not-spam). the binary logistic regression maps the regression lines onto the interval [0,1], which is compatible with the logical range of probabilities. the core function of the model is logistic function or sigmoid function that can hold any real value and map it between 0 and 1. a simple logit model is shown in equation 4: in ( π 1 −π ) = log(odd s) = logit = α + βx (4) π = probability(y = outcome o f interest|x = x) = eα+βx 1 + eα+βx , (5) where, π is the probability of the outcomes of interest, or the event, under variable y , α is the y intercept, and β is the slope parameter and within the inferential framework, the null hypothesis states that β is equal to zero in the population. as a result, by rejective such a null hypothesis signifies that a relation exists between x and y . x can be categorical or continnous whereas y is always categorical. by taking the antilog of equation 4 on both sides, an equation to predict the probability of the occurrence of the outcome of interest can be derived as shown in equation 5. 4. experimental model with proposed testbed and simulation this section discusses the experimental model with the proposed testbeds as well as the simulation environment used in the implementation. 4.1. proposed itfidf-lr model the proposed improved term frequency-inverse document frequency with optimised logistic regression (itfidf-lr employ both itfidf and the optimised lr model for vectorisation. in the default tfidf equation as shown in equation 3, n √ t f is substituted for t f, to eliminate excessive influence of table 1. sample dataset sample (sentence) percentage (%) sqli attack query 62 normal sql query 38 term frequency and reflect the balance of weight [36]. similarly, the word class weight coefficient pt and that of the position weight coefficient bt are incorporated into the traditional tf-idf formula, leading to the formation of an itf-idf shown in equation 6: w(t, d) = n √ t f(t, d) × log ( n nt ) × pt × bt (6) an optimised logistic regression model is then put forward to maximise several predictors as well the likelihood of obtaining data given its parameter estimates. the optimised lr model is shown in equation 7: in ( π 1 −π ) = α + βx1 + β2x2+, . . . , +βkxk (7) π = probability (y = outcomeo f interest | x = x1, x ) = eα+β1x1+β2x2,+...,+ βkxk 1 + eα+β1x1+β2x2+,...,+ βkxk (8) 4.2. datasets this research make use of sqli attack dataset [37] from a popular dataset repository known as kaggle. the instances of the sqli attack dataset contain 34,084 instances. the sample dataset is shown in table 1. two features from the historic dataset was chosen; namely; sentence and class. the sentence needs to be detected as either normal or sqli attack query while the second feature which is the class stands for a numeric value to determine whether it is a normal sentence or sqli query. in our experiment, the value 1 has been used to represent the sentence as a sqli query and 0 has been used to represent the sentence as normal statement. 4.3. trained model in the detection method, the proposed sqlias is based on itfidf-lr, comprising of data pre-processing phase and optimised logistic regression model training and detecting phase. the pre-processing phase involves first getting the dataset. then each data in the dataset is marked. the statement is marked as 1 when it is an attack sample and is marked as +1 when it is not. then, a word segmentation tool (happierfuntokenising) is used to segment sql statements in the dataset. in addition, in order to reduce the difference of different features during the experiment, normalisation is carried out to bring all values to a limited range. that is, the min-max method is used to normalise the data using equation 9: xnorm = x − xmin xmax − xmin (9) 4 shagari et al. / j. nig. soc. phys. sci. 4 (2022) 832 5 figure 1. flowchart of the proposed algorithm where, x denotes the current sample data value, xmin denotes the minimum of the sample data, xmax denotes the maximum of the sample data, and x norm denotes the normalised value. the feature dataset is then grouped and divided into training set and testing set in the model training and detecting phase with a composition of 70:30, that is 70% for training and 30% for testing the model. grid search optimisation algorithm was deployed to enhance the parameters of logistic regression model. the training set is used to train the optimised logic regression classifier model after which the testing set is used to verify the generated model and results of the classification generated are then evaluated. the flowchart for the proposed algorithm is shown in figure 1. 4.4. simulation environment the detail experimental hardware environments used are shown in table 2. likewise, grid search optimisation algorithm was deployed to enhance the parameters of logistic regression model after a complex computational operation was performed on the following defined ranges; the cost (c), random state, penalty and solver parameters, optimised logistic regression parameter, and default logistic regression parameter as shown in table 3. figure 2. comparative analysis of optimised and default lr based itfidf 5. performance evaluation metrics after the data pre-processing and the implementing stages, the proposed approach is evaluated against the benchmarked scheme based on the metrics shown in table 4. that is, accuracy, precision, sensitivity, specificity, f1 and fpr as used in [38, 39]. 5.1. results and discussion this section presents an analysis of optimised lr based itfidf and the default lr based itfidf, table 5 while figure 2 show graphical presentation as achieved based on the analysis performed in this research, a clear cut view gives insight of the enhanced performance and the capability strength of an optimised model compared to using a default parameter of a model, in the developed model for detection and prevention of sqlia, there is a distinct outperformance record achieved across the board for performance metric evaluation analysis, such as outlined in succeeding subsections. the accuracy score variation between the optimised and default lr based itfidf is 0.04847 which is quite significant in this analysis, 0.99781 was achieved through optimised lr based itfidf against 0.94934 achieved through the default lr based itfidf, this indicates the superiority of the optimised lr based itfidf model against the training of the model with default lr based itfidf model. the developed research 5 shagari et al. / j. nig. soc. phys. sci. 4 (2022) 832 6 table 2. properties of the experimental environment items properties window windows 10 operating system processor intel (r) core (tm) i7-7300hq cpu@2.5ghz software visual studio 2016, programming language python table 3. defined lr parameter for optimisation optimised lr parameter default lr parameter cross validation (cv) 5 5 cost(c) 100.0 100.0 random state 7 7 penalty ‘l1’ ‘l2’ solver ’liblinear’ ’libfgs’ table 4. performance evaluation metrics metric formula accuracy = true positves +true negstotal number of examples precision = true positvestrue positves +true negatives sensitivity / recall = true positvestrue positves + false negatives specificity = true negativestrue negatives+true positves false positive rate (fpr) = fp(tn+fp) f1 score = 2 ∗ tp2 ∗ tp + fn + fp model for detection and prevention of sqlia analysis evaluation of precision score reveals a definitive outperformance score with an excess score of 0.04623, the following precision score was achieved; 0.99782 and 0.951596 respectively for optimised and default lr based itfidf model. recall rate of 0.99781 and 0.94956 was recorded in this research for optimised and default lr based itfidf developed model, the result proves the enhancement performance rating of the optimised model with 0.04825 superiority. the following scores was achieved from the result analysis of the developed model for sqlia, 0.99781 and 0.94866 for the optimised and default lr based itfidf f-score performance, this is an indication outperformance of 0.04915 greater than and against default lr based itfidf model when compared to the optimised lr based itfidf model. a record of 0.99409 and 0.87659 was achieved for sensitivity score, this is an outperformance experienced with about 0.100 depicting the optimality of the tuned parameters of lr based itfidf in this research. the result obtained from this research based on the optimised lr based itfidf proves the promising feature expectant of a tuned model, the optimised lr based itfidf fpr achieved 0.00591 against the default lr based itfidf with a score of 0.12341, this presents an obvious width margin that depicts the efficiency of an enhanced lr based itfidf based on the excellent result achieved. the analysis of the developed detection and prevention model for sqlia against few other approaches in the literatures is presented to establish strength of the developed model in the detection and prevention of sqlias in terms of optimal performance, to this end the performance of the developed model was compared against the existing sqlia detection techniques based on relevant performance evaluation metrics found in literatures, metrics such as accuracy, precision, recall, f1-score, specificity and fpr. table 6 present scores of the performance metrics achieved in this research alongside that of baseline literatures, the performance record from this research outperformed other techniques in respect to accuracy, precision, recall, f1score, specificity and fpr. in addition, deducing from table 5, accuracy record score achieved is the must employed performance evaluation metric in the field of sqlia detection involving machine learning based models, this is based on the fact that it is most common in the baseline reviewed literatures. the developed model for detection and prevention of sqlia achieved an optimal accuracy of 0,99781, which is followed by the technique employed by [27], [28] and [29] with a distinct wide margin in accuracy of 0.9934, 0.98 and 0.95 respectively while [29] have the least accuracy record of 0.95. the significant performance recorded by this research reflects how correctly the developed model can detect and prevent sqlias in dbms environment, a low fpr of 0.00591 was recorded in the developed model, although baseline literature reviewed in this research did not capture fpr, fpr is an important evaluation metric in detection of attacks in computing environment as it is recorded for literatures earlier than 2021 articles reviewed. the precision score of 0.99 was achieved for the developed model in this research and the research by [27], however, the precision that determine the exactness of the model developed in this research for the detection and prevention of sqlia in dbms environment outperformed that of [27] as well as [28] with a score of 0.99782 superseding 0.993 and 0.97 for [27] and [29] respectively. the recall performance score recorded in this research which entails the measure of the completeness of the performance of detection of sqlia achieved 0.99781, against that of [27], f1-score of 0.99781 with a significant difference against [23] as well as [27], though, [27] score 0.9934 that is slightly above 0.99, [29] had the worst performance score of 0.989, the robustness of the developed detection and prevent model for sqlia in dbms environment have established its 6 shagari et al. / j. nig. soc. phys. sci. 4 (2022) 832 7 table 5. results of the optimised lr based itfidf and the default lr based itfidf metrics / approach accuracy precision recall f-score specificity false positive rate default lr based itfidf 0.94934 0.95159 0.94956 0.94866 0.9487 0.1234 optimised lr based itfidf 0.99781 0.99782 0.99781 0.99409 099409 0.00591 table 6. comparative analysis of optimised lr based itfidf with baseline literatures reference approach accuracy precision recall f1-score specificity fpr developed detection and prevention model optimised lr based itfidf 0.99781 0.99782 0.99781 0.99781 0.99409 0.00591 [28] ensemble learning model 0.98 0.989 [29] random forest model 0.95 0.97 [27] ensemble machine learning model 0.9934 0.9934 0.9934 0.9934 n.b: (-) means the metric value is not reported in the existing approach. optimality capability across all performance metrics relevant in the field of this research area. though specificity and fpr was not recorded for the baseline journal model, this research used the performance metrics based on the fact that they are being employed for analysis engaging detection-based machine learning models, the developed detection and prevention model achieved the scores of 0.99409 and 0.00591 for specificity and fpr respectively, showing the efficient capability in detection of sqlia in dbms environment 6. conclusion the new generation of security threats have been promoted by real-time applications, where several users develop new ways to communicate on the internet via web applications. the internet is witnessing growth with several threats on a daily basis to the web application. sqlia is one of a kind of these threat, since it serves as a medium for other several severe attacks. sqlia, which is currently tagged as one of the most notorious means of attacking database of a system continue to haunt the securities of websites from multiple angles. this can only be checkmate through consistently deploying an evolving defense technique. hence, in this paper, we proposed an itfidf-lr model to serve as a potential solution. the developed model for detection and prevention of sqlia employed itfidf with an optimised logistic regression model to addressed threats that springs up from sqlia in web application. the optimal performance as revealed from the analysis performed in this paper shows performance record of 0.99781 for accuracy, recall and f1-score respectively as well as 0.99782, 0.99409 and 0.00591 for precision, specificity and fpr respectively as 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[39] l. wahab & h. jiang. “a comparative study on machine learning based algorithms for prediction of motorcycle crash severity,” plos one 14 (2019) 1. 8 j. nig. soc. phys. sci. 2 (2020) 12–25 journal of the nigerian society of physical sciences original research block third derivative trigonometrically-fitted methods for stiff and periodic problems r. i. abdulganiya,∗, o. a. akinfenwab, o. a. yusuffb, o. e. enobaborc, s. a. okunugab adistance learning institute, university of lagos, nigeria bdepartment of mathematics, university of lagos, nigeria cdepartment of mathematics, yaba college of technology, lagos, nigeria abstract this article constructed and implemented a family of a third derivative trigonometric fitted method of order k + 3 whose coefficients are functions of frequency and step size for the integration of systems of first-order stiff and periodic initial value problems. the block third derivative trigonometric fitted methods (btdtfms) are constructed via multistep collocation technique and applied in block form as simultaneous numerical integrators which make them self-starting. the basic properties of the btdtfms are analyzed and presented. the accuracy and efficiency of the methods based on number of function evaluation are established through some standard numerical examples which are either stiff or periodic in nature. keywords: convergence, frequency, stiff, trigonometrically-fitted. article history : received: 02 january 2020 received in revised form: 16 february 2020 accepted for publication: 17 february 2020 published: 28 february 2020 c©2020 journal of the nigerian society of physical sciences. all rights reserved. communicated by: t. latunde 1. introduction a-stable methods are of great importance in solving stiff problems but put stark limitations on the choice of suitable methods for stiff problems. dahlquist [1] established that the most accurate a-stable linear multistep method has order 2. the search for higher-order a-stable multistep method is carried out in the following three major directions (biala et al. [2]): 1. use of higher derivatives of the solution; 2. incorporating additional stages and hybrid points; ∗corresponding author tel. no: +2348035195870 email addresses: profabdulcalculus@gmail.com (r. i. abdulganiy), akinolu35@yahoo.com (o. a. akinfenwa), olaoluwa.yusuff@gmail.com (o. a. yusuff), enobaborosaretin@yahoo.com (o. e. enobabor), waleokunuga@gmail.com (s. a. okunuga) 3. incorporating super future points. many numerical methods have been developed along these three directions which include but not limited to gear [3], enright [4], cash [5], wu [6], hojjati [7], jator [8], sahi et al. [9], ngwane and jator [10], akinfenwa et al. [11,12,13], mehdizadeh et al [14] and abdulganiy et al. [15]. method based on simpson’s was explored by akinfenwa et al. [16] while methods bases on exponential fitting were considered in okunuga [17], vaquero and vigo-aguiar [18], abhulimen et al. [19,20], ehigie et al. [21] and adesanya et al. [22] to numerically approximate stiff problems. although the exponential fitting methods for solving stiff problems were easy to implement, tedious mathematical analyses are involved in obtaining the a-stability. periodic problems, on the other hand, have received much attention in the past few decades. the methods for solving peri12 abdulganiy et al. / j. nig. soc. phys. sci. 2 (2020) 12–25 13 odic problems according to yakubu et al., [23] can be classified into two: methods with constant coefficients and methods having coefficients depending on the frequency of the problems. if a good estimate of the frequency or some suitable substitute is known in advance, a linear multistep method can be used for the integration of periodic problems (vanden berghe et al., [24] and van de vyver [25]). the idea of using basis function which integrates a set of linearly independent function exactly other than polynomial can be traced to the work of gautchi [26] and lyche [27]. many of such extension has been discussed in coleman and duxbury [28], ixaru et al. [29], vanden berghe and his collaborators ([24], [30,31]), simos [32,33], tsitouras and simos [34], nguyen et al. [35], senu et al. [36], jator and his collaborators [37-41], jator [42] and abdulganiy and his collaborators [43-45]. in what follows, a continuous third derivative method with trigonometric coefficients (ctdmtc) is derived via a multistep collocation technique for which the approximate interpolating function is a linear combination of polynomial and trigonometric terms. the ctdmtc is used to generate a family of third derivative trigonometrically-fitted method (tdtfm) and some other discrete methods as by-products that are combined together into a block third derivative trigonometricallyfitted method (btdtfm). the btdtfm is applied as a simultaneous numerical integrator for the integration of the system of first-order ordinary differential equations (odes) y ′ (t) = f (t, y (t)) , y (t0) = y0, t ∈ [t0, tn ] (1) where f satisfies the lipschitz condition with a periodic solution whose frequency is known in advance. stiff or/and periodic differential equation of the form in equation (1) frequently arises in many area of science and engineering, a list of such are provided in jator and agyingi [46], varden berghe et al. [24] and ndukum et al. [47]. the methods in this paper is different from the methods in jator [48,49] and akinfenwa [50] in that they are used to solve system of first-order odes while the methods in jator [48,49] are direct numerical integrators of general second-order odes and the methods in akinfenwa [50] are hybrid in nature respectively. some of the advantages of continuous methods include but not restricted to providing defect control (enright [51]) and ability to produce complementary methods which are combined and applied in block-by-block fashion (onumanyi et al. [52], akinfenwa et al. [11,13]). the foremost block methods are credited to milne [53], rosser [54], shampine and watts [55] and chu and hamilton [56] and are implemented in predictorcorrector modes that reduce the stability properties of the methods. the block method in this paper is implemented without the use of predictors. according to jator and agyingi [46], for a block method, it is needless to make a function evaluation at the preliminary part of the original block since at all blocks (with exception of the first block) the first function evaluation is previously known from the previous block. the rest of this paper is arranged as follows: the theoretical and basic elements of the btdtfm is discussed in section 2. the basic properties of the methods are discussed in section 3. in section 4, the estimation of the computational frequency is discussed while numerical experiments are presented in section 5. finally, section 6 concludes the paper. 2. theoretical and basic elements of the btdtfm the k−step third derivative method has the form yn+k−yn+k−1 = h k∑ j=0 β j (u) fn+ j+h 2δk (u) gn+k+h 3γk (u) ln+k(2) where u = ωh ,ω is the frequency, β j,δk,γk which depends on frequency and step size are parameters to be determined uniquely. also, yn+k is the numerical approximation to exact solution y (xn+k) and fn+ j = f ( xn+ j , yn+ j ) , gn+k = d f (x, y (x)) d x | x = xn+k y = yn+k ln+k = d2 f (x, y (x)) d x2 | x = xn+k y = yn+k in order to obtain (2), the exact solution y (x) is approximated on the interval [xn , xn+k] by the interpolating function i (x) given by i (x) = k+1∑ j=0 a j x j + ak+2sin(ωx) + ak+3cos(ωx) (3) imposing conditions in equations (4) and equation (4) below on equation (3) i ( xn+ j ) = yn+ j (4)  i ′ ( xn+ j ) = fn+ j j = 0(1)k i ′′ ( xn+ j ) = gn+ j j = k i ′′′ ( xn+ j ) = ln+ j j = k (5) lead to a system of (k + 4) equations which is solved using a computer algebra system (cas) such as maple. the derivation of the methods manually becomes more difficult for k ≥ 2. hence, the explorations of cas become necessary. the continuous form is developed by substituting the values of a j, j = 0 (1) (k + 3) into equation equation (3). after some algebraic manipulations, the continuous form is obtained as the expression equation (6). i (x) = yn+k−1+h k∑ j=0 β j (u, x) fn+ j+h 2δk(u, x)gn+k+h 3γk(u, x)ln+k(6) the continuous method in equation (6) is evaluated at x = xn+k to generate the principal method given by 13 abdulganiy et al. / j. nig. soc. phys. sci. 2 (2020) 12–25 14 yn+k = yn+k−1 + h k∑ j=0 β j (cos (u) , sin(u)) fn+ j+h 2δk (cos (u) , sin (u) ) gn+k + h 3γk(cos (u) , sin(u))ln+k (7) while the (k − 1) secondary methods are obtained as yn+i = yn+k−1 + h k∑ j=0 β j(cos (u) , sin(u)) fn+ j+h 2δk(cos (u) , sin(u))gn+k + h 3γk(cos (u) , sin(u))ln+k (8) by evaluating equation (6) at x = xn+i , i = 1, 2, . . . , k − 1. following the steps discussed above, the secondary and principal methods for k = 2 and k = 3, their coefficients and the corresponding power series conversion up to o ( u10 ) are given as follows for k = 2, yn = yn+1 + hβ0,1 (cos(u) , sin(u) ) fn + hβ1,1 (cos(u) , sin(u) ) fn+1 +h2δ2,1 (cos(u) , sin(u) )gn+1 + h3γ2,1 (cos(u) , sin(u) )ln+1 (9) yn+2 = yn+1 + hβ0 (cos(u) , sin(u) ) fn + hβ1 (cos(u) , sin(u) ) fn+1 + hβ2 (cos(u) , sin(u) ) fn+2 +h2δ2 (cos(u) , sin(u) ) gn+2 + h3γ2 (cos(u) , sin(u) ) ln+2 (10)  β0,1 = 1 6 ( −14sin (u) u3 + 5u4 − 12cos (u) u2 − 6cos (2u) u2 + 12sin (2u) u + 6u2 − 24cos (u) + 12cos (2u) + 12 ) / (( −2u − 2cos (2u) u − sin (2u) u2 − 4sin (u) + 2sin (2u) + 4cos (u) u + 4sin (u) u2 − 2u3 ) u ) β1,1 = 1 6 (( −7u3 + 18u ) sin(2u) + ( −21u2 + 6 ) cos (2u) − 4u4 + 12cos (u) u2 − 3u2 − 12sin (u) u − 6 ) / ((( 1 2 u 2 − 1 ) sin (2u) + u3 − 2sin (u) u2 − 2cos (u) u + cos (2u) u + u + 2sin(u) ) u ) β2,1 = 1 6 (( 8u3 − 12u ) sin(2u) + u4 + 10sin(u) u3 − 12cos (u) u2 + 24cos (2u) u2 − 24cos (u) + 24 ) / (( −4sin (u) u2 + sin (2u) u2 + 2u3 − 4cos (u) u + 2cos (2u) u + 4sin (u) − 2sin(2u) + 2u ) u ) δ2,1 = 1 6 ( 8sin (u) u3 + 5sin(2u) u3 − 12cos (u) u2 − 12cos (2u) u2 − 12sin (u) u + 6sin (2u) u − 48cos (u) + 12cos (2u) + 36 ) / (( −2u − 2cos (2u) u − sin (2u) u2 − 4sin (u) + 2sin (2u) + 4cos (u) u + 4sin (u) u2 − 2u3 ) u ) γ2,1 = (( −5u2 + 18 ) cos (2u) − 8cos (u) u2 + u2 − 16sin (u) u + 20sin (2u) u − 48cos (u) + 30 ) / ( 12cos (2u) u2 + 6 ( 2u3 − 4sin (u) u2 + sin (2u) u2 − 4cos (u) u + 2u + 4sin (u) − 2sin (2u) ) u ) (11)  β0,1 = − 49 160 − 13 1680 u 2 − 277 1728000 u 4 − 14911 13970880000 u 6 + 3775394358914560000 u 8 β1,1 = − 13 10 + 67 2100 u 2 + 289378000 u 4 + 7703873180000 u 6 − 334687 1362160800000 u 8 β2,1 = 97 160 − 29 1200 u 2 − 7309 12096000 u 4 − 108337 13970880000 u 6 + 346729721794572800000 u 8 δ2,1 = − 33 80 + 23 1400 u 2 + 179403200 u 4 + 155712328480000 u 6 − 263267 3632428800000 u 8 γ2,1 = 23 240 − 1 2100 u 2 − 373 6048000 u 4 − 15901 6985440000 u 6 − 11203 222393600000 u 8 (12)  β0 = 1 6 (( −2u3 + 24u ) sin(u) − u4 − 12cos (u) u2 + 24cos (u) − 24 ) / (( −4sin (u) u2 + sin (2u) u2 + 2u3 − 4cos (u) + 2cos (2u) u + 4sin (u) − 2sin (2u) + 2u ) u ) β1 = 1 6 (( u3 − 6u ) sin(2u) + ( 3u2 − 6 ) cos (2u) + 4 u4 + 12cos (u) u2 − 3u2 − 12sin (u) u + 6 ) / ( u (( 1 2 u 2 − 1 ) sin (2u) + u3 − 2sin (u) u2 − 2cos (u) u + cos (2u) u + u + 2sin (u) )) β2 = 1 6 ( 22sin(u) u3 + sin(2u) u3 − 5u4 + 36cos (u) u2 − 6cos (2u) u2 − 24sin (u) u − 18u2 + 24cos (u) − 12cos (2u) − 12 ) / (( −2u − 2cos (2u) u − sin (2u) u2 − 4sin (u) + 2sin (2u) + 4cos (u) u + 4sin (u) u2 − 2u3 ) u ) δ2 = 1 6 ( −8sin (u) u3 + sin (2u) u3 − 12cos (u) u2 − 12sin (u) u + 6sin (2u) u + 12u2 − 48cos (u) + 12cos (2u) + 36 ) / (( −2u − 2cos (2u) u − sin (2u) u2 − 4sin (u) + 2sin (2u) + 4cos (u) u + 4sin (u) u2 − 2u3 ) u ) γ2 = 1 6 ((−u2 +6)cos(2u) +8cos(u) u2 +5u2−32sin(u) u+4sin(2u) u−48cos(u) +42) ((−4sin(u) u2 +sin(2u) u2 +2u3−4cos(u)u +2cos(2u) u+4sin(u) )−2sin(2u) +2u)u (13) 14 abdulganiy et al. / j. nig. soc. phys. sci. 2 (2020) 12–25 15 β0 = − 1 160 − 1 1680 u 2 − 53 1728000 u 4 − 15119 13970880000 u 6 − 115693 4358914560000 u 8 β1 = 3 10 + 1 700 u 2 + 41378000 u 4 + 3847873180000 u 6 + 1658171362160800000 u 8 β2 = 113 160 − 1 1200 u 2 − 941 12096000 u 4 − 46433 13970880000 u 6 − 2074607 21794572800000 u 8 δ2 = 17 80 + 1 4200 u 2 + 19403200 u 4 + 52192328480000 u 6 + 831191210809600000 u 8 γ2 = 7 240 + 1 2100 u 2 + 436048000 u 4 − 269 6985440000 u 6 − 84803 10897286400000 u 8 (14) for k = 3, yn=yn+2+hβ0,1 (ucos (u) , sin(u)) fn+hβ1,1(cos (u) , sin(u)) fn+1+hβ2,1(cos (u) , sin(u)) fn+2 +hβ3,1(cos (u) , sin(u)) fn+3+h2δ3,1(cos (u) , sin(u))gn+3+h3γ3,1(cos (u) , sin(u))ln+3 (15) yn+1=yn+2+hβ0,2 (cos (u) , sin(u)) fn+hβ1,2 (ucos (u) , sin(u)) fn+1+hβ2,2(cos (u) , sin(u)) fn+2 +hβ3,2(cos (u) , sin(u)) fn+3+h2δ3,2(cos (u) , sin(u))gn+3+h3γ3,2(cos (u) , sin(u))ln+3 (16) yn+3=yn+2+hβ0 (cos (u) , sin(u)) fn+hβ1 (cos (u) , sin(u)) fn+1+hβ2 (ucos (u) , sin(u)) fn+2 +hβ3 (cos (u) , sin(u)) fn+3+h2δ3(cos (u) , sin(u))gn+3+h3γ3(cos (u) , sin(u))ln+3 (17)  β0,1 = 1 3 (−16sin(u)u 3 + 17sin(2u)u3 + 6u4 − 30cos(u)u2 + 6cos(3u)u2 + 48cos(2u)u2 + 90sin(u)u −18sin (3u) u − 54sin (2u) u + 12u2 + 21cos (u) − 21cos (3u) + 24cos (2u) − 24)/ρ β1,1 = 1 3 (−81sin (u) u 3 − 17sin (3u) u3 + 24u4 − 117cos (u) u2 − 75cos (3u) u2 + 30sin (u) u + 126sin (3u) u −60sin (2u) u + 48u2 − 78cos (u) + 78cos (3u) − 84cos (2u) + 84)/ρ β2,1 = 1 3 (16sin (3u) u 3 + 81sin (2u) u3 + 6u4 − 54cos (u) u2 + 78cos (3u) u2 + 144cos (2u) u2 + 150sin (u) u −102sin (3u) u − 120sin (2u) u + 12u2 + 57cos (u) + 57cos (3u) + 24cos (2u) − 24)/ρ β3,1 = 1 3 (−11sin (u) u 3 − 11sin (3u) u3 − 44sin (2u) u3 + 6u4 + 21cos (u) u2 − 45cos (3u) u2 − 48cos (2u) u2 +12sin (u) u + 36sin (3u) u + 12sin (2u) u + 36cos(2u) + 12u2 + 57cos (u) + 57cos (3u) + 24cos (2u) − 24)/ρ δ3,1 = ( ( −24u3 − 48 ) cos(2u) + ( −6u2 + 33 ) cos (3u) − 6cos (u) u2 − 71sin (u) u + 28sin (2u) u + 29sin (3u) u −33cos (u) + 48)/ρ γ3,1 = (29 sin (3 u) u + 28 u sin (2 u) − 6 u2 cos (u) − 24 cos (2 u) u2 − 6 cos (3 u) u2 − 71 sin (u) u − 33 cos (u) − 48 cos (2 u) +33 cos (3 u) + 48)/( ( −6 u3 + 21 u ) sin (3 u) + 18 (−cos (3 u) u + ( 3/2 u2 − 173 ) sin (2 u) + u3 − 3 u2 sin (u) −5 u cos (u) + 4 cos (2 u) u + 2 u + 47 sin(u)6 )u) (18)  β0,1 = − 121 405 − 289 42525 u 2 − 4397 53581500 u 4 + 518879123773265000 u 6 + 1540154587405481216140000 u 8 β1,1 = − 23 15 + 101 3150 u 2 + 18733969000 u 4 − 12401 833490000 u 6 − 48315783 300356456400000 u 8 β2,1 = 1 3 − 23 315 u 2 − 619 396900 u 4 + 5353916839000 u 6 + 7851770930035645640000 u 8 β3,1 = − 203 405 + 4061 85050 u 2 + 125353107163000 u 4 + 1200029247546530000 u 6 − 11234074463 8109624322800000 u 8 δ3,1 = 10 27 − 83 2835 u 2 − 3079 3572100 u 4 − 93587 8251551000 u 6 + 1115045320793908520000 u 8 γ3,1 = − 4 25 + 2 675 u 2 + 6112976750 u 4 + 548236876292500 u 6 + 45298619225267342300000 u 8 (19) 15 abdulganiy et al. / j. nig. soc. phys. sci. 2 (2020) 12–25 16 β0,2 = 1 24 (44sin (u) u 3 + 17sin (2u) u3 − 6u4 + 60cos (u) u2 + 6cos (3u) u2 + 102cos (2u) u2 + 132sin (u) u −210sin (2u) u − 18u2 + 384cos (u) − 168cos (2u) − 216)/ρ β1,2 = 1 24 (−351sin (u) u 3 − 17sin (3u) u3 + 78u4 − 486cos (u) u2 − 54cos (3u) u2 − 216cos (2u) u2 − 18sin (u) +66sin (3u) u + 486sin (2u) u + 108u2 − 1248cos (u) + 600cos (2u) + 684)/ρ β2,2 = 1 24 ( ( −351u3 + 1278u ) sin(2u) + ( 44u3 − 102u ) sin (3u) − 78u4 + 432cos (u) u − 1134cos (2u) u2 +108cos (3u) u2 − 126u2 − 810sin (u) u + 384cos (u) + 246cos (2u) − 648)/ρ β3,2 = 1 24 (−125sin (u) u 3 + 13sin (3u) u3 − 152sin (2u) u3 − 6u4 + 138cos (u) u2 + 18cos (3u) u2 −444cos (2u) u2 + 204sin (u) u + 186sin (3u) u + 1248sin (2u) u − 168cos(2u) − 1080)/ρ δ3,2 = 1 4 (13sin (u) u 3 − sin (3u) u3 + 13sin (2u) u3 − 24cos (u) u2 + 24cos (2u) u2 − 87sin (u) u − 3sin (3u) u +48sin (2u) u − 288cos (u) + 72cos (2u) + 216/ρ γ3,2 = 1 24 (78cos (u) u 2 − 6cos (3u) u2 + 78cos (2u) u2 + 353sin (u) u + 13sin (3u) u − 340sin (2u) u − 6u2 +960cos (u) − 312cos (2u) − 648)/ρ (20)  β0,2 = 1 90 + 1 800 u 2 + 1571984500 u 4 + 1053089293388480000 u 6 + 1456666991201425825600000 u 8 β1,2 = − 61 160 − 331 67200 u 2 − 14369 42336000 u 4 − 6318043 391184640000 u 6 − 611857117 1067934067200000 u 8 β2,2 = −1 + 19 3360 u 2 + 6131058400 u 4 + 63165719559232000 u 6 + 672527513429840000 u 8 β3,2 = 533 1440 − 19 9600 u 2 − 40501 127008000 u 4 − 23157647 1173553920000 u 6 − 8248324979 961140660480000 u 8 δ3,2 = − 11 48 − 1 2240 u 2 + 5834233600 u 4 + 42094139118464000 u 6 + 169075937320380220160000 u 8 γ3,2 = 11 240 + 47 33600 u 2 + 70321168000 u 4 + 491195592320000 u 6 − 1081123 19776556800000 u 8 (21)  β0 = 1 24 (−20sin (u) u 3 + sin (2u) u3 − 6u4 − 132cos (u) u2 + 6cos (2u) u2 + 324sin (u) u − 18sin (2u) u − 18u2 +384cos(u) − 24cos (2u) − 360)/ρ β1 = 1 24 (81sin (u) u 3 − sin (2u) u3 + 30u4 + 522cos (u) u2 − 6cos (2u) u2 − 1218sin (u) u + 18sin (3u) u +6sin (2u) u + 60u2 − 1272cos (u) + 24cos (3u) − 24cos (2u) + 1272)/ρ β2 = 1 24 (20sin (u) u 3 − 81sin (2u) u3 − 114u4 − 432cos (u) u2 + 84cos (3u) u2 − 306cos (2u) u2 + 714sin (u) u −186sin (3u) u + 642sin (2u) u − 66u2 − 192cos (u) − 192cos (3u) + 840cos (2u) − 456)/ρ β3 = 1 24 (371sin (u) u 3 + 29sin (3u) u3 + 136sin(2u)u3 − 54u4 + 762cos (u) u2 + 66cos (3u) u2 − 276cos (2u) u2 −948sin (u) u + 186sin (2u) − 264u2 + 1080cos (u) + 168cos (3u) − 792cos (2u) − 456)/ρ δ3 = 1 4 (−19sin (u) u 3 − sin (3u) u3 + 5sin (2u) u3 − 24cos (u) u2 − 31sin (u) u − 11sin (3u) + 32sin (2u) + 24u2 −264cos (u) − 24cos (3u) + 120cos (2u) + 168)/ρ γ3 = 1 144 (116sin(2u) + ( 6u2 − 48 ) cos (3u) + 114cos(u)u2 − 30cos (2u) u2 + 54u2 − 433sin (u) u − 29sin (3u) u −912cos (u) + 264cos (2u) + 696)/ρ (22)  β0 = 1 810 + 229 1360800 u 2 + 35326790750 u 4 + 60449877921488960000 u 6 + 114774361732438497291200000 u 8 β1 = − 7 480 − 97 201600 u 2 − 6403 127008000 u 4 − 3738641 1173553920000 u 6 − 1490068837 9611406604800000 u 8 β2 = 1 3 − 1 1440 u 2 + 1513175200 u 4 + 2864595867769000 u 6 + 1711626160071291280000 u 8 β3 = 8813 12960 + 5483 5443200 u 2 − 35383 3429216000 u 4 − 77924501 31685955840000 u 6 − 42892337857 259507978329600000 u 8 δ3 = − 83 432 − 209 181440 u 2 − 1571 114307200 u 4 + 8447031056198528000 u 6 + 7008125718650265944320000 u 8 γ3 = 17 720 + 167 302400 u 2 + 3383190512000 u 4 + 8740511760330880000 u 6 + 12076795714417109907200000 u 8 (23) 16 abdulganiy et al. / j. nig. soc. phys. sci. 2 (2020) 12–25 17 where ρ = u(18u2sin(u) + 2sin(3u)u2 − 9u2sin (2u) −6u3 + 30ucos (u) + 6cos (3u) u − 24cos (2u) u −47sin (u) − 7sin (3u) + 34sin(2u) − 12u) (24) it is interesting to note that as u → 0 in the power series expansion of the btdtfms, classical third derivative methods based on polynomial basis are recovered. 3. basic properties of btdtfm the basic properties of btdtfm which includes the local truncation error (lte), order, error constant, zero stability, convergence and region of stability are discussed. 3.1. local truncation error (lte) and order of btdtfm theorem 1. the btdtfm has a local truncation error of the form lt e = ck+4hk+4(ω2yk+2(xn) − y(k+4)(xn)) + o(hk+5). proof 1 since the btdtfm is made up of generalized linear multistep methods with trigonometric coefficients, we associate the btdtfm with a linear operator lω [ y (xn) ; h ] for the principal method and lω [ y (xn) ; h ] for the secondary methods defined respectively by  lω [ y (xn) ; h ] = y (xn + kh) − ( yn+k−1 + h ∑k j=0 β j (u)y ′ (xn + kh) +h2δky ′′ (xn + kh) + h3γky ′′′ (xn + kh) lω [ y (xn) ; h ] = y (xn + ih) − ( yn+k−1 + h ∑k j=0 β j (u)y ′ (xn + kh) +h2δk(u)y ′′ (xn + kh) + h3γk(u)y ′′′ (xn + kh) (25) assuming that y(xn) is sufficiently differentiable, expanding with taylor series expansions of y(xn +kh), y(xn +ih), y′(xn +kh), y′′(xn + kh) and y′′′(xn + kh) about the point xn. substituting the coefficients β j (u) ,δk (u) ,γk (u) ,β j (u) δk(u) and γk(u) into equations (7) and (8) respectively, after simplification, we obtain the order, the local truncation error as presented in table 1. � 3.2. consistency of the btdtfms since each btdtfm is of order p ≥ 5 > 1, (for each k )as shown in table 1, then it is consistent (lambert [57] and fatunla [58]). 3.3. stability of the btdtfm the combined methods (principal and secondary) are assemble in the block form given by a1yµ+1 = a0yµ+hb0 fµ+hb1 fµ+1+h 2gµ+1c1+h 3 d1 lµ+1(26) where yµ+1= (yn+1, yn+2, yn+3) t , yµ = (yn−2, yn−1, yn) t , fµ+1 = ( fn+1, fn+2, fn+3) t , fµ = ( fn−2, fn−1, fn) t , gµ+1 = (gn+1, gn+2, gn+3) t , lµ+1 = (ln+1, ln+2, ln+3) t . a0, a1, b0, b1, c1 and d1 are k×k matrices defined in canonical form as follows: for k= 2, we have a1= ( −1 0 −1 1 ) , a0= ( 0 −1 0 0 ) , b0= ( 0 β0,0 0 β0 ) , b1= ( β1,0 β2,0 β1 β2 ) , c1= ( 0 δ2,0 0 δ2 ) and d1= ( 0 γ2,0 0 γ2 ) for k = 3, we have a1 =  1 −1 0 0 −1 0 0 −1 1  , a0 =  0 0 0 0 0 −1 0 0 0  , b0 =  0 0 β0,1 0 0 β0,0 0 0 β0  , b1 =  β1,1 β2,1 β3,1 β1,0 β2,0 β3,0 β1 β2 β3  , c1 =  0 0 δ3,1 0 0 δ3,0 0 0 δ3  and d1 =  0 0 γ3,1 0 0 γ3,0 0 0 γ3  3.4. zero stability definition 2. a block method is zero stable if the roots of the first characteristic polynomial have modulus less than or equal to one and those of modulus one do not have multiplicity greater than 2. i.e. ρ(r) = det(ra(1) − a(0)) = 0 satisfies |ri| ≤ 1 and for those roots with |ri| = 1, the multiplicity does not exceed 2 fatunla [58]. proposition 3. the btdtfm is zero stable. proof 2 from the normalized first characteristic polynomial of btdtfm, we have in canonical form that ρk (r) = det [∑1 i=0 a1−ir i ] . so that ρk (r) = 0 =⇒ −rk−1 (1 + r) = 0. consequently, the roots r j, j = 1, 2, . . . k of ρk(r) satisfy ∣∣∣r j∣∣∣ = 1, the roots are simple. hence for each k = 2 and k = 3, the btdtfm is zero stable � 17 abdulganiy et al. / j. nig. soc. phys. sci. 2 (2020) 12–25 18 table 1: local truncation error of btdtfm k lte order ( p) error constant (c p+1) 2  ( − 1 1800 d (6) (y) (x) − 11800 ω 2 d(4) (y)(x) ) h6( − 1 200 d (6) (y) (x) − 1200 ω 2 d(4) (y)(x) ) h6  [ 5 5 ] − 1 1800 − 1 200 3  ( − 11 50400 d (7) (y) (x) − 1150400 ω 2 d(5) (y)(x) ) h7( 29 6300 d (7) (y) (x) + 296300 ω 2 d(5) (y)(x) ) h7( − 59 50400 d (7) (y) (x) − 5950400 ω 2 d(5) (y)(x) ) h7   6 6 6  − 1150400 296300 − 5950400 3.5. convergence of btdtfm theorem 4. let y be an approximation of the solution vector y for the system obtained from btdtfm given by equations (6) and (7) respectively, if en = |y (xn) − yn|, where the exact solution is several times differentiable on [x0, xn ] and if ‖e‖ =∥∥∥y − y∥∥∥, then for sufficiently small h,btdtfm is a (k + 3)thorder convergent method. that is ‖e‖ = o(hk+3). proof 3 the proof can be readily obtained, similarly to the one given in abdulganiy et al. (2017). � 3.6. linear stability and region of absolute stability of btdtfm according to ndukum et al. [47], linear stability analysis of trigonometrically fitted methods is more complicated than those of the corresponding polynomial-based methods for which λandh occur only in the combination z = λh . for trigonometrically fitted methods, three parameters are considered since the step length occurs in both z = λh and u = ωh . to analyze the linear stability of btdtfm, the block method (25) is applied to the test equations y ′ = λy , y ′′ = λ2y and y ′′′ = λ3y . after simple algebraic simplification and letting z = λh , we obtain yw+1 = m(z, u)yw where m (z, u) = [ a1 − b1z − c1z 2 − d1z 3 ]−1 [a0 + b0z)] (27) the rational function m (z, u) is called the amplification matrix which determines the stability of the method. definition (coleman and ixaru, [59]: a region of stability is a region in the zu−plane throughout which |ρ (z, u)| ≤ 1, where ρ (z, u) is the spectral radius of m (z, u). since the stability matrix depends on two parameters z and u, we plot the stability regions in the (z, u)−plane for both k = 2 and k = 3 respectively in figures 1 and 2. definition (ndukum etal., [47]): a btdtfm with variable coefficients (a0 (u) , a1 (u) , b0 (u) , b1(u), c1(u), d1(u)) with the stability function ρ (z, u) is said to be a-stable at u = u0 if |ρ (z, u0)| < 1 for all z ∈ c−. we plot the region |ρ (z, u)| < 1 (for each k) in the complex plane via boundary locus method and look for the interval of figure 1: the z − u plot for btdtfm k = 2 figure 2: the z − u plot for btdtfm k = 3 u for which z ∈ c−. for k = 2, we notice that the values of u ∈ [2π,∞) are satisfactory and for k = 3, the values of u ∈ [2.96π, 2.70π] are satisfactory. figures 3 and 4 show the plot of |ρ (z, u)| < 1 at u0 = 2π for k = 2 and u0 = 2.96π for k = 3 respectively. it is also worthy to note that the stability function of the btdtfm for each k satisfies |ρ (z, u)| < 1 for all z ∈ c−, for some u and limz→∞ ρ (z, u) = 0 is a property analogous to l−stability for the classical methods which is essential for a method to perform well on highly stiff problems (ndukum etal., [47]). hence the btdtfms are l−stable. 18 abdulganiy et al. / j. nig. soc. phys. sci. 2 (2020) 12–25 19 figure 3: ras for btdtfm k = 2 figure 4: ras for btdtfm k = 3 4. estimation of computational frequency in fitted methods for solving periodic problems, the choice of estimating computational frequency remains a hard nut to crack. a rigorous theory for the exact computation of the frequency is yet to be developed. however, some attempts have been made by a number of researchers. for exponentially fitted methods, the procedure for estimating frequency is given in ixaru et al. ([29], [60,61]), vanden berghe et al. ([30], [62,63]) and van de vyve.r [25]. the estimation of the computational frequency of trigonometrically-fitted methods can be found in ramos and vigo-aguiar [64], vigo-aguiar and ramos [65], ngwane and jator [42], jator [48,49] and ndukum et al., [47]. it is interesting to note that all the aforementioned methods focus on the optimization of the local truncation error (lte). in the present study, the approach described in vanden berghe et al. [30] for the exponentially fitted methods is adapted to the btdtfm with trigonometric coefficients. it is observed that the lte of the principal and secondary methods of the btdtfm consist of a product of three factors. the lte of the principal method of k=2, for instance, consists of the product of a general factor, a numerical factor in rational form ( − 1 200 ) , and a factor which involves two derivatives of the solution vector. for easy reference, we introduce the following functional d [ y (x) ; ω ] = ( y(6) (x) + ω2y(4) (x) ) . assume ω exists such that d is identically vanishes on the given interval, then the principal method corresponding to that ω will be exact. the reason according to vanden berghe et al. [30] is equivalent to solving the differential equation y(6) (x) + ω2y(4) (x) = 0 and, indeed, sin (ωx) and cos (ωx) are solutions. in general, no constant ω can be found such that d is identically vanished but it makes logic to discourse the problem of obtaining ω for which d are kept close to zero as possible for x in the given interval with the result that upon the taylor series expansion of ω2 = − y(6) (xn ) y(4) (xn ) about xn given by d [ y (x) ; ω ] = − (x − xn) ( y(7) (xn) −ω2y(5) (xn) ) +o ( (x − xn) 2 ) . this implies that the bound of d behaves as h and consequently, the bound of lte for k = 2 in table 1 behaves as h7 and hence increase the order by 1. remark 1. we remark that other steps and the technical details for implementations are as described in section 3 of varden berghe et al. [30]. 5. numerical experiments a number of numerical examples are provided in this section to illustrate the accuracy and computational efficiency of btdtfm. we have calculated the absolute error at the end point for stiff problem as |y (t) − y| and the maximum absolute error of the approximate solution as err = max |y (t) − y| for periodic problems and the efficiency is plotted as number of function evaluation (nfe) against max |y (t) − y| . 5.1. stiff problems example 5.1 we consider as our first example a non-linear system of first order differential equations in the range 0 ≤ t ≤ 10 y ′ 1 = µy1 + y 2 1 , y1 (0) = − 1 µ + 2 y ′ 2 = −y2 , y2 (0) = 1 whose solution in closed form is given by y1 = − e−2t µ+2 , y2 = e−2t , where µ = 10000 . the new btdtfm is used to solve this problem with ω = 1 and compared with the second derivative multistep method (sdmm) in hojjati et al. [7], simpson’s 38 method ( s 3 8 ) in akinfenwa et al. [16], multiderivative hybrid implicit rungekutta method (mhirk) in akinfenwa et al. [13], l stable method (lm) in mehdizabeh and molayi [14], third derivative block hybrid method (tdbhm) in akinfenwa [50] and the 19 abdulganiy et al. / j. nig. soc. phys. sci. 2 (2020) 12–25 20 figure 5: numerical solution of example 5.1 for h = 0.1 results are displayed in table 2 for the accuracy while figure 5 show the numerical solution of btdtfm against the time t for h = 0.1 and ω = 1 . it is evident from table 2 that the newly constructed method is more accurate than some of the previously known method in the literature. in particular, btdtfm k = 2 is far more accurate with large step size compare to ssdm with smaller step size and compete favorably with s 3 8 with same step size, notwithstanding s 3 8 is of order 8. example 5.2 consider the non-linear stiff system given by y ′ 1 (t) = −1002y1 + 1000y 2 2 , y1 (0) = 1 y ′ 2 (t) = y1 − y2 (1 + y2) , y2 (0) = 1 whose exact solution is given as y1 (t) = y22 , y2 (t) = e −t we solved this problem with ω = 1 . the results of btdtfm k = 2 for this problem in comparison with exponentially fitted gauss (ef-gauss), gauss-2s ,methods in vaquero and vigo-aguiar [18] and hybrid backward differentiation formula (hbdf) of jator and agyingi [46] for h = 0.1 and h = 0.01 are displayed in table 3. table 4 presents the results of btdtfm k = 2 for different values of h as compared to extended continuous block backward differentiation formula (ecbbdf) of akinfenwa and jator [12] and continuous block backward differentiation formula (cbbdf) of akinfenwa et al. [11] for 0 ≤ t ≤ 10 while table 5 shows the results of comparison with the lm method in mehdizabeh and molayi [14] and the method in wu [6] for h = 0.05 in the interval 0 ≤ t ≤ 50 respectively. the graphical representation of the numerical results of btdtfm at h = 0.05 is displayed in figure 6. it is seen from tables 3-5 that for various values of h considered, btdtfm does better in terms of accuracy than the methods in [18], [46], [12], [11], [14] and [6] respectively. figure 6: numerical solution of example 5.2 for h = 0.05 example 5.3 consider a classical four-dimensional problem given by y′1 (t) y′2 (t) y′3 (t) y′4 (t)  =  −104y1 (t) + 100y2 (t) − 10y3 (t) + y4 (t) −1000y2 (t) + 10y3 (t) − 10y4 (t) −y3 (t) + 10y4 (t) −0.1y4 (t)  ,  y1 (0) y2 (0) y3 (0) y4 (0)  = 1 1 1 1  the eigenvalue of the jacobian matrix λ1 = −0.1, λ2 = −1.0, λ3 = −1000 andλ4 = −10000 . the exact solution of example 5.3 is given as y1 (t) = − 89990090 8999010009 e−0.1t + 818090 89901009 e−t + 9989911 899010090 e−1000t + 89071119179 89990100090 e−10000t y2 (t) = 9100 89991 e−0.1t − 910 8991 e−t + 9989911 9989001 e−1000t, y3 (t) = 100 9 e−0.1t − 91 9 e−t, y4 (t) = e −1.0t this problem is considered to show the performance of btdtfm on a 4 × 4 stiff problems. table 6 shows the numerical results of btdtfm with ω = 1 in comparison with the second derivative extended backward differentiation formula (sdebdf) of ehigie et al. [21] and the method nmtd in adesanya et al. [22] for h = 0.05 and 0.1 at t = 20 while table 7 displays the results btdtfm as it compared with ab7 method in abhulimen and otunta [19], cege method of abhulimen and omeike [20] for h = 0.05 and 0.1 at t = 1 . the graphical representation of the numerical results of btdtfm at h = 0.05 is shown in figure 7. the results in tables 6 and 7 showed the superiority of btdtfm in terms of accuracy over the methods it compared with 20 abdulganiy et al. / j. nig. soc. phys. sci. 2 (2020) 12–25 21 table 2: comparison of results for example 5.1 t h=0.0001 ssdm h=0.1 s 3 8 h=0.1 mhirk h = 0.0001 lm h = 0.1 t dbh m h = 0.1 bt dt f mk = 2 err (y1) err (y2) 3 2.48e-11 2.47e-06 2.03e-19 1.44e-14 3.06e-15 3.08e-10 1.78e-20 2.08e-12 1.34e-17 2.98e-13 6.54e-19 6.56e-14 err (y1) err (y2) 5 3.45e-14 2.30e-08 1.20e-20 3.21e-15 9.35e-17 6.94e-11 2.49e-19 4.66e-13 2.94e-18 1.79e-13 2.00e-20 1.48e-14 err (y1) err (y2) 10 3.46e-18 3.15e-11 1.11e-20 4.38e-17 8.49e-21 9.36e-13 5.74e-20 6.35e-12 3.82e-20 2.84e-14 1.81e-24 2.00e-16 table 3: comparison of errors at t = 5 for example 5.2 h gauss-2s ef-gauss hbdf btdtfm k = 2 0.1 5.12e − 06 8.36e − 07 5.49e − 11 1.82e − 11 0.01 6.46e − 12 4.24e − 12 6.07e − 15 2.01e − 16 table 4: comparison of methods at t = 10 for example 5.2 h y c bbdf4 c bbdf5 ec bbdf4 ec bbdf5 btdtfm k = 2 0.02 y1 y2 4.88e-16 5.39e − 12 8.37e-18 9.16e − 14 2.48e-19 3.75e − 16 1.33e-20 1.35e − 16 5.76e-19 6.34e − 15 0.01 y1 y2 3.13e-17 3.45e − 13 3.39e-21 1.23e − 17 2.68e-19 2.93e − 15 2.87e-22 2.93e − 15 1.82e-20 2.00e − 16 table 5: comparison of result of at t = 50 for example 5.2 h y lmm wu method btdtfm k = 2 0.05 y1 y2 4.13e − 25 1.29e − 22 2.3e − 21 1.4e − 18 4.89e − 51 1.27e − 29 table 6: results comparison at t = 20 for example 5.3 h method y1 − error y2 − error y3 − error y4 − error 0.05 s de bdf n mt d bt dt f m k = 2 5.31e-12 − 3.71 × e − 17 7.27e-11 − 3.75e − 16 5.90e-09 − 4.13e − 17 1.34e-09 − 3.41e − 15 0.1 de bdf n mt d btdtfm k = 2 2.25e-10 8.96e-07 1.18e − 15 2.29e-09 2.06e-08 1.19e − 10 2.50e-07 8.96e-10 1.31e − 12 2.06e-08 8.06e-11 1.19e − 13 in the reviewed literature. 5.2. problems with periodic solutions example 5.4 we consider the following system of couple differential equations which is well known as the two body problem y ′′ 1 (t) = − y1 r3 , y1 (0) = 1, y ′ 1 (0) = 0 y ′′ 2 (t) = − y2 r3 , y2 (0) = 0, y ′ 2 (0) = 0 21 abdulganiy et al. / j. nig. soc. phys. sci. 2 (2020) 12–25 22 table 7: results comparison at t = 1 for example 5.3 h method y1 − error y2 − error y3 − error y4 − error 0.05 ab7 cege n mt d bt dt f m k = 2 3.2e-02 3.5e-05 7.5e-11 2.2e − 12 3.2e-02 3.8e-04 8.6e-10 2.5e − 12 3.3e-01 3.5e-07 8.3e-08 2.5e − 09 3.7e-05 3.7e-08 2.9e-09 1.3e − 11 0.1 ab7 cege n mt d btdtfm k = 2 2.5e-02 2.9e-05 1.4e-08 6.7e − 11 2.1e-01 2.7e-04 1.3e-08 7.5e − 10 2.4e-03 2.6e-06 2.1e-09 7.5e − 08 2.7e-05 2.6e-08 2.7e-11 4.0e − 14 figure 7: numerical solution of example 5.3 for h = 0.05 where r = √ y21 + y 2 2 and whose analytic solution is given by y1 (t) = cos (t) , y2 (t) = sin (t) the problem was considered in senu et al. [36] and abdulganiy et al. [44] in the interval 0 ≤ t ≤ 10 with ω = 1. the btdtfm is compared with block hybrid trigonometry method (bhtm) of abdulganiy et al. [44] and a new diagonally implicit runge-kutta nystrom method for periodic ivps (dirknnew) of senu et al. [36] and the numerical results are displayed in table 8 below. while the graphical illustration of btdtfm is shown in figure 8, figure 9 shows the efficiency of btdtfm. it is evident from table 8 that the newly developed method btdtfm uses fewer number of function evaluation and consequently produces good results and gives a better approximation when compared with bhtm and dirknnew. also, figure 9 hows that the method in this paper is efficient. example 5.5: nonlinear strehmel-weiner problem we consider the nonlinear second order ivp which was also solved by nguyen et al. [35] and jator [37] in the interval 0 ≤ t ≤ 10 respectively y”1 (t) = (y1 (t) − y2 (t)) 3 + 6368y1 figure 8: numerical solution of example 5.4 figure 9: efficiency curve of example 5.4 × (t) − 6384y2 (t) + 42 cos (10t) , y1 (0) = 0.5, y ′ 1 (0) = 0 y”1 (t) = − (y1 (t) − y2 (t)) 3 + 12768y1 (t) − 12784y2 (t) +42 cos (10t) , y2 (0) = 0.5, y ′ 2 (0) = 0 with analytic solution in closed form given by y1 (t) = y2 (t) = cos (4t) − cos(10t)2 . numerical results of the maximum global errors of btdtfm 22 abdulganiy et al. / j. nig. soc. phys. sci. 2 (2020) 12–25 23 table 8: comparison of numerical results for example 5.4 n btdtfm k = 2 bhtm dirknnew error nfe error nfe error nfe 100 2.84e − 29 201 5.13e − 27 1600 7.49 e − 4 5000 200 1.92e − 28 401 1.30e − 29 3200 5.62e − 7 10000 400 1.18e − 27 801 4.0e − 30 6400 1.00e − 9 25000 800 2.47e − 27 1601 7.43e − 28 12800 1.78e − 7 60000 table 9: comparison of maximum errors and number of function evaluation btdtfm k = 2 svbm tirkn 3 n f e err n f e err n f e err 751 2.25e-7 801 2.6e-7 907 2.5e-4 1126 2.95e-8 1201 1.6e-8 1288 6.6e-6 1501 7.01e-9 1601 2.8e-9 1682 7.0e-6 table 10: comparison of number of steps used in the computation for ω = 10 steps btdtfm k = 2 err vigo-aguiar and ramos err news5(4)-2 err 898 5.5 2.5 4.9 1344 6.6 3.2 6.5 2123 7.8 3.4 7.8 2990 8.7 4.6 8.9 4690 9.9 5.3 9.9 7215 11.0 6.1 11.0 k = 2 were compared with symmetric boundary value method (sbvm) of jator [37] and trigonometric implicit runge-kutta methods (tirkm) of nguyen [35] are presented in table 9 while figure 10 is the graphical representation of the numerical results of btdtfm. also, figure 11 shows that the method in this paper is more efficient in table 9 and figure 11, we show that a fifth order btdtfm has almost the same maximum errors with svbm of order 6 but uses fewer number of function evaluation and consequently a more accurate and more efficient integrator for the nonlinear strehmel-weiner problem. example 5.6: a nonlinear oscillator lastly, we consider the initial-value problem ẍ = −100x + sin (x) , x (0) = 0, ẋ (0) = 1, t ∈ [0, 20π] the analytical solution is not known but for comparison purposes we use that x (20π) = 0.000392823991 given by tsitouras and simos [34]. the comparison of different number of steps of btdtfm k = 2 with new5(4)-2 of tsitouras and simos [34] and method in vigo-aguiar and ramos [65] with ω = 10. the number of steps are obtained as contained in the references from literature. the error is calculated as err = figure 10: numerical solution of example 5.5 −log ( |x ( t f − x f ) | ) where x f is the approximation value obtained at the final point t f . table 10 revealed that btdtfm compete favourably with methods in tsitouras and simos [34] and method in vigo-aguiar and ramos [65]. 23 abdulganiy et al. / j. nig. soc. phys. sci. 2 (2020) 12–25 24 figure 11: efficiency curve of example 5.5 figure 12: numerical solution of example 5.6 6. conclusion a family of order k+3 block third derivative trigonometricallyfitted methods are considered and implemented in this paper for stiff and periodic problems. the stability properties of the method was analyzed and was found to be l-stable which in fact is a requirement for an algorithm to solving stiff problems. numerical results on the representative examples show the accuracy of the btdtfm on linear and nonlinear stiff problems and its efficiency on nonlinear periodic problems compared to some accurate and efficient methods respectively in the literature. the present study is limited to trigonometric basis. our future research will consider functionally fitted fitted form as suggested in nguyen et al. 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[65] j. vigo-aguiar & h. ramos, “on the choice of the frequency in trigonometrically fitted methods for periodic problems”, journal of computational and applied mathematics 277 (2015) 94. 25 j. nig. soc. phys. sci. 4 (2022) 843 journal of the nigerian society of physical sciences forecasting of the epidemiological situation: case of covid-19 in morocco el mehdi chouita,∗, mohamed rachdib, mostafa bellafkiha, brahim raouyanec araiss laboratory, department of mathematics and computer science, national institute of posts and telecommunications (inpt), rabat, morocco bschool of engineering school of management (isga), casablanca, morocco. cfaculty of sciences ain chock, university hassan ii (fsac), casablanca, morocco. abstract since the coronavirus pandemic started, many people have died due to the disease. the epidemic has been challenging to predict, as it progresses and spreads throughout the world. we used auto-regressive integrated moving average (arima) models to predict the outbreak of covid-19 in the upcoming months in morocco. in this work, we measured the effective reproduction number using the real data and the forecasted data produced by the two commonly used approaches, to reveal how effective the measures taken by the moroccan government have been in controlling the covid-19 outbreak. the prediction results for the next few months show a strong evolution in the number of confirmed and death cases in morocco. we study the spread of covid-19 in morocco to see how many cases are discovered, recovered, and dead, and the forecasting of further cases is used as a basic novel method. it is based on time series models. we used coronavirus outbreak data from march 02, 2020, to august 04, 2021. arima (autoregressive integrated moving average) and prophet time-series models are used to forecast the development of covid-19, which is not a novel method. the mean absolute error, root mean square error, and coefficient of determination r2 were computed to assess the model’s performance. our study aims to provide a better understanding of the infectious disease outbreak that affected morocco. it also provides information on the disease outbreak’s epidemiology. our study shows that the fbprophet model is more accurate in predicting the prevalence of covid-19. it can help guide the government’s efforts to prevent the virus’ spread. doi:10.46481/jnsps.2022.843 keywords: arima, covid-19, pandemic, fbprophet, time series forecasting article history : received: 30 may 2022 received in revised form: 11 september 2022 accepted for publication: 11 september 2022 published: 11 october 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: b. j. falaye 1. introduction covid-19 is a new respiratory virus that emerged in china in 2019. it has a high mutation and is easily spreadable. people with compromised immune systems or chronic illnesses such as diabetes or heart disease can develop severe respiratory conditions if they become infected with this virus [1]. in ∗corresponding author tel. no: +212 662 331 895 email address: elmehdi.chouit@gmail.com (el mehdi chouit ) this paper, we are going to forecast the further covid cases in morocco. so, before time, we can save the lives of people. the outbreak of covid-19 was designated a pandemic by the world health organization (who) on march 11, 2020. looking into these numbers of confirmed and death cases, we know the extreme danger of this pandemic situation. in morocco, the covid-19 pandemic was confirmed start to propagate on 2 march 2020, when the first confirmed covid-19 case was found in casablanca. a few weeks later, many activities and 1 chouit et al. / j. nig. soc. phys. sci. 4 (2022) 843 2 shops were closed by the moroccan authorities to contain the spread of this new virus. infections can be difficult to contain since individuals may not exhibit symptoms for a long time. this can be achieved through social isolation, and the only way to stop the spread of covid-19 is by quarantining or isolation of infected individuals. covid-19’s spread can be divided into three stages [2]: 1. local outbreak: at this stage, the virus’s spread can be tracked, and it can be identified as the source of infection. most cases are mild or moderate. the ongoing coronavirus (covid-19) pandemic has underscored the importance of working together, in partnership, to respond to outbreaks of infectious disease. it has highlighted the importance of having a clear plan and of everyone being aware of each others roles and responsibilities in the event of a pandemic 2. community transmission: the source of the transmission chain can’t be determined at this time. symptoms of the illness can be found in different parts of the country. the risk of breathing these in is highest when people are near, but they can be inhaled over longer distances, particularly indoors. transmission can also occur if splashed or sprayed with contaminated fluids in the eyes, nose or mouth, and, rarely, via contaminated surfaces. 3. large-scale transmission:due to large-scale uncontrolled human mobility, the virus is spreading rapidly in various parts of the country at this stage. close-contact settings, especially where people have conversations very near each other. confined and enclosed spaces with poor ventilation. because of the spread of the coronavirus, which has a global reach, national governments have imposed a lockdown to prevent its infection. as of dec. 24, 2021, 278,588,451 cases had been confirmed, 249,356,570 cases had been recovered, 5,402,557 fatalities had been recorded, and 23,829,324 active cases had been detected throughout the globe [3] was used to compile the statistical data. 1.1. key motivation the key motivation behind the current research work is: to accurately forecast the spread of covid-19 in morocco which could help the govt officials better plan to minimize its impact. the goal of this research is to model and predict the effects of coronavirus transmission on various public services and resources. various studies [4-8] have been published on the use of statistical analysis and artificial intelligence to predict the spread of the covid19 virus and its effects. the findings are based on the limited data that was available during the outbreak. covid-19 predictions are used to help predict the spread of infectious diseases. they are also used to prepare for and limit the spread of diseases [9]. in this paper, we study the capacities of the arima [10] and fbprophet [11] models for predicting future outcomes. many prediction models are used to forecast the impact of covid-19 on morocco. this paper aims to analyze these models and find out their applicability to predicting the virus impact in the country. covid-19 trend analysis and model performance are analyzed using various metrics. these include mean absolute error (mae), root mean square error (rmse) and coefficient of determination (r2). we produce predicting results for covid-19 confirmed, recovered, and death cases. 1.2. general context because of the geographic position of morocco and its proximity is near to europe, where the virus is still extensively spreading, the infection spread has been discovered. on march 2, 2020, morocco became aware of the first coronavirus case, with 954 199 persons infected with covid-19, resulting in 937,122 people being cured and 14,823 fatalities, while 8,801,829 cases were ruled out due to negative laboratory testing (according to the official portal of the coronavirus in morocco www.covidmaroc.ma on 24/12/2021). to control and limit the spread of the pandemic. morocco has taken the following measures and measures: 1. all air and marine flights to or from countries with epidemic hotspots will be suspended on march 13, 2020. 2. on march 15, 2020, the king ordered the government to quickly create a special fund to deal with the epidemic 3. morocco declares a health emergency on march 19, 2020; 4. on march 12, 2020, the covid-19 is given access to field hospitals and private clinics. 5. on april 7, 2020, the mask is made mandatory for the whole country. given the economic and societal consequences of the pandemic, statistical analyzes can be used to predict the number of people affected. we used machine learning models such as the time series model arima and fbprophet to predict the daily recorded cases (confirmed, recovered, and deaths) of covid-19 during this study to get an idea of the possible scenarios soon to take the necessary measures and educate people and decisionmakers about the consequences of this epidemic. 1.3. related works works can also be found on the application of sir (susceptible infectious recovered) models [12], this last study conducted by scientists revealed that the sir model can predict the epidemic trend based on the diseases spread. different diseases, such as sars, ebola, pandemic influenza and dengue fever, are usually predicted using time series analyses. a. g. amiranashvili [13] conducted research using the arima model to project the number of covid-19 cases in india in the future, and they determined that cases in india would increase exponentially, with cases decreasing relative to expected forecasts. they have also emphasized the importance of social separation and sanitizing to reduce the spread of the human virus. the goal of their research was to assist the government and medical community in preparing for the crisis. they failed to account for the effect of covid-19 tests on the number of cases where positive results were recorded. 2 chouit et al. / j. nig. soc. phys. sci. 4 (2022) 843 3 a similar study was conducted by tyagi [14], who estimated the number of covid-19 cases in india using the arima model. these efforts are directed toward the goal of developing effective methods to reduce the impact of infectious diseases on public health. these include mean absolute error (mae), root mean square error (rmse) and coefficient of determination (r2). according to dost khan [15] and t. latunde [16], they represent the popular autoregressive integrated moving average (arima) will be used to forecast the cumulative number of confirmed, recovered cases, and the number of deaths from covid-19 for the specific country. further, miksic [17] presents the model that method considers expected recoveries and deaths, and it determines maximally allowed daily growth rates that lead away from exponential increase toward stable and declining numbers. modelling and forecasting the total number of cases and deaths due to the pandemic is represented by n. khan [18] with the use of a polynomial regression model. with data intelligence, the model should be able to assess the probability of pandemic disease. according to abonazel [19] used the arima model based on the box-jenkins approach was used to predict the confirmed, recovered cases and deaths of covid-19 in egypt. further, he researches suitable statistical prediction models to be meaningful in forecasting and controlling this global pandemic threat, especially after the genetic mutation of the virus in 2021. the results indicated that the estimated arima models have a high ability to predict the number of confirmed cases, recovered covid-19 cases and death in egypt. on the other side, f. o. awedaa [20] and awwad [21], used two models that are autoregressive integrated moving average (arima) model and the spatial time-autoregressive integrated moving average (starima) model the measure the impact of the curfew on the spread of covid-19 in ksa. he concluded that starima models are more reliable in forecasting future epidemics of covid-19 than arima models due to the inclusion of spatial information i.e. neighboring effect in the form of the spatial weight matrix. unlike the univariate arima models, starima models have a smaller number of parameters. for the illustrated dataset, the starima model has only three parameters for the three locations, whereas arima model has numerous parameters, under such cases parameterization may prompt to lower sum of squares of residuals. in conclusion, they studied the trend pattern of the covid19 outbreak in makkah, jeddah, and taif in saudi arabia. finally, they found out that the best prediction model for forecasting the trend of daily confirmed cases in makkah, jeddah, and taif is starima. 2. method of data analysis a time series model is used to predict future outcomes based on historical data. for our study, we used the arima and facebook prophet models. 2.1. autoregressive integrated moving average (arima) arima(p,d,q) is composite of autoregressive (ar) model, moving average (ma) model, and ’i’ stands for integration; where p is order of autoregression, d is order of differencing, q is order of moving average. the ar(p) model is defined as a linear process given as the following equation. zt = α + φ1zt−1 + φ2zt−2 + · · · + φpzt−p + wt, (1) where zt−1, zt−2, . . . , zt−p are the lags (past values); and φ1,φ2, . . . ,φp are lag coefficients which are estimated by the model, wt is the white noise, and is defined as follows. α = 1 − p∑ i=1 φi µ, (2) where µ is mean of the process. the ma(q) model is defined as the following equation. zt = α + wt + θ1wt−1 + θ2wt−2 + · · · + θqwt−p, (3) where wt, wt−1, . . . , wt − q error terms of the model for the respective lags i.e. zt−1, zt−2, . . . , zt−p arima is able to fit if the data is stationary i.e. data mean and standard deviation is constant. the differencing parameter d is the order of transformation to make dataset stationary. second order differencing is shown in the following equation. zt = (zt − zt−1) − (zt−1 − zt−2) = zt − 2zt−1 + zt−2 (4) finally the equation for the arima(p,d,q) is defined as follows. zt = α + p∑ i=1 φizt−i + wt + q∑ j=1 θ jwt− j (5) 2.2. facebook prophet taylor et al. [11] presented the facebook prophet (fbprophet), a model that combines many non-linear and linear approaches with time as a regressor. prophet was developed by facebook’s data science team and released as open-source library. the model assumes that the data’s time dependency is not significant, and that training is treated as if it’s a curvefitting operation. as a result, irregular observations are permitted in a dataset. the approach has few benefits, including the ability to handle several periods of seasonality, as well as unique and well-known holidays; it provides flexibility by offering two options for trend: 1. a piecewise linear model, 2. a saturating growth model; and the model fits very fast. the model includes another vacation term as part of the time series, so that a time series can be defined by the following equation. zt = tt + s t + ht + �t, (6) where tt is trend, s t is seasonality, ht is holiday, and �t is error term. 3 chouit et al. / j. nig. soc. phys. sci. 4 (2022) 843 4 3. methods and analyzes to predict coronavirus disease (coivd-19), we are using the time series analysis using facebook prophet model and arima, designed for making forecasts for time series datasets and using forecasting tools available in python. this enables us to observe and predict the spread of the coronavirus pandemic on a daily or weekly basis. then we started with reading the whole dataset and cleaning it of missing values and outliers. we then analysis the data into five python libraries are ‘numpy’, ‘pandas’, ‘matplotlib’, ‘datetime’, and ‘sklearn.metrics’. 3.1. modeling dataset the repository by the center for systems science and engineering [22] provided the data on which the resulting model was built and tested. from march 02, 2020 to august 04, 2021, this data shows the daily values since the first case of covid19 in morocco. figure 1 depicts the daily cumulative confirmed, fatalities, and recovered cases from covid-19 in morocco during this time period. the total number of confirmed figure 1. daily cumulative confirmed, deaths and recovered cases covid-19 in morocco. cases in the dataset was 523 620, the number of recovered cases was 510 958, and the number of death cases was 9 207 for morocco as a whole, as shown in table 1 table 1. number of covid-19 cases in morocco till 04 august 2021. confirmed recovered deaths total 653,286 582,692 10,015 our study is based on a well-structured approach to prediction: analysis, cleaning and configuration of data, then prediction. all the steps are mentioned in figure 2. figure 2 depicts the basic phases involved in the prediction process. the architecture used to predict and analyze instances of covid 19 using the arima and fbprophet models is shown in figure 3. the confirmed, recovered, and dying datasets patients were separated into training (80%) and testing (20%) groups (the remaining 20%). figure 2. prediction process diagram. figure 3. framework to evaluate the forecasting models. 3.2. performance measures the following statistical metrics are used to evaluate the prediction models: • mean absolute error (mae): mae = 1 n n∑ i=1 |yi − ŷ| , (7) where, ŷ predicted value of y yi mean value of y 4 chouit et al. / j. nig. soc. phys. sci. 4 (2022) 843 5 • root mean square error (rmse): ms e = 1 n n∑ i=1 (yi − ŷ) 2 (8) • r-squared score: rms e = √ ms e = √√√ 1 n n∑ i=1 (yi − ŷ) 2 (9) 4. results and discussion in python 3.8, we employed the arima and prophet models from the stats-models and fbprophet open-source packages, respectively. our testing were performed on a machine with a 2.80 ghz intel r© core i7 processor, 16 gb of ram, and an 8gb intel r© hd graphics 620 gpu. we’ll discuss how accurate adopted models are in forecasting confirmed, recovered, and mortality cases in this section. 4.1. forecasting of confirmed cases we employ arima and fbprophet models to anticipate future instances. arima may be used to create predictions if the data is stationary. the fbprophet is a tool for analyzing real data. the forecasting accuracy statistics for verified moroccan instances are shown in table 2. in terms of various sorts table 2. table of different metrics (confirmed cases). mae rmse r2 arima 2,761.00 3,096.79 0.76 fbprophet 638.287 1,157.446 1.00 of error metrics, such as mae, rmse, and r2, the findings clearly show that fbprophet performs considerably better than the arima model. figure 4 displays both the actual and predicted confirmed cases. it is obvious from the graph that the predicted cumulative confirmed cases are rising more quickly, but the curve still has to be flattened. figure 4. fbprophet forecasting for morocco confirmed cases. 4.2. forecasting of recovered cases we performed an evaluation of the suggested models using data from cured cases in morocco to predict and assess the cure rate of the disease. we have applied fbprophet directly on actual data to fit the model and generated the forecasting results. the accuracy results of the models for the recovered cases are shown in table 3. best results are by fbprophet. the results table 3. table of different metrics (recovered cases). mae rmse r2 arima 3,301.43 3,882.269 0.72 fbprophet 474.333 803.590 1.00 reveal that fbprophet predicts values that are almost identical to the actual values, however arima does not. figure 5 also shows actual and anticipated recovered cases, which demonstrates that covid-19 recovered cases will rise over the next few days. figure 5. fbprophet forecasting for morocco recovered cases. 4.3. forecasting of death cases since the coronavirus has caused many deaths, it’s important to study its mortality rate to predict its future cases. this can help plan governments for its spread. we assessed the predicting models for death cases in morocco in this area. table 4 indicates the models’ accuracy in predicting fatality cases. we table 4. table of different metrics (death cases). mae rmse r2 arima 20.83 25.41 0.67 fbprophet 7.359 11.364 1.00 can observe that the arima prediction errors are quite high, but the prediction results of fbprophet have a low error factor. according to the findings, fbprophet may be used to predict occurrences in real time and structure services accordingly. figure 6 also displays the actual and anticipated mortality cases, which demonstrates that the covid-19 death rate is lower than the virus’s propagation. aside from predicting, some statistical and mathematical modeling studies have been available since the beginning of the 5 chouit et al. / j. nig. soc. phys. sci. 4 (2022) 843 6 figure 6. fbprophet forecasting for morocco death cases. covid-19 pandemic to anticipate national and worldwide epidemics by varying the degrees of accuracy. the accuracy of predictions is subject to the assumptions that have been made based on the available data. because of the differences in input value parameters and assumptions, the forecasting results may vary. during new pandemics, such as covid-19, the quality and availability of information vary as the epidemic progresses, causing uncertainty in early-stage forecasts and improving as the epidemic progresses. 5. conclusion the patterns from the current data reveal that prompt and effective measures taken by moroccan authorities to contain the pandemic are showing a positive impact when compared with other countries. based on the obtained data, we predicted the evolution of covid-19 using the arima and prophet models. although the results generated by these models are quantitatively different, both models have predicted an important increase in confirmed cases and deaths in morocco in the next few months. the finding indicates that the transmission of covid-19 in morocco will continue to expand in the next few months. fortunately, both models predict that the situation will not be considered too dramatic and, therefore, may give hope for controlling the disease. finally, we hope that these results will help the moroccan government to better control and 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[online]. available: https://github.com/cssegisanddata/covid-19 7 j. nig. soc. phys. sci. 4 (2022) 835 journal of the nigerian society of physical sciences synthesis, characterization and antimicrobial activities of copper-tea leaves (camellia sinensis) extract nanoparticles. b. t. iorhunaa,∗, t. t. awuheb, i. c. azuagaa, e isaaca, f. shuaibua, b. yohannaa a chemical sciences department, taraba state university, jalingo, taraba state, nigeria. b science department, federal polytechnic of oil and gas, bonny island, rivers state, nigeria. abstract nanoparticles, nps synthesis has gained attention recently due to their ease of preparation (especially green synthesis), availability of raw materials and usefulness. the green synthesis of copper nanoparticles (cunps) was done using tea leaves extract-harvested from the mambilla in taraba state-nigeria. the phytochemical analysis of the tea leaves extract was done and found to contain phenols, steroids and saponins which could have caused colour change, reduction of copper ions, capping and stabilisation of the synthesized cunps. the presence of nps in the mixture was identified by the change in colour of the mixture, λmax of the ultraviolet-visible spectrophotometry and spectra of the fourier transform infrared, ftir spectrophotometry on the mixture. antimicrobial studies of the synthesized cunps on the bacterial (escherichia coli) showed effective toxic effects. this study showed that, tea leaves extract is good for synthesizing copper nanoparticles that can be used as antimicrobial agents. doi:10.46481/jnsps.2022.835 keywords: nanoparticles, characterisation, antimicrobial, copper, tea leaves. article history : received: 28 may 2022 received in revised form: 19 july 2022 accepted for publication: 27 july 2022 published: 10 november 2022 c©2022 journal of the nigerian society of physical sciences. all rights reserved. communicated by: b. j. falaye 1. introduction diseases causing organisms like fungi, bacteria, and viruses which are resistant to many drugs have caused a lot of issues worldwide. this could be due to the unprescribed use of antimicrobial drugs that have led to increasing cases of drug resistance, thus bringing down the effectiveness of these drugs when used for treatment. therefore, new procedures such as the development of alternative antimicrobial drugs or modifying the existing drugs are being sorted with high interest [1]. ∗corresponding author tel. no: +2347030479355 email address: boniface.tersoo@tsuniversity.edu.ng (b. t. iorhuna ) nanoparticles could occur naturally or they can be engineered. they are very small in size and their particles sizes vary between 1 100 nm. they are unique in their properties and are highly valuable. nanoscaled particles, display better properties (such as; catalytic, magnetic, electrical, mechanical, optical, chemical, biological, medical, etc). because these particles have high surface/volume ratio, they show higher reactivity, mobility, dissolution properties, and strength compare to other materials [2]. the design, synthesis and manipulation of nanoparticles structure is an upcoming and relevant field of research and technology [3]. nanoparticles obtained from metals have found a very good number of applications in nature, they can be applied in many industries (including catalysis) and medicine including; drug delivery, cancer treatment, wastewater treatment and dna analysis, as 1 b. t. iorhuna et al. / j. nig. soc. phys. sci. 4 (2022) 835 2 antibacterial agents, as biosensors, in solar power generation, and many others [4]. many methods (including chemical, physical and biological) can be used in the synthesis of nanoparticles. although chemical method of synthesis takes a shorter period of time to synthesize a large quantity of nanoparticles, this method uses capping agents to stabilize the nanoparticles, and most of these chemicals are toxic and can result to by-products that are non-ecofriendly. because we are all concerned with the use of environmentally friendly synthetic approaches for nanoparticles synthesis, this leads to the developing interest in biological approaches which reduce the use of toxic chemicals as by-products [5]. we shall prepare nanoparticles by green synthesis of metallic nanoparticles using locally available materials as a cost-effective and environmentally benign alternative to chemical and physical methods. copper nanoparticles (cunps) research has been of great interest in recent years due to its applications in various fields of endeavour. tea leaves and many other plants (contain amino acids, alkaloids, proteins, polysaccharides, phenolics, terpenoids, flavones, and other active biomolecules) play important role in nanoparticles synthesis, these plants’ phytochemicals act as both reducing and capping agents [6]. amongst the biomolecules, phenols and flavonoids have unique chemical properties and can reduce and wrap nps. this could be because of the presence of hydroxyl and carboxyl functional groups that can bind to the metal [7]. tea is a drink that is an infusion of processed leaves and flowers of one of the varieties of an evergreen shrub botanically called camellia sinensis. tea is a common beverage that is drunk by many. tea leaves (camellia sinensis) is rich in polyphenolic compounds, and has been reported as a natural source in the synthesis of nanoparticles [3]. green tea is mainly composed by epigallocatechin, epigallocatechin-3gallate (egcg), epicatechin and epicatechin-3-gallate which act as reducing and capping agents [8, 9]. the green tea has many medical benefits and the phytochemicals obtained from tea leaves have been used previously in cosmetic and therapeutic applications. the phytochemicals are mainly catechins and caffeine which are water-soluble and can act as reducing agents of metal ions, leading to the formation of capped metallic nanoparticles [9]. 2. materials and methods 2.1. materials cu(no3)2. 2.5h2o, naoh and iron(iii) chloride (lobaar chemie pvt ltd), h2so4 (jhdar), ethanol and iodine solution (fenxichunar), nutrients agar and acetic anhydrate (qualikemsar), deionized water, tea leaves (camellia sinensis), sieve (cole parmer-typed sieve of 0.80 mm mesh), ph meter (hanna instrument h19024) , uv-visible (uv apel 3000 uv usa) and ftir spectrophotometers (agilant technologies microlab pc). 2.2. methods 2.2.1. harvest and preparation of tea leaves extract fresh tea leaves were harvested around the premises of kakara beverage of sardauna local government area of taraba state of nigeria. it was dried at room temperature and grounded into a powder form through sieve. the tea leaves extract was prepared by taking 10 g of grounded tea leaves and dissolving it in a 500 ml beaker with the addition of 200 ml of deionized water and stirring for 25 minutes, figure 1. the mixture was left to stand in a cupboard for 2 hours at room temperature. the extraction was done by filtration using whatman filter paper [10]. 2.2.2. phytochemical analysis of tea leaves extract test for steroids: this was done using the general procedure; 5 ml of chloroform was added to 1 ml of the tea leaves extract, then 5 ml of concentrated h2so4 was then slowly added. the upper layer turned red and the acid layer turned yellow-green which showed the presence of steroids [11]. test for carbohydrates: to the fresh tea leaves extract were added a few drops of iodine solution, and no reaction was observed indicating the absence of carbohydrates [10, 11]. test for flavonoids: these are tested according to this method; to the 4 ml of the tea leaves extract were added 1 ml of a 2 n naoh(aq), no reaction was observed (no yellow colour formation) indicating the absence of flavonoids [10, 11]. test for saponins: a test-tube containing 4 ml of aqueous extract of tea leaves was vigorously shaken, and a 1 cm foam layer was formed which indicated the presence of saponins [11]. test for phenols: to a 2.5 ml of ethanol was added equal volume of tea leaves extract and a few drops of iron(iii) chloride. a yellow-greenish precipitate was formed indicating the presence of phenols [10]. 2.2.3. synthesis of copper nanoparticles, cunps nanoparticles derived from the tea leaves extract were synthesized using the methodology according to a previous study with some modifications [10]. 2.2.4. characterization of the tea leaves extract and cunps the tea leaves extract and cunps functional (chromophore) groups were characterized by uv-visible absorption and fourier-transform infrared (ftir) spectrophotometry. in addition, the formation of the cunps was done with continuous observation of the change in colour of the mixture [12]. 2.2.5. antibacterial activity of the cunps agar disc-diffusion method was used to determine the antibacterial activity of our prepared cunps against escherichia coli, a gram-negative bacteria. the microbial strain was obtained from federal medical center jalingo, taraba state, and 2 b. t. iorhuna et al. / j. nig. soc. phys. sci. 4 (2022) 835 3 figure 1: (a) fresh tea leaves, (b) dried tea leaves and (c) grounded tea leaves was cultured in a nutrient broth for 24 hours to obtain a growth phase of the strain bacteria. this actively growing bacterial culture was inoculated/spread into the muller hinton agar (mha). the cunps solutions were prepared at varying concentrations; 0.031, 0.063, 0.125, 0.250 and 0.500 mg/ml (dissolved in dimethyl sulfoxide, dmso). the whatman filter paper disks were punched at 6 mm diameter and spread through with the dissolved cunps and then placed to the mha surface. streptomycin disc was used as a positive control while dmso was taken as the negative control. the plates were incubated at 37◦c for 34 and 48 hours. the effect of synthesized cunps on the bacterial was evaluated in terms of zones of inhibition measured, and recorded in millimeters using a ruler [10, 12]. 3. results and discussion 3.1. phytochemical analysis of the tea leaves extract the phytochemical analysis of the tea leaves extract is as presented in table 1. table 1: phytochemical analysis of tea leaves extract. phytochemicals present absent phenols ++ steroids ++ carbohydrates anthocyanins ++ flavonoids ++ saponins aqueous tea leaves extract was analysed and the results are presented in table 1. this analysis showed phytochemicals such as phenols, steroids, flavonoids, anthocyanins are present in the tea leaves extract while, carbohydrates and saponins are absent in the tea leaves extract. this is in line with the branded; ’highland tea’ (a product of kakara group of companies, taraba state-nigeria) which uses the tea leaves for its products, and the other findings [13]. the phytochemicals present in the tea leaves extract could be responsible for the reduction of cu2+ ions and capping of the cunps. 3.2. characterization of tea leaves extract and cunps by uvvisible spectrophotometry the interaction of nps with wavelengths of light is dependent on the size, shape and cluster state of nps. thus, uvvisible spectroscopy is used for confirming the formation of nps [13]. in our synthesis, the reduction of cu ions to cunps by tea leaves extract was observed by the change in colour of the reaction mixture (from red to brown then darkened), figure 2. the mixture (synthesized cunps) showed an absorption peak at 400 nm (figure 3a), confirming the presence of cunps, which is in line with other works [12]. the shift in the absorption band from 440 nm of the tea leaves extract (figure 3b) to 400 nm of cunps (figure 3a) further supports this claim [1, 14]. figure 2: synthesis of copper nanoparticles. the functional groups in the tea leaves extract and cunps were also analyzed by ftir spectroscopy. the ftir spectrum of the tea leaves extract (figure 4) shows various peaks at; 3200 cm−1 (corresponding to o-h), 1540 and 1544 cm−1 (corresponding to c=c), while those at 1160 and 1087 cm−1 correspond to stretching vibrations of hydroxyl and carboxyl functional groups of phenols and carboxylic acids respectively. the band at 1396 cm−1 corresponds to c−o−h bending [15, 16]. the ftir spectrum of the synthesized cunps (figure 5) on the other hand shows a shift in some of the peaks and the resulting peaks are narrower and less intense, because some of the active compounds in the extract took part in the reduction of cu2+ 3 b. t. iorhuna et al. / j. nig. soc. phys. sci. 4 (2022) 835 4 (a) (b) figure 3: (a) uv-visible spectrum of synthesized copper nanoparticles, (b) uvvisible spectrum of tea leaves extract. figure 4: ftir spectrum of tea leaves extract figure 5: ftir spectrum of synthesized copper nanoparticles ions and capping of the cunps. the reduction and capping of the cunps prepared could be due to the presence of phenols, flavonoids, steroids or anthocyanins groups, which were seen in the ir spectrum (figure 4). previous reports on green synthesis of nps have shown the presence of these phytochemicals [10]. 3.3. antibacterial activity of the cunps the antibacterial activity was done in terms of zone of inhibition measured, and was recorded in millimeters using a ruler as contained in table 2. table 2: antibacterial activity of cunps on escherichia coli. microorganism nanoparticles (mg/ml) sensitivity (34hrs) (mm) sensitivity (48hrs) (mm) escherichia coli 0.500 5.00 7.00 0.250 2.00 4.00 0.125 0.00 1.50 0.063 0.00 0.00 0.031 0.00 0.00 bacterial strain, escherichia coli was used to study the antimicrobial activity of the cunps obtained from tea leaves extract. table 2 presents the antibacterial activity of the cunps on the bacteria in a well diffusion assay. the results showed that cunps have very good antibacterial activity against escherichia coli. the inhibition zone diameter is related to the magnitude of activity of cunps on microbes. it was observed that, the cunps exhibited antibacterial activity at a concentration similar to that of the reference drug (streptomycin). the effects of the cunps against bacterial could be due to the dual activity nature of the nano-sized copper and the bioactive agents of the tea leaves extract on the surface of the cunps [17, 18]. 4. conclusion due to the phytochemicals present in tea leaves (camellia sinensis) extract; phenols, steroids, flavonoids, anthocyanins etc, which are responsible for the reduction and capping of cunps, it use in the synthesis of cunps has been developed. tea leaves were used in producing copper nanoparticles. the cunps synthesized were characterized by uv-visible and ftir spectrophotometry. the formation of cunps was observed by colour change from red to brown (and darkened). the synthesized cunps displayed excellent antibacterial activity against escherichia coli. the bioreduction process of the cunps is an economic and ecofriendly, simple one-step method, and this green synthetic protocol would be a better alternative to the existing methods. the use of camellia sinensis in the synthesis of nps could also expand its market beyond mere tea production/consumption, providing a means of livelihood for the mambilla people of taraba state, of nigeria. 4 b. t. iorhuna et al. / j. nig. soc. phys. sci. 4 (2022) 835 5 acknowledgements we greatly appreciate the editor in chief of this journal for his patience and encouragement references [1] r. humaira, s. a. mona, a. a. hadeel, a. a. manal, a. s. dina & s. b. ramesa, “green synthesis, characterization, and antimicrobial activity of silver nanoparticles prepared using trigonella foenum-graecum l. leaves grown in saudi arabia”, green processing and synthesis 10 (2021) 421. 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[18] k. s. kumar, g. r. venkatakrishnan, r. rengaraj, p. k. gayathri, g. lavanya & d. hemapriya, “antimicrobial activity of green synthesized trimetallic oxide ni/cr/cu nanoparticles”, journal of the nigerian society of physical sciences 3 (2021) 144. 5 j. nig. soc. phys. sci. 2 (2020) 26–35 journal of the nigerian society of physical sciences original research on fuzzy soft set-valued maps with application mohammed shehu shagari∗ department of mathematics, faculty of physical sciences, ahmadu bello university, nigeria abstract soft set and fuzzy soft set theories are proposed as mathematical tools for dealing with uncertainties. there has been tremendous progress concerning the extensions of these theories from different point of views of researchers so as to accommodate more robust and expressive applications in everyday life. in line with this development, in this paper, we combine the two aforementioned notions to initiate a novel concept of set-valued maps whose range set is a family of fuzzy soft sets. the later idea is employed to define suzuki-type fuzzy soft (e, k)-weak contractions, thereby establishing some related fuzzy soft fixed point theorems. as a consequence, several well-known suzuki-type fixed point theorems are derived as corollaries. examples are also provided to validate the new concepts and to support the authenticity of the obtained results. moreover, an application in homotopy is considered to show the usability of the obtained results. keywords: soft set, fuzzy soft set, fuzzy soft set-valued map, fuzzy soft fixed point, weak contraction article history : received: 30 january 2020 received in revised form: 14 february 2020 accepted for publication: 15 february 2020 published: 28 february 2020 c©2020 journal of the nigerian society of physical sciences. all rights reserved. communicated by: b. j. falaye 1. introduction and preliminaries the classical banach contraction theorem [1] is one of the extremely useful results in nonlinear functional analysis. in the setting of a metric space , this theorem is stated as follows. theorem 1.1. [1] let (x, d) be a complete metric space and t : x −→ x be a contraction, i.e., there exists an α ∈ (0, 1) such that d(t x, t y) ≤ αd(x, y), for all x, y ∈ x. (1) then (i) t has a unique fixed point u in x; ∗corresponding author tel. no: email address: ssmohammed@abu.edu.ng (mohammed shehu shagari) (ii) the picard iteration {xn}∞n=0 , given by xn+1 = t xn, n = 0, 1, 2, · · · converges to u, for any x0 in x. theorem 1.1 has several generalizations, see, for example,[2, 3, 4, 5, 6, 7] and the references therein. the two well-known drawbacks of theorem 1.1 are: (drb1) the contractive condition in (1) compels t to be continuous; (drb2) the theorem cannot characterize metric completeness of x. problems (drb1)-(drb2) were resolved affirmatively by kannan [18]. recall that a mapping t (not necessarily continuous) on a metric space x is said to be a kannan contraction if there exists an α ∈ [ 0, 12 ) such that d(t x, t y) ≤ α [ d(x, t x) + d(y, t y) ] , for all x, y ∈ x. (2) 26 m. s. shagari / j. nig. soc. phys. sci. 2 (2020) 26–35 27 this achievement in kannan contraction was first noted by subrahmanyam [28] that a metric space x is complete if and only if every kannan contraction on x has a fixed point. following (2), more than a handful of papers were devoted to studying fixed point theorems for classes of contractive type conditions that do not require the continuity of t ; see, for instance,[8, 9, 10, 11, 12]. however, researchers noticed that kannan’s theorem is not a generalization of theorem 1.1. along the way, the notion of weak contraction was introduced by berinde [13]. the idea generalized the well-celebrated fixed point theorems due to banach [1], chatterjea [9], zamfirescu [12], and many others. definition 1.2. let (x, d) be a metric space. a mapping t : x −→ x is called weak contraction if there exists a constants α ∈ (0, 1) and k ≥ 0 such that d(t x, t y) ≤ αd(x, y) + kd(y, t x), for all x, y ∈ x. (3) thereafter, in 2007, m.berinde and v. berinde [14] extended the concept of weak contraction from the case of single-valued mappings to multi-valued mappings and established some convergence theorems for the picard iteration in connection with multi-valued weak contraction. definition 1.3. let (x, d) be a metric space and t : x −→ c b(x) be a multi-valued mapping. then t is called a multivalued weak contraction or multi-valued (θ, l)-weak contraction if and only if there exist constants θ ∈ (0, 1) and k ≥ 0 such that for all x, y ∈ x, h(t x, t y) ≤ θd(x, y) + ld(y, t x). (4) furthermore, a notable attempt at resolving problems (drb1)(drb2) was presented in 2008 by suzuki [29]. the following is known in the literature as suzuki fixed point theorem. theorem 1.4. let (x, d) be a complete metric space and t be a mapping on x. define a nondecreasing function r from [0, 1) onto ( 1 2 , 1 ] by r(t) =  1, if 0 ≤ t ≤ √ 5−1 2 (1−t) r2 , if √ 5−1 2 ≤ r ≤ 1 √ 2 1 (1+t) , if 1 √ 2 ≤ r < 1. assume that there exists t ∈ [0, 1) such that r(t)d(x, t x) ≤ d(x, y) implies d(t x, t y) ≤ rd(x, y), for all x, y ∈ x. then t has a unique fixed point u in x. moreover, t n x −→ u as n −→∞ for all x ∈ x. an interesting improvement of suzuki fixed point theorem in the setting of multivalued mappings is due to djoric and lazovic [16]. in subsequent section, we shall derive the main result in [16] as a consequence of our result. the real world is filled with uncertainty, vagueness and imprecision. the notions we meet in everyday life are vague rather than precise. in recent time, researchers have taken keen interests in modelling vagueness due to the fact that many practical problems within fields such as biology, economics, engineering, environmental sciences, medical sciences involve data containing various forms of uncertainties. to handle these complications, one cannot successfully employ classical mathematical methods due to the presence of different kinds of incomplete knowledge typical of these mix-ups. earlier in the literature, there are four known theories for dealing with imperfect knowledge, namely probability theory (pt), fuzzy set theory (fst)[30], interval mathematics and rough set theory (rst)[27]. all the aforementioned tools require pre-assignment of some parameters; for example, membership function in fst, probability density function in pt and equivalent relation in rst. such pre-specifications, viewed in the backdrop of incomplete knowledge, give rise to every day problems. in this concern, molodstov [26] initiated the concept of soft set theory (sst) with the aim of handling phenomena and notions of ambiguous, undefined and imprecise environments. hence, sst does not need the pre-specifications of a parameter, rather it accommodates approximate descriptions of objects. in other words, one can use any suitable parametrization tool with the help of words, sentences, real numbers, mappings, and so on. thereby, making sst an adequate formalism for approximate reasoning. in the pioneer work of molodstov [26], several potential applications of sst have been pointed out in the areas of riemann integration, smoothness of functions, theory of probability, theory of measurement, game theory and operation research . interestingly, molodstov [26] emphasized that the models by fuzzy sets and soft sets are interrelated. yang et al [22] emphasized that sst needed to be expanded in different directions to extend its applications to other fields. along the lane, by combining the ideas of soft sets and fuzzy sets, maji et al [21] initiated the concept of fuzzy soft sets and discussed its various properties. recent researches [20, 21, 22] have shown that both theories of fst and sst can be combined to have a more flexible and expressive framework for modelling and processing data and information possessing nonstatistical uncertainties. in this paper, by combining the ideas of soft sets and fuzzy soft sets, the concept of fuzzy soft set-valued maps is initiated; that is, a map whose range set is a family of fuzzy soft sets. thereafter, motivated by the work of suzuki [29], djoric and lazovic [16], we define the notion of suzuki-type fuzzy soft (e, k)-weak contraction in the setting of fuzzy soft sets, where e is an element of the parameter set e and k ≥ 0. as a result, fuzzy soft fixed point theorem of suzuki-type is proved and some consequences are obtained thereafter. in addition, an application in homotopy result is established to show the usability of our obtained results. the rest of this paper is organized as follows. in subsection 1.1, specific concepts of fuzzy sets and multivalued mappings are recalled. subsection 1.2 briefly reviews some background of soft sets and fuzzy soft sets. section 2 contains the novel concepts of fuzzy soft set-valued maps. in subsection 2.1, a 27 m. s. shagari / j. nig. soc. phys. sci. 2 (2020) 26–35 28 few fuzzy soft fixed point theorems of suzuki-type (e, k)-weak contractions are established. subsection 2.2 gives some consequences of subsection 2.1. an application in homotopy result is presented in subsection 2.3. 1.1. fuzzy sets and multivalued mappings let (x, d) be a metric space. we denote by c b(x) the class of all nonempty, closed and bounded subsets of x. let h(., .) be the hausdorff metric on c b(x) induced by d, that is, h(a, b) = max { sup a∈a d(a, b), sup b∈b d(a, b) } , for a, b ∈ c b(x), where d(x, a) = inf{d(x, a) : a ∈ a}. a point u in x is a fixed point of a multi-valued mapping t : x −→ c b(x) if u ∈ t u. let x be an initial universe. recall that an ordinary subset a of x is determined by its characteristic function χa, defined by χa : a −→{0, 1}: χa(x) = 1, if x ∈ a0, if x < a. the value χa(x) specifies whether an element belongs to a or not. this idea is used to define fuzzy sets by allowing an element x ∈ a to assume any possible value in the interval [0, 1]. thus, a fuzzy set a in x is a set of ordered pair given as a = {(x,µa(x)) : x ∈ x}, where µa : x −→ [0, 1] = i and µa(x) is called the membership function of x or the degree to which x ∈ x belongs to the fuzzy set a. if a is a fuzzy set in x, the (crisp) set of elements in x belonging to a at least of degree α ∈ (0, 1] is called the α-level set, denoted by [a]α. that is, [a]α = {x ∈ x : µa(x) ≥ α}. on the other hand, [a]∗α = {x ∈ x : µa(x) > α}, is called the strong α-level set or strong α-level cut. denote by i x , the family of all fuzzy sets in x. a mapping t : x −→ i x is called fuzzy mapping. an element u in x is said to be a fuzzy fixed point of a fuzzy mapping t if there exists an α ∈ (0, 1] such that u ∈ [t u]α. if there exists α ∈ [0, 1] such that [a]α, [b]α ∈ c b(x), then d∞(a, b) = sup α h ([a]α, [b]α) . 1.2. soft sets and fuzzy soft sets let e be the parameter set, a ⊆ e and p(x) represents the power set of an initial universe of discourse x. in this section, some basic concepts and examples of soft sets and fuzzy soft sets are recalled. molodstov [26] established the concept of soft sets with the following definition. definition 1.5. [26] a pair (f, a) is called a soft set over x under e, where a ⊆ e and f is a mapping given by f : a −→ p(x). in other words, a soft set over x is a parameterized family of subsets of x. for each e ∈ e, f(e) is considered as the set of e-approximate elements of (f, a). example 1.6. [25] suppose the following: xis the universal set of all students in a certain university, eis the set of parameters, given as: e = {intelligent, hardworking, dull, hardworking and intelligent}. assume that they are one hundred students in the university x given as x = {x1, x2, x3, x4, x5 · · · x100}, and e = {e1, e2, e3, e4}, where e1= intelligent, e2 = hardworking, e3= dull, e4= hardworking and intelligent. then f : e −→ p(x) defined by f(e1) = {x1, x2, · · · x10} means that x1, x2, · · · x10 are intelligent , f(e2) = {x11, x12, · · · x30} means that x11, x12, · · · x30 are hardworking, f(e3) = ∅ means that there is no dull student in the university in question, f(e4) = {x15, x81} means that the students x15 and x81 are both intelligent and hardworking. then we can view the soft set (f, e) describing the “kind of students” as the following approximations: (f, e) = { (intelligent students, {x1, x2, · · · x10}), (hardworking studnts, {x11, x12, · · · x30}) (dull,∅), (intelligent and hardworking students, {x15, x81}) } . many researchers carried out formal studies of these basic ideas of soft sets and related notions . for example, maji etal [24] developed these notions and established other concepts such as soft subsets and supersets, intersections and unions, soft equalities and so on. for soft set-based decision making approach, the interested reader may consult cagman and enginoglu [15], feng and zhou[17] and maji et al [23]. remark 1.7. (see [26]) every fuzzy set is a special kind of soft set. in other words, let a be a fuzzy set with membership function µa. consider the family of α-level sets given by ω(α) = {x ∈ x : µa(x) ≥ α}, α ∈ [0, 1]. if the family ω is known, then we can find µa(x) via the formula: µa(x) = sup{α : α ∈ [0, 1]}, x ∈ ω(α). this shows that in particular, every fuzzy set is a soft set (ω, [0, 1]). in order to model more general scenarios, maji et al [21] defined the notion of fuzzy soft sets in the following manner. 28 m. s. shagari / j. nig. soc. phys. sci. 2 (2020) 26–35 29 definition 1.8. [21] a pair (f, a) is a fuzzy soft set over x when a ⊆ e and f : a −→ i x . clearly, every soft set can be thought of as a fuzzy soft set. following example 1.6, fuzzy soft sets allow the investigation of some more intriguing properties such as “how much time each student works” in which case partial memberships are indispensable. example 1.9. [25] consider example 1.6. the fuzzy soft set (f, e) describing the “kind of students” under fuzzy circumstances may be given as (i) f(e1) = {x1/0.5, x2/0.1, x4/0.7}, f(e2) = {x3/0.6, x9/0.7, x20/0.1} (ii) f(e3) = {x19/0.8, x25/0.1, x4/0.7}, f(e4) = {x13/0.4, x91/0.3, x94/0.1, x97/0.3}. in what follows, we initiate the concept of fuzzy soft setvalued maps. 2. main results let x be an initial universe of discourse, e the parameter set, i = [0, 1] and i(e;x) denotes the family of fuzzy soft sets over x under e. by a ∈ i(e;x), we mean the mapping a : e −→ i x , where x is the set of all fuzzy sets in x. let n, z, and r denote the set of nonnegative integers, set of integers and the set of real numbers, respectively. definition 2.1. a mapping t : x −→ i(e;x) is called fuzzy soft set-valued map. notice that if t : x −→ i(e;x) is a fuzzy soft set-valued map, then for x ∈ x, t x ∈ i(e;x) implies that t x : e −→ i x is a fuzzy soft set. this further implies that for any e ∈ e, t (x)(e) : x −→ [0, 1] is a fuzzy set. hence, (t x)(e)(t) is the degree of membership of t in (t x)(e). in the remaining part of this paper, we shall write t [x; e] and t [x; e](t) to represent (t x)(e) and (t x)(e)(t), respectively. example 2.2. let x = [−4, 4] and e = [0, 5]. define t : x −→ i(e;x) by t [x; e](t) = t2 x2 + e + t2 . then t is a fuzzy soft set-valued map. notice that t [x; e](t) ∈ [0, 1] for all x, t ∈ x and e ∈ e . definition 2.3. let t : x −→ i(e;x) be a soft set-valued map. then the e-level set of t is defined by [t x]ee = ⋃ e∈e [t [x; e]]ee , where [t [x; e]]ee = {t ∈ x : t [x; e](t) ≥ e}, e ∈ e. for simplicity, we shall also denote the e-level set of t by [t ]ee , whenever the domain of t is obvious. proposition 2.4. every α-level set is an e-level set. proof. let a be a fuzzy set in x and e = (0, 1] be a set of parameters. consider a fuzzy soft set-valued map t : x −→ i(e;x) defined by t [x; e](t) = a(x), for all x, t ∈ x and e ∈ e. then for α ∈ (0, 1] = e, we have⋃ α∈(0,1] [a]α = ⋃ α∈(0,1] {t ∈ x : a(x) ≥ α} = ⋃ α∈(0,1] {t ∈ x : t [x; e](t) ≥ α = e} = ⋃ e∈e {t ∈ x : t [x; e](t) ≥ e} = [t ]ee . remark 2.5. every fuzzy mapping υ : x −→ i x is a fuzzy soft set-valued map tυ : x −→ i(e;x), defined by tυ[x; e](t) = {t ∈ x : (υx)(t) ≥ α}, α ∈ (0, 1]. notice that i x 7−→ i(e;x) is embedding by λ −→ θλ, for every λ in i x ; where θλ[x; e](t) = {t ∈ x : λ(t) ≥ e}. let x be a metric space. for x ∈ x, consider two fuzzy soft sets a, b ∈ i(e;x) and e ∈ e. assume that [a]ee, [b] e e ∈ c b(x). then define pee (x, a) = inf { d(x, a) : a ∈ [a]ee } ; pee (a, b) = inf { d(a, b) : a ∈ [a]ee, b ∈ [b] e e } ; p(a, b) = sup e pee (a, b); dee (a, b) = d h e ([a] e e, [b] e e ); where dhe ([a] e e, [b] e e ) = max  supa∈[a]ee d(a, [b]ee ), supb∈[b]ee d(b, [a]ee )  . define a distance function d∞e : i (e;x) × i(e;x) −→ r by d∞e ([a] e e, [b] e e ) = sup d h e ([a] e e, [b] e e ). let g : x −→ x be a single-valued mapping and t : x −→ i(e;x) be a fuzzy soft set-valued map. an element x ∈ x is said to be: (i) a fixed point of g if x = gx. denote by fix(g), the set of all fixed points of g. (ii) fuzzy soft fixed point of t if x ∈ [t x]ee for some e ∈ e. 29 m. s. shagari / j. nig. soc. phys. sci. 2 (2020) 26–35 30 for x, y ∈ x, define∐ (x, y) = max { d(x, y), d(x, [t x]ee ), d(y, [t y] e e ), d(x, [t y]ee ) + d(y, [t x] e e ) 2 , d(x, [t x]ee )d(y, [t y] e e ) 1 + d(x, y) } g∐ (x, y) = max { d(x, y), d(x, gx), d(y, gy), d(x, gy) + d(y, gx) 2 , d(x, gx)d(y, gy) 1 + d(x, y) } .∏ (x, y) = min { d(x, [t x]ee ), d(y, [t x] e e ), d(x, [t x]ee )d(y, [t y] e e ) 1 + d(x, y) . g∏ (x, y) = min { d(x, gx), d(y, gx), d(x, gx)d(y, gy) 1 + d(x, y) } . throughout this paper, for e = (0,σ), σ ≥ 1, the function r : e −→ ( 1 2 , 1 ) is defined by r(e) = 1, if 0 < e < 121 − e, if 12 ≤ e < 1σ. motivated by definitions 1.2 and 3 as well as theorem 1.4, we give the next definition. definition 2.6. a fuzzy soft set-valued map t : x −→ i(e;x) is called suzuki-type soft (e, k)-weak contraction if for all x, y ∈ x with x , y, there exist some e ∈ e and k ≥ 0 such that r(e)d(x, [t x]ee ) ≤ d(x, y) implies dhe ([t x] e e, [t y] e e ) ≤ e ∐ (x, y) + k ∏ (x, y). 2.1. fuzzy soft fixed points of suzuki-type soft (e, k)-weak contractions in this subsection, we first present fuzzy soft fixed theorem of suzuki-type fuzzy soft set-valued (e, k)-weak contractions and then obtain some associated consequences. theorem 2.7. let (x, d) be a complete metric space and t : x −→ i(e;x) a suzuki-type fuzzy soft set-valued (e, k)-weak contraction. if for some x ∈ x, there exists e ∈ e such that [t x]ee is a nonempty closed and bounded subset of x, then fix(t ) , ∅. proof. let e1 ∈ e be such that 0 < e ≤ e1 < 1 σ , x1 ∈ x and ρ = 1√ e . then by hypothesis, [t x1]ee is nonempty. therefore, we can find x2 ∈ x such that x2 ∈ [t x1]ee . since ρ > 1, choose x3 ∈ [t x2]ee such that d(x2, x3) ≤ ρd h e ([t x1] e e, [t x2] e e ). if x1 = x2, then x1 ∈ [t x1]ee for some e ∈ e, and the theorem is proved. assume that x1 , x2 . since r(e) ≤ 1, therefore, r(e)d(x1, [t x1] e e ) ≤ d(x1, [t x1] e e ) ≤ d(x1, x2) implies (5) if d(x1, x2) ≤ d(x2, x3), then from (5), we have d(x2, x3) ≤ √ ed(x2, x3) < d(x2, x3), a contradiction. hence, d(x1, x2) > d(x2, x3) and (5) becomes d(x2, x3) ≤ √ ed(x1, x2) ≤ √ e1d(x1, x2). continuing in this fashion, we generate a sequence {xn}n∈n in x such that xn+1 ∈ [t xn]ee and d(xn, xn+1) ≤ √ e1d(xn−1, xn), from which we have ∞∑ n=1 d(xn, xn+1) ≤ ∞∑ n=1 (√ e1 )n−1 d(x1, x2) < ∞. by a standard argument, we conclude that {xn}n∈n is a cauchy sequence in x. the completeness of x implies that there exists u ∈ x such that xn −→ u as n −→∞. claim: for all u , z,we have d(u, [t z]ee ) ≤ e max { d(u, z), d(z, [t z]ee ) } . (6) since xn −→ u as n −→ ∞, there exists a positive integer m such that d(xn, u) ≤ 1 3 d(u, z), for all n ≥ m. given that xn+1 ∈ [t xn]ee , we get r(e)d(xn, [t xn] e e ) ≤ d(xn, [t xn] e e ) ≤ d(xn, xn+1) ≤ d(xn, u) + d(u, xn+1) ≤ 2 3 d(u, z). thus, for n ≥ m, we have (7) hence, r(e)d(xn, [t xn] e e ) ≤ d(xn, z) (8) implies (9) from (9), we have (10) as n −→∞ in (10), we obtain d(u, [t z]ee ) ≤e max { d(u, z), d(z, [t z]ee ), d(u, [t z]ee ) + d(u, z) 2 } + k min { 0, d(u, z) } . ≤e max { d(u, z), d(z, [t z]ee ) } . 30 m. s. shagari / j. nig. soc. phys. sci. 2 (2020) 26–35 31 now, to show that u ∈ [t u]ee for some e ∈ e, we consider the following two possible cases: case(i): 0 < e < 12 . suppose that for all e ∈ e, u , p, u < [t u]ee and p ∈ [t u] e e such that d( p, u) < d(u, [t u]ee ). setting z = p in (6), we have d(u, [t p]ee ) ≤ e max { d(u, p), d( p, [t p]ee ). } (11) now, r(e)d(u, [t u]ee ) ≤ d(u, [t u] e e ) ≤ d(u, p) implies (12) suppose that d(u, p) ≤ d( p, [t p]ee ), then from (12), we have d( p, [t p]ee ) ≤ ed( p, [t p] e e ) < d(p, [t p] e e ), a contradiction. hence, d(u, p) > d( p, [t p]ee ), and d( p, [t p]ee ) ≤ ed(u, p) < d(u, p). (13) therefore, (11) becomes d(u, [t p]ee ) ≤ ed(u, p). consequently, d(u, [t u]ee ) ≤ d(u, [t p] e e ) + d h e ([t p] e e, [t u] e e ) ≤ d(u, [t p]ee ) + e max { d(u, p), d( p, [t p]ee ) } ≤ ed(u, p) + ed(u, p) = 2ed(u, p) < d(u, p) < d(u, [t u]ee ),( in view of assumption ) yields a contradiction. therefore, u ∈ [t u]ee for some e ∈ e. case(ii): 12 ≤ e < 1 σ . for this, we shall prove that (14) holds for allz ∈ x with z , u. now, for all m ∈ n , there exists vm ∈ [t z]ee such that d(u, vm) ≤ d(u, [t z] e e ) + 1 5m d(u, z). thus, we have d(z, [t z]ee ) ≤ d(z, vm) ≤ d(z, u) + d(u, vm) ≤ d(z, u) + d(u, [t z]ee ) + 1 5m d(z, u). using (6), we have (15) if d(z, u) > d(z, [t z]ee ), then from (15), we get d(z, [t z]ee ) ≤ d(u, z) + ed(u, z) + 1 5m d(u, z) = [ (1 + e) + 1 5m ] d(u, z), from which we have[ 1 1 + e ] d(z, [t z]ee ) ≤ [ 1 + 1 5(1 + e)m ] d(u, z). using r(e) = 1 − e, we get (16) as m −→∞ in (16), we have r(e)d(z, [t z]ee ) ≤ d(u, z). on the other hand, if d(u, z) < d(z, [t z]ee ), then (15) gives d(z, [t z]ee ) ≤ d(u, z) + ed(z, [t z] e e ) + 1 5m d(u, z), which yields (1 − e)d(z, [t z]ee ) ≤ ( 1 + 1 5m ) d(u, z). (17) as m −→∞ in (17), we have (1 − e)d(z, [t z]ee ) ≤ d(u, z). this shows that r(e)d(z, [t z]ee ) ≤ d(u, z), which, by definition 2.6, implies (14). moreover, since xn+1 , xn for all n ∈ n, then u , xn+1 . therefore, setting xn = z in (14), we have (18) as n −→∞ in (18), we have (19) since 1 − e > 0, therefore, (19) implies that d(u, [t u]ee ) = 0, and consequently, u ∈ [t u]ee , for some e ∈ e. example 2.8. let x = [−40, 60], e = (0, 30], e ∈ e be arbitrary and d : x × x −→ [0,∞) be defined by d(x, y) = |x − y|, for all x, y ∈ x. define the mapping t : x −→ i(e;x) by t [x; e](t) =  ∆1, if 0 ≤ t ≤ x 30 ∆2, if x 30 < t ≤ x 10 ∆3, if x 10 < t ≤ 60, where ∆1 =  0.2, if 0 < e1 ≤ 10 0.5, if 10 < e2 ≤ 20 0.8, if 20 < e3 ≤ 30. ∆2 =  0.7, if 0 < e1 ≤ 10 0.9, if 10 < e2 ≤ 20 0, if 20 < e3 ≤ 30. ∆3 =  0.4, if 0 < e1 ≤ 10 0.6, if 10 < e2 ≤ 20 0.2, if 20 < e3 ≤ 30. 31 m. s. shagari / j. nig. soc. phys. sci. 2 (2020) 26–35 32 if e = 0.8, then [t [x; e1]] 0.8 e = {t ∈ x : t [x; e1](t) ≥ 0.8} = ∅. [t [x; e2]] 0.8 e = {t ∈ x : t [x; e2](t) ≥ 0.8} = ( x 30 , x 10 ] . [t [x; e3]] 0.8 e = {t ∈ x : t [x; e3](t) ≥ 0.8} = [ 0, x 30 ] . therefore, [t x]ee = ⋃ e∈e [t [x; e]]ee = [ 0, x 10 ] . note that t is a suzuki-type fuzzy soft set-valued (e, k)-weak contraction with e = 0.8 and k = 0. in particular, for x = 40 and y = 50, r(e)d(x, [t x]ee ) = r(0.8)d(40, [t 40] 0.8 e ) = (0.2)d(40, [0, 4]) = 7.2 ≤ d(40, 50) = 10 implies dhe ([t 40] 0.8 e , [t 50] 0.8 e ) = d h e ([0, 4], [0, 5]) = 1 ≤ 118 = (0.8) max { d(40, 50), d(40, [t 40]0.8e ), d(50, [t 50] 0.8 e ), d(40, [t 40]0.8e ) + d(50, [t 40] 0.8 e ) 2 , d(40, [t 40]0.8e )d(50, [t 50] 0.8 e ) 1 + d(40, 50) } . consequently, theorem 2.7 can be applied to find 0 ∈ x such that 0 ∈ [t 0]0.8e . corollary 2.9. let (x, d) be a complete metric space and t : x −→ i(e;x) a fuzzy soft set-valued map. assume that for x, y ∈ x with x , y, there exists e ∈ e such that [t x]ee is a nonempty closed and bounded subsets of x. if r(e)d(x, [t x]ee ) ≤ d(x, y) implies d∞e ([t x] e e, [t y] e e ) ≤ e ∐ (x, y) + k ∏ (x, y), then fix(t ) , ∅. proof. since dhe ([t x] e e, [t y] e e ) ≤ d ∞ e ([t x] e e, [t y] e e ), then by theorem 2.7, the conclusion holds. corollary 2.10. let (x, d) be a complete metric space and t : x −→ i(e;x) a fuzzy soft set-valued map. assume that for x, y ∈ x with x , y, there exists e ∈ e such that [t x]ee is a nonempty closed and bounded subsets of x. if r(e)d(x, [t x]ee ) ≤ d(x, y) implies dhe ([t x] e e, [t y] e e ) ≤ e ∐ (x, y), then fix(t ) , ∅. corollary 2.11. let (x, d) be a complete metric space and t : x −→ i(e;x) a soft set-valued map. if there exist positive constants a, b, c with e = a + b + c < 1, k ≥ 0 such that for x, y ∈ x with x , y, [t x]ee is a nonempty closed and bounded subsets of x and r(e)d(x, [t x]ee ) ≤ d(x, y) implies dhe ([t x] e e, [t y] e e ) ≤ ad(x, y)+bd(x, [t x] e e )+cd(d(y, [t y] e e ))+k ∏ (x, y). then fix(t ) , ∅. corollary 2.12. let (x, d) be a complete metric space and g : x −→ x a single-valued mapping. assume that there exists 0 < e < 1 and k ≥ 0 such that for x, y ∈ x with x , y, g(x) is a nonempty closed and bounded subset of x . if r(e)d(x, gx) ≤ d(x, y) implies d(gx, gy) ≤ e g∐ (x, y) + k g∏ (x, y), then g has a unique fixed point. proof. the existence of the fixed point of g follows from theorem 2.7. for uniqueness, suppose that there exist u1, u1 ∈ x with u1 , u2 such that gu1 = u1 and gu2 = u2. then r(e)d(u1, gu1) ≤ d(u1, gu1) = d(u1, u1) = 0 ≤ d(u1, u2) implies d(u1, u2) = d(gu1, gu2) ≤ e max { d(u1, u2), d(u1, gu1), d(u2, gu2), d(u1, gu2) + d(u2, gu1) 2 , d(u1, gu1)d(d(u2, gu2)) 1 + d(u1, u2) } +k min { d(u1, gu1), d(u2, gu1), d(u1, gu1)d(u2, gu2) 1 + d(u1, u2) } ≤ e max { d(u1, u2), d(u1, u1), d(u2, u2), d(u1, u2) + d(u1, u2) 2 , d(u1, u1)d(u2, u2) 1 + d(u1, u2) } +k min { d(u1, u1), d(u1, u2), d(u1, u1)d(u2, u2) 1 + d(u1, u2) } ≤ e max{d(u1, u2), 0} + k(0) ≤ ed(u1, u2), a contradiction. therefore u1 = u2. next, we present a local fuzzy soft fixed point theorem for suzuki-type soft (e, k)-weak contractions. first, recall that an open ball with radius r > 0 , centered at x0 in a metric space x, is given by br (x0) = {x ∈ x : d(x, x0) < r}. 32 m. s. shagari / j. nig. soc. phys. sci. 2 (2020) 26–35 33 theorem 2.13. let (x, d) be a complete metric space, t : br (x0) −→ i(e;x) be a suzuki-type fuzzy soft set-valued (e, k)-weak contraction. assume that for some x ∈ x, there exists e ∈ e such that [t x]ee is a nonempty closed and bounded subset of x and d(x0, [t x0] e e ) < (1 − e)r. then fix(t ) , ∅. proof. let 0 < η < r be such that 0 < (1 − e)(1 + √ e) ≤ 1 1+η , b ∗ η(x0) ⊂ br (x0) and d(x0, [t x0] e e ) < (1 − e)η. therefore, (1 − e)η − d(x0, [t x0]ee ) > 0. choose γ = (1 − e)η − d(x0, [t x0]ee ) > 0; then there exists x1 ∈ [t x0] e e such that d(x0, x1) < d(x0, [t x0]ee ) + γ. thus, d(x0, x1) < (1 − e)η. now, for ρ = 1√ e and x1 ∈ [t x0]ee , there exists x2 ∈ [t x1] e e such that d(x1, x2) ≤ ρdhe ([t x0] e e, [t x1] e e ). noting that r(e)d(x0, [t x0] e e ) ≤ r(e)d(x0, x1) ≤ d(x0, x1), we have (20) if d(x0, x1) ≤ d(x1, x2), then (20) gives d(x1, x2) ≤ √ ed(x1, x2) < d(x1, x2), a contradiction. hence, d(x0, x1) > d(x1, x2) and d(x1, x2) < √ ed(x0, x1) < √ e(1 − e)η. notice that by assumption and the triangle inequality, we have d(x0, x2) ≤ d(x0, x1) + d(x1, x2) < (1 − e)η + √ e(1 − e)η ≤ η 1 + η < η. hence, x2 ∈ bη(x0). continuing this process recursively, we generate a sequence {xn}n∈n such that (a) xn ∈ bη(x0) for all n ∈ n; (b) xn ∈ [t xn]ee for each n ∈ n; (c) d(xn, xn+1) ≤ ( √ e)n(1 − e)η for all n ∈ n. from (c), it follows by usual arguments that {xn}n∈n is a cauchy sequence and converges to some u ∈ br (x0). from here, following the steps in the proof of theorem 2.7, we conclude that fix(t ) , ∅. 2.2. consequences here, we show that there is a link between multi-valued mappings and fuzzy soft set-valued maps by obtaining some existing results as consequences of our result. corollary 2.14. (see [16, theorem 2.1]) let (x, d) be a complete metric space and t : x −→ c b(x) be a multi-valued mapping. assume that there exists an α ∈ [0, 1) such that for all x, y ∈ x, r(α)d(x, t x) ≤ d(x, y) implies h(t x, t y) ≤ αd(x, y); where the mapping r : [0, 1) −→ (0, 1] is defined by r(α) = 1, if 0 ≤ α ≤ 11 −α, if 12 ≤ α < 1. then there exists u ∈ x such that u ∈ t u. proof. let e := (0, 1) be a parameter set and e ∈ e be arbitrary. consider a mapping $ : x −→ e and a fuzzy soft set-valued map z : x −→ i(e;x) defined by z[x; e](t) = $(x), if t ∈ t x0, if t < t x. then [zx]ee = ⋃ e∈e {t ∈ x : z[x; e](t) ≥ $(x) = e} = t x. therefore, d(x, y) ≥ r(α)d(x, t x) = r(α)d(x, [zx]ee ). thus, theorem 2.7 can be applied to find u ∈ x such that u ∈ t u. from theorem 2.7 and using the method of proof of corollary 2.14, we can also derive the following results. corollary 2.15. (see [19]) let (x, d) be a complete metric space and t : x −→ c b(x) be a multi-valued mapping. assume that there exists α ∈ [0, 1) such that r(α)d(x, t x) ≤ d(x, y) implies h(t x, t y) ≤ α max {d(x, y), d(x, t x), d(y, t y)} for all x, y ∈ x, where the function r is defined as in corollary 2.14. then, there exists u ∈ x such that u ∈ t u. corollary 2.16. (see [9]) let (x, d) be a complete metric space and t : x −→ c b(x) be a multi-valued mapping. let the function r : [0, 1) −→ [0, 1) be as defined in corollary 2.14. assume that there exists a(x, y) ∈ [0, 1) such that r(α)d(x, t x) ≤ d(x, y) implies h(t x, t y) ≤ a(x, y) [ d(x, t y) + d(y, t x) ] for all x, y ∈ x. then there exists u ∈ x such that u ∈ t u. corollary 2.17. let (x, d) be a complete metric space and t : x −→ c b(x) a multi-valued mapping. assume that there exist constants α ∈ [0, 1) and k ≥ 0 such that for all x, y in x with x , y, if r(α)d(x, t x) ≤ d(x, y) implies h(t x, t y) ≤ α ∐ (x, y) + k ∏ (x, y), where the function r is as defined in corollary 2.14, then, there exists u ∈ x such that u ∈ t u. 33 m. s. shagari / j. nig. soc. phys. sci. 2 (2020) 26–35 34 corollary 2.18. let (x, d) be a complete metric space and t : x −→ c b(x) a multi-valued mapping. assume that there exist constants α ∈ [0, 1) and k ≥ 0 such that for all x, y in x with x , y, if r(α)d(x, t x) ≤ d(x, y) implies d∞(t x, t y) ≤ α ∐ (x, y) + k ∏ (x, y), where the function r is as defined in corollary 2.14. then, there exists u ∈ x such that u ∈ t u. in corollary 2.17, if t is taken as a single-valued mapping, then we have the next result. corollary 2.19. let (x, d) be a complete metric space and t : x −→ x be a single-valued mapping. assume that there exist constants α ∈ [0, 1) and k ≥ 0 such that for all x, y in x with x , y, if r(α)d(x, t x) ≤ d(x, y) implies d(t x, t y) ≤ α ∐ (x, y) + k ∏ (x, y), where the function r is as defined in corollary 2.14, then, there exists a unique u ∈ x such that u = t u. 2.3. application in homotopy in this section, we apply theorem 2.13 to prove a homotopy result. first, for convenience, we recall the following familiar definitions. definition 2.20. a relation ≤ is a total order on a set u if for all s, t, u ∈ u, the following conditions hold: (i) reflexivity: s ≤ s; (ii) antisymmetry: if s ≤ t and t ≤ s, then s = t; (iii) transitivity: if s ≤ t and t ≤ u, then s ≤ u; (iv) comparability: for every s, t ∈ u, either s ≤ t or t ≤ s. recall that if the set u satisfies only the axioms (i) − (iii), then it is said to be partially ordered. in what follows, we shall call a totally ordered set a chain. lemma 2.21. (kuratowski-zorn’s lemma ) if u is any nonempty partially ordered set in which every chain has an upper bound, then u has a maximal element. definition 2.22. let x1 and x2 be any two topological spaces and π,ω : x1 −→ x2 a continuous functions. a function h : x1 × [0, 1] −→ x2 such that if x ∈ x1, then h(u, 0) = π(u) and h(u, 1) = ω(u), is called a homotopy between π and ω. we shall denote the boundary of a set u by bd(u). theorem 2.23. let (x, d) be a complete metric space and u be an open subset of x. if m : u × [0, 1] −→ i(e;x) satisfies the following conditions: (hom-1) u < m(u, t) for every u ∈ bd(u) and t ∈ [0, 1]; (hom-2) m(., t) : u −→ i(e;x) is a suzuki-type soft set-valued (e, k)-weak contraction for all t ∈ [0, 1]; (hom-3) there exist a nondecreasing function f : [0, 1] −→ r such that dhe (m(u, t), m(u, s)) ≤ | f (t) − f (s)| for all s, t ∈ [0, 1] and each u ∈ u; (21) (hom-4) m : u × [0, 1] −→ i(e;x) is closed and bounded. then m(., 0) has a fixed point. proof. assume p is a fixed point of m(., 0). then by (hom-1), p ∈ u. consider the set ∧ given by∧ = {(t, u) ∈ [0, 1] × u : u ∈ m(u, t)} . notice that (0, p) ∈ ∧ ; hence ∧ , ∅. let e ∈ e := (0, 1) and define a partial order ≤ on ∧ as follows: (t, u) ≤ (s, v) if and only if t ≤ s and d(u, v) ≤ 2 1 − e [ f (s) − f (t) ] . suppose ω is a chain of ∧ and t∗ := sup{t : (t, u) ∈ ω}. assume that {tn, un} is a sequence in ω such that (tn, un) ≤ (tn+1, un+1) and tn −→ t∗ as n −→∞. then for all positive integers m, n(m > n), d(um, un) ≤ 2 1 − e | f (tm) − f (tn)| . (22) as m, n −→ in (22), we get d(um, un) −→ 0. hence, {un}n∈n is a cauchy sequence and converges to some u∗ ∈ x. since m is closed and un ∈ m(un, tn), therefore, u∗ ∈ m(u∗, t∗). from condition (hom − 1), u∗ ∈ u. thus, (t∗, u∗) ∈ ∧ . since ω is a chain, therefore, (t, u) ≤ (t∗, u∗) for all (t, u) ∈ ω. in other words, (t∗, u∗) is an upper bound of ω. thus, by kuratowskizorn lemma’s , ∧ has a maximal element (t0, u0). next, we show that t0 = 1. suppose on the contrary that t0 < 1. let r = 21−e | f (t) − f (t0)| > 0 with t ∈ (t0, 1] such that br (u0) ⊂ u. notice that by (hom-3), d(u0, m(u0, t)) ≤ d(u0, m(u0, t0)) + d h e (m(u0, t0), m(u0, t)) ≤ | f (t) − f (t0)| = (1 − e)r 2 < (1 − e)r. therefore, m(., t) : br (u0) −→ i(e;x) satisfies all the hypotheses of theorem 2.13 for every t ∈ [0, 1]. consequently, there exists u ∈ br (u0) such that u ∈ m(u, t), which implies that (t, u) ∈ ∧ for all t ∈ [0, 1]. now, d(u0, u) ≤ r = 2 1 − e | f (t) − f (t0)|, yields (t0, u0) < (t, u), a contradiction to the fact that (t0, u0) is maximal. conversely, assume that m(., 1) has a fixed point; then on similar steps as above, one can show that m(., 0) has a fixed point. 34 m. s. shagari / j. nig. soc. phys. sci. 2 (2020) 26–35 35 competing interests the author declares that there is no competing interests. acknowledgments the author is thankful to the editors and the anonymous reviewers for their valuable suggestions and comments on the manuscript. references [1] s. banach,“ sur les opérations dans les ensembles abstraits et leur application aux équations intégrales”, fund. math. 3 (1922) 133. 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[30] l. a. zadeh, “fuzzy sets”, information and control 8 (1965) 338. 35 j. nig. soc. phys. sci. 4 (2022) 848 journal of the nigerian society of physical sciences the type ii topp-leone-g power series distribution with applications on bladder cancer boikanyo makubate∗, broderick o. oluyede, marang pearl matsuokwane, lesego gabaitiri, simbarashe chamunorwa botswana international university of science and technology, p. bag 16, palapye, botswana abstract statistical distributions are important in modeling the real life of an item and therefore proper distributions that provide useful information for sound conclusions and decisions are needed. for that reason, the demand for developing new generalized distributions have become appropriate for data that have both monotonic and non-monotonic hazard rate functions. in this paper, we develop a new distribution called the type ii topp-leone-g power series (tiitlgps) distribution by compounding the type ii topp-leone-g (tiitlg) distribution with the power series distribution. statistical properties of the tiitlgps distribution are obtained. a variety of shapes for the densities and hazard rate are presented of the considered special case. a simulation study to examine the efficiency of the maximum likelihood estimates is also conducted. finally, the bladder cancer data example is analyzed for illustrative purposes, it is displayed that the introduced distribution provides better fit when compared to other non-nested distributions considered in this work. doi:10.46481/jnsps.2022.848 keywords: type ii topp-leone-g distribution, power series distribution, maximum likelihood estimation. article history : received: 12 may 2022 received in revised form: 16 july 2022 accepted for publication: 20 july 2022 published: 15 august 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: b. j. falaye 1. introduction statistical distributions are of tremendous importance in real lifetime analysis in various fields such as clinical studies, medical studies, biological studies and environmental studies, just to mention few. several classical distributions have been derived over past years for modeling data. however, data arising from public health and other areas may not fit the classical distributions. therefore, modifications of the well-established distributions are highly recommended to obtain more flexible new families of distributions. these generalized distributions ∗corresponding author tel. no: email address: makubateb@biust.ac.bw. (boikanyo makubate ) have proved to be crucial in capturing heavily tailed, and where the hazard rate function is non-monotonic (uni-modal, bathtub, upside bathtub or upside bathtub followed by bathtub) and improving the goodness-of-fit in empirical distribution. thus, increased demand in generating new families of distributions that provide flexibility in lifetime phenomenon data modeling. the topp-leone (tl) distribution, proposed by topp and leone [1] as a lifetime model, is one of the distributions used within the theory and practice of statistics. it has proven to be a useful lifetime model through its applicability in different fields such as medical and actuarial sciences. ghitany et al. [2] studied the tl distribution by providing some reliability measures while vicaria et al. [3] introduced a two-sided generalized version of the tl distribution. other various families of extended 1 makubate et al. / j. nig. soc. phys. sci. 4 (2022) 848 2 distributions have been introduced. for instance, the toppleone-gompertz-g family by oluyede et al. [4], the exponentiated odd weibull-topp-leone-g family by chamunorwa et al. [5] and the type ii generalized topp-leone family by hassan et al. [6]. for any baseline distribution, g(x; ψ), the cdf of the type ii topp-leone-g (tiitlg)[7, 8] family of distributions is given by ft i it lg (x; b,ψ) = 1 − [1 − g 2(x; ψ)]b, (1) where b > 0 and ψ is a parameter vector. the survival function associated with 1 is given by s t i it lg (x; b,ψ) = [1 − g 2(x; ψ)]b. (2) suppose that a series system has n components at a given time where n is a discrete random variable with a power series distribution (truncated at zero) and probability mass function (pmf) p(n = n) = anθn c(θ) , n = 1, 2, ..., where an ≥ 0 depends only on n, c(θ) = ∑ ∞ n=1 anθ n and θ ∈ (0, s) (s can be ∞) is chosen such that c(θ) is finite and its three derivatives with respect to θ are defined and given by c′(.), c′′(.) and c′′′(.), respectively. the power series family of distributions [9, 10, 11] includes binomial, poisson, geometric and logarithmic distributions. given n, suppose (x1, x2, ...xn ) ∼ t i it lg are the independent and identically distributed failure times of the n components. let x=min(x1, ..., xn ), then the conditional cdf of x|n = n is given by fx|n=n(x) = 1 − [s t i i−t l−g (x; ψ)] n, x > 0. (3) the type ii topp-leone-g power series (tiitlgps) class of distributions is defined by the marginal cdf of x. the cdf of x ∼ t i it lgps is given by ft i it lgps (x; θ, b,ψ) = 1 − c(θ(1 − g2(x; ψ))b) c(θ) . (4) table 1 shows some sub-classes of the tiitlgps distribution. the density function associated with equation (4) is given by ft i it lgps (x; θ, b,ψ) = 2bθg(x; ψ)g(x; ψ) ×[1 − g2(x; ψ)]b−1 c′ ( θ[1 − g2(x; ψ)]b ) c(θ) , (5) where c′(θ) = ∑ ∞ n=1 nanθ n−1. the hazard function (hrf) is given by ht i it lgps (x; θ, b,ψ) = 2bθg(x; ψ)g(x; ψ)[1 − g2(x; ψ)]b−1 c(θ(1 − g2(x; ψ))b) ×c′ ( θ[1 − g2(x; ψ)]b ) . the quantile function of the tiitlgps distribution is given by q(x) = g−1 (1 − [ c−1(c(θ)(1 − u)) θ ]1/b )1/2 , (6) where g−1 is the inverse of the baseline distribution. note that, the tiitlg family of distribution defined in equation (1) is a limiting special case of the tiitlgps class of distributions when θ → 0+. using the binomial expansion, we express the pdf in 5 as ft i it lgps (x; θ, b,ψ) = ∞∑ m=0 vmgm(x; ψ), (7) where gm(x; ψ) = (2m + 2)g(x; ψ)g 2m+1(x; ψ) (8) is the exp-g distribution with power parameter 2m + 2, and vm = ∞∑ n=0 (−1)m2b(n + 1)a(n+1)θ(n+1) c(θ)(2m + 2) ( b(n + 1) − 1 m ) . (9) thus, the statistical properties of the tiitlgps class of distributions including rth moment, incomplete moment, the moment generating function, the rth moment of residual life, the rth moment of reversed residual life, mean deviations, bonferroni curves and lorenz curves can be obtained directly from those of the exp-g family of the distributions. for, more information regarding the properties of exp-g distributions the reader is refereed to [12-21]. the primary motivation for developing the type ii topp-leoneg power series (tiitlgps) family of distributions is the versatility and flexibility derived from compounding continuous distributions in public health field. the proposed distribution exhibit both monotonic, non-monotic hazard rate functions and heavy tailed data sets, which is a common phenomenon with medical data especially cancer data. figure 1b shows that the special case tiitlwp distribution can capture upside-down bathtub followed by bathtub and uni-modal shapes hazard rate function(s). furthermore, from bladder cancer data modeling example presented, the introduced distribution captures well heavy tailed data and has better fit compared to some of the selected generalizations in this work. this paper is organized as follows. in section 2 some mathematical properties of the new model are presented. parameter estimation via the method of the maximum likelihood is given in section 3. some special models of tiitlgps class of distributions are presented in section 4. in section 5, a simulation study is done. section 6 contain applications to bladder cancer data set. concluding remarks are given in section 7. 2. some mathematical properties of the tiitlgps 2.1. moments and moment generating function this section discusses the rth moment, mth incomplete moment, moment generating function and residual and reversed residual life of the tii-tl-gps. the rth moment of the tii-tlgps can be represented as µr = e(x r ) = ∞∑ m=0 vm ∫ ∞ 0 xr gm(x; ψ)d x (10) 2 makubate et al. / j. nig. soc. phys. sci. 4 (2022) 848 3 table 1. sub-classes of the tiitlgps distribution distribution an c(θ) cdf type ii topp-leone g poisson (n!)−1 eθ − 1 1 e θ ( [1−g2 (x;ψ)]b ) −1 eθ−1 type ii topp-leone g geometric 1 θ(1 − θ)−1 1 ( [1−g2 (x;ψ)]b ) (1−θ) 1−θ ( 1−g2 (x;ψ) )b type ii topp-leone g logarithmic n−1 − log(1 − θ) 1 log ( 1−θ[1−g2 (x;ψ)]b ) log(1−θ) type ii topp-leone g binomial ( m n ) (1 + θ)m − 1 1 ( 1+θ[1−g2 (x;ψ)]b )m −1 (1+θ)m−1 where gm(x; ψ) and vm are defined in (8) and (9) respectively and 2p+2 is the power parameter. the mth incomplete moment can be obtained in the following way µm(y) = ∞∑ p=0 vp ∫ y 0 xr gp(x; ψ)d x. (11) the moment generating function of the tii-tl-gps is represented as: mx(t) = e(e tx) = ∞∑ p=0 vp ∫ ∞ 0 etxgp(x; ψ)d x. (12) we state the following two theorems and provide the proofs at https://drive.google.com/file/d/1wmwisk60cj0unyrqa– pwfkbbwgap9ta/view?usp=sharing. theorem 2.1. let x1, x2, .., xm be a random sample of size m from the tiitlgps class of distributions and x1:m < x2:m < ... < xm:m denote the corresponding ordered random sample of size m. the pdf of the ith order statistic, xi:m is given by fi:m(x) = ∞∑ s=0 ηsgs(x; ψ), where gs(x; ψ) = (2s + 2)g(x; ψ)g2s+1(x; ψ) is the exp-g distribution with power parameter 2s + 2 and ηs = ∞∑ j,n,k,z=0 m! (i − 1)!(m − i)! × 2b(n + 1)a(n+1)θz+n+1(−1) j+k+sdz,k ck+1(θ) × ( m − i j )( j + i − 1 k )( b(z + n + 1) − 1 s ) (2s + 2). theorem 2.2. the rényi entropy for tiitlgps class of distributions is given by ir(ν) = 1 1 −ν log  ∞∑ s=0 wse (1−ν)ireg  , where ireg = 1 1 −υ log ( ∫ ∞ 0 [ ( 2s ν + 2)g(x; ψ)g 2s ν +1(x; ψ) ]ν d x ) is the rényi entropy of exp-g distribution with power parameter ( 2s ν + 2) and ws = ∞∑ z=0 (2b)ν dz,νθ z+ν(−1)s ( b(ν + z) −ν s ) 1 ( 2s ν + 2)ν . 3. maximum likelihood estimation this section uses the method of maximum likelihood to estimate unknown parameters of the tiitlgps class of distributions. let x1, x2, ..., xn be a random sample from a tiitlgps distribution given in equation (5), then the log-likelihood of the parameter vector φ = (b,θ,ψ)t is given by: `(φ) = nlog2 + nlogb + nlogθ + n∑ i=1 logg(xi; ψ) +(b − 1) n∑ i=1 log[1 − g2(xi; ψ)] −nlogc(θ) + n∑ i=1 logc′ ( θ[1 − g2(xi; ψ)] b ) . the components of the score function are obtained by finding the partial derivatives (with respect to the parameters θ, b and ψ, respectively). they are given by: uθ = n θ − nc′(θ) c(θ) + n∑ i=1 ( c′′ ( θ[1 − g2(xi; ψ)]b )) [1 − g2(xi; ψ)] c′ ( θ[1 − g2(xi; ψ)]b ) , ub = n b + n∑ i=1 [1 − g2(xi; ψ)] + n∑ i=1 ( c′′ ( θ[1 − g2(xi; ψ)]b )) c′ ( θ[1 − g2(xi; ψ)]b ) [1 − g2(xi; ψ)]b ×log[1 − g2(xi; ψ)] 3 makubate et al. / j. nig. soc. phys. sci. 4 (2022) 848 4 and uψ = (b − 1) n∑ i=1 1 [1 − g2(xi; ψ)] ∂[1 − g2(x; ψ)] ∂ψ + n∑ i=1 ∂g(xi ; psi) ∂ψ g(xi; ψ) + n∑ i=1 ∂g(xi ; psi) ∂ψ g(xi; ψ) + n∑ i=1 ( c′′ ( θ[1 − g2(xi; ψ)]b )) c′ ( θ[1 − g2(xi; ψ)]b ) ×2bθ[1 − g2(xi,ψ)] b−1g(xi; ψ) ∂g(xi; ψ) ∂ψ . note that these equations are non-linear and can not be solved analytically, but can be solved numerically using software like r language. 4. some special models this section considers and presents some special cases of the tiitlgps class of distributions when the baseline distributions are weibull and burr xii distributions. 4.1. type-ii-topp-leone-weibull-poisson (tiitlwp) distribution the cdf and pdf of the tiitlwp distribution are given by ft i it lw p(x; θ, b,α,λ) = 1 − e ( θ [ 1− ( 1−e−λx α )2]b) −1 eθ − 1 and ft i it lw p(x; θ, b,α,λ) = 2αbθλx α−1e−λx α ( 1 − e−λx α ) [ 1 − ( 1 − e−λx α )2]b−1 × e ( θ [ 1− ( 1−e−λx α )2]b) eθ − 1 , respectively, for θ, b,α,λ and x > 0. figures 1(a) and 1(b) show the plots of the pdfs and hrfs, respectively, for the tiitlwp distribution for selected parameter values. plots of the tiitlwp pdf exhibit different shapes including almost symmetric, left-skewed, right-skewed, and reversej shapes. plots of the hrf of the tiitlwp distribution show different shapes including increasing, decreasing, upside-down bathtub and uni-modal shapes. the cdf, pdf and plots (pdf and hrfs plots) of typeii-topp-leone-weibull-geometric (tiitlwg), type-ii-toppleone-burr xii-poisson (tiitlbxiip), type-ii-topp-leoneburr xii-geometric (tiitlbxiig) distributions can be seen at https://drive.google.com/file/d/1wmwisk60cj0unyrqa– pwfkbbwgap9ta/view?usp=sharing. (a) (b) figure 1. pdfs and hrfs plots for the tiitlwp distribution 5. simulation study in this section, we conduct monte carlo simulation study to assess the performance of maximum likelihood estimates of the tiitlwp distribution. we generated n=1000 samples of size n= 100, 200, 400 and 800 with the help of the r package. we consider simulations for the following sets of initial parameters values (i: α = 1.0, b = 0.5,λ = 1.5,θ = 1.0), (ii: α = 0.5, b = 0.5,λ = 0.5,θ = 1.0), (iii: α = 1.5, b = 0.5,λ = 0.5,θ = 1.0), and (iv: α = 0.5, b = 4 makubate et al. / j. nig. soc. phys. sci. 4 (2022) 848 5 table 2. parameter estimates and goodness-of-fit statistics for various models fitted for cancer patients data set estimates statistics model α b λ θ −2 log l aic aicc bic w∗ a∗ k-s p-value tiitlwp 0.7361 3.0338 0.0749 3.6985 819.4 827.4 827.8 838.9 0.0226 0.1495 0.0371 0.9945 (0.1340) (8.2303) (0.1021) (2.4628) α λ γ θ owtlllogl 1.0089 0.4757 9.7751 1.2905×10−9 826.4 834.4 834.7 845.8 0.0925 0.6058 0.0574 0.7930 (0.3164) (0.1060) (3.1140) (0.0033) p α β λ egel 1.0729×10−8 0.8922 1.1774 1.3292 826.2 834.2 834.6 845.6 0.1135 0.6819 0.0707 0.5442 (0.0095) (0.6760) (0.1426) (10.0710) a b α β eplp 5.8448×10−8 0.5647 0.82441 2.7914 820.9 828.9 829.2 840.3 0.0391 0.2563 0.0428 0.9733 (0.0196) (0.1021) (0.3150) (1.3114) c s α λ ebxiip 1.0223 1.5133 1.9338 7.8251×10−9 871.7 879.7 880.0 891.1 0.2452 1.600 0.2481 2.872×10−7 (0.0902) (0.2426) (0.2105) (0.0085) a b λ θ bol-u 1.1799 2.8446 3.2800 ×105 7.6115 ×106 826.4 834.4 834.7 845.8 0.1186 0.7117 0.0734 0.4955 (0.1329) (0.3599) (3.1630 ×10−6) (1.3631 ×10−7) β λ θ γ elollogw 6.1173×10−5 0.0378 2.9053 1.0478 828.2 836.2 836.5 847.6 0.1314 0.7864 0.0700 0.5573 (0.4015) (0.0061) (8.4383 ×10−5) (0.0676) (a) (b) figure 2. fitted densities and probability plots for cancer patients data 1.0,λ = 1.5,θ = 0.5). simulation results are shown at https://drive.google.com/file/d/1wmwisk60cj0unyrqa– pwfkbbwgap9ta/view?usp=sharing. the results show that as the sample size increases, the estimates approach the true parameter values, since root mean square errors (rmse) and average bias decays toward zero for all the parameters. 6. applications and discussion in this section, we demonstrate the applicability of the tiitlwp distribution to bladder cancer data set. we compared the tiitlwp distribution to six non-nested models. we use the nlm function in r software to estimate model parameters. we present the model parameters estimates (standard errors in parenthesis) and the goodness-of-fit-statistics. we assessed model performance using -2loglikelihood (-2 log l), akaike information criterion (aic), consistent akaike information criterion (aicc), bayesian information criterion (bic), cramer-von-mises (w∗), and andersen-darling (a∗) (see chen and balakrishnan [23] for details). these statistics are used to verify which model fits the given data well. the smaller the values of these statistics, the better the model. kolmogorov-smirnov (k-s) statistic (and its p-value) and sum of squares (ss) from probability plots were also used to assess the fit of the model. the smaller the ks value and the higher the p-value for the k-s statistic the better the model. tables 2 shows model parameters estimates (standard errors in parentheses) and several goodness-of-fit statistics. figure 2 shows fitted densities and probability plots (as described by chambers et al. [22]) to demonstrate how well our model fits the selected data sets. the non-nested models considered in this paper are the exponentiated generalized logarithmic (egel) distribution by 5 makubate et al. / j. nig. soc. phys. sci. 4 (2022) 848 6 oluyede et al. [24], odd weibull-topp-leone-log logistic logarithmic (owtlllogl) distribution by oluyede et al. [25], exponentiated power lindley poisson (eplp) distribution by pararai et al. [21], exponentiated burr xii poisson (ebxiip) distribution by da silva et al. [12], beta odd lindley-uniform (bol-u) distribution by chipepa et al. [26] and exponential lindley odd log-logistic weibull (elollogw) distribution by korkmaz[15]. the pdfs of the non-nested models can be seen at https://drive.google.com/file/d/1wmwisk60cj0unyrqa– pwfkbbwgap9ta/view?usp=sharing. for the ebxiip and elollw distributions we consider k = 1 and α = 1, respectively. 6.1. bladder cancer patients data this data set represents remission times of a random sample of 128 bladder cancer patients. for more information regarding the bladder cancer data set, lee and wang [27] is recommended. from the values of the goodness-of-fit statistics a∗, w∗, k-s and the p-value of the k-s statistic as shown in table 2, we conclude that the tiitlwp model performs better than the nonnested models considered in this paper. figures 2(a) and 2(b) show the fitted densities and probability plots for the tiitlwp model. we observe that the tiitlwp model has better fit to extreme tailed data compared to the selected competitive models. 7. conclusions the demand of developing new families of distributions from classical ones has been of interest among authors in the past years. a new distribution called the tiitlgps distribution is developed which combines the tiitl-g with the power series to provide compound tiitl-gps family of distributions with better performance compared to selected models. we study some of mathematical properties of the new family of distributions, such as ordinary moment, quantile functions, order statistics and rényi entropy. we discuss maximum likelihood estimates of the model parameters for bladder cancer data. additionally, the performance of the mles of the two selected members is assessed via monte carlo simulation studies based on two criteria; bias and rmse. the study exhibits a good performance when estimating the parameters of the proposed family using the maximum likelihood method. finally, the special case of the new distribution applied to the bladder cancer data set and a simulation study demonstrate its usefulness and potentiality to analysis of lifetime data. we hope that the proposed distribution will attract wider application in various fields such as health, survival and lifetime data, finance, insurance, among others. future work • take into consideration bivariate extensions of proposed model via copulas like the farlie-gumbel-margestern copulas, ali-mikhail-haq copula, clayton copula and rényi entropy copula. • apply different parameter estimation techniques such as bayesian technique, weighted least squares (wls), maximum product of spacings (mps) on the proposed models. limitations • we derive mathematical and statistical properties from the exponentiated g distribution cause its a bit complicated to derive them straight from the tiitlgps pdf. • the model can not perform well on some data sets. references [1] c. w. topp & f. c. leone, ”a family of j-shaped frequency functions”, journal of the american statistical association 50 (1955) 209. doi: http://dx.doi.org/10.1080/01621459.1955.10501259. 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[27] e. t.lee & j. w. wang, statistical methods for survival data analysis, 3rd edn., john wiley and sons, new york, isbn: 9780471458555 (2003) 534. 7 j. nig. soc. phys. sci. 2 (2020) 166–169 journal of the nigerian society of physical sciences original research measurement of the excited energies identified in 42ca using the rosphere gamma-ray arrays t. daniela,∗, a. a. mcasuleb adepartment of physics, benue state university, p. m. b. 102119, makurdi, benue state bdepartment of physics, federal university of agriculture, p. m. b. 2373, makurdi, benue state abstract excited energy states totalling 11 in number identified to be associated with the 42ca were detected via the rosphere gamma-ray array detectors of ifin-hh bucharest from a 28si(18o, 2p2n), where a multi-particle of 2 protons and 2 neutrons were evaporated in a fusion reaction. the excited energies were identified using γ-ray coincidences. all detected gamma energies (γ) recorded were compared with various literatures from nndc and this shows an excellent agreement with each and the results are presented together with their calculated relative intensities. keywords: absolute relative error, accuracy, comparative study, stability. article history : received: 05 february 2020 received in revised form: 20 april 2020 accepted for publication: 03 may 2020 published: 01 august 2020 c©2020 journal of the nigerian society of physical sciences. all rights reserved. communicated by: w. a. yahya 1. introduction calcium-42 is one of the isotopic-nucleus of calcium with 22 neutrons and 20 protons. the two excess neutrons present in the nucleons distinguish it from the first stable nucleus of 40ca with either protons or neutrons being magic. the heaviest of the isotopes is 48ca with 8-neutrons higher than 40ca [1]. the measured gamma energies of the nucleus have been reported in several articles with some of them showing the relative intensities of the measured gamma transitions. in this, the relative intensities of these gamma energies have been measured after they were identified to be associated with 42ca through the instrumentalities of coincidence analysis [2]. ∗corresponding author tel. no: +2348167598988 email address: tdaniel@bsum.edu.ng, terver.daniel@yahoo.co.uk (t. daniel ) 2. experimental details in the coulomb unsafe reaction of 192os target with the 18o beam at 80 mev using the 9 mv tandem accelerator of the horia hulubei national institute of physics and nuclear engineering (known as ifin-hh) in bucharest [3], a fusion multiparticle of 2n2p was evaporated from the fusion of 28si target with 18o – beam. 28si was found to be one of the impurities in the 192os target foil material bought from the trace company in usa. this contaminated percentage of 42ca and other chemical impurities from the target were observed to have reacted during the experiment when the 18o beam delivered the laboratory energy of 80 mev on the 192os target in the chamber of the accelerator. details of the purity composition of the 192os are as shown in table 1. this nuclear fusion evaporation reaction produced 42ca as 166 daniel & mcasule / j. nig. soc. phys. sci. 2 (2020) 166–169 167 the product of the reaction after evaporating 2n2 p particles from the compound with the beam current of 20 pna with a hardware trigger condition of either labr3(ce) – labr3(ce) or hpge – hpge [4, 5]. the predicted fusion-evaporation cross-section for the observed channels (that is, from 60 mev to 100 mev) have 42ca nucleus dominating the entire laboratory energy region as chosen in this work. the relative cross-section measured for the 42ca nucleus is ≤ 100 mb [4, 5]. the experiment for the 18o beam on 192os target was conducted for a period of 9 consecutive days at ifin-hh bucharest within which an approximate count of 1,200,000.00 was recorded using the romanian spectroscopy in heavy reactions, rosphere gamma-ray array of 14 high purity germanium, hpge and 11 lathanum-bromide doped cerium , labr3(ce) detectors [4, 5, 6]. table 1. the 192os target with the observed chemical impurities [5] element symbol impurity measurement (ppm) aluminium al 500 calcium ca 100 copper cu 70 iron fe 500 magnesium mg 50 manganese mn 50 nickel ni 100 lead pb 50 platinum pt 50 silicon si 500 tungsten w 100 3. data analysis coincidence analysis was performed on the detected gamma energies from the ropshere gamma-ray array detectors of the ifin-hh bucharest and these were identified to be associated with 42ca. doing this, a gating condition was used where a particular gamma transition was selected as a reference energy and all other energies that were seen to be in coincidence, that is, appearing with the selected gate at the same time were recorded. this was to further confirm that the measured gamma transitions were indeed associated with 42ca [1, 7, 8]. table 2 shows all the measured gamma transitions in the current work. the partial energy level scheme of 42ca was obtained as shown in figure 1 using radware. table 2. details of the observed transitions in the current work as detected from the 28si(18o,2n2p)42ca eγ(kev ) energy level (kev) relative intensity transition ei → e f iγ 146 6555 → 6409 23 9− → 8− 264 6409 → 6146 7 8− → 7− 437 3190 → 2752 50 6+ → 4+ 810 6555 → 5747 28 9− → 7− 815 7369 → 6555 20 10− → 9− 910 4099 → 3190 36 5− → 6+ 917 6409 → 5492 18 8− → 6− 1227 2752 → 1525 78 4+ → 2+ 1347 4099 → 2752 7 5− → 4+ 1525 1525 → 0 100 2+ → 0+ 1645 5745 → 4099 28 7− → 5− figure 1. symmetrical hpge γ – γ matrix showing gates on 1227 kevand 1525 kev (left panel). the right panel of the figure shows the same gates on 1227 kev and 1525 kev gamma energies but with a different folding coincidence condition of γ – γ γ (triple) trigger in hpge detectors using radware and gaspware softwares. 4. the measured energy levels associated with 42ca in the current work the 1525-kev level. the 1525 kev level is fed with a 1227 kev transition from the yrast iπ = 4+ state to the iπ = 2+. this energy level is assigned a spin of 2 with a positive parity. it decays to the ground state 0+ with the gamma energy of 1525 kev. the relative intensities of the gamma transitions have been determined in this current work as shown in table 2. the 2752-kev level. this energy level is populated by 437 kev gamma transition which is a decay from the 3190 kev level to a spin and positive parity of 4+. the relative intensity is calculated to be 50 (even though the directional correlations from 167 daniel & mcasule / j. nig. soc. phys. sci. 2 (2020) 166–169 168 oriented states dco’s calculation is not shown here). the multipolarity is dominantly an e2, but can possibly be an m3 transition using the expression in equation (1):∣∣∣ii − i f ∣∣∣≤ ∆l ≤ ∣∣∣ii + i f ∣∣∣ , (1) where the largest possible value of ∆l is ii + i f and the lowest possible value ii − i f . the parity change in the transition is given by the selection rules [9]. the 3190-kev level. as shown in figure 2 of the current work, more than two gamma transitions populated this energy level but with different experimental branching ratios, b(m1)/b(m2) (not determined here). apart from 910 kev transition energy populating this state, both 2956 kev and 2302 kev gamma transitions are not observed in this current work. this also forms the basis for which the gamma transitions are shown as broken lines and under brackets. the observed multipolarity is m1 which is the likely dominant one. this is so because there is a strong dependence of the transition rate on multipolarity where the lowest multipolarities are most likely to occur [9]. the relative intensity of the 910 kev gamma transition is calculated as 18 and this is shown in table 2. 5. discussion and conclusion the coincidence spectra on selected gamma transition energies in the left panel of figure 1 is a double fold: that is, γ – γ matrix sorting condition in the master trigger of rosphere. in this sorting, the gamma energy generally has more counts as recorded for same gates in triple fold gating condition. on the 1227 kev gate, gamma transitions of 145-, 264-, 382-, 437, 810-, 815-, 910-, 917-, 1347-, 1525-, and 1645-kev were observed to be in coincidence with one another with 437-kev having the highest counts of approximately 2000 counts/kev. looking at both figures 1.0 and 2.0, 1227 kev and 1525 kev are coincident with each other. this implies that, a gate on either one should produce the other one in coincidence. on 1525 kev gate, gamma transitions found to be in coincidence are the 145-, 437-, 507-, 810-, 910-, 1227-, and 1645-kev [7, 8, 5, 10]. the coincidence analysis performed in figure 1 further affirms the assigned nucleus of 42ca in the current work where the gamma transitions have already been reported by other researchers [1]. acknowledgments td thanks in a special way his phd supervisors, profs. patrick regan and zsolt podolyak for their mentorship during his programme at the university of surrey, guildford, uk. td thanks dr. stanimir kisyov of the ifin-hh bucharest and all other staff of the institute for their various contributions during the 192os(18o,16o)194os experiment late 2015. figure 2. partial energy level scheme of 42ca showing the measured gamma transitions up to higher spin and parity of iπ = 10-in the current work. references [1] evaluated nuclear data file (ensdf), http://www.nndc.bnl.gov/ensdf. retrieved (2020). 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[10] h. h. eggenhuisen, i. o. ekstrom, g. a. p. engelbertink & h. j. m. aarts, “high spin states of 39k and 42ca”, nuclear physics a 305 (1978) 245. 169 j. nig. soc. phys. sci. 4 (2022) 123–129 journal of the nigerian society of physical sciences influence of precursor temperature on bi doped znse material via electrochemical deposition technique for photovoltaic application imosobomeh l. ikhioyaa, eli danladib,∗, okoli d. nnanyerec, abdulazeez o. salawud adepartment of physics and astronomy, faculty of physical sciences, university of nigeria, nsukka, enugu state, nigeria bdepartment of physics, faculty of science, federal university of health sciences, otukpo, benue state, nigeria cdepartment of physics and industrial physics, faculty of physical sciences, nnamdi azikiwe university, awka, anambra state, nigeria ddepartment of computer science, nile university of nigeria abstract in this study, bismuth (bi) doped znse thin films were deposited on conducting glass substrates by electrochemical deposition technique and the influence of precursor temperature (room, 50, 55, 60 oc) on their optical and structural properties were systematically studied using the combined effect of x-ray diffraction (xrd), scanning electron microscope (sem) and uv-vis spectrophotometer. the xrd patterns show a face-centred cubic structure indexed with peaks at (220), (221) and (300). the grain size was in the range of 3.24056 to 4.60481 nm with a lattice constant of 7.189å. the material deposited at room, 500c, 550c, and 600c reveals agglomeration of particle on the surface of the substrate indicating uniform deposition. the optical spectra show that at different temperature (say room, 50oc, 55oc and 60oc), the absorbance and reflectance of biznse thin films decreases with increase in wavelength of the incident radiation while the transmittance shows direct proportionality with the increase in wavelength. the bandgap demonstrated an increase in the range 1.75-2.25 ev with increase in temperature. doi:10.46481/jnsps.2022.502 keywords: znse, doping, precursor temperature, photovoltanic, electrochemical deposition article history : received: 05 december 2021 received in revised form: 03 february 2022 accepted for publication: 15 february 2022 published: 28 february 2022 c©2022 journal of the nigerian society of physical sciences. all rights reserved. communicated by: e. etim 1. introduction among the various ii–vi semiconductors, zinc selenide displays unique optical properties that makes it found application in light-emitting diodes [1,2], magneto-optical devices [3], ultraviolet lasers [4,5], gas sensors [6,7], solar cells [8,9], photocatalysis to mention but a few. the band gap tunability of znse makes its optical properties outstanding and it can be used as ∗corresponding author tel. no: +2348063307256 email address: danladielibako@gmail.com (eli danladi ) transparent conductive oxide thin films. many methods, both chemical and physical, including sputtering [10], thermal evaporation [11], electro-deposition [12, 20, 21], spray pyrolysis [13] and chemical vapour deposition [14] have been used to produce doped and undoped semiconductor thin films. so far, some of the methods have drawbacks such as toxic reducing agents, and organic solvents, or may require special conditions such as high temperature or low pressure, and sometimes can require costly and time consuming procedure. among them, the electrochemical deposition of thin films is a viable alternative to vacuum-based deposition process. its 123 ikhioya et al. / j. nig. soc. phys. sci. 4 (2022) 123–129 124 figure 1: schematic diagram of electrochemical deposition technique major advantages are that processing can take place at room temperatures and pressures and thin film properties can be controlled [4,10]. generally speaking, adding an appropriate amount of dopant can change the physical properties of a semiconductor film. the effects of sulfur (s), iron (fe), chromium (cr) [15], copper (cu) [16], ag [17], mn [18], etc. have been systematically studied. the semi-metal element bismuth (bi) is used as a suitable doping material for electronic applications because of its high anisotropic electrical properties and high electron mobility [10,19], which is rarely reported in published works. bidoped znse thin film has a transparent nature, and its band gap nature results to potential support for heterojunction layers in solar cells. this work aims to use x-ray diffraction (xrd), scanning electron microscopy (sem), energy dispersive spectroscopy (eds), ultraviolet-visible absorption spectroscopy and four point probe to explore the structural, optical and electrical properties of the as-deposited thin film. 2. experimental details 2.1. material and methods the chemicals used in this work were analytical grade and purchased from sigma-aldrich. they include: zinc tetraoxosulphate (vi) heptahydrate (znso4.7h2o), bismuth nitrate pentahydrate (bi(no2)3.5h2o), selenium metal powder (se), and hydrogen chloride (hcl). in this work, electrochemical deposition technique (ecd) is used, which involves the deposition of any substance on electrodes due to electrolysis. the electrochemical bath system consists of cation source (ie bi(no2)3.5h2o, znso4.7h2o for bi2+, zn2+), anion source (i.e. selenium metal powder se2−) and deionized water, all in a 100 ml beaker. the reaction bath was stirred using a magnetic stirrer, and a powder supplied was used to supply an electric field (dc voltage), a conductive glass (fto) with electrical resistance of 16.6 ω for the cathode, and a carbon electrode for the anode. finally, the uniform deposition of the thin film was achieved through electrochemical deposition technique. 2.2. substrate cleaning procedure the substrates were conducting glass materials. acetone was used to cleaned the substrates. it was therefore rinsed with distilled water and ultrasonicated for 30 minutes in acetone solution. final rinsing was done in distilled water and the substrates were dried using an oven. 2.3. growth of znse and bi/znse films the growth of znse and bi/znse thin film materials was carried out using an aqueous solution of 0.1 mol of znso4.7h2o as the cationic precursor while the anionic precursor was 0.1 mol solution of selenium metal powder dissolved in 5 ml of hydrochloric acid (hcl) to ensure uniform deposition. the electrochemical bath system was composed of sequential variations of the zinc selenide molar concentrations. the varied precursor temperature of the solution are presented in table 1. the deposited films were afterward annealed at 300 oc. 2.4. characterization and measurement the grown films were characterized for their morphological, structural, elemental, optical and electrical properties using zeiss scanning electron microscope (sem), bruker d8 advance x-ray diffractometer with cu-kα line (λ =1.5405 å) in 2θ range from 10 ◦ 90 ◦, energy dispersive x-ray spectroscope (edx), uv-1800 visible spectrophotometer and a fourpoint probe device respectively. 3. results and discussion 3.1. xrd analysis the xrd pattern of biznse thin films deposited on fto substrates at different temperatures is as shown in figure 2. from the figure, the diffraction peaks were at (220), (221), (300) which correspond to the following angles (26.000), (31.98o), (32.01o) respectively. from the pattern, biznse was confirmed to be a face-centred cubic structure which agrees with the jcpds card no. 01-088-2400. the un-indexed peaks could have possibly resulted from the fto substrates used for deposition. the lattice constant a = 7.1890 å was obtained using equation (1). from the pattern the higher peaks could be due to the fact that the thickness of the film increases with an increase in dopant concentration, thus creating larger surface areas for photovoltaic devices and solar cell activities. the average crystallite size of the film is determined using equation (1), and the value of the scherrer constant k is estimated to be 0.94 of the crystallite size using equation (2). table 2 shows the calculated crystallite or grain size and dislocation density of the films deposited at different temperature room, 50oc, 55oc and 60oc which agrees to similar studies by other researchers [20-23]. d = nλ 2 sin θ (1) d = kλ b cos θ (2) 124 ikhioya et al. / j. nig. soc. phys. sci. 4 (2022) 123–129 125 table 1: varied molar concentration of growth materials temperature (degree) bi(no2)3.5h2o (ml) znso4.7h2o (ml) se(ml) temperature (degree) time (s) voltage (v) znse 10 20 20 00 5 10 bi/znse 50 10 20 20 50 5 10 bi/znse 55 10 20 20 55 5 10 bi/znse 60 10 20 20 60 5 10 table 2: structural parameters of biznse thin film sample 2θ (degree) d (spacing) å lattice constant (å) (β) fwhm (hkl) grain size(d) nm dislocation density,σ lines/m2 znse room 27.07 3.52076 7.1890 0.32717 220 4.54275 0.04845 38.98 2.87509 0.32717 221 4.60447 0.04716 51.01 2.87246 0.32717 300 4.60481 0.04716 bi/znse 500c 27.07 3.52076 7.1890 0.40959 220 3.62863 0.07594 38.98 2.87509 0.40959 221 3.67793 0.07392 51.01 2.87246 0.40959 300 3.67820 0.07391 bi/znse 550c 27.07 3.52076 7.1890 0.45864 220 3.24056 0.09522 38.98 2.87509 0.45864 221 3.28459 0.09269 51.01 2.87246 0.45864 300 3.28483 0.09267 bi/znse 600c 27.07 3.52076 7.1890 0.36444 220 4.07818 0.06012 38.98 2.87509 0.36444 221 4.13358 0.05852 51.01 2.87246 0.36444 300 4.13389 0.05851 figure 2: xrd pattern of biznse 3.2. surface morphology analysis using scanning electron microscopy (sem) of biznse figure 3 shows the micrograph of znse and biznse. the materials were deposited at room, 500c, 550c, and 600c. the morphologies of the samples at different temperatures are smooth with bigger grains with decrease in temperature. when the deposition temperature increased to 600c the surface of the as deposited film resulted to formation of islands with reduced grains due to smaller crystallites formed. the large grains formed with reduced precursor temperature was due to agglomeration of the crystallites. it is evidenced that the crystallites get bigger as the deposition temperature reduces which agrees with similar studies [20-23]. 3.3. elemental composition analysis using energy dispersive x-ray (edx) for biznse utilizing energy dispersive x-ray technique, the formation of znse and biznse are evident in figure 4a&b and the other elements which include si, ca, and o resulted from the elemental composition of the (fto) substrate used for the deposition of the films. 3.4. optical analysis the optical absorbance-wavelength spectra of biznse thin films is shown in figure 5a. the films grown at different temperatures (say, room, 50oc, 55oc and 60oc) reveals that, the absorbance has an inverse relationship with its corresponding wavelength (in another word, as the wavelength of the incident radiation increases the absorbance of biznse thin films decreases). the material grown at 50oc, 55oc and 60oc exhibited the same trend and reveals the highest behavior and the material grown at room reveals lowest which shows that znse and biznse will be a good material that will absorb energy from the sun and can serve as a good device for photovoltaic applications [24-26]. the optical transmittance spectra of biznse thin films in figure 5b shows that the transmittance of the films grown at 125 ikhioya et al. / j. nig. soc. phys. sci. 4 (2022) 123–129 126 figure 3: sem micrograph of znse and biznse figure 4: edx spectra of znse and biznse different temperatures (room, 50oc, 55oc and 60oc) increases with increased in wavelength, which means that, the transmittance is directly related with the wavelength of biznse thin films which agrees with similar studies [24-26]. similarly, the optical reflectance spectra of biznse thin films in figure 5c shows an inverse relationship with wavelength which means that, at different temperatures (room, 50oc, 55oc and 60oc) as the wavelength of the incident radiation increase the reflectance of biznse thin films decreases. the energy gap (eg) was determined based on the wavelength of maximum absorption (λ) according to the formula eg = 1240/λ. here, the maximum absorption wavelength is obtained from the absorption wavelength data. the eg was obtained in the formula describes as tauc plot [27]. the eg was obtained from extrapolation of the linear part (αhv)2 versus hv. the optical band gap energy of biznse thin films in figure 6 shows an increase in band gap energy with increase in deposition temperature. the bandgap demonstrated an increase in the range 1.75-2.25 ev with increase in temperature (room, 50oc, 55oc and 60oc). figure 7a shows the optical refractive index of biznse thin films grown at different temperature (room, 50oc, 55oc and 126 ikhioya et al. / j. nig. soc. phys. sci. 4 (2022) 123–129 127 figure 5: absorbance (a), transmittance (b) and reflectance (c) versus wavelength figure 6: plot of absorption coefficient square versus photon energy 60oc). the result shows that, as the photon energy of the material increases the refractive index also increases. the increase in the refractive index as a result of photon energy increase was due to larger crystallites as observed in the sem micrograph while the refractive index increases with increasing photon energy as a result of the increase in grain size observed in the sem analysis. the refractive index (n) is the range of frequencies in which films are weakly absorbing. the optical extinction coefficient which is the measure of the fraction of light lost due to scattering and absorption per unit distance of the penetration medium is as shown in figure 7b. the increase in the extinction coefficient indicates the scattering loss of light while travelling through the medium with high absorption. the optical conductivity of biznse thin films is shown in figure 7c. the optical conductivity has been found to increase with increase photon energy. this increase in optical conductivity attributed to increase in density of localized states in the energy band itself is due to the rise in new defect states [28]. the real and imaginary dielectric constant of biznse thin films in figure 8a&b has a direct relationship with the photon energy. this shows that, as the photon energy of the material increases, the real and imaginary dielectric constants increases. 3.5. electrical properties of biznse the material deposited at room, 50◦c, 55◦c and 60◦c reveals an increase in thickness from 117.19 – 132.21 nm with increase in the resistivity of the deposited material from 5.3210× 103 −−6.2943 × 103(ω.cm) which result to the decrease in the conductivity of the deposited material from 1.8793 × 1011 − 1.5887 × 1011(s/m) (see figure 9). the high resistance value allows the film to be used in solar cell applications to improve conversion efficiency, because it can reduce the inevitable defects in solar cell manufacturing during the actual production process. therefore, biznse thin 127 ikhioya et al. / j. nig. soc. phys. sci. 4 (2022) 123–129 128 figure 7: refractive index (a), extinction coefficient (b) and optical conductivity (c) versus photon energy figure 8: plot of real dielectric (a) imaginary dielectric constant (b) versus photon energy table 3: electrical properties of biznse samples thickness, t(nm) resistivity, ρ (ω.cm) conductivity, σ (s/m) znse room 117.19 5.3210x103 1.8793x1011 bi/znse 50◦c 122.15 6.7231x103 1.4874x1011 bi/znse 55◦c 128.15 6.9874x103 1.4311x1011 bi/znse 60◦c 132.21 6.2943x103 1.5887x1011 film resistivity is very suitable for use as a buffer layer in solar cells, photovoltaic panels and photovoltaic devices. 4. conclusion undoped znse and bismuth (bi) doped znse thin films have been prepared and deposited using electrochemical de128 ikhioya et al. / j. nig. soc. phys. sci. 4 (2022) 123–129 129 figure 9: resistivity and conductivity versus thickness position technique at room, 50, 55, 60 oc. the xrd patterns show a face-centred cubic structure indexed with peaks at (220), (221) and (300). the grain size was in the range of 3.24056 to 4.60481 nm with a lattice constant of 7.189å. the material deposited at room, 500c, 550c, and 600c reveals agglomeration of particle on the surface of the substrate indicating uniform deposition. the optical spectra show that at different temperature (say room, 50oc, 55oc and 60oc), the absorbance and reflectance of biznse thin films decreases with increase in wavelength of the incident radiation while the transmittance shows direct proportionality with the increase in wavelength. the bandgap demonstrated an increase in the range 1.75-2.25 ev with increase in temperature. acknowledgements we acknowledge nanosciences african network (nanoafnet), ithemba labs national research foundation as well as the staff of nano research group, university of nigeria, nsukka for their research support references [1] s. j. pearton & f. ren, “advances in zno-based materials for light emitting diodes”, current opinion in chemical 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[28] n. f. mott & e. a. davis, electronic processes in non-crystalline materials, 2nd edition clarendon, oxford (1979). 129 j. nig. soc. phys. sci. 4 (2022) 874 journal of the nigerian society of physical sciences application of hourglass matrix in goldreich-goldwasser-halevi encryption scheme olayiwola babarinsaa,∗, olalekan ihinkalub, veronica cyril-okemea, hailiza kamarulhailic, arif mandangand, azfi zaidi mohammad sofie, akeem b. disuf adepartment of mathematics, federal university lokoja, kogi state, nigeria bdepartment of computer sciences, federal university lokoja, kogi state, nigeria cschool of mathematical sciences, universiti sains malaysia, 11800 pulau pinang, malaysia dfaculty of science and natural resources, universiti malaysia sabah, 88400 kota kinabalu, malaysia efaculty of bioengineering & technology, universiti malaysia kelantan, 16100 kota bharu, malaysia fdepartment of mathematics, national open university of nigeria, abuja, nigeria abstract goldreich-goldwasser-halevi (ggh) encryption scheme is lattice-based cryptography with its security based on the shortest vector problem (svp) and closest vector problem (cvp) with immunity to almost all attacks, including shor’s quantum algorithm and nguyen’s attack of higher lattice dimension. to improve the efficiency and security of the ggh scheme by reducing the size of the public basis to be transmitted, we use an hourglass matrix obtained from quadrant interlocking factorization as a public key. the technique of quadrant interlocking factorization to yield a nonsingular hourglass matrix compensates the encryption scheme with better efficiency and security. doi:10.46481/jnsps.2022.874 keywords: goldreich-goldwasser-halevi encryption scheme, hourglass matrix, quadrant interlocking factorization. article history : received: 18 june 2022 received in revised form: 25 july 2022 accepted for publication: 27 july 2022 published: 08 october 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: s. fadugba 1. introduction cryptography aims to achieve information security in confidentiality, data integrity, authentication, and non-repudiation [1]. integer factorization problem (ifp), elliptic curve discrete logarithm problem (ecdlp), and discrete logarithm problem (dlp) are number theoretic hard problems established in cryptographic schemes based on the hardness of their security, which ∗corresponding author tel. no: +2348060032554 email address: olayiwola.babarinsa@fulokoja.edu.ng (olayiwola babarinsa ) is mostly deployed in rivest-shamir-adleman (rsa), el-gamal and elliptic curve cryptosystems [2-4]. notwithstanding, the security goals of the schemes can be attacked by a powerful algorithm, for instance, shor’s quantum algorithm, to compute the problems in less amount of time, see [5-7]. the immunity of some lattice problems such as the shortest vector problem (svp) and closest vector problem (cvp) against shor’s quantum algorithm is exploited by the idea behind lattice-based cryptography [8,9]. the security of ggh cryptosystem relies on the smallest-basis problem (sbp) and cvp [10]. the earliest lattice-based encryption scheme which was the 1 babarinsa et al. / j. nig. soc. phys. sci. 4 (2022) 874 2 most considered practical scheme is goldreich-goldwasser-halevi encryption scheme or ggh scheme [11]. the security of this scheme lies in the hardness of the underlying ggh-cvp instance which is proven to be np-hard [12]. however, the security of ggh scheme has been compromised by nguyen’s attack [13]. there are significant efforts to improve the efficiency of the ggh scheme, such as the application of hermite normal form (hnf) or jensen-based cryptographic scheme as the public key [14, 15]. the ggh scheme survives against nguyen’s attack when being implemented in lattice dimensions above 400 [16]. however, the implementation of the scheme in lattice dimensions beyond 400 immediately makes the ggh scheme inefficient, impractical, and uncompetitive compared to other existing encryption schemes. this is due to the large key sizes involve in the scheme once it is implemented in a large lattice dimensions. this is because the keys of this scheme are bases of lattice which could be represented in matrices form. that means the implementation of the scheme in a large lattice dimension requires the submission of a large lattice basis as a public key from alice to bob. for instance, in a lattice dimension of 400, the public basis can be represented as a 400 × 400 matrix with 4002 = 160000 entries. the transmission of a matrix with 160000 entries from alice to bob requires a high computational cost. by transforming the underlying ggh-cvp instance into its simpler form, nguyen’s attack successfully breaks the security of the ggh scheme when being implemented in lattice dimensions smaller than 400. from the simplified ggh-cvp instance, nguyen’s attack derives an easier svp instance which could be solved by using lattice reduction methods such as lll and bkz algorithms. in low lattice dimension, these algorithms efficiently work for solving the derived svp-instance which makes nguyen’s attack succeeds. as the lattice dimension increases, the efficiency of these algorithms declines significantly. consequently, the attack failed to break the security of the ggh scheme in a lattice dimension of 400 and above [17]. to systematically reduce the number of non-zero elements in the public basis while maintaining all the required properties of the public basis, especially the linear independency and orthogonality properties, we use a nonsingular hourglass matrix h as a public basis and its corresponding factorization matrix r as a private basis of the ggh scheme. these matrices are related as r = u h where u is unimodular matrix. section 2 gives the detail of an hourglass matrix and its factorization algorithm, while section 3 entails the application of hourglass matrix and its factorization technique in ggh encryption scheme. 2. hourglass matrix babarinsa and kamarulhaili [18] gave details on hourglass matrix and its factorization algorithm by restricting the computed entries of the factorization to be nonzero in comparison with an hourglass device. they also suggested its applications in mathematics, graph theory, statistics, and computer science, see [19-24]. an hourglass matrix is defined as a nonsingular matrix of order n (n ≥ 3) with nonzero entries from the ith to the (n− i + 1) element of the ith and (n− i + 1) row of the matrix, 0’s otherwise for i = 1,2,...,bn+12 c [18]. unlike z-matrix with nonzero restricted entries, hourglass matrix conforms with the shape of an hourglass device, see figure 1 which illustrates the structural comparison between the hourglass device and hourglass matrix with nonzero elements denoted with black dots. to buttress the shape of hourglass matrix, figure 1 figure 1. structural comparison between hourglass device and hourglass matrix. for the factorization algorithm of hourglass matrix, we compute w(k)i,k and w (k) i,n−k+1 from a dense square matrix r by solving 2×2 linear systems in equation (1) using cramer’s rule to generalize for every update of r to h and proceed similarly for the inner square matrices of size (n −2k) and so on, for k = 1,2,...,bn−12 c.{ h(k−1)k,k w (k) i,k −h (k−1) n−k+1,kw (k) i,n−k+1 = h (k−1) i,k h(k−1)k,n−k+1w (k) i,k −h (k−1) n−k+1,n−k+1w (k) i,n−k+1 = h (k−1) i,n−k+1 (1) then we compute for kth steps of h(k)i, j as: h(k)i, j = h (k−1) i, j + w (k) i,k h (k−1) k, j + w (k) i,n−k+1h (k−1) n−k+1, j (2) where i, j = k + 1,...,n−k. from equation (2), if one of the computed entries is zero, then apply possible row-interchange in no more than (n−2k) times in h(k−1) and re-factorize, else the factorization breakdown to produce h. from every successful loop for each stage (with bn−12 c total stages in the factorization), there are bn2−1c ∑ k=1 (n−2k) of 2×2 linear systems to be solved during the factorization using cramer’s rule. hourglass matrix (h-matrix) is nonsingular and its w -matrix is a unimodular matrix with det(w ) = (−1)pn =±1, where pn is the number of permutation matrix in the factorization. based on the structure of hourglass matrix, the matrix could be potentially used as the key (basis) in the ggh encryption scheme. the usage of hourglass matrix is expected to be able to reduce the size of bases, especially the public key. almost half of the entries of the hourglass matrix are zero entries, which means the size of the public key can be reduced if the public key is generated in the form of hourglass matrix. this reduction will allow the ggh scheme to be implemented in a higher lattice dimensions while still being able to be efficient and practical. 2 babarinsa et al. / j. nig. soc. phys. sci. 4 (2022) 874 3 hourglass matrix has linearly independent columns forming the basis of a lattice, which makes it suitable for ggh scheme. the fixed zero entries in hourglass matrix will not only minimize the memory cache used but also reduce computational time. in addition, the generation of hourglass matrix from qif can be executed in polynomial time. 3. ggh scheme with hourglass matrix consider (n,σ) ∈ n to be the security parameter, where n is a lattice dimension and σ is a threshold parameter. denote ~m,~e ∈ z n as the message vector and error vector respectively, where the entries of ~e are ei ∈{−σ,+σ}. let i = 1,2,...,n, then denote h as hourglass matrix such that h = [~h1,~h2,...,~hn], where ~hi ∈ z n are the column vectors of h, denote r as a nonsingular matrix such that r = [~r1,~r2,...,~rn] where~ri ∈zn are the column vectors of r, and denote u as a unimodular matrix such that u = [~u1,~u2,...,~un], where ~ui ∈ zn are the column vectors of u and det(u) =±1. proposition 3.1. [22] any two bases for a lattice l are related by a unimodular matrix u that has integer coefficients and det(u) =±1. definition 3.1. [23] let ~b1,~b2,...,~bn be n linearly independent vectors of rm with n ≤ m. the set of all integer linear combinations of the vectors ~b1,~b2,...,~bn is called lattice and can be denoted in the form l (~b1,,...,~bn) = { n ∑ i=1 mi~bi|mi ∈z } (3) the linearly independent vectors ~b1,~b2,...,~bn form the columns of the basis for the lattice l (b). suppose that b,h ∈ rn×n be nonsingular square matrices with linearly independent vectors ~b1,~b2,...,~bn and ~h1,~h2,...,~hn as their columns respectively. the lattice l(b) ⊂ rn that is spanned by the basis b is defined as follows [22]. l(b) = { n ∑ i, j=1 µi, j~bi | ~bi ∈ b and µi, j ∈z, ∀i, j = 1,2,...,n } (4) and the lattice l(h) ⊂ rn that spanned by the basis and the lattice l(b) ⊂ rn that spanned by the basis b is defined as follows l(h) = { n ∑ i, j=1 τi, j~hi | ~hi ∈ h and τi, j ∈z, ∀i, j = 1,2,...,n } (5) to ensure that the bases b and h are spanning the same lattice (i.e l(b) = l(h)), the matrix w is required to be a unimodular matrix with det(u) =±1. the desired properties for the public and private basis are as the following: 1. both h and r a) two different bases span the same lattice l, i.e., l (r) = l = l (h). b) to be a lattice basis, both matrices h and r must satisfy the following conditions: i. the vectors {~h1,~h2,...,~hn} and {~r1,~r2,...,~rn} are linearly independent, where the only solution for the following equations α1~h1 + α2~h2 +...+ αn~hn =~0 (6) and β1~r1 + β2~r2 +...+ βn~rn =~0 (7) are the trivial solutions, i.e., αi = 0 and βi = 0 ∀,i = 1,2,...,n. ii. the vectors {~h1,~h2,...,~hn} and {~r1,~r2,...,~rn} span the whole space z, i.e., span{~h1,~h2,...,~hn}= z (8) and span{~r1,~r2,...,~rn}= z (9) c) since both h and r are spanning the same lattice l, these bases have the same determinant, i.e., det(h) = det(r). d) both h and r are mathematically related by unimodular matrix u , as r = u h. 2. the public basis h the column vectors {~h1,~h2,...,~hn} are long vectors from the origin and highly nonorthogonal vectors where the hadamard ratio of the matrix h is far from 1 and closer to 0. 3. the private basis r a) the column vectors {~r1,~r2,...,~rn} are short vectors from the origin and reasonably orthogonal vectors where the hadamard ratio of the matrix r is far from 0 and closer to 1. b) the rounding vector br~ee=~0 to ensure the decryption process succeeds. proposition 3.2. [19] let b,h ∈ rn×n be nonsingular square matrices such that b = hw , where b is a basis for lattice l(b), h is a basis for lattice l(h). if w is a unimodular matrix, then l(b) = l(h) where w ∈zn×n. 3.1. ggh scheme algorithm using hourglass matrix the algorithm of the ggh scheme by using hourglass matrix is as follows: 1. key generation by alice (recipient) • sets the security parameters n,σ ∈ n where n is an even number. • generates a non-singular n×n -matrix r with the following properties: a) represent the matrix r as r = [~r1 ~r2 ... ~rn] (10) where the vectors~ri ∈z are the columns of r. these vectors are required to be linearly independent. 3 babarinsa et al. / j. nig. soc. phys. sci. 4 (2022) 874 4 b) compute the hadamard ratio of the matrix r as h (r) =  |det(r)|n ∏ i=1 ‖~ri‖   1 n (11) where ‖~ri‖ is the euclidean norm of the vectors ~ri. the hadamard ratio is required to measure the orthogonality of the vectors~ri. the closure the hadamard ratio to 1, the more orthogonal the vectors ~ri are. we accept r to be the private basis if h (r)∈ [0.7,1) (12) to make sure that the private basis r is a reasonably orthogonal basis. • computes the factorization of the matrix r as follows r = u h where u ∈zn×n is a unimodular matrix with det(u) = ±1 and h ∈ zn×n is an hourglass matrix. for the hourglass matrix h, a) represent the matrix h as h = [~h1 ~h2 ... ~hn] (13) where the vectors ~hi ∈ zn are the columns of h. these vectors are required to be linearly independent. b) since h and r are related by a unimodular matrix u as r = u h, then det(u h) = det(r) =±det(h) (14) c) compute the hadamard ratio of the matrix h as h (h) =  |det(h)|n ∏ i=1 ‖~hi‖   1 n (15) where ‖~hi‖ is the euclidean norm of the vectors ~hi. the closure of the hadamard ratio to 0, the more highly non-orthogonal the vectors ~hi are. we accept h to be the public basis if h (h)∈ (0,0.3] (16) to make sure that the public basis h is a highly nonorthogonal basis. • keeps the matrix r as her private basis and u as her private matrix. • sends the hourglass matrix h as her public basis to bob together with her security parameters n,σ ∈ n. 2. encryption by bob (sender) • sets the message as ~m =   m1 m2 ... mn  ∈zn • generates the error vector as ~e =   e1 e2 ... en   where ei ∈{±σ} • encrypts the message ~m as follows ~c = h~m +~e (17) where ~c ∈zn is the ciphertext vector. • sends the ciphertext ~c to alice. 3. decryption by alice (recipient) • computes a vector t ∈rn as follows t = r−1~c (18) • rounds each entry of the matrix t to the nearest integer, i.e., btie∈z where ti ∈ t for all i = 1,2,...,n. • decrypts the ciphertext as follows ~m = ubte (19) where bte is a matrix t with rounded entries. decryption succeeds if ~m = ~m (20) now, consider the following scenario. suppose that bob wants to send a secret message to alice. as the recipient, alice generates the private basis r as a good basis. then, she derived the public basis h as r = u h. the public basis h is accepted if the basis h is a highly non-orthogonal basis. otherwise, the key generation process will be re-initiated. once the proper bases r and h are completely generated, alice sends the public basis h, the security parameter {n,α} to bob and keeps the other basis and parameters secret. upon receiving the information from alice, bob encoded his secret message in a vector ~m ∈ zn. then, he generates the error vector ~e ∈{−α,α}n and then proceeds to the encryption process which is done as~c = h~m+~e. the ciphertext ~c ∈ zn is then sent to alice. upon receiving ~c from bob, alice proceed with the decryption process which can be done by solving the underlying cvp instance using babai’s round-off method. she computes ~m = ubr−1~ce. proposition 3.3. suitability of hourglass matrix in ggh scheme is attainable if the encrypted message m is successfully decrypted to message m, such that ~m = ~m. proof. since r = u h, then u = h−1r. thus, ~m = ubte = ubr−1~ce 4 babarinsa et al. / j. nig. soc. phys. sci. 4 (2022) 874 5 = ubr−1(h~m +~e)e = ubr−1h~m + r−1~ee =bu r−1h~m +u r−1~ee =bh−1rr−1h~m +u r−1~ee =b~m +u r−1~ee =b~me+bu r−1~ee. since ~m ∈ zn, bme = m. for the decryption to succeed, the rounding vector is required to be br−1~ee=~0. therefore, ~m = ~m +ubr−1~ee if all the entries of the rounding vector br−1~ee are smaller than 1 2 , then we have br −1~ee=~0. therefore, we have ~m = ~m +u~0 ~m = ~m 3.2. a numerical example of hourglass matrix in ggh scheme we consider a numerical example of 4×4 hourglass matrix in ggh scheme. let r, u, h be the hourglass matrix, unimodular matrix, nonsingular matrix, error, and encrypted message to be sent from alice to bob, and e, m are vectors. h =   143 123 −211 103 0 −14 33 0 0 −14 −124 0 −211 −14 −122 213  , ~m =   5 6 7 8  , ~e =   −3 3 −3 −3  . then det(h) = 114,718,016 and the hadamard ratio of h is 0.4495. then u is given as u =   −1 2 1 −2 2 −1 1 2 1 −1 1 2 1 1 1 −1   where det(u) = 1. thus, r is computed as r = h ·u =   −5 477 158 −565 5 −19 19 38 −152 138 −138 −276 274 −73 −134 −63   vividly, det(r) = 114,718,016. thus, the hadamard ration of r is 0.2606. now compute h−1 as h−1 =   0.00408 0.033828 0.004000 −0.001973 0 −0.056415 −0.015014 0 0 0.006370 −0.006370 0 0.00404 0.033451 −0.000673 0.002740   and h−1 ·e =   −0.107163 0.214286 0 −0.077988   the round-off bh−1 ·ee gives a zero vector, ~0 =   0 0 0 0  . the message m sent by alice to bob be m =   5 6 7 8   r ·m =   −577 348 −3106 −510   c = r·m + e =   −574 345 −3109 −507   t = h−1c =   −2.107163 27.214286 22.000000 9.922012   the round-off bte to nearest integers gives bte=   −2 27 22 10   hbte=   −577 348 −3106 −510   the decryption m is m = r−1ht =   4.9999999994 5.9999999780 6.9999999926 8.0000000102   the round-off decrypted message bme by bob is bme=   5 6 7 8   hence, ~m−~m =~0. 5 babarinsa et al. / j. nig. soc. phys. sci. 4 (2022) 874 6 4. conclusion the advantage of the hourglass matrix as a public key has been explored in the encryption scheme to enhance the efficiency of the ggh. the orthogonal columns of hourglass matrix keep the ggh encryption scheme efficient and practical by reducing the number of non-zero elements on the public basis while maintaining all the required properties of the public basis, especially the linear independency and orthogonality properties. with better security and efficiency, the scheme is expected to be highly competitive in the post-quantum era. due to the simplicity and practicality that can be offered by the scheme, it may be widely adopted for providing security in devices with small computing capacity. references [1] b. schneier, applied cryptography: protocols, algorithms, and source code in c, john wiley and sons, 2007. 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[24] o. babarinsa, ”graph theory: a lost component for development in nigeria”, journal of the nigerian society of physical sciences 4 (2022) 844. doi:https://doi.org/10.46481/jnsps.2022.844. 6 j. nig. soc. phys. sci. 2 (2020) 134–140 journal of the nigerian society of physical sciences original research linear stability analysis of runge-kutta methods for singular lane-emden equations m. o. ogunnirana,∗, o. a. tayoa, y. harunab, a. f. adebisia adepartment of mathematical sciences, osun state university, osogbo bdepartment of mathematics, saadatu rimi college of education kumbotso, kano state abstract runge-kutta methods are efficient methods of computations in differential equations, the classical runge-kutta method of order 4 happens to be the most popular of these methods, and most times it is attached to the mind when runge-kutta methods are mentioned. however, there are numerous forms of them existing in lower and higher orders of the classical method. this work investigates the linear stabilities and abilities of some selected explicit members of these runge-kutta methods in integrating the singular lane-emden differential equations. the results obtained established the ability of the classical runge-kutta method and why is mostly used in computations. keywords: classical, stability, explicit, lane-emden equations, ordinary differential equations article history : received: 13 april 2020 received in revised form: 30 may 2020 accepted for publication: 01 june 2020 published: 01 august 2020 c©2020 journal of the nigerian society of physical sciences. all rights reserved. communicated by: t. latunde 1. introduction the most popular of the methods of runge-kutta is the classical runge-kutta method of order 4, ’classical’ as related to the method obtained in the pre-computer era. however, many other forms of runge-kutta methods have been derived by numerous authors in the past. existing methods of runge-kutta are of orders 2, 3, and orders greater than 4. methods of orders greater than 4 are regarded as higher-order methods of rungekutta. the numerical solutions of singular initial value problems (ivp) and boundary value problems (bvp) of secondorder ordinary differential equations (odes) have been studied in this work. existing methods for the singular problems are ∗corresponding author tel. no: +2347039104606 email address: muideen.ogunniran@uniosun.edu.ng (m. o. ogunniran ) series, analytical methods and of recent non-series numerical methods as obtained by various authors. the general form of second-order singular differential equations is denoted as: y′′(x) + p(x) q(x) y′(x) + f (x, y) = g(x, y); a ≤ x ≤ b. (1) to solve equation (1), any of the conditions stated below need to be imposed. y(a) = α, y′(a) = β (2) y(a) = α1, y ′(b) = β1 (3) equation (1) together with (2) are called initial value problems (ivps) while equations (1) and (3) are called boundary value problems (bvps). equation (1) is singular at q(x) = 0, and f (x, y), g(x, y) are nonlinear continuous functions. it is well known that some of these 134 ogunniran et al. / j. nig. soc. phys. sci. 2 (2020) 134–140 135 problems have proved to be either difficult to solve or cannot be solved analytically due to the singularity as the approximate solutions lost their accuracy in the neighbourhood of the singular points. this discontinuous property has raised more interest in many applied mathematicians and physicists in the studies of this problem. huta [1] developed the 8-stage runge-kutta method of order 6 in solving differential equation problems, qu and agwarwal [2] employed collocation method for solving a class of singular non-linear two-point bvp. koch, peter & ewa [3] evaluated the approximate solution of the singular ivp by implicit euler method and finally used an acceleration technique known as the iterated defect correction to improve the approximations. wazwaz [4, 5] presented series and exact solution of lane-emden and emden-fowler type of problems based on adomian decomposition and modified adomian decomposition methods. a simplified derivation and analysis of fourth order runge kutta method was presented by musa, ibrahim, & waziri [6]. roul [7] presented a new efficient recursive technique for solving singular boundary value problems arising in various physical models. abbas al-shimmary [8] used the runge-kutta method of order 6 to solve the initial value problem of ordinary differential equations. roul [9] also presented a fourth-order b-spline collocation method and its error analysis for bratu-type and lane-emden problems. a fastconverging iterative scheme for solving a system of lane–emden equations arising in catalytic diffusion reactions was presented by madduri and roul [10]. roul [11] presented a fast converging iterative approach for the solution of doubly singular bvp with derivative dependent source foundation. ogunniran, haruna and adeniyi [12] developed an efficient k−derivative method for lane-emden equations and related stiff problems, the paper focuses on the exploration of the possibility of a class of multi-derivative methods for approximating its solution. a new basic rational approximation method was developed for solving singular initial value problems of ordinary differential equations by ogunniran & adeniyi [13]. ogunniran [14] also developed a class of block multi-derivative numerical integrator for singular advection equations. the solution of emdenfowler equations was solved using the variational iteration method by olayiwola [15]. olayiwola and adegoke [16] presented the approach of homotopy perturbation with laplace transform method in solving singular initial value problems. an optimal sixth-order quartic b-spline collocation method was developed by roul, thula and goura [17] for solving bratu-type and lane-emden–type problems. roul, madduri and agarwal [18] developed a fast-converging recursive approach for lane–emden type initial value problems arising in astrophysics. singh, garg, kanwar & ramos [19] developed an efficient optimized adaptive step-size hybrid block method for integrating differential system. this study aims at the exploration of the possibility of the classes of explicit runge-kutta methods in the solution of singular lane-emden problems of ordinary differential equations. 2. method the general m-stage runge-kutta method is defined by yr+1 − yr = hθ(xr, yr, h), (4) θ(x, y, h) = m∑ i=1 qiµi (5) µ1 = f (x, y) (6) µr = f (x + hpi, y + h i−1∑ s=1 bisµs); i = 2, 3, · · · , m (7) pi = i−1∑ s=1 bis, i = 2, 3, · · · , m; h = xi − xi−1 (8) it can be observed that an m-stage runge-kutta method involves m-function evaluations per step. each of the functions µi(x, y, h), i = 1, 2, 3, · · · , m may be interpreted as an approximation of the derivative yi(x) and the function θ(x, y, h) as a weighted mean of these approximations. there is a great deal of tedious manipulation involved in deriving runge-kutta methods of the higher order. however, the forms of runge-kutta methods in the scope of this work shall be given in the following section. (i) 2-stage runge-kutta method of order 2 (rk2) yi+1 − yi = hµ2 µ1 = f (xi, yi) µ2 = f (xi + 1 2 h, yi + 1 2 hµ1) (9) (ii) 3-stage runge-kutta method of order 3 (rk3) yi+1 − yi = 1 6 h(µ1 + 4µ2 + µ3) µ1 = f (xi, yi) µ2 = f (xi + 1 2 h, yi + 1 2 hµ1) µ3 = f (xi + h, yi − h(µ1 − 2µ2)) (10) (iii) 4-stage runge-kutta method of order 4 (rk4) yi+1 − yi = 1 6 h(µ1 + 2µ2 + 2µ3 + µ4) µ1 = f (xi, yi) µ2 = f (xi + 1 2 h, yi + 1 2 hµ1) µ3 = f (xi + 1 2 h, yi + 1 2 hµ2) µ4 = f (xi + h, yi + hµ3) (11) 135 ogunniran et al. / j. nig. soc. phys. sci. 2 (2020) 134–140 136 (iv) 6-stage kutta-nyström method of order 5 (rk nyström) yi+1 − yi = 1 192 h(23µ1 + 125µ3 − 81µ5 + 125µ6) µ1 = f (xi, yi) µ2 = f (xi + 1 3 h, yi + 1 2 hµ1) µ3 = f (xi + 2 5 h, yi + 1 25 h(4µ1 + 6µ2)) µ4 = f (xi + h, yi + 1 4 h(µ1 − 12µ2 + 15µ3)) µ5 = f (xi + 2 3 h, yi + 181 h(6µ1 + 90µ2 − 50µ3 + 8µ4)) µ6 = f (xi + 4 5 h, yi + 175 h(6µ1 + 36µ2 + 10µ3 + 8µ4)) (12) (v) 8-stage huta method of order 6 (rk 8) yi+1 − yi = 1 840 h(4µ1 + 216µ3 + 27µ4 + 272µ5 + 27µ6 + 216µ7 + 41µ8 ) µ1 = f (xi, yi ) µ2 = f (xi + 1 9 h, yi + 1 9 hµ1 ) µ3 = f (xi + 1 6 h, yi + 1 24 h(µ1 + 3µ2 )) µ4 = f (xi + 1 3 h, yi + 1 6 h(µ1 − 3µ2 + 4µ3 )) µ5 = f (xi + 1 2 h, yi + 1 8 h(−5µ1 + 27µ2 − 24µ3 + 6µ4 )) µ6 = f (xi + 2 3 h, yi + 1 9 h(221µ1 − 9813µ2 + 867µ3 − 102µ4 + µ5 )) µ7 = f (xi + 5 6 h, yi + 1 48 h(−183µ1 + 678µ2 − 472µ3 − 66µ4 + 80µ5 + 3µ6 )) µ8 = f (xi + h, yi + 1 82 h(716µ1 − 2079µ2 + 1002µ3 + 834µ4 − 454µ5 − 9µ6 + 72µ7 )) (13) for purpose of completion and without loss of generality, special runge-kutta methods which were developed to exhibit certain properties were also considered. (vi) 5-stage merson’s method of order 4 (rk merson) yi+1 − yi = 1 6 h(µ1 + 4µ4 + µ5) µ1 = f (xi, yi) µ2 = f (xi + 1 3 h, yi + 1 3 hµ1) µ3 = f (xi + 1 3 h, yi + 1 6 h(µ1 + µ2)) µ4 = f (xi + 1 2 h, yi + 1 8 h(µ1 + 3µ3)) µ5 = f (xi + h, yi + 1 2 h(µ1 − 3µ3 + 4µ4)) (14) (vii) 5-stage scraton’s method of order 4 (rk scraton) yi+1 − yi = h ( 17 162 µ1 + 81 170 µ3 + 32 135 µ4 + 250 1377 µ5 ) µ1 = f (xi, yi) µ2 = f (xi + 2 9 h, yi + 2 9 hµ1) µ3 = f (xi + 1 3 h, yi + 1 12 h(µ1 + 3µ2)) µ4 = f (xi + 3 4 h, yi+ 3 128 h(23µ1 − 81µ2 + 90µ3)) µ5 = f (xi + 9 10 h, yi+ 9 10000 h(−345µ1 + 2025µ2 − 1224µ3 + 544µ4)) (15) (viii) 6-stage england’s method of order 4 (rk england) yi+1 − yi = 1 6 h(µ1 + 4µ3 + µ4) µ1 = f (xi, yi) µ2 = f (xi + 1 2 h, yi + 1 2 hµ1) µ3 = f (xi + 1 2 h, yi + 1 4 h(µ1 + µ2)) µ4 = f (xi + h, yi − h(µ2 − 2µ3)) µ5 = f (xi + 2 3 h, yi+ 1 27 h(7µ1 + 10µ2 + µ4)) µ6 = f (xi + 1 5 h, yi+ 1 625 h(28µ1 − 125µ2 + 546µ3 + 54µ4 − 378µ5)) (16) figure 1. stability region for runge-kutta method of order 2 2.1. linear stability this is a behaviourial property related to h > 0. as in most literature, the linear stability will be analyzed using the dalquist’s test y′(t) = γy(t), <(γ) < 0. (17) applying methods (9)-(16) on (17), we have obtained a recurrence equation yi = m(z)yi−1 (18) where m(z = hγ) for each of the method is given in table 1 below the contour for regions of absolute stability as obtained from their characteristics equations are given in figure 1 8. 3. numerical experiment this section contains the numerical example considered and their results presented in tables of absolute errors. example 1: variable coefficient non-homogeneous singular ivp [3] y′′(x) = −2x y ′(x) − n2cos(nx) − 2 x nsin(nx); y(0) = 2, y ′(0) = 0 theoretical solution : y(x) = 1 + cos(nx)  (19) * absolute error=|e xact solution − numerical solution|, numerical solution is obtained using the runge-kutta methods. example 2: variable coefficient homogeneous singular initial bvp [6] (x2y′)′ = β(xy′ + y(α + β− 1))xα+β−2, 0 ≤ x ≤ 1. y(0) = 1, y(1) = ex p(1) theoretical solution = exp(x4) (20) 136 ogunniran et al. / j. nig. soc. phys. sci. 2 (2020) 134–140 137 table 1. table of characteristics equations for runge-kutta methods method m(z), z = hλ (9) 14 z 2 + 12 z + 1 (10) 16 z 3 + 12 z 2 + z + 1 (11) 124 z 4 + 16 z 3 + 12 z 2 + z + 1 (12) 1120 z 5 + 124 z 4 + 16 z 3 + 12 z 2 + z + 1 (13) z 8 483840 + z7 4480 + z6 720 − 179 z5 4032 − 3049 z4 3360 − 3119 z3 420 − 3119 z2 140 + 803 z 840 + 1 (14) 136 z 4 + 16 z 3 + 12 z 2 + z + 1 (15) 196 z 5 + 124 z 4 + 16 z 3 + 12 z 2 + z + 1 (16) 124 z 4 + 16 z 3 + 12 z 2 + z + 112 table 2. table of absolute errors for example 1, n = 3 at different h h x y(x) rk2 rk3 rk4 rk nyström rk merson rk scraton rk england rk8 10−1 0.2 9.1982 × 10−2 9.0000 × 100 5.0392 × 10−2 4.5167 × 10−2 1.3355 × 10−1 7.61289 × 100 5.0392 × 10−2 1.1722 × 100 0.5 7.7896 × 10−2 9.0000 × 100 2.0474 × 10−2 7.5533 × 10−2 1.6385 × 10−1 8.3700 × 100 2.04738 × 10−2 1.2308 × 100 0.7 8.1030 × 10−2 9.0000 × 100 1.4680 × 10−2 8.1382 × 10−2 1.6970 × 10−1 7.4117 × 100 1.4680 × 10−2 1.2624 × 100 10−2 0.04 7.0148 × 10−4 9.0000 × 100 2.5148 × 10−4 7.0046 × 10−4 1.5808 × 10−3 9.7960 × 10−1 2.5148 × 10−4 1.2152 × 10−2 0.4 5.7493 × 10−4 9.0000 × 100 2.5188 × 10−5 9.2689 × 10−4 1.8072 × 10−3 2.7924 × 100 2.5188 × 10−5 3.9599 × 10−2 0.9 7.5147 × 10−4 9.0000 × 10+1 1.1203 × 10−5 9.4089 × 10−4 1.8212 × 10−3 1.200 × 10−2 1.1203 × 10−5 9.5724 × 10−2 10−3 0.09 4.7178 × 10−6 9.0000 × 100 1.1179 × 10−7 9.4074 × 10−6 1.8210 × 10−5 8.2736 × 10−1 1.1179 × 10−7 1.6996 × 10−3 0.19 4.8424 × 10−6 9.0000 × 10+1 5.2955 × 10−8 9.4662 × 10−6 1.8269 × 10−5 1.5242 × 100 5.2955 × 10−8 7.0497 × 10−3 0.69 6.7846 × 10−6 9.0000 × 10+1 1.4584 × 10−8 9.5046 × 10−6 1.8307 × 10−5 1.2185 × 100 1.4584 × 10−8 6.5798 × 10−2 cmpt time (s) 0.0025 0.0022 0.0035 0.0059 0.0039 0.0077 0.0065 0.021 table 3. table of absolute errors for example 2 using α = 2, β = 4 h x y(x) rk2 rk3 rk4 rk nyström rk merson rk scraton rk england rk8 1 10 0.2 1.5913 × 10−3 1.5954 × 10−3 1.5745 × 10−3 1.5610 × 10−3 1.5706 × 10−3 7.0036 × 10−4 1.5745 × 10−3 1.5705 × 10−3 0.5 5.8226 × 10−2 5.8697 × 10−2 5.7050 × 10−2 5.7009 × 10−2 5.7014 × 10−2 2.5489 × 10−2 5.7050 × 10−2 5.7175 × 10−2 0.8 3.8263 × 10−1 3.8559 × 10−1 3.7177 × 10−1 3.7160 × 10−1 3.7166 × 10−1 5.7962 × 10−1 3.7177 × 10−1 3.7544 × 10−1 1 16 0.125 2.4357 × 10−4 2.4382 × 10−4 2.4258 × 10−4 2.4231 × 10−4 2.4234 × 10−4 1.5826 × 10−4 2.4258 × 10−4 2.4233 × 10−4 0.5 5.7534 × 10−2 5.7757 × 10−2 5.7026 × 10−2 5.7019 × 10−2 5.7020 × 10−2 2.5563 × 10−2 5.7026 × 10−2 5.7237 × 10−2 0.75 2.6863 × 10−1 2.8731 × 10−1 2.8304 × 10−1 2.8302 × 10−1 2.8303 × 10−1 3.7761 × 10−1 2.8304 × 10−1 2.8600 × 10−1 1 32 0.25 3.8056 × 10−3 3.8090 × 10−3 3.7977 × 10−3 3.7976 × 10−3 3.7976 × 10−3 1.1239 × 10−3 3.7977 × 10−3 3.8001 × 10−3 0.5625 9.0304 × 10−2 9.0405 × 10−2 9.0077 × 10−2 9.0076 × 10−2 9.0077 × 10−2 5.8501 × 10−2 9.0077 × 10−2 9.0639 × 10−2 0.875 5.5592 × 10−1 5.5618 × 10−1 5.5376 × 10−1 5.5538 × 10−1 5.5376 × 10−1 1.0939 × 100 5.5376 × 10−1 5.6323 × 10−1 cmpt time (s) 0.0023 0.0026 0.0042 0.0055 0.0033 0.0061 0.0055 0.011 137 ogunniran et al. / j. nig. soc. phys. sci. 2 (2020) 134–140 138 figure 2. stability region for runge-kutta method of order 3 figure 3. stability region for runge-kutta method of order 4 figure 4. stability region for kutta-nyström method example 3: variable coefficient non-linear homogeneous singular ivp [3] y′′(x) = −2x y ′(x) − y5(x); x ∈ [0, 1] y(0) = 1, y′(0) = 0 theoretical solution : y(x) = 1√ 1+ x 2 3 (21) figure 5. stability region for runge-kutta method of order 6 figure 6. stability region for merson’s method figure 7. stability region for scraton’s method example 4: van der pol singular problem [11] y′′ = y ′(1−y′2 )−y µ ; y(0) = 2, y′(0) = −23 + 10 81 µ− 292 2187 µ 2 − 1814 19683 µ 3; µ = 10−1. (22) the methods considered in this work were used to approximate the problem over the interval [0, 0.55139] for h = 10−3. 138 ogunniran et al. / j. nig. soc. phys. sci. 2 (2020) 134–140 139 table 4. table of absolute errors for example 3 using different h h x y(x) rk2 rk3 rk4 rk nyström rk merson rk scraton rk england rk8 1 7 0.2857 2.4565 × 10−2 9.7182 × 10+6 2.7524 × 10−2 3.4496 × 10−2 4.1169 × 10−2 3.5739 × 10−2 2.7525 × 10−2 1.1722 × 10−1 0.5714 2.8265 × 10−2 9.7182 × 10+6 3.1850 × 10−2 3.8832 × 10−2 4.54985 × 10−2 3.9917 × 10−1 3.1851 × 10−2 1.2281 × 10−1 0.8571 1.0144 × 10−2 9.7182 × 10+6 1.4411 × 10−2 2.1093 × 10−2 2.7780 × 10−2 4.0945 × 10−1 1.4112 × 10−2 1.0723 × 10−1 1 14 0.2857 2.9836 × 10−2 1.2148 × 10+6 3.0700 × 10−2 3.2486 × 10−2 3.4150 × 10−2 2.1129 × 10−1 3.0701 × 10−2 5.4061 × 10−2 0.4286 3.4167 × 10−2 1.2148 × 10+6 3.5075 × 10−2 3.0686 × 10−2 3.8524 × 10−2 2.3129 × 10−1 3.5075 × 10−2 5.9065 × 10−2 0.6429 2.8969 × 10−2 1.2148 × 10+6 2.9944 × 10−2 3.1729 × 10−2 3.3393 × 10−2 2.4973 × 10−1 2.9944 × 10−2 5.5316 × 10−2 1 21 0.4286 3.4837 × 10−2 3.5993 × 10+5 3.5246 × 10−2 3.6042 × 10−2 3.6782 × 10−2 1.7714 × 10−1 3.5246 × 10−2 4.6596 × 10−2 0.6667 2.8266 × 10−2 3.5999 × 10+5 2.8707 × 10−2 2.9504 × 10−2 3.0243 × 10−2 1.9637 × 10−1 2.8707 × 10−2 4.1652 × 10−2 0.9048 9.3712 × 10−3 3.5993 × 10+5 9.8446 × 10−3 1.0641 × 10−2 1.1138 × 10−2 1.9780 × 10−1 9.8446 × 10−3 2.4823 × 10−2 cmpt time (s) 0.0025 0.0021 0.0035 0.0059 0.0045 0.0077 0.0051 0.021 figure 8. stability region for england’s method table 5. numerical results for example 4 at x = 0.55139 h method y(x) cmpt. time (s) rk2 0.625821430245524 0.0025 rk3 0.625821606669667 0.0019 rk4 0.625821602543592 0.0065 10−3 rk nyström 0.625821602551168 0.018 rk merson 0.625821602551009 0.0028 rk scarton -0.031597285014076 0.025 rk england 0.625821602543570 0.0021 rk 8 0.642280546551478 0.063 4. discussion of results and conclusion it is worthy to note that numerical methods were programmed via matlab 9.2 version on a personal computer with the following specifications: • system namehp pavilion x360 convertible • processorintel(r) core(tm) i3-7100u cpu @ 2.40ghz • installed memory (ram)8.00gb • system type64-bits operating system, x64-based processor • operating systemwindows 10 cmpt time is the computer computation time measured in seconds(s). this paper presents the performances of different orders of runge-kutta methods in the integration of singular lane-emden equations. from tables 2-4, it could be observed that the runge-kutta method of order 4 performs the best in terms of accuracy. the methods of order 3, scraton’s, and order 8 were found to have failed in the solution of singular problems in ordinary differential equations. the england’s method performs more satisfactorily as it shows some competitive strength against the order 4 method. it could be concluded that the runge-kutta method of order 4 outperforms all other methods under consideration and this property makes it the most stable and accurate of the runge-kutta methods. acknowledgments the authors thank the referees and editor for their valuable comments and suggestions. references [1] a. huta, “contribution à la formule de sixieme ordre dans la methode de runge-kutta-nyström”, acta fac rerum natu univ. comenian math. 2 (1957) 21. 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[20] j. d. lambert, computational methods in ordinary differential equations, john wiley and sons, new york, 1991. 140 j. nig. soc. phys. sci. 2 (2020) 186–196 journal of the nigerian society of physical sciences original research portfolio strategy for an investor with logarithm utility and stochastic interest rate under constant elasticity of variance model edikan e. akpanibaha,∗, udeme o. inib adepartment of mathematics and statistics, federal university otuoke, bayelsa, nigeria bdepartment of mathematics and computer science, niger delta university, bayelsa, nigeria abstract this paper is aim at maximizing the expected utility of an investor’s terminal wealth; to achieve this, we study the optimal portfolio strategy for an investor with logarithm utility function under constant elasticity of variance (cev) model in the presence of stochastic interest rate. a portfolio comprising of a risk free asset and a risky asset is considered where the risk free interest rate follows the coxingersoll-ross (cir) model and the risky asset is modelled by cev. using power transformation, change of variable and asymptotic expansion technique, an explicit solution of the optimal portfolio strategy and the value function are obtained. furthermore, numerical simulations are presented to study the effect of some parameters on the optimal portfolio strategy under stochastic interest rate. keywords: keywords: stochastic interest rate, optimal portfolio strategy, asymptotic technique, constant elasticity of variance, logarithm utility article history : received: 12 april 2020 received in revised form: 04 july 2020 accepted for publication: 15 july 2020 published: 01 august 2020 c©2020 journal of the nigerian society of physical sciences. all rights reserved. communicated by: t. latunde 1. introduction in the study of optimal portfolio strategy in a financial market, volatility plays a vital role in influencing the behaviour of the risky assets due to its fluctuating nature as a result of different information available in the financial market. for an investor to make relatively right choice when investing in risky assets, there is need to consider stochastic volatility models and not constant volatility in order to understand the fluctuating na∗corresponding author tel. no: +2347030811189 email address: edikanakpanibah@gmail.com (edikan e. akpanibah ) ture of the risky assets. one of such stochastic volatility model is the cev model. the cev model was developed by [1] and is an extension geometric brownian motion (gbm). according to [2], the model is capable of capturing the implied volatility skew. a lot of researchers such as ([3],[4]) studied utility maximization under constant elasticity model in defined contribution (dc) pension scheme. in [5], optimal investment and reinsurance problem of utility maximization under cev model was studied. the authors in [2] studied optimal investment problem with taxes, dividend and transaction cost using cev model and logarithm utility function. the optimal portfolio strategy with multiple 186 akpanibah & ini / j. nig. soc. phys. sci. 2 (2020) 186–196 187 contributors in a dc pension fund using legendre transformation method was studied by [6]. the authors in [7] solved the optimal portfolio problem with default risk and refund of premium clause in a dc plan; in their work, the stock market price followed the cev model. in [8], the effect of additional voluntary contribution on the investment strategies under cev model; they used power transformation method in solving their problem. the study of optimal portfolio with stochastic interest have been investigated by many authors though the risky assets were modelled by gbms; they include [9], who studied optimal portfolio management with stochastic interest rate for a protected case of dc fund. in ([10]-[11]), stochastic interest rate model was used to obtain optimal portfolio allocation in a dc plan. also, the authors in ([12]-[13]) considered investment strategy with interest rate of vasicek type while the authors in ([14]-[16]), studied the optimal portfolio problem when the interest rate is of affine interest. from the available literatures, most of the authors that worked on optimal portfolio strategies under cev model considered cases where the risk free interest rate is constant except for [17] who studied the optimal portfolio strategies under cev model with stochastic interest rate. they pointed out that one of the main reasons why other authors could not combined cev model and stochastic interest in finding the optimal portfolio strategies was in the difficulty to find a closed form solution of the optimal portfolio strategies analytically. also they pointed out that in financial market, interest rate is not constant rather a fluctuating processes and that the interest rate volatility presents another source of risks in financial market; in other words, when this risk is not taken into consideration, we are actually undermining the risk generated by this interest rate which is crucial in affecting the prices of various assets available in financial market. in their work, they used the exponential utility to optimize the expected utility of an insurer with exponential utility function and used the legendry transformation method and asymptotic expansion method to obtain solutions for the optimal portfolio strategies. in this work, we maximize the expected utility of an insurer’s wealth by studying the optimal portfolio strategies of an insurer with logarithm utility function whose risky asset is model by cev model and the risk free interest is stochastic and follows the cir model. furthermore, we use the power transformation, change of variable and asymptotic approach to derive an asymptotic solution of the optimal portfolio strategies and value function. also some numerical simulations to explain our results are given. the main difference between our work and that of [17] is that we consider an investor with logarithm utility instead of exponential utility and apply power transformation method and change of variable method instead of legendre transformation method. 2. preliminaries for a financial market with portfolio comprising of a risk free asset (treasury security) and a risky asset (marketable security) which is open continuously for a period of t > 0 representing the expiring date of the investment. let (ω, f, p) be a probability space which is complete, ω and p are real space probability measure respectively, {br (t) , bs (t) : t ≥ 0} is a set of brownian motions and f the filtration representing information produced by the brownian motions. let the risk free asset price c (t) at time t be given as dc(t) c(t) = r (t) dt, c (0) > 0 (1) where r (t) is the short interest rate and follows the (cir) model whose dynamics is{ dr (t) = (b − cr (t)) dt − a1 √ r (t)dbr (t) r (0) = r0 > 0 (2) and b , c and a1 are positive numbers such that the following condition holds a21 < 2b (feller’s condition). [17] let s (t) be the price of the marketable security whose price process follows the cev model which is an extension of the gmb. the cev as earlier stated has the capability to capture the volatility skew of the marketable security unlike the gbm whose volatility is constant. from the work of [4, 6, 8, 17], the dynamic of the price process of the marketable security is given by the stochastic differential equation when t ≥ 0 as follows ds (t) s (t) = µdt + σs β (t) dbs (t) (3) where µ , σ and β are positive constant and represent the instantaneous expected rate of return, instantaneous volatility and elasticity parameter respectively see [2−4]. also, br (t) and bs (t) are assumed to be correlated instantaneous correlation coefficient ρ = 1 such that dbr (t) dbs (t) = dt see [17]. note, when β = 0 in equation (3), the model in (3) reduce to that of gmb see [2]. 3. wealth formulations and methodology let z (t) be the investor’s wealth at time t. also, let ϕ and ϕ1 be the proportions of the investor’s wealth to be invested in risky asset and risk-free asset respectively such that ϕ1 = 1−ϕ. since the investor’s total wealth is summation of his investments in the two assets hence the differential form of the investor’s total wealth is dz (t) = z (t) ( ϕ1 dc (t) c (t) + ϕ ds (t) s (t) ) (4) 187 akpanibah & ini / j. nig. soc. phys. sci. 2 (2020) 186–196 188 substituting equations (1) and (3) into (4), we have{ dz(t) = z(t)((ϕ(µ− r) + r)dt + ϕσs β (t) dbs (t)) z (0) = z0 (5) next, let the investor’s utility at any given state z at time t be given as jϕ (t, r, s2, z) = eϕ [ u (z (t )) r (t) = r, s (t) = s, z (t) = z ] (6) with boundary condition j (t, r, s2, z) = u (z) where r is the risk free interest rate and z is the wealth. the objective here is to determine the optimal portfolio strategies and the optimal value function of the investor given as ϕ∗ and j (t, r, s2, z) = supϕ jϕ (t, r, s2, z) such that jϕ∗ (t, r, s2, z) = j (t, r, s2, z) (7) the value function jϕ∗ (t, r, s2, z) can be considered as a kind of utility function. from the maximum principle and ito’s lemma, the hamilton jacobi bellman (hjb) equation which is a nonlinear partial differential equation associated with (6) is obtained by maximizing the expected utility jϕ (t, r, s2, z) subject to the investor’s wealth in as follows jt + µsjs + 1 2 σ 2 s2β+2 jss +rzjz + (b − cr) jr + 1 2 ra 2 1 +σa1 √ rsβ+1 jsr + sup  1 2 ϕ 2σ2z2 s2β jzz +ϕ  (µ− r) zjz +σ2zs2β+1 jzs +σa1 √ rzsβ jzr    = 0 (8) differentiating (8) with respect to ϕ , we obtain the first order maximizing condition as( ϕσ2z2 s2βjzz + (µ− r) zjz + σ2zs2β+1 jzs +σa1 √ rzsβjzr ) = 0 (9) solving (9) for ϕ we have ϕ∗ = − [ (µ− r) jz + σ2 s2β+1 jzs + σa1 √ rsβjzr ] zσ2 s2βjzz (10) where ϕ∗ is the optimal portfolio strategy of the risky asset. substituting (10) into (8), we have jt + µsjs + 1 2 σ 2 s2β+2 jss + rzjz + (b − cr) jr + 12 ra 2 1 jrr + σa1 √ rsβ+1 jsr + 12 [(µ−r)jz +σ2 s2β+1 jzs +σa1 √ rsβ jzr ]2 σ2 s2β jzz  = 0 (11) 4. optimal portfolio strategies for an investor with logarithm utility in this section, we consider an investor with utility function which exhibit constant relative risk aversion (crra) different from the one in [17] where the investor exhibits constant absolute risk aversion (cara). since our interest here is to determine the optimal portfolio strategies for the investor with crra utility, we choose the logarithm utility function similar to the one in [2]. from [2, 14], the logarithm utility function is given as u (z) = lnz (12) next, we conjecture a solution to (11) similar to the one in [2] with the form:{ j (t, r, s, z) = w (t, r, s) + v (t, r, s) ln z w (t, r, s) = 0, v (t, r, s) = 1, (13)  jt = wt + vt ln z, js = ws + vs ln z , jss = wss + vss ln z,jr = wr + vr ln z, jrr = wrr + vrr ln z, jsr = wsr + vsr ln zjsr = wsr + vsr ln z, jzz = − vz2 , jzr = vrz , jzs = vsz  (14) substituting (14) into (11), we have  vt + µsvs + 1 2 σ 2 s2β+2vss + (b − cr) vr + 1 2 ra 2 1vrr +σa1 √ rsβ+1vsr  ln z +  wt + µsws + 1 2 σ 2 s2β+2wss + (b − cr) vr + rv + 1 2 ra 2 1wrr +σa1 √ rsβ+1wsr + 12 [(µ−r)v+σ2 s2β+1 vs +σa1 √ rsβvr ]2 σ2 s2βv   = 0 (15) splitting (15) we have( vt + µsvs + 1 2 σ 2 s2β+2vss + (b − cr) vr + 12 ra 2 1vrr + σa1 √ rsβ+1vsr ) = 0 (16)  wt + µsws + 1 2 σ 2 s2β+2wss + (b − cr) vr +rv + 12 ra 2 1wrr + σa1 √ rsβ+1wsr + 12 [(µ−r)v+σ2 s2β+1 vs +σa1 √ rsβvr ]2 σ2 s2βv  = 0 (17) taking the boundary condition v (t, r, s) = 1 into consideration, we solve (16) as follows using power transformation and change variable approach. proposition 4.1 the solution of equation (16) is given as v (t, r, s) = h (t, r, q) = 1 where h (t, r, q) = h1 (t, r, q) + √ αh2 (t, r, q) + αh3 (t, r, q) and h1 (t, r, q) = 1, h2 (t, r, q) = 0, and h3 (t, r, q) = 0 188 akpanibah & ini / j. nig. soc. phys. sci. 2 (2020) 186–196 189 proof assume{ v (t, r, s) = h (t, r, q) , q = s−2β h (t, r, s) = 1 } (18) then vt = ht, vs = −2βs−2β−1hq , vss = 2β (2β + 1) s−2β−2hq + 4β2 s−4β−2hqq, vr = hr, vrr = hrr, vrs = −2βs−2β−1hrq  (19) substituting (19) into (16), we have[ ht − 2µβqhq + σ2β (2β + 1) hq + 2β2σ2qhqq + (b − cr) hr + 1 2 ra 2 1hrr − 2βσa1 √ r √ qhrq ] = 0 (20) we can rewrite (20) as (a + b + c) h = 0 (21) where a = [ (b − cr) hr + 1 2 ra21hrr ] h (22) b = [ ht + β ( σ2 (2β + 1) − 2µq ) hq + 2β 2σ2qhqq ] h (23) c = [ −2βσa1 √ r √ qhrq ] h (24) next we follow the approach in [17] by applying the asymptotic expansion method to solve the problem in (21). assume that the volatility follows a slow fluctuating process, we attempt to determine an asymptotic solution for (21) by using the slow-fluctuating process rα to replace r in (2), such that 0 < α � 1 is a small positive integer: drα (t) = (b − crα (t)) dt − a1 √ rα (t)dbr (t) (25) substituting (25) into (21) and also replacing b − cr by α (b − cr) and √ r by √ α √ r, we will have( αa + b + √ αc ) hα = 0 (26) next, we conjecture a solution for (26) as follows hα (t, r, q) = h1 (t, r, q) + √ αh2 (t, r, q) + αh3 (t, r, q) (27) substituting (27) into (26) ( αa + b + √ αc ) ( h1 (t, r, q) +√αh2 (t, r, q) +αh3 (t, r, q) ) = 0 simplifying the above equation, we arrive at ( bh1 (t, r, q) + [ bh2 (t, r, q) + ch1 (t, r, q) ]√ α + [ ah1 (t, r, q) + bh3 (t, r, q) + ch2 (t, r, q) ] α ) = 0 this implies that{ bh1 (t, r, q) = 0 h1 (t, r, s) = 1 (28) { bh2 (t, r, q) + ch1 (t, r, q) = 0 h1 (t, r, s) = 1 , h2 (t, r, s) = 0 (29) { ah1 (t, r, q) + bh3 (t, r, q) + ch2 (t, r, q) h1 (t, r, s) = 1, h2 (t, r, s) = 0 h3 (t, r, s) = 0 (30) to obtain the solution of (27), we solve (28), (29) and (30). recall from (22), (23) and (24), equation (28), (29) and (30) can be expressed as h1t + β ( σ2 (2β + 1) − 2µq ) h1q + 2β2σ2qh1qq = 0 h1 (t, r, q) = 1 (31)  h2t + β ( σ2 (2β + 1) − 2µq ) h2q +2β2σ2qh2qq − 2βσa1 √ r √ qh1rq = 0 h2 (t, r, q) = 0 (32)  h3t + β ( σ2 (2β + 1) − 2µq ) h3q + 2β2σ2qh3qq + (b − cr) h1r + 1 2 ra 2 1h1rr − 2βσa1 √ r √ qh2rq h3 (t, r, q) = 0 (33) let h1 (t, r, q) = d0 (t, r) + qd1 (t, r) (34) and( h1t = d0t + qd1t, h1q = d1, h1qq = 0 ) (35) substituting (35) in (31), we have d0t + βσ 2 (2β + 1) d1 = 0, d0 (t, r) = 1 (36) d1t − 2µβd1 = 0, d1 (t, r) = 0 (37) solving (37), we have d1 (t, r) = 0 (38) putting (38) into (36) solving it, we have d0 (t, r) = 1 (39) hence from (34) h1 (t, r, q) = 1 (40) next, we solve (32), by assuming a solution of the form h2 (t, r, q) =  d2 (t, r) + q 12 d3 (t, r) + qd4 (t, r) +q 3 2 d5 (t, r)  (41) 189 akpanibah & ini / j. nig. soc. phys. sci. 2 (2020) 186–196 190 and h2t = d2t + q 1 2 d3t + qd4t + q 3 2 d5t , h2q = 1 2 q − 1 2 d3 + d4 + 3 2 q 1 2 d5 h2qq = − 1 4 q − 3 2 d3 + 3 4 q − 1 2 d5, h1rq = 0  (42) substituting (42) into (32), we have d2t + βσ 2 (2β + 1) d4 = 0, d2 (t, r) = 0 (43) d4t − 2µβd4 = 0, d4 (t, r) = 0 (44) d5t − 3µβd5 = 0, d5 (t, r) = 0 (45) d3t −µβd3 + 3 2 βσ2 (3β + 1) d5 = 0, d3 (t, r) = 0 (46) solving (43), (44), (45) and (46) with their boundary conditions, we have d2 (t, r) = 0, d3 (t, r) = 0, d4 (t, r) = 0, d5 (t, r) = 0 hence from (41) h2 (t, r, q) = 0 (47) next , we attempt to solve (33), by assuming a solution of the form h3 (t, r, q) =  d6 (t, r) + q 12 d7 (t, r) + qd8 (t, r) +q 3 2 d9 (t, r) + q2 d9 (t, r)  (48)  h3t = d6t + q 1 2 d7t + qd8t + q 3 2 d9t + q2 d10t , h3q = 1 2 q − 1 2 d7 + d8 + 3 2 q 1 2 d9 + 2qd10 h2qq = − 1 4 q − 3 2 d7 + 3 4 q − 1 2 d9 + 2d10h1r = 0, h1rr = 0, h2rq = 0  (49) substituting into (33), we have d6t + βσ 2 (2β + 1) d8 = 0, d6 (t, r) = 0 (50) d7t −µβd7 + 3 2 βσ2 (3β + 1) d9 = 0, d7 (t, r) = 0 (51) d8t − 2µβd8 + 2βσ 2 (4β + 1) d10 = 0, d8 (t, r) = 0(52) d9t − 3µβd9 = 0, d9 (t, r) = 0 (53) d10t − 4µβd10 = 0, d10 (t, r) = 0 (54) solving (50), (51), (52), (53) and (54), we obtain d6 (t, r) = 0, d7 (t, r) = 0, d8 (t, r) = 0, d9 (t, r) = 0, d10 (t, r) = 0 hence from (48) h3 (t, r, q) (55) therefore, from (27), we have hα (t, r, q) = h1 (t, r, q) + √ αh2 (t, r, q) + αh3 (t, r, q) = 1 hence from (18), v (t, r, s) = 1 proposition 4.2 the solution of equation (17) is given as w (t, r, s) = f1 (t, r, q) + √ α f2 (t, r, q) + α f3 (t, r, q) where  f1 (t, r, q) = [ (2β+1)(µ−r)2 8βµ + r ] (t − t) + ( q (µ−r) 2 4µβσ2 − (2β+1)(µ−r)2 8βµ2 ) [ 1 − e2µβ(t−t ) ] f2 (t, r, q) = k2 (t, r) + q 1 2 k3 (t, r) + qk4 (t, r) f3 (t, r, q) = k6 (t, r) + q 1 2 k7 (t, r) + qk8 (t, r)  and  k2 (t, r) = [ (2β+1)(µ−r)2 8βµ + r ] (t − t) − (2β+1)(µ−r)2 8βµ2 [ 1 − e2µβ(t−t ) ] k3 (t, r) = a1 √ r(µ−r) σβµ2 [ 1 − 2eµβ(t−t ) + e2µβ(t−t ) ] k4 (t, r) = (µ−r)2 4µβσ2 [ 1 − e2µβ(t−t ) ] k6 (t, r) =   (2β+1)(µ−r)(b−cr)[1−e2µβ(t−t ) ] 8βσ2µ2 + (2β+1)(µ−r)2 4σ2µ + r r(2β+1)(µ−r)(b−cr) 4βµ2 + ra21 (2β+1)(µ−2r) 8βµ2 + (µ−3r)a21 2µ2  (t − t) +  r(2β+1)(µ−r)(b−cr) 4µ − (b−cr) 2 + ra21 (2β+1)(µ−2r) 8µ  (t2 − t 2) r(2β+1)(µ−r)(b−cr) 8β2µ3 + (2β+1)(µ−r)(ra21 +µ−r) 8βσ2µ2 + ra21 (2β+1)(µ−2r) 8µ + (µ−3r)a21 4βµ3  ( 1 − e2µβ(t−t ) ) (µ−3r)a21 4βµ3 ( 1 − eµβ(t−t ) )  k7 (t, r) = a1 √ r(µ−r) σβµ2 [ 1 − 2eµβ(t−t ) + e2µβ(t−t ) ] k8 (t, r) =  (µ−r)2 4µβσ2 [ 1 − e2µβ(t−t ) ] +  ra21 −2 (b − cr) (µ− r) 4µβσ2   12µβ [ 1 − e2µβ(t−t ) ] + (t − t ) e2µβ(t−t )    proof substitute (t, r, s) = 1, vs = 0, vr = 0 into (17), we have wt + µsws + 12 σ2 s2β+2wss + (b − cr) wr + r + 12 ra21wrr +σa1 √ rsβ+1wsr + 1 2 (µ−r)2 σ2 s2β  = 0(56) w (t, r, s) = 0 assume{ w (t, r, s) = f (t, r, q) , q = s−2β f (t, r, q) = 0 (57) then wt = ft, ws = −2βs−2β−1 fq , wss = 2β (2β + 1) s−2β−2 fq + 4β2 s−4β−2 fqq, wr = fr, wrr = frr, wrs = −2βs−2β−1 frq (58) 190 akpanibah & ini / j. nig. soc. phys. sci. 2 (2020) 186–196 191 substituting (58) into (56), we have ft − 2µβq fq + σ2β (2β + 1) fq + 12 (µ−r)2 q σ2 + r + 2β2σ2q fqq + (b − cr) fr + 12 ra 2 1 frr − 2βσa1 √ r √ q frq  = 0 (59) we can rewrite (59) as (e + f + g) f = 0 (60) where e = [ (b − cr) fr + 1 2 ra21 frr ] f (61) f =  ft + β ( σ2 (2β + 1) − 2µq ) fq +2β2σ2q fqq + 1 2 (µ−r)2 q σ2 + r  f (62) g = [ −2βσa1 √ r √ q frq ] f (63) similarly, we apply the same approach used in solving (21) to determine the solution of (60) as follows drα (t) = (b − crα (t)) dt − a1 √ rα (t)dbr (t) (64) substituting (64) into (60) and also replacing b − cr by α (b − cr) and √ r by √ α √ r, we will have( αe + f + √ αg ) fα = 0 (65) next, we conjecture a solution for (65) as follows fα (t, r, q) = f1 (t, r, q) + √ α f2 (t, r, q) + α f3 (t, r, q) (66) substituting (66) into (65) and simplifying it, we have f f1 (t, r, q) + [ f f2 (t, r, q) + g f1 (t, r, q) ]√ α + [ e f1 (t, r, q) + f f3 (t, r, q) + g f2 (t, r, q) ] α = 0 this implies that{ f f1 (t, r, q) = 0 f1 (t, r, s) = 0 (67) { f f2 (t, r, q) + g f1 (t, r, q) = 0 f1 (t, r, s) = f2 (t, r, s) = 0 (68) { e f1 (t, r, q) + f f3 (t, r, q) + g f2 (t, r, q) f1 (t, r, s) = f2 (t, r, s) = f3 (t, r, s) = 0 (69) to obtain the solution of (66), we solve (67), (68) and (69). recall from (61), (62) and (63), equation (67), (68) and (69) can be expressed as f1t + β ( σ2 (2β + 1) − 2µq ) f1q + 2β2σ2q f1qq + 12 (µ−r)2 q σ2 + r = 0 f1 (t, r, q) = 1 (70)  f2t + β ( σ2 (2β + 1) − 2µq ) f2q + 2β2σ2q f2qq + 12 (µ−r)2 q σ2 + r − 2βσa1 √ r √ q f1rq = 0 f2 (t, r, q) = 0 (71)  f3t + β ( σ2 (2β + 1) − 2µq ) f3q + 2β2σ2q f3qq + 12 (µ−r)2 q σ2 + r + (b − cr) f1r + 1 2 ra 2 1 f1rr − 2βσa1 √ r √ q f2rq f3 (t, r, q) = 0 (72) let f1 (t, r, q) = k0 (t, r) + qk1 (t, r) (73) and( f1t = k0t + qk1t, f1q = k1, f1qq = 0 ) (74) substituting (74) in (70), we have k0t + βσ 2 (2β + 1) k1 + r = 0, k0 (t, r) = 0 (75) k1t − 2µβk1 + 1 2 (µ− r)2 σ2 = 0, k1 (t, r) = 0 (76) solving (76), we have k1 (t, r) = (µ− r)2 4µβσ2 [ 1 − e2µβ(t−t ) ] (77) putting (77) into (75) solving it, we have k0 (t, r) =  [ (2β+1)(µ−r)2 8βµ + r ] (t − t) − (2β+1)(µ−r)2 8βµ2 [ 1 − e2µβ(t−t ) ]  (78) hence from (34) f1 (t, r, q) =  [ (2β+1)(µ−r)2 8βµ + r ] (t − t) + ( q (µ−r) 2 4µβσ2 − (2β+1)(µ−r)2 8βµ2 ) [ 1 − e2µβ(t−t ) ] (79) next, we solve (71), by assuming a solution of the form f2 (t, r, q) = k2 (t, r)+q 1 2 k3 (t, r)+qk4 (t, r)+q 3 2 k5 (t, r)(80) and f2t = k2t + q 1 2 k3t + qk4t + q 3 2 k5t , f2q = 1 2 q − 1 2 k3 + k4 + 3 2 q 1 2 k5 f2qq = − 1 4 q − 3 2 k3 + 3 4 q − 1 2 k5, k1r = (µ−r) 2µβσ2 [ e2µβ(t−t ) − 1 ]  (81) substituting (81) into (71), we have k2t + βσ2 (2β + 1) k4 + r = 0, k2 (t, r) = 0( k3t −µβk3 + 3 2 βσ 2 (3β + 1) k5 −2βσa1 √ rk1r = 0 ) , k3 (t, r) = 0 k4t − 2µβk4 + (µ−r)2 2σ2 = 0, k4 (t, r) = 0 k5t − 3µβk5 = 0, k5 (t, r) = 0  (82) 191 akpanibah & ini / j. nig. soc. phys. sci. 2 (2020) 186–196 192 solving (82) with their boundary conditions, we have k2 (t, r) =  [ (2β+1)(µ−r)2 8βµ + r ] (t − t) − (2β+1)(µ−r)2 8βµ2 [ 1 − e2µβ(t−t ) ]  k3 (t, r) = a1 √ r(µ−r) σβµ2 [ 1 − 2eµβ(t−t ) + e2µβ(t−t ) ] k4 (t, r) = (µ−r)2 4µβσ2 [ 1 − e2µβ(t−t ) ] k5 (t, r) = 0  (83) hence from (80) f2 (t, r, q) =  k2 (t, r) + q 12 k3 (t, r) + qk4 (t, r) +q 3 2 k5 (t, r)  (84) where k2, k3, k4, and k5 are given in equation (83) next, we attempt to solve (72), by assuming a solution of the form f3 (t, r, q) =  k6 (t, r) + q 12 k7 (t, r) + qk8 (t, r) +q 3 2 k9 (t, r) + q2 k10 (t, r)  (85)  f3t = k6t + q 1 2 k7t + qk8t + q 3 2 k9t + q2 k10t , f3q = 1 2 q − 1 2 k7 + k8 + 3 2 q 1 2 k9 + 2qk10 f3qq = − 1 4 q − 3 2 k7 + 3 4 q − 1 2 k9 + 2k10, f1r = k0r + qk1r, f1rr = k0rr + qk1rr f2rq = 1 2 q − 1 2 k3r + k4r + 3 2 q 1 2 k5r  (86) substituting into (72), we have ( k6t + βσ2 (2β + 1) k8 + r + (b − cr) k0r −βσa1 √ rk3r + 1 2 ra 2 1 k0rr ) = 0 k6 (t, r) = 0 k7t −µβk7 + 3 2 βσ 2 (3β + 1) k9 − 2βσa1 √ rk4r = 0, k7 (t, r) = 0 k8t + 2βσ2 (4β + 1) k10 + (b − cr) k1r −2µβk8 − 3βσa1 √ rk9r + 1 2 ra 2 1 k1rr + (µ−r)2 2σ2  = 0 k8 (t, r) = 0 k9t − 3µβk9 = 0, k9 (t, r) = 0 k10t − 4µβk10 = 0, k10 (t, r) = 0  (87) solving (87), we obtain k6 (t, r) =   (2β+1)(µ−r)(b−cr)[1−e2µβ(t−t ) ] 8βσ2µ2 + r(2β+1)(µ−r)(b−cr) 4βµ2 + (2β+1)(µ−r)2 4σ2µ + r + ra21 (2β+1)(µ−2r) 8βµ2 + (µ−3r)a21 2µ2  (t − t) +  r(2β+1)(µ−r)(b−cr) 4µ − (b−cr) 2 + ra21 (2β+1)(µ−2r) 8µ  (t2 − t 2) r(2β+1)(µ−r)(b−cr) 8β2µ3 + (2β+1)(µ−r)(ra21 +µ−r) 8βσ2µ2 + ra21 (2β+1)(µ−2r) 8µ + (µ−3r)a21 4βµ3  ( 1 − e2µβ(t−t ) ) (µ−3r)a21 4βµ3 ( 1 − eµβ(t−t ) )  k7 (t, r) = a1 √ r(µ−r) σβµ2 [ 1 − 2eµβ(t−t ) + e2µβ(t−t ) ] k8 (t, r) =  (µ−r)2 4µβσ2 [ 1 − e2µβ(t−t ) ] + ra21−2(b−cr)(µ−r) 4µβσ2  12µβ [ 1 − e2µβ(t−t ) ] + (t − t ) e2µβ(t−t )   k9 (t, r) = 0 k10 (t, r) = 0  (88) hence from (85) f3 (t, r, q) =  k6 (t, r) + q 12 k7 (t, r) + qk8 (t, r) +q 3 2 k9 (t, r) + q2 k10 (t, r)  (89) with k6, k7 , k8, k9 , k10 given in (88) therefore, from (66), we have fα (t, r, q) = f1 (t, r, q) + √ α f2 (t, r, q) + α f3 (t, r, q) with f1 (t, r, q) , f2 (t, r, q) and f3 (t, r, q) given in equation (79), (84) and (89) respectively. hence from (57), w (t, r, s) = f1 (t, r, q) + √ α f2 (t, r, q) + α f3 (t, r, q) proposition 4.3 the optimal portfolio strategies and optimal value function are given as ϕ∗ = (µ− r) σ2 s2β (90) ϕ∗1 = (µ− r) σ2 s2β (91) and j (t, r, s, z) = ( ln z + f1 (t, r, q) + √ α f2 (t, r, q) +α f3 (t, r, q) ) (92) where  f1 (t, r, q) =  [ (2β+1)(µ−r)2 8βµ + r ] (t − t) +  q (µ−r) 2 4µβσ2 − (2β+1)(µ−r)2 8βµ2  [1 − e2µβ(t−t )]  f2 (t, r, q) = k2 (t, r) + q 1 2 k3 (t, r) + qk4 (t, r) f3 (t, r, q) = k6 (t, r) + q 1 2 k7 (t, r) + qk8 (t, r)  and 192 akpanibah & ini / j. nig. soc. phys. sci. 2 (2020) 186–196 193  k2 (t, r) =  [ (2β+1)(µ−r)2 8βµ + r ] (t − t) − (2β+1)(µ−r)2 8βµ2 [ 1 − e2µβ(t−t ) ]  k3 (t, r) =  a1√r(µ−r)σβµ2  1 − 2eµβ(t−t ) +e2µβ(t−t )   k4 (t, r) = (µ−r)2 4µβσ2 [ 1 − e2µβ(t−t ) ] k6 (t, r) =   (2β+1)(µ−r)(b−cr) [ 1−e2µβ(t−t ) ] 8βσ2µ2 + (2β+1)(µ−r)2 4σ2µ + r r(2β+1)(µ−r)(b−cr) 4βµ2 + ra21 (2β+1)(µ−2r) 8βµ2 + (µ−3r)a21 2µ2  (t − t) +  r(2β+1)(µ−r)(b−cr) 4µ − (b−cr) 2 + ra21 (2β+1)(µ−2r) 8µ  (t2 − t 2) r(2β+1)(µ−r)(b−cr) 8β2µ3 + (2β+1)(µ−r) ( ra21 +µ−r ) 8βσ2µ2 + ra21 (2β+1)(µ−2r) 8µ + (µ−3r)a21 4βµ3  ( 1 − e2µβ(t−t ) ) (µ−3r)a21 4βµ3 ( 1 − eµβ(t−t ) )  k7 (t, r) = a1 √ r(µ−r) σβµ2 [ 1 − 2eµβ(t−t ) + e2µβ(t−t ) ] k8 (t, r) =  (µ−r)2 4µβσ2 [ 1 − e2µβ(t−t ) ] + ra21−2(b−cr)(µ−r) 4µβσ2  12µβ [ 1 − e2µβ(t−t ) ] + (t − t ) e2µβ(t−t )    proof 1. recall that equation (11) is given as ϕ∗ = − [(µ−r)jz +σ2 s2β+1 jzs +σa1 √ rsβ jzr ] zσ2 s2β jzz . from proposition 1 and equation (14) v (t, r, q) = 1 jz = 1 z , jzz = − 1 z2 , jzr = vr z = 0 , jzs = vs z = 0 substituting the above equation into (11), we have ϕ∗ = − [ (µ− r) 1z ] zσ2 s2β ( − 1 z2 ) = (µ− r) 1z σ2 s2β 1z = (µ− r) σ2 s2β 2. recall that from equation (13), j (t, r, s, z) = w (t, r, s) + v (t, r, s) ln z from proposition 1 and 2, the proof is completed. remark 4.1 from proposition 4.3, we observed that our result is similar to the one in [3] but the difference between our result and theirs is that their interest rate is a constant while ours is stochastic. 5. numerical simulations here, the numerical simulations are used to study the effect of parameters on the portfolio strategies under logarithm utility. to achieve this, the following data will be use unless stated otherwise: σ = 1, β = −1, µ = 0.4, r (0) = 0.05, s (0) = 1.5, t = 3 6. discussion figure 1, present a simulation of optimal portfolio strategies against time. the graph shows that the investor will invest more in marketable security and gradually increases investment in treasury security to balance with the marketable security and as expiry date draws closer; there is a continuous decrease in investment in risky asset and a continuous increase in that of risk free asset. this is so because investors like avoiding risk toward the end of investment especially for highly risky assets. figure 2, shows a plot of optimal portfolio strategies against time with different values of the elasticity parameter β. the graph shows that as the elasticity parameter decreases, the insurer is more scared to invest in marketable security as expiration date approaches. furthermore, we observed a very sharp decline when β = −2, showing how volatile the risky asset can be hence discouraging for investors with high risk aversion coefficient but when β = 0, the decline is almost unnoticeable, showing that the risky asset is not volatile. this is shows that geometric brownian motions i.e when β = 0 may lead an investor astray while taking investment decisions. figure 3, present a simulation of optimal portfolio strategies against time with different values of µ; the graph shows that as µ increases, the investment strategy for the marketable security decreases continuously while that of treasury security increases continuously with time. the reason being the interest rate is not constant. furthermore, we observe that as expiration date of investment draws closer, the risk-free interest may increase faster than µ , thereby making µ− r < 0. hence a drastic decrease in optimal portfolio strategy of marketable security. figure 4, shows a simulation of the optimal portfolio strategies against time with different values of σ, it is observed that as σ increases the optimal investment in marketable security decreases while that of risk-free asset increases. recall that σ is the instantaneous volatility representing the risk coefficient of marketable security. therefore, for risk averse investors, bigger σ , implies less investment in marketable security. 7. conclusion this paper considers optimal portfolio strategy for an insurer with logarithm utility using cev to model the risky asset in the presence of stochastic interest rate. investment in treasury security and marketable security were considered such that the risk free interest rate follows the cir model. the asymptotic solutions of the the optimal portfolio strategies and it value function was found using power transformation, change of variable and asymptotic expansion technique. furthermore, we present some numerical simulations to study the effect of some parameters on the optimal portfolio strategy under stochastic interest rate. 193 akpanibah & ini / j. nig. soc. phys. sci. 2 (2020) 186–196 194 figure 1. time evolution of optimal portfolio strategies. figure 2. time evolution of optimal portfolio strategy with different β. 194 akpanibah & ini / j. nig. soc. phys. sci. 2 (2020) 186–196 195 figure 3. time evolution of optimal portfolio strategy with different µ. figure 4. time evolution of optimal portfolio strategy with different σ. 195 akpanibah & ini / j. nig. soc. phys. sci. 2 (2020) 186–196 196 references [1] j. c. cox & s. a. ross, “the valuation of options for alternative stochastic processes”, journal of financial economics 3 (1976) 145 [2] d. li, x. 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[17] y. he & p. chen, “optimal investment strategy under the cev model with stochastic interest rate”, mathematical problems in engineering 2020 (2020) 1 196 j. nig. soc. phys. sci. 4 (2022) 183–192 journal of the nigerian society of physical sciences optimization by box behnken design for eosin yellow dye removal from aqueous medium using date palm seeds-porous carbon@tio2 blend samsudeen olanrewaju azeeza,∗, akeem adebayo jimoha, ismaila olalekan saheeda, kabir opeyemi otuna, aliru olajide mustaphaa, folahan amoo adekolab adepartment of chemistry and industrial chemistry, faculty of pure and applied sciences, kwara state university, 241103, malete, nigeria bdepartment of industrial chemistry, faculty of physical sciences, university of ilorin, 240003, ilorin, nigeria abstract biological stains are potentially harmful compounds present in the environment, in which eosin yellow dye (eyd) is one of the most commonly applied stains. in this research, date palm seeds-porous carbon (dpsc) and its tio2 blend (tio2-dpsc) were prepared and their efficiency on the removal of eyd from an aqueous medium was investigated. characterization by sem, edx, ftir and bet surface area was performed on the materials. the bet surface area (542.63 m2/g) and pore diameter (2.02 nm) of tio2-dpsc were found to be higher than that of dpsc (332.74 m2/g and 1.85 nm) indicating that tio2-dpsc is mesoporous while dpsc is microporous. the major and interactive impacts of the adsorption parameters: initial eyd concentration, ph, adsorbent dose, and time of contact were examined by box behnken design in response surface methodology. the high r2 values 0.9658 and 0.9597 for dpsc and tio2-dpsc agreed with the adjusted r2 values suggesting the quadratic model sufficiently interprets the adsorption data. the optimum removal efficiency of eyd onto dpsc and tio2-dpsc was 34.63 mg/g and 55.34 mg/g which are in agreement with the predicted removal of 34.75 mg/g and 50.11 mg/g respectively at the center point values of co=300 mg/l, ph 2, 362.5 min and 0.1 g adsorbent dose. the results also showed the acceptability of the box behnken design in response surface methodology for the optimization of eyd removal from aqueous media using dpsc and tio2-dpsc blends. hence, better eyd removal reported in tio2-dpsc compared to dpsc was due to its improved adsorptive features. doi:10.46481/jnsps.2022.533 keywords: porous carbon, tio2, adsorption, eosin yellow dye, box behnken design article history : received: 21 december 2021 received in revised form: 17 february 2022 accepted for publication: 25 february 2022 published: 29 may 2022 c©2022 journal of the nigerian society of physical sciences. all rights reserved. communicated by: e. etim 1. introduction in recent years, industrial evolution has left an indelible mark on the environment. several industrial, domestic, and laboratory activities employ dyes for coloring [1,2]. and most of ∗corresponding author tel. no: email address: samsudeen.azeez@kwasu.edu.ng (samsudeen olanrewaju azeez ) these dyes are non-biodegradable, highly hazardous, toxic, and carcinogenic which poses a great risk to health [3,4]. also, the discharge of wastewaters containing dyes into the environment aids the pollution of rivers and greatly affects both human and aquatic lives negatively [1,3,5]. biological stains are organic dyes commonly applied for the determination of biological tissues. these stain substances have been reported as potentially carcinogenic and mutagenic com183 azeez et al. / j. nig. soc. phys. sci. 4 (2022) 183–192 184 figure 1. structure of eyd pounds, therefore their presence in the wastewaters being discharged into the environment is a threat to human and aquatic lives [2,4,6]. eosin yellow dye (eyd) is a reactive, anionic water-soluble acid dye commonly used as a stain. it is heterocyclic with the molecular formula of c20h6br4na2o5 and iupac name, 2-(2,4,5,6-tetrabromo-6-oxido-3-oxido-3h xanthenes9-yl) benzoate disodium salt as shown in fig. 1. eyd is pink in color and belongs to the fluorescein class of dye with a λmax of 517 nm. it is commonly used in gram staining [4,6]. exposure to eyd may result in severe eye and skin irritation and adversely affects vital organs such as kidney, liver, etc. it also reduces the pulmonary gas exchangeability of the lungs if inhaled [6,7]. the treatment of wastewater containing dyes before release into the environment becomes germane as a small quantity of dye in water even at part per billion level can be toxic [8]. and government policy requires that wastewater containing dye must be treated before discharge, consequently, a need to establish an effective technique that can efficiently remove these dyes from aqueous media has been the concern of the researcher. before now, various techniques have been used to treat and remove organic pollutants in wastewater [9] which include chemical precipitation, membrane filtration, catalytic, photocatalytic oxidation methods, ion exchange, electrochemical methods, adsorption technology, etc [1,4,7,10]. the adsorption technique is widely used now for wastewater treatment due to its efficiency, cheapness, ease to handle, and simplicity of its regeneration procedure [1,10]. the search for a low-cost and effective adsorbent prepared from agricultural waste to adsorb dyes and other organics from wastewater is of interest to researchers [7,11–14]. lately, in an attempt to fabricate an adsorbent with improved adsorptive characteristics for better removal of contaminants, research towards the development of composite materials that are more effective and of low cost to treat wastewater pollutants is increasing, viz; date palm seeds, goethite, and their composite was used to remove acid dye from wastewater [4], novel magnetic/activated charcoal/βcyclodextrin/alginate polymer nanocomposite was utilized in the removal of cationic [10], adsorption of organic dye by nanoporous composites of activated carbon-metal organic frameworks [8], activated carbon prepared from wild date stones was employed in the adsorption of acid dye [1], the adsorption of methyl orange by fegrafting sugar beet bagasse was investigated [12] and a new type of porous zn (ii) metal-organic gel was designed for effective adsorption of methyl orange [9]. tio2 is a non-toxic cheap and biocompatible semiconductor. it is the most commonly used photocatalyst, due to its high stability and potential in the decomposition of many organic pollutants from aqueous media [15,16]. loading tio2 onto activated carbon as support has been reported [17,18]. the date palm (phoenix dactylifera) is one of man’s earliest plants. it originated from north africa and the middle east and was brought to nigeria. it is a palm tree from the palmae ( arecaceae ) family that grows in the tropics [16]. experimentally, date palm seeds were known to contain about 55-65% carbohydrate, which makes it a suitable agricultural waste precursor for the preparation of high-grade activated carbon [19]. previously, in adsorption studies, the optimization of process variables was usually investigated individually while keeping the other parameters constant. however, this method does not give the simultaneous effect of all parameters [20]. this approach is time-wasting, requires a lot of experiments, and consumes large reagents and chemicals. these setbacks can be prevented by optimizing the independent variables concurrently employing a statistical experimental design like the box behnken design (bbd) under response surface methodology (rsm) [1,13]. therefore, this research is concerned with investigating the optimization by box behnken design for eyd removal from aqueous medium using date palm seeds-porous carbon and tio2composite. 2. experimental 2.1. reagents and chemicals all the reagents used in this investigation are of analytical grade and were used with no additional purification. these include eosin yellow dye (eyd), titanium(iv)oxide (tio2), phosphoric acid (h3po4), ethanol (c2h5oh), sodium hydroxide (naoh), and hydrochloric acid (hcl) which were purchased from sigma aldrich, usa. deionized water was used to prepare all the solutions. 2.2. sample collection and preparation of date palm seeds-porous carbon (dpsc) the date palm fruits were gotten from kaduna, nigeria. the seeds were removed from the fruits, washed thoroughly with deionized water, and sundried for 2 days. the seeds were then carbonized in the furnace at 500 ◦c for 120 min. after which it was activated with phosphoric acid. the activation involved 184 azeez et al. / j. nig. soc. phys. sci. 4 (2022) 183–192 185 immersing 100 g of the carbonized date palm seeds in 500 ml of 1 m h3po4 for 48 h. it was then filtered, washed to neutral ph with hot and cold deionized water, and oven-dried at 105 ◦c for 2 h. subsequently, it was reactivated for 2 h at a temperature of 300 ◦c then cooled, ground, and sieved with < 63 µm mesh. this product was then labeled date palm seeds-porous carbon (dpsc) [19,21]. 2.3. preparation of tio2-dpsc blend the titanium (iv) oxide-date palm seeds-porous carbon (tio2dpsc) was prepared by the impregnation method as follows: 10 g of dpsc was added to 100 ml of deionized water andtreatment in an autoclave at 120 ◦c for 2 h. the stirred for 1 h. 5 g of commercial nanoparticle tio2 was then added and stirred for another 2 h at 40 ◦c. subsequently, a 1:1 ratio of ethanol and water was added to the resulting tio2-dpsc mixture and then subjected to hydrothermal treatment in an autoclave at 120 ◦c for 2 h. the resultant mixture was centrifuged at 4000 rpm for 15 min and the blend solid particles obtained were oven-dried at 110 ◦c for 1 h. it was then calcined at 400 ◦c for 3 h. after cooling it was ground to fine particles and stored for further use [18,22,23]. 2.4. characterization of dpsc and tio2-dpsc the adsorbents were characterized using an ftir spectrophotometer (perkin-elmer spectrum gx, uk) to ascertain the functional groups present on the adsorbents. the surface morphology and elemental composition of the material were obtained with a scanning electron microscope coupled with energy dispersive spectroscopy (jeol jsm-6510lv, japan). the surface area of both adsorbents was determined by bet analysis via n2 adsorption-desorption at 77 k using micromeritics asap 2020 v3.02h model. 2.5. preparation of eyd solution a 1000 mg/l of eyd solution was prepared by dissolving 1 g of eyd in 1 l of deionized water and lower concentrations of eyd were then prepared by serial dilution [7]. 2.6. adsorption experiments the removal of eyd by dpsc and tio2-dpsc blends were determined in batch mode. the initial eyd concentration, ph, contact time, and adsorbent dosage were investigated. the adsorption study involves adding 0.1 g of the adsorbents (dpsc or tio2-dpsc blend) to 20 ml of different initial concentrations (50 to 300) mg/l of eyd solution in a 100 ml conical flask. the flasks containing the solutions were then placed in a temperature-controlled water bath shaker and were agitated for 120 min at 30±2 ◦c and 200 rpm. the ph of the solutions was controlled using 0.1 m naoh and 0.1 m hcl solutions. the samples were then centrifuged for 10 min at 4000 rpm and filtered. after which, the filtrate was analyzed for change in concentration of eyd using uv/visible spectrophotometer (microprocessor uv-vis double beam avi-2802) at a λmax table 1. process parameters and their coded levels for bbd in rsm range of coded values parameters units factors -1 0 +1 initial conc mg/l a 50 175 300 ph b 2 7 12 adsorbent dose g c 0.1 0.3 0.5 time min d 5 326.5 720 of 517 nm. the quantity of eyd removal was calculated with eq. (1) [1,4,24,25]: qe = (c0 − ce) v w (1) where, qe is the quantity of eyd removed, co is the initial eyd concentration (mg/l), ce is the equilibrium eyd concentration (mg/l), v is the volume of eyd solution (l) and w is the weight of adsorbent (g). 2.7. box behnken experimental design and statistical analysis optimization of the process parameters was achieved with box behnken design (bbd) under rsm. this was employed to determine the best interaction between the process parameters in the removal of eyd from aqueous media by dspc and tio2dpsc. the design was used to obtain sets of designed experiments by design expert 11.1.2.0 with four factors; initial concentration, ph, adsorbent dose, and time. to study the influence of operating variables on the quantity of eyd removal, the four variables: concentration (a), ph (b), adsorbent dose (c), and time (d) each at three levels were selected as presented in table 2. the number of experimental runs could as well be calculated using eq. (2): n = k2 + k + c p (2) where n is the number of experimental runs, k is the number of factors and cp is the replicate number of central points. consequently, a total of 29 experimental runs was gotten as a function of the four factors on a three-level design, that is (-1. 0, +1) as presented in table 1 [13,20,26]. the responses were presented as the quantity of eyd removed (mg/g) as in table 3. the experimental data were studied and fitted well with the quadratic polynomial equation model which describes the nature of the process as expressed by eq. (3): y = β0 + k∑ i=1 βi xi + k∑ i=1 βii x 2 i + k∑ 1≤i≤ j βi j xi x j + � (3) where, y is the response (quantity of eyd removed); xi = variables; k = number of variables; β0 = constant term; βi = coefficients of linear parameters, βi j = coefficient of the interaction parameters; βii =coefficient of the quadratic parameter; ε = residual associated to the experiments. 185 azeez et al. / j. nig. soc. phys. sci. 4 (2022) 183–192 186 table 2. characteristics of the adsorbents dpsc tio2-dpsc bet surface area (m2g−1) 332.744 542.634 average pore volume (cm3g−1) 0.188 0.271 average pore diameter (nm) 1.853 2.015 %c 70.70 26.71 %o 29.17 44.42 %ti 28.87 %ca 0.13 3. results and discussion 3.1. adsorbents characterization 3.1.1. bet surface area analysis and elemental composition the brunauer, emmett and teller (bet) analysis results of both dpsc and tio2-dpsc are given in table 2. the bet surface area of both adsorbents is large which are attributed to carbon materials used as a precursor and the preparation procedure utilized, but the surface area of tio2-dpsc is higher than dpsc. the enhancement of tio2-dpsc surface area is due to tio2 incorporated with the carbon. the result also showed that the pore volume and pore diameter of tio2-dpsc is greater than dpsc. according to iupac classification, it is perceived that tio2-dpsc is mesoporous and dpsc microporous [27]. this result suggests that both adsorbents are materials with a good potential for the removal of eyd. although, tio2-dpsc would be a better adsorbent for removal eyd than dpsc owing to its higher surface area and mesoporous surface [21,28]. the edx results give the elemental composition of the adsorbents. the major constituents of dpsc are carbon and oxygen with calcium as a minor element while tio2-dpsc exhibited a high percentage of titanium with lower percentages of carbon and oxygen as compared to dpsc (table 2 & fig. 2.). the relatively high weight percentage of ti in tio2-dpsc indicates successful incorporation of tio2 into the date palm seedsporous carbon. 3.1.2. scanning electron microscopy (sem) the sem micrograph (fig. 3a) of dpsa showed an irregular coarse surface morphology with many micropores which exhibited a high probability for eyd to be adsorbed. also, the sem image of tio2-dpsa (fig. 3b) revealed an improvement on the surface of the adsorbent due to agglomeration of tio2 particles on the surface of dsac indicating its porous nature and supported by the increase in the bet surface area and pore diameter of tio2-dpsa. 3.1.3. ftir analysis of the adsorbents the ftir spectra of dpsc and tio2-dpsc are given in fig. 4a. the absorption bands at ∼ 3400 cm−1 observed for both adsorbents are characteristics of o-h stretching vibrations of alcohol [5]. the band at ∼ 1600 cm−1 shows the presence of the c=c aromatic functional group, while the peak at ∼ 1233 cm−1 is related to the c-oh band. a band at 495.88 cm−1 on the tio2-dpsc spectrum is attributed to ti-o-ti bending vibrations that do not exist on dpsc. all these functional groups figure 2. edx spectra of (a) dpsc, (b)tio2-dpsc figure 3. sem micrograph of (a) dpsa and (b) tio2-dpsa figure 4. ftir spectra of dspc and tio2-dpsc (a) before adsorption (b) after adsorption will take part in the adsorption process [1,4,19]. following adsorption of eyd (fig. 4b.), there were differences in intensities of the absorption peak indicating the involvement of the functional groups in the adsorption process as a similar observation was reported by [21]. 186 azeez et al. / j. nig. soc. phys. sci. 4 (2022) 183–192 187 3.2. optimization of eyd onto dpsc and dpsc-tio2 by bbd the optimum conditions for eyd removal onto dpsc and dpsctio2 were established by bbd under rsm. the matrix of experimental design of the four process variables each at three levels with their equivalent actual and predicted responses (quantity adsorbed in mg/g) are given in table 3. design expert 11.1.2.0 was used to obtain the second-order polynomial equations that describe the adsorption responses (quantity adsorbed) with the four operating variables where the insignificant terms (p?0.05) are not included as given by eq. (4) and eq. (5) for dpsc and dpsc-tio2 respectively: ydps c = +2.90 + 8.38a − 5.02b− 9.41c + 1.84d − 9.21ac − 5.43bd + 4.34b2 + 5.48c2 + 7.77d2 (4) yt io2−dps c = +9.82 + 10.14a − 4.49b − 14.13c + 1.63d − 4.15ab − 8.64ac − 4.79bd + 7.90c2 (5) equations (4) and (5) depict how the interactive model terms impacted the removal of eyd from an aqueous solution by both adsorbents. the synergistic effect of the parameters is indicated by a positive sign while a negative sign represents an antagonistic effect [1,29]. 3.2.1. anova results the adequacy of the proposed model was analyzed using the anova. the results obtained are shown in table 4. high f-values of the models 28.20 and 23.74 for dpsc and tio2dpsc respectively with their p-value less than 0.05 suggested that the quadratic models are significant. the high correlation coefficient (r2) values of 0.965 and 0.960 with adjusted r2 values of 0.932 and 0.919 for dpsc and tio2-dpsc respectively agreed with their corresponding predicted r2 values of 0.806 and 0.785 implying that the model is adequate to describe the adsorption process. also, the low values of percentage coefficient of variation (%cv) 29.35% and 26.80% for dpsc and tio2-dpsc respectively signify good reliability and high precision of the experimental values [13]. it is obvious in table 4 that the linear terms a, b, c, and d, the interaction term ac and bd, and quadratic terms b2, c2, and d2 are statistically significant (p< 0.05) for eyd onto dspc while for eyd on tio2dpsc, the linear term d is not significant (p>0.05), the interaction terms ab, ac and bd, and quadratic terms c2 are significant. this infers that the number of experiments performed is acceptable to explain the effects of the process parameters on the amount of eyd removal from aqueous solution by both adsorbents [30–32]. furthermore, table 4 shows that the lack of fit (f-value = 19.25) with p-value < 0.05 indicates the lack of fit is significant for eyd removal onto dpsc while the lack of fit (f-value = 3.42) and p-value > 0.05 for eyd adsorbed by tio2dpsc is not significant compared to the pure error, suggesting that the mathematical model proposed was well explained the graph of actual versus predicted values of eyd removal onto both adsorbents is given in fig. 5. it can be seen that the figure 5. the plot of actual versus predicted values for eyd removal by (a) dpsc, (b)tio2-dpsc points on the plots were well grouped on the straight line; signifying a good agreement between the actual and predicted values of the responses for both adsorbents [29]. this indicates that the proposed statistical model is satisfactory for the optimization of eyd removal by both dpsc and tio2-dpsc [33,34]. dpsc: mean = 9.92; c.v.% = 29.35; r2 = 0.9658; adjusted. r2 = 0.9315; predicted. r2 = 0.8057; adeq. precision = 19.2583 tio2-dpsc: mean = 14.19; c.v.% = 26.80; r2 = 0.9597; adjusted. r2 = 0.9192; predicted. r2 = 0.7850; adeq. precision = 18.3713 3.3. simultaneous effects of process parameters on the removal of eyd the interactions between the process parameters on the quantity of eyd removal from aqueous solution by dpsc and tio2dpsc are illustrated by 3d response surface plots (figs. 6 – 11). these involve the simultaneous effects of any two of the process variables while keeping the other factors at the central level. 3.3.1. simultaneous effect of initial eyd concentration and ph the simultaneous effect of the initial eyd concentration and ph on the amount of eyd removal by dpsc and tio2-dpsc is given in fig. 6. it can be seen (fig. 6.) that increase in initial eyd concentration at low ph increases the amount of eyd removal per gram for both adsorbents. this dual interaction is not significant for eyd on dpsc but significant for eyd onto tio2-dpsc. the increase in the quantity of eyd adsorbed as the adsorbate concentration increases could be as a result of more eyd ions in solution as its concentration increases. at low ph, the removal of eyd was at its optimum as a result of more availability of h+ ions in the solution. this may be due to electrostatic attraction between the anionic species of eyd in solution and the protonated surface of the adsorbents. while, at higher ph, the adsorption of eyd by the two adsorbents decreased, because more oh− ions exist in the solution and electrostatic repulsion results between the molecules of eyd in solution and the negatively charged surface of the adsorbents. this trend corresponds to the observations reported by mittal et al. [6], bello et al. [21] and abdus-salam et al. [4]. 187 azeez et al. / j. nig. soc. phys. sci. 4 (2022) 183–192 188 table 3. box behnken experimental design for eyd removal onto dpsc & tio2-dpsc experimental run operating variables responses (mg/g) dpsa responses (mg/g) tio2-dpsa initial conc (mg/l) ph adsorbent dose (g) time (min) actual quantity adsorbed predicted quantity adsorbed actual quantity adsorbed predicted quantity adsorbed 1 50 2 0.3 362.5 2.15 2.01 0.89 0.54 2 175 7 0.5 720 3.86 6.64 3.86 6.21 3 175 12 0.3 720 8.43 6.40 8.43 5.35 4 175 7 0.1 5 24.82 21.77 31.91 31.21 5 175 7 0.1 720 30.4 29.35 34.4 36.78 6 175 12 0.3 5 9.79 13.59 9.15 11.67 7 50 7 0.3 720 1.39 3.17 1.39 2.09 8 175 2 0.3 5 10.64 12.77 11.58 11.08 9 175 7 0.5 5 6.09 6.87 6.00 5.28 10 175 7 0.3 362.5 2.8 2.90 8.50 9.82 11 175 7 0.3 362.5 3.05 2.90 6.54 9.82 12 300 12 0.3 362.5 8.86 8.73 9.82 11.83 13 175 7 0.3 362.5 2.95 2.90 12.03 9.82 14 175 2 0.5 362.5 6.92 6.22 6.87 8.61 15 300 7 0.3 5 17.87 16.26 17.89 19.11 16 50 12 0.3 362.5 -2.39 -5.57 2.12 -0.15 17 175 12 0.5 362.5 -0.26 0.41 -0.02 0.62 18 50 7 0.1 362.5 -4.05 -0.45 10.46 12.55 19 50 7 0.3 5 2.18 0.13 2.39 0.56 20 50 7 0.5 362.5 -0.81 -0.83 -0.08 1.58 21 175 12 0.1 362.5 14.12 14.99 27.70 27.88 22 175 2 0.1 362.5 29.77 29.27 36.58 37.86 23 300 7 0.3 720 18.35 20.57 20.35 24.09 24 300 2 0.3 362.5 18.33 21.24 25.18 29.11 25 175 7 0.3 362.5 3.95 2.90 11.71 9.82 26 300 2 0.1 362.5 34.63 34.75 55.34 50.11 27 300 7 0.5 362.5 1.01 -2.50 10.24 4.57 28 175 2 0.3 720 31.01 27.31 30.01 23.91 29 175 7 0.3 362.5 1.77 2.90 10.31 9.82 figure 6. 3d response surface plots for the simultaneous effect of initial eyd concentration and ph at constant 0.1 g dosage and 362.5 min contact time on the quantity of eyd adsorbed onto (a) dpsc, (b) tio2-dpsc 3.3.2. simultaneous effect of initial eyd concentration and adsorbent dose the 3d response surface plots presented in fig. 7 depict the simultaneous effects of initial eyd concentration and adsorbent dose on the quantity of eyd removal per gram of the adsorbents (dpsc and tio2-dpsc) at a fixed time of 362.5 min and ph 2. these binary interaction terms are significant in achieving the maximum amount of eyd removal by both adsorbents. the increase in initial eyd concentration greatly impacts the removal of eyd onto dpsc and tio2-dpsc. the uptake of eyd was at the highest at a low adsorbent dose. the vigorous adsorption of eyd observed at a lower dose could be due to many vacant adsorption sites on the adsorbents’ surfaces. however, at a higher dosage, the free adsorption sites reduce because of overlapping of the active adsorption sites [4,11,31] 3.3.3. simultaneous effect of initial eyd concentration and time fig. 8 represents the effect of combined interaction between initial eyd concentration and time of contact on the 3d response surface plots for eyd removal from an aqueous solution at constant ph and adsorbent dose. as seen from the plots, the removal of eyd increases immensely with increasing initial 188 azeez et al. / j. nig. soc. phys. sci. 4 (2022) 183–192 189 table 4. anova analysis using bbd under rsm for eyd removal by dpsc & tio2dpsc source sum of squares df mean square f-value p-value remarks %c dpsc tio2dpsc dpsc tio2dpsc dpsc tio2dpsc dpsc tio2dpsc dpsc tio2dpsc dpsc tio2dpsc model 3345.55 4809.09 14 238.97 343.51 28.2 23.74 < 0.0001 < 0.0001 significant significant aconc 843.03 1233.23 1 843.03 1233.23 99.5 85.24 < 0.0001 < 0.0001 significant significant 25.20 25.64 b-ph 302.71 242.19 1 302.71 242.19 35.73 16.74 < 0.0001 0.0011 significant significant 9.05 5.04 cdosage 1061.82 2394.75 1 1061.82 2394.75 125.32 165.53 < 0.0001 < 0.0001 significant significant 31.74 49.80 dtime 40.52 31.75 1 40.52 31.75 4.78 2.19 0.0462 0.1606 significant 1.21 0.66 ab 6.08 68.81 1 6.08 68.81 0.7171 4.76 0.4113 0.0467 significant 0.18 1.43 ac 339.66 298.6 1 339.66 298.6 40.09 20.64 < 0.0001 0.0005 significant significant 10.15 6.21 ad 0.4032 2.99 1 0.4032 2.99 0.0476 0.2069 0.8305 0.6562 0.01 0.06 bc 17.94 0.99 1 17.94 0.99 2.12 0.0684 0.1677 0.7974 0.54 0.02 bd 118.05 91.68 1 118.05 91.68 13.93 6.34 0.0022 0.0246 significant significant 3.53 1.91 cd 15.25 5.36 1 15.25 5.36 1.8 0.3704 0.2011 0.5525 0.46 0.11 a2 2.67 1.71 1 2.67 1.71 0.3151 0.1179 0.5834 0.7365 0.08 0.04 b2 122.16 6.84 1 122.16 6.84 14.42 0.4731 0.002 0.5028 significant 3.65 0.14 c2 194.86 404.41 1 194.86 404.41 23 27.95 0.0003 0.0001 significant significant 5.82 8.41 d2 391.83 30.22 1 391.83 30.22 46.24 2.09 < 0.0001 0.1704 significant 11.71 0.63 residual 118.62 202.55 14 8.47 14.47 lack of fit 116.21 181.35 10 11.62 18.13 19.25 3.42 0.0059 0.1235 significant not significant pure error 2.41 21.2 4 0.6036 5.3 cor total 3464.17 5011.64 28 std. dev. 2.91 3.80 figure 7. 3d response surface plots for the simultaneous effect of initial eyd concentration and adsorbent dose at affixed ph 2 and 362.5 min contact time on the quantity of eyd adsorbed onto (a) dpsc, (b) tio2-dpsc eyd concentration. this may be due to the low ph of the solution and increase in the initial eyd concentration. however, an increase in the time of contact between the adsorbents and initial eyd concentration in solution has a small influence on the amount of eyd adsorbed at the start of the adsorption process. this low initial eyd removal result may be from several vacant active sites originally available for adsorption. therefore, eyd molecules quickly filled the active sites. as adsorption advances, the repulsive forces between the eyd molecules on the adsorbents surfaces and in the solution becomes larger needing more time to fill the remaining vacant active sites on the adsorbents surfaces [4,21,31]. this is why there were no considerable differences in the amount of eyd removed over the time investigated. 3.3.4. simultaneous effect of ph and adsorbent dose the binary effect of interaction between ph and adsorbent dose on the response surface for the removal of eyd from solution at constant initial eyd concentration and time of contact is shown in fig. 9. the optimum removal of eyd per gram was observed at low ph and low adsorbent doses indicating a synergistic effect between the parameters. the quantity of eyd removal was 189 azeez et al. / j. nig. soc. phys. sci. 4 (2022) 183–192 190 figure 8. 3d response surface plots for the simultaneous effect of initial eyd concentration and time of contact at constant 0.1 g dosage and ph 2 on the quantity of eyd adsorbed onto (a) dpsc, (b) tio2-dpsc figure 9. 3d response surface plots for the simultaneous effect of ph and adsorbent dosage at affixed 300 mg/l initial eyd concentration and 362.5 min contact time on the quantity of eyd adsorbed onto (a) dpsc, (b) tio2-dpsc 18.33 mg/g and 25.18 mg/g for dpsc and tio2-dpsc respectively. 3.3.5. simultaneous effect of time and ph fig.10 illustrates the simultaneous effect of ph and time of contact on the response of eyd removal from aqueous solution at constant initial eyd concentration and adsorbents dose. the binary effect was highly significant for both adsorbents analyzed (table 4). these dual interaction terms were used in establishing the optimum conditions for this study. according to this figure, the optimum (actual) removal of eyd was found to be 34.63 mg/g and 55.34 mg/g and predicted removal 34.75 mg/g and 50.11 mg/g for dpsc and tio2-dpsc respectively at the center point values of 300 mg/l initial eyd concentration, ph 2, 362.5 min time of contact and 0.1 g adsorbent dose for the two adsorbents. an additional increase in these two parameters (ph and time) resulted in a decrease in the quantity of eyd adsorbed [31,34,35]. 3.3.6. simultaneous effect of time and adsorbent dose the simultaneous effect between the time of contact and adsorbent dose on the response surface for the amount of eyd uptake per gram by dpsc and dpsc-tio2 at affixed ph and initial eyd concentration is given in fig. 11. these binary interaction terms have no significant impact on the quantity of eyd removed by both adsorbents (table 4). it is seen that the quantity of eyd removal was at the highest at a low dose of 0.1 g for both adsorbents and very rapid at the start of the adsorption process. however, an increase in time of contact and figure 10. 3d response surface plots for the simultaneous effect of ph and contact time at constant 300 mg/l initial eyd concentration and 0.1 g dosage on the quantity of eyd adsorbed onto (a) dpsc, (b) tio2-dpsc figure 11. 3d response surface plots for the simultaneous effect of contact time and adsorbent dose at affixed 300 mg/l initial eyd concentration and ph 2 on the quantity of eyd adsorbed onto (a) dpsc, (b) tio2-dpsc table 5. comparison of adsorption efficiencies of dpsc and tio2-dpsc with other adsorbents for eyd removal adsorbents adsorption efficiency (mg/g) reference pineapple peels 11.76 [7] oil bean acid activated carbon 26.32 [36] teak leaf litter powder 31.64 [37] polyaniline saw dust 5.90 [38] goethite 27.78 [4] composite of goethite and thermally activated carbon 30.30 [4] chemically activated carbon 3.13 [4] thermally activated char-coal 1.98 [4] dpsc 34.63 this investigation tio2-dpsc 55.34 this investigation adsorbent dose does not influence the quantity of eyd removed as the adsorption progresses. this is due to the availability of various vacant sites ready for adsorption in the early stage of the process. thus, an increase in adsorbent doses led to a decrease in the quantity of eyd removed while an increase in time do not have a meaningful effect on the removal. 3.4. interactive effects of the process variables generally, the perturbation graph is utilized to examine the interactive influence of all factors concurrently. the response is designed by changing one of the variables while other parame190 azeez et al. / j. nig. soc. phys. sci. 4 (2022) 183–192 191 figure 12. perturbation graph for eyd removal onto (a) dpsc, (b) tio2dpsc ters are kept constant at the center of the plot. therefore, a perturbation plot is employed to determine the factor that greatly impacts the adsorption process. fig. 12a exhibited that initial eyd concentration, ph, adsorbent dose, and time have a meaningful influence on the quantity of eyd removal per gram by dpsc due to their relatively straight line. however, the amount of eyd removal onto tio2-dpsc was largely influenced by three of the parameters except time due to its curvature (fig. 12b) and this revealed that eyd removal is less sensitive to this parameter. this is following the report of rahman &nasir [35]. 3.5. contribution of process parameters on the response surface plots the contribution (c) of each parameter in percentage on the response model was studied by eq. 6 [35,39]: %c = s s i s s m × 100 (6) where ssm is the sum of squares for the model and ssi is the sum of squares for the individual parameter. the results are presented in table 4. the results indicated that the adsorbent dose, initial eyd concentration, and ph of solution immensely contributed to eyd removal by both adsorbents with %c values of 31.74, 25.20, and 9.05 for dpsc and 49.80, 25.64 5.04 for tio2-dpsc respectively. the percentage contribution of a time of contact is the lowest among the individual parameters for both adsorbents in the response model. the interactive term ac gives a %c value of 10.15 and 6.21 for dpsc and tio2-dpsc respectively while the other dual terms have lesser contribution in eyd removal from aqueous solution. 4. conclusion in this research, dpsc and tio2-dpsc blends were successfully prepared for the removal of eyd from an aqueous medium. the surface area and pore diameter of tio2-dpsc is higher than dpsc. the major and interactive impacts of the adsorption parameters which include initial eyd concentration, ph, adsorbent dose, and time of contact were examined by box behnken design based on response surface methodology. the high r2 values of the models are in agreement with the adjusted r2 values. the second-order regression model sufficiently interprets the adsorption data. the adsorbent dose delivered the largest percentage contribution of 49.80% and 31.74% for tio2-dpsc and dpsc respectively to the response parameters while the dual interaction terms of initial eyd concentration and adsorbent dose had the highest percentage contribution of 10.15% and 6.21% for dpsc and tio2-dpsc respectively to eyd removal per gram compared to other interaction terms. the optimum amount of eyd removal onto dpsc and tio2-dpsc were discovered to be 34.63 mg/g and 55.34 mg/g respectively. the results also showed the reliability of the box behnken design in response surface methodology for the optimization of eyd removal from aqueous media. based on the observed trends, it can be concluded that tio2-dpsc is a better adsorbent material in the removal of eyd from an aqueous solution than dpsc. acknowledgments the authors are highly appreciative of chemistry and industrial chemistry department, kwara state university, malete, nigeria for giving 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[39] f. bandari, f. safa & s. shariati, “application of response surface method for optimization of adsorptive removal of eriochrome black t using magnetic multi-wall carbon nanotube nanocomposite”, arab j sci eng. 40 (2015) 3363, https://doi.org/10.1007/s13369-015-1785-8. 192 j. nig. soc. phys. sci. 4 (2022) 884 journal of the nigerian society of physical sciences masses and thermal properties of a charmonium and bottomonium mesons e. p. inyanga,∗, e. o. obisungc, p. c. iwujic, j. e. ntibib, j. amajamac, e. s. williamb a department of physics, national open university of nigeria, jabi, abuja, nigeria b theoretical physics group, department of physics, university of calabar, pmb 1115, calabar, nigeria c department of physics, university of calabar, pmb 1115, calabar, nigeria abstract in this research, we model hulthén plus generalized inverse quadratic yukawa potential to interact in a quark-antiquark system. the solutions of the schrödinger equation are obtained using the nikiforov-uvarov method. the energy spectrum and normalized wave function were obtained. the masses of the heavy mesons for different quantum states such as 1s, 2s , 1p, 2p 3s, 4s, 1d, and 2d were predicted as 3.096 gev, 3.686 gev, 3.327 gev, 3.774gev, 4.040 gev, 4.364gev, 3.761 gev, and 4.058 gev respectively for charmonium (cc̄). also, for bottomonium (bb̄) we obtained 9.460 gev, 10.023 gev, 9.841 gev, 10.160 gev, 10.345 gev, 10.522 gev, and 10.142gev for different states of 1s , 2s , 1p , 2p , 3s , 4s , 1d respectively. the partition function was calculated from the energy spectrum, thereafter other thermal properties were obtained. the results obtained showed an improvement when compared with the work of other researchers and excellently agreed with experimental data with a percentage error of 1.60 % and 0.46 % for (cc̄) and (bb̄), respectively. doi:10.46481/jnsps.2022.884 keywords: schrödinger equation; nikiforov-uvarov method; heavy mesons; thermal properties; hulthén plus generalized inverse quadratic yukawa potential. article history : received: 23 june 2022 received in revised form: 9 july 2022 accepted for publication: 11 july 2022 published: 19 august 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: c. edet, p. amadi & r. khordad 1. introduction in non-relativistic quantum mechanics, researchers work mostly on the solutions of the schrödinger equation (se) to obtain the eigenvalues and eigenfunction. these eigenvalues and eigenfunction can be used to study the mass spectra (ms) of the heavy mesons, and theoretic measures, among other things [1, 2, 3, 4]. in describing the interaction of charmonium ∗corresponding author tel. no: +2348032738073 email address: etidophysics@gmail.com, einyang@noun.edu.ng (e. p. inyang ) (cc̄) and bottomonium (bb̄), a cornell potential (cp) is used because of the coulomb interaction and a confining term [5]. more so, in solving the se with any potential of choice, analytical methods are employed. most of the frequently used analytical methods are as follows, the nikiforov-uvarov (nu) method [6-12], the nikiforov-uvarov functional analysis (nufa) method [13-16], the series expansion method (sem) [17, 18, 19], laplace transformation method (ltm) [20], asymptoptic iteration method [21], wkb approximation method [22-24], among others [25-31]. the investigation of the ms with cp has gained remarkable interest, and has 1 inyang et al. / j. nig. soc. phys. sci. 4 (2022) 884 2 attracted the attention of many scholars [32-35]. for instance, mutuk [36] solved the se with cp using a neural network approach. the bottomonium, charmonium, and bottom-charmed spin-averaged spectra were calculated. using, the vibrational method (vm) and supersymmetric quantum mechanics vega and flores, [37] obtained the analytical solutions of the se with cp. the eigenvalues were used to calculate the ms of the hms. also, kumar et al., [38] used the nufa method to solve the se with generalized cornell potential. the result was used to predict the ms of the hms. furthermore, hassanabadi et al. [39], used the vm to solve the se with cp. the eigenvalues were used to calculate the mesonic wave function. in recent times, the study of ms of the hms with exponential-type potentials has aroused the interest of scholars [40, 41]. potential models such as yukawa potential [42], varshni [43], screened kratzer potential [44], and so on have been used in the prediction of the ms of the hms. for instance, purohit et al [45] combined linear plus modified yukawa potential to obtain the masses of the hms through the solutions of the klein-gordon equation. the se for most of the potentials with spin addition cannot be solved analytically; hence, numerical solutions such as runge-kutte approximation [46], numerov matrix method [47], fourier grid hamiltonian method [48], and so on [49] are employed. also, adding spin enables one to determine other properties of the mesons like decay properties and root mean square radii. however, we have considered our mesons as spinless particles for easiness [1, 40, 50, 51, 52]. recently, researchers have studied thermal properties (tps) of the hms [40, 53]. for instance, abu-shady et al., [54] studied the tps of hms with extended cornell potential. also inyang et al., [55] obtained the tps of the hms with exponential potential. abu-shady and ezz-alarab [40], solved the n-radial se with the trigonometric rosen-morse potential, and the obtained energy equation was used in studying tps and masses of heavy and heavy-light mesons. cao et al., [56], studied the tps of light mesons, including the temperature dependence of their masses by solving the spatial correlation. in recent times scholars, have proposed new potentials through the combination of two potentials [57, 58]. it’s observed that the combination of at least two potential models has a propensity to fit experimental data more than a single potential [55]. okoi et al.,[59] proposed the hulthén plus generalized inverse quadratic yukawa potential (hgiqyp) to obtain the bound state energy and expectation values. hulthén potential [60] and generalized inverse quadratic yukawa potential [61], is applied to diverse areas of physics such as nuclear and particle and so on. the aim of this study is in three folds. first, the hgiqyp will be modeled to fit in the cp, secondly, the se will be solved using the well-known nu method, and finally to predict the masses of charmonium and bottomonium mesons and calculate its tps. the hgiqyp takes the form [60]: v (q) = − r0eθq 1 − eθq − r1 ( 1 + eθq q )2 (1) where r0 and r1 are the strength of the potential, θ is the screening parameter and q is inter-nuclear distance. we model equation (1) to interact in the quark-antiquark system by expanding the exponential terms in a series of powers up to order three and have, v (q) = − g0 q + g1q + g2q 2 + g3 q2 + g4 (2) where g0 = r0 σ + 2r1 − 2r1α, g1 = − αr0 6 − r1α 2 (3) g2 = r1α3 3 , g3 = −r1, g4 = r0 2 − r1 + 2αr1 − 2r1α 2 2. the solutions of the schrödinger equation with hgiqyp in this research, the nu method is employed. the details are found in [62]. the se is given as [63] d2u(q) dq2 + [ 2µ ~2 (en` − v (q)) − `(` + 1) q2 ] u(q) = 0 (4) where ` is the angular momentum quantum number, µ is the reduced mass for the quark-antiquark particle, q is the interparticle distance, and ~ is reduced plank constant. then, we substitute equation (2) into equation (4), and got d2u(q) dq2 + [ 2µen` ~2 + 2µg0 ~2q − 2µg1q ~2 − 2µg2q2 ~2 − 2µg3 ~2q2 (5) − 2µg4 ~2 − `(` + 1) q2 ] u(q) = 0 transformation of q (in equation (8) to w coordinates yields equation (6), w = 1 q , q > 0 (6) the second derivative of equation (6) is given as d2u(q) dq2 = 2w3 du(w) dw + w4 d2u(w) dw2 (7) replacing equations (6) and (7) in equation (8) gives; d2u(w) dw2 + 2w w2 du dw + 1 w4 [ 2µenl ~2 + 2µg0w ~2 − 2µg1 ~2w (8) − 2µg2 ~2w2 − 2µg3w2 ~2 − 2µg4 ~2 − l (l + 1) w2 ] u(w) = 0 the approximation scheme (as) is introduced by assuming that there is a characteristic radius r0 of the meson. the as is achieved by the expansion of g1w and g1 2 w2 in a power series 2 inyang et al. / j. nig. soc. phys. sci. 4 (2022) 884 3 around r0 ; i.e. around φ ≡ 1 r0 , up to the second-order [64]. by setting y = w − φ and around y = 0 we expand it in powers of series as: g1 w = g1 y + φ = g1 φ ( 1 + y φ ) = g1 φ ( 1 + y φ )−1 (9) equation (9) yields g1 w = g1 ( 3 φ − 3z φ2 + z2 φ3 ) (10) similarly, g2 w2 = g2 ( 6 φ2 − 8w φ3 + 3w2 φ4 ) (11) placing equations (10) and (11) into equation (13), we obtain: d2u(w) dw2 + 2w w2 du(w) dw + 1 w4 [ −ε + x0w − x1w 2 ] u(w) = 0 (12) where −ε = ( 2µenl ~2 − 6µg1 ~2φ − 12µg2 ~2φ2 − 2µg4 ~2 ) , (13) x0 = ( 2µg0 ~2 + 6µg1 ~2φ2 + 16µg2 ~2φ3 ) x1 = ( 2µg1 ~2φ3 + 6µg2 ~2φ4 + 2µg3 ~2 + γ ) , γ = l(l + 1) connecting equation (12) and equation (1) of [62], we obtain τ̃(w) = 2w, σ(w) = z2 (14) σ̃(w) = −ε + αw −βw2 σ′(w) = 2w, σ′′(w) = 2 plugging equation (23) into equation (8) of [56], gives π(w) = ± √ ε− x0w + (x1 + k) w2 (15) to determine k, in equation (15), the discriminant of the function equation (16) and equation (17) are obtained as k = x02 − 4x1ε 4ε (16) π(w) = ± ( x0w 2 √ ε − ε √ ε ) (17) for a physically acceptable solution, the negative part of equation (17) is required, upon differentiating we get. π′−(w) = − x0 2 √ ε (18) placing equation (23) and equation (17) into equation (6) of ref. [62] gives τ(w) = 2w − x0w √ ε + 2ε √ ε (19) equation (19) gives τ′(w) = 2 − x0 √ ε (20) using equations (19) and equation (21) of ref. [62], we have the following, λ = x02 − 4x1ε 4ε − x0 2 √ ε (21) λn = nx0 √ ε − n2 − n (22) equating equations (21) and (21), followed by the substitution of equations (5) and (14) yielded the energy spectrum of the hgiqyp enl = r0 2 − r1 + 2ϑr1 − 2r1ϑ 2 − ϑr0 2φ − 3r1ϑ2 φ + 2r1ϑ3 φ2 (23) − ~2 8µ  6µ ~2φ2 ( − ϑr0 6 − r1ϑ 2 ) + 2µ ~2 ( r0 ϑ + 2r1 − 2r1ϑ ) + 16µr1ϑ3 3~2φ3 n + 12 + √( 1 2 + l )2 + 2µ ~2φ3 ( − ϑr0 6 − r1ϑ 2 ) + 2µr1ϑ3 ~2φ4 − 2µr1 ~2  2 the wave function, is obtained by putting equations (23) and (17) into equation (4) of ref. [62] dφ φ = ( ε w2 √ ε − x0 2w √ ε ) dw (24) integration of equation (24) gives φ(w) = w− x0 2 √ ε e− ε w √ ε (25) putting equations (23) and (19) into equation (26) of ref. [62] we have ρ(w) = w− x0√ ε e− 2ε w √ ε (26) the substitution of equations (23) and (26) into equation (5) of ref.[62] gives yn(w) = bne 2ε w √ ε z x0√ ε dn dwn [ e− 2ε w √ ε wn− x0√ ε ] (27) the rodrigues’ formula of the associated laguerre polynomials is l x0√ ε n ( 2ε w √ ε ) = 1 n! e 2ε w √ ε w x0√ ε dn dwn ( e− 2ε w √ z wn− x0√ ε ) (28) 3 inyang et al. / j. nig. soc. phys. sci. 4 (2022) 884 4 where bn = 1 n! . hence, yn(w) ≡ l x0√ ε n ( 2ε w √ ε ) (29) the substitution of equations (25) and (29) into equation (2) of ref. [62], gives the wave function in terms of associated laguerre polynomials as ψ(w) = nnlw − x0 2 √ ε e− ε w √ ε l x0√ ε n ( 2ε w √ ε ) (30) where nnl is normalization constant, which can be obtained from ∞∫ 0 |ψnl(r)| 2dr = 1 (31) inserting equation (30) into (31) with w = 1/r gives n2nl ∞∫ 0 rx0/ √ ε e−2 √ εr [ lx0/ √ ε n ( 2 √ εr )]2 dr = 1 (32) by using the transformation x = 2 √ εr we obtained the well-known standard integral of the laguerre polynomials n2nl( 2 √ ε )x0/√ε+1 ∞∫ 0 xx0/ √ ε e−x [ lx0/ √ ε n (x) ]2 d x = 1 (33) the solution of the standard integral [65] is given as ∞∫ 0 xx0/ √ ε e−x [ lx0/ √ ε n (x) ]2 d x = γ ( n + x0/ √ ε + 1 ) γ (n + 1) (34) comparing equations (33) and (34) we obtained the normalization factor such that the total wave function of the mesons can be written in closed form as ψnl (r) = √√√√√ ( 2 √ ε )x0/√ε+1 γ (n + 1) γ ( n + x0/ √ ε+ 1 ) (35) × rx0/2 √ ε e− √ εr lx0/ √ ε n ( 2 √ εr ) 3. thermal properties of the se with hgiqyp to obtain the tps of the heavy mesons, we first calculate the partition function. equation 26 is written in the form enl = q1 − ~2 8µ [ q2 (n + θ) ]2 (36) where θ = 1 2 + (37)√( 1 2 + l )2 + 2µ ~2φ3 ( − ϑr0 6 − r1ϑ2 ) + 2µr1ϑ3 ~2φ4 − 2µr1 ~2 q1 = r0 2 −r1 +2ϑr1−2r1ϑ 2 − ϑr0 2φ − 3r1ϑ2 φ + 2r1ϑ3 φ2 (38) q2 = 6µ ~2φ2 ( − ϑr0 6 − r1ϑ 2 ) (39) + 2µ ~2 ( r0 ϑ + 2r1 − 2r1ϑ ) + 16µr1ϑ3 3~2φ3 3.1. partition function z(β) the partition function (pf) takes the form [54]: z(β) = λ∑ n=0 e−βenl (40) where, β = 1k0 t , k0 is the boltzmann constant, t is the absolute temperature, and λ is the maximum quantum number. swapping equation (36) into equation (40) we obtain z(β) = λ∑ n=0 e −β ( q1− ~2 8µ [ q2 (n+θ) ]2) (41) in the classical limit, the summation is replaced by an integral, z(β) = λ+θ∫ θ e m1β+ n1β ρ2 dρ (42) where n + θ = ρ m1 = −q1 n1 = ~2 q22 8µ (43) from equation (42), the pf is obtain as, z(β) = em1β  ρe n1β ρ2 − n1β √ πer f i ( √ n1β ρ ) √ n1β  , (44) θ ≤ ρ ≤ λ + θ other tps can be obtained as follows: 3.2. mean energy u(β) u(β) = − ∂ ∂β inz(β), (45) 3.3. free energy f(β) f(β) = −k0t ln z(β) (46) 3.4. entropy s (β) s (β) = k0 ln z(β) − k0β ∂ ∂β ln z(β) (47) 3.5. specific heat capacity c(β) c(β) = ∂u ∂t = −k0β 2 ∂u ∂β (48) 4 inyang et al. / j. nig. soc. phys. sci. 4 (2022) 884 5 figure 1: wave function of charmonium against the radius at different s state figure 2: probability density function of charmonium against the radius at different s state figure 3: wave function of bottomonium against the radius at different s state 4. results and discussion the prediction of the ms of the hms is carried out using the relation [66, 67] m = 2m + enl (49) figure 4: probability density function of bottomonium against the radius at different s state figure 5: partition function against β for different values of λ where m is quark-antiquark mass and enl is energy eigenvalues. plugging equation (26) into equation (49) gives: m = 2m + r0 2 − r1 + 2ϑr1 − 2r1ϑ 2 − ϑr0 2φ − 3r1ϑ2 φ + 2r1ϑ3 φ2 (50) − ~2 8µ  6µ ~2φ2 ( − ϑr0 6 − r1ϑ 2 ) + 2µ ~2 ( r0 ϑ + 2r1 − 2r1ϑ ) + 16µr1ϑ3 3~2φ3 n + 12 + √( 1 2 + l )2 + 2µ ~2φ3 ( − ϑr0 6 − r1ϑ 2 ) + 2µr1ϑ3 ~2φ4 − 2µr1 ~2  2 the numerical values of charmonium (cc̄) and bottomonium (bb̄), are mb = 4.823 gev and mc = 1.209 gev , and the corresponding reduced mass are µb = 2.4115 gev and µc = 0.6045 gev , respectively [68]. the potential parameters were calculated by fitting with experimental data (ed) [69]. 5 inyang et al. / j. nig. soc. phys. sci. 4 (2022) 884 6 figure 6: mean energy function against β for different values of λ figure 7: specific heat against β for different values of λ we observed that the results obtained from the prediction of ms of (cc̄) and (bb̄) mesons for different quantum states are in agreement with ed and are seen to be improved when compared with other theoretical predictions with different analytical methods from literature as shown in tables 1 and 2. the absolute deviation in the predicted results and ed is calculated using the following relation σ = 100 n ∑ ∣∣∣∣∣ tp − texp.texp. ∣∣∣∣∣, (51) where texp. is the ed, tp is the predicted results and n is the number of ed and, also the percentage error (pe) is computed with the given relation [70] error = ∑ ard∑ texp. × 100%. (52) figure 8: entropy against β for different values of λ figure 9: free energy against β for different values of λ the absolute deviation was calculated using equation (51) and the values for (cc̄) and (bb̄) are 1.601 gev and 0.457 gev, respectively. also, the pe of the predicted results to the ed, was calculated using equation (52) and the results show that for (cc̄) we have 1.60 % while that of the (bb̄) is 0.46 %. in figures 1−4 we plotted the radial wave functions and probability densities for (cc̄) and (bb̄) mesons. generally, the probability densities decay with the radial distance but the decay happens faster at a short distance for the low-lying quantum states compare to the high-lying states. also, the peaks of the (cc̄) mesons are higher than the corresponding (bb̄) mesons and they tend towards shorter distances. the observed high peaks may be ascribed to their heavy masses. the variation of the pf with respect to temperature (β) for different values of maximum quantum number(λ) is as shown 6 inyang et al. / j. nig. soc. phys. sci. 4 (2022) 884 7 table 1: mass spectra of charmonium (in gev) ( mc = 1.209 gev , µ = 0.6045 gev , θ = 0.06, r1 = −2.31507 gev , r0 = −0.15571 gev , φ = 0.67 ) state our work [1] [64] [44] experiment [69] absolute relative deviation (ard) 1s 3.096 3.096 3.096 3.095922 3.096 0.000 2s 3.686 3.686 3.686 3.685893 3.686 0.000 1p 3.327 3.214 3.433 3.525 0.198 2p 3.774 3.773 3.910 3.756506 3.773 0.001 3s 4.040 4.275 3.984 4.322881 4.04 0.000 4s 4.364 4.865 4.150 4.989406 4.263 0.101 1d 3.761 3.412 3.767 3.770 0.009 2d 4.058 4.159 0.101 table 2: mass spectra of bottomonium (in gev) ( mb = 4.823 gev , µ = 2.4115 gev , r1 = −0.2542 gev , r0 = −0.37717 gev , φ = 0.67, θ = 0.06, ~ = 1 ) state our work [1] [64] [44] experiment [69] absolute relative deviation (ard) 1s 9.460 9.46 9.460 9.515194 9.460 0.000 2s 10.023 10.023 10.023 10.01801 10.023 0.000 1p 9.841 9.492 9.840 9.899 0.058 2p 10.160 10.038 10.16 10.09446 10.260 0.100 3s 10.345 10.585 10.28 10.44142 10.355 0.010 4s 10.522 11.148 10.42 10.85777 10.580 0.058 1d 10.142 9.551 10.14 10.164 0.022 in figure 5. in figure 5, the pf decreases as λ is increased. the mean energy of the mesons is depicted in figure 6, we found that as β increases, the mean energy decreases monotonically. the variation of heat capacity of the mesons is presented in figure 7. we observed an increase in the heat capacity asβincreases for different valves of λ, and later begin to decrease and converges at a point. the plots of the behavior of the entropy as a function ofβ is presented in figure 8. the entropy increases to a peak value before decreasing and converges. the free energy of the mesons is plotted against β in figure 9. it was noticed that the free energy decreases at different values of λ. 5. conclusion in this study, the hgiqyp is model to interact in quarkantiquark system and the solutions were obtained using the nu method. the energy spectrum and normalized wave function were obtained. the energy spectrum was used to predict the ms of the hms. also, the pf was calculated from the energy spectrum, thereafter other tps were obtained. the results obtained showed an improvement when compared with the work of other researchers and excellently agreed with ed with a pe of 1.60 % and 0.46 % for (cc̄) and (bb̄), respectively. the pe shows that the hgiqyp is fitted in the prediction of the ms of the hms since the pe is less. acknowledgements the authors appreciate the referee’s for the careful reading and the suggestions that improved the manuscript. references [1] h. ciftci & h.f. kisoglu, “nonrelativistic-arbitrary l-states of quarkonium through asymptotic iteration method”, advances in high energy physics 45 (2018) 49705. 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[70] e.p. inyang, p.c. iwuji, j.e. ntibi, e. omugbe, e.a. ibanga & e.s. william, “quark-antiquark study with inversely quadratic yukawa potential using nikiforov-uvarov-functional analysis method”, east european journal of physics 2 (2022) 51. https://doi.org/10.26565/2312-43342022-2-05 9 j. nig. soc. phys. sci. 5 (2023) 865 journal of the nigerian society of physical sciences hybrid block methods with constructed orthogonal basis for solution of third-order ordinary differential equations folake lois josepha, adeyemi sunday olagunjub,∗, emmanuel oluseye adeyefac, adewale adeyemi jamesd adepartment of mathematical sciences, bingham university karu,nigeria bdepartment of mathematics, federal university of lafia, nigeria cdepartment of mathematics, federal university oye-ekiti, nigeria ddepartment of mathematics and statistics, american university of nigeria, yola, nigeria abstract in this work, an orthogonal polynomial with a weight function w(x) = x2 + x+1 in the interval [-1, 1] was constructed and used as the basis function to develop block methods, using collocation and interpolation approach. an efficient class of continuous and discrete numerical integration schemes of implicit hybrid form for third-order problems were developed and successfully implemented. three different problems were solved with these schemes, and they performed favourably. the investigation, using the appropriate existing theorems, shows that the methods are consistent, zero-stable, and hence, convergent. doi:10.46481/jnsps.2023.865 keywords: hybrid block, collocation, interpolation, third-order ode, integration scheme article history : received: 14 june 2022 received in revised form: 27 october 2022 accepted for publication: 27 october 2022 published: 22 december 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: j. ndam 1. introduction there are no analytic solutions for some initial value problems of third-order ordinary differential equations (odes) of the form y′′′ = f (x, y, y′, y′′); y(a) = α, y′(a) = β, y′′(a) = γ, x ∈ (a, b) (1) on the other hand, numerical methods for solving such involves reducing the equation to systems of first order differential equa∗corresponding author tel. no: +2348032515655 email address: olagunjuas@gmail.com (adeyemi sunday olagunju) tions, this technique increases the dimension of the real problem, thus involves more computations, [1]. also, linear multistep methods (lmms) implemented by the predictor-corrector mode have been found to be prone to error propagation and are also known to be very expensive to implement in terms of the number of function evaluations per step. the predictors often have a lower order of accuracy than the correctors, especially when all the grid and off-grid points are used for collocation and interpolation. this disadvantage led to the development of block methods from lmms. apart from being self-starting, a block method does not require the development of a predictor separately. the development of block methods for solving ivps in odes is the central concern of this work. many researchers 1 joseph et al. / j. nig. soc. phys. sci. 5 (2023) 865 2 have used this block method to integrate ivps, among them are [2], [3], [4], and [5]. some researchers also constructed orthogonal polynomials using different weight functions; see [6]. these constructed polynomials were employed as basis function for the derivation of numerical schemes. they include; [7] who used orthogonal polynomials with weight function w(x) = x2 for interval [−1, 1]. [8] employed orthogonal polynomials with weight function w(x) = x2 in the interval [0, 1], while [9] engaged orthogonal polynomials with weight function w(x) = x for interval [0, 1]. virtually all the a forementioned contributions are for first order differential equations. [10] constructed an orthogonal basis in the interval [-1, 1] with respect to the weight function w(x) = 1 + x2 , to solve the initial value problem of a third-order differential equation. [11] constructed a block method to integrate third-order ode, [12] developed harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems, while [13] formulated a block algorithm for general third-order ordinary differential equations. several methods have been formulated using different polynomials and targeting varying orders of odes (see [14], [15]). the need to further explore this area of research with the view to solving higher order differential equations effectively is the major focus of this work. to this end, orthogonal polynomials were constructed to serve as basis functions for the development of hybrid block methods for the class of problem (1). the polynomial is characterized with w(x) = x2 + x+1 in the interval [-1, 1]. 2. formulation of the orthogonal polynomial we formulate here, the basis function for the numerical scheme to be developed by constructing an orthogonal polynomial qn(x) of degree n. as established in literature, orthogonality of two polynomials is attained if their inner product varnishes in the interval within which they are defined [9]. i. e. the inner product yields:∫ b a w(x)φm(x)φn(x)d x = hnδmn for δmn = { 0,m,n 1,m=1 where w(x) is a continuous and positive weight function in the range [a,b], and the moments µ = ∫ b a w(x)xnd x, n = 0, 1, 2, ... exist. the integral 〈φm,φn〉 = ∫ b a w(x)φm(x)φn(x)d x . is the inner product of polynomials φm and φn. when it comes to orthogonality, 〈φm,φn〉 = ∫ b a w(x)φm(x)φn(x)d x = 0, f or m , n for the construction of a befitting approximating polynomial, consider the orthogonal polynomial class {φn(x)}, which is defined as φn(x) = n∑ r=0 c(n)r x r which requires that: φn(1) = 1 and < φm(x),φn(x) >= 0 (m , n) . let w(x) = 1 + x + x2 and [a, b] = [−1, 1]. then, for n = 0, we have φ0(x) = c (0) 0 and φ0(1) = 1 = c (0) 0 hence, φ0(x) = 1 for n = 1, we have two equations φ1(1) = c (1) 0 + c (1) 1 = 1 < φ0(x),φ1(x) >= 8 3 c(1)0 + 2 3 c(1)1 = 0 these two equations yield: φ1(x) = − 1 3 + 4 3 x for n = 2, we solve c(1)0 + c (2) 1 + c (2) 2 = 1 = φ2(1) 8 3 c(1)0 + 2 3 c(2)1 + 16 5 c(2)2 = 0 =< φ0(x),φ2(x) > 6 5 c(2)1 + 8 45 c(2)2 = 0 =< φ1(x),φ2(x) > which gives the polynomial φ2(x) = − 49 66 − 10 33 x + 45 22 x2 following this approach, we have a class of orthogonal polynomials given as follows: 2 joseph et al. / j. nig. soc. phys. sci. 5 (2023) 865 3 q0(x) = 1 q1(x) = 4 3 x − 1 3 q2(x) = 45 22 x2 − 10 33 − 49 66 q3(x) = 1484 419 x3 − 483 838 x2 − 930 419 x + 213 838 q4(x) = 49427 7880 x4 − 994 985 x3 − 4347 788 x2 + 2002 2995 x + 13609 23640 q5(x) = 169587 14882 x5 − 108625 59528 x4 − 13735 1063 x3 + 50445 29764 x2 + 127805 44646 x − 5285 25512 q6(x) = 21640593 1029296 x6 − 865683 257324 x5 − 29995065 1029296 x4 + 511695 128662 x3 + 112595805 11322256 x2 − 2700945 2830564 x − 5509685 11322256 q7(x) = 912217735 23254947 x7 − 2332825495 372079152 x6 − 663966303 10335532 x5 + 3345316975 372079152 x4 + 1383496345 46509894 x3 − 134149785 41342128 x2 − 315107135 93019788 x + 66696665 372079152 q8(x) = 206238477159 2794077824 x8 − 4632559789 392917194 x7 − 876515334695 6286675104 x6 + 867587721 43657466 x5 + 1026917752861 12573350208 x4 − 3774934163 392917194 x3 − 10578992655 698519456 x2 + 462916223 392917194 x + 10792421983 25146700416 q9(x) = 7592947736959 54306345568 x9 − 4849132400307 21722582272 x8 − 4061887568739 13576586392 x7 + 2347732919259 54306345568 x6 + 5760948177621 27153172784 x5 − 2848740151257 108612691136 x4 − 750682757643 13576586392 x3 + 278681373243 54306345568 x2 + 208683420559 54306345568 x − 34811913843 217225382272 3. numerical block algorithms for third-order problems the analytical solution of (1) is approximated via experimental solution of the form: y (x) = v+w−1∑ j=0 a jφ j(x) (2) where x ∈ [a, b], v and w are the number of collocation and interpolation points respectively. the function φ j(x) is the jth degree orthogonal polynomial valid in the range of integration of [a, b]. the third derivative of (2) is given by y′′′(x) = v+w−1∑ j=3 a jφ ′′′ j (x) = f (x, y, y ′, y′′) (3) to estimate the solution of the problem given in (1), interpolation must be done at least three times. as a result, equation (2) is interpolated at (xn+w) points, and equation (3) is collocated at (xn+v) points, yielding a system of equations to be solved using the gaussian elimination method. we will use a hybrid approach to apply this concept. 3 joseph et al. / j. nig. soc. phys. sci. 5 (2023) 865 4 3.1. development of one-step method with xn+ 14 , xn+ 12 and xn+ 34 as the off-step point the new off-step points are xn+ 14 , xn+ 12 and xn+ 34 . interpolating (2)at x = xn+w, w = 0, 1 4 and 1 2 and collocating (3) at x = xn+v, v = 0, 14 , 1 2 , 3 4 and 1 ; yields the system of equations: 1 −53 53 33 −689 419 983 591 −519 310 631 375 −10287 6091 1 −1 −788 823 1059 −400 641 −159 2278 593 1002 −201 409 1 −13 −49 66 213 838 1297 2253 −301 1453 −382 785 197 1099 0 0 0 35701210 −246792 197 140399 27 −112087 7 203929 5 0 0 0 35701210 −24072 37 30440 33 8344 135 −38642 17 0 0 0 35701210 −2761 57 −17986 29 11263 59 39979 28 0 0 0 35701210 21595 39 25171 44 −30707 63 −26670 13 0 0 0 35701210 9247 8 17997 4 105317 8 64401 2   a0 a1 a2 a3 a4 a5 a6 a7  =  yn yn+ 14 yn+ 12 h3 fn h3 fn+ 14 h3 fn+ 12 h3 fn+ 34 h3 fn+1  (4) the solution of (4) gives the values: a0 = 21 20 yn + 51 20 yn+ 12 − 13 44645 yn+ 14 + 4 44645 fnh 3 + 23 2378 fn+ 12 h 3 + 23 3087 h3 fn+ 14 + 35 36069 h3 fn+ 34 + 3 64369 h3 fn+1 a1 = 35 36 yn + 89 36 yn+ 12 − 31 9 yn+ 14 + 67 928956 h3 fn + 93 8017 h3 fn+ 12 + 325 39192 h3 fn+ 14 + 89 62116 h3 fn+ 34 + 54 971405 h3 fn+1 a2 = 44 45 yn + 44 45 yn+ 12 − 88 45 yn+ 14 + 15 148586 h3 fn + 126 14065 h3 fn+ 12 + 70 11661 h3 fn+ 14 + 6 3259 h3 fn+ 34 + 6 187829 h3 fn+1 a3 = h 3 ( 3 290044 fn + 71 21932 fn+ 12 + 55 50968 fn+ 14 + 32 20807 fn+ 34 + 8 456815 fn+1 ) a4 = h 3 ( −1 96160 fn − 12 71539 fn+ 12 − 22 30633 fn+ 14 + 33 37892 fn+ 34 + 4 157477 fn+1 ) a5 = h 3 ( 1 56502 fn − 20 35299 fn+ 12 + 14 50845 fn+ 14 + 13 55005 fn+ 34 + 7 188168 fn+1 ) a6 = h 3 ( −7 233230 fn + 4 220991 fn+ 12 + 8 148167 fn+ 14 − 13 166396 fn+ 34 + 4 110967 fn+1 ) a7 = h 3 ( 5 494258 fn + 15 247129 fn+ 12 − 10 247129 fn+ 14 − 10 247129 fn+ 34 + 5 494258 fn+1 ) (5) substituting (5) in (3) yields the continuous implicit one-step method: ȳ(x) = 1∑ j=0 α jyn+ j + α 1 4 (x)yn+ 14 + α 12 (x)yn+ 12 + α 34 (x)yn+ 34 +h  1∑ j=0 β j(x) fn+ j + β 1 4 (x) fn+ 14 + β 12 (x) fn+ 12 + β 34 (x) fn+ 34  (6) where α j(x) and β j(x) are continuous coefficients. in (6) the parameters, α j(x) and β j(x), are obtained as : α0(t) = 2t 2 + t α 1 4 (t) = −4t2 − 4t α 1 2 (t) = 2t2 + 3t + 1 β0(t) = h 3 ( t7 2520 − t6 720 + t5 180 − t4 144 − t3 19622435024537215000 − t2 23040 4 joseph et al. / j. nig. soc. phys. sci. 5 (2023) 865 5 + 13t 161280 + 1 903542973820909160000 ) β 1 4 (t) = h3 ( − t7 630 + t6 720 + t5 180 − t4 144 − t3 119312808645680590000 + 199 7103 t2 + 47 6501 t − 1 34323495691077853000 ) β 1 2 (t) = h3 ( t7 420 + t6 6654769540940698600 − t5 96 + t4 1318120659808086000 + t3 48 + 21 1280 t2 + 44 12193 − 1 917808302492149500 ) β 3 4 (t) = −h3 ( t7 630 + t6 720 − t5 180 − t4 144 + t3 4590274775345433600 + 5t2 2304 + 47t 80640 − 1 13479442579811103000 ) β1(t) = h 3 ( t7 2520 + t6 1440 − t5 2880 − t4 1152 − t3 14793289125000270000 + 7t2 23040 + 7t 86843 + 1 115497606639823160000 ) (7) where t = 2x−2xn−hh . by evaluating (6) at xn+ 34 and xn+1, the main methods are obtained as: yn+ 34 = yn − 3yn+ 14 + 3yn+ 12 + h 3 ( 1 15360 fn + 21 2560 fn+ 12 − 1 3840 fn+ 34 + 1 15360 f1 ) (8) yn+1 = 3yn − 8yn+ 14 + 6yn+ 12 + h 3 ( 1 3840 fn + 43 1920 fn+ 14 + 21 640 fn+ 12 + 13 1920 fn+ 34 + 1 3820 f1 ) (9) differentiating (6) yields the continuous coefficients: α′0(t) = 8t + 2 h α 1 4 (t) = − 16t + 8 h α′1 2 (t) = 8t + 6 h β′0(t) = h 2 ( − t6 180 + t5 120 + t4 288 − t3 144 + t2 5537761027586915300 + t 5760 − 13 86040 ) β′1 4 (t) = h2 ( − t6 45 + t5 60 + t4 18 − t3 18 − t2 2305645383593181400 + 193t 2880 + 94 6501 ) β′1 2 (t) = h2 ( t6 30 + t5 495329736356353410 − 5t4 48 + t3 164765082476010720 + t2 8 + 21t 320 + 97 13440 ) β′3 4 (t) = h2 ( t6 45 + t5 60 − t4 18 − t3 18 + t2 686884395823310340 + 5t 576 + 47 40320 ) β′1(t) = h 2 ( t6 180 + t5 120 − t4 288 − t3 144 − t2 2514196338285239800 + 7t 5760 + 13 80640 ) (10) the second derivatives of the continuous functions are given as α′′( t) = 16 h2 5 joseph et al. / j. nig. soc. phys. sci. 5 (2023) 865 6 α′′1 4 (t) = − 32 h2 α′′1 2 (t) = 16 h2 β′′0 (t) = −h ( − t5 15 + t4 12 + t3 36 − t2 24 + t 1384440256896728800 + 1 ) β′′1 4 (t) = −h ( 4t5 15 − t4 6 − 4t3 9 + t2 3 + t 576411345898295360 − 193 1440 ) β′′1 2 (t) = h ( 2t5 5 + t4 40538871860771056 − t3 12000000000 + t2 29029771673218900 + t 2 21 160 ) β′′3 4 (t) = h ( − 4t5 15 + t4 6 − 4t3 9 − t2 3 + t 171721098955827550 − 5 288 ) β′′1 (t) = h ( − t5 15 − t4 12 + t3 36 + t2 24 + t 628549084571309950 − 7 2880 ) (11) the additional methods to be coupled with the main method are obtained by evaluating the first and second derivatives of (6) at xn, xn+ 14 , xn+ 12 , xn+ 34 and xn+1 respectively to give: hy′n + 6yn − 8yn+ 14 + 2yn+ 12 = h3 ( 76 19963 fn + 245 12457 fn+ 14 − 19 4480 fn+ 12 + 29 14801 fn+ 34 − 13 36149 fn+1 ) (12) hy′ n+ 14 + 2yn + 0yn+ 14 − 2yn+ 12 = h3 ( − 76 19963 fn − 62 6527 fn+ 14 − 3 8960 fn+ 12 − 1 8064 fn+ 34 − 1 32256 fn+1 ) (13) hy′ n+ 12 − 2yn + 8yn+ 14 − 6yn+ 12 = h3 ( 13 80640 fn + 94 6501 fn+ 14 + 97 13440 fn+ 12 − 47 40320 fn+ 34 + 13 80640 fn+1 ) (14) hy′ n+ 34 − 6yn + 16yn+ 14 − 10yn+ 12 = h3 ( 15 27182 fn + 209 4679 fn+ 14 + 117 1792 fn+ 12 + 58 14347 fn+ 34 + h 32256 fn+1 ) (15) hy′n+1 − 10yn + 24yn+ 14 − 14yn+ 12 = h3 ( − 11 16128 fn + 273 3596 fn+ 14 + 324 2551 fn+ 12 + 220 3527 fn+ 34 + 104 21449 fn+1 ) (16) h2y′′n − 16yn + 32yn+ 14 − 16yn+ 12 = h3 ( − 233 2880 fn − 101 480 fn+ 14 + 31 480 fn+ 12 − 41 1440 fn+ 34 + 1 192 fn+1 ) (17) h2y′′ n+ 14 − 16yn + 32yn+ 14 − 16yn+ 12 = h3 ( 1 60 fn + 1 72 fn+ 14 − 13 480 fn+ 12 + 1 120 fn+ 34 − 1 720 fn+1 ) (18) h2y′′ n+ 12 − 16yn + 32yn+ 14 − 16yn+ 12 = h3 ( − 1 2880 fn + 193 1440 fn+ 14 + 21 160 fn+ 12 − 5 288 fn+ 34 + 7 2880 fn+1 ) (19) 6 joseph et al. / j. nig. soc. phys. sci. 5 (2023) 865 7 h2y′′n+1 − 16yn + 32yn+ 14 − 16yn+ 12 = h3 ( − 1 320 fn + 209 1440 fn+ 14 + 19 96 fn+ 12 + 157 480 fn+ 34 + 239 280 fn+1 ) (20) equations (8), (9) and (12)-(20) are solved using [22] block formula defined as aym = hbf(ym) + e(yn) + hd fn (21) a = (ai j), b = (bi j), column vectors d = (e1...er )t ,d = (d1...dr )t , ym = (yn+1...yn+r )t and f (ym) = ( fn+1, ..., fn+r )t thus, a, b, d and e are obtained as follows: a =  3 −3 1 0 0 0 0 0 0 0 0 0 8 −6 0 1 0 0 0 0 0 0 0 0 −8 2 0 0 0 0 0 0 0 0 0 0 0 −2 0 0 1 0 0 0 0 0 0 0 8 −6 0 0 0 1 0 0 0 0 0 0 16 −10 0 0 0 0 1 0 0 0 0 0 24 −14 0 0 0 0 0 1 0 0 0 0 32 −16 0 0 0 0 0 0 0 0 0 0 32 −16 0 0 0 0 0 0 1 0 0 0 32 −16 0 0 0 0 0 0 0 1 0 0 32 −16 0 0 0 0 0 0 0 0 1 0 32 −16 0 0 0 0 0 0 0 0 0 1  , b =  29 3840 21 2560 −1 3840 1 15360 43 1920 21 640 13 1920 1 3840 245 12457 −19 4480 29 14801 −13 36149 −62 6527 −3 8960 −1 8064 1 32256 94 6501 97 13440 −47 40320 13 80640 209 4679 117 1792 58 14347 1 32256 273 3596 324 2551 220 3527 104 21449 −101 480 31 480 −41 1440 1 192 1 72 −13 480 1 120 −1 720 193 1440 21 160 −5 288 7 2880 13 120 139 480 37 360 −1 240 209 1440 19 96 157 480 239 2880  , d =  1 15360 1 3840 76 19963 −27 55121 13 80640 15 27182 11 16128 −233 2880 1 160 −1 2880 1 288 −1 320  , e =  1 0 0 3 0 0 −6 −1 0 −2 0 0 2 0 0 6 0 0 10 0 0 16 0 −1 16 0 0 16 0 0 16 0 0 16 0 0  . substituting a, b, d and e into (21) yields the explicit schemes: yn+ 14 = yn + 1 4 y′n + h2 32 y′′n + h 3 ( 116 73583 fn + 83 50042 − 60 62633 fn+ 12 + 15 37507 fn+ 34 − 17 233341 fn+1 ) (22) yn+ 12 = yn + 1 2 y′n + h2 8 y′′n + h 3 ( 347 42269 fn + 83 5040 − 1 168 fn+ 12 + 13 5040 fn+ 34 − 19 40320 fn+1 ) (23) yn+ 34 = yn + 3 4 y′n + 9h2 32 y′′n + h 3 ( 409 70578 fn + 164 3155 − 49 7227 fn+ 12 + 45 7168 fn+ 34 − 16 14159 fn+1 ) (24) yn+1 = yn + y ′ n + h2 2 y′′n + h 3 ( 31 840 fn + 34 315 + 1 210 fn+ 12 + 2 105 fn+ 34 − 1 504 fn+1 ) (25) y′ n+ 14 = y′n + h 4 y′′n + h 2 ( 222 1393 fn + 3 128 fn+ 14 − 47 3840 fn+ 12 + 29 5760 fn+ 34 − 7 7680 fn+1 ) (26) y′ n+ 12 = y′n + h 2 y′′n + h 2 ( 53 1440 fn + 1 10 fn+ 14 − 1 48 fn+ 12 + 1 90 fn+ 34 − 1 480 fn+1 ) (27) y′ n+ 34 = y′n + 3h 4 y′′n + h 2 ( 147 2560 fn + 117 640 fn+ 14 + 27 1280 fn+ 12 + 3 128 fn+ 34 − 9 2560 fn+1 ) (28) 7 joseph et al. / j. nig. soc. phys. sci. 5 (2023) 865 8 y′n+1 = y ′ n + hy ′′ n + h 2 ( 7 90 fn + 4 15 fn+ 14 + 50793 761894 fn+ 12 + 4 45 fn+ 34 + 0 ) (29) y′′ n+ 14 = y′′n + h ( 251 2880 fn + 323 1440 fn+ 14 − 11 120 fn+ 12 + 53 1440 fn+ 34 − 19 2880 fn+1 ) (30) y′′ n+ 12 = y′′n + h ( 29 360 fn + 31 90 fn+ 14 + 1 15 fn+ 12 + 1 90 fn+ 34 − 1 360 fn+1 ) (31) y′′ n+ 34 = y′′n + h ( 27 320 fn + 51 160 fn+ 14 + 9 40 fn+ 12 + 21 160 fn+ 34 − 3 320 fn+1 ) (32) y′′n+1 = y ′′ n + h ( 7 90 fn + 16 45 fn+ 14 + 2 15 fn+ 12 + 16 45 fn+ 34 7 90 fn+1 ) (33) 3.2. development of two-steps method with xn+ 13 and xn+ 23 as the off-step point in this section, we introduce xn+ 13 and xn+ 23 as the new off-step point. interpolating (2) at x = xn+s, s = 0, 1 3 , 2 3 and collocating (3) at x = xn+r , r = 0, 1 3 , 2 3 , 1 and 2 yields the system: 1 −53 53 33 −689 419 983 591 −519 310 631 375 −10287 6091 1 −119 73 198 236 551 −1033 1308 841 1390 −153 1999 −1835 4279 1 −79 −41 99 4586 5741 −199 1342 −397 704 852 1805 281 1441 0 0 0 8904419 −30849 197 138449 213 45884 9 0 0 0 8904419 −7449 70 13033 51 −37143 107 0 0 0 8904419 −4555 81 6221 477 12701 90 −455 44 0 0 0 8904419 −3203 529 −8993 116 3985 167 11958 67 0 0 0 8904419 13726 95 17997 32 31266 19 68426 17   a0 a1 a2 a3 a4 a5 a6 a7  =  yn yn+ 13 y 2 3 h3 fn h3 fn+ 13 h3 f n + 23 h3 fn+1 h3 fn+2  (34) solving this system gives the values of the unknown parameters a j, j = 0(1)7 as: a0 = 157 40 yn − 48y 5 yn+ 13 + 267 40 yn+ 23 +h3  25342525 fn + 3563685 fn+1 + 3117539 fn+2 + 1871851 fn+ 13 + fn+ 232479  a1 = 31 8 yn − 10yn+ 13 + 49 8 yn+ 23 −h3 ( 241 28690 fn − 406 3287 fn+1 − 85 34601 fn+2 − 449 3939 fn+ 13 − 247 6237 fn+ 23 ) a2 = 11 5 yn − 22 5 yn+ 13 + 11 5 yn+ 23 −h3 ( 146 15675 fn − 680 5903 fn+1 − 46 16677 fn+2 + − 166 1999 fn+ 13 + 71 4953 fn+ 23 ) a3 = 264 4033 − 79 11205 fn+1 + 8 3683 fn+2 + 253 7124 fn+ 13 − 79 1611 fn+ 23 a4 = 332 15051 − 41 11313 fn+1 + 48 35893 fn+2 + 381 25870 fn+ 13 − 220 6377 fn+ 23 a5 = 37 25260 − 31 31135 fn+1 + 1 1636 fn+2 + 63 1067 fn+ 13 − 135 19337 fn+ 23 a6 = 11 135050 − 19 9761 fn+1 − 11 58979 fn+2 − 13 9535 fn+ 13 + 34 11177 fn+ 23 a7 = 47 172075 − 41 75054 fn+1 + 10 366177 fn+2 − 121 123056 fn+ 13 + 15 122039 fn+ 23 (35) substituting (35) in (2) yields a continuous implicit two-steps method in the form: y(x) = 2∑ j=0 α j(x)yn+ j + α 1 3 (x)yn+ 13 + α 23 (x)yn+ 23 + h  1∑ j=0 β j(x) fn+ j + β 1 3 (x) fn+ 13 + β 23  (36) 8 joseph et al. / j. nig. soc. phys. sci. 5 (2023) 865 9 where α j(x) and β j(x) are continuous coefficients, and the parameters α j(x) and β j(x) are given by: α0(t) = 9t2 2 + 9t 2 + 1 α 1 3 (t) = −9t2 − 12t − 3 α 2 3 (t) = 9t2 2 + 15t 2 + 3 β0(t) = fnh 3 ( 3 280 t7 − 1 1811776004788461600 t6 − 7 240 t5 − 1 48 t4 + 1 523827403997699330 t3 + 7 2160 t2 + 9 9349 t + 1 9720 ) β 1 3 (t) = fn+ 13 h 3 ( − 27 700 t7 − 9 40 t6 27 200 t5 + 9 80 t4 − 1 119422617446588190 t3 + 61 900 t2 + 601 7560 t + 49 2700 ) β 2 3 (t) = fn+ 23 h 3 ( 27 560 t7 + 9 160 t6 − 27 160 t5 − 9 32 t4 + 1 321376444486800900 t3 + 29 144 t2 + 701 6048 t + 41 2160 ) β1(t) = fn+1h 3 ( − 3 140 t7 − 3 80 t6 + 7 120 t5 + 3 16 t4 + 1 6 t3 + 11 180 t2 + 258 35179 t − 1 4860 ) β2(t) = fn+2h 3 ( 3 2800 t7 + 3 800 t6 + 11 2400 t5 + 1 480 t4 + 1 2162636848303812900 t3 − 1 5400 t2 + 1 272160 t + 1 97200 ) (37) where t = x−xn−hh . by evaluating (36) at xn+1 and xn+2 the main methods are obtained as: yn+1 = yn − 3yn+ 13 + 3yn+ 23 + h 3 ( 1 9720 fn + 49 2700 fn+ 13 + 41 2160 fn+ 23 − 1 4860 fn+1 + 1 97200 fn+2 ) (38) yn+2 = 10yn − 24yn+ 13 + 15yn+ 23 + h 3 ( − 17 486 fn + 19 54 fn+ 13 − 1 108 fn+ 23 + 205 486 fn+1 + 11 972 fn+2 ) (39) the first derivatives of continuous functions (36) are given as: α′0(t) = 9t + 9 2h α′1 3 (t) = −18t + 12 h α′2 3 (t) = 9t + 15 2h β′0(t) = h 2 ( 3 40 t6 − 1 353797023374953600 t5 − 7 48 t4 − 1 12 t3 + 1 150940880005095680 t2+ 7 1080 t + 9 9349 ) β′1 3 (t) = h2 ( − 27 100 t6 − 27 200 t5 + 27 40 t4 + 9 20 t3 − 1 36491974884133584 t2 + 61 450 t + 601 7560 ) β′2 3 (t) = h2 ( 27 80 t6 + 27 80 t5 − 27 32 t4 − 9 8 t3 + 1 90911263767416096 t2 + 29 72 t + 701 6048 ) 9 joseph et al. / j. nig. soc. phys. sci. 5 (2023) 865 10 β′n+1(t) = h 2 ( − 3 20 t6 − 9 40 t5 + 7 24 t4 + 3 4 t3 + 1 2 t2 + 11 90 t + 258 35179 ) β′n+2(t) = h 2 ( 3 400 t6 + 9 400 t5 + 11 480 t4 + 1 120 t3 + 1 758390757276836610 t2 − 1 2700 t + 1 272160 ) (40) the second derivatives are given as follows: a′′0 (t) = 9 h2 a′′1 3 (t) = − 18 h2 a′′2 3 (t) = 9 h2 β′′0 (t) = h ( 9 20 t5 − 1 109969890506378530 t4 − 7 12 t3 − 1 4 t2 + 1 75470440002547824 t + 7 1080 ) β′′1 3 (t) = −h ( 81 50 t5 + 27 40 t4 − 27 10 t3 − 27 20 t2 + 1 18245987442066792 t − 61 450 ) β′′2 3 (t) = h ( 81 40 t5 + 27 16 t4 − 27 8 t3 − 27 8 t2 + 1 45455631883708040 t + 29 72 ) β′′1 (t) = h ( − 9 10 t5 − 9 80 t4 + 7 6 t3 + 9 4 t2 + t + 11 90 ) β′′2 (t) = h ( 9 200 t5 + 9 80 t4 + 11 120 t3 + 1 40 t2 + 1 379195378638418240 t − 1 2700 ) we obtain the additional methods to be be coupled with the main methods by evaluating the first and second derivatives of (36) at xn, xn+ 13 , xn+ 23 , xn+1 and xn+2 respectively to get: hy′n + 9 2 yn − 6yn+ 13 + 3 2 yn+ 23 = h3 ( 95 13608 fn + 119 3506 fn+ 13 − 17 3024 fn+ 23 + 35 19681 fn+1 − 14 328469 fn+2 ) (41) hy′ n+ 13 + 3 2 yn − 3 2 yn+ 23 = h3 ( 43 48359 fn − 127 7560 fn+ 13 − 23 30240 fn+ 23 − 1 13608 fn+1 + 1 272160 fn+2 ) (42) hy′ n+ 23 − 3 2 yn + 6yn+ 13 − 9 2 yn+ 23 = h3 ( 13 68040 fn + 186 7109 fn+ 13 181 15120 fn+ 23 − 89 68040 fn+1 + 2 1046770 fn+2 ) (43) hy′n+1 − 9 2 yn + 12yn+ 13 − 15 2 yn+ 23 = h3 ( 9 9349 fn + 601 7560 fn+ 13 + 701 6048 fn+ 23 + 258 35179 fn+1 + 193 3170 fn+2 ) (44) hy′n+2 − 27 2 yn + 30yn+ 13 − 33 2 yn+ 23 = h3 ( − 385 26240 fn + 6979 1047 fn+ 13 − 1175 1516 fn+ 23 + 652 503 fn+1 + 1 2160 fn+2 ) (45) h2y′′ n+ 13 − 9yn + 18yn+ 13 − 9yn+ 23 = h3 ( 29 3240 fn + 7 450 fn+ 13 − 11 360 fn+ 23 + 1 162 fn+1 − 1 8100 fn+2 ) (46) 10 joseph et al. / j. nig. soc. phys. sci. 5 (2023) 865 11 h2y′′ n+ 23 − 9yn + 18yn+ 13 − 9yn+ 23 = h3 ( − 1 648 fn + 331 1800 fn+ 13 + 119 720 fn+ 23 − 47 3240 fn+1 + 489 32400 fn+2 ) (47) h2y′′n+1 − 9yn + 18yn+ 13 − 9yn+ 23 = h3 ( 7 1080 fn + 61 450 fn+ 13 + 29 72 fn+ 23 + 11 90 fn+1 − 1 2700 fn+2 ) (48) h2y′′n+2 − 9yn + 18yn+ 13 − 9yn+ 23 = h3 ( − 407 1080 fn + 1261 667 fn+ 13 − 1897 720 fn+ 23 + 181 72 fn+1 + 489 1786 fn+2 ) (49) equations (38) -(39) and (42) (50) are solved using equation (21). a, b, d and e are obtained from the equations as: a =  3 −3 1 0 0 0 0 0 0 0 0 0 24 −15 0 1 0 0 0 0 0 0 0 0 −6 32 0 0 0 0 0 0 0 0 0 0 0 −32 0 0 1 0 0 0 0 0 0 0 6 −92 0 0 0 1 0 0 0 0 0 0 12 −152 0 0 0 0 1 0 0 0 0 0 30 −332 0 0 0 0 0 1 0 0 0 0 18 −9 0 0 0 0 0 0 0 0 0 0 18 −9 0 0 0 0 0 0 1 0 0 0 18 −9 0 0 0 0 0 0 0 1 0 0 18 −9 0 0 0 0 0 0 0 0 1 0 18 −9 0 0 0 0 0 0 0 0 0 1  b =  49 2700 41 2160 −1 4860 1 97200 19 54 −1 108 205 486 11 972 119 3506 −17 3024 35 19681 −14 328469 −127 7560 −23 30240 −1 13608 1 272160 186 7109 181 15120 −89 68040 2 104677 601 7560 701 6048 258 35179 1 272160 979 1047 −1175 1516 652 503 193 3170 −97 360 47 720 −7 360 1 2160 7 450 −11 360 1 162 −1 8100 331 1800 119 720 −47 3240 7 32400 61 640 29 72 11 90 −1 2700 1261 667 −1897 720 181 72 489 1786  d =  1 9720 −17 486 95 13608 −43 48359 13 68040 9 9349 −385 2624 −119 1080 29 3240 1 648 7 1080 −407 1080  e =  1 0 0 10 0 0 −9 2 −1 0 −3 2 0 0 3 2 0 0 9 2 0 0 27 2 0 0 9 0 −1 9 0 0 9 0 0 9 0 0 9 0 0  inputting a, b, d and e into (21) yields: yn+ 13 = yn + 1 3 y′n + 1 18 y′′n + h 3 ( 69 18185 fn + 259 70858 fn+ 13 − 53 30240 fn+ 23 + 11 22566 fn+1 − 2 173719 fn+2 ) (50) yn+ 23 = yn + 2 3 y′n + 2 9 y′′n + h 3 ( 233 11749 fn + 176 4725 fn+ 13 − 61 5670 fn+ 23 + 16 5103 fn+1 − 2 255150 fn+2 ) (51) yn+1 = yn + y ′ n + 1 2 y′′n + h 3 ( 27 560 fn + 333 2800 fn+ 13 − 9 1120 fn+ 23 + 13 1680 fn+1 − 1 5600 fn+2 ) (52) yn+2 = yn + 2y ′ n + 2y ′′ n + h 3 ( 6 35 fn + 144 175 fn+ 13 − 9 70 fn+ 23 + 16 35 fn+1 + 11 1050 fn+2 ) (53) y′ n+ 13 = +y′n + 1 3 y′′n + h 2 ( 187 6480 fn + 211 5400 fn+ 13 − 73 4320 fn+ 23 + 1 216 fn+1 − 7 64800 fn+2 ) (54) 11 joseph et al. / j. nig. soc. phys. sci. 5 (2023) 865 12 y′ n+ 23 = y′n + 2 3 y′′n + h 2 ( 1 15 fn + 116 675 fn+ 13 − 7 270 fn+ 23 + 4 405 fn+1 − 1 4050 fn+2 ) (55) y′n+1 = y ′ n + y ′′ n + h 2 ( 5 48 fn + 63 200 fn+ 13 + 9 160 fn+ 23 + 1 40 fn+1 − 1 2400 fn+2 ) (56) y′ n+ 13 = y′n + 2y ′′ n + h 2 ( 1 15 fn + 36 25 fn+ 13 − 9 10 fn+ 23 + 4 3 fn+1 + 3 50 fn+2 ) (57) y′′n = y ′′ n + h ( 193 1620 fn + 57 200 fn+ 13 − 23 240 fn+ 23 + 83 3240 fn+1 − 19 32400 fn+2 ) (58) y′′ n+ 13 = y′′n + h ( 181 1620 fn + 34 75 fn+ 13 + 1 10 fn+ 23 + 2 405 fn+1 − 1 4050 fn+2 ) (59) y′′n+1 = y ′′ n + h ( 7 60 fn + 81 200 fn+ 13 + 27 80 fn+ 23 + 17 120 fn+1 − 1 1200 fn+2 ) (60) y′′n+2 = y ′′ n − h ( 4 15 fn + 54 25 fn+ 13 − 27 10 fn+ 23 + 35 15 fn+1 + 41 150 fn+2 ) (61) 4. analysis of the methods in this section, a detailed examination of the performance of the discussed methods is carried out. these involve basic properties of the numerical integration scheme for odes, which include the order, error constant, zero stability, and consistency. the main methods derived are discrete schemes belonging to the class of lmms of the form: k∑ j=0 α jyn+ j = h 3 k∑ j=0 β j fn+ j (62) following [16] and [18], the local truncation error(lte) associated with (63) is defined by difference operator; l[y(x) : h] = k∑ j=0 [α jy(xn + jh) − h 3β j f (xn + jh)] (63) where y(x) is an arbitrary function, continuously differentiable on [a, b]. expanding (63) in taylor’s series about the point x, we obtain the expression l[y(x) : h] = coy(x) + c1hy ′ (x) + ...cp+3h p+3yp+3(x) (64) where the co, c1, c2...cp...cp+3 are obtained as follows c0 = k∑ j=0 α j (65) c1 = k∑ j=1 jα j (66) c3 = 1 3! k∑ j=1 j3α j (67) cq = 1 q! [ k∑ j=1 jqα j − q(q − 1)(q − 2)(q − 3) k∑ j=1 β j j q−3] (68) in the sense of [18], equation (63) is of order p if co = c1 = c2 = c2 = ...cp = cp+1 = cp+2 = 0 and cp+3 , 0. the cp+3 , 0 is called the error constant and cp+3hp+3yp+3(xn) is the principal local truncation error at the point xn. equations (8),(9) and (36) all have order p= 8 with error constants c p+3 = 1493 845571686400 , 1 1410533428 , and −8 97197 respectively, 12 joseph et al. / j. nig. soc. phys. sci. 5 (2023) 865 13 4.1. consistency of the methods the lmm is said to be consistent if it has order p ≥ 1 and the first and second characteristic polynomials, which are defined, respectively, as ρ(z) = k∑ j=0 α jz j (69) and σ(z) = k∑ j=0 β jz j (70) where z is the principal root, satisfy the following conditions: k∑ j=0 α j = 0 (71) ρ(1) = ρ′(1) = 0 (72) and ρ′′′(1) = 3!σ(1) (henrichi, 1962) (73) the schemes (8), (9), and (36) are of order ρ = 8 > 1 and they have been investigated to satisfy conditions (i)-(iii). hence, the schemes are consistent. 4.2. zero stability of the methods the lmm is said to be zero-stable if no root of the first characteristic polynomial ρ(r) has modulus greater than one and if and only if every root of modulus one has multiplicity not greater than the order of the differential equation. to analyze the zero-stability of the methods we present equations (22) (25) and (51)-(54) in block form as: a0ym = hb f (ym) + a ′ynhd fn where h is a fixed mesh size within a block. in line with this, the zero stability of equations (24)-(25) gives: a0 =  1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1  a′ =  0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1  b =  83 5004 −60 62633 15 3750 −17 233341 83 5040 −1 168 13 5040 −19 40320 164 3155 −49 7227 45 7168 −16 14159 34 315 1 210 2 105 −1 504  d =  0 0 0 11673583 0 0 0 34742269 0 0 0 140970578 0 0 0 31840  ρ(r) = det ( ra0 − a′ ) (74)  r 0 0 0 0 r 0 0 0 0 r 0 0 0 0 r  −  0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1  det  r 0 0 −1 0 r 0 −1 0 0 r −1 0 0 0 r − 1  e(r) = r4 − r3 = 0 r3(r − 1) = 0 r3 = 0 or r − 1 = 0 r = 0 or r = 1 substituting a0 and a′ in equation (75) and solving for r, the values of r are obtained as 0 and 1. the block method equations (24)-(33) are zero-stable, since from (75), ρ(r) = 0, satisfy |r j| ≤ 1, j = 1 and for those roots with |r j| = 1,the multiplicity does not exceed 3. the zero stability of equation(51)(54) a0 =  1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1  a′ =  0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1  b =  259 70858 −53 30240 11 22566 −2 173719 176 4725 −61 670 16 5103 −19 255150 333 2800 −9 1120 13 1680 −1 5600 144 175 −9 70 16 35 11 1050  d =  0 0 0 6918185 0 0 0 23311749 0 0 0 27560 0 0 0 635  ρ(r) = det ( ra0 − a′ ) (75)  r 0 0 0 0 r 0 0 0 0 r 0 0 0 0 r  −  0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1  det  r 0 0 −1 0 r 0 −1 0 0 r −1 0 0 0 r − 1  e(r) = r4 − r3 = 0 r3(r − 1) = 0 13 joseph et al. / j. nig. soc. phys. sci. 5 (2023) 865 14 figure 1. region of absolute stability for one-step with off-step points 14 , 1 2 and 34 r3 = 0 or r − 1 = 0 r = 0 or r = 1 substituting a0 and a′ in the equation (76) and solving for r, the values of r are obtained as 0 and 1. the block method equation (51)-(54) are zero-stable, since from (76), ρ(r) = 0, satisfy |r j| ≤ 1, j = 1 and for those roots with |r j| = 1, the multiplicity does not exceed 3. 4.3. convergence of the methods by the theorem of dahlguist in [17], the necessary and sufficient condition for an lmm to be convergent, is that, it is consistent and zero-stable. the methods satisfy the two conditions stated in (70)-(74), thus the methods are convergent. 4.4. region of absolute stability (ras) for the one-step with off-step points 14 , 1 2 and 3 4 , we have y n+ 34 + 3y n+ 14 − 3y n+ 12 − yn = h3 15360 ( fn + 126 fn+ 12 − 4 f n+ 34 + fn+1 ) h(z) = 15360 ( z 3 4 − 1 + 3z 1 4 − 3z 1 2 ) z + 126z 1 2 − 4z 3 4 + 1 h(θ) = 15360 ( ei 3 4 θ − 3ei 1 2 θ + 3ei 1 4 θ ) 126ei 1 2 θ − 4ei 3 4 θ + eiθ + 1 the ras is shown in figure 1 for the two-steps with off-step points 13 and 2 3 , we have yn+1 + 3yn+ 13 − 3y n+ 23 − yn = h3 97200 ( 10 fn + 1764 fn+ 13 + 1845 f n+ 23 − 20 fn+1 + fn+2 ) h(z) = 97200(z + 3z 1 3 − 3z 2 3 − 1) 10 + 1764z 1 3 + 1845z 2 3 − 20z + z2 h(θ) = 97200(eiθ + 3ei 1 3 θ − 3ei 2 3 θ − 1) 10 + 1764ei 1 3 θ + 1845ei 2 3 θ − 20eiθ + ei2θ figure 2. region of absolute stability for two-steps with off-step points 13 and 23 the ras is shown in the figure below 5. application of the methods problem 1 consider the nonlinear genesio equation that governs chaotic system [12]. x′′′(t) + ax′′(t) + bx′(t) = x2(t) − c x(t) x(0) = 0.2, x′(0) = −0.3, x′′(0) = 0.1, t ∈ [0, 1] where a = 1.2, b = 2.29 and c = 6 are positive constants that satisfy the condition: ab < c for the existence of the solution. problem 2 we consider the problem of thin film flow earlier studied by [11] and sourced from [19]: y′′′ = y−k, y(0) = y′(0) = y′′(0) = 1 for case k = 2, 3. we choose h = 0.1 for its solution. problem 3 here, the constant coefficient homogeneous problem sourced from [13] is considered. y′′′ + y′ = 0 y(0) = 0, y′(0) = 1, y′′(0) = 2 whose analytic solution is y(x) = 2(1 − cos x) + sin x is solved with step size h = 0.1. 6. discussion of results the tables shown above present the numerical solutions of the proposed methods. problem one, considered by [12], has 14 joseph et al. / j. nig. soc. phys. sci. 5 (2023) 865 15 table 1. results for problem 1 analytical one-step with two-steps with solution v = 14 , 1 2 , 3 4 v = 1 3 , 2 3 0.1 0.170440346269364 0.170440347007720 0.170440345967497 0.2 0.141582173138664 0.141582172011890 0.141582152176304 0.3 0.113282963581607 0.113282961614997 0.113282616429441 0.4 0.085554524922736 0.085554528032768 0.085553513248216 0.5 0.058543682864593 0.058543687069665 0.058541676426054 0.6 0.032510877478247 0.032510880258777 0.032507616958020 0.7 0.007806854082744 0.007806857068552 0.007802314622899 0.8 -0.015152336804258 -0.015152330175052 -0.015158060443852 0.9 -0.035911645118586 -0.035911630900664 -0.035918123711900 10 -0.054004107797261 -0.054004083268490 -0.054010808999158 table 2. results for problem 2 analytical one-step with two-steps with [11] solution v = 14 , 1 2 , 3 4 v = 1 3 , 2 3 0.2 1.22121001337746 1.221210004612670 1.221210004446210 1.221216123 0.4 1.48883473296637 1.488834780302660 1.488834776064700 1.488884596 0.6 1.80736134919721 1.807361398646870 1.807361383887600 1.807527232 0.8 2.17981922624938 2.179819235535390 2.179819203068140 2.180203564 10 2.60827491835218 2.608274869934920 2.608274812039350 2.609006355 table 3. results for problem 3 exact one-step with two-steps with solution v = 14 , 1 2 , 3 4 v = 1 3 , 2 3 0.1 0.109825086090778 0.109825086090808 0.109825086091021 0.2 0.238536175112581 0.238536175016773 0.238536175109910 0.3 0.384847228410130 0.384847228107163 0.384847228371941 0.4 0.547296354302880 0.547296353668227 0.547296354184755 0.5 0.724260414823453 0.724260413721916 0.724260414556519 0.6 0.913971243575675 0.913971241864159 0.913971243082311 0.7 1.114533312668710 1.114533310199220 1.114533311853540 0.8 1.323942672205190 1.323942668827970 1.323942670968180 0.9 1.540106973086150 1.540106968652950 1.540106971317480 1.0 1.760866373071620 1.760866367438980 1.760866370661570 table 4. error for problem 1 one-step with two-steps v = 14 , 1 2 , 3 4 with v = 1 3 , 2 3 0.1 7.383560000 ×10−10 3.01867000 ×10−10 0.2 1.126774000 ×10−9 2.09623600 ×10−8 0.3 1.96661000 ×10−9 3.471521660 ×10−7 0.4 3.1100320 ×10−9 1.011674520 ×10−6 0.5 4.20507200 ×10−9 2.006438539 ×10−6 0.6 2.78053000 ×10−9 3.260520227 ×10−6 0.7 2.98580800×10−9 4.539459845 ×10−6 0.8 6.62920600 ×10−9 5.723639594 ×10−6 0.9 1.42179220 ×10−8 6.478593314 ×10−6 1.0 2.45287710 ×10−8 6.701201897 ×10−6 no exact solution and, the solution of [12] is not available for comparison. however, exact numerical solutions were generated via maple 18 [23], and the solutions of the two proposed methods and their errors are compared in tables 1 and 4 respectively. tables 2 and 3 present the solutions to problems 2 and 3, respectively, and in tables 5 and 6, error comparisons with existing methods are displayed. it is obvious that the proposed methods performed favourably. 7. conclusion two unique hybrid algorithms have been proposed as initial value problem solvers. these self-starting algorithms effectively and accurately handle third-order ivps. on investigation, the schemes were found to be convergent and accurate. the approaches are highly recommended for solving third-order ivps 15 joseph et al. / j. nig. soc. phys. sci. 5 (2023) 865 16 table 5. error for problem 2 one-step with two steps error in [13] v = 14 , 1 2 , 3 4 withv = 1 3 , 2 3 order p = 4 0.2 8.7647900 ×10−9 8.9312500 ×10−9 1.070000 ×10−6 0.4 4.73362900 ×10−8 4.30983000 ×10−8 4.130000 ×10−7 0.6 4.9449660 ×10−8 3.46903000 ×10−8 8.5100000 ×10−7 0.8 9.2860100 ×10−9 2.31812400 ×10−8 1.7100000 ×10−6 1.0 4.8417260 ×10−8 1.063128300 ×10−7 3.8600000 ×10−6 table 6. error for problem 3 one-step two-steps error in [13] v = 14 , 1 2 , 3 4 with v = 1 3 , 2 3 0.1 −3.000000 × 10−14 −1.3100000 × 10−14 1.60880000 × 10−9 0.2 9.580800000 × 10−11 1.44400000 × 10−12 1.03870000 × 10−8 0.3 3.029670000 × 10−10 2.07030000 × 10−11 2.95720000 × 10−8 0.4 6.346530000 × 10−10 6.4005000 × 10−11 2.31470000 × 10−7 0.5 1.101537000 × 10−9 1.430840000 × 10−10 4.54200000 × 10−7 0.6 1.711516000 × 10−9 2.621400000 × 10−10 1.47460000 × 10−6 0.7 2.469490000 × 10−9 4.259500000 × 10−10 2.87340000 × 10−6 0.8 3.37722000 × 10−9 6.357000000 × 10−10 4.68260000 × 10−6 0.9 4.4332000000 × 10−9 8.89120000 × 10−10 6.92170000 × 10−6 1.0 5.6326400000 × 10−9 1.184590000 × 10−9 9.59740000 × 10−6 as a numerical scheme. the development of multi-order ivp solvers will be considered in a future study. references [1] r. a. bun & y. d. vasil’yev, “a numerical method for solving differential equations of any orders”, journal of computer mathematical physics 32 (1992) 3. 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[14] e. o. adeyefa & j. o. kuboye, “derivation of new numerical model capable of solving second and third order ordinary differential equations directly”, iaeng international journal of applied mathematics 50 (2020) 2. [15] j. o. kuboye & e. o adeyefa, “new development numerical formula for solution of first and higher order ordinary differential equations”, journal of interdisciplinary mathematics 25 (2022) 2. https://doi.org/10.1080/09720502.2021.1925453. [16] s. o. fatunla, numerical methods for initial value problems in ordinary differential equations, academic press incorporation, harcourt brace, jovanovich publishers, new york (1988). [17] g. dahlquist, some properties of linear multistep and one leg method for ordinary differential equations. department of computer science, royal institute of technology, stockholm (1979). [18] j. d. lambert, computational methods in ordinary differential equation, john wiley and sons, new york (1973). https://doi.org/10.1002/zamm.19740540726 [19] k. lee, i. fudziah & s. norazak, “an accurate block hybrid collocation method for third order ordinary differential equations”, journal of applied mathematics, hindawi publishing corporation 549597 (2014) 9, https://doi.org/10.1155/2014/549597 [20] g. dahlquist, “on one-leg multistep method”, siam journal on numerical analysis 20 (1983) 1130. [21] p. henrichi, discrete variable methods in ode, john wiley and sons, new york (1962). [22] l. f. shampine & h. a. watts, “block implicit one-step methods”, journal of numerical mathematics 23 (1972) 731. https://doi.org/10.1007/bf01932819 [23] a. s. olagunju & f. l. joseph, “construction of functions for fractional derivatives using matlab”, journal of advances in mathematics & computer science 36 (2021) 6. 16 j. nig. soc. phys. sci. 4 (2022) 205–213 journal of the nigerian society of physical sciences analysis of adenanthera pavonine l. (febaceae) pod and seed as potential pyrolysis feedstock for energy production olugbenga oludayo oluwasina∗ department of chemistry, federal university of technology akure, nigeria abstract though countless possible bioenergy feedstocks are available, the lack of information on their characteristics has made them unusable for industrial purposes. this study revealed the bioenergy potential of seed and pod of adenanthera pavonine by analyzing their physicochemical, ultimate, proximate, kinetic, thermodynamic, thermal, and higher heat value. the seed presented 19.90%, 2.12%, 24.40% and 14.73% cellulose, hemicellulose, lignin and extractive respectively, while the pod has 21.35%, 25.15%, 23.50% and 11.63%. from the proximate analysis the pod has higher volatile matter (92.79%), and fixed carbon (1.40%), while the seed has higher moisture (6.36%), ash (0.84%), and higher heat value (18.63 mj kg−1). the kinetic and thermodynamics results present the seed with ea 23.73 kjmol-1, ∆h 14.06 kjmol-1, ∆g 10.74 kjmol-1 and ∆s -78 jmol-1, while the pod has 21.3 kjmol-1, ∆h 12.20 kjmol-1, ∆g 10.98 kjmol-1 and ∆s -83 jmol-1. the probable energy blockade between ea and ∆h for the seed and pod was 9.72. the high value of h: c and low o: c, with the higher heating values recorded for the pod and seed, presented them as better biofuel candidates. the study results have supplied necessary information for the industrial utilization of adenanthera pavonine seed and pod as valuable feedstocks for bioenergy conversion. doi:10.46481/jnsps.2022.555 keywords: adenanthera pavonine, composition analysis, ultimate analysis, proximate analysis, kinetic, thermodynamics article history : received: 30 december 2021 received in revised form: 25 february 2022 accepted for publication: 25 februrary 2022 published: 29 may 2022 c©2022 journal of the nigerian society of physical sciences. all rights reserved. communicated by: e. a. emile 1. introduction an increase in world population coupled with rapid industrialization and urbanization is causing a reduction in fossil fuel energy resources. also, the negative environmental greenhouse gas effect of fossil fuel is calling for alternative replacement for fossil fuel energy. plant biomass is renewable, abundant, and easily adaptable is being considered as the capable replacement for fossil fuel ∗corresponding author tel. no: email address: oooluwasina@futa.edu.ng (olugbenga oludayo oluwasina ) energy. it can supply a hygienic, renewable, dependable, and low-carbon print fuel. about thirteen percent of the world’s energy can be derived from biomass, is converted directly to heat, biofuels, and value-added chemicals through various reaction pathways [1]. nevertheless, using biomass as a raw material for fuel generation has its drawback, there are numerous character differences between and within various biomass types and species. therefore, knowledge of the physicochemical properties and the energy value of biomass are very vital for its proper utilization [2]. the chemical constituent of biomass such as moisture, cellulose, lignin, hemicellulose, extractive, and ash play different vital roles in determining the suitability of the mate205 oluwasina / j. nig. soc. phys. sci. 4 (2022) 205–213 206 rial for particular end-use. the moisture content of biomass can affect the higher heating value if too high and also impact negatively the bio-oil properties. the elemental composition of fuel-derived biomass is very important for the control and prevention of ash fouling and slag formation which could affect thermochemical processing delivery pipes. biomass with high cellulose or hemicellulose content would produce a higher yield of bio-oil, while those with higher lignin content will give rise higher yield of char [3]. to obtain the inherent energy content of biomass, a thermochemical processing system is needed, this cannot be constructed without the knowledge of its thermal behaviour and chemical composition [4]. thus, information on the pyrolytic behavior of the biomass from its thermogravimetric (tg) characterization is essential before the industrial utilization in a thermochemical plant. parameters such as rate of biomass decomposition and the activation energy valuable for engineering design of thermochemical conversion unit are obtained from tg using coats and redfern model [5]. thermodynamic parameters; entropy (∆s), enthalpy (∆h), and gibbs free energy (∆g) are useful to verify the visibility of the pyrolysis occurrence, operation condition, and possible products proximate analysis is a valuable means of defining and envisaging the heating values of biomass, it helps to reveal the fuel behavior of the biomass at various stages during combustion [6]. whereas, the biomass fuel efficiency and cleanness can be predicted using both the proximate and ultimate analysis [7]. higher heating value (hhv) is used to quantify the energy worth of biomass fuel. experimentally, hhv of biomass fuel can be determined using a bomb calorimeter, however, because of the cost and time-consuming nature of the procedure, various authors have utilized data obtained from the proximate and ultimate analysis to determine the hhv of different biomass through the use of empirical models [8-12]. adenanthera pavonina l. is a wild underutilized plant that can be exploited for firewood, shade, ornamental, medicinal, and dyeing purposes [13]. the ability of adenanthera pavonine as low-cost biosorption for pb (ii) and cd (ii) from wastewater has been demonstrated and it was reported that the seed has a good sorption capacity for the two metallic ions [14]. extracted galactomannan from adenanthera pavonina l has been reported as a good binder in cement-based hydroxyapatite composite [15]. medicinal importance of the various parts of the plant has been documented by various authors; anti-inflammatory [16], anti-blood pressure [17], and anti-diabetic [18]. due to the lack of data on the biofuel potential of pod and seed of adenanthera pavonine, this research work is aimed at the exhaustive analysis of the pod and seed of adenanthera pavonina to reveal their bioenergy possibility and provide data for designing a thermochemical conversion system for the industrial utilization. 2. material and method 2.1. material samples were obtained from adenanthera pavonine tree at the federal university of technology, akure, ondo state, nigeria. contaminants were sorted out, the seeds were removed from the pods and the materials were sundry (72 h). the materials were pulverized and sieved to obtain 450 m particle size and then stored in an airtight container at room temperature. 2.2. extractive free sample a biomass (b1) was soxhlet extracted (4 h) serially with (1: 2) ethanol (985%): toluene, ethanol (4 h), distilled water (500ml x 3), then dried (105 oc) to a constant weight (b2) [19]. percentage extractive was calculated thus after a triplicate experiment: extractive content (%) = [ (b1 − b2) b1 ] × 100 (1) 2.3. acid-insoluble lignin to the extracted biomass (e) was added sulphuric acid (72%, 15 ml) at 20 ◦c (2 h). the experiment was stopped by adding deionized water (560 ml), before heating (95 ◦c, 4 h) and then allowed to cool (24 h). the filtered was obtained as acid-soluble lignin and the solid as the acid-insoluble lignin (washed to ph 7) after oven-dried to a known weight (l) [20]. a triplicate experiment was done and acid-insoluble lignin was calculated thus: klason lignin, (%) = l × 100 e (2) 2.4. acid-soluble lignin to acid-soluble lignin filtrate volume (v = 575 ml) was diluted with sulphuric acid (3%) and the ultraviolet absorbance (ab) read at 205 nm, with sulphuric acid (3%) being the reference sample [21]. lignin content in the filtrate (b in g/1000 ml) was calculated as follows: b = ad 100 (3) where: d= the dilution factor of the filtrate, 110 = absortivity or extinction coefficient. the acid acid-soluble lignin was calculated as follows: lignin (%) = 110bv 1000w (4) 2.5. holocellulose a mixture of extractive free biomass (we), distilled water (150 ml), glacial acetic acid (0.2 ml), and sodium chlorite (1 g) was subjected to heat treatment (70 ◦c). the procedure was repeated three times but without the addition of distilled water and biomass. the solid sample was washed free of chlorine, oven-dried (105 oc) to weight (wh) [22]. a triplicate experiment was performed and the holocellulose content was determined as: holocellulose content (%) = [ we wh ] × 100 (5) 206 oluwasina / j. nig. soc. phys. sci. 4 (2022) 205–213 207 2.6. alpha-cellulose the obtained holocellulose was treated with sodium hydroxide (17.5%, 50 ml, 20oc), distilled water (50 ml) was added after 29 min. the solid particle obtained was washed to neutral ph with deionized water [22] and then oven-dried (105 oc) to constant weight (w00). a triplicate experiment was done and cellulose was determined with the following formula: cellulose (%) = [ w9 woo ] × 100 (6) 2.7. instrumental analysis perkin elmer sta 6000, was used for the determination of the sample proximate composition (moisture content, volatile matter, ash content, and fixed carbon). scanning electron microscope (sem) coupled with-energy dispersed x-ray (edx) (fei fib/sem nova 600 nanolab) was used for the morphological characteristic and elemental composition of the samples using the ash component. thermogravimetry analysis (tg/dtg) was conducted (tga ta std q6000) in a nitrogen atmosphere, flow rate 20 ml min−1 at a temperature (30 and 900 oc), and heating rate 10 oc min−1, about 10.0 mg sample was used for the thermal characterization. perkin elmer chns was employed for ultimate analysis determination of percentage carbon, hydrogen and sulfur content, while oxygen (%) was calculated by difference {(o =100 – (c+h+n+s+ash)}[23]. the h/c and o/c ratios were determined using empirical methods [24, 25] as follows: h c = number of h atoms number of c atoms = %h/1 %c/12 (7) o c = number of o atoms number of c atoms = %o/16 %c/12 (8) 2.8. higher heating value higher heating value (hhv) or gross calorific value were using the empirical formula [26] as follows; hhv (m j/kg−1) = 0.0877l + 16.4951 (9) 2.9. kinetic and thermodynamic parameters modified coats and redfern model (equation 10) [5] was employed for the determination of the activation energy (ea), pre-exponential factor (a), and regression coefficient (r2) through the graph plot of ln [ln (1x)] against 1000/t. thermodynamics parameters such as entropy (∆s), enthalpy (∆h), and gibbs free energy (∆g) were calculated using equations (11-14) [27,28]. different factors such as boltzmann constant kb (1.381 x10−23 j/k), plank constant h (6.626 x 10−34 j/s), and tg peak temperature (tm/k) are usually used in the calculation and other necessary values during the calculation are obtained from the tg data. ln [−ln (1 − x)] = ln art 2 βea − ea rt (10) a = [ β∗ e x p ( e rtm )] rt 2m (11) figure 1: percentage composition of biomass ∆h = ea − rt (12) ∆g = ea + rtmln (kbtm/ha) (13) ∆s = ∆h − ∆g/tm (14) 3. results and discussions 3.1. composition of biomass the seed and pod of adenanthera pavonine displayed distinct characters. the extractive results revealed 13.73 % for the seed while the pod recorded 11.63 %. this is expected because the chemical composition of biomass can be influenced by the nature of such material. it has been reported that both saturated and unsaturated fatty acids are abundant in adenanthera pavonine seed [29]. the percentage extractive presented in this study is higher than 7.78% and 9.75%, but lower than 13.82% to 32.86% reported for different date palm residue [30]. the high extract content of the seed suggests that it can support biomass burning and ignition ability, an added advantage if used in composite briquette compounding. the pod recorded 21.14% cellulose and 23.50% hemicellulose, both are higher than the 19.90% and 21.12% for cellulose and hemicellulose of the seed. on the other hand, the 24.40% lignin content of the seed was higher than 23.50% of the pod. the cellulose content of the seed and pod are lower than 45-50 % for hardwood -and soft wood-cellulose [31], but compared favourably with 12-29% reported for lignocellulosic residue [32], while the hemicellulose was in agreement with 15-35% for hardwood and 20-31% for softwood [31]. however, the lignin content of both the seed and pod was higher than 11-22 reported for lignocellulosic biomass [32]. it can be inferred from the results of the hemicellulose and cellulose of the pod and seed that biomass will experience faster thermal degradation, thus saving energy cost during pyrolysis, though at the expense of higher heating value that the lignin would have supplied. 3.2. proximate analysis of biomass the pod has a lower moisture content of 5.22% against 6.36% of the seed. the higher moisture content of the seed 207 oluwasina / j. nig. soc. phys. sci. 4 (2022) 205–213 208 might be due to its high extractive content (fatty acid, resin, gum) which could have prevented the evaporation of bond water. the reported moisture contents (pod and seed) were lower than 8.00% [33] and 7.02% [34] for lantana camara by different authors. the low moisture content of the pod agreed with the low value reported for the pod acacia mangium amongst all its various parts [35]. the moisture contents of the pod and seed are lower than fifteen percent recommended for pyrolysis required biomass [36]. higher moisture content is expected to engineer microbial degradation, which will invariably affect the fuel quality of the biomass. thus, the seed and the pod would be good fuel oil-producing feedstock [37]. high volatile matter is desirable in pyrolysis feedstock because it would aid ignition and support the biomass burning ability. after all, they are expected to produce a high quantity of bio-oil [38]. there is no significant difference between the volatile matter of the seed (91.79%) and pod (92.79%), suggesting that those materials would have good ignition ability. the higher volatile matter reported in this study was not an isolated case, higher volatile values have been reported for the pod of acacia mangium [35] and bauhinia monandra [39]. the volatile matter the seed and pod here reported is higher than 69.82 to 74. 85% for various part acacia mangium [35], 74.30% to 87.50% for date palm [30]. the difference in the volatile matter of the different biomass could be linked to the plant types, nature of the biomass, and the state of the biomass before analysis. the oxides of the elements present in the biomass represent the ash content, which can be determined as the solid inorganic component after the thermal degradation of the biomass. the worth and durability of thermochemical products and plant-unit are mostly influenced by the ash feedstock content. slag development during the thermochemical process would corrode the system [4], the corrosion indirectly affects the quality and quantity of the biofuel, increasing the energy need and cost of system maintenance. thus, feedstock with minima ash content is desirable for thermochemical conversion. it has been affirmed that biomass feedstock with ash values ranging from 1.41 to 2.69% is a good candidate for bioenergy utilization [35]. hence, ash values of 0.84% (seed) and pod 0.59% (pod) make these materials better feedstock and it is envisaged that the associated ash problem of biomass would be eliminated or minimized if these materials are utilized as thermochemical feedstock. the pod and the seed ash values are lower than 4.5 to 10.5% switchgrass [40] and 2.68% grapevine pruning, 1.94% olive pruning, and 2.25% riverbank residue [41]. the ash differences of the various biomass can be ascribed to various factors such as soil type, ecological reason, plant types, and part. the thermochemical undecomposed part of the biomass is the fixed carbon [35], which can be employed in biochar preparation. however, low biochar feedstock is preferable for thermochemical biofuel application because of the expected high bio-oil yield. the two materials examined in this study; pod and seed presented 1.01% and 1.40% fixed carbon respectively, probably the reason for the high volatile matter content of those materials. it is therefore assumed that the pod and seed of adenanthera pavonine, would be a good candidate for bioenergy production. the fixed carbon of this study is very low compared with 14.47 to 18.31% from different parts of acacia mangium [35], but are higher than 0.34% and 0.623% for delonix regia seed and pod. from the hhv analysis of the biomass using [26] formula for non-wood lignocellulosic fuels. it was revealed that the pod and the seed have higher but different values; the seed has 18.63 hhv values, while the pod recorded 18.56. the higher value of the seed could be the effect of its high lignin and extractive values, both of which have been documented to increase hhv [26]. the hhv reported in this study are lesser than 20-25 mj kg-1 of sweet sorghum [42], but are higher 15.00 mj kg1 for camel grass [43], 15.10 mj kg−1 of para grass [44], and 17.20 mj kg-1 of a. donax [45], all of which are notable fuel biomass. thus, the seed and pod of adenanthera pavonine can be adjudged to be good energy biomass. 3.3. ultimate analysis of biomass the chns carbon, hydrogen, sulphur, and oxygen values for the seed are 38.26%, 6.30%, 1.92%, and 52.68% against 43.01%, 5.785. 1.63% and 48.99% of the pod. all the obtained values are in agreement with literature values for various biomass [26]. the pod is richer in carbon and oxygen, while the seed is richer in hydrogen and sulphur. the fuel potential of the biomass can be predicted from carbon, oxygen, and hydrogen, making those elements important in the biomass thermochemical process. the knowledge of sulphur content would help in preventing possible environmental pollution from its gaseous products. the h: c ratio of 1.97 and 1.61 were recorded for the seed and the pod respectively. these values are within the range of 1.46 to 2.18 for different non-edible biomass [46] and 1.54 to 1.67 for various walnut shells [47]. also, 1.03 and 0.85 recorded for the o: c ratio is within the range of 0.72 to 0.99 for different date palm residues [30]. the higher h: c fraction and lower o: c fraction of the pod and seed, indicate them as better biofuel candidates [48] 3.4. elemental analysis of biomass the mineral content of biomaterial for bio-energy research is very essential, considering the negative effect those metals could have on the desired products and the processing unit. the edx analysis of the ash presents the presence of k, na, ca, mg, p, and o in the seed while na was not detected in the seed, si and other elements in the seed are detected. the seed has a higher content of na (1.330), mg (7.36), and p (1.81), while the pod has k (44.71), and si (0.93). the low level of silica in the pod and its absence in the seed, coupled with the absence of heavy metals in both the seed and the pod signifies the safeness of these materials as a good feedstock for bioenergy production. the ash is rich in essential minerals (k, na, mg, and p) for plant growth; therefore, the ash can be utilized as a biofertilizer. however, alkali earth metals are known to induce in-situ reactions such as cracking which often led to low oil yield and high gas production [49]. also, those metals can inhibit microorganisms if those biomasses are used biochemical for purposes [50]. hence, there is a need for pretreatment of the biomass to reduce the metal content during thermochemical conversion 208 oluwasina / j. nig. soc. phys. sci. 4 (2022) 205–213 209 figure 2: proximate analysis of biomass (fixed carbon calculated by difference) figure 3: ultimate analysis of biomass (o* calculated by difference) 3.5. thermogravimetry analysis the major chemical composition of biomass is cellulose, hemicellulose, lignin, extractive, and ash. the chemical compositional similarity and the proximity in the proximate, ultimate and elemental analyses of the two analyzed materials in this study, could be responsible for the sameness of their different tg/dtg curves. the curves presented the initial weight loss from zero to 150 ◦c ascribed to moisture loss. the second weight loss (150 400 ◦c) is due to the decomposition of hemicellulose and cellulose. while the pod presented a single peak at this temperature, the seed presented about two peaks, which suggests that there is hemicellulose/cellulose decomposition overlap in the pod but this was not so in the seed that presented two distinct peaks as revealed by the dtga. lignin being thermally stable than cellulose and hemicellulose were the last to be degraded. this experienced a long duration from 400 900◦c. the seed experienced higher degradation energy at 357.36 ◦c, while that of the pod was at 353.28◦c, indicating 209 oluwasina / j. nig. soc. phys. sci. 4 (2022) 205–213 210 figure 4: tg/dtg of pod (a) and seed (b) table 1: edx analysis result element (%) seed pod potassium 36.66 44.71 sodium 1.33 not detected calcium 6.11 4.5 magnesium 7.36 1.98 silica not detected 0.93 phosphorous 10.81 1.00 oxygen 37.73 46.83 that higher energy is required for degradation of the seed. 3.6. morphology characterization the sem investigation revealed a noticeable variance in the morphology of the seed and the pod of adenanthera pavonine. when the pod appeared as a smooth surface, ball-like, and not too densely packed material, the seed presents a rough surface, flat-like, densely packed, and irregular shape material. though, the two materials are composed of the same chemical composition the differences in the amount present and their location could have influenced the surface morphology 3.7. kinetic and thermodynamics activation energy signifies the minimum energy that biomass must acquire before its main constituents (such as cellulose, hemicellulose, and lignin) can be converted to various pyrolytic products such as gas, liquid, and solid char. the higher activation energy (23.78kjmol−1) of the seed over that of the pod (21.92kjmol−1) can be linked to its high lignin content. thermally lignin is stable than cellulose and hemicellulose, and higher energy is required in breaking the lignin bond. the high ea of the seed agreed with the literature finding that the higher ea of kaner seed was as a result of its volatile constituents such as fatty acid [51]. ea observed in this study are lower than 221-229 kj mol−1 (rice husk) [52] and tobacco residue [53]. the difference between the ea and ∆h values for the seed and pod was found to be 9.72kjmol−1 which is higher than approximately 5 kjmol−1 reported to signify fastness in converting the biomass to products [54]. the higher ea of the seed can be supported from the thermal analysis in which the seed’s highest degradation temperature was higher (357.36 ◦c) than that of the pod (353.28 ◦c), revealing that much energy was consumed by the seed in breaking its components. the energy released by biomass during pyrolysis is quantified by gibbs free energy and values of 10.74 kjmol-1 (seed) and 10.98 kjmol−1 (pod) suggest a non-spontaneous process. the result implies that the pod will supply more bioenergy than the seed. the values obtained in this study were lower than those reported by [54]. the results of ∆h follow the same pattern as that of ea, the seed has the higher value. though, the positive ∆h predicts that heat would be absorbed during the breaking down of the precursor components. the lower ∆h value of the pod indicates that its products can be easily separated during pyrolysis. the higher ∆h recorded by the seed might be associated with its chemical composition because it contains high lignin and extractive contents. both of which will require extra energy in bond breaking. the ∆h reported in this study was lower than those reported for kaner seed, flaxseed, and microalgae with a 5 to 20 ◦c/min heat rate [51]. the entropy ∆s reflects the disorderliness the biomass precursor underwent under chemical or physical treatment owing to thermal treatment [55]. the higher ∆s of the seed implies that it would be more reactive and require less reaction time to reach the activation complex, while the pod will experience the opposite [28]. undoubtedly, the reactivity of the seed is not unconnected with its higher ea and ∆h. 4. conclusion the bioenergy ability of the seed and pod of adenanthera pavonine has been evaluated. the chemical composition of both ma210 oluwasina / j. nig. soc. phys. sci. 4 (2022) 205–213 211 (a) figure 5: sem of the pod (a) and seed (b) table 2: kinetics and thermodynamics results seed pod ea-∆h ea 23.78 kj mol−1 21.92 kj mol−1 9.72 a 0.01648985sec 0.0106400833sec r2 0.9432 0.9433 ∆s -79.82 j mol−1 -83.466 j mol−1 ∆h 14.06 kj mol−1 12.20 kj mol−1 9.72 ∆g 10.74 kj mol−1 10.98 kj mol−1 terials revealed the presence of lignocellulose constituents that can support 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[55] y. xiang, y. xiang, y. & l. wang, “kinetics of the thermal decomposition of poplar sawdust. energy sources, part a: recovery”, util environ eff 39 (2017) 213. 213 j. nig. soc. phys. sci. 4 (2022) 891 journal of the nigerian society of physical sciences levels of zinc (zn), copper (cu), iron (fe), and cadmium (cd) in soil, rice stalk, and oryza sativa grain in ishiagu rice field, ebonyi state, nigeria; human health risk d. n. ajaha, e. agboeze id a,∗, j. n. ihediohab, e. chukwudi-madua, c. c. chimea adepartment of industrial chemistry, enugu state university of science and technology, enugu state, nigeria bdepartment of pure and industrial chemistry, university of nigeria nsukka, enugu state, nigeria abstract levels of heavy metals (zn, cu, fe, cd) were determined in soil, rice grain, and rice stalk from federal college of agriculture ishiagu rice field, ebonyi state, nigeria. the dried samples were digested with a 1: 3 (hno3: hcl) mixture and analyzed with atomic absorption spectrophotometer (aas). the mean concentration of the metals in the soil before planting, soil after harvest, and rice grain were as follows: zn (7.28, 11.33 and24.90); cu (3.40, 4.64 and 4.14); fe (803.04, 735.47 and 107.78); cd (1.14, nd and nd) and were all within fepa and fao/who limits. the daily intake values for a 60 kg adult were zn (0.04), cu (0.01), and fe (0.18) and were all below the recommended limits by codex alimentarius standards. the target hazard quotient (thq) for zn, cu, and fe was less than one (1<), and the total hazard index was less than 1, indicating that the population will not be exposed to the potential health risk from these metals. however, the metal levels should be monitored to ensure they stay at harmless levels. doi:10.46481/jnsps.2022.891 keywords: heavy metals, rice, soil, health risk, environmental pollution article history : received: 26 june 2022 received in revised form: 28 july 2022 accepted for publication: 14 august 2022 published: 25 september 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: e. a. emile 1. introduction advancement and mechanization of urban growth have promoted socio-economic development in developing countries like nigeria and the world at large. but besides all the positive effects, they cause ”environmental pollution”. soil contamination ∗corresponding author tel. no: +234 9039239802 email address: emmanuel.agboeze@gmail.com (e. agboeze id ) by heavy metals is a significant environmental concern worldwide, as it concerns human health and food security/quality [15]. according to yap et al. [5], the significant sources of heavy metals include anthropogenic environmental activities like mining, smelting processes, steel and iron industries, chemical industries, agriculture, and domestic activities [4]. these sources outweigh natural sources like weathering of the parent material and volcanic eruptions [5]. recently, concern has been raised about possible contamination of the crop (rice) by heavy metals. the plants can absorb these heavy metals in the soil through 1 https://orcid.org/0000-0001-5337-117x https://orcid.org/0000-0001-5337-117x ajah et al. / j. nig. soc. phys. sci. 4 (2022) 891 2 their roots, stems, or leaves and accumulate in their organs [6, 7]. in specific concentrations, some of these heavy metals are essential to plants, but in higher concentrations can become toxic. these metals are a health hazard to living organisms due to their persistence, non-biodegradable, and non-thermal degradable environmental characteristics [5]. the uptake of metals in excessive amounts may either cause harm to the plant or enter the food chain and accumulate when these plants are taken up. metals accumulated in the human body through the food chain cause diseases like lung cancer and damage the central nervous system, kidney, and liver [8-11]. heavy metals, which can pose severe hazards to humans and the environment, are increasingly being found in the environment due to the growth of mining, smelting, and other industrial operations. the quality of the surrounding air, soil, and water bodies is affected by pollution from heavy metals like lead (pb), arsenic (as), nickel (ni), cadmium (cd), copper (cu), and zinc (zn), which endangers the lives of both animals and humans through the food chain [12]. evaluation of potential impacts on human health in contaminated environmental media is part of assessing the risk to human health [4]. human exposure to contaminants, the type of contaminants, and the affected person’s susceptibility all determine how they affect human health [4]. health impacts may include increased risk of cancer, high blood pressure, acute neurological abnormalities in fetuses, organ dysfunction, respiratory issues, physical and mental illness, shortened life expectancy, and immune system deterioration [12]. this study is intended to determine the levels of zinc, copper, iron, and cadmium in soil, rice grain, and stalk obtained from a rice field in a local area in nigeria (federal college of agriculture ishiagu, ebonyi state) through the determination of the physicochemical properties of the soil, evaluation of the level of contamination, and health risk assessment. 2. experimental 2.1. area of study see figure 1. 2.2. sample collection the land (federal college of agriculture, ishiagu, rice field) is about 6 hectares, i.e. (240×75m). the land was divided into 4parts (a, b, c, and d). 2.3. soil sample collection eight (8) composite soil samples were collected from four sections of the rice farm before plantation. each composite is made of 5 grab samples randomly collected from each area of the rice farm. in the same way, twelve (12) composite soil samples were collected after the rice harvest. four (4) grab samples were randomly collected from different parts of the land which has not undergone any agricultural practice (a fallow land) as the soil control. these soil samples were taken from 0-20cm depth from the surface of the ground with a clean machete, and the collected samples were placed in well-labeled polyethylene bags. 2.4. rice sample collection twelve (12) composite rice plant samples were collected from four different sections of the farm. each composite consists of five (5) randomly selected rice plants concerning the soil already sampled. the rice grains from each composite were separated from the stalk. the rice grains and stalks were each placed in polyethylene bags. 2.5. sample preparation 2.5.1. soil preparation the soil samples were air-dried, crushed, and passed through a 0.16mm sieve to remove gravel-sized materials and then homogenized with a mortar and pestle. the samples were then stored in polyethylene bags with labels for analysis. 2.5.2. rice preparation the rice grain samples were refined to remove the husk using mortar and pestle and winnowed with a tray pan. the polished rice grains were then stored in polyethylene bags well labeled. the rice stems were finely chopped with the knife on a wooden platform and stored in labeled polyethylene bags. 2.5.3. ash two (2) ash samples were collected from 2 different abattoir sites and were mixed to get a homogenous sample. the ash was used to mix the rice seeds before broadcasting to scare birds away or prevent them from eating them. 2.5.4. sample size the overall sample for analysis became: 8 for soil before planting + 12 for soil after harvest + 12 for rice grain +12 for rice stem +4 for soil control, and + 1 for ash, totaling 49 samples. 2.5.5. digestion of soil samples, rice grain samples, and rice stalk samples 10g each of the soil and rice grain samples, 5g of the rice stalk were measured into different digestion flasks, and 40ml of aquaregia (nitric acid and hcl in the ratio of 1:3) was added. the mixtures were then heated on a heating mantle to boil until the fume became clear and were allowed to cool. each of the various digested solutions was diluted with deionized water and filtered through a filter paper into 100ml volumetric flasks and were made up to the mark with deionized water. the solutions were finally kept in test tubes. concentrations of zn, cu, fe, and cd were determined using the aa-7000 atomic spectroscopy. statistical analysis of the data obtained was done using spss version 17 for windows and palisade analysis. 2.6. quality assurance the precision of the analytical procedure was investigated by carrying out recovery experiments. this was done by determining the metals’ concentration in duplicate spiked and unspiked rice samples. spiking was done by adding 1 ml of 2ppm and 2 ajah et al. / j. nig. soc. phys. sci. 4 (2022) 891 3 figure 1. area of study 4ppm metal solution to 2 g of samples, which were later subjected to the digestion procedure. % recovery = a − b × 100 c , (1) where a denotes concentration in the spiked sample, b represents concentration in the unspiked sample and c denotes concentration of the metal ion added. 2.6.1. risk assessment this process evaluates the potential effects of a contaminant on humans from doses received through one or more exposure pathways. the health risk from rice consumption was assessed using the target hazard quotient (thq) and total hazard index (thi) [13]. the target hazard quotient (thq) is the ratio of the determined dose of a pollutant to a reference dose level. in contrast, the total hazard index (thi) evaluates the potential risk of adverse health effects from a mixture of chemical constituents in rice. if this ratio is less than 1, the exposed population is unlikely to experience noticeable adverse effects [14-16]. the following equation calculated the thq and thi values for the metals: t hq = ef × ed × ir × c r f d × bw × at × 10−3 (2) t hi = t hq1 + t hq2 + · · · + t hqn (3) where thq is the target hazard quotient, ef is the exposure frequency (365 days /year), ed is the exposure duration (54 years), ir is the rice ingestion(kg/person/day), and c is the metal concentration in rice (mg/kg), rfd is the oral reference dose (mg/kg/day) and at is the average time for non–carcinogens (365 days /year× ed). the oral reference dose for metals (mg/kg/day) were: cu(0.040), zn(0.300), fe(0.700) [17]. the calculated thq value is also shown in table 2. it is less than 1, thus indicating that the nigerian population has not been exposed to the potential health risk of dietary copper via rice consumption. 2.7. data analysis data obtained were analyzed with spss 17 for windows. correlation coefficient analysis was carried out to establish the cor3 ajah et al. / j. nig. soc. phys. sci. 4 (2022) 891 4 relation pattern of various metal pairs in soil and rice grain samples. the student’s t-test was carried out to determine any significant difference in the metal concentration in the soil before planting and after harvest at p<0.05. 3. results and discussion table 1 shows recoveries ranging from 84% to 128% with a mean of 96% and precision of 5.7, a value lower than 10% indicating high accuracy. 3.1. heavy metals concentration in soil and rice grain 3.1.1. copper in soil the concentrations of copper (table 2) in the soil samples from the control portion ranged from 1.67 mg/kg 3.33 mg/kg. the lowest value (1.67 mg/kg) was recorded at control2, while the highest (3.33 mg/kg) was at control4. these concentrations gave a mean ± sd of 2.22 ± 0 .75 (table 3). as in table 2, the concentration of copper in the ”soil before planting” ranged between 2.39 mg/kg 5.18 mg/kg. the lowest value (2.39 mg/kg) was recorded at d2 and the highest (5.18 mg/kg) at a1. these gave a mean ± sd of 3.40 ± 0.91 (table 3). also, the concentrations of copper (table 2) in ”soil after harvest” ranged from 3.70 mg/kg to 6.07mg/kg. a3 had the lowest value (3.70mg/kg), and b2 had the highest value (6.07 mg/kg). the mean ± sd was 4.64 ± 0.69 (table 3). it compared the mean concentrations of (”soil before planting” and ”soil after harvest”) (3.40 mg/kg and 4.64 mg/kg, respectively) with the mean concentration of the ”soil control” (2.22 mg/kg). it can be seen that the soil was slightly contaminated with copper. this contamination could be attributed to the use of pesticides or herbicides. this possibility was supported by the fact that most pesticides used in agricultural soils were based on compounds containing cu, hg, mn, pb, or zn [18]. the results compared with other researchers; zhuang reported 502 mg/kg as the mean concentration of copper in agricultural soils around the daboshan mine in guandong, china. also, song recorded 8.41-148.73 mg/kg as the mean concentration of copper in agricultural soils of suxian county, south china. this variation of results might be due to the copper concentrations in irrigation water and other agronomic practices in the respective areas [16, 19]. however, the mean concentrations of cu in (soil before planting (3.40) and soil after harvest (4.64) is below the permissible limits; 1. 150 mg/kg as with chinese environmental quality standards (1995) for soils [9]. 2. (70-80) mg/kg as with fepa (1991) guidelines for heavy metals in soil. the low concentrations recorded in this study may be attributed to the continuous removal of copper by rice grown in the field. therefore, the concentration of copper in soil may not harm mice and humans when used for rice production [3]. 3.1.2. copper in rice the concentration of copper in “rice stalk” as in table 2 above ranged between 0.74mg/kg-11.06mg/kg with a1& b3 having the lowest value (0.74 mg/kg) and a3having the highest value(11.06mg/kg). these concentrations gave a mean ± sd of 2.46 ± 2.95, as in table 3. as in table 2 above, the concentration of copper in ”rice grain” ranged between 1.84mg/kg-14.81mg/kg, with a2 having the lowest value (1.84 mg/kg) and c3 having the highest value (14.81 mg/kg). the concentrations gave a mean ± sd value of 4.14 ± 3.92as seen in table 3. in tanzania, machiwa reported the mean concentration of cu in rice collected from different locations to be 3.7mg/kg. also, in guandong, china, zhuang reported the mean concentration of copper in rice as 6.34 mg/kg [16, 20]. however, the mean cu concentrations of the rice samples in this study are within china’s maximum permissible limits (10 mg/kg) [21]. they are also within the fao/who recommended limits (20 mg/kg) for copper in rice grains, which means that the rice is suitable for consumption [21]. 3.1.3. relationships between copper concentration in the soil before planting and soil after harvest statistical analysis with student t-test indicates a significant difference in the metal (copper) concentration of the soil before planting and after harvest at p=0.05 with a calculated value of 3.27. table 3 shows the results of the student t-test for metal concentrations in the ”soil before planting and soil after harvest.” 3.1.4. zinc in soil the zinc concentration in the soil samples collected from the ”control portion” as in table 2 ranged between 3.73 mg/kg 8.43 mg/kg. the lowest value (3.73 mg/kg) was recorded in control2, while the highest (8.43 mg/kg) was in control2. these concentrations gave a mean ± sd of 5.26 ± 2.2, as shown in table 3. also, the concentration of zinc in the “soil before planting” (table 2) ranged between 2.75mg/kg 13.17mg/kg with the lowest value (2.75mg/kg) at a1 and the highest value (13.17mg/kg) at a2. these gave a mean ± sd of 7.28 ± 3.90 (table 3). as in table 2, the zinc concentration in ”soil after harvest” ranged between 6.88mg/kg-15.15mg/kg. the lowest value was c3(6.88 mg/kg), while the highest was d3(15.15 mg/kg). the mean ± sd is 11.33 ± 2.51, as in table 3. comparing the mean concentration of zinc in the ”soil before planting and soil after harvest” (7.28 mg/kg and 11.33 mg/kg respectively) with that of the soil control (5.26 mg/kg) as seen in table 2, it can be seen that their values are more than that in the ”soil control” suggesting that the soil is contaminated with zinc. the contamination could be attributed to pesticides and herbicides on the field, as zinc is one of the constituents of most pesticides [18]. also, we compared the results with those reported by researchers like; 1. zhuang reported that the zn concentrations of the agricultural soils around the daboshan mine in guangdong, 4 ajah et al. / j. nig. soc. phys. sci. 4 (2022) 891 5 china is 498 mg/kg [14]. 2. machiwa reported the mean zinc concentration as 65.46mg/kg in the agricultural soils of lake victoria basin, tanzania [20]. 3. kibassa reported the average range concentration of zinc in agricultural soils collected from six sites in daressalan to be 33.18 mg/kg [22]. the mean concentrations gotten from the different soil portions (soil before planting and soil after harvest, 7.28 mg/kg and 11.33 mg/kg, respectively) in this study did not exceed that recorded by the researchers mentioned earlier and also are far below the maximum permissible limits (300-400 mg/kg) for zinc in soils by (fepa) and (300 mg/kg) as with grade ii environmental quality standards for agricultural soils in china. therefore, the soil may not be harmful to rice production for consumption by human beings. 3.1.5. zinc in rice the concentration of zinc in ”rice stalks” (table 2) ranged between 15.14 mg/kg and 69.03 mg/kg, with d2 having the lowest value (15.14 mg/kg) and a1 having the highest value (69.03mg/kg). these concentrations gave a mean ± sd of 45.14 ± 13.73, as shown in table 3. also, the concentration of zinc in ”rice grain” ranged between 14.78 mg/kg-32.89mg/kg, as shown in table 2, with d1 having the lowest value (14.78 mg/kg) and b2 has the highest value (32.89 mg/kg). the mean ± sd value was 24.90 ± 6.06, as shown in table 3. it compared the mean concentration of zinc in rice grains (24.90 mg/kg) in this study with those reported by [20] (21.7 ug/g) and that reported by [23] (21.5 ug/g). it was found that they are still within the range, although most of the concentrations of zinc in rice individually in this study fall within these limits except for a few which are a bit higher in the rice grains in areas a2, b2, b3, and c1(31.37,32.89,30.81, and 29.48 mg/kg respectively). the concentrations in the rice stalk were relatively high. this could be attributed to the ash from the abattoir (rubber tire), which was used to mix the rice grains before broadcasting on the field to avoid /scare birds from eating the broadcasted rice. this ash contains zinc, which was made highly available to the rice stalk then, followed by the little the rice grain absorbed. however, the mean concentrations are still within the chinese maximum permissible limits for zinc in rice (50 mg/kg) and the fao/who (2002) recommended limits for zinc in rice grains. therefore, rice is suitable for consumption [17]. 3.1.6. relationships between zinc concentrations in the soil before planting and soil after harvest the statistical analysis with the student t-test indicated a significant difference in the soil’s metal (zinc) concentration before and after planting at p=0.05 with a calculated value of 2.60. table 3 shows the results of the student t-test for metal concentration in the soil before and after planting. 3.1.7. iron in soil the concentration of iron in ”soil control” ranged between 805.26 mg/kg 823.69 mg/kg, as shown in table 2, with control1 having the lowest value (805.26) and control4having the highest value (823.69 mg/kg) and a mean ± sd of 814.40 ± 7.52 as in table 3. as in table 2, the iron concentrations in the ”soil before planting” ranged between 764.29 mg/kg 845.59 mg/kg. d2 had the lowest value (764.29 mg/kg) while a1 had the highest value (845.59mg/kg), and the mean ± sd value was 803.04 ± 26 as in table 3. also, the iron concentrations in ”soil after harvest” ranged between 611.14-838.81 mg/kg, as in table 2. the lowest value (611.14 mg/kg) was recorded at c2 and the highest (838.81 mg/kg) at d2. the mean ± sd value was 735.4 ± 73.20, as shown in table 3. comparing the mean concentration of iron in (soil before planting and soil after planting) (803.04 mg/kg and 735.47 mg/kg, respectively) with soil control (814.40mg/kg). these values are lower than the soil control value meaning the soil is not contaminated with iron. however, the mean concentrations in soils (before planting (803.04mg/kg) and after harvest (735.47 mg/kg) are still within the maximum permissible limits (7000-550000) as with chine’s grade ii environmental quality standards [24]. 3.1.8. iron in rice the iron concentrations in ”rice stalk” ranged between 56.61mg/kg-300.12mg/kg, as shown in table 2. the lowest value (56.61 mg/kg) was recorded at c2, and the highest 300.12 mg/kg value was recorded at b2. the mean ± sd value was 150.02 ± 85.20as shown in table 3. the iron concentrations in ”rice grain,” as shown in table 2, ranged between 0.66mg/kg-392 mg/kg. the lowest value (0.66 mg/kg) was recorded at b3 and the highest value (392.35 mg/kg) at d2. the mean ± sd value was 107.78 ± 138.15, as in table 3. 3.1.9. relationships between the iron concentration in the soil (before planting) and soil (after harvest) the statistical analysis with the student t-test indicated a significant difference in the soil’s metal (iron) concentration before and after planting at p=0.05 with a calculated value of 2.92. table 3 shows the results of the student t-test for metal concentrations in the soil before and after planting. 3.1.10. cadmium in soil cadmium concentrations were detected in only one portion; in the ”soil before planting” area a1, with a value (1.14 mg/kg), as shown in table 2. 3.1.11. cadmium in rice the concentrations of cadmium in ”rice stalk” were scantily detected at a1, b1, b2, c3 & d1 with the range (0.02mg/kg-2.03mg/kg) as shown in table 2. the lowest value (0.02 mg/kg) was recorded at b2 and the highest value at d1. the mean ±sd was 0.53±0.85, as shown in table 3. the mean 5 ajah et al. / j. nig. soc. phys. sci. 4 (2022) 891 6 cadmium concentration in rice stalk was higher than the codex standards (1993-1995) (0.1mg/kg) for cadmium in rice stalk. this high value could be attributed to the ash (from the abattoir (rubber tire) that was used to mix the rice seeds before broadcasting on the rice field [25]. however, none of these concentrations were made available to the rice grain, which leaves the conclusion that the rice grain is free from cadmium contamination and is, therefore, fit for consumption. 3.1.12. zinc concentrations in soil zinc concentration in the ”soil before planting” ranged between 2.75mg/kg and 13.17, with a mean of 7.28. zinc concentration in ”soil after harvest” ranged between 6.88mg/kg and 15.15 mg/kg, with a mean of 11.33. the mean concentrations of zinc from the different soil portions (soil before planting and soil after harvest) in this study are below the 33.18 mg/kg and 65.46 mg/kg reported by kibassa and machiwa as average range concentration of zinc in agricultural soils of daressalan and lake victoria basin, tanzania respectively [20, 22]. moreover, the mean zinc concentration in the soil did not exceed the maximum permissible limits (300-400 mg/kg) for zinc in soils by fepa and (300 mg/kg) as with grade ii environmental quality standards for agricultural soils in china [24]. 3.1.13. copper concentration in soil copper concentration in ”soil before planting” ranged between 2.39mg/kg and 5.18 mg/kg, with a mean of 3.40. the concentrations of copper in the soil after harvest ranged between 3.70 mg/kg and 6.07 mg/kg, with a mean of 4.64. these mean concentrations are below the permissible limits of 150 mg/kg as with chinese environmental quality standards for soil and 70-80 mg/kg as with epa guidelines for heavy metals in soil. the low concentrations are in line with 8.41-148.73mg/kg as reported by song as the mean concentration of copper in agricultural soils of suxian county, south china [19]. 3.1.14. iron concentration in soil iron concentration in ”soil before planting” ranged between 764.29 mg/kg and 845.59 mg/kg, with a mean of 803.04. iron concentrations in ”soil after harvest” ranged between 611.14 and 838.81 mg/kg with a mean of 735.4. however, the mean concentrations are still within the maximum permissible limits (7000-550000) as with chinese grade ii environmental quality standards [24]. 3.1.15. heavy metals in rice grain zinc in rice grain zinc concentration in rice grain ranged between 14.78 mg/kg and 32.89 mg/kg, with a mean of 24.90. this mean value is within the 21.5g/g reported by herawati and the 21.7g/g reported by machiwa as the mean concentration of zinc in rice collected from different locations in tanzania [23, 20]. however, the zinc mean concentration in this study is below the 50mg/kg recommended limits by fao/who (2002) for zinc in rice grains. hence it is worth noting that the rice may not pose any kind of danger to human health as a result of zinc contamination [26]. copper in rice grain the concentration of copper in rice grain ranged between 1.84 mg/kg-14.81mg/kg with a mean of 4.14. this mean value is not far from the 3.7 mg/kg reported as a copper concentration in rice grain collected from different locations in tanzania by machiwa and the 6.34 mg/kg reported as the mean concentration of copper in rice from guangdong, china, by zhuang [16]. however, the mean copper concentration of the rice grain samples in this study is within the maximum permissible limits,10 mg/kg by china and 20 mg/kg by fao/who (2002), which shows that the rice is good for consumption. [20, 14]. iron in rice grain iron concentrations in ”rice grain” ranged from 0.66mg/kg-392mg/kg with a mean of 107.78. cadmium in rice grain cadmium was not detected in ”rice grain” for all the rice grain samples. correlation matrix correlation is a statistical technique that shows whether and how strongly pairs of variables are related. table 4 presents the correlation matrix for the metals in the soil before planting. a strong positive and significant correlation was observed between copper and iron with an r-value of 0.787. the strong positive correlation suggests the similar origin of the metal pairs, probably from agrochemicals used on the farm. liu reported strong correlations among cu, ni, and cr in the soil around an electroplating plant and have implied that the metals have the same pollution sources [27, 28]. a weak positive correlation was observed between zinc and iron (r=0.345). however, a weak negative correlation (r=-0.030) was observed between copper and zinc, showing that an increase in the concentration of one metal results in a decrease in the concentration. table 5 presents the correlation matrix for the metals in the soil after harvest. weak positive correlations were observed between cu-fe, cu-zn, and fe-zn pairs. table 6 presents the correlation matrix for the metals in the rice grain. a weak positive correlation was observed between cu-zn, showing that an increase in the concentration of one metal results in a decrease in the concentration of another. 3.1.16. dietary intake of heavy metals and potential health risks via rice consumption table 7 shows the metal concentrations (mg/kg) in rice, provisional maximum tolerable daily intake (pmtdi) (mg/kg bw) with estimated daily intake (edi)(mg/kg bw/day), and target hazard quotient (thq) data for a 60 kg adult. the intake of heavy metals was estimated by multiplication of daily consumption rate with metal content in rice divided by the bodyweight edi = mc ×ir bw (4) where; mc= concentration of the heavy metal in contaminated rice, ir= ingestion rate or average rice consumption in the study region, and bw= bodyweight. 6 ajah et al. / j. nig. soc. phys. sci. 4 (2022) 891 7 table 1: recoveries and precision (%) of metals cu, zn, fe, and cd from spiked soil, ash, rice grain, and stalk samples after digestion element sample spike (g/ml) conc in unspiked sample (g/ml) conc in spiked sample (g/ml) recovered conc. (g/m l) recovery (%) precision zinc soil before planting a 2ppm 4ppm 2.000 2.000 3.8182 5.9827 1.8182 3.9827 92 100 6.3 soil before planting b 2ppm 4ppm 0.2251 0.2251 1.9134 3.8009 1.6883 3.5758 84 89 4.1 control 1 2ppm 4ppm 0.3723 0.3723 2.1126 3.9307 1.7403 3.5584 87 89 2.5 control 2 2ppm 4ppm 0.2078 0.2078 2.1039 4.0260 1.8961 3.8182 95 95 0 ash 2ppm 4ppm 5.3506 5.3506 7.4632 9.2987 2.1126 3.9481 106 99 4.8 soil after harvest a 2ppm 4ppm 0.7100 0.7100 2.4242 4.5801 1.7142 3.8701 86 97 8.5 soil after harvest b 2ppm 4ppm 0.6494 0.6494 2.4242 4.4675 1.7748 3.8181 89 95 4.6 grain 1 2ppm 4ppm 0.9524 0.9524 2.7965 4.6667 1.8441 3.7143 92 93 1.1 grain 2 2ppm 4ppm 0.7013 0.7013 2.5108 4.5628 1.8095 3.8615 90 97 5.3 stalk 1 2ppm 4ppm 2.6753 2.6753 4.4242 6.554 1.7489 3.8787 87 97 3.5 stalk 2 2ppm 4ppm 1.7489 1.7489 3.6190 5.6190 1.8701 3.8701 94 97 2.3 iron soil before planting a 2ppm 4ppm 111.2664 111.2664 113.4672 115.4236 2.2008 4.1572 110 103 4.7 soil before planting b 2ppm 4ppm 128.2620 128.2620 129.9738 132.5546 1.7118 4.524 86 113 14.3 control 1 2ppm 4ppm 132.7860 132.7860 134.2533 137.5546 1.4673 4..7686 72 119 33.9 control 2 2ppm 4ppm 128.5066 128.5066 131.0742 131.9301 2.5676 3.4235 128 86 27.8 ash 2ppm 4ppm 106.8646 106.8646 108.5764 110.2882 1.7118 3.4236 86 86 0 soil after harvest a 2ppm 4ppm 134.7424 134.7424 136.8210 138.5328 2.0786 3.7904 104 95 6.4 soil after harvest b 2ppm 4ppm 132.4192 132.4192 134.1310 136.0873 1.7118 3.6681 86 92 4.7 grain 1 2ppm 4ppm 1.9563 1.9563 3.6681 5.9913 1.7118 4.035 86 101 11.3 grain 2 2ppm 4ppm 1.2227 1.2227 3.1790 5.5022 1.9563 4.2795 99 107 5.5 stalk 1 2ppm 4ppm 6.6026 6.6026 8.6812 10.5153 2.0786 3.9127 104 98 4.2 stalk 2 2ppm 4ppm 10.0262 10.0262 11.9825 14.0611 1.9563 4.0349 99 101 1.4 copper soil before planting a 2ppm 4ppm 0.0323 0.0323 2.0693 4.0416 2.037 4.0093 102 100 1.4 soil before planting b 2ppm 4ppm 0.0000 0.0000 1.9723 3.9769 1.9723 3.9769 99 99 0 7 ajah et al. / j. nig. soc. phys. sci. 4 (2022) 891 8 control 1 2ppm 4ppm 0.0000 0.0000 1.8430 3.9769 1.8430 3.9769 92 99 5.2 control 2 2ppm 4ppm 0.0647 0.0647 1.9400 3.8799 1.8753 3.8152 94 95 1.1 ash 2ppm 4ppm 5.2702 5.2702 7.3395 9.2148 2.0693 3.9446 103 99 2.8 soil after harvest a 2ppm 4ppm 0.2587 0.2587 2.0370 3.8476 1.7783 3.5889 89 90 1.1 soil after harvest b 2ppm 4ppm 0.2587 0.2587 2.2633 4.0739 2.0046 3.8152 100 95 3.7 grain 1 2ppm 4ppm 0.0000 0.0000 1.7460 4.1709 1.7460 4.1709 87 104 12.5 grain 2 2ppm 4ppm 0.0647 0.0647 2.0693 3.9446 2.0046 3.8799 100 97 2.2 stalk 1 2ppm 4ppm 0.0647 0.0647 1.9400 3.8476 1.8753 3.7829 94 95 1.1 stalk 2 2ppm 4ppm 0.0970 0.0970 1.8753 3.7182 1.7783 3.6212 89 91 x=96 1.6 x=5.7 table 2: metals concentrations (mg/kg) sample copper zinc iron cadmium soil control 1 1.85 3.81 814.32 2 1.67 3.73 814.31 3 2.04 5.06 805.26 4 3.33 8.43 823.69 soil before planting a1 5.18 2.75 845.59 1.14 a2 4.05 13.17 821..89 b1 3.32 9.11 812.46 b2 3.30 7.48 803.78 c1 3.68 6.98 785.79 c2 2.59 3.99 774.81 d1 2.68 11.68 815.74 d2 2.39 3.09 764.29 ash 1 156.81 42.07 848.28 soil after harvest a1 5.55 13.74 789.75 3-.68 a2 4.25 12.51 761.50 a3 3.70 12.16 761.35 b1 4.25 9.49 746.86 b2 6.07 9.00 759.38 b3 4.04 8.29 736.97 c1 4.25 10.21 703.58 c2 4.26 12.96 611.14 c3 4.60 6.88 659.74 d1 5.32 13.11 625.99 d2 4.80 12.41 838.81 d3 4.59 15.15 830.55 rice stalk a1 0.74 69.03 61.86 0.20 a2 5.15 46.61 86.51 a3 11.06 58.33 144.68 8 ajah et al. / j. nig. soc. phys. sci. 4 (2022) 891 9 b1 1.48 33.16 280.15 0.16 b2 1.85 39.05 300.12 0.02 b3 0.74 41.26 185.20 c1 1.48 46.10 197.25 c2 1.85 41.01 56.61 c3 1.11 41.74 60.60 0.22 d1 1.47 51.36 126.97 2.03 d2 1.11 15.44 213.29 d3 1.48 58.62 86.95 rice grain a1 2.03 23.06 15.13 a2 1.84 31.37 10.49 a3 2.03 28.46 61.16 b1 1.85 19.88 2.63 b2 2.03 32.89 213.71 b3 2.04 30.81 0.66 c1 7.92 29.48 22.92 c2 8.86 24.62 16.42 c3 14.81 23.78 22.37 d1 2.21 14.78 289.63 d2 2.04 24.59 392.35 d3 2.03 15.07 245.83 table 3. mean ± sd of metal concentrations (mg/kg) sample copper zinc iron cadmium soil control 2.22 ± 0.75 5.26 ± 2.2 814.40 ± 7.52 soil b/4 planting 3.40 ± 0.91 7.28 ± 3.90 803.04 ± 26.76 1.14 ash 156.81 ± 156.81 42.07 ± 42.07 848.28 ±848.28 soil after harvest 4.64 ± 0.69 11.33 ± 2.51 735.47 ± 73.20 rice stalk 2.46 ± 2.95 45.14 ± 13.73 150.02 ± 85.20 0.53±0.85 rice grain 4.14 ± 3.92 24.90 ± 6.06 107.78 ± 138.15 table 4. pearson’s correlations between different metals in the soil before planting cu zn fe cu 1 zn -0.030 1 fe 0.787 0.345 1 table 5. pearson’s correlations between different metals in the soil after harvest cu zn fe cu 1 zn 0.071 1 fe 0.079 0.274 1 table 6. pearson’s correlations between different metals in rice grain cu zn fe cu 1 zn 0.031 1 fe -0.338 -0.373 1 9 ajah et al. / j. nig. soc. phys. sci. 4 (2022) 891 10 table 7. metal concentrations (mg/kg) in rice, pmtdi (mg/kg /person/day) with edi (mg/kg bw/day) and thq data for a 60 kgadult metal mean ± sd pmtdi edi thq copper 4.14±3.92 0.05-0.5 0.01 0.00017 zinc 24.90±6.06 0.3-1 0.04 0.00014 iron 107.78±138.15 0.8 0.18 0.00026 cadmium provisional maximum tolerable daily intake (pmtdi) by jecfa according to the international rice research institute (2001), the average nigerian consumes 24.8kg of rice per year, equivalent to 0.1kg per person/day. the daily intakes for a 60 kg adult were compared to the provisional maximum tolerable daily intakes as stipulated by joint fao/who expert committee food additive (jecfa). it was found that the edis of the metals falls within the range of the safe values stipulated by jecfa [29]. the target hazard quotient (thq) of heavy metals from rice consumption is in decreasing order: fe>cu>zn and are all less than 1 indicating there will be no health risk. the total hazard index (thi) for rice consumption for a 60-kg adult is 0.00057, which is less than 1. thus, the consumption of rice from this field will show no adverse effects from the metals. 4. conclusion this research demonstrated that zn, cu, and fe concentrations in soil and rice grains did not exceed the threshold set by several international organizations. the preliminary maximum tolerated daily intake (pmtdi) established by jecfa was less than the metals’ estimated daily intake (edi). the total hazard index was also less than one, indicating no potential health risk associated with the intake of rice in this field. the metal’s target hazard quotient (thq) was less than one. the result from the physicochemical analysis of the soil showed that the soil is a type suitable for rice production. however, we recommend that the field be monitored continuously to ensure that the metals stay at harmless levels. acknowledgment the authors appreciate the editor and the anonymous reviewers for their valuable comments towards the improvement of this article. references [1] y. jin, l. wang, y. song, j. zhu, m. qin, l. wu & d. hou, “integrated life cycle assessment for sustainable remediation of contaminated agricultural soil in china”, environmental science & technology 55 (2021) 12032. 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[29] w. h. organization, safety evaluation of certain food additives: prepared by the eighty-ninth meeting of the joint fao/who expert committee on food additives (jecfa), (2022). 11 introduction experimental area of study sample collection soil sample collection rice sample collection sample preparation soil preparation rice preparation ash sample size digestion of soil samples, rice grain samples, and rice stalk samples quality assurance risk assessment data analysis results and discussion heavy metals concentration in soil and rice grain copper in soil copper in rice relationships between copper concentration in the soil before planting and soil after harvest zinc in soil zinc in rice relationships between zinc concentrations in the soil before planting and soil after harvest iron in soil iron in rice relationships between the iron concentration in the soil (before planting) and soil (after harvest) cadmium in soil cadmium in rice zinc concentrations in soil copper concentration in soil iron concentration in soil heavy metals in rice grain dietary intake of heavy metals and potential health risks via rice consumption conclusion j. nig. soc. phys. sci. 4 (2022) 621 journal of the nigerian society of physical sciences validation of tritium calibration curve in ciemat/nist activity measurement using non linear least squared fittings and calculations of the half-life and decay constant of potassium-40 s. adamsa,∗, e. josepha, g. kamalb a department of physics, federal university dutsin-ma, katsina state-nigeria. b department of physics, federal university, lafia, nasarawa state-nigeria. abstract non-linear curve fitting of tracer efficiency is one of the uncertainty contributors in ciemat/nist method for specific activity measurement and half-life evaluation of radionuclide. this study applied least squared fitting of four different polynomials to validate the tracer efficiency calibration curve for the specific activity measurements of nine potassium chloride samples from which half-life and decay constants were computed. all samples were measured using tr1000 liquid scintillation counter and the results of the relative standard uncertainties in tracer interpolated efficiencies associated with the least square fit analysis were found to be 8.363%, 8.076%, 7.941% and 8.767% for polynomials of n = 2, n = 3, n = 4 and n = 5, respectively. the corresponding values of 40k specific activity, from the application of empirical efficiencies generated with these polynomials were found to be (16.541, 16.540, 16.537 and 16.548) bq/g respectively. from these measured specific activity values, the computed half-life and decay constants were found to be (1.2518, 1.2519, 1.2521 and 1.2513) ×109 y (for n = 2, n = 3, n = 4, and n = 5), respectively, and (5.365, 5.5364, 5.5354 and 5.5391) ×10−10y−1 respectively. all values were found to be in good agreement with results of other literatures. however to minimize uncertainty associated with empirically generated tracer interpolation efficiencies, least squared fit analysis of tracer calibration curve should be done, with control trials using different polynomials so as to obtain the best fit. doi:10.46481/jnsps.2022.621 keywords: ciemat/nist method, liquid scintillation counting, least square fit, analysis, tracer calibration curve article history : received: 28 january 2022 received in revised form: 3 may 2022 accepted for publication: 5 may 2022 published: 20 august 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: w. a. yahya 1. introduction non-linear curve fitting is a very important component in nuclear metrology [1]. in fact, it is applied in analytical formula for half-life evaluation of radionuclide [2, 3] and one of the ∗corresponding author tel. no: +2348132888358 email address: silasadams50@gmail.com (s. adams ) uncertainty contributors in ciemat/nist method for specific activity measurement [4]. this is because the ciemat/nist method relies on tracer non-linear curve fitting for accurate interpolation of the relative counting efficiency of the radionuclide under investigation [5, 6]. the choice of both tritium and potassium-40 (40k) in this study was because of the following reasons: (i) tritium suitability for use in ciemat/nist 1 s. adams et al. / j. nig. soc. phys. sci. 4 (2022) 621 2 method as a pure beta emitting radionuclide (ii) significance of potassium-40 (40k) in radiometric age determination [4] which depends solidly on accurately determined half-life and decay constants [7]. 40k is a naturally occurring radionuclide aside uranium and thorium [8], whose decay to calcium-40 (one of the stable isotopes of calcium) [9, 10] and especially to argon-40 (figure 1) is widely used in radiometric dating [4]. unfortunately, uncertainty in half-life and decay constant values have limited present day radiometric dating [4]. more so, the recent attribution of disagreement in the values of decay constants to the method used in determining half-lives by mc donough et al, [11] serves as a trigger for this study as slight uncertainty of only one percent in decay constant could lead to significant disagreements in the ages of radioisotopes [7]. unlike, geochronology, nuclear physics experiments determine halflife through activity measurement. ciemat/nist efficiency tracing method is one of such methods for specific activity measurement whose worth has been proven in the activity measurements of many radionuclide and employs fitting as the best means of interpolation [4, 5, 6]. in partial response to the call made by half-life evaluators for in-depth assessment of individual uncertainty components for good half-life measurement [12], this paper therefore shows how to validate the tracer (tritium) calibration curve through controlled trials of non-linear least square fit analysis with different polynomials to determine the relative uncertainty associated with interpolated counting efficiencies of tracer (tritium). this was achieved with the use of ciemat/nist method. tr 1000 liquid scintillation analyzer was used to measure the specific activity of 40k in potassium chloride (kcl) samples from which the half-life and decay constant were evaluated. 2. materials and methods 2.1. sample preparation and counting a total of nine (9) samples were prepared as follows: 15 ml of ultima-gold ab scintillation cocktail was added each to 9 glass vials(20 ml capacity) of low potassium content. in a similar manner, 1 ml of potassium chloride (kcl) solution (5g in 25 ml of water) each was again added to all the glass vials. all weighing with mettler ac100 were gravimetrically controlled. samples were gradually quenched by adding varying but increasing amount of the nitromethane as quenching agent then vigorously shaken and kept for stability. a blank sample was also prepared without the addition of kcl salt. all samples including blanks were carefully labelled and counted using tricarb tr 1000 lsa each for 60 minutes to obtain good counting statistics. records of count rates in counts per minutes (cpm) and spectral index of the samples (sis) as the quench indicating parameter (qip) were taken. the total counting time for potassium chloride samples was more than 30 days in order to obtain stable counts rates. figure 1: 4019 k simplied decay scheme [4] 2.2. liquid scintillation techniques radiation emitted from dissolved radionuclide samples in scintillation cocktails transferred energy to the organic scintillator that in turn emits light photons. this way each emission result is a pulse of light in form of digit called counts [13]. 2.3. ciemat/nist model the ciemat/nist model used in this work for the calculation of 40k specific activity was modified from broda et al., [14] as shown in figure 2. the slight modification (improvement) is clearly reflected in figure 2 where four different polynomials were used to generate the empirical efficiencies of the tracer radionuclide (tritium-3) using a method of interpolation called least squared fit analysis as stated below on non-linear least squared fittings of tritium calibration curve. the usage of these polynomials for the fittings is to account for the least uncertainty associated with the empirically generated tracer efficiency as a result of the least squared fit analysis. broda et al description of ciemat/nist method can be summarized into five steps as follows: (i) theoretical computation of detection efficiencies of both tracer and the radionuclide under study (40k in this case) . this is done via computer program so as to establish an efficiency calibration curve (a plot of nuclide efficiency as function of tracer efficiency) [14]. (ii) experimental determination of tracer efficiency from lsc measurement (cpm and qips). since the activity (dpm) of the quench set is known, the efficiency of tracer is computed as (cpm/dpm). (iii) measurement of samples of radionuclide under study for determination of tracer efficiencies and the corresponding radionuclide efficiencies. by applying the qips obtained from measured samples to equation (1) the tracer efficiencies are obtained. similarly, by applying the obtained tracer efficiencies to equation (10), the corresponding radionuclide efficiencies are obtained. (iv) conversion of the efficiency of the radionuclide so determined and the measured count rate (rn) in cmp to the activity of the radionuclide [14]. rn denotes sample count rate, µ is the free parameter, sis is the spectral index of the sample as quench indicating parameter, m, is the mass of kcl salt in-cooperated into the 1 ml 2 s. adams et al. / j. nig. soc. phys. sci. 4 (2022) 621 3 figure 2: illustration model for ciemat/nist method [14]. figure 3: efficiency calibration curve (a plot of efficiency of 40k versus tracer efficiency) of the sample εk−40and εh−3 efficiencies of potassium-40 and tritium-3 radionuclides, respectively. normally, fittings of tracer calibration curves in cimat/nist method are done using a single order polynomial [15]. however this study applied four (4) polynomials of different order to validate the tracer calibration curve hence generating four different sets of 40k efficiencies. this served as the slight variation from other ciemat/nist method. 2.4. analysis the significant analysis in this study is based on the nonlinear least squared fittings at different polynomial orders for tracer (tritium) calibration curve as well as the evaluation of 40k specific activity and half-life. 2.5. non-linear least squared fittings of tritium calibration curve four least squared fitted calibration curves for tracer efficiency were obtained at varying polynomial order (n = 2, 3, 4, and 5) for tracer calibration curve (a plot of tracer efficiency εh−3versus quench indicating parameter). the tracer calibration curve upon which the current least square is fitted was obtained from radiation analysis [16]. the general form of polynomial used to generate the empirical tracer efficiencies for the least square fit analysis is given as equation (1) [17] while equations (2), (3), (4), and (10a) were used for the four different non-linear least square fittings in accordance with the polynomial order. all fittings were done using excel spread sheet including the calculations of relative standard uncertainties in the interpolated tracer efficiency associated with the least squared fittings using equations (7), (8) and (9) as presented in table 1 3 s. adams et al. / j. nig. soc. phys. sci. 4 (2022) 621 4 (a) (b) (c) (d) figure 4: least squared fitted tritium calibration curve for polynomials of order (a) n = 2; (b) n = 3; (c) n = 4 and (d); n = 5 respectively. figure 5: combined least squared tritium calibration curve for polynomials (n = 2, n = 3, n = 4, n = 5) and figures 3, 4, 5, and 6. εh−3 = n∑ i=0 ki(s is ) i (1) for n = 2, the least squared theory becomes: εh−3 = k0 + k1 (s is ) + k2(s is ) 2 (2) for n = 3 the least squared theory for fitting the tritium calibration curve becomes εh−3 = k0 + k1 (s is ) + k2(s is ) 2 + k3(s is ) 3 (3) similarly for n = 4 and n = 5 the theory for the least squared fitting of the calibration curve is given by åh−3 = k0 +k1 (s is )+k2(s is ) 2 +k3(s is ) 3 +k4(s is ) 4(4) 4 s. adams et al. / j. nig. soc. phys. sci. 4 (2022) 621 5 table 1: least squared fit analysis, for polynomials (n = 2, n = 3, n = 4, n = 5) experiment predicted least squared predicted predicted predicted least squared predicted least squared n = 2 n = 3 n = 4 n = 5 sis exp. ε(h-3)% ε(h-3)% delta ε(h-3)% delta squared ε(h-3)% delta squared ε(h-3)% delta squared 18.6 68 68.39 0.15217 67.939 0.00361 67.962 0.0014 67.949 0.00259 16 64 62.936 1.13039 63.507 0.24285 63.197 0.64414 63.865 0.01806 14.8 58 58.607 0.36942 59.058 1.12063 59.028 1.05738 58.71 0.50466 13.6 52 53.134 1.28618 53.286 1.65585 53.457 2.12396 52.707 0.50069 12 48 44.055 15.55986 43.775 17.84978 43.928 16.57536 43.708 18.41697 11 38 37.347 0.42523 36.909 1.18919 36.913 1.18076 37.343 0.43059 10.5 29 33.696 22.05335 33.242 17.9988 33.16 17.31165 33.863 23.65655 9.2 23 23.271 0.07371 23.066 0.0044 22.878 0.01482 23.585 0.34245 8.5 18 17.101 0.80676 17.249 0.56265 17.183 0.6673 17.151 0.71978 8 13 12.456 0.2955 12.971 0.00082 13.124 0.01553 12.108 0.79495 sum 42.15261 40.628616 39.59230 45.38733 standard uncertainty(su) in ε (h-3)% 0.684370 0.671884 0.663260 0.710143 order of polynomial coefficients k0 k1 k2 k3 k4 k5 n = 2 -88.89410 15.84820 -0.39742 n = 3 -61.11043 9.00344 0.14012 -0.013502 n = 4 -3.136618 -10.32740 2.48410825 -0.1360546 0.002333 n = 5 -104.8710 10.70044 2.43550 -0.387136 0.021350 -0.000419 figure 6: relative uncertainty in tritium interpolated efficiencies for different polynomial and εh−3 = k0 + k1 (s is ) + k2(s is ) 2 (5) + k3(s is ) 3 + k4(s is ) 4 + k5(s is ) 5, where k0, k1, k2, k3, k4, and k5, are the coefficients of the least squared fittings respectively, sis= spectral index of the sample for tracer, and εh−3 = tritium counting efficiency delta = e x periment − t heory (6) sum = sum o f delta squared (7) standard error = s qrt ( sum n ∗ (n − 1) ) (8) from the standard error, the relative standard uncertainties in empirically generated tritium efficiencies was obtained using equation (9) [18]. u(εh−3) εh−3 = √∑ ( s ε εh−3 )2 (9) 2.6. calculation of efficiencies all theoretical efficiencies for tracer (3h) and 40k were calculated using cn2003 code. the range of the calculation chosen for the tracer (tritium) efficiency was between 25% to 65% for ionization quench constant of 0.0075 cm/mev. the interpolated efficiency (εk−40)40k was obtained using a fitted polynomial given by equation (10) from the general form of equation given by [17] for some selected values of free parameter as shown in figure 3. the cn2003 visual basic code takes into account the following: statistics of nuclear decay, detector response as well as physics of the nuclear decay processes. εk−40 = n∑ i=0 ki(εh−3) i (10a) ε(k − 40)% = 2e − 05ε(h − 3)3 − 0.001ε(h − 3)2 (10b) + 0.081ε(h − 3) + 88.39 the regression value of 1 on figure 3 indicates a perfect polynomial fitting. 2.7. 40k specific activity and half-life the measured sis for potassium samples were applied to the least squared fitted calibration curves to obtain the corresponding tracer (tritium) efficiencies. the corresponding values of 40k efficiencies (εk−40 ) were obtained by fitting the empirically generated tracer efficiencies to equation (10b). equation (10b) is the efficiency calibration curve extracted from ciemat/nist calculation for some selected values of free parameter (µ). with the efficiencies of 40k (εk−40 ) obtained using equation (10b) and data of 40k measurement from liquid scintillation counter (count rates and sis), the specific activities of 40k in all the measured samples were computed using equation (11) as follows [4]: a = rs − rb mεk−40 , (11) 5 s. adams et al. / j. nig. soc. phys. sci. 4 (2022) 621 6 table 2: specific activities, half-lives, decay constants and efficiencies for all kcl samples order of polynomial sample id efficiency specific activity half-life ( t1/2 ) decay constant (λ) �(h − 3)% �(k − 40)% bq/g ×109 y ×10−10y−1 n = 2 gvt1 60.6755 94.0907 12.916 1.6031 4.3234 gvt2 49.764 92.4092 16.247 1.2745 5.4344 gvt3 46.6121 92.0183 17.011 1.2173 5.6908 gvt4 46.7852 92.0388 17.651 1.1731 5.9048 gvt5 50.6193 92.5219 15.839 1.3073 5.2981 gvt6 46.2459 91.9753 18.741 1.1049 6.2697 gvt7 49.2336 92.3407 16.556 1.2507 5.5381 gvt8 47.2002 92.0884 16.637 1.2447 5.5555 gvt9 48.5306 92.2517 17.376 1.1917 5.8125 mean 16.541 1.2518 5.5365 n = 3 gvt1 61.2099 94.1879 12.91 1.6047 4.3189 gvt2 49.7335 92.4052 16.236 1.2754 5.4346 gvt3 46.432 91.9971 17.005 1.2177 5.692 gvt4 46.6126 92.0184 17.645 1.1736 5.9061 gvt5 50.6334 92.5237 15.828 1.3083 5.298 gvt6 46.0501 91.9525 18.736 1.1052 6.271 gvt7 49.1763 92.3334 16.546 1.2514 5.5385 gvt8 47.046 92.0699 16.63 1.2452 5.5562 gvt9 48.4386 92.2402 17.367 1.1923 5.8132 mean 16.540 1.2519 5.5364 n = 4 gvt1 61.0546 94.1595 12.907 1.6043 4.3204 gvt2 49.2672 92.345 16.232 1.2757 5.4331 gvt3 46.168 91.9662 17.002 1.2179 5.6908 gvt4 46.336 91.9858 17.651 1.1738 5.9047 gvt5 50.1258 92.4565 15.834 1.3086 5.2965 gvt6 45.8133 91.9251 18.731 1.1055 6.2699 gvt7 48.7391 92.2779 16.542 1.2518 5.537 gvt8 46.7397 92.1912 16.545 1.2478 5.5547 gvt9 48.04352 92.1912 17.363 1.1926 5.8117 mean 16.537 1.2521 5.5354 n = 5 gvt1 61.1382 94.1748 12.905 1.6045 4.3195 gvt2 49.2672 92.34508 16.247 1.2745 5.4388 gvt3 46.168 91.96626 17.011 1.2173 5.6941 gvt4 46.336 91.98589 17.651 1.1731 5.9082 gvt5 50.1258 92.45652 15.839 1.3073 5.3019 gvt6 45.8133 91.92513 18.741 1.1049 6.2731 gvt7 48.7391 92.27796 16.556 1.2507 5.5418 gvt8 46.7397 92.03347 16.637 1.2447 5.5592 gvt9 48.0435 92.19121 17.376 1.1917 5.5816 mean 16.548 1.2513 5.5391 where rs =sample count rate, rb = background count rate, m = mass of potassium salt incorporated into the sample, εk−40 = counting efficiency for 40 k. from the specific activity obtained using equation (8) the half-life of 40 k was computed using equation (12) [4]: t1/2 = (ln 2) nam p a , (12) where na = avogadro constant, m is the relative molar mass of kcl, p = n(4019 k)/n (k) = 0.01167%. following the relationship between half-life and decay constant, the decay constant was computed using equation (13) expressed as [11]: λ = ln 2 t1/2 (13) 3. results and discussion results of least squared fitted tritium calibration curves for all the polynomials is presented in figures 4 and 5. from figure 4 (a, b, c and d), the least squared fitted curves of the experimental tritium calibration curve it can be observed 6 s. adams et al. / j. nig. soc. phys. sci. 4 (2022) 621 7 table 3: 40khalf-life and decay constant values from source and some calculated decay constants reference 40k (t1/2) ×10 9y 40k (λ) ×10−10y−1 original original calculated steiger and jägar, (1977) [20] 1.250 5.543 min et al. (2000) [19] 1.269 5.463 grau malonda and grau carles (2002) [21] 1.248 5.554 kossert and günther,(2004) [4] 1.248 5.554 this study (polynomial order) n = 2 1.2519 5.5367 5.5367 n = 3 1.2518 5.5365 5.5365 n = 4 1.2521 5.5354 5.5354 n = 5 1.2513 5.5391 5.5391 that not all points of empirically generated efficiencies are perfectly fitted to the experimental data. this is because each fitted polynomial has an associated level of uncertainty in determining the empirical efficiencies for interpolation with the spectral index of potassium chloride (kcl) measured samples as clearly shown in figure 5. the magnitude of the standard uncertainties associated with each polynomial including the relative standard interpolated efficiencies for the least squared analysis is shown is table 1. all least squared fit analysis in this study and the estimation of standard and relative interpolated efficiencies of tritium were carried out using excel spread sheet. from table 1 it can be seen that the standard uncertainties and relative standard uncertainties in tritium interpolated efficiencies due to the least squared fitted polynomials ranges from (0.663260 to 0.710143)% and (7.941 to 8.767)% respectively. polynomial of order (n = 4) has the lowest interpolation uncertainties while the highest uncertainty was associated to polynomial of n=5 as clearly shown in figure 6. it can also be deduced from figure 5 that the uncertainties associated with non-linear least square fit analysis of the interpolated efficiencies does not completely depend on the polynomial order. 3.1. 40k specific activity, half-life and decay constant the values of 40k specific activity, as well as the computed half-life and decay constants obtained in this study for the four (4) different polynomials are presented in table 2. from results presented in table 2, it can be seen that the mean obtained values of 40k specific activity, half-life and decay constant for the four least squared fitted polynomials varied slightly ranging from (16.537 to 16.548) bq/g, (1.2513 to1.2521) ×109 y and (5.5354 to 5.5392)×10−10y−1 respectively. this slight variation can be linked to the relative standard uncertainties associated with empirically generated interpolation efficiencies of the tracer (tritium) from the least squared fitted polynomials. polynomial of n=4 has the lowest specific activity value of 16.537 bq/g followed by polynomial of n = 3 with 16.540 bq/g, then polynomial of n = 2 (which is basically quadratic) with 16.541 bq/g and lastly polynomial of n = 5 with the highest value of16.548.this results shows that the higher the relative uncertainty associated with the least squared polynomial fitting, the higher the values of specific activities and decay constants. on the contrary, higher values of half-life are linked to lower relative standard interpolated uncertainties of tracer (tritium) associated with the least squared fitted polynomials. however, all values of specific activities in this current study are in good agreement with values of (16,594 to 16.616) bq/g obtained by kossert and günther, [4] while the half-life and decay constant values are in good agreement with the values obtained by steiger and jäger, min et al., grau malonda and grau carles, kossert and güther, [4, 19, 20, 21] as shown in table 3. 4. conclusion the relative standard uncertainties associated with the interpolated tracer efficiencies from the least squared fitting of the four different polynomials used in this study revealed that the choice of the prefer polynomial fit does not necessary depend on the order of polynomial but on the coefficients of the polynomial that will produce the lowest standard error. this is obtained through control trials starting from the lowest order. the validity of this is seen in the slight variation of 40k specific activity, upon which half-life is computed as well as decay constant. the slight variation in these values is as a result of the closed agreement in the relative standard uncertainties of the fitted polynomials. however a more noticeable variation is observed with least squared fitted polynomial of n=5. the choice of the best fit will go a long way in further reducing the relative uncertainty associated with interpolated tracer efficiency, which could improve the accuracy of ciemat/nist method in the activity determination of radionuclide and strengthen its acceptance. also, further non-linear least square fit analysis of tracer (tritium) calibration curve should be carried out with polynomials of higher order to confirm the independence of the best fit on the order of polynomial. references [1] s. m. craig, “least square fitting algorithms of the nist testing system”, j. res. natl. inst. stand. techno. 103 (1998) 633 [2] w. a. yahya, “alpha decay half-lives of 171−189hg isotopes using modified gamow-like model and temperature dependent proximity potential”, j. nig. soc. phys. sci. 2 (2020) 250. [3] t. daniel, s. kisyov, p.h. regan, n. marginean, zs. podolyak r. marginean, k. nomura, m. rudigier, r mihai, v. werner, r.j. carroll, l.a.gurgi, a. oprea, t. berry, a. serban, c.r. nita, c. sotty, r. suvaila, a. turturica, c. costache, l. stan, a. olacel, m. boromiza, s. toma, s.j. germanam, f. gbaorun, i, ochala, e. c hemba “ prompt response function (prf) of lifetime measurement in the 2+ state of 192os nuclei energy levels from triple-gamma coincidence techniques.”, j. nig. soc. phys. sci. 2 (2020) 257. [4] k. kossert, and e. günther, “lsc measurement of the half-life of 40 k”, applied radiation and isotopes 60 (2004) 459. 7 s. adams et al. / j. nig. soc. phys. sci. 4 (2022) 621 8 [5] a. grau malonda, “ciemat/nist method”, laboratoire national henri becquerel (lnhb) www.lnhb.fr/icrm lsc ciemat-nist method (2001). [6] i.r. dobrin, m. pavellescu, and c.n. dulama, “measurements of beta ray emitters in lsc using efficiency tracing ciemat/nist method”, paper presentation at the 11th international balkan workshop on applied physics, rom. journal physics 56 (2010) 1148. [7] a. snelling, “ determination of the decay constant and half-lives of uranium-238 (238u) and uranium-235 (235u), and the implications for u-pb and pb-pb radioisotope dating methodologies”, answers research journals 10 (2017) 1. [8] a.m. asere, t.o. owolabi, b.d. alafe, m.b. alimi, “assessment of excess gamma dose exposure level in typical nigeria commercial building materials distribution outlets”, j. nig. soc. phys. sci. 3 (2021) 216223. [9] b. m. dyavapa, “mass resolution of ca, k isotopes and co, n2 and c2 h4 isobars in isotopes separators on-line trap mass spectrometry”, j. nig. soc. phys. sci. 2 (2020) 180. [10] t. daniel, a.a. mc asule, “measurement of the excited energies identified in 42ca using rosphere gamma-ray arrays”, j. nig. soc. phys. sci. 2 (2020) 166. [11] w.f. mc donough, o. sramek, s. a. wipperfurth, “radiogenic power and geoneutrino luminosity of terrestrial bodies through time”, geochemistry, geophysics, geosystems 21 (2020) 1. [12] l.a. nichols, “nuclear decay data: on-going studies to address and improve radionuclide decay characteristics”, cp769 international conference on nuclear data for science and technology american institute of physics (2005) 242. [13] e. joseph, t. atsue, and s. adams, “assessment of radon-222 in selected water sources at dutsin-ma town, dutsin-ma local government area, katsina state”, journal of science and engineering research 5 (2018) 49. [14] r. p. cassette, k. kossert, “radionuclide metrology using liquid scintillation counting”, metrologia 44 (2007) s36. [15] e. w. günther, “ a simple method for transferring the tritium calibration curve of an lsc system to other radionuclides”, in liquid spectrometry 1994 (g.t cook, d.d. harkness, a.b mackenzie, b.f miller, & e.m. scott, ed) radiocarbon (1996) 373. [16] m. j. l’annuziata, “hand book of radioactive analysis” liquid scintillation analysis: principles and practice, montage group, usa (2003). [17] e. günther, “standardization of the ec nuclides 55fe and 65zn with the ciemat/nist lsc tracer method”, appl. radiat. isot. 49 (1998) 1055. [18] marlap, “multi-agency radiological laboratory analytical protocols manual”, 1 (2004) 1. [19] k. min, r. mundil, r.p. renne, and k.r. ludwig, “a test for systematic errors in 40ar/39ar geochronology through comparism with u/pb analysis of a 1.1-ga rhyolite” geochim cosmochim acta. 64 (2000) 73. [20] r.h. steiger, & e. jägar, “ subcommission on geochronology convention on the use of decay constants in geo and cosmochronology earth planet”, earth and planetary sctence letters 36 (1977) 359. [21] a. grau malonda, & a. grau carles, “half-life determination of 40k by lsc”, appl. radiat. iso. 56 (2002) 153. 8 j. nig. soc. phys. sci. 5 (2023) 527 journal of the nigerian society of physical sciences benchmark studies on the isomerization enthalpies for interstellar molecular species e. e. etim∗ department of chemical sciences, federal university wukari, katsina-ala road, p.m.b. 1020 wukari, taraba state, nigeria & department of chemistry, indian institute of technology bombay, powai, mumbai 400 076, india. abstract with the well-established correlation between the relative stabilities of isomers and their interstellar abundances coupled with the prevalence of isomeric species among the interstellar molecular species, isomerization remains a plausible formation route for isomers in the interstellar medium. the present work reports an extensive investigation of the isomerization energies of 246 molecular species from 65 isomeric groups using the gaussian-4 theory composite method with atoms ranging from 3 to 12. from the results, the high abundances of the most stable isomers coupled with the energy sources in interstellar medium drive the isomerization process even for barriers as high as 67.4 kcal/mol. specifically, the cyanides and their corresponding isocyanides pairs appear to be effectively synthesized via this process. the following potential interstellar molecules; cnc, nccn, c-c5h, methylene ketene, methyl ketene, ch3sch3, c5o, 1,1-ethanediol, propanoic acid, propan-2-ol, and propanol are identified and discussed. the study further reaffirms the importance of thermodynamics in interstellar formation processes on a larger scale and accounts for the known isomeric species. in all the isomeric groups, isomerization appears to be an effective route for the formation of the less stable isomers (which are probably less abundant) from the most stable ones that are perhaps more abundant. doi:10.46481/jnsps.2023.527 keywords: isomers, energy barrier, interstellar chemistry, astrochemistry, hydrogen bonding. article history : received: 19 december 2021 received in revised form: 28 january 2023 accepted for publication: 12 february 2023 published: 18 april 2023 © 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: b. j. falaye 1. introduction that the interstellar medium (ism) is chemically rich is not an argument with the discoveries of over 200 different molecular species in this thin space between the stars which was earlier regarded as a vacuum dotted with stars, black holes, and other celestial bodies [1-6]. these molecules are important in various fields such as atmospheric chemistry, astrochemistry, prebiotic chemistry, astrophysics, astronomy, astrobiology, etc, ∗corresponding author tel. no: +2349079192231 email address: emmaetim@gmail.com (e. e. etim ) and in our understanding of the solar system, “the world around us” with each successfully detected interstellar molecule telling the story of the chemistry and physics of the environment from where it was detected. they serve as the most important tools for probing deep into the interior of the molecular clouds and the molecular clouds are significant because it is from them that stars and consequently new planets are formed. the symmetric rotors serve as interstellar thermometers while the metal-bearing species provide useful information regarding the depletion of these molecular species into the molecular dust grains. understanding how the simple molecules that were present on the early earth may have given rise to the com1 e. e. etim / j. nig. soc. phys. sci. 5 (2023) 527 2 plex systems and processes of contemporary biology is one question the biologically related interstellar molecules can be used to address. molecules also provide the cooling mechanism for the clouds through their emission [3, 7, 8]. a careful look at the list of known interstellar and circumstellar molecules reveals some interesting chemistries among these molecular species. tables 1 and 2 contain the list of the currently (as of june 2021) known interstellar and circumstellar species arranged according to the number of atoms making up the molecules. some of the chemistries existing among the interstellar molecular species include the dominance of the linear carbon chain species of the form cn, h2cn, hcnn, ch3(cc)nh, ch3(c≡c)ncn and cnx (x=n, o, si, s, h, p, n-, h-) which account for more than 20% of all the known interstellar and circumstellar molecular species; the presence of about 30 alkyl group containing interstellar molecules mostly observed from the same or similar molecular clouds; periodic trends in which elements from the same group appear to form similar compounds with similar properties as in the case of oxygen and sulphur (group 6/16 elements) in which of the over 25 s-containing molecules observed in ism, about 20 have the corresponding o-containing molecules uniquely detected in ism and the abundance of s-compound relative to its o-analogue is approximately equal to the cosmic s/o ratio, 1/42 as seen in methyl mercaptan, thioisocyanic acid, etc, except where the effect of interstellar hydrogen bonding dominates [9, 10]; successive hydrogen addition where larger species are believed to be formed from the smaller unsaturated species via successive hydrogen addition in which both species could be shown to be chemically and spatially related [11]. isomerism is yet another prevailing chemistry among the interstellar molecular species. apart from the diatomics and a few species which cannot form isomers, about 40% of all the known interstellar molecules have isomeric analogues ranging from the isomeric pairs to the isomeric triads which are believed to have a common precursor for their formation process [12]. table 3 lists some of the known isomeric species and their isomeric groups. because of the conditions (low temperature and pressure) in the interstellar medium, there is hardly a consensus as to how these molecules are formed but some of the chemistries listed above and the effect of thermodynamics serve as clues as to how these molecules could be formed in ism. focusing here on isomerization, the prevalence of isomeric species among the known astromolecules coupled with the energy sources in ism such as the shock waves (which could arise from the interaction of the earth’s magnetic field with the solar wind, molecular outflows during star formation, supernova blasts and galaxies colliding with each other) which provide energy for both the formation and distribution of large interstellar species, place isomerization as one of the most plausible routes for the formation of interstellar molecules [13]. as observed in some studies, the most stable isomers are found to be more abundant than their less stable analogues except where other factors dominate; thus, the isomerization of the most stable isomer (which is probably the most abundant) to the less stable isomers can be a very effective and efficient formation mechanism in ism. also, apart from the energy sources in ism, the high abundance of the most stable isomer can drive the isomerization process irrespective of the energy barrier. according to the minimum energy principle [14], isomerization is the most important process in determining the relative abundances of the isomers in ism. isomers of the same generic formula are said to have a common intermediate in their formation and destruction routes. after reaching the generic formula, the equilibration process is said to occur. this implies an internal isomerization with a low activation barrier, assisted isomerization, or catalytic isomerization at the grain/ice surface [14]. the present work aims at estimating accurate isomerization enthalpies for 246 different molecular species from 65 isomeric groups using the gaussian 4 theory composite method [15]. the molecules range from the 3 atomic species to those with 12 atoms with at least one known interstellar molecule from each isomeric group. the results account for the extent and effectiveness of isomerization as a plausible formation route in ism; and the rationale behind the successful observation of the known species. among other things, potential candidates for astronomical searches are highlighted and discussed. 2. computational details a concerted effort between theory and experiment is found in the heart of some of the most successful scientific studies. considering the large range of molecules examined in this study, only very few of them have experimentally measured enthalpies of formation while others are so unstable that they cannot be probed experimentally in the terrestrial laboratory but all of these can be comfortably handled computationally. the gaussian 4 (g4) theory composite method is employed in estimating accurate standard enthalpies of formation (∆fho) for all the molecular systems investigated in this work. the g4 composite method is very accurate for several systems and benchmark studies; in its release note, it has an average absolute deviation of 0.8 kcal/mol from experimental values for the enthalpies of formation of 270 molecular species. the g4 method is a modification of the g3 method which has an average absolute deviation of 1.19 kcal/mol for the same number of molecular systems [15, 16]. in the g4 method, geometry optimization and zero point energy (zpe) are carried out at the b3lyp level of theory using the 6-31g(2df,p) basis set. the zpe is scaled by 0.9854. single-point calculations and energy are done using the mp4/6-31g(d) method modified by corrections from additional calculations (with mp4 and other methods) while core correlation is obtained via higher-level correction terms. the gaussian 09 suite of programs is used for all the computational studies reported here. only stable geometries with no imaginary frequencies are considered. the standard enthalpies of formation are obtained from the atomization energies using the approach described in previous studies [4-6, 17-23]. 2 e. e. etim / j. nig. soc. phys. sci. 5 (2023) 527 3 table 1: interstellar molecules between 2 and 7 atoms 2 atoms 3 atoms 4 atoms 5 atoms 6 atoms 7 atoms h2, co, csi, h2o, h2s, nh3, h2co ch4, sih4 ch3oh, ch3sh ch2choh cp hcn, tio2, hnc, co2, h2cs, c2h2 ch2nh, c2h4, hc4h c-c2h4o cs, no, ns, so2 nh2cn so hncs ch3cn, ch3nc hc(o)ch3 mgcn, ch2co, hcl, nacl, h3o+, sic3 hcooh hconh2, h3ccch nacn , c3s, h2cn hcccn, hc2c(o)h ch3nh2 fecn, kcn, alcl, alf, pn hccnc c-c3h, lhc3nh+ ch2chcn aloh, sin, sio, sis c3h c-c3h2, lh2cl, hc4n hc4cn nh, oh, c2 h2o+, hccn, ch3 c3h2 c5n, c5h c6h h2cl+, ch2cn, cn, hf, feo n2o, nh2, c2cn, c3o h2coh+ h2c4, h2ccnh ch3nco ocs lih, ch, ch+ hcnh+, c4si c5nhc5o, hoco+ ch2, hco, co+, so+, sh, c3 c5 c3n-, c-h2c3o hoch2cn c2h, c2o, hnccc hnchcn, c5s, hnchcch, o2, n2, cf+ c2s , alnc, sih3cn, hscn, c3n, c4h hno hc4nc, po, hd ph3 z-hnchcn, sicn, n2h+ c4hc-c3hcch, sih, alo, hmgnc sinc, cmgc4h hc(o)cn, arh+ , oh+, sic2 hcco, ch3o, cn-, sh+, nccp, hncnh, ch3co+, hcl+, tio, mgcch, hco+, h2nco+, ch2cch, cro, ns+, hocn, nh3d+, h2cccs, ch3cl, hcs+, h3+ vo, heh+ cncn ocn-, hcp, ccp, sicsi , h2o2 , mgc3n, s2h, hcs, nh2oh, hsc, nco, t-hono hc3o+, canc, hc3s+, ncs, ho2 c4s, t-hc(o)sh, h2ccs, nccnh+ 3. results and discussion the standard enthalpies of formation (∆fho) for all the 246 molecular species from 65 isomeric groups examined in this study are contained in the supporting information. the relative enthalpies for each isomeric group are presented and discussed in this section. the different isomeric groups investigated are 3 e. e. etim / j. nig. soc. phys. sci. 5 (2023) 527 4 table 2: interstellar molecules between 8 and >12 atoms 8 atoms 9 atoms 10 atoms 11 atoms 12 atoms > 12 atoms ch3cooh (ch3)2o (ch3)co hc9n c6h6 hc11n hcooch3 ch3ch2cn hoc2h4oh ch3c6h c3h7cn c60 hoch2cho ch3ch2oh h3cch2coh hcooc2h5 c2h5och3 c60+ h3c3cn, ch3ch2sh ch3c4cn, ch3ococh3 branchedc70 c7h, ch3chcho, c3h7cn , (nh2)2co ch3c4h ch3coch2oh cch3och2oh c-c5h5cn c6h5cn h2c6 hc7n c-c5h6 c10h7cn h(cc)3h c8h nh2ch2ch2oh c-c9h8 h2cchcho ch3conh2 ch2cchcn c8hh2nch2cn, ch3chch2, ch3chnh ch3nhcho, ch3sih3, hc7o (nh2)2co hccchchcn hccch2cn, h2cchc3n hc5nh+, h2ccchcch ch2chcch, table 3: known interstellar isomeric pairs, triads and their isomeric groups isomeric group astronomically observed isomers chn hcn, hnc cnmg mgcn, mgnc cnsi sicn, sinc cho+ hoc+ , hco+ chsn hscn, hncs hc3 c-c3h , l-c3h h2c3 c-c3h2 , l-c3h2 cn2h2 nh2cn, hncnh c3h2o hcccho, c-h2c3o c4h2 h(cc)2h, h2c4 c6h2 h2c6, h(cc)3h c4h3n ch2cchcn, ch3cccn c2h6o (ch3)2o, ch3ch2oh c3h6o (ch3)2co, ch3ch2c(o)h c3h6o2 hc(o)och2ch3, ch3oc(o)ch3 c4h7n n-ch3ch2ch2cn , b-ch3ch2ch2cn chon hnco, hcno, hocn c2h4o2 ch3cooh, hc(o)och3, hoch2c(o)h c2h4o ch2ch(oh) c-c2h4o, hc(o)ch3 c2h3n ch3cn, ch3nc, h2ccnh c3hn hc2cn, hc2nc, hnccc grouped according to the number of atoms beginning from 3 to 12. 3.1. isomers with 3 atoms in table 4, the relative enthalpies or isomerization energies of the 26 molecular species from 13 isomeric groups with 3 atoms are shown alongside the current astronomical status of these species. at least one isomer from each of these isomeric groups is a known interstellar molecule [24]-[43]. the isomerization enthalpies range from 0.2 to 41.5 kcal/mol for groups where both isomers have been detected. this range illustrates the effectiveness of the isomerization mechanism as a plausible route for the formation of these molecular species in ism; it also shows how far or the extent to which the energy sources within the ism can drive some chemical processes. for the isomeric groups where only one isomer has been detected, the observed range of the relative enthalpy of formation suggests that species like nanc, alcn, oncand even hoc can be formed from their most stable isomers that have been detected already. the high abundances of the stable isomers coupled with the energy sources in ism imply the possibility of these less stable isomers being formed from their most stable isomers via the isomerization mechanism. with the exception of c2n group where only the less stable isomer is been detected, in all other cases where only one isomer is detected, it is the most stable isomer which supports the fact that the most stable isomer is probably the most abundant and the most abundant species is easily detected compared to the less stable isomer. where both isomers have been detected, the most stable isomer is found to be the most abundant except where other processes dominate. cnc is more stable than ccn but the less stable isomer has been detected while the most stable isomer is yet to be astronomically observed. from literature perusal, there is no information regarding the spectroscopic parameters of cnc that would have warranted it astronomical searches. thus, the detection of cnc awaits the availability of accurate spectroscopic parameters. 4 e. e. etim / j. nig. soc. phys. sci. 5 (2023) 527 5 table 4: isomerization enthalpies for isomers with 3 atoms isomeric group isomers relative ∆fho (kcal/mol) astronomical status ref. cnh hcn 0.0 observed [24] hnc 13.5 observed [25] cnna nacn 0.0 observed [26] nanc 2.5 not observed cnmg mgnc 0.0 observed [27] mgcn 0.7 observed [28, 29, 30] cnal alnc 0.0 observed [31] alcn 7.3 not observed cnsi sinc 0.0 observed [32] sicn 0.2 observed [33] cho hco 0.0 observed [34] hoc 38.2 not observed chp hcp 0.0 observed [35] hpc 75.7 not observed c2n cnc 0.0 not observed ccn 2.1 observed [36] c2p ccp 0.0 observed [37] cpc 85.3 not observed cho+ hco+ 0.0 observed [38] hoc+ 37.3 observed [39, 40] chs+ hcs+ 0.0 observed [41] hsc+ 94.1 not observed cnoocn0.0 observed [42] onc17.1 not observed chs hcs 0.0 observed [43] hsc 41.5 observed [43] ions (both cations and anions) play important role in the formation processes of interstellar molecules. under the conditions of the ism, neutral atoms and molecules tend to be unreactive toward molecular hydrogen but in the presence of ions, most of the reactions become very efficient since the ions can easily react without having to overcome the reaction barrier [7, 44]. this can be seen in the case of the cho+ group where hoc+ with a relative enthalpy of formation of 37.3 kcal/mol has been detected. theoretical calculations and laboratory experiments have shown that the isomerization process in the cho+ isomers has essentially no barrier [14]. thus, with the availability of accurate spectroscopic parameters, the hsc+ and oncions (where their most stable isomers are already detected) could be successfully detected. table 5: isomerization enthalpies for isomers with 4 atoms isomeric group isomers relative ∆fho (kcal/mol) astronomical status reference chon isocyanic acid 0.00 observed [45] cyanic acid 29.0 observed [46, 47] fulminic acid 67.4 (0.0) observed [48] isofulminic acid 86.1 (18.7) not observed chsn hncs 0.0 observed [49] hscn 11.2 observed [50] hcns 40.6 (0.0) not observed hsnc 43.3 (2.8) not observed c3h c-c3h 0.0 observed [51] l-c3h 3.1 observed [52] c3n l-c3n 0.0 observed [53, 54] c2nc 22.7 not observed c-c3n 28.8 not observed c2nh hc2n 0.0 observed [55] hcnc 28.0 not observed c3o l-c3o 0.0 observed [56, 57] c-c3o 21.6 not observed c3s l-c3s 0.0 observed [58, 59, 60] c-c3s 26.3 not observed sic3 c-c3si 0.0 observed [61] l-c3si 49.8 not observed c2np nccp 0.0 observed [62] cncp 24.1 not observed c2n2 nc2n 0.0 not observed cncn 24.7 observed [63] isomerization appears to be a favourable route for the cyanide/isocyanide pair. table 4 contains 7 cyanide/isocyanide pairs of which both cyanide and isocyanide have been detected in three groups with isomerization energy ranging from 0.2 to 13.5 kcal/mol. except for the oncion with a relative enthalpy of formation of 17.1kcal/mol; the remaining members of the cyanide/isocyanide pairs have relative enthalpy of formation in the range of 0.0 to 7.3 kcal/mol which is well within the range of those already detected. thus, nanc, alcn 5 e. e. etim / j. nig. soc. phys. sci. 5 (2023) 527 6 figure 1: optimized structures of cyclic c5n isomers. figure 2: optimized structures of cyclic c5h isomers. figure 3: optimized structures of cyclic c5s isomers. figure 4: optimized structures of cyclic sich3n isomers. figure 5: optimized structures of cyclic c5o isomers. and cnc are most likely to be formed from their corresponding analogues and could be successfully detected. 3.2. isomers with 4 atoms for the isomers with 4 atoms, 25 molecular species from 10 isomeric groups are examined. table 5 contains the isomerization energies of these molecular species and their 6 e. e. etim / j. nig. soc. phys. sci. 5 (2023) 527 7 table 6: isomerization enthalpies for isomers with 5 atoms isomeric group isomers relative ∆fho (kcal/mol) astronomical status reference c3hn hcccn 0.0 observed [65] hccnc 24.1 observed [66] hnccc 45.4 observed [67] cc(h)cn 50.5 not observed hcncc 72.6 not observed c2h2o ketene 0.0 observed [68] ethynol 38.8 not observed oxirene 81.9 not observed c3h2 c-c3h2 0.0 observed [69] l-c3h2 13.7 observed [70] n2h2c nh2cn 0.0 observed [71] ch2nn 26.3 not observed nh2nc 43.4 not observed c2h2n ch2cn 0.0 observed [72] ch2nc 22.2 not observed c2hno cncho 0.0 observed [73] hconc 12.9 not observed c-c2nho 30.2 not observed hncco 69.2 not observed hc2no 83.3 not observed current astronomical status. fourteen of these molecular species have been detected from different astronomical sources [45]-[63]. in the chon, chsn, and c3h groups where more than one isomer has been detected, the relative enthalpy of formation ranges from 3.1 to 67.4 kcal/mol. for the remaining 7 isomeric groups with only one known molecular species from each, the relative enthalpy of formation ranges from 21.6 to 49.8 kcal/mol. this range falls within that of the known species, thus, pointing to the possibility of detecting the less stable isomers of these groups that could be formed via the isomerization process. among the isomers with 4 atoms examined here, there are eight cyanide/isocyanide pairs. in two of these pairs, both cyanide and isocyanide have been detected with a relative enthalpy of formation in the range of 2.8 to 29.0 kcal/mol. for the other cyanide/isocyanide pairs where only the cyanides are observed, the relative enthalpy of formation ranges from 18.7 to 24.7 kcal/mol which is below the 29.0 table 7: isomerization enthalpies for isomers with 6 atoms isomeric group isomers relative ∆fho (kcal/mol) astronomical status reference c5n l-c5n 0.0 observed [74] c-c5n* 17.4 not observed c4nc 20.7 not observed c-c5n** 21.0 not observed c-c5n*** 100.1 not observed c5h l-c5h 0.0 observed [75, 76, 77] c-c5ha 1.0 not observed c-c5hb 34.7 not observed c-c5hc 60.8 not observed c2h3n methyl cyanide 0.0 observed [78] methyl isocyanide 23.0 observed [79] ketenimine 23.1 observed [80] ethynamine 42.3 not observed 2h-azirine 46.5 not observed 1h-azirine 80.8 not observed sich3n sih3cn 0.0 observed [62] sih3nc 4.5 not observed c-sich3nd 33.4 not observed h2sicnh 46 not observed c-sich3ne 50.2 not observed h2ncsih 57.0 not observed c4hn hc4n 0.0 observed [81] hc3nc 17.4 not observed h2c3o methylene ketene 0.0 not observed propynal 7.3 observed [82] cyclopropenone 10.6 observed [83] h3con formamide 0.0 observed [84] hydroxymethylimine 12.6 not observed nitrosomethane 60.8 not observed c5s l-c5s 0.0 observed [62, 85] c-c5s# 26.9 not observed c-c5s## 41.8 not observed c-c5s### 84.1 not observed c4sc 114.2 not observed c5o l-c5o 0.0 not observed c-c5oσ 28.6 not observed c-c5oσσ 46.3 not observed c-c5oσσσ 84.5 not observed c4oc 114.3 not observed *figure 1a, **figure 1b, ***figure 1c; afigure 2a, bfigure 2b, cfigure 2c; #figure 3a, ##figure 3b, ###figure 3c; dfigure 4a, efigure 4b; σfigure 5a, σσfigure 5b, σσσfigure 5c. kcal/mol relative enthalpy of formation calculated for the known cyanide/isocyanide pairs. thus, isomerization remains a plausible mechanism for the formation of the less stable isocyanides from their corresponding cyanides which are probably more abundant. the c2n2 isomeric group is the isoelectronic analogue of the c2np isomeric group. just as there is interesting chemistry between the o and s-containing interstellar molecular species; 7 e. e. etim / j. nig. soc. phys. sci. 5 (2023) 527 8 such also exists for the n and p-containing species, though it is not well explored as that of o and s. over 80% of all the known interstellar and circumstellar molecules have been detected via their rotational spectral features because, at the low temperature of the ism, rotational excited states are easily populated. nc2n (the most stable isomer of the c2n2 group) is microwave inactive meaning that it cannot be astronomically detected via radio astronomy. there are reasons to believe the presence and detectability of nc2n; its protonated analogue, nc2nh+ has been detected [64]. nccp, the isoelectronic analogue of nc2n is also known [62]. both the protonated analogues of nc2n and its isoelectronic analogue that have been observed are likely to be less abundant than nc2n because of the reactive nature of the ion and the less cosmic abundance of p as compared to n. thus, infrared astronomy of nccn or radio astronomy of its isotopologues (which are microwave active) is likely to be successful. the observed isomers in all the groups shown in table 5 are also the most stable isomers. 3.3. isomers with 5 atoms isomerization energies for 20 molecular species from 6 isomeric groups are presented in table 6 alongside their current astronomical status. nine of these species are known astromolecules [65]-[73]. in the c3hn and c3h2 groups where more than one isomer has been detected, the isomerization energy ranges from 13.7 to 45.4 kcal/mol, and the isomerization energies for some of the isomers with five atoms whose most stable isomers have been detected fall within this range. these include; ethynol, ch2nn, nh2nc, ch2nc, hconc and c-c2nho. these species can as well be formed from their most stable isomers via the isomerization mechanism. again, isomerization appears to be a favourable route for the formation of isocyanides from their corresponding cyanides. as shown in table 6, an isomerization enthalpy as high as 45.4 kcal/mol is noted for an isocyanide formation. this strongly supports the formation of other isocyanides from their corresponding cyanides. the barrier for other isocyanides whose corresponding cyanides have been detected ranges from 12.9 to 43.4 kcal/mol which is within the limit of the one that has been observed. the fact that the most stable isomers are probably the most abundant and are easily detected in ism is well demonstrated among these isomers. from table 6, all the observed isomers are the most stable ones in their respective isomeric groups as compared to the ones that have not been detected. 3.4. isomers with 6 atoms the isomerization energies and the current astronomical status of the different isomeric species with 6 atoms investigated in this study are presented in table 7. figures 1 to 5 display some of the cyclic isomers highlighted in table 7. one-third of all the molecular species presented in table 7 have all been detected from several astronomical sources [74]-[85]. in the c2h3n and h2c3o groups where more than one isomer has been detected, the isomerization enthalpies range from 7.3 to 23.1 kcal/mol. except for some of the cyclic isomers which are highly unstable, most of the unknown isomers (c-c5n*, c4nc, c-c5n**, c-c5ha, sih3nc, hc3nc, methylene ketene, hydroxymethylimine) have isomerization enthalpies in the range of the known isomers, suggesting their possible astronomical observation if they could be formed via isomerization as the ones that have been observed. ketenimine for instance is reported to be formed from methyl cyanide via tautomerization (i.e., an isomerization pathway in which the migration of hydrogen atom from the methyl group to the nitrogen atom is accompanied by a rearrangement of bonding electrons) and this process is said to be driven by shock waves that provide the energy for both the formation and distribution of large interstellar species [80]. the c5h isomers; l-c5h and c-c5ha are almost identical in energy. if the spectroscopic parameters of c-c5ha (figure 2a) are accurately probed, either experimentally or theoretically; this species (c-c5h) could be detected in ism. the four cyanide/isocyanide pairs among the isomers with 6 atoms presented in table 7 have isomerization enthalpies in the range of 4.5 to 23.0 kcal/mol. interestingly, the c2h3n group where methyl cyanide and methyl isocyanide have been detected has the highest relative enthalpy of formation of 23.0 kcal/mol which implies the possible detection of the other isocyanides with lower barriers (4.5 to 20.7kcal/mol) assuming isomerization as their plausible formation route. methylene ketene and methyl ketene have been proposed as potential interstellar molecules for many reasons. the ketenes are found to be more stable than their corresponding isomers (tables 6, 7, and 9); they are less affected by interstellar hydrogen bonding assuming surface formation processes; ketene and ketenyl radical (h2c2o and hc2o respectively) are known interstellar molecules [64, 65, 66, 67, 68]. just as every known o-containing interstellar molecule points to the presence and the detectability of the s-analogue. for every known s-species, the o-analogue is not only present in detectable abundance, but it can also be said to have even been overdue for astronomical detection. c5o is the oanalogue of c5s that has been detected. thus, it is an important potential interstellar molecule. 3.5. isomers with 7 atoms table 8 shows the isomerization energies and the current astronomical status of different isomeric species with 7 atoms investigated in this study. of the 22 molecular species in the table, 7 have been astronomically observed [86]-[93]. in the h4c2o isomeric group where all the isomers have been detected, the isomerization energies range from 12.2 to 27.8 kcal/mol. all the non-observed isomers in the h3c3n and c3h4 isomeric groups including, cyanomethanol, iminoacetaldehyde, and methyl cyanate (from c2h3no isomeric 8 e. e. etim / j. nig. soc. phys. sci. 5 (2023) 527 9 group) have isomerization enthalpies in the range calculated for the h4c2o isomers which have all been detected. this points to the possibility of detecting these molecules in ism with relative enthalpy of formation within those of the known interstellar molecules. there is no cyanide/isocyanide pair observed among these systems, however, the relative enthalpy of formation in the range of 19.9 to 27.3 kcal/mol is within the range of those noted for other observed cyanide/isocyanide pairs. as in the previous examples, all the observed isomers are the most stable ones in their respective groups. in the h4c2o isomeric group where all the stable isomers have been observed, the most stable isomer; acetaldehyde was first observed before the other isomers. as would be expected, the most stable isomer (acetaldehyde) is present in high abundance in the different astronomical sources where it has been detected as compared to the abundances of the less stable isomers in the same sources [94, 95, 96]. table 8: isomerization enthalpies for isomers with 7 atoms isomeric group isomers relative ∆fho (kcal/mol) astronomical status reference h4c2o acetaldehyde 0.0 observed [86, 87] vinyl alcohol (syn) 12.2 observed [88] vinyl alcohol (anti) 13.9 observed [88] ethylene oxide 27.8 observed [89] h3c3n acrylonitrile 0.0 observed [90] isocyanoethene 19.9 not observed c3h4 ch3c2h 0.0 observed [91] h2ccch2 7.8 not observed c-c3h4 23.6 not observed c2h3no methyl isocyanate 0.0 observed [92, 93] cyanomethanol 13.6 not observed iminoacetaldehyde 20.1 not observed methyl cyanate 27.3 not observed 2-aziridinone 36.0 not observed 2-oxiranimine 40.8 not observed methyl fulminic acid 56.5 not observed 2-iminoethenol 60.9 not observed nitrosoethene 69.3 not observed 2h-1,2-oxazete 79.8 not observed methyl isofulminic acid 84.8 not observed nhydroxyacetylenamin 93.6 not observed (aminooxy)acetylene 94.7 not observed 3.6. isomers with 8 atoms table 9 contains the isomerization energies for 33 isomeric species from 7 groups containing eight atoms each. the astronomical statuses of these species are also shown in the table. figure 6 highlights the cyclic isomers from the c2h5n and ch4n2o groups. as shown in table 9, 11 of these species have been astronomically detected [97]-[109]. the isomerization energies of these species range from 3.1 to 50.0 kcal/mol. except for 1,2-dioxetane, 1,3-dioxetane, and epoxyproprene, the isomerization energies for all the unknown isomers with 8 atoms fall within the range (of 13.0 to 46.9 kcal/mol) of the known isomers (3.1 to 50.0 kcal/mol). thus, some of the unknown isomers in table 9 could be formed from their most stable isomers (which are probably more abundant) via the isomerization process. the three cyanide/isocyanide pairs among these isomers have isomerization enthalpies ranging from 19.0 to 25.6 kcal/mol. though no cyanide/isocyanide pair has been detected among these molecular species, the relative enthalpy of formation is within the range of those where the pairs have been detected. ch3ccnc and h2nch2nc could be formed from their corresponding cyanides that have been detected, thus, they could also be detected provided their spectroscopic parameters are accurately known. methyl ketene, the most stable isomer of the h4c3o group is yet to be astronomically detected. the reasons for its presence and detectability in ism are the same as discussed for methylene ketene (among the isomers with 6 atoms). table 9: isomerization enthalpies for isomers with 8 atoms isomeric group isomers relative ∆fho (kcal/mol) astronomical status reference c4h3n ch3cccn 0.0 observed [97] ch2cchcn 3.1(0.0) observed [98, 99] hccch2cn 13.0 not observed ch3ccnc 25.6 not observed ch2cchnc 25.7 (22.6) not observed c2h4n2 h2nch2cn 0.0 observed [100] h2nch2nc 19.0 not observed c2h4o2 acetic acid 0.0 observed [101] methylformate 17.7 observed [102, 103] glycolaldehyde 33.2 observed [104] 1,3-dioxetane 52.9 not observed 1,2-dioxetane 103.0 not observed h4c3o methyl ketene 0.0 not observed propenal 2.3 observed [105] cyclopropanone 17.3 not observed propynol 30.8 not observed propargyl alcohol 35.2 not observed methoxy ethyne 42.0 not observed 1-cyclopropenol 44.0 not observed 2-cyclopropenol 45.5 not observed epoxypropene 65.8 not observed h2c6 hc6h 0.0 observed [106] h2c6 50.0 observed [107] c2h5n ch3chnh 0.0 observed [108] h2cchnh2 2.8 not observed ch3nch2 8.1 not observed c-c2h5nm 19.8 not observed ch4n2o h2nconh2 0.0 observed [109] h2nnhcho 11.0 not observed hn2ch2oh 35.0 not observed cch4n2on 42.4 not observed ch3nhno 43.4 not observed h2nchnho 46.9 not observed mfigure 5a; nfigure 5b. 9 e. e. etim / j. nig. soc. phys. sci. 5 (2023) 527 10 figure 6: optimized structures of cyclic c2h5n (a) and ch4n2o (b) isomers. table 10: isomerization enthalpies for isomers with 9 atoms isomeric group isomers relative ∆fho (kcal/mol) astronomical status reference c2h6o ethanol 0.0 observed [110, 111] dimethyl ether 12.6 observed [112] c2h6s ch3ch2sh 0.0 observed [113] ch3sch3 1.4 not observed c2h5on acetamide 0.0 observed [114] nmethylformamide 9.7 not observed nitrosoethane 64.7 not observed 1-aziridnol 77.9 not observed cyanoethoxyamide 134.4 not observed c3h5n cyanoethane 0.0 observed [115] isocyanoethane 20.8 not observed propylenimine 22.0 not observed 2-propen-1-imine 25.3 not observed n-methylene ethenamine 26.0 not observed azastene 32.2 not observed cyclopropanimine 37.2 not observed methylene azaridine 43.8 not observed propargylamine 45.1 not observed 1azabicyclo(1.1.0)butane 53.5 not observed c5h4 ch3c4h 0.0 observed [116] h2c3hc2h 5.2 not observed h2c5h2 8.1 not observed cc5h4 x 26.1 not observed cc5h4 y 31.6 not observed c3h6 ch3chch2 0.0 observed [117] cc3h6 z 8.2 not observed c7hn hc7n 0.0 observed [118][120] hc6nc 27.1 not observed xfigure 6a; yfigure 6b; zfigure 6c. figure 7: optimized structures of cyclic c5h4 (a and b) and c3h6 (c) isomers. 3.7. isomers with 9 atoms in table 10, 28 isomeric species from 7 groups are presented with their isomerization enthalpies and current astronomical status. eight of these species are known interstellar molecular species [110]-[120]. figure 7 shows some table 11: isomerization enthalpies for isomers with 10 atoms isomeric group isomers relative ∆fho (kcal/mol) astronomical status reference c2h6o2 1,1-ethanediol 0.0 not observed ethylene glycol 6.1 observed [123] methoxy methanol 15.9 not observed ethyl hydroperoxide 51.9 not observed dimethane peroxide 57.7 not observed c3h6o propanone 0.0 observed [124, 125] propanal 8.1 observed [126] propen-2-ol 13.8 not observed 1-propen-1-ol 18.6 not observed methoxy ethene 25.6 not observed 2-propene-1-ol 27.1 not observed 1,2-epoxypropane 30.0 not observed cyclopropanol 31.2 not observed oxetane 33.2 not observed c6h3n ch3(cc)2cn 0.0 observed [127] ch3(cc)2nc 25.3 not observed table 12: isomerization enthalpies for isomers with 11 atoms isomeric group isomers relative ∆fho (kcal/mol) astronomical status reference c3h6o2 propanoic acid 0.0 not observed ethylformate 11.8 observed [128] methyl acetate 14.3 observed [129] lactaldehyde 28.1 not observed dioxolane 36.1 not observed glycidol 52.1 not observed dimethyldioxirane 81.7 not observed c9hn hc9n 0.0 observed [130] hc8nc 27.2 not observed figure 8: optimized geometries of cnc and ccn at mp2(full)/6-311++g** level. of the cyclic molecules with 9 atoms. among the species shown in table 10, only ethanol and dimethyl ether from the c2h6o group have been observed in more than one isomeric form with a relative enthalpy of formation of 12.6 kcal/mol. ch3sch3, the s-analogue of dimethyl ether remains a potential candidate for astronomical detection for many reasons; the well-established chemistry of s and o-containing interstellar molecules, the high abundance of ch3ch2sh (the most stable isomer of the group which can isomerize to ch3sch3) and the low relative enthalpy of formation (1.4 kcal/mol) as compared to that of dimethyl ether (12.6 kcal/mol). assuming accurate spectroscopic parameters are available, interstellar ch3sch3 10 e. e. etim / j. nig. soc. phys. sci. 5 (2023) 527 11 table 13: isomerization enthalpies for isomers with 12 atoms isomeric group isomers relative ∆fho (kcal/mol) astronomical status reference c6h6 benzene 0.0 detected [137] fulvene 34.2 not detected 3,4dimethylenecyclopropene 62.0 not detected 1,5-hexadiene-3-yne 62.3 not detected 2,4-hexadiyne 65.2 not detected 1,2,3,4-hexateraene 71.7 not detected 1,3-hexadiyne 73.2 not detected trimethylenecyclopropane 79.8 not detected 1,4-hexadiyne 82.3 not detected 1,5-hexadiyne 85.8 not detected c3h8o propan-2-ol 0.0 not observed propanol 3.7 not observed ethyl methyl ether 8.2 observed [139] c4h7n isopropyl cyanide 0.0 observed [138] propyl cyanide 0.4 observed [140] 2-isocyanopropane 18.2 not observed 2-aminobutadiene 18.2 not observed 3-pyrroline 20.2 not observed 2,2-dimethylethylenimine 20.5 not observed but-1-en-1-imine 22.6 not observed 2,3-butadiene-1-amine 38.1 not observed n-vinylazaridine 40.1 not observed n-methyl-1-propyn-1amine 41.4 not observed 3-butyn-1-amine 43.2 not observed n-methyl propargylamine 49.3 not observed 2azabicyclo(2.1.0)pentane 51.9 not observed table 14: relative energies of cnc and ccn method relative energy (kcal/mol) cnc ccn ccsd/6311++g** 0.0 1.5 mp2(full)/6311++g** 0.0 6.2 b3lyp/6311++g** 0.0 2.0 g4 0.0 3.7 will soon become a reality. isomerization remains one of the plausible formation routes table 15: relative energies of linear and cyclic stable isomers of c5h method relative energy (kcal/mol) dipole moment (debye) c-c5h l-c5h c-c5h l-c5h mp2(full)/6311++g** 0.0 18.7 3.4 4.2 mp2/aug-ccpvtz 0.0 17.5 3.4 4.2 g4 10.9 0.0 3.7 4.6 ccsd/6311++g** 3.0 0.0 3.4 4.6 b3lyp/6311++g** 10.4 0.0 3.7 5.2 b3lyp/aug-ccpvtz 10.3 0.0 3.7 5.2 for the formation of the less stable isomers of the different groups here whose most stable isomers have been detected. the isomerization enthalpies for the two cyanide/isocyanide pairs range from 20.8 to 27.1 kcal/mol. though none of these pairs has been detected, the possibility of their formation from their corresponding cyanides (via isomerization) which could lead to their successful detection cannot be ruled out. hc6nc is the highest member of the hc2nnc linear chains with experimentally measured rotational transitions that can be used for its astronomical search [121]. however, a recent study using a combined experimental and theoretical approach has provided accurate rotational constants for higher members of the hc2nnc linear chains [6]. hnc remains the only member of this series that has been detected in ism [122]. 3.8. isomers with 10 atoms isomerization energies and astronomical statuses for 16 molecular species from three isomeric groups comprising 10 atoms each are presented in table 11. of these three groups, only the c3h6o group contains isomers (propanone and propanal) that have been detected in more than one isomeric form with a relative enthalpy of formation of 8.1 kcal/mol [123, 124, 125, 126, 127]. ch3(cc)2nc is the isocyanide analogue of ch3(cc)2cn that has been detected. it has a relative enthalpy of formation of 25.3 kcal/mol which is within the range of the known cyanide/isocyanide pairs. however, there is little or no information regarding the rotational spectrum of ch3(cc)2nc that could warrant its astronomical search. thus, the availability of accurate spectroscopic parameters for this species remains the starting point in the steps toward its astronomical detection. 1,1-ethanediol; the most stable isomer of the c2h6o2 group has not been detected whereas ethylene glycol, the next stable isomer of the group with an isomerization enthalpy of 6.1kcal/mol has been detected in good abundance. the delayed astronomical detection of 1,1-ethanediol is directly linked to a lack of spectroscopic parameters for this molecule. the rotational spectrum of 1,1-ethanediol is yet to be probed either experimentally or theoretically. once this is done, this molecule could be successfully observed. 11 e. e. etim / j. nig. soc. phys. sci. 5 (2023) 527 12 3.9. isomers with 11 atoms there are currently seven known interstellar molecules (hc9n, ch3c6h, ethyl formate, methyl acetate, hydroxyacetone, cyclopentadiene and ethanolamine) containing 11 atoms [128]-[134]. in table 12, the 9 isomeric species from two groups containing 11 atoms are shown with their isomerization enthalpies and current astronomical status. for the known isomers in the c3h6o2 group, the isomerization energy ranges from 11.8 to 14.3 kcal/mol. the non-detection or delayed detection of propanoic acid (the most stable isomer of the group) has been traced to the effect of interstellar hydrogen bonding on the surface of the dust grains which reduces the overall abundance of this species in the gas phase, thus, influencing its successful astronomical observation. hc9n is the second largest member of the cyanopolyyne chain that has been detected in ism. as seen in other cyanide/isocyanide pairs, the isomerization of hc9n to hc8nc with the relative enthalpy of formation of 27.2 is achievable. an accurate rotational constant for hc8nc is now available from a recent study [6, 17]. thus, hc8nc could be astronomically searched. 3.10. isomers with 12 atoms only very few cyclic molecules are known among the interstellar molecular species [17, 20, 21]. from the different isomeric groups considered in this study, it is clear that the cyclic isomers are found to be among the less stable isomers in their respective groups as compared to their corresponding straight-chain analogues. this low stability could affect their interstellar abundance and influences their astronomical observation. the c6h6 isomeric group is one of the few groups where the cyclic molecules are found to be the most stable species. apart from cyanocyclopentadiene and 2-cyanocyclopentadiene that have been recently observed in the interstellar medium [135, 136], the other four known interstellar molecules with 12 atoms, [137, 138, 139, 140] their corresponding isomers, and isomerization enthalpies are presented in table 13. the only known branched-chain interstellar molecule; isopropyl cyanide is almost equivalent in energy to its linear chain analogue; propyl cyanide with a relative enthalpy of formation of 0.4kcal/mol. propan-2-ol and propanol (the two most stable isomers of the c3h6o group) are yet to be astronomically observed. these species have also been shown to be strongly affected by interstellar hydrogen bonding as compared to ethyl methyl ether that has been detected [6, 18, 19, 20, 21]. the stability of ethyl methyl ether is also enhanced by the stabilizing effect of the two alkyl substituents while propanol and propan2-ol have only one alkyl substituent each. 3.11. potential interstellar molecules knowing the right candidates for astronomical searches is vital in reducing the number of unsuccessful astronomical searches considering the time, energy and resources involved in these projects. from the present study, few molecular species have been identified as potential interstellar molecules. these are briefly summarized below: isocyanomethylidyne, cnc: this is the isocyanide analogue of c2n that has been detected [141]. cnc is found to be more stable than its cyanide analogue. table 14 contains the relative energies of cnc and ccn from different quantum chemical calculation methods while figure 8 shows the optimized geometries of these isomers at the mp2(full)/6-311++g** level with the bond angle in degrees and bond distance in angstroms. as discussed under the isomers with 3 atoms, all the observed species are the most stable ones and these stable species are also found to be more abundant. where both species have been detected, the most stable is reported to present in high abundance than the less stable. for instance, hcn is found to be more abundant than hnc in different molecular clouds and this is also the case for the mgnc/mgcn abundance ratio measured in the asymptotic giant branch (agm) stars [142, 143, 144, 145]. cnc is more stable than ccn which implies that cnc should be present in high abundance in ism than ccn that has been detected. thus, cnc remains a potential candidate for astronomical detection. cnc is microwave inactive with a zero-dipole moment; thus, infrared astronomy remains the best approach for its astronomical observation. nccn: the rationale for the choice of nccn as a potential interstellar molecule is well discussed under the isomers with 4 atoms. c-c5h: with the limited number of known cyclic interstellar molecules, it is always exciting finding rings that are as stable as their corresponding chains. c-c5h (figure 2a) is the cyclic analogue of c5h that has been detected. from table 7, the relative enthalpy of formation between the linear chain and this cyclic analogue is just 1 kcal/mol. table 15 shows the relative energies of these species at different quantum chemical calculation methods. while the mp2 method predicts the cyclic isomer to be more stable than the linear, others methods predict the reverse but the magnitude of the difference in energy is much at the mp2 level as compared to other methods. figure 9 displays the optimized structures of these species. as shown in table 15, c-c5h is microwave active with a very good dipole moment making its astronomical searches in the radio frequency possible. if the spectroscopic parameters of c-c5h can be accurately probed, either experimentally or theoretically, the possibility of its astronomical observation is very high. figure 9: optimized structures of c5h most stable isomers. methylene and methyl ketenes: as highlighted in the text, the ketenes are found to be more stable than their corresponding isomers in all the isomeric 12 e. e. etim / j. nig. soc. phys. sci. 5 (2023) 527 13 groups examined in this study (tables 6, 7 and 9) and as such be present in detectable amounts in ism. ketenes have also been shown to be less affected by hydrogen bonding on the surface of the interstellar dust grains as compared to their corresponding isomers. also, the fact that ketene (h2c2o), ketenyl radical (hc2o) and isomers of ketenes (like propynal, cyclopropenone, and propenal) are known interstellar molecular species further supports the presence and detectability of methyl ketene and methylene ketene in ism. thus, methyl ketene and methylene kenetne remain potential interstellar molecules pending their astronomical searches. no unsuccessful astronomical search has been reported for any of these ketenes. c5o: this is the oxygen analogue of c5s which is a known interstellar molecule. without any exception, an interstellar o-containing molecular species is more abundant than its sanalogue. this could simply be traced to the cosmic abundance of o and s. thus, for every known s-species, the o-analogue is not only present in detectable abundance, but it can also be said to have even been overdue for astronomical detection because for sure the o-species are more abundant than their s-analogue and as such could be detected with less difficulty as compared to its s-analogue. the column density of 2.14x 1012cm-2 reported for c5s suggests thehigh abundance of c5o in ism [62, 85]. the microwave spectrum of c5o that could guide its successful astronomical observation is available [146]. interstellar c5o is just a matter of time. ch3sch3: this is the s-analogue of dimethyl ether; a known interstellar molecule. the interstellar chemistry of sand o-containing species is well established. every known o-containing interstellar molecule points to the presence and detectability of the s-analogue. as earlier mentioned, except where the effect of interstellar hydrogen bonding dominates, the ratio of an interstellar sulphur molecule to its oxygen analogue is close to the cosmic s/o ratio. assuming dimethyl ether is formed from ethanol via isomerization with a barrier of 12.6 kcal/mol, then the isomerization of c2h5sh to ch3sch3 is a much more feasible process with a relative enthalpy of formation of just 1.4 kcal/mol (nine times lower than that of ch3och3) compared to that of the o-analogue. thus, because of the unique chemistry of sand o-containing interstellar molecules and the low relative enthalpy of formation, ch3sch3 is considered a potential candidate for astronomical observation assuming its spectroscopic parameters are accurately known. 1,1-ethanediol, propanoic acid, propan-2-ol, and propanol: these molecules are found to be the most stable isomers in their respective groups (tables 11, 12 and 13). whereas their less stable isomers have been astronomically detected; successful detection of these most stable isomers is highly feasible. the spectroscopic parameters for propanoic acid, propan-2-ol, and propanol are known but those of 1,1-ethanediol are yet to come by. though the delayed astronomical observations of some of these species have been linked to the effect of interstellar hydrogen bonding; there are high chances for their successful astronomical detection. 4. conclusion an extensive investigation of the isomerization energies of 246 molecular species from 65 isomeric groups is reported in this study. from the results, isomerization is found to be one of the plausible mechanisms for the formation of molecules in ism. the high abundances of the most stable isomers coupled with the energy sources in ism drive the isomerization process even for barriers as high as 67.4 kcal/mol. however, the chemical reaction pathways can also influence the final abundance of any species in the ism. specifically, the isomerization process is found to be very effective in converting cyanides to their corresponding isocyanides and vice versa. thus, for every cyanide or isocyanide, the corresponding isocyanide or cyanide could be synthesized via isomerization. also, from the results, the following potential interstellar molecules; cnc, nccn, cc5h, methylene ketene, methyl ketene, ch3sch3, c5o, 1,1ethanediol, propanoic acid, propan-2-ol, and propanol are highlighted and discussed. acknowledgement eee acknowledges a research fellowship from the indian institute of science, bangalore, and fruitful discussions with prof. e. arunan of the same institute. the visiting position from the indian institute of technology bombay where the final part of 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[146] t. ogata, y. ohshima, y., endo, “rotational spectra and structures of carbon monoxides c5o, c7o and c9o”, j. am. chem. soc 117 (1995) 3593. 16 j. nig. soc. phys. sci. 4 (2022) 242–250 journal of the nigerian society of physical sciences approximate solutions of the schrödinger equation for a momentum-dependent potential c. a. onatea,d,∗, i. b. okonb, m. c. onyeajuc, a. d. antiab adepartment of physical sciences, landmark university, omu-aran, nigeria btheoretical physics group, department of physics, university of uyo, nigeria cdepartment of physics, university of port harcourt, choba, nigeria dlandmark university sdg 4 (quality education) abstract the solution of one-dimensional schrödinger equation for a newly proposed potential called modified shifted deng-fan momentum-dependent potential is obtained via supersymmetric approach. the expectation values of momentum and position were calculated using hellmann feynman theorem. the effects of momentum-dependent parameter on the solutions of the system as well as the expectation values were studied. finally, the special cases of the interacting potential were obtained. doi:10.46481/jnsps.2022.653 keywords: eigensolutions, wave equation, potential models, momentum dependent article history: received: 10 february 2022 received in revised form: 13 april 2022 accepted for publication: 14 april 2022 published: 29 may 2022 c©2022 journal of the nigerian society of physical sciences. all rights reserved. communicated by: c. o. edet, p. amadi & r. khordad 1. introduction in both the relativistic and nonrelativistic mechanics, wave equations play some significant roles. the wave functions embedded information that describes a complete quantum system. it is also noted that the solutions of a complicated physical systems are checked and improved through numerical method by wave equations [1, 2]. however, the major objective of the wave equations is the determination of the eigenvalue and its wave function. the solution of this wave equation can be studied via h |ψ〉 = e |ψ〉, (in this case, h is hamiltonian and e is energy). the wave function ψ is expanded as |ψ(r, e)〉 =∑ n fn(e) |ψn(r)〉, the term r stands for set of coordinates for real space. ∗corresponding author tel. no: +2348056985024 email address: oaclems14@physicist.net (c. a. onate ) in most of these cases, several potentials such as hyperbolic potential [3, 4], modified rosenmorse [5], hulthén potential [6, 7], manning-rosen potential [8-10], kratzer potential [11], potential family [12, 13], and others, were used. in recent time, a greater interest has been focused on the wave equation with position dependent-potential [14], constant mass-dependent potential [15], and energy-dependent potential for both relativistic and nonrelativistic wave equations [16]. motivated by these, we want to study the non-relativistic equation with a modified shifted dengfan momentum-dependent potential. in the present study, the effect of the momentum-dependent parameter on energy of the system will be examined under the proposed potential and its subset potentials. the modified shifted dengfan potential is a combination of deng-fan and hulthén like potentials. the deng-fan potential also known as improved manning-rosen potential, is an empirical and diatomic molec242 onate et al. / j. nig. soc. phys. sci. 4 (2022) 242–250 243 table 1. bound states of the modified shifted deng-fan momentum-dependent potential model with µ = ~ = 1, α = 0.15 å, re = 0.4 å, de = 5 ev and d0 = 0.2 ev for various n and l. n l ρ = 1 ρ = 0 0 0 2.4925255 4.0257731 1 0 3.9418902 4.6881046 1 4.2651097 4.8366869 2 0 4.5080914 4.8958752 1 4.6571299 4.9527631 2 4.8134283 4.9944769 3 0 4.7739363 4.9739034 1 4.8496210 4.9946798 2 4.9305476 5.0055476 3 4.9863491 5.0015209 ular potential function proposed in 2012 [17]. this potential was used to examine the molecular vibrations and the energy eigenvalues under relativistic and nonrelativistic wave equation [18]. in ref. [19], it was theoretically used to study hf and information entropy. the deng-fan potential model is given by v (r) = de ( 1 − eαre − 1 eαr − 1 )2 . (1) where, de is the dissociation energy, re is the equilibrium bond length, α is the screening parameter and r is the internuclear separation. the scheme of our presentation is as follows: in section 2, we presented the bound state solutions, results and discussion are given in section 3 while conclusion is given in section 4. 2. bound state solutions in the nonrelativistic quantum mechanics, the radial schröd− inger equation with an interacting potential v (r) together with a nonrelativistic energy en`, is given by − ~2 2µ d2un,`(r) dr2 + [ v (r) − en,` + ~2`(` + 1) 2µr2 ] un,`(r) = 0, (2) where, l is the angular quantum number, r is the internuclear distance, n is the quantum number, ~ is the reduced planck’s constant and un,`(r) is the wave function. in this study, we shall solve the radial schrödinger equation above for modified shifted deng-fan momentum-dependent potential. thus, the potential function for momentum dependent is given as v (r) = de ( 1 − eαre − 1 eαr − 1 )2 − d0 ( 1 eαr − 1 ) (1 + ρ). (3) in the equation above, de is the dissociation energy with different values for various diatomic molecules, d0 is the potential strength whose value is arbitrary chosen and re is the bond separation. the schrödinger equation in equation (2) has centrifugal term which needs to be approximated. several approximation scheme have been used for different potential [20]. in this study, the centrifugal term will be approximated using 1 r2 ≈ α2 ( c0 + eαr (eαr − 1)2 ) , (4a) where c0 is a dimensionless constant obtained by using the following power series α2 ( c0 + eαr (eαr − 1)2 ) = α2 [ c0 + 1 (αr)2 − 1 12 + (αr)2 240 − (αr)4 6048 + o((αr)6) ] . (4b) this finally give the dimensionless constant as c0 = 1/12. with equation (3) and equation (4), the radial schrödinger equation in equation (2) becomes d2un,`(r) dr2 =  ( 2µdeβ2 b2 e−αr ~2 + `(` + 1)α 2 ) e−αr (1 − e−αr )2 − 2µβ2 (2de b+d0 )e−αr ~2 1 − e−αr + e2  un,`(r), (5) where, β2 = 1 + ρ, (6) e2 = 2µ(deβ2 − en,`) ~2 + `(` + 1)c0α 2, (7) b = eαre − 1. (8) in the present study, the authors adopt supersymmetric approach (susyqm) to solve equation (5). to proceed using the basic concept and formalism of susyqm, we write the ground state wave function as u0,`(r) = ex p ( − ∫ w(r)dr ) , (9) where w(r) is defined as a superpotential that is propose based on the interacting potential [21]. on the basis of this work, the superpotential is given as w(r) = ρ0 + ρ1 eαr − 1 , (10) where the constants ρ0 and ρ1 will be determine later. to relate the superpotential function in equation (10) to equation (5), we establish a reccati differential equation of the form w 2(r) − dw(r) dr = ( 2µdeβ2 b 2 e−αr ~2 +`(`+1)α2 ) e−αr (1−e−αr )2 (11) − 2µβ2 (2de b+d0 )e −αr ~2 1−e−αr + er. relating equation (11) with equation (5), we can now determine the two parametric constants in the superpotential function as ρ20 = er, (12) 243 onate et al. / j. nig. soc. phys. sci. 4 (2022) 242–250 244 figure 1. variation of energy en,`(ev ) against the screening parameter α(å) with momentum dependent potential ρ1(a) and without momentum dependent potential ρ0 (b) for three different quantum states with µ = ~ = l = 1, re = 0.2 å, de = 3 ev and d0 = 6 ev . table 2. ro-vibrational energy spectra in (ev ) for 2p, 3p, 3d, 4p, 4d, and 4f for deng-fan and hulthèn potential with three different values of the screening parameter for various quantum and angular quantum states. ρ = 0 ρ = 1 state α de = 5, d0 = 0 de = 0, d0 = 5 de = 5, d0 = 0 de = 0, d0 = 5 2p 0.05 3.7357639 -1252.5010 1.0959759 -5005.0010 0.15 3.8435208 -141.39826 1.2510768 -560.56493 0.25 3.9472499 -52.526042 1.4029712 -205.02604 3p 0.05 4.3380223 -558.05774 2.6082849 -2227.2244 0.15 4.4738042 -64.248083 2.8289266 -251.93327 0.25 4.5993040 -24.776910 3.0417528 -93.943576 4p 0.05 4.6792822 -315.00479 3.7798457 -1255.0048 0.15 4.8154953 -37.265347 4.0535781 -143.93201 0.25 4.9170844 -15.119792 4.2949652 -55.119792 4d 0.05 4.6480486 -315.00438 3.6133017 -1255.0044 0.15 4.7878709 -37.261597 3.8809305 -143.92826 0.25 4.9000869 -15.109375 4.1249749 -55.109375 4f 0.05 4.6313806 -315.00375 3.5085755 -1255.0038 0.15 4.7760037 -37.255972 3.7735201 -143.92264 0.25 4.8997892 -15.093750 4.0224272 -55.093750 ρ1 = − α 2 1 ± √ (1 + 2l)2 + 8µdeβ2b2 α2~2  , (13) ρ0 = 2µdeβ2 b(b+2) ~2 2µd0β2 ~2 −ρ 2 0 2ρ1 . (14) the bound state solution requires that the wave function satisfies the boundary conditions un,`(r)/r = 0 as r −→ ∞ and un,`(r)/r is finite at r = 0. the regularity conditions enable us to determine ρ0 > 0 and ρ1 < 0 as can be justify in equation (13) and eq. (14). by using equation (10), a pair of partner potentials v±(r) = w 2 ± dw(r)/dr is constructed as: v+(r) = ρ 2 0 + ρ1(2ρ0 −ρ1)e−αr 1 − e−αr + ρ1(ρ1 −α)e−αr (1 − e−αr )2 , (15) v−(r) = ρ 2 0 + ρ1(2ρ0 −ρ1)e−αr 1 − e−αr + ρ1(ρ1 + α)e−αr (1 − e−αr )2 , (16) the two partner potentials satisfied the relationship v+(r, a0) = v−(r, a1) + r(a1), (17) where a0 is an old set of parameters and a1 is a new set of parameters uniquely determined from a0 , the r(a1) is a reminder term that is independent of the variable r. however, 244 onate et al. / j. nig. soc. phys. sci. 4 (2022) 242–250 245 figure 2. variation of energy en,`(ev ) against the screening parameter α(å) with momentum dependent potential ρ1 (a) and without momentum dependent potential ρ0 (b) for three different quantum states with µ = ~ = l = 1, re = 0.2 å, de = 3 ev and d0 = 6 ev . as ρ1 −→ ρ1 + α for a0 = ρ1, we can write a recurrent relation of the form ρ2 = a0 + 2α, ρ3 = a0 + 3α, ρ4 = a0 + 4α and consequently, ρn = a0 + nα. for a proper relation, eq. (17) can be written using the above recurrence relation as r(a1) = ( 2µdeβ2 b(b+2)−2µd0β2 ~2 −a20 2a0 )2 − (18)( 2µdeβ2 b(b+2)−2µd0β2 ~2 −a21 2a1 )2 , figure 3. variation of energy en,`(ev ) against the screening parameter α with momentum dependent potential ρ1 (a) and without momentum dependent potential ρ0 (b) for three different quantum states with µ = ~ = l = 1, re = 0.2 å, de = 0 ev and d0 = 8 ev . r(a2) = ( 2µdeβ2 b(b+2)−2µd0β2 ~2 −a21 2a1 )2 − (19)( 2µdeβ2 b(b+2)−2µd0β2 ~2 −a22 2a2 )2 , r(a3) = ( 2µdeβ2 b(b+2)−2µd0β2 ~2 −a22 2a2 )2 − (20)( 2µdeβ2 b(b+2)−2µd0β2 ~2 −a23 2a3 )2 , 245 onate et al. / j. nig. soc. phys. sci. 4 (2022) 242–250 246 figure 4. wave function and probability density for the modified shifted deng-fan momentum-dependent potential. r(an) = ( 2µdeβ2 b(b+2)−2µd0β2 ~2 −a2n−1 2an−1 )2 − (21)( 2µdeβ2 b(b+2)−2µd0β2 ~2 −a2n 2an )2 . using all desirable results inconjuction with the negative partner potential, we finally deduce the energy equation of the nonrelativistic equation for modified shifted deng-fan momentum figure 5. wave function and probability density for the shifted dengfan momentum-dependent potential. dependent potential as en,` = de + α2~2 2µ [`(` + 1)c0− 2µβ2 [de b(b+2)−d0 ] α2~2 1 + 2n + √ (1 + 2l)2 + 8µdeβ2 b 2 α2~2 − 1 + 2n + √ (1 + 2l)2 + 8µdeβ2 b 2 α2~2 4  2 . (22) the radial wave function was obtained using the parametric nikiforov-uvarov method. defining a variable of the form y = e−αr we obtain the wave function in terms of jacobi polynomial as 246 onate et al. / j. nig. soc. phys. sci. 4 (2022) 242–250 247 figure 6. wave function and probability density for the hulthèn momentum-dependent potential. un,`(y) = ny √ er (1 − y) 1 2 + 1 2 √ (1+2l)2 + 8µdeβ2 b 2 α2~2 (23) ×p 2√er, √ (1+2l)2 + 8µdeβ2 b 2 α2~2  n (1 − 2y). detail procedure on how to calculate the wave function can be found in ref. [22]. 2.1. expectation values in this section, we compute the expectation values using hellmann-feynman theorem [23-27]. given a formula ∂e(q) ∂q = 〈 ψ(q) ∣∣∣∣∣∂h(q)∂q ∣∣∣∣∣ ψ(q) 〉 . (24) figure 7. variation of the expectation values 〈 p2 〉 (ev ) and 〈 r−2 〉 (ev ) against the momentum dependent parameter ρ with µ = ~ = l = 1, re = 0.2 å, de = 3 ev , α = 0.8 å and d0 = 2 ev . equation (24) only holds if the normalized eigenfunction is continuous with respect to the parameter. given the hamiltonian of potential (1) as h = − ~2 2µ d2 dr2 + de ( 1 − eαre − 1 eαr − 1 )2 − d0 ( 1 eαr − 1 ) (1 + ρ), (25) the time and momentum expectation values can now be calculated by transforming the parameter q. to calculate the momentum expectation value 〈 p2 〉 , we set q = de and have 247 onate et al. / j. nig. soc. phys. sci. 4 (2022) 242–250 248 table 3. comparison of bound states for deng-fan potential with re = 0.4 å, µ = ~ = 1 and de15 ev for 2p, 3p and 3d at various states. state present [18] [28] 2p 0.05 7.86080 7.86080 7.8628 0.10 7.95330 7.95330 7.95537 0.15 8.04510 8.04510 8.04724 0.20 8.13620 8.13620 8.13842 0.25 8.22663 8.22663 8.22892 3p 0.05 10.99776 10.9978 10.9998 0.10 11.16256 11.1626 11.1647 0.15 11.32425 11.3242 11.32647 0.20 11.48284 11.4828 11.48513 0.25 11.63834 11.6383 11.64068 3d 0.05 10.21598 10.21598 10.21651 0.10 10.35354 10.35354 10.35409 0.15 10.48935 10.48935 10.48992 0.20 10.62346 10.62346 10.62403 0.25 10.75591 10.75591 10.75645 table 4. comparison of bound states for hulthèn potential with µ = ~ = 1 and d0 = −α for 2p, 3p and 3d at various states. state present [29] [30] 2p 0.025 0.1128125 0.1127605 0.1127605 0.050 0.1012500 0.1010425 0.1010425 0.075 0.0903125 0.0898478 0.0898478 0.100 0.0800000 0.0791794 0.0791794 0.150 0.0612500 0.0594415 0.0594415 3p 0.025 0.0437587 0.0437069 0.0437069 0.050 0.0333681 0.0331645 0.0331645 0.075 0.0243837 0.0239397 0.0239397 0.100 0.0168056 0.0160537 0.0160537 0.150 0.0058681 0.0044663 0.0044663 3d 0.025 0.0437587 0.0436030 0.0436030 0.050 0.0333681 0.0327532 0.0327532 0.075 0.0243837 0.0230307 0.0230307 0.100 0.0168056 0.0144842 0.0144842 0.150 0.0018681 0.0013967 0.0013966 〈 p2 〉 = 1 − 1 µ [ α2~2 2µ ( 2λ0 − n − 1 2 − λ1 2 ) × ( λ0 − 2µd0β2 − µβ2b2 α2~2λ1 ( 4λ0 1 + 2n + λ1 ))] , (26) λ0 = 2µβ2(deb2 + 2deb − d0) α2~2(1 + 2n + λ1) , λ1 = √ (1 + 2l)2 + 8µβ2 deb2 α2~2  (27) to calculate the position expectation value 〈 r−2 〉 we set q = l and obtain 〈 r−2 〉 = 1 (2l + 1)~2 [ α2~2 ((1 + 2l)c0 −2 ( 2λ2 α2~2λ3 − 1 4 − n 2 − λ1 4 ) × −4λ2(1 + 2l) α2~2λ1λ23 − 1 + 2l 2λ1  , (28) λ2 = µβ2(deb(b + 2) − d0), λ3 = 1 + 2n + λ1  (29) 2.2. special cases of the potential the two special cases of the potential are the deng-fan potential and the hulthén potential. when we put d0 = 0, the potential becomes the deng-fan potential of the form v (r) = de ( 1 − eαre − 1 eαr − 1 )2 (1 + ρ), (30) with the energy equation as enl = de + α2~2 2µ [`(` + 1)c0 −  2µβ2 de b(b+2) α2~2 1 + 2n + √ (1 + 2l)2 + 8µdeβ2 b 2 α2~2 − 1 + 2n + √ (1 + 2l)2 + 8µdeβ2 b 2 α2~2 4  2 . (31) if we put de = 0 the interacting potential reduces to hulthén potential of the form v (r) = −d0 ( 1 eαr − 1 ) (1 + ρ), (32) with the energy equation of the form en,` = α2~2 2µ `(` + 1)c0 − − 2µβ2 d0 α2~2 − (1+n+l)2 2 2(1 + n + l)  2 . (33) 3. results and discussion in figure 1, we examined the energy of the system with the screening parameter for the modified shifted deng-fan potential. the energy of the system goes down while the screening parameter increases. at some point, the energy of the system has a turning point even when the screening parameter continues its linear increase. with momentum-dependent parameter, the energy goes beyond -15 before the turning point while without the momentum-dependent parameter, the energy has the turning point before -13. the energy with momentum dependent parameter for the ground state and first excited has their turning point before the screening parameter equals 2 while 248 onate et al. / j. nig. soc. phys. sci. 4 (2022) 242–250 249 without the momentum parameter, all the turning points are obtained above screening parameter equals 2. the turning point physically shows that there is a revised in the direction in the system. in figure 2, we examined the energy of the system with the screening parameter for the deng-fan potential. the variation of energy against the screening parameter in the presence of the momentum dependent parameter at the ground state and at the excited states differs. the energy at the excited states decreases and then have a turning point while the screening parameter increases but for the ground state, the revise case is obtained. this indicates that the effect of momentum-dependent parameter on the energy at the ground state is insignificant but highly significant for the excited states. in the absence of momentum dependent parameter, the energy at different quantum states has the same variation with the screening parameter. for both presence and absence of the momentum dependent parameter, the energy of the system at various state are equal for the screening parameter less than 0.1. in figure 3, we examined the energy of the system with the screening parameter for hulthén potential. the energy of the system with and without momentum dependent parameter at various quantum states decreases monotonically as the screening parameter increases. however, with momentum dependent parameter, the energy of the system are lower than their counterpart without momentum dependent parameter. it is noted that the energy of the system at different quantum states without momentum dependent parameter are closer compared to the energy of the system at different quantum states with momentum dependent parameter. in figures 4, 5 and 6, we plotted the wave function and probability density for modified shifted deng-fan momentum-dependent potential, deng-fan momentum-dependent potential and hulthén momentum-dependent potential respectively. the wave function for the modified shifted deng-fan momentum-dependent potential has the highest pick both at the negative and positive vertical component, followed by deng-fan momentumdependent potential and lastly, hulthén momentum-dependent potential. the probability densities are seen to be positive for the three potentials. however, the same trend observed for the wave function are also observed in the probability density for the three potentials. variation of the expectation values against the momentum dependent parameter ρ with momentum dependent potential are figure 7. the position expectation value and the momentum expectation value respectively rises as the momentum dependent parameter increases. a clear observation shows that the position expectation value for various quantum states are higher than their counterpart in the momentum expectation value. in table 1, we presented the numerical results for modified shifted deng-fan momentum-dependent potential and without momentum-dependent potential. the numerical values without momentum-dependent potential are higher than their counterpart with momentum-dependent potential. this simply shows that the present of momentum-dependent potential reduces the energy of the system. the numerical values for deng-fan potential (d0 = 0) and hulthén potential (de = 0) in the presence and absence of the momentum dependent parameter are presented in table 2. in both the presence and absence of the momentum dependent parameter, the variation of energy with the screening parameter, quantum and angular quantum numbers are the same. however, the presence of the momentum dependent parameter reduces the energy of the system for both potentials. for hulthén potential, there are similarities in the numerical values for 3p and 3d as well as 4p, 4d and 4f. the discrepancy for the numerical values rises as the screening parameter increases. in table 3 and table 4, we compared the results for deng-fan potential and hulthén potential respectively with existing results. the two results aligned with the previous results. 4. conclusion the solutions for modified shifted deng-fan potential was obtained with a momentum-dependent potential in the present study. the results showed that the present of the momentumdependent parameter 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[30] y. p. varshni, “eigen-energies and oscillator strengths for the hulthèn potential”, physical review a 41 (1990) 4682. 250 j. nig. soc. phys. sci. 4 (2022) 214–222 journal of the nigerian society of physical sciences the corrosion characteristics of ss316l stainless steel in a typical acid cleaning solution and its inhibition by 1-benzylimidazole: weight loss, electrochemical and sem characterizations k. k. adamaa, i. b. onyeachub,∗ a department of chemical engineering, faculty of engineering, edo state university uzairue, edo state, nigeria. b department of industrial chemistry, faculty of science, edo state university uzairue, edo state, nigeria. abstract acid cleaning, an inevitable industrial practice used to descale chemical reactors, usually causes serious corrosion attack on underlying alloy substrates. ameliorating this phenomenon requires the addition of effective corrosion inhibitors into the acid solution. current global regulations encourage environmentally-benign molecules as corrosion inhibitors. consequently, 1-benzylimidazole has been investigated for its inhibitive characteristics against the corrosion of ss316l stainless steel in a typical acid cleaning solution containing 2 % hcl + 3.5 % nacl. weight loss measurements confirm that the corrosion inhibition property of 1-benzylimidazole increases with concentration but depreciates with increased temperature. electrochemical impedance spectroscopy (eis) and potentiodynamic polarization (pdp) measurements confirm that 1-benzylimidazole adsorb on the stainless steel surface to isolate its surface from the acid solution. 1-benzylimidazole is a mixed-type inhibitor with greater anodic influence, and its adsorption enhances the formation and protectiveness of a passive film. weight loss and the electrochemical measurements agree to an average inhibition efficiency > 70 % at 1000 ppm. the inhibitor adsorbs via physisorption and obeys the temkin isotherm model. sem surface characterization confirm the ability of 1-benzylimidazole to protect the surface microstructure of the stainless steel during the corrosion. doi:10.46481/jnsps.2022.596 keywords: corrosion inhibitor; stainless steel; acid cleaning; imidazole; polarization. article history : received: 18 january 2022 received in revised form: 17 march 2022 accepted for publication: 19 march 2022 published: 29 may 2022 c©2022 journal of the nigerian society of physical sciences. all rights reserved. communicated by: edward anand emile 1. introduction stainless steel is heavily deployed in the construction of many reactors in the chemical industry. it exhibits good corrosion resistance because it readily forms an iron/chromiumenriched passive layer [1]. unfortunately, the passive film ∗corresponding author tel. no: +2348034917997 email address: onyeachu.benedict@edouniversity.edu.ng (i. b. onyeachu ) formation is usually truncated by chloride ion-containing corrosion environments because its products are transformed into soluble chlorides [2]. the resultant effect is a localized form of corrosion which diminishes the structural integrity of many chemical reactors and shortens their span of usage. acid cleaning is a typical industrial process which exposes stainless steel surfaces to chloride ion-induced corrosion. it involves the use of dilute mineral acid (especially 2-5 % hcl) to remove inorganic scales which grow on the surfaces of industrial 214 adama and onyeachu / j. nig. soc. phys. sci. 4 (2022) 214–222 215 metallic equipment and diminish their performance. while the metal surface is descaled, a simultaneous acid corrosion attack on the underlying substrate usually ensues. the extent of this corrosion is even aggravated by the seawater sources used to prepare the acid solutions, since seawater is naturally enriched with chloride ions, especially as nacl [3, 4]. a practical industrial approach to ameliorate this corrosion attack is to add highly effective corrosion inhibitors into the acid cleaning solution. currently, most industries apply propargyl alcohol-based inhibitors which are very toxic, both, to humans and other environmental species [5]. imidazole and its derivatives are nitrogen-based heterocyclic compounds largely used for the production of numerous cheap environmentally-safe drugs [6, 7]. in the past decade, imidazole derivatives have been reported as formidable corrosion inhibitors for various metals. singh et al. [8] reported that 2-(4-methoxyphenyl)-4,5-diphenyl-imidazole exhibited 93 % efficiency for the inhibition of j55 steel corrosion in co2-saturated brine solution. ting et al. [9] reported that 4-phenylimidazole acted as a mixed-type corrosion inhibitor for copper in sulphuric acid and provided 95.84 % efficiency; adsorbing both by physical and chemical mechanisms. in other reports [10], thiazole-4-carboxylic acid, 2-methyl-1,3thiazole-4-carboxylic acid, and some imidazole zwitterions have been reported as highly efficient for carbon steel during acid corrosion [11]. particularly, dimethylbenzodiimidazole (2,7-dimethyl3,6-dihydrobenzo[1,2-d;3,4-d’] diimidazole acted as an anodic inhibitor for ss316l stainless steel in 0.5 m h2so4 and provided 77 % at optimum concentration [12]. also, some synthesized dicationic imidazolium ionic liquids were also reported for ss304 stainless steel in 0.5 m h2so4 solution whereby 92 % efficiency was recorded [13]. apparently, not much work has been reported for imidazole derivatives as stainless steel corrosion inhibitors. even so, the reported derivatives are expensively synthesized compounds with cumbersome molecular structures that disturb their adsorption. identifying a cheap and simple-structure imidazole derivative which can deliver high inhibition efficiency will be very beneficial to the industry. in the present work, 1-benzylimidazole has been identified as a promising imidazole derivative for the inhibition of stainless steel acid corrosion. 1-benzylimidazole is a major pharmacological ingredient with environmentally-safe characteristics, according to the paris commission (parcom) standard [14], because it exhibits log po/w = 1.93 and ld50 = 75 mg/kg [15]. as figure 1 shows, 1-benzylimidazole has a simple structure whereby a phenylmethyl group is attached to the imidazole nitrogen. computational simulation previously revealed that this simple structure enables 1-benzylimidazole to inherently orient in a flat position and achieve good metal surface coverage during corrosion inhibition [16]. the most visible reports of 1benzylimidazole as corrosion inhibitor have been presented for carbon steel [16, 17, 18] in acid solutions. the inhibitor exhibited more anodic behaviour on the carbon steels, causing us to suspect its ability to enhance anodic passivation when applied for stainless steel. furthermore, the inhibitor efficiency has not been reported in a practical acid cleaning solution which contains hcl + nacl, and is usually conducted at elevated temperatures [3]. filling this research gap will be highly beneficial towards the development of greener corrosion inhibitors for industrial acid cleaning. consequently, we apply weight loss technique to screen the effect of concentration and temperature on the performance of 1-benzylimidazole as inhibitor against the corrosion of ss316l stainless steel in 2 % hcl + 3.5 % nacl solution. thereafter, we employ electrochemical techniques like electrochemical impedance spectroscopy (eis) and potentiodynamic polarization (pdp) to provide mechanistic interpretation of 1-benzylimidazole performance at maximum concentration. scanning electron microscopy (sem) was employed to probe the extent of stainless steel surface damage by corrosion and its mitigation by the 1-benzylimidazole. 2. materials and method 2.1. materials all chemicals and reagents used in this work were purchased from sigma aldrich and fischer scientific, respectively. stainless steel ss316l, having the elemental composition previously reported [19], was used. the bulk alloy bar was machined into corrosion test coupons with total surface area 24.89 cm2 and 1 cm2 , respectively, exposed for the weight loss and electrochemical experiments. the test solution (2 % hcl + 3.5 % nacl) was prepared by, firstly, diluting concentrated hcl (37 w/v%) with distilled water to obtain a 2 % hcl solution and, secondly, preparing 3.5 % nacl solution using the 2 % hcl as solvent. stock solution of 1-benzylimidazole was prepared in isopropanol and calculated volumes were subsequently retrieved and made up to 200 ml with the acid solution, so as to obtain working inhibitor concentrations of 200, 400, 800 and 1000 ppm. a thermostatic water bath was employed to maintain experimental temperatures. 2.2. methods prior to any experiment, the surface of the stainless steel coupon was polished with #400, #600, #800 and #1000 grit silicon carbide papers. this was performed with simultaneous washing under running tap water, degreasing with acetone and mechanical drying with warm air. weight loss experiments, after 24 h immersion in the uninhibited and inhibited acid solutions, were conducted, initially, at 25◦c. using the optimum inhibitor concentration, the effect of temperature (45 and 65◦c) was investigated. the stainless steel coupons were retrieved after 24 h immersion and cleaned according to the astm g103 standard procedure [20]. the coupons were then re-weighed and the weight difference (∆w) before and after immersion was calculated based on equation (1). from the weight loss, the corrosion rate (cr) in millimeter per year (mpy) was calculated, based on equation (2); where a = total surface area of coupon 215 adama and onyeachu / j. nig. soc. phys. sci. 4 (2022) 214–222 216 figure 1: structure of 1-benzylimidazole. (24.89 cm2 ), ρ = density of ss316l stainless steel (8.0 g cm−3 ), t = immersion time (24 h). ∆w = wbefore − wafter (1) cr(mpy) = ∆w × 3.45 × 106 a ×ρ× t (2) the electrochemical experiments such as eis and pdp were conducted on a gamry 600 potentiostat equipped with the echem analyst software for data analysis. the electrochemical set-up was a 3-electrode system which contained the stainless steel as the working electrode, a graphite rod as counter electrode and ag/agcl as the reference electrode. all measurements were performed at the end of 1 hour free immersion to ensure that the corrosion coupons attained steady-state open circuit potential [21]. the eis measurement was conducted by applying ± 10 mv sinusoidal signal amplitude perturbation over a frequency range of 105 hz to 10−1 hz. the pdp experiment was conducted by polarizing the stainless steel from −0.25 v (vs. open circuit potential) to +1.2 v vs. ref. at a rate of 0.2 mv/s . the sem surface characterization for the polarized inhibited and uninhibited electrode was achieved using the jeol jsm-6610 lv scanning electron microscope. 3. results and discussion 3.1. weight loss results the weight loss, and associated corrosion rate, experienced after 24 h immersion of the stainless steel coupons in 2 % hcl + 3.5 % nacl solution at 25◦c without and with different 1benzylimidazole concentrations are presented in figures 2a and 2b. in the acid solution without inhibitor, the stainless steel loses an average of 0.0411 g after 24 h immersion, which corresponds to a corrosion rate of 30.20 mpy. in the presence of 1-benzylimidazole, the weight loss diminishes continually with increasing inhibitor concentration. the least average weight loss of 0.0109 g (or 8.01 mpy corrosion rate) was observed in the presence of 1000 ppm, and approximated to 73.48 %, following equation (3) and figure 2c. figure 2: results of (a) weight loss (b) corrosion rate and (c) inhibition efficiency after 24 immersion of ss316l stainless steel in 2 % hcl + 3.5 % nacl solution without and with different concentration of 1-benzylimidazole at 25◦c %i e∆w = 1 − crwith inhibitor crwithout inhibitor × 100% (3) this result can be interpreted by considering that the weight loss experienced by the immersed coupons is caused by the loss of surface atoms like cr and fe via oxidation into species like cr3+ , fe2+ and fe3+. on the other hand, organic inhibitors impede corrosion by interacting the reactive electrons n their heteroatoms and c = c functional groups with the d-orbitals of metal atoms. these electron-rich sites donate electrons that fill the d-orbitals in the metal atoms, thus, supressing their oxidation. the resultant inhibitor adsorption increases the hydrophobicity of the metal surface and isolates it from the corrosion environment, as the schematic in figure 3 elucidates. the more inhibitor in the acid solution, the more metal surface 216 adama and onyeachu / j. nig. soc. phys. sci. 4 (2022) 214–222 217 is covered and protected. as temperature increases to 45 and 65◦c, table 1 shows that the weight loss and corrosion rate increase for, both, the uninhibited and inhibited acid solutions. the corrosion inhibition efficiency provided by the maximum concentration of 1000 ppm, initially, decreased rather moderately to 67.80 % (45◦c) and, thereafter, quite drastically to 37.61 % (65◦c). although the presence of 1-benzylimidazole, somewhat, protected the stainless steel at these elevated temperatures, the depreciation in inhibitor efficiency may be attributed to the competitive mechanisms of increased surface dissolution and weakened inhibitor adsorption. increased temperature promotes the transfer of corrosive agents, especially the chloride ions, from solution bulk to the metal surface, thus, promoting surface dissolution rather than inhibitor adsorption. also, elevated temperatures could induce desorption of already adsorbed inhibitors from the metal surface, indicating that 1-benzylimidazole adsorbs via a physiosorption mechanism [22]. 3.2. adsorption isotherm adsorption isotherms provide adequate understanding of the mode of interaction between an adsorbing corrosion inhibitor molecule and the metal substrate. they describe whether inhibitor molecules adsorb on the metal surface as a single layer of film, as multilayer, or if any repulsion exists among inhibitor molecules [23]. in this research, the langmuir, temkin and freundlich isotherms were employed to test the mode of interaction of 1-benzylimidazole on the stainless steel surface. the langmuir isotherm predicts that the inhibitor adsorbs on the stainless steel surface as a monolayer. the temkin isotherm predicts that the inhibitor forms a multilayer with plausible inhibitor-inhibitor interaction. on the other hand, the freundlich isotherm depicts that the inhibitor is adsorbing on a rough and heterogeneous surface. the linear forms of the langmuir, temkin and freundlich isotherms are, respectively, presented as follows: c θ = 1 kad s + c (4) θ = − 1 2α ln c − 1 2α kad s (5) log θ = log kad s + 1 n log c (6) θ = surface coverage = %i e100 , kad s = adsorption-desorption equilibrium constant, c = inhibitor concentration (in ppm) and the factor, α, accounts for the lateral interaction between inhibitor and metal surface. the value of n describes the feasibility of inhibitor adsorption; which is highly feasible when 0 < 1n < 1 but, respectively, moderate or difficult when 1n = 1 or > 1 [24]. a plot of versus c will yield the langmuir isotherm curve, as shown in figure 4a. the temkin isotherm, figure 4b, is described by the plot of θ against ln c whereas the plot of log θ versus log c gives figure 3: schematic representation to show how 1-benzylimidazole adsorption increases the hydrophobicity of the stainless steel surface the freundlich isotherm, figure 4c. given that it displayed the closest agreement between experimental plots and fitting line, and because its correlation coefficient (r2) value was closest to unity, the freundlich isotherm most appropriately describes the adsorption of 1-benzylimidazole on the ss316l stainless steel. the adsorption equilibrium constant was calculated as 1.025 ppm mol−1 and, from this value, the gibbs’ free energy (∆gad s = −rt ln(106 kad s) ) was deduced as -34.29 kj/mol and the negative value of ∆g confirms that the inhibitor adsorption was spontaneous and feasible. 3.3. electrochemical results electrochemical measurements are capable of providing mechanistic explanations to the corrosion inhibition behaviour of 1-benzylimidazole on the ss316l stainless steel in the acid solution. at the open circuit potential, a quasi-equilibrium is established between the rates of oxidation and reduction reactions taking place in the micro-cells forming anodic and cathodic active sites on the alloy surface. by displacing the alloy from its equilibrium potential, relevant electrochemical parameters could be measured and related to the corrosion phenomenon. steady-state electrochemical techniques like eis can provide accurate information about the dielectric and charge transfer phenomena at the alloy-acid solution interface. a more aggressive technique like the pdp is able to provide details about the corrosion potential, corrosion kinetics, passivation and passivation breakdown/pitting corrosion phenomena. the eis results are presented in figure 5, as the plots of nyquist (figure 5a), absolute impedance (figure 5b) and phase angle (figure 5c). the characteristic arcs displayed by the stainless steel appear larger in the presence of 1benzylimidazole, compared with the size in the blank acid solution. larger arcs usually depict greater resistance to charge transfer at the alloy-solution interface, i.e. greater corrosion resistance. the result confirms the inhibitive property of 1-benzylimidazole for the stainless steel. the trend in the nyquist arc sizes is confirmed by the values of low-frequency absolute impedance in figure 5b. clearer mechanism is 217 adama and onyeachu / j. nig. soc. phys. sci. 4 (2022) 214–222 218 table 1: weight loss, corrosion rate and inhibition efficiency of 316 stainless steel during corrosion after 24 h in 2 % hcl + 3.5 % nacl solution at 25◦c, 45◦c and 65◦c parameter 25◦c 45◦c 65◦c weight loss (g) corrosion rate (mpy) weight loss (g) corrosion rate (mpy) weight loss (g) corrosion rate (mpy) without inhibitor 0.041 g 30.20 mpy 0.059 g 43.35 mpy 0.109 g 80.71 mpy with 1000 rpm 0.011 g 8.01 mpy 0.019 g 13.96 mpy 0.068 g 50.35 mpy inhibition efficiency (% ie) 73.48 % 67.80 % 37.61 % figure 4: (a) langmuir (b) temkin and (c) freundlich isotherm plots for the adsorption of 1-benzylimidazole on 316 stainless steel during corrosion in 2 % hcl +3.5 wt% nacl solution deduced from the phase angle plots in figure 5c, whereby the stainless steel clearly displayed two time constants at high and low frequencies, compared with the single time constant exhibited by the uninhibited alloy. the two time constants depict two phenomena at the alloy-solution interface; namely the phenomenon at a corrosion layer-solution interface (at high 218 adama and onyeachu / j. nig. soc. phys. sci. 4 (2022) 214–222 219 figure 5: eis plots in formats of (a) nyquist (b) absolute impedance (c) phase angle and equivalent circuit for ss316l stainless steel during corrosion in 2 % hcl + 3.5 % nacl (d) without and (e) with 1-benzylimidazole. table 2: eis parameters for corrosion of 316 stainless steel in 2 % hcl + 3.5 % nacl solution without and with 1000 ppm of 1-benzylimidazole at 25◦c. cpedl cpe f conc. rs( ωcm2 ) ydl µfcm−2 sn−1 αdl rct( ωcm2 ) y f µfcm−2 sn−1 α f r f( ωcm2 ) (rp = r f + rct)( ωcm2 ) goodness of fit( ×10−3 ) % ie without inhibitor 0.719 374 0.75 212 212 3.92 with 1000 rpm 0.877 110 0.85 685 57 0.83 80 765 2.29 72.29 219 adama and onyeachu / j. nig. soc. phys. sci. 4 (2022) 214–222 220 table 3: polarization parameters for corrosion of 316 stainless steel in 2 % hcl 3.5 % nacl solution without and with 1000 ppm of 1-benzylimidazole at 25◦c. sample ecorr (mv) icorr ( µa/cm2 ) βa (mv/decade) -βc (mv/decade) % ie without inhibitor -277 161.30 89.73 132.60 with 1000 rpm -258 45.90 68.84 136.10 71.54 figure 6: pdp plots for ss316l stainless steel during corrosion in 2 % hcl + 3.5 % nacl without and with 1-benzylimidazole. figure 7: sem surface morphology of ss316l stainless steel after polarization in 2 % hcl + 3.5 % nacl (a) without and (b) with 1-benzylimidazole. frequency) and at a corrosion layer-substrate interface (at low frequency) [25]. compared with the single time constant for the uninhibited stainless steel, the two time constants in the presence of 1-benzylimidazole is due to the adsorbed corrosion inhibitor layer which isolates the stainless steel substrate from the acid solution. consequently, the figure 5d and 5e were adopted as electrical models to interpret the corrosion phenomena for the uninhibited and inhibited stainless steel, respectively, after fitting experimental results using the echem analysts software. the values of respective electrical elements are detailed in table 2. the elements rs, rct and cpedl represent the solution resistance, charge transfer resistance and double layer capacitance, respectively. the use of constant phase element (cpe), rather than pure capacitor, in the model is to account for surface roughness of the corroding alloy [26]. the rct is equivalent to the corrosion resistance of the corroding metal while the cpedl elements, ydl and n, describe the admittance of charge into the double layer and the roughness of the alloy surface (αdl ), respectively. in figure 5e, the characteristic of the adsorbed inhibitor layer is characterized by the r f and cpe f elements (y f and α f ) which, respectively, describe the porosity of the adsorbed layer and the charge capacitance at the layer-solution interface. from table 2, the lower ydl values obtained in the presence of 1-benzylimidazole confirms that the inhibitor adsorption reduces the accumulation of corrosion-inducing aqueous species at the stainless steel surface and, thus, lowers the rate of charge transfer at the alloy surface-solution interface. as table shows, the inhibitor adsorption impacts a total resistance (rct + r f ) of 765 ωcm2, compared with the value of 212 ωcm2 recorded for the blank solution. this gives an inhibition efficiency of 72.29 %, based on the equation (7). furthermore, the αdl value is also higher in the presence of inhibitor, which indicates that the inhibitor adsorption minimizes the roughness of the alloy surface due to the corrosion attack. %i eeis = 1 − rtotal(with inhibitor) rtotal(without inhibitor) × 100% (7) %i epdp = 1 − icorr(with inhibitor) icorr(without inhibitor) × 100% (8) in figure 6, each pdp curve shows an activation-controlled cathodic arm representing the dominant h+ reduction in the acid solution. the anodic arm is characterized by an active region, a region of passivation and a region of passivation breakdown. the active region is attributed to the oxidation and dissolution of surface elements like fe and cr, the passivation potential region depicts the formation of protective oxides and hydroxides of fe and cr [27], whereas the region of passivation breakdown indicates potential regions where protective passive layer is broken, especially locally by chloride ions. extrapolating the pdp plots around ± 10 mv of the cathodic-anodic potential transition enabled the deduction of corrosion current density (icorr ), corrosion potential (ecorr ), anodic (βa ) and cathodic (βc) tafel constants [28]. the e corr indicates the thermodynamic tendency to corrode in the solution and the most preferable active sites of corrosion inhibitor interaction with the corroding surface, while the icorr is a measure of the corrosion rate. 220 adama and onyeachu / j. nig. soc. phys. sci. 4 (2022) 214–222 221 by table 3, the presence of 1-benzylimidazole in the acid solution slightly shifts the e corr towards more anodic potential and lowers the i corr value from 161.30 µ a cm−2 to 45.90 µacm−2; which amounted to an inhibition efficiency of 71.54 % according to equation (8). it is reported that 1-benzylimidazole exists as a cationic species in acidic solution [17]. therefore, the greater inclination of 1-benzylimidazole to adsorb on anodic sites of the stainless steel strongly suggests the existence of pre-adsorbed anions on the anodic sites containing fe2+ and cr3+, which eventually attract the cationic inhibitor towards the anode. such phenomenon was also reported elsewhere for 1benzylimidazole in co2 + h2s-saturated nacl solution [16]. this anodic preference is a stronger inclination towards more protective passivation, as confirmed by figure 6; where the inhibitor shifts passivation current to lower values. although the inhibitor does not affect the passivation potential (and passivation mechanism), it strongly modifies the passive film formed into a more protective one. this means that 1-benzylimidazole adsorption encourages the formation of fe and cr oxides and hydroxides. nevertheless, the inhibitor appears not too protective once the passive film is deteriorated. this is feasible because the adsorbing inhibitor molecules and nucleating corrosion products would experience some structural mismatch which are easily targeted for localized attack and collapse. 3.4. sem results the surface morphologies of the stainless steel, after polarization in the acid solution without and with 1000 ppm of 1-benzylimidazole, are presented in figure 7a and 7b, respectively. in the absence of inhibitor, figure 7a, the stainless steel surface exhibits a rough morphology with patches of passive products around the surface. in the presence of inhibitor, figure 7b, the passive products appear to agglomerate into a more continuous and denser layer which provides more surface coverage for the alloy surface. this justifies the findings from electrochemical measurements that the adsorption of 1benzylimidazole could facilitate the formation of passive layer. at some points, however, some shallow cracks could be observed in the figure 7b. these are likely points where localized corrosion occurs when the alloy is polarized beyond the passivation region. 4. conclusion the corrosion characteristics of a chemical reactor alloy ss316l stainless steel, and its inhibition by 1-benzylimidazole, have been investigated in 2 % hcl + 3.5 % nacl solution (a typical solution used for industrial acid cleaning). the addition of 1-benzylimidazole into the test solution significantly lowered the rate of corrosion of the alloy, wherein an optimum inhibitor concentration of 1000 ppm yielded inhibition efficiency greater than 70 %. the 1-benzylimidazole acted more like an anodic-type inhibitor which improved the characteristics of the passive film formed on the alloy surface. the adsorption of 1benzylimidazole followed the temkin isotherm and protected the 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[28] m. stern, a.l. geary, “electrochemical polarization, 1. a theoretical analysis of the shape of polarization curves”, the electrochemical society 104 (1957) 751. 222 j. nig. soc. phys. sci. 2 (2020) 128–133 journal of the nigerian society of physical sciences original research dissolution of a nigerian sourced muscovite ore for use as an ingredient in paint production daud t. olaoluwaa,∗, abdulhadi e. abdulmalika, taoreed a. murainaa, sadisu girigisub, ayo f. balogunc adepartment of science laboratory technology, the federal polytechnic, p.m.b. 231, ede, nigeria. bdepartment of science laboratory technology, the federal polytechnic, p.m.b. 420, offa, nigeria cdepartment of chemistry, kogi state college of education (technical), kabba. p.m.b. 242, nigeria abstract dissolution and characterization studies on the purification of muscovite ore in hydrochloric acid for use in paint production was investigated. specific dissolution parameters including the effects of acid concentration as well as temperature on the dissolution of muscovite ore were studied. important instrumentation techniques such as x-ray fluorescence (xrf), x-ray diffraction (xrd) and scanning electron microscopy (sem) were employed for the better explanation of the dissolution process so as to fathom the availability of elements and compounds within the ore. the results revealed that the dissolution rates were considerably influenced as the acid concentration and temperature increased, while at optimal leaching conditions, about 85% of the ore was found to have been reacted by 2.5 mol/l at 75◦c temperature and at 120 minutes of leaching time. the reaction order for the dissolution can be deduced to be half order reaction as the value obtained was in the bracket of 0.50. the reaction kinetic data revealed the dissolution mechanism to involve diffusion and surface chemical mechanisms as the rate controlling mechanisms while the different instrumentation techniques corroborated the dissolution as well as purification of the muscovite ore as an ingredient for possible use in paint production. keywords: characterization, diffusion, dissolution kinetics, hydrochloric acid, leaching, muscovite ore. article history : received: 17 april 2020 received in revised form: 09 may 2020 accepted for publication: 09 june 2020 published: 01 august 2020 c©2020 journal of the nigerian society of physical sciences. all rights reserved. communicated by: w. a. yahya 1. introduction muscovite with the chemical formula kal2(alsi3o10)(oh)2, is an important mineral amongst the mica group that is characterized by layered crystal structure, a density range of 2.77 to 2.88 g/cm3 and majorly described predominantly as of metamorphic origin but also found in igneous and sedimentary rocks as a light coloured phyllosilicate [1]. it is made up of paral∗corresponding author tel. no: +2348032064189 email address: daudolaoluwa@gmail.com (daud t. olaoluwa ) lel sheets of silicate tetrahedrons that are weakly bonded together by a layer of potassium ions [2]. these sheets are characterized as being chemically inactive, water absorbing, prevents loss of heat, flexible, dielectric, lightweight, reflective, resilient, refractive, transparent to opaque and stable when exposed to water, electricity, light and extreme temperatures. as a result of these its superior electrical properties as well as its abundance, muscovite has been employed as the major mica used by industries as insulation materials in high voltage and high power electrical machines, electronic instruments, water 128 olaoluwa et al. / j. nig. soc. phys. sci. 2 (2020) 128–133 129 columns, condensers and radio tubes, high-duty boilers and in other industrial products and instruments as well as its extensive use as a pigment extender in paints where it serves purposes such as; chalking reduction, water penetration and weathering reduction, tone of coloured pigments brightener, sustains pigments in suspension as well as in the reduction of shrinking and shearing of the finished surfaces [3, 4, 5]. muscovite, despite its high density is still not readily and easily separated from quartz using heavy liquids as a result of its high surface energy [6] which allows it float and also as a result of the presence of elements such as aluminium, potassium, iron and titanium etc. which serves as impurities and could apparently reduce the quality of the muscovite [7, 8]. muscovite ore processing methods mainly include shape separation; a method where particles are separated based on shapes required for the improvement in quality of powder products, pneumatic separation; using aerodynamics properties of particles to heave light, dusty and husky materials out of grains while heavier materials settle, and flotation; a process for selectively separating hydrophobic materials from hydrophilic substances [9, 10, 11, 12]. purification studies of muscovite involves fluoric acid leaching which has been found to show great effects on processing of muscovite but also found to lead to severe environmental pollution [13, 14]. other purification studies such as the oxygen pressure acid leaching involves high acid consumption and an elongated leaching or reaction time which ends up leading to consequential declines in equipment life, production output as well as raise in production costs [15, 16]. therefore, this research sets out to determine the purification activities of muscovite ore in hydrochloric acid while critically examining the effects and extents acid concentrations, reaction temperature and particle size might have on the ore dissolution for better interpretation of the optimal purification process as well as its possible use as a pigment extender in paint production. 2. materials and methods for this study, the muscovite sample was sourced from mangu local government area of plateau state, nigeria and the sample was crushed and milled to yield a fine powder and then sifted into different particle mesh sizes (53 µm, 63 µm and 90 µm) using the american society for testing and materials (astm) standard sieve fractions, with the size fraction with the largest surface area used for characterization and leaching studies. the empyrean x-ray diffractometer equipped with a bruter x-flash detector using esprit 1.82 software x-ray diffraction spectroscopy (xrd) was employed in determining the material purity assessment as well as to identify compounds of interest present in the ore sample while the mini pal 4 edxrf spectrometer was used in determining the elemental composition of the muscovite sample. the samples were carbon coated and viewed at 5.0kv , 26mm working distance using sem model vega3 tescan with lab6 filament to study the structural morphology of the ore sample. physicochemical parameters such as moisture content, loss of mass on ignition as well as the surface ph of the raw ore were adequately appraised using standard procedures. 2.1. leaching process acid dissolution experiments was carried out in a 250 ml glass beaker erected on a hot plate with automatic temperature control and magnetically stirred for 120 minutes. the setup was equipped with a watch glass on the beaker to avoid evaporation, thereby, enabling the condensation of the leach solution. dissolution parameters including hydrochloric acid (hcl) concentration and leaching temperatures were studied to ascertain the optimum leaching conditions. known concentrations of acid solutions were prepared and placed in the glass beaker till the solution reached the desired temperature, then the ore sample was introduced into the beaker and magnetically stirred as the dissolution proceeded. the amount of ore sample introduced for the reaction/dissolution was weighed to make up a pulp of 10 g/l. the residue after leaching was filtered, washed, dried and weighed and the fraction of the ore reacted was calculated from the initial difference in weight of amount dissolved and undissolved at various time intervals. appropriate kinetic curve was examined using the established shrinking core models for the establishment of the dissolution mechanism while selected residues after appropriate treatments were fully characterized by sem and xrf [16]. 3. results and discussion the elemental composition (table 1) of the raw muscovite ore sample by edxrf and confirmed by xrd spectrum shows that the ore contains silicon, aluminium, potassium, rubidium and tin, occurring as major elements while other elements such as calcium, phosphorus, manganese, iron, nickel, copper, zinc, and molybdenum occur from low to trace levels (≤ 0.5%). the xrd pattern (figure 1) as shown confirms the raw ore sample to be predominantly muscovite, although with traces of albite and quartz. the xrd also gave the formulas of each of these components as muscovite (k3.60na0.28al9.60fe0.88mg0.66ti0.08si2.80 o48.00h8.00), albite (na2.60al2.00si6.00o16.00) and quartz (si3.00o6.00). the scanning electron image (sei) at different magnifications to investigate the structural morphology of the raw muscovite ore samples is as shown in figure 2 and it was observed that 129 olaoluwa et al. / j. nig. soc. phys. sci. 2 (2020) 128–133 130 table 1. edxrf result of the raw ore elements si al k rb sn s ca p mn fe ni cu zn mo % composition 10.2 4.2 9.3 3.55 1.2 0.54 0.1 0.22 0.41 0.43 0.05 0.05 0.08 0.22 figure 1. identified compounds as shown by xrd with their respective joint committee on powder diffraction standard file number for peaks assignments showing muscovite (k3.60na0.28al9.60fe0.88mg0.66ti0.08si2.80o48.00h8.00) 96-900-5188, albite (na2.60al2.00si6.00o16.00) 96-900-1633 and quartz (si3.00o6.00) 96-901-2604 throughout the wide size range of the samples, the ore particles has irregular shapes. figure 2. sei of raw muscovite ore at different magnifications on ignition in a muffle furnace, the ore sample and its weight was estimated and affirmed in percentages with an average loi value of 1.5 suggesting that the organic component of the ore is just in low volume while the moisture content obtained was an average of 2.5%. the ph range of the muscovite ore solution suspension taken in a period of five (5) days gave values between 7.35 and 7.47 with the mean ph being 7.42, this affirmed that the surface of the mineral is neutral. 3.1. purification studies extent of hcl concentration. muscovite ore dissolution was carried out using hcl solution in the concentration range of 0.1 mol/l to 3.0 mol/l at 55◦c and at different leaching times while being moderately stirred at about 300 r pm. the fraction of the ore dissolved at these different leaching times is plotted for the separate hcl concentrations as evidenced in figure 3. figure 3 shows that as the hcl concentration at various leaching times increases, the rate of dissolution increases, therefore, 2.5mol/l hcl is considered to be the optimum acid concentration and hence, for economic consideration, the optimum hcl concentration for this study was 2.5 mol/l and hitherto used in the advancement of some other leaching parameters such as reaction temperature. effect of reaction temperature. the optimal acid concentration obtained was employed to study the extent of reaction temperature on the dissolution rate of muscovite ore dissolution over the temperature ranges of 28 − 75◦c and its result is as shown in the figure 4. from the plot, it can be observed that the dissolution of muscovite is positively influenced as the leaching time rises at 130 olaoluwa et al. / j. nig. soc. phys. sci. 2 (2020) 128–133 131 figure 3. extent of hcl concentrations on muscovite ore dissolution per different leaching time (min.). experimental conditions: hcl concentration = 0.1 − 3.0 mol/l; temperature = 55◦c, particle size = −90 + 53 µm; solidliquid ratio = 10 g/l with moderate stirring figure 4. extent of temperature on hcl muscovite ore dissolution. experimental conditions: hcl = 2.5 mol/l; temperature = 25 − 75◦c, particle size = −90 + 53 µm; solid-liquid ratio = 10 g/l with moderate stirring different temperature. in other words, muscovite dissolution rate is greatly affected by reaction temperature as the amount of muscovite ore leached increased at 75◦c and 120 minutes of leaching which could be attributable to the certainty of particles reacting faster when they collide with a rapid rise in temperature. 3.2. dissolution kinetics analysis the reaction between a solid and fluid can be represented as presented in equation (1) where the rate of reaction may be controlled by either of the mechanisms namely: diffusion through fluid films, diffusion through ash/product layer, or the chemical reaction at the surface of the core of unreacted materials [17]. aa(fluid) + bb(solid) → products (1) data collected from leaching with various acid concentrations and at optimal dissolution conditions was used to estimate the leaching kinetics as well as the reaction order and activation energy for better understanding of the dissolution mechanisms while employing the following shrinking core models to describe the kinetic models [18, 19]. 1 − (1 −α) 1 3 = kct (2) 1 − 2 3 α− (1 −α) 2 3 = kd t, (3) where α is the fraction reacted, k is the reaction rate constant and t is the leaching time. equation (2) is used to predict the chemical reaction as the leaching rate controlling step occurring at the mineral particle surface while equation (3) predicts the diffusion through the product layer on the particle surface as the rate controlling step. therefore, the hcl solution treatment was subjected to the two aforementioned mathematical model equations. thus, treating figures 3 and 4 accordingly using the above kinetic equations as a function of reaction time gives a perfectly straight line for equation (3) only. figure 5. plot of 1 − 23 α − (1 −α) 2 3 versus leaching time at different hcl concentrations. experimental conditions: same as in figure 3 the results of the influence of hcl concentration (figure 3) were also subjected to this kinetic models with the experimental rate constant, kd values and correlation coefficients for each concentration evaluated. the graph of ln kd versus ln[hcl] was then plotted to give the reaction order of 0.4057 as presented in figure 6. as evidenced in figure 6, the slope of the resulting plot estimated the reaction order to be 0.4057 with respect to h+ ion and a correlation coefficient of 0.9157. the data from figure 4 was also linearized using equation (3) to obtain the result represented in figure 7. the apparent rate constant kd calculated from the slopes of the straight lines in figure 7 was then used to obtain arrhenius relation as a function of time as shown in figure 8 below. from the plot (figure 8), the calculated activation energy ea was 53.38 kjmol−1 which clearly suggests a surface chemical 131 olaoluwa et al. / j. nig. soc. phys. sci. 2 (2020) 128–133 132 figure 6. ln kd against ln[hcl] figure 7. plot of 1− 23 α−(1 −α) 2 3 versus reaction time at varying temperatures. experimental conditions: same as in figure 4 figure 8. ln kd versus 1/t controlled reaction for the dissolution process as proposed by several investigators [20, 21]. the calculated value of the activation energy obtained in the present study is above 40 kj/mol; which supports that leaching is controlled by surface chemical mechanism, however, the high ea could be as a result of 2-stage mechanism which involves diffusion and surface chemical concurrently as all the subjected dissolution kinetics data gave a perfect fit for diffusion controlled mechanism. 3.3. residual product analysis the residual product formed after leaching at optimal conditions of 2.5 mol/l hcl, 75◦c and 120 minutes of dissolution was analyzed using xrf and the result shows the presence of 5.7% al, 18.3% si, 7.8% k, 3.1% rb, 3.4% s and 0.67% sb. the xrf result suggests that there was an increase in the amount of these elements as compared to the raw ore, therefore, it could be said that this process purified the muscovite ore and increased the presence of other elements which could also be used to produce these compounds. from the sem images of the leached muscovite ore particles presented in figure 9 below, it can be deduced that the sem photograph shows that the residual products were closely packed with shining surface which suggests that the ore was greatly enhanced by the hydrochloric acid processing. it also shows charged surfaces which corroborates its use in literature as a cosmetic ingredient [4, 5]. figure 9. sem images of hcl residual products 4. conclusion this research studied the dissolution of muscovite ore to obtain a high grade material which could be used as extender for paint production in acidic solution hcl at a set of optimal experimental conditions of 2.5 mol/l hcl solution, within 120 minutes of agitation/dissolution and at 75◦c. the effects of solution concentrations and reaction temperatures were investigated to ascertain the level of ore dissolution while comparing the detailed characterization of the raw ore sample with the products obtained after dissolution. on this basis, these subsequent conclusions can be drawn: • the dissolution rate increases as the leachant concentration and temperature are increased as the rate of dissolution of muscovite ore in hcl solution was found to enhance the ore dissolution. 132 olaoluwa et al. / j. nig. soc. phys. sci. 2 (2020) 128–133 133 • the calculated activation energy obtained in the present study is above 40 kj/mol; which supports the fact that the leaching process is controlled by a 2-stage mechanism which involves diffusion and surface chemical concurrently. • the residual product analysis by xrf and sem confirms the purification of the ore and an increased elemental constituents in the products. references [1] h.-p. blume, g.w. brümmer, h. fleige, r. horn, e. kandeler, i. kögel-knabner, r. kretzschmar, k. stahr & b.-m. wilke, scheffer/schachtschabel soil science, 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[21] e. h. oelkers, j. schott, j. m. gauthier & t. herrero-roncal, “an experimental study of the dissolution mechanism and rates of muscovite”, geochim. cosmochim. acta 72(20) (2008) 4948. 133 j. nig. soc. phys. sci. 4 (2022) 588 journal of the nigerian society of physical sciences statistical analysis and distribution of global solar radiation and temperature over southern nigeria m. a. okonoa, e. p. agbob,∗, b. j. ekaha, u. j. ekahb, e. b. ettahb, c. o. edetb,c,d adepartment of physics, university of calabar, nigeria bcross river university of technology, calabar, cross river state, nigeria cfaculty of applied and human sciences, universiti malaysia, perlis, malaysia dfaculty of electronic engineering technology, universiti malaysia, perlis, malaysia abstract the intensity of solar energy that is received by a particular location is affected by most meteorological conditions including, the solar irradiance received by the location, precipitation, extreme heat as a result of the surface or ambient temperature, etc. we obtain the monthly global solar irradiation and ambient temperature for the three (3) eco-climatic zones in the south of nigeria (17 locations) for 12 years (2005 2016) from the photovoltaic geographical information system (pvgis) satellite. the goal of this study is to understand how regional meteorological conditions affect radiation and temperature reception. monthly and annual trends were plotted and compared for both variables in each region to show the similarity or dichotomy in their trends. the mann-kendall (m-k) trend test has been adopted to reveal the changes in the variations on an annual basis, and results showed that the trend were not significant for both variables. box plots have been used to give a better description of the data, and compared to show similarities and differences. finally, we adopted the gaussian (normal) distribution to show, understand and compare the data distribution. linear regression plots for each zone shows that the relationship between the solar irradiation and temperature is high. results show that the climate and vegetation of a region contributes majorly to the variation of radiation and temperature. inhomogeneity of data or results for locations in the same zones may be attributed to local meteorological conditions. the results obtained here will prove vital in decision making relating to the adoption of solar energy technologies in the region. results show that the climate and vegetation of a region contributes majorly to the variation of radiation and temperature. inhomogeneity of data or results for locations in the same zones may be attributed to local meteorological conditions. doi:10.46481/jnsps.2022.588 keywords: global solar radiation, temperature, box plots, mann-kendall test, sen’s slope estimate, gaussian distribution, linear regression article history: received: 15 january 2022 received in revised form: 24 june 2022 accepted for publication: 12 july 2022 published: 14 august 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: s. j. adebiyi 1. introduction careful study of the world’s economy shows the importance of solar energy (directly related to global solar radiation ∗corresponding author tel. no: +234(0)7018774580 email address: emmanuelpaulagbo@gmail.com (e. p. agbo) and temperature) to economic growth [1, 2]. many developing countries in africa have this flaw of not being able to develop systems to put them in the position of producing enough less expensive energy despite the potential of africa as a continent [3]. this inefficiency in solar energy conversion causes the poverty and under development activities for low gdp countries [4]. 1 okono et al. / j. nig. soc. phys. sci. 4 (2022) 588 2 usage of fossil fuels and wood for example, causes harm on a long term to these regions [5, 6]. this shows that global solar radiation is one of the environmentally friendly source of energy for the climate system and has an amount that varies with respect to the geographical location, altitude, atmospheric conditions, among others [4]. this study seeks to unravel the ambiguity of the variation of these most important variables in solar energy. these are arguably the climatic variables with the most effect on a region’s solar energy variation [7–11]. a good understanding of climate change and projected trends of these two variables is very important for policymakers and/or industry managers to undertake project planning and/or installing solar cells in a considered locality. many studies across the world have been carried out for highlighting the changes, trends and variability analysis of the solar irradiance and temperature. [12-18], but the problem is that little or no study has focused on the regional variations of a country, this study in particular is unique in that it focuses on all major cities (stations) in a country like nigeria. the assumption is that locations in the same region have the same variation, but this may not be entirely so as latitudes, elevation, and other local meteorological factors vary. abdulkarim et al. [19] analysed solar radiation and climate data’s effects generally (including on economics) on the development of solar photovoltaic (pv) systems in nigeria. with one location for each radiation region, results showed that the southern station of port harcourt had the lowest viability for solar energy conversion when compared to maiduguri and minna. their results also showed a positive relationship between solar radiation and temperature for all stations, while showing a negative linear relationship between solar radiation and relative humidity. the horizontal surface solar radiation (or global horizontal radiation) and temperature levels for stations in southern nigeria were evaluated using nasa sse data by dike et al. [20]. results summarized that there is a potential for more adoption of solar energy in the region in spite of recorded lower levels. ohunakin et al. [21] developed six (6) empirical models (linear and quadratic) based on ambient temperature and sunshine parameters for the southern region of osogbo. results showed that the temperature based models had the lowest statistical error, and therefore showed the best agreement with measured data, making it suitable for the estimation of global solar radiation. they went further to conclude that this is so for locations with the same latitude. this clearly proves the connection between the two parameters. another study which applied solar radiation models was that of okundamiya et al. [22]. using twenty-two (22) years data for abuja, sokoto and the southern station of benin city, confirmed results by ohunakin et al. [21] after applying the multivariate regression relationship deduced after estimations has been made that air temperature ratio, maximum air temperature, among others performed better than other relationships. this study however is focused on analyzing the global solar radiation and temperature variations in 17 locations of the southern eco-climatic zones of nigeria (the guinea savannah, the tropical rainforest and the mangrove swamp. monthly trends for radiation and temperature for all locations (17) in each zone of the southern region will plotted and compared with each other, the same will be done for the annual trends. the mannkendall (m-k) trend test has been applied to discern the significance of the increasing or decreasing radiation and temperature trends. two-dimensional gaussian distributions as well as box plots were used to show the nature of the distribution. results can be applied to aid in the knowledge of these variables in the region to better understand its climate dynamics like understanding the potential of global solar radiation reception for renewables and vegetation. 2. methodology various methods have been utilized to analyse the data obtained. these methods include the use of box plots to describe data and linear regression (lr) plots between radiation and temperature to give a better understanding of the relationship between the variations of both. kernel density estimation (kde) plots have also been used to better explain the distribution of data. 2.1. theoretical review box plots shows a graphical summarization of data. unlike the kde, it does not give a detailed representation of the data distribution, but gives an indication of the skewness of data. box plots summarizes data in the following way; lower extreme/minimum (q1 − 1.5 × i qr), lower quartile (q1), median (q2), upper quartile (q3) and upper extreme (q4 = q3 + 1.5 × i qr). we use the (kde) to examine the features and distributions of observations. the kde takes into account the probability density function (pdf) for a normal distribution. for a univariate kde, the pdf is given as [23]. f (x|µ, σ2) = 1 √ 2πσ2 ex p [ − 1 2σ2 (x −µ)2 ] (1) where, σ is the standard deviation and µ is the mean. from equation (1), the ‘argument’ exponential function 12σ2 (x− µ)2 is a quadratic function of x. this can be seen as a parabola which points downward (as evident from gaussian/normal distributions); this is because the quadratic function has a negative coefficient [24]. the term in front of the exponential function 1√ 2πσ2 from equation (1) is the coefficient of the function (a constant) which does not depend on x. in simple terms, this is a normalization factor that warrants that; 1 √ 2πσ2 ∫ ∞ −∞ ex p [ − 1 2σ2 (x −µ)2 ] = 1. (2) we can simplify equation (1) by assuming a standard deviation (σ) of 1 and a mean (µ) of 0. f (x) = 1 √ 2π ex p ( − x2 2 ) . (3) the equation (3) is a simplification of the one-dimensional (1d) gaussian kernel function which has been employed in this study. 2 okono et al. / j. nig. soc. phys. sci. 4 (2022) 588 3 the linear regression function is used to show the relationship between two variables (dependent and independent). in the case of a simple linear regression we have one independent and one dependent variable [23]. it is given by y = ±mx ± c + e, (4) where y is the dependent variable, x is the independent variable, m is the value of the slope, c is the intercept and e is the residual error. 2.2. test for trend (the mann-kendall trend test) we use the m-k test when analysing time-series data. the test is non-parametric and does not require the data to conform to a particular distribution. [10, 25-26]. we apply this test when a given range of data x j agrees with the relation; x j = f (ti) + �i, (5) f (ti) here is a function of continuously increasing or decreasing monotonically, �i are the range residuals. in the m-k test, we test the null hypothesis ho which says that there is no trend, and the alternative hypothesis h1 which means that there is a trend in the series. if the results gotten from the test agrees with the null hypothesis (meaning that there is no trend), it means that the given data xi is randomly ordered in time (t), while the alternative hypothesis says that there is either an increasing monotonic or a decreasing monotonic trend [27-29]. the m-k test uses the statistic s , calculated using; s = n−1∑ k=1 n∑ j=λ=1 sgn(x j − xk), (6) where; sgn(x j − xk) =  +1; if(x j − xk) > 0 0; if(x j − xk) = 0 −1; if(x j − xk) < 0 (7) the number of data values is represented by n. a positive s value indicates an upward variation, and a negative s value characterizes a downward variation. the normal approximation (z statistic) is always used if the number of data values n is from 10 and above. we should also note that when there are tied/equal values, the accuracy of using the z statistic will reduce if the data values are close to 10. to compute the value of the z statistic, the variance of s ′v ar(s )′ is used [25] v ar(s ) = 1 18 n(n − 1)(2n + 5) − g∑ p=1 tp(tp − 1)(2tp + 5) (8) g represents the number of tied groups in the series (showing that the test takes the tied or equal values into consideration. the number of data values in the pth group is represented by tp. the test statistic z is now obtained using the values of v ar(s ) and s [25]; z =  s−1 √ v ar(s ) ; s > 0 0; s = 0 s +1 √ v ar(s ) ; s < 0 (9) the decreasing variation is discerned from a negative kendall z value, and an increasing trend is seen from a positive z value. both interpretations can be concluded to have a significant trend if and when the data’s p-value is lower than the significance level (< 5% = 0.05 in this case). the trend is not significant if the p-value is higher than the level of significance [30]. 3. study location nigeria is lies between longitude 2 o e and 15 o e and latitude 4 o n and 14o n, having two major seasons, wet and dry [31]. the country is situated just above the geographical equator and this results to high sunshine intensity. this study is focused on the analysis of temperature and global solar radiation for the southern eco-climatic region, and figure 1a shows all states in the southern region and 1b shows the eco-climatic region as well as the locations in study. the region is characterized by her proximity to the atlantic ocean [32]. table 1 gives more information about each location. 3.1. data source and method of analysis monthly data for average ambient temperature in celsius and the total monthly gsr (global global solar radiation) in kwh/m2/mo were obtained from the pvgis (photovoltaic geographical information system) for twelve years (2005-2016) at https://re.jrc.ec.europa.eu/pvg tools/en?tools.html#pvp. the total monthly ghi (kwh/m2/month) for all southern locations converted to average daily ghi in mega joules (m j/m2/day) and further analyzed with the average daily temperature (in celsius). the analysis tool for this study is the microsoft excel software, which was used for the data conversion, storage and wrangling. data visualization was carried out by the python programming language. all graphical representations and some statistical analysis were implemented. packages in python programming like sklearn, seaborn, matplotlib, pandas, and numpy etc., were utilized for representations including box plots, linear regression, 1d kde plots. all were compared for all locations in each zone. the 2d kde employed the two (2) random variables (radiation and temperature) for its implementation. temperature represented the x-axis and radiation represented the y-axis. 3 okono et al. / j. nig. soc. phys. sci. 4 (2022) 588 4 figure 1. : (a) map of nigeria showing the southern region (b) map of nigeria showing the distribution of al eco-climatic zones, with the exact locations of the stations being studied. table 1. elevation and location (coordinates) of all study locations. location(state) elevation latitude longitude (m) (o n) (o e) abeokuta (ogun) 30 7.155 3.339 ado-ekiti (ekiti) 375 7.601 5.273 akure (ondo) 354 7.258 5.218 ibadan (oyo) 217 7.371 3.964 ikeja (lagos) 41 6.626 3.361 osogbo (osun) 352 7.754 4.581 asaba (delta) 44 6.025 6.693 calabar (cross river) 13 4.977 8.313 benin city (edo) 88 6.325 5.607 port harcourt (rivers) 10 4.814 7.080 uyo (akwa ibom) 73 5.048 7.907 yenagoa (bayelsa) 16 4.904 6.234 abakiliki (ebonyi) 46 6.312 8.107 awka (anambra) 103 6.218 7.083 enugu (enugu) 199 6.463 7.542 owerri (imo) 73 5.490 7.034 umuahia (abia) 154 5.525 7.517 4. results and discussion 4.1. monthly trends figures 2a, 2b, and 2c show the monthly variation of radiation for the guinea savannah, mangrove swamp and tropical rain forest regions respectively. they all have their lowest values between the months of july and august (july mostly). this variation corresponds approximately with the month (s) that have the lowest recorded temperature (figures 2d, 2e, and 2f) in the study areas. the radiation trend steadily reduces from january till it reaches its minimum value in july, and then rises steadily between september and december. these radiation trends (figures 2a, 2b, and 2c) correlates positively with that of the ambient temperature (figures 2d, 2e, and 2f). the significant difference between the monthly trends of radiation and temperature for all locations is that while the peak radiation value for the locations are observed to be in january (on average), the peak ambient temperatures are observed to be in the month of march (on average). this may be attributed to the inclusion or the increased longwave radiation around the months of march with in agreement with soneye et al. [33]. the sky is clear and have no clouds from december through february, but the clouds begin to form in the month is march when rainfall begins. this increases the diffused radiation as the clouds acts as a greenhouse gas [34]. figures 3 and 4 shows the mean radiation and temperature trends respectively for the 3 zones in study. they show that the mangrove swamp zone has lower values of temperature in the dry months (january april), and relative to other zones, her temperature value is higher in the wet months. this could be attributed to the fact that the mangrove swamp zone has a closer proximity to the ocean than other regions in the south. figures 2b and 2e proves this as it shows that calabar, asaba, yenegoa, and uyo all have low values due to their close proximity to the atlantic ocean. the hottest locations from the monthly trends include abakiliki and lagos, having monthly temperature trends that are far above others; this can be attributed to the industrial activities in those locations [35]. from figure 4, we can see that the months with the lowest temperature values corresponds to months in the rainy/wet season which runs between the months of june and october approximately. 4.2. annual trends results were gotten for the annual global solar radiation variations for the guinea savannah, mangrove swamp and tropical rain forest regions in figures 5a, 5b and 5c; and for the ambient temperature variations for the guinea savannah, mangrove swamp and tropical rain forest regions in figures 5d, 5e and 5f respectively. sen’s slope is represented for all the aforementioned figures in table 2 and table 3. from just mere observation, we can see from the figures that the annual radiation trends steeply increased in the latter parts of the trend. the trends for the average ambient temperature are relatively stable across all years. figures 6 and 7 show the radiation and ambient temperature trends respectively for all southern zones. from figure 6, we can see that the radiation variation representation for the mangrove swamp zone is the lowest among all zones. if we compare this variation to that of ambient temperature in figure 7, we will observe that all temperature trends for all zones are 4 okono et al. / j. nig. soc. phys. sci. 4 (2022) 588 5 table 2. the m-k trend test results for global solar radiation showing the nature of each of their across all years (2005 2016). mann pkendall’s kendall’s test value sen’s location tau statistic statistic (twointercept slope test trend (s) (s) tailed) (q) interpretation abakiliki -0.212 -14 -0.89 0.37 19.38 -0.008 false nst (↓) awka -0.212 -14 -0.89 0.37 18.95 -0.026 false nst (↓) enugu -0.273 -18 -1.17 0.24 19.24 -0.025 false nst (↓) adoekiti -0.061 -4 -0.21 0.84 19.54 -0.005 false nst (↓) umuahia -0.273 -18 -1.17 0.24 18.49 -0.030 false nst (↓) owerri -0.212 -14 -0.89 0.37 18.44 -0.024 false nst (↓) port harcourt -0.030 -2 -0.07 0.95 16.68 -0.009 false nst (↓) osogbo 0.061 4 0.21 0.84 18.92 0.019 false nst (↑) benin city 0.091 6 0.34 0.73 18.19 0.018 false nst (↑) abeokuta 0.182 12 0.75 0.45 18.47 0.036 false nst (↑) akure 0.091 6 0.34 0.73 19.06 0.014 false nst (↑) ibadan 0.061 4 0.21 0.84 18.79 0.018 false nst (↑) asaba -0.121 -8 -0.48 0.63 19.04 -0.016 false nst (↓) calabar 0.091 6 0.34 0.19 18.19 0.018 false nst (↑) ikeja -0.091 -6 -0.34 0.73 17.92 -0.029 false nst (↓) uyo -0.152 -10 -0.62 0.54 17.18 -0.009 false nst (↓) yenagoa -0.333 -22 -1.77 0.15 17.33 -0.050 false nst (↓) *note: significance level = 5% (0.05); (↑) indicates an increasing variation and (↓) indicates a decreasing variation; nst no significant trend table 3. the m-k trend test results for average ambient temperature showing the nature of each of their across all years (2005 2016). mann pkendall’s kendall’s test value sen’s location tau statistic statistic (twointercept slope test trend (s) (s) tailed) (q) interpretation abakiliki -0.20 -13 -0.82 0.41 26.45 -0.049 false nst (↓) awka -0.24 -16 -1.03 0.30 25.36 -0.038 false nst (↓) enugu -0.21 -14 -0.89 0.37 25.57 -0.052 false nst (↓) adoekiti 0.02 1 0.00 1.00 24.16 2.22×−16 false nst (↑) umuahia -0.33 -22 -1.44 0.15 24.70 -0.036 false nst (↓) abeokuta 0.08 5 0.27 0.78 25.40 0.007 false nst (↑) port harcourt -0.18 -12 -0.75 0.45 25.12 -0.018 false nst (↓) owerri -0.18 -12 -0.75 0.45 24.82 -0.020 false nst (↓) akure -0.08 -5 -0.27 0.78 24.91 -0.005 false nst (↓) ibadan 0.00 0 0.00 1.00 25.38 0.001 false nst (↑) benin city -0.30 -20 -1.30 0.19 25.17 -0.020 false nst (↓) osogbo 0.03 2 0.07 0.95 24.09 0.004 false nst (↑) asaba -0.27 -18 -1.17 0.24 25.86 -0.025 false nst (↓) calabar -0.29 -19 -1.24 0.22 25.59 -0.023 false nst (↓) ikeja -0.29 -19 -1.24 0.22 26.79 -0.017 false nst (↓) uyo -0.24 -16 -1.03 0.30 25.08 -0.025 false nst (↓) yenagoa -0.21 -14 -0.89 0.37 25.46 -0.023 false nst (↓) *note: significance level = 5% (0.05); (↑) indicates an increasing variation and (↓) indicates a decreasing variation; nst no significant trend relatively similar, with highest values for both the radiation and temperature observed in its most recent year (2016). the hottest locations in the mangrove swamp region for the annual trends from figure 5e are ikeja and asaba, and this corresponds to the locations with the highest radiation in figure 5b. the same could be said for abakiliki in figures 5a and 5d for 5 okono et al. / j. nig. soc. phys. sci. 4 (2022) 588 6 figure 2. monthly trend of radiation for all eco-climatic locations (a) guinea savannah (b) mangrove swamp and (c) rainforest; average ambient temperature trends (d) guinea savannah (e) mangrove swamp and (f) rainforest. figure 3. monthly trend of radiation for all eco-climatic zones in southern nigeria. the guinea savannah zone, where the location of highest temperature corresponds to that of highest global solar radiation. the elevation height of the locations is a contributing factor to the temperature values as observed from the figures. it extension, ikeja shows that a low radiation value is contributed by a figure 4. monthly trend of temperature for all eco-climatic zones in southern nigeria. figure 5. annual trend of radiation for all eco-climatic locations (a) guinea savannah (b) mangrove swamp and (c) rainforest; average ambient temperature trends (d) guinea savannah (e) mangrove swamp and (f) rainforest. location’s proximity to a water body, directly related to the sea level. 6 okono et al. / j. nig. soc. phys. sci. 4 (2022) 588 7 figure 6. annual trends of radiation for all southern regions. figure 7. annual trends of average ambient temperature for all southern regions. mann-kendall trend test the m-k test has been adopted to analyze the significance of the trends of radiation and average ambient temperature along with sen’s slope (q). results presented in table 2 shows that the results for all locations in each vegetation zone for global solar radiation and in table 3 for ambient temperature. significance level of 5% (0.05) was used for the test. for the global solar radiation trends from table 2, we can see that all locations in the mangrove swamp vegetation region all have reducing trends apart from calabar, characterized by their negative sen’s slope. apart from calabar, they all have negative sen’s slope value. the same reducing radiation trends can be observed in owerri and port harcourt for the rain forest vegetation zone after showing negative z values also. five of the six locations that have increasing trends are in the rain forest vegetation region, these locations are characterized by their positive sen’s slope and kendall z values. they include abeokuta, akure, ibadan, osogbo and benin city for the sw region having z values of 0.75, 0.34, 0.21, 0.21 and 0.34 respectively. this shows that the rain forest vegetation region is majorly characterized by increased radiation trends. for the ambient temperature trends from table 3, as was similar with that of the radiation results, all locations in the mangrove swamp vegetation region have reducing trends, characterized by their negative sen’s slope. they have a z-statistic value of -1.17, -1.24, -1.24, -1.03, -0.89 for asaba, calabar, ikeja, uyo, yenagoa respectively. figure 8. box plots showing radiation distribution for all eco-climatic locations (a) guinea savannah (b) mangrove swamp and (c) rainforest; average ambient temperature distributions (d) guinea savannah (e) mangrove swamp and (f) rainforest. in the same way as the mangrove vegetation region, the guinea savannah vegetation region also shows reducing temperature trends. this shows the similarity between the mangrove vegetation region and the guinea savannah region. the rain forest vegetation region has three (3) locations has increasing trends for ambient temperature, these locations are characterized by their positive sen’s slope and kendall z values. they include abeokuta, ibadan and osogbo of the rain forest vegetation zone having z values of 0.27, 0, and 0.07. the results for majority of these locations for temperature in table 3 approximately corresponds relatively to that of global solar radiation in table 2. at 5% significance level, all radiation and temperature variations accepted the null hypothesis ho and rejected the alternative hypothesis h1 after its probability value (p-value) of these series (increasing or decreasing) was found to be more than the significance level α = 0.05. this means that although most figures reducing or increasing variations, this increase or decrease aren’t occurring with much significance. 4.3. box plots to discern the distribution of radiation and temperature for all locations, box plots have presented in figures 8 (a – f) and 9 7 okono et al. / j. nig. soc. phys. sci. 4 (2022) 588 8 table 4. linear regression results for all mangrove swamp locations. asaba calabar ikeja uyo yenagoa correlation (r) 0.95 0.92 0.89 0.92 0.88 slope 1.75 2.65 1.87 2.46 2.36 error in slope 0.19 0.36 0.30 0.34 0.40 r2 0.89 0.84 0.80 0.84 0.78 intercept (c) -26.07 -50.53 -32.05 -44.30 -42.77 error in c 4.89 9.32 7.90 8.51 10.13 sd 0.75 1.27 1.18 1.15 1.37 figure 9. box plots showing the average (a) radiation and (b) temperature distribution over each southern eco-climatic zone. table 5. linear regression results for all guinea savannah locations. abakiliki awka enugu ado-ekiti umuahia correlation (r) 0.95 0.95 0.93 0.93 0.94 slope 1.45 1.76 1.55 1.74 2.04 error in slope 0.15 0.18 0.19 0.22 0.23 r2 0.91 0.90 0.86 0.86 0.89 intercept (c) -18.59 -25.49 -20.08 -22.47 -31.82 error in c 3.85 4.65 4.91 5.28 5.67 sd 0.72 0.79 0.89 0.93 0.86 table 6. linear regression results for all sw locations. abeokuta benin city akure ibadan owerri osogbo port harcourt correlation (r) 0.91 0.96 0.93 0.94 0.95 0.95 0.90 slope 1.94 1.96 1.61 2.13 2.18 2.17 2.55 error in slope 0.29 0.18 0.20 0.25 0.23 0.21 0.38 r2 0.82 0.92 0.87 0.87 0.90 0.91 0.82 intercept (c) -30.67 -30.71 -21.02 -35.19 -35.82 -33.22 -47.34 error in c 7.33 4.57 4.94 6.47 5.62 5.15 9.63 sd 1.12 0.69 0.92 0.98 0.79 0.88 1.27 figure 10. kde plots showing radiation distribution in m j/m2/day for all locations in the mangrove swamp zone (a) asaba (b) calabar (c) ikeja (d) uyo (e) yenagoa. (a and b). a careful observation of the box plots show that their distribution for some locations is almost similar, characterized by a median that is almost uniform. figures 8a and 8d shows the radiation and temperature distribution respectively for all locations in the guinea savannah zone. similar to that of locations in the mangrove swamp zone, all boxes representing radiation in the guinea savannah zone (from figure 8a), shows that the peak radiation value for all locations is approximately equal. however, for the temperature distribution in figure 8d, abakiliki has the highest temperature value for all locations in the se, with umuahia and ado-ekiti having the lowest. the representation for the radiation and temperature in figures 8b and 8e respectively for the mangrove swamp zone shows that asaba has the highest radiation and temperature value; this corresponds to results of our monthly and annual line plots. the 8 okono et al. / j. nig. soc. phys. sci. 4 (2022) 588 9 figure 11. kde plots showing temperature distribution in celsius for all locations in the mangrove swamp zone (a) asaba (b) calabar (c) ikeja (d) uyo (e) yenagoa. figure 12. kde plots showing radiation distribution in m j/m2/day for all locations in the guinea savannah zone (a) abakiliki (b) awka (c) enugu (d) ado-ekiti (e) umuahia. figure 13. kde plots showing temperature distribution in celsius for all locations in the guinea savannah zone (a) abakiliki (b) awka (c) enugu (d) adoekiti (e) umuahia. temperature representations in the figure 8e for each location has relatively smaller ranges than that of the radiation plot in figure 8e. this could be attributed to the fact that it takes a certain threshold of temperature to bring about a notable shift in radiation [36]. all location in the mangrove swamp zone all have approximately the same peak radiation values from figure 8b, but the same cannot be said for the temperature plot in figure 8e. figures 8c and 8f represents the distribution of radiation and temperature for all locations in the rain forest zone. this region seems to have the widest range of radiation distribution for all her locations from figure 8e. the data distribution range for the temperature however, has some variations. as it was earlier explained in the monthly and annual line plots, abeokuta in ogun state has the highest temperature for all locations this region. the range of this box represented in figure 8f is not also large; hence, the temperature is constantly high. although scattered across different peak values, all locations in this zone have about the same distribution range from the highest to the lowest value. all plots in figure 8 shows that the peak radiation is closely similar which is in contrast with that of the ambient temperature. this observation shows that changes in the peak value of temperature to only results in a radiation peak value that cannot rise above a particular threshold value. this crystalizes the argument that the radiation value of a location is affected by the geographical location (latitude, elevation height) also. figure 9 shows the collation of radiation and temperature 9 okono et al. / j. nig. soc. phys. sci. 4 (2022) 588 10 figure 14. kde plots showing radiation distribution in m j/m2/day for all locations in the rain forest zone (a)abeokuta (b)awka (c) ibadan (d) osogbo (e) owerri (f) benin city (g)port harcourt. box plots respectively for all zones in the south. the average values of temperature and radiation are well represented in the figure. the global solar radiation representation from figure 9a shows that the southern locations have between 14 and 20 m j/m2/day. due to the fact that the global solar radiation variation depends majorly on the variation of ambient temperature, we observe higher values of global solar radiation for the guineas savannah and rainforest regions. this is not so for the mangrove swamp zone. all southern regions had their values within the range of 23 oc to 26.5 oc from the temperature box plot in figure 9b. relating to the radiation plot in figure 9, the mangrove swamp region’s close proximity to the atlantic ocean bring about the low temperature ranges, even though judging from the latitude of the region (closest to the equator), it should be slightly hotter. 4.4. univariate kde distribution plots for temperature and global solar radiation for all locations being studied has been well represented (figures 10 16). the univariate gaussian (normal) distribution was carried out to show the density of data distribution as well as to show the modal peaks and skewness of the distribution. the purpose of these plots is to show how the temperature and figure 15. kde plots showing temperature distribution in celcuis for all locations in the rain forest zone (a)abeokuta (b)awka (c) ibadan (d) osogbo (e) owerri (f) benin city (g)port harcourt. radiation data is being distributed across the data range, revealing the median, modal peak(s)/densities, etc. figures 10 show the radiation distribution for locations in the mangrove swamp zone. all locations seem to have at least a modal peak at about 20 m j/m2/day. yenagoa have 2 modal peaks while calabar and uyo have the same distribution and density from the figure. the similarity between these pair of regions arises from the fact that they are in close proximity to each other. the temperature distributions for all locations in the mangrove swamp zone (figure 11) all have the same gaussian distribution. their densities peak at approximately 0.27 and they all have 2 modal peaks. the uniformity in the distributions of all temperature representations for the ss zone shows the uniform climatic conditions in the region. from figure 12, we see the radiation distributions for all locations in the guinea savannah locations. similar to the mangrove swamp locations, the modal peak lies round 20 m j/m2/day, showing the similarity between the locations. in particular, the density/gaussian distributions for abakiliki, enugu and umuahia are more similar with each other, having a peak density value of about 0.15 around their modal peak. the temperature distribution locations in the guinea savannah zone (figure 13) like that of the mangrove swamp zone. is 10 okono et al. / j. nig. soc. phys. sci. 4 (2022) 588 11 figure 16. kde plots showing distributions for all eco-climatic regions in the south for global solar radiation (a) mangrove swamp (b) rain forest (c) guniea savannah; and temperature (d) mangrove swamp (e) rain forest (f) guniea savannah. similar across all locations in the same zone. the distribution is the same with a gaussian distribution that has 2 close modal peaks; this shows the data spread across range of values. the homogeneity in the distributions of all temperature representations for the guinea savannah zone shows the uniform climatic conditions in the region just as was in that of the mangrove swamp. figure 14 shows the radiation distribution for all locations in the rain forest region. contrary to other zones (mangrove swamp and guinea savannah), the locations in the rain forest have a more similar density (gaussian) distribution. the temperature distribution for all locations in the rainforest zone (figure 15) is not a similar as of that for all locations in the mangrove swamp and guinea savannah zones. all locations show relatively similar gaussian distributions. figure 16 shows the combined distribution plots for temperature and radiation for all the southern zones. the density (kde) plots for the global solar radiation for the mangrove swamp, rain forest, and guniea savannah (figures 16a, 16b and 16c respectively) follows similar curves with that of the ambient temperature (figures 16d, 16e and 16f respectively). this shows a strong positive coefficient of correlation between the global solar radiation and temperature for each zone. figure 17. regression plots showing the relationship between radiation and temperature for all locations in the mangrove swamp zone. figure 18. regression plots showing the relationship between radiation and temperature for all locations in the guinea savannah zone. 4.5. linear regression the goal of the linear regression to understand the relationship between the variation of ambient temperature and global11 okono et al. / j. nig. soc. phys. sci. 4 (2022) 588 12 figure 19. regression plots showing the relationship between radiation and temperature for all locations in the rain forest zone. solar radiation for all locations in each zone. the same scale was used for the plots to show the discrepancies in the trend lines. the independent variable was the temperature and the radiation was the dependent variable. results from the regression analysis have also be well represented in tables including the coefficient of correlation (r), coefficient of determination (r2), slope, error in slope, intercept, error in intercept. standard deviation. from figure 17, we can see the regression plots for all locations in the mangrove swamp zone. table 4 shows the results from the regression. asaba has the strongest relationship between radiation and temperature; characterized by its strong positive r value (the strongest for the zone), lowest error in slope and standard deviation from the mean. other locations in the mangrove swamp zone from the figure all have strong positive values of r also. since we used the same scale to plot for all locations in the region, we can see the differences in the radiation and temperature spread across the y and x axes respectively. this spread agrees with results from the box plots for the zone (figures 8b and 8e). yenagoa, although having a strong positive r value (0.88), represents the lowest in the mangrove swamp zone. figure 18 represents the regression plots for all locations in the guinea savannah region. table 5 shows the results from this regression. we can observe from the trend that all locations have an almost similar relationship between radiation and temperature. this is characterized by the similar r value for all 5 locations in the region (r values ranges between 0.93 and 0.95). this very strong positive r value corresponds to lower errors in the slope and intercept of the trends, as well as the standard deviation (table 5). this means that the guinea savannah zone constitutes the strongest relationship between global solar radiation and temperature for all zones. figure 19 represents the regression plots for all locations in the sw region. table 6 shows the results from this regression. from our results, benin city has the strongest positive r value (0.96), followed by owerri and osogbo ibadan (0.95 each). the lowest r value is observed in port harcourt (0.90). this is so because from our box plot results for radiation and temperature in figure 8c and 8f respectively, a high temperature value for port harcourt does not bring about a high radiation value for the same location in the region. in conclusion, all southern locations have strong positive relationships between radiation and temperature. this proves that for any change in the temperature variations in the region, there is a corresponding strong similar change in the global solar radiation variation [37, 38]. 5. conclusion we have shown the relationship between the variation of solar irradiation and average ambient temperature for locations in the eco-climatic zones in the south of nigeria. this region is made up of 17 locations and 2 eco-climatic zones (mangrove swamp, guinea savannah and tropical rainforest). the study was done to have a better understanding of the trends, distributions and relationships between the two variables. the mannkendall trend test has been used to study the significance of the variations of both variables in all regions and in general the following results and inferences were outlined. • the trends of solar irradiance and ambient temperature for all stations and eco-climatic zones are not increasing or decreasing annually with significance from the results of the mk test. • linear regression, and distribution plots depict the positive relationship between both variables (global solar radiation and temperature) across all stations, albeit, some variations in the strong positive correlation for some locations were observed. this may be attributed to the differences in elevation height, climatic conditions, and local weather, etc. • results show that the climate and vegetation of a region contributes majorly to the variation of radiation and temperature. inhomogeneity of data or results for locations in the same zones may be attributed to local meteorological conditions and effects like carbon emission from industries, proximity to major water bodies, etc. • although 12 years’ data might not be sufficient to conclude in terms of climatological conditions, the results show that meteorological conditions of the study locations are almost uniform in variation. this proves that latitudinal locations are one of the major contributors to 12 okono et al. / j. nig. soc. phys. sci. 4 (2022) 588 13 global solar radiation and temperature variations, especially for locations that are relatively close in geographical proximity. acknowledgments we thank the referees for the positive enlightening comments and suggestions, which have greatly helped us in making improvements to this paper. code availability: the code associated with this analysis can be assessed from https://github.com/emmaestro001/jnspsarticle-analysis. data availability: the data associated (temperature and global solar radiation) with this study was obtained from the pvgis (photovoltaic geographical information system) for twelve years (2005-2016) at https://re.jrc.ec.europa.eu/pvg tools/en?tools. html#pvp. references [1] m. irfan, r. m. elavarasan, y. hao, m. feng, & d. sailan, “an assessment of consumers’ willingness to utilize solar energy in china: endusers’ perspective”, journal of cleaner production, 292 (2021) 126008. https://doi.org/10.1016/j.jclepro.2021.126008. 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[38] d. j. mildrexler, m. zhao, and s. w. running. ”a global comparison between station air temperatures and modis land surface temperatures reveals the cooling role of forests”, journal of geophysical research: biogeosciences, 116 (2011). https://doi.org/10.1029/2010jg001486. 14 j. nig. soc. phys. sci. 1 (2019) 30–34 journal of the nigerian society of physical sciences original research eigenvalue elasticity and sensitivity analyses of the transmission dynamic model of corruption adeyemi olukayode binuyo∗ department of mathematical sciences, ajayi crowther university, oyo, nigeria abstract in this paper, the eigenvalue elasticity and sensitivity values of the mathematical model of transmission dynamics of corruption were obtained and presented using the eigenvalue elasticity and sensitivity analysis methods. the parameter with the greatest impact on the mathematical model was determined using the methods. this parameter will assist the government on how to reduce and provide measures to eradicate corrupt practices among the populace. from the mathematical model of corruption presented, it was obtained that the effective contact rate of corruption has the highest value when using the eigenvalue elasticity and sensitivity analysis. keywords: corruption, mathematical model, population, eigenvalue elasticity analysis c©2019 journal of the nigerian society of physical sciences. all rights reserved. communicated by: b. j. falaye 1. introduction corruption is a form of dishonesty or criminal activity undertaken by a person or organization entrusted with a position of authority often to acquire illicit benefits [1]. in addition, it is the misuse of public power usually by elected politician or appointed civil servants for private gain. it is clear to every citizen that the level of corruption in the country is so high. corruption is found in every sector of the society [2]. be it a small or big sector, there is every possibility of observing corrupt practices when critically examined. forms of corruption include: taking gifts or money to speed up problem solving, extortion usually common among civil servants illegally set a fee for services or artificially create a deficit, bribery, fraud, embezzlement, falsification, forgery, nepotism, favoritism and a host of others [3]. corruption in nigeria wears ∗author’s tel. no: +2348056681580 email address: oa.binuyo@acu.edu.ng (adeyemi olukayode binuyo ) many kinds of unattractive and dirty clothes. the situation has made so many people feel a lot of pains as the money which would have been used to reduce poverty in the country are being channeled into the pockets of a small group of persons [1]. efforts are being made by the government of the day to reduce the rate of corruption in the country such that there will be growth and national development. in this paper, it is intended to determine the parameter that has the greatest impact on the mathematical model of corruption which incorporates immune class and jailed class on the transmission dynamics of corruption in a certain population of the society. therefore, the study of the transmission dynamics of an sicjr model is presented. the parameter is determined using the eigenvalue elasticity and sensitivity analyses [4] and this parameter will assist the government to reduce the rate of corrupt practices in the society. 30 article history : received: 13 march 2019 received in revised form: 29 april 2019 accepted for publication: 03 may 2019 published: 29 may 2019 a. o. binuyo / j. nig. soc. phys. sci. 1 (2019) 30–34 31 2. mathematical formulation in order to formulate the mathematical model of corruption, the total population is divided into five compartments namely: susceptible class (s), immune class (i), corrupt class (c), jailed class (j) and the reformed class (r) with the following definitions: a) susceptible class: this class consists of individuals who have never been involved in any corrupt practices that will have harmful effects on the countrys national growth and development but likely vulnerable to being infected by the corrupt practices in the society. b) immune class: this class consists of individuals who can never be involved in corrupt practices irrespective of the circumstances around them. c) corrupt class: this class consists of individuals who are often involved in corrupt practices and are capable of influencing the susceptible and immune individuals to become corrupt. d) jailed class: this class involves individuals who have been convicted or punished of corrupt practices and imprisoned for a specific period of time during which he/she cannot be involved in any corrupt acts and cannot influence others during the imprisonment. e) reformed class: this class consists of the ex-convicts who have been reformed while serving their jailed term and can become susceptible to corruption after a while. the susceptible class is generated from daily recruitment of individuals born into homes with good moral standards and can likely be vulnerable to being infected by the corrupt practices at a rate θβ while the immune class are those who has moral standards from their homes and can never become corrupt at a rate (1 − θ)β [5]. susceptible individuals acquire corruption infection from corrupt individuals and become corrupt at a rate α while the corrupt individuals are jailed at a rate δ. corrupt and jailed individuals become reformed in the reformed class while serving their jail terms at rates τ and ρ respectively. individuals in the reformed class become susceptible after a while at a rate ω while susceptible individuals prone to corruption become immune at a rate ν due to moral and religious beliefs as well as public enlightenment campaign. all the classes are subjected to natural death at a rate µ. the above explanations result in the system of non-linear ordinary differential equations given as follows; s ′ (t) = θβ− αs (t)c(t) n(t) − (µ + ν)s (t) + ωr(t) (1) i ′ (t) = (1 − θ)β + νs (t) − (µ + γ)i(t) (2) c ′ (t) = αs (t)c(t) n(t) + γi(t) − (µ + τ + δ)c(t) (3) j ′ (t) = δc(t) − (µ−ρ)j(t) (4) r ′ (t) = ic(t) + ρj(t) − (µ + ω)r(t) (5) the fractional system (1 5) is given thus i.e. s ′ (t) = θβ−αs(t)c(t) − (µ + ν)s(t) + ωr(t) (6) i ′ (t) = (1 − θ)β + νs(t) − (µ + γ)i(t) (7) c ′ (t) = αs(t)c(t) + γi(t) − (µ + τ + δ)c(t) (8) j ′ (t) = δc(t) − (µ−ρ) j(t) (9) r ′ (t) = τc(t) + ρ j(t) − (µ + ω)r(t) (10) the corruption transmission dynamic model is shown in figure 1 3. parameter with the greatest impact 3.1. eigenvalue elasticity and sensitivity analysis eigenvalue elasticities measure the transient response sensitivities of the model to parameters [6] and since the values of elasticities are dimensionless, they can be compared with each other. this can aid us identifying the parameters which could greatly influence the mathematical model [7]. a) eigenvalue sensitivity with respect to a parameter: this is defined as the partial derivative of the eigenvalue with respect to that parameter [7]. the eigenvalue sensitivity s i(i = 1, . . . , n and n is the dimension of the state vector) with respect to the jth parameter of the system p j( j = 1, 2, 3, 4, 5, 6, 7) is given in the form; s i( p j) = lim ∆p j → 0 ∆λi ∆p j = ∂λi ∂p j = iti ∂j ∂p j ri. (11) b) eigenvalue elasticity with respect to a parameter: this is defined as the partial derivative of the eigenvalue with respect to that parameter normalized for the size of the parameter and the size of the eigenvalue [4]. this could also be described as the product of the eigenvalue sensitivity and the ratio of the eigenvalue and parameter [6]. thus, it is given in the form; ei( p j) = lim ∆p j → 0 ∆λi λi ∆p j p j = ∂λi λi ∂p j p j = ∂λi ∂pi · p j λi = iti ∂j ∂p j ri · p j λi . (12) with these equations (11) and (12), the eigenvalue elasticity and sensitivity with respect to a parameter can be computed using the left eigenvectors (ii) and the right eigenvectors (ri) with the partial derivatives of the linearized jacobian matrix (j) with respect to a parameter ( p j). because j and ∂j ∂p j can often be easily determined symbolically and because the eigenvalues can be computed for particular parameters values and points in 31 a. o. binuyo / j. nig. soc. phys. sci. 1 (2019) 30–34 32 figure 1: diagram of the corruption transmission dynamic model. table 1: definitions of parameters used in the mathematical model parameters description values source θ proportion of individuals not borne immune 0.00403 eguda et al.[5] β birth rate of individuals borne into the population 0.042 eguda et al.[5] α effective corruption contact rate 0.8 assumed γ rate at which susceptible individuals become immune to corruption 0.0004 eguda et al. [5] ω rate at which reformed individuals become susceptible to corruption 0.0021 eguda et al. [5] ν rate at which immune individuals are susceptible to corruption 0.143 eguda et al. [5] δ rate at which prosecution and imprisonment of corrupt individuals occur 0.0001 eguda et al. [5] τ rate at which corrupt individuals become reformed due to public enlightenment 0.001 assumed ρ rate at which jailed individuals become reformed 0.0001 assumed µ natural death rate 0.0189 eguda et al. [5] s (t) susceptible individual class at time t. 79,640 assumed i(t) immune individual class at time t. 16,635 assumed c(t) corrupt individual class at time t. 72,375 assumed j(t) jailed individual class at time t. 20,695 assumed r(t) reformed individual class at time t. 10,655 assumed n(t) number of individuals in the population at time t. 200,000 assumed s(t) fraction of susceptible individual at time t. 0.3982 estimated i(t) fraction of the immune individual at time t. 0.0832 estimated c(t) fraction of the corrupt individual at time t. 0.3619 estimated j(t) fraction of the jailed individual at time t. 0.1035 estimated r(t) fraction of the reformed individual at time t. 0.0533 estimated time, both eigenvalue elasticity and sensitivity with respect to a parameter can be computed without the need to either compute closed form expressions for eigenvalues nor to perform numeric differentiation [4]. using the matlab software package [8], the computer program was written for the evaluation of the values of eigenvalue elasticity and sensitivity of the mathematical model given in equations (6)–(10). definitions of parameters used in the mathematical model are shown in table 1 the results obtained are shown in the table 2 and the computer program is shown under the appendix. 32 a. o. binuyo / j. nig. soc. phys. sci. 1 (2019) 30–34 33 table 2: the values of the eigenvalue sensitivity and elasticity analysis parameters definition of parameters eigenvalue sensitivity eigenvalue elasticity α effective corruption contact rate 1.000064625 1.025787253 µ natural death rate -1.0000 -0.002423266 ν rate at which immune individuals are susceptible to corruption -0.0003606 -0.000066117 γ rate at which susceptible individuals become immune to corruption -0.151919916 -0.000007791372 ω rate at which reformed individuals become susceptible to corruption 0.00000009464 0.000000002548 τ rate at which corrupt individuals become reformed due to public enlightenment -1.0001404 -0.00128233 δ rate at which prosecution and imprisonment of corrupt individuals occur -1.0001406 -0.01283315 ρ rate at which jailed individuals become reformed 0.00000000024938 0.031973813 4. discussion and recommendations table 2 shows that the parameter alpha (α) which is defined as the effective corruption contact rate has the highest eigenvalue elasticity and eigenvalue sensitivity values. this means that the parameter has the greatest impact on the formulated mathematical model of the corruption model. this is to show that the parameter should be thoroughly investigated and worked upon as a possible policy lever such that the rate of corrupt transmission will be reduced to the barest minimum by the public officials especially all the government agencies that deal with corrupt practices (efcc, icpc, etc). in order to reduce the rate of effective corruption contact among the populace, the following recommendations are necessary: 1. self satisfaction: this means that an average human being should be contented with whatever he or she has. when the leaders are satisfied with the salary they are being paid and use them in the right way, the issue of embezzlement and money laundering will be stopped. managers, civil servants, etc. who are satisfied with what they are paid will not have time to indulge in corruption to make more money. 2. creating strong and efficient anti corruption institutions: another arsenal to win the fight against corruption and the reduce the effective rate of corruption contact is for the government to establish more anti corruption agencies in addition to the already established ones. these agencies will work independently with the government to ensure transparency. anyone who is caught in corrupt practices by any of the agencies should experience the consequences decided by the anti corruption agencies. 3. equal treatment: treating any corruption offender equally will help to reduce the rate of effective corruption contact. nobody is above the law and any who acts contrary to the rule of law should be given the adequate punishment that he or she deserves. 5. conclusion in this paper, a five compartmental model has been presented to gain insight on the parameter that has the greatest impact on the mathematical model of the transmission dynamics of corruption. the result from the research has shown that the rate of effective corruption contact among the populace has the greatest impact. with this result, when necessary and adequate measures are taken by the government, it is enough and expedient to reduce to the barest minimum the spread of corruption among the populace. references [1] u. m. p. okwuagbala, “corruption in nigeria: review, causes and solutions”, 2018. [2] o. a. oyinlola, corruption eradication in nigeria: an appraisal, library philosophy and practice, 2011. [3] s. abdulrahman, “stability analysis of the transmission dynamics and control of corruption”, the pacific journal of science and technology, 15 (2014) 99. [4] a. o. binuyo, “eigenvalue elasticity analysis of an seirc epidemic model for an infectious disease”, elixir electrical engineering online journal 53 (2012) 11928. [5] f. y. eguda, f. oguntolu & t. ashezua, “understanding the dynamics of corruption using mathematical modelling approach”, international journal of innovative science, engineering & technology, 4 (2017) 190. [6] b. guneralp, progress in eigenvalue elasticity analysis as a coherent loop dominance analysis tool, in proc. the 23rd international conference of the system dynamics society, boston, ma, 2005. [7] q. zhang, application and evaluation of local and global analysis for dynamic models of infectious disease spread, master’s thesis project, department of computer science, university of saskatchewan, saskatoon, 2008. [8] r. h. brian, a guide to matlab for beginners and experienced users, cambridge university press, webnote, 2001. 33 a. o. binuyo / j. nig. soc. phys. sci. 1 (2019) 30–34 34 appendix: eigenvalue elasticity and sensitivity analysis program syms mu nu alpha omega gamma tau delta rho; mu = 0.0189 nu = 0.143 alpha = 0.8 omega = 0.0021 gamma = 0.0004 tau = 0.001 delta = 0.0001 rho = 0.0001 mat31 = [-(mu+nu) 0 -alpha 0 omega; nu -(mu+gamma) 0 0 0; 0 gamma alpha-(mu+tau+delta) 0 0; 0 0 delta -(mu+rho) 0; 0 0 tau rho -(mu+omega)] [wmat31, dmat31] = eig(mat31) vmat31 = conj(inv(wmat31)) lambdiag = diag(dmat31) lambvec = sort(abs(lambdiag)) dim = size(lambvec,1) lambda1 = lambvec(dim) pd_mu = diff(mat31, mu) pd_nu = diff(mat31, nu) pd_alpha = diff(mat31, alpha) pd_omega = diff(mat31, omega) pd_gamma = diff(mat31, gamma) pd_tau = diff(mat31, tau) pd_delta = diff(mat31, delta) pd_rho = diff(mat31, rho) mat31 = subs(mat31) pd_mu = subs(pd_mu) pd_nu = subs(pd_nu) pd_alpha = subs(pd_alpha) pd_omega = subs(pd_omega) pd_gamma = subs(pd_gamma) pd_tau = subs(pd_tau) pd_delta = subs(pd_delta) pd_rho = subs(pd_rho) s_mu = sum(sum(pd_mu.*sens)) s_nu = sum(sum(pd_nu.*sens)) s_alpha = sum(sum(pd_alpha.*sens)) s_omega = sum(sum(pd_omega.*sens)) s_gamma = sum(sum(pd_gamma.*sens)) s_tau = sum(sum(pd_tau.*sens)) s_delta = sum(sum(pd_delta.*sens)) s_rho = sum(sum(pd_rho.*sens)) e_mu = mu/lambda1*s_mu e_nu = nu/lambda1*s_nu e_alpha = alpha/lambda1*s_alpha e_omega = omega/lambda1*s_omega e_gamma = gamma/lambda1*s_gamma e_tau = tau/lambda1*s_tau e_delta = delta/lambda1*s_delta e_rho = rho/lambda1*s_rho 34 j. nig. soc. phys. sci. 4 (2022) 682 journal of the nigerian society of physical sciences effect of treatment parameter on oscillatory flow of blood through an atherosclerotic artery with heat transfer r. r. hanveya,∗, k. w. bunonyob adepartment of mathematics & statistics, faculty of science, shuats, india bdepartment of mathematics & statistics, federal university, otuoke, nigeria abstract this research work has been carried out to investigate the influence of treatment parameter on flow of blood in a stenosed artery in the presence of magnetic field with heat transfer. the momentum equation governing the flow field has been solved by scaling it to dimensionless structure with the aid of some dimensionless parameters. the equations have been analytically solved using modified bessel equation and by the method of undetermined coefficients in order to obtain the temperature profile and velocity profile of the blood flow. the characteristics of the flow have been derived for a certain set of values rt , da, θ, gr, re, pr, ω, δ involved in the model analysis and are presented graphically with the help of software mathematica. moreover the velocity of the blood is adopting a wavy pattern as the values of the parameters vary. the study can be useful in providing a perception of the treatment caused by the superfluous consumption of fatty foods hence decreasing the risk of cancer, hypertension and many heart related diseases. doi:10.46481/jnsps.2022.682 keywords: blood flow, heat transfer, atherosclerosis, treatment, darcy number article history : received: 27 february 2022 received in revised form: 01 july 2022 accepted for publication: 01 july 2022 published: 15 august 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: t. latunde 1. introduction the blood is regarded as a thick red liquid comprising of red blood cells, white blood cells and platelets which are circulating in through arteries and veins in a human body. it has a strong nourishing effect and serves as one of the most important substance constituting the human body. as now-a-days, heart diseases due to thickening of blood carrying artery is very common which causes the accumulation of fatty substances in the lumen which in turn tends to block the side walls of the veins ∗corresponding author tel. no: +91-9580493319 email address: rishab.rhanvey@shiats.edu.in (r. r. hanvey) and artery which is due to the pulsating nature of the heart. furthermore, the most important function of the heart is pumping the blood throughout the circularity system and to supply oxygen to organs of the body. severe stenosis may lead to critical flow conditions of blood by reducing the blood supply and resulting in serious consequences called carotid artery blockage which is one of the eminent factors contributing to fatal hemorrhages and hypertension. regarding this, atherosclerosis, the leading killer disease of the 21st century which is due to the hardening of the artery caused by the waxy substances, cellular wastes, calcium and cholesterol that leads to narrowing of the arterial walls resulting in brain damage or even death. nevertheless, the significance of 1 hanvey & bunonyo / j. nig. soc. phys. sci. 4 (2022) 682 2 additional factors cannot be neglected but the diet contributing to atherosclerosis is turning out to be a grave concern as the uncontrolled intake of greasy and oily food enhances the risk of developing such chronic diseases. over the decades, a lot of mathematicians and scientists have done ample of research on the blood flow with heat transfer in human circulatory system and have discovered a handful of useful results. some of them supporting this research work are bunonyo et al. [1] studied the impact of treatment parameter on blood flow in an atherosclerotic artery. bhatti et al. [2] examined the heat transfer analysis on peristaltic induced motion of particle fluid suspension with variable viscosity: clot blood model. the influence of blood flow in large vessels on temperature distribution in hyperthermia was analyzed by lagendijk [3]. srivastava [4] discussed the analysis of flow characteristics of blood flowing through an inclined tapered porous artery with mild stenosis under the influence of inclined magnetic field. bunonyo and amos [5] did the research on treatment and radiation effects on oscillatory blood flow through a stenosed artery. the study of slip effects on the unsteady mhd pulsating blood flow through porous medium in an artery under the effect of body acceleration was carried out by eldesoky [6]. makinde and mhone [7] studied the heat transfer to mhd oscillatory flow in a channel filled with porous medium. unsteady heat transfer to oscillatory flow through a porous medium under slip condition was investigated by hamza et al. [8]. choudhury and das [9] analyzed the heat transfer to mhd oscillatory visco elastic flow in a channel filled with porous medium. biswas and chakraborty [10] gave the pulsatile flow of blood in a constricted artery with body acceleration. hanvey et al. [11] developed a model on heat and mass transfer in oscillatory flow of a non-newtonian fluid between two inclined porous plates placed in a magnetic field. plourde et al [12] looked into alterations of blood flow through arteries following atherectomy and the impact on pressure variation and velocity. mathematical modeling of blood flow through vertebral artery with stenosis was studied by ali et al. [13]. mathur and jain [14] discussed the mathematical model of nonnewtonian blood flow through artery in the presence of stenosis. ali and asghar [15] studied the oscillatory channel flow for non-newtonian fluid. the study of heat transfer in the flow of blood has gained much importance due to its significant role in medical sciences and in the research field as well. many researchers such as bunonyo, cookey and amos [16] studied the modeling of blood flow through stenosed artery with heat in the presence of magnetic field. ali and ahmad [17] gave an analytical solution of unsteady mhd blood flow and heat transfer through parallel plates when lower plate stretches exponentially. the effect of radiative heat and magnetic field on blood flow in an inclined tapered stenosed porous artery was studied by abubakar and adeoye [18]. lavanya [19] made an analytical study on mhd rotating flow through a porous medium with heat and mass transfer. rawat et al. [20] considered the effect of magnetic field on oscillatory blood flow in multi stenosed artery. varshney, katiyar and kumar [21] studied the effect of magnetic field on the blood flow in artery having multiple stenosis. elangovan and selvaraj [22] gave the study of multiple stenosed artery with periodic body acceleration in presence of magnetic field. transport of mhd couple stress fluid through peristalsis in a porous medium under the influence of heat transfer and slip effects was investigated by sankad and nagathan [23]. tripathi and sharma [24] studied the effects of variable viscosity on mhd inclined arterial blood flow with chemical reaction. modelling of arterial stenosis and its applications to blood flow was given by pralhad and schultz [25]. kot and elmaboud [26] studied the unsteady pulsatile fractional maxwell viscoelastic blood flow with cattaneo heat flux through a vertical stenosed artery with body acceleration. the study of efficient hybrid block method for numerical solution of second order partial differential problems via the method of lines was done by olaiya, azeez and modeebei [27]. in the present paper, the influence of treatment parameter on the blood flow with heat transfer is examined. it has been seen that the unsteady flow of blood in stenosed artery with heat transfer placed in magnetic field has gained much importance as it can be a treatment to the fatty substances stuck at the walls of the arteries which increases the risk of having atherosclerosis, hence giving rise to such research using mathematical model. after studying the available literature, we have considered the effect of treatment parameter on the flow of blood through an atherosclerotic artery in the presence of magnetic field and heat transfer. the equations which govern the flow field are solved by using non dimensional parameters and a graphical approach has been studied which can be very useful in detection of heart related diseases which can be treated an early stage to avoid serious medical condition. 2. material and methods figure 1. geometry of the problem let us consider blood to be newtonian, unsteady, electrically conducting, incompressible and viscous flowing past an atherosclerotic artery that is presumed to be a cylindrical polar channel w′r′, x′), where r′ and x ′ are the direction of the flow. this type of flow arises due to the pumping of the blood in the region of the heart (especially when the contraction of the heart takes place). furthermore, the pressure gradient is in the horizontal direction and the magnetic field is taken perpendicularly to the direction of the flow of blood. the flow field is governed by the following equations which were given by bunonyo and 2 hanvey & bunonyo / j. nig. soc. phys. sci. 4 (2022) 682 3 amos [1] are: ρ ∂w′ ∂t′ = − ∂p′ ∂x′ + µ r′ ∂ ∂r′ ( r′ ∂w′ ∂r′ ) − µ k′ w′ −σb20w ′ + ρβt (t ′ − t8)g (1) ∂t′ ∂t′ = kt ρc p ( ∂2t′ ∂r′2 + 1 r′ ∂t′ ∂r′ ) − q′r ρc p (t′ − t′8) (2) the atherosclerosis region is supposed to be: r′ = r0 − δ′ 2 ( 1 + cos2 πx′ λ ) (3) where x′ = d0 + l0 2 (4) the boundary conditions are: w′ = 0 , t′ = tw at r ′ = 0 w′ = 0 , t′ = t8 at r ′ = r (5) in order to write the governing equations in dimensionless form, some non-dimension variables are being introduced: w = w′ u0 , t = t′u0 r0 , da = k′ r20 , re = u0r0 v , m = b0r0 √ σ µ , p = r20 p ′ ρvλu0 , x = x′ λ , δ′ = 2δr0 rt , r = r′ r0 , gr = gβt r20 vu0 t8, rd = q′r r 2 0 µc p , pr = µc p kt (6) the equations (1) to (4) are solved using dimensionless parameter in (6), hence the following equations are obtained after dropping the primes: re ∂w ∂t = − ∂p ∂x + 1 r ∂ ∂r ( r ∂w ∂r ) − 1 da w − m2w + grθ (7) pr ∂θ ∂t = ( ∂2θ ∂r2 + 1 r ∂θ ∂r ) − rdprθ (8) r = 1 − δ rt (1 + cos2πx ) (9) where x = 1 λ ( d0 + l0 2 ) the boundary conditions in dimensionless form are: w = 0, θ = 0 at r = 0 w = 0, θ = 1 at r = h (10) where re, da, m, rd, gr, pr are reynolds number, darcy’s number, hartmann number, radiation parameter, grashof number and prandtl number respectively. 3. method of solution the flow of blood through the arteries and veins is generally governed by the pumping action of the heart which in turn gives rise to oscillatory flow of blood, therefore the solution can be assumed to be of the form: w = w0e iωt (11) θ = θ0e iωt (12) the pressure gradient can be represented by: ∂p ∂x = −p0e iωt (13) where p0 is the pressure constant and ω represents the angular frequency of the oscillations. now putting equations (10-12) into equations (7-9), we get: 1 r ∂ ∂r ( r ∂w0 ∂r ) − ( 1 da + m2 + reiω ) w0 = −p0 − grθ0(14) ( ∂2θ0 ∂r2 + 1 r ∂θ0 ∂r ) −η2θ0 = 0 (15) where η2 = (rd + iω)pr the corresponding boundary conditions can be given as: w0 = 0 , θ0 = 0 at r = 0 w0 = 0, θ0 = e −iωt at r = h (16) where β2 = h2 (rd + iω) pr applying the transformation ξ = r/h in the equation (15) and (16), we have: ∂2θ0 ∂ξ2 + 1 ξ ∂θ0 ∂ξ −β2θ0 = 0 (17) w0 = 0, θ0 = 0 at ξ = 0 w0 = 0, θ0 = e −iωt at ξ = 1 (18) hence equation (17) can be re-written as: ξ2 ∂2θ0 ∂ξ2 + ξ ∂θ0 ∂ξ −β2ξ2θ0 = 0 (19) the above equation represents a modified bessel differential equation therefore its solution can be given as: θ0 (ξ) = a i0 (βξ) + b k0 (βξ) (20) where i0 and k0 are modified bessel function of zero order. here b = 0, as the solution is bounded at r = 0, hence otherwise θ0 (ξ) will not be finite. therefore equation (20) will be reduced to: θ0 (ξ) = a i0 (βξ) (21) 3 hanvey & bunonyo / j. nig. soc. phys. sci. 4 (2022) 682 4 solving (21) subject to conditions given in the equations (18), we have: θ0 (ξ) = i0 (βξ) i0(β) e−iωt (22) hence substituting equation (22) into equation (12), we have: θ (ξ) = i0 (βξ) i0(β) (23) again, applying the transformation ξ = r/h in the equation (14) and using (23), we have: ∂2w0 ∂ξ2 + 1 ξ ∂w0 ∂ξ −ϕ w0 = χ (24) where ϕ = h2 ( 1 da + m 2 + reiω ) and χ = h2 ( −p0 − gr i0 (βξ) i0 (β) e−iωt ) hence the homogeneous solution of equation (24) is: ∂2w0 ∂ξ2 + 1 ξ ∂w0 ∂ξ −ϕ w0 = 0 (25) equation (25) can be re-written as: ξ2 ∂2w0 ∂ξ2 + ξ ∂w0 ∂ξ −ϕ ξ2w0 = 0 (26) equation (26) represents a modified bessel differential equation therefore its solution can be given as: w0h (ξ) = c1 i0 (√ ϕ ξ ) + c2 k0 ( √ ϕ ξ) (27) here c2 = 0, as the solution is bounded at ξ = 0, therefore equation (27) will be reduced to: w0h (ξ) = c1 i0 (√ ϕ ξ ) . (28) now solving equation (24) to the particular solution by using the method of undetermined coefficients, we obtain: w0p (ξ) = h2 p0 ϕ + gr h2 i0 (βξ) ϕi0(β) e−iωt (29) further the general solution is: w0g (ξ) = w0h (ξ) + w0p (ξ) (30) putting the values of equations (28) and (29) into (30), we get: w0g (ξ) = c1 i0 (√ ϕ ξ ) + h2 p0 ϕ + gr h2 i0 (βξ) ϕi0(β) e−iωt (31) substituting from (18) into (31) to calculate the value of c1, we have: c1 = − p0h2 ϕ − grh2e−iωt ϕi0(β) (32) keeping the value of c1 in equation (31), we have: w0 = p0h2 ϕ {( 1 − i0 (√ ϕ ξ )) + grh2e−iωt ϕi0(β) ( i0 (βξ) − i0 (√ ϕ ξ ))} (33) the final form of general equation of the velocity profile by using equation (33) in (11), we obtain: wξ =[ p0h2 ϕ {( 1 − i0 (√ ϕ ξ )) + grh2e−iωt ϕi0(β) ( i0 (βξ) − i0 (√ ϕ ξ ))}] eiωt (34) the expressions representing the profile for temperature and velocity are given by the equations (23) and (34) respectively. 4. results and discussion this study is carried out in view of combined effects of various parameters on blood flow flowing past an atherosclerotic artery. authors have derived the numerical results for velocity profile and temperature profile. the results calculated above are obtained for different values of parameters like radiation parameter, prandtl number, treatment parameter, reynolds number, darcy number, grashoff number, height of the stenosis and hartmann number. in this paper, the velocity profile is calculated for rd, rt , pr, m, t, gr, da, δ,ω and re as it is shown in figures 1 to 10. it is clear from figure 1 that the impact of radiation parameter dominates the flow of blood and as the value of rd increases the velocity takes a decreasing pattern. in addition, the velocity increases and attains a maximum position for different values of radiation parameter, viz, rd = 0.2, 0.4, 0.6, 0.8, 1.0 and then it starts to decrease in a converging manner and comes down to zero as r approaches 1. figure 2 is obtained for various values of prandtl number. on increasing the prandtl number (pr = 3, 5, 7, 9, 11) an increase in the blood velocity is observed. furthermore, the amplitude of the velocity profile keeps on increasing and reaches a maximum point and then it starts to decrease and converges at a same point. in figure 3, the treatment parameter highly impacts the flow of blood as at first the velocity shows pulsating character for rt = 0.1 and then starts decreasing as the parameter rt increases in the arterial channel. the treatment parameter controls the plague that is build up in the arterial section due to the consumption of fatty and oily food. the results obtained shows an obvious reading according to the physical behavior of human body and the velocity of the blood depends on the doses of treatment parameter values. figure 4 expresses the curve for various values of reynolds number (re = 5, 10, 15, 20, 25). the velocity of the blood takes a curvilinear pattern and the velocity decreases in a regular trend as the reynolds number increases. the graph attains a maximum position at r = 0 and gradually starts to drop and further converges to a point r = 1. figure 5 illustrates that the growth of plague matter can cause many cardiovascular diseases which is quite clear from 4 hanvey & bunonyo / j. nig. soc. phys. sci. 4 (2022) 682 5 figure 2. effect of radiation rd on blood velocity w(r, t) with other parameters values pr = 21, re = 2, rt = 0.5, gr = 15 and variation of r figure 3. effect of prandtl number pr on blood velocity w(r, t) with other parameters values rd = 2, re = 2, rt = 0.5, gr = 15, t = 2 and variation of r figure 4. effect of treatment parameter rt on blood velocity w(r, t) with other parameters values rd = 2, re = 2, m = 2, gr = 15, t = 2 and variation of r the graph as the height of the stenosis increases the blood velocity declines. this retardation of blood in arteries and veins gives figure 5. effect of treatment parameter rt on blood velocity w(r, t) with other parameters values rd = 2, rt = 0.5, m = 2, gr = 15, t = 2 and variation of r figure 6. effect of height of steonsis δ on blood velocity w(r, t) with other parameters values rd = 2, rt = 0.5, m = 2, gr = 15, t = 2 and variation of r figure 7. effect of darcy number da on blood velocity w(r, t) with other parameters values rd = 2, rt = 0.5, m = 2, gr = 15, t = 2 and variation of r a crystal clear proof that the glands, organs and tissues of the human body gets famished from oxygen rich blood which ultimately results in hypertension and other cardiovascular prob5 hanvey & bunonyo / j. nig. soc. phys. sci. 4 (2022) 682 6 figure 8. effect of hartmann number m on blood velocity w(r, t) with other parameters values rd = 2, rt = 0.5, pr = 21, gr = 15, t = 2 and variation of r figure 9. effect of oscillatory frequency ω on blood velocity w(r, t) with other parameters values rd = 2, rt = 0.5, pr = 21, gr = 15, t = 2 and variation of r lems. the result comes out be very obvious from the fact that velocity becomes maximum at the midpoint and minimal at both the ends. figure 6 explains the effect of darcy’s number of the flow of blood as the presence of porosity directly effects the flow. as the value of the parameter increases (da = 0.05, 0.06, 0.07, 0.08, 0.09) the velocity shows an increasing pattern which reaches to an amplitude and then starts to converge at r = 1. the application of applied magnetic field holds utmost importance in the study of blood flow as it can be a cure to many heart related diseases. it is examined in figure 7 that as the intensity of the magnetic is increased the blood velocity is decreased. it is because when the magnetic field is applied on the blood flow which is electrically conducting in nature, generates lorentz force which makes the velocity of blood in arteries and veins to retard. figure 8 represents the effects of oscillatory frequency pafigure 10. effect of grashoff number gr on blood velocity w(r, t) with other parameters values rd = 2, rt = 0.5, pr = 21, da = 0.05, t = 2 and variation of r figure 11. effect of time parameter t on blood velocity w(r, t) with other parameters values rd = 2, rt = 0.5, pr = 21, da = 0.05, gr = 15 and variation of r figure 12. effect of prandlt number pr on temperature profile θ(r, t) with other parameters values rd = 2, rt = 0.5, t = 2, δ = 0.2 and variation of r rameter of the velocity of blood. as the frequency parameter increases (ω = 0.5, 0.6, 0.7, 0.8, 0.9) the velocity also adopts an increasing pattern and every time the amplitude keeps on escalating for higher values of frequency parameter. in figure 9, the influence of grashoff number is observed and its impact on 6 hanvey & bunonyo / j. nig. soc. phys. sci. 4 (2022) 682 7 figure 13. effect of treatment parameter rt on temperature profile θ(r, t) with other parameters values rd = 2, , t = 2,ω = 2 δ = 0.2 and variation of r figure 14. effect of radiation parameter rd on temperature profile θ(r, t) with other parameters values rt = 0.5, δ = 2, t = 2,ω = 2, and variation of r figure 15. effect of height of stenosis δ on temperature profile θ(r, t) with other parameters values rt = 0.5, rd = 2, t = 2,ω = 2, and variation of r the velocity of blood is perceived. as the grashoff number increases (gr = 5, 10, 15, 20, 25), the velocity takes an increasing trend and the amplitude uniformly increases. figure 10 explains the effect of time ‘t′ on the flow of blood as the value of t is increased the value of velocity also increases which further takes a maximum value and then converges to a point at r = 1. the influence of other parameters on the temperature profile can be seen from figure 11 to figure 16. the parameters figure 16. effect of oscillatory frequency ω on temperature profile θ(r, t) with other parameters values rt = 0.5, rd = 2, t = 2 and variation of r affect the temperature profile by increasing and decreasing it at different levels. 5. conclusion in the present study, we have investigated the influence of treatment parameter on the blood flowing past an atherosclerotic artery with heat transfer. on solving the mathematical model and carrying out the graphical study using the software mathematica and furthermore the results have been discussed. the main conclusions of this work are as follows: it is observed that the velocity takes a oscillatory pattern with increasing values of radiation parameter rd, however the effect of other parameters can’t be ignored. the effect of prandlt number causes the velocity of blood to increase as pr increases. the treartment parameter causes the flow of blood to retard which is due to the magnetic field that is applied on the surface portion however the contribution of other parameters cannot be neglected. the height of the stenosis is responsible for the decreases in the blood flow through arteries which inturns lowers the oxygen level in the human body leading to plethora of cardiovascular issues. the velocity of the blood decreases as the reynolds number increases while other parameters are held. the magnetic field intensity decelerates the blood flow in the human system which can be very useful in the detection of formation of plagues in arteries and veins, detection of tumors through mri scan and detects the diseases in liver and bile duct. the porosity factor holds great importance in circulation of blood flow through arteries as the porosity factor increases the velocity of blood increases and healthy oxygenated blood can reach different organs and tissues. therefore the level of porosity factor can be maintained by implication of treatment parameter. the velocity of the blood increases as frequency ω, grashof number gr, and time t increases. we can also conclude that the curvilinear pattern of the figures is because of the oscillatory 7 hanvey & bunonyo / j. nig. soc. phys. sci. 4 (2022) 682 8 nature of the blood flow in arteries and veins; however the presence of some prominent parameter affecting the flow of blood cannot be 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[24] b. tripathi & b. k. sharma, “effects of variable viscosity on mhd inclined arterial blood flow with chemical reaction”, international journal of applied mechanics and engineering 23 (2018) 767. [25] r. n. pralhad & d. h. schultz, “modelling of arterial stenosis and its applications to blood flow”, mathematical biosciences 190 (2004) 203. [26] m. a. e. kot & y. a. elmaboud, “unsteady pulsatile fractional maxwell viscoelastic blood flow with cattaneo heat flux through a vertical stenosed artery with body acceleration”, journal of thermal analysis and calorimetry (2021), https://doi.org/s10973-021-10822-2. [27] o. o. olaiya, r. a. azeez & m. i. modeebei, “efficient hybrid block method for numerical solution of second order partial differential problems via the method of lines”, journal of the nigerian society of physical sciences 3 (2021) 26, https://doi.org/10.46481/jnsps.2021.140. 8 j. nig. soc. phys. sci. 4 (2022) 223–230 journal of the nigerian society of physical sciences statistical modelling by topological maps of kohonen for classification of the physicochemical quality of surface waters of the inaouenwatershed under matlab r. el chaal∗, m. o. aboutafail engineering sciences laboratory. data analysis, mathematical modeling, and optimization team, department of computer science, logistics and mathematics, ibn tofail university national school of applied sciences ensa, kenitra 14 000 morocco abstract self-organizing maps (soms) and other artificial intelligence approaches developed by kohonen can be used to model and solve environmental challenges. to emphasize the classification of physico-chemical parameters of the inaouen watershed, we presented a classification strategy based on a self-organizing map (som) artificial neural network in this study. the use of a self-organizing map to classify samples resulted in the following five categories: low quantities of sodium na (mg/l), potassium k(mg/l), magnesium mg(mg/l), calcium ca(mg/l), sulfates so4(mg/l), and total dissolved solids tds (mg/l) distinguish classes 2 and 3. bicarbonate hco3 (mg/l), total dissolved solids tds (mg/l), total alkalinity caco3(mg/l), mg(mg/l), calcium ca (mg/l), and electrical conductivity cond (s/cm) are slightly greater in classes 1 and 4. except for dissolved oxygen d.o. (mg/l) and nitrate no3(mg/l), class 5 has exceptionally high values for all metrics. the results suggest that kohonen’s self-organizing maps (som) classification is an outstanding and fundamental tool for understanding and displaying the spatial distribution of water physicochemical quality. doi:10.46481/jnsps.2022.608 keywords: classification, self-organizing maps, som, physical-chemical parameters, cluster article history : received: 24 january 2022 received in revised form: 29 march 2022 accepted for publication: 30 march 2022 published: 29 may 2022 c©2022 journal of the nigerian society of physical sciences. all rights reserved. communicated by: b. j. falaye 1. introduction self-organizing maps (som) are artificial neural network techniques based on unsupervised learning algorithms [1]. because of their classification capabilities and visualization performance, they have been successfully used in environmental fields (soil,water, air, etc.). shen et al. [2] studied the groundwater chemistry and quality of a multilayered groundwater system in the northwest china coal an approach based ∗corresponding author tel. no: +212710115110 email address: rachid.elchaal@uit.ac.ma ( r. el chaal ) on the som technique combined with multivariate statistical tools. bigdeli et al. [3] applied self-organizing map (som) technique and k-means clustering algorithms to describe geochemical anomalies in moalleman district, northeast iran. santos et al. [4] investigated the regional hydrogeochemical spatialization and controls of the serra geral aquifer system in the southern region of brazil, using kohonen’s self-organizing maps (som) in combination with k-means clustering. amiri et al. [5] conducted a spatio-temporal assessment of groundwater quality in a coastal aquifer, based on kohonen’s linear discriminant analysis (lda) and self-organizing maps (som). 223 el chaal & m.o.aboutafail / j. nig. soc. phys. sci. 4 (2022) 223–230 224 the goal of this study is to use a statistical technique (som) to display and analyze the spatial distribution of water samples and their physico-chemical properties at the level of the inaouen watershed. to emphasize the distinct classes and detect the spatial fluctuations of the physico-chemical characteristics of the examined watershed, som’s hierarchical classification (som-cha) is utilized. 2. materials and methods 2.1. sample location and description the oued inaouene watershed is located in northeastern morocco, between the middle atlas and the pre-rif, and covers approximately 5109 km2 (figure 1. the watershed has a mediterranean climate with oceanic influences, with great seasonal variations and very evident abnormalities in rainfall due to its geographic location [6]. with an average annual rainfall of roughly 600 mm/year, this region is known for its very varied rainfall from month to month. the rainfall distribution reveals two bands: the first below 500 meters with an annual rainfall of roughly 800 mm, and the second between 500 and 1000 meters with an annual rainfall of 800 to 1500 mm. the rainy season in the study region lasts from november to april, with december and january being the wettest months. july and august, on the other hand, are the driest months of the year. the oued inaouène watershed is distinguished by a lithological contrast between the two banks. the pre-rif, whose outcrops are marly and support a clear sown flora, is on the right bank. the left bank corresponds to the middle atlas’ northern boundary, with outcrops ranging from tazzeka’s paleozoic formations to the triasi formations [6]. figure 1: location and dtm of the study area. 2.2. data source the dataset for this study consists of 16 physico-chemical characteristics measured on 100 surface water samples taken in the inaouen watershed between 2014 and 2015. the processes for collecting, transporting, and stocking water samples is done by our team according to the protocol of the national office of drinking water. a portion of the analyses was completed on-site, while the remainder was completed at the curi laboratory in fezby our team [7]. the parameters used are ph,bicarbonate (hco3), dissolved oxygen (oxy. diss), conductivity (cond),temperature (t◦c), total dissolved solids (tds), total alkalinity (caco3),potassium (k), magnesium (mg), sodium (na), chlorides (cl), calcium (ca), sulfates (so4), nitrate (no3), phosphorus (p) and ammoniac (nh3). 2.3. self-organizing maps (som) kohonen, who was looking for a technique to represent huge multidimensional data, was the first to introduce topological maps. to do this, kohonen uses machine learning[8]to divide data into ”similar” groups whose neighborhood structure can be realized and represented using a discrete low-dimensional (1, 2, or 3d) space called a ”topological map” [9]. topological map methods allow data to be projected onto a low-dimensional environment while revealing the data’s inherent structures. as a result, the som technique preserves both the topology of the data and the distance relationship between them (figure 2). unsupervised learning artificial neural network methods[10,11], often known as som maps, are a type of neural network [12]. the samples (input vectors) are supplied to a grid of d neurons (or nodes, or units) in these networks [13]. the choice of the parameter d (the map’s dimension) is chosen ahead of time. the input vectors are connected to each grid neuron via d synapses and d weight vectors w. the map neuron in data space is represented by the vector w, which is also known as the prototype or referent vector [14]. the bmu is the closest data referent (the best matching unit). the quantization and topological preservation capabilities of a self-organizing map are assessed. topological error (te) and quantization error (qe) are commonly used to validate the som categorization [15,16]. quantization error qe: (or resolution measure) the average quantization error, which is defined by the average distance of the data from their referents(bmu), is used to determine the degree of deployment of the map on the data or the degree of quantization [1,14]. the better the quality of the som algorithm, the lower the value of qe. it is expressed as follows: qe= 1 n n∑ k=1 ∥∥∥∥x(k)−w (x(k))∥∥∥∥2 (1) where n is the number of data, x(k) is the k-th individual, and w ( x(k) ) is the bmu of the individual x(k) topographic error te [16]: (or a measure of topology preservation) this criterion assesses how well the som preserves the data set’s topology [17]. which is the percentage of data for which the two nearest referents do not correspond to adjacent 224 el chaal & m.o.aboutafail / j. nig. soc. phys. sci. 4 (2022) 223–230 225 map units [18]. in contrast to the quantification mistake, te considers the som card’s structure [19]. the criterion used to measure the number of observations where the first winning neuron (ci) and the second winning neuron (si) are not neighbours on the map. the second winning neuron of observation has its closest referent vector to this observation after the first winning neuron [20]. the topographic error is a metric that is determined as follows: t e= ∑n i=1 e n (2) e= { 1 si rci−rsi 2,1 0 si rci−rsi 2= 1 where rc and rs are respectively the locations of neuron c and neuron s on the map. the topology is perfectly preserved when this criterion is 0. figure 2: structure of a topological map the topological map has several advantages over the linear and classification methods[21] usually used to extract groups of collected samples, such as principal component analysis (pca) [22], correspondence analysis (c.a.) and hierarchical clustering (h.c.)[23]. their limitations are well known. for example, for each of them, a strong distortion is observed when there are non-linear relationships between the variables [24]. 2.4. hierarchical clustering by som (som-cha) like other data analysis methods, of which it is a part, the som-cha classification aims to obtain a simple schematic representation [25]. it consists in calculating a matrix expressing the mutual distances between the points to be classified, which are the nodes of the map, and then, based on this matrix, grouping together the closest points. this method allows the construction of a hierarchical tree [26], which reveals several possible partitions, where each point is assigned to one of the groups of a given partition. the choice of the best partition is made once the hierarchical classification is completed [27]. 2.5. algorithm: kohonen maps (som) [16] let ( wt1, . . . ,w t n ) ∈(rn)n be neurons of the vector space rn. we designate by v ( w j ) the set of neighbouring neurons ofw j for this kohonen card. by definition, we have w j∈v ( w j ) .let (x1, . . . ,xk )∈(rn)k a cloud of points. we use a sequence of positive real numbers (αt) checking ∑ t≥0 α 2 t < ∞ and ∑ t≥0 αt < ∞. initialization the neurons ( w01, . . . ,w 0 n ) are distributed in the space rn in a regular way according to the shape of their neighbourhood. t←−0. closest neuron we choose a point of the cloud xi randomly; then, we define the neuron wtk∗, so that: w t k∗−xi =min16 j6n w t j−xi . update for each wtj in v ( wtk∗ ) wt+1j ←−w t j+αt ( xi−w t+1 j ) (3) t←−t+1 as long as the algorithm has not converged, return to the nearest neuron step. the update step can be modified to improve the convergence speed [28] : wt+1j ←−w t j+αth ( wtj, w t k∗ ) wk ( xi−w t+1 j ) (4) where h is a function with value in the interval [0,1] which is 1 when wtj=w t k′ . and that decreases when the distance between these two neurons increases. a typical function is : hck(σ(t)) = exp ( − d22 (rc,rk ) 2σ2 (t) ) = exp ( − ‖rc−rk‖2 2σ2 (t) ) (5) where rc and rk are respectively the locations of neuron c and neuron k on the map, and σ(t) is the radius of the neighbourhood at iteration t of the learning process. kohonen maps are used in data analysis to project a point cloud into a two-dimensional space in a non-linear manner using a rectangular neighbourhood. they are also used to perform unsupervised classification by clustering neurons where the points are concentrated. the edges connecting the neurons or vertices of the kohonen map are either narrowed to indicate that two neurons are neighbours or distended to indicate a separation between classes. 3. results and discussion 3.1. classification of surface water samples by som the concept of the som algorithm is to conduct a nonlinear classification of complicated datasets by recognizing similar patterns. in this work, the input layer consists of vectors representing individuals., each of which contains 16 components representing the 16 physico-chemical parameters of the surface waters studied. the output layer is composed of 225 el chaal & m.o.aboutafail / j. nig. soc. phys. sci. 4 (2022) 223–230 226 100 neurons (10 rows × 10 columns). this size was chosen for the output map because it minimizes the two error criteria (qe=0.268 and te=0.03). the som component planes of the data set allow distinguishing two types of colors; dark red cells represent high values, while blue cells represent low values (figure 3) [16]. the similar colors between the variables correspond to a positive correlation; this can be illustrated between the variables bicarbonates (hco3), chlorides (cl), magnesium (mg), sodium (na), sulfates (so4), conductivity (cond), nitrate (no3), ammoniacal nitrogen (nh3), calcium (ca), total alkalinity (in caco3), potassium (k) and total dissolved solids (tds). on the other hand, dissolved oxygen (oxy. diss) and p.h., total alkalinity (caco3), and phosphorus (p) show a negative correlation. the other variables, especially t(◦c),p, and no3, show neither positive nor negative correlations and vary autonomously fromthe others (figure 3). figure 3: gradient of values of physico-chemical parameters on the kohonen 3.2. principal component analysis (pca) the pca result shows the score road composed of the two components, pc1 and pc2, which are regarded as the most informative ones since they contribute to most of the variance. in our case, pc1 and pc2 respectively contribute 51.1% and 11.2% of the total variance. therefore, the first two components of the circle of correlations between variables in the subspace pc1 vs pc2 contain 62.3% of the data. figure 4 illustrates the circle of correlations between the variables in the factor plane (pc1 × pc2). the correlation circle between variables in the factorial diagram (pc1 × pc2) shows that the variables mg, hco3, caco3, k, so4, na, cl, nh3, and tds are positively correlated with the pc1 axis with coefficients above 0.6. however, the element oxy diss is negatively correlated with the pc2 axis, with coefficients greater than −0.6. the elements mg, hco3, caco3, k, so4, na, cl, nh3, cond and tds are therefore positively correlated with each other. the other variables do not have positive or negative correlations and are poorly represented in the circle, in particular the parameters t◦, p, no3 and p.h. that vary autonomously. figure 4: pca circles of correlation (pc1 × pc2) 3.3. som-cha hierarchical classification once the kohonen map is obtained, we use a hierarchical classification based on the ward method [29,30]and euclidean distance. the hierarchical classification by som allows grouping the cells of the som map into groups of physico-chemical parameters of the inaouen watershed. the dendrogram obtained by som-cha suggests that the 100 neurons should be grouped into five classes (figure 5). the first class contains 13 samples and represents 13% of the total data, it includes waters with average chemical element concentration respectively (hco (105.39 mg/l), tds (100.54 mg/l) caco3(86.38 mg/l), mg (4.41mg/l), ca (24.69 mg/l) and electrical conductivity (201.08 µs/cm)) which are a little high and (nh3 (31.23 µg/l) and (na (11.60 mg/l) which are low. the second class includes the largest number of samples (48) and represents 48 % of the total database. it is characterized mainly by low concentrations of chemical elements such as: na (9.00 mg/l), k (1.02 mg/l), so4(4.61 mg/l) and tds (58.33 mg/l) and by high dissolved oxygen concentration (6.24 mg/l). the third class contains 13 samples and represents 13% of the total database. it is characterized mainly by low concentrations of chemical elements (so4 (4.87 mg/l), cl (3.19 mg/l) and na (4.91 mg/l)), mg (1.81 mg/l), k (0.84 mg/l), ca (9.95 mg/l), nh3 (17.69 mg/l), and tds (43.38 mg/l) and a very high concentration of p(225.39mg/l) and dissolved oxygen(6.30 mg/l). the fourth class contains 23 samples that represent 23 % of the database. it is characterized by medium high concentrations of chemical elements hco3 (117.76 mg/l), cl (47.98 mg/l), mg (4.85 mg/l), caco3 (96.52 mg/l), ca (29.83 mg/l), na (38.90 226 el chaal & m.o.aboutafail / j. nig. soc. phys. sci. 4 (2022) 223–230 227 figure 5: classification tree (dendrogram) of the physico-chemical parameters of the surface waters of inaouen watershed accessed with the method of topological maps (som) mg/l), nh3(74.17 mg/l), electrical conductivity (375.87 µs/cm) and tds (188.74 mg/l) finally, the fifth class includes the smallest number with 3 samples that represent 3% of the database. it is characterized by very high concentrations of chemical elements nh3(770 mg/l), electrical conductivity (1801.33 µs/cm), tds (901 mg/l), hco3 (435.13 mg/l), na (303.33 mg/l), cl (47.98 mg/l), mg(4.85 mg/l), caco3 (356.67 mg/l), ca (31.40 mg/l), k (9.70 mg/l), so4 (59.67 mg/l) (table 1 and figures 6-8. this enrichment would probably have a triple origin, on the one hand, the geological nature of the crossed grounds, the difference of altitude and on the other hand, the domestic discharges of the neighbouring communes. figure 6: picture of the five clusters with the samples accessed by the som hierarchical clustering. figure 7: projection of the samples relative to the five clusters on the plan factorial pc1 × pc2 figure 8: the five clusters generated by the som hierarchical clustering are illustrated. the number of patterns given to each neuron is indicated. table1.statistics on-base quantities (min, mean, maximum) for physico-chemical parameters, respectively, for the whole database and for classes 1, 2, 3, 4 and 5. 227 el chaal & m.o.aboutafail / j. nig. soc. phys. sci. 4 (2022) 223–230 228 table 1: statistics on-base quantities (min, mean, maximum) for physico-chemical parameters,respectively, for the whole database and for classes 1, 2, 3, 4 and 5. stat of all data stat of data from cluster 1 stat of data from cluster 2 name min mean max min mean max min mean max t ◦c 16,3 19,9 23 17,1 18,9846 21 16,3 19,7042 23 ph 5,21 7,1923 9,61 6,36 7,56462 9,42 5,21 7,04458 9,61 oxydiss 0,09 5,0408 14,01 0,08 2,85 7,76 0,08 6,24229 13,26 cond 20 233,69 2959 134 201077 315 20 116729 629 tds 45 117,15 1480 67 100538 157 10 58,3333 314 hco3 3,66 80,6298 634,4 68,32 105389 170,8 6,1 45,1654 170,8 caco3 3 66,09 520 56 86,3846 140 5 37,0208 140 mg 0,2 3,1735 25 1 4,40769 11 0,2 1,82542 6,8 na 1 24513 540 2,4 11,6 31 1 9,00208 76 k 0,1 1,7211 13 0,34 1,63615 4 0,1 1,01563 6,4 cl 0,4 25,8394 550 0,6 5,98462 22 0,4 8,62792 99 ca 1 17794 110 3 24,6923 54 1 11,4333 45 so4 275 7,83175 150 1,65 7,08462 13 275 4,60781 15 no3 0,02 0,4805 4,8 0,1 1,15846 4,8 0,02 0,464167 4,5 p 10 58,5 900 20 30,7692 80 10 33,3333 80 nh3 11 62,93 1000 11 31,2308 130 11 34,1875 130 stat of data from cluster 3 stat of data from cluster 4 stat of data from cluster 5 name min mean max min mean max min mean max t ◦c 18 20,8231 23 17,5 20,3174 23 19 19,8 21 ph 5,41 6,70692 8,86 5,63 7,44826 9 6,62 8,08333 9,04 oxydiss 1,25 6,3 14,01 1.32 3,59913 10,24 0,7 0,906667 2,2 cond 0.09 84,8462 198 100 375,87 1118 980 1801,33 2959 tds 23 43,3846 99 50 188739 559 490 901 1480 hco3 7,32 39,3215 82,96 3,66 117757 329,4 231,8 435133 634,4 caco3 6 32,2308 68 3 96,5217 270 190 356667 520 mg 0,53 1,80923 4,5 0,61 4,84826 23 3,1 12,4667 25 na 1,5 4,90769 13 1,4 38,8957 180 140 303333 540 k 0,16 0,84 2,4 0,43 2,6987 7,6 6,8 9,7 13 cl 0,7 3,19231 11 0,4 47,9783 260 77 315667 550 ca 1,9 9,95385 30 14 29,8261 110 2,4 31,4 86 so4 1,65 4,86538 18 1,65 9,89783 37 11 59,6667 150 no3 0,03 0,133077 0,4 0,02 0,364348 2,4 0,1 0,2 0,4 p 20 225385 900 20 34,7826 80 20 40 70 nh3 11 17,6923 50 11 74,1739 250 530 770 1000 figure 9: variation of chemical elements in surface waters of the inaouen watershed figure 10: variation of the total dissolved solids and conductivity in surface waters of inaouen watershed 228 el chaal & m.o.aboutafail / j. nig. soc. phys. sci. 4 (2022) 223–230 229 figure 11: variation of the na, cl and so4 in surface waters of inaouen watershed. 3.4. the u matrix classification: for the u matrix classification map, the hexagonal topology was chosen to achieve a better resolution and speedierresults. the rectangular topology needed many fewer neurons to get a small quantization and topography error. the result of the umatrix (figure 12) from the selected som parameters. the precision of the classification via the bmus in the u-matrix is considered almost exact and produces a high quality and very smooth mapping. figure 12: u-matrix representation of som. the training samples are the 16 physico-chemical parameters taken from 100 inaouen watershed sources. 4. conclusion statistical analysis based on kohonen’s self-organizing map (som) approach was applied to a database consisting of 16 physico-chemical parameters carried out on 100 samples of the surface waters of the inaouen watershed between february 2014 and december 2015. it highlighted the different positive, and 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[29] m. roux, “a comparative study of divisive and agglomerative hierarchical clustering algorithms,” j. classif. 35 (2018) 345. doi: 10.1007/s00357-018-9259-9. [30] n. randriamihamison, n. vialaneix, and p. neuvial, “applicability and interpretability of ward’s hierarchical agglomerative clustering with or without contiguity constraints,” j. classif. 38 (2021) 363. doi: 10.1007/s00357-020-09377-y. 230 j. nig. soc. phys. sci. 4 (2022) 712 journal of the nigerian society of physical sciences effect of ph on the leaching of potentially toxic metals from different types of used cooking pots o. t. fatunsin∗, o. f. adeyeye, k. o. olayinka, t. o. oluseyi department of chemistry, university of lagos, akoka, yaba, lagos, nigeria. abstract humans are exposed to potentially toxic metals (ptms) through many routes. cooking foods in cookwares which are prone to material leaching can be an exposure route to ptms. this study assessed the effect of ph on the leaching of some ptms from used cooking pots into deionized water. series of deionized waters were prepared from ph 3 to 7. each water was brought to boil in clay, non-stick, stainless steel, cast aluminum, pressed aluminum and glass pots respectively. the ptms leached from each sample pot were determined by inductively couple plasma-optical emission spectrophotometer (icp-oes) (agilent nu7m technologies 700 series). the deionized water from the aluminum cast pot and nonstick pot gave the highest concentration of aluminum (2273 µg/l) and zinc (24.39 µg/l) respectively. while that from the clay pot gave the highest concentrations of chromium and nickel, (7.27 and 22.63 µg/l) and that from the stainless-steel pot gave the highest concentration of iron (237 µg/l) and lead (24.39 µg/l). no ptm was found in the deionized water from the glass pot. the results from this study showed more leaching of ptms into deionized water occurred more at lower phs (ph 3 to 5) than at neutral ph for almost all the pots. thus, cooking of acidic foods in pots except when glass pots are used should be avoided. the results of this study therefore reveal the health implications associated with using metal pots for cooking slightly acidic foods as metals can be easily leached from the pots into the foods. doi:10.46481/jnsps.2022.712 keywords: leaching, potential toxic metals, aluminum pot, clay pot, non stick pot, aluminum cast pot, stainless steel pot, glass pot. article history : received: 17 march 2022 received in revised form: 08 september 2022 accepted for publication: 22 october 2022 published: 27 november 2022 c©2022 journal of the nigerian society of physical sciences. all rights reserved. communicated by: edward anand emile 1. introduction potentially toxic metals (ptms) are significant environmental pollutants which by ingestion, inhalation and dermal contacts they may enter the human body. ingested metal are hazardous to humans because they tend to bio-accumulate and cause harm to internal organs [1]. their concentrations increase in biological systems over time because they are slowly metabolized or excreted and are therefore stored in the ∗corresponding author tel. no: +2348027178550. email address: ofatunsin@unilag.edu.ng (o. t. fatunsin ) system [2]. metals, unlike organic molecules, do not require bio-activation or undergo enzymatic modification to produce reactive chemical species for detoxification process [3, 4] but use other mechanisms, such as long-term storage (for example cadmium and iron), biliary and/or urinary excretion. though metal toxicity to a biological system depends on the amount ingested, exposure to certain metals, such as cadmium (cd) and lead (pb), may lead to severe toxic effects even at low concentrations [5, 6]. there are several sources/ routes of exposure to ptms [7]. leaching of ptms into food and water from cookware 1 o. t. fatunsin et al. / j. nig. soc. phys. sci. 4 (2022) 712 2 during cooking presents additional exposure to ptms. foods are usually cooked /processed in pots to make them edible, this process may leach metals into food [8]. different types of pots are currently available for cooking. they vary from locally fabricated cast pot commonly called ikoko irin and clay pots, to industrially produced aluminum pots, non-stick pots, stainless steel pots and glass-pyrex pots. the fabricated cast pots in nigeria are produced from aluminum scraps (such as cans from food packaging, roofing sheets among others) by melting and molding them in sand moulds [9] while the clay pots are molded from wet clay and fired to make them strong. they are not glazed when they are to be used for cooking [10]. street et al., [11] in south africa investigated the risk of metal exposure from the use of artisanal cookware. these cook wares were produced from metal scraps and e-waste. using xrf and inductive coupled plasma mass spectroscopy, total and leached silver (ag), asernic (as), baron (ba), cd, colbalt (co),chromium (cr), copper (cu), iron (fe), mecury (hg), manganese (mn), molybedium (mo), nickel (ni), pb, antimony (sb), selenium (se), tin (sn), vanadium (v), al and zinc (zn) were evaluated. the mean total al was 509 mg/ l and was over 100 times the eu maximum permissible level allowed for cookware. pb was detected in the leachates of the pot samples with some concentration higher than the maximum eu permissible level (10 µg/l) for 1st, 2nd and 3rd migrations respectively of pb. cd and hg were also detected in the leachates from the pots. chagas et al., [12] investigated the leaching of pb from clay pot of brazil origin using inductively coupled plasma optical emission spectroscopy (icp-oes). the concentration of pb leached was found to be higher than 2.0 mg/kg, the value regulated by the brazilian health regulatory agency and the concentration increased with increase in contact time of food with the pot. in another study by odularu et al., [13] al concentration of rice cooked in old and new aluminum, stainless steel, steel and clay pot respectively showed that al was leached from all their pot samples. the al concentrations varied between 186.83 ± 75.18 and 350 ± 130 µg/g. in 2016, ojezele et al., [14] also analyzed the level of metals (fe, zn, cd, ni, mn, cr, co, pb, cu and al) in rice cooked in iron, stainless steel, aluminum and clay pots. they also observed increased levels of metals in rice cooked in pots compared to the all glass pot used as control. kamerud et al., [15] in 2013 also found stainless steel pots leached metals. nickel and chromium increased by 24 and 35 fold respectively in tomato sauce cooked in stainless steel pots, however the amount leached reduced with subsequent use and was dependent on factors like grade of pot, length of cooking and cookware usage. lar et al., [16] found that varying amount of metal (al, ca, fe, mg and na) leached from clay pots and cast pots made in different parts of nigeria (clay-ife, ilorin, minna, sokoto, makurdi, lokoja, plateau and calabar; cast potsplateau, nassarawa, anambra , kaduna) when distilled water is boiled in them. previous researches conducted on the leaching of metals from different cook wares used specific food in their studies such as tomatoes sauce [15, 17], fish stew [12], 3% acetic acid solution [11], rice [13], vegetables [18], fruit juices [19], beans [20], except lar et al.,[16], alabi et al., [21] alabi and adeoluwa, [22] and noemie et al [17] used water. with food substances, parameters such as ph cannot be easily monitored or varied. though, lar et al., [16] used water in their study they only determined metals leached from cast and clay pots while alabi et al., [21], alabi and adeoluwa, [22] only studied aluminum pots. noemie et al [17] studied the effect of salty water and tomato juice on leaching from cast aluminum pots. so far no study has been carried out on the amount of ptms leached nor from nonstick pots and at varying ph’s in nigeria despite the presence of these pots in virtually every home. ingestion is the main route of toxic metal exposure to the human body [23] thus there is a need to study the leaching of metals to foods from cooking pots. therefore the aim of this study is to determine the effect of ph on the amount of ptms leached from different cookware pots into food during cooking. 2. materials and methods 2.1. sampling samples of used pressed aluminum, clay, stainless steel, non-stick and casted aluminum cooking pots of similar sizes (1 liter) were randomly collected from different households in lagos state, nigeria (plate 1). the pots were properly washed with soap, rinsed with tap water followed by distilled water, allowed to air dry. an all glass pyrex glass pot was used as the control for this study. 2.2. sample preparation one liter of distilled, de-ionized water were boiled in each pot (clay, non-stick, stainless steel, aluminum cast pot (koko-irin), aluminum and glass pots ) and aliquots stored in cleaned plastics storage bottles (plastics that have been soaked in 0.1 m nitric acid overnight) and analyzed within 24 hours by an icp-oes (agilent nu7m technologies 700 series) for aluminum (al), chromium (cr), iron (fe), nickel (ni), lead (pb), and zinc (zn). ph values of water samples were adjusted using a 1 molar sodium hydroxide and 1 molar hydrochloric acid respectively with a calibrated ph meter. the water samples were adjusted to ph values of 3, 4, 5, 6, and 7 respectively. these ph values were chosen to stimulate ph of food usually eaten since there is no standardized test for the study of metals leached for pot thus to simulate the cooking condition, one liter of water at ph of 3, 4, 5, 6, and 7 were brought to boil at 100◦c for 30 minutes in the different pots. 2.3. icp -oes analysis extracts from the study were kept at 4◦c for metallic element analysis using the inductively couple plasma-optical emission spectrophotometer (icp-oes) (agilent nu7m technologies 700 series). all operating parameters for the icpoes were optimized for the sample solutions which base on 2 o. t. fatunsin et al. / j. nig. soc. phys. sci. 4 (2022) 712 3 plate 1: pictures of the pots used in this study de-ionized water. plasma observation axial, nebulizer licht emodified, spray chamber cyclonic, torch injector quartz diameter, plasma power, coolant gas, auxiliary gas, nebulizer gas were all optimized. the uptake rate for solutions were set at 2.0 ml/ min, replicate read time was 45s, pre-flush time was set at 60s [24]. the analytical method was validated by using instrumental detection limit (idl), limit of detection (lod), limit of quantification (loq), precision, and accuracy studies. in this study, the idl for each metal was calculated from the analysis of seven replicates of the blank concentration. idl = 3×sbl, where sbl is the standard deviation of the seven calibrated blank solutions, lod for each metal was determined using seven replicates of method blank which were digested using the same procedures as the samples. lod = 3×sbl, where sbl is the standard deviation of the seven method blank solution and the loq is the lowest concentration of an analyte in the sample which can be quantitatively determined with some levels of uncertainty. this can be achieved in triplicate of seven method blanks which are digested as the actual samples. loq = 10×sbl), where sbl is the standard deviation of the seven method blank solution [25]. 2.4. quality control appropriate measures were taken to prevent contamination and ensure reliability of data. they include actions such as the use of glass pot as control. deionized water that was initially distilled was used throughout the experiment as blank. all glass wares were soaked overnight and rinsed with 0.1 mole hno3 and allowed to dry. they were also rinsed with the solutions to be measure in them prior to use. recovery studies were carried out using standards to spike de-ionized water for analyses and values between 75 and 100 % were obtained. 3. results and discussion the glass, aluminum, cast, stainless steel, clay and nonstick pots were used to boil distilled water at ph 3, 4 5, 6 and 7. foods are usually within the ph of 3 to 7. the amounts of ptms leached were quantified as described in the methodology and the results are as shown in table 1 and figures 1 to 5 (including tables 1s to 5s; supplementary data). 3.1. effect of ph on the amount of ptms leached from glass pot glass pot was used as control to compare with other pots which were all made of metals and the results are shown in table 1. no metal was found to leach from the glass pots (as the leached metals from the glass pot were all below the limit of detection for each metal). this may be due to the type of glass used in this study. the glass material was made of pyrex-type sodium borosilicate known to have improved thermal shock resistance, excellent weathering resistance, very low solubility, resistant to chemicals (inert) and excellent chemical durability against most leaching solution. they are commonly used for 3 o. t. fatunsin et al. / j. nig. soc. phys. sci. 4 (2022) 712 4 cookware, chemical laboratory ware, flat panel devices and fluorescent lamps and are considered as one of the best glasses. they are usually made of 80.6% sio2, 12.6% b2o3 , 4.2% na2o, 2.2% al2o3, 0.1% cao, 0.1% cl, 0.05% mgo, and 0.04% fe2o3 [26]. table 1: concentration of leached ptms from glass pot (µg/l). metals ph 3 ph 4 ph 5 ph 6 ph 7 al ≤ 0.03 ≤ 0.03 ≤ 0.03 ≤ 0.03 ≤ 0.03 cr ≤ 0.02 ≤ 0.02 ≤ 0.02 ≤ 0.02 ≤ 0.02 fe ≤ 0.02 ≤ 0.02 ≤ 0.02 ≤ 0.02 ≤ 0.02 ni ≤ 0.05 ≤ 0.05 ≤ 0.05 ≤ 0.05 ≤ 0.05 pb ≤ 0.10 ≤ 0.10 ≤ 0.10 ≤ 0.10 ≤ 0.10 zn ≤ 0.01 ≤ 0.01 ≤ 0.01 ≤ 0.01 ≤ 0.01 3.2. effect of ph on the amount of ptms leached from aluminum pot aluminum pot sometimes referred to as pressed aluminum pots was used to boil distilled water of phs 3 to 7 and the results of analyses of the water samples for leached metals are as shown in table 1s (supplementary data) and figure 1. at ph 3 and 4, zn was found to leach but from ph 5 to 7, no zn leached. other elements monitored did not leach from ph 3 to ph 7. aluminum may not have been found in the leachate because of the quality of aluminum pot used and the remediating effect of hot water on aluminum pot. karbouj et al., [27] discovered in their study on aluminum cook ware that boiling water in the cookware prior to cooking decreases the amount aluminum thus remediated aluminum leaching. in this study, the leaching of aluminum pot and other pots at the different phs were done using the same pot with deionised water after each other.this process must have inhibited the leaching of aluminum. however, alabi et al., [21] found that when water was boiled in new aluminum pots for 1 hour it leached 0.023 mg/l of aluminum. while three year old pressed aluminum pots leached 0.029 mg/l and 6 years old aluminum pots leached 0.048 mg/l of aluminum. lomolinon et al., [24] also found that the amount of aluminum leached from aluminum cooking materials depends on quality of aluminum material it is made of. figure 1: concentration of leached ptms from aluminum pot (µg/l) 3.3. effect of ph on the amount of ptms leached from cast aluminum pot aluminum cast pots are produced by the informal sector (artisanal sector) in many african countries including nigeria from recycling of scrap metal and e-waste. the cast pot is neither as smooth nor compacted in appearance as pressed aluminum pot usually industrially manufactured. cast pots can be cracked if smatched unlike the pressed pot and it also flakes when water is stored in it for a while. probably due to the temperature of smelting, type of mould and the purity or type of the starting material. temperature employed in the informal sector is less, sand moulds are used and starting materials are aluminum scraps. cast aluminum pot (koko irin), was used to boil de-ionised water of phs 3 to 7 and the results of analyses of the water samples for leached metals are as shown in figure 2 (table 2s of supplementary data). aluminum leached increased from ph 3-5 and then reduced at ph 6 before increasing slightly at ph 7 which was still lower than at ph 3. thus the ph 5 leached the highest amount of al (2273 µg/l). in a study by verissimo et al., [8] when food samples were cooked with different acidic additives (lemon juice, wine vinegar and cider apple vinegar) and low ph in cast aluminum pot, increased leaching of al was observed. similar trend was observed in this study. amount of aluminum leached at ph 3 to 5 were higher than amounts leached at ph 6 to 7. cr was only leached at ph 7 while other metals were not leached at all ph tested. in the study by street et al., [11] the average concentration of al leached from cast pot was 509,000 µg/l which was above the eu maximum permissible limit of 5000 µg/kg for cook ware and pb was detected in all their samples with some instances at concentrations higher than the 10 ug/l. weidenhamer et al., [28] also detected lead in the extract from leached casted aluminum pots made from scraps in cameroun. however, in this study the concentration of al leached from cast pots were between 1147 and 2273 µg/l which are lower than the values reported in street et al., [11] and lower than the eu permissible value for pots. figure 2: concentration of leached ptms from cast pot (koko irin) (µg/l) 3.4. effect of ph on the amount of ptms leached from clay pot deposits of clay exist in many parts of nigeria and are usually used in pottery [29]. in the clay pot, pb was not detected 4 o. t. fatunsin et al. / j. nig. soc. phys. sci. 4 (2022) 712 5 at all the ph tested. the concentration of al gradually reduced from ph of 3 to ph 4 and increased till ph 7 as shown in figure 3 (table 3s supplementary data). clay is known to be of a source of aluminum [30]. the concentration of fe gradually increased from ph 3 to 5 and at ph 6 it decreased and increased at ph 7. boisa et al., [31] observed that the concentration of fe leached upon exposure of ara-ekiti clay pot to water of acidic ph, neutral ph, alkaline ph were (1.15 mg/l 0.16 mg/l, and 0.08 mg/l respectively). aleksanyan et al., [32] reported the tendency of fe to leach at near neutral to neutral ph conditions. the concentration of zn was higher at ph 5 than at ph 7, this varying trend in concentration of zn leached at the different ph conditions was also recorded by boisa et al. [31]. pb was not detected at all ph conditions. however, chagas et al., [12] in their study of pb leached from clay pot found high levels higher than 2.0 mg/kg which they attributed to the substance used for glazing the pots. the pot used in this study was not glazed but was an oven baked hardened pot. the high level of fe recorded in this study may be attributed to the availability of the metal in the earth’s crust. soils from nigeria are known to be rich in iron [31, 33] and the pot used was made in nigeria. figure 3: concentration of leached ptms from clay pot (µg/l) 3.5. effect of ph on the amount of ptms leached from non-stick pot the concentration of al, fe, cr, ni and pb leached were all below the limit of detection, meanwhile the zn (24.39 µg/l) was only detected at ph 3 as shown in figure 4 (table 4s). the zn may have been from the exposed part of the handle inside the pot that had contact with water. most part of non-stick pot are coated with polytetrafluoroethylene (ptfe, teflon) or its other substitutes which are usually organic in nature [34]. 3.6. effect of ph on the amount of ptms leached from stainless steel pot stainless steel cookware is made from metal alloy primarily made of mostly iron and chromium along with differing percentages of molybdenum, nickel, titanium, copper and vanadium. stainless steel is known to exhibit good thermal conductivity and good corrosion resistance due to its ability to readily form an iron/chromium-enriched passive layer [35]. figure 4: concentration of leached ptms from nonstick pot (µg/l) the result obtained in this study shows that stainless steel pot leached all the metals of interest at ph 3 only (fe 237, al 53.18, pb 29.87 and zn 1.09 ug/l) except ni and cr which were below the detection limit as shown in figure 5 (table 5s supplementary data). okazaki et al., [36] conducted a seven day immersion test using several solutions on stainless steel and it was found that the quantities of fe and ni released gradually decreased with increasing ph from 2 to 7.5. hedberg et al., [37] stated that acidic ph causes changes n the ionic strength, affects corrosion and dissolution processes, affects ligand conformation and their adsorption behavior, changes the surface hydroxide which all results into increased corrosion of stainless steel. dan and ebong, [18] also found that stainless steel pot leached more fe than those cooked with aluminum pot, this tallied with the result obtained from this study. they noted that the fe content in cooked food was higher than in uncooked food, and attributed it to the over fifty percent iron content of stainless steel. in the study by herting, [38] cr and pb leached into acetic acid solutions which are also acidic solutions from stainless steel surface. in this study pb was also detected at an acidic ph of 3 for solutions in stainless steel pot. lead is a very dangerous heavy metal poison that can accumulate in the body over time to cause neurological and behavioral, renal, cardiovascular dysfunction. thus, it is important that its levels are well monitored and are way below toxic levels in foods at all times. the concentration of zn (1.09 µg/l) suggests that the stainless steel may not directly cause or be involved in the toxic effects of zinc in the body as the recommended intake values for zinc range zinc is between 7 and 16 mg/day for adults, pregnant and lactating women, depending on sex and dietary phytate intake [39] while for infants and children its is from 2.9 to 14.2 mg/day [40]. the aluminum cast pot koko irin (2273 µg/l) leached the highest concentration of aluminum while the clay pot leached the highest concentration of cr and ni, (7.27 and 22.63 µg/l). nonstick pot leached the highest concentration of zn (24.39 µg/l) and stainless steel pot leached the highest concentration of fe (237 µg/l) and pb (24.39 µg/l). no ptm was leached from the glass pot. 5 o. t. fatunsin et al. / j. nig. soc. phys. sci. 4 (2022) 712 6 figure 5: concentration of leached ptms from stainless steel pot (µg/l) the type of food with respect to the ph was found to be an important factor during the study of metal leachability from cooking pots. semwal et al., [41] also found that migration of metals into the food was higher with acidic foods than with alkaline foods. they found that more metals leached when food of low (ph of 4.25) was cooked in aluminum cooking pot and that the concentration of al metal increased by 20.3 mg/kg [8] and also reported that red cabbage samples cooked with different acidic additives (lemon juice, wine vinegar and cider apple vinegar) showed increased leaching of aluminum. similarly, this study showed that leaching ptms were considerably higher at more acidic phs than at neutral ph for most of the pots. thus cooking of acidic foods should generally be avoided. longterm exposure to ptms can lead to gradually progressing physical, muscular, and neurological degenerative processes that imitate diseases such as multiple sclerosis, parkinson’s disease, alzheimer’s disease and muscular dystrophy [42]. repeated long-term exposure of some metals and their compounds may even cause cancer [43]. 4. conclusion the results obtained from this study shows that the pots (aluminum, cast aluminum, clay, nonstick, stainless steel) used for the analysis leached considerable concentrations of ptms into the distilled water on varying ph especially at acidic phs while no ptm was leached from the glass pot. more leaching were observed between ph 3 to 5 than between ph 6 to 7 for most the pot types. since ptms are known to bioaccumulate, there are health implications associated with the amount leached from the pots. thus the public should be enlightened about exposure to metals from foods. it should be a public health priority acknowledgement this study was supported by the allan ure busary awarded to oluwatoyin t. fatunsin references [1] r. van-der-oost, j. beyer & p.e.n. vermeulen, “fish bioaccumulation and 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[34] m. sajid, & m. ilyas, “ptfe-coated non-stick cookware and toxicity concerns: a perspective”, environmental science and pollution research 24 (2017) 23436. https://doi.org/10.1007/s11356-017-0095-y [35] k. adama & i. onyeachu, “the corrosion characteristics of ss316l stainless steel in a typical acid cleaning solution and its inhibition by 1-benzylimidazole: weight loss, electrochemical and sem characterizations” journal of the nigerian society of physical sciences 4 (2022) 214. https://doi.org/10.46481/jnsps.2022.596 [36] y, okazaki & e. gotoh, “metal release from stainless steel, co-cr-moni-fe and ni-ti alloys in vascular implants. corrosion science 50 (2008) 3429. https://doi.org/10.1016/j.corsci.2008.09.002 [37] y.s. hedberg, &w.i odnevall, “metal release from stainless steel in biological environments: a review. biointerphases 11 (2015) 34. https://doi.org/10.1116/1.4934628 [38] g. herting, w.i. odnevall, & c. leygraf, “corrosion-induced release of chromium and iron from ferritic stainless steel grade aisi 430 in simulated food contact”, journal of food engineering 87 (2008) 291. https://doi.org/10.1016/j.jfoodeng.2007.12.006 [39] h. haase, s. ellinger, j. linseisen, m . neuhäuser-berthold & m. richter, “revised d-a-chreference values for the intake of zinc”, journal of trace elements in medicine and biology 61 (2020) 126536. https://doi.org/10.1016/j.jtemb.2020.126536. [40] efsa-nda (european food safety authority (efsa) panel on dietetic products, nutrition and allergies (nda)). scientific opinion on dietary reference values for zincefsa journal 12 (2014) 3844. https://doi.org/10.2903/j.efsa.2014.3844. [41] a.d. semwal, a. padmashree, m.a. khan, k. gopal, g.k. sharma & a .s. bawa, “leaching of aluminum from utensils during cooking of food”, journal of the science of food and agriculture 86 (2016) 2425. https://doi.org/10.1002/jsfa.2635 [42] m. jaishankar, t. tseten, n . anbalagan, b.b. mathew & k.n. beeregowda, “toxicity, mechanism and health effects of some heavy metals”, interdisciplinary toxicology 7 (2014) 60. https://doi.org/10.2478/intox2014-0009. [43] l. jarup, “hazards of heavy metal contamination”. british medical bulletin 68 (2003) 167. https://doi.org/10.1093/bmb/ldg032. supplementary data table 1s: concentration of leached ptms from aluminum pot (µg/l). metals ph 3 ph 4 ph 5 ph 6 ph 7 al ≤ 0.03 ≤ 0.03 ≤ 0.03 ≤ 0.03 ≤ 0.03 cr ≤ 0.02 ≤ 0.02 ≤ 0.02 ≤ 0.02 ≤ 0.02 fe ≤ 0.02 ≤ 0.02 ≤ 0.02 ≤ 0.02 ≤ 0.02 ni ≤ 0.05 ≤ 0.05 ≤ 0.05 ≤ 0.05 ≤ 0.05 pb ≤ 0.10 ≤ 0.10 ≤ 0.10 ≤ 0.10 ≤ 0.10 zn 11.66 1.75 ≤ 0.01 ≤ 0.01 ≤ 0.01 table 2s: concentration of leached ptms from cast pot (koko irin) (µg/l). metals ph 3 ph 4 ph 5 ph 6 ph 7 al 1462 1657 2273 1147 1440 cr ≤ 0.02 ≤ 0.02 ≤ 0.02 ≤ 0.02 1.99 fe ≤ 0.02 ≤ 0.02 ≤ 0.02 ≤ 0.02 ≤ 0.02 ni ≤ 0.05 ≤ 0.05 ≤ 0.05 ≤ 0.05 ≤ 0.05 pb ≤ 0.10 ≤ 0.10 ≤ 0.10 ≤ 0.10 ≤ 0.10 zn ≤ 0.01 ≤ 0.01 ≤ 0.01 ≤ 0.01 ≤ 0.01 7 o. t. fatunsin et al. / j. nig. soc. phys. sci. 4 (2022) 712 8 table 3s: concentration of leached ptms from cast pot (koko irin) (µg/l). metals ph 3 ph 4 ph 5 ph 6 ph 7 al 65.43 ≤ 0.03 30.84 48.25 462.69 fe 4.03 6.61 32.55 ≤ 0.02 43.68 cr 4.42 7.27 0.62 ≤ 0.02 1.88 ni 0.80 7.42 22.63 1.35 7.81 pb ≤ 0.10 ≤ 0.10 ≤ 0.10 ≤ 0.10 ≤ 0.10 zn ≤ 0.01 ≤ 0.01 12.73 ≤ 0.01 10.36 table 4s: concentration of leached ptms from nonstick pot (µg/l). metals ph 3 ph 4 ph 5 ph 6 ph 7 al ≤ 0.03 ≤ 0.03 ≤ 0.03 ≤ 0.03 ≤ 0.03 fe ≤ 0.02 ≤ 0.02 ≤ 0.02 ≤ 0.02 ≤ 0.02 cr ≤ 0.02 ≤ 0.02 ≤ 0.02 ≤ 0.02 ≤ 0.02 ni ≤ 0.05 ≤ 0.05 ≤ 0.05 ≤ 0.05 ≤ 0.05 pb ≤ 0.10 ≤ 0.10 ≤ 0.10 ≤ 0.10 ≤ 0.10 zn 24.39 ≤ 0.01 ≤ 0.01 ≤ 0.01 ≤ 0.01 table 5s: concentration of leached ptms from stainless steel pot (µg/l). metals ph 3 ph 4 ph 5 ph 6 ph 7 al 53.18 ≤ 0.03 ≤ 0.03 ≤ 0.03 ≤ 0.03 fe 237 ≤ 0.02 ≤ 0.02 ≤ 0.02 ≤ 0.02 cr ≤ 0.02 ≤ 0.02 ≤ 0.02 ≤ 0.02 ≤ 0.02 ni ≤ 0.05 ≤ 0.05 ≤ 0.05 ≤ 0.05 ≤ 0.05 pb 29.87 ≤ 0.10 ≤ 0.10 ≤ 0.10 ≤ 0.10 8 j. nig. soc. phys. sci. 4 (2022) 231–241 journal of the nigerian society of physical sciences computational investigation of free convection flow inside inclined square cavities mariya helen mercy jk, prabhakar.v∗ division of mathematics, school of advanced sciences, vellore institute of technology, chennai, tamil nadu, india. abstract the temperature and fluid profiles of flow inside tilted square cavities are analysed with two different cases of thermal boundary conditions, (1) isothermally cold sidewalls of the cavity and the hot bottom wall kept parallel to the insulated top wall, (2) hot left wall, cold right wall, insulated top and bottom walls. the galerkin finite element method with penalty parameter is used to solve the nonlinear coupled system of partial differential equations governing the flow and thermal fields. the method is further used to solve the poisson equation for stream function. the results are presented in terms of isotherms and streamlines. the gaussian rule with the hybrid function formed from the block-pulse function and lagrange polynomial is implemented for the evaluation of the definite integrals present in the residual equations. attempting to affix the hybrid methods in the integration part for solving finite element method (fem) turned efficacious as the convergence is achieved for streamlines and isotherms with the existing results. the tilted square cavities with inclination angles 0 ◦ ≤ φ ≤ 90 ◦ and rayleigh number ranging between 103 ≤ ra ≤ 105 for pr = 0.71 (air) are investigated. the source code for the finite element analysis is written in mathematica. the results thus obtained are found to be competent with those of comsol, the software for multiphysics simulation. doi:10.46481/jnsps.2022.637 keywords: penalty finite element analysis, inclined square cavities, thermal distribution, fluid distortion, mathematica, hybrid function, numerical integration, comsol article history : received: 04 february 2022 received in revised form: 07 april 2022 accepted for publication: 07 april 2022 published: 29 may 2022 c©2022 journal of the nigerian society of physical sciences. all rights reserved. communicated by: t. latunde 1. introduction the term ‘convection’ is derived from the latin convectare, which means to take to a place. during the transmission of heat, convection can be defined as the drift of heat energy from or to a solid because of nearby fluid in motion, in the existence of a temperature gradient. further classified into two major types depending upon the source of the fluid motion: forced convection, where the fluid movement happens due to an external ∗corresponding author tel. no: +914439931239 email address: prabhakar.v@vit.ac.in (prabhakar.v) influence, such as a fan, extractor, pump etc. it is called as a natural convection, when the fluid movement is solely due to the characteristics of the fluid between two determined points of the process. the fact that the velocity and the temperature equations are solved all together makes the study of natural convection seemingly complex. the natural convection within a cavity plays important roles in many science and engineering applications namely heat exchangers [1], solar stills [2], cooling of electric and electronic components [3], nanoscience [4]-[6], etc. square cavity with obstacles, lid-driven, heated walls are also in the scope of the researchers [7]-[16]. navier stokes equations governs the fluid flow, whereas the en231 mariya & prabhakar / j. nig. soc. phys. sci. 4 (2022) 231–241 232 ergy balance equations regulate the nature of the temperature. among the various methods utilized to handle square cavities, fem, a very competent method in dealing complex geometries and unstructured domains, is employed in this work. as the complexity increases, the integration process to calculate the stiffness and mass matrix becomes more complicated. the penalty finite element method [17] is involved in the nonlinear system of dimensionless governing equations of the 2d flow problem to waive the pressure constant. two non-dimensional numbers, prandtl number (pr), signifying how much heat is carried away by how much fluid, transferring from one point to another and rayleigh number (ra), describing the relationship between buoyancy and viscosity within the fluid, are explored for the convectional fluid flow. the tilted square cavities with inclination angles 0 ◦ ≤ φ ≤ 90 ◦ and rayleigh number ranging between 103 ≤ ra ≤ 105 for pr = 0.71 (air) are investigated. the rayleigh number above 103 will display a distinctive change in the convective heat transfer rate as the inclination angle is increased. at ra = 103 and ra = 105 gives the minimum and maximum heat transfer rate as the angle increases [8]. literature provides with various methods for solving odes and pdes and differential equations are solved using block methods, step points, numerical methods, etc., [18]-[22].in common practice, gaussian quadrature method is used for evaluating the integrals present in the finite element equations. among the numerical methods available in the literature, a technique based on the hybrid function approximation for solving nonlinear initial-value problems with applications to lane-emden type equations was suggested [23]. they made use of the properties of block-pulse functions and lagrange interpolating polynomials to reduce the nonlinear initial-value problem to a system of non-algebraic equations to arrive at the accurate results. the same method was also used to provide accurate results for solving nonlinear integro-differential equations such as in volterrra’s population model [24] and volterra-fredholm equation, and also for variational problems. a comparative study of quadrature rules based on haar wavelet and hybrid function of block-pulse and legendre polynomial, for finding the approximate value of the definite integrals, was made and the procedure was extended to the numerical solution of double and triple integrals with variable limits [25, 26]. it was observed that the hybrid method provides faster convergence when related to haar wavelet and that the orders of the block-pulse function and legendre polynomial can be attuned to attain very precise solution. hybrid of block-pulse function using lagrange polynomial was considered for evaluation by [27] for the evaluation of general double and triple integrals with variable limits that shows better accuracy over haar wavelet. solution of variational problems are also derived using the same [28]. variable weights are used for the study in contrast with the constant weights considered in [25]. the square cavities inclined at angles 0 ◦ ≤ φ ≤ 90 ◦ are studied with the fluid (pr = 0.71) flowing within a two dimensional, steady and laminar flow across the calculation domain with two different cases of thermal boundary conditions (bc), (case 1) isothermally cold sidewalls of the cavity and the hot bottom wall kept parallel to the insulated top wall, figure 1. square tilted at angle (case 1) φ (case 2) hot left wall, cold right wall, insulated top and bottom walls. are featured in the present work. an endeavour is made to incorporate this hybrid method in the integration part for solving fem turned efficacious as the convergence is achieved for streamlines and isotherms with the existing results in the literature [10, 12]. the source code for the finite element analysis is written in mathematica and the results obtained are compared with comsol. mathematical procedure involving the governing differential equations (gdes) and finite element formulation is discussed in the first section, hybrid method for the numerical integration is explained in the next section, and the results are detailed in the last section. 2. methodology 2.1. computational technique the physical domain (d) of square cavity (pqrs) with angles of inclination 0 ◦ ≤ φ ≤ 90 ◦ for both thermal boundary conditions is considered in the present study. the range of rayleigh’s numbers falls between 103 ≤ ra ≤ 105 for the study. figure 1, shows a clear picture of the geometry considered, incorporating case 1 boundary conditions. physical properties are kept constant during the calculation except the density in buoyancy term, where change in density due to temperature variation is estimated using boussinesq approximation. the phenomena of thermal and fluid flow inside the domain are governed by the energy balance and navier–stokes equations, respectively. the governing equations for steady natural convection flow using conservation of mass, momentum and energy in dimensionless form are given below: ∂u ∂x + ∂v ∂y = 0 (1) u ∂u ∂x + v ∂u ∂y = − ∂p ∂x + pr ( ∂2u ∂x2 + ∂2u ∂y 2 ) (2) 232 mariya & prabhakar / j. nig. soc. phys. sci. 4 (2022) 231–241 233 u ∂v ∂x + v ∂v ∂y = − ∂p ∂y + pr ( ∂2v ∂x2 + ∂2v ∂y 2 ) + ra pr θ (3) u ∂θ ∂x + v ∂θ ∂y = ∂2θ ∂x2 + ∂2θ ∂y 2 (4) the penalty finite element method [17] is used to eliminate the pressure term by the inclusion of the penalty parameter (γ) in the equations (2) and (3) with the following relationship involving the incompressibility. p = −γ ( ∂u ∂x + ∂v ∂y ) (5) generally, (γ = 107) is considered for reliable solutions [10]. applying (5) to (2) and (3) yields u ∂u ∂x + v ∂u ∂y = −γ ∂ ∂x ( ∂u ∂x + ∂v ∂y ) + pr ( ∂2u ∂x2 + ∂2u ∂y 2 ) (6) u ∂v ∂x + v ∂v ∂y = −γ ∂ ∂y ( ∂u ∂x + ∂v ∂y ) + pr ( ∂2v ∂x2 + ∂2v ∂y 2 ) + ra pr θ (7) the whole domain is discretized into bi-quadratic elements. galerkin finite element method is applied to solve the system of governing differential equations (6), (7) and (4). a numerical integration method involving hybrid functions explained in the section 2.3 is used for obtaining the finite element solutions from the residual equations. the thermal (θ) and the velocity components (u&v ) are expanded through basis set {∑9 j=1 n j } given in equation (8). n j s are the shape functions. the procedure to obtain n j s are detailed in [29] and those functions considered for this study is provided in the appendix. u ≈ 9∑ j=1 u j n j(x, y ), v ≈ 9∑ j=1 v j n j(x, y ), θ ≈ 9∑ j=1 θ j n j(x, y ) (8) the residual equations for an element are obtained by using (8) in the governing equations, resulting in the following system of nonlinear residual partial differential equations. r(1)i = 9∑ j=1 u j × ∫ d   9∑ j=1 u j n j  ∂n j∂x +  9∑ j=1 v j n j  ∂n j∂y  nidxdy + γ  9∑ j=1 u j ∫ d ∂ni ∂x ∂n j ∂x dxdy + 9∑ j=1 v j ∫ d ∂ni ∂x ∂n j ∂y dxdy  + pr 9∑ j=1 u j ∫ d ( ∂ni ∂x ∂n j ∂x + ∂ni ∂y ∂n j ∂y ) dxdy (9) r(2)i = 9∑ j=1 v j × ∫ d   9∑ j=1 u j n j  ∂n j∂x +  9∑ j=1 v j n j  ∂n j∂y  nidxdy + γ  9∑ j=1 u j ∫ d ∂ni ∂y ∂n j ∂x dxdy + 9∑ j=1 v j ∫ d ∂ni ∂y ∂n j ∂y dxdy  + pr 9∑ j=1 v j ∫ d ( ∂ni ∂x ∂n j ∂x + ∂ni ∂y ∂n j ∂y ) dxdy − ra pr 9∑ j=1 θ j ∫ d n jdxdy (10) r(3)i = 9∑ j=1 θ j × ∫ d   9∑ j=1 u j n j  ∂n j∂x +  9∑ j=1 v j n j  ∂n j∂y  nidxdy + 9∑ j=1 θ j ∫ d ( ∂ni ∂x ∂n j ∂x + ∂ni ∂y ∂n j ∂y ) dxdy (11) for nine-noded biquadratic element (figure 2) with three degrees of freedom, residual equations (9)− (11) consists of 27 unknowns in 27 equations. 2.2. stream function the stream function is used to display the fluid flow and is acquired from velocity components u and v . the relationships between stream function, ψ and velocity components for 2d flows are u = ∂ψ ∂y and v = − ∂ψ ∂x (12) on differentiating (12), the governing equation for stream function is attained. ∂2ψ ∂x2 + ∂2ψ ∂y 2 = ∂u ∂y − ∂v ∂x (13) expanding the stream function ψ using the basis set ψ ≈ 9∑ j=1 ψ j n j(x, y ) (14) and the relation for u, v from (8), the residual equations for (13) is, r(ψ)i = 9∑ j=1 ψ j ∫ d ( ∂ni ∂x ∂n j ∂x + ∂ni ∂y ∂n j ∂y ) dxdy − ∫ γ nin.∇ψdγ + 9∑ j=1 u j ∫ d ∂ni ∂n j ∂y dxdy − 9∑ j=1 v j ∫ d ∂ni ∂n j ∂x dxdy (15) applying no-slip boundary condition (ψ = 0), ψ’s is obtained from the linear system (15). stream functions (ψ′s) thus obtained might be positive or negative denoting the anticlockwise and clockwise circulation respectively. the integrands of the definite integrals appearing in (9)-(11) and (15) are functions of the global coordinates x and y . figure 2 shows the co-ordinate transformation for the discretized elements from the x − y plane to s − t the plane [11]. the 233 mariya & prabhakar / j. nig. soc. phys. sci. 4 (2022) 231–241 234 figure 2. coordinate transformation of the discretized element form x − y to s − t plane table 1. isotherms of ra = 103 ; pr = 0.71; ps & qr-cold; sr-adiabatic; pq-hot φ mathematica results comsol results 15 ◦ 45 ◦ 75 ◦ integrals appearing in the transformed residual equations are evaluated using hybrid function of block pulse function and lagrange polynomial. the transformed residual equations solved for every node in the domain provides the thermal and velocity components. finite element procedure to solve this is briefed in the appendix. it was shown that hybrid function based on block-pulse and lagrange polynomial gives better accuracy than haar wavelets and other hybrid functions [26], with comparatively lesser nodal points. motivated by the accuracy of this method in solving the differential equations (both linear and nonlinear), an integration scheme based on hybrid functions for definite single, double and triple integrals was presented by [27]. an attempt has been made in applying hybrid functions to obtain the finite element solutions. details regarding the hybrid function and the integration scheme are explained in the following section. table 2. isotherms of ra = 104 ; pr = 0.71; ps & qr-cold; sr-adiabatic; pq-hot φ mathematica results comsol results 15 ◦ 45 ◦ 75 ◦ table 3. isotherms of ra = 105 ; pr = 0.71; ps & qr-cold; sr-adiabatic; pq-hot φ mathematica results comsol results 15 ◦ 45 ◦ 75 ◦ 2.3. hybrid function definition 1: block pulse functions: a set of block-pulse function φ j(t), j = 1, 2, ...j defined on the interval [0, 1) are denoted as φ j(t) = 1, t j−1 ≤ t ≤ t j0, otherwise (16) where j represent the number of partitions of [0, 1) or the order 234 mariya & prabhakar / j. nig. soc. phys. sci. 4 (2022) 231–241 235 table 4. streamlines of ra = 103 ; pr = 0.71; ps & qr-cold; sr-adiabatic; pq-hot φ mathematica results comsol results 15 ◦ 45 ◦ 75 ◦ table 5. streamlines of ra = 104 ; pr = 0.71; ps & qr-cold; sr-adiabatic; pq-hot φ mathematica results comsol results 15 ◦ 45 ◦ 75 ◦ of the block pulse function in [0, 1). definition 2: hybrid functions of block pulse and lagrange polynomial functions: a set of hybrid functions h jk(t), j = 1, 2, ...j, table 6. streamlines of ra = 105 ; pr = 0.71; ps & qr-cold; sr-adiabatic; pq-hot φ mathematica results comsol results 15 ◦ 45 ◦ 75 ◦ table 7. isotherms of ra = 103 ; pr = 0.71; ps-hot; qr cold; pq and sr -adiabatic; φ mathematica results comsol results 0 ◦ 30 ◦ 50 ◦ k = 0, 1, ...k − 1 on the interval [0, 1) are denoted as h jk(t) = lk(2jt − 2 j + 1), t ∈ [ j−1 j , j j ] 0, otherwise (17) where lk(t) denoting the lagrange polynomial of order k. as the block pulse functions and lagrange interpolating polynomials are complete and orthogonal, { h jk } is a complete orthogonal set in the hilbert space l2 [23]. 235 mariya & prabhakar / j. nig. soc. phys. sci. 4 (2022) 231–241 236 table 8. isotherms of ra = 104 ; pr = 0.71; ps-hot; qr cold; pq and sr -adiabatic; φ mathematica results comsol results 0 ◦ 30 ◦ 50 ◦ table 9. isotherms of ra = 105 ; pr = 0.71; ps-hot; qr cold; pq and sr -adiabatic; φ mathematica results comsol results 0 ◦ 30 ◦ 50 ◦ 2.3.1. approximation of function of a single variable: a function f (t) ∈ l2 [ 0, 1 ) can be approximated, in terms of the hybrid functions as f (t) = ∞∑ j=1 ∞∑ k=0 c jkh jk(t) (18) for an interval [0, 1) divided into j partitions and k internal table 10. streamlines of ra = 103 ; pr = 0.71; ps-hot; qr cold; pq and sr -adiabatic; φ mathematica results comsol results 0 ◦ 30 ◦ 50 ◦ table 11. streamlines of ra = 104 ; pr = 0.71; ps-hot; qr cold; pq and sr -adiabatic; φ mathematica results comsol results 0 ◦ 30 ◦ 50 ◦ nodes in each partition, f (t) is taken in the form f (t) ≈ j∑ j=1 k−1∑ k=0 c jkh jk(t) (19) considering k nodes in the jth partition [ j−1 j , j j ] , denoted by r jk, it can be shown that c jk = f (r jk) [26]. 236 mariya & prabhakar / j. nig. soc. phys. sci. 4 (2022) 231–241 237 table 12. streamlines of ra = 105 ; pr = 0.71; ps-hot; qr cold; pq and sr -adiabatic; φ mathematica results comsol results 0 ◦ 30 ◦ 50 ◦ 2.3.2. approximation of function of two variables: extending the function approximation of a single variable in terms of hybrid functions, the function f (t, u) defined in [0, 1) × [0, 1) can be approximated as f (t, u) ≈ j′∑ j=1 k′−1∑ k=0 j∑ j=1 k−1∑ k=0 c jklmh jk(t)hlm(u) (20) by considering j partitions and k internal nodes in each partition for [0, 1) in the t-direction, and j′ partitions and k′ internal nodes in each partition for [0, 1) in the u-direction. it can be shown that c jklm = f ( r jk, rlm ) (21) where gaussian nodes along the t and u directions are and r jk and rlm respectively [25]. 2.3.3. integration scheme for the integrals in residual equations: using the function approximation from equations (19) and (20), the integrals (both single and double) of the residual equations can be easily evaluated as follows. ∫ 1 0 f (t)dt = j∑ j=1 k−1∑ k=0 c jkw jk (22) where w jk = ∫ 1 0 h jk(t)dt are the weights. as the hybrid function h jk(t) is a polynomial, its weight can be easily calculated. similarly, ∫ 1 0 ∫ 1 0 f (t, u)dtdu = j′∑ j=1 k′−1∑ k=0 j∑ j=1 k−1∑ k=0 c jklmw jkwlm (23) where w jk = ∫ 1 0 h jk(t)dt and wlm = ∫ 1 0 hlm(u)du are the weights. for single integral j = 1 and k = 2; for double integrals j = 2, k = 4, j′ = 3 and k′ = 5 are well accommodated for the integrals present in the residual equations. 3. results and discussion we consider a two dimensional, steady and laminar flow across the inclined square cavity (pqrs) under two different thermal boundary conditions as mentioned in section 1. the physical domain (d) of square cavity with angle of inclination (φ), inclined with the x-axis within the acute angles of 15 ◦ , 45 ◦ and 75 ◦ for case 1 and 0 ◦ , 30 ◦ and 50 ◦ for case 2 thermal boundary conditions is considered for the present study. pr = 0.7, and ra varied from 103 to 105 for both the boundary conditions are investigated. in order to assess the accuracy of our numerical procedure, we have tested our algorithm for the results presented in [10] for 28 × 28 elements and found to be exactly matching. in this paper, however, we restrict our study considering 20 × 20 elements (figure 3) with the grid size of 51 × 51, which were also found to be in good agreement with the isotherms and streamlines reported in [10]. distributions of isotherms and streamlines of case (1) are portrayed in tables 1-3 and 4-6 respectively; similarly tables 7-9 and 10-12 illustrates the heat and fluid contours of case (2). observing the isotherms for case (1), when ra is 103, however inclined the square cavity is, there is not much disruption, indicating heat transfer through conduction as depicted in table 1, whereas, when ra is increased, isotherms are concentrated at the hot and cold walls. both clockwise and anticlockwise circulations seen in all the rayleigh numbers are pondered at φ = 15 ◦ . when ra is 103, the secondary cells are comparatively lesser in size but when it is 104 the cells are slightly bigger and when the value of ra is 105 the two axisymmetric flow exquisitely occupies the entire cavity. as φ increases to 45 ◦ , the strength of anticlockwise circulation cells increases (table 5). as the angle of inclination increases, there is a significant push of isotherms towards the right wall. at φ = 75 ◦ , the isotherms are found to be qualitatively similar to those of φ = 45 ◦ as the temperature contours are piling towards the right wall (tables 2 and 3). the overall amount of heat transfer along the right wall increases with inclination angle and decreases along the left wall. in table 6 at 45 ◦ , strong primary anticlockwise circulation cells occupy almost the entire part of cavity except top corner of the wall qr. as inclination angle φ further increases to 75 ◦ , the primary circulation cells span the entire cavity whereas secondary circulation cells completely disappear. it is interesting to observe that the centre of left vortex for the fluid 237 mariya & prabhakar / j. nig. soc. phys. sci. 4 (2022) 231–241 238 circulations is pushed towards the bottom wall and centre of right vortex is pushed towards the top wall due to enhanced tangential components of buoyancy force along the bottom wall which tends to impose an anticlockwise fluid circulation. exploring case (2), the temperature contours exhibit a gentle migration from left to right side of the calculation domain for all the investigated. in table 7, dealing with the ra = 103 , the contours are observed to exhibit an elegant protuberance toward the right wall as there is a surge in the inclination angles. streamlines in these conditions are not much varied but are laminar in nature as in tab.12. with increase in ra, heat transfer due to convection is clearly visible (table 9). the difference is observed as φ is gradually amplified. the fluid flow is more unruffled than in the case 1. only when ra = 105 and φ = 0 ◦ , the centre of the vortex is not circular in shape as in table 12. the flow in the circulation cells are almost circular in shape near the core and further, they swell and take the shape of the cavity near the walls due to strong convection only increasing the angle of inclination. from what is seen in the isotherms the convection mode of heat transfer dominates over the conduction, as the inclination angle increases. evident picture of the conduction effect can be seen in the contour for temperature. the movement of fluid backs up the convection, due to the bulk movement of air triggers the convective heat transfer moves the constant temperature line in the direction of its motion. hence, the circulation cells are strained along these walls, prevailing in the sizeable portion of the central region. the temperature layer in the core breaks down because of the good mixing occurring in the core. the inclined square is the calculation domain, hence the effect of tangential and normal components of buoyancy force relative to hot wall gains importance for flow and thermal phenomena. stronger anticlockwise fluid circulations are witnessed when increasing the angle of inclination as the buoyancy force along the hot wall also grows. results indicate that the fluid circulations, isotherms are strongly dependent on the inclination angle of the square cavity. the heat and the flow contours obtained by using the hybrid function are in good agreement with [10] for case (1) and [12] for case (2) where the fem with traditional gaussian quadrature were used to carry out the calculation. 3.1. comparison using comsol the comsol model wizard was used to create a stationary, 2d multiphysics model. the computation was carried out in a physical machine with intel i7 64-bit processor with 8 gb of memory and ssd for storage, loaded with windows 10 operating system. the time taken for computation for a particular pr and various ra is 20 seconds. the 2d geometry is mathematically extended to infinity in both directions along the z-axis, assuming no variation along that z − axis. a square with 1metre on all sides is built, inclined to specific angles for needed cases. different components with different inclination angles are created in the same model. ‘parameters’ nodes are present under global definitions, that are used for creating and figure 3. 20 × 20 meshing of the domain defining the values of the non-dimensional temperature boundary conditions for hot, cold and adiabatic walls, rayleigh number, prandtl number and the reference pressure. coupling of the fields are at ease even with more than one physics interface. all applicable fields that can be used as inputs in one physics interface is spontaneously found in the other physics interface’s settings. laminar flow (sp f ) takes care of fluid properties, initial conditions, volume force and the pressure of the fluid. heat transfer in fluids (ht) handles the nature of the heat transfer and the temperature boundary conditions. the mesh nodes are generated by meshing the domain, which enables the discretization of the geometry into mesh elements. the parameter ra varied is listed for the given range using auxiliary sweep. the set-up is computed. comsol multiphysics generates plot windows for displaying convergence results and isothermal contours and streamlines. the results from comsol are compared with those of mathematica for every boundary condition and ra found to be in good agreement. 4. conclusion in this paper, we considered a two dimensional, steady and laminar flow across the tilted square cavity under two different thermal boundary conditions: case (1) isothermally cold sidewalls of the cavity and the hot bottom wall kept parallel to the insulated top wall and case (2) hot left wall, cold right wall, insulated top and bottom walls. in case 1, it is found that two axisymmetric flows dwell in the entire cavity, whereas, as the angle of inclination increases, the primary circulation cells occupy the entire cavity and the secondary circulation cells disappear completely. in case 2, only at ra = 105, the streamlines are circular only on enlarging inclination of the square cavity. in both the cases as inclination increases, buoyancy force along the hot wall gradually increases leading to stronger anticlockwise fluid circulations corresponding to the rayleigh number. results indicate that the streamlines and isotherms are strongly dependent on the inclination angle of the cavity. an attempt has been made to evaluate the definite integrals of the residual equations of the finite element equations using gaussian quadrature with hybrid functions of block-pulse function and lagrange polynomial as orthogonal polynomials, and the results are found 238 mariya & prabhakar / j. nig. soc. phys. sci. 4 (2022) 231–241 239 to be in accordance with [10] for case (1) and [12] for case (2). with hybrid functions, desired accuracy can be achieved with the flexibility of choosing the orders of block pulse function and lagrange polynomial independently. finite element results presented with wolfram mathematica were found to be in good agreement with comsol multiphysics. the model can be applied for a field problem like plastic injection mould flow, porous media flow etc., by varying the variables such as fluid temperature, fluid velocity, mass weight and flow rate. acknowledgments we thank the referees for the positive enlightening comments and suggestions, which have greatly helped us in making improvements to this paper. appendix a.notations used: x = x l ,y = y l , (x and y distance in dimensionless form) u = ul α , y = vl α , (u and v velocity components in dimensionless form) θ = t − tc th − tc (θ temperature component in dimensionless form; th&tc-temperature at hot and cold) p = pl3 ρα2 (p pressure in the dimensionless form) pr = ν α (pr prandtl number) ra = gβ(th − tc)l3 pr ν (pr rayleigh number) α = k ρc p (α thermal diffusivity) b.the shape functions of the transformed residuals from x − y to s − t plane with reference to the local numbering in the figure 2. n1 = (1 − 3s + 2s2)(1 − 3t + 2t2); n2 = (1 − 3s + 2s2)(4t − 4t2); n3 = (1 − 3s + 2s2)(−t + 2t2); n4 = (4s − 4s2(1 − 3t + 2t2); n5 = (4s − 4s2)(4t − 4t2); n6 = (4s − 4s2)(−t + 2t2); n7 = (−s + 2t2)(1 − 3t + 2t2); n8 = (−s + 2t2)(4t − 4t2); n9 = (−s + 2t2)(−t + 2t2); c.the nonlinear residual equations (9) (11) are solved by the following procedure grouping the co-efficient of ∑9 j=1 u j from all three residues s11 = ∫ d   9∑ j=1 u j n j  ∂n j∂x +  9∑ j=1 v j n j  ∂n j∂y  nidxdy + γ ∫ d ∂ni ∂x ∂n j ∂x dxdy + pr ∫ d ( ∂ni ∂x ∂n j ∂x + ∂ni ∂y ∂n j ∂y ) dxdy (c.1) s12 = γ ∫ d ∂ni ∂y ∂n j ∂y dxdy (c.2) s13 = 0 (c.3) grouping the co-efficient of ∑9 j=1 v j from all three residues s21 = γ ∫ d ∂ni ∂y ∂n j ∂x dxdy (c.4) s22 = ∫ d   9∑ j=1 u j n j  ∂n j∂x +  9∑ j=1 v j n j  ∂n j∂y  nidxdy + γ ∫ d ∂ni ∂y ∂n j ∂y dxdy + pr ∫ d ( ∂ni ∂x ∂n j ∂x + ∂ni ∂y ∂n j ∂y ) dxdy (c.5) s23 = −ra pr 9∑ j=1 θ j ∫ d n jdxdy (c.6) grouping the co-efficient of ∑9 j=1 θ j from all three residues s31 = 0 (c.7) s32 = 0 (c.8) s33 = ∫ d   9∑ j=1 u j n j  ∂n j∂x +  9∑ j=1 v j n j  ∂n j∂y  nidxdy + ∫ d ( ∂ni ∂x ∂n j ∂x + ∂ni ∂y ∂n j ∂y ) dxdy (c.9) rewriting the governing equations, with the above substitutions, in matrix form r(1)i r(2)i r(3)i  = 9∑ j=1  si, j11 s i, j 12 s i, j 13 si, j21 s i, j 22 s i, j 23 si, j31 s i, j 32 s i, j 33   u j v j θ j  (c.10) { ri } = [ ki j ] { u j } , 1 ≤ i ≤ 9 (c.11) the integrands, which leads to the element stiffness matrix, are functions of the global coordinates x and y . the functions n j can be expressed in terms of the local coordinates s and t. figure 2 gives an idea of co-ordinate transformation for the discretized elements in the x − y plane to the s − t plane. x = 9∑ j=1 x j n j and y = 9∑ j=1 y j n j (c.12) the integrand contains not only functions but also derivatives with respect to the global coordinates (x j, y j). therefore, ∂n j ∂x and ∂n j ∂y must be related to ∂n j ∂s and ∂n j ∂t . applying the shape functions to equation (c.10), the transformed finite element equation is obtained. element stiffness matrices thus obtained are assembled using the element connectivity (table 13) resulting in a global stiffness matrix. the global stiffness matrix is solved to get the thermal and the velocity components. on applying of the boundary conditions to the walls, matrix reduces to 9 nodes (internal), i.e., 27 239 mariya & prabhakar / j. nig. soc. phys. sci. 4 (2022) 231–241 240 table 13. element connectivity involving nodes (n) and elements (e) ne 1 2 3 4 1 1 3 11 13 2 2 4 12 14 3 3 5 13 15 4 6 8 16 18 5 7 9 17 19 6 8 10 18 20 7 11 13 21 23 8 12 14 22 24 9 13 15 23 25 figure 4. isotherms contour of 4 elements inclined φ = 15 ◦ ; ra = 104; pr = 0.71 obtained from mathematica unknowns and equations, for the entire domain discretized to for 2 × 2. the temperature contours of 4 elements for a tilted square cavity with ra = 104; pr = 0.71 is displayed in figure 4. the values of the velocity (u ,v) and thermal components (θ) for the internal nodes of the discretized domain with 2 × 2 (4 elements) are given below. u7 = −0.0000228252, v7 = −0.0000176029,θ7 = 0.414016, u8 = 4.69382e-6, v8 = −1.88969e-6,θ8 = 0.208614, u9 = 0.0000131575, v9 = −1.06333e-6,θ9 = 0.121948, u12 = −0.0000119197, v12 = −0.000053752,θ12 = 0.575621, u13 = 0.0000149714, v13 = −0.0000638262,θ13 = 0.271557, u14 = 0.0000157566, v14 = −0.000030133,θ14 = 0.171982, 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[29] larry j.segerlind, applied finite element analysis, united states of america, john wiley & sons, new york (1984). 241 j. nig. soc. phys. sci. 2 (2020) 61–68 journal of the nigerian society of physical sciences original research mathematical modelling of concentration profiles for species transport through the single and the interconnected multiple-compartment systems o. adedire∗, j. n. ndam department of mathematics, university of jos, nigeria. abstract in this research work, we investigate the concentration profiles in the single and the interconnected multiple-compartment systems with sieve partitions for the transport of chemical species with second order chemical reaction kinetics. with assumption of unidirectional transport of chemical species and constant physical properties with same equilibrium contant, the developed partial differential equations representing the two systems are spatially discretised using the method of lines (mol) technique and the resulting semi-discrete system of odes are solved using matlab ode15s solver. the results show that the interconnected multiple-compartment system has lower concentration profile than the single-compartment system for different values of diffusivities. keywords: concentration profile, species, single-compartment, interconnected, multiple-compartment article history : received: 17 march 2020 received in revised form: 02 may 2020 accepted for publication: 04 may 2020 published: 14 may 2020 c©2020 journal of the nigerian society of physical sciences. all rights reserved. communicated by: t. latunde 1. introduction chemical reactor engineers have certain desired concentration profiles for optimum yield of products. while high concentration profile is preferable for some chemical species to yield adequate product, low concentration profile is desired for some other chemical species. the type of chemical reactor used is a factor that may determine the concentration profiles in such systems due to pressure, heat and possible consumption of the chemical species as a result of the presence of other reactions. different chemical species have different diffusivities in the medium through which they are being transported. the medium through which a chemical species is being transported could be solid, liquid or gas medium. ∗corresponding author tel. no: email address: dharenss@gmail.com (o. adedire ) in literature, researchers have modelled high and low concentration profiles for different diffusion values in different media. gaiseanu presented analysis on diffusion of boron in silicon at high surface concentration beginning from some diffusion constants and his conclusion is centred on the dependence of diffusion coefficient on concentration for some given parameters whose details can be found in his work [1]. chepurniy and savage also investigated the effect of diffusion on concentration profile in a solar pond and concluded that constant value of diffusivity existed in the medium considered [2]. while investigating probe effects on concentration profiles in the diffusion layer, critelli et al. [3] solved a pure diffusion problem for a onecomponent two dimensional axisymmetric model. they concluded that positioning of a probe like a microelectrode close to a working electrode interferes with local potential and concentration distributions. 61 o. adedire & j. n. ndam / j. nig. soc. phys. sci. 2 (2020) 61–68 62 yang and lue examined coupled concentration-dependent diffusivities of ethanol and water mixture through a polymeric membrane. while determining effects on pervaporative flux and diffusivities profiles, they concluded that the concentration profiles exhibited a linear to concave response as a function of depth in the considered membranous medium [4]. there have been various works on concentration profiles and diffusivities of chemical species in different media [5-11]. zhou et al. [12] generated complex concentration profile by partial diffusive mixing in multi-stream laminar flow. one of the key insights in their work is that details in the concentration profile can be tuned with geometrical and operating parameters. adedire and ndam investigated model of chlorine decay through water and intermediate pseudomonas aeruginosa in multiple-compartment isothermal system [13]; they however did not compare their work with single compartment system. researchers in literature have not compared the concentration profiles between the single and the interconnected multiplecompartment systems for the transport of chemical species with second order chemical reaction kinetics. the research question is to determine whether single compartment system will have higher concentration profile than interconnected multiple-compartment system for chemical species with second order chemical reaction kinetics as time progressively increases. would there be higher concentration profile for chemical species with higher diffusivities than for those with lower diffusivities? in this study, we intend to compare the concentration profiles in both the single-compartment system and the interconnected multiple-compartment system for the transport of chemical species with second order chemical reaction kinetics with respect to diffusivities. the interconnected multiple-compartment system is such that each compartment is separated by sieve partitions allowing transport of chemical species through each compartment of the system. the subsequent part of this paper is organised as follows: section 2 deals with model development. section 3 deals with numerical simulation of the model equations. while in section 4, results and discussion from the perspective of the considered systems will be addressed, conclusion comes up in section 5. 2. model development the development of the model in this work is centred on treating fluid as a continuum [14]. let any chemical species be a substance distributed in rn with boundary ∂ω and let n̂ be an outward normal to ∂ω and define a function ψϕ(x, t) : rn × r → r (1) where ψϕ(x, t) is the concentration of any chemical species ϕ in some units of measurements having x ∈ rn with x = x1, x2, ...xn at any time t. let q ∈ n be total number of compartments and γ ∈ {1, 2, 3, 4, 5} be compartment 1, 2, 3, 4 and 5 respectively. for q = 1,γ = 1, n = 1, x1 = x we set up mass balance equations assuming that convection and diffusion are taking place along the x-axis as shown in figure 1. for chemical species ϕ with flux qϕ and reaction term rϕ over infinitesimal thickness ∆x in x -direction of γth -compartment of the reactor, the mass balance gives (∆x) ψϕ(x, t) ∣∣∣ t − (∆x)ψϕ(x, t) ∣∣∣ t+∆t = vψϕ(x, t) ∣∣∣ x ∆t − v ψϕ(x, t) ∣∣∣ x+∆x ∆t + qh1 x ∣∣∣ x ∆t − qh1 x ∣∣∣ x+∆x ∆t + rϕ∆x∆t (2) division of (2) by (∆x)(∆t) gives ψϕ(x,t)|t−ψϕ(x,t)|t+∆t ∆t = −v ψϕ(x,t)|x−ψϕ(x,t)|x+∆x ∆x − qh1 |x−qh1 |x+∆x ∆x + rϕ (3) take the limit of (3) as ∆x → 0, ∆t → 0 to get ∂ψϕ(x, t) ∂t = −v1 ∂ψϕ ∂x − ∂qϕ ∂x + rϕ (4) assuming that the rate of consumption of the chemical species follows second order chemical reaction kinetics and choosing the flux qϕ to follow fick’s law of diffusion [15], we have ∂ψϕ(x, t) ∂t = −v ∂ψϕ(x, t) ∂x + ∂ ∂x [ d ∂[ψϕ(x, t) ∂x ] , −kϕ [ ψϕ(x, t) ]2 xl0 6 x 6 xµ, t > 0 (5) with initial and boundary conditions given as ψflϕ(x, 0) = ψflϕ0(x) (6) ψflϕ(x = xl0, t) = ψflϕ(t) (7) ψflϕ(x = xµ, t) = ψflϕe (x = xµ, t) (8) following the same procedure used to obtain equations (5), (6), (7) and (8), we consider the following figure 2 . for the parameters q = 5; γ = 1, 2, 3, 4, 5; n = 1, x1 = x, we obtain the following system of partial differential equations with assumption that convection and diffusion take place along x -axis as ∂ψ1ϕ(x, t) ∂t = −v ∂ψ1ϕ(x, t) ∂x + ∂ ∂x [ d1 ∂ψ1ϕ(x, t) ∂x ] −kϕ [ ψ1ϕ(x, t) ]2 , xl 6 x 6 x1, t > 0 (9) ∂ψ2ϕ(x, t) ∂t = −v ∂ψ2ϕ(x, t) ∂x + ∂ ∂x [ d2 ∂ψ2ϕ(x, t) ∂x ] −kϕ [ ψ2ϕ(x, t) ]2 , x1 ≤ x ≤ x2, t ≥ 0 (10) ∂ψ3ϕ(x, t) ∂t = −v ∂ψ3ϕ(x, t) ∂x + ∂ ∂x [ d3 ∂ψ3ϕ(x, t) ∂x ] −kϕ [ ψ3ϕ(x, t) ]2 , x2 ≤ x ≤ x3, t ≥ 0 (11) ∂ψ4ϕ(x, t) ∂t = −v ∂ψ4ϕ(x, t) ∂x + ∂ ∂x [ d4 ∂ψ4ϕ(x, t) ∂x ] 62 o. adedire & j. n. ndam / j. nig. soc. phys. sci. 2 (2020) 61–68 63 figure 1: schematic representation of species transport through (q=1) single-compartment chemical reactor system. figure 2: schematic representation of species transport through interconnected (q=5)multiple-compartment chemical reactor system. −kϕ [ ψ4ϕ(x, t) ]2 , x3 ≤ x ≤ x4, t ≥ 0 (12) ∂ψ5ϕ(x, t) ∂t = −v ∂ψ5ϕ(x, t) ∂x + ∂ ∂x [ d5 ∂ψ5ϕ(x, t) ∂x ] −kϕ [ ψ5ϕ(x, t) ]2 , x4 6 x 6 xµ, t > 0 (13) equation (9) has initial and boundary conditions ψ1ϕ(x, 0) = ψ1ϕ0(x) (14) ψ1ϕ(x = xl0, t) = ψ1ϕ(t) (15) ψ1ϕ(x = x1, t) = k[ψ2ϕ(x = x1, t)] (16) equations (10), (11) and (12) for γ = 2, 3 and 4 have the initial and boundary conditions ψγϕ(x, 0) = ψγϕ0(x) (17) ∂ψflϕ(x = xγ, t) ∂x = d(γ−1)ϕ dγϕ ∂ψ(γ−1)ϕ(x = x(γ−1), t) ∂x (18) ψγϕ(x = xγ, t) = k [ ψ(γ+1)ϕ(x = xγ, t) ] (19) equation (13) for γ = 5 has the initial and boundary conditions given as ψ5ϕ(x, 0) = ψ5ϕ0(x) (20) ∂ψ5ϕ(x = x4, t) ∂x = d4ϕ d5ϕ ∂ψ4ϕ(x = x4, t) ∂x (21) ψ5ϕ(x = x5, t) = ψ5ϕ0(t) (22) while the neumann boundary conditions (18) and (21) show continuity of flux of species ϕ from γth compartment to the adjacent (γ + 1)th compartment, the dirichlet boundary conditions (16), (19) and (22) show the link between the concentrations in one compartment with the concentration in the preceding compartment in a way showing relationship at the boundaries. 2.1. existence and uniqueness of solution the solutions of the governing equations (5)-(8) and (9)-(22) which are pdes and their auxiliary conditions exist and are unique; hence they are well–posed system of pdes [16]. detailed proofs of the existence and uniqueness of the governing model equations leading to their well-posedness will not be considered here to avoid repetition as they have been extensively proved elsewhere. for further analysis and detailed proofs, the reader is referred to [16] and to [17] and references contained therein. 3. numerical simulation the governing model equation (5) together with its auxiliary conditions (6)-(8) for (q=1) single-compartment system 63 o. adedire & j. n. ndam / j. nig. soc. phys. sci. 2 (2020) 61–68 64 and the system of equations (9)-(13) together with their auxiliary conditions (14)-(22) for interconnected multiple-compartment system are solved using method of lines (mol) technique. we performed numerical simulation after only the spatial derivatives of the governing pdes have been discretised with finite difference approximations. to this end, let x = xi such that i ∈ n be an index representing positions on grids in x. we consider first order approximations to ∂ψ ∂x for the convective part of the pdes as ∂ψ(x, t) ∂x ≈ ψi − ψi−1 ∆x + o(∆x) (23) also, for the diffusive part, second order finite difference approximations to the second order derivative ∂ 2 ψ ∂x2 are considered as ∂2ψ(x, t) ∂x2 ≈ ψi+1 − 2ψi + ψi−1 ∆x2 + o(∆x2) (24) the terms o(∆x) and o(∆x2) are the truncation error of approximation of finite difference scheme obtained from taylor series. the number of points on the grids in x is chosen as m such that at the first boundary (left-end) of x a value of i=1 is assigned and at the last boundary (right-end) of x a value of i=m is chosen. for q = 1,γ = 1 , n = 1 substitution of (23) and (24) into (5) results in dψ(t)(ϕ) i dt = −v ψ(t)(ϕ) i−ψ(t)(ϕ) i−1 ∆x + d [ ψ(t)(ϕ) i+1−2ψ(t)(ϕ) i +ψ(t)(ϕ) i−1 ∆x2 ] −kϕ [ ψ(t)(ϕ) i ]2 , 1 6 i 6 m (25) and (6), (7) and (8) are semi-discretised as ψ(t = 0)(ϕ) i = ψϕ0(x(i)) (26) ψ(t)(ϕ) 1 = ψϕ(t) (27) ψ(t)(ϕ) m = ψ(t)(ϕ)0 m (28) for q = 5; γ = 1, 2, 3, 4, 5 , substitution of (23) and (24) into (9), (10), (11), (12) and (13) results in dψ(t)(1ϕ) i dt = −v1 ψ(t)(1ϕ) i − ψ(t)(1ϕ) i−1 ∆x +d1 [ ψ(t)(1ϕ) i+1 − 2ψ(t)(1ϕ) i + ψ(t)(1ϕ) i−1 ∆x2 ] −k1ϕ [ ψ(t)(1ϕ) i ]2 , 1 6 i 6 m(1) (29) dψ(t)(2ϕ) i dt = −v2 ψ(t)(2ϕ) i − ψ(t)(2ϕ) i−1 ∆x +d2 [ ψ(t)(2ϕ) i+1 − 2ψ(t)(2ϕ) i + ψ(t)(2ϕ) i−1 ∆x2 ] −k2ϕ [ ψ(t)(2ϕ) i ]2 , 1 6 i 6 m(2) (30) dψ(t)(3ϕ) i dt = −v3 ψ(t)(3ϕ) i − ψ(t)(3ϕ) i−1 ∆x +d3 [ ψ(t)(3ϕ) i+1 − 2ψ(t)(3ϕ) i + ψ(t)(3ϕ) i−1 ∆x2 ] −k3ϕ [ ψ(t)(3ϕ) i ]2 , 1 6 i 6 m(3) (31) dψ(t)(4ϕ) i dt = −v4 ψ(t)(4ϕ) i − ψ(t)(4ϕ) i−1 ∆x +d4 [ ψ(t)(4ϕ) i+1 − 2ψ(t)(4ϕ) i + ψ(t)(4ϕ) i−1 ∆x2 ] −k4ϕ [ ψ(t)(4ϕ) i ]2 , 1 6 i 6 m(4) (32) dψ(t)(5ϕ) i dt = −v5 ψ(t)(5ϕ) i − ψ(t)(5ϕ) i−1 ∆x +d5 [ ψ(t)(5ϕ) i+1 − 2ψ(t)(5ϕ) i + ψ(t)(5ϕ) i−1 ∆x2 ] −k5ϕ [ ψ(t)(5ϕ) i ]2 , 1 6 i 6 m(5) (33) where m(1), m(2), m(3), m(4), m(5) are the number of points on the grid in x in each compartment γ = 1, 2, 3, 4, 5. the initial and boundary conditions (14),(15) and (16) are semi-discretised as ψ(t = 0)(1ϕ) i = ψ1ϕ0(x(i)) (34) ψ(t)(1ϕ) 1 = ψ1ϕ(t) (35) ψ(t)(1ϕ) m(1) = kψ(t)(2ϕ) m(1) (36) and equations (17), (18) and (19) are semi-discretised for (30), (31) and (32) as ψ(t = 0)(γϕ) i = ψγϕ0(x(i)) (37) ψ(t)(γϕ) m(γ−1) − ψ(t)(γϕ) m(γ−1)−1 ∆x = d(γ−1)ϕ dγϕ ψ(t)[(γ−1)ϕ] m(γ−1) − ψ(t)[(γ−1)ϕ] m(γ−1)−1 ∆x (38) ψ(t)(γϕ) m(γ) = kψ(t)[(γ+1)ϕ] m(γ) (39) finally, equations (20),(21) and (22) are semi-discretised as ψ(t = 0)(5ϕ) i = ψ5ϕ0(x(i)) (40) ψ(t)(5ϕ) m(4) − ψ(t)(5ϕ) m(4)−1 ∆x = d(4)ϕ d5ϕ ψ(t)4ϕ m(4) − ψ(t)4ϕ m(4)−1 ∆x (41) ψ(t)(5ϕ) m(5) = kψ(t)5ϕ0 m(5) (42) 64 o. adedire & j. n. ndam / j. nig. soc. phys. sci. 2 (2020) 61–68 65 figure 3: species concentration profile for diffusivity 5.4 × 101 m2/min γ = 1 (q=1) single-compartment system at time t = 9 minutes. figure 4: species concentration profile for diffusivity 5.4×101 m2/min γ = 1, 2, 3, 4, 5 of interconnected (q=5) multiple-compartment system at time t = 9 minutes. 4. results and discussion for parameters q = 1,γ = 1, n = 1 , the total length of the system is 600m with a constant boundary value ψϕ0 = 70mg/l. the second order reaction rate constant kϕ is 1.2 × 101 l mg min−1, the diffusivities dϕ are taken to be 5.4×101 m2/min and 5.4 × 102 m2/min respectively with initial concentration of the chemical species taken as ψaϕ0 = 70mg/l. for parameters q = 5; γ = 1, 2, 3, 4, 5; n = 1 , the length of each compartment is 120m so that the total length gives 600m. a constant boundary value ψ1ϕ0 = 70mg/l is used, second order reaction rate constants kiϕ(i = 1, 2, 3, 4, 5) are taken to be 1.2 × 101 l mg min−1 with diffusivities diϕ(i = 1, 2, 3, 4, 5) also 65 o. adedire & j. n. ndam / j. nig. soc. phys. sci. 2 (2020) 61–68 66 figure 5: species concentration profile for diffusivity 5.4 × 102 m2/min γ = 1 of (q = 1) single-compartment system at time t = 9 minutes. figure 6: species concentration profile for diffusivity 5.4 × 102 m2/min γ = 1, 2, 3, 4, 5 of interconnected (q = 5) multiple-compartment system at time t = 9 minutes. taken to be 5.4×101 m2/min and 5.4×102 m2/min respectively with the initial concentration of the species in the system taken to be 70mg/l . after spatial discretisation of the resulting pdes and their auxiliary conditions using mol techniques as shown in section 3 with grid points m(1), m(2), m(3), m(4), m(5) taken to be 101 in each compartment, we obtained 505 semi-discrete systems of odes in one independent variable t for (q=5)-compartment system and 101 semi-discrete systems for (q=1)-compartment system. we use matlab ode15s solver due to its efficient way 66 o. adedire & j. n. ndam / j. nig. soc. phys. sci. 2 (2020) 61–68 67 of handling such systems of odes with variable step size, the results are displayed for t = 9 minutes as shown in figures 3 and 4. analogously, the same set of parameters used to obtain figures 3 and 4 but with different diffusivities dϕ and diϕ(i = 1, 2, 3, 4, 5) taken to be 5.4 × 102 m2/min are used and the results are displayed in figures 5 and 6 at t = 9 minutes for (q=1) single-compartment system and interconnected (q=5) multiplecompartment system respectively. simulation of the governing model equations for different values of diffusivities are used to investigate effects of diffusivity on the concentration profile of chemical species undergoing second order chemical reaction kinetics in each compartment of interconnected (q=5)multiple-compartment system. the results from figure 4 show that for diffusivities diϕ(i = 1, 2, 3, 4, 5) given as 5.4×101 m2/min at t = 9 minutes, the values of the concentration at the end of the boundary of each compartment of the system has the concentration values of 32.87 mg/l, 18.19 mg/l, 10.49 mg/l, 4.99 mg/l and 0.00 mg/l respectively. on the other hand, for diffusivities diϕ(i = 1, 2, 3, 4, 5) given as 5.4 × 102 m2/min measured at the same time t=9 minutes, the values of concentration at the end of each compartment are obtained as 49.32 mg/l, 34.43 mg/l, 22.23 mg/l, 11.01 mg/l and 0.00 mg/l for γ = 1, 2, 3, 4, 5 respectively as shown in figure 6. the mid-point of the multiple-compartment system of length 600m is 300m, so that interconnected (q=5)compartment system has its mid-point in the third compartment at the point x = 60m which is equivalent to 300m of the entire system. while concentration of the species at 300m of the system is 13.90 mg/l for diffusivity of 5.4 × 101 m2/min, the concentration of the species is 28.13 mg/l for diffusivity of 5.4×102 m2/min at the mid-point of the system at t=9 minutes. from the results, there is higher concentration profile when diffusivities diϕ(i = 1, 2, 3, 4, 5) are given as 5.4 × 102 m2/min than when diffusivities diϕ(i = 1, 2, 3, 4, 5) are given as 5.4×101 m2/min at t=9 minutes. this means that chemical species with higher diffusivities in (q=5)-compartment system have higher concentration profile than those with lower diffusivities. as time t progressively increases, the concentration profile of chemical species with higher diffusivity still shows higher values than the one with lower diffusivity in (q=5)-compartment system. we also made comparison between (q=1)-compartment systems for different values of diffusivities in order to ascertain whether the behavior exhibited by (q=5)-compartment is consistent with that of (q=1)-compartment system. from the results shown in figure 3 at t=9 minutes, the concentration at the mid-point of the system is 34.88 mg/l when diffusivity is taken to be 5.4 × 101 m2/min and 34.99 mg/l when diffusivity is 5.4 × 102 m2/min. from the comparison made between the (q=1)-compartment system and the interconnected (q=5)-compartment system for different values of diffusivities, there is higher concentration profile in chemical species with higher diffusivity than with the one with lower diffusivity in each system. however, the concentration profile in interconnected (q=5) multiple-compartment system is lower than that of (q=1) single-compartment system. 5. conclusion in this paper, we compared the concentration profile between (q=1)single–compartment and interconnected (q=5)multiple-compartment systems for a chemical species with second order chemical reaction kinetics. from the results, the (q=5)multiple-compartment system has 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[17] h. kreiss & j. lorenz, ”initial-boundary value problems and the navierstokes equation”, academic press, inc, san diego. 68 j. nig. soc. phys. sci. 4 (2022) 740 journal of the nigerian society of physical sciences cathodic corrosion inhibition of steel by musa paradisiaca leave extract titus o. martinsa, edwin a. ofudjea,∗, abimbola a. ogundiranb, ojo a. ikeoluwab, osipitan a. oluwatobib, ezekiel f. sodiyaa, opeyemi ojoa adepartment of chemical sciences, mountain top university, prayer city, ogun state, nigeria bdepartment of chemical sciences, tai solarin university of education, ijebu-ode, ogun state abstract it is reported here that the phytochemicals present on the surface of the musa paradisiaca (mpl) prevent water and other corroding agents from having direct access to the surface of mild steel. these phytochemicals were extracted from the mpl using 70% ethanol solution and weight loss experiment was carried out with variation of temperature, time and concentration hcl and that of the mpl extract in % v/v. the inhibitive effect of m. paradisiaca leaves of mild steel in aqueous solutions of hydrochloric acid were investigated at 25, 35, 45 and 60 oc being immersed simultaneously and independently in the acid medium over a period of 12, 24, 48, and 72 hours. a protecting film appeared on the metal surface by the mpl extract via electron donation, hence, acting as the cathode. the temperature and immersion time were inversely proportional to inhibition efficiency while concentration of mpl is directly proportional. ft-ir of the extract showed oxygen and nitrogen containing functional groups which are the general characteristics of a typical corrosion inhibitor, while the gas chromatography–mass spectrometry (gc–ms) investigation revealed different biomolecules thus suggesting that the plant extract consists of different molecules. doi:10.46481/jnsps.2022.740 keywords: corrosion, corrosion inhibitor, desorption, green chemistry, m. paradisiaca, mild steel article history: received: 27 march 2022 received in revised form: 25 june 2022 accepted for publication: 09 september 2022 published: 05 november 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: e. a. emile 1. introduction corrosion of metals has been around us even before industries came to existence and the need for the prevention or cure has been a major problem. aggressive acids such as hcl are frequently utilized to eliminate mild scale and rust found in iron and steel in chemical industries [1]. these inhibitors were utilized by the industries for their anti-corrosive properties but ∗corresponding author tel. no: +234(0)8060981250 email address: eaofudje@mtu.edu.ng (edwin a. ofudje) they cause some adverse effects on the environment. this has resulted in the synthesis of corrosion inhibitors that are environmentally affable and are called organic corrosion inhibitors (oci) [2]. corrosion inhibition can be carried out by the use of organic or inorganic approach. the organic approach is also known as green corrosion inhibition which is one of the bestknown corrosion safety techniques [3-5]. the use of extract of natural product as inhibitor of corrosion was first evidence in 1930 whilst extracts from chelidonium majus and some other plants were utilized in sulfuric acid, h2so4 medium for the first time [6-7]. with this, there is no or minimal amount of 1 martins et al. / j. nig. soc. phys. sci. 4 (2022) 740 2 pollution caused and it is cost efficient since these are extracts from plants that can be easily isolated. organic compounds with effective corrosion inhibitors often contain conjugated systems, and conjugated aliphatic bonds due to the presence of lone pairs of electrons (n, o, s and p), in aromatic rings, and pi-electrons, often have ionizable parts which are either hydrophilic or hydrophobic in nature [6, 8-10]. more research and developments are emerging in the study of natural corrosion inhibitors as need for environmentally friendly inhibitors are gaining ground. for example, cr2o3 which is insoluble in aqueous media that span across acidic and basic ph range and its susceptible to the underlying metal, it may serve as a barrier to corrosion [11]. hexavalent chromium has shown to persuade irrevocable health challenges through skin irritation, nose, throat and eye and most importantly raises the alarm of lung cancer [12]. for application in coatings and primers, a comparable universal anti-corrosion compound has yet to be found, so that the need for chromate substitutes and ways of deployment of corrosion protection spans many industries. drugs had been studied as corrosion inhibitor at some time [13]. over the decades, green chemistry has shown how basic scientific approaches can benefit from the conservation of the human health and that of the environment by combining areas such as polymers [14-15]. awareness to corrosion and the adaptation of fast, appropriate and accurate control measures holds the key in the eradication of corrosion failures. heterocyclic molecules show essential behaviour as anti-corrosion owning to the available of sulfur, oxygen, and nitrogen atoms in either open or in their ring structure. however, no much works had provided detailed study on the mechanism of corrosion inhibition by these heteroatoms or heterocyclic molecules. in this work, extracted phytochemicals from mpl were used as green corrosion inhibition to decrease the rate of corrosion of steel in acidic medium. gc-ms and ft-ir investigations were carried out to examine the composition of the corrodent. the concept of crystal field splitting was deployed to provide lucid interpretation of the corrosion mechanism. isotherms and thermodynamics were deployed to investigate corrodent sorption onto the steel surface. 2. materials and methods 2.1. preparation of metal coupons the specimen of mild steel was sliced into rectangular coupons by lathe machine into dimension of 20.21 mm by 14.63 mm. the coupons with its edge were polished with emery paper of 600 grades. the thickness was examined with mitutoyo brand of analog micrometer screw gauge and the dimension by mitutoyo digital venier gauge. the surface of the coupons was degrease through immersion in absolute ethanol and then removed for rinse with double distilled water and acetone and thereafter stored in moisture free desiccators before use [16]. 2.2. preparation of plant extract and cold extraction process fresh musa paradisiaca leaves (mpl) were gathered in large quantities, selected from around same source, to ensure no contaminated leaves of different plant or specie. the impurities associated with the leaves were removed through tap water washing and thereafter air-dried to constant weight. it was shredded into smaller pieces to increase drying rate and thereafter dried for 3 weeks. using 70% ethanol for solvent extraction at room temperature, the pulverized sample of mpl was soaked for 48 hours until a dark-green color was observed indicating that extraction has taken place. the darkish-green solution was filtered out which contains the plant extract while the residue is the chaff from pulverized leaves. the procedure was repeated on the chaff to ensure maximum extraction. the filtrate was refluxed to ensure homogeneity of the sample. the concentration of the extract of plants was carried out using a rotary evaporator which removes solvents from the extract thereby making it slurry. the concentrated and purified extract was then stored in desiccator. 2.3. plant extract structural elucidation 2.3.1. gas chromatography mass spectroscopy (gc-ms) investigation a 7820a gas chromatograph coupled to 5975c inert mass spectrometer was used for this analysis. scanning of potential compounds was at 2.62 s/scan rate and was recognized by comparing detected spectral data with nist 14 mass spectral library standard [17]. 2.3.2. ft-ir analysis the functional group evaluation was done using fourier transform infrared spectroscopy (ft-ir) model 8400s (shimadzu, japan). a mortar and pestle were used to mixed the mlp and kbr in the ratio of 1:99% and pestle and compressed into pellets. the ft-ir spectrums were determined from 400 cm−1 to 4000 cm−1 at a resolution and times scanning of 4 and 64 respectively. 2.3.3. weight loss experiments mild steel of 20.21 mm by 14.63 mm were used for weight loss measurement. the total geometric of the coupon’s surface area available to the corroding agent was about 295.67 mm2 and average weight of coupons was 13.18 to 15.61 grams. the weighed coupons were immersed in 20 ml sample bottles containing concentration of corroding agent, hcl to that of the extract (musa paradisiaca) as: 20:0 v/v, 15:5 v/v, 10:10 v/v, 5:15 v/v at 25 ± 2 oc and 60 oc consecutively for 72 hrs. this procedure is repeated further for 1.0 m and 2.0 m of hcl at 60 oc. experiment on weight loss was used to estimate the inhibition efficiency, rate of corrosion and also the half-life of the metal coupon. 3. results and discussion 3.1. fourier transform infrared (ft-ir) analysis investigation on the various functional groups in the plant extract was determined using ft-ir. in the ft-ir spectra shown in figure1 below, it revealed absorption bands belonging to the n-h and o-h bands which occured between 3325 3567 2 martins et al. / j. nig. soc. phys. sci. 4 (2022) 740 3 figure 1. musa paradisiacal leaves ft-ir spectrum cm−1, while the c-h stretching vibrations was noted between 2378 2987 cm−1. the vibration of c=c stretching occured at around 1645 cm−1, while that corresponding to c-n vibration stretching was observed at 1248 cm−1. the absorption vibrations which occurred between 1098 and 1102 cm−1 are due to c-o functional group. ft-ir of the extract showed that oxygen and nitrogen atoms were present in the plant extracts which is the general characteristics of a typical corrosion inhibitor. table 1 shows the different functional groups present on the plant extract. table 1. musa paradisiacal leaves extract ft-ir analysis wave number (cm−1) vibration mode 3325 3567 o-h and n-h 2987 c-h 2378 c-h 1645 c=c 1462 methyl c-h 1248 c-n 1098 1102 c-o gc–ms examination of the corrosion inhibitor revealed different biomolecules thus suggesting that the plant extract consists of different molecules as listed in table 2 with the microgram shown in figure 2. as identified from the gc-ms analysis, the compounds which could be responsible for the weight loss are 6-amino-1,3,5-triazine-2,4(1h,3h)dione, 9octadecenamide, (z)ole amide, polygalitol, cyclotetrasiloxane, octamethyl-, thiophene, 2,3-dihydroand ginsenol repectively. it should be noted that o, n or s and/or p-electrons are present in most of the biomolecules identified from the extracts of the plant. from the gc-ms investigation, the compound which mostly has more inhibition is ginsenol with time retention of 24.063. the inhibition of the rate of corrosion by mpl extract could be assigned to the presence of these phytochemicals which are made up of s, n, o, and other heteroatoms in their structures which are responsible for donating electrons to inhibit the corrosion process and thus serving as adsorption points on the surface of the metal. 3.2. weight loss determination of extract inhibition the metal weight loss during the experiment can be calculated as shown in equation 1 [18]: ∆w = (wi − w j)g (1) given that ∆w is the coupon weight loss, the initial coupon weight is denoted as wi, the final coupon weight is given as w f . the corrosion rate is estimated as: cr,ρ = weight loss (s ur f ace area) × t ime = ∆wt s (2) where cr= ρ and is corrosion rate gcm−2 min−1 or gcm−2 hr−1, ∆w is weight loss, s is surface area, and t is immersion period. the inhibition efficiency can be obtained by using equation 3a or 3b i.e.% = [ wu − wi wu ] × 100 (3a) i.e.% = [ ρ1 −ρ2 ρ1 ] × 100 (3b) such that i.e. is inhibition efficiency, wu is weight of uninhibited mild steel surface, wi = weight of inhibited mild steel surface, ρ1 is rate of rate of uninhibited corrosion, and ρ2 is the rate of inhibited corrosion. for calculating percentage extract yield, %e xtraction y ield = w1 − w2 w1 × 100 (4) where w1 is weight of extract before extraction and w2 is weight of extract after extraction. surface coverage determination for isotherm calculations is given as: θ = i.e 100 (5) given θ is the surface coverage and i.e. is inhibition efficiency. 3.3. effect of mpl extract concentration the mpl extract concentration is the variation in volume of the corrodent to the acid used, hcl, to make a volume of 20 ml. the experiment indicates a reduction in rate at which corrosion occur, thereby increasing inhibition efficiency (figure 3). this also was calculated by comparing a medium with only the acidic medium and another containing the mpl extract, even with temperature variation, the result was still relative. it was established that the best inhibition efficiency was gotten from 5-15% v/v which is 5 ml of the acid and 15 ml of the plant extract. and the highest corrosion rate was seen on 20-0% v/v. as shown in figure 3 below, the corrosion rate is inversely proportional to the concentration of the mpl extraction. moreover, corrosion rate is the inverse of inhibition efficiency. 3 martins et al. / j. nig. soc. phys. sci. 4 (2022) 740 4 figure 2. gc-ms chromatogram of mpl extract table 2. molecules extracted from the gc-ms chromatogram of mpl retention name formula molecular estimated time weight conc. (g.mol−1) 6.841 6-amino-1,3,5-triazinec3h4n4o2 127.29 27 2,4(1h,3h)-dione 6.841 9-octadecenamide, (z)c18h35no 281.477 98 ole amide 12.726 polygalitol c6h12o5 164.16 49 9.324 cyclotetrasiloxane, c8h24o4si4 296.6152 87 octamethyl8.475 thiophene, 2,3-dihydroc4h6s 86.16 50 24.063 ginsenol c15h26o 222.3663 38 3.4. effect of immersion time the variation of inhibition performance of musa paradisiacal leave extract as well as the corrosion rate with immersion time at concentration of 5-15% v/v is shown figure 4. it was observed that the corrosion rate increases with the immersion time until 72 hour. a best immersion time of 12 hours was perceived for the whole concentration gradient of the inhibitor. marginal inhibition sets in after the 48 hours due to desorption which occurs with prolonged exposure of the inhibitor to the acid medium, the corrosion rate tend to decline more rapidly than normal. and i.e. was noticed to have decreased with increased immersion time, this phenomenon could be blamed on desorption of the plant extract which initially prevent corrosion from taking place and also, at initial stage of the immersion, the inhibitory efficiency of the active components of the plant is high but as the reaction proceed, these active components are used up or destroyed; thus giving rise to increase in corrosion rate. more also, the presence of the corroding agent over long time made the corrosion abundant. not forgetting that adsorption is by van der waal forces of attraction, and they are easily reversible [19], by stress. this stress may include pressure, temperature, current, etc. but on this case, it is caused by temperature and long exposure to the corroding agent. according to ascencio et al. [20], he reported the influence of immersion time on corrosion mechanism and there is an inclination from this study. 4 martins et al. / j. nig. soc. phys. sci. 4 (2022) 740 5 figure 3. effect of musa paradisiaca concentration on the inhibition efficiency and corrosion rate figure 4. effect of time of immersion 3.5. effect of temperature inhibitors desorption or decomposition of the inhibitor with increase in temperature are some of the changes that occur on the metal surface [21-23]. variation in cr at different concentration of mpl extract was examined, while putting into constant immersion time at 12 hours and concentration as indicated in figure 5. it was observed that inhibitor efficiency (ie) decreased with a rise in temperature and the best inhibitor efficiency obtained at a temperature of 25 oc. the reduction in ie may be as a result of desorption at higher temperatures of the molecules of inhibitors from the surface of the metal [2425]. corrosion rate (cr) rises as temperature rose which is as a result of desorption of bio-molecules of mpl extract from the surface of the metal, resulting in greater portions of the surface of the metal being exposed to acidic medium. an increase in temperature also brings about increase in weight loss and increase in corrosion rate, cr [24, 26]. when the inhibitor covers the surface of the metal, it hinders corrosion process to set in since the surface of the metal has been coated with the inhibitor. it takes time for the inhibition coating to wear off and consequently, this slows down corrosion [27]. but when temfigure 5. effect of temperature perature increases, it causes the inhibitor to desorb from the surface of the metal thereby, causing a sharp rise in corrosion rate, and resulting in the decrease of the inhibitory efficiency. the reduction in the physical adsorption at higher temperature agrees that inhibition on the metal surface is physical in nature. table 3 shows that the mpl compete favorably well with other plant extracts in literature. 3.6. thermodynamic study thermodynamic properties such as activation energy (ea), free energy change (∆g), entropy change (∆s ) and enthalpy change (∆h) were estimated to further examine the impact of temperature on the inhibitory efficiency of the inhibitor. the arrhenius equation was used to estimate the activation energy: logcr = loga − ea 2.303r · 1 t (6) such that cr denotes corrosion rate, the activation energy is given as ea, the molar gas constant is represented as r and t stands for temperature. then, the slope of the graph of logcr against 1/t is shown in figure 6 from where the activation energy was determined as presented in table 4. the activation energy determined is 0.014 kj/mol/k. all chemical reactions need a minimum amount of energy for the reaction to proceed which is known as activation energy. low activation energy implies that the reaction is likely to occur spontaneously or without the intervention of catalyst or increase in temperature and thus favours the inhibition process. the ∆h and ∆s estimated are listed in presented in table 4 using the slope and intercept of equation 7. ln( cr t ) = ln( r nh ) + ∆s θ r − ∆hθ rt (7) such that h is the planck’s constant, avogadro’s number is given as n, r, t and cr are as previously defined above. the free energy change is concerned with how feasible and spontaneously a reaction will occur and this was obtained using equation 3 below: ∆g = −rt ln(55.5k) (8) 5 martins et al. / j. nig. soc. phys. sci. 4 (2022) 740 6 table 3. comparative inhibition efficiencies as reported in literature with mlp plants inhibition references efficiency (%) neem ark extract 91.73 desai [28] leaves extract of 97 abiola et al. [29] gossipium seed extract of 94 abiola et al. [29] gossipium khillah seed 71 el-etre [30] (ammi visnaga) justicia gendarussa 93 satapathy et al. [31] pennyroyal oil 80 bouyanzer et al. [32] occidimum 69 oguzie [33] leaves of ankado 90.09 desai [34] ark extract adathoda vasica 98.1 shyamala and kasthuri [35] eclipta alba 99.6 shyamala and kasthuri [35] centella asiatica 85 shyamala and kasthuri [35] bitter leaf 90 caroline et al. [36] root extract cassia italica 87.4 al-otaibi et al. [37] tripleurospermum 83.4 al-otaibi et al. [37] auriculatum artemisia sieberi 86.2 al-otaibi et al. [3] gentian olivieri extract 93 evrim et al. [38] extract musa paradisiacal 99.3 present study leave extract and k was estimated as: k = θ c(1 − θ) . (9) all the parameters in equation 9 above are as previously defined above. a negative value of ∆g was obtained, showing that it is a spontaneous reaction. this is also used to determine the pathway of the experiment to either be physical adsorption or chemisorption. if ∆g is negative, then the process is more of physical adsorption, while a positive value of ∆g or relatively higher is chemisorption [39]. from this study, value of ∆g from the study of mpl extract inhibit on mild steel ranges between -5.57 and -4.96 kjmol−1 which further indicate that the reaction pathway of corrosion is somewhat physical adsorption. since van der waal force is responsible for adsorption, the process happens spontaneously i.e. it requires no activation energy. enthalpy, on the other hand is seen to be positive, depicting an endothermic reaction. corrosion is known to be exothermic reaction, because it increases with increase in temperature and this temperature comes from the surrounding. the value of enthalpy obtained from this study is 6.72 kjmol−1 k−1. this indicates that the inhibition process of the metal surface by the inhibitor (mpl) is endothermic in nature. it has been noted from figure 6. a plot to determine activation energy figure 7. a plot to determine entropy and enthalpy experiment that positive enthalpy (endothermic) value and low entropy (-) is best for inhibition which is in line with what was observed in this study and that a rise in temperature reduces inhibition and increase the corrosion rate [27]. entropy depicts the degree of disorderliness of a slope and this is brought about by high temperature that causes the molecule to move randomly due to gained energy which explains desorbing of the mpl extract from the surface of the metal and this will leads to low inhibitor efficiency. thus, a relatively low temperature is required for an effective and efficient inhibition process. so, for corrosion process, it always leads to the evolution of heat, while for the adsorption of inhibitor onto the metal surface, the process is always endothermic because higher temperature will results in desorption of the inhibitor from the surface. so, while corrosion is favoured by exothermic process, inhibition process (adsorption on metal surface) is favoured by endothermic process. 3.7. inhibitor mechanism it is important to note that steel is an alloy of carbon and iron where the carbon atom is bonded to iron to reduce corrosion and the formation of steel is a function of fe − c phase present i.e., ferrite (αfe), austenite (γfe), cementite (fe3c), 6 martins et al. / j. nig. soc. phys. sci. 4 (2022) 740 7 table 4. thermodynamic properties of musa paradisiacal temperature ∆g ∆h ∆s ea (k) (kj/mol) (kj/mol) (kj/mol) (kj/mol) 298 -5.57 308 -5.10 6.72 -197.53 0.014 318 -4.99 333 -4.96 pearlite (αfe+fe3c) [40]. the bonding of c with fe in cementite made the 3d6 electrons of fe to remain in t2g that is low spin [41]. consequently, the four electrons of the carbon are made readily available to the eg-orbital for bonding. the ferrite regions are the most reactive site and tend to corrode easily [42]. the iron particles become oxidized in water when they are lost to the water’s acidic electrolytes, resulting in the formation of fe2+. the formed fe ion at the anode migrates to the cathode. the hydroxyl ions formed from reaction of the hydrogen ion produced with oxygen react with fe+ to produce hydrous iron oxide (feoh), also which is often called rust [43]. when inhibitors are added into corrosive medium in minute quantity, they retard the corrosion process. the inhibition of corrosion process is successful when fe2+ is stabilized from further oxidation to fe3+. either physical adsorption or chemisorption is the process through which inhibitors should readily be adsorbed to the metal surface so as to be an effective shield opposing the metal corrosion [44]. the inhibitor group’s physicochemical properties, like the functional group, the donor atom electronic density and molecular structure depend primarily on one of these adsorption processes. corrosion inhibition mechanism is also accomplished in aromatic rings, nitrogen and oxygen containing molecules which donate their lone pair of electrons to the metal and thus favoring the uptake of these molecules onto the surface of the metal [45]. furthermore, molecules having large structure occupy a larger surface area and thus developing a protective coating. plantain leaves on this note have inhibitive properties as they have been identified by gc-ms analysis to contain compounds such as 6-amino-1,3,5-triazine2,4(1h,3h)dione, 9-octadecenamide, (z)ole amide, polygalitol, cyclotetrasiloxane, octamethyl-, thiophene, 2,3-dihydroand ginsenol which contain elements such as o, n or s which serves as electrons donors. these molecules of inhibitors contain strong ligands such as cn−, n h3−, ocn −, s cn−, etc, which causes low spin forcing the 3d6 electrons into t2g orbital (dxy, dxz and dyz). the degree of splitting is the inverse of the strength of ligands. this disrupts the degeneracy and increasing the energy of the eg orbital which allows for the donating of electron by the ligand to the empty eg orbital (d x2 − y2 and dz2) [46]. when this donated electron by the inhibitor molecules or ions (ligand) is dative covalent bond between the inhibitor and metal, weak bond or van der waal force of attraction is produced, which is physical adsorption (physical adsorption). while an ionic bonding gives rise to chemisorption due to strong electrostatic bond created [47]. figure 8. fe2+ ion and energy levels of d orbital in octahedral field (low spin) figure 9. isotherm plot of langmuir isotherm 3.8. isotherms study adsorption isotherm can reveal information relating to the adsorption mechanism, surface coverage, and adsorption equilibrium constant. to verify the isotherm, the linear relation that exists between θ and c was established using the value of the correlation coefficients (r2) of langmuir isotherm as illustrated in equation 10: log c θ = logc − logb, (10) where b stands for inhibitor adsorption equilibrium constant. figure 9 depicts the plots of logc/θ against the inhibitor concentrations (c) at 303 k. from the calculated value of correlation coefficients (r2) which is 0.971, it can be said that the langmuir isotherm can also be deployed to explain the inhibition process which assumed a physical reaction with weak intermolecular forces. 4. conclusion this work examined the potency of leaf extract of musa paradisiaca (mp) as inhibitor against the corrosion of mild steel 7 martins et al. / j. nig. soc. phys. sci. 4 (2022) 740 8 in acidic medium. an immersion period of 12 hours was perceived as the best performance for mpl for the whole concentration of the inhibitor examined. inhibitor efficiency (ie) reduced with rise in temperature. the reduction in i.e. is as a result of desorption of the molecule of the inhibitor at higher temperatures 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[47] c. alfred, “x the influence of ligand symmetry on chemisorption”, phy. chem. 32 (1974) 177. 9 j. nig. soc. phys. sci. 4 (2022) 945 journal of the nigerian society of physical sciences efficient and intelligent decision support system for smart irrigation monika sainia, ashish kumara,∗, vijay singh maana, deepak sinwarb adepartment of mathematics & statistics, manipal university jaipur, jaipur-303007, india bdepartment of computer and communication, manipal university jaipur, jaipur-303007, india abstract the main aim of present analysis is to develop a novel efficient and intelligent irrigation system (eiis). the proposed irrigation system configured using five components arranged in a series configuration along with the internal cold standby redundancy on sensor unit. the failure and repair rates are exponentially distributed. by using the markovian birth-death process differential difference equations of the model are developed to derive the availability expressions and estimation of parameters. the availability of the system is optimized by employing grey-wolf optimization (gwo) and dragon fly algorithm (da) for efficiency and performance evaluation. the derived results are helpful for the system designers. doi:10.46481/jnsps.2022.945 keywords: markov model, availability, cold standby redundancy, intelligent irrigation system article history : received: 20 july 2022 received in revised form: 28 september 2022 accepted for publication: 14 october 2022 published: 11 november 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: t. latunde 1. introduction agriculture is one of the oldest occupations done by human being. it is one of the strong pillars that contribute to the development of the economic development of any country. a big portion of the population of any country depends on the agriculture for survival. in most of the countries traditional methods of irrigation are adopted for farming but these methods suffer due to several drawbacks. the wasteful use of water and inappropriate irrigation are the major drawbacks while smart irrigation technology ensures the efficient use of water in irrigation. the traditional method of irrigation is very challenging due to several factors like soil nature, crop requirement, etc. while in ∗corresponding author tel. no: +917725922864 email address: ashish.kumar@jaipur.manipal.edu (ashish kumar) smart irrigation system based on soli and environmental measurements irrigation decision can be made. several researchers like morais et al. [1], vellidis et al. [2], kehui et al. [3], giusti and marsili-libelli [4], navarro-hellı́n et al. [5], and sinwar et al. [6] proposed model for smart irrigation systems in various aspects. nasikou et al. [7] proposed a model for smart energy utilization in smart irrigation systems. gu et al. [8] developed a software for irrigation scheduling that work on the crop water stress. shekhar et al. [9] developed an automated irrigation system using intelligent iot. goap et al. [10] used machine learning and open source technologies in development of a internet based irrigation management systems. nawandar and satpute [11] devloped a smart irrigation system based on iot . a lost cost intelligent module has beem utilized in the development of the system. wang et al. [12] proposed a decision support system to manage the canal irrigation. it is observed 1 saini et al. / j. nig. soc. phys. sci. 4 (2022) 945 2 that the performance aspects of these smart system has not been extensively explored so far. reliability and availability are the major concern with the perofrmance of these systems. many researchers worked in the direction of reliability evaluation and optimization of performance of systems. maihulla and yusuf [13] examined the reliability, availability, maintainability and dependability to check the sensitivity effect in a grid-connected photovoltaic systems. this sensitivity analysis shows that close attention and close monitoring are needed to ensure the distribution board’s reliability of the system. venkatakrishnan et al. [14] compare the effect of modified differential evolution algorithm (mde) on wind turbine system along with ga and pso techniques. it concludes that in addition to meet energy demand, the modified de algorithm assists in finding the system’s most cost-effective solution. kumar et al. [15] proposed an efficient model for availability optimization of cooling tower using metaheuristic algorithms. saini et al. [16] optimized the performance of a biological and chemical processing unit using genetic algorithm and particle swarm optimization. saini et al. [17] proposed a stochastic model for availability optimization of condenser used in steam turbine power plants using ga and pso. though area of reliability optimization of smart irrigation systems is still untouched. but efficient and intelligent irrigation-based systems are necessary nowadays for optimum utilization of fresh water. so, the work is proposed develop such an efficient and intelligent irrigation system which ensures the availability of the system as and when required for irrigation. the rest of the work is arranged in six sections: section 2 devoted to the novelty claims by the investigators, section 3 explained the notations and section 4 briefed the system description. mathematical model proposed in section 5 and in section 6 numerical discussion and graphical representation is made followed by concluding section 7. a very less efforts have been made so far for development of efficient and intelligent irrigation systems. so, here an effort is made to develop and efficient and intelligent irrigation system. the efficiency of the system is evaluated in terms of availability of the system for use. by probabilistic arguments and markov methodology, a mathematical model is developed and optimized using grey-wolf optimization (gwo) and dragon fly algorithm (da). the state transition diagram of proposed system is shown in figure 2. the blow mentions points are claimed as the novelty by the inventors. 1. design: a novel design of the proposed system is developed for efficiency improvement. the cold standby redundancy is used at sensor level, and it is verified that it performed better. 2. efficient: the availability of the proposed irrigation system is optimized using various algorithms like grey-wolf optimization (gwo) and dragon fly algorithm (da). in literature so far these algorithms are not employed on availability optimization of irrigation systems. the system is proved efficient when it is operated with the parameters estimated by gwo parameters. 3. intelligent: the proposed system is capable of taking intelligent decisions for irrigation based on the data captured by sensors in real time. 2. notations the nomenclatures used in the development of the model are as follows: s i: ithstate of smart irrigation system pi(t): probability that smart irrigation system is in state i at time t p ′ i (t): derivative of first order of pi(t) a, b1, b2, c, d, e: description of the operative states of smart irrigation system a, b1, b2, c, d, e: description of the failed states of smart irrigation system αi/βii = 1, 2, 3, 4, 5: failure/repair rates of a, b1, b2, c, d, e respectively. 3. system description in this section, the configuration of the efficient and intelligent irrigation system is presented. a smart irrigation system is presented in figure 1 that depicts five main components viz. power unit, active and cold standby sensor unit, raspberry pi, water pump, and irrigation unit. as the name suggest, power unit is needed to provide the power supply to the smart irrigation systems. the power may come directly or through solar power supply. on the other hand, sensor unit indicates a collection of sensors (i.e., moisture, temperature, water, humidity, etc.) needed for decision making by the control unit. the heart of this smart irrigation system is raspberry pi that is responsible for taking efficient irrigation decisions based on data captured by sensor unit. in addition, raspberry pi is utilized to trigger the water pump whenever required. once the water pump is turned on, the irrigation unit (i.e., sprinklers) start working towards irrigation of the field. upon reaching the threshold values by sensor units, raspberry pi initiates actions to turn off the water pump. it is evident that sensor unit is more prone to failures, that’s why the redundancy has been utilized in sensor unit of the system. if primary sensor failed in fetching information any one or more of these than standby sensor starts immediately and failed sensor undergoes for repair. the concept of cold standby redundancy and exponential distributed random variable have been utilized in development of the stochastic model. the repair and switch devices are perfect and sufficient repair facility is available with system. the architecture and state transition diagram of smart irrigation system is depicted in figure 1 and figure 2 respectively. 4. mathematical modelling and analysis here, mathematical model for the smart irrigation system is developed using markov birth-death process. the chapmankolmogorov differential difference equations derived based on figure 2. p0 (t+∆t) = (1−α1∆t−α2∆t−α3∆t−α4∆t−α5∆t) p0 (t) 2 saini et al. / j. nig. soc. phys. sci. 4 (2022) 945 3 figure 1. framework of intelligent decision support system for smart irrigation figure 2. state transition diagram +β1p1 (t) ∆t+β2p2 (t) ∆t+β3p3 (t) ∆t+β4p4 (t) ∆t+β5p5(t)∆t lim ∆t→0 p0 (t+∆t)−p0 (t) ∆t = − (α1+α2+α3+α4+α5) p0+β1p1 (t) +β2p2 (t) +β3p3 (t) +β4p4 (t) +β5p5(t) p ′ 0 (t) = − (α1+α2+α3+α4+α5) p0+β1p1 (t) +β2p2 (t) +β3p3 (t) +β4p4 (t) +β5p5(t) 3 saini et al. / j. nig. soc. phys. sci. 4 (2022) 945 4 lim t→∞ p ′ 0 (t) = − (α1+α2+α3+α4+α5) p0+β1p1+β2p2 +β3p3+β4p4+β5p5 − (α1+α2+α3+α4+α5) p0+β1p1+β2p2 +β3p3+β4p4+β5p5 = 0 (1) p1 = α1p0 β1 (2) p2 = β1p6+β2p7+β3p8+β4p9+β5p10 + α2p0( α1+α2+α3+α4+α5 + β2 ) (3) p3 = α3p0 β3 (4) p4 = α4p0 β4 (5) p5 = α5p0 β5 (6) p6 = α1p2 β1 (7) p7 = α2p2 β2 (8) p8 = α3p2 β3 (9) p9 = α4p2 β4 (10) p10 = α5p2 β5 (11) the initial conditions are as follows: p0 (0) = 1 pi (0) = 0 where i = 1 to 10 (12) the set of linear equations [1]-[11] along with initial conditions [12] constitute the mathematical model for smart irrigation system. after simplification the probabilities at respective states are derived as follows: p0 = p0, p1 = α1 β1 p0, p2 = α2 β2 p0, p3 = α3 β3 p0, p4 = α4 β4 p0, p5 = α5 β5 p0, p6 = α1 β1 α2 β2 p0, p7 = α2 β2 α2 β2 p0, p8 = α3 β3 α2 β2 p0, p9 = α4 β4 α2 β2 p0, p10 = α5 β5 α2 β2 p0 (13) by normalizing criteria that sum of all are transition probabilities is equal to 1 10∑ i=0 pi = 1 (14) we get p0 = [ 1 + ( 1 + α2 β2 ) ( α1 β1 + α2 β2 + α3 β3 + α4 β4 + α5 β5 )]−1 (15) the system availability function is defined as: a0 = p0 + p2 = ( 1 + α2 β2 ) [ 1 + ( 1 + α2 β2 ) ( α1 β1 + α2 β2 + α3 β3 + α4 β4 + α5 β5 )]−1 (16) 5. numerical results and discussion in this section, parameter estimation of the failure and repair rates of the smart irrigation system is done using swarmintelligence based algorithms namely gray wolf optimization (gwo) and dragon fly algorithm (da). the possible search space of the decision variables is appended in table 1. the estimated values of the parameters with respect to 30 iteration levels at several population sizes is derived and appended in table 2. the availability of smart irrigation system derived at various population sizes are appended in figure 3-6. the execution time of the program taken by algorithms are appended in table 6. the simulation study is performed using r software on windows 10 64-bit operating system having 8 gb of ram and intel core i5 8th generation cpu. the range of these decision variables is provided in table 1 as follows: here parameters of failure and repair rates for gwo are constant for different population sizes and system attains its maximum availability in early stage of simulation process. parameters values change rapidly when using da algorithm. it is seen that failure rate of controller arduino uno increase rapidly for population size 600 and water pump failure increase with population size 400 and 1000. table 3 appended the various parameters values after 50 iterations corresponding to various population sizes. it also shows that by using gwo, parameters for failure rates and repair rates have its minimum and maximum value simultaneously and system attains its maximum availability. for da, range of parameters of failure rates increase for sub-system sensor/standby sensor unit, controller arduino uno and water pump for population size 800 and 1000. table 4 highlights the estimated values after 70 iterations on different population sizes. for gwo, system attains its maximum availability. while using da, range of parameters of failure rates increase rapidly for sub-system sensor/standby sensor unit for population size 800. table 5 reported the estimated parametric values of failure and repair rates after 90 iterations at various population sizes. for gwo, system attains its maximum availability. while using da, range of parameters of failure rates increase rapidly for sub-system controller arduino uno for population size 600 and 1000. the best parameter values of failure and repair rates, derived by simulation process done in r studio for gwo and da optimizations. parameters obtained for different population sizes and various iterations. these are the best parameters 4 saini et al. / j. nig. soc. phys. sci. 4 (2022) 945 5 table 1. range of the decision variables sub-system range of failure-rate (α) range of repair-rate (β) power unit α1 = [0.000003, 1.99] β1 = [0.000009, 2.11] sensor unit &stand by sensor unit α2= [0.000005, 1.89] β2 = [0.000006, 2.30] controller arduino uno α3 = [0.000001, 1.53] β3 = [0.000008, 2.57] water pump α4 = [0.000004, 1.24] β4 = [0.000007, 2.08] smart valve α5 = [0.000002, 1.32] β5 = [0.000019, 2.34] table 2. parameter estimation of various failure and repair rates after 30 iterations and different population sizes by using gwo and da iter\np 400 600 800 1000 gwo α1 0.0000030 0.0000030 0.0000030 0.0000030 α2 0.0000050 0.0000050 0.0000050 0.0000050 α3 0.0000010 0.0000010 0.0000010 0.0000010 α4 0.0000040 0.0000040 0.0000040 0.0000040 α5 0.0000020 0.0000020 0.0000020 0.0000020 β1 1.86464 2.11 2.11 2.11 β2 2.3 2.3 2.3 2.3 β3 2.57 2.57 2.57 2.57 β4 2.08 2.08 2.08 2.08 β5 2.34 2.34 2.34 2.34 da α1 0.0000030 0.0000030 0.0000030 0.0000030 α2 0.0000050 0.0000050 0.243093 0.0000050 α3 0.0000010 1.53 0.0000010 0.0000010 α4 1.24 0.0000040 0.0000040 1.24 α5 0.0000020 0.0000020 0.0000020 0.0000020 β1 2.11 2.11 2.11 2.11 β2 0.6341965 2.118504 2.3 2.3 β3 2.57 2.57 1.130091 2.57 β4 2.08 2.08 2.08 2.08 β5 2.34 2.34 1.496042 2.34 table 3. parameter estimation of various failure and repair rates after 50 iterations and different population sizes by using gwo and da iter\np 400 600 800 1000 gwo α1 0.0000030 0.0000030 0.0000030 0.0000030 α2 0.0000050 0.0000050 0.0000050 0.0000050 α3 0.0000010 0.0000010 0.0000010 0.0000010 α4 0.0000040 0.0000040 0.0000040 0.0000040 α5 0.0000020 0.0000020 0.0000020 0.0000020 β1 2.11 2.11 2.11 2.11 β2 2.3 2.3 2.3 2.3 β3 2.57 2.57 2.57 2.57 β4 2.08 2.08 2.08 2.08 β5 2.34 2.34 2.34 2.34 da α1 0.0000030 0.0000030 0.0000030 0.0000030 α2 0.0000050 0.0000050 1.047884 1.414718 α3 0.0000010 0.8860956 1.190195 0.02381295 α4 0.0000040 0.0000040 0.0000040 1.24 α5 0.0000020 0.0000020 0.2853557 0.0355639 β1 0.239885 2.11 0.7081718 2.11 β2 2.3 2.3 2.3 2.3 β3 0.6369218 1.704359 2.57 2.57 β4 2.08 0.1690187 2.08 2.08 β5 2.34 1.701183 2.34 2.34 5 saini et al. / j. nig. soc. phys. sci. 4 (2022) 945 6 table 4. parameter estimation of various failure and repair rates after 70 iterations and different population by using gwo, da iter\np 400 600 800 1000 gwo α1 0.0000030 0.0000030 0.0000030 0.0000030 α2 0.0000050 0.0000050 0.0000050 0.0000050 α3 0.0000010 0.0000010 0.0000010 0.0000010 α4 0.0000040 0.0000040 0.0000040 0.0000040 α5 0.0000020 0.0000020 0.0000020 0.0000020 β1 2.11 2.11 2.11 2.11 β2 2.3 2.3 2.3 2.3 β3 2.57 2.57 2.57 2.57 β4 2.08 2.08 2.08 2.08 β5 2.34 2.34 2.34 2.34 da α1 0.0000030 0.0000030 0.0000030 0.0000030 α2 0.0000050 0.05620236 1.89 0.0000050 α3 0.0000010 0.0000010 0.0000010 0.0000010 α4 0.4775554 0.1183404 0.0000040 0.1965477 α5 0.0000020 0.8200995 0.0000020 0.0000020 β1 2.11 0.00640018 2.11 2.11 β2 0.1106202 2.3 2.3 2.3 β3 2.57 0.7383446 1.424582 2.267976 β4 1.718693 1.26267 2.08 0.8067474 β5 1.063321 2.34 2.34 2.34 table 5. parameter estimation of various failure and repair rates after 90 iterations and different population sizes by using gwo and da iter\np 400 600 800 1000 gwo α1 0.0000030 0.0000030 0.0000030 0.0000030 α2 0.0000050 0.0000050 0.0000050 0.0000050 α3 0.0000010 0.0000010 0.0000010 0.0000010 α4 0.0000040 0.0000040 0.0000040 0.0000040 α5 0.0000020 0.0000020 0.0000020 0.0000020 β1 2.11 2.11 2.11 2.11 β2 2.3 2.3 2.3 2.3 β3 2.57 2.57 2.57 2.57 β4 2.08 2.08 2.08 2.08 β5 2.34 2.34 2.34 2.34 da α1 1.059087 0.0000030 0.5533655 0.0000030 α2 0.0000050 0.4501209 0.0000050 0.0000050 α3 0.6671488 1.53 0.0000010 1.53 α4 0.0000040 0.4808177 0.0000040 0.1965477 α5 0.0000020 0.1602266 0.0000020 0.0000020 β1 2.11 1.725929 2.11 2.11 β2 0.5978837 1.181086 2.3 2.3 β3 1.702016 2.57 2.57 2.57 β4 1.457232 1.438981 0.2638387 2.08 β5 2.34 2.34 2.34 2.34 table 6. elapsed time (in seconds) of the gwo and da algorithms used in finding the optimum availability with respect to iterations at various population size iteration population size 400 600 800 1000 gwo da gwo da gwo da gwo da 30 3.96 9.37 3.72 7.01 4.5 8.52 4.32 7.89 50 4.18 7.75 4.61 7.82 7.98 7.69 4.2 12.61 70 3.81 9.8 3.55 7.1 3.79 7.05 3.58 7.39 90 3.37 7.17 3.92 7.75 3.63 7.2 4.35 7.84 6 saini et al. / j. nig. soc. phys. sci. 4 (2022) 945 7 figure 3. availability of smart irrigation system at various iterations at pop. size =400 figure 4. availability of smart irrigation system at various iterations at pop. size =600 figure 5. availability of smart irrigation system at various iterations at pop. size =800 for which system attains its maximum availability. for gwo, parameters for failure rates and repair rates have its minimum and maximum value simultaneously and system attains its maximum availability at early stage of simulation. for da the values of parameters are fluctuating rapidly, and sub-systems need adequate maintenance at time. in figure 3, graphical representation of the optimum availability of system using gwo and da with respect to various iterations with population size 400 is shown. it is seen that at iteration 30 and 50 both techniques attain the same and maximum availability after that availability for da is decreasing with increasing the iteration size and for gwo it remains same at 0.9999954. in figure 4, graphical representation of the optimum availability of system using gwo and da with respect to various iterations with population size 600 is shown. here availability for gwo remains constant at 0.9999954 for each iteration. availability for da slightly increase from 0.6268276 to 0.6919241 for iteration 30 and 70 respectively but iteration 90 availability decreases rapidly to 0.4754869. figure 5 shows the graphical representation of the optimum availability of system using gwo and da with respect to various iterations with population size 800 is shown. availability for gwo remains constant at 0.9999954 for each iteration. figure 6. availability of smart irrigation system at various iterations at pop. size =1000 availability for da decreases rapidly from 0.9899925 to 0.5788153 in-between iteration 30 to 50. then after it increases to 0.7922205 for iteration 90. figure 6 is the representation of the optimum availability of system using gwo and da with respect to various iterations with population size 1000. system attains its maximum availability 0.9999954 in early stage of population size and iteration by using gwo and then remain constant for higher population size and iterations. for da availability varies between 0.804096 at iteration 70 to 0.5391205 at iteration 50. table 6 show elapsed time taken by optimization techniques which includes time taken by system and optimization technique and provide the total execution time. the variation in time taken to complete the task by using gwo and da can be easily seen. time taken by gwo is almost half of the time taken by da. it is also seen that gwo attains the maximum availability at early stage with respect to da. 6. conclusion in present work, a novel efficient and intelligent irrigation system is developed using the concept of cold standby redundancy. the availability expression for the model is derived and optimized using the grey-wolf optimization (gwo) and dragon fly (da) algorithms. the parameters of all failure and repair rates are estimated by both algorithms. the maximum availability derived is 0.9999954 in the search space by both the algorithms. it is observed that dragon fly algorithm is not given sustain results with the increase of number of iterations. though the elapse time taken by grey-wolf optimization (gwo) is sufficiently less in comparison to dragon fly (da) algorithm. so, it is recommended that the system when operated according to the parameters estimated by grey wolf optimization perform more efficiently. the present work may be further extended to the component level investigation by using some other nature inspired algorithms. the concept of simultaneous failure and redundancy can be involved in further study. the 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[17] m. saini, n. gupta, v. g. shankar & a. kumar, “stochastic modeling and availability optimization of condenser used in steam turbine power plants using ga and pso”, quality and reliability engineering international 38 (2022) 2670. 8 j. nig. soc. phys. sci. 4 (2022) 926 journal of the nigerian society of physical sciences a comparison of the performance and energy resolution of cdte and si detectors in the x-ray fluorescence studies of metal samples and alloys iniobong p. etima,b,∗, rebecca emmanuel mfonc adepartment of physics, university of surrey guildford surrey gu2 7xh, united kingdom bdepartment of physics, university of calabar, p.m.b 1115 calabar, cross river state, nigeria cdepartment of physics federal university of lafia, p.m.b 146 lafia, nasarawa state, nigeria abstract x-ray fluorescence provides a powerful means of non-destructively determining the elemental composition of a sample. x-rays from a molybdenum (mo) source was fired on copper, molybdenum, lead, steel and brass samples to determine their composition and relative abundance of their constituent elements. two different detectors : the cadmium telluride (cdte) and silicon(si) detectors were used to pick up the signals from the scattering of the x-rays at the sample surfaces and their energy resolutions as well as efficiencies were compared. with a non-noisy amplifier, the si detector had a higher resolution (0.27 % ) when compared to the 0.38 % for the cdte detector but it had a lower efficiency when compared to that of the cdte detector. it was also discovered that higher energies produced lower detector efficiencies. doi:10.46481/jnsps.2022.926 keywords: x-ray fluorescence, cadmium telluride detector, silicon detector, energy resolution, energy efficiency article history: received: 09 july 2022 received in revised form: 10 october 2022 accepted for publication: 11 november 2022 published: 27 november 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: edward anand emile 1. introduction x-ray fluorescence (xrf) is a technique which non-destructively analyses a material to identify the elements that make it up [1]. a material bombarded with high-energy x-rays absorbs it and emits characteristic secondary x-rays of lower energy but higher energy x-rays can also be produced using an x-ray tube [2]. usually a heated cathode emits electrons, which are accelerated using an electric field to hit a metal target called an anode. when these electrons interact either with the orbit ∗corresponding author tel. no: +234(0)8034451806 email address: ini2etim@unical.edu.ng (iniobong p. etim ) electrons of the atoms, they produce characteristic x-rays but when they interact with the nuclei of the target, they generate a continuum spectrum (bremsstrahlung radiation) [3]. the continuum spectrum can have energies between zero and the maximum energy of the incident electrons. the x-rays produced are then incident on a sample and if the incident photons have enough energy, they excite or ionize electrons of the inner orbits (photoelectric effect), producing a hole in the inner orbit and making the atom unstable. an electron from the outer orbits moves in to fill that hole resulting in the emission of a photon with energy equal to the energy difference of the initial and final states producing x-ray fluorescence (xrf) [4]. the energy of xrf is measured using a semiconductor de1 etim & mfon / j. nig. soc. phys. sci. 4 (2022) 926 2 tector. the signal from the detector is transferred to an amplifier and then to a multi-channel analyzer (mca). the spectrum produced is visible and can be displayed on a computer screen. xrf is used for monitoring compliance with limits set by monitoring bodies [5] and for investigating soil samples to ascertain the extent of chemical weathering and possible vulnerability to gully erosion [6]. xrf is also used for forensic geoscience investigations and is a tool for environmental and criminal investigations [7]. xrf can be used to analyze elements in a material as the produced x-rays characterize each element and directly relates its amount in the material. that method is mainly used for metals, ceramics and glasses and apart from that, it is used in archaeology and geochemistry. in this experiment, xrf produced from different metals and alloy samples are measured using two types of detectors. the characteristic x-ray peaks emanating from the samples aid the identification of the metals it contains and for alloys, the ratio of the different elements it contains was calculated. the energies of the characteristic x-rays from these samples were also calculated and obtained results were used to identify the metal/metals present in the investigated samples. the resolution and efficiencies of the two detectors as the angle the sample makes with the x-ray beam direction was changed to minimize self-absorption, were obtained and compared . the implication of all obtained results are discussed. 2. background theory 2.1. x-ray fluorescence the process of xrf is realized through the shell model of the atom. this model consists of a series of electron orbits (or shells) surrounding an atomic nucleus as shown in figure 1. the proximity of a particular shell to the atomic nucleus reflects how tightly bound the electrons it houses are to that nucleus. thus the k-shell would then be the shell with the most tightly bound electrons. if an atom is excited from its ground state from an impinging photon via the photoelectric effect, the electrons would rearrange themselves to de-excite the atom resulting in the emission of a fluorescent x-ray photon [4]. to conserve momentum and energy, an incident photon is most likely to create a vacancy in the k-shell. consequently, an electron from less bound shells (l, m etc) will be transferred to the vacancy emitting the fluorescent x-ray photon in the process with energy equal to the difference in binding energy between the two shells. the possible origins of a fluorescent xray are illustrated in figure 2 [4]. competing with xrf is the auger process which dominates for lower atomic number (z) atoms. this is where the atom de-excites through the emission of an outer shell electron as opposed to an x-ray photon [4]. mosely’s law describes how the energy of the emitted fluorescent x-ray is proportional to z2 with a proportionality constant that depends on the shells involved in the emission as shown in figure 2. (z is the atomic number of an element) more specifically the relationship for the k and l line’s x-ray frequency with z is [8]: kα : vx−ray ∼ (z − 1) 2 (1) figure 1. the atomic shell structure. (http://earthsci.org/education/teacher/basicgeo/miner/electshells.gif) figure 2. an illustration of possible sources of fluorescent x-rays. lα : vx−ray ∼ (z − 7.4) 2 (2) the additional constants of 1 and 4.5 account for the screening effect from outer shell electrons on the fluorescent x-ray. the xrf is considered a powerful tool for identifying elemental composition in materials because of the x-ray’s energy strong dependence on the z of its origin atom [8]. 2.2. self-absorption and secondary emission self-absorption a frequent occurence in fluorescent x-ray spectroscopy is where x-rays produced from one atomic class in an alloy are absorbed by an atom of different atomic numbers. fernandez et al. [9] further studied this phenomenon and sought for a means of correcting the intensity of x-rays for secondary emission. they suggested that the total intensity it of a characteristic line from an atom is a combination of xrf from the ‘primary’ excitation source, i1 and the absorption of characteristic x-rays from other atoms which may then emit ‘secondary’ x-rays, i2 [9]. thus: it = i1 + i2 (3) by considering the propagation plane of the incident and the fluoresced beam of photons as shown in figure 3 the intensity of secondary emission dependence on the geometry can be appreciated. the effects of secondary emission can be removed by noting that i2 → 0 as the angle of the propagation plane α → 90o. 2 etim & mfon / j. nig. soc. phys. sci. 4 (2022) 926 3 figure 3. propagation plane of incident and take off x-rays varied with α [9 ]. figure 4. the i2 probability bubble model for (c) α = 0o, (a) 0o < α < 90o and (b) α = 90o [9]. the mechanism of this process is described by a probability bubble by considering an infinitely long sample. primary fluorescent x-ray photons are produced at the very center of the sphere and travel isotropically across the radius of the bubble and are absorbed producing a subsequent secondary emission at the surface of the sphere. when α = 0o the position of the bubble is such that it is completely immersed in the material producing a large number of secondary photons which are emitted normal to the surface of the sample. figure 4 shows that by increasing α, the probability bubbles begin to leave the material meaning that less secondary x-rays are produced. figure 5 shows the results of a monte carlo simulation (with 50,000 photons) published by fernandez et al. [9] which shows the relationship between the intensity of secondary photons as a function of angle α for a binary. note how it falls to zero at 90o as in accordance with the probability bubble model. throughout this process, the intensity of the primary photons remains invariant to α, hence when α = 90o, it = i1 [9]. 2.3. semiconductor detectors when a photon impinges on a semiconductor detector, electronhole pairs are produced. electrons move from the valence band to the conduction band and due to the electric field across the detector, they move to the positive electrode. a hole that is produced in the first position of the electron moves to the negative electrode. that charge created is measured across the detector. figure 5. secondary emission dependence on α from a monte carlo simulation developed in pascal with 50,000 initial photons [9]. in this experiment, two different semiconductor detectors namely the silicon (si) and the cadmium telluride (cdte) detectors were used. in the si detector, 3.6 ev is required to produce one electron-hole (e/h) pair while in the cdte detector, 4.43 ev is required [4, 10]. in both amptek detectors used, there is a high voltage across them to help in the collection of the charge and produce an increase in their efficiency. conversely because that voltage is high, the resolution of the detector reduces and the effect from leakage current becomes more significant. that is why a thermoelectric cooler is installed just next to the detector and the pre-amplifier so that the electronic noise of the system is reduced and the resolution of the detector is enhanced as the peaks seem narrower. apart from the cooler, there is a small monitor to read the temperature directly and the beryllium window is used for the low energy photons (x-rays) [10]. at the range of low x-rays energies that were measured in this experiment, the si detector resolution was compared to that of the cdte detector. thus the resolution of a peak given as a percentage is: resolution(%) = fw h m centroid × 100 (4) (where fwhm is the full width at half maximum of the peak and ‘centroid’ is the centroid of the peak defined by the software). at that chosen energy range, the cdte detector has an extremely high efficiency. thus to find the efficiency of the si detector, which was assumed should be lower than that of the cdte detector, the cdte detector was assumed to be 100% efficient and the ratio of the two intensities gives the efficiency of the si detector[10]. the error in resolution can be calculated using equation 5 : σr = r · √( σfw h m fw h m )2 + ( σcentroid centroid )2 (5) 3 etim & mfon / j. nig. soc. phys. sci. 4 (2022) 926 4 figure 6. experimental setup showing the copper collimator, lead shield and detector. 3. methodology 3.1. materials an x-ray tube with an air-cooled mo k as a primary x-ray source was used. si and cdte detectors with gains 100 and 3 respectively were used. other instruments included an amplifier, a dp4 digital pulse processor, a computer with software maestro which was used for data acquisition and samples studied were single element metals and alloys in plate form. 3.2. system geometric setup/calibration of cdte detector a copper sample was placed 30 cm from the x-ray source and the sample plate was turned at 45o to the primary x-ray beam. the detector was placed at 45o to the sample and 25 cm from it such that the angle between the x-ray beam and the detector axis is 90o. in this setup, there was no beam collimator nor a shield between the collimator and the detector and so the detector energy calibration was done using americium241(241am) .with the x-ray set at 50 kv, the sample was scanned for 500 s and a spectra was obtained. the spectra obtained had a large continuum background with no identifiable characteristic x-ray peaks and so another set up which was a modification of the first was created. a copper collimator was placed in front of the x-ray source to bring to a focus on a limited area of the target, the primary xray beam and prevent them from spreading to hit the material around the system instead of the target. furthermore, a lead shield was placed between the collimator and the detector to prevent the secondary x-rays from the collimator getting to the detector. the distance between the x-ray source and the sample was reduced to 20 cm while that between the sample and the detector was 10 cm. the angle between the primary x-ray beam and the detector axis was maintained at 45o (figure 6). with the x-ray source at 50 kv and for 500 s, another spectra of the calibration sample was obtained. the calibration plot is as shown in figure 7. the spectra for single-element samples like copper and lead as well as for alloys (brass and steel) were obtained and analysed. the effects of secondary emission from the sample were also investigated using si detector as illustrated in figure 3. with the brass sample placed in the sample holder such that the incident x-ray beam was normal to the plane and with a brass collimator in place, the spectra for copper after 500 seconds was obtained. for x-rays of intensity 8.048 kev, the characteristic x-ray peak for copper was obtained. this process was repeated for various values of α ranging from 0o to 90o. using the same setup but without a sample, the background was measured and obtained results are presented in table 2. using the cdte detector and the setup described above, the spectra of different samples were also collected in order to identify their composition from their characteristic x-ray peaks. obtained results are presented in table 2. for the alloys, the ratio of their constituent elements was calculated by assuming that the amount of each element in the alloy sample is proportional to the number of counts in the peak of the characteristic lines of each element. the most abundant element was used as a reference to determine the abundance of the other constituent elements in the sample. as was done with the cdte detector, the spectra of the same samples were analysed with the si detector to get each sample composition. the resolution of the si detector was measured and obtained results were compared with those for the cdte detector. the efficiency of the si detector was also calculated and the ratio of the si/cdte intensities was measured and plotted as a function of the theoretical photon energy. the error for the ratio was calculated using: σi = √√( σis i icdt e )2 +  is i ·σicdt e i2cdt e 2 (6) the uncertainties for the intensities, of si and cdte detectors: σis i = σicdt e = σn t (7) the error of the number of counts was found from the poisson statistics. the theoretical energy assumed that it does not have an error. furthermore, for the brass sample, the effect of varying the propagation plane angle α for the cu 8.048 kev peak was also studied and the result is presented in figure 12. 4. results and discussion the calibration plot for the cdte detector performed using the am source is shown in figure 7. a first-order polynomial fit is visible above using the equation y = ax + b, where a = (0.017 ± 0.003) kev and b = (−0.061 ± 0.006) kev . the values and errors for a and b were calculated from the software “originpro8” using the least-squares method. the error bars of the channel number are very small and therefore were not visible in the graph. the slope of the above plot gave the gain of the system as (0.017 ± 0.003) kev/channel. the cdte detector resolution was calculated using the spectrum for the calibration and equation 4 and the error for each peak was calculated using equation 5. obtained results are as presented in table 1. the errors were very small so more decimal places were introduced and the cdte detector resolution was high at 0.38 % with the poorest resolution being 3.37 %. 4 etim & mfon / j. nig. soc. phys. sci. 4 (2022) 926 5 figure 7. theoretical energy against the channel number (cdte detector). table 1. energy resolution of the cdte detector. peak energy error in energy (kev ) resolution (%) resolution (%) 8.01 0.38 0.0001 13.90 3.37 0.0013 17.70 2.04 0.0005 59.50 0.99 0.0001 the energy of the characteristic x-rays of each sample was measured using the maestro software which translated the obtained results to the peak centroid energy. the uncertainty for the energy was calculated using the standard error in the mean. the energies of the xrf of the samples in all cases except for lead were derived from the main peaks found in the spectra due to kα characteristic line. kα x-rays from lead have higher energies when compared with the range of energies measured (∼ 79 kev ), hence only lα and lβ transition were seen in the spectrum and recorded in table 2. table 2 shows the characteristic x-ray energies, the errors as well as the percentage differences between the obtained values and the published values for the other samples. the errors as in the first case are very small. for lead, more than one transition was presented and the x-rays used were of high intensity and they were easily distinguished. the percentage difference was calculated using equation 8 : %di f f erence =( energyt heoretical − energymeasured energyt heoretical ) × 100 (8) the background measurements show two peaks which are present in all spectra. the first peak at 16.61 kev is for the xrf produced from the x-ray tube. the x-ray tube is made of mo and thus the visible peak in the spectrum has the energy of the kα x-ray of mo. furthermore, in all spectra, a peak with energy figure 8. spectra of characteristic x-rays of lead and brass. 7.80 kev was found. that peak is due to the xrf produced from the collimator which is made of copper. consequently, samples with a small amount of copper were not measured as the peak from copper was seen as a background peak. for the brass sample, the most abundant element is copper. with this as the reference, the ratio of the counts for zn peak to the counts of the cu peak was calculated and found to be 1:0.25. by the same reasoning, for steel, the main element is iron (fe) but nikel(ni) and copper (cu) are also available. thus the ratio of fe : ni : cu was found to be 1:0.20:0.03. the spectra showing the characteristic fluorescent x-rays for steel and brass are shown in figure 8. 4.1. silicon (si ) detector calibration plot the calibration plot for the si detector is as shown in figure 9. a first-order polynomial fit is visible above using the equation y = ax + b, where a = (0.038 ± 0.002) kev and b = (−0.064 ± 0.002) kev . the values and errors for a and b were calculated from the software “originpro8” using the least-squares method. the error bars of the channel number are very small and therefore not visible in the graph. the slope of the above plot gave the gain of the system which was (0.038±0.002) kev/channel. the energy resolution of the si detector was measured and are as presented in table 3. 5 etim & mfon / j. nig. soc. phys. sci. 4 (2022) 926 6 figure 9. calibration plot of the silicon detector. table 3. energy resolution of the si detector. peak energy error in energy (kev ) resolution (%) resolution (%) 13.90 4.29 0.0014 17.70 3.62 0.0010 59.50 0.27 0.0000 the errors for the si detector as in the case of cdte were very small so more decimal numbers were used in order to give a more accurate value. the peaks measured from the background were the same peaks found when the cdte detector was used. a comparison of the energy resolution of the si detector (table 3) with that of the cdte detector (table 1) shows that the resolution of the si detector is lower. that was strange because at lower energy ranges, the si detector has a higher resolution compared with the cdte detector. this discrepancy was attributed to the noisy amplifier which may have reduced the resolution of the system. as before, the errors were very small so more decimal numbers were introduced. the ratio of the elements in brass, cu:zn was found to be 1:0.70. for the steel, as it is shown from table 4, one peak was visible so the ratio of the different elements could not be calculated. this perhaps was due to the lower resolution of the si detector using that amplifier. for a visible difference in the resolution of the si detector, another amplifier was used (with shaping time equal to 12 µs). a new calibration plot (figure 10) was done for the si detector with the new amplifier and the obtained results were found to be completely different. a first-order polynomial fit is visible above using the equation y = ax + b, where a = (0.014 ± 0.003) kev and b = (0.014±0.001) kev . the values and errors for a and b were calculated as above. the error bars of the channel number are very figure 10. new calibration plot for si detector used with new amplifier. figure 11. silicon detector efficiency relative to cdte detector. figure 12. total intensity dependence on α for a cu 8.048 kev peak. small and therefore cannot be visible in the graph. the slope of the above chart gives the gain of the system. therefore, it can be seen that the gain is equal to (0.014 ± 0.003) kev/channel. the resolution was then calculated and compared with the previous one (table 5). 6 etim & mfon / j. nig. soc. phys. sci. 4 (2022) 926 7 table 4. showing the sample composition and identified elements with their characteristic x-ray energies and associated uncertainties for si detector sample element characteristic uncertainty published percentage x-ray energy of the x-ray x-ray difference (kev) energy (kev) energy (kev) background mo 16.41 0.00319 17.479 6.116 cu 7.44 0.00681 8.048 7.555 copper cu 8.00 0.00009 8.048 0.596 mo mo 17.33 0.00029 17.479 0.852 lead pb 14.73 0.00093 14.765 0.237 pb 12.53 0.00026 12.614 0.666 pb 10.48 0.00067 10.551 0.673 brass cu 7.96 0.00014 8.048 1.093 zn 8.57 0.00013 8.639 0.799 steel fe 8.53 0.00012 6.404 33.198 ni 8.53 0.00012 7.478 14.068 cu 8.53 0.00012 8.048 5.989 table 5. new energy resolution for si detector with new amplifier peak energy energy error in new resolution error in new (kev) resolution (%) resolution (%) (%) resolution (%) 13.90 4.29 0.0014 2.13 0.0005 17.70 3.62 0.0010 1.65 0.0004 59.50 0.27 0.0001 0.04 0.0001 table 6. showing the new sample composition for brass and steel with their characteristic x-ray energies and associated uncertainties for si detector with new amplifier sample element characteristic uncertainty published percentage x-ray energy of the x-ray x-ray difference (kev) energy (kev) energy (kev) brass cu 6.29 0.00066 6.404 1.780 zn 8.61 0.00037 8.048 6.983 fe 8.00 0.00037 6.404 1.780 steel cu 7.98 0.00464 8.048 0.845 ni 7.01 0.00084 7.478 6.258 fe 6.36 0.00032 6.404 0.687 with the new amplifier, a higher resolution of the si detector was obtained from the new spectra produced. for the errors as before were very small and so more decimal numbers were used in order to give a more accurate. to test how efficient the si detector was with new amplifier, the spectra for brass and steel were acquired and the results are displayed in table 6. the errors were small and the peaks were easily distinguished. the ratios of the constituent elements in these two alloys were calculated and for brass, a peak due to iron (fe) though of a lower intensity could be distinguished because now the detector resolution was higher (recall that with the cdte detector that peak was not visible). the ratio cu:zn:fe in brass was found to be 1:0.53:0.01 while the ratio fe:ni:cu for steel was 1:0.14:0.01. the efficiency of the cdte detector was assumed to be 100% and thus the ratio of the intensities gives the efficiency of the si detector and high energies were found to produce lower detector efficiencies. this is because photons with higher energies move faster in the detector and intereact less with the sample so the number of counts measured is less and resulted in a lower efficiency. the graph which shows the total number of counts from fluorescent x-rays in the 8.048 kev cu characteristic x-ray peak of the brass sample for different inclination angles α shows that the intensity decreases as α increases suggesting a reduction in the enhancement from self-absorption and secondary x-ray emission as predicted by fernandez’s bubble model. for this data to correctly fit the model, the photons from the x-ray tube have to be heavily collimated to produce a thin beam. the monte carlo simulation results in figure 5 was produced from modeling a point beam but the same level of collimation could not be achieved experimentally as the risk of no xrf hitting the detector at all and also the scattering continuum would dwarf any characteristic x-ray peaks present. moreover, the model assumes an infinitely long and thick sample, although 7 etim & mfon / j. nig. soc. phys. sci. 4 (2022) 926 8 due to the probabilistic nature of photon, not all the x-rays will be absorbed, and the sample thickness will have a direct consequence on the number of counts. 5. conclusion this project developed a means of non-destructively determining the composition of metals through a study of the energy of their emitted characteristic x-rays. a mo source x-ray tube collimated to scatter off a sample at a 45o incidence and takeoff angle to a semiconductor detector was used. the intensity of the characteristic x-ray energy peaks from different samples was used for estimating the abundance of particular elements in each sample. the silicon detector superior resolution over that of the cdte detector proved to be a much better tool for correctly identifying energy peaks. however, its reduced efficiency translated to poor counting statistics at higher energies as seen from the reduced energy resolution for a 241 am γ − peak at 59.5 kev for the silicon detector. thus higher energies were found to produce lower detector efficiencies. the peak intensities from steel and brass alloys were used to estimate element composition and these produced reasonable abundances of zinc compared to copper. the investigation into the correction of secondary emission was done using the ‘probability bubble’ model described by fernandez et al. [9]. experimentally the dependence on total intensity on the angle α compares well with the model. the limiting factors in this technique however lie in the level of collimation that can be placed on the beam without reducing counting statistics to an unacceptable level and preventing the scattering component from dwarfing any characteristic lines present. to extend this study, it might be necessary to develop a setup that can measure the angle α much more accurately. acknowledgments we thank the referees for the positive enlightening comments and suggestions, which have greatly helped us in making improvements to this paper. the authors would also like to thank dr. annika lohstroh, prof. paul sellin, gary strudwick and john-william brown of the department of physics university of surrey for their guidance and technical assistance. references [1] h. ait bouh, “x-ray fluorescence analysis techniques:principles and instrumentations”, lap lambert academic publishing (2020) isbn 978620-2-78771-0. [2] p. mudgal, d.j. bell, “x-ray production/radiology reference article/”, radiopaedia.org, https://doi.org/10.53347/rld-25428. [3] g. lloyd-jones, “basics of x-ray physics-x-ray production”, https://www.radiologymasterclass.co.uk/tutorials/physics. [4] g. f. knoll, “radiation detection and measurement”, wiley publishers, 4th edition, (2010) isbn 978-0-470-13148-0. [5] m. alghamdi , m a-e. abdallah, s. harrad, “the utility of xray fluorescence spectrometry as a tool for monitoring compliance with limits on concentrations of halogenated flame retardants in waste polymers: a critical review”, emerging contaminants 8 (2022) 9, https://doi.org/10.1016/j.emcon.2021.12.002. [6] d. j. kalnicky, r. singhvi, “field portable xrf analysis of environmental samples”, journal of hazardous materials 83 (2001) 93, https://doi.org/10.1016/s0304-3894(00)00330-7.pmid:11267748. [7] f. l. melquiades, c. appoloni, “application of xrf and field portable xrf for environmental analysis”, journal of radioanalytical and nuclear chemistry 262 (2004) 533, https://doi.org/10.1023/b:jrnc.0000046792.52385.b2. [8] h. k. herglotz & l. s. birks, “x-ray spectrometry”, isbn 0-8247-66253, new york. [9] j. e. fernandez, “monte carlo computer simulation of the xrf intensity dependence on the propagation plane inclination”, computer physics communications (1989): 54211, https://doi.org/10.1016/00104655(89)90083-0. [10] laboratory script: rdi23-si and cdte x-ray spectroscopy, professor paul sellin (2009). 8 j. nig. soc. phys. sci. 4 (2022) 751 journal of the nigerian society of physical sciences pollution status of groundwater resources through hydrochemical characteristics a case study from southern india s. ponsadai lakshmia,∗, r. deepaa, s. ganapathy sankarib, m. jeyachandranb adepartment of science and humanities, e.g.s. pillay engineering college, nagapattinam, tamil nadu, india bpg & research department of chemistry, sri paramakalyani college, alwarkurichi, manonmaniam sundaranar university, abishiekapatti, tirunelveli, tamil nadu, india abstract mayiladuthurai is situated on the banks of river cauveri. this research work aims to determine the quality of water which were collected from twenty groundwater sources. from the identified twenty locations, water samples were collected. the sampling stations include three taluks in the mayiladuthurai district. the samples were analysed for various physicochemical qualities of water and compared with bis & who standards. frequency histogram and statistical analysis were applied to analyse the obtained data. results of hydrochemistry revealed the dominance of hardness, magnesium and fluoride ions. the present investigation of hydrochemistry concludes that the drinking nature of the analyzed samples was very weak quality-wise and could not be used for drinking as such. besides, fluoride ion related health problems were raised in some packets of the study area. the analytical report reveals that the quality of water has deteriorated and this may have a severe impact on human beings and other organisms in the study area. doi:10.46481/jnsps.2022.751 keywords: mayiladuthurai district, physicochemical parameters, bis & who standards, frequency histogram article history : received: 04 april 2022 received in revised form: 17 july 2022 accepted for publication: 03 august 2022 published: 07 november 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: e. etim 1. introduction the only source of fresh water and safety drinking water is ground water due to industrialization and modernization activities. naturally groundwater is present in the body segments of rocks. according to the geological nature of the region, the chemical salts present in the ground water differ. due to the accumulation time length differs, the salt content available in ground water becomes lower or higher [1-2]. ∗corresponding author tel. no: +91 9443493398 email address: lakshmiabinav@gmail.com (s. ponsadai lakshmi) to identify the quality of ground water resources of the samples, hydro chemical analyses was carried out. the statistical tools help to get a better understanding of the real nature of ground water resources. then to identify various reasons caused for the changes of the quality of ground water, ground water chemistry is highly supported. it is also helpful to study the origin and possible mechanisms of contamination processes [3]. then to draw up adequate management plans to guard the contamination of resources, hydrochemistry is highly supportive. therefore, it is completely examined to know the status and identify the pollutants available in the ground water of the 1 lakshmi et al. / j. nig. soc. phys. sci. 4 (2022) 751 2 study area. 2. materials and methods 2.1. about the study area mayiladuthurai is a new district of tamil nadu is one of the commercial places and fast developing cities in tamil nadu. its surroundings are mostly cultivated lands. the residents mostly depend on groundwater for their cultivation, domestic and industrial purposes. in this study area 19 bore wells and one open well were selected for collecting water samples during the study period. latitudes 11◦ 6′35′′ of north and longitudes 79◦ 39′0′′of east covers the three taluks namely sirkazhi, kuttalam and mayiladuthurai. all the 20 groundwater sampling points were given in fig. 1. the names of sample collecting stations were tabulated in table 1. they were separated into three, namely profile a, profile b and profile c. in profile a, 6 sampling stations were selected from kuthalam and mayiladuthurai taluks and in profile b, 7 sampling stations were selected from mayiladuthurai taluk. in profile c, 7 sampling stations were selected from mayiladuthurai and sirkazhi taluks. the study and analyses were carried out from january 2021 to december 2021. during the study period, every month a study was conducted. totally 240 samples were received from the stations. the physical qualities of all the 240 samples were appraised immediately after collecting the water with the help of some calibrated digital equipment. the chemical characteristics of water samples were determined by using methods recommended by apha(1995) and bis(1991). acid titration was used to identify the total alkalinity amount. elements like calcium, magnesium and total hardness were measured with the help of the edta complexometric titration method. argentometric titration was utilized to calculate chloride ions. the turbidity spectrometric technique was applied for measuring sulphate ions. spadns spectrometric technique was used to measure fluoride content. brucine sulphate spectrometric tool was taken into account for measuring nitrate ion concentration. flame photometry has been utilized to identify sodium and potassium ions. 2.2. description of the profiles 2.2.1. profile a profile a is located on the cauvery river bed which goes north up to kollidam then it flows clockwise towards poompuhar, were situated in the bay of bengal. several small scale industries, rubber industries, brick manufacturing and oil and natural gas corporation(ongc) power plant are situated around profile a. a few kilometres away from profile a many silk industries are situated. the effluents of these industries are directly discharged into the land and finally, it reaches the river. 2.2.2. profile b profile b is positioned towards the northeast of profile a. in all these areas cauvery river plays the main source for agricultural purposes and on the river bed found the gas plants at different stages. municipal solid waste dump sites are located, table 1. details of groundwater sampling stations in the study area s. no sampling stations source of water profile a s-1 kuthalam open well s-2 sethrabalapuram bore well s-3 arayapuram bore well s-4 malliyam borewell s-5 mahadhanapuram bore well s-6 moovalur borewell profile b s-7 sitharkadu bore well s-8 mayiladuthurai pookadai street borewell s-9 mayiladuthurai koranadu borewell s-10 mayiladuthurai mahadhana street borewell s-11 thiruvazhandur borewell s-12 mayiladuthurai coconut tree street borewell s-13 senthangudi bore well profile c s-14 nagangudi bore well s-15 lakshmipuram bore well s-16 uluthukuppai bore well s-17 s.s. nallur borewell s-18 thirunanriyur borewell s-19 keezha athukudi borewell s-20 mela athukudi bore well from which open-air combustion is taking place regularly. the area is a flourishing commercial central place in the district. this profile has many temporary water storage ponds. these are depending on the rainy season and rain water. similarly, surface water channels get a large huge amount of discharges and sewage unwanted water all through the year. these ponds receive impurities from the city, and industrial left-over. the nearby agricultural excess water is also mixing these ponds during the rainy season. 2.2.3. profile c profile c is located again towards the northeast of profile b and it is also on the river bed of cauvery. so mostly fertile lands are found here. to the inhabitants of this area agricultural activity supports a lot. the leading industrial setups in these places are weaving, rubber, brick making, earthenware, and food and soap manufacturing. sawmills, gas stations and auto-repair workshops are located frequently in the research paper region. and in this region, many ponds are found and they are seasonal. open sewage water is discharged into the ponds through canals. 2 lakshmi et al. / j. nig. soc. phys. sci. 4 (2022) 751 3 figure 1. the map of the study area with groundwater sampling stations 2.3. accuracy of analysis the ionic balance is one of the most common ways to check for analytical errors. water is electrically neutral, so the addition of major cations should equal the addition of the anions. major ions like ca2+, mg2+, na+, k+,(hco3− + co32−), cl−, so42− and no3− were taken into account in this analysis since these ions contribute more than 95% of ions in water. 2.4. statistical analysis a statistical tool namely ms excel-2007 is used to analyze the obtained results of the analysis. 2.5. descriptive statistics minimum, maximum, median, standard deviation and coefficient of variation values are helpful, to sum up, the obtained results. 2.6. graphical representation in this paper, the characteristics results of groundwater samples are graphically represented with a percentage frequency histogram which is to interpret the quality of samples in high quantum. in the present study, the data of each parameter are categorized into class intervals based on their values or permissible limits. the frequency was enumerated for those class intervals. the frequency of each class interval was further changed into frequency percentage to represent the data. from the percentage frequency value, a histogram was formed for each parameter to interpret the quality based on their guideline value. figure 2. scatter plot represents the relationship of total cations with total anions 3. results and discussion 3.1. accuracy of analysis the accuracy of this chemical analysis in the hydro-chemistry of groundwater samples was checked with ionic balance error (ibe) and relation of total cations (tz+) & anions (tz−) with the measured electrical conductance (ec) which is presented in fig. 2. from the obtained values, the ionic balance errors were within ± 10 %. a perusal of fig. 2. indicates that total cations values were increased with the increasing values of total anions with a correlation coefficient (r2) value of 0.9782. 3.2. physical and chemical properties of the selected resources analytical results are obtained for the physical and chemical properties of the selected groundwater samples were put into 3 lakshmi et al. / j. nig. soc. phys. sci. 4 (2022) 751 4 comparison with guideline values, the guideline value is given by [4], and [5] for identifying the samples for drinking purpose usage. (table 2.) describes the guideline values. the statistical summary of the analyzed physical and chemical parameters were tabulated in tables 3 -5 3.3. physical characteristics of groundwater resources 3.3.1. ph – from the percentage frequency histogram [fig. 3] of the physical property ph, all of the samples were alkaline and desirable for drinking purposes in the range of 6.5 – 8.5. the data on ph values (tables 3 5) indicated that groundwater samples were collected from all the profiles namely a, b and c. median value obtained for ph is7.6. groundwater should meet the standards and guideline values to make it suitable for drinking. because when it is used as drinking water, it gives excellent support for functioning good metabolic activities in human beings. the quality speaks in the industrial usage too. generally, ph values of natural water lie between 4.5 and 9.0. in this study, ph values in the most of samples oscillated between the ph values of 6.5 – 8.5. as per the guidelines, this range is suitable for drinking purposes. 3.3.2. electrical conductance (ec) – the obtained results (tables 3-5) revealed that groundwater samples collected in this study were a fresh category, which fluctuated between 0.36 and 1.93 ds/m in the profile a, 0.36to 2.39 ds/m in the profile b and 0.45 to 2.20 ds/m in the profile c. the results of ec values are plotted in a percentage frequency histogram [fig. 4]. according to fig. 4. about 45.8% of groundwater samples from profile a, 53.6% of groundwater samples from profile b and 50.0% of groundwater samples from profile c were in the desirable range (<1 ds/m). about 54.2% of groundwater samples from profile a, 46.4% of groundwater samples from profile b and 50.0% of groundwater samples from profile c were in the acceptable range of ec (1 3 ds/m). the presence of salt content reflects in the form of electrical conductivity. usually, the salts are present in the form of ions. the ionic strength of water is greatly influencing the electrical conductivity of water. the electrical conductivity value easily informs the number of salts present in the particular sample. from the results of the physical and chemical investigation in the research work, it is noted that most of the samples used in the study are within the acceptable limit of ec value, and suitable for drinking purposes. from the obtained results, s-8 of profile b showed a higher value of ec (2.39 ds/m) and this may be due to the surface contamination, farming activity and developed engineering activities which release massive quantities of their waste products into the ground. 3.3.3. total dissolved solids – tds values give better support to classifying the samples into different types. for classification, the physical and chemical properties of samples are very much important [2, 6]. fig. 5 explains the percentage frequency histogram for tds values. figure 3. percentage frequency histogram of ph figure 4. percentage frequency histogram of ec values in the groundwater samples collected from the study area according to fig. 5, 33.3% of the water samples from profile a, 50.0% of the water samples from profile b and 42.9% of the water samples from profile c exhibited a desirable level (< 500 mg/l) of tds. 66.7% of the groundwater samples from profile a, 50.0% of water samples from profile b and 57.1% of water samples from profile c were in the acceptable range (5001500 mg/l) of tds. in addition, all samples collected for this study did not exceed the allowed scale of guideline value (>1500 mg/l) for drinking water quality. it means that the collected water samples can be used for drinking purposes according to the tds value. numerous salts are present in the naturally available water. the figure 5. percentage frequency histogram of tds 4 lakshmi et al. / j. nig. soc. phys. sci. 4 (2022) 751 5 table 2. guideline values for drinking water quality substance or characteristics bis (2003) who (2006b) desirable max. acceptable permissible max. acceptable ph 6.5 – 8.5 no relax. 6.5 – 8.5 6.5 – 9.2 ec (ds/m) 1 3 tds (mg/l) 500 1500 500 1500 total alkalinity (as caco3), mg/l 200 600 500 total hardness (as caco3), mg/l 300 600 200 500 calcium (as ca), mg/l 75 200 magnesium (as mg), mg/l 30 100 sodium (as na), mg/l 200 potassium (as k), mg/l chloride (as cl), mg/l 250 1000 250 600 fluoride (as f), mg/l 1.0 1.5 1.5 sulphate (as so4), mg/l 200 400 250 500 nitrate (as no3), mg/l 45 no relax. 50 phosphate (as po4), mg/l table 3. summary statistics of physico-chemical characteristics of groundwater samples collected in the profile a parameters unit n a min. max. mean median sd cv ph 240 7.4 8.0 7.7 7.6 0.17 0.022 electrical conductivity (ds/m) 240 0.36 1.93 1.12 1.08 0.54 0.482 tds (mg/l) 240 230.33 1233.33 719.50 693.25 347.22 0.483 sodium (mg/l) 240 18.00 35.00 27.09 27.00 4.09 0.151 potassium (mg/l) 240 0.12 0.27 0.22 0.22 0.04 0.174 carbonate (mg/l) 240 0.00 0.40 0.05 0.00 0.10 2.085 bicarbonate (mg/l) 240 125.50 400.80 262.30 257.90 82.37 0.314 total alkalinity (mg/l) 240 125.70 400.80 262.35 257.98 82.34 0.314 total hardness (mg/l) 240 206.07 828.90 512.45 525.17 194.83 0.380 calcium (mg/l) 240 38.00 155.00 104.31 110.75 36.68 0.352 magnesium (mg/l) 240 27.00 110.33 61.19 59.38 26.08 0.426 chloride (mg/l) 240 56.00 195.80 138.07 148.00 46.47 0.337 sulphate (mg/l) 240 22.33 139.80 76.29 68.63 42.04 0.551 fluoride (mg/l) 240 1.96 5.13 3.08 3.01 1.12 0.363 nitrate (mg/l) 240 0.06 0.61 0.26 0.22 0.18 0.687 phosphate (mg/l) 240 0.02 0.07 0.05 0.04 0.02 0.407 sar (mg/l) 240 0.36 0.83 0.55 0.54 0.15 0.263 a n – no. of samples [(20 sampling stations × 12 months) = 240], sd – standard deviation, cv – co-efficient of variation, sarsodium adsorption ratio 5 lakshmi et al. / j. nig. soc. phys. sci. 4 (2022) 751 6 table 4. summary statistics of physico-chemical characteristics of groundwater samples collected in the profile b parameters unit n a min. max. mean median sd cv ph 240 7.2 7.8 7.6 7.6 0.16 0.021 electrical conductivity (ds/m) 240 0.36 2.39 1.07 0.81 0.64 0.594 tds (mg/l) 240 228.33 1314.00 678.95 517.33 393.01 0.579 sodium (mg/l) 240 17.30 38.00 25.56 24.75 5.80 0.227 potassium (mg/l) 240 0.13 0.27 0.20 0.19 0.04 0.189 carbonate (mg/l) 240 0.00 0.10 0.00 0.00 0.02 5.292 bicarbonate (mg/l) 240 135.80 380.70 231.89 212.00 81.14 0.350 total alkalinity (mg/l) 240 135.80 380.70 231.89 212.00 81.14 0.350 total hardness (mg/l) 240 243.69 875.98 494.44 423.73 222.09 0.449 calcium (mg/l) 240 55.00 183.00 102.55 89.05 42.65 0.416 magnesium (mg/l) 240 25.10 111.00 57.89 49.08 28.51 0.493 chloride (mg/l) 240 63.00 243.80 140.54 128.20 62.27 0.443 sulphate (mg/l) 240 20.67 165.00 77.57 61.50 44.76 0.577 fluoride (mg/l) 240 2.09 3.46 2.84 2.86 0.48 0.169 nitrate (mg/l) 240 0.05 0.46 0.19 0.15 0.13 0.687 phosphate (mg/l) 240 0.02 0.08 0.05 0.04 0.02 0.435 sar (mg/l) 240 0.35 0.89 0.53 0.50 0.13 0.254 a n – no. of samples [(20 sampling stations × 12 months ) = 240], sd – standard deviation, cv – co-efficient of variation, sarsodium adsorption ratio table 5. summary statistics of physico-chemical characteristics of groundwater samples collected in the profile c parameters unit n a min. max. mean median sd cv ph 240 7.1 7.8 7.5 7.6 0.18 0.024 electrical conductivity (ds/m) 240 0.45 2.20 1.16 1.08 0.56 0.240 tds (mg/l) 240 286.00 1408.00 744.29 690.38 355.84 0.478 sodium (mg/l) 240 17.00 40.00 27.99 26.72 6.16 0.220 potassium (mg/l) 240 0.12 0.25 0.19 0.20 0.04 0.185 carbonate (mg/l) 240 0.00 0.00 0.00 0.00 0 0.00 bicarbonate (mg/l) 240 124.00 398.90 248.60 246.68 86.69 0.349 total alkalinity (mg/l) 240 124.00 398.90 248.60 246.68 86.69 0.349 total hardness (mg/l) 240 279.09 918.22 532.72 466.50 234.35 0.440 calcium (mg/l) 240 55.70 199.33 112.92 95.75 48.80 0.432 magnesium (mg/l) 240 31.10 111.00 60.89 55.23 28.31 0.465 chloride (mg/l) 240 65.33 276.80 151.49 138.28 67.69 0.447 sulphate (mg/l) 240 31.25 143.00 85.01 72.75 33.77 0.397 fluoride (mg/l) 240 1.79 4.00 2.91 2.97 0.67 0.230 nitrate (mg/l) 240 0.05 0.35 0.15 0.11 0.10 0.681 phosphate (mg/l) 240 0.02 0.10 0.05 0.05 0.03 0.505 sar (mg/l) 240 0.36 0.95 0.56 0.50 0.16 0.281 a n – no. of samples [(20 sampling stations × 12 months ) = 240], sd – standard deviation, cv – co-efficient of variation, sarsodium adsorption ratio presence of the total concentration of cation and anion gives us the tds value. high tds content in water increases its osmoregulatory behaviour which in turn decreases the solubility of oxygen in the water and minimizes the water quality for drinking purposes [7]. based on the presence of tds quantity, most of the samples analyzed in this study exceeded the desirable level (500 mg/l) of drinking water quality limit but were within the upper limit (1500 mg/l). total dissolved solids in drinking water make it unfit, and not advisable for consumption if the consumer is suffering from kidney-related and heart issues. 3.4. chemical characteristics of groundwater resources 3.4.1. sodium and potassium the values of sodium are given in fig.6 as a percentage frequency histogram, which revealed that all the analyzed samples were not exceeding the desirable level of sodium (<200 mg/l). 6 lakshmi et al. / j. nig. soc. phys. sci. 4 (2022) 751 7 figure 6. percentage frequency histogram of sodium values in the groundwater samples collected from the study area figure 7. percentage frequency histogram of potassium the presence of salt makes the water saline. salts such as sodium and potassium are naturally available in groundwater. besides, wastes from industries also supportive for the presence of higher levels of these salt in groundwater. the consumption of water with high ec, sodium and chloride may not cause any health-related problems, but can impair the potability of water and can give an unacceptable taste to the consumers [8]. from the results of the hydro-chemical investigation in the study area, the sodium content levels of all samples were within the desirable level drinking limit of sodium (<200 mg/l). therefore all the collected samples are good for drinking according to sodium content. the highest sodium concentration 40 mg/l (s-20) was found in profile c during rainy seasons and this may be due to heavy rain. the element potassium is considered a minor element in groundwater. it varied from 0.12 to 0.27 mg/l in profile a, 0.13 to 0.27 mg/l in profile b and it fluctuated from 0.12 and 0.25 mg/l in profile c. no guideline value for potassium was proposed for drinking purposes. the frequency percentage histogram of the potassium profile was given in fig.7. the highest potassium concentration was found in profile b of sample numbers s-9 (0.27 mg/l) and the lowest recorded in the sample collected from the sampling station s-6 & s-17 (0.12 mg/l). higher potassium is likely due to silicate minerals and agricultural activities. here the study area is covered from a large number of agricultural lands. 3.4.2. total hardness, calcium and magnesium – hardness is the property of water which does not produce lather with soap. cations like calcium and magnesium and anions like carbonate, bicarbonate, chloride and sulphate ions are figure 8. percentage frequency histogram of total hardness values in the groundwater samples collected from the study area responsible for hardness. according to the presence of hardness causing ions, hard water is generally categorized into soft (<75 mg/l), moderately hard (75 – 150 mg/l), hard (150 – 300 mg/l) and very hard (> 300 mg/l) [9]. a perusal of hardness values depicted in the percentage frequency histogram (fig. 8), shows that all the groundwater samples are hard. i.e moderately hard to very hard in nature. from fig. 8, it is clear that about 37.5% of samples from profile a, 25% of samples from profile b and 32.1% of samples from profile c exhibited higher hardness values (>600 mg/l), not acceptable for consumption. a higher concentration of hardness was recorded in the present work. a higher concentration of hardness affects the groundwater and it is also harmful to human beings. direct release of waste from small scale industries into land, agricultural waste and largescale human used wastes [10] may be the reason for higher values of hardness. fig. 9 depicted the calcium ion of the study area as a percentage frequency histogram which denoted that about 25% of water samples from profile a, 39.3% of water samples from profile b and 32.1% of samples from profile c were in the desirable range (<75 mg/l) of calcium. then, 75% of samples from profile a, 60.7% of samples from profile b and 67.9% of samples from profile c were in the acceptable range (75200 mg/l). magnesium ion concentration fluctuated from 27 to 110.33 mg/l with a median of 59.38 mg/l in the profile a, 25 to 111 mg/l with a median of 49.08 mg/l in the profile b and the values extended between 31.10 and 111 mg/l with a median value of 55.23 mg/l in the groundwater samples of the profile c. the profile of magnesium values was presented in fig. 10, as a percentage frequency histogram which denoted that 8.3% of the samples from the profile a and 10.7% of samples from profile b and none of the water samples from the profile c exhibited in the desirable range of magnesium (<30 mg/l). then, 79.2% of water samples from profile a, 78.6% of samples from profile b and 82.1% of samples from profile c were in the acceptable range (30-100 mg/l). this could be used as drinking water when alternate water was not availble. besides, 12.5% of samples from profile a, 10.7% of samples from profile b and 17.9% of samples from profile c went beyond the allowable range. bedrock minerals with magnesium salts are responsible for the concentration of magnesium ions in groundwater, higher 7 lakshmi et al. / j. nig. soc. phys. sci. 4 (2022) 751 8 figure 9. percentage frequency histogram of calcium figure 10. percentage frequency histogram of magnesium values in the groundwater samples collected from the study area value of magnesium ions in water may alter the taste of water. the sampling stations s-19 (918 mg/l) and s-20 (199 mg/l) in profile c recorded the highest value of total hardness and calcium during the summer season respectively. but the sampling station s-5 (110 mg/l) in profile a recorded the highest value of magnesium during the summer season. a higher concentration of hardness related parameters (th, calcium and magnesium) during the summer season may be due to the large scale human use, regular addition of large quantities of waste from industry and agricultural activity. the high concentration of hardness related parameters can impair the potability of water [11]. the presence of a large amount of calcium and magnesium may be due to the presence of limestone in the bedrock. 3.4.3. total alkalinity – the presence of hydroxide, carbonate and bicarbonate ions makes the water neutral to alkaline nature. here bicarbonate alkalinity is reported for all samples but carbonate alkalinity is below the detectable limit. the measured alkalinity of this study is presented in fig. 11, as a percentage frequency histogram. from the observation, 25% of the groundwater samples from profile a, 35.7% of the samples from profile b and 35.7% of the samples from profile c fell within the desirable range (<200 mg/l), the remaining samples from all the three profiles were in the acceptable limit (200-600 mg/l) of guideline value for drinking water quality. according to tables 3-5, groundwater samples collected from profile a exhibited higher alkalinity with the obtained median value of 257.98mg/l when compared with profiles b and c. figure 11. percentage frequency histogram of alkalinity figure 12. percentage frequency histogram of chloride values in the groundwater samples collected from the study area during biodegradation of natural material carbon dioxide is released into the soil, mixed with water and converting the water into carbonate-rich. in this study, many samples crossed the desirable limit of alkalinity but were within the acceptable limit. high alkaline nature can impair the potability of water. groundwater samples collected from the sampling station s-5 in profile a recorded the highest value of total alkalinity (400.8 mg/l). this high value in s-5 may be due to the percolation of water from sewage water and industrial waste in the land [12]. 3.4.4. chloride – the chloride values of this study is given in fig. 12, as percentage frequency histogram. according to the fig. 12, all of the groundwater samples from profiles a, b and 89.3 % of the collected samples from profile c were less than (<250 mg/l), which is the desirable limit for chloride. the remaining samples from profile c were within the unacceptable limit (2501000 mg/l). due to high attraction of chloride towards sodium, chloride ion is abundantly present in water. next the high temperature and low rainfall also enhances its presence in ground water. besides porous nature of soil also increases its presence in water [13]. based on the low results obtained for chloride, the study area samples can be used for drinking purpose. 3.4.5. sulphate – the data on sulphate values are given in (tables 3 5).the sulphate values in the groundwater samples of profile a fluctuated between 22.33 and 139.80 mg/l with a median value of 8 lakshmi et al. / j. nig. soc. phys. sci. 4 (2022) 751 9 figure 13. percentage frequency histogram of sulphate figure 14. percentage frequency histogram of fluoride values in the groundwater samples collected from the study area 68.63mg/l, profile b ranged between 20.67 to 165 mg/l with a median value of 61.5 mg/l while in the profile c, it changed from 31.25 to 143 mg/l with a median value of 72.75 mg/l. the sulphate values of this study area were given in fig. 13 as frequency percentage histogram. based on the type of bedrock present sulphates are present in different forms in soil. its quantity varied from minor to major. but higher sulphate concentration along with the presence of magnesium ion i.e. exceeding 1000mg/l may cause laxative effect on human beings [14-15]. but in this study, from fig. 13 it was observed that all the analyzed samples were below the desirable level (<200 mg/l) of sulphate. therefore, the water of this study area can be used for drinking purpose. the obtained data reveal that the groundwater samples collected from the sampling station s-13 in profile b recorded the highest value of sulphate (165 mg/l) during summer. this highest level of sulphate may be due to the waste received from municipality and sewage water. 3.4.6. fluoride – fluoride concentrations in the profile a varied between 1.96 to 5.13 mg/l, 2.09 to 3.46 mg/l in the profile b whereas in the profile c region, it ranged from 1.79 to 4.00 mg/l. fluoride values were presented in fig.14. from the diagram it is clear that all the analyzed samples exceeded the desirable limit of fluoride (>1.0mg/l). besides all the analyzed samples were crossed the maximum acceptable limit (>1.5mg/l), which is recommended for drinking water quality. fluoride ion is the required ion for human beings but if it exceeds it creates dangerous effects on humans [16]. for proper functioning of teeth and bone, the presence fluoride ion is the important one. human health and fluoride ion present in the environment are interrelated to each other [17, 37]. deficiency of fluoride ion may be harmful to teeth. it creates dental carries on human teeth. excess quantities of fluoride ion consumption make it unfit for human health because it creates the disease fluorosis on humans [18]. drinking water with high levels of fluoride ion triggers the disease skeletal fluorosis, which deforms the bone structure on humans [19-21]. hence, usage of water with allowable limit of fluoride prescribed by various organizations namely the bureau of indian standards (bis) and world health organization (who) is the most important. 1.5 mg/l, is the acceptable limit for the ion fluoride in drinking water. but in water scarce areas this limit can be increased [36]. drinking water limit for fluoride ion is 1.0 and 1.5 mg/l. but [22] has prescribed the limiting values on the basis of the climatic conditions and quantity of water consumed by individual. in this study, the collected samples crossed the upper fluoride limit. bedrock, temperature of the soil, amount of rainfall [23] may influence the presence of fluoride ion in water. here the presence of fluoride supportive rock was responsible for high levels of fluoride ion in the samples [24-25]. rocks with fluoride ion minerals may increase the fluoride ion concentration in surface and ground water [26-27]. the presence of fluoride in groundwater was also supported by the factors such as alkali enduring of silicate rocks, igneous rocks and sedimentary rocks. except natural cause none is the cause for the presence of fluoride ion in water. high concentration of fluoride causes dental fluorosis [28-30]. water with greater fluoride concentration in drinking water developed the symptoms of dental fluorosis and skeletal fluorosis in the residents of the study place. 3.4.7. nitrate – fig. 15 represents the analyzed nitrate values of the study area as percentage frequency histogram. from the analyzed values, nitrate levels present in the samples were below the desirable limit (45 mg/l). this limit has been suggested by [4-5] for drinking water. from the (tables 3 5), the nitrate values in the groundwater samples of profile a varied in the range of 0.06 0.61 mg/l with a median value of 0.22 mg/l, profile b varied in the range of 0.05 to 0.46 mg/l with a median of 0.15 mg/l while in the profile c, it oscillated between 0.05 and 0.35 mg/l with a median value of 0.11 mg/l. several sources are responsible for the presence of nitrate content in water. prime factor is usage of fertilizers in the nearby agricultural field. decay of died plants and animals are also the major cause for the presence of nitrogen in the water samples. the releasing nitrogen from decay process is converted into nitrate. natural nitrate concentrations in groundwater span are from 0.1 to 10 mg/l [31]. the continuous use of water containing higher concentration of nitrate can cause blue baby syndrome, cancer in gastric system, birth deformations and hypertension in human system [15, 32, 41-42]. here nitrate ion concentration in the analyzed samples were within the desirable and it was suitable for drinking. 9 lakshmi et al. / j. nig. soc. phys. sci. 4 (2022) 751 10 figure 15. percentage frequency histogram of nitrate figure 16. percentage frequency histogram of phosphate values in the groundwater samples collected from the study area from the obtained analytical results groundwater samples collected from the sampling station s-3 in profile c recorded the highest value of nitrate (0.61 mg/l) during monsoon season. this highest value of nitrate in sampling station-3 may be due to the filtration of nitrogen content from the nearby municipal waste dumping sites. besides the surrounding agricultural field was also reason for the nitrogen content in water [12, 33, 3940]. 3.4.8. phosphate– the phosphate concentrations in water were from 0.02 to 0.07 mg/l in the groundwater samples of profile a, 0.02 to 0.08 mg/l in profile b and 0.02 to 0.10 mg/l in profile c. fig. 16 showed the percentage frequency histogram of the phosphate values in the study area. the observation from the percentage frequency histogram clearly revealed that the phosphate values for all the analyzed samples were lesser than 1 mg/l. according to phosphate values, groundwater samples collected from the sampling stations s-18, s-19 & s-20 recorded the utmost high value of phosphate i.e., 0.10 mg/l. the usage phosphatic fertilizers for agricultural activity, filtration of agricultural wastes into the soil and human bone excreta may cause this ion concentration in water. no guideline value was proposed for drinking water quality. 3.4.9. sodium adsorption ratio (sar) – sodium adsorption ratio measure the comparative amount of the ions such as sodium, magnesium and calcium which affects penetration power of water into the soil. the change is penetration power of water is also caused by electrical conductance property. sar value of irrigation water is computed as figure 17. percentage frequency histogram of sar values in the groundwater samples collected from the study area per [34, 38]. [eq. 1] sar = na+√ (ca2+ + mg2+)/2 (1) where the concentrations are reported in meq/l. sar indicates irrigation water quality which tends to go in for the reactions which are involving with cation-exchange in soil. soil which is exchanging sodium ion with calcium and magnesium ions, damages the soil structure. water with more salt content increases infiltration and vice versa. this dual phenomenon may function simultaneously. if the land is continuously irrigated with water with rich sodium ion lowers the soil [35-37]. the data on sar values were given in fig. 17 as percentage frequency histogram. from the fig. 17, the sar values for all the analyzed samples were within the low category (1-10). thus all the collected samples which were analyzed were appropriate for irrigation. 4. conclusion groundwater samples collected from the mayiladuthurai district were generally characterized by fresh, moderately hard water to strongly hardness in nature. the samples were alkaline in nature. from the obtained results of hydrochemistry, it was noticed that a number of groundwater samples from all the three profiles exhibited higher levels of hardness and magnesium. these samples exceeded the desirable level for drinking water quality suggested by bis (2003) and who (2006b). besides, almost all the sites fluoride ion concentration went beyond the upper limit of its concentration value (>1.5 mg/l). fluoride ion is a health wise important parameter. its higher concentration greatly affects the quality of water, which in turn the quality of water for drinking purpose was greatly impaired. the present investigation of hydrochemistry concludes that drinking nature of the analyzed samples were very weak in quality wise and could not be used for drinking as such. besides, fluoride ion related health problems raised in some packets of the study area. although many parameters were within the allowable limit, but most of the health oriented parameters were crossed the permissible limit. hence the present investigation confirmed that the quality of water was deteriorated and this may have severe impact on human beings and other organism in the study area. 10 lakshmi et al. / j. nig. soc. phys. sci. 4 (2022) 751 11 acknowledgment the authors will like to appreciate the handling editor and the reviewers for their valuable comments that improved the quality of this paper. references [1] r. a. freeze & j. a. cherry, groundwater, prentice-hall inc., englewood cliffs, new jersey 604 (1979). 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[41] n. kure, h. i. daniel, c. g. afuwai, e. j. adoyi & i. a. bello, “the delineation of ground water and geotechnical parameters within marmara area of chikun local government of kaduna state, nigeria”, journal of the nigerian society of physical sciences 1 (2019) 1. 11 j. nig. soc. phys. sci. 4 (2022) 706 journal of the nigerian society of physical sciences a higher-order block method for numerical approximation of third-order boundary value problems in odes adefunke bosede familuaa, ezekiel olaoluwa omoleb,c,∗, luke azeta ukpebord adepartment of mathematics and statistics, first technical university, ibadan, oyo state, nigeria bdepartment of mathematics and statistics, joseph ayo babalola university, ikeji arakeji, osun state, nigeria cdepartment of mathematics, federal university oye-ekiti, ekiti state, nigeria ddepartment of mathematics, ambrose alli university edo state, nigeria abstract in recent times, numerical approximation of 3rd-order boundary value problems (bvps) has attracted great attention due to its wide applications in solving problems arising from sciences and engineering. hence, a higher-order block method is constructed for the direct solution of 3rd-order linear and non-linear bvps. the approach of interpolation and collocation is adopted in the derivation. power series approximate solution is interpolated at the points required to suitably handle both linear and non-linear third-order bvps while the collocation was done at all the multiderivative points. the three sets of discrete schemes together with their first, and second derivatives formed the required higher-order block method (hbm) which is applied to standard third-order bvps. the hbm is self-starting since it doesn’t need any separate predictor or starting values. the investigation of the convergence analysis of the hbm is completely examined and discussed. the improving tactics are fully considered and discussed which resulted in better performance of the hbm. three numerical examples were presented to show the performance and the strength of the hbm over other numerical methods. the comparison of the hbm errors and other existing work in the literature was also shown in curves. doi:10.46481/jnsps.2022.706 keywords: convergence analysis, linear & nonlinear problems, ordinary differential equations, power series basic function, third-order boundary value problems article history : received: 13 march 2022 received in revised form: 28 may 2022 accepted for publication: 30 may 2022 published: 25 august 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: j. ndam 1. introduction numerous common happenings in connection with physical sciences, and engineering are modeled in form of linear and nonlinear bvps. although, some modeled problems do not have theoretical solution or closed form. consequently, numer∗corresponding author tel. no: +2348065086963 email address: eoomole@jabu.edu.ng, omolez247@gmail.com (ezekiel olaoluwa omole) ical method is employed to solve such class of modeled problems. in this article, the numerical solution of third-order bvps of the type y′′′(x) = v(x, y, y′, y′′), (1) with the initial conditions, boundary conditions or any other form as follow, y(a) = αa, y ′(a) = βa, y ′(xb) = νb, (2) y(xa) = α0, y ′(a) = βa, y(b) = νb, (3) 1 familua et al. / j. nig. soc. phys. sci. 4 (2022) 706 2 y(xa) = αa, y ′(b) = βb, y(b) = νb, (4) the constants parameters in equations (2) (4) are taken to be continuous functions and v fulfils the condition for the existence and uniqueness of the problem. it follows that (1) is a third-order ordinary differential equations with initial, and boundary conditions (3) (4). modeled equation (1) is of great importance to scientists and engineers due to its numerous usage in sciences and engineering. scholars have developed numerous techniques for solving (1). the conventional methods of solving (1) could be; by reducing it to the system of first-order ordinary differential equations and a suitable numerical methods for first-order odes would be applied to solve the system of equations. the shooting method, and finite different method. the limitations of these methods have been discussed by numerous scholars [1–4]. for instance, the reduction approach have been reported by various literatures to have alot of limitations such computational burden, lots of human efforts, requires lots of time for the computation, and complexity in the computer computation which affects the accuracy and efficiency of the method in terms of error and time of execution. on the other hand, the shooting method suffers inaccuracy and instability while the finite different method is very demanding and does not give a satisfactory results. other numerical methods for solving ordinary differential equations also exist in literature [5–15]. in other to improve the limitations and the weakness associated with the conventional methods, we present a higher-order block method capable of solving (1) directly, accurately and reduces computational time. the new method namely hbm is expected to improve the accuracies of the existing methods in the literature. 2. methodology in this section, a new method capable of handing (1) is constructed. we considered an approximate equation as the power series of the form, y(x) = 3k+1∑ z=0 mz x z (5) where the step-number (k = 3) is taken into consideration, 3k+1 is equivalent to r + 2s − 1, where r is the interpolation points and s is the collocation points. the third and fourth derivative of (5) yields y′′′(x) = 3k+1∑ z=3 z(z − 1)(z − 2)mz x z−3 y(iv)(x) = 3k+1∑ z=4 z(z − 1)(z − 2)(z − 3)mz x z−4 (6) now, interpolating (5) at xn+r, r = 0(1)k − 1 and collocating both the third and fourth derivatives in (6) at xn+s, s = 0(1)k to generate the system of ten by ten equations written in matrix form as x1 a1 = b1 (7) x1 =  1 xn x2n x 3 n x 4 n x 5 n · · · x 9 n x 10 n 0 1 2xn 3x2n 4x 3 n 5x 4 n · · · 9x 8 n 10x 9 n 0 0 2 6xn 12x2n 20x 3 n · · · 72x 7 n 90x 8 n 0 0 0 6 24xn 60x2n · · · 504x 6 n 720x 7 n 0 0 0 6 24xn+1 60x2n+1 · · · 504x 6 n+1 720x 7 n+1 0 0 0 6 24xn+2 60x2n+2 · · · 504x 6 n+2 720x 7 n+2 0 0 0 6 24xn+3 60x2n+3 · · · 504x 6 n+3 720x 7 n+3 0 0 0 0 24 120xn · · · 3024x5n 5040x 6 n 0 0 0 0 24 120xn+1 · · · 3024x5n+1 5040x 6 n+1 0 0 0 0 24 120xn+2 · · · 3024x5n+2 5040x 6 n+2 0 0 0 0 24 120xn+3 · · · 3024x5n+3 5040x 6 n+3  b1 = (yn, yn+1, yn+2, vn, vn+1, vn+2, vn+3, wn, wn+1, wn+2, wn+3) t a1 = (m0, m1, m2, m3, m4, m5, m6, m7, m8, m9, m10) t solving the system of equations above, we obtained the values of the coefficients mn, n = 0, ..., 10 as follows. after obtaining the values of these coefficients and changing the variable, x = xn + lh using the appropriate transformation, the polynomial in (5) may be written as, y (l) = a0yn + a1yn+1 + a2yn+2 + h 3 (b0vn + b1vn+1 + b2vn+2 + b3vn+3) +h4(c0wn + c1wn+1 + c2wn+2 + c3wn+3) (8) 2 familua et al. / j. nig. soc. phys. sci. 4 (2022) 706 3 where a0 = de 2 , a1 = −el, a2 = dl 2 , d = (l − 1), e = (l − 2) b0 = h2 544320 de(77 l7 − 1059 l6 + 5699 l5 − 14625 l4 + 15749 l3 + 3165 l2 − 22003 l + 18381) b1 = lh3 20160 de(7 l7 − 79 l6 + 304 l5 − 370 l4 − 206 l3 + 122 l2 + 778 l + 2090) b2 = − lh3 20160 de(7 l7 − 89 l6 + 409 l5 − 755 l4 + 319 l3 + 199 l2 − 41 l − 521) b3 = − lh3 544320 de(77 l7 − 789 l6 + 2864 l5 − 4230 l4 + 1574 l3 + 1086 l2 + 110 l − 1842) c0 = lh4 181440 de(7 l7 − 99 l6 + 559 l5 − 1581 l4 + 2245 l3 − 1191 l2 − 503 l + 873) c1 = lh4 20160 de(7 l7 − 89 l6 + 424 l5 − 878 l4 + 550 l3 + 382 l2 + 46 l − 626) c2 = lh4 20160 de(7 l7 − 79 l6 + 319 l5 − 517 l4 + 205 l3 + 137 l2 + l − 271) c3 = lh4 181440 de(7 l7 − 69 l6 + 244 l5 − 354 l4 + 130 l3 + 90 l2 + 10 l − 150) (9) remark 1. the variable coefficients functions in (8) are continuous and differentiable within the interval of solution of [a, b] with a step size given by the function h = b−an . it follows that n is the number of sub-interval of the solution. the continuous function (8) and its first y′(x) and second derivatives y′′(x) were used to generate the main and auxiliary methods which produces a sum of nine equations jointed together to supply the entire approximations on the interval for the direct solution of third-order bvps of the type (1). furthermore, by evaluating (8), its first and second derivatives at xn+l, l = 0(1)k, the following nine equations or otherwise called hbm were acquired. yn+1 = yn + y ′ nh + 1 2 y′′n h 2 + 62387 544320 h3vn + 89 3360 h3vn+1 + 439 20160 h3vn+2 + 1031 272160 h3vn+3 + 1879 181440 h4wn − 359 10080 h4wn+1 − 13 960 h4wn+2 − 17 18144 h4wn+3 yn+2 = yn + 2y ′ nh + 2 y ′′ n h 2 + 5048 8505 h3vn + 164 315 h3vn+1 + 4 21 h3vn+2 + 244 8505 h3vn+3 + 172 2835 h4wn − 4 15 h4wn+1 − 34 315 h4wn+2 − 4 567 h4wn+3 (10) yn+3 = yn + 3y ′ nh + 9 2 y′′n h 2 + 657 448 h3vn + 2187 1120 h3vn+1 + 2187 2240 h3vn+2 + 117 1120 h3vn+3 + 351 2240 h4wn − 729 1120 h4wn+1 − 729 2240 h4wn+2 − 27 1120 h4wn+3 y′n+1 = y ′ n + h y ′′ n + 19519 68040 h2vn + 1301 10080 h2vn+1 + 181 2520 h2vn+2 + 3329 272160 h2vn+3 + 371 12960 h3wn − 313 2520 h3wn+1 − 89 2016 h3wn+2 − 137 45360 h3wn+3 y′n+2 = y ′ n + 2h y ′′ n + 5731 8505 h2vn + 296 315 h2vn+1 + 109 315 h2vn+2 + 344 8505 h2vn+3 + 206 2835 h3wn − 20 63 h3wn+1 − 52 315 h3wn+2 − 4 405 h3wn+3 (11) 3 familua et al. / j. nig. soc. phys. sci. 4 (2022) 706 4 y′n+3 = y ′ n + 3h y ′′ n + 603 560 h2vn + 2187 1120 h2vn+1 + 729 560 h2vn+2 + 27 160 h2vn+3 + 27 224 h3wn − 243 560 h3wn+1 − 243 1120 h3wn+2 − 9 280 h3wn+3 y′′n+1 = y ′′ n + 6893 18144 hvn + 313 672 hvn+1 + 89 672 hvn+2 + 397 18144 hvn+3 + 1283 30240 h2wn − 851 3360 h2wn+1 − 269 3360 h2wn+2 − 163 30240 h2wn+3 y′′n+2 = y ′′ n + 223 567 hvn + 20 21 hvn+1 + 13 21 hvn+2 + 20 567 hvn+3 + 43 945 h2wn − 16 105 h2wn+1 − 19 105 h2wn+2 − 8 945 h2wn+3 (12) y′′n+3 = y ′′ n + 93 224 hvn + 243 224 hvn+1 + 243 224 hvn+2 + 93 224 hvn+3 57 1120 h2wn − 81 1120 h2wn+1 + 81 1120 h2wn+2 − 57 1120 h2wn+3 3. the properties of the hbm in this section, it is important to examine and discuss the convergence analysis of the hbm such as the order & the error constants, convergence, zero stability, and convergence. 3.1. order & error constant of the hbm according to [16–18], the linear difference operator l in respect to equation (10) is defined by l [ y (x) ; h ] = ∑k j=0 { a jy (xn + jh) − h3v jy′′′ (xn + jh) − h4w jy′′′′ (xn + jh) } (13) y (x) is assumed to be continuously differentiable function. therefore function (13) can be maximize in taylor series about x to obtain l (y (x) ; h) = d0y (x) + d1hy ′ (x) + d2h 2y′′ (x) + ... + dqh qy(q) (x) (14) where dq, q = 1, 2, ... are constants in such that, d0 = d1 = ... = dq = 0, dp+3 , 0 (15) the method (10) is of uniform order 11 with the following error constants dq+3 = (−5.143239 × 10−10,−1.312531 × 10−09,−2.394622 × 10−09)t . 3.2. consistency as stated by [18–20], a lmm of the form (10) is said to be consistent if it has order greater than or equals to one. the hbm satisfies the condition for consistency since its order, is 11 which is greater than one. 3.3. zero-stability of the hbm taking into consideration method (10) could be written in matrix difference form as , a(0)ym = a (1)ym−1 + h 3 [ c(0)gm + c (3)gm−3 ] + h4 [ d(0) hm + d (4) hm−3 ] (16) the matrix parameter a(0), a(1), c(0), c(3), d(0), d(4), h(0), h(1) are the square matrices whose arrays are the coefficients (10) and are defined as below. 4 familua et al. / j. nig. soc. phys. sci. 4 (2022) 706 5 a(0) =  1 0 0 0 1 0 0 0 1  , a(1) =  0 0 1 0 0 1 0 0 1  , c(0) =  89 3360 439 20160 1031 272160 164 315 4 21 244 8505 2187 1120 2187 2240 117 1120  c(3) =  0 0 62387544320 0 0 50488505 0 0 657448  , d(0) =  − 359 10080 − 13 960 − 17 18144 − 4 15 − 34 315 − 4 567 − 729 1120 − 729 2240 − 27 1120  d(4) =  0 0 1879181440 0 0 1722835 0 0 3512240 , the limit of (16) is taken as h → 0, to obtain the difference system a(0)ym − a (1)ym−1 = 0 (17) the first characteristics of (17) is given by ρ (f) = det ( f a(0) − a(1) ) = f2 (f − 1) = 0 (18) hence, f = 0, 0, 1 the block method of the form (10) is said to be zero stable if as ρ (f) = 0, then ∣∣∣f j∣∣∣ ≤ 1, j = 0, 1, ... for those roots with∣∣∣f j∣∣∣ = 1, the multiplicity does not exceed 1 [21–26]. also the block method (10) is consistent since p>1. since (10) is consistent and zero stable, it is also convergence [27]. 3.4. convergence in respect to the claim of lambert [18] which also corroborates with proof of adogbe & omole, and adogbe et al. [28, 29] that any numerical method belonging to a class of lmm must satisfies the fundamental and adequate conditions. it follows that for such class of method to be convergent it must be consistent and zero-stable. consequently, the hbm satisfies the conditions for consistency and zero-stability, so it is convergent. 4. implementation tactics in this part, we present the comprehensive procedure for the implementation of the new proposed method tagged higherorder block method (hbm). the hbm is implemented in block method together with the aid of newton-raphson approach via a mathematica 11.0 code which uses f-solve for linear and findroot for non-linear to simultaneously generate the solution at the initial point to the terminal point while adjusting for boundary conditions. meanwhile, each block integrators in (10), (11) and (12) forms a system of equations which is applied along with the newton’s method. the starting values in the application of the newton’s raphson method which are considered as the approximations provided by the taylor series expansion formulas yn+i = yn + ihy ′ n + ( (ih)2 2 ) y′′n + ( (ih)3 6 ) vn + ( (ih)4 24 ) wn, i = 0 (1) ..., k y′n+i = y ′ n + ihy ′′ n + ( (ih)2 2 ) vn + ( (ih)3 6 ) wn, i = 0 (1) ..., k y′′n+i = y ′′ n + ihvn + ( (ih)2 2 ) wn, i = 0 (1) ..., k. (19) the wn+i, i = 0 (1) ..., k appearing in (19) connotes the fourth derivative at xn+i, i = 0 (1) ..., k. in other to get a closed form solution of (20) which is expanded in (19), it is important to calculate the values of wn+i, i = 0 (1) ..., k. for more clarification, y(iv)(x) = ( dv(x, y, y′, y′′, y′′′) d x ) , i = 0 (1) ..., k. (20) y(iv)(x) = dv d x + dv dy y′ + dv dy′ y′′ + dv dy′′ y′′′ + v dv dy′′′ v, i = 0 (1) ..., k. (21) consequently, the solution of (1) is simultaneously generate on the entire interval of integration by using the main method (10) for n = 0, 1, ..., n2 to obtain 3n −2 equations and the additional method (11) and (12) which are employed to complete the set of equations needed to simultaneously solve 3n by 3n system of equations for solving (1) directly. 5. the numerical experiments it is very important to test the accuracy and usefulness of the hbm by applying the hbm to solve modeled problems. three standard third-order bvps varying from linear to nonlinear was computed and the results were presented and discussed extensively. all computations and programming were carried out using maple 18.0 and mathematica 11.0. 5.1. test problem 1 first test problem, third-order linear boundary value problem solved by ahmed [30]. y′′′−xy+(x2−2x2−5x−3)ex, y(0) = 0, y′(0) = 1, y′(1) = −e(22) with theoretical solution as y(x) = x(1 − x)ex (23) 5 familua et al. / j. nig. soc. phys. sci. 4 (2022) 706 6 table 1. numerical results of hbm, ae in hbm and ae in [30] for problem 1 using n = 10 or h = 0.1 x y-exact solution y-computed solution ae in hbm ae in [30] 0.1 0.09946538262680829 0.09946538262680911 8.18789 × 10−16 1.36200 × 10−10 0.2 0.19542444130562720 0.19542444130563258 5.38458 × 10−15 1.90000 × 10−12 0.3 0.28347034959096070 0.28347034959097905 1.83742 × 10−14 1.10000 × 10−12 0.4 0.35803792743390490 0.35803792743392630 2.14273 × 10−14 7.00000 × 10−12 0.5 0.41218031767503205 0.41218031767503670 4.66294 × 10−15 1.00000 × 10−11 0.6 0.43730851209372210 0.43730851209368804 3.40838 × 10−14 1.70000 × 10−12 0.7 0.42288806856880007 0.42288806856871670 8.33777 × 10−14 8.80000 × 10−11 0.8 0.35608654855879485 0.35608654855866470 1.30174 × 10−13 9.42000 × 10−11 0.9 0.22136428000412547 0.22136428000395672 1.68754 × 10−13 1.36800 × 10−10 1.0 0.00000000000000000 1.83070 × 10−13 1.83070 × 10−13 0.00000 × 10−00 table 2. comparison of maximum absolute error in hbm with other existing methods for problem 1 references maximum absolute error current method-hbm 1.8307 × 10−13 [30] 1.3700 × 10−10 [31] 5.3000 × 10−07 [32] 1.8400 × 10−06 [33] 2.3700 × 10−07 [34] 8.1200 × 10−04 [35] 1.6400 × 10−02 [36] 2.6400 × 10−07 [37] 8.2900 × 10−09 table 3. numerical results of hbm and absolute error for problem 2 taking n = 100 or h = 0.01 x y-exact solution y-computed solution ae in hbm 0.01 −0.009998833350833248 −0.009998833350833078 1.70003 × 10−16 0.02 −0.019990667226655746 −0.019990667226655066 6.80012 × 10−16 0.03 −0.029968504252313413 −0.029968504252311883 1.53003 × 10−15 0.04 −0.039925351251935550 −0.039925351251932820 2.72699 × 10−15 0.05 −0.049854221347501636 −0.049854221347497375 4.26048 × 10−15 0.06 −0.059748136056118590 −0.059748136056112450 6.14092 × 10−15 0.07 −0.069600127385578860 −0.069600127385570460 8.39606 × 10−15 0.08 −0.079403239927770000 −0.079403239927759020 1.09773 × 10−14 0.09 −0.089150532949507160 −0.089150532949493310 1.38500 × 10−14 0.10 −0.098835082480359870 −0.098835082480342770 1.70974 × 10−14 table 4. numerical results of hbm and absolute error for problem 2 taking n = 10 or h = 0.1 x y-exact solution y-computed solution ae in hbm 0.1 −0.098835082480359870 −0.09883508248034277 1.70974 × 10−14 0.2 −0.190722557563258760 −0.19072255756318998 6.87783 × 10−14 0.3 −0.268923388061819000 −0.26892338806166494 1.54043 × 10−13 0.4 −0.327111407539266430 −0.32711140753900390 2.62512 × 10−13 0.5 −0.359569153953152255 −0.35956915395277234 3.79918 × 10−13 0.6 −0.361371182972822670 −0.36137118297233617 4.86500 × 10−13 0.7 −0.328551020491222400 −0.32855102049065060 5.71820 × 10−13 0.8 −0.258248192723828200 −0.25824819272318880 6.39433 × 10−13 0.9 −0.148832112829221850 −0.14883211282853925 6.82593 × 10−13 1.0 0.0000000000000000000 6.98759 × 10−13 6.98759 × 10−13 in table 1, the theoretical solution, approximate solution, absolute error in hbm and the absolute error in other existing method were presented. on the other hand, table 2 shows the comparison of maximum absolute error in hbm against nu6 familua et al. / j. nig. soc. phys. sci. 4 (2022) 706 7 table 5. comparison of maximum absolute error in hbm with other existing methods for problem 2 references maximum absolute error current method-hbm 6.98759 × 10−13 [34] 8.55940 × 10−05 [35] 8.88390 × 10−03 [36] 2.15720 × 10−08 table 6. numerical results of hbm and absolute error problem 3 with using (n = 10) or (h = 0.1) x y-exact solution y-computed solution ae in hbm 0.1 0.09531017980432493 0.09531017980433536 1.04222 × 10−14 0.2 0.18232155679395460 0.18232155679398550 3.09197 × 10−14 0.3 0.26236426446749106 0.26236426446754463 5.35683 × 10−14 0.4 0.33647223662121290 0.33647223662128860 7.57172 × 10−14 0.5 0.40546510810816440 0.40546510810826103 9.66449 × 10−14 0.6 0.47000362924573563 0.47000362924585110 1.15463 × 10−13 0.7 0.53062825106217040 0.53062825106230150 1.31117 × 10−13 0.8 0.58778666490211910 0.58778666490226180 1.42775 × 10−13 0.9 0.64185388617239470 0.64185388617254560 1.50879 × 10−13 1.0 0.69314718055994530 0.69314718056009890 1.53655 × 10−13 table 7. comparison of the numerical errors in hbm and other existing methods for problem 3 with n = 10 x ae in hbm ae in [37] ae in [38] 0.1 1.04222 × 10−14 2.22000 × 10−06 2.947641 × 10−10 0.2 3.09197 × 10−14 4.67000 × 10−06 2.084566 × 10−11 0.3 5.35683 × 10−14 1.890000 × 10−06 4.147355 × 10−11 0.4 7.57172 × 10−14 3.560000 × 10−06 2.208859 × 10−12 0.5 9.66449 × 10−14 7.430000 × 10−07 2.777215 × 10−11 0.6 1.15463 × 10−13 1.220000 × 10−06 2.647594 × 10−11 0.7 1.31117 × 10−13 2.780000 × 10−06 1.190750 × 10−11 0.8 1.42775 × 10−13 3.74000 × 10−06 1.816331 × 10−12 0.9 1.50879 × 10−13 4.11000 × 10−06 7.666364 × 10−10 1.0 1.53655 × 10−13 0.00000 × 10−00 0.00000 × 10−00 merous methods proposed by various authors in the literatures. it is very clear that the efficiency and accuracy of hbm is established comprehensively. 5.2. test problem 2 consider the third-order boundary value problem solved by abdullah et al. [34]. y′′′ − y + (7 − x2)cosx + (x2 − 6x − 1)sinx, y(0) = 0, y′(0) = −1, y′(1) = 2sin1 (24) with the analytical solution given as y(x) = (x2 − 1)sin(x) (25) here, the interpretation of tables 3, 4 and 5 are made. the computational results of problem 2 using the proposed method named hbm with h = 0.01 is shown, while in table 2, we presents the computational results of problem 2 using hbm with h = 0.1. it could be observed that as the value of h decreases, the accuracies also increases whereas as the h increases, the accuracies of the method decreases as demonstrated in tables 3 and 4. furthermore, we computed the maximum absolute error of the hbm for problem 2 and compared with other numerical methods in the cited literature. this is illustrated in the table 5. the hbm gives a minimal error when compared with other existing method. 5.3. test problem 3 lastly, a non-linear third-order problem studied by akram et al. [37] and, hossain et al. [38] is taking into consideration. y′′′ + 2e3y = −4(1 + x)−3, y(0) = 0, y′(0) = 1, y′(1) = in(2) (26) with the exact solution. y(x) = in(1 + x) (27) finally, we considered a non-linear third-order bvp in other to determine the strength and advantage of the hbm. in table 6, 7 familua et al. / j. nig. soc. phys. sci. 4 (2022) 706 8 figure 1. comparison of absolute error in hbm with absolute error in akram et al. [37] figure 2. comparison of absolute error in hbm with absolute error in hossain et al. [38] the computational results and absolute error on hbm is presented while table 7 presented the comparison of absolute error of the non-linear problem using hbm and compared with other similar methods in the cited work. in addition, the comparison of errors in curves for the non-linear problem were also presented in figure 1 and 2. without any iota of doubt, we have been able to demonstrate the convergence, efficiency and accuracy of the new method namely hbm over other techniques. 6. conclusion this study has successfully presented a construction of new numerical method (hbm), analyse and implemented for solving numerous type of third-order bvps directly without utilizing the conventional method such as shooting method, reduction to first-order system of equations, finite difference method. in the derivation, the method make use of power series basic function due to its great stability and convergence attributes. the multi-collocation points was introduced for the purpose of improving the order of the work, which also enhances good accuracy, the points of interpolation and collocation were also chosen strategically in order to obtain a desired results. the method was applied to solve linear and non-linear boundary value problems in other to test its efficiency and accuracy. the results was also compared with numerous methods in the literatures as shown in tables 1, 3, 4 and 6. the results shows that the hbm is more efficient and accurate than other methods in the literature. the method is of order 11 with a smaller error terms. the discussion of hbm were discussed in details in chapter three. both the absolute error and maximum absolute error were also tested in other to ascertain the uniqueness and reliability of the hbm. hence we conclude that hbm is a best candidate in solving a class of such problems. acknowledgements the authors are much grateful to the reviewer for their recommendations and constructive comments. references [1] s. n. jator, “multiple finite difference methods for solving third order ordinary differential equations”, international journal of pure and applied mathematics 43 (2007) 253. 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[38] m. b. hossain, m. j. hossain, s. m. galib & m. m. h. sikdar, “the numerical solution of third order boundary value problems using piecewise bernstein polynomials by the galerkin weighted residual method”, international journal of advanced research in computer science 7 (2016). 9 j. nig. soc. phys. sci. 5 (2023) 917 journal of the nigerian society of physical sciences dirac equation for energy-dependent potential with energy-dependent tensor interaction c. a. onatea, m. o. oluwayemib,∗, i. b. okonc adepartment of physics, kogi state university, anyigba, nigeria b department of physical sciences, landmark university, omu-aran, nigeria. cphysics department, university of uyo, uyo, nigeria. abstract the relativistic symmetries of the dirac equation were investigated with an energy-dependent tensor potential interaction for two different energydependent potentials under parametric nikiforov-uvarov method and supersymmetric quantum mechanics and shape-invariance method. it is observed that the energy-dependent tensor interaction has stronger removal effect of the energy degeneracies in both the spin and pseudospin symmetries than the non-energy-dependent tensor interaction. doi:10.46481/jnsps.2022.917 keywords: bound state, dirac equation, eigen solutions, wave equation, potential function. article history : received: 05 july 2022 received in revised form: 31 august 2022 accepted for publication: 01 september 2022 published: 14 january 2023 c© 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: oluwatimilehin oluwadare 1. introduction the relativistic spin-1/2 particles in quantum mechanics is usually described by dirac equation. the subject has drawn attention of physicist in the theoretical domain over the years. the theoretical physicists under the dirac equation have analysed the characteristic features of deformed nuclei, effective shell models [1-6], etc for the pseudospin symmetry (ps) for different physical potential models. the identical bands and mesons were also studied in details for the spin symmetry (ss). the theoretical reports of these studies showed that energy doublets were produced under the ss and ps for different levels. very recently, tensor potential interaction was introduced to the ∗corresponding author tel. no: +2348032652678 email address: oluwayemimatthew@gmail.com (m. o. oluwayemi ) dirac equation to reduce the energy degeneracy. the reduction of the degeneracy depends on the applied tensor potential. the coulomb tensor potential for instance, reduced some degeneracies leaving some doublets unbroken. the application of yukawa tensor potential also breaks some degeneracy doublets and produced another form of degeneracies. onate et al. in their recent study applied hellmann tensor potential and they found out that the whole degeneracies were broken even when the tensor strength is taken as small as 0.2. owing to the application of dirac equation, different authors have studied the dirac equation in diverse areas using different traditional techniques [7-13]. however, it is very clear that the dirac equation under ss and ps for energy-dependent potential (edp) has not received attention to the best of our knowledge. hence, the call for this study. motivated by the application of relativistic wave equations particularly the dirac equation, this study wants 1 c. a. onate et al. / j. nig. soc. phys. sci. 5 (2023) 917 2 to examine the effect of edp potential on the eigenvalues of the ss and ps. this study will consider two sets of potentials using two different traditional methodologies. the two potential models are modified mie-type-constant edp and kratzer edp respectively in the presence of an energy-dependent tensor (edt) interaction via parametric nikiforov-uvarov method and supersymmetry quantum mechanics and shape invariance method. the major aim of this study is to determine the production of energy degenerate doublets by edp and its removal by edt interaction. the modified mie-type-constant edp and the kratzer edp respectively are given as v (r, e) = λ1(1 + a1 e) r2 + λ2(1 + a2 e) r−2 + λ3(1 + a3 e) (1) v (r, e) = λ4(1 + a4 e) r2 − λ5(1 + a5 e) r . (2) here, λi(i = 1, 2, . . . ) are potential strengths and ai(i = 1, 2, . . . ) are potential parameters. since this paper aim at determining the production and removal of energy degenerate state in the presence and absence of energy-dependent tensor interaction, we propose a coulomb-constant energy-dependent tensor interaction of the form u(r, e) = h1(1 + b1 e) r + h2(1 + b2 e)r (3) 2. dirac equation (ss and ps) the dirac equation with spin-1/2 particles under the potentials s (r) and v (r) as attractive scalar potential and repulsive vector potential is of the form [14, 15, 16] [cα ·ρ + β ( mc2 + s (r) + v (r) − e ] ψnκ(r) = 0, (4) with e and m as energy and particle mass, ρ = −i~∇ defines momentum operator with α and β as 4 × 4 dirac matrices, i.e. α = ( 0 σ1 σ1 0 ) , β = ( i 0 0 i ) (5) and σ1 = ( 0 1 1 0 ) ,σ2 = ( 0 −i i 0 ) ,σ3 = ( 1 0 0 −1 ) . (6) here, i represents the 2×2 matrix identity and, σi are the pauli 3-vector spin matrices. in the nuclei spherical symmetry, the angular momentum operator j and spin-orbit matrix operator κ = −β(σl + 1) commute with the dirac hamiltonian, where l is the total orbital angular momentum operator. the spinor wave functions can be classify following the radial quantum number n and the spin-orbit quantum number κ and can be expressed according to the pauli-dirac representation [14, 15, 16]. ψnκ(r) = ( fnκ(r) gnκ(r) ) = 1 r  fnκ(~r)y`jmκ(θ,ϕ)ignκ(~r)y ¯̀jm(−κ)(θ,ϕ)  (7) where the upper and lower spinor components fnκ(r) and gnκ(r) are the real square-integral radial wave functions. y`jmk(θ,φ) and y ¯̀jm−k(θ,φ) are the spin spherical harmonic functions coupled to the total angular momentum j and its projection m on the z axis for κ(κ+1) = `(`+1) and κ(κ−1) = i(`+1). the quantum number κ is related to the quantum number ` for spin and pseudospin symmetries as κ =  −(` + 1) = −( j + 12 ), (s1/2, p3/2, etc), j = ` + 1 2 , aligned spin (κ < 0) +`( j + 12 ), ( p1/2, d3/2, etc), j = `− 1 2 , unaligned spin (κ > 0) (8) the quasi-degenerate doublet structure can be expressed in terms of pseudospin angular momentum s̃ = 1/2 and pseudoorbital angular momentum ˜̀ which is defined as κ =  − ˜̀ = (− j + 12 ), (s1/2, p3/2, etc), j = ˜̀ − 12 , aligned spin (κ < 0) +( ˜̀ + 1) = ( j + 12 ), (d3/2, f5/2, etc), j = ˜̀ + 12 , unaligned spin (κ > 0) (9) where κ = ±1,±2, . . . . upon direct substitution of equation (7) into equation (4), we can obtain two radial coupled dirac equation for the two symmetry components as follows:( d dr + κ r ) fnκ (r) = [ mc2 + enκ − ∆(r) ] gnκ(r) (10) ( d dr − κ r ) gnκ (r) = [ mc2 − enκ + ∑ (r) ] fnκ(r). (11) for the spin symmetry, ∆(r) = cs = constant. then, we obtain a second-order differential equation for upper-spinor component as[ − d2 dr2 + κ (κ + 1) r2 + 1 ~2c2 (mc2 + enκ − cs) ∑ (r) ] fnκ(r) = 1 ~2c2 [ (e2nk − m 2c4 + cs) ( mc2 − enκ )] fnκ(r), (12) and the lower-spinor component is given by gnκ(r) = 1 mc2 + enκ − cs ( d dr + κ r ) fnκ(r) (13) it is only the real positive energy states that exist when cs = 0. however, under the pseudospin symmetry, σ(r) = c p = constant, one can have from equation (10) a second-order differential equation for the lower-spinor component as [14, 15][ − d2 dr2 + κ (κ− 1) r2 − 1 ~2c2 (mc2 − enκ + c ps)∆(r) ] gnκ(r) = 1 ~2c2 [ e2nκ − m 2c4 − c ps)(mc 2 − enκ) ] gnκ(r), (14) and the upper-spinor component fnκ(r) as fnκ(r) = 1 mc2 − enκ + c ps ( d dr + κ r ) gnκ(r) (15) it is only real negative energy states that exist when c p = 0. if we now include tensor interaction, then we obtain an equation in each case for both spin and pseudospin symmetries as 2 c. a. onate et al. / j. nig. soc. phys. sci. 5 (2023) 917 3 follows[ d2 dr2 − κ (κ + 1) r2 + 2κ r u(r) − du(r) dr − u 2(r) + d∆(r) dr m + enκ − ∆(r) ( d dr + κ r − u(r) ) fnκ(r) = [ (m + enκ − ∆(r)) ( m − enκ + ∑ (r) )] fnκ(r) (16) [ d2 dr2 − κ (κ− 1) r2 + 2κ r u(r) + du(r) dr − u 2(r) + d ∑ (r) dr m − enκ + ∑ (r) ( d dr − κ r + u(r) ) gnκ(r) = [ (m + enκ − ∆(r)) ( m − enκ + ∑ (r) )] gnκ(r). (17) 2.1. parametric nikiforov-uvarov method (pnum) the pnum is one of the analytical techniques of mathematical physics that solves second-order differential equations in quantum mechanics. this method has a general form of the schrödinger-like equation [17, 18, 19, 20][ d2 dr2 + c1 − c2 s s(1 − c3 s) d d s + −ξ1 s2 + ξ2 s − ξ3 s2(1 − c3 s) 2 ] ψ(s) = 0 (18) according to the pnum, the eigenvalues and eigenfunctions can be obtain following the condition [17, 18, 19, 20, 21] c2n − (2n + 1)c5 + [ n2 − n + 2c8 ] c3 + √ c8 (√ c9 + 2nc3 + c3 ) + √ c9 ( 2n + 1 + √ c8 ) = −c7 (19) ψ(s) = sc12 (1 − s) −c12− c13 c3 p (c10−1, c11 c3 c10−1) n (1 − 2c3 s) (20) the values of the parametric constants in equations (19) and (20) are obtained as follows: c4 = 0.5(1 − c1), c5 = 0.5(c2 − 2c3), c6 = c 2 5 + ξ1, c7 = 2c4c5 − ξ2, c8 = c 2 4 + ξ3, c9 = c3(c7 + c3c8) + c6, c10 = 1 + 2 ( c4 + √ c8 ) , c11 = c2 − 2c5 + 2c3 √ c8 + 2 √ c9, c12 = c4 + √ c8, c13 = c5 − c3 √ c8 − √ c9. (21) 3. solutions of dirac equation 3.1. ss limit the ss limit occurs when d∆(r)/dr = 0 and ∆(r) = cz with σ(r) = v (r, e). plugging equation (1) and equation (3) into equation (16), we have d2 fnκ(r) dr2 + [ χ1s r2 + χ2s r 2 + χ3s ] fnκ(r) = 0 (22) where χ1s = h1(1 + b1 enκ) −λ1(1 + a1 enκ)βs + 2κh1(1 + b1 enκ) − h21 (1 + b1 enκ) 2 − κ(κ + 1), (23) χ2s = −λ2(1 + a2 enκ)βs − h 2 2 (1 + b2 enκ) 2, (24) χ3s = h2(1 + b2 enκ) [2κ− 1 − 2h1(1 + b1 enκ)] −βs [λ3(1 + a3 enκ) + m − enκ] , (25) βs = m + enκ − cs. (26) using a transformation of the form y = r2 in equation (22), we obtain d2 fnκ(y) dy2 + 1 2y dfnκ(y) dy + χ2s y 2 + χ3s y + χ 1 s y2 fnκ(y) = 0. (27) comparing equation (27) with (18), we obtain the values of the parametric constants in equation (21) as follows c1 = 0.5, c2 = c3 = 0, c4 = 0.25, c5 = 0, c6 = −0.25χ 2 s, c7 = −0.24χ 3 s, c8 = 0.25(0.25 −χ 1 s ), c9 = −0.25χ 2 s, c10 = 1 + √ 0.25 −χ1s, c11 = √ −χ2s, c12 = 0.5 ( 0.5 + √ (0.25 −χ1s ) ) , c13 = − √ −0.25χ2s. (28) plugging equation (28) into equation (19) and equation (20), respectively, gives λ1(1 + a1 enκ)βs 4 + h1(1 + b1 enκ) 2 [ h1(1 + b1 enκ) − 1 2 − κ ] +( 1 + 2κ 4 )2 + βs [λ3 (1+a3 enκ)+m−enκ] 4 + h2 (1+b2 enκ) 2 [ h1(1 + b1 enκ) − κ− 1 2 ] + ℵs 1 + 2n + √ h22 (1 + b2 enκ) 2 + λ2(1 + a2 enκ)βs  2 = 0. (29) fnκ(y) = y 0.5(0.5+ √ 0.25−χ1s e− √ −0.25χ2s y l √ 0.25−χ1s n ( √ −χ2s y ) (30) ℵs = ( n + 1 2 ) √ h22 (1 + b2 enκ) 2 + λ2(1 + a2 enκ)βs. (31) 3.2. ps limit the pseudospin symmetry limit occurs when dς(r)/dr = 0 and σ(r) = c p. in this symmetry limit, the potential is taken as ∆(r) = v (r, e). now, substituting equations (1) and (3) into equation (17) and by using the same transformation as before, 3 c. a. onate et al. / j. nig. soc. phys. sci. 5 (2023) 917 4 we have the parametric constants as c1 = 0.5, c2 = c3 = 0, c4 = 0.25, c5 = 0, c6 = 0.25 [ h22 (1 + b2 enκ) 2 −λ2(1 + a2 enκ)βp ] , c7 = 0.5h2(1 + b2 enκ) [h1(1 + b1 enκ) − 0.5 − κ] + 0.25 [m + enκ −λ3(1 + a3 enκ)] βp, c8 = 0.5h1(1 + b1 enκ) [0.5h1(1 + b1 enκ) + 0.5 − κ] + (0.25 − 0.5κ)2 −λ1(1 + a1 enκ)βp, c9 = 0.25 [ h22 (1 + b2 enκ) 2 −λ2(1 + a2 enκ)βp ] , c10 = 1+√ 1 + h1(1 + b1 enκ) [ h1(be ) + 1 − 2κ ] + κ(κ− 1) −λ1(ae )βp, c11 = √ h22 (1 + b2 enκ) 2 −λ2(1 + a2 enκ)βp, c12 = 0.25+√ 0.5h1(be ) [ 0.5h1(be ) + 0.5 − κ ] + (0.25 − 0.5κ)2 −λ1(ae )βp, c13 = −0.5 √ h22 (1 + b2 enκ) 2 −λ2(1 + a2 enκ)βp, (32) where be = 1 + b1 enκ, ae = 1 + a1 enκ substituting equation (32) into equations (19) and (20), the energy for ps limit and its corresponding wave function are given as h1(1 + b1 enκ) 2 [ h1(1 + b1 enκ) + 1 2 − κ ] + ( 1 − 2κ 4 )2 − λ1(1 + a1 enκ)βs 4 + βp [m−λ3 (1+a3 enκ)+enκ] 4 + h2 (1+b2 enκ) 2 [ h1(1 + b1 enκ) − κ− 1 2 ] + ℵp 1 + 2n + √ h22 (1 + b2 enκ) 2 −λ2(1 + a2 enκ)βp  2 = 0. (33) fnκ(y) = y 0.25+ηp1 e0.5ηp2 y l ηp3 n (η4y) . (34) ℵp = ( n + 1 2 ) √ h22 (1 + b2 enκ) 2 −λ2(1 + a2 enκ)βp (35) ηp1 = √ 0.5h1(be ) [ 0.5h1(be ) + 0.5 − κ ] + γ1, (36) where γ1 = (0.25 − 0.5κ) 2 −λ1(1 + a1 enκ)βp. ηp2 = √ h22 (1 + b2 enκ) 2 −λ2(1 + a2 enκ)βp, (37) ηp3 = √ 1 + h1(be ) [ h1(be ) + 1 − 2κ ] + κ(κ− 1) − γ2, (38) where γ2 = λ1(1 + a1 enκ)βp. ηp4 = √ h22 (1 + b2 enκ) 2 −λ2(1 + a2 enκ)βp (39) 3.3. solutions of the ss and ps via supersymmetric approach in this section, we obtain the solutions of the spin and pseudospin symmetry limits for kratzer energy-dependent potential via susyqm. this method involves the proposition of superpotential function which is the solution of the non-linear riccati equation. 3.3.1. solution of the ss limit to obtain the solution of the spin symmetry limit of the dirac equation with kratzer energy-dependent potential, we substitute equations (2) and (3) into equation (16) to have a second-order differential equation of the form d2 fnκ(r) dr2 =[ χs1 r2 − λ5(1 + a5 enκ)βs r + h22 (1 + b2 enκ) 2r2 + χs2 ] fnκ(r) (40) where we have defined the following for mathematical simplicity χs1 = κ(κ + 1) + λ4(1 + a4 enκ)βs+ h1(1 + b1 enκ) [h1(1 + b1 enκ) − 2κ− 1] (41) χs2 = h2(1 + b2 enκ) [2h1(1 + b1 enκ) − 2κ + 1] + (m − enκ)βs (42) for a non-energy-dependent potential in the absence of tensor interaction, the energy eigenvalues in equation (40) purely depends on the quantum number n and the spin-orbit coupling term κ. this is physically related to energy as enκ = (n,κ(κ+1)). this shows that for κ , 0, the states are degenerate. to solve equation (40) using susy approach [22, 23, 24, 25], we can write f0κ(r) = ex p ( − ∫ w(r)dr ) , (43) where w(r) is a superpotential which determines the solution of equation (40). to proceed to the next level, it is necessary to propose a superpotential function [22, 23, 24, 25, 26]. in this work, our superpotential function is proposed as w(r) = δ0 −δ1r −1 (44) where δ0 and δ1 are two different constants. for equation (44) to determine the solution of equation (40), the following condition must be satisfied w 2(r) − dw(r) dr = χs1 r2 − λ5(1 + a5 enκ)βs r + h22 (1 + b2 enκ) 2r2 + χs2 (45) substituting equation (44) into equation (45), we easily determine the values of the two constants in equation (44) as δ20 = χ s 2 (46) δ1 = 1 2 ± √ 1 + 4 [ χs1 + h 2 2 (1 + b2 enκ) 2 ] 2 (47) 4 c. a. onate et al. / j. nig. soc. phys. sci. 5 (2023) 917 5 δ0 = λ5(1 + a5 enκ)βsδ−11 2 (48) to test the correctness of the superpotential function, we construct the partner potentials of the supersymmetric quantum mechanics and examine the shape invariance condition [27, 28, 29]. our partner potentials are obtain in the following forms v+(r) = w 2(r) + dw(r) dr = δ20 − 2δ0δ1 r + δ1(δ1 − 1) r2 (49) v−(r) = w 2(r) − dw(r) dr = δ20 − 2δ0δ1 r + δ1(δ1 + 1) r2 . (50) equations (49) and (50) satisfied the shape-invariance condition and so, the following relationship exist v+(r, a0) = v−(r, a1) + r(a1) (51) in equation (51), a0 = δ1 and the hamiltonian is shapeinvariant. however, a1 = f (a0) = a0 + 1, which simply means that the potentials v±(r) are the same apart from a constant and the residual term r(a1) is independent of the variable r. in this case, a1 is uniquely determined from an old set of parameter a0. from the mapping, we now established that δ1 → δ1 + 1 using the negative partner potential. in terms of the newly introduced parameters, we express the residual term as r(a1) = λ25(1 + a5 enκ) 2β2s 4a20 − λ25(1 + a5 enκ) 2β2s 4a21 (52) r(a2) = λ25(1 + a5 enκ) 2β2s 4a21 − λ25(1 + a5 enκ) 2β2s 4a22 (53) r(a3) = λ25(1 + a5 enκ) 2β2s 4a22 − λ25(1 + a5 enκ) 2β2s 4a23 (54) the energy of the system is obtained using the above summation given by σnk=1r(ak) which is generalized as r(an) = λ25(1 + a5 enκ) 2β2s 4a2n−1 − λ25(1 + a5 enκ) 2β2s 4a2n (55) in view of the negative partner potential, the complete energy equation for the spin symmetry is given as h2(1 + b2 enκ) [1 − 2κ + h2(1 + b2 enκ)] + (m − enκ)βs = λ25 [ (1 + a5 enκ)βs ]2 4(δ1 + n) 2 (56) 3.3.2. solution of the ps limit to obtain the solution of the pseudospin symmetry limit, we substitute equations (2) and (3) into equation (17) to have d2gnκ(r) dr2 = χ p 1 + λ5(1 + a5 enκ)βpr + χ p 2 r 2 + h22 (1 + b2 enκ) 2r4 r2 gnκ(r) (57) table 1: energies in the ss limit for modified mie-type-constant edp with λ1 = λ2 = λ3 = 0.5, a1 = 3, a2 = 2, a3 = 1, b1 = b2 = 0.1, m = 5 f m−1 and cs = 1 f m−1. ` κ n (`, j) h1,2 = 0 h1,2 = 0.1 h1,2 = 0.2, 0.1 h1,2 = 0.1, 0.2 0 -1 0 0s1/2 1.942623900 1.958361278 1.959096533 1.972020529 0 -1 1 1s1/2 2.941736416 2.957567872 2.958305747 2.971859998 0 -1 2 2s1/2 4.060873315 4.076229954 4.076927679 4.090458326 0 -1 3 3s1/2 5.254527544 5.269270999 5.269915044 5.283193841 1 -2 0 0p3/2 1.841748059 1.861208075 1.850731135 1.889789645 1 -2 1 1p3/2 2.850509762 2.869662294 2.859097611 2.898566721 1 -2 2 2p3/2 3.981140297 3.999437112 3.989244626 4.027491284 1 -2 3 3p3/2 5.185188130 5.202574765 5.192859711 5.229500168 2 -3 0 0d5/2 1.644091503 1.670587775 1.650604087 1.715582858 2 -3 1 1d5/2 2.667778395 2.692515093 2.671366252 2.737514242 2 -3 2 2d5/2 3.820578636 3.843285779 3.822525569 3.886288212 2 -3 3 3d5/2 5.045509610 5.066514491 5.046609652 5.107241634 3 -4 0 0f7/2 1.362484903 1.400794769 1.374384444 1.463630855 3 -4 1 1f7/2 2.394333292 2.428432855 2.397977358 2.491841820 3 -4 2 2f7/2 3.577245062 3.606844862 3.576008806 3.666695125 3 -4 3 3f7/2 4.833507250 4.859747029 4.829865691 4.915639885 1 1 0 0p1/2 1.841748059 1.852413864 1.877371357 1.836808899 1 1 1 1p1/2 2.850509762 2.861449107 2.885908697 2.847162481 1 1 2 2p1/2 3.981140297 3.991828450 4.015024293 3.978925935 1 1 3 3p1/2 5.185188130 5.195485760 5.217335080 5.183790476 2 2 0 0d3/2 1.644091503 1.649346874 1.685727304 1.616913066 2 2 1 1d3/2 2.667778395 2.674654649 2.711126809 2.644310749 2 2 2 2d3/2 3.820578636 3.827991306 3.862736003 3.800294001 2 2 3 3d3/2 5.045509610 5.052997276 5.085717151 5.027654662 3 3 0 0f5/2 1.362484903 1.357450308 1.402754356 1.305640324 3 3 1 1f5/2 2.394333292 2.393986861 2.442017313 2.344787927 3 3 2 2f5/2 3.577245062 3.579492175 3.625897479 3.534961395 3 3 3 3f5/2 4.833507250 4.837006867 4.880792058 4.796634088 table 2: energies in the ps limit for modified mie-type-constant edp with λ1 = λ3 = 0.5, λ2 = −0.5, a1 = −3, a2 = −2, a3 = −1, b1 = b2 = 0.1, m = 5 f m−1 and cs = 1 f m−1. ` κ n (`, j) h1,2 = 0 h1,2 = 0.1 h1,2 = 0.2, 0.1 h1,2 = 0.1, 0.2 1 -1 1 1s1/2 -1.758132998 -1.769902692 -1.779164454 -1.773124361 2 -2 1 1p3/2 -1.891711144 -1.914557485 -1.929357426 -1.923214633 3 -3 1 1d5/2 -2.100102133 -2.132843969 -2.153277701 -2.145586292 4 -4 1 1f7/2 -2.390428405 -2.431076159 -2.456965418 -2.446048454 1 -1 2 2s1/2 -2.424883237 -2.435163923 -2.443110934 -2.438169927 2 -2 2 2p3/2 -2.549727011 -2.569554464 -2.582073017 -2.577422600 3 -3 2 2d5/2 -2.740445644 -2.768710697 -2.785650672 -2.780425981 4 -4 2 2f7/2 -3.000000000 -3.035085686 -3.056131285 -3.049319087 1 2 1 0d3/2 -1.758132998 -1.735243670 -1.728251600 -1.719908479 2 3 1 0f5/2 -1.891711144 -1.858269052 -1.845486264 -1.837996526 3 4 1 0g7/2 -2.100102133 -2.341193985 -2.038745712 -2.316883988 4 5 1 0h9/2 -2.390428405 -2.341193985 -2.316253094 -2.316883988 1 2 2 1d3/2 -2.424883237 -2.404762492 -2.398620806 -2.391343780 2 3 2 1f5/2 -2.549727011 -2.956867588 -2.509496082 -2.502502503 3 4 2 1g7/2 -2.740445644 -2.956867588 -2.936575208 -2.682113098 4 5 2 1h9/2 -3.000000000 -2.956867588 -2.936575208 -2.934010648 where χ p 1 = κ(κ− 1) + h1(1 + b1 enκ) [h1(1 + b1 enκ) − 2κ + 1]− λ4(1 + a4 enκ)βp (58) χ p 2 = h2(1 + b2 enκ) [2h1(1 + b1 enκ) − 2κ− 1] + (m + enκ)(m − enκ + c p) (59) to avoid repetition of works and algebra, we follow the same steps as in the spin symmetry and obtain the energy equation for the pseudospin spin symmetry as h2(1 + b2 enκ) [2h1(1 + b1 enκ) − 2κ− 1] + (m + enκ)βp = [ −λ5(1 + a5 enκ)βp 2(δp + n) ]2 (60) 5 c. a. onate et al. / j. nig. soc. phys. sci. 5 (2023) 917 6 δp = 1 2 + 1 2 √ 4χp1 + 1 + 4h 2 2 (1 + b2 enκ) 2 (61) table 3: energies in the ss limit for modified mie-type-constant edp with λ1 = λ2 = λ3 = 0.5, a1 = 3, a2 = 2, a3 = 1, b1 = b2 = 0.1, m = 5 f m−1 and cs = 1 f m−1 ` κ n (`, j) h1,2 = 0.2, 0 h1,2 = 0, 0.2 0 -1 0 0s 1/2 1.942079678 1.968171442 0 -1 1 1s 1/2 2.941066860 2.967805266 0 -1 2 2s 1/2 4.060044875 4.086322787 0 -1 3 3s 1/2 5.253529416 5.279016863 1 -2 0 0p3/2 1.818436560 1.897264650 1 -2 1 1p3/2 2.826966136 2.905851918 1 -2 2 2p3/2 3.958335296 4.034259115 1 -2 3 3p3/2 5.163329221 5.235688819 2 -3 0 0d5/2 1.601252028 1.732796555 2 -3 1 1d5/2 2.622584840 2.755428278 2 -3 2 2d5/2 3.776272981 3.903614939 2 -3 3 3d5/2 5.003004662 5.123603755 3 -4 0 0f7/2 1.306444952 1.487803464 3 -4 1 1f7/2 2.329887999 2.519191159 3 -4 2 2f7/2 3.512241944 3.694086997 3 -4 3 3f7/2 4.770643993 4.941943853 1 1 0 0p1/2 1.889417309 1.808576037 1 2 1 1p1/2 2.897131599 2.819215988 1 1 2 2p1/2 4.025207202 3.952158620 1 3 3 3p1/2 5.226530876 5.158310400 2 4 0 0d3/2 1.713977782 1.577278752 2 2 1 1d3/2 2.737952422 2.604208929 2 2 2 2d3/2 3.887416000 3.761839331 2 2 3 3d3/2 5.108364728 4.991200184 3 3 0 0f5/2 1.449429077 1.257493042 3 3 1 1f5/2 2.486841044 2.293004999 3 3 2 2f5/2 3.666840421 3.484650816 3 3 3 3f5/2 4.918103438 4.748960521 4. discussion in table 1, we presented the energy eigenvalues of the spin symmetry for equal and unequal values of h1 and h2. for h1 = h2 = 0, which is the solution without energy-dependent tensor interaction, the following degeneracies were produced: 0p3/2 = 0p1/2, 1p3/2=1p1/2, 3p3/2=3p1/2, 0d5/2, 0d3/2, 1d5/2=1d3/2, 2d5/2 = 2d3/2, 3d5/2=3d3/2, 0f7/2=0f5/2, 1f7/2=1f5/2. 2f7/2=2f5/2 and 3f7/2=3f5/2/. these degeneracies are the usual degeneracies obtained with non-energy-dependent potentials. however, for h1 = h2 = 0.1, which is a solution with energy-dependent tensor interaction, there are no degeneracies. this shows that the energy-dependent tensor potential has broken the energy degenerate doublets in the system. for a non-energy-dependent tensor potential, even at h = 0.5 and h = 1, there are still degenerate doublets. for h1 = 0.2, h2 = 0.1 and h1 = 0.1, h2 = 0.2, there are no degeneracy production. in table 2, we presented energy eigenvalues of the pseudospin symmetry for equal and unequal values of the tensor strengths i.e. h1 table 4: energies in the ps limit for modified mie-type-constant edp with λ1 = λ3 = 0.05, λ2 = −0.05, a1 = 6, a2 = −4, a3 = 5, b1 = b2 = 0.1, m = 15 f m−1 and cs = 5 f m−1. ` κ n (`, j) h1,2 = 0.2, 0 h1,2 = 0, 0.2 1 -1 1 1s 1/2 -1.775253455 -1.763815218 2 -2 1 1p3/2 -1.920130164 -1.908460034 3 -3 1 1d5/2 -2.140140406 -2.125315642 4 -4 1 1f7/2 -2.441826999 -2.420424044 1 -1 2 2s1/2 -2.439648344 -2.430168824 2 -2 2 2p3/2 -2.573814603 -2.564901362 3 -3 2 2d5/2 -2.773637600 -3.028396662 4 -4 2 2f7/2 -3.041719317 -3.028396662 1 2 1 0d3/2 -1.742804990 -1.726917318 2 3 1 0f5/2 -1.865069919 -1.850942038 3 4 1 0g7/2 -2.061722917 -2.053329000 4 5 1 0h9/2 -2.340384372 -2.342315770 1 2 2 1d3/2 -2.411514069 -2.397477706 2 3 2 1f5/2 -2.526953758 -2.513529136 3 4 2 1g7/2 -2.708412767 -2.698017234 4 5 2 1h9/2 -2.959189164 -2.954567108 table 5: energies in the ss limit for kratzer edp with λ4 = λ5 = 0.5, a1 = 4.0, a2 = 2, b1 = b2 = 0.1, m = 10 f m−1 and cs = 5 f m−1 ` κ n (`, j) kratzer potential coulomb potential h1,2 = 0 h1,2 = 0.1 h1,2 = 0.1 h1,2 = 0.1, 0 0 -1 0 0s1/2 0.370204340 0.393187068 1.252040915 1.230390156 0 -1 1 1s1/2 0.945739430 0.963640344 2.192321102 2.167520674 0 -1 2 2s1/2 1.435407674 1.453860405 2.839097594 2.811476651 0 -1 3 3s1/2 1.899432950 1.919827326 3.304673071 3.274484128 1 -2 0 0p3/2 0.568272837 0.605461699 2.196388371 2.167520674 1 -2 1 1p3/2 1.067819409 1.102125131 2.847354128 2.811476651 1 -2 2 2p3/2 1.530395283 1.565774885 3.316370162 3.274484128 1 -2 3 3p3/2 1.979996399 2.018417622 3.663421017 3.616377889 2 -3 0 0d5/2 1.262754541 0.865450254 2.858804254 2.811476651 2 -3 1 1d5/2 0.820311004 1.309314656 3.330517856 3.274484128 2 -3 2 2d5/2 1.694588465 1.744582496 3.679846038 3.616377889 2 -3 3 3d5/2 2.125227188 2.180498632 3.943595506 3.873809868 3 -4 0 0f7/2 1.085509689 1.138674624 3.345044630 3.274484128 3 -4 1 1f7/2 1.493098313 1.550825160 3.696546742 3.616377889 3 -4 2 2f7/2 1.901400337 1.965254589 3.962133536 3.873809868 3 -4 3 3f7/2 2.315910246 2.387891090 4.166042378 4.070830840 1 1 0 0p1/2 0.568272837 0.542744504 1.976844362 1.975929806 1 3 1 1p1/2 1.067819409 1.048345302 2.667541872 2.668584697 1 1 2 2p1/2 1.530395283 1.513218985 3.163567616 3.166216009 1 5 3 3p1/2 1.979996399 1.963746819 3.529456299 3.533423105 2 6 0 0d3/2 0.820311004 0.786755216 2.654220464 2.668584697 2 2 1 1d3/2 1.262754541 1.231133014 3.147935985 3.166216009 2 2 2 2d3/2 1.694588465 1.663175926 3.511836761 3.533423105 2 2 3 3d3/2 2.125227188 2.092809055 3.785235098 3.809600325 3 3 0 0f5/2 1.085509689 1.044886371 3.132908744 3.166216009 3 3 1 1f5/2 1.493098313 1.451243785 3.494674460 3.533423105 3 3 2 2f5/2 1.901400337 1.857234266 3.766256136 3.809600325 3 3 3 3f5/2 2.315910246 2.268233378 3.973416820 4.020621390 and h2. for h1 = h2 = 0, the following degenerate doublets are obtain: 1s1/2 = 0d3/2, 1p3/2 = 0f5/2, 1d5/2 = 0g7/2, 1f7/2 = 0h9/2, 2s1/2 = 1d3/2, 2p3/2 = 1f5/2, 2d5/2 = 1g7/2 and 2f7/2 = 1h9/2. these degeneracies are also equal to the degeneracies produced for non edp for non-tensor interaction. for h1 = h2 = 0.1, there are no degeneracies. similarly, for h1 > h2 and h1 < h2, there are no degenerate doublets. this also shows that the inclusion of the edt term breaks the whole degeneracies even at small values of the tensor strengths. in 6 c. a. onate et al. / j. nig. soc. phys. sci. 5 (2023) 917 7 table 6: energies in the ps limit for kratzer edp with λ4 = λ5 = 0.5, λ2 = 0, a1 = 4.0, a2 = 2, b1 = b2 = 0.1, m = 10 f m−1 and cs = 5 f m−1 ` κ n (`, j) kratzer potential coulomb potential h1,2 = 0 h1,2 = 0.1 h1,2 = 0.1 h1,2 = 0.1, 0 1 -1 1 1s1/2 -9.096854737 -9.097243710 -3.957853178 -3.956166280 2 -2 1 1p3/2 -9.103820973 -9.104900620 -4.705516764 -4.702953988 3 -3 1 1d5/2 -9.114041578 -9.115790810 -5.338447790 -5.334597330 4 -4 1 1f7/2 -9.127260481 -9.129650770 -5.877946235 -5.872766190 1 -1 2 2s1/2 -9.165271209 -9.165625395 -4.704351342 -4.702953988 2 -2 2 2p3/2 -9.171400175 -9.172390830 -5.337077070 -5.334597330 3 -3 2 2d5/2 -9.180401886 -9.182010810 -5.876538030 -5.872766190 4 -4 2 2f7/2 -9.192060938 -9.194263445 -6.338969120 -6.333932324 1 2 1 0d3/2 -9.096854737 -9.095134090 -3.852996864 -3.855104284 2 3 1 0f5/2 -9.103820973 -9.101415860 -4.624203482 -4.628695316 3 4 1 0g7/2 -9.114041578 -9.110975965 -5.272927745 -5.279249573 4 5 1 0h9/2 -9.127260481 -9.123565500 -5.823169340 -5.831036969 1 2 2 1d3/2 -9.165271209 -9.163683540 -4.625917968 -4.628695316 2 3 2 1f5/2 -9.171400175 -9.169181270 -5.274472190 -5.279249573 3 4 2 1g7/2 -9.180401886 -9.177572380 -5.824661165 -5.831036969 4 5 2 1h9/2 -9.192060938 -9.188647735 -6.294424900 -6.302161924 note: hi, j = 0 means hi = h j = 0. hi, j = 0.1, 0.2 means hi = 0.1, h j = 0.2. figure 1: energies in the ss limit against mass m for modified mie-typeconstant edp with λ1 = λ2 = λ3 = 1, a1 = 3, a2 = 2, a3 = 1, b1 = b2 = 1, and cs = 5 f m−1. tables 3 and 4, we presented the energy for ss and ps respectively for coulomb energy-dependent tensor potential ( h2 = 0) and constant energy-dependent tensor potential ( h1 = 0). in both cases, there are no degeneracies. to check the accuracy and correctness of the energy-dependent tensor potential, we also studied the solutions of the spin and pseudospin symmetries with the same energy-dependent tensor potential with the kratzer energy-dependent potential. the special cases of this potential was studied numerically. the results of the two symmetries are given in tables 5 and 6. in table 5, the energy for spin symmetry is given for both kratzer and coulomb energydependent potentials. for kratzer energy-dependent potential, the degeneracies obtained in table 1 were equal obtained. for coulomb energy-dependent potential ( λ4 = 0), it was considered for coulomb-constant energy-dependent tensor potential and coulomb energy-dependent tensor potential ( h2 = 0). for h1 = h2 = 0.1, the same energy degeneracies obtained figure 2: energies in the ps limit against mass m for modified mie-typeconstant edp with λ1 = λ2 = λ3 = 1, a1 = 3, a2 = 2, a3 = 1, b1 = b2 = 1, and cs = 5 f m−1. in table 1 were also obtained, but for h2 = 0, which reduces the coulomb-constant energy-dependent tensor potential to coulomb energy-dependent tensor potential, a new set of energy degeneracies were formed. the degeneracies formed are 1s1/2 = 0p3/2, 2s1/2 = 1p3/2 = 0d5/2, 3s1/2 = 2p3/2 = 1d5/2 = 0f7/2, 3p3/2 = 2d5/2 = 1f 7/2, 3d5/2 = 2f7/2, 1p1/2 = 0d3/2, 2p1/2 = 1d3/2 = 0f5/2, 3p1/2 = 2d3/2 = 1f5/2 and 3d3/2 = 2f5/2. these are new degeneracies different from the degeneracies obtained with ordinary coulomb tensor potential. for the pseudospin symmetry in table 6, the following degeneracies were formed with coulomb energy-dependent tensor potential for coulomb energy-dependent potential: 1d5/2 = 2p3/2, 1f7/2 = 2d5/2, 0f5/2 = 1d3/2, 0g7/2 = 1f5/2 and 0h9/2 = 1g7/2. for ordinary coulomb tensor potential with h = 0.5 and h = 1, the degeneracies formed in each case are different from those formed in the present work. figure 1 and figure 2 showed the variation of the energy of ss and ps respectively with the mass m for the modified mie-type potential. the energy of the ss increases with the mass while that of the ps decreases with the mass. 5. conclusion in this work, we have employed two traditional techniques to solve dirac equation with two energy-dependent potential and an energy-dependent tensor interaction without the use of any approximation scheme to the centrifugal term. the degeneracies formed in our results 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[29] c. a. onate, g. o. egharevba, & d. t. bankole, “eigensolution to morse potential for scandium and nitrogen monoiodides”, j. nig. soc. phys. sci. 3 (2021) 286. 8 j. nig. soc. phys. sci. 4 (2022) 699 journal of the nigerian society of physical sciences assessing the need for radiation protection measures in artisanal and small scale mining of tantalite in oke-ogun, oyo state, nigeria a. e. ajetunmobia,∗, a. o. musthaphab, i. c. okeyodeb, a. m. gbadeboc, d. al-azmid, t. w. davida a department of physics, olabisi onabanjo, ago-iwoye, ogun state, p m b 2002, ago-iwoye, ogun state. b department of physics, federal university of agriculture, abeokuta, p m b 2240, abeokuta, nigeria c department of environmental management and toxicology, federal university of agriculture, abeokuta, p m b 2240, abeokuta, nigeria d department of applied sciences, college of technological studies, public authority for applied education and training, shuwaikh, p o box 42325, code 70654, kuwait abstract there is concern that work scenarios on the tantalite mining sites in oke-ogun, oyo state, nigeria may cause occupational radiation exposure of workers due to enhanced concentrations of naturally occurring radionuclides in some process materials. a prior radiological assessment of the mining activities was carried out to determine if and which exposure scenarios may require radiation protection measures. samples of the materials involved, comprising tantalite (tantalum ores), soil and waste rock were collected and analyzed for activity concentrations of 40k, 226ra, 238u and 232th using a hyper pure germanium detector gamma-ray spectrometer. radon concentrations in the mines were also measured using a continuous radon monitor – radon-scout plus (sarad, gmbh). activity concentrations of 40k are below 10 bqg−1 in all the samples but all the tantalite samples contain more than 1 bqg−1 of 226ra and 238u. hence tantalite is regarded as naturally occurring radioactive material (norm) and the mining activity as a practice. the requirements for planned exposure situations apply to all the mining sites but, on the basis of graded approach, the optimum radiation protection measures vary from one mine to another, ranging from exemption to authorization. exposures to radon in the underground mines pose the greatest radiological risks and portend the greatest need for regulatory control in the mining operations. the results further underscore the need to integrate radiation protection with the other health and safety measures in the mining sector. doi:10.46481/jnsps.2022.699 keywords: radiation protection, tantalite, germanium detector article history : received: 7 march 2022 received in revised form: 27 april 2022 accepted for publication: 29 april 2022 published: 16 august 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published articleâs title, journal citation, and doi. communicated by: w. a. yahya ∗corresponding author tel. no: +2348158593532 email address: abayomi.ajetunmobi@oouagoiwoye.edu.ng (a. e. ajetunmobi ) 1. introduction minerals and raw materials normally contain moderate to elevated concentrations of naturally occurring radionuclides, mainly 40k and those in the decay series of 238u and 232th [16]. radiation safety regulations are well established in uranium 1 a. e. ajetunmobi et al. / j. nig. soc. phys. sci. 4 (2022) 699 2 mining and processing where uranium ores contain up to several thousand bqg−1 of radionuclides in the 238u and 232th decay series [2]. there are many other industrial activities where uranium is not the main product but a contaminant or a secondary mineral [2]. the radiological hazards in such industrial activities may be significantly reduced and the need for regulatory control may not be clear. the international atomic energy agency (iaea) has provided some guidance but the decision on which of the industrial activities to regulate is normally left for the regulatory body [2, 4, 5]. this includes establishing activity concentrations of the relevant radionuclides in the materials involved, determination of doses and appropriately designating exposure situations based on the graded approach and other requirements of the basic safety standards [4]. the effective dose received by a worker exposed to naturally occurring radioactive material is reliably assessed by conducting a properly developed monitoring programme in the relevant workplaces [5, 6]. it can also be derived from model calculations using appropriate operational quantities and dose coefficients [3]. however, if the work scenarios and the activity concentration of radionuclides in relevant process materials are well known, it is possible to obtain, in advance, a broad indication of the expected dose to workers from exposure to gamma radiation and exposure due to airborne dust. the latter approach is described in the iaea safety series report no. 49 [2] as a prioritization tool for identifying the types of industrial operations and scenarios having the greatest need for radiation protection measures. this approach was adopted in the present study. artisanal and small-scale mining (asm) have been practiced over several decades in many parts of nigeria, including oke-ogun, south western nigeria. it is an informal mining sector where rudimentary techniques or machinery with small degree of mechanization are used to exploit small and shallow mineral resources. although asm provides livelihood alternative for some people, it contributes very little to nigeria’s economy and it has been associated with various environmental degradation and health hazards. formalization and reform of the asm are now being carried to improve the mining sector as part of the national policy to diversify from the oil and gas sector in nigeria [7]. the formalization is expected to institutionalize best practices including those that address the associated environmental and health hazards, both radiological and nonradiological. in this regard, it is anticipated that the outcome of is study will provide some insights into the potential radiological exposures associated with asm operations in general and tantalite mining operations in particular. the report will also serve as practical guide on how to determine the need for radiation protection measures in industrial activities that involve naturally occurring radioactive materials (norm) based on the graded approach. 2. materials and methods 2.1. geographic and geological features of the study area the mines are located in komu, sepeteri, gbedu and eluku villages in itesiwaju, saki east, iwajowa and saki local government areas respectively, all in oke-ogun, oyo state, southwest of nigeria. oke-ogun spans over latitudes 80 00’ – 80 39’ n and longitudes 20 56’ – 30 46’ e, and it is averagely at an elevation of 188m above the sea level. it has a population of about 1.4 million people according to the latest population census. the main occupations of residents in the area are artisanal mining and farming. the area is well drained by rivers oyan, ofiki, olori and many other tributaries of ogun river, forming dendritic drainage patterns in many parts of the area (figure 1). the major geological formations in the area include undifferentiated schist and gneiss that are widely distributed around sepeteri in north eastern and gbedu in south western regions. alternate layers of granitic gneiss, biotite hornblende granite, migmatite, migmatic granite gneiss, amphibole schist and porphyroblastic gneiss run parallel from northwest to southeast, with discontinuities occasioned by minor geological formations including porphyritic granite in the southwest, quartz veins in the east, pegmatite occurrence around sepeteri, komu, gbedu and iwere-ile in the south and southwest, and syenite and pyroxene diorite in the north. parts of komu in the west and eluku in the northwest are also both underlain by granitic gneiss and biotite hornblende granite, respectively. oke-ogun is endowed with a variety of minerals and precious stones, including tantalite and iron ore, marble, talc, beryl, etc. (figure 2). 2.2. determination of activity concentrations of radionuclides in process materials 2.2.1. types of operations and processes involved the study covered eight mining sites, comprising one each in gbedu and sepeteri, two in eluku and four in komu. the four mines in komu are underground while the rest are open pit mines. the entire mining operations were divided into two broad types, namely artisanal mining operations which involve digging manually and blasting to extract ore-bearing rocks, and processing of tantalum ore (tantalite) involving work scenarios such as crushing (dry or wet) and grinding to separate the orebearing rock from waste rock, physical separation of tantalite from other metallic ores using gravity, electrostatic or electromagnetic processes, and working or staying close to stock-piles of tantalite. these two types of operations are among those that have earlier been identified as being likely to require regulatory control [2]. 2.2.2. types of materials involved and sample preparation the three involved materials considered are residual soil, waste rock and tantalite or tantalum ore. forty samples, comprising twelve ores, seventeen soils and eleven waste rocks were collected from all the selected mining sites. the samples were air dried, pulverized and homogenized. aliquots were sealed in 500 ml cylindrical plastic containers for a month to ensure secular equilibrium between 226ra and its decay products. 2.2.3. gamma-ray spectrometry the samples were analyzed using a hyper pure germanium (hpge) detector gamma-ray spectrometer at the national institute of radiation protection and research, university of 2 a. e. ajetunmobi et al. / j. nig. soc. phys. sci. 4 (2022) 699 3 figure 1: maps of nigeria, oyo state and oke-ogun showing the selected mining areas ibadan, nigeria. a multi-radionuclides standard source was used for energy and efficiency calibrations. the radionuclides of interest are 40k, 226ra 232th and 238u, and the gamma-ray energies used to estimate the activity concentrations are 609.3 kev of 214bi for 226ra; 911 kev of 228ac for 232th; 1001 kev of 234mpa for 238u; and 1460 kev for 40k. 3 a. e. ajetunmobi et al. / j. nig. soc. phys. sci. 4 (2022) 699 4 figure 2: map of oke-ogun in oyo state nigeria showing major geological and mineral deposits 2.3. measurement of radon concentration the health effect as a result of exposure to 222ra is radiation induced [8], hence the need to measure radon concentration in the study. radon concentrations were measured in the open pit and underground mines using a continuous radon monitoring device: radon scout plus (sarad, gmbh). it is portable (of dimension 175 × 135 × 55 mm3), battery operated and capable of continuous cyclic data logging of radon concentration, air temperature, relative humidity and atmospheric pressure after every predetermined period. during the measurement, the instrument was held above the floor and away from the walls in underground mines. the sampling time varied depending mainly on the time permitted by the mine operators. the logged data were assessed by connecting the radon monitor to a personal computer through a usb connector for graphical display and further data analysis through the dedicated software sarad radon vision 4.0.7 [9]. 4 a. e. ajetunmobi et al. / j. nig. soc. phys. sci. 4 (2022) 699 5 table 1: relationship between dose and activity concentration for occupational exposure to gamma radiation and to dust [2] category of material broad estimate of annual effective dose per unit activity concentration (msv/y per bq/g) individual radionuclide activity concentration above which the expected dose may exceed 10% of the dose limit (bq/g) minimum maximum large quantity, e.g. ore body, large stockpile 0.02 0.4 5 small quantity, e.g. mineral concentrate, scale, sludge 0.008 0.04 50 volatilized: furnace fume and precipitator dust 0.0006 0.003 500a athis value refers to the activity concentration in the precipitator dust with exposure to fume having been accounted for by assuming an equivalent dust loading of 1 mg/m3 at the same activity concentration (i.e. a concentration of 0.05 bq/m3 in fume) and an activity media aerodynamic diameter (amad) of 1 µm. 2.4. exposure scenarios and pathways many work scenarios are involved in the two operations as mentioned in section 2.2.1. each of these work scenarios may constitute an exposure scenario, but in this screening radiological assessment the exposure pathways considered for workers effective dose are: i. external exposure due to whole body irradiation by gamma-rays emitted by materials associated with the work, ii. internal exposure due to intake (ingestion and inhalation) of radionuclides in dust of the materials, iii. inhalation of radon released into air from the surrounding materials 2.5. assessment of effective doses to artisans the effective dose to workers due to exposure to gamma radiation and to airborne dust was calculated from the measured activity concentration of radionuclides in 238u and 232th decay series, and dose coefficients, i.e. indicative relationships between dose and activity concentration in the iaea safety series report 49 [2], (table 1). the annual effective dose (e) from exposure to 222rn in the mine was estimated from the measured 222rn concentrations (crn) using the conventional equation [1]: e = crn × f × t × d, (1) where f is an equilibrium factor between 222rn and its decay products (0.4 for indoor/underground and 0.6 for outdoor), t is hours in a year (hy−1) spent by miners at the site and d is the dose conversion factor (9.0 × 10−6 msv per bq h m−3), which is the effective dose received by an adult per unit 222rn exposure. 3. results and discussion 3.1. activity concentration of radionuclides in process materials the ranges of activity concentrations of 40k, 226ra, 232th and 238u in the tantalite, soil and rock samples are( 0.0031.183) bq/g, (1.231-140.982) bq/g, (0.753-4.808) bq/g and (2.314-173.827) bq/g; (0.003-1.139) bq/g, (0.008-0.369) bq/g, (0.003-0.253) bq/g and (0.003-0.761) bq/g; and ( 0.071-1.709) bq/g, (0.003-0.261) bq/g, (0.003-0.318) bq/g and (0.0030.297) bq/g1, respectively (table 2). all the materials contain less than 10 bqg−1 of 40k and all soil and rock samples contain less than 1 bqg−1 of 226ra, 232th and 238u. concentrations of 226ra and 238u exceed 1 bq/g in all the tantalite samples, while those of 232th exceed 1 bqg−1 in about 75% of the tantalite samples. the implication of these results is that tantalite qualifies to be regarded as norm, its mining in all the mine sites is regarded as a practice and the requirements for planned exposure situations apply as stated in the iaea gsr-3 [4]. on the other hand, soil and rock in the mine sites do not require further radiological considerations unless there are possibilities of being used as building materials. the results also show disequilibrium between 238u and 226ra in all the tantalite samples, but only in a few soil and rock samples. this was also reported by [3] in columbite-tantalite from the democratic republic of congo and by [10] in niobium ore from brazil. the disequilibrium between 238u and 226ra in the tantalum ores may indicate mobilization into other materials not considered in this study, such as waste water, and this may have radiological environmental impact around the mines as discussed by [3]. 5 a. e. ajetunmobi et al. / j. nig. soc. phys. sci. 4 (2022) 699 6 table 2: activity concentration of radionuclides in samples of materials involved in the mining operations and 222rn concentration in the mine air location and material range of activity concentrations (bq/g) average 222rn concentration (bqm−3) 40k 226ra 232th 238u eluku i 92 tantalite 0.113 20.821 1.473 23.979 rock 0.262 0.047 0.005 bdl soil 0.071 0.029 0.009 bdl eluku ii 55.5 tantalite 0.13 44.674 1.489 51.01 rock 0.296 0.062 0.007 bdl soil 0.13 0.032 0.024 bdl gbedu 56 tantalite 0.162 24.277 2.786 27.867 rock 1.111 0.01 bdl bdl soil 0.676 0.188 0.107 0.249 komu i 180335.3 tantalite bdl 127.048 1.583 155.724 rock bdl 0.17 0.008 0.23 soil 1.709 0.035 0.003 bdl komu ii 228.4 tantalite 0.171 95.663 2.917 118.903 rock bdl 0.178 0.008 0.259 soil 1.638 bdl bdl bdl komu iii 4387.6 tantalite 0.006 95.327 1.509 123.56 rock 0.621 369 0.253 0.761 soil 1.652 0.035 0.003 bdl komu iv 2565.8 tantalite bdl 140.982 2.056 173.827 rock 0.213 0.239 0.09 0.417 soil 1.666 bdl bdl bdl sepeteri 55 tantalite 0.743 18.254 1.966 55.783 rock 0.812 0.023 0.007 0.021 soil 0.656 0.073 0.016 0.152 bdl=below detection limit; detection limits are 3.1, 2.8, 2.9 and 2.9 (10−3 bq/g) for 40k, 226ra, 232th and 238u activity concentrations, respectively. 3.2. radon concentration in the mines the mean 222rn concentrations vary widely from 55 to 180335 bqm−3 (table 2). the values recorded in the open pit mines are higher than the global average outdoor value (10 bqm−3) but within the global range (1-100 bqm−3) [1]. the values recorded in the underground mines are all significantly higher than the action level for workplaces, which is 1000 bqm−3 annual average [4, 5]. the operations in underground mines are further subject to the requirements for planned exposure situations on account of significantly high 222rn concentrations. this decision is based on the assumption that the average radon concentration during the short-term monitoring is representative of the annual average, which could be more or less, subject to atmospheric, diurnal and seasonal variations. unfortunately, long term or repeated measurements were not allowed by the mine owners. presently, they seem unaware of occupation exposure to radiation or the needs for radiation protection in their industry. 3.3. determination of optimum regulatory control options using the graded approach a graded approach to regulation, as embodied in the iaea basic safety standards [4], states that the application of the requirements in planned exposure situations “shall be commensurate with characteristics of the practice or the source within a practice, and with the magnitude and likelihood of the exposures”. in the case of exposure to norm, it will involve more than just establishing that the activity values of the relevant radionuclides are exceeded, but the particular types of operation, process and material will also be considered in more detail, including screening dose assessments. the outcome will determine the optimum regulatory option. 6 a. e. ajetunmobi et al. / j. nig. soc. phys. sci. 4 (2022) 699 7 table 3: expected effective dose to workers, operation/scenario with the greatest need for radiation protection and recommended measure based on the graded approach mine location mine type workers dose (msv/y) operation/scenario with greatest need for radiation protection measure recommended radiation protection measure exposure to 222rn exposure to gamma radiation and dust eluku i open pit 0.73 2.040 tantalite processing exemption eluku ii open pit 0.44 0.959 tantalite processing exemption gbedu open pit 0.22 1.115 tantalite processing exemption komu i underground 5690.0 2.231 underground mining authorization komu ii underground 7.20 6.229 underground mining notification komu iii underground 80.71 4.756 underground mining authorization komu iv underground 94.40 4.942 underground mining authorization sepeteri open pit 0.11 6.953 tantalite processing notification the two types of operations considered in the study are mining of tantalite and processing of tantalite, and tantalite (tantalum ore) is the norm involved in both operations, i.e. material in which concentrations of 226ra and 238u exceed 1 bq/g (table 2). tantalite processing involves exposure scenarios, e.g. magnetic separation of tantalite from other metallic ores, where workers come in contact with small quantities of materials. therefore, the expected effective doses due to workers exposed to gamma radiation and to airborne dust of tantalite were estimated using the published annual effective dose per unit activity concentration for small quantities of materials (table 1). the estimated values range from about 1 to about 7 msv/y as shown in table 3. according to the graded approach to regulation, the operation could be exempted if the effective dose received by a worker from this pathway does not exceed 1 – 2 msv/y. therefore the operations in the mines in eluku i, eluku ii and gbedu could be considered for exemption, meaning there is no need to impose any regulatory requirements on them. after exemption the next level in the graded approach is notification, which is the requirement for the legal person (which in this case is the mine owner) to submit a formal notification to the regulatory body informing it of the operation. notification alone could be sufficient provided the exposures do not exceed a small fraction of the relevant limits. consequently, the open pit mine in sepeteri and one of the underground mines in komu (komu ii) could be considered for notification. authorization is the next to notification when there is need to place higher obligations on the legal person. according to the graded approach, authorization may take the form of registration or licensing, depending on how stringent the regulation is required to be. registration is the less stringent of the two and it places only limited obligations on the legal person, but “it may provide sufficient level of control in many operations involving significant, but nevertheless moderate exposures to norm and/or radon”. licensing is the highest level in the graded approach to regulation. it is the more appropriate authorization where optimized protection can only be achieved through the enforcement of specific exposure control measures. in this regard, the high exposures to radon in the remaining three underground mines in komu may require authorization. while registration may provide sufficient level of control for the operations in komu iii and komu iv, licensing may be more appropriate for the operation in komu i where workers may experience significant exposures to radon. specific control measures such as installation of ventilation system and other remedial actions to drastically reduce exposure to radon would be required, considering that any situation that may result in a annual dose greater than 100 msv is unacceptable, except under emergency exposure situation [4]. there may also be need to establish suitable radiation protection programmes that entail monitoring and dose assessments. 4. conclusion an assessment was carried out to evaluate the need for radiation protection measures in artisanal and small scale mining of tantalite in oke-ogun, southwest, nigeria. the results generally indicate there is need for radiation protection measures in the mining operations. two types of operations were considered, namely tantalite mining in both open pit and underground mines and tantalite processing. three of the materials involved in the operations, namely soil, rock and tantalite, were 7 a. e. ajetunmobi et al. / j. nig. soc. phys. sci. 4 (2022) 699 8 considered but only tantalite (tantalum ore) contains more than 1 bq/g of 226ra and 238u and qualifies as norm. high concentrations of 222rn were measured in mine air, particularly in the underground mines. doses to workers due to exposures to gamma radiation and due to dust, and exposure to 222rn show that the mining operations are services and the requirements for planned exposure situations apply. the graded approach to regulation was applied to optimize the regulatory options for each of the mines. the optimized regulatory options range from exemption to licensing of the operations. but the greatest need for radiation protection measures in the tantalite mining operations is workers exposure to radon in underground mines. it is recommended that the regulatory body should liaise with other relevant stakeholders and seize the opportunity of the ongoing formalization and reform of the asm sector in nigeria to integrate radiation protection with the general ohs measures in the mining industry. references [1] unscear, “dose assessment methodologies”, new york: united nations scientific committee on the effect of atomic radiation (2000). report to the general assembly unscear) [2] iaea, “assessing the need for radiation protection measures in work involving minerals and raw materials”, safety report series no. 49 (2006),(vienna: iaea). [3] a.o. mustapha, p. mbuzukongira , & m. mangala, “occupational radiation exposure of artisanal mining of columbite-tantalite in the eastern democratic republic of congo”, journal radiological protection 27 (2007) 187. [4] iaea. “radiation protection and safety of radiation sources”. (international atomic energy agency) international basic safety standards, general safety requirements part 3, no. gsr (2004) (vienna). [5] iaea, “occupational radiation protection”, general safety guide no. gsg-7 (2018) (vienna). [6] iaea, “occupational radiation protection in the uranium mining and processing industry”, safety report series no. 100(2020) (vienna: iaea) [7] the world bank, nigeria, “mineral sector support for economic diversification project (mindiver)”, implementation status and results report (2019). [8] o. a. ndubisi, b. k. m. apaemi, s. f. barikpe & i. t. ari”, analysis of indoor radon level and its health risks parameters in three selected towns in port harcourt, rivers state, nigeria”, j. nig. soc. phys. sci. 3 (2021) 181. [9] sarad, “user manual, radon scout, version 10/2009, sarad gmbh”, wiesbadener strabe 10. (2009) d-01159 dresden. [10] d. o. pires, m. a. rio, e. c. s. amaral, h. m. fernandes & e. r. r. rochedo, “environmental radiological impact associated with nonuranium mining industries: a proposal for screening criteria”, j. environ. radioact. 59 (2002) 1. 8 j. nig. soc. phys. sci. 2 (2020) 149–159 journal of the nigerian society of physical sciences original research modelling and forecasting climate time series with state-space model a. f. adedotuna, t. latundeb,∗, o. a. odusanyac adepartment of computer science, caleb university, imota, ikorodu, lagos state, nigeria bdepartment of mathematics, federal university oye-ekiti, oye-ekiti, nigeria cdepartment of computer science and statistics, d.s adegbenro ict polytechnic, itori abeokuta, ogun state, nigeria abstract this study modelled and estimated climatic data using the state-space model. the study was specifically to identify the pattern of the trend movement i.e., increase or decrease in the occurrence of the climatic change; to use of univariate kalman filter for the computation of the likelihood function for climatic projections; to modelling the climatic dataset using the state-space model and to assess the forecasting power of the state-space models. the data used for the work includes temperature and rainfall for periods january 1991 to december 2017. the data are tested for normality. shapiro-wilk, anderson-darling and kolmogorov-smirnov test of normality for the climatic data all showed that the variables are not normally distributed. the work spans the use of breaking trend regression model to fit climatic data to estimate the slopes which show much increase in climatic data has been recorded from the initial time data collection until the present. investigations and diagnostic are carried out by checking for corrections in the residuals and also checking for periodicity in the residuals. the results of this investigation show significant autocorrelation in the residuals indicating the presence of underlying noise terms which is not accounted for. by treating the residual as an autoregressive moving average (arma) process whereby we can obtain its spectral density, the result from the parametric spectral estimate shows underlying periodic patterns for monthly data, thus, leads to a discussion on the need to treat climatic data as a structural time series model. we select appropriate models by considering the goodness of fit of the model by comparing the akaike information criterion (aic) values. parameters are estimated and accomplished with some measures of precision. keywords: rainfall, temperature, kalman filter, state-space models, residuals article history : received: 25 april 2020 received in revised form: 03 july 2020 accepted for publication: 06 july 2020 published: 01 august 2020 c©2020 journal of the nigerian society of physical sciences. all rights reserved. communicated by: b. j. falaye 1. introduction largely, state-space models (ssms) have been used in several areas of applied statistics. in specifically, there are some desirable properties of the linear state-space models and also, they have vast potential in time series modelling that incorporates latent processes. when a model is found to be in the linear ∗corresponding author tel. no: +2348032801624 email address: tolulope.latunde@fuoye.edu.ng (t. latunde ) state-space form, the most used algorithm to predict the latent process, the state, is the kalman filter algorithm. this algorithm is a technique for computing, at each time (t = 1, 2, ...), the optimal estimator of the state vector based on the existing information until t and its success lies on the fact that is an online estimation procedure. this predicting structure, whose performance was in recent times compared to other numerous forecasting methods across thousands of time series [1], adapts to underlying alterations 149 adedotun et al. / j. nig. soc. phys. sci. 2 (2020) 149–159 150 in series dynamics and automatically revises forecasts as new observations. in agreement with the above, the study adopts the state-space model to analyse and forecast for the temperature and rainfall data in nigeria. the purpose of this study is to model and estimate the climate using the state-space model and the specific objectives are to: identify the pattern of the trend movement i.e. increase or decrease in the occurrence of the climatic change; use of univariate kalman filter for the computation of the likelihood function for climatic projections; modelling the climatic dataset using the state-space model and assess the forecasting power of the state-space models. shamshad et al. [2] show a comparison of artificial neural networking multilayer perceptron (ann-mlp) with the automatic exponential smoothing algorithm (ets) and the autoregressive integrated moving average (arima) models for forecasting lahore, pakistan’s main weather parameter. models are built by taking into account average monthly maximum and minimum temperature, relative humidity, wind speed and precipitation amount. data covering thirty years (1987-2016) was used to build the models. ann-mlp is a mathematical method, and the computational methods are arima and ets. they divide the thirty years data (1987-2018) data into training (1987 till 2016) and test (2017 till 2018) set to ensure the efficiency and reliability of all these models along with the performance criteria of the estimates. the research explained in brief how the various methods of learning can be used to formulate annmlp. deciding the most appropriate model and network configuration is based on their performance forecast. mae (moving average error), rmse (root-mean-square error), me (mean error), mase (mean absolute scaled error) suggest better results for ann-mlp. afsar et al. [3] investigated variability in temperature and precipitation in the gilgit-baltistan region. they used regression and stochastic models to show temperature and rainfall predictions. we observed precipitation prolonged with temperature increasing. a decrease in the amount of precipitation is observed from 2007 to 2011 with an increase in the monthly average maximum temperature. they considered ar(1) ideally suited to temperature forecasting. in the study of faisal and ghaffar [4] on the thiessen polygon technique to test pakistan’s 56 stations for 50 years (19612010) weighted rainfall area (awr). month-to-month precipitation records of fifty-six stations, storm measurements for the fifty-year season (1961-2010) and a standard precipitation size relationship were used by the theissen system. yusof et al. [5] used an amount of rainfall to be categorised into seven categories (extremely wet to extremely dry) to analyze dry and wet events using peninsular malaysia data. they used precipitation index (spi) standardizes to model the best fit distribution to reflect the rainfall. in comparison with gamma and weibull distributions, the lognormal distribution is found to be better matched to the daily rainfall in the area. extreme pakistan temperature events and rainfall for the period 1965-2009 were examined in zahid and rasul work [6] to quantify pakistan frequency. they used f-test to determine the country minimum and maximum extreme temperature events. we pointed out that all over the country, certain extreme events are increasing. regarding extreme rainfall events, they used the k-s method at a confidence interval of 95 per cent and concluded that the southern half of pakistan faces more wet spells due to global warming and climate change. in the same vein, rainfall data in queensland, australia including climate indices, monthly rainfall and temperature was surveyed in abbot and marohasy [7]. they brought ann into the area to predict monthly rainfall. they suggested there is scope for improvement in this product design. analysis of ann to forecast lasting monthly temperature and rainfall from 76 stations in turkey at any point for the period 1975-2006 was also carried out in bilgili and sahin’s work [8] based on knowledge from neighbouring stations. they divided 76 measuring stations into training sets and test sets. the fitted model was satisfactory because the errors are within reasonable limits. 2. materials and methods 2.1. state-space model a state-space representation in control engineering is a mathematical model of a physical system as a collection of input, output and state variables similar to first-order differential equations or difference equations. state variables are variables whose values change over time in a way that depends on the values they have at any given time and often depends on the values of input variables placed externally. the values of the output variables depend on the values of the state variables. ”state-space” is the euclidean space, where the variables on the axes are the variables of the body. the state of the system within that space can be expressed as a vector. the state-space method is characterized by significant algebraization of general system theory, which makes it possible to use kronecker vector-matrix structures. the ability of these structures can be efficiently used to study systems with modulation or without modulation. the state-space representation (also called the ”time-domain method”) gives a convenient and compact way to model and analyze systems with multiple inputs and outputs. with p and q outputs, we would otherwise have to write down q × p laplace transforms to encode all the information about a system. 2.2. state-space representation in the time domain, a system can be described in general by a set of linear differential and algebraic equations (i.e., state150 adedotun et al. / j. nig. soc. phys. sci. 2 (2020) 149–159 151 space model): d x dt = ax + bu (1) y = c x + du (2) where xis the vector of the state variable u is the vector of inputs (manipulated variables) y is the vector of outputs (controlled/measured variables), and a, b, c, and d are constant matrices of appropriate dimensions. taking laplace transform of equations (1) and (2) using zero initial conditions, and re-arranging, the system transfer function matrix g(s), the relationship between the inputs uand the outputsyrelates the state model equations (1) and (2) as: g (s) = y(s) u(s) = c(si − a)−1 b + d (3) the representation of the (ssm) was developed based on graph theory principles. two major types are used: 1. the signal-flow-graph (sfg) type. 2. the matrix representation. the first type is easier to use and hence has been adopted by many researchers [9-10], and it has also been adopted in this research for the same reason. each sfg consists of directed branches interconnected at nodes. in the state-space model, the nodes represent the variables (signals), and the branches connecting the nodes indicate that these nodes are related. each branch is assigned a numerical value or a function, which quantifies the relationship between the two variables in terms of a gain factor or a transfer function. 2.3. identification and estimation of a state-space model process before the analysis can be processed, identification of the statespace model would be carried out in the following ways: the measurement equation has the form yt = ztαt + dt + ett = 1. . . t (4) to accommodate data properties as temporal correlation and the periodic behaviour, a periodic state-space model is proposed. this model is defined as ys,n = [s (n − 1) + s] xs,n + ds,n + es,n (5) xs,n = µs + s(xs−1,n −µs−1) + es,n (6) where s is the season of the year with s = 1, 2, .., s ; n is the year with n = 1, 2, ...,n; ys,n represents the time series observation in the sth season of the nth year; [s (n − 1) + s]th is the observation of the time series; [1, 2, 3, 4, . . ., s ] are the unknown parameters representing the fixed effects in the model; ds,n is a 1 × s matrix of known values, a design matrix; the error process (es,n) is a white noise disturbance, which is assumed to have var (es,n) = 2 e µs is the mean of the process (xs,n) for the sth season, s is the autoregressive parameter for season s and es,n is the white noise disturbance; the means, standard deviations and variances of the different series gotten for each sector would be examined where the series with a lesser variance being the most efficient. their skewness and kurtosis will also be determined. also, the existence of stationarity, unit root and long memory properties would be determined. 2.4. univariate kalman filter for the computation of the likelihood function for climatic projection parameter estimation that specifies the state-space model is very important in analyzing various components of the time series model. the idea is that θ = {µ0, ∑ 0,ϕ, q, r, y, γ} used to represent the vector of unknown parameters containing the elements of the initial mean µ0, the covariance ∑ 0, the transition matrix, θ, the state and observation covariance matrices, q and r are inputs y and γ are estimated using the maximum likelihood estimation. the maximum likelihood, for a time series model where the observations y1, y2, . . . , yn are not independent, is defined as a conditional probability density function to write the joint density function l(y : θ) = n∏ t=1 p(yt|yt−1) (7) where p(yt|yt−1) is the distribution of yt conditional on the information set at time t − 1 that is yt−1 = {y1, y2, . . . , yn}. the maximum likelihood is used under the assumption that the initial state is normal x0 ∼ n(µ0 : ∑ 0) and the error v1, v2..vn and w1, w2..wn are jointly normal and uncorrelated vector variables. wt ∼ n(0 : q)andvt ∼ n(0 : σ2). hence, the likelihood is derivedbyusing the innovations εt = yt − at x t−1 t − γµt (8) which are independent normal where e(εt) = 0 and the covariance matrix∑ 0 = at p t−1 t at + r (9) where the dependence of the innovation on the parameter θ has been emphasized using the kalman filter for given θ, this can be done as follow; 1. select initial and starting values for the parameter (θ0) 151 adedotun et al. / j. nig. soc. phys. sci. 2 (2020) 149–159 152 2. for θ0, compute the likelihood l(y : θ0) using the kalman filter 3. apply a numerical optimization algorithm to l(y : θ0) 4. repeat this process for n steps until the value of θ corresponding to the maximum likelihood is found. 2.5. forecasting with kalman filter an m-period-ahead forecast of the state vector can be calculated from the equation below: ξt+k = f kξt+f k−1vt+1+f k−2vt+2+. . .f 1vt+k−1 +vt+k for k = 1, 2, 3. . .(10) lead to equation ξ̂t+k|t = e(ξt+k|, yt, yt−1, . . .y1, xt, xt−1, . . .x1) = f kξ̂t|t (11) the error of this forecast can be found by subtracting ξt+k from ξ̂t+k|t ξt+k−ξ̂t+k|t = f k(ξt−ξ̂t|t)+f k−1vt+1+f k−2vt+2+. . .+f 1vt+k−1 +vt+k(12) from which it follows that the mean squared error of the forecast is pt+k|t = e[(ξt+k − ξ̂t+k|t)(ξt+k − ξ̂t+k|t)] (13) = fk pt|t(f k)′+fk−1 q(fk−1)′+fk−2 q(fk−2)′+. . .+f qf′+q(14) these results can also be used to describe m-period-ahead forecasts of the observed vector yt+m, provided that {x,} is deterministic. applying the law of iterated expectations to e(yt+k jξt,ξt−1, . . .yt, yt−1, . . .) e[(a′xt+k, +h ′ξt+k + wt+k) jξt,ξt−1, . . .yt, yt−1, . . .) (15) results in ŷt+k|t = e(yt+k jyt, yt−1. . ., y1) = a ′xt+k + h ′fkξ̂t|t (16) the error of this forecast is yt+k−ŷt+k|t = (a ′xt+k+h ′ξt+k+w t+k)−(a ′xt+k+h ′fmξ̂t|t)(17) = h′(ξt+m − ξ̂t|t) + w t+m (18) with mean squared error e[(yt+k − ŷt+k|t)(yt+k − ŷt+k|t) ′] = h′pt+k|t h + r (19) 3. presentation of results and discussion figures 1-2 represent the time plot of temperature and rainfall; figure 1. the time plot of the monthly temperature series shows the underlying trend and possible seasonal and cyclic patterns as well figure 2. the time plot of the monthly rainfall series shows the underlying trend and possible seasonal and cyclic patterns as well 3.1. fourier analysis table 1 shows the values of the various components of the spectral analysis for temperature. the numbers in parentheses, (d, d, s, m, t ), are defined as follows: d is the regular differencing order, d is the seasonal differencing order, s is the number of seasons (ignored if d is 0), m is 1 if the mean is subtracted, 0 otherwise, t is 1 if the trend is subtracted, 0 otherwise. (0, 0, 12, 1, 0) indicates that there is no regular differencing. the seasonal differencing is zero, while the number of seasons is zero. the value indicates that the mean is subtracted, while the trend is not subtracted. figures 3-6 show that there exists an underlying periodic component in the residuals obtained by fitting the smoothed and filtered data to the observation. we can see that the first peak looking at the monthly series corresponding to the fitted data is at a frequency ω = 0.389, corresponding to a period of 50 months. and the second peak is at a frequency of ω = 0.35, with a period of approximately 12 months. therefore, the need to extend the structural model to contain a seasonal and cyclic component is important. 152 adedotun et al. / j. nig. soc. phys. sci. 2 (2020) 149–159 153 table 1. fourier analysis of temp (0,0,12,1,0) frequency wavelength period cosine(a’s) sine(b’s) spectrum 0.2010619 31.25 17392.62 21.48521 -14.59315 9522.42 0.2764601 22.72727 2180.31 0.8657146 -9.155004 7795.009 0.3518584 17.85714 3812.099 11.42719 -4.155941 4270.688 0.4272566 14.70588 6819.656 -14.50404 -7.357564 15927.96 0.5026549 12.5 37152.14 -6.556379 37.38934 17404.13 0.5780531 10.86957 8240.605 -13.99402 -11.12565 15948.32 0.6534513 9.615385 2452.211 -7.719598 5.959619 3608.136 0.7288495 8.620689 131.5911 2.229034 -0.3676695 2495.332 0.8042477 7.8125 4902.192 -12.52921 -5.757686 6569.818 0.8796459 7.142857 14675.67 4.816339 23.36665 7431.676 0.9550442 6.578948 2717.165 8.031762 6.393456 15443.23 1.030442 6.097561 28936.85 22.1179 25.16181 12334.13 1.105841 5.681818 5348.37 3.409208 -13.99337 11674.39 1.181239 5.319149 737.9613 -3.06099 4.387738 5851.817 1.256637 5 11469.12 13.69089 16.04339 5366.608 1.332035 4.716981 3892.745 2.951285 -11.92772 5620.186 1.407434 4.464286 1498.693 0.0307705 7.624041 3149.379 1.482832 4.237288 4056.698 12.02642 -3.564355 2910.082 1.55823 4.032258 3174.855 10.01725 4.774079 4193.151 1.633628 3.846154 5347.899 -13.94717 3.591002 4181.128 1.709026 3.676471 4020.631 -9.593548 7.994023 4142.489 1.784425 3.521127 3058.936 -7.539809 7.860816 2993.722 1.859823 3.378378 1901.597 2.08475 -8.331114 1728.495 1.935221 3.246753 224.9497 2.751325 1.074666 1347.075 2.010619 3.125 1914.677 8.535864 1.183197 1418.189 2.086018 3.012048 2114.941 3.061299 -8.523886 4419.094 2.161416 2.906977 9227.663 -9.289092 -16.48055 4187.763 2.236814 2.808989 1220.684 -6.871478 0.3565736 6609.169 2.312212 2.717391 9379.16 -9.712505 16.41459 5221.976 2.38761 2.631579 5066.084 11.07367 8.594324 5238.45 2.463009 2.55102 1270.106 4.720753 5.19381 2286.001 2.538407 2.475248 521.812 0.8846894 4.410879 1232.752 2.613805 2.403846 1906.337 -8.432693 1.681413 1483.803 2.689203 2.336449 2023.261 -6.378614 6.147003 1774.629 2.764601 2.272727 1394.288 -0.8783695 7.3011 2174.107 2.84 2.212389 3104.772 -10.88803 -1.367346 4679.055 2.915398 2.155172 9538.105 -18.3482 5.768869 4346.808 2.990796 2.10084 397.5458 -3.743783 1.184457 3586.218 3.066195 2.04918 823.0026 -5.125026 2.377886 668.0311 3.141593 2 783.5449 -5.5127 -2.484305e-12 809.85 153 adedotun et al. / j. nig. soc. phys. sci. 2 (2020) 149–159 154 table 2. fourier analysis of rainfall (0,0,12,1,0) frequency wavelength period cosine(a’s) sine(b’s) spectrum 0.2010619 31.25 3505336 -155.2216 334.4563 1.340589e+07 0.2764601 22.72727 2.376145e+07 -473.745 834.9586 1.081362e+07 0.3518584 17.85714 5174079 204.4676 398.5852 1.242262e+07 0.4272566 14.70588 8332332 372.0547 429.8204 8652136 0.5026549 12.5 1.245e+07 -522.3141 458.3256 1.045092e+07 0.5780531 10.86957 1.057043e+07 496.6129 -404.1668 2.557089e+07 0.6534513 9.615385 5.369223e+07 347.9105 -1400.506 2.371912e+07 0.7288495 8.620689 6894689 310.2988 -413.6736 2.459778e+07 0.8042477 7.8125 1.320643e+07 271.7906 -662.0743 1.099229e+07 0.8796459 7.142857 1.287576e+07 -420.4702 -567.9723 9935063 0.9550442 6.578948 3723000 -203.048 -321.1984 1.345724e+07 1.030442 6.097561 2.377295e+07 -334.7887 -899.9736 1.160267e+07 1.105841 5.681818 7312064 418.9944 328.6986 1.113207e+07 1.181239 5.319149 2311208 -185.0102 235.3968 5046145 1.256637 5 5515162 108.9928 -449.4738 3876995 1.332035 4.716981 3804614 -342.3183 174.3 4267821 1.407434 4.464286 3483686 -122.2476 -346.6563 5960738 1.482832 4.237288 1.059391e+07 -237.493 -595.3847 2.304692e+07 1.55823 4.032258 5.506316e+07 -1014.657 -1051.713 2.313139e+07 1.633628 3.846154 3737106 182.7847 -333.9667 1.996097e+07 1.709026 3.676471 1082651 77.12722 -189.8476 1866007 1.784425 3.521127 778264.9 -13.22315 173.2346 1592170 1.859823 3.378378 2915595 331.0794 -58.89026 1575621 1.935221 3.246753 1033002 79.65166 183.632 2011053 2.010619 3.125 2084562 141.3623 246.7116 3220544 2.086018 3.012048 6544069 -469.2179 183.4309 5135841 2.161416 2.906977 6778893 -507.325 -74.44065 5237774 2.236814 2.808989 2390359 -264.1649 151.4172 4376703 2.312212 2.717391 3960858 -391.2391 23.54103 2330373 2.38761 2.631579 639903.6 54.25442 -147.9026 2294082 2.463009 2.55102 2281486 -274.1098 115.5485 979001.7 2.538407 2.475248 15615.94 5.175655 -24.05989 2194616 2.613805 2.403846 4286746 -333.2503 -234.9598 1575710 2.689203 2.336449 424768.6 -121.5072 -41.36044 1572592 2.764601 2.272727 6261.016 -3.307111 -15.22817 806171.2 2.84 2.212389 1987484 64.74786 269.9861 1018778 2.915398 2.155172 1062588 -102.2991 175.3496 1285118 2.990796 2.10084 805281.5 176.6789 4.182919 2102421 3.066195 2.04918 4439392 126.9668 -395.0464 2849186 3.141593 2 3302885 -357.9144 2.442917e-10 4060556 154 adedotun et al. / j. nig. soc. phys. sci. 2 (2020) 149–159 155 figure 3. periodogram of temperature (frequency) figure 4. periodogram of temperature (frequency) figure 5. spectral analysis of temperature (frequency) table 2 shows the values of the various components of the spectral analysis for rainfall. the numbers in parentheses, (d, d, s, m, t ), are defined as follows: d is the regular differencing order, d is the seasonal differencing order, s is the number of seasons (ignored if d is 0), m is 1 if the mean is subtracted, 0 otherwise, t is 1 if the trend is subtracted, 0 otherwise. (0, 0, 12, 1, 0) indicates that there is no regular differfigure 6. spectral analysis of temperature (wavelength) figure 7. periodogram of rainfall (frequency) figure 8. periodogram of rainfall (wavelength) encing. the seasonal differencing is zero, while the number of seasons is zero. the value indicates that the mean is subtracted, while the trend is not subtracted. figures 7-10 show that there exists an underlying periodic component in the residuals obtained by fitting the smoothed and filtered data to the observation. we can see that the first peak looking at the monthly series corresponding to the fitted data 155 adedotun et al. / j. nig. soc. phys. sci. 2 (2020) 149–159 156 figure 9. spectral analysis of rainfall (frequency) figure 10. spectral analysis of rainfall (wavelength) is at a frequency ω = 0.25, corresponding to a period of 50 months. and the second peak is at a frequency of ω = 0.23, with a period of approximately 12 months. therefore, the need to extend the structural model to contain a seasonal and cyclic component is important. 3.2. state-space model from table 3, the ar coefficients with respect to the mles are given as 1.271 and -0.274, respectively also, the roots of the characteristic equation φ = 1 − 1.271z + 0.274z2 = 0 are 0.542 and 1.998, respectively. since these values are greater than 1, the ar component is covariance stationary. the variance of the permanent component is 0.000000, and the variance of the transitory component is 1.107762854. hence, the variance of the transitory component is higher than the variance of the permanent component and the ratio of the variance of the permanent component to the variance of the stationary component is 0.000. this shows that the stationary component is almost twice as important as the permanent component for explaining the variation of temperature. from table 4, the mles for the ar coefficients are 1.311 and -0.456, respectively. the roots of the characteristic equation φ = 1 − 1.311z + 0.456z2 = 0 are 1.779 and 1.137, respectively. since these values are greater than 1, the ar component is covariance stationary. the variance of the permanent component is 0.0000000, and the variance of the transitory component is 2.648752906. hence, the variance of the transitory component is higher than the variance of the permanent component and the ratio of the variance of the permanent component to the variance of the stationary component is 0.000. this shows that the stationary component is almost twice as important as the permanent component for explaining the variation of temperature. 3.3. forecasting the in-sample performance of the state-space model for forecasting the rainfall and temperature series appears to be more favourable to the model identified by shittu and yemitan [11]. this is evident in tables 5 and 6. given the relative accuracy of the model by shittu ad yemitan [11], the improvement achieved by the kalman filter method is mainly as a result of the built-in the specification for updating the estimation based on latest available information. hence, the rainfall and temperature exhibit elements of time-varying or regime-switching characteristics over the study period. as a result, it is an indication and a signal that the climatic data may be best estimated using non-linear time series methodologies. 4. conclusion this study modelled and estimated climatic data using the state-space model. the study was specifically to identify the pattern of the trend movement, model the dataset using the state-space model and to evaluate the forecasting power of the state-space models. the data used for the study include temperature and rainfall for periods january 1991 to december 2017. the data were tested for normality. the study showed that the average temperature is 27.3◦cand standard deviation is 1.87◦c. the maximum temperature is 31.5◦c and minimum temperature is 23.3◦c.the average rainfall is 94.5mm with a standard deviation of 86.3mm. the maximum rainfall is 314.3mm and the minimum rainfall is 0.2mm. shapiro-wilk, anderson-darling and kolmogorov-smirnov test of normality for the climatic data all showed that the variables are not normally distributed. the plot of the monthly temperature series shows the underlying trend and possible seasonal and cyclic patterns as well. the plot of the monthly rainfall series shows the underlying trend and possible seasonal and cyclic patterns as well. the mles for the ar coefficients are 1.271 and -0.274, respectively. the roots of the characteristic equation φ = 1 − 1.271z + 0.274z2 = 0 are 0.542 and 1.998, respectively. the 156 adedotun et al. / j. nig. soc. phys. sci. 2 (2020) 149–159 157 table 3. state-space model (temperature) sspace: ss03 method: maximum likelihood (marquardt) date: 09/07/19 time: 15:37 sample: 1991m01 2017m12 included observations: 324 estimation settings: tol= 0.00010, derivs=accurate numeric initial values: c(1)=7.79271, c(2)=1.31153, c(3)=-0.45685 failure to improve likelihood after 12 iterations coefficient std. error z-statistic prob. c(1) 0.818740 0.079268 10.32875 0.0000 c(2) 1.271171 0.055552 22.88260 0.0000 c(3) -0.274151 0.056246 -4.874137 0.0000 final state root mse z-statistic prob. sv1 22.24512 1.505869 14.77229 0.0000 sv2 23.23000 0.000000 na 0.0000 log likelihood -600.6427 akaike info criterion 3.726190 parameters 3 schwarz criterion schwarz criterion 3.761197 diffuse priors 0 hannan-quinn criter. hannan-quinn criter. 3.740162 table 4. state-space model (rainfall) sspace: ss01 method: maximum likelihood (marquardt) date: 09/07/19 time: 01:26 sample: 1991m01 2017m12 included observations: 324 estimation settings: tol= 0.00010, derivs=accurate numeric initial values: c(1)=0.85106, c(2)=1.29335, c(3)=-0.29596 convergence achieved after 42 iterations coefficient std. error z-statistic prob. c(1) 7.792711 0.067286 115.8147 0.0000 c(2) 1.311527 0.049124 26.69830 0.0000 c(3) -0.456845 0.052524 -8.697765 0.0000 final state root mse z-statistic prob. sv1 -1.907737 49.22274 -0.038757 0.9691 sv2 2.567010 0.000000 na 0.0000 log likelihood -1723.221 akaike info criterion 10.65568 parameters 3 schwarz criterion schwarz criterion 10.69069 diffuse priors 0 hannan-quinn criter. hannan-quinn criter. 10.66966 157 adedotun et al. / j. nig. soc. phys. sci. 2 (2020) 149–159 158 table 5. foreast for temperature period forecast (ssm) 95% limits lower upper 2018 24.6816 21.8312 27.5320 2018 25.6077 22.2266 28.9888 2018 26.1985 22.6239 29.7730 2018 26.5754 22.9250 30.2258 2018 26.8158 23.1351 30.4966 2018 26.9692 23.2762 30.6623 2018 27.0671 23.3690 30.7652 2018 27.1295 23.4294 30.8296 2018 27.1694 23.4684 30.8703 2018 27.1948 23.4935 30.8960 2018 27.2110 23.5096 30.9124 2018 27.2213 23.5199 30.9228 2019 27.2279 23.5264 30.9294 2019 27.2321 23.5306 30.9336 2019 27.2348 23.5333 30.9363 2019 27.2365 23.5350 30.9380 2019 27.2376 23.5361 30.9391 2019 27.2383 23.5368 30.9398 2019 27.2388 23.5373 30.9403 2019 27.2391 23.5375 30.9406 2019 27.2392 23.5377 30.9407 2019 27.2393 23.5378 30.9409 2019 27.2394 23.5379 30.9409 2019 27.2395 23.5380 30.9410 table 6. foreast for rainfall period forecast (ssm) 95% limits lower upper 2018 21.9472 -83.629 127.524 2018 37.1513 -97.037 171.340 2018 49.0790 -100.014 198.172 2018 58.4364 -99.130 216.003 2018 65.7775 -96.785 228.340 2018 71.5366 -94.025 237.099 2018 76.0547 -91.327 243.436 2018 79.5992 -88.892 248.091 2018 82.3799 -86.791 251.551 2018 84.5614 -85.027 254.149 2018 86.2728 -83.571 256.117 2018 87.6154 -82.386 257.617 2019 88.6687 -81.430 258.767 2019 89.4950 -80.663 259.653 2019 90.1433 -80.051 260.338 2019 90.6518 -79.565 260.869 2019 91.0508 -79.180 261.282 2019 91.3638 -78.876 261.603 2019 91.6094 -78.635 261.854 2019 91.8020 -78.446 262.050 2019 91.9531 -78.297 262.203 2019 92.0717 -78.180 262.323 2019 92.1647 -78.087 262.417 2019 92.2377 -78.015 262.490 158 adedotun et al. / j. nig. soc. phys. sci. 2 (2020) 149–159 159 mles for the ar coefficients are 1.311 and -0.456, respectively. the roots of the characteristic equation φ = 1−1.311z + 0.456z2 = 0 are 1.779 and 1.137, respectively. investigations and diagnostic were carried out by checking for correlations in the residuals and also checking for periodicity in the residuals. the results of this investigation show significant autocorrelation in the residuals indicating the presence of underlying noise terms which is not accounted for. also, the result from the parametric spectral estimate shows underlying periodic patterns for monthly data, thus, leads to a discussion on the need to treat climatic data as a structural time series model. we selected the appropriate models by considering the goodness of fit of the models and by comparing the aic values. parameters were estimated and accomplished with some measures of precision. an important aspect of fitting structural models is the underlying changes in the unobserved trend component. for the case of state-space models with correlated errors, further work can be done here by taking into consideration correlation between the error terms in state equation when considering structural models of the form trend, seasonal, cycle and an autoregressive process. this type of results is important since they show how well the kalman filter for correlated errors carries out the filtering of the unobserved component as compared to the case when errors are not correlated. references [1] s. makridakis & m.hibon, “the m3-competition: results, conclusions and implications”, international journal of forecasting, 16(2000), 451. [2] b. shamshad, m. z khan & z. omar, “modeling and forecasting weather parameters using ann-mlp, arima and ets model: a case study for lahore, pakistan”, journal of applied statistics 5 (2019) 388. [3] s. afsar, n. abbas,& b.jan, “comparative study of temperature and rainfall fluctuation in hunza-nagar district”, journal of basic and applied sciences 9 (2013) 151. [4] n. faisal & a. ghaffar, “development of pakistan’s new area weighted rainfall using thiessen polygon method”, pakistan journal of metrology 17 (2012) 107. [5] f. yusof, i. l. kane & z. yusop, “structural break or long memory: an empirical survey on daily rainfall data sets across malaysia”, hydrology and earth system sciences 17 (2013) 1311. [6] m. zahid & g. rasul, “frequency of extreme temperature and precipitation events in pakistan 1965-2009”, science international 23 (2011) 313. [7] j. abbot, & j. marohasy, “application of artificial neural networks to rainfall in queensland, australia”, advances in atmospheric sciences 29 (2012) 717. [8] m. bilgili & b. sahin, “prediction of long-term monthly temperature and rainfall in turkey, energy sources 1 (2010) 60. [9] r. d. johnston, “steady-state closed-loop structural interaction analysis”, international journal of control 6 (1990) 1351. [10] j. wang, & k. m. cameron, “a method of constructing a modified signalflow-graph”, international journal of control 12 (1993) 305. [11] o. i. shittu & r. a. yemitan, “inflation targeting with dynamic time series modelling the nigerian case”, international journal of scientific and engineering research 9 (2014) 206. 159 j. nig. soc. phys. sci. 5 (2023) 931 journal of the nigerian society of physical sciences influence of bath ph values on the structural and optical properties of electrodeposited mgo thin films for optoelectronic applications s. c. onuegbua,∗, s. s. oluyamob, o. i. olusolab a department of physics, alex ekwueme federal university, ndufu-alike, ebonyi state, nigeria bdepartment of physics, federal university of technology, akure, nigeria abstract this paper investigates the influence of bath ph values on the structural and optical properties of magnesium oxide (mgo) thin films synthesized using electrodeposition technique. the films were deposited on conductive fluorine tin oxide (fto) substrates using magnesium nitrate hexahydrate as a precursor material. the structural, morphological and optical properties of the electrodeposited films were examined by scanning electron microscopy (sem), x-ray diffraction and uv-vis spectrophotometer. the morphology and optical properties of the films were found to vary with bath ph values. the band gap decreased as the bath ph values increased. the deposited mgo films exhibited average transmittance of 80%, 50%, and 25% with thicknesses 400 nm, 480 nm, and 540 nm for bath ph values of 2.0, 5.0, and 9.0, respectively. the results obtained indicate that bath ph values play significant role in the formation of mgo films and can be used to tune the material into useful optoelectronic applications. doi:10.46481/jnsps.2023.931 keywords: mgo thin films, bath ph, electrodeposition. article history : received: 11 july 2022 received in revised form: 26 august 2022 accepted for publication: 27 august 2022 published: 18 march 2022 © 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: dr. k. sakthipandi 1. introduction magnesium oxide (mgo) has received considerable attention due to their unique properties such as chemical inertness, electrical insulation, optical transparency, high temperature stability, high thermal conductivity, high dielectric constant and secondary electron emission [1, 2, 3]. these excellent characteristics make mgo nanomaterial a promising material for several industrial applications. magnesium oxide ex∗corresponding author tel. no: +2347062916216 email address: onuegbus.c@yahoo.com (s. c. onuegbu ) hibits the cubic crystalline structure and has interesting properties because of its thermal and chemical stability properties [4]. it is non-toxic and has been employed as a layer material for the fabrication of thin film transistors (tfts) for nextgeneration of electronic devices owing to its large dielectric constant (9.8)[5, 6, 7, 8]. magnesium oxide materials are the best candidates for dielectric layer applications because of its high dielectric constant, large band gap energy, and breakdown voltage (12 mv/cm) when compared to the commonly used silicon dioxide (sio2) [9]. magnesium oxide possess a direct large band gap energy in the range of 3.5-5.7ev, large exciton binding energy 1 s. c. onuegbu et al. / j. nig. soc. phys. sci. 5 (2023) 931 2 (80mev) [10], and high transmission value above 80% [11]. mgo thin films have been utilized as buffer layer for superconducting and ferroelectric thin film fabrications; however, its wide band gap, low optical loss and relatively low refractive index have reported improved the optical modes in many ferroelectric materials [12]. moreover, due to their biocompatibility, mgo nano-materials find several bio-applications in the areas of antibacterial and anticancer agents, biosensors, reactive oxygen species, and bone regeneration [13, 14, 15]. magnesium oxide films continue to find numerous applications in various technologies such as reflecting and anti-reflecting coatings [16, 17], light emitting diodes (leds) [18], solar cells [19], laser diodes [20] and even as a protective layer to shield electrodes from ionizing radiations especially in secondary electron emission by minimizing the discharge voltage layer in ac plasma display panels. magnesium oxide (mgo) exists in many nanostructures such as nanotube, nanocrystals, nanowires, nanoparticles, nanofloweres, and even broken-floor like structures [21, 22]. these nanostructures are heavily influenced by the type of deposition technique. the mgo thin films have been synthesized by various techniques such as pulsed laser deposition [23], chemical vapour deposition, cvd [24], rf magnetron sputtering [25], sol-gel synthesis [26], spray pyrolysis [27], and electrodepostion [28][31]. among these deposition techniques, electrodeposition has become the most fascinating deposition techniques due to the controllable morphological and structural growth of nanomaterial with optimization of deposition parameters; the technique also allows the fabrication of thin films with high surface area to volume ratio, it is simple, does not required any sophisticated equipment and films can be deposited at low temperatures [32]-[39]. generally, bath conditions have been noted to play vital role in the optimum deposition of thin films. in particular, less attention is usually placed on the bath ph values which to a large extent enhance transport of charges within the medium of deposition. in addition, the knowledge of the variation of properties of mgo with preparation conditions would assist in the fabrication process for optimum deposition, characterization and application. in this paper we electrodeposited mgo thin films at different bath ph values and investigated the effect of the bath ph values (2, 5 and 9) on the morphological, structural and optical properties of mgo thin films. 2. materials and method the materials used in this work are magnesium nitrate hexahydrate mg(no3)2.6h2o of purity 99% which was purchased from sigma-aldrich, sodium hydroxide pellets (naoh) and hydrochloric acid (hcl) purity over 98% were purchased from thomas baker, were used to adjust the bath ph values; deionized water was used as the solvent throughout the experiments. all chemicals and salts were analytical reagent grade and were used as received without further purifications. fto coated glass substrates of sheet resistance 13-15 ohm/square and dimension 20x40x2.2 mm were purchased from brotain hong kong co, ltd, china. before the electrodeposition, the fto coated glass substrates were degreased and cleaned ultrasonically in acetone solution to remove any contaminants from its surface. the mgo thin films were potentiostatically electrodeposited onto well cleaned fto coated glass substrates from a bath solution containing 40mm magnesium nitrate hexahydrate mg(no3)2.6h2o. the bath solution of 40mm was prepared by dissolving 2.6 g of mg(no3)2.6h2o in 250 ml of deionized water in 300 ml beaker, the solution was stirred magnetically for 2hr before deposition. the bath ph values was adjusted to 2.0, 5.0, and 9.0 using naoh or/and hcl. a metrohm autolab pgstat302n of three electrodes system configuration comprising fto, platinum plate wires and ag/agcl as working (we), counter (ce) and reference (re) respectively was employed to perform the electrodeposition. the mgo thin films were deposited at cathodic potential of 1.2 v, temperature 80oc in 120 minutes, without stirring the solution during deposition. immediately after electrodeposition, the mgo films were rinsed in deionized water to remove loosely bound particles. the electrodeposited mgo films were annealed at 500o for 1hr and allowed to cool down gradually in the annealing chamber before characterization. the annealed mgo films were characterized for morphological, structural and optical properties using tescan vega3 microscope at an accelerated voltage of 20kv, x-ray diffractometer d2 phaser-e and agilent cary 60 uv-vis spectrophotometer (tshwane university of technology, pretoria, south africa) respectively. the film’s crystallite size of the films was calculated using the equation: d = kλ β cos θ , (1) where d is the crystallite size (in nm), k is a scherrers constant 9 (k = 0.9), λ is the wavelength of x-ray radiation λ = 1.5418 å, β is the full width at half maximum (fwhm), and θ is the bragging angle. the thickness of the electrodeposited mgo films was determined using gravimetric method given as: t = ∆m ρa , (2) where ∆m is the mass of deposited mgo thin film, the mass of the film was calculated as the difference between the bare fto and electrodeposited mgo thin films, ρ is the density of bulk mgo, and a is the surface area of deposited film. the optical conductivity was calculated using the equation: σ = αnc 4π , (3) where α is the absorption coefficient, n is refractive index, c is the speed light. the band gap energy (eg) is estimated from the tauc plots of (αhν)1/n versus photon energy (hν) by extrapolating the linear portion where (αhν)n approaches zero on the horizontal axis in the equation: αhν = a ( hν− eg )1/n (4) 2 s. c. onuegbu et al. / j. nig. soc. phys. sci. 5 (2023) 931 3 where a is a constant, the value of n is dependent on the type of electronic transition; it is 1/2 for direct allowed transition, n = 2 for indirect allowed transition, n = 3 for direct forbidden transition, and n = 3/2 for indirect forbidden transition. here the value of n is taken as 1/2. the refractive index (n) of the deposited zno-mgo was calculated using the equation: n = (1 + √ r) (1 − √ r) , (5) where r is the reflectance of the film. the optical conservation of energy is given as t + r + a = 1, (6) where t = transmittance, r = reflectance, and a = absorbance. the voltammetry study was conducted on the mg(oh)2.6h2o at different bath ph values (2, 5 and 9) on a bare fto to determine the suitable deposition potentials for the fabrication of mgo thin films . it was observed from the voltammogram (figure 1) that the possible region of potentials lies within -1.0 to -1.5 v. therefore, the mgo thin films in this study were deposited at -1.2 v being the potential of the optimum deposition for all the bath ph values considered. figure 1: cyclic voltammograms recorded on fto coated glass substrate in a 40mm magnesium nitrate solution at various bath ph (a) 2.0, (b) 5.0 and (c) 9.0 at scan rate was of 20mv/s. the plots indicate a reverse (cathodic) scan. 3. results and discussion 3.1. structural characterization of electrodeposited mgo thin films figure 2 shows the x-ray diffraction spectra of mgo thin films grown at different bath ph values 2.0, 5.0 and 9.0. the result revealed three basic peaks of mgo films which are localized at 2θ = 42.87o, 62.1o, and 78.34o corresponding to (200), (220) and (222) respectively [40]. with increase in the bath ph values, the peaks became more intense, especially at bath ph value 9.0, and were found to shift towards the lower diffraction angles. the broadness of the peaks decreased which depicts improvement in the crystal quality of the films [1]. the films showed preferred orientation along (220) plane with high intensity. furthermore, secondary phases relating to mg (oh)2 and fto glass substrate were observed in the xrd spectra. interestingly, aside these known phases no other peaks relating to impurities were found [8]. the presence of mg (oh)2 was localised at angle (2θ) = 37.89◦ and 51.49◦ which correspond to 101 and 102 planes. the presence of fto substrate in the xrd spectra could be due to the thinness of the films deposited [18]. the calculated crystallite sizes are 22.18 nm, 21.11 nm, and 23.31 nm corresponding to thickness of 400 nm, 480 nm and 540 nm for mgo films deposited at bath ph values 2.0, 5.0 and 9.0, respectively. figure 2: xrd diffraction patterns of the electrodeposited mgo thin films grown at bath ph (a) 2.0, (b) 5.0, and (c) 9.0 3.2. morphology and energy-dispersive x-ray spectroscopy mgo thin films table 1: edxs analysis of electrodeposited mgo thin films at different bath ph values sample weight percentage of the element (at. %) o mg mg/o ph: 2.0 55.9 28.2 0.50 ph: 5.0 56.2 24.8 0.44 ph: 9.0 55.9 29.5 0.52 figure 3 (a-c) presents the sem images of electrodeposited mgo thin films at different bath a ph values 2.0, 5.0 and 9.0. the films deposited at bath ph value of 2.0 revealed net-like structures with plenty of white dots (figure 3a). as the bath ph value increased to 5.0, the morphology of film significantly 3 s. c. onuegbu et al. / j. nig. soc. phys. sci. 5 (2023) 931 4 figure 3: sem with the energy-dispersive x-ray spectroscopy spectra of electrodeposited mgo thin films; films grown at various bath ph (a, d) 2.0, (b, e) 5.0 and (c, f) 9.0. table 2: average results of absorbance, reflectance and transmittance of electrodeposited mgo at different bath ph values bath ph values absorbance (a) (arb. unit) reflectance (r) (arb. unit) transmittance (t) (arb.unit) a+r+t 2.0 0.038 0.111 0.851 1 5.0 0.375 0.087 0.538 1 9.0 0.614 0.056 0.330 1 changed to sheet-like structures (figure 3b). these sheet-like structures were agglomerated in high density and developed into orderly thick flower-like shaped structures. furthermore, when the bath ph was raised to higher value of 9.0, the films drastically changed entirely into broken floor-like structures with plenty of heavy dark dots (figure 3c). this result therefore demonstrates that the bath ph values can be used to tune the morphological properties of the mgo thin films. the pattern of this morphology had previously been reported [1, 3, 18]. the peaks of mg and o elements in mgo films are displayed in energy dispersive spectra shown in figure 3(d-f). for accuracy, the edxs spectrum of each sample was recorded at two different locations on the mgo film surface and the average value of element concentration recorded as shown in table 1. it was observed that the edxs spectrum of mgo films consists of only mg, o and peaks relating to the fto substrate. this indicates that the electrodeposited mgo film is of high purity. additionally, the quantitative edxs data indicated that the (mg/o at. %) ratio for all the samples has values less than one as seen in table 1. the result therefore reveal inhomogeneous distribution of mg and o atoms on the film surface with a deficiency of mg atoms (that is high percentages of magnesium vacancies). figure 4: transmittance (a), absorbance (b), and reflectance (c) versus wavelength of electrodeposited mgo thin films. 4 s. c. onuegbu et al. / j. nig. soc. phys. sci. 5 (2023) 931 5 figure 5: plots of optical conductivity (a), refractive index (b) against bath ph values and (c) energy band-gap estimation of mgo thin films. 3.3. optical characterization of electrodeposited mgo thin films table 2 shows the absorbance, reflectance and transmittance of the electrodeposited mgo thin films. the summation of these three basic optical properties agrees with the theoretical conservation of energy (i.e., unity) for all the bath ph values considered. the optical properties were examined using uv-vis spectrophotometry; the result showed that the absorbance increased with increase in bath ph values while the reflectance and transmittance decreased for all the bath ph values. research has shown that high absorbance films are useful materials for absorber layers in solar cell device material applications [49]. consequently, the thin films in the research can find useful applications as absorber layer in solar cell fabrication when deposited at high bath ph values. the increase in optical absorption permits more charge carriers to be transited from the valence band into the conduction band, thereby enhancing the band gap energy of the materials (this was observed in the values of the band gap). the plots of transmittance, absorbance and reflectance against wavelength for mgo films at different bath ph values are shown in figure 4. the transmittance spectra (figure 4a) indicate that the film’s transmittance values vary significantly with increase in the wavelength of radiation as the bath ph value increased. the mgo films deposited at bath ph value of 2.0 has transmittance value from 24% to 75% in the ultra violet (uv) region of the electromagnetic spectrum, increased from 75% to 91% in visible (vis) region and maintains the same value in the near infra-red (nir) region. as the bath ph value increased to 5.0, the films exhibited transmittance value from 3% to 19% in the uv region, 19% to 63% in vis region, and 63% to 68% in nir region. furthermore, the mgo films deposited at 9.0 bath ph showed least optical transmittance in all the spectra regions i.e., the films had a transmittance value of 4% in uv region, 4% to 22% in vis region, and 22% to 28% in nir region. these results in table 2 showed that the transmittance values decreased with increase in bath ph values; this could be due to higher defect concentration occasioned by increased bath ph value [41]. however, it can be observed that the films maintained transmittance above 70 % in the vis and nir regions [18] suggesting that the films can find application as an absorbent of visible and nir radiations making it fit as a protective coating to replace the extensive ito [42]. the uv-vis absorbance spectra of mgo film deposited at different bath ph values 2.0, 5.0 and 9.0 and annealed at 500oc for 1hr is presented in figure 4(b). the spectra were recorded within a wavelength range of 300-900 nm. the absorption spectra were classified into three regions depending on the amount of absorptions. in the highest absorption region 5 s. c. onuegbu et al. / j. nig. soc. phys. sci. 5 (2023) 931 6 where λ ≤ 400 nm the electron of the material absorbs higher energy from the incident photon to migrate into the conduction band, creating a hole (positive charge carrier) behind in the valence band thereby forming an exciton; a peculiarity of mgo material. at moderate absorption region, where 400 nm ≤ λ ≥ 600 nm, the electron tends to cross over to shallow states below the conduction band. finally the least absorption region occurs beyond 600 nm wavelength [18]. from the figure 4(b), it obvious that the film deposited at bath ph value of 2.0 exhibited almost zero absorbance in the wavelength range 500-900 nm [43, 44, 45]. the uv-vis reflectance spectra of mgo films deposited at different bath ph values is presented in figure 4(c). the film showed high reflectance in uv region, while the films deposited at 5.0 and 9.0 bath ph values showed almost zero reflectance in the uv region. as the bath ph value increased to 9.0, the reflectance attained maximum value above 30% in the nir region [46]. the plots of optical conductivity, refractive index against the bath ph values and band gap energy are shown in figure 5 (a), (b) and (c) respectively. figure 5(a) presents the optical conductivity of electrodeposited mgo films plotted against the bath ph values. it can be seen from this figure that the optical conductivity increased in great magnitude with increase in bath ph values. the increase in optical conductivity could be as a result of the increase in absorption coefficient of the films. the values of the optical conductivity calculated are 2.952 x1011, 14.120 x1011 and 21.527 x1011 s-1 corresponding to bath ph values 2.0, 5.0 and 9.0 respectively. figure 5(b) shows that the refractive index increased as the bath ph values increased. the calculated values of refractive index are 2.08, 2.32 and 3.69 for mgo thin films deposited at 2.0, 5.0 and 9.0 bath ph values respectively. increase in refractive index with increase in deposition condition is in agreement with period research [47]. figure 5(c) presents the graph of (αhν)2 versus photon energy (hν) for the mgo thin films. the band gap energy was estimated by extrapolating the linear region of the absorption curve to the point where (αhν)2 tends to zero [48, 49, 50, 51]. from the graph, the band gap values of the mgo thin films were calculated to be 3.88 ev, 3.67 ev, 3.42 ev for the films deposited at bath ph values 2.0, 5.0, and 9.0 respectively. it was observed that with increase in bath ph value, the band gap decreased gradually. the variation in the values of band gap of mgo nanomaterial could be attributed to recrystallization of atoms into the crystal lattice sites of the material [48] as the bath ph values increased. however, the remarkable band gap of 3.42 ev achieved at bath ph value 9.0 indicates that the films can serve as a suitable alternative to sio2 for device fabrication and utilization especially in capacitor applications [30]. 4. conclusion mgo thin films have been successfully fabricated on fto glass substrate by electrodeposition technique at different bath ph values. the results of the sem, xrd, and optical properties showed good agreement with previous reports. the bath ph values contributed significantly to the growth, structural, morphology and optical properties of the electrodeposited mgo thin films. the thickness of the film increased from 400 to 540 nm as the bath ph values increased from 2 to 9. the morphology of the films transited from flower-like structure to a broken floor-like shaped structure with increase in the bath ph. the film deposited at bath ph value 9.0 was found to exhibit better morphological, structural, and optical properties. the band gap energy was found to decrease from 3.67 to 3.42 ev while the optical transmittance values decreased from 85% to 33%, and the optical conductivity increased significantly as the bath ph value increased from 2.0 to 9.0 respectively. the improvement in the optical properties with respect to bath ph values indicates that the electrodeposited mgo films can be employed as a suitable material for solar cells and protective layer applications. acknowledgements the authors sincerely appreciate the financial support of tertiary education trust fund (tetfund) of nigeria through alex ekwueme federal university, ndufu-alike, ebonyi state, nigeria, and tshwane university of technology, pretoria, south africa. references [1] h. zulkefle, l. n. ismail, r. abu bakar, and m. r. mahmood, ”molar concentration effect on mgo thin films properties”, ieee symp. ind. electron. appl. isiea 2011 2011 (2011) 468. doi:10.1109/isiea.2011.6108754. 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[51] daniel, t., v. balasubramanian, j. henry, g. sivakumar, and k. mohanraj. ”electrochemical performance of green stabilizer-and biomoleculeassisted pbwo 4 nanoparticles.” journal of electronic materials 49 (2020) 4680. 8 j. nig. soc. phys. sci. 2 (2020) 115–119 journal of the nigerian society of physical sciences original research a pursuit differential game problem on a closed convex subset of a hilbert space abbas ja’afaru badakayaa,∗, bilyaminu muhammadb adepartment of mathematical sciences, bayero university, kano bdepartment of mathematics,federal college of eduction(technical), gusau, zamfara state abstract in this paper, we study a pursuit differential game problem with finite number of pursuers and one evader on a nonempty closed convex subset of the hilbert space l2. players move according to certain first order ordinary differential equations and control functions of the pursuers and evader are subject to integral constraints. pursuers win the game if the geometric positions of a pursuer and the evader coincide. we formulate and prove theorems that are concerned with conditions that ensure win for the pursuers. consequently, winning strategies of the pursuers are constructed. furthermore, illustrative example is given to demonstrate the result. keywords: pursuit, integral constraint, closed convex set. article history : received: 01 december 2019 received in revised form: 07 april 2020 accepted for publication: 11 april 2020 published: 01 august 2020 c©2020 journal of the nigerian society of physical sciences. all rights reserved. communicated by: t. latunde 1. introduction differential game has been an area of great interest to many applied mathematicians due its numerous applications in solving many real life problems. it was birthed as a result of interfield research activities in game theory and optimal control.. thus, many research articles have been devoted to this field and a lot of results were published ( see for example, [1-16]). in some of these research works, players move according to the following differential equations:{ ẋ = a(t)u, x(0) = x0, ẏ = b(t)v, y(0) = y0 (1) ∗corresponding author tel. no: email addresses: ajbadakaya.mth@buk.edu.ng (abbas ja’afaru badakaya ), bilyamamman@gmail.com (bilyaminu muhammad) where u(t) and v(t) are control functions of the players which are either subject to integral or geometrical constraints and a(·), b(·) are scalar measurable functions. the problems considered in [1, 7, 12, 13, 18, 19] involve players’ motion described by the differential equations (1), where a(t) = b(t) = 1. whereas, in the problems considered in [6, 8, 9, 10, 14, 17], players move according to the differential equations (1), where a(t) = b(t) , 1. problem in which players move according to (1), with a(t) , b(t) are investigated in [2]. in this work, control functions of the players are subject to integral constraints. optimal strategies of the players are constructed and value of the game is found. in all of the above cited works, only in [1, 8, 10, 18] constraints on the state variables are considered. the paper [1] reports study of pursuit problem on a closed convex subset of rn with control functions of players subject to coordinate-wise in115 abbas ja’afaru badakaya & bilyaminu muhammad / j. nig. soc. phys. sci. 2 (2020) 115–119 116 tegral constraints. some conditions under which pursuit can be completed from any position of the players in the given set are obtained. moreover, strategies for the pursuers are constructed. the work [8] is concerned with pursuit problem in which players control functions are subject to integral constraints. players are not allowed to move out of a closed convex subset of rn. optimal time of pursuit is found and optimal strategies for the players are constructed. ibragimov and satimov in [10] studied pursuit differential game problem on a nonempty convex subset of rn. in the game, both pursuers and evaders are not allowed to leave the given set and their control functions subject to integral constraints. sufficient conditions for completion of pursuit are obtained. a differential pursuit problem of an evader by finite number of pursuers on a closed convex set of l2, was studied by leong and ibragimov in [18]. the authors showed that an evading player cannot avoid an exact contact with any finite number of pursuing players whose individual resources are less than that of this evading players. in the present paper, pursuit differential game problem with finite number of pursuers and one evader on a nonempty closed convex subset of l2 is investigated. control functions of the pursuers and evader are subject to integral constraints. during the game players are to stay within a closed convex subset of l2. players move according to (1), where a(t) , b(t). 2. statement of the problem consider the hilbert space l2 = α = (α1,α2, ...,αk..., ) | ∞∑ k=1 α2k < ∞  , with the inner product (α,β) = ∞∑ k=1 αkβk and norm ‖α‖ = (∑ ∞ k=1 α 2 k )1/2 . let n be closed convex subset of l2 and define a ball in l2, with center at x0 and radius r by h(x0, r) := {x ∈ l2 : ‖x − x0‖ ≤ r}. let c(0,θ; l2) denotes space of continuous functions f (t) = ( f1(t), f2(t), . . . ) defined on the interval [0,θ] with absolutely continuous coordinates. the following definitions are important in the paper: definition 1. let (x,µ1) and (y,µ2), where µ1 and µ2 are σalgebra on x and y respectively, be a measurable spaces . the function f : (x,µ1) → (y,µ2) is borel measurable if f −1(e) ∈ µ1 for all e ∈ µ2 [20]. definition 2. let a be a subset of a hilbert space x , if every point of the hilbert space has exactly one projection onto a, then a is a chebyshev set [3]. definition 3. let x and y be normed linear spaces and let t : x → y be a linear map, then t is said to be lipschitz, if there exists a constant k > 0 such that, for each x ∈ x ‖t x‖ ≤ k‖x‖ [4]. we define a pursuit differential game problem in which countable but finite number of pursuers p j, j ∈ j = {1, 2, . . . , m} and evader e move according to the following equations:{ p j : ẋ j = a(t)u j, x j(0) = x j0, e : ẏ = b(t)v, y(0) = y0, (2) where x j, x j0, u j, y, y0, v ∈ l2, u j = (u j1, u j2, . . . ) is a control parameter of the pursuer p j and v = (v1, v2, . . . ) is that of the evader e. additionally, a(t) and b(t) are scalar measurable functions such that 1 ≤ b(t) ≤ a(t) for all t ∈ [0,θ]. the positive number θ is denoting duration of the game. definition 4. a borel measurable function u j(·); u j : [0,θ] → h(0,ρ j) such that(∫ ∞ 0 ||u j(t)|| 2dt )1/2 ≤ ρ j, (3) is called admissible control of the pursuer p j. definition 5. a borel measurable function v(·); v : [0,θ] → h(0,σ) such that(∫ ∞ 0 ‖v(t)‖2dt )1/2 ≤ σ, (4) is called admissible control of the evader. definition 6. a function u(t, x j, y, v), u j : [0,θ) × l2 × l2 × h(0,σ) → h(0,ρ j), such that the system{ ẋ j = u j(t), x j(0) = x0 j, ẏ = v(t), y(0) = y0, has a unique solution (x j(·), y(·)) with x j(·), y(·) ∈ c(0,θ; l2), for an arbitrary admissible control v = v(t), 0 ≤ t ≤ θ, of the evader e is called strategy of the pursuer p j. a strategy u j is said to be admissible if each control formed by this strategy is admissible. in what follows, we refer the game described by (2) in which the control functions u(·) and v(·) satisfying (3) and (4) respectively, as game g1. definition 7. pursuers win the game g1, if there exist pursuer’s strategy u j that ensure the equality x j(θ) = y(θ) for some j ∈ j. when the pursuer p j and evader e use admissible controls u j(t) = (u j1(t), u j2(t), . . . ) and v(t) = (v1(t), v2(t), . . . ) respectively, then from (2) their corresponding motions is given by x j(t) = (x j1(t), x j2(t), . . . ), y(t) = (y1(t), y2(t), . . . ), 116 abbas ja’afaru badakaya & bilyaminu muhammad / j. nig. soc. phys. sci. 2 (2020) 115–119 117 where x j(t) = (x j1(t), x j2(t), . . . , x jk(t), . . . ), x jk(t) = x j0k+ ∫ t 0 a(t)u jk(s)d s; y(t) = (y1(t), y2(t), . . . , yk(t), . . . ), yk(t) = yk0 + ∫ t 0 b(t)vk(s)d s. it can easily be shown that xi(·), y(·) ∈ c(0,θ; l2). problem. what are the sufficient conditions for pursuer to win the game g1? 3. results to present our result, we need the following notations: ρ2 := m∑ j=1 ρ2j and σ j := σ ρ ρ j, f or j ∈ j. then, it is easy to see that σ2j < ρ 2 j and σ 2 = ∑m j=1 σ 2 j. the following lemma is useful in the presentation of our results lemma 8. let 1 ≤ p ≤ ∞. if f, g ∈ lp(x,µ) then f + g ∈ lp(x,µ) and ‖ f + g‖ ≤ ‖ f‖p + ‖g‖p [5]. the theorem below gives sufficient conditions for pursuers to win the game when the players move freely without state constraint in the space l2. theorem 1. . if ρ2j < σ 2 for all j ∈ j and m∑ j=1 ρ2j > σ 2 , then for any initial positions of the players, pursuers win the game g1. proof: if y0 = x j0, for some j ∈ j, then the proof is trivial. therefore, let y0 , x j0, for all j ∈ j. we construct the strategies of the pursuers as follows: in the first phase of the game, we allow only one pursuer to move using a define strategy. without loss of generality, we allow the first pursuer to move and others to stay static. that is, the pursuers to use the strategies define byu1(t) = y0−x10 a(t)θ1 + b(t) a(t) v(t), j = 1 u j(t) = 0, j , 1, (5) where x10 and y0 is the initial position of the first pursuer and the evader respectively; θ1 = ( ||y0−x10|| ρ1−σ1 )2 . now we show that the strategy (5) is admissible whenever ∫ θ1 0 ||v(t)||2dt ≤ σ1. indeed, (∫ θ1 0 ‖u1(t)‖ 2dt )1/2 = (∫ θ1 0 ∥∥∥∥∥ y0 − x10a(t)θ1 + b(t)a(t) v(t) ∥∥∥∥∥2 dt )1/2 ≤ (∫ θ1 0 ∥∥∥∥∥ y0 − x10a(t)θ1 ∥∥∥∥∥2 dt )1/2 + (∫ θ1 0 ∥∥∥∥∥ b(t)a(t) v(t) ∥∥∥∥∥2 dt )1/2 ≤ (∫ θ1 0 ‖y0 − x10‖ 2 a2(t)θ12 dt )1/2 + (∫ θ1 0 ‖v(t)‖2dt )1/2 ≤ ‖y0 − x10‖ θ1 (∫ θ1 0 1 a2(t) dt )1/2 + σ1 ≤ ‖y0 − x10‖ θ1 (∫ θ1 0 dt )1/2 + σ1 = ‖y0 − x10‖ θ 1/2 1 + σ1 = ‖y0 − x10‖(( ||y0−x10|| ρ1−σ1 )2)1/2 + σ1 ≤ ρ1 −σ1 + σ1 = ρ1. now if the pursuer p1 uses the strategy (5) then, x1(θ1) − y(θ1) = x10 − y0 + ∫ θ1 0 a(t)u1(t)dt − ∫ θ1 0 b(t)v(t)dt = x10 − y0 + ∫ θ1 0 a(t) ( y0 − x10 a(t)θ1 + b(t) a(t) v(t) ) dt − ∫ θ1 0 b(t)v(t)dt = x10 − y0 + ∫ θ1 0 y0 − x10 θ1 dt + ∫ θ1 0 b(t)v(t)dt − ∫ θ1 0 b(t)v(t)dt = x10 − y0 + y0 − x10 θ1 θ1 = x10 − y0 + y0 − x10 = 0. this means that x1(θ1) = y(θ1) and implies that pursuers win the game. however, if x1(t) , y(t) for all t ∈ [0,θ1], then energy expanded by the evader in the time interval [0,θ1] is more than the energy of the first pursuer. that is∫ θ1 0 ‖v(t)‖2dt > ρ21. but from the fact that ρ21 > σ 2 1, then we have∫ θ1 0 ‖v(t)‖2dt > σ21. (6) if the first pursuer cannot win the game for the pursuers then the game will proceed into the second phase. in the second phase, the second pursuer will move using the strategy similar to (5) and the remaining pursuers p j, j = 1, 3, 4, . . . , m remain static. in general, during the kth phase of the game, pursuers p j, j = 1, 2, ...k − 1, k + 1, . . . , m, remain static and only the kth pursuer moves. that is, the strategies of the pursuers in the kth phase of the game are given byuk(t) = y(τk−1 )−xk0 a(t)θk + b(t) a(t) v(t) j = k u j(t) = 0, j , k; (7) 117 abbas ja’afaru badakaya & bilyaminu muhammad / j. nig. soc. phys. sci. 2 (2020) 115–119 118 where t ∈ [τk−1,τk], k = 1, 2, . . . , m,τ0 = 0 ; τk = k∑ j=1 θ j, and θk = ( ||y(τk−1 )−xk0|| ρk−σk )2 . again, for any admissible control of the evader v(·) the constructed strategy (5) is admissible and pursuers win the game. now, if in each phase of the game the equation x j(t) = y(t) does not hold for some t ∈ [0,τm], then the following inequalities must hold∫ τ1 0 ‖v(t)‖2dt >σ21, ∫ τ2 τ1 ‖v(t)‖2dt > σ22 , ..., + ∫ τm τm−1 ‖v(t)‖2dt > σ2m. consequently,∫ ∞ 0 ‖v(t)‖2dt ≥ ∫ τ1 0 ‖v(t)‖2dt + ∫ τ2 τ1 ‖v(t)‖2dt+ . . . + ∫ τm τm−1 ‖v(t)‖2dt > σ21 + σ 2 2 + . . . + σ 2 m = σ 2. this contradicts (4). therefore, we must have the equality x j(t) = y(t) holding for some j ∈ j and for some t ∈ [0,τm]. this completes the proof. now, consider the game g1 in which x j, x0, y, y0 ∈ n ⊂ l2 and all player cannot move out of the closed convex set n. then we have the following theorem: theorem 2. if ρ2j < σ 2 for all j ∈ j and ∑m j=1 ρ 2 j > σ 2, then for any initial positions of the players, pursuers win the game g1. proof: for purpose of the proof of this theorem, we introduce m number of dummy pursuers p̄ j, j = 1, ..., m which moves according to the following their equations p̄ j : ẇ = a(t)ū j(t), j = 1, ..., m; w j(0) = w j0. (8) where the controls ū j is such that∫ ∞ 0 ||ū j(t)|| 2dt ≤ ρ2j. the dummy pursuers has no restriction in their movements. that is, they can move outside the set n. therefore, dummy pursuers can win the game if we define the strategy of each of the dummy pursuer p̄ j, j ∈ i (same with the pursuers’ strategies define in the proof of theorem (1) as follows: ū j(t) = y(τ j−1) − z j0 a(t)θ j + b(t) a(t) v(t), τ j−1 ≤ t ≤ τ j; ūk(t) = 0,∀k = 1, 2, ..., j − 1, j + 1, ..., m. define by fn (x) the projection of a point x ∈ l2 onto n : |x − fn (x)| = min y∈n |x − y|. since n is closed convex subset of l2, it follows that n is a chebyshev set and the inequality |fn (x) − fn (y)| ≤ |x − y| (9) holds for any x, y ∈ l2, that is fn (·) is lipschitz continuous with 1. the operator fn (·) associates each of the absolute continuous function w j(t), 0 ≤ t ≤ τm, to an absolute continuous function x j(t). that is, fn : l2 → n, such that fn (w j(t)) = x j(t) i f w j(t) , n,w j(t) = x j(t), i f w j(t) ∈ n, (10) where t ∈ [0,τm]. we define the strategies of the real pursuers by u j(t) = ū j(t), i f w j(t) ∈ n1 a(t) ḟ(w j(t)), i f w j(t) < n. (11) we now show that this strategy ensure win for the pursuers and is admissible. by theorem (1) we can infer that the equality wi(t∗) = y(t∗) holds for some time t∗ ∈ [0,τm], for an index i ∈ i. since y(t) ∈ n, then using (10) yields xi(t∗) = fn (wi(t∗)) = wi(t∗) = y(t∗). lastly, we show the admissibility of the strategy (11). indeed, if w j(t) ∈ n then∫ ∞ 0 ‖u j(t)‖ 2dt = ∫ τ j τ j−1 ‖ū j(t)‖ 2dt ≤ ρ2j. in the other hand, we use the fact that fn (w j(·)) = x j(·) ∈ n; n is closed convex and the inequality (9) to deduce that ‖fn (w j(t + h)) − fn (w j(t)‖ ≤ ‖w j(t + h)) − w j(t)‖. in view of this inequality and for w j(t) < n, we have∫ ∞ 0 ‖u j(t)‖ 2dt = ∫ τ j τ j−1 1 a2(t) ‖ḟ(w j(t))‖ 2dt = ∫ τ j τ j−1 1 a2(t) ∥∥∥∥∥∥limh→0 fn (w j(t + h)) − fn (w j(t)h ∥∥∥∥∥∥ 2 dt = ∫ τ j τ j−1 1 a2(t) lim h→0 ‖fn (w j(t + h)) − fn (w j(t)‖2 h2 dt ≤ ∫ τ j τ j−1 1 a2(t) ∥∥∥∥∥∥limh→0 w j(t + h) − w j(t)h ∥∥∥∥∥∥ 2 dt = ∫ τ j τ j−1 1 a2(t) ‖ẇ j(t))‖ 2dt = ∫ τ j τ j−1 1 a2(t) ‖a(t)ū j(t)‖ 2dt = ∫ τ j τ j−1 ‖ū j(t)‖ 2dt ≤ ρ2j. 118 abbas ja’afaru badakaya & bilyaminu muhammad / j. nig. soc. phys. sci. 2 (2020) 115–119 119 4. illustrative example consider the game in which five dynamical object(pursuers) versus one dynamical object(evader) confined in a circular arena with radius 3 and whose dynamics described by{ p j : ẋ j = (2 + t)u j, x j(0) = x j0, j = 1, 2, 3. e : ẏ = (1 + t)v, y(0) = y0, where x10 = (1, 0, 0, . . . ); x20 = (0, 1, 0. . . . ); x30 = (0, 0, 1, . . . ); y0 = (0, 0, 0, . . .); ρ1 = 3, ρ2 = 4, ρ3 = 5,σ = 7. observe that 1 < (1 + t) < (2 + t); ρ2j < σ 2, ∀ j = 1, ..., 3 and ∑3 j=1 ρ 2 j > σ 2. this means that hypothesis of our theorems is satisfied, therefore pursuers win this game. 5. conclusion we studied pursuit problem in a closed convex subset of a hilbert space in which energy of each of pursuer is less than that of the evader. we were able to show that pursuers win the game when the total energy of the pursuers is greater than that of the single evader. furthermore, pursuers’ wining time is at least min j∈j θ j and at most τm. acknowledgments the valuable comments and observations of the reviewers of this article are acknowledged. indeed, their contributions added quality to the article. references [1] i. d. alias , g. i. ibragimov , m. ferrra, m. salimi, & m. monsi , “differential game of many pursuers with integral constraints on a convex set in the plane” (2015) arxiv:1505.00054v1[math.oc]. 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[15] r. p. ivanov, & yu. s. ledyaev, “optimality of pursuit time in a simple motion differential game of many objects”, trudy matematicheskogo instituta imeni v. a. steklova 158 (1981) 87. [16] a. b. ja’afaru, & g. i. ibragimov, ”on some pursuit and evasion differential game problems for an infinite numberof first-order differential equations”, journal of applied mathematics, 12 (2012) 13. doi:10.1155/2012/717124. [17] r. juwaid, & a. j. badakaya, “pursuit differential game problem with integral and geometric constraints in a hilbert space”, journal of the nigerian mathematical society 37 (2018) 203. [18] w. j. leong, & i. g. ibragimov, “a multiperson pursuit problem on a closed convex set in hilbert space”, far east journal of applied mathematics 33 (2008) 205. [19] a. yu. levchenkov, & a. g. pashkov, “differential game of optimal approach of two inertial pursuers to a noninertial evader”, journal of optimization theory and applications 65 (1990) 501 . [20] j. yeh, “real analysis theory of measure and integration”, world scientific publishing,(2006). 119 j. nig. soc. phys. sci. 4 (2022) 796 journal of the nigerian society of physical sciences biochemical synthesis, characterization and electrodeposition of silver nanoparticles on a gold substrate r. e. mfona,b,∗, j. j. deshic, z. al amria,d, j. s. madugue a school of physics, h.h wills physics laboratory tyndall avenue, university of bristol, bristol bs8 1tl united kingdom b department of physics, federal university of lafia, p.m.b 146 lafia, nasarawa state, nigeria c chemistry department, modibbo adama university of technology, p.m.b. 2076,yola, adamawa state, nigeria d engineering department, university of technology and applied sciences, salalah dhofar region, thumrait rd, 211 salalah, sultanate of oman e department of pure and applied physics, adamawa state university,p.m.b 25, mubi, adamawa state, nigeria abstract silver nanoparticles (agnps) antibacterial and antimicrobial properties have made them useful in the fields of medicine, for health care, consumer products, industrial purposes and more specifically food packaging industries. though agnps can be synthesized by various methods, the more environmentally friendly option was adopted. available literature shows that agnps can be infused into plastic and polyethylene containers and used for packaging foods and drinks to shield them from fungal or bacterial decay thereby extending their shelf lives. tests to ascertain the concentration and rate of migration of the agnps from the packaging to the food are deemed necessary. in this research ocimum gratissimum (og) and vernonia amygdalina (va) silver nanoparticles were biosynthesized, and were of varied sizes with some agglomeration with mean sizes 41 nm and 28 nm, respectively. their surface plasmon resonance (spr) occurred in the range 432 nm − 442 nm. electrodeposition of these nanoparticles on a gold substrate from an acidic medium was done and afm images show that the va-silver nanoparticles had small grains and provided a better surface coverage than the larger round flakes of the og-silver nanoparticles. the nanoparticles were found to have diffusion coefficient values which tallied with their sizes. thus for the smaller va-silver nanoparticles it was 1.76 × 10−7 cm2/s, while for the og silver nanoparticles it was 3.94 × 10−7 cm2/s showing that the migration rate of the ogsilver nanoparticles was higher than that of the va-silver nanoparticles. hence for faster nanoparticle migration, the og-nanoparticles is ideal but for a uniform, and even surface coverage, the va-silver nanoparticles should be employed. doi:10.46481/jnsps.2022.796 keywords: silver nanoparticles, characterisation, electrodeposition, diffusion coefficient, afm. article history : received: 3 may 2022 received in revised form: 8 june 2022 accepted for publication: 8 june 2022 published: 14 august 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published articleâs title, journal citation, and doi. communicated by: edward anand emile 1. introduction silver nanoparticles have become popular due to their optical, antibacterial and antimicrobial properties. they are ∗corresponding author tel. no: +2348034451806 email address: mfonbecca2000@yahoo.com (r. e. mfon ) used in electronics [1, 2] as sensors [3], in medicine [4] as drug carriers [5], as coating agents for dressing wounds [6], for food preservation and for treating contaminated water [7]. agnps have been used in textiles to kill bacteria and eliminate body odours and can be used for extending the shelf life of fruit juices [8, 9]. 1 r. e. mfon et al. / j. nig. soc. phys. sci. 4 (2022) 796 2 most recent studies on antimicrobial nanocomposite in food packaging are based on silver nanoparticles and it is reported that using nanoparticles in food packaging industries could tackle the problem of short shelf-life for products that can decay [10, 11]. silver nanoparticles can be synthesized using chemical methods, but its biosynthesis is more environmentally friendly. [12].the use of leaf extracts is applauded because they have in them stabilizers or capping agents. biosynthesis also produces biocompatible nanoparticles said to be useful in the field of medicine [13]. for effective application of silver nanoparticles, there is need to develop cheap, eco-friendly methods of synthesizing them and the electrodeposition technique is one of those methods. electrodeposition of a metal on a foreign substrate produces a thin film of the metal on the substrate and modifies the surface properties of that substrate and the thin film acquires physical and chemical properties which are different from the bulk of that metal. electrodeposition technique for nanoparticle production is fast and cheap as reported by mohanty [14] and improves the electrical properties of nanoparticles [15]. furthermore; it gets the nanoparticles straight onto the substrates offering effective control of its size, form and structure. silver nanoparticles act against pathogenic bacteria and spoilage fungi and can be used in biodegradable and nonbiodegradable polymers for making food packages [16] however, a migration test of the silver nanoparticles from the packages to food is of utmost importance and should be done to ascertain if the obtained values fall within the safe limits. silver and zinc nanoparticles have been infused into thins films which were used to form polyethylene packaging for storing orange fruit juice. the packaging material was reported to have reduced the pasteurization temperature of the juice, improving on its shelf life [10]. this study reports that though agnps are considered safe as far as us food and drug administration is concerned, there is a growing concern about health issues which may be triggered by high intake levels of the nanoparticles as a result of their migration from the packaging to the food they contain. studies on silver-lined packaging have been done and main concerns were also on the sizes of the nanoparticles as well as its rate of transfer to foods [17]. another research addressed the migration of silver in the form of nanoparticles into food [18]. the study which investigated four types of plastic containers, reported that concentration and amount of silver which migrated into food content varied from 13 to 42 µg/g. it was concluded that silver easily migrates into food especially if in contact with acidic substances. the above review stresses the need to make silver nanoparticles using environmentally friendly methods. it also reports that thin films of silver nanoparticles can be infused into plastics or polyethylene containers used for food packaging. major concerns in each of the listed cases are the nanoparticles figure 1: (a) electrodeposition process of silver from the silver nanoparticles on a gold substrate using a three-electrode cell (b) and the schematic illustration of the stripping process. figure 2: time dependent uv-vis absorbance spectra of the og and va silver nanoparticles. figure 3: ftir spectra of the ogand va-silver nanoparticles. sizes, concentration and migration of the silver nanoparticles from the packaging to the food it contains. in this present research, two species of silver nanoparticles were synthesized using aqueous silver nitrate and two types of plant leaf extracts. in this work, ocimum gratissimum (og) and vernonia amygdalina (va) plant leaf extracts were used for the silver nanoparticles synthesis. the nanoparticles were characterised using op2 r. e. mfon et al. / j. nig. soc. phys. sci. 4 (2022) 796 3 table 1: variation of peak current with scan rate for og-silver nanoparticles scan rate v/s √ scan rate (v/s)−1/2 peak current ip (a cm−2) 0.01 0.1 0.000422 0.02 0.141 0.000354 0.05 0.223 0.000272 0.1 0.316 0.000184 0.15 0.387 0.000117 (a) (b) figure 4: (a) tem of og-silver nanoparticles and the histogram showing size distribution and (b) tem of va-silver nanoparticles and the histogram showing size distribution figure 5: cyclic voltammograms of the ogand vasilver nanoparticles tical spectroscopy, electron microscopy and by using electrodeposition techniques (cyclic voltammetry (cv) and chronoamperometry (ca). this work did not study the antibacterial or antimicrobial properties of the nanoparticles but rather the behaviour of the nanoparticles in an acidic medium. furthermore, the sizes, structure, surface morphology, surface coverage abilities and diffusion coefficients of the two silver nanoparticles species were obtained. 2. materials and methods ogand vasilver nanoparticles (agnps) were synthesized using aqueous and ocimum gratissimum (og) and vernonia amygdalina (va) plant leaf extracts as presented by mfon et al. [18]. the agnps were characterised using uv-vis, ftir, and tem. a mixture of 50 ml of the ogor va -5 mm silver nanoparticles solution and 50 ml of 0.1 m h2s o4 was poured into a three electrode electrochemical cell (figure 1a) with silver as the reference electrode (re), platinum as counter electrodes (ce) and “gold-on-glass” as the working electrode (we) was used for the experiment. platinum wire was chosen as the counter electrode (ce) because it is stable under varying conditions and will not negatively affect coupled redox systems [19, 20, 21]. the working electrode was gold thin films evaporated on glass thermally treated to have the (111) dominant surface orientation [22] and a potentiostat (bio logic sp-150 science instruments) was used for the electrochemical deposition process. all chemicals used were of analytical grade and with > 99% purity and purchased from sigma aldrich. the silver deposition on the gold substrate was done with de-aeration to reduce the background current from oxygen reduction reaction and minimize any interference with the deposition kinetics. the h2s o4 acted as the charge bearing group to enhance conductivity and the electrodeposition was done at an assumed temperature of 25 ◦c [23]. multistep chronoamperometry (ca) was used for the deposition of the nanoparticles on the gold substrate and the derived graph was integrated to get q/a which was substituted in equation (1) to obtain the film thickness for different deposition times t. the thin films of the silver nanoparticles electrochemically deposited on a gold substrate was scanned using the nanosurf easyscan 2 afm and the tapping mode was employed to avoid destroying the surface of the thin films. the probe tip used was nclr and with its voltage set at 1v and free vibration 3 r. e. mfon et al. / j. nig. soc. phys. sci. 4 (2022) 796 4 amplitude at 200 mv, the surface morphology of the silver nanoparticles deposit on the gold surface was studied. the average thickness of the silver film deposits on the gold (from the electrolyte) for varying deposition times: 5, 10, 20 and 30 mins was calculated using equation (1). the thickness of the silver (deposit) film is: t = qm zfρa . (1) q/a = charge per unit area derived from the integration of the chronoamperometry (ca) graph, z = 1 because one electron is involved in the reaction), f = faraday’s constant (9.6 × 104), ρ = density of silver 10.5 g/cm3, a = 0.98 cm2. cyclic voltammetry (cv) of pt/ag+ on gold electrode for bulk deposition of the silver nanoparticles was studied. the cv of the og and va silver nanoparticles (agnps) was done in the range +500 mv and -520 mv. the cathodic peak due to the reduction of the silver and the diffusion of the agnps to the gold electrode as well as the anodic peak attributed to the stripping of the electrodeposited silver from the gold surface were noted. with different scan rates ranging from 10 mv/s to 150 mv/s the variation of the oxidation/reduction peak potentials with the peak height of the current density for each scan rate was also noted. from the obtained data, a graph of ip against v1/2 was plotted and the value of the slope was substituted into equation 2 (randles-sevcik equation with an assumed temperature of 25◦c ) and the diffusion coefficients of the two species (ogagnps and va-agnps) under investigation were determined. ip = 2.69 × 10 5n3/2 ad1/2cv1/2 (2) where ip = the peak current in amps, n = the number of electrons transferred in the redox event (usually 1), a = electrode area in cm2, d = diffusion coefficient in cm2/s, c = concentration in mol/cm3, v = scan rate in v/s. 3. results and discussion the uv-vis absorbance spectra (figure 2), shows that the surface plasmon resonance for the ogsilver nano-particles occurred between 440 nm and 442 nm while that for the vasilver nanoparticles was between 432 nm and 438 nm. the ftir scan of the silver nanoparticles (figure 3) done in the range 600−4000 cm−1 identified the o-h functional group which represents flavonoids and phenols which may have aided in the nanoparticles synthesis. other functional groups present include the o-h stretch for carboxylic acid and c-h for alkane. the tem images show silver nanoparticles of a wide range of sizes with some agglomeration. the size distribution for the two agnps species is as shown on the histograms. for the og silver nanoparticles, the mean size was calculated to be 41.4 ± 0.02 nm while that for the va silver nanoparticles was 28 ± 0.01 nm (figures 4a and 4b). the cv done in the range +500 mv to -550 mv (figure 5) showed that the difference between the anodic and cathodic (a) (b) figure 6: (a) cv of og-agnps with different scan rates (b) cv of va-agnps for different scan rates peaks for both samples ∆e was minimal and confirmed the reversibility of the process. for the og silver nanoparticles, the anodic and cathodic peaks were found to be -0.035 v and -0.150 v while that for the va silver nanoparticles, they were -0.058 v and -0.250 v, respectively. the cyclic voltammetry (cv) scan of the electrolyte (agnps +0.1m h2s o4 ) from 10 mv/s to 150 mv/s (figures 6a and 6b) shows that for the og agnps, the shift of the peak potential was almost 20 mv with the lowest scan rate producing the high est current density. for the va silver nanoparticles, the cv scan showed that the process was reversible but with very 4 r. e. mfon et al. / j. nig. soc. phys. sci. 4 (2022) 796 5 table 2: variation of peak current with scan rate for va-silver nanoparticles scan rate v/s √ scan rate (v/s)−1/2 peak current ip (a cm−2) 0.01 0.100 0.001513 0.02 0.141 0.00145 0.05 0.223 0.00140 0.10 0.316 0.00134 0.15 0.387 0.00131 figure 7: surface morphology of the deposited silver nanoparticles and the 3d images of the silver nanoparticle deposits on the gold . figure 8: thickness of the og and va silver nanoparticles with time on the gold electrode insignificant changes in the peak current height with scan rate with the 20 mv/s scan rate producing the highest peak current. the slope of the ip versus v s−1/2 for the og-silver nanoparticles is 8.28417 × 10−4 acm−2v s−1/2 while that for the va-silver nanoparticles is 5.542 × 10−4 acm−2v s−1/2 these results when substituted into equation (2) (randles sevcik equation) gave the diffusion coefficient for the va silver nanoparticles as 1.76 × 10−7 cm2/s, while that of the og silver nanoparticles was found to be 3.94 × 10−7 cm2/s. this shows that the migration rate of the og-silver nanoparticles was higher than that of the va-silver nanoparticles. the afm images (figure 7) show the morphology and surface coverage of the silver nanoparticle deposits on gold. the ognanoparticles, grains looked like well-defined round flakes though the surface coverage was uneven as the 3d image shows. in contrast, the vananoparticles produced a better and evenly distributed surface coverage with the grains looking like spikes as the 3d image shows. this feature suggests that the oxidation was more anodic for the va sample. though the og agnps got loaded on the gold surface, they either got depleted from the electrolyte with time or while covering the surface did not allow more nanoparticles to settle on already deposited ones leading to a halt in further deposition. the z-height of the ogsample was also found to be higher than that of the va-sample (figure 7). the thickness of silver film that was deposited on the gold substrate with time was found to increase steadily with time until saturation was attained after a self-limiting time of 20 mins for both nanoparticles (figure 8). for the og-sample, it is either that the nanoparticles in the electrolyte solution were depleted and further deposition of the og-silver nanoparticles was halted or an electrochemical oxidation of the agnps caused a decrease in particle loading on the gold substrate. this was in contrast to the va-silver nanoparticles whose deposition continued steadily to a maximum value after 30 mins. 4. conclusion ogand vasilver nanoparticles were biochemically synthesized and characterised. the og-nanoparticles had a mean size of 41 nm, with spr at 440 nm-442 nm, while the vananoparticles had a 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[23] a. j. bard, & l. r. faulkner, “electrochemical methods: fundamentals and applications, (2nd ed. john wiley & sons, isbn 0-471-04372-9 (2001). 6 j. nig. soc. phys. sci. 4 (2022) 287–296 journal of the nigerian society of physical sciences implicit four-point hybrid block integrator for the simulations of stiff models j. sundaya,∗, g. m. kumlenga, n. m. kamoha, j. a. kwanamub, y. skwameb, o. sarjiyusc adepartment of mathematics, university of jos, jos 930003, nigeria bdepartment of mathematics, adamawa state university, mubi 650001, nigeria cdepartment of computer science, adamawa state university, mubi 650001, nigeria abstract over the years, the systematic search for stiff model solvers that are near-optimal has attracted the attention of many researchers. an attempt has been made in this research to formulate an implicit four-point hybrid block integrator (fphbi) for the simulations of some renowned rigid stiff models. the integrator is formulated by using the lagrange polynomial as basis function. the properties of the integrator which include order, consistency, and convergence were analyzed. further analysis showed that the proposed integrator has an a-stability region. the a-stability nature of the integrator makes it more robust and fitted for the simulation of stiff models. to test the computational reliability of the new integrator, few well-known technical stiff models such as the pharmacokinetics, robertson and van der pol models were solved. the results generated were then compared with those of some existing methods including the matlab solid solvent, ode 15s. from the results generated, the new implicit fphbi performed better than the ones with which we compared our results with. doi:10.46481/jnsps.2022.777 keywords: hybrid block integrator, implicit, lagrange polynomial, simulation, stiff model article history : received: 22 april 2022 received in revised form: 22 may 2022 accepted for publication: 22 may 2022 published: 29 may 2022 c©2022 journal of the nigerian society of physical sciences. all rights reserved. communicated by: t. latunde 1. introduction in this research article, an a-stable implicit fphbi is formulated for the simulations of stiff models. the concept of astability was introduced by [1] to address the effect of stiffness in differential equations. the desire for this special property (a-stability) implies that suitable methods have to be derived for the simulations of stiff models, [2]. this has, therefore, motivated us to formulate an implicit fphbi for the simulations of ∗corresponding author tel. no: +2348026322933 email address: joshuasunday2000@yahoo.com, sundayjo@unijos.edu.ng (j. sunday ) stiff models of the form, y′ = (ay + ϕ) = f (x, y), y(a) = µ, x ∈ [a, b] (1) where yt = (y1, y2, ..., ym), µ t = (µ1,µ2, ...,µm), a is an m × m matrix and ϕ is an m-dimensional vector. equation (1) is assumed to satisfy the lipschitz conditions, [3]. equations of the form (1) often arise in many applications in engineering and sciences. suffice to say that most of these equations often results to stiff differential equations which in some cases do not have closed form solutions. stiff models are found in description of pharmacokinetics, chemical reactions, molecular dynamics, control systems, mechanics, electronic circuits, lasers, [4-8]. 287 sunday et al. / j. nig. soc. phys. sci. 4 (2022) 287–296 288 in general, equation (1) may be stiff or non-stiff. a nonstiff differential equation is one in which there is a simultaneous evolution of solution components and also has time-scales that are comparable. they are often solved using explicit methods with some error control [9]. on the contrary, stiff differential equations do not have a universally accepted definition. the term ‘stiff’ first appeared in [10]. equation (1) is called stiff if it satisfies any of the following definitions: 1. the stability step-size is much more smaller than the accuracy step-size [11]. that is, the step-size depends on stability requirement rather than the accuracy requirement [12], 2. the explicit method performs slowly or do not even work on it [13], 3. it has time scales that vary widely. in order words, some solution components decay more rapidly than others [12], 4. all its eigenvalues have negative real parts with large stiffness ratio [12], 5. real(λi) < 0, i ∈ [1, m], where λi is the eigenvalues of the matrix j = ∂ f /∂y called the jacobian matrix of equation (1) [14] and 6. maximum i |real(λi)| ≥ minimum i |real(λi)|, where s = maximum i |real(λi )| minimum i |real(λi )| is often called the stiffness ratio. the stiffness ratio measures the degree of stiffness of the system [14]. conditions (5) and (6) as given by [14] shall be adopted as the definition of the stiff models that shall be considered in this research. the expression, y(x) = m∑ i=0 ∈i e λi xki + σ(x) (2) is the general solution of the stiff model (1) where ki are the eigenvectors corresponding to the eigenvalues λi, ∈i are arbitrary constants and σ(x) is a particular integral. taking x as time, the first term of y(x) = ∑m i=0 ∈i e λi xki is denoted as transient solution and the second term σ(x) as steady-state solution. assuming the stiff model (1) satisfies condition (5), then the term y(x) = ∑m i=0 ∈i e λi xki → 0 as x → ∞. suppose |real(λu)| and |real(λv)| further satisfies the condition |real(λu)| ≥ |real(λi)| ≥ |real(λv)| , i = 1, 2, ..., m so that fastest and slowest transients are ∈i eλu xki and ∈i eλv xki respectively. if the stiff model (1) is solved numerically and aimed at achieving a steady-state phase, continuing integration is needed until the slowest transient is negligible [2]. thus, the smaller |real(λv)| is, the longer the integration time. on the other hand, for the larger|real(λu)|, a sufficiently small step is required so that h will lie within the stability region of the method [12, 15]. many authors have made several attempts at solving stiff models. for instance, the authors in [16] proposed k-step hybrid methods for solving stiff models arising from chemical reactions. the authors also proved some relevant theorems regarding the regions of stability of the methods. the authors in figure 1: physical illustration of the implicit fphbi [2] developed a block backward differentiation formula that is diagonally implicit for the approximation of stiff pharmacokinetics models. they further carried out convergence and stability analysis of the formula. also, the authors in [17] derived an optimized hybrid block adams method for the solution of first order differential equations. the a-stable method which is of order six was found to be convergence, zero-stable and consistent. the following authors also developed hybrid methods for solving different types of stiff differential equations [2, 8, 9, 16, 18-27]. furthermore, these authors also proposed different approaches to solving some special differential equations [28-35]. 2. formulation and implementation of the implicit fphbi 2.1. formulation in this section, an implicit fphbi is formulated for the simulations of stiff models of the form (1). the values in the previous block (i.e. xn−1 and xn) are employed in computing the stiff model (1) at the points xn+1, xn+2, xn+ 52 , xn+3, xn+ 72 and xn+4 (see figure 1). to formulate the implicit fphbi at the points xn+r, r = 1, 2, 5 2 , 3, 7 2 and 4, we integrate equation (1) in the interval (xn, xn+r ),∫ xn+r xn y′d x = ∫ xn+r xn [ ay + ϕ ] d x (3) the function f (x, y) = ay + ϕ in (1) is then approximated by lagrange polynomial pq(x) of the form pq(x) = k∑ j=0 lq, j(x) f (xn+4− j) (4) where lq, j(x) = k−1∏ i = 0 i , j x − xn+4−i xn+4− j − xn+4−i , j = 0, 1 2 , 1, 2, ..., k therefore, the lagrange interpolation polynomial associated with the interpolating points (xn−1, yn−1), (xn, yn),(xn+1, yn+1), (xn+2, yn+2), ( xn+ 52 , yn+ 52 ) , (xn+3, yn+3), ( xn+ 72 , yn+ 72 ) and (xn+4, yn+4) is given by, 288 sunday et al. / j. nig. soc. phys. sci. 4 (2022) 287–296 289 p4(x) = [ (x − xn−1)(x − xn)(x − xn+1)(x − xn+2)(x − xn+5/2)(x − xn+3)(x − xn+7/2) (xn+4 − xn−1)(xn+4 − xn)(xn+4 − xn+1)(xn+4 − xn+2)(xn+4 − xn+5/2)(xn+4 − xn+3)(xn+4 − xn+7/2) ] f (xn+4) + [ (x − xn−1)(x − xn)(x − xn+1)(x − xn+2)(x − xn+5/2)(x − xn+3)(x − xn+4) (xn+7/2 − xn−1)(xn+7/2 − xn)(xn+7/2 − xn+1)(xn+7/2 − xn+2)(xn+7/2 − xn+5/2)(xn+7/2 − xn+3)(xn+7/2 − xn+4) ] f (xn+7/2) + [ (x − xn−1)(x − xn)(x − xn+1)(x − xn+2)(x − xn+5/2)(x − xn+7/2)(x − xn+4) (xn+3 − xn−1)(xn+3 − xn)(xn+3 − xn+1)(xn+3 − xn+2)(xn+3 − xn+5/2)(xn+3 − xn+7/2)(xn+3 − xn+4) ] f (xn+3) + [ (x − xn−1)(x − xn)(x − xn+1)(x − xn+2)(x − xn+3)(x − xn+7/2)(x − xn+4) (xn+5/2 − xn−1)(xn+5/2 − xn)(xn+5/2 − xn+1)(xn+5/2 − xn+2)(xn+5/2 − xn+3)(xn+5/2 − xn+7/2)(xn+5/2 − xn+4) ] f (xn+5/2) + [ (x − xn−1)(x − xn)(x − xn+1)(x − xn+5/2)(x − xn+3)(x − xn+7/2)(x − xn+4) (xn+2 − xn−1)(xn+2 − xn)(xn+2 − xn+1)(xn+2 − xn+5/2)(xn+2 − xn+3)(xn+2 − xn+7/2)(xn+2 − xn+4) ] f (xn+2) + [ (x − xn−1)(x − xn)(x − xn+2)(x − xn+5/2)(x − xn+3)(x − xn+7/2)(x − xn+4) (xn+1 − xn−1)(xn+1 − xn)(xn+1 − xn+2)(xn+1 − xn+5/2)(xn+1 − xn+3)(xn+1 − xn+7/2)(xn+1 − xn+4) ] f (xn+1) + [ (x − xn−1)(x − xn+1)(x − xn+2)(x − xn+5/2)(x − xn+3)(x − xn+7/2)(x − xn+4) (xn − xn−1)(xn − xn+1)(xn − xn+2)(xn − xn+5/2)(xn − xn+3)(xn − xn+7/2)(xn − xn+4) ] f (xn) + [ (x − xn)(x − xn+1)(x − xn+2)(x − xn+5/2)(x − xn+3)(x − xn+7/2)(x − xn+4) (xn−1 − xn)(xn−1 − xn+1)(xn−1 − xn+2)(xn−1 − xn+5/2)(xn−1 − xn+3)(xn−1 − xn+7/2)(xn−1 − xn+4) ] f (xn−1) (5) substituting x = sh + xn+4 in equation (5) gives, p4(sh + xn+4) = [ (sh+5h)(sh+4h)(sh+3h)(sh+2h)(sh+ 32 h)(sh+h)(sh+ 12 h) (5h)(4h)(3h)(2h)( 32 h)(h)( 12 h) ] f (xn+4) + [ (sh+5h)(sh+4h)(sh+3h)(sh+2h)(sh+ 32 h)(sh+h)(sh) ( 92 h)( 72 h)( 52 h)( 32 h)(h)( 12 h)(− 12 h) ] f (xn+7/2) + [ (sh+5h)(sh+4h)(sh+3h)(sh+2h)(sh+ 32 h)(sh+ 12 h)(sh) (4h)(3h)(2h)(h)( 12 h)(− 12 h)(−h) ] f (xn+3) + [ (sh+5h)(sh+4h)(sh+3h)(sh+2h)(sh+h)(sh+ 12 h)(sh) ( 72 h)( 52 h)( 32 h)( 12 h)(− 12 h)(−h)(− 32 h) ] f (xn+5/2) + [ (sh+5h)(sh+4h)(sh+3h)(sh+ 32 h)(sh+h)(sh+ 12 h)(sh) (3h)(2h)(h)(− 12 h)(−h)(− 32 h)(−2h) ] f (xn+2) + [ (sh+5h)(sh+4h)(sh+2h)(sh+ 32 h)(sh+h)(sh+ 12 h)(sh) (2h)(h)(−h)(− 32 h)(−2h)(−52 h)(−3h) ] f (xn+1) + [ (sh+5h)(sh+3h)(sh+2h)(sh+ 32 h)(sh+h)(sh+ 12 h)(sh) (h)(−h)(−2h)(− 52 h)(−3h)(− 72 h)(−4h) ] f (xn) + [ (sh+4h)(sh+3h)(sh+2h)(sh+ 32 h)(sh+h)(sh+ 12 h)(sh) (−h)(−2h)(−3h)(− 72 h)(−4h)(− 92 h)(−5h) ] f (xn−1) (6) equation (6) is further simplified as, p4(sh + xn+4) = ( 1 360 ) [(s + 1)(s + 2)(s + 3)(s + 4)(s + 5)(2s + 1)(2s + 3)] fn+4 − ( 32 945 ) [s(s + 1)(s + 2)(s + 3)(s + 4)(s + 5)(2s + 3)] fn+ 72 + ( 1 24 ) [s(s + 2)(s + 3)(s + 4)(s + 5)(2s + 1)(2s + 3)] fn+3 − ( 32 315 ) [s(s + 1)(s + 2)(s + 3)(s + 4)(s + 5)(2s + 1)] fn+ 52 + ( 1 36 ) [s(s + 1)(s + 3)(s + 4)(s + 5)(2s + 1)()2s + 3] fn+2 − ( 1 180 ) [s(s + 1)(s + 2)(s + 4)(s + 5)(2s + 1)(2s + 3)] fn+1 + ( 1 840 ) [s(s + 1)(s + 2)(s + 3)(s + 5)(2s + 1)(2s + 3)] fn − ( 1 7560 ) [s(s + 1)(s + 2)(s + 3)(s + 4)(2s + 1)(2s + 3)] fn−1 (7) integrating the polynomial (7) with respect to s and replacing d x with hd s in equation (3) gives, y(xn+r ) = y(xn) + h ∫ xn+r xn p4(sh + xn+4)d s (8) in order to obtain a zero-stable and computationally robust implicit fphbi, the points (-4, -3), (-4, -2), (-4, -3/2), (-4, -1), (-4, -1/2) and (-4, 0) are chosen as the limits of integration. this gives the new implicit fphbi formulae for yn+1, yn+2, yn+ 52 , yn+3, yn+ 72 and yn+4 as follows, yn+1 = yn + h ( − 965 127008 fn−1 + 1681 4704 fn + 149 144 fn+1 − 21859 15120 fn+2 + 4384 2205 fn+ 52 − 4397 3360 fn+3 + 8816 19845 fn+ 72 − 631 10080 fn+4 ) yn+2 = yn + h ( − 29 4410 fn−1 + 251 735 fn + 191 135 fn+1 − 9 35 fn+2 + 1408 1323 fn+ 52 − 169 210 fn+3 + 128 441 fn+ 72 − 8 189 fn+4 ) yn+ 52 = yn + h ( − 107725 16257024 fn−1 + 206015 602112 fn + 25975 18432 fn+1 − 13375 387072 fn+2 + 19765 14112 fn+ 52 − 75125 86016 fn+3 + 38975 127008 fn+ 72 − 11425 258048 fn+4 ) yn+3 = yn + h ( − 31 4704 fn−1 + 2679 7840 fn + 113 80 fn+1 − 41 560 fn+2 + 416 245 fn+ 52 − 687 1180 fn+3 + 208 735 fn+ 72 − 47 1120 fn+4 ) yn+ 72 = yn + h ( − 245 36864 fn−1 + 4207 12288 fn + 77861 55296 fn+1 − 343 10240 fn+2 + 6811 4320 fn+ 52 − 14063 61440 fn+3 + 707 1440 fn+ 72 − 27097 552960 fn+4 ) yn+4 = yn + h ( − 128 19845 fn−1 + 50 147 fn + 64 45 fn+1 − 136 945 fn+2 + 4096 2205 fn+ 52 − 64 105 fn+3 + 4096 3969 fn+ 72 + 34 315 fn+4 )  (9) equation (9) is the new implicit fphbi for the simulations of stiff models of the form (1). 289 sunday et al. / j. nig. soc. phys. sci. 4 (2022) 287–296 290 the predictor formulae that compute the previous block at the points xn−1 and xn is derived in similar fashion like the implicit fphbi. it is given by the formulae, ypn+1 = yn + h 2 (3 fn − fn−1) ypn+2 = yn + h (4 fn − 2 fn−1) yp n+ 52 = yn + 5h 8 (9 fn − 5 fn−1) ypn+3 = yn + 3h 2 (5 fn − 3 fn−1) yp n+ 72 = yn + 7h 8 (11 fn − 7 fn−1) ypn+4 = yn + h (12 fn − 8 fn−1)  (10) 2.2. implementation the newly formulated implicit fphbi in equation (9) will be implemented in a predictor-corrector mode. firstly, we set the data inputs namely, the problem to be solved, initial conditions, step size and tolerance level. the predictor in equation (10) is used to evaluate the previous block at the points xn−1 and xn. the implicit fphbi which serves as the corrector and main method is then employed in solving the required stiff models. the codes for the implementation of the method were written in matlab 2021a while the method was derived using scientific workplace 5.5. 3. analysis of the implicit fphbi the analysis of the new implicit fphbi shall be carried out in this section. 3.1. order definition 3.1 [36] a method and its associated difference operator l given by l {y(x); h} = k∑ j=0 [ α jy(x + jh) − hβ jy ′(x + jh) ] (11) are said to be of order p if c0 = c1 = c2 = ... = cp = 0, cp+1 , 0. the component cp+1 , 0 is called the error constant of the method. the general form for the constant cqis defined as c0 = ∑k j=0 α j c1 = ∑k j=0 ( jα j −β j ) . . . cp = ∑k j=0 [ 1 p! j pα j − 1 (p−1)! j p−1β j ] , p = 2, 3, ..., q + 1  (12) applying equation (12) on the implicit fphbi (9), we obtain c0 = c1 = c2 = c3 = c4 = c5 = c6 = c7 = c8 =  0 0 0 0 0 0  , while c9 =  4.9730 × 10−4 3.8179 × 10−4 3.8942 × 10−4 3.8305 × 10−4 3.9424 × 10−4 3.5021 × 10−4  therefore, the implicit fphbi is of uniform order 8 with the error constant c9 =  4.9730 × 10−4 3.8179 × 10−4 3.8942 × 10−4 3.8305 × 10−4 3.9424 × 10−4 3.5021 × 10−4  3.2. consistency definition 3.2 [36] a method is called consistent if it is of order p ≥ 1. since the new implicit fphbi is of order 8, it implies that it is consistent. 290 sunday et al. / j. nig. soc. phys. sci. 4 (2022) 287–296 291 3.3. zero-stability definition 3.3 [12] if no root of the characteristic polynomial has a modulus greater than one and every root with modulus one is simple, then such a method is called zero-stable. [1] proposed scalar test to ascertain the zero-stability status of a method. thus substituting the scalar test problem y′ = f = λy (13) into the implicit fphbi equation (9), we obtain, yn+1 = yn + h ( − 965 127008 λyn−1 + 1681 4704 λyn + 149 144 λyn+1 − 21859 15120 λyn+2 + 4384 2205 λyn+ 52 − 4397 3360 λyn+3 + 8816 19845 λyn+ 72 − 631 10080 λyn+4 ) yn+2 = yn + h ( − 29 4410 λyn−1 + 251 735 λyn + 191 135 λyn+1 − 9 35 λyn+2 + 1408 1323 λyn+ 52 − 169 210 λyn+3 + 128 441 λyn+ 72 − 8 189 λyn+4 ) yn+ 52 = yn + h ( − 107725 16257024 λyn−1 + 206015 602112 λyn + 25975 18432 λyn+1 − 13375 387072 λyn+2 + 19765 14112 λyn+ 52 − 75125 86016 λyn+3 + 38975 127008 λyn+ 72 − 11425 258048 λyn+4 ) yn+3 = yn + h ( − 31 4704 λyn−1 + 2679 7840 λyn + 113 80 λyn+1 − 41 560 λyn+2 + 416 245 λyn+ 52 − 687 1180 λyn+3 + 208 735 λyn+ 72 − 47 1120 λyn+4 ) yn+ 72 = yn + h ( − 245 36864 λyn−1 + 4207 12288 λyn + 77861 55296 λyn+1 − 343 10240 λyn+2 + 6811 4320 λyn+ 52 − 14063 61440 λyn+3 + 707 1440 λyn+ 72 − 27097 552960 λyn+4 ) yn+4 = yn + h ( − 128 19845 λyn−1 + 50 147 λyn + 64 45 λyn+1 − 136 945 λyn+2 + 4096 2205 λyn+ 52 − 64 105 λyn+3 + 4096 3969 λyn+ 72 + 34 315 λyn+4 )  (14) equation (14) is then written compactly in matrix form as,  1 − 149144 hλ 21859 15120 hλ − 4384 2205 hλ 4397 3360 hλ − 8816 19845 hλ 631 10080 hλ − 191 135 hλ 1 + 9 35 hλ − 1408 1323 hλ 169 210 hλ − 128 441 hλ 8 189 hλ − 25975 18432 hλ 13375 387072 hλ 1 − 19765 14112 hλ 75125 86016 hλ − 38975 127008 hλ 11425 258048 hλ − 113 80 hλ 41 560 hλ − 416 245 hλ 1 + 687 1120 hλ − 208 735 hλ 47 1120 hλ − 77861 55296 hλ 343 10240 hλ − 6811 4320 hλ − 14063 61440 hλ 1 − 707 1440 hλ 27097 552960 hλ − 64 45 hλ 136 945 hλ − 4096 2205 hλ 64 105 hλ − 4096 3969 hλ 1 − 34 315 hλ   yn+1 yn+2 yn+ 52 yn+3 yn+ 72 yn+4  =  0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1   yn− 72 yn−3 yn− 52 yn−2 yn−1 yn  + h  0 0 0 0 − 965127008 λ 1681 4704 λ 0 0 0 0 − 294410 λ 251 735 λ 0 0 0 0 − 10772516257024 λ 206015 602112 λ 0 0 0 0 − 314704 λ 2679 7840 λ 0 0 0 0 − 24536864 λ 4207 12288 λ 0 0 0 0 − 12819845 λ 50 147 λ   yn− 72 yn−3 yn− 52 yn−2 yn−1 yn  (15) equation (15) can be written as aym = (b + ch)ym−1 (16) where a =  1 − 149144 hλ 21859 15120 hλ − 4384 2205 hλ 4397 3360 hλ − 8816 19845 hλ 631 10080 hλ − 191 135 hλ 1 + 9 35 hλ − 1408 1323 hλ 169 210 hλ − 128 441 hλ 8 189 hλ − 25975 18432 hλ 13375 387072 hλ 1 − 19765 14112 hλ 75125 86016 hλ − 38975 127008 hλ 11425 258048 hλ − 113 80 hλ 41 560 hλ − 416 245 hλ 1 + 687 1120 hλ − 208 735 hλ 47 1120 hλ − 77861 55296 hλ 343 10240 hλ − 6811 4320 hλ − 14063 61440 hλ 1 − 707 1440 hλ 27097 552960 hλ − 64 45 hλ 136 945 hλ − 4096 2205 hλ 64 105 hλ − 4096 3969 hλ 1 − 34 315 hλ  b =  0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1  , c =  0 0 0 0 − 965127008 λ 1681 4704 λ 0 0 0 0 − 294410 λ 251 735 λ 0 0 0 0 − 10772516257024 λ 206015 602112 λ 0 0 0 0 − 314704 λ 2679 7840 λ 0 0 0 0 − 24536864 λ 4207 12288 λ 0 0 0 0 − 12819845 λ 50 147 λ  , ym =  yn+1 yn+2 yn+ 52 yn+3 yn+ 72 yn+4  291 sunday et al. / j. nig. soc. phys. sci. 4 (2022) 287–296 292 figure 2: stability region of the implicit fphbi and ym−1 =  yn− 72 yn−3 yn− 52 yn−2 yn−1 yn  let h = hλ, then the stability polynomial r(t, h)of the implicit fphbi is, r(t, h) = det (ta − (b + ch)) = t6 ( 896369 25401600 h 6 − 222166877 889056000 h 5 + 26657599495334336000 h 4 + 4555297714817600 h 3 + 30028514116000 h 2 − 196033 88200 h + 1 ) −t5 ( 384922243 45519667200 h 6 + 324280887154623600640 h 5 + 52444936612167603200 h 4 + 2600241151336578304000 h 3 + 76767644775120962560 h 2 + 94984435017600 h + 1 ) +t4 ( 2689 6502809600 h 6 + 78911571365590016000 h 5 + 496972131365590016000 h 4 + 94811746496000 h 3 + 440765318289152000 h 2 + 1593781285120 h ) (17) substituting h = 0 into (17) gives, r(t, 0) = t6 − t5 (18) therefore, the zeros of equation (18) are t1 = t2 = t3 = t4 = t5 = 0 and t6 = 1. since all the zeros lie within |t| ≤ 1, the implicit fphbi is said to be zero-stable. 3.4. convergence theorem 3.1 [12] consistence and zero-stability are the necessary and sufficient conditions a method must satisfy in order to be convergent. thus, the fphbi is convergent since it satisfies the conditions for consistency and zero-stability. 3.5. stability regions 3.5.1. stability region of the implicit fphbi the part of the complex plane where a method is absolutely stable when applied to the scalar test equation y′ = λy is called its stability region. definition 3.4 [14] if the stability region of a method contains the whole left half-plane re(hλ) < 0, such a method is referred to as a-stable. the graphical plot of the implicit fphbi is presented in figure 2. 3.5.2. comparison of stability regions and intervals of instability the following figure shows the stability region of the diagonally implicit block backward differentiation formula (dibbdf) derived by [2]. 292 sunday et al. / j. nig. soc. phys. sci. 4 (2022) 287–296 293 figure 3: stability region of dibbdf by [2] table 1: juxtaposition of instability intervals of fphbi and dibbdf method interval of instability fphbi (0, 3.8135) dibbdf (0, 15.333) the stability of the new implicit fphbi (see figure 2) shall be compared with that of the dibbdf (in figure 3). while both methods are a-stable, it is obvious from the two figures that the implicit fphbi has a larger stability region than that of dibbdf. thus, the fphbi is expected to be more accurate and efficient than the dibbdf, since the larger the stability region of a method, the better the performance. note that the stable region is the region that lies outside the closed curve. the interval of instability of the new implicit fphbi is compared with that of dibbdf as shown in table 1. from table 1, it is clear that the implicit fphbi has a larger stability region than the dibbdf. 4. test problems the implicit fphbi derived in equation (9) shall be used to simulate some rigid well-known stiff models in order to test its accuracy and efficiency. numerical solution, maximum error, computation time, efficiency curves and solution curves of the said problems shall be presented. 4.1. problem 1 (pharmacokinetics model) pharmacokinetics is a phenomenon that describes how an administered drug behaves in the body over a period of time. consider the pharmacokinetics model (see figure 4) composed of two components; that is, the drug concentration in the gastrointestinal (gi) tract y1(t)and drug concentration in the bloodstream y2(t) as a function of time t given by, y ′ 1 = −k1y1 + d, y1(0) = 1 y ′ 2 = k1y1 − key2, y2(0) = 0 } (19) where ke(> 0) and k1 are the clearance rate constant and rate constants from one compartment to another respectively. the parameter d is the regimen of drug intake. taking k1 = 2(ln 2)and ke = (ln 2)/5, the exact solution of equation (19) for t ∈ [0, 6] are given by figure 4: schema of drug intake from the stomach to bloodstream y1(x) = e−k1 t y2(t) = − k1 (k1−ke ) ( e−k1 t − e−ke t ) , k1 , ke } (20) note that y1(x) = e−k1 t is the drug’s exponential decay in terms of absorption [37]. this implies that the behaviour in the long term of the concentration of the drug in the gi tract will reduce to level zero. equation (18) is a two-compartmental pharmacokinetics model formulated by [38] to model the drug flow via the compartments of the body namely the gi tract and the circulatory system. the drug moves from the gi tract compartment into the bloodstream compartment at a rate relative to the drug’s concentration in the gi tract [2]. the drug is finally metabolized and cleared from the bloodstream at a rate relative to its concentration there [39]. for more details on general two-compartment pharmacokinetics models, see [37, 40, 41]. 4.2. problem 2 (robertson model) according to [4], the robertson model defines an autocatalytic kinetics reaction. the robertson problem, popularly called the robber according to [13] is a set of three ordinary differential equations that can be put up under some ideal conditions. it is mathematically given by the stiff system y ′ 1 = −k1y1 + k3y2y3 y ′ 2 = k1y1 − k2y 2 2 − k3y2y3 y ′ 3 = k2y 2 2  (21) with (y1(0), y2(0), y3(0)) t = (y01, y02, y03) t , where y1, y2 and y3 are the concentrations of a, b and c respectively and y01, y02 and y03 are concentrations over a period of time t = 0. this problem was proposed by [42] and the file rober.f containing the software of the problem can be found in [43]. the robertson problem is composed of the following reactions a k1 −→ b b + b k2 −→ c + b b + c k3 −→ a + c where a, b and care chemical species and k1, k2 and k3are rate constants. it is important to state that the cause of stiffness of this problem is the substantial variation in the reaction rate constants. in the test problem, we shall assume k1 = 0.04, k2 = 3×107 and k3 = 1 × 104 as the numerical values of rate constants and 293 sunday et al. / j. nig. soc. phys. sci. 4 (2022) 287–296 294 y01 = 1, y02 = 0 and y03 = 0 as the initial concentrations. the stiff model (21) is thus given as y ′ 1 = −0.04y1 + 10 4y2y3, y1(0) = 1 y ′ 2 = 0.04y1 − 1 × 10 4y2y3 − 3 × 107y22, y2(0) = 0 y ′ 3 = 3 × 10 7y22, y3(0) = 0 (22) 4.3. problem 3 (van der pol model) the van der pol equation proposed originally in 1920s by the dutch electrical engineer balthazar van der pol (1889-1959) has been used to model systems that have self-excited oscillations of limit cycles. the equation depicts oscillatory processes in physics, electronics, biology, neurology, etc.[44]. van der pol himself used the equation to model the range of stability of the human heart dynamics. furthermore, coupled neurons in gastric mill circuits of lobsters in the field of biology were successfully modelled using the van der pol equations [45, 46]. the equation has also been used as a model in seismology for developing a model in a geological fault that shows interaction of two plates [47]. the generation of spike in giant squid axons can also be modelled using van der pol equation [48]. it is therefore important to explore ways of developing a deep understanding of the van der pol equation in view of its widespread applications. one of the ways to do this is by modeling and simulating it. the van der pol equation is a stiff nonlinear equation expressed as, y′′ + µ ( 1 + y2 ) y′ + y = 0 (23) equation (23) is transformed to its equivalent system of first order system of the form, y ′ 1 = y2, y1(0) = 2 y ′ 2 = [ −y1 + (1 − y21)y2 ] / ∈, y2(0) = 0 } (24) this transformation is achieved by substituting y1 = ϕ(x), y2 = µϕ′(x) and x = x/µ, where ∈= 1 / µ2 is a parameter that controls stiffness. in this paper, the value of ∈= 500. 5. results and discussions in this section, the implicit fphbi is employed in simulating stiff models presented in section 4. this is aimed at testing the accuracy, efficiency and computational reliability of the proposed integrator. the following abbreviations shall be used in the tables 2-4. x: point of evaluation h: step size t /s: computation/execution time in seconds abse: absolute error maxe: maximum error numsol: numerical (approximate) solution dibbdf : diagonally implicit block backward differentiation formula developed by [2] kshm: k-step hybrid method derived by [16] ode 15s: matlab inbuilt stiff solver table 2: juxtaposition of maximum error for problem 1 h method maxe t /s 10−2 fphbi 2.14126 × 10−6 3.17921 × 10−7 dibbdf 3.09796 × 10−4 1.48973 × 10−5 ode 15s 7.75030 × 10−3 3.40630 × 10−2 10−4 fphbi 1.24178 × 10−10 2.81291 × 10−5 dibbdf 3.26669 × 10−8 4.80924 × 10−4 ode 15s 1.53220 × 10−4 6.09380 × 10−2 10−6 fphbi 1.12471 × 10−12 5.21799 × 10−3 dibbdf 5.29902 × 10−11 2.34869 × 10−2 ode 15s 2.44190 × 10−6 9.37500 × 10−1 figure 5: efficiency curves for problem 1 fphbi: newly derived implicit four-point hybrid block integrator absolute error (abse) is defined as abs e = |y(x) − yn(x)| on the other hand, maximum error (maxe) is defined as maxe = max 0≤n≤ns |y(x) − yn(x)| where ns is the total number of steps, y(x)is the exact solution and yn(x) is the computed (approximate) solution. in table 2, it is obvious that the fphbi performed better than the dibbdf developed by [2] and the ode 15s solver as the new method has lesser maximum error for problem 2. it is also important to state that the fphbi is more efficient in terms of computation time than those of dibbdf and ode 15s. this is evident in table 2 and the efficiency curves presented in figure 5. table 3 presents the approximate solutions of problem 2 at points x = 0.4,x = 40 and x = 4000for the three solution components y1(x), y2(x) and y3(x). the results obtained clearly show that the implicit fphbi effectively approximates problem 2. to further buttress this point, accuracy (solution) curves were plotted for the problem, see figure 6. table 4 presents the numerical solution of the van der pol model in (24). since the problem does not have exact solution, the approximate solution using the newly derived implicit fphbi is compared with that of matlab in built stiff solver (ode 15s) at the end points x = 1, x = 5, x = 10 and x = 20. from the solution curve obtained, it is clear that the result of the fphbi converges to that of the ode 15s solver, see figure 7. thus, the fphbi is computationally reliable. 294 sunday et al. / j. nig. soc. phys. sci. 4 (2022) 287–296 295 table 3: juxtaposition of numerical solution for problem 2 at h = 0.1 x yi numsol in fphbi numsol in kshm numsol in ode 15s 0.4 y1 9.8517211401 × 10 −1 9.8517211386 × 10−1 9.8517211213 × 10−1 y2 3.3863953813 × 10 −5 3.3863953789 × 10−5 3.3863953666 × 10−5 y3 1.4794022199 × 10 −2 1.4794022185 × 10−2 1.4794022107 × 10−2 40 y1 7.1582706966 × 10 −1 7.1582706871 × 10−1 7.1582706714 × 10−1 y2 9.1855347724 × 10 −6 9.1855347646 × 10−6 9.1855347554 × 10−6 y3 2.8416375888 × 10 −1 2.8416375746 × 10−1 2.8416375614 × 10−1 4000 y1 1.8320204207 × 10 −1 1.8320204187 × 10−1 1.8320204126 × 10−1 y2 8.9423584099 × 10 −7 8.9423584000 × 10−7 8.9423583987 × 10−7 y3 8.1679706453 × 10 −1 8.1679706389 × 10−1 8.1679706305 × 10−1 figure 6: accuracy (solution) curves for problem 2 table 4: juxtaposition of numerical solution for problem 3 at h = 0.1 x yi numsol in fphbi numsol in ode 15s 1 y1 -1.8650950931 -1.8650950571 y2 0.7524845342 0.7524845299 5 y1 1.8985234584 1.8985234421 y2 -0.7289532571 -0.7289532451 10 y1 1.7865365214 1.7865365103 y2 -0.8156276595 -0.8156276438 20 y1 1.5075643297 1.5075643177 y2 -1.1911230041 -1.1911230003 figure 7: accuracy (solution) curves for problem 3 6. conclusion in this paper, an implicit fphbi was derived for the simulations of first order stiff models. special rigid stiff problems like the pharmacokinetics, robertson and the van der pol models were considered. the results obtained clearly showed that the new integrator is computational reliable, as it performed creditably well. the paper further analysed the integrator on the basis on consistency, zero-stability and convergence. the stability region of the integrator was plotted and the plot generated shows that the integrator is a-stable, thus making it fit for simulating stiff models. finally, the authors are optimistic that this study has added to the collective understanding of the nature of solutions of these stiff models. 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[48] r. fitzhugh, “impulses and physiological state in theoretical models of nerve membrane”, biophysics journal 1 (1961) 445. 296 j. nig. soc. phys. sci. 4 (2022) 820 journal of the nigerian society of physical sciences modeling and forecasting selected meteorological parameters for the environmental awareness in sub-sahel west africa stations f. o. awedaa,∗, j. a. akinpelua, t. k. samsonb, m. sannic, b. s. olatinwoa aphysics and solar energy programme, college of agriculture, engineering and science, bowen university, iwo, osun state, nigeria bstatistics programme, college of agriculture, engineering and science, bowen university, iwo, osun state, nigeria cdepartment of science technology, federal polytechnic, offa, kwara state abstract the monthly air temperature, rainfall, air pressure, and wind speed direction for the environmental time series recorded between january 1, 1980 and december 31, 2020 in six african stations from different climatic zones were modeled and forecasted. in the forecasting, augmented dickey fuller test, arima models, auto correlation function (acf) and partial autocorrelation function (pacf) were used. result showed that in most of the fitted models, the moving average terms both seasonal and nonseasonal were also significant (p < 0.05) indicating that the previous day value of the stochastic term also had a significant effect on the present value of meteorological parameters in the environment. it was observed that in all the fitted models except for wind direction in conakry and rainfall in abidjan have all their autoregressive term of order 1 significant (p < 0.05) which implies that previous day value of these meteorological parameter had a significant effect on the present day value of the parameters. therefore, the forecast model indicates that maximum temperature are expected in february, march, april, and june while minimum temperatures in january, august, december. although, the selected models cannot forecast the precise air temperature, this can also provide information that can be of help to create tactics for appropriate preparation of farming which can be used as tools for effective environmental preparation and policymaking. doi:10.46481/jnsps.2022.820 keywords: meteorological parameters, arima models, acf, pacf, forecast article history: received: 18 may 2022 received in revised form: 06 july 2022 accepted for publication: 25 july 2022 published: 12 august 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: o. j. abimbola 1. introduction the forecasting of future weather sequence variables for environmental awareness is based on the spell of historical categorization, which is known as an essential metric for atmospheric ∗corresponding author tel. no: +234(0)8032566053 email address: aweda.francis@bowen.edu.ng (f. o. aweda ) parameter forecasting and modeling [1-9]. furthermore, agricultural products are subject to meteorological, ecological, and environmental variation [8, 10-14], wherein planetary and chronological qualities have a very important link over the environment by applying time series modeling i.e arima model [5, 15-18]. international heating impacts, according to [19-22], contribute to the development of the atmospheric structure; nevertheless, this effect is on the upcoming alterations in the cli1 aweda et al. / j. nig. soc. phys. sci. 4 (2022) 820 2 matic growth of the environment, where the occurrence and size of diverse happenings affect the growth of any region [6, 23]. according to the international heating effect, an increase in temperature and a decrease in rainfall lead to famine in any environment; nevertheless, this contributes to multiple pressures on food as a result of reduced rainfall and an increase in environmental temperature [17]. the impact of weather quantity forecasting on environmental consciousness is critical for agricultural products employing statistical models for forecasting [4, 6, 24, 25]. statistical time series offered historical information on the modeling of the atmospheric awareness of the atmospheric parameter, according to several studies [6]. various writers have noticed that data series patterns play a role in forecasting based on the environment, which has an impact on the climatological influence of environmental degradation and development [4, 25]. the autoregressive integrated moving average (arima) time series modeling has been found to be a good method for forecasting and predicting some atmospheric characteristics that is widely accepted [7, 24]. furthermore, arima models have been shown to be one of the most widely utilized models for a variety of applications, including science, technology, economy, market, commerce, and industry[4, 6, 7]. arima models, in particular, have been found to be effective in forecasting various weather conditions [26]. [20] revealed linear arima model and quadratic model as the overall best performing model for the prediction of the monthly and annual temperature in libya. following [27], the use of the arima model for 50years (1955 2005) in shiraz, south of iran was found to be a good model in the future forecasting of temperature. moreover, [27, 28] use another model kwon as a seasonal autoregressive integrated moving average (serima) to forecast the mean monthly average of the maximum temperature experienced in the india sub-region. according to the above researchers, their results revealed that maximum temperature forecast was a good parameter for guiding agricultural producers. in another vein, [29] used different arima models and linear trends models to forecast the temperature and precipitation in the afyonkarahisar area in turkey and found that there would be an increase in temperature till the year 2025. according to pieces of literatures, [30] using statistical properties studied the historical effect of temperature in canada for the period 1913-2013 whereby determine a seasonal arima model to forecast forthcoming temperature records. in their research [31] implemented the use of the serima model as proposed by different researchers on the frequency analysis and forecasting monthly rainfall in umuahia. [7, 32, 33] predicted once-a-month rainwater across several areas within ghana using different models such as serima and arima models. weekly and monthly rainfall using time series with the help of serima models over selected weather stations in malaysia were erected by [34] and in india according to [34, 35]. various authors, including ([25, 36-40]), worked on atmospheric and meteorological modeling using various statistical models. the stated arima versions take a good quality postsample predicting performing for yearly and monthly agrometeorological time series [6]. however, for this study trend parameters shall be fitted for the polynomial function, and the period constraint will be projected along with fourier sequences for the environmental awareness. furthermore, the major purpose of this research was to examine statistical modelling of the monthly atmospheric parameters’ perception utilizing time series for the environmental awareness from six distinct stations in sub-sahel african cities over four decades. 2. methodology 2.1. study area this study was conducted for locations in the west african sub-region as shown in figure 1. the selected locations for this research were: daka (17.366 ow, 14.765 o n), conakry (13.578 ow, 09.641 o n), abijan (04.008 ow, 05.360 o n), bamako (08.003 ow, 12.639 o n), niamey (12.125 ow, 13.512 o n), abuja (07.399 ow, 09.077 o n). this study used the data from 1980 to 2020 for all the stations considered. 2.2. data collection and analysis the data for this study was collected from the archive of the modern-era retrospective analysis for research and applications, version 2 (merra-2) web services. the data were accessed on 5th of august, 2021 and the method of data collection followed what was done by [41] as reported by [24, 42, 43]. 2.3. statistical analysis modeling the stationarity of each of these meteorological parameters in each station was tested using the augmented dickey fuller test and p-value less than 0.05 indicates stationarity. the tentative arima models were identified based on autocorrelation function (acf) and partial autocorrelation function (pacf) plots and the estimation of parameters of the arima were facilitated using the statistical package for social sciences (spss version 20.0). after the estimation of the model parameters, model diagnostic checking principally aimed at checking the fitness of the models was determined based on ljung-box test with p-value greater than 0.05 indicating a good fit and hence one year forecast of these meteorological parameters were provided for the different locations. 3. results and discussion 3.1. the variation of the meteorological parameters over the environment the results of the monthly variation of air temperature, relative humidity, pressure, wind speed, wind direction and rainfall for the african stations considered as related to temperature are shown in figures 2, 3, 4, 5, 6, and 7. abidjan figure (aa, ba, ca, da, ea.) revealed that air temperature had a significant effect on the other parameters considered, as shown in figure 2(aa), the temperature got to it minimum t = 24.4 oc in august, while relative humidity got to it minimum rh = 71.9% in january and maximum temperature t = 27.9 oc was recorded in february while the maximum relative humidity 2 aweda et al. / j. nig. soc. phys. sci. 4 (2022) 820 3 table 1. the division of the studied african stations into the hinterland and coastal regions. station division country longitude latitude period of data (ow) (o n) dakar coastal region senegal 17.366 14.765 1980 2020 conakry coastal region guinea 13.578 09.641 1980 2020 abidjan coastal region cote d’ivoire 04.008 05.360 1980 2020 bamako hinterland region mali 08.003 12.639 1980 2020 niamey hinterland region niger 12.125 13.512 1980 2020 abuja hinterland region nigeria 07.399 09.077 1980 2020 figure 1. map of africa showing the stations for the research. rh = 85.2% was recorded in june. figure 2(ba) shows the correlation between temperature and pressure. the results revealed that the maximum temperature t = 27.9 oc was recorded in february while the maximum air pressure p = 1009.9 hpa has been recorded in july. however, the minimum temperature was recorded in august t = 24.4 oc and the minimum air pressure was recorded in february p = 1006.0 hpa. however, the maximum and minimum temperature of the station does not change for the plot across all other parameters. figure 1(ca) showed that wind speed had a minimum value in january (ws = 2.2 m/s), while the maximum value was recorded in july (ws = 3.9 m/s). this correlation shows that, when the temperature is low, there is an increase in the wind speed of the station. figure 2(da) shows that the wind direction got to its minimum in november (w d = 189.1o), this indicates that the direction towards which the wind is moving was low in november, maximum in september (w d = 209.0o). maximum rainfall was recorded in may (rf = 253.8 mm) while the minimum rainfall was recorded in august (rf = 84.0 mm). the result demonstrates that temperature has a significant influence on the other parameters, as the higher the temperature, the greater the variation in the other parameters. this applies to all of the stations in this investigation. figure 3 (ab, bb, cb, db, eb) shows the correlation between temperature and other parameters. the result ab shows that maximum temperature t = 27.7 oc was observed in march, while minimum temperature t = 23.8 oc was observed in august. this shows that august is more humid than other months. however, maximum relative humidity rh = 85.8% was observed in august while minimum relative humidity rh = 43.6% was observed in february. the correlation between the two parameters shows that when the temperature is at maximum, relative humidity at minimum, vice vasa. figure 3(bb) shows the relationship between temperature and air pressure, this revealed 3 aweda et al. / j. nig. soc. phys. sci. 4 (2022) 820 4 figure 2. the environmental variation of atmospheric parameters for abidjan. table 2. summary result for augmented dickey fuller (adf) test for stationarity of the meteorological parameters. station pressure rainfall rh temp wd ws dakar 0.0054* 0.0013* 0.0009* 0.0020* 0.0006* 0.0001* conakry 0.0011* 0.0036* 0.0023* 0.0000** 0.0000* 0.0104* abidjan 0.002* 0.0023* 0.0015* 0.0048* 0.0227* 0.0082* bamako 0.0104* 0.0044* 0.0000** 0.0000** 0.0159* 0.0001* niamey 0.0003* 0.0051* 0.003* 0.0000* 0.0000* 0.0001* abuja 0.0007* 0.0010* 0.0088* 0.0213 0.0001* 0.0001* rhrelative humidity, wdwind direction, wswind speed, values reported are the p-values, **stationary after first differencing, *stationary at level that as the temperature increases air pressure decreases. however, air pressure (p = 961.4 hpa) got to its minimum in march and maximum (p = 964.7 hpa), this shows that air pressure is more pronounced in march at abuja than in other months. the correlation of wind speed and temperature as shown in figure 3(cb), revealed that wind speed was at minimum level in oc4 aweda et al. / j. nig. soc. phys. sci. 4 (2022) 820 5 figure 3. the environmental variation of atmospheric parameters for abuja. tober and maximum in april as shown. more so, for wind direction, the maximum value was recorded in august and the minimum in january. this show that the wind direction is high in august than in other months as revealed by figure 3(db). for figure 3(eb), the rainfall got to its maximum (rf = 556.5 mm) in august with a sharp increase, whereas there was no record of rainfall or zero rainfall in january, november, and december. this was also reported by [31]. these months signify the winter period of the station, where no record of rainfall was observed. the results of abuja revealed that temperature follows a sinusoidal pattern with a minimum in august and a maximum in march. this shows the transition between the winter to the rainy season of the station. the results of the station bamako as presented in figures ac, bc, cc, dc, ec and fc, revealed that temperature has a minimum and maximum values as t = 32.5 oc and t = 23.4 oc 5 aweda et al. / j. nig. soc. phys. sci. 4 (2022) 820 6 figure 4. the environmental variation of atmospheric parameters for bamako. in january and april respectively. this signifies the winter and the summer month for the station. the result also shows that relative humidity has minimum and maximum values as rh = 18.3 % in february and rh = 81.5 % in august, this shows that the water content in the atmosphere is low in february and high in august. thereby, august is classified to be the rainy season for the station. the air pressure was observed to be in the range (965.5 968.4) hpa, this shows that the air pressure was observed to be significant in the station. wind speed was observed to be in the range (0.6 and 3.9) m/s, this shows that 6 aweda et al. / j. nig. soc. phys. sci. 4 (2022) 820 7 figure 5. the environmental variation of atmospheric parameters for conakry. the wind speed for the station is low to power any wind turbine machine in the station. the direction of the wind shows that the wind moves with the value (54.0 240.8) o. rainfall is at maximum of 227 mm in august and a minimum of 0.6 mm in december. this shows that the rate of rainfall in august is as high as the other months. august is observed to be the peak of rainfall in the station, and it signifies the summer month. the station conakry shows that temperature ranges between 7 aweda et al. / j. nig. soc. phys. sci. 4 (2022) 820 8 figure 6. the environmental variation of atmospheric parameters for dakar. t = (25.7 − 27.4) oc, the relative humidity is at its minimum 62.6 % in january and maximum in august 86.5 %, this shows that the water content in the atmosphere is more in august and low in january. the air pressure of the station showed that june 1009.5 hpa had the maximum value as compared to the other months, whereby, march had the minimum 1006.5 hpa. the wind speed and wind direction revealed that december had the minimum values with may and august as the highest. the rainfall of 798.0 mm in july was the highest and 6.6 mm was the lowest in the station. 8 aweda et al. / j. nig. soc. phys. sci. 4 (2022) 820 9 figure 7. the environmental variation of atmospheric parameters for niamey. figure 6(ae, be, ce, de, ee.) revealed that air temperature has a significant effect on other parameters considered, as shown in figure 6(ae), the temperature got to it minimum t = 20.0 oc in january, while relative humidity got to it minimum rh = 12.4% value in april and maximum temperature t = 34.3 oc was recorded in june while maximum relative humidity rh = 41.6% was recorded in september. figure 6(be) shows the correlation between temperature and pressure. the results revealed maximum air pressure p = 980.5 hpa was recorded in january and minimum air pressure was recorded in april p = 974.2 hpa. however, the maximum and minimum temperature of the station does not change for the plot across all other parameters. figure 6(ca) shows that wind speed has a minimum value in june (ws = 12.2 m/s), while the maximum value was recorded in february (ws = 5.2 m/s). this correlation shows that, when the temperature, there is an increase in 9 aweda et al. / j. nig. soc. phys. sci. 4 (2022) 820 10 table 3. summary result for arima models for the different meteorological parameters in the various stations. stations models constant ar1 ar2 ma1 mma2 ma3 sar1 sma1 ljung box test p-value temp ari ma(1,0,1)(1,0,1)12 28.091 0.677 0.406 1.000 0.903 0.288 (0.000)** (0.000)** (0.000)** (0.000)** (0.000)** rh ari ma(1,0,0)(1,0,1)12 25.893 0.622 0.998 0.874 0.507 (0.000)** (0.000)** (0.000)** (0.000)** p ari ma(1,0,1)(1,0,1)12 977.164 0.274 0.999 0.921 0.195 dakar (0.000)** (0.000)** (0.000)** (0.000)** ws ari ma(1,0,1)(1,0,1)12 3.610 1.115 0.999 0.920 0.492 (0.000)** (0.011)* (0.000)** (0.000)** wd ari ma(0,0,0)(1,0,1)12 92.466 0.999 0.898 0.127 (0.000)** (0.000)** (0.000)** rf ari ma(1,0,0)(1,0,1)12 6.140 0.238 0.992 0.857 0.908 (0.103) (0.000)** (0.000)** (0.000)** temp ari ma(1,1,1)(1,0,1)12 0.001 0.599 0.9999 0.9999 0.935 0.691 (0.000)** (0.000)** (0.000)** (0.000)** (0.000)** rh ari ma(1,0,1)(1,0,1)12 1007.848 0.838 0.563 0.999 0.926 0.227 (0.000)** (0.000)** (0.000)** (0.000)** (0.000)** p ari ma(1,0,1)(1,0,1)12 -0.005 0.365 0.053 -0.439 0.990 0.634 conakry (0.651)** (0.008)** (0.718)** (0.000)** (0.004)** ws ari ma(1,0,1)(1,0,1)12 2.533 0.204 0.998 0.845 0.053 (0.000)** (0.000)** (0.000)** (0.000)** wd ari ma(1,0,1)(1,2,1)12 -0.017 -0.229 -0.068 -0.477 0.990 0.634 (0.784) (0.432) (0.818) (0.000)** (0.000)** rf ari ma(1,0,0)(1,0,1)12 289.068 0.119 0.999 0.960 0.430 (0.000)** (0.009)** (0.000)** (0.001)** temp ari ma(1,0,1)(1,0,1)12 26.442 0.620 0.023 0.999 0.895 0.432 (0.000)** (0.000)** (0.765) (0.000)** (0.000)** rh ari ma(1,0,1)(1,0,1)12 1007.461 0.802 0.515 0.999 0.920 0.065 (0.000)** (0.000)** (0.000)** (0.000)** (0.000)** p ari ma(1,0,1)(1,0,1)12 80.66 0.516 0.044 0.995 0.883 0.265 abidjan (0.000)** (0.000)** (0.643) (0.000)** (0.000)** ws ari ma(1,0,1)(1,0,1)12 3.060 0.390 0.100 0.997 0.885 0.750 (0.000)** (0.006)** (0.508) (0.000)** (0.000)** wd ari ma(1,0,1)(1,0,1)12 203.989 0.978 0.918 0.999 0.967 0.145 (0.000)** (0.000)** (0.508) (0.000)** (0.000)** rf ari ma(1,0,0)(1,0,1)12 154.539 0.264 0.033 0.982 0.832 0.145 (0.000)** (0.1620) (0.867) (0.000)** (0.000)** temp ari ma(1,1,1)(1,1,1)12 -0.000286 0.2910 0.924 0.023 0.998 0.143 (0.467) (0.000)** (0.000)** (0.662) (0.158) rh ari ma(1,1,1)(1,1,1)12 0.004 0.5290 0.955 0.139 0.996 0.894 (0.467) (0.000)** (0.000)** (0.008)** (0.000)** p ari ma(1,0,1)(1,0,1)12 967.146 0.7430 0.440 0.999 0.939 0.245 bamako (0.000)** (0.000)** (0.000)** (0.008)** (0.000)** ws ari ma(2,0,3)(1,0,1)12 2.129 1.711 -0.951 1.548 -0.711 -0.084 0.999 0.917 0.052 (0.000)** (0.000)** (0.000)** (0.000)** (0.000)** (0.087) (0.008)** (0.000)** wd ari ma(1,0,0)(1,0,1)12 141.407 0.104 0.995 0.766 0.131 (0.000)** (0.022)** (0.000)** (0.000)** rf ari ma(1,0,0)(1,0,1)12 56.648 0.148 0.999 0.969 0.131 (0.012)** (0.001)** (0.000)** (0.000)** temp ari ma(1,0,1)(1,0,1)12 28.674 0.555 0.254 0.999 0.903 0.511 (0.000)** (0.000)** (0.048)* (0.000)** (0.000)** rh nsm p ari ma(1,0,1)(1,0,1)12 982.790 0.627 0.353 0.999 0.964 0.905 niamey (0.000)** (0.000)** (0.004)** (0.000)** (0.000)** ws ari ma(0,0,0)(1,0,1)12 0.999 0.9160 0.561 (0.000)** (0.000)** wd ari ma(0,0,0)(1,0,1)12 141.247 0.034 0.999 0.983 0.300 (0.000)** (0.460)** (0.000)** (0.000)** rf ari ma(1,0,0)(1,0,1)12 25.059 0.201 0.9930 0.813 0.568 (0.019)* (0.000)** (0.000)** (0.000)** temp ari ma(1,1,1)(1,0,1)12 25.090 0.810 0.217 0.997 0.869 0.187 (0.000)** (0.000)** (0.001)** (0.000)** (0.000)** rh ari ma(1,0,1)(1,0,1)12 68.759 0.710 0.182 0.998 0.885 0.360 (0.000)** (0.016)* (0.000)** (0.000)** (0.000)** p ari ma(1,0,1)(1,0,1)12 963.248 0.770 0.485 0.998 0.900 0.387 abuja (0.000)** (0.000)** (0.000)** (0.000)** (0.000)** ws ari ma(1,0,1)(1,0,1)12 1.498 0.461 0.283 0.999 0.972 0.750 (0.000)** (0.022)* (0.192) (0.000)** (0.000)** wd ari ma(1,0,0)(1,2,1)12 169.613 -0.022 -0.178 0.999 0.931 0.142 (0.000)** (0.000)* (0.000)** (0.000)** (0.000)** rf ari ma(1,0,0)(1,0,1)12 158.924 0.999 0.909 0.182 (0.000)** (0.000)** (0.000)** *significant at 5% (p < 0.05), **significant at 1% ( p < 0.01), nsmno suitable model the wind speed of the station. figure 6(da) shows that the wind direction got to its minimum in february (w d = 31.8o), this indicates that the direction towards which the wind is moving was low in february. however, it got to its maximum in july 10 aweda et al. / j. nig. soc. phys. sci. 4 (2022) 820 11 table 4. summary of forecast of meteorological parameters in 2021. months january february march april may june july august september october november december dakar 20.33 23.64 28.21 31.95 34.21 34.60 32.57 29.92 30.80 29.36 24.78 20.82 conakry 28.23 28.13 27.96 27.99 27.91 27.23 26.45 26.12 26.46 27.11 27.73 27.68 temperature abidjan 27.31 27.86 27.71 27.51 27.06 26.01 24.94 24.63 25.30 26.16 26.92 27.16 bamako 23.91 26.46 29.74 32.27 31.88 29.18 26.45 25.23 25.57 26.54 25.48 23.31 niamey 23.42 26.54 30.50 33.79 34.23 32.11 29.52 27.62 28.07 29.18 26.96 23.93 abuja 25.41 27.67 29.11 28.54 26.92 25.64 24.62 24.15 24.60 25.13 25.31 24.40 dakar 19.33 15.11 12.45 11.72 16.38 26.09 40.40 55.84 45.09 28.65 20.63 22.02 conakry 63.49 65.15 67.34 70.61 77.84 81.63 85.64 86.36 84.90 81.98 74.65 67.44 relative abidjan 75.69 77.61 80.70 82.84 84.22 85.06 84.80 83.46 83.65 83.51 82.04 77.92 humidity bamako 28.73 22.87 23.32 34.08 46.91 64.15 78.73 86.56 84.18 71.25 46.92 35.20 niamey abuja 41.16 40.71 51.24 64.31 76.49 81.21 83.92 85.26 83.98 79.43 61.78 46.55 dakar 980.41 978.62 976.49 974.67 974.69 975.38 976.17 977.12 977.06 977.17 978.65 980.39 conakry 1007.48 1006.82 1006.68 1006.75 1007.79 1009.06 1009.37 1009.15 1008.56 1007.75 1007.29 1007.61 pressure abidjan 1006.70 1006.01 1005.97 1006.12 1007.28 1008.89 1009.50 1009.38 1008.56 1007.47 1006.65 1006.77 bamako 968.63 967.15 965.69 964.76 965.75 967.45 968.06 967.94 967.99 967.37 967.43 968.52 niamey 985.04 983.42 981.33 979.79 980.66 982.48 983.30 983.55 983.46 982.72 983.30 984.76 abuja 963.25 962.08 961.55 961.69 963.03 964.43 964.75 964.60 964.28 963.37 962.84 963.32 dakar 4.98 5.15 4.89 4.04 2.29 1.40 3.42 3.00 1.53 3.01 4.70 5.06 conakry 1.12 2.27 3.29 3.45 3.13 3.11 4.36 4.60 2.93 1.61 0.69 0.87 wind abidjan 2.13 2.98 3.17 2.96 2.89 3.50 3.80 3.87 3.59 3.07 2.36 1.86 speed bamako 3.75 3.61 2.68 1.02 1.79 2.01 1.64 1.47 0.66 0.71 2.27 3.61 niamey 4.13 3.83 2.85 1.38 3.12 3.61 3.15 2.22 1.68 1.41 2.84 3.91 abuja 1.56 1.11 1.24 1.90 1.73 1.68 1.91 1.84 1.09 0.82 1.28 1.75 dakar 34.25 33.26 35.77 52.80 66.87 166.12 217.38 209.63 178.39 67.88 43.54 37.41 conakry 172.18 245.84 250.73 253.05 260.33 243.58 242.35 230.74 249.20 228.12 224.98 288.50 wind abidjan 207.00 213.61 214.48 212.81 206.76 206.54 211.48 213.79 214.00 208.84 194.38 198.62 direction bamako 60.00 57.05 51.42 82.24 191.22 210.36 227.14 242.40 165.71 130.56 67.29 63.67 niamey 54.18 53.37 53.18 165.19 213.17 214.98 217.19 217.75 206.62 162.16 73.92 59.08 abuja 66.78 122.09 193.64 217.06 220.62 222.25 231.96 230.92 192.51 153.75 92.82 71.67 dakar 0.26 0.34 0.47 0.78 3.02 5.61 19.43 49.03 12.15 1.95 0.35 0.38 conakry 10.30 9.09 14.28 29.71 181.35 466.29 799.55 766.82 608.51 426.43 135.00 19.42 rainfall abidjan 71.46 111.98 149.54 147.04 200.59 225.24 127.42 98.04 189.13 247.20 201.00 123.49 bamako 2.83 1.75 3.04 12.42 29.57 58.56 145.64 227.11 135.31 47.73 4.33 1.32 niamey 0.75 5.04 3.41 4.09 15.44 23.05 48.94 142.67 57.58 22.88 0.96 0.92 abuja 2.26 7.53 17.90 71.01 157.21 207.67 407.30 503.16 295.58 129.21 8.07 2.16 (w d = 219.9o). maximum rainfall was recorded in august (rf = 36.4 mm) while zero rainfall was recorded in february (rf = 0.0 mm). the maximum rainfall indicates that there is a low temperature during the period, this signifies increases in rainfall rate due to the low temperature for the station. the result shows that temperature has a significant effect on other parameters because of the more the temperature the higher the variation of other parameters. figure 7 (af, bf, cf, df, ef) shows the correlation between temperature and other parameters. the result af shows that maximum temperature t = 34.1 oc was observed in april, while minimum temperature t = 23.2 oc was observed in january. maximum relative humidity rh = 70.3% was observed in august while minimum relative humidity rh = 18.0% was observed in january. the correlation between the two parameters shows that when the temperature is at maximum, relative humidity at minimum, vice vasa. figure 7(bf) shows the relationship between temperature and air pressure, this revealed that as the temperature increases air pressure increases. however, air pressure (p = 979.7 hpa) got to its minimum in february and maximum (p = 984.9 hpa), this shows that air pressure is more pronounced in january at niamey than in other months. as shown in figure 7(cf), the correlation of wind speed and temperature, revealed that wind speed was at minimum in april and maximum in january. more so, for wind direction, the maximum value recorded was in august and the minimum in march. this show that the wind direction was high in august than in other months as revealed by figure 7(df). for figure 3(ef), the rainfall got to its maximum (rf = 93.1 mm) in august with a sharp increase, whereas there was no record of rainfall or zero rainfall in december. the results of niamey revealed that temperature follows a sinusoidal pattern with the minimum in january and a maximum in april. this shows the transition between the dry to rainy seasons of the station. 3.2. statistical analysis table 2 shows summary result of the stationarity of the series using augmented dickey fuller test for each of the meteorological parameter in the different locations. the result reveals that in all stations, all the parameters were stationary at level with exception of temperature in conakry, rh and temperature in bamako which were only stationary after first differencing meaning that they are integrated of order 1. the estimates of the different autoregressive integrated moving average (arima) models fitted to these meteorological parameters are as shown in table 3. the ljung box test was applied to diagnose the different arima model fitted and the pvalues obtained for all models were greater than 0.05 (p > 0.05) which indicates that these models were of good fit. in all the fitted models except for wind direction in conakry and rainfall in 11 aweda et al. / j. nig. soc. phys. sci. 4 (2022) 820 12 abidjan have all their autoregressive term of order 1 significant ( p < 0.05) which implies that previous day value of these meteorological parameter has a significant effect on the present day value of the parameters. this indicates as the previous day value of these parameters increases significantly, there is a corresponding significant increase in the present day value of the meteorological parameters. result also shows that in most of the fitted models, the moving average terms both seasonal and nonseasonal were also significant (p < 0.05) indicating that the previous day value of the stochastic term also has a significant effect on the present value of meteorological parameters in the study area. since the different arima models were found to be of good fit, these models were used in forecasting the future values of the meteorological parameters in each of the selected locations and the summary result of the forecast are presented in table 4. the forecast indicates the maximum temperature are expected in june in daka while in conakry, abidjan, bamako, niamey and abuja, maximum temperature are expected in january, february, april, may and march respectively. also, minimum temperatures were predicted in daka, conakry, abidjan, bamako, niamey and abuja in the months of january, august, december, january and august respectively. in daka, conakry, bamako and abuja, maximum relative humidity was predicted in august while in abidjan maximum relative humidity was predicted in june. for other these meteorological parameters, the predicted values for 2021 were presented in table 4. 4. conclusion the environment and statistical analysis of the meteorological parameters shows that the monthly mean variations of air temperature and other parameters considered for african stations exhibit similarity behaviour and dynamics, however, some statistical parameters fluctuate significantly between the stations studied. air temperature, rainfall, air pressure, wind speed and direction modelling and its predicting posture a stimulating mission aimed at treatment slightly on monthly time series over the environment. in this study, it was observed that arima model for the environmental study can proficiently get the reason of the increase in the meteorological parameter study by producing the minimum prediction of the root mean squared error which could be better forecast in the long time seasonal time series of the high frequency. for this research, the best model for prediction depends on the stations. the results revealed different time series (arima 101, arima 111, arima 110, arima 011) which are appropriate for the west african continental data sets. however, investigative inspection settles the competence of the models. therefore, the forecast model indicates that maximum temperature are expected in june while minimum temperatures in january, august, december. although, the selected models cannot forecast the precise air temperature, this can also provide information that can be of help to create tactics for appropriate preparation of farming which can be used as tools for effective environmental preparation and policymaking. acknowledgments we thank the referees for the 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[43] s. adebayo, f. o. aweda, i. a. ojedokun & o. t. olapade, ”refractive index perception and prediction of radio wave through recursive neural networks using meteorological data parameters”, 35 (2022) 810. 13 j. nig. soc. phys. sci. 4 (2022) 954 journal of the nigerian society of physical sciences theoretical air requirement and combustion flue gases analysis for indigenous biomass combustion m. a. lalaa,∗, o. a. adesinaa, o. j. odejobib, j.a. sonibareb adepartment of chemical and petroleum engineering, afe babalola university, ado-ekiti, nigeria bdepartment of chemical engineering obafemi awolowo university, nigeria abstract rice-husk is one of the most abundant and commonly utilized biomass. it can be used directly to generate primary heating agent in combustion plants. however, emission characteristic of biomass usually vary depending on factors such as the variety of the parent plant, region of cultivation and climatic condition. moreover, gases emission from biomass combustion could be detrimental to health; hence there is a need for information on the theoretical volume of air and flue gases required for optimal operation of combustion equipment. thus, this work analyzed the theoretical combustion air requirement and the resulting flue gas volume of the abundant nerica-rice-husk variety. rice-husk samples were collected from southwestern part of nigeria. the samples were characterised for their ultimate and proximate content. afterward, the combustion analysis was carried out using the elemental composition obtained from the ultimate analysis to determine the theoretical combustion air requirement and the resulting flue gas volume. the proximate analysis result showed the moisture content level, fixed carbon content and ash content value of 10.93%, 12.59% and 18.26% respectively for sample a and 9.80%, 20.76% and 14.81% for sample b. this reveals that nerica-rice-husk is advisable for use as an alternative energy source in a combustion process. the calculated theoretical volume of air for sample a was 3.69596 (n m3/kg fuel), while 3.53947 (n m3/kg fuel) was obtained for sample b. this signifies a slight different between the combustion properties of rice-husk of same variety. it also showed that more quantity of theoretical air will be required to burn sample a compared to sample b, and that sample a will generate little more emissions compared to sample b. hence, this study reveals the volume of air required for low emission combustion of nerica-rice-husk in combustion plants. doi:10.46481/jnsps.2022.954 keywords: rice-husk, biomass combustion, theoretical combustion air, ultimate analysis, proximate analysis article history : received: 25 july 2022 received in revised form: 16 august 2022 accepted for publication: 17 august 2022 published: 19 september 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: e. e. anand 1. introduction national ambient air quality standards (naaqs) by government regulatory agencies are available for the allowable concentration of pollution in the air. these standards apply to ∗corresponding author tel. no: +234 7031549936; +234 8163280469 email address: lalam@abuad.edu.ng (m. a. lala) emissions from combustion reactors such as fireplace, stoves, furnaces, woodchip and pellet furnace, furnaces with boilers, industrial combustion systems and many others [1]. biomass such as rice-husk is a good alternative source of energy which is now mostly used in combustion plants to generate primary heating agent in boiler [2]. emission of more toxic pollutants from the combustion of fossil fuels together with the abundant nature of biomass has switched people attention to burning cleaner 1 lala et al. / j. nig. soc. phys. sci. 4 (2022) 954 2 and renewable biomass [3]. nevertheless, analysis of flue gases from biomass-fired combustion reactors is highly important in ensuring proper compliance of combustion operation with standards set by the environmental pollution regulatory agencies [2]. operations where the volume of the emitting flue gases is resulting in high pollutant concentration can be turn off or modified since it could be detrimental to people within the proximity of the emission. excessive flue gas can also indicate a need for maintenance on the combustion equipment to reduce the emission of toxic gases. the presence of gaseous emission such as carbon monoxide (co) signifies incomplete combustion and the process can be modified to operate more effectively [1]. combustion analysis which involves the determination and monitoring of flue gases from solid fuels combustion system is extremely significant in designing and operating combustion equipment. it also ensures safe operating procedure and maximum combustion efficiency [4]. therefore, this work analyzed the theoretical combustion air requirement and the resulting flue gas volume for the abundant and indigenous rice-husk as an alternative energy source. the rice-husk samples were collected from two different geographical locations in southwestern nigeria. the samples were characterised for their elemental composition, energy content and proximate content (moisture, volatile, ash, fixed carbon and total carbon content) prior to the combustion analysis. characteristic properties of biomass usually vary depending on factors such as the variety of the parent plant, region of cultivation, climatic condition and postharvest processing method. the proximate analysis gives the moisture, ash, volatile and fixed carbon content; ultimate analysis provide information (kg component/ kg fuel) on carbon (%c), hydrogen (%h), nitrogen (%n), sulphur (%s), oxygen (%o) and other elementary chemical composition, and; calorimetry presents the energy content or the heating value of the fuel. the heating value measured in (j/g) is the heat released by the complete combustion of a unit mass of fuel under normal physical conditions, p0 = 1.013 bar and t = 00c. heating value is either high heating value (hhv) or low heating value (lhv) depending on the amount of water vapour available in the combustion products. the water vapour is a result of hydrogen oxidation, fuel moisture content and combustion air moisture content. all these characteristic properties determine the combustion characteristic of biomass. similarly, factors such as the origin of biomass, variety of the parent plant, region of cultivation and climatic condition also affect biomass compositions [5-7]. the use of agricultural waste biomass as fuel has been broadly studied by many researchers [5]. rice-husk is one of the most abundant and commonly utilized biomass and can be classified as combustible solid waste [5, 6]. during combustion process, variations in the composition of biomass will influence the volume of air required for burning and the volume of the emitted flue gases. hence, it is important to know the optimum quantity of air that will be needed for maximum combustion efficiency, thus permitting appropriate control of the combustion process. this will culminate in optimum functioning of the equipment, reduce emissions and reduce heat losses [4]. combustion is a form of chemical reaction process during which combustible elements in fuel react with oxygen in the air to produce large quantity of heat energy. heat has various applications in industrial processes such as; expansion of gases in cylinder and piston push, environmental heating, boilers, furnaces, power plants and many other systems [4, 7, 8]. fuel can be fossil or organic and must be able to produce a reasonable amount of heat during combustion with oxygen in the air. a good fuel, under normal condition, will react exothermically with oxygen at high speed and temperature to produce environmentally friendly and non-corrosive flue gases. also, such fuel must be cheap and abundant in nature. biomasses are good alternative source of energy which can burn readily in combustion plants to generate primary heating agent in boiler [2]. burning of biomass and determination of the theoretical combustion requirements necessitate analysis of the biomass to establish: • the combustible mass, • ballast, that does not participate in combustion, and • the energy content of the fuel. in industrial boiler and many other industrial and household applications, the combustion process involves the reaction between the two major combustion reactants (fuel and air) in a combustion chamber to produce new chemical substances called combustion gases [9]. in most combustion processes, air is the most common and readily available oxidizing agent because it contains free and cheap oxygen while metallurgy and some other specialized system use high-purity oxygen. the oxy-fuel combustion technology is suitable for co2 emission reduction [4, 10-12]. air ratio or excess air coefficient (γ) indicates the nature of the combustion and this determine the composition of the combustion flue gases [4]: • γ < 1, indicates incomplete combustion and the products may contain co, co2, so2, h2o and n2. • γ = 1, indicates theoretical or stoichiometric combustion and the products may contain co2, so2, h2o and n2. • γ > 1, indicates excess air combustion and the products may contain co2, so2, h2o, n2 and o2. 1.1. theoretical combustion theoretical or stoichiometric air is the minimum amount of air necessary for the complete combustion of a fuel. this result in no uncombined oxygen in the flue gas and no combustible chemical element such as soot or carbon particles especially for solid fuel. complete combustion is otherwise called theoretical or stoichiometric combustion. excess combustion occurs when the amount of air supplied for combustion is more than the theoretical amount needed. a combustion process is termed incomplete when the amount of air or oxygen supplied is less than the theoretical requirement, or when there is an inadequate degree of mixing of sufficient or even excess oxygen. incomplete combustion can either be mechanical or chemical. mechanical incomplete combustion is identified by the presence of 2 lala et al. / j. nig. soc. phys. sci. 4 (2022) 954 3 combustible elements such as soot or carbon in the combustion gases while chemical incomplete combustion contains only the flue gases [4]. combustion process calculation is mainly aimed at determining: • the thermal energy resulting from the combustion process, • the volume of air required for the chemical reactions of the combustible elements in fuel, and • the volume of flue gas resulting from the combustion process. monitoring the volume of air supplied for combustion is highly important. it helps in ensuring that the process is operating with a sufficient quantity of air. insufficient air will lead to incomplete combustion and unregulated excess air will decrease combustion temperature. this will increase the resulting flue gas volume and then culminates into heat loss [13]. moreover, monitoring the volume of flue gas will assist in the [4]: • design and appropriate sizing of the exhaust pipes, • design and appropriate sizing of the chimney, and • heat recovery system. in this work, the combustion air volume and the resulting flue gas volume for a unit mass of rice-husk obtained from two different locations in southwestern nigeria were determined. this was done by assuming complete combustion and without taking the complex exothermic oxidation of the fuel into consideration. combustion analysis of the rice-husk samples was achieved using the formulations described under theoretical calculations. these formulations are developed into a web application by paraschiva et al. [4, 14]. the method is simple and straightforward and also avoids cumbersome calculation that may bring errors. 1.2. rice-husk rice-husk also known as rice hull, can be described as the coating on a grain of rice. it protects the rice grain or seed during the growing season and is formed from materials, such as silica and lignin. rice-husk is an agricultural by-product of rice-milling industry and belongs to the category of biomass/bioresource material. each kilogram of milled rice paddy results in roughly 0.28 kg of rice-husk as a by-product. this is about 20-25% of the rice paddy. singh et al., and muhammad et al., reported rice to husk ratio to be 0.20 [15, 16]. therefore, abundance of rice-husk is a function of rice production and the proportion of husk in a paddy. husk generation rate in nigeria is high because rice (oryza sativa) is one of the most important staple foods in the country [17]. about 70% of nigerians feed directly on rice and about 30% feeds on cereal-based foods which are derived from rice [18]. in west africa, nigeria is ranked highest as both the producer and consumer of rice [19, 20]. statistic shows that there is a continuous increase in rice production in nigeria, especially from the year 2011 due to the government agricultural reform program put in place. this includes agricultural transformation agenda (ata) and agriculture promotion policy (app) [21]. according to gems4/coffey international development ltd report [21], rice is produced in almost all the 36 states of nigeria and also in the federal capital territory (fct), abuja. but majorly, only 18 states represent cumulatively about 80% of the aggregate domestic paddy output in the country [21]. among these 18 states, preference is given to ekiti and ogun state because of their local rice varieties such as ofada rice. figure 1 shows the continuous increase in the production of rice in nigeria from the year 2011. rice-husk has numerous uses and among many others, it is a lignocellulosics biomass. it can be used directly in its raw form or converted either into briquette, liquid fuel, tar, and ash before use [22]. utilization of rice-husk as a feedstock in a biomass combustion system requires prior ultimate, proximate and energy content analysis. that is, in other to optimize the combustion process in adequate reactors, a comprehensive study of the characterization of biomass fuel properties is needed because it remains one of the factors that heavily influence the decomposition of biomass under oxidative condition. 2. materials and methods 2.1. materials and instrumentations two (2) rice-husk samples of the same paddy variety but from different geographical locations were considered for this research work. sample a was collected from a local rice mill in igbemo town, ekiti state and sample b was obtained from another local rice mill in ofada town, ogun state. both states located in southwestern nigeria. the rice-husks considered are from nerica variety. the test materials and instrumentations include; bomb calorimeter, oven; muffle furnace; gas desiccator; digital weighing balance; cal 2k-eco calorimeter assembly, pulveriser with shaker, and atomic absorption spectrometer (aas). 2.2. proximate analysis of rice-husk the rice-husk samples were analyzed for their proximate moisture, volatile, ash, fixed carbon and total carbon content, using american standard testing methods (astm) d 3173-87 [1, 20, 21]. the percentage of total carbon in the samples was determined directly by adding the volatile matter and the fixed carbon content. 2.3. chemical analysis of rice-husk the chemical analysis considered for this research work is the ultimate analysis. this determines the elemental composition of c (carbon), h (hydrogen), n (nitrogen), s (sulphur) and o (oxygen) in the rice-husk samples (chnso test). the information obtained was sufficient to determine the combustion air volume and the resulting flue gas volume for a unit mass of rice-husk. 3 lala et al. / j. nig. soc. phys. sci. 4 (2022) 954 4 figure 1. rice production trends in nigeria from 1960-2018 1. ultimate analysis: ultimate analysis was carried out to determine the percentage of c (carbon), h (hydrogen), o (oxygen), n (nitrogen), and s (sulphur) in the rice-husk samples. i determination of carbon and hydrogen: the percentage composition of carbon and hydrogen in the rice-husk samples were determined using astm d: 5868-10a procedure [23, 24]. here, a known quantity of rice-husk sample was burnt in the presence of dry oxygen to ensure that carbon and hydrogen content in the biomass was converted into carbon (iv) oxide (co2) and water (h2o) respectively. combustion gas from the combustion process was passed over a weighed tubes of anhydrous calcium chloride (cacl2) and potassium hydroxide (koh), which absorb the h2o and co2 present in the combustion gas respectively. to this effect, the weight of water (h2o) and carbon (iv) oxide (co2) present in the combustion gas was determined as the increase in weight of cacl2 and koh in the tube respectively. ii determination of nitrogen: the percentage composition of nitrogen in the rice-husk samples was determined using kjeldahl’s method [23, 24]. here, a known quantity of pulverized rice-husk sample was heated together with concentrated tetraoxosulphate (vi) acid (h2so4) in a kjeldahl’s flask, in the presence of potassium tetraoxosulphate (vi) salt solution (k2so4) and copper (ii) tetraoxosulphate (vi) salt solution (cuso4). the reaction converted the nitrogen present in the rice-husk sample to ammonium sulphate and the solution became clear after the entire nitrogen was converted into ammonium sulphate. this clear solution was then treated with 50% naoh solution to form ammonia which was distilled and absorbed over a given quantity of standard h2so4 solution. unused volume of h2so4 acid was determined by titrating against standard naoh solution. this gives the amount of acid neutralized by liberated ammonia in the rice-husk sample. thus the percentage of nitrogen was determined. iii determination of sulphur: the percentage composition of sulphur in the rice-husk samples was determined using astm d: 5868-10a procedure [23]. here, a known quantity of rice-husk sample was heated with eschka mixture, a combination of 2 parts of magnesium oxide (mgo) and 1 part of anhydrous sodium trioxocarbonate (iv) (na2co3) at 8000c. this formed a sulphate solution and later precipitates barium tetraoxosulphate (vi) salt (baso4) after treating with bacl2. iv determination of oxygen: the percentage of oxygen in the sample was calculated indirectly by subtracting the sum of the percentage content of carbon, hydrogen, nitrogen and sulphur and from 100%. this is calculated without considering the percentage composition of ash and is similar to the procedure used by ghetti’s [7, 25]. 2. high heating value of rice-husk energy content of each sample of rice hush was determined by bomb calorimeter (cal 2k-eco calorimeter), 4 lala et al. / j. nig. soc. phys. sci. 4 (2022) 954 5 in afe babalola university, ado-ekiti laboratory (abuad) as described by rominiyi and adaramola, 2020. here, 0.5g of dry pulverized rice-husk sample was measured into a clean crucible whose weight has been corrected to zero on the digital weighing balance. and just before the determination of the energy content by the calorimeter, the identification and corresponding weight of the sample were entered into the calorimeter through the connected keyboard. to determine the energy content, a precut of firing cotton was looped over the firing wire of the setup in such a way that the firing wire touches the weighed crucible (containing the sample) placed in the crucible holder. the lid assembly was inserted into the vessel body and the cap down screwed until it touches the top of the lid. this vessel was then placed in the vessel holder in an upright position together with the filling station and filled with oxygen to 3000kpa. after this, the entire vessel was inserted into the measuring chamber and then the lid was closed. this was then followed by the temperature stabilization phase which was carried out for about 10 minutes until the vessel fires automatically at the initial condition of 220 v. the heating value was calculated automatically every 6 seconds taking into account the calibration curve, heating value corrections and sample mass. this also takes place for 10 minutes. finally, the heating value was displayed on the screen of the calorimeter and taken. 2.4. theoretical calculations stoichiometry combustion air volume and the resulting flue gas volume for a unit mass of rice-husk were calculated according to the procedures given by paraschiva et al. [4]. these procedures have been developed into a user-friendly web application for easy analysis of the combustion fuel [14]. utilization of this web application requires knowledge of the analyzed fuel elemental composition, excess air coefficient, absolute humidity of air and the solid fuel flow rate. 1. stoichiometric combustion air i stoichiometric volume of oxygen: stoichiometric volume of oxygen required for combustion when the minimum quantity of air required for combustion is used (γ = 1) is given as equation 1: v so2 = 22.41 100 ( %c 12 + %h 4 + %s − %o 32 ) ( nm3o2 kg fuel ) (1) ii stoichiometric oxygen flow rate: equation 2 describes the stoichiometric flow rate of oxygen required for combustion qso2 = • mrh · v s o2 ( nm3o2 h ) (2) iii stoichiometric volume of dry air: stoichiometric volume of dry air required to burn one kilogram of fuel with air having 21% oxygen volume participation is given as equation 3: v sair = v so2 0.21 ( nm3air kg fuel ) (3) iv dry combustion air flow rate: equation 4 presents the flow rate of dry combustion air. qsair = • mrh · v s air ( nm3air h ) (4) the correlation given in equation 1 to 4 is applicable only to dry air. the stoichiometric air volume for wet air is greater than the stoichiometric air volume for dry air due to water vapour content [1]. therefore, the correlation for wet air involves taking into consideration, absolute humidity, x (kg h2o/kg dry air) of the air; air temperature, tair (oc) and the relative humidity of air, φair (%). the absolute humidity, x is usually giving as 0.01 kg h2o/kg for combustion calculation and this corresponds to air at tair =25 oc and φair =50% according to paraschiva et al.. but for more accuracy, the absolute humidity can be obtained at known valves of tair andφair , using molliere diagram for moist air. the more accurate absolute humidity can be calculated using equation 5 and the volume of water vapour in wet air can be determined using equation 6. x = 0.622φair ps (pb −φair ps) ( kg h2o kg dry air ) (5) v sh2 o = xρair v sair ρh2 o ( nm3 h2o kg fuel ) (6) v stoichiometric volume of wet air: equation 7 gives the stoichiometric volume of wet air required for combustion. v sairw = v sair + xρair vair ρh2 o = (1 + xρair ) ρh2 o · v sair = (1 + 1.61x) · v sair ( nm3 kg fuel ) (7) vi wet combustion air flow rate: the required wet combustion air flow rate for complete combustion can be calculated using equation 8. qsairw = • mrh · v s airw ( nm3airw h ) (8) where • mrh = rice-husk (rh) mass flow rate ( kg fuel h ) %c, %h, %s and %o= elemental composition of rh (ultimate analysis). ρair = density of air, usually 1.2925 kg/m3 at normal state. ρh2 o = density of water vapour, usually 0.804 kg/m 3 at normal state. 5 lala et al. / j. nig. soc. phys. sci. 4 (2022) 954 6 2. stoichiometric combustion products the following minimum volume of combustion gases will be obtained when the minimum quantity of air required for combustion ′γ = 1′is used. i volume of carbon dioxide in flue gas: the volume of carbon dioxide obtainable when minimum quality of dry air requirement is used is given by equation 9. vco2 = 22.41 12 · %c 100 ( nm3co2 kg fuel ) (9) ii volume of sulphur dioxide in flue gas: the volume of sulphur dioxide obtainable when minimum quality of dry air requirement is used is given by equation 10. vs o2 = 22.41 32 · %s 100 ( nm3s o2 kg fuel ) (10) iii volume of water vapour: the volume of water vapour in the flue gases is the summation of the volume of water vapour from combustion hydrogen and volume of water vapour due to combustion air humidity. the dew point temperature of cooled flue gases depends on the volume of water vapour present in the flue gases. • volume of water vapour from combustion hydrogen: v ′ h2 o = 22.41 100 · ( %h 2 + %w 18 ) ( nm3 h2o kg fuel ) (11) • volume of water vapour due to combustion air humidity: v ′′ h2 o = 1.61 · x · v s air ( nm3 h2o kg fuel ) (12) therefore, the total volume of water vapour in flue gas: v sh2o = v ′ h2 o + v ′′ h2 o = 22.41 100 · ( %h 2 + %w 18 ) ( nm3 h2o kg fuel ) (13) iv volume of nitrogen in flue gas: the volume of nitrogen obtainable from minimum quality of dry air and wet air requirement is given by equation 14 and equation 15 respectively. v sn2 = 22.41 28 · %n 100 + 0.79 · v sair ( nm3 n2 dry kg fuel ) (14) v sn2 wet = 22.41 28 · %n 100 + 0.79 · v sairw ( nm3 n2 wet kg fuel ) (15) nitrogen in flue gas is a result of the elemental flue composition of nitrogen and also the 79% nitrogen composition in the combustion air. v total stoichiometric volume of flue gas: equation 16 presents the total stoichiometric volume of flue gas obtainable from dry stoichiometric combustion air while equation 17 present total stoichiometric volume of flue gas obtainable from wet stoichiometric combustion air. v sf g = vco2 + vs o2 + v s n2 ( nm3dry fuel gas kg fuel ) v sf g = 22.41 100 · ( %c 12 + %s 32 + %n 28 ) + 0.79 · v sair ( nm3dry fuel gas kg fuel ) (16) v sf gw = vco2 + vs o2 + v ′ h2 o + v sn2 wet ( nm3wet fuel gas kg fuel ) v sf gw = v s f luegas + v s h2 o ( nm3fuel gas kg fuel ) (17) vi flue gas flow rate: the total wet flue gas flow rate is given by equation 18. q f gw = • mrh · v f gw ( nm3fuel gas h ) (18) this research work used paraschiva et al web application [14] to calculate the theoretical volume of air required for the chemical reactions of the combustible elements in the sampled rice-husk. also, the application was used to evaluate the volume of flue gas that will result from the combustion process. the elemental composition of the solid fuel samples was obtained from the ultimate analysis, excess air coefficient ‘γ = 1’ was considered for theoretical or stoichiometric combustion and rice-husk flow rate of 1 kg/hr was used as the calculation basis. the list of all parameters is given in appendix a. 3. results and discussion 3.1. proximate analysis moisture content of biomass is expected to be below 10% for a pre-dried sample and about 50% if not dried [7, 14]. moreover, the volatile matter is expected to be between 65% and 85%, fixed carbon from 7% to 20% and ash content from below 5% to 20% [7, 26]. the details in table 1 shows that the moisture content of the studied samples is almost 10% which is considered suitable for biomass combustion. sample b has 9.8% moisture content while sample b has a moisture level that is slightly above 10%. this potential biofuel can be considered 6 lala et al. / j. nig. soc. phys. sci. 4 (2022) 954 7 suitable for use in biomass boilers because of the low moisture levels. furthermore, from the analysis result shown in table 1, sample a and sample b has 18.26% and 14.81% ash content which are within the recommended range. these low values indicated that at normal combustion conditions the level of particulate matter emission will be low or negligible and the emission from sample b is expected to be slightly below that of sample a [7]. the values obtained for volatile matter, 58.22% and 54.63% for sample a and b respectively are slightly below the range given by so researchers, therefore their reactivity may not be as high as biomass with a volatile matter in the range of 65% to 85% [7, 26]. the higher the volatile matter content the higher the reactivity of biomass [7, 27]. the fixed carbon content of the samples was determined empirically and not experimentally mainly because this study focused more on the elemental composition for the combustion gases analysis. the values obtained for sample a and b as shown in table 1 is 12.59% and 20.76%. biomass with such fixed carbon value is advisable for use in combustion process [4]. the moisture content values and the low fixed carbon values obtained indicate that the level of cxhy emission will be low and will be generated at a trivial level during theoretical combustion, under proper conditions [7]. 3.2. ultimate analysis results obtained for ultimate analysis of the rice-husk samples are shown in table 2. the major components in solid fuels combustion are carbon, nitrogen and oxygen. carbon and oxygen react exothermically to generate co2 and h2o, and they also contribute positively to the hhv of fuel [7]. according to studies [7, 26], carbon, hydrogen, oxygen and nitrogen content is expected to range from 47% to 54%, 5.6% to 7%, 40% to 44% and 0.1% to 0.5% respectively. sulphur is expected to have a value of about 0.1%. as shown in table 2, carbon content is 45.08% and 43.99% for sample a and b respectively. these values are approaching the expected value of between 45% and 48% given in the literature. also, the value obtained for oxygen is 49.27% and 50.64% for sample a and b respectively. these are slightly above the expected value of 40% to 44%. the values of carbon and oxygen obtained for the characterised rice-husk samples are in contrast with the values obtained by garcı́a et al [7]. garcı́a et al have carbon and oxygen content to be 26.69% and 70.05% which indicate poor energy density, thus showing that biomass characteristic properties vary depending on factors such as the variety of the parent plant, region of cultivation, climatic condition and postharvest processing method. nitrogen content (0.84%) in sample a presented in table 2 is slightly higher than the recommended value of nitrogen in biomass [26], while sample b is within the recommended value of 0.1% to 0.5%. but as the case may be, both analyzed samples are expected to produce a very little or negligible amount of n2 and nox emissions during the combustion process. the majority of the nox in waste gases can be attributed to nitrogen present in the combustion air [7]. a low level of sulphur content is highly recommended for biomass combustion because they generate so2 which leads to sulphates. sulphates constitute great problem in heat exchangers. the result presented in table 2 showed little and negligible amount of sulphur in the samples, thus making the samples suitable as biofuel. hydrogen content of both samples as presented in table 2 slightly agrees with literature. reports from studies show that hydrogen content of biomass is expected to be between 5.6% and 7% [7, 26]. the energy content of the two samples considered is also similar to those in literature [7]. hhv of sample a, 14.112 mj/kg is slightly higher than hhv of sample b, 13.326 mj/kg. the low hydrogen content of sample a (4.76%) and sample b (4.93%) compared with the higher carbon content of the samples (45.08 for sample a and 43.99 for sample b) suggested that carbon contributes more to rice-husk hhv. 3.3. theoretical calculations the theoretical volume of air needed for the chemical reactions of the combustible elements in samples of rice-husk, and also the volume of flue gas that will be produced from the combustion process were calculated using paraschiva et al web application [14]. excess air coefficient of ′γ′ = 1 and 1 kg/hr biofuel flow rate were considered for the calculation. the results obtained are shown in table 3. the stoichiometric volume of oxygen required for sample a is slightly greater than sample b. sample a requires 0.76385 (n m3/kg fuel) while sample b requires 0.74329 (n m3/kg fuel). the contrast is due to the slight variation in the elemental composition of the two samples, and this variation may be as a result of the difference in the area of cultivation and postharvest processing method. the contrast is slight because both samples are of the same variety and are both from southwestern part of nigeria. the trend observed in the stoichiometric volume of oxygen required for the samples is also noticed for the stoichiometric combustion air required and wet combustion air flow rate. this is because stoichiometric combustion air required and wet combustion air flow rate both depend on the stoichiometric volume of oxygen. as presented in table 3, stoichiometric volume of dry air, stoichiometric volume of wet air and wet combustion air flow rate are 3.63740 (n m3/kg fuel), 3.69596 (n m3/kg fuel) and 3.69596 (n m3/h) respectively for sample a, and 3.53947 (n m3/kg fuel), 3.59645 (n m3/kg fuel) and 3.59645 (n m3/h) for sample b. the difference between the stoichiometric volume of dry air and the stoichiometric volume of wet air accounts for the water vapour present in the combustion air. the combustion air flow rate is for a unit mass of fuel per hour. analyzed results of the major stoichiometric combustion products are given in table 3. carbon dioxide is majorly from the fuel carbon, and the volumes evaluated are 0.84187 (n m3/kg fuel) and 0.82151 (n m3/kg fuel) for sample a and b respectively. the volume of sulphur dioxide in the flue gas estimated for both samples are very small, 0.00035 (n m3/kg fuel) for sample a and 0.00021 (n m3/kg fuel) for sample b. these quantities are proportional to the level of sulphur in the fuel. 7 lala et al. / j. nig. soc. phys. sci. 4 (2022) 954 8 table 1. proximate analysis of rice-husk (dry weight basis %) rice-husk moisture content volatile matter ash content fixed carbon total carbon sample a 10.93 58.22 18.26 12.59 70.81 sample b 9.80 54.63 14.81 20.76 75.39 table 2. ultimate and calorific value of rice-husk (dry weight basis %) rice-husk c h n s oa hhv (mj/kg) sample a 45.08 4.76 0.84 0.05 49.27 14.112 sample b 43.99 4.93 0.41 0.03 50.64 13.326 a % of o calculated from the difference of c, h, n and s. table 3. rice-husk combustion analysis rice husk stoichiometric combustion air a,b, c (n m3/kg fuel) stoichiometric combustion products a,b, c (n m3/kg fuel) qso2 v s air v s airw q s airw (n m3/h) vco2 vs o2 v s h2o v sn2 v s f g v s f gw q f gw (n m3/h) sample a 0.76385 3.63740 3.69596 3.69596 0.84187 0.00035 0.59192 2.88027 4.31441 4.31441 4.31441 sample b 0.74329 3.53947 3.59645 3.59645 0.82151 0.00021 0.60939 2.79946 4.23058 4.23058 4.23058 aγ = 1, b • mrh = 1 kg/hr, c x = 0.01 water vapour and nitrogen are inert gas. water vapour volume is 0.59192 (n m3/kg fuel) and 0.60939 (n m3/kg fuel) for sample a and b respectively. nitrogen gas is 2.88027 (n m3/kg fuel) and 2.79946 (n m3/kg fuel) for sample a and b respectively. it is noticed that the volume of nitrogen in the flue gas is greatly influenced by the 79% composition of nitrogen in the combustion air. finally, the stoichiometry volume of flue gas and the flue gas flow rate is proportional to the stoichiometry volume of wet air and the wet combustion air flow rate. therefore as presented in table 3, the stoichiometry volume of flue gas for sample a is 4.31441 (n m3/kg fuel) and its flue gas flow rate is 4.31441 (n m3/h), while and sample b has stoichiometry volume of flue gas of (n m3/kg fuel) and flue gas flow rate of 23058 (n m3/h). 4. conclusion rice-husks of one of the most commonly planted and abundant rice variety in nigeria (nerica-rice-husk) were characterised and evaluated for theoretical combustion air requirement and potential combustion flue gases. the sampled husks were obtained from two different geographical locations within the southwestern part of nigeria. the characterization gives insight into the elemental composition and the energy content of the samples considered. also, the proximate moisture, volatile, ash, fixed carbon and total carbon content of the samples were determined using standard methods. it was revealed that the difference in the region of cultivation slightly influenced the characteristic properties of the rice-husk. the low proximate moisture content level of the studied samples which is almost 10% indicated that nerica-rice-husk is suitable for use in biomass combustion chambers. ash content value of 18.26% and 14.81% obtained for sample a and sample b respectively showed that at proper combustion condition the level of particulate emission will be low or negligible. the low fixed carbon content of 12.59% and 20.76% obtained for sample a and b respectively also indicated that nerica-rice-husk is advisable for use in the combustion process. the proximate volatile matter of the two nerica-rice-husk samples slightly fall below the recommended range of 65% to 85%, therefore the reactivity of the biomass may not be very high during combustion process. the characterization revealed low level of nitrogen and sulphur in both samples, thus at normal combustion conditions the level of nox and so2 will be low. moreover, there will be a trivial level of cxhy emission during proper theoretical combustion of nerica-rice-husk. therefore, effective utilization of the abundant nerica-rice-husk as biofuel will help reduce our dependence on fossil fuels. results obtained for the volume of air required for the chemical reactions of the combustible elements in the samples are slightly different. sample a requires 0.76385 (n m3/kg fuel) and 3.69596 (n m3/kg fuel) stoichiometric oxygen and air respectively while sample b requires 0.74329 (n m3/kg) and 3.53947 (n m3/kg fuel). this showed that more quantity and higher flow rate of theoretical air will be required to burn sample a in combustion reactors such as in steam boiler compared to sample b. the result of the flue gas analysis shows that combustion of sample b will generate less emission compared to sample a. also, at theoretical combustion of nerica-rice-husk, the ratio and the amount of emitted sulphur dioxide in the flue gas will be negligible due to low sulphur content of the husk. also, the ratio and the amount of nitrogen dioxide in the flue gas will be high. the high nature of nitrogen dioxide, 2.88027 (n m3/kg fuel) and 2.79946 (n m3/kg fuel) for sample a and 8 lala et al. / j. nig. soc. phys. sci. 4 (2022) 954 9 b are due to the inert nitrogen present in the combustion air. moreover, the combustion of sample a will emit more nitrogen dioxide due to the 0.84% nitrogen content which is more than the recommended nitrogen content for biomass. this research work indicates a better way of managing the abundant nerica-rice-husk as an alternative energy source. it can be burn readily in combustion plants to generate primary heating agent. the discrepancies noticed in the results of the two samples considered in this work simply illustrate potential intrinsic risk that is associated with using generally assumed values for the design and sizing of combustion reactors. such can lead to high calculation errors, thus creating financial and infrastructural problems. also, the results obtained for the volume of air required for the chemical reactions of the combustible elements in nericarice-husk together with the volume of flue gas resulting from the combustion process will assist in investigating the operating conditions for minimizing air emissions from rice-husk fired combustion reactors. this will ensure proper compliance of the combustion operation with the standards set by the environmental pollution regulatory agencies. it will also assist in reducing thermal losses from any rice-husk fired combustion system, as theoretical combustion provides the perfect fuel to air ratio, which lower losses and extracts all the energy from a given fuel. in this work, stoichiometry combustion air volume and the resulting flue gas volume for a unit mass of nerica rice-husk variety were determined by assuming complete combustion and without taking the complex exothermic oxidation of the fuel into consideration. this can be improved upon in future works by considering the exothermic oxidation of the fuel. also, a study on excess air combustion analysis of rice-husk which is highly important in attaining better combustion efficiency is recommended. acknowledgment we thank the referees for the positive enlightening comments and suggestions, which have greatly helped us in making improvements to this paper. references [1] fsa-1056, “biomass combustion, agriculture and natural resources” https://www.uaex.uada.edu/environment-nature/energy/docs/fsa1056.pdf. 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[27] m. m. küçük & a. demirbas, “biomass conversion processes”, energy convers. manage 38 (1997) 151. appendix a: parameters p0 = normal atmospheric pressure t = temperature γ= air ratio or excess air coefficient 9 lala et al. / j. nig. soc. phys. sci. 4 (2022) 954 10 x = absolute humidity tair = air temperature φair = relative humidity of air ps= water vapour pressure in saturated wet air pb= water vapour partial pressure ρair = density of air ρh2 o = density of water vapour • mrh = rice-husk mass flow rate qso2 = stoichiometric oxygen flow rate in dry air qsairw = wet combustion air flow rate q f gw = wet flue gas flow rate v so2 = stoichiometric volume of oxygen in dry air v sh2 o = volume of water vapour in wet air v sair = stoichiometric volume of dry air v sairw = stoichiometric volume of wet air vco2 = volume of carbon dioxide in dry flue gas vs o2 = volume of sulphur dioxide in dry flue gas v ′ h2 o = volume of water vapour from combustion hydrogen v ′′ h2 o = volume of water vapour due to combustion air humidity v sh2o = total volume of water vapour in stoichiometric dry flue gas v sn2 = volume of nitrogen in stoichiometric dry flue gas v sn2 wet = volume of nitrogen in stoichiometric wet flue gas v sf g = total stoichiometric volume of dry flue gas v sf gw = total stoichiometric volume of wet flue gas %c, %h, %s and %o= elemental composition of rh (ultimate analysis) nm3 = normal cubic meter (amount of gas present in a volume of 1m3 at a temperature of 0 0c and pressure of 1013 mbar) 10 j. nig. soc. phys. sci. 4 (2022) 834 journal of the nigerian society of physical sciences collocation approach for the computational solution of fredholm-volterra fractional order of integro-differential equations g. ajileyea,∗, a.a. jamesb, a. m. ayindec, t. oyedepod adepartment of mathematics and statistics, federal university wukari, taraba state, nigeria. bdepartment of mathematics, american university of nigeria, yola, adamawa state, nigeria. cdepartment of mathematics, moddibo-adama university, yola nigeria. dfederal college of dental technology and therapy, enugu, nigeria. abstract in this work, a collocation technique is used to determine the computational solution to fractional order fredholm-volterra integro-differential equations with boundary conditions using caputo sense. we obtained the linear integral form of the problem and transformed it into a system of linear algebraic equations using standard collocation points. the matrix inversion approach is adopted to solve the algebraic equation and substituted it into the approximate solution. we established the uniqueness and convergence of the method and some modelled numerical examples are provided to demonstrate the method’s correctness and efficiency. it is observed that the results obtained by our new method are accurate and performed better than the results obtained in the literature. the study will be useful to engineers and scientists. it is advantageous because it addresses the difficulty in tackling fractional order fredholm-volterra integro-differential problems using a simple collocation strategy. the approach has the advantage of being more accurate and reducing computer running time. doi:10.46481/jnsps.2022.834 keywords: collocation, fredholm-volterra, integro-differential, fractional derivatives, approximate solution. article history : received: 27 may 2022 received in revised form: 13 august 2022 accepted for publication: 18 august 2022 published: 03 october 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: j. ndam 1. introduction fractional calculus is an aspect in mathematics that studies the properties of integrals combined with noninteger order derivatives.the concept and method of solving differential equations containing fractional derivatives of unknown functions are covered in this field (fractional differential ∗corresponding author tel. no: +2348077092831 email address: adewale.james@aun.edu.ng (g. ajileye ) equations). many prominent mathematicians, such as liouville, grunwald, riemann, euler, langrange, heaviside, fourier, abel, and others, established fractional calculus on a formal foundation[1]. very recently, scholars developed huge interest in fractional calculus due to its relevance and application to many areas of scientific endeavors. fractional differential equations, or those containing real and complex order derivatives have been more significant in describing the most broad fields of science and technology with peculiar dynamics of various processes involving complex systems [2]. 1 ajileye et al. / j. nig. soc. phys. sci. 4 (2022) 834 2 many different approaches have been adopted to investigate the solution of fractional integrodifferential equations, such as adomian decompositions method [3-5], collocation method [6, 7], laplace decomposition method [8, 9], taylor expansion method [10], least square method [9], differential transform method [11], homotopy perturbation method [12-18], sinc-collocation method [15-17] and variational iteration method [12, 13]. linna et al. [20] considered a numerical method to solve fractional variational problem. they simplified the fractional variational problems by the operational matrices. the operational matrices are based on the chelyshkov polynomials. the fractional variational problem was transformed into a set of algebraic equations. the unknown coefficients were solved using the lagrange multiplier techniques. avcı & mahmudov [19] proposed a numerical method based on the fractional taylor vector for solving multi-term fractional differential equations. the main idea of this method was to reduce the given problems to a set of algebraic equations by utilizing the fractional taylor operational matrix of fractional integration. some numerical examples were given to demonstrate the accuracy and applicability. the results show that the presented method was efficient and applicable. fadugba [21] presented the mellin transform for the solution of the fractional order equations. the mellin transform approach occurs in many areas of applied mathematics and technology. the mellin transform of fractional calculus of different flavours; namely the riemann-liouville fractionalderivative, riemann-liouville fractional integral, caputo fractional derivative and the miller-ross sequential fractional derivative were obtained. in this study, we consider fredholm-volterra integrodifferential equation of fractional order of the the form: d α xy(x) − p1y ′ (x) − p2y ′′ (x) − p0y(x) = g (x) + λ1 ∫ x 0 k1 (x, t) y (t) dt +λ2 ∫ 1 0 k2 (x, t) y (t) dt, (1) with the given boundary conditions y (a) = 0, y(b) = 0, a < x < b, (2) where y(x) is to be determined, dαx is the caputo’s derivative, k1 (x, t) and k2 (x, t) are the fredholm and volterra integral kanel function respectively. p j, pα,λ j are known constants. g(x) is the known function 2. basic definitions under this section, we present some definitions and basic concepts of fractional calculus for the formulation of the given problem definition 2.1: the caputo derivative with order α > 0 of the given function f (x), x ∈ (a, b) is defined as [ litfi, dehghan and yousefi, 2011] c x d α a y(x) = 1 γ(m −α) ∫ x a (x − s)m−α−1y(m)(s)d s, (3) where m − 1 ≤ α ≤ m, m ∈ n, x > 0 definition 2.2: let (an) , n ≥ 0 be a sequence of real numbers. the power series in x with coefficients an is an expression [ edward, ford and simpson, 2002] y(x) = a0 + a1 x + a2 x 2 + a3 x 3 + · · ·an x n = n∑ n=0 an x n = φ(x) a, (4) where φ(x) = [1 x x2 · · · xn ], a = [a0 a1 · · · an ]t , then y(x, n) = xna, n = 0(1)n, n ∈ z+. definition 2.3: standard collocation method (scm). this approach is used to find the collocation points that are desired within a certain interval. i.e [a,b] and is given by xi = a + (b − a)i n , i = 1, 2, 3, ...n. (5) definition 2.4: a metric on a set m is a function d : m×m −→ r with the following properties. for all x, y ∈ m (a) d(x, y) ≥ 0; (b) d(x, y) = 0 ⇐⇒ x = y (c) d(x, y) = d(y, x) (d) d(x, y) ≤ d(x, z) + d(x, y) if d is a metric on m, then the pair (m, d) is called a metric space. definition 2.5: let (x, d) be a metric space, a mapping t : x −→ x is lipschitzian if ∃ a constant l > 0 such that d(t x, t y) ≤ ld(x, y) ∀ x, y ∈ x. definition 2.6: let y(x) be a continuous function, then 0 i β x ( c 0 d β xy(x) ) = y(x) − n∑ k=0 y(k)(0) k! xk, (6) where m − 1 < β < 1. let p(s) be an integrable function, then 0 i β x ( p(s)) = 1 γ(β) ∫ x 0 (x − s)β−1 p(s)d s. (7) 3. mathematical background in this section, combination of collocation method and power series approximation is employed for the computational solution of (fvid) of fractional order. theorem 3.0: banach’s fixed point theorem let (x, d) be a complete metric space, then each contraction mapping t : x −→ x has a unique fixed point x of t in x, such that t (x) = x lemma (3.1). let y(x) be the solution to (1) subject to (2) , the integral form y (x) = w(x) + 1 γ(α) ∫ x 0 (x − s)α−1 ( p2y ′′ (s) ) d s (8) 2 ajileye et al. / j. nig. soc. phys. sci. 4 (2022) 834 3 + 1 γ(α) ∫ x 0 (x − s)α−1 ( p1y ′ (s) ) d s + 1 γ(α) ∫ x 0 (x − s)α−1 (p0y(s)) d s − 1 γ(α) ∫ x 0 (x − s)α−1 [ λ1 ∫ s 0 k1 (s, t) y (t) dt + λ2 ∫ 1 0 k2 (s, t) y (t) dt ] where w(x) = n∑ k=0 y(k)(0) k! xk − 1 γ(α) ∫ x 0 (x − s)α−1 g(s)d s. multiply equation (1) by 0 i β x (.) gives 0 i α x (d αy(x)) = 0 i β x ( p2y ′′ (x) ) + 0 i β x ( p1y ′ (s) ) +0 i β x (p0y(s)) −0 i β x (g(x)) − 0 i β x [ λ1 ∫ s 0 k1 (s, t) y (t) dt + λ2 ∫ 1 0 k2 (s, t) y (t) dt ] . (9) using(6) on equation (9)gives y(x) = n∑ k=0 y(k)(0) k! xk + 0 i β x ( p2y ′′ (x) ) + 0 i β x ( p1y ′ (s) ) +0 i β x (p0y(s)) (10) 0 i β x (g(x)) + 0 i β x (∫ b 0 k1(x, t)y(t)dt ) + 0 i β x (∫ s 0 k2(s, t)y(t)dt ) applying equation (4) and (7) to (10) gives y(x) = n∑ k=0 y(k)(0) k! xk + 1 γ(α) ∫ x 0 (x − s)α−1 p2 ( φ(s) ′′ ) d sa + 1 γ(α) ∫ x 0 (x − s)α−1 p1 ( φ(s) ′ ) d sa + 1 γ(α) ∫ x 0 (x − s)α−1 p0 (φ(s)) d sa − 1 γ(α) ∫ x 0 (x − s)α−1 (g(s)) d s − 1 γ(α) ∫ x 0 (x − s)α−1 [ λ1 ∫ s 0 k1 (s, t) φ(t) dt (11) +λ2 ∫ 1 0 k2 (s, t) φ(t) dt ] d sa . 3.1. method of solution collocatiing at xi in equation (11) gives y(xi) = n∑ k=0 y(k)(0) k! xk + 1 γ(α) ∫ xi 0 (xi − s) α−1 p2 ( φ(s) ′′ ) d sa + 1 γ(α) ∫ xi 0 (xi − s) α−1 p1 ( φ(s) ′ ) d sa + 1 γ(α) ∫ xi 0 (xi − s) α−1 p0 (φ(s)) d sa − 1 γ(α) ∫ xi 0 (xi − s) α−1 (g(s)) d s − 1 γ(α) ∫ xi 0 (xi − s) α−1 [ λ1 ∫ s 0 k1 (s, t) φ(t) dt (12) +λ2 ∫ 1 0 k2 (s, t) φ(t) dt ] d sa applying (4) on (12) gives φ(xi)a = w(xi) +  1 γ(α) ∫ xi 0 (xi − s) α−1 p2 ( φ(s) ′′ ) d s+ 1 γ(α) ∫ xi 0 (xi − s) α−1 p1 ( φ(s) ′ ) d s+ 1 γ(α) ∫ xi 0 (xi − s) α−1 p0 (φ(s)) d s − 1 γ(α) ∫ xi 0 (xi − s) α−1 ( λ1 ∫ xi 0 k1 (xi, t) φ(t) dt +λ2 ∫ 1 0 k2 (xi, t) φ(t) dt ) d s  a, (13) where w(xi) = n∑ k=0 y(k)(0) k! xk − 1 γ(α) ∫ xi 0 (xi − s) α−1 g(s)d s. factorise the values of a from equation (13) gives w(xi) =  φ(xi) − 1 γ(α) ∫ xi 0 (xi − s) α−1 p2 ( φ(s) ′′ ) d s− 1 γ(α) ∫ xi 0 (xi − s) α−1 p1 ( φ(s) ′ ) d s− 1 γ(α) ∫ xi 0 (xi − s) α−1 p0 (φ(s)) d s + 1 γ(α) ∫ xi 0 (xi − s) α−1( λ1 ∫ s 0 k1 (s, t) φ(t) dt + λ2 ∫ 1 0 k2 (s, t) φ(t) dt ) d s  a. (14) equation 14 can be in the form τn(xi)a =w(xi), (15) where τn(xi) =  φ(xi) − 1 γ(α) ∫ xi 0 (xi − s) α−1 p2 ( φ(s) ′′ ) d s− 1 γ(α) ∫ xi 0 (xi − s) α−1 p1 ( φ(s) ′ ) d s− 1 γ(α) ∫ xi 0 (xi − s) α−1 p0 (φ(s)) d s + 1 γ(α) ∫ xi 0 (xi − s) α−1( λ1 ∫ s 0 k1 (s, t) φ(t) dt + λ2 ∫ 1 0 k2 (s, t) φ(t) dt ) d s,  and a = [a0 a1 · · · an ]t . multiply both sides of equation(15) byτ−1n (xi) gives a =τ−1n (xi)w(xi). (16) substituting a into the approximate solution (4) gives y(x) = φ(xi)τ −1 n (xi) w(xi) (17) 3 ajileye et al. / j. nig. soc. phys. sci. 4 (2022) 834 4 4. uniqueness of the method in order to establish the uniqueness of the method, we introduce the following hypothesis h1 : k ∗ 1 = maxx∈[0,1] ∫ x 0 λ1 |k1(x, t)|dt h2 : k ∗ 2 = maxx∈[0,1] ∫ 1 0 λ2 |k2(x, t)|dt h3 : ∣∣∣y(m)1 − y(m)2 ∣∣∣ ≤ lm |y1 − y2| . lemma (4.1) (q-contraction) let t : x −→ x be a mapping defined by theorem (3.0) for y1, y2 ∈ x. t is q-contraction if and only if  ∑2 n=0 ln − ( k∗1 + k ∗ 2 ) γ(α + 1)  < 1, then there exist a unique solution of t. (t y1) (x) = w(x) + 1 γ(α) ∫ x 0 (x − s)α−1 ( p2y ′′ 1 (s) ) d s + 1 γ(α) ∫ x 0 (x − s)α−1 ( p1y ′ 1(s) ) d s + 1 γ(α) ∫ x 0 (x − s)α−1 (p0y1(s)) d s − 1 γ(α) ∫ x 0 (x − s)α−1 [ λ1 ∫ s 0 k1 (s, t) y1 (t) dt +λ2 ∫ 1 0 k2 (s, t) y1 (t) dt ] d s and (t y2) (x) = w(x) + 1 γ(α) ∫ x 0 (x − s)α−1 ( p2y ′′ 2 (s) ) d s + 1 γ(α) ∫ x 0 (x − s)α−1 ( p1y ′ 2(s) ) d s + 1 γ(α) ∫ x 0 (x − s)α−1 (p0y2(s)) d s − 1 γ(α) ∫ x 0 (x − s)α−1 [ λ1 ∫ s 0 k1 (s, t) y2 (t) dt +λ2 ∫ 1 0 k2 (s, t) y2 (t) dt ] d s. thus, |(t y1) (x) − (t y2) (x)| = 1 γ(α) ∫ x 0 (x − s)α−1 p2 ∣∣∣y′′1 (s) − y′′2 (s)∣∣∣ d s + 1 γ(α) ∫ x 0 (x − s)α−1 p1 ∣∣∣y′1(s) − y′2(s)∣∣∣ d s + 1 γ(α) ∫ x 0 (x − s)α−1 p0 |y1(s) − y2(s)|d s − 1 γ(α) ∫ x 0 (x − s)α−1  ∫ s 0 λ1 |k1(s, t)| |y1(t) − y2(t)|dt + ∫ 1 0 λ1 |k2(s, t)| |(y1(t) − y2(t))|dt  d s. taking maximum of both sides and using h1 to h3 gives d (t y1(x), t y2(x)) ≤  l2 + l1 + l0 − ( k∗1 + k ∗ 2 ) γ(α + 1)  d(y1, y2) since t is a contraction ∑2 n=0 ln − ( k∗1 + k ∗ 2 ) γ(α + 1)  < 1. we can conclude that t has a unique solution. 5. convergence analysis we establish the convergence of the method yn (x) = w(x) + 1 γ(α) ∫ x 0 (x − s)α−1 ( p2y ′′ n (s) ) d s + 1 γ(α) ∫ x 0 (x − s)α−1 ( p1y ′ n (s) ) d s + 1 γ(α) ∫ x 0 (x − s)α−1 (p0yn (s)) d s (18) − 1 γ(α) ∫ x 0 (x − s)α−1 [ λ1 ∫ s 0 k1 (s, t) yn (t) dt +λ2 ∫ 1 0 k2 (s, t) yn (t) dt ] d s. subtract (8) from (18) gives en (x) = yn (x) − y(x). hence |en (x)| = 1 γ(α) ∫ x 0 (x − s)α−1 |en (s) (p2 + p1)|d s − 1 γ(α) ∫ x 0 (x − s)α−1[ |λ1| ∣∣∣∣∣ ∫ s 0 k1 (s, t) en (t)dt ∣∣∣∣∣ + |λ2| ∣∣∣∣∣∣ ∫ 1 0 k2 (s, t) en (t)dt ∣∣∣∣∣∣d s. (20) therefore ‖en (x)‖∞ ‖en (t)‖∞ ≤ 1 γ(α) ∫ x 0 (x − s)α−1 ×  |en(s) (p2 + p1)| − 1 γ(α) ∫ x 0 (x − s)α−1( |λ1| ∣∣∣∫ s 0 k1 (s, t) en(t)dt ∣∣∣ + |λ2| ∣∣∣∣∫ 10 k2 (s, t) en(t)dt∣∣∣∣  d s. the method converges 4 ajileye et al. / j. nig. soc. phys. sci. 4 (2022) 834 5 6. numerical examples in this section, two numerical problems with boundary conditions are presented to text the efficiency and simplicity of the method. all computations are done with the aid of maple 18. let yn(x) and y(x) be the approximate and exact solution respectively. errorn = |yn(x) − y(x)| example 1: considering fractional integro-differential equation [17] y′′ (x) + 1 x d0.5x y(x) + 1 x2 y(x) = f (x) + ∫ x 0 sin (x − t) y(t)dt + ∫ 1 0 cos (x − t) y (t) dt, subject to the boundary conditions y(0) = 0, y(1) = 0, where f (x) = 5 + 1.50451x0.5 − 13x − 1.80541x1.5 −x2 + x3 − 2.067x cos(x) + 5.95385 sin(x). exact solution y (x) = x2 (1 − x) solution 1 we solve this problem at n = 3 and 5 but we use n = 3 for demonstration. integral form of example 1 is y′′ (x) + 1 x ( 1 γ(1 − 0.5) ∫ x 0 (x − t)−0.5 y(1)(t)dt ) + 1 x2 y(x) = f (x) + ∫ x 0 sin (x − t) y(t)dt + ∫ 1 0 cos (x − t) y (t) dt.(21) using approximate solution (4) on (14) gives φ ′′ (x) + 1x ( 1 γ(1−0.5) ∫ x 0 (x − t)−0.5 dd x (φ(t))dt ) + 1 x2 φ ′′ (x) − ∫ x 0 sin (x − t) φ(t)dt − ∫ 1 0 cos (x − t) φ (t) dt  a = f (x) (22) equation (15) gives τ(x)a = f (x), (23) where τ(x) = φ ′′ (x) + 1x ( 1 γ(1−0.5) ∫ x 0 (x − t)−0.5 dd x (φ(t))dt ) + 1x2 φ ′′ (x) − ∫ x 0 sin (x − t) φ(t)dt − ∫ 1 0 cos (x − t) φ (t) dt. collocating at x3 = [ 1 3 2 3 1 ] and substituting the boundary conditions gives τ(x)∗a = f (x)∗, (24) where τi (x) =  8.1837497830 4.5298367530 3.5647166670 2.4321976680 1.0664602830 2.3357941530 3.8888778550 5.4288519160 -0.3011686789 1.5101524580 4.1068429140 8.5147669310 1 0 0 0 1 1 1 1  table 1. exact and approximate values, example 1 x exact n = 3 n = 5 0.2 0.032000000000 0.031999821270 0.031999851830 0.4 0.096000000000 0.095999891010 0.095999961860 0.6 0.144000000000 0.143999795600 0.143999892900 0.8 0.128000000000 0.127999546700 0.127999648400 1.0 0.000000000000 -8.444000000000e-7 -7.382000000000e-7 table 2. absolute error for example 1 x error3 error5 error [17]=32 0.2 -1.787300000000e-7 -1.481700000000e-7 2.048e-5 0.4 -1.089900000000e-7 -3.814000000000e-8 2.503e-5 0.6 -2.044000000000e-7 -1.071000000000e-8 1.789e-5 0.8 -1.071000000000e-7 -3.516000000000e-8 7.682e-6 1 -8.444000000000e-7 -7.382000000000e-8 3.034e-6 f (x) = [ 1.1325152410 -1.5399784720 -4.4079289640 0 0 ] . we now solve for the unknown values a (17) making use of matrix inversion results in; y3 = ( −4.2497275388 × 10−7 + 0.16917e − 5x+ 0.9999976497x2 − 0.9999997608x3 ) . applying the same procedure for n = 5 gives y5 = ( −3.4773620430 × 10−7 + 8.9350760390 × 107 x + 1.0000017198x2 −1.0000070682x3 + 0.57092e − 5x4 − 0.16454e − 5x5 ) example 2 considering fractional integro-differential equation [17] y′′ (x)+ dαx y(x) = f (x)+2 ∫ x 0 (x − t) y(t)dt + ∫ 1 0 ( x2 − t ) y (t) dt with the given boundary conditions y(0) = 0, y(1) = 0 where f (x) = − 1 30 − 6x + 181x2 20 + 4x3 − x5 10 + x6 15 and α = 1 exact solution y (x) = x3 (x − 1) solution 2 we solve this problem at n = 4 and 5, however, make use of n = 4 for demonstration. integral form of example 2 is y′′ (x) + ( 1 γ(2 − 1) ∫ x 0 (x − t)0 y(2)(t)dt ) (25) = f (x) + 2 ∫ x 0 (x − t) y(t)dt + ∫ 1 0 ( x2 − t ) y (t) dt 5 ajileye et al. / j. nig. soc. phys. sci. 4 (2022) 834 6 table 3. exact and approximate values, example 2 x exact n = 4 n = 5 0.2 -0.6400000000e-2 -0.6400000000e-2 -0.6400000000e-2 0.4 -0.3840000000e-1 -0.3840000000e-1 -0.3840000000e-1 0.6 -0.864000000e-1 -0.8640000000e-1 -0.8640000000e-1 0.8 -0.1024000000 -0.1024000000 -0.1024000000 1.0 -0.9000000000e-3 -0.900000000e-3 -0.900000000e-3 table 4. absolute error for example 2 x error4 error5 error [17]=64 0.2 0.0 0.0 2.931e-7 0.4 0.0 0.0 3.915e-7 0.6 0.0 0.0 2.696e-7 0.8 0.0 0.0 1.011e-7 1 0.0 0.0 3.596e-8 using approximate solution (4) on (18) gives φ′′(x) + ( 1 γ(2−1) ∫ x 0 (x − t)0 d 2 d x2 (φ(t))dt ) −2 ∫ x 0 (x − t) φ(t)dt − ∫ 1 0 ( x2 − t ) φ (t) dt  a = f (x). (26) equation (6.6) gives τ(x)a = f (x) (27) where τ(x) = φ ′′ (x) + ( 1 γ(2−1) ∫ x 0 (x − t)0 d 2 d x2 (φ(t))dt ) −2 ∫ x 0 (x − t) φ(t)dt − ∫ 1 0 ( x2 − t ) φ (t) dt. collocating at x4 = [ 0 14 2 4 3 4 1 ] and substituting the boundary conditions gives τ(x)∗a = f (x)∗ (28) where τi (x) =  1 2 0.3333333333 2.250000000 0.2000000000 0.1666666667 1 2 59 192 4193 1536 19169 10240 59393 61440 1 2 1 4 305 96 249 64 3473 960 1 2 37 192 1851 512 64211 10240 522457 61440 1 2 1 6 49 12 181 20 481 30 1 0 0 0 0 1 1 1 1 1  f (x) = [ 130 55621 61440 131 480 137191 61440 419 60 0 0 ] we solve for the unknown values a using matrix inversion, results; y4 =  −2.2211399390 × 10−13 +3.0783153800 × 10−12 x −1.4125589590 × 10−11 x2 −1.0000000000x3 + 1.0000000000x4  applying the same procedure for n = 5 gives y5 =  5.329070518000 × 10−15 +1.781685910000 × 10−12 x −1.580957587000 × 10−11 x2 −1.000000000000x3 +1.000000000000x4 +1.062971933000 × 10−11 x5  7. conclusion collocation approach is utilized to solving the fractional order fredholm-volterra integro-differential problems in this paper. the applied method is consistent, efficient and easy to compute. the results of the numerical example 1 as shown in table 1 shows that the approximate solution at n = 3 gives y3(x) = −4.2497275388 × 10−7 + 0.16917e − 5x + 0.9999976497x2 − 0.9999997608x3 and solving at n = 5 indicates that as the value of n increases, the error becomes smaller. we also compare our absolute errors with alkan et al (2017) as shown in table 2, this clearly shows that our method performs better. the results of the numerical example 2 as shown in table 3 shows that the approximate solution obtained at n = 4 and 5 converges to the exact solution. table 4 compares the errors in the method proposed by alkan et al (2017) at n = 64 and errors in our new method which converges to the exact solution at n = 4. references [1] p. kamal, higher order numerical methods for fractional order differential equations, doctoral dissertation, university of chester, united kingdom, 2015. http://hdl.handle.net/10034/613354. 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[21] s. e. fadugba, “solution of fractional order equations in the domain of the mellin transform”, journal of the nigerian society of physical sciences 4 (2019) 138. https://doi.org/10.46481/jnsps.2019.31 7 j. nig. soc. phys. sci. 4 (2022) 961 journal of the nigerian society of physical sciences on lemniscate of bernoulli of q-janowski type afis saliua,b,∗, semiu oladipupo oladejob adepartment of mathematics, university of the gambia, mdi road, kanifing p.o. box 3530, serrekunda, the gambia bdepartment of mathematics, gombe state university p.m.b 127, tudun wada, gombe, gombe state, nigeria abstract in this article, we introduce the q-analogue of functions characterized by the lemniscate of bernoulli in the right-half plane and define the class l∗q(a, b). furthermore, we study the geometric properties of this class, which include coefficient inequalities, subordination factor sequence property, radii results and fekete-szegö problems. some deductions of our results show relevant connections between this present work and the existing ones in many literature. it is worthy of note that some of our results are sharp. doi:10.46481/jnsps.2022.961 keywords: univalent functions, schwarz functions, lemniscate of bernoulli, subordination, janowski functions. article history : received: 28 july 2022 received in revised form: 12 september 2022 accepted for publication: 13 september 2022 published: 08 october 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published articleâs title, journal citation, and doi. communicated by: p. thakur 1. introduction and preliminaries the theory of univalent functions had its history from the riemann mapping theorem [1]. this theorem aroused the interest of many researchers to start working in this field. for example, bieberbach [1] proved that for every univalent function f (z) = z + ∞∑ n=2 anz n (z ∈ u := {z ∈ c : |z| < 1}) , (1) |a2| ≤ 2 and conjectured that, in general, |an| ≤ n , n ≥ 2. ∗corresponding author tel. no: +2348066221155, +22941154 email address: asaliu@utg.edu.gm, saliugsu@gmail.com (afis saliu ) this conjecture stood for more than fifty years until it was finally settled by de branges [2] in 1985. geometric function theory (gft) has witnessed many developements and introduction of new subfamilies of univalent functions within these intervening years. in particular, ma and minda [3] gave a comprehensive classification of classes s ∗ and c of starlike and convex functions [1], respectively. for this purpose, they considered the class a of normalized analytic functions (1) and a univalent function ϕ (which maps u onto domains symmetric with the real axis and starlike with respect to 1) normalized such that ϕ(0) = 1 and ϕ′(0) > 0. thus, the classes are respectively defined as follows: s ∗(ϕ) = { f ∈ a : z f ′(z) f (z) ≺ ϕ } and c(ϕ) = { f ∈ a : z f ′ ∈ s ∗(ϕ) } . in particular, if we chose ϕ = (1 + z)/(1 − z), s ∗ and c respectively become the usual classes s ∗ and c of starlike and convex 1 saliu & oladejo / j. nig. soc. phys. sci. 4 (2022) 961 2 functions. sokół and stankiewicz [4] introduced the class l∗ by setting ϕ(z) = √ 1 + z. this class consists of those functions that map u onto the lemniscate of bernoulli. they obtained radius of convexity, growth and distortion results for this class. also, early coefficients of functions in l∗ were derived in [5]. furthermore, ali et al. [6] and sokół [7] determined various radii results associated with this class. moreover, raza and malik [8] proved the third hankel determinant associated with l∗. for other related works, see [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20] for details. one of the most recent approaches in gft is the concept of calculus without the notion of limits known as q-calculus, or quantum calculus [21, 22]. this idea was first discussed in the theory of univalent functions by ismail et al. [23]. they introduced the class s ∗q of q-starlike functions, which have led to many new results and techniques in quantum calculus associated with gft. recently, khan et al. [24] gave a q-analogue of √ 1 + z and introduced the class l∗q. moreover, they proved few coefficient bounds and obtained the third hankel determinant for the class. motivated by these works and in particular, [24, 8] we introduce a new class l∗q(a, b) and establish many coefficient inequalities, subordination factor sequence, radii results and fekete-szegö bounds for this class. 2. definitions and lemmas let w be the class of analytic functions w(z) = ∞∑ n=1 wnz n, z ∈ u (2) such that w(0) = 0 and |w(z)| < 1. these functions are known as schwarz functions. if f (z) and g(z) are analytic functions in u, then f (z) is subordinate to g(z) (written as f (z) ≺ g(z)) if there exists w(z) ∈ w such that f (z) = g(w(z)), z ∈ u. in addition, if g is univalent in u, then f (0) = g(0) and f (u) ⊂ g(u). an analytic function p(z) = 1 + ∞∑ n=1 cnz n (3) is a function with positive real part. the class of all such functions is denoted by p. more generally, for −1 ≤ b < a ≤ 1, the class p(a, b) consists of functions p(z) of the form (3) satisfying the subordination condition p(z) ≺ 1 + az 1 + bz , z ∈ u. on the other hand, p ∈ p(a, b) is called a janowski function [25] if and only if p(z) = (1 + a)h(z) + (1 − a) (1 + b)h(z) + (1 − b) , −1 ≤ b < a ≤ 1, where h ∈ p. definition 2.1 (subordinating factor sequence). [26] a sequence {bn}∞n=1 of complex numbers is called a subordinating factor sequence if, whenever f (z) of the form (1) is analytic, univalent and convex in u, we have the subornation given by ∞∑ n=1 anbnz n ≺ f (z) (z ∈ u, a1 := 1). definition 2.2. [27] let q ∈ (0, 1). then the q-number [n]q is given as [n]q =   1−qn 1−q , n ∈ c, n−1∑ ι=0 qι = 1 + q + q2 + · · · + qn−1, n ∈ n, n, as q → 1−. (4) and the q-derivative of a complex valued function f (z) in u is given by dq f (z) =   f (qz)− f (z) (q−1)z , z , 0 f ′(0), z = 0, f ′(z), as q → 1−. (5) from the above explanation, it is easy to see that for f (z) given by (1), dq f (z) = 1 + ∞∑ n=2 [n]qanz n−1. (6) using the idea of q-calculus, khan et al. [24] extended the work of raza and malik [8] by presenting the following class: definition 2.3. [24] a function of the form (1) belongs to the class l∗q if and only if∣∣∣∣∣ ( zdq f (z) f (z) )2 − 1 1 − q ∣∣∣∣∣ < 11 − q, (7) or equivalently,( zdq f (z) f (z) )2 ≺ 2(1 + z) 2 + (1 − q)z , z ∈ u, q ∈ (0, 1). (8) inspired by the work of khan et al. [24] and raza and malik [8], and in view of the janowski functions, we introduce the following class. definition 2.4. a function of the form (1) belongs to the class l∗q(a, b) if and only if∣∣∣∣∣∣∣ (b − 1) ( zdq f (z) f (z) )2 − (a − 1) (b + 1) ( zdq f (z) f (z) )2 − (a + 1) − 1 1 − q ∣∣∣∣∣∣∣ < 1 1 − q , (9) or equivalently,( zdq f (z) f (z) )2 ≺ (ao1 + o3) z + 4 (bo1 + o3) z + 4 , zu, q ∈ (0, 1), (10) where o1 = 1 + q, o3 = 3 − q, −1 ≤ b < a ≤ 1. (11) 2 saliu & oladejo / j. nig. soc. phys. sci. 4 (2022) 961 3 remark 2.5. for a = 1 and b = −1, we are back to the class in definition 2.3. in addition to that, if q → 1−, definition 2.4 reduces to the one given in [4, 7]. to prove our main findings, we need the following results lemma 2.6. [28] if w ∈ w is of the form (2), then for a real number σ, |w2 −σw 2 1| ≤   −σ, for σ ≤−1, 1, for − 1 ≤ σ ≤ 1, σ for σ ≥ 1. when σ < −1 or σ > 1, equality holds if and only if w(z) = z or one of its rotations. if −1 < σ < 1, then equality holds if and only if w(z) = z2 or one of its rotations. equality holds for σ = −1 if and only if w(z) = z(x+z)1+xz (0 ≤ x ≤ 1) or one of its rotations while for σ = 1, equality holds if and only if w(z) = −z(x+z)1+xz (0 ≤ x ≤ 1) or one of its rotations. also, the sharp upper bound above can be improved as follows when −1 < σ < 1: |w2 −σw 2 1| + (1 + σ)|w1| 2 ≤ 1 (−1 < σ ≤ 0) and |w2 −σw 2 1| + (1 −σ)|w1| 2 ≤ 1 (0 < σ < 1). lemma 2.7. [26] the sequence {bn}∞n=1 is a subordinating factor sequence if and only if re ( 1 + 2 ∞∑ n=1 bnz n ) > 0, (zu). in the next section, we present our main findings. 3. main results theorem 3.1. let f ∈ a and suppose ∞∑ n=3 { n−1∑ j=1 [ 2 ( [ j]q[n − j]q − 1 ) (12) + ∣∣∣∣(b + 1)[ j]q[n − j]q − (a + 1) ∣∣∣∣]∣∣∣a jan− j∣∣∣ } < ∣∣∣b − a∣∣∣. (13) then f ∈ l∗q(a, b) proof. suppose condition (12) is satisfied. then it suffices to prove (9). now, ∣∣∣∣∣∣∣ (b − 1) ( zdq f (z) f (z) )2 − (a − 1) (b + 1) ( zdq f (z) f (z) )2 − (a + 1) − 1 1 − q ∣∣∣∣∣∣∣ ≤ ∣∣∣∣∣∣∣ (b − 1) ( zdq f (z) f (z) )2 − (a − 1) (b + 1) ( zdq f (z) f (z) )2 − (a + 1) − 1 ∣∣∣∣∣∣∣ + q 1 − q =2 ∣∣∣∣∣ ( zdq f (z) )2 − ( f (z))2 (b + 1) ( zdq f (z) )2 − (a + 1)( f (z))2 ∣∣∣∣∣ + q1 + q =2 ∣∣∣∣∣∣∣∣∣∣ ∞∑ n=2 ( n−1∑ j=1 a jan− j[ j]q[n − j]q ) zn − ∞∑ n=2 ( n−1∑ j=1 a jan− j ) zn (1 + b) ∞∑ n=2 ( n−1∑ j=1 a jan− j[ j]q[n − j]q ) zn − (a + 1) ∞∑ n=2 ( n−1∑ j=1 a jan− j ) zn ∣∣∣∣∣∣∣∣∣∣ + q 1 − q =2 ∣∣∣∣∣∣∣∣∣∣ ∞∑ n=3 [ n−1∑ j=1 a jan− j ( [ j]q[n − j]q − 1 )] zn (b − a)z2 + ∞∑ n=3 { n−1∑ j=1 a jan− j [ (b + 1)[ j]q[n − j]q − (a + 1) ]} zn ∣∣∣∣∣∣∣∣∣∣ ≤ 2 ∞∑ n=3 [ n−1∑ j=1 ∣∣a jan− j∣∣([ j]q[n − j]q − 1) ] |b − a| + ∞∑ n=3 [ n−1∑ j=1 ∣∣a jan− j∣∣ ∣∣∣∣(b + 1)[ j]q[n − j]q − (a + 1) ∣∣∣∣ ] + q 1 − q . 3 saliu & oladejo / j. nig. soc. phys. sci. 4 (2022) 961 4 this last expression is bounded by 11−q provided (12) is satisfied. hence, we have the required result. we observe that theorem 3.1 implies the following result. corollary 3.2. let f ∈ a. then f ∈ l∗q(a, b) if ∞∑ n=2 [ 2q[n−1]q+ ∣∣(b + 1)[n]q − (a + 1)∣∣] |an| < |b − a| .(14) proof. let φ j,n = [ 2 ( [ j]q[n − j]q − 1 ) + ∣∣∣∣(b + 1)[ j]q[n − j]q − (a + 1) ∣∣∣∣ ]∣∣∣a jan− j∣∣∣ < |b − a| in (12). then ∞∑ n=3 ( φ1,n + φ2,n + φ3,n + · · · + φn−1,n ) < |b − a| , which implies that ∞∑ n=3 φn−1,n < |b − a| . that is ∞∑ n=3 [ 2 ( [n−1]q−1 ) + ∣∣(b + 1)[n − 1]q − (a + 1)∣∣] |an−1| < |b − a| or ∞∑ n=2 [ 2q[n − 1]q + ∣∣(b + 1)[n]q − (a + 1)∣∣] |an| < |b − a| for a = 1, b = −1 in corollary ??corollary1, we are led to the following assertion. corollary 3.3. let f ∈ a. then f ∈ l∗q(1,−1) if ∞∑ n=2 [n]q |an| < 1. as q → 1− in corollary ??corollary2, we have corollary 3.4. let f ∈ a. then f ∈ l∗ if ∞∑ n=2 n |an| < 1. by corollary 3.2, the class l∗q(a, b) is assumed to be a subclass of l∗q(a, b) consisting of f (z) of the form (14) and satisfying the condition (14). therefore, by the coefficient inequality for the class l∗q(a, b), we have the following results. theorem 3.5. let −1 ≤ b1 ≤ b2 < a1 ≤ a2 ≤ 1. if f ∈ a, then l∗q(a2, b2) ⊂ l ∗ q(a1, b1). proof. by the hypothesis of the theorem, we have that ∞∑ n=2 [ 2q[n − 1]q + ∣∣(b1 + 1)[n − 1]q − (a1 + 1)∣∣] |an| ≤ ∞∑ n=2 [ 2q[n − 1]q + ∣∣(b2 + 1)[n − 1]q − (a2 + 1)∣∣] |an| ≤ |b2 − a2| ≤ |b1 − a1| . thus, f ∈ l∗q(a1, b1). theorem 3.6. let f j ∈ l∗q(a, b) and be of the form f j(z) = ∞∑ n=1 an, jz n, j = 1, 2, . . . m, a1, j = 1. then g(z) ∈ l∗q(a, b), where g(z) = m∑ j=1 |c j| f j(z) with m∑ j=1 |c j| = 1. proof. in view of (14), we have ∞∑ n=2 [ 2q[n − 1]q + ∣∣(b + 1)[n]q − (a + 1)∣∣ |b − a| ]∣∣an, j∣∣ < 1. therefore, g(z) = m∑ j=1 |c j| f j(z) = m∑ j=1 |c j| ( z + ∞∑ n=2 an, jz n ) = m∑ j=1 ∞∑ n=1 |c j|an, jz n = ∞∑ n=1 ( m∑ j=1 |c j|an, j ) zn. but ∞∑ n=2 [ 2q[n − 1]q + ∣∣(b + 1)[n]q − (a + 1)∣∣ |b − a| ]∣∣∣∣ m∑ j=1 |c j|an, j ∣∣∣∣ ≤ m∑ j=1 { ∞∑ n=2 [ 2q[n − 1]q + ∣∣(b + 1)[n]q − (a + 1)∣∣ |b − a| ]∣∣an, j∣∣}|c j| < m∑ j=1 |c j| = 1. hence, g(z) ∈ l∗q(a, b). 4 saliu & oladejo / j. nig. soc. phys. sci. 4 (2022) 961 5 theorem 3.7. let g(z) = z + ∞∑ n=2 bnzn. if f, g ∈ l∗q(a, b), then their weighted mean f j(z) = (1 − j) f (z) + (1 + j)g(z) 2 (15) also belongs to l∗q(a, b). proof. from (15), we have f j(z) = z + ∞∑ n=2 [ (1 − j)an + (1 + j)bn 2 ] zn. to show that f j(z) ∈ l∗q(a, b), we need to show that ∞∑ n=2 [ 2q[n − 1]q + ∣∣∣(b + 1)[n]q − (a + 1) ∣∣∣]∣∣∣∣(1 − j)an + (1 + j)bn2 ∣∣∣∣ < |b − a| . for this, we have ∞∑ n=2 [ 2q[n − 1]q + ∣∣∣(b + 1)[n]q − (a + 1) ∣∣∣]∣∣∣∣(1 − j)an + (1 + j)bn2 ∣∣∣∣ ≤ ( 1 − j 2 ) ∞∑ n=2 [ 2q[n − 1]q + ∣∣(b + 1)[n]q − (a + 1)∣∣] |an| + ( 1 + j 2 ) ∞∑ n=2 [ 2q[n − 1]q + ∣∣(b + 1)[n]q − (a + 1)∣∣] |bn| < |b − a|. theorem 3.8. suppose f j(z) ∈ l∗q(a, b). then the arithmetic mean f(z) of f j(z) given by f(z) = 1 n n∑ j=1 f j(z) is also in l∗q(a, b). proof. we have f(z) = 1 n n∑ j=1 f j(z) = 1 n n∑ j=1 ( z + ∞∑ k=2 ak, jz k ) = z + ∞∑ k=2  1 n n∑ j=1 ak, j   zk. since f j(z) ∈ l∗q(a, b), then ∞∑ k=2 [ 2q[k − 1]q + ∣∣(b + 1)[k]q − (a + 1)∣∣] ∣∣∣∣∣∣1n n∑ j=1 ak, j ∣∣∣∣∣∣ ≤ 1 n n∑ j=1 { ∞∑ k=2 [ 2q[k − 1]q + ∣∣(b + 1)[k]q − (a + 1)∣∣] }∣∣ak, j∣∣ < 1 n n∑ j=1 |b − a| = |b − a| . thus, f ∈ l∗q(a, b). in the next theorem, we provide a sharp subordination result involving class l∗q(a, b). theorem 3.9. let f ∈ l∗q(a, b). then for every convex function g(z) in u, we have 2q + |(1 + b)q + b − a| 2 [ 2q + |b − a| + |(1 + b)q + b − a| ] ( f ∗ g) (z) ≺ g(z)(16) and re f (z) > − [ 2q + |b − a| + |(1 + b)q + b − a| ] 2q + |(1 + b)q + b − a| . (17) the constant 2q + |(1 + b)q + b − a| 2 [ 2q + |b − a| + |(1 + b)q + b − a| ] cannot be replaced by any larger one. proof. let f ∈ l∗q(a, b) and g(z) = z + ∞∑ n=2 cnzn be convex in u. then 2q + |(1 + b)q + b − a| 2 [ 2q + |b − a| + |(1 + b)q + b − a| ] ( f ∗ g) (z) = 2q + |(1 + b)q + b − a| 2 [ 2q + |b − a| + |(1 + b)q + b − a| ] (z + ∞∑ n=2 anbnz n ) . therefore, by definition 2.1, the assertion of our theorem holds if the sequence{ 2q + |(1 + b)q + b − a| 2 [ 2q + |b − a| + |(1 + b)q + b − a| ]an}∞ n=1 is a subordinating factor sequence, with a1 = 1. in view of lemma 2.7, this will hold true if and only if re [ 1 + ∞∑ n=1 2q + |(1 + b)q + b − a|[ 2q + |b − a| + |(1 + b)q + b − a| ]anzn ] > 0 (z ∈ u). since 2q[n − 1]q + ∣∣(b + 1)[n]q − (a + 1)∣∣ is an increasing function of n (n ≥ 2), we have 5 saliu & oladejo / j. nig. soc. phys. sci. 4 (2022) 961 6 re [ 1 + 2 ∞∑ n=1 2q + |(1 + b)q + b − a| 2 [ 2q + |b − a| + |(1 + b)q + b − a| ]anzn ] = re [ 1 + 2q + |(1 + b)q + b − a|[ 2q + |b − a| + |(1 + b)q + b − a| ]z + ∞∑ n=2 2q + |(1 + b)q + b − a|[ 2q + |b − a| + |(1 + b)q + b − a| ]anzn] ≥ 1 − 2q + |(1 + b)q + b − a|[ 2q + |b − a| + |(1 + b)q + b − a| ]r − ∞∑ n=2 2q + |(1 + b)q + b − a|[ 2q + |b − a| + |(1 + b)q + b − a| ] |an|rn > 1 − 2q + |(1 + b)q + b − a|[ 2q + |b − a| + |(1 + b)q + b − a| ] − ∞∑ n=2 2q + |(1 + b)q + b − a|[ 2q + |b − a| + |(1 + b)q + b − a| ] |an| ≥ 1 − 2q + |(1 + b)q + b − a|[ 2q + |b − a| + |(1 + b)q + b − a| ] − 1[ 2q + |b − a| + |(1 + b)q + b − a| ] ∞∑ n=2 [ 2q[n − 1]q + ∣∣(b + 1)[n]q − (a + 1)∣∣] |an| > 1 − 2q + |(1 + b)q + b − a|[ 2q + |b − a| + |(1 + b)q + b − a| ] − |b − a|[ 2q + |b − a| + |(1 + b)q + b − a| ] = 0, where we have used condition (12). this proves (16). furthermore, condition (17) is achieved by setting g0 (z) = z 1 − z = z + ∞∑ n=2 zn to prove the sharpness of the constant 2q+|b−a|4(q+|b−a|) , we consider the function given by f (z) = z − |b − a| 2q + |(1 + b)q + b − a| z2. evidently, f ∈ l∗q(a, b) and from (16), we have 2q + |(1 + b)q + b − a| 2 [ 2q + |b − a| + |(1 + b)q + b − a| ] f (z) ≺ z 1 − z (zu). it is observed that min { re ( 2q + |(1 + b)q + b − a| 2 [ 2q + |b − a| + |(1 + b)q + b − a| ] f (z))} = −1 2 . this completes the proof. for a = 1, b = −1 in theorem 3.9, we have the following result. corollary 3.10. let f ∈ l∗q ⊂ l ∗ q. then for every convex function g(z) in u, we have 1 + q 2(q + 2) ( f ∗ g)(z) ≺ g(z) and re f (z) > − 2 + q 1 + q . moreover, the constant 1+q2(2+q) cannot be a larger one. as q → 1− in corollary 3.10, we obtain the following assertion. corollary 3.11. let f ∈ l∗ ⊂ l∗. then for every convex function g(z) in u, we have 1 3 ( f ∗ g)(z) ≺ g(z) and re f (z) > − 3 2 . the constant factor 13 cannot be replaced by a larger one. in the following theorems, we present some radii results associated with the class l∗q(a, b). theorem 3.12. l∗q(a, b) ⊂ l ∗ q in the disc |z| < rq(a, b), −1 ≤ b < a ≤ 1, where rq(a, b) = min { 4 2(a − b) + [ 2(1 + b) + (1 − q)(1 − a) ] , 1}.(18) proof. to determine l∗q radius, we need to find r such that 0 < r < 1 and h(zr) := (ao1 + o3) z + 4 (bo1 + o3) z + 4 ≺ 2(1 + z) 2 + (1 − q)z := q(z) (z ∈ u), where o1 and o3 are given by (11). this is equivalent to |q−1(h(rz))| ≤ 1. it is easy to see that |q−1(h(rz))| = ∣∣∣∣∣ 2(a − b)rz4 + [(2b − (1 − q)a) + o3] rz ∣∣∣∣∣ ≤ 2(a − b)r 4 − [ 2(1 + b) + (1 − q)(1 − a) ] r . the last expression is bounded by 1 if 2(a − b)r ≤ 4 − [ 2(1 + b) + (1 − q)(1 − a) ] r. this completes the proof. theorem 3.13. let f ∈ s ∗q. then f ∈ l ∗ q in the disc |z| < rq, where rq = 1 3 + √ 11 − 2q + |2q − 1| . (19) 6 saliu & oladejo / j. nig. soc. phys. sci. 4 (2022) 961 7 proof. since f ∈ s ∗q, then zdq f (z) f (z) ≺ 1 + z 1 − qz (z ∈ u, see [29, 30, 31]) , and by subordination property, we have( zdq f (z) f (z) )2 ≺ ( 1 + z 1 − qz )2 := s (z) (z ∈ u). we need to determine the smallest positive radius r such that s (rz) ≺ q(z) (z ∈ u), which is equivalent to showing that∣∣q−1(s (rz))∣∣ ≤ 1 (z ∈ u). now, we can obviously see that ∣∣q−1(s (rz))∣∣ = ∣∣∣∣∣ 2 ( 2r + (1 − q)r2z2 ) 1 − 2rz − (2q − 1)r2z2 ∣∣∣∣∣ ≤ 2 ( 2(1 − q) + r2 ) 1 − 2r − |2q − 1|r2 . this expression is bounded by 1 if (2(1 − q) + |2q − 1|) r2 + 6r − 1 ≤ 0. let t (r) = (2(1 − q) + |2q − 1|) r2 +6r−1. then t (0)t (1) < 0. therefore, there exists r ∈ (0, 1) such that t (r) = 0. hence, we have the result. as q → 1− in theorem 3.13, we get the following result. corollary 3.14. let f ∈ s ∗. then f ∈ l∗ in the disc |z| < 1 3+ √ 10 . in the next theorem, using lemma 2.6, we present the feketeszegö inequality for the class l∗q(a, b). theorem 3.15. let f ∈ l∗q. then for a real number µ, |a3−µa 2 2| ≤ ( a − b 8q )   − qv (q)+2µ(a−b)(1+q)2 16q , for µ ≤− q(16+v (q))2(a−b)(1+q)2 := ρ1, 1, for ρ1 ≤ µ ≤ q(16−v (q)) 2(a−b)(1+q)2 := ρ2, qv (q)+2µ(a−b)(1+q)2 16q , for µ ≥ ρ2, . it is asserted also that |a3 −µa 2 2| + [ µ + q(16 + v (q)) 2(a − b)(1 + q)2 ] |a2| 2 ≤ a − b 8q , − q(16 + v (q)) 2(a − b)(1 + q)2 < µ ≤− qv (q) 2(a − b)(1 + q)2 and |a3 −µa 2 2| − [ µ− q(16 − v (q)) 2(a − b)(1 + q)2 ] |a2| 2 ≤ a − b 8q , − qv (q) 2(a − b)(1 + q)2 < µ ≤ q(16 − v (q)) 2(a − b)(1 + q)2 where v (q) = (a + 3b − 4)q2 − (a − 5b − 12)q − 2(a − b).(20) these bounds are sharp. proof. let f ∈ l∗q(a, b). then zdq f (z) f (z) = √ 1 + a1w(z) 1 + b1w(z) , where w(z) = ∞∑ n=1 wnz n ∈w, a1 = ao1 + o3 4 , and b1 = bo1 + o3 4 . therefore, 1 + qa2z + q ( [2]qa3 − a 2 2 ) z2 + . . . = 1 + w1o1(a − b) 8 z − o1(a − b) 8 [ 1 16 ( ao1 + 4o3 + 3bo1 ) w21 − w2 ] z2 + . . . . on comparing coefficients, we have a2 = (a − b)o1w1 8q and a3 = a − b 8q ( w2 − v (q) 16 w21 ) , so that a3 −µa 2 2 = a − b 8q (w2 −σw 2 1), where σ = qv (q) + 2µ(a − b)o21 16q . then we have the required result by applying lemma 2.6. these inequalities are sharp for the functions  λ f1(zλ), for µ ∈ (−∞,ρ1) ∪ (ρ2,∞), λ f2(zλ), for ρ1 ≤ µ ≤ ρ2, λ f3(zλ), for µ = ρ1, λ f4(zλ), for µ = ρ2, where |λ| = 1 and zdq f1(z) f1(z) = √ 1 + a1z 1 + b1z zdq f2(z) f2(z) = √ 1 + a1z2 1 + b1z2 zdq f3(z) f3(z) = √ 1 + a1wx(z) 1 + b1wx(z) zdq f4(z) f4(z) = √ 1 − a1wx(z) 1 − b1wx(z) with wx(z) = z(x + z) 1 + xz , 0 ≤ x ≤ 1. 7 saliu & oladejo / j. nig. soc. phys. sci. 4 (2022) 961 8 remark 3.16. for a = 1, b = −1, theorem 3.15 reduces to the fekete-szegö inequalities presented by khan [24]. further, as q → 1− we are led to the result of raza and malik [8]. 4. conclusion in this work, we introduced the class l∗q(a, b) and obtained coefficient inequalities, subordination factor sequence property radii results and fekete-szegö inequality for it. overall, we presented many consequences of our investigation. these results are fascinated essentially by their particular cases and consequences. also, to have more new hypotheses under present assessments, new extension and applications can be investigated with some positive and novel results in different fields of science, particularly, in gft. for more details about the applications, one may go through [32, 33]. acknowledgments we thank the referees for the positive enlightening comments and suggestions, which have greatly helped us in making improvements to this paper. references [1] a. w. goodman, univalent functions, vols. i-ii, united states of america, mariner publishing company, tempa. florida (1983). 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[33] l. k. alzaki, & h. k. jassim, “time-fractional differential equations with an approximate solution”, j. nig. soc. phys. sci. 4 (2022) 818. https://doi.org/10.46481/jnsps.2022.818 8 j. nig. soc. phys. sci. 4 (2022) 812 journal of the nigerian society of physical sciences characterisation of singular domains in threshold dependent biological networks benitho a. ngwua,∗, godwin c. e. mbahb, chika o. mmaduakora, sunday isienyic, oghenekevwe r. ajewolea, felix d. ajibadea adepartment of mathematics, federal university oye-ekiti, oye/ekiti, nigeria bdepartment of mathematics, university of nigeria, nsukka/enugu, nigeria cdepartment of mathematics & statistics,akanu ibiam federal polytechnic unwana, afikpo/ebonyi, nigeria abstract threshold-dependent networks which behave like piecewise smooth systems and belong to a class of systems with discontinuous right hand side are studied with piecewise linear differential equations. at threshold values and their intersections, known as switching boundaries and surfaces, the state of such networks is not defined because of singularity at such points. this study characterises, in terms of number, singular domains of any order in a network and the total number of such domains; and also proposes new definitions for walls, using filippov’s first order theory on characterisation of (sliding) wall. the finding of this study is presented as propositions i, ii and iii respectively. in particular, using proposition ii the study identified two white walls previously considered transparent. introducing monotonicity to definition of transparent wall is also seen to affect qualitative dynamics like source, sink and cycles. doi:10.46481/jnsps.2022.812 keywords: discontinuous system, singular domain, threshold-dependent network, biological system, piecewise linear models article history : received: 12 may 2022 received in revised form: 16 july 2022 accepted for publication: 20 july 2022 published: 20 august 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: t. latunde 1. introduction differential equations with discontinuous right-hand-side are applied in different areas of study [1, 2]. they include but are not limited to control problems [3], neural networks [4] and gene regulatory networks [5, 6]. due to its importance, different types of solutions exist for this class of problems [5] of which filippov’s is the most widely invoked [6-12]. thresholddependent networks from biology, such as gene regulatory net∗corresponding author tel. no: +2348067876254 email address: benitho.ngwu@fuoye.edu.ng; benamaobi@gmail.com (benitho a. ngwu ) works can be considered as piecewise linear differential model with discontinuous right hand side [13]. the discontinuity observed in this class of threshold-dependent networks results from threshold boundaries known as the switching boundaries. switching boundaries can be of different orders depending on the number of variables at play in the network. intersection of two or more different switching boundaries is known as switching surfaces. due to the wide application and acceptance of filippov’s approach to this class of problems, the threshold boundaries (or surfaces) are called filippov’s boundary (or surface). in biological networks, they represent variables’ concentration levels (and may be referred to as concentration thresholds) and 1 mbah et al. / j. nig. soc. phys. sci. 4 (2022) 812 2 their intersections respectively. at these boundaries, the state of the variable is assumed to change. for instance, above a certain concentration level (that is, threshold value) a variable may be activated or deactivated depending on the effect of control of the activity of all other variables regulating its function [14, 15]. these switching boundaries, which can also be called (switching) hyperplanes, partition the state space of the network or systems into rectangular boxes [16] and segments [17] when step functions are used to describe the regulatory control resulting from the threshold. within these boxes, the state of the network is known and can be defined as a linear function because of the use of step functions. on the contrary, the state of the network is considered not known at the threshold boundaries yielding the discontinuity observed in networks of such nature. one of the major challenges of this class of network is defining a solution along the switching boundary. filippov’s approach is used to study the behaviour of this class of network [7, 9, 10, 12], especially at the threshold boundary. one of the results of filippov is defining what is called sliding vector field at a (sliding) threshold boundary, where discontinuity exists. this provides an insight into the behaviour of solutions of the system in the neighbourhood of the threshold boundary. within the vicinity of the threshold, a trajectory approaching a discontinuity boundary can either glide through, move away from, or slide on, it. this is how a threshold boundary is classified. using filippov approach, differential models with discontinuous right hand side have been investigated for different qualitative properties such as orbits [18-20], limit cycles [1] and sliding motions and solutions on threshold boundaries [11, 13]. as noted earlier, discontinuous boundaries have been selectively investigated for existence of solutions due to a conventional definition that is based on the flow of trajectory towards such domains. for instance, sari and gouze in [13] conclude that transparent walls are not candidate for sliding vector motion. their conclusion is premised on the direction of flow of solution trajectories towards the wall. however, they failed to investigate if such walls are actually transparent let all providing a criterion for testing the transparency of a wall beyond direction of trajectories of focal points. in recent times this class of problem was considered as fastslow systems with delay [22-25]. to analyse the behaviour of solution at the switching boundary regularisation approach was adopted. in [21]the qualitative behaviour of the system is analysed in the neighbourhood of the threshold boundary. in the use of regularisation approach, two types of hyperplane are considered, sewing and sliding manifolds. these definitions were made using singular solution of the switching manifold [23, 24]. from these studies, it is obvious that transparent walls are not studied for sling vector field or motions because vector field cannot be defined on them. noted also is the fact that the definition of the nature of these walls have not been reviewed since its introductions. encouraged by the definition of sewing and sliding manifolds in [23, 24] with respect to singular solution an attempt is made to redefine the nature of these walls using focal points (as a limit solution) in the network. if transparent walls are not candidates for sliding motion and there is no concrete criterion for determining transparency as noted earlier, then the need arises to provide for such. secondly, if switching manifolds (walls) a more robust classification of the nature of walls can be made using focal points which are limit solutions. within the limit of our knowledge, only regular domains have been characterised in terms of number in this class of network; so, a need arises to characterise singular domains in terms number. the objective of the study is to use filippov’s first order theory to provide a criterion for transparency and in addition introduce monotonicity to check behaviour of variables at such transparent walls; derive new definitions that shall capture some intrinsic behaviour in threshold-dependent systems, especially biological systems where relapse is known to occur and characterise singular domains in terms of number. to achieve its objective, the paper is organized as follows: introduction is contained in section 1 while the system under study and the method of study are presented in section 2. in section 3, the result of the network is presented as propositions. analysis of the results is verified in this section too to demonstrate the usefulness and applicability of the study. discussion of the result of the study is done in section 4 whereas conclusion of the work is in section 5. 2. materials and methods this section is to introduce discontinuous system of the nature to be considered in this work. let σ = {x ∈<n : h(x) = 0} (1) and define a smooth function b : <n → < with the property that bx(x) , 0 on σ, where x ∈ <n. then consider a discontinuous system given as x′(t) ∈ f(x(t)) (2) with x0(0) = x0 ∈<n, where f(x(t)) satisfies f(x) =  f1(x) if x ∈ h1f2(x) if x ∈ h2 (3) and h1 = {x ∈< n : h(x) > 0} (4) h2 = { x ∈<n : h(x) < 0 } (5) . equation (1) defines the discontinuity boundary which partitions the state space into h1 , the upper half plane and h2 the lower half plane. equation (1) shall be referred to as hyperplane or wall in this work. equation (2) is known as differential inclusion problem in control problems. within h1 and h2, the state of the system is known but not on σ as defined in equation (1). at the threshold boundary (that is, the wall), define fw = c̄o{ f1(x), f2(x)} = (1 −α) f j + α fi (6) where α ∈ [0, 1] and c̄o(v ) is the smallest closed convex set containing v [9, 10]. this is the nature of the class of problems 2 mbah et al. / j. nig. soc. phys. sci. 4 (2022) 812 3 to be discussed here. definition 1. ([9, 13]). an absolutely continuous function xt : [0,τ) → <n is said to be the solution of equation (2) in the sense of filippov if for almost all t ∈ [0,τ) it holds that x0(0) = x0 and x′t (t) ∈ f(x(t)), where f(x(t)) is the closed convex hull defined as given in equation(6). the reader is referred to [13] for more on this. one of the many challenges of this class of problem is defining a solution on the threshold hyperplane with respect to x0. as a way out, closed convex hull defined as the smallest set containing a set, which is equation (6), is proposed for defining vector motion on the threshold hyperplane [13]. it uses convex combination of f1(x) and f2(x) to make definition of a vector field on the threshold hyperplane possible [12, 13]. when such vector fields are defined, then the behaviour of its trajectory in the neighbourhood of the hyperplane is used to characterise the nature of the hyperplane. this characterisation determines whether a hyperplane shall be investigated for existence of solution or not. for instance, transparent walls are considered asymptotically stable but not white walls. black walls can be stable or not [25, 26, 13]. the interest in this study is not on the stability of domains in this network but characterisation of the threshold hyperplane and singular domains respectively. however, the discussion of the results shall make use of stability of domains to highlight the importance of this study and to show that the findings of this study, with respect to nature of walls, performs better than existing ones in [13, 25]. therefore, we refer to [13] (theorems 1 and 6) for stability of domains and [17] for stability of regular steady point (rsp) and singular steady point (ssp). the stability used in this paper is in line with stability in [13] 2.1. gene regulatory networks let zi : <+ → [0, 1] be a regulatory function, where i = 1, 2, ..., n and suppose the system given below (see [32]) exists ẋi(t) = gi(x, z) = fi(z) − gi(z)xi (7) where xi = (x1, x2, ..., xn)t ∈<n represents the concentration of the ith variable, fi and gi are bounded multilinear polynomial functions defining the production and degradation rates respectively. most often, the regulatory function z is used to define the switching behaviour in gene regulatory networks. as such, either step function is used as z in which case zi : <+ →{0, 1}, or a sigmoid function is used so that zi : <+ → (0, 1). the focus of this study is on the use of step function as switching function. so, consider z(x,θ) = 1 if x > θ0 if x < θ (8) where θ is a threshold for the variable x. with equation (8), the discontinuity at the threshold boundary becomes clear. it then follows that the hyperplane of interest coincides with the threshold value of the variable of interest. definition 2 [8]: let ω = ω1 ×ω2 × ...×ωn be the partitioning of phase space of (7) by the threshold hyperplanes where ωi = {x ∈ <+ : 0 ≤ x ≤ maxi}. a domain d is defined to be a set d = d1 × d2 × ...× dn where di is one of the following: di = {xi ∈ ωi : 0 ≤ xi < θ 1 i } (9) di = {xi ∈ ωi : θ j i < xi < θ j+1 i }, j = {1, 2, ..., pi − 1}(10) di = {xi ∈ ωi : θ pi i < xi ≤ maxi} (11) di = {xi ∈ ωi : xi = θ i j}, j = {1, 2, ..., pi} (12) the threshold hyperplane partitions the state space of the network into two different domains known as regulatory and switching [8],or regular and switching [27]. regulatory (or regular) domain refers to boxes where the state of the network is known. this is given by any of equations (9) to (11). it then follows that a domain is regulatory or regular if none of its variables’ concentration is at a threshold value which can be interpreted as di = {xi ∈ ωi : xi , θi, } or di = {xi ∈: ωi : xi < θi, xi > θi} switching (or singular) domain, similarly, is where the state of the network cannot be determined because at least one variable assumes a threshold value. so, d is called a switching (or singular) domain of order k ≤ n, denoted dk, if exactly k variables have threshold values in dk. xs in this case is called a switching variable. this is an intersection of a box and hyperplane and given by a combination of equation (12), with or without, any or some of equations (9) to (11). mathematically speaking, a singular domain is defined as d = ( xs = θs, xr , θr ) where subscripts s and r denote switching (or singular) and regulatory (or regular) variables respectively. example 1 to illustrate these domains, consider a two dimensional network (that is, a network with two variables only) whose variables have one threshold each. the state space of the network is shown in figure 1. there are four regular boxes in this network given as bi i = 1, 2.3, 4 these boxes are defined as follows b1 = {x1 < θ1, x2 < θ2} b2 = {x1 > θ1 x2 < θ2} b3 = {x1 > θ1, x2 > θ2} b4 = {x1 < θ1, x2 > θ2} the switching domains for example 1 are the following d1 1 = {x1 < θ1, x2 = θ2} d2 1 = {x1 > θ1 x2 = θ2} 3 mbah et al. / j. nig. soc. phys. sci. 4 (2022) 812 4 d3 1 = {x1 = θ1, x2 < θ2} d4 1 = {x1 = θ1, x2 > θ2} d5 2 = {x1 = θ1, x2 = θ2} as the superscript denotes, d11 to d 1 4, known as walls, are of order 1 because only one variable switch at a time in them whereas d25 , called centre, is a switching domain of order two because the two variables switch simultaneously. total number of regulatory domains in a network where each xi has mi thresholds have been obtained for this class of network [28, 8]. it is given as n∏ i=1 ( mi + 1 ) the number of switching domains in this network has not been given in literature and as such, it is one of the main result of this study shall present in section 3. consider the piecewise linear model presented below ẋi(t) = βis ±(x,θ) −γi xi (13) where s ±(x,θ) = 1 − s ∓(x,θ) is same as equation (8). within a box bi, equation of motion given by (7) becomes ẋi = fi bi −γi xi (14) in vector form, equation (14) is given as ẋ = f b − γx (15) where f b and γ are diagonal matrices which give the production and degradation rate function in the system. within a box, bi, the solution of (14) is given as xi(t) = fi γi + ( x0 − fi γi ) e−γi (t−t0 ) (16) and in finite time, t →∞, the solution trajectory approaches φ bi i = fi γi (17) which is called the focal point of the box bi. the set of all focal points in the network is given in vector form as φ = {φ bi i } n i=1 it is obvious from equation (16) that solution curves of equation (14) are straight lines inside a box. these curves, which are directed towards the focal point of their respective boxes, originate from a different point on a threshold hyperplane and might show corners [29]. as observed earlier, if the solution trajectory hits a threshold wall one may be at a loss for what to do. sequel to this, a threshold wall is said to be transparent if solutions trajectory can be extended through it. that is to say, if trajectory can cross through it. a threshold wall is said figure 1: network partitioned by two thresholds θ1 and θ2 for x1 and x2 respectively to be white if solution trajectory rebounds from it while a wall is said to be black if trajectories slide on it. these definitions are good but fail to capture qualities such as relapse in biological network, where a variable can regain or loss its position in the curse of discharging its duty. for black walls, filippov proposes a way to define a vector field on it. this class of walls are considered stable [13]. but white walls are never stable. though the interest of this study is not in the stability or otherwise of these classes of domains, it is important to note that a definition that captures the actual happening within a threshold wall can really affect the stability properties of walls. for results on stability (of box and wall), see theorems 1 and 6 (respectively) in [13]. the definition of the threshold hypeprlane, depending on the behaviour of trajectory in the vicinity of the hyperplane, is of any of the following • if a threshold hyperplane is such that trajectories depart from it and enter the adjacent boxes, then it is said to be white • if a threshold hyperplane is such that trajectories approach it from either adjacent box, then it is said to black • a threshold wall that is neither black nor white is said to transparent as shall be shown later, this classification in a ’conventional sense’ does not capture intrinsic behaviours in some systems, and as a result, this study shall propose new definitions that can capture such qualities. 2.2. filippov’s first order theorem ([12]) let x ∈ σ and let n(x) be the normal to σ at x. let nt (x) f1(x) and nt (x) f2(x) be the projections of f1(x) and f2(x) onto the normal to the hypersurface σ, where nt (x) is the transpose of n(x) . (a) transversal intersection exists if, at x ∈ σ , [nt (x) f1(x)][n t (x) f2(x)] > 0 (18) (b) sliding mode exists if, at x ∈ σ , [nt (x) f1(x)][n t (x) f2(x)] < 0 (19) 4 mbah et al. / j. nig. soc. phys. sci. 4 (2022) 812 5 from equation (18), it is evident that either [nt (x) f1(x)] > 0 and [nt (x) f2(x)] > 0 or [nt (x) f1(x)] < 0 and [nt (x) f2(x)] < 0. as explained in [14] the flow will enter h1 or h2 respectively in each of those cases. for equation (19) to be satisfied, then [nt (x) f1(x)] < 0 (20) and [nt (x) f2(x)] > 0 (21) or [nt (x) f1(x)] > 0 (22) and [nt (x) f2(x)] < 0 (23) equations (18), (20) and (21), and (22) and (23) are used to characterise the hyperplane σ as transparent, black and white respectively. transparent hyperplane is same as transverse intersection given by equation (18), equations (20) and (21) gives the condition that guarantee a hyperplane to be black known as sliding hyperplane and equations (22) and (23) produces white wall called repulsive sliding hyperplane, see [14]. on the threshold hyperplane, filippov’s convex function has been defined as follows [12, 13] fw = (1 −α) f j + α fi (24) where α(x) = nt (x) f j(x) nt (x)( f j(x) − fi(x)) (25) and 0 ≤ α(x) ≤ 1. w stands for wall, i : xi < θi and j : x j > θ j filippov’s first order theory shall be used to propose new characterisation or definition for the nature of threshold hyperplanes in networks that exhibit the properties under study. these definitions shall focus on the existence of convex constant α(x) given by equation (25) which is verifiable rather than the definitions presented earlier in subsection 2.1. 3. result and analysis this section presents the result of this study. the results come in the form of proposition that will give new characterisation of walls and their nature. examples of these walls are the threshold hyperplane, x1 = θ1 and x2 = θ2 shown in figure 1 and defined as before. sequel to the presentation shall be analyses of some networks (using the new results from this study) to expose the weakness in the existing definitions thereby highlighting the significance of our study. 3.1. proposition i: nature of switching hyperplane a characterisation of switching hyperplane (that is wall) is presented using the vector field equation (14) in the boxes adjacent to the dswitching hyperplane with the assumption that nt as defined in subsection 2.2 is non-negative. let b j = {x ∈ xn : xi < θi} and bk = {x ∈ xn : xi > θi} be two boxes adjacent to a threshold wall xi = θi. given that the conditions contained in equations (24) and (25) are satisfied by (20) at the threshold hyperplane σ where xi = θi, then the nature of the hyperplane σ can be defined as any of the following • σ is said to be transparent if f b j i = f bk i (26) • σ is said to be white if f b j i > f bk i (27) • σ is said to be black if f b j i < f bk i (28) where f bτi refers to vector field equation (14) for xi in the box τ = i, j 3.2. proposition ii: nature of walls another way of characterising a switching hyperplane using the focal points of boxes adjacent to the hyperplane is presented here. let σ be a hyperplane where only one variable, xi, is singular (that is where xi = θi) and b j = {x ∈ xn : xi < θi} and bk = {x ∈ xn : xi > θi} be two boxes adjacent to σ with φ j and φk the respective focal points of the boxes adjacent to σ. then • σ is said to be transparent if xi < θi ⇒ φ j > θi and xi > θi ⇒ φk > θi (29) or xi < θi ⇒ φ j < θi and xi > θi ⇒ φk < θi (30) in the event of (29), the new definitions which this study proposes is that the wall is transparently increasing while for (30) it is said to be transparently decreasing. • σ is said to be white if xi < θi ⇒ φ j < θi and xi > θi ⇒ φk > θi (31) • σ is said to be black if xi < θi ⇒ φ j > θi and xi > θi ⇒ φk < θi (32) 3.3. proposition iii: number of switching hyperplanes the stability of network of the nature considered here has been discussed [30, 31, 13]. the equilibrium of this network revolves around the focal points in the two domains regulatory and switching respectively [13, 30]. the total number of regulatory domains has been obtained [28]. if the stability of this system can be discussed with respect to its switching domain, it will not be out of place to characterise singular domains of all orders. so, this subsection is for such result. first, number of walls is presented followed by pencils, centres and number of regulatory domains where k variables switch. 5 mbah et al. / j. nig. soc. phys. sci. 4 (2022) 812 6 3.3.1. walls let (13) be such that only one variable out of n variable switch at a time, where each variable has mi thresholds. then the number of walls belonging to xi = θi is mi n−1∏ j=1 ( m j + 1 ) (33) where j = 1, 2, · · · , i − 1, i + 1, · · · , n the total number of walls in such network is given n∑ i=1 [ mi n−1∏ j=1 ( m j + 1 )] (34) 3.3.2. pencils let equation (13) be such that each variable xi has mi thresholds at which it interacts with other variables in the network. the number of pencils in such a network where k variables switch at a time is k∏ s=1 ms n∏ r=k+1 ( mr + 1 ) (35) the total number of such pencils is n∑ i=1 [ k∏ j=1 m j n−k∏ j=k+1 ( m j + 1 )] (36) 3.3.3. centres the number of centres in a network with n variables is n∏ s=1 ms (37) 3.3.4. regular domains where k variables switch the total number of regular domains in a network where k variables switch at a time is n−k∏ i=1 ( mi + 1 ) (38) 3.4. verification of results three networks to illustrate the effectiveness and advantage of our results over existing results are presented here. to achieve this, the following network examples from [25, 30, 32] are considered. the walls of the network from [32] agrees completely with our result on the definition of walls. the second example from [25] showed a wall as decreasing instead of increasing which changes and affects the qualitative properties of the network. in the third example [32], two new non-transparent walls were discovered. these were obtained from the result of this study. 3.4.1. example networks the three example networks are as follows example 2 [28] ẋ1 = z1 + z2 − 2z1z2 −γ1 x1 ẋ2 = 1 − z1z2 −γ2 x2 example 3 [26] ẋ1 = k1z̄2z3 −γ1 x1 ẋ2 = k2z2z3 −γ2 x2 ẋ3 = k3(z̄1 + z2 − z̄1z2) −γ3 x3 example 4 [13, 27] ẋ1 = k1z 1 1 + k3z 2 2 − x1 ẋ2 = k2z 2 1 + k4z 2 2 − x2 example 2 is a network that has two variables with a threshold associated with each of the variables. using the result of equation (33), the number of thresholds walls associated with these thresholds is 1(1 + 1) × 1(1 + 1) = 4. this is as shown in figure 1. the network of this example has no pencil and just a centre. with the use of equation (8) in this example, the vector equation and their focal points inside the boxes bi, i = 1, 2, 3 and 4 as defined by equations (14) and (17) respectively are as given below. b1 : ẋ1 = −γ1 x1; ẋ2 = 1 −γ2 x2 and (0,γ −1 2 ) (39) b2 : ẋ1 = 1 −γ1 x1; ẋ2 = 1 −γ2 x2 and (γ −1 1 ,γ −1 2 ) (40) b3 : ẋ1 = −γ1 x1; ẋ2 = −γ2 x2 and (0, 0) (41) b4 : ẋ1 = 1 −γ1 x1; ẋ2 = 1 −γ2 x2 and (γ −1 1 ,γ −1 2 ) (42) threshold hyperplanes (walls) of this network as described in example 1 has the following characterisations: d1 1 = {x1 < θ1, x2 = θ2} is transparent. d2 1 = {x1 > θ1 x2 = θ2} is black. d3 1 = {x1 = θ1, x2 < θ2} is white. d4 1 = {x1 = θ1, x2 > θ2} is black. d11 is described as transparent in [32]. the focal points of the switching variable x2 in boxes adjacent to d11 ,which are b1 and b4, given by equations (40) and (41) satisfies propositions i and ii. for instance, x2 has the same focal points (which is greater than zero) in each of the said boxes. inside b1 , x2 < θ2 but φ2 > θ2 (see [32] for the value of the constant) satisfying proposition ii. again, the production function for x2, the switching variable, in the two boxes are the same (1 in each case). this satisfies proposition i as well. the same analysis can be carried out for all the other threshold hyperplanes, 6 mbah et al. / j. nig. soc. phys. sci. 4 (2022) 812 7 d21, d31 and d11 to see that the proposition is effective in characterizing threshold hyperplanes. example 3 has three (3) variables with a threshold each. the use of proposition iii shows that there are a total of twelve (12) threshold hyperplanes in the network. each of the thresholds for the variable has four (4) walls associated with it. when the result of monotonicity of transparent walls is applied to these walls, it contradicts the result presented in [25].to see this, we refer to figure 2 shown below. the use of (29) or (30) on these walls given as the edges of the cube in figure 2 reveals that the wall between the box b2 := x1 < θ1, x2 > θ2, x3 < θ2 and b3 := x1 > θ1, x2 > θ2, x3 < θ2 is not increasing in nature as shown in the flow diagram in [26]. our result shows that the wall is actually transparently decreasing which produces figure 2. from this it can be seen that the box (node), indicated as a source in [26] is not so. this node is the box b2 represented as (010) on the flow diagram shown in figure 2. the node that is a source from the result of this study is the box b3 represented as (110) on figure 2. to see clearer the usefulness of the results of this study, consider the focal point of these two boxes obtainable as (0, 0, k3γ−13 ). this means that the two boxes have the same focal point. from this focal point one sees that x1 < θ1 ⇒ φ1 = 0 < θ1 and x1 > θ1 ⇒ φ1 = 0 < θ1 which satisfies equation (30) and shows that the said wall is transparently decreasing. again, the focal point of the box b2 defined as a source by the flow diagram of [25] is 0 and contradicts that box as a source because the switching variable x1 is trapped within that box at all times. the box b3 , on the other hand, has the focal points of all the variables exiting the domain in finite time as each variable switch, showing that the said box is a source indeed. the focal points of these boxes as obtained agree quite well with those reported in [13] as well. finally, example 4 is a two-dimensional network whose variables have two thresholds each. the phase space diagram of the network is presented in figure 3. as is evident from figure 3, this network has eight boxes denoted b1 to b9. there are twelve one-dimensional domains (that is, walls) which are the edges of the corresponding boxes bounded by the threshold values, θ1 and θ2 respectively. the intersection points of the thresholds, (θi1,θ i 2) where i = 1, 2, is the centre and there are four of these centres in the network presented in example 4. see figure 3. this can be verified using equations (36) and (37) of proposition iii. each threshold has three walls associated with it while each variable has six walls associated with it. testing these walls to characterize them as black, white or transparent, using proposition i, revealed that there are two walls previously considered transparent in [13] which are actually white in nature. these are the walls between b3 and b4 on one side, and b7 and b8 on the other. to see this, observe that the vector motion for x2 which switches between the boxes b3 and b4 in these boxes are respectively ẋ2 = k2 + k4 − x2; ẋ2 = k2 − x2 (43) while for the wall belonging to x1 between the boxes b7 and b8 figure 2: flow diagram of example 3 network figure 3: phase space diagram of example 4 network the vector equations are respectively ẋ1 = k1 + k3 − x1; ẋ1 = k1 − x1 (44) using proposition i shows f b42 − f b2 2 = k4 and k4 , 0, meaning that the walls is not transparent. as such α defined by equation (25) can be found such that equation (24) can be defined on the wall. similar thing can be conducted on equation (44) to show that the wall is not transparent. this notwithstanding, proposition ii can be used to characterize the said wall. for this, the focal point of x1 in the boxes of interest are φ b8 1 = k1 + k3; φ b7 1 = k1 (45) apply proposition ii to equation (45) to see that x1 < θ11 ⇒ φ b7 1 = k3 < θ 1 1 and x1 > θ 1 1 ⇒ φ b7 1 = k1 + k3 > θ 1 1 , (see [28, 27] for details on the relationship between rate constants and threshold values). thus, a conclusion can be reached that the wall is white and not transparent. these walls though white can have filippov vector motion de7 mbah et al. / j. nig. soc. phys. sci. 4 (2022) 812 8 fined on them using equations (24) and (25). this non-unique vector motion can be obtained as follows w1 : (ẋ1, ẋ2) = (0, k4 − x2) (46) w2 : (ẋ1, ẋ2) = (k4 − x2, 0) (47) using the characterisation in propositions ii, the flow diagram shown in figure 4 is obtained. 4. discussion a regulatory domain (that is, box) in piecewise linear models of the nature discussed here can be stable or not. it is said to be asymptotically stable if its focal point is contained in the box as t →∞, otherwise it is said to be unstable. the same concept was adopted in [13] to study the stability of non-transparent walls. that is, a threshold wall is said to be stable if the focal point of the vector defining motion on it belongs to the set defined by the closed convex hull of equation (6) in finite time. if it is such that it resides on the wall as t → ∞, then it is asymptotically stable. based on this, it is expected that the qualitative properties of the network in example 3 should change due to the wrong classification of the walls described above. the stability analysis of the walls of the network which depends on the nature of walls is seen to be greatly affected. this will in turn affect cycles and some other properties of the network. it then follows that the definition proposed in this network is better and should be preferable as it can actually bring out some hidden qualities in the networks. from these qualitative properties, one can study the behaviour of the variables. as seen from the result some walls were defined in a way that does not capture the intrinsic behaviour associated with biological regulatory networks. such behaviours include relapses in elements involved in the regulatory activities. this is one of the reasons the two walls in example 4 previously thought to be transparent were not but white. our conjecture in this case is that the focal point of the switching variable could not attain the threshold boundary let alone crossing it. as it approaches the threshold vicinity, relapse occurs in the production activity which decreases rise in variable’s concentration that result in failure to stimulate switching which caused its focal point to drift away from the wall. knowledge of such properties can help in identifying where there should be search for failures in such systems the implication of equation (26) which is one of the results of this work is that if no α exists with respect to the wall of interest such that equation (24) holds, then the wall is transparent, a criterion for searching for transparent walls. also by providing equation (35), one can know switching domain of any order k < n, in terms of number. the study as presented used step function as threshold function which is limiting because it is a crude approximation of happenings within the vicinity of thresholds. smooth functions such as hill function is known to be better in studying systems with steep behaviour. we limited this study to step function because of the interest of the work which is to classify the threshold hypeplanes. it is the reason we proposed new definition figure 4: flow diagram across the walls of example 4 network based on the position of focal points in adjacent boxes to threshold walls to take care of relapses and rebounds in the vicinity of the threshold. one of the questions that arise from this study is how the new definition affects stability of singular solutions at threshold boundaries. for instance, solutions are not expected to stay on transparent wall. the two walls identified as white here were not studied for stability in [13]. also the qualitative properties too have to change. currently a work on the development of the propositions into theorems that can take care of stability and related qualitative properties is under way. furthermore, with knowledge of how many singular domains are obtained in a network, one can study the stability of the entire system by investigating the relationship between the stability of regular and singular domains. 5. conclusion the work presented in this paper is piecewise linear models of threshold dependent networks. such network is associated with rectangular regions bounded by the threshold hyperplane of the network. dynamics of the system inside these rectangular regions can be given by piecewise linear differential equations (plde). these threshold hyperplanes as defined before failed to capture the real behaviour of the network in their vicinity. by using filippov’s method we proposed a definition which identified correctly some hyperplanes not properly defined before. the consequence of our condition for transparent walls is that the first task in walls analysis should be to test for transparency or not. by so doing, one can then know which walls to investigate for qualitative properties. classifying transparent wall as decreasing or increasing reveals the behaviour of variable in the vicinity of threshold walls. this classification means that one can easily identify which variable is gaining or losing in function at the walls and as such take appropriate decision on what to do to obtain a better result at such places. the characterisation of nature of walls presented in proposition i and ii is considered from the point of view of focal 8 mbah et al. / j. nig. soc. phys. sci. 4 (2022) 812 9 point and vector field of regulatory domains adjacent to the wall respectively. it is derived from filippov’s first order theory presented in section two. furthermore, the result on switching domain obtained has not been given before to the best of our knowledge. it then follows that the results presented in this work, which is novel and 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[32] e. plahte & s. kjoglum, “analysis and generic properties of gene regulatory networks with graded response functions”, physica d 201 (2005) 155 9 j. nig. soc. phys. sci. 4 (2023) 878 journal of the nigerian society of physical sciences on nonexpansive and expansive semigroup of order-preserving total mappings in waist metric spaces i. f. usamota, o. t. wahabb,∗, s. m. alatac, k. r. tijanid a department of mathematics, university of ilorin, ilorin, nigeria bdepartment of mathematics and statistics, kwara state university, malete, nigeria c department of computer science, college of arabic and islamic legal studies, ilorin, nigeria ddepartment of mathematical sciences, osun state university, osogbo, nigeria abstract in this paper, we introduce nonexpansive and expansive semigroup of order-preserving total mappings (ontn) and (oetn), respectively, to prove some fixed point theorems in waist metric spaces. we examine the existence of mappings that satisfy the conditions ontn and oetn. we also prove that every semigroup of order-preserving total mappings otn has fixed point properties and that the set of fixed points is closed and convex. the present study generalised many previous results on semigroup of order-preserving total mappings otn. efficacy of the results was justified with some practical examples. doi:10.46481/jnsps.2022.878 keywords: fixed point, semitopological semigroup, order-preserving total mappings, waist metric space, nonexpansive map. article history : received: 20 june 2022 received in revised form: 08 november 2022 accepted for publication: 09 november 2022 published: 22 december 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: joel ndam 1. introduction in the last four decades, semigroup of mappings is one of the areas of application in fixed point theory. in algebra, a semigroup is simply a set s with an associative binary operation. a subset p ⊂ s is called a subsemigroup of s if it is closed under the binary operation on s . let xn be an ordered finite set in a standard way and let α : dom(α) ⊆ xn → xn be a self-map. the map α is called a full or total transformation of xn if dom(α) = xn. it is said to be partial if dom(α) ⊆ xn. otherwise, it is called partial one-to-one or strictly partial. the set of full transformations on ∗corresponding author tel. no: +2348035224754. email address: taofeek.wahab@kwasu.edu.ng ( o. t. wahab ) xn, denoted by tn, forms a semigroup under the composition of mappings called the full transformation semigroup. the set of partial and partial one-to-one transformations on xn, denoted by pn and in, respectively, also form a semigroup under the composition of mappings. the semigroup of order-preserving full transformation of xn is defined by otn = {α ∈tn : x ≤ y ⇒ xα ≤ yα, for all x, y ∈ xn} . let α be a transformation in otn. a point x∗ ∈ xn is said to be fixed if it coincides with the image α. for α ∈ otn, the fix of α is given by fix(α) = {x ∈ xn : xα = x}. let octn and oc∗tn be subsemigroups of otn, then a map1 usamot et al. / j. nig. soc. phys. sci. 4 (2023) 878 2 ping α ∈ octn is called a ’contraction’ if |xα− yα| ≤ |x − y| for all x, y ∈ xn (1) and a mapping α ∈ oc∗tn is called ’contractive’ if |xα− yα| ≥ |x − y| for all x, y ∈ xn (2) it is worthy to note there is a difference between ’contraction’ and ’contractive’ transformations in semigroup and deterministic fixed points. we refer to the following references for standard concepts and terminologies in the semigroup, (see [1, 2, 3, 4, 5]). if α preserves only distance (with no order), then it satisfies both (1) and (2). this is called an isometry semigroup, |xα− yα| = |x − y| for all x, y ∈ xn. (3) several fixed point results have been proved on semigroup for the family of isometries under the asymptotic nonexpansive operators [6, 7, 8, 9], lipschitzian semigroup of mappings [10, 11], commutative semigroup [12] etc. worthy to mention a few recent studies of fixed points when the parameter set of semigroups is equal to {0, 1, 2, 3, . . .} and tn = t n is the nth iterate of asymptotic pointwise contractions and asymptotic nonexpansive mappings in metric spaces (see [9, 13]). also, a procedure for constructing and finding the cardinality of orderpreserving total transformations with finite fixed points have been considered in [14, 15]. however, we observed through a survey that few or no record of results concerning the existence of the fixed points of a semigroup of order-preserving total mappings. in this respect, existence of semigroup of orderpreserving full transformation (which double as a nonlinear operator on xn) is studied in the present paper. the intuitive notion of semigroup is integrated into a more robust geometric structure to unify some results in the semigroup theory. in concrete, the paper introduces some fixed point theorems for the nonexpansive (and expansive) semigroup of order-preserving mappings to prove some existence of fixed points of the elements of subsemigroups octn (and oc∗tn). we recall banach’s contraction mapping principle [16] which has been used in many areas of applied sciences to study the existence properties of nonlinear operators. definition 1. let (e, d) be a metric space. a map t : e → e is called contraction on e if there exists a constant λ ∈ [0, 1) such that for all x, y ∈ e, d(t x, t y) ≤ λd(x, y) (4) if the condition (4) is weakened, that is λ = 1, then it reduces to a nonexpansive mapping d(t x, t y) ≤ d(x, y) (5) otherwise, it is an expansive mapping. we note that every mapping t ∈ octn is a nonexpansive mapping and every mapping t ∈ oc∗tn is an expansive mapping. the inclusion in both cases are strict. for few older results on the family of nonexpansive mappings, see [6, 17, 18, 19, 20, 21]. 2. waist metric space let α : dom(α) → im(α) be a map in otn, where dom(α), im(α) ⊂ x. the right waist and left waist of dom(α) are given, respectively, by w+(dom(α)) = max {|x| : x ∈ dom(α)} and w−(dom(α)) = min {|x| : x ∈ dom(α)} . similarly, the right and left waist of im(α) are, respectively, given by w+(α) = max {|y| : y ∈ im(α)} and w−(α) = min {|y| : y ∈ im(α)} . in view of the above, we introduce a notion of distance function with the left waist ω−(·, ·) and right waist ω+(·, ·) terms as follow: definition 2. let m be a non-empty ordered set and x be a finite subset of m. a function ω : x × x → x ∪{0} is called a right and left waist metric if for given transformation α and for each x, y ∈ dom(α) ⊆ x, the following conditions hold: w1: ω+(x, y) and ω−(x, y) are finite and nonnegative integer; w2: ω+(x, x) = 0 and ω−(x, x) = 0; w3: ω+(x, y) = ω+(y, x) and ω−(x, y) = ω−(y, x); w4: ω+(x, y) + ω+(y, z) ≥ ω+(x, z) and ω−(x, y) + ω−(y, z) ≥ ω−(x, z) for x, y, z ∈ x. the pair (m,ω)α is called a waist metric space (wms). wms is a weakening form of the canonical metric space and it is classified as pseudometric space. example 1. let x = {1, 2} ⊂ m be endowed with the waist distance ω+x (x, y) = max {|x − y| : x, y ∈ x} and α = (1)(2), then ω + x (x, y) is a waist metric on x. similarly for ω−x (x, y). example 2. let α be a total map on set x = {1, 2, 3, 4, 5} ⊂ m such that α = (1)(4)(2 1](3 4](5 4] ∈ otn, observe that im(α) = {1, 4} and dom(α) = {1, 2, 3, 4, 5}. the following are verifiable: i w+(dom(α)) = 5 and w−(dom(α)) = 1. ii both ω+x (x, y) and ω − x (x, y) are waist metric on x. remark 1. if α ∈ otn for any given set x, then ω−(x, y) = ω+(x, y). on the other hand, this is not so if α is a partial map ptn. since the main focus of this present study is on the mappings in otn, we denote ω(x, y) by a waist metric with no emphasis on left or right waist metric. 2 usamot et al. / j. nig. soc. phys. sci. 4 (2023) 878 3 2.1. completeness of (m,ω)α let {xk} be a sequence in x ⊂ (m,ω)α. since x is a finite set, the convergent of {xk} is vacuously satisfied. we present the following useful lemmas. lemma 1. a finite set x ⊂ m is a closed set. proof: let x = {x1, x2, . . . , xn} be a finite set and let x∪x′ = m, where x ∩ x′ = ∅. for each xi ∈ m, there is an ε−net such that x j ∈ b(xi,ε) ⊆ x′ for i , j. observe that each b(xi,ε) is an open ball in m. let x′ = ∪i∈∆ {b(xi,ε)}, then x′ is the union of open balls which itself is an open set in m. now, m\x′ = ∩i∈∆ {m\b(xi,ε)} = {x1, x2, . . . , xn}. that is, x = m\x′ is a complement of an open set. thus, x is closed. remark 2. in lemma 1, observe that for each y ∈ x, b(x,ε) ∩ x = {x}. this means that no point in x is an accumulation point but every point in x is an isolated point. more so, any metric on a finite space induces a discrete topology (see [22]). definition 3. let (m,ω)α be a wms and x ⊂ m. a sequence xk ∈ x is said to be a cauchy sequence in x if for given ε−net, there exist l and k with l ≥ k such that xl ∈ b(xk,ε). lemma 2. any convergent sequence in any metric space is a cauchy sequence. definition 4. a waist metric space (m,ω)α is said to be complete if every cauchy sequence in m converges to an element in m. theorem 1. let (m,ω)α be a complete waist metric space and x ⊂ m. the subspace (x,ω)α is complete if and only if x is a closed subset of m. the proof follows from lemma 1 and 2. the following concepts are versions of some results in [23, 24]. definition 5. let (m,ω)α be a waist metric space. a mapping u : m × m ×[0, 1] → m is called a convex structure on m if for all x, y ∈ m and λ ∈ [0, 1] ω(z, u(x, y,λ)) ≤ λω(z, x) + (1 −λ)ω(z, y) holds for all z ∈ m. the waist metric space (m,ω)α together with a convex structure uλ = u(x, y,λ) is called a convex waist metric space. in definition 5, a convex waist metric space (m,ω, u)α satisfies the following: ω(u(x, p,λ), u(y, p,λ)) ≤ λω(x, y), x, y, p ∈ m, λ ∈ [0, 1] ω(x, y) = ω(x, u(x, p,λ)) + ω(u(y, p,λ), y), x, y ∈ m, λ ∈ [0, 1] definition 6. a nonempty subset x of a convex waist metric space (m,ω, u)α is said to be convex if u(x, y,λ) ∈ x for all x, y ∈ x and λ ∈ [0, 1]. definition 7. a nonempty subset x is said to be p-starshaped, where p ∈ x, provided u(x, p,λ) ∈ x for all x ∈ x and λ ∈ [0, 1], that is, the segment [ p, x] = {u(x, p,λ) : 0 ≤ λ ≤ 1} joining p to x is contained in x for all x ∈ x. the set x is said to be starshaped if it is p-starshaped for some p ∈ x. clearly, each convex waist metric space is starshaped but not conversely. lemma 3. let (m,ω)α be a waist metric space. then ω2(z, u 1 2 ) ≤ 1 2 ω2(z, x) + 1 2 ω2(z, y) − 1 4 ω2(x, y) (6) for all x, y, z ∈ m. the proof follows from the parallelogram law. inequality (6) is similar to the (cn) inequality of bruhat and tits [26]. 2.2. nonexpansive semigroup of order-preserving maps let s be a semitopological semigroup and x be a nonempty closed subset of a waist metric space (m,ω)α. a family ℘ = {αs : s ∈ s} of mappings of x into itself is called a semigroup if it satisfies the following: s1: xαst = xαsαt for all s, t ∈ s and x ∈ x; s2: for every x ∈ x the mapping s → αs x from s into x is continuous. the set of all fixed points of semigroup mappings is denoted by f(αs) = {x ∈ x : xαs = x for s ∈ s}. more so, any map αs ∈ octn or αs ∈ oc ∗tn possesses the properties as stated in section one. note that (i) the set of fixed points of order-preserving maps, denoted by fo(αs ), is a subset of f(αs). (ii) fo(αs ) ⊂ f n o (αs ) for all n > 1. (iii) if f n o (αs ) is singleton for some n, so does fo(αs ). without loss of generality, the notion ’nonexpansive’ connotes ’contraction’ while ’expansive’ connotes ’contractive’. in view of the above, we present some definitions of nonexpansive semigroup of order-preserving mappings in wms under the property that both octn and oc∗tn have no common maps. definition 8. let x be a closed subset of (m,ω)α and let s be a semitopological semigroup. the map αs : x → x is called a nonexpansive semigroup of order-preserving total map ontn if for x, y ∈ x and s ∈ s , ω(xαs, yαs ) ≤ ω(x, y). (7) definition 9. let x be a closed subset of (m,ω)α and let s be a semitopological semigroup. the map αs : x → x is called an expansive semigroup of order-preserving total map oetn if for x, y ∈ x and s ∈ s , ω(xαs, yαs ) > ω(x, y). (8) 3 usamot et al. / j. nig. soc. phys. sci. 4 (2023) 878 4 3. main results the following lemma is useful in the proof of the main results. lemma 4. if ℘ = {αs : s ∈ s} is a semigroup of continuous mappings of x into itself and ω(αs x, y) → 0 as s → ∞r for x, y ∈ x, then y ∈ fo(αs) ⊂ x. proof: let ε > 0 be given. by the continuity of αt for t ∈ s , there exists τ > 0 such that ω(αt x,αty) < ε 2 whenever ω(x, y) < τ for x, y ∈ x. also, since ω(αs x, y) → 0 as s → ∞r, then there exists u ∈ s such that ω(αau x, y) < min{ ε 2 ,τ} for each a ∈ s . thus, ω(αtau x,αty) < ε 2 . now, ω(y,αty) ≤ ω(y,αtau x) + ω(αtau x,αty) < ε 2 + ε 2 = ε since ε > 0 is arbitrary, it follows that y ∈ fo(αs). remark 3. if s = n, then the hypothesis on αs would include asymptotic regularity condition. for example, see theorem 6.7. in [25]. in the next theorem, we let a(x, c̄o{αs x}) denote the asymptotic center and r(x0, c̄o{αs x} is the asymptotic radius. theorem 2. let x be a nonempty closed subset of a convex waist metric space (m,ω)α and let s be a semitopological semigroup. suppose αs ∈ ℘ is a nonexpansive semigroup of orderpreserving total mapping (7) of x into itself, that is, αs ∈ ontn for s ∈ s . if the set {αs x, s ∈ s} is bounded for some x ∈ x and y ∈ a(x, c̄o{αs x}), then y ∈ fo(αs). proof: let {αs x, s ∈ s} be a bounded net. define r := r(y, c̄o{αs x}) for y ∈ a(x, c̄o{αs x}) with the property that ω(x, y) < r. if r = 0, then lim sup ω(αs x, y) = 0 and by lemma 4, the proof is complete. on the other hand, suppose r > 0 and y < fo(αs), then for given ε > 0 and a subnet {sβ} in s , we have ω(αsβ y, y) > ε, for sβ ∈ s. also, since ω(αs x, y) → 0 as s → ∞r, then there exists γ ∈ s such that, for choosing ν ≥ 0, ω(αγ x, y) < r + ν. (9) moreover, we have by hypothesis that ω(αs x,αsy) ≤ lim sup β ω(αβx,αβy) + ν ≤ lim sup ω(x, y) + ν = r + ν (10) by lemma 3, (9) and (10), we have ω2(u,αs x) ≤ 1 2 ω2(u,αs x) + 1 2 ω2(αs x,αsy) − 1 4 ω2(y,αsy) ≤ 1 2 (r + ν)2 + 1 2 (r + ν)2 − 1 4 ε2 ≤ (r −ν)2 thus, ω(u,αs x) < r −ν which implies that r(u,αs x) < r(y, c̄o{αs x}). is a contradiction. hence, y ∈ fo(αs). necessary and sufficient result for theorem 2 is presented as follow: theorem 3. let x be a nonempty closed subset of a convex waist metric space (m,ω)α and let s be a semitopological semigroup. suppose αs ∈ ℘ is a nonexpansive semigroup of orderpreserving total mapping (7) of x into itself, that is, αs ∈ ontn for s ∈ s . the set {αs x, s ∈ s} is bounded for some x ∈ x if and only if fo(αs) is nonempty. proof: assume that {αs x, s ∈ s} is bounded for some x ∈ x, there is a unique element y ∈ x for which y ∈ a(x, c̄o{αs x}). by theorem 2, fo(αs) is nonempty. the converse is obvious. remark 4. if in theorem 2, the boundedness assumption on {αs x, s ∈ s} is dropped, then another suitable concept is stated in the next theorem. theorem 4. let x be a closed subset of a complete waist metric space (m,ω)α and let s be a semitopological semigroup. suppose αs ∈ ℘ is a nonexpansive semigroup of order-preserving total mapping (7) of x into itself, that is, αs ∈ ontn for s ∈ s . then, αs has at least one fixed point. proof: for δ ∈ (0, 1], set tδ = (1 − δ)αs. it follows that tδ is a δ-contraction on x and by the banach fixed point theorem, there exists xδ for δ ∈ (0, 1] such that tδxδ = xδ. now, we show that for δk → 0, the net xδk converges to p, where p is a fixed point of αs. indeed, for any arbitrary u ∈ x, we have ω2(xδk, u) = ω 2(xδk − p, u − p) = ω2(xδk, p) + ω 2( p, u) + 2ω(xδk − p, u − p) ≤ ω2(xδk, p) + ω 2( p, u) by setting u = αs p, we have lim sup ( ω2(xδk,αs p) −ω 2(xδk, p) ) ≤ ω2( p,αs p) (11) also, since tδk xδk = xδk and δk → 0 as k →∞, then lim k→∞ ω(xδk,αs xδk ) = lim k→∞ [ω(xδk, tδk xδk ) + δkω(0,αs xδk )] = lim k→∞ ω(xδk, tδk xδk ) → 0 (12) on the other hand, since αs ∈ ontn, then ω(αs p,αs xδk ) ≤ ω(p, xδk ) (13) we have from (12) and (13) that ω(xδk,αs p) ≤ ω(xδk,αs xδk ) + ω(xδk, p) which further implies lim sup[ω(xδk,αs p) −ω(xδk, p)] ≤ lim k→∞ ω(xδk,αs xδk ) = 0 (14) 4 usamot et al. / j. nig. soc. phys. sci. 4 (2023) 878 5 also, from (11) and (14), we obtain lim sup ( ω2(xδk,αs p) −ω 2(xδk, p) ) = lim sup ( ω(xδk,αs p) −ω(xδk, p) ) × ( ω(xδk,αs p) + ω(xδk, p) ) thus, lim sup ( ω2(xδk,αs p) −ω 2(xδk, p) ) = ω2(p,αs p) = 0 and hence, p is the fixed point of αs theorem 5. let x be a closed subset of a complete waist metric space (m,ω)α and let s be a semitopological semigroup. suppose αs ∈ ℘ is a mapping satisfying (7), that is, αs ∈ ontn for s ∈ s . then, the set fo(αs) ⊂ x is a nonempty closed convex set. proof: since αs satisfies (7), by theorem 2, αs has fixed point in x. it is left to show that fo(αs) is closed and convex. firstly, we show that fo(αs) is closed. let {xt} be a net in fo(αs) such that xt → x, then by hypothesis: ω(αt x, x) ≤ ω(αt x, xt) + ω(xt, x) → 0 this implies αt x → x ∈ fo(αs). hence, fo(αs) is closed. also, let x, y ∈ fo(αs) and λ ∈ [0, 1], we have uλ = u(x, y,λ) ∈ fo(αs). indeed, ω(αsuλ, x) = ω(αsuλ,αs x) ≤ ω(uλ, x) similarly, ω(αsuλ, y) ≤ ω(uλ, y). thus, ω(x, y) ≤ ω(x,αsuλ) + ω(αsuλ, y) ≤ ω(x, y). this shows that for some a, b with 0 ≤ a, b ≤ 1, we have ω(x,αsuλ) = aω(x, uλ) and ω(y,αsuλ) = bω(y, uλ) from which it follows that αsuλ ∈ fo(αs). by letting s = n in theorems 4 and 5, we obtain the existence property of a nonexpansive semigroup of order-preserving total mapping in waist metric spaces as follow: corollary 1. let x be a closed subset of a convex waist metric space (m,ω). suppose α ∈ ℘ = {αn : n ∈ n} is a nonexpansive semigroup of order-preserving total mapping (7) of x into itself. then. fo(αs) , ∅ if and only if {αn x : n ∈ n} is bounded for some x ∈ x. furthermore, fo(αs) is closed and convex. theorem 6. let x be a closed subset of a complete waist metric space (m,ω)α and let s be a semitopological semigroup. suppose αs ∈ ℘ = {αs : s ∈ s} is a mapping satisfying (8). then, i. the map αs exists; ii. fo(αs) is nonempty. proof: i. let x1, x2 ∈ x with x1 < x2 . suppose, on contrary, that αs < oetn, then αs is an order-reversing map, that is, x1αs ≥ x2αs. by induction, xkαs ≥ xk+1αs, for k = 1, 2, 3, . . . , n − 1 define xk+1 = xkαs. using condition (8), there gives ω(xk+1, xk+2 ) ≥ ω +(xk+1αs, xk+2αs) > ω+(xk+1, xk+2 ) this is a clear contradiction. hence, αs ∈ oetn. ii. let αs ∈ oetn. suppose that fo(αs) is empty, that is, there is no fixed β such that β ∈ fo(αs), this implies that ω+(β,βαs) > ε, for ε > 0. let ω+(xkαs,β) = 0, by condition (8), there results ω(xk,β) < ω(xkαs,βαs) ≤ ω(xkαs,β) + ω(β,βαs) = ω(β,βαs) on the other hand, since αt is continuous on x for each t ∈ s . then, for xk ∈ x and τ > 0, ω(xk,β) > τ implies that ω(xkαt,βαt) > ε 2 . since ω+(xkαs,β) = 0, then there exists b ∈ s such that ω+(xkαab,β) > max{ ε 2 ,δ} for each a ∈ s . intuitively, ω+(xkαtαab,βαt) > ε 2 . but then, ω+(βαt,β) ≤ ω +(βαt, xkαtab) + ω +(xkαtab,β) and ω+(βαt, xkαtab) + ω +(xkαtab,β) > ε from the last two inequalities, we obtain ω+(βαt,β) = ε. this is a contradiction. therefore, ω+(βαs,β) = 0 for each s ∈ s and β ∈ fo(αs). remark 5. the convergence theorems (strong convergence and ∆-convergence) of the map αs ∈ ℘ satisfying the nonexpansive semigroup of order-preserving total mapping (7) in waist metric spaces can be routinely proved using the next lemma and their left. lemma 5. let x be a closed subset of a convex waist metric space (m,ω). suppose αs is a nonexpansive semigroup of order-preserving total mapping (7) of x into itself with fo(αs) , ∅. then lims ω(αs x, y) exists for each y ∈ fo(αs) and s ∈ s . 4. practical examples the following three examples are considered to justify the theorems in the main results. the first two examples are from the same family for n = 2 and n = 3, respectively, while the third example is given for n = 3. example 3. let s be a semigroup and x = {1, 2}. let αs : {1, 2}→ {1, 2} be the mapping αs = (1)(2) in ot2, where s ∈ s , associated to xαs = x2 − 2x + 2 in x ⊂ (m,ω)α. the map αs is a nonexpansive order-preserving total mapping. hence, it satisfies all hypotheses of theorem 2 and 4 and the set fo(αs) = {1, 2}. 5 usamot et al. / j. nig. soc. phys. sci. 4 (2023) 878 6 example 4. let αs be a mapping in ot3 define by α : {1, 2, 3} → {1, 2, 3} as α = (1)[3 2 1) which is equivalent to the map xαs = x2−3x 2 + 2. here, the map αs is also a nonexpansive order-preserving total mapping, that is, αs ∈ ont3. thus, it satisfies all hypotheses of theorem 2 and 4. the set fo(αs) = {1}. example 5. let αs : {1, 2, 3} → {1, 2, 3} be given by the mapping α = (1)[2 1)(3) in ot3 corresponding to the map xαs = x2 − 3x + 3. in example 5, all hypotheses of theorem 6 are satisfied since αs ∈ oetn with fixed point fo(αs) = {1, 3}. 5. concluding remark this study introduced the nonexpansive and expansive semigroup of order-preserving total mappings to prove some fixed point theorems in complete waist metric spaces. the study also considered some examples (see examples 3, 4 and 5) on semigroup of order-preserving total mappings to validate the hypotheses of theorems 4, 5 & 6. results show that the nonexpansive (and expansive) semigroup of order-preserving mappings compares favorably with the semigroup of mappings octn (and oc∗tn). in fact, every mapping αs ∈ ℘ under the action of subsemigroups octn and oc∗tn is also in ontn and oetn, respectively. however, this study only features some elements of subsemigroups of order-preserving full mappings in [27]. therefore, future studies would be to establish some notions to study other elements of subsemigroups in ℘ such as order-preserving partial mappings [1, 28, 15], order decreasing full mappings [29], symmetric inverse semigroups [5], orientation-preserving mappings [30], fence-preserving mappings [31] among others. acknowledgements the authors wish to thank prof. k. rauf for his mentor-ship. references [1] g. u. garba, “nilpotents in semigroups of partial one-to-one orderpreserving mappings”, semigroup forum 41 (1991) 1. 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[27] a. laradji & a. umar, “combinatorial results for semigroups of orderpreserving full transformations”, semigroup forum 72 (2006) 51 springer-verlag. [28] a. laradji, a. umar, “combinatorial results for semigroups of orderpreserving partial transformations”, journal of algebra, 278 (2004), 342. [29] a. laradji, a. umar, “on certain finite semigroups of order-decreasing transformations i”, semigroup forum 69 (2004) 184. [30] i. dimitrova, j. koppitz, “on relative ranks of the semigroup of orientation-preserving transformations on infinite chains”, asianeuropean journal of mathematics, 14 (2021) 2150146. [31] r. tanyawong, r. srithus, r. chinram, “regular subsemigroups of the semigroups of transformations preserving a fence”, asian-european journal of mathematics 9 (2016) 1650003. 6 j. nig. soc. phys. sci. 4 (2022) 844 journal of the nigerian society of physical sciences graph theory: a lost component for development in nigeria olayiwola babarinsa∗ department of mathematics, federal university lokoja, kogi state, nigeria abstract graph theory is one of the neglected branches of mathematics in nigeria but with the most applications in other fields of research. this article shows the paucity, importance, and necessity of graph theory in the development of nigeria. the adjacency matrix and dual graph of the nigeria map were presented. the graph spectrum and energies (graph energy and laplacian energy) of the dual graph were computed. then the chromatic number, maximum degree, minimum spanning tree, graph radius, and diameter, the eulerian circuit and hamiltonian paths from the dual graph were obtained and discussed. doi:10.46481/jnsps.2022.844 keywords: adjacency matrix, laplacian matrix, dual graph, graph spectrum, graph energy. article history : received: 31 may 2022 received in revised form: 10 july 2022 accepted for publication: 11 july 2022 published: 20 august 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: s. fadugba 1. introduction the mathematical structures used to model pairwise relationships between things are studied in graph theory. graph theory has applications in natural science, management and social sciences, arts and agricultural science, engineering and medicine, and information systems [1, 2]. it is the key subject essentially in mastering data science and is widely adopted by tech giants like google, amazon, meta, apple, and microsoft. the considerable origin of graph theory emanated from the problem of the konigsberg bridges [3]. the city konigsberg (now kaliningrad in western russia) has two islands and mainlands with seven bridges connecting the lands. the challenge was to leave a point (home) and traverse each bridge exactly once and return to the same point [4, 5], see figure 1. ∗corresponding author tel. no: +2348060032554 email address: olayiwola.babarinsa@fulokoja.edu.ng (olayiwola babarinsa ) figure 1. the bridges of konigsberg. in 1736, swiss mathematician leonhard euler took up the problem of the seven bridges of konigsberg and reconstructed it in a dual graph where the two mainlands and the two islands represent vertices and the seven bridges represent the edge (see figure 2) and concluded that the task was not possible to do [6, 7]. his method of the proof of konigsberg bridges leads to the concept of the eulerian graph and to the introduction of a new branch of mathematics called graph theory. 1 babarinsa / j. nig. soc. phys. sci. 4 (2022) 844 2 figure 2. a graph representation of the bridges of konigsberg. in 1840, the development of graph theory started when german mathematician ferdinand mobius presented the idea of a complete graph and bipartite graph [8]. in 1852, a scottish divine francis guthrie found the famous four-colour theorem [9]. over two decades, the term ”graph” was coined in 1878 by english mathematician joseph sylvester. he represented it as partitions of numbers by nodes placed in order at the points of a rectangular lattice [9]. this paper is organized into sections. in section 2, basic graph terminologies and illustrations are given, and in section 3, we discuss the impact of graph theory and its applications with examples emanating from nigerian settings. then the adjacency matrix and laplacian matrix, and representation of the (nigeria) map graph were computed and visualized, and the results were discussed. 2. preliminary and background in graph theory graphs harmonize vertices and edges that connect these vertices. this definition and its terminologies will be meaningless to researchers who want to pick up research interest in graph theory or to apply its usefulness in other research areas without understanding the rudimentary subject matter. it is important to note that an edge is an ambiguous word that may refer to both undirected edges and directed edges. readers should be aware that in this paper whenever an ”edge” and a ”graph” are mentioned, they are referring to an undirected edge and a simple graph respectively, except otherwise stated. 2.1. graph models to model a graph, there must exist an interaction. the interaction may be a mutually exclusive interaction (called an undirected edge) or just one-way interaction (called a directed edge or arc). now, consider two or more objects (we say they are ”vertices” or ”nodes”) and the interactions or connections between these objects are referred to as edges. we are much interested, in the diagrams, whether two given vertices are joined by a line (edge or arc) but the way they are joined is less important. for instance, facebook users are vertices and their friendships are the edges, proteins are vertices and their interactions are edges, two or more elements are vertices and their bonds are the edges, particles are vertices and their collisions are the edges, footballers on the pitch are vertices and the passes made with the ball are the edges, dating apps/websites connecting people (vertices) by their interests (edges). problems in almost every conceivable discipline can be solved using graph models. graphs are widely used to describe the acquaintanceships between people, phone calls between two or more lines, links between websites, a collaboration between researchers, roadmaps, assignment of jobs, coloring (minimum color) the regions of a map, differentiating between isomers, food chain and food web, sequencing of dna, finding the shortest path between two cities, communication link between computers (mesh network), scheduling exams and assigning channels to television stations [2]. some of the modeled graphs are acquaintanceship and friendship graphs, influence graphs, collaboration graphs, call graphs, molecular graphs, web graph and citation graphs, module dependency graphs, precedence graphs, concurrent processing, airline routes and road networks, tournaments, niche overlap graphs in ecology and protein interaction graphs [2, 10]. these graph models can be created and visualized using graph software such as networkx, gephi, graphviz, sage, or matlab. 2.2. graphs definition to define a simple graph to a non-graph theorist one needs to ”unconventionally” illustrate it. for instance, a tug of war between two people is a simple graph with two vertices (people) and an edge (rope), see figure 3. figure 3. a simple graph: tug of war between two people. a simple graph is an undirected (non-directed) graph but if the edges have direction, then it is called a directed graph, mostly known as a digraph [11]. for example, see figure 4, only the rag doll (v1) exerts a greater force (arc) to pull a box (v2). figure 4. a directed graph: a rag doll pulling a box. the directed edges (arcs or arrows) of a digraph have head ends and tail ends, see figure 5. an in-degree, deg−(v), is the number of arcs (head ends ) towards a particular vertex, otherwise, it is an out-degree, deg+(v). a vertex that consists of zero deg−(v) is termed a source and a sink is a vertex with zero deg+(v) [12]. figure 5. the head and tail end of an arc. 2 babarinsa / j. nig. soc. phys. sci. 4 (2022) 844 3 in addition, if there exist both undirected edges and directed edges in a graph then the graph is referred to as a mixed graph [13]. let us consider a weight lifter (v3) with his hands exerting pressure (arcs) in balancing (edge) two ends (v1 and v2) of weight, see figure 6. figure 6. a mixed graph: a man lifting weight. mathematically, a simple graph g = (v,e) consists of a set of vertices v and a set of undirected edges e [14]. a directed graph (or a digraph) is a graph that contains only a set of directed edges with the set of vertices v [2]. a mixed graph g = (v,e,a) is an ordered triple consisting of a set of vertices v = {v1,v2...vn}, a set of undirected edges e = {e1,e2...en}, and a set of directed edges a [13, 15]. 2.3. graphs classification in considering the type of graph to use, one must know if the graph is directed or undirected, weighted or not weighted, and sometimes if it is cyclic or acyclic. another thing to consider about a graph is whether loops are permitted. a loop (or buckle) is an edge that links a vertex to itself. a pendant is a vertex that joins exactly one other vertex. an edge connects the vertices that define it. a graph with a countable number of vertices is termed a finite graph, and if the vertices are infinitely many is called an infinite graph [2]. a weighted graph or edge-labeled graph has edges (directed or undirected) having a non-negative value called weight. an unweighted graph can be considered a weighted graph if each of the edges has a numerical value of weight 1 [16]. the number of vertices (nodes or points) in g, written as v(g), is the order of a graph while the number of edges in graph g, written as e(g), is the size of a graph [17]. a vertex of degree zero is called isolated. with the terms and definitions, graphs can be classified based on the types, see table 1 and figure 7. simple graph. multigraph. pseudograph. directed graph. directed multigraph. mixed graph. figure 7. graph classification. 2.4. types of graphs many complex graphs (such as network graph) have fundamental underlaying structure the following are the types of graphs in graph theory: complete graph. cyclic graph. wheel graph. planar graph. bipartite graph. regular graph. star graph. path graph. weighted graph. mixed hourglass. acyclic (tree). acyclic (forest). trivial graph. euler graph. halmitonian graph. null graph. hypercube graph. infinite graph. connected graph. friendship graph. figure 8. basic graph types. 2.5. matrices: graph representations two types of matrices are mostly used to represent graphs; the adjacency matrix and incidence matrix. the adjacency matrix, a(g) = [ai, j], is an n×n matrix indexed by the vertices {v1,v2...vn} given as, see for instance ai, j = { 1 if viv j is an edge of g 0 otherwise. (1) the adjacency matrix can determine if the graph is connected or not by checking the shortest path for every pair of vertices. the incidence matrix: let g = (v,e) such that {v1,v2...vn} are the set of vertices and {e1,e2...em} the set of edges, then m = [mi, j] is an n×m matrix defined as [18] mi, j = { 1 when edge e j is incident with vi 0 otherwise. (2) graph spectrum is the set of graph eigenvalues of a(g) [8]. this spectrum can be used to obtain graph energy. ivan gutman introduced the concept of graph energy of a simple graph 3 babarinsa / j. nig. soc. phys. sci. 4 (2022) 844 4 table 1. summary of graph classification graph classification graph type edges multiple edges? loops? weighted or unweighted simple graph undirected no no multigraph undirected yes no directed or undirected pseudograph undirected yes yes directed graph directed no no cyclic or acyclic directed multigraph directed yes yes mixed graph directed and undirected yes yes from theoretical chemistry [19]. nikiforov [20] gave the extended concept of graph energy to any matrix. therefore, energy of a graph e(g) is the sum of absolute values of eigenvalues, λi(g); i = 1,2,...,n, of a(g) [21]. that is, e(g) = n ∑ i=1 |λi(g)|. (3) some of the forms of matrices used in graph theory are the degree matrix and (signless) laplacian matrix [22]. a degree matrix, d(g) = [di, j], is an n×n diagonal matrix defined as di, j = { 1 deg(vi) if i = j 0 otherwise. (4) furthermore, the laplacian matrix, l(g) = d(g)−a(g), has spectrum that consist of µi; i = 1,2,...,n such that µ1(l(g)) ≥ µ2(l(g)) ≥ ... ≥ µn(l(g)) ≥ 0, where l(g) is symmetric and positive semidefinite [23]. the laplacian energy is defined as le(g) = n ∑ i=1 ∣∣∣∣µi − ( 2×edges vertices )∣∣∣∣. (5) a graph g with n vertices is hyperenergetic if e(g) > 2n−2, hypoenergetic if e(g) < n and non-hypoenergetic if e(g) ≥ n [24, 10]. applications of graph energies are pattern recognition, object identification, face recognition, image analysis and processing which are widely adopted by military and police [25, 26]. in the area of medicine, attempts has been made to use it in neuronal network and epidemics see [27] and the references therein. 3. graph theory and nigeria in this section, the paucity of graph theory and representation of the nigeria map (all 36 states and f.c.t) are presented, and the results are discussed. 3.1. the paucity of graph theory applications in nigeria a graph theorist, a boundless researcher, has the most edge for collaboration. thus, it will be difficult, if not impossible, to mention fields where graph theory is not applicable. in nigeria, graph theory is one of the aspects of discrete mathematics that has not been considered beyond undergraduate and postgraduate studies, as an elective course, in mathematics, computer science, and engineering. the few published articles from nigerian researchers that mentioned graph theory only made references to the terms that apply to the research interest of the authors. presently, no nigerian mathematician has taken graph theory, especially graph energy, as his/her research interest, except perhaps the author of this paper whose erdos number and dijkstra number are 4 and 5 respectively, see reference in [28]. in the nigeria setting, for instance, linking bvn to many bank accounts or nin to sim registration is achievable by graph theory. in today’s nigerian politics, graph theory is an essential tool for political analysis. we unconsciously apply graph theory in our daily activities such as the shortest distance we take to get to our destinations. the shortest distance usually has weights, and the weighted graph has huge applications in transportation systems. almost every ministry in nigeria such as ncc, ncdc, nipost, waec, nhis, cbn, faan, fha, inec, nbs, npa, and npc can deploy graph theory for the effective performance of the agency in terms of data collection, analysis, and prediction. besides, one of the greatest uses of graph theory in nigeria is in town planning but it is not fully utilized. the paucity of graph usage can be seen in almost all sectors and places in nigeria. for example, let us consider the chaotic urbanization of lokoja (with an area of 3,180km2) where two major equipped government hospitals (kogi state specialist hospital and federal medical centre lokoja) are 2km apart of 3 minute’s drive. it is obvious that in the planning of the establishment of these hospitals, a sudden cardiac arrest of a patient was not considered for the outskirt residents of lokoja nor of other bordering local government areas. another chaotic planning in lokoja is the two tertiary institutions (kogi state polytechnic and federal university lokoja), where the two institutions are about 8.3km apart on 6 minute’s drive. furthermore, emergency exits during natural disasters (such as a flood) or terrorist attacks are not considered in planning the state capital with three entrance/exit roads that consist of only two roads (lokoja-okenne/abuja-lokoja road and ganaja/lokoja-ajaokuta road) which lead to lokoja. thus, the cons of the urbanization of lokoja outweigh its pros. this catastrophic planning can be found in each state of the country. another deficiency of graph theory in nigeria can be noticed in the examination timetable scheduling or lecture timetable in the nigerian tertiary institutions. students often complain of having two or three examinations in a day and sometimes two examinations at the same time. to solve the problem of scheduling examination timetables, graph coloring should be applied. 4 babarinsa / j. nig. soc. phys. sci. 4 (2022) 844 5 table 2. adjacency matrix of states (and f.c.t) in nigeria. v∗1 v ∗ 2 v ∗ 3 v ∗ 4 v ∗ 5 v ∗ 6 v ∗ 7 v ∗ 8 v ∗ 9 v ∗ 10 v ∗ 11 v ∗ 12 v ∗ 13 v ∗ 14 v ∗ 15 v ∗ 16 v ∗ 17 v ∗ 18 v ∗ 19 v ∗ 20 v ∗ 21 v ∗ 22 v ∗ 23 v ∗ 24 v ∗ 25 v ∗ 26 v ∗ 27 v ∗ 28 v ∗ 29 v ∗ 30 v ∗ 31 v ∗ 32 v ∗ 33 v ∗ 34 v ∗ 35 v ∗ 36 v ∗ 37 v∗1 0 0 1 1 0 0 0 0 1 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 v∗2 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 v∗3 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 v∗4 1 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 v∗5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 v∗6 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 v∗7 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 v∗8 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 v∗9 1 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 v∗10 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 v∗11 1 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 v∗12 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 v∗13 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 v∗14 1 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 v∗15 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 v∗16 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 v∗17 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 v∗18 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 1 v∗19 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 v∗20 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 v∗21 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 v∗22 0 0 0 1 0 0 1 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 0 0 0 0 0 0 0 1 v∗23 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 v∗24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 v∗25 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 v∗26 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 v∗27 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 0 0 v∗28 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 v∗29 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 0 0 0 0 0 0 0 v∗30 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 v∗31 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 v∗32 1 0 1 1 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 v∗33 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 v∗34 0 1 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 v∗35 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 v∗36 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 v∗37 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 3.2. representation of nigeria map in graph the nigeria map, including all the states and federal territory capital (f.c.t), can be represented in a graph, see figure 9. first, let the border between states be represented by edges and the states be vertices. the connection between the borders (edges) and the states (vertices) is the graph obtained from the nigeria map. figure 9. the map of nigeria with states (and f.c.t). now, let abia −v∗1, adamawa −v ∗ 2, akwa ibom −v ∗ 3, anambra −v∗4, bauchi −v ∗ 5, bayelsa −v ∗ 6, benue −v ∗ 7, borno −v∗8, cross river −v ∗ 9, delta −v ∗ 10, ebonyi −v∗11, edo −v ∗ 12, ekiti −v ∗ 13, enugu −v ∗ 14, gombe −v∗15, imo −v ∗ 16, jigawa −v ∗ 17, kaduna −v ∗ 18, kano −v∗19, katsina −v ∗ 20, kebbi −v ∗ 21, kogi −v ∗ 22, kwara −v∗23, lagos −v ∗ 24, nasarawa−v ∗ 25, niger −v ∗ 26, ogun −v∗27, ondo −v ∗ 28, osun −v ∗ 29, oyo −v ∗ 30, plateau −v ∗ 31, rivers −v∗32, sokoto −v ∗ 33, taraba −v ∗ 34, yobe −v ∗ 35, zamafara −v∗36 and f.c.t −v ∗ 37. thus, the representation of vertices and edges gives the adjacency matrix of the states (and f.c.t) in nigeria, see table 2. using the sage and graphviz packages, the representation of table 2 in voronoi diagram and dual graphs are given in figure 10, 11 and 12 respectively. 3.3. results and discussions the followings are the results and useful analysis obtained from figure 12. there is no presence of a loop since no state can share a border with herself. lagos (v∗24) is a pendant because it borders almost entirely the ocean and is suitable for a seaport. the colour number indicated on the vertices is the chromatic number. thus, the chromatic number χ(g) (the minimum number of colours required for colouring the regions of a graph) is 4. readers are advised to colour the nigeria map with 4 colours such that no two states (vertices) with the same colour share a border (edge). the largest number of neighbors of a vertex in the graph, the maximum degree, is 11 which corresponds to v∗22. that means kogi state (v∗22) shares a border with 11 other states, which makes it a commercial and geographical state of nigeria. for the commercial purpose, amazon (cadabra inc) can establish a warehouse in kogi state to minimize the cost of transportation. 5 babarinsa / j. nig. soc. phys. sci. 4 (2022) 844 6 figure 10. the voronoi diagram of nigeria map. figure 11. the dual graph of nigeria map. using kogi state as a reference, the other warehouses can be situated in kaduna state (v∗18), ogun state (v ∗ 27), abia state (v ∗ 1) and gombe state ( v∗15 ) which will not only cover five geopolitical zone of the country but also saves the company the cost to build minimum number of warehouses require to serve nigeria and neighbouring countries. these strategic states will serve other neigbouring countries, except kogi state, if the needs arise. for instance, ogun state warehouse can directly serve benin republic, kaduna state can serve part of niger republic, abia state serves part of cameroon, and gombe state can serve both chad republic and part of cameroon. figure 12. the graph of nigeria map. the connected component of the graph is 1. the weight of the minimum spanning tree (mst) is 36. the graph radius (minimum eccentricity of any vertex in the graph) and graph diameter (maximum eccentricity) from the figure are 4 (v∗7 ⇒ v∗22 ⇒ v ∗ 28 ⇒ v ∗ 27 ⇒ v ∗ 24 ) and 7 ( v ∗ 8 ⇒ v ∗ 2 ⇒ v ∗ 34 ⇒ v ∗ 7 ⇒ v ∗ 22 ⇒ v∗28 ⇒ v ∗ 27 ⇒ v ∗ 24 ) respectively. figure 12 has no euler circuit (simple circuit containing every edge) because there are more than two states( v∗2,v ∗ 3,v ∗ 8,v ∗ 16,v ∗ 21 and v ∗ 30) bordering exactly three states, or two states ( v∗14,v ∗ 23,v ∗ 29 and v ∗ 36 ) bordering exactly five states. this implies euler circuit cannot exist since the number of odd degrees is higher than two. the hamiltonian path from the dual graph indicates that a nigerian who is willing to visit all states without traversing any state twice can do so by following these paths: v∗1 ⇒ v ∗ 3 ⇒ v ∗ 9 ⇒ v ∗ 7 ⇒ v ∗ 11 ⇒ v ∗ 14 ⇒ v ∗ 4 ⇒ v∗16 ⇒ v ∗ 32 ⇒ v ∗ 6 ⇒ v ∗ 10 ⇒ v ∗ 12 ⇒ v ∗ 22 ⇒ v ∗ 25 ⇒ v ∗ 37 ⇒ v ∗ 18 ⇒ v ∗ 5 ⇒ v∗31 ⇒ v ∗ 34 ⇒ v ∗ 2 ⇒ v ∗ 8 ⇒ v ∗ 15 ⇒ v ∗ 35 ⇒ v ∗ 17 ⇒ v ∗ 19 ⇒ v ∗ 20 ⇒ v ∗ 36 ⇒ v∗33 ⇒ v ∗ 21 ⇒ v ∗ 26 ⇒ v ∗ 23 ⇒ v ∗ 13 ⇒ v ∗ 28 ⇒ v ∗ 29 ⇒ v ∗ 30 ⇒ v ∗ 27 ⇒ v ∗ 24. from table 2, the determinant of the adjacency matrix is −6822 (clockwise orientation) and the spectrum of the adjacency matrix are λ ∗ 1 =−3.0149, λ ∗ 2 =−2.5616, λ ∗ 3 =−2.3765, λ ∗ 4 =−2.3457, λ ∗ 5 =−2.1655, λ ∗ 6 =−2.1054, λ ∗ 7 =−2.0253, λ ∗ 8 =−1.9140, λ ∗ 9 = −1.7695, λ ∗ 10 = −1.7151, λ ∗ 11 = −1.5978, λ ∗ 12 = −1.4870, λ ∗ 13 = −1.3401, λ ∗ 14 = −1.3257, λ ∗ 15 = −1.2377, λ ∗ 16 = −1.1903, λ ∗ 17 = −0.7940, λ ∗ 18 = −0.6964, λ ∗ 19 = −0.5332, λ ∗ 20 = −0.3504, λ ∗ 21 =−0.2638, λ ∗ 22 = 0.0641, λ ∗ 23 = 0.1458, λ ∗ 24 = 0.2270, λ ∗ 25 = 0.6558, λ ∗ 26 = 0.6833, λ ∗ 27 = 0.9538, λ ∗ 28 = 1.0645, λ ∗ 29 = 1.4782, λ ∗ 30 = 1.8905, λ ∗ 31 = 2.2471, λ ∗ 32 = 2.6553, λ ∗ 33 = 3.0397, λ ∗ 34 = 3.3508, λ ∗ 35 = 4.0026, λ ∗ 36 = 4.7088, λ ∗ 37 = 5.6430. thus, the energy of the graph is e(g) = 65.6201, 6 babarinsa / j. nig. soc. phys. sci. 4 (2022) 844 7 which is non-hypoenergetic. to compute the laplacian energy (see equation 5) from table 2, we evaluate the degree matrix via equation 4. then its spectrum are µ ∗ 1 = 0.0000, µ ∗ 2 = 0.2841, µ ∗ 3 = 0.4619, µ ∗ 4 = 0.6405, µ ∗ 5 = 1.0892, µ ∗ 6 = 1.3870, µ ∗ 7 = 1.4319, µ ∗ 8 = 1.9546, µ ∗ 9 = 2.3059, µ ∗ 10 = 2.5968, µ ∗ 11 = 2.8290, µ ∗ 12 = 2.9777, µ ∗ 13 = 3.2076, µ ∗ 14 = 3.3923, µ ∗ 15 = 3.5828, µ ∗ 16 = 4.0672, µ ∗ 17 = 4.1943, µ ∗ 18 = 4.5491, µ ∗ 19 = 4.5880, µ ∗ 20 = 5.0092, µ ∗ 21 = 5.1446, µ ∗ 22 = 5.4446, µ ∗ 23 = 5.6693, µ ∗ 24 = 5.8715, µ ∗ 25 = 5.9542, µ ∗ 26 = 5.9889, µ ∗ 27 = 6.1590, µ ∗ 28 = 6.7064, µ ∗ 29 = 7.2008, µ ∗ 30 = 7.4258, µ ∗ 31 = 7.4721, µ ∗ 32 = 7.6544, µ ∗ 33 = 8.0111, µ ∗ 34 = 8.3096, µ ∗ 35 = 8.6034, µ ∗ 36 = 9.5428, µ ∗ 37 = 12.2923 and its laplacian energy is le(g) = 87.6227. the energies of this graph could be used by the security agencies (such as nia, dia and dss) to counter terrorism in nigeria, since the country has (land and coastline) border of about 19,382km with 1400 illegal entry points. 4. conclusion the retrogressive development in nigeria is shown by the paucity of application of graph theory. the dual graph of the nigeria map shows the connectivity between states which can be used as a transportation heuristic for road, water, and rail systems. further research implementations and applications of graph theory to the ministries, agencies, or sectors in nigeria will be important for the country to be the giant of africa. acknowledgments the author thanks the anonymous referees for positive suggestions in improving this paper. references [1] g. chartrand, p. zhang, a first course in graph theory, courier corporation, 2013. [2] k. h. rosen, k. krithivasan, discrete mathematics and its applications, mcgraw-hill education, singapore, 2015. [3] f. harary, a seminar on graph theory, courier dover publications, 2015. [4] h. sachs, m. stiebitz, r. j. wilson, an historical note: euler’s königsberg letters, journal of graph theory 12 (1988) 133. doi:https://doi.org/10.1002/jgt.3190120114. [5] s. m. cioabă, a first course in graph theory and combinatorics, vol. 55, springer, 2009. [6] j. a. bondy, u. s. r. murty, et al., graph theory with applications, vol. 290, macmillan london, 1976. [7] d. b. west, et al., introduction to graph theory, vol. 2, prentice hall, 2001. [8] a. e. brouwer, w. h. haemers, spectra of graphs, springer science & business media, 2011. [9] s. shirinivas, s. vetrivel, n. elango, applications of graph theory in computer science an overview, international journal of engineering science and technology 2 (2010) 4610. 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[22] o. babarinsa, h. kamarulhaili, on determinant of laplacian matrix and signless laplacian matrix of a simple graph, in: international conference on theoretical computer science and discrete mathematics, vol. 10398, springer, 2017, pp. 212–217. [23] d. kiani, m. mirzakhah, on the laplacian characteristic polynomials of mixed graphs, electronic journal of linear algebra 30 (2015) 135. doi:https://doi.org/10.13001/1081-3810.2959. [24] s. majstorovic, a. klobucar, i. gutman, selected topics from the theory of graph energy: hypoenergetic graphs, applications of graph spectra, math. inst., belgrade (2009) 65–105. [25] z. huigang, b. xiao, z. huaxin, z. huijie, z. jun, c. jian, l. hanqing, hierarchical remote sensing image analysis via graph laplacian energy, ieee geoscience and remote sensing letters 10 (2012) 396. [26] e. s. leni, et al., a technique for classification of high resolution satellite images using object-based segmentation., journal of theoretical & applied information technology 68 (2014) 275. [27] i. gutman, b. furtula, graph energies and their applications, bulletin (académie serbe des sciences et des arts. classe des sciences mathématiques et naturelles. sciences mathématiques) 44 (2019) 29 [28] i. gutman, h. ramane, research on graph energies in 2019, match commun. math. comput. chem 84 (2020) 277. 7 j. nig. soc. phys. sci. 5 (2023) 983 journal of the nigerian society of physical sciences health risk assessment of heavy metals in sediment of tropical freshwater stream godwin o. olutonaa,b,∗ aindustrial chemistry programme, college of agriculture, engineering and science, bowen university, iwo, nigeria bschool of basic science, kampala international university, western campus, ishaka, uganda abstract an investigation of the heavy metals in the bed sediment of asunle stream was carried out to assess how seriously the sediment is polluted using inductively coupled plasma optical emission spectroscopy (icp-oes). the potential health risk assessment was calculated for a lifetime exposure (ingestion) based on the united state environmental protection agency (usepa) models to determine the carcinogenic and noncarcinogenic risks for children and adults. the range of values (mg/kg) of heavy metals in bed sediment were: fe (2850 – 7260), mn (58 – 209), co (0.7 – 33), ti (21.6 – 67), ba (1.61 – 9.81), zn (7.5 – 79), cu (5.6 – 25), as (8 – 137), al (273 – 2160), y (24 – 49), and sr (0.10 – 5.3). as and sr, values were below the background values for typical soil. the health risk assessment of heavy metals in the bed sediments revealed that carcinogenic risk was almost insignificant while the non-carcinogenic risk was significant since their values were above the recommended minimal risk level. the results also revealed that children are more vulnerable to hazards than adults. the chronic hazard quotient index for exposure to these metals through ingestion exceeded the acceptable usepa value of 1.0. doi:10.46481/jnsps.2023.983 keywords: contamination, toxicology, pollution, public health, asunle stream article history : received: 10 august 2022 received in revised form: 02 december 2022 accepted for publication: 19 december 2022 published: 24 february 2023 © 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: t. owolabi 1. introduction over time pollution from both point and non-point sources have been a great challenge to water bodies [1] all over the world, and sediment has been the reservoir of these pollutants. the obafemi awolowo university dumpsite had been reported to be polluted with the heavy metals and non-metals, metalloids, actinides and rare earth metals at varying degrees [2]. leaching of these elements into the adjoining stream is inevitable the distribution of metals in the sediment of a tropical stream ∗corresponding author tel. no: +234 8132406932; +256 726393978 email address: delog2@gmail.com (godwin o. olutona) adjoining a university dumpsite, and runs through human settlements with heavy agricultural activities can provide researchers with proof of the anthropogenic impact on the ecosystem, and thus aid in assessing the ecological risk to the aquatic habitats and toxic risks on terrestrial habitat. the build-up of heavy metals in sediment has significant environmental implications on water quality and local inhabitants [3]. the impact of open dumpsite on groundwater contamination with heavy metals was investigated by coker et al. [4]; their results revealed contamination of groundwater as a result of leachate from the dumpsite.. ogunfowokan et al. [5] earlier conducted an early wet speciation study of heavy metals in the water and sediment of 1 olutona / j. nig. soc. phys. sci. 5 (2023) 983 2 the asunle stream using atomic absorption spectroscopy (aas). contamination factors of the study area were not fully explored. this study is a follow-up monitoring of the perennial stream with a comprehensive seasonal study and the use of icp-oes to further established the contamination levels of the adjoining stream to the obafemi awolowo university dumpsite. the study further fully explored the pollution status of the stream. health risk assessment of the stream had not been conducted in the previous study. the objectives of this study was to investigate the metal concentrations of the bottom sediments using icp-oes; their contamination factors and conduct the health risk assessment. 2. materials and methods 2.1. study area asunle, an adjoining stream is a periodic stream that has its spring located about 100 m uphill from the obafemi awolowo university (oau), ile-ife refuse dumpsite (figure 1).the university dumpsite falls between latitude 07° 32´n and longitude 4° 31´e. geologically, the study area falls within the basement complex of southwestern nigeria. it forms part of african crystalline shield with which consists predominantly of dolerites, apitite, microgranite genesis, granite genesis, ultrasonic rocks, mica and banded genesis. the characteristic of wastes generated in this dumpsite had been reported by olutona et al[6] the stream was about 100 m away from the major road that runs a stretch of more than 10 km, cutting across three human settlements [5]. it has the highest vegetation cover (about 90%) of all the sites. noticeable human activities along the stream are rigorous farming activities where cash crops (such as palm trees, cola nut, and cocoa) and various food crops were planted. oil palm processing took place in all the human settlements along this stream. along the course of the stream, water from the stream is used by the inhabitants who are predominantly farmers for palm oil processing, irrigation of fruits and vegetables, mixing and dilution of agrochemicals used for spraying of both cash and food crops. downstream, the water from the stream is utilized for household purposes [6-7]. 2.2. sample collection and analysis five sites were chosen for this study along the course of the stream. sediment samples were collected on a monthly basis for the period of eight (8) months (4 months dry and 4 months wet). sediment sampling protocol described by iaea [8] was employed in this study. sediment samples were collected from the upstream (source as control), point of discharge and downstream sampling points of receiving stream (figure 1). composite sediment samples (two samples) were randomly collected (10-20 m apart) at 0-3 cm depth each month from each location. sediment samples were collected using stainless steel scoop facing upstream. excess water was drained from the scoop. black cellophane nylon beforehand cleaned with pure acetone was used to collect the sediment samples. the samples were air dried for about five days and further dried at 50 ºc in a vacuum oven to ensure that the samples were completely figure 1. a map of asunle an adjoining stream to obafemi awolowo university dumpsite dried. the dried samples were crushed, ground and sieved with 500-micron plastic sieve. the samples were stored in clean 250 ml capacity amber bottles kept in the refrigerator until further analysis was required. one (1 g) each of the samples was acid digest using nitric acid (5 ml) and perchloric acid (1 ml) acid. the digested samples were concentrated to a volume (< 2 ml) transferred into a volumetric flask (25 ml) and topped up with distilled water. the digested solutions were analysed using icp-oes. the agilent, varian 710 icp-oes equipped with axiallyviewed plasma available at the department of nano science, university of the western cape, cape town, south africa was used for metal analysis in this study. the axially-viewed plasma cover all-important wavelength in the visible region from 177785 nm. the icp-oes is equipped with ccd detector and optimized optical design that give excellent signal-to-noise performance, ensuring low detection limit. the ccd features the clocked recombination system (crs) for anti-blooming protection. to enhance the performance of the 710 series are the accessories such as vga for mercury and hydride forming elements, the fast sps auto sampler for unattended automation, the svs switch valve for rinsing and improve productivity, the agm for organic matrices and the usn for lower detection limit with environmental sample. the choice of icp-oes is due to its analytical advantage over other excitation sources originates from its capability for efficient and reproducible vaporization, atomization, excitation and ionization for a wide range of elements in various sample matrices. this is basically due to high temperature 60007000k, in the observation zones of the icp which is much higher than the maximum temperature of flames or furnaces (3300k). this instrument also makes it capable of exciting refractory elements and renders it less prone to matrix interferences. similarly, the choice of axial rather than radial view icpoes is due to the fact that the axial view provides better lods than radial view. this may be attributed to longer viewing path 2 olutona / j. nig. soc. phys. sci. 5 (2023) 983 3 available down the axis of the plasma, thus a better sensitivity of 5-10 fold improvement in the lod can be achieved. 2.3. calculation of pollution assessment indices in this study, geo-accumulation index, enrichment factor, contamination factor and pollution index have been applied to assess heavy metals distribution and contamination in the sediment samples from asunle stream. 2.3.1. enrichment factor (ef) enrichment factor (ef) analysis is a method used to differentiate between the metals originating from anthropogenic activities and those from natural sources and to assess the degree of anthropogenic influence [9]. the ef is defined as follows: ef = [ cx cal ] sample[ c x cal ] background (1) where, [cx/cal] sample is the ratio of metal [cx] to that of al [cal] in the soil /sediment sample and [cx/cal] background is the ratio of metal and al concentration of the geochemical background. the geochemical background values of metals are not available. thus, the geochemical average shale values given by turekian and wadepohl [10] were adopted. seven contamination categories are recognized based on enrichment factor as follows [11]: 2.3.2. geo-accumulation index the geo-accumulation index (i-geo) values were calculated for different metals as introduced by muller[12] as follows: i-geo = log2{ cn [1.5 bn] } (2) where cn is the measured concentration of the heavy metal ‘n’ in the sample and bn is the geochemical background value of element ‘n’ and 1.5 is the background matrix correction factor. 2.3.3. contamination index the calculation of contamination index of metals in the sediment samples was done using the relationship: contamination index (ci) = metal concentration in the soil background value o f the metal (3) 2.3.4. pollution load index each location was evaluated for the extent of metal pollution by employing the method based on the pollution load index (pli) developed by thomilson et al. [13] as follows: pli = n √ cf1 × cf2 × cf3 × cf4 . . . . . . .cfn (4) where n is the number of metals studied and cf is the contamination factor calculated as described in equation 3. 2.4. health risk assessment the basic formulas and values used for the calculation of ingestion and inhalation of soil as described by grezetic et al. [14] are shown below. chronic daily intake (cdi) for carcinogenic risk (ingestion of soil): carcinogenic cdi (mg/kg/day) = cs xi f x ef at (5) where if = iradult× edadult bwadult + irchild × edchild bwchild (6) lnon-carcinogenic: cdi (mg/kg/day) = cs × i n × ef × ed bw × at (7) 3. results and discussion 3.1. validity of analytical method adopted for heavy metal analysis the linear calibration curve for each metal was plotted and each near unity. blank and internal standard were also conducted to authenticate the results and to check for background contaminants. table 1 presents the limits of detection (lod) and quantification (loq) of the elements. 3.2. heavy metals in asunle stream sediment this section of the study presents the metal content in the asunle stream bed sediment. generally, the concentrations of metals in the bed sediment were significantly lower at p < 0.05 when compared with either those of the dumpsite soils or the lateral soil sampling towards the receiving stream. the data obtained (table 2) were subjected to duncan multiple range test to establish possible differences in all the sampling periods. the statistical analysis showed that except for fe, mn, ba, cu, and y, all other metals had significant difference in all the sampling periods. biologically, iron (fe) plays a crucial role in the transport and storage of oxygen, and also in electron transport [15 ]. it is safe to say that, with only a few possible exceptions in the bacterial world, there would be no life without iron [16]. the monthly mean level of fe ranged from 2850 ± 1600 mg/kg in august to 7260 ± 5400 mg/kg in february. these values were below the background value of 47,200 mg/kg in shales [10] and 26,000 mg/kg elemental concentration of typical soil [17]. manganese is an indispensable element in human food with a normal nutritional intake considered to be nearly 2 –5 mg/day [18]. it is a constituent of certain enzymes and can also activate many enzymes [19]. manganese monthly mean levels ranged from 58 ± 50 mg/kg in august to 210 ± 210 mg/kg in february. these values were below the background value of 850 mg/kg in shales [10] and 550 mg/kg elemental concentration of typical soil [17]. cobalt means levels ranged from 0.73 ± 1.74 mg/kg in december to 33 ± 21 mg/kg in august. the values obtained in the 3 olutona / j. nig. soc. phys. sci. 5 (2023) 983 4 table 1. validation of analytical methods fe mn cr ni co ti zr zn cu be λ (nm) 238.2 257.6 267.7 221.6 238.89 336.12 343.82 213.86 324.75 313.04 %rsd 0.71 1.41 1.05 1.07 0.01 0.84 0.67 0.59 0.67 0.86 lod 0.006 0.003 0.0009 0.03 0.018 0.0012 0.015 0.00 0.002 0.001 loq 0.06 0.03 0.009 0.3 0.18 0.012 0.15 0.00 0.02 0.01 dry season were low when compared to the background level of 19 mg/kg in shales [10] and 9.1 mg/kg elemental concentration of typical soil [17]. the monthly mean levels of ti ranged from 21.6 ± 5.6 mg/kg in november to 67 ± 77 mg/kg in february. these values were below the background value of 4600 mg/kg in shales [10]. barium monthly mean values ranged from 1.61 ± 3.94 mg/kg in july to 10 ± 16 mg/kg in february. these values obtained were below the background value of 580 mg/kg in shales [10]. zinc is an indispensable element essential for the life processes of several enzymes. zn impedes at diverse levels in the endocrine system and lipids and carbohydrate metabolism [20]. at higher levels zinc may be carcinogenic [21]. the monthly mean values of zn ranged from 7.53 ± 9.31 mg/kg in august to 79 ± 46 mg/kg in january. these values were below the background value of 95 mg/kg in shales [10] and 60 mg/kg elemental concentration in typical soil except values obtained in december and january that were above the elemental concentration in typical soil[17]. copper is a vital metal to human life at modest levels operational as part of some enzymes e.g., tyrosine (necessary for the formation of melanin pigments), cytochrome oxidase, superoxid dismutase, and amine oxidases. it is essential for the utilization of iron in the formation of haemoglobin [22]. cu monthly mean values ranged from 5.63 ± 7.16 mg/kg in july to 25 ± 43 mg/kg in february. the values were below 45 mg/kg background value in shales7 and 25 mg/kg elemental concentration in atypical soil [17]. arsenic is a metalloid, poisonous [23] classified as carcinogenic, and harmful to human healthiness. in addition to natural origin, it can also be predominant in the soil in an area where there are mining-related activities; municipal sewage, coal burning, agrochemicals, fertilizers, vehicular emissions and wood preservative chemicals and industrial effluents [24-26]. the monthly mean values of as ranged from 8.42 ± 21 mg/kg in december to 137 ± 100 mg/kg in august. except for december, the values obtained were above the background value of 13 mg/kg in shales7 and 7.2 mg/kg elemental concentration in atypical soil [17]. the monthly mean values of al ranged from 273 ± 288 mg/kg in august to 2160 ± 610 mg/kg in january. the values were below 80,000 mg/kg background value in shales [10]. yttrium was only above the detection limit in january and february and ranged between 24 ± 60 mg/kg and 49 ± 120 mg/kg. the values were above the background value of 26 mg/kg in shales7. strontium ranged from 0.10 ± 0.17 mg/kg in june to 5.25 ± 2.45 mg/kg in january. the values were below the background value of 300 mg/kg in shales [10]. figure 2. seasonal levels of heavy metals in sediment of asunle stream 3.3. seasonal levels of heavy metals of asunle stream sediment seasonal variation of heavy metals of asunle stream sediment is presented in figure 2. the data obtained revealed that all the metals with exception of co and as were generally higher in the dry season. moreover, the concentrations of fe and al were exceptionally higher than all other metals in both seasons. higher levels of these metals in dry season might be due to the slower movement of water in the stream during the dry season resulting in greater settlement chances of the metalbound sediment particles. also, the lower levels of the metals in the bed sediment during the wet season compared to the dry season might be due to dilution arising from high precipitation. 3.3.1. principal component analysis of heavy metals in sediment of asunle stream the principal component analysis (pca) of heavy metals in the bed sediment (figure 3), the total variance for the two components was explained by 55.32%. the first component accounted for 39.23% of the explained variance in which fe, al, ba and mn recorded high positive loadings of 0.86, 0.79, 0.78 and 0.72, respectively. the second component accounted for 16.09% of the explained variance and only as had the highest loading 0.74. in terms of association, four groups can readily be identified. these are zn, sr, cu and al; mn, ba, fe and ti; as and co; and y. the metals that are associated probably have a common origin or source. 3.4. pollution assessment indices 3.4.1. geo-accumulation index of heavy metals in sediment of asunle stream the geo-accumulation index of heavy metals in the bed sediment of asunle stream is presented in table 3. the results ob4 olutona / j. nig. soc. phys. sci. 5 (2023) 983 5 table 2. monthly levels of heavy metals in sediment of asunle stream fe mn co ti ba zn cu as al y sr nov 3530a ±1900 68a ±45 1.87a ±3.67 21.6a ±5.6 bdl 49abc ±31 8.87a ±11 14bc ±34 1270d ±650 bdl 2.53ab ±1.31 dec 5130a ±3100 110a ±65 0.7a ±1.7 32ab ±12 2.9a ±7.0 69bc ±64 18a ±32 8bc ±21 1650a ±770 bdl 3.9ab ±3.8 jan 6850a ±3600 124a ±95 2.9a ±5.3 31ab ±12 5.96a ±12 79c ±46 14a ±25 15c ±37 2160a ±610 24a ±60 5.3b ±2.5 feb 7260a ±5400 209a ±210 5.0a ±8.6 67b ±77 9.81a ±16 31ab ±49 25a ±43 33ab ±46 1070ab ±740 49a ±120 2.7ab ±4.5 may 4970a ±3900 96a ±118 25b ±14 59ab ±43 7.73a ±19 12a ±16 24a ±36 114bc ±39 1430bc ±1300 bdl 0.44ab ±1.10 jun 4970a ±3000 124a ±150 21b ±13 28ab ±15 3.1a ±7.6 16a ±16 13a ±15 46ab ±86 997ab ±550 bdl 0.10a ±0.17 jul 3320a ±2500 74a ±68 31b ±18 25ab ±17 1.6a ±3.9 8.6a ±9.5 5.6a ±7.2 62ab ±95 701abc ±600 bdl bdl aug 2850a ±1600 58a ±50 33b ±21 27.4ab ±4.8 bdl 7.5a ±9.3 13a ±14 137a ±100 273c ±290 bdl 3.2ab ±8.1 the alphabet in each column denotes mean that are significantly different (p < 0.05) bdl = below detection limit figure 3. pca of heavy metals in sediment of asunle stream tained revealed that the sediments were practically unpolluted with the metals during both seasons. arsenic, however, demonstrated a moderate to heavily pollution status of the sediments during the dry season. 3.4.2. enrichment factor/index of heavy metals in sediment of asunle stream the enrichment factor (ef) for the heavy metals in the bed sediments is presented in table 4. these values were obtained using the geochemical background values in average shale [10]. the enrichment status of the soil based on taylor [11] proposition revealed that the sediment was not enriched with ti, ba and sr. on the other hand the enrichment of the sediment with fe, mn during the two seasons and co in dry season were moderately severe. enrichment of zn in wet season and cu in dry season were severe. similarly, enrichment of the sediment with zn and y in dry season and cu in wet season were very severe while those of co in wet season and as in both seasons were extremely severe. 3.4.3. contamination factor with respect to temporal variation of heavy metals in sediments of asunle stream the monthly and seasonal contamination factors (cfs) for each metal were calculated according to equation 3. and presented in table 5. four groups of classification of cf were described by nase et al. [27]; and mmolawa et al.,[28]. these are: cf < 1 (low contamination); 1≤ cf < 3 (moderate contamination); 3 ≤ cf < 6 (considerable contamination), and cf > 6 (very high contamination). the results of contamination factor of temporal variation of metals in bed sediment of asunle river revealed that the mean value varied between 0.0007(ti) and 0.60 (zn) in wet and 0.003(sr) and 6.9(as) in dry. these results indicated that the contamination due to fe, mn, ti, ba, zn, cu, al, sr in both seasons and co in the dry season were low. the sediment was moderately contaminated with co in the wet season and as in the dry season while the contamination of the sediment with respect to as in the wet season stood between considerable contamination and high contamination. the contamination status of the sediments with respect to the metals in decreasing order in dry season was: as > y > zn > cu > mn > co > al > sr > ba > ti while in wet season, the order was: as > co > cu > zn> mn > fe > al > ti > ba > sr. 3.4.4. contamination factor with respect to temporal variation of heavy metals in sediments of asunle stream the monthly and seasonal contamination factors (cfs) for each metal were calculated according to equation 3. and presented in table 5. four groups of classification of cf were described by nase et al. [27]; and mmolawa et al.,[28]. these are: cf < 1 (low contamination); 1≤ cf < 3 (moderate contamination); 3 ≤ cf < 6 (considerable contamination), and cf > 6 (very high contamination). the results of contamination factor of temporal variation of metals in bed sediment of asunle river revealed that the 5 olutona / j. nig. soc. phys. sci. 5 (2023) 983 6 table 3. geo accumulation index of heavy metals in the sediment of asunle stream cn i-geo class pollution intensity dry wet bn dry wet dry wet dry wet fe 5690 4030 47200 -3.64 -4.14 0 0 pu pu mn 128 87.9 850 -3.32 -3.86 0 0 pu pu co 2.64 27.7 19 -3.43 -0.04 0 0 pu pu ti 38 34.8 4600 -7.50 -7.63 0 0 pu pu zn 57.2 11.2 95 -1.40 -2.10 0 0 pu pu cu 16.6 13.9 45 -2.03 -2.28 0 0 pu pu ba 4.66 3.11 580 -2.27 -8.13 0 0 pu pu al 1540 850 80000 -6.29 -7.14 0 0 pu pu as 17.8 89.8 13 -0.13 2.20 0 3 pu mp-hp sr 3.58 0.97 300 -6.97 -8.86 0 0 pu pu y 18.32 26 -1.09 0 0 pu key pu= practically unpolluted, mp-hp = moderately to heavy polluted table 4. enrichment factor of heavy metals in sediment of asunle stream cn enrichment factor dry wet bn dry wet fe 5690 4030 47200 6.27 8.03 mn 128 87.9 850 7.83 9.74 co 2.64 27.7 19 7.22 137 ti 38 34.8 4600 0.43 0.71 ba 4.66 3.11 580 0.42 0.50 zn 57.2 11.2 95 31.3 11.1 cu 16.6 13.9 45 19.1 29 as 17.8 89.8 13 71 650 y 18.3 26 36.6 nd sr 3.58 0.97 300 0.62 0.30 key enrichment status: < 1 = no enrichment, 1-3 = minor, 3-5 moderate, 5-10 = moderately severe, 10-25= severe, 25-30 = very severe, and > 50 = extremely severe. mean value varied between 0.0007(ti) and 0.60 (zn) in wet and 0.003(sr) and 6.9(as) in dry. these results indicated that the contamination due to fe, mn, ti, ba, zn, cu, al, sr in both seasons and co in the dry season were low. the sediment was moderately contaminated with co in the wet season and as in the dry season while the contamination of the sediment with respect to as in the wet season stood between considerable contamination and high contamination. the contamination status of the sediments with respect to the metals in decreasing order in dry season was: as > y > zn > cu > mn > co > al > sr > ba > ti while in wet season, the order was: as > co > cu > zn> mn > fe > al > ti > ba > sr. 3.5. spatial variation of heavy metals in sediment of asunle stream the spatial variation of metals in bed sediments of the asunle stream is presented in table 6. the results obtained when subjected to anova showed that all the metals in all locations were significantly different (p < 0.05) from one another with respect to co, ti, as, and sr. this clearly indicated the nonuniformity in the levels of the metals found in the bed sediment. this could be due to uneven anthropogenic inputs of these metals, differential distribution by the flowing stream, the unequal sequestering influence of sediment components to precipitate the metals along the watercourse, etc. similarly, the data were also subjected to duncan multiple range tests to establish possible differences in the mean values of metals in each location. the statistical analysis showed that except for co, as and sr, virtually all the metals had significant differences in all the locations investigated. the total metal burden at the control point (location 0) was generally low compared to other locations. this could be since this location is situated upstream some distance away from the dumpsite. the source of metal pollution at location 0 might be traced to natural sources, agro-allied chemicals used by the farmers and deposition of metal containing fly ash coming from incineration of solid waste. location 1 is the closest point on the stream bank from the dumpsite. it is at a lower gradient with respect to the dumpsite. this could explain why the potentially 6 olutona / j. nig. soc. phys. sci. 5 (2023) 983 7 table 5. contaminant factor of heavy metals in sediment season fe mn co ti ba zn cu as al y sr nov dry 0.07 0.08 0.10 0.005 0.52 0.20 1.08 0.02 0.008 dec 0.11 0.13 0.04 0.007 0.005 0.73 0.41 0.65 0.02 0.01 jan 0.15 0.15 0.15 0.007 0.01 0.83 0.32 1.18 0.03 0.94 0.02 feb 0.15 0.25 0.27 0.01 0.02 0.33 0.55 2.57 0.01 1.88 0.009 mean 0.12 0.15 0.14 0.007 0.009 0.60 0.37 1.37 0.02 0.71 0.01 may wet 0.11 0.11 1.34 0.01 0.01 0.13 0.53 8.78 0.02 0.001 jun 0.11 0.15 1.09 0.006 0.005 0.17 0.29 3.53 0.01 0.0003 jul 0.07 0.09 1.64 0.005 0.003 0.09 0.13 4.77 0.009 aug 0.06 0.07 1.76 0.006 nd 0.08 0.28 10.52 0.003 0.01 mean 0.09 0.11 1.46 0.007 0.005 0.12 0.31 6.9 0.01 0.003 toxic metals burden of this location had generally higher values than other locations. from location 1, there was a general decrease in concentrations of the metals downward the stream with exception of location 4 which had a higher metal concentration than location 1. this occurrence might be due to the flat topography of location 4 which could encourage better and more effective sedimentation of metal containing particulates. the fe concentrations having a range of 1620 ± 630 mg/kg at location 0 to 8310 ± 2700 mg/kg at location 4 were below the background value of 47200 mg/kg in shales [10], and 26000 mg/kg elemental concentration of a typical soil[17]. manganese is a vital nutrient and naturally occurring element, useful in steelmaking, fireworks, fertilizers, chemicals, glass, and drycell batteries production, textile and leather industries. its presence in soil results in vegetable and animal foods reliably containing varying amounts of the mineral [29]. manganese mean concentration ranged from 18 ± 18 mg/kg at location 0 to 244 ± 110 mg/kg at location 4. the values obtained were below the background value of 850 mg/kg in shales [10], and 550 mg/kg elemental concentration in a typical soil[17]. cobalt in sediment could be of natural and anthropogenic origins. the anthropogenic sources could be as a result of phosphatebased fertilizer application, smelting, sewage sludge, alloys, and mining. in water, cobalt is basically settled in the bottom sediment while some may be adsorbed by suspended solids in the water column [29]. cobalt ranged from 10 ± 11 mg/kg at location 3 to 23 ± 22 mg/kg at location 2. these values were very low with exception of location 1 and 2 compared to 19 mg/kg in shales [10]. the mean concentrations of co in all the locations were above 9.1 mg/kg elemental concentration of a typical soil [17]. the mean range value of ti was between 18.3 ± 7.1 mg/kg at location 5 and 59 ± 67 mg/kg at location 2. these values were below the 4600 mg/kg background value in shales [10]. barium levels ranged from 0.65 ± 1.9 mg/kg at location 5 to 16 ± 17 mg/kg at location 4. these values were below the background value of 580 mg/kg in shales [10]. zinc is abundant in the environment, instituting 20–200 ppm (by weight) of the earth’s crust [29]. the mean concentration of zn ranged from 13 ± 16 mg/kg at location 0 to 84 ± 69 mg/kg at location 1; the values were below the background value of 95 mg/kg in shales [10], and 60 mg/kg elemental concentration of a typical soil [17] with exception of location 1. copper found their way into water bodies by natural weathering of soil and rocks or anthropogenic sources [29]. copper mean concentrations ranged from 5.9 ± 4.3 mg/kg at location 3 to 17 ± 30 mg/kg at location 0. the values of cu at all locations were lower except at location 1 when compared with the background value of 45 mg/kg in shales[10], and 25 mg/kg elemental concentration in a typical soil[17].the levels of as in the bed sediment ranged from 9 ± 26 mg/kg at location 2 to 83 ± 80 mg/kg at location 4. the mean values of as were higher than the background value of 13 mg/kg in shales[10] except at location 2 where the value was higher. the mean concentrations of al ranged from 667 ± 400 mg/kg at location 3 to 1590 ± 970 mg/kg at location 1. the mean values of al were low compared with the background value of 80,000 mg/kg in shales[10]. yttrium was only above the detection limit at location 0 and the value was above the background value of 26 mg/kg in shales [10] strontium in bed sediment ranged from 0.8 ± 1.3 mg/kg at location 0 to 4.4 ± 4.9 mg/kg at location 1. the mean values of sr obtained in this study were far below the background value of 300 mg/kg in shales[10]. generally, the metals contamination levels in the bed sediment were wide-ranging significant among the stream site sampling locations. potential toxic metals in bed sediments are either lithogenic or anthropogenic. the total metal burden at the control point was generally low compared to other locations. however, this could not be totally attributed to the lithogenic effect since there was the possibility of atmospheric distribution as a result of incineration of dumped solid waste. the total metal burden in the bed sediments of locations 1 to 5 showed the influence of anthropogenic inputs because of the leaching of these metals from the dumpsite into the receiving stream. since bed sediment acts as both carrier and source of contamination in the aquatic environment. high contaminations of the bed sediment with metals may have adverse effects on aquatic habitats; hence, remediation of the bed sediment is highly required. 3.5.1. contaminant factor of heavy metals with respect to their spatial variation in sediment of asunle stream table 7 is presentation of monthly and seasonal contamination factors (cfs) for each heavy metal. the results revealed 7 olutona / j. nig. soc. phys. sci. 5 (2023) 983 8 table 6. spatial variation of heavy metals in sediment of asunle stream site fe mn co ti ba zn cu as al y sr total metal burden 0 1620a ±630 18a ±18 10a ±14 40ab ±19 bdl 13a ±16 17a ±30 41a ±73 1280abc ±730 55a ±110 0.8a ±1.3 3090 1 7440cd ±3400 117bc ±100 21a ±24 59b ±67 6.54a ±13 84b ±69 51b ±32 61a ±100 1590bc ±970 bdl 4.4a ±4.9 9430 2 3030ab ±1700 43ab ±300 23a ±22 29ab ±14 bdl 32a ±26 8.50a ±14 9.31a ±26 896ab ±580 bdl 3.4a ±6.9 4070 3 4960bc ±2900 172cd ±130 10a ±11 25ab ±10 bdl 31a ±39 5.9a ±4.3 70a ±77 667a ±400 bdl 0.99a ±1.18 5940 4 8310d ±2700 244d ±110 10a ±14 47ab ±36 16b ±17 25a ±20 6.63a ±11 83a ±80 2020c ±960 bdl 2.00a ±2.54 10800 5 3820ab ±3100 54ab ±52 17a ±16 18.3a ±7.1 0.7a ±1.9 21a ±29 2.1a ±5.2 58a ±75 722a ±760 bdl 2.01a ±2.9 4710 the alphabet in each column denotes a mean value that is significantly different at p < 0.05 from each other. that the contamination of bed sediment with respect to fe, mn, ti, ba, zn, al, sr, and zn were low. co at locations 1 and 2 as well as cu in location 2 exhibited moderate contamination. similarly, y had moderate contamination. arsenic contamination varied between considerable and high contaminations except location 2 which exhibited low contamination. 3.5.2. pollution load index (pli) of the sediments pli was employed to adequately compare whether all the locations suffer contamination or not. the pli was intended at providing an extent of the degree of the total contamination of the sampling locations along the course of the stream. table 8 shows the result of the pli for the eleven metals studied at the various locations. based on the results, the overall degree of contamination in all the locations was of the order: location 1 > location 4 > location 3 > location 0 > location 5 > location 2. results of pli indicated that no location was polluted with fe, mn, ti, ba, zn, al and sr. however, location 0 was polluted with as and y; location 1 with co, cu and as; location 2 with cu only; and location 3, 4 and 5 with as only. 3.6. correlation analysis of heavy metals in sediment of asunle stream the determination of the correlation analysis is to quantify the strength of association observed between two variables. this association is likely to better illustrate the causal relationship between the variables. two-tailed correlation analysis between various metals detected in the lateral sampling of sediment toward the receiving stream is presented in table 9. the result obtained revealed that fe was positively correlated with mn, ti, ba, zn, cu, al and sr. manganese positively correlated with ti, ba and as, ti positively correlated with ba, cu, al and sr; ba correlated positively with al; zn positively correlated with cu and al; cu and al and sr are positively correlated, and fe and sr are positively correlated. however, co negatively correlated with zn and al while zn negatively correlated with as and as in turn negatively correlated with sr. correlation analysis obtained in this study was similar to the values obtained from the dumpsite soil [2]. the association between these metals could be attributed to a common source, hence, positive correlations among these metals were controlled by features such as, anthropogenic factors, properties and soil genesis [5]. the strong association among fe, ba, zn, al could be attributed to the degradation of e-waste materials in the dumpsite that was leached into the stream. 3.7. health risk assessment of heavy metals in sediment of asunle stream the model used to calculate the exposure of humans to potentially toxic metals in the sediments is based on those developed by the united states environmental protection agency. the chronic daily intake for carcinogenic risk (oral) in the bottom sediment of the asunle stream is showed in table 10. the chronic daily intake for fe ranges from 0.31 – 3.87 mg/kg/day. there is no available mrl data for fe. manganese is an essential component of steel. inorganicmn is equally used in the manufacture of dry cells, fireworks, and glass various chemicals, leather and textile, and fertilizer. manganese found their way into water bodies majorly via the erosion of rocks and soils, mining works, and leaching e-waste material dumped in landfills [29]. the chronic daily intake of mn for carcinogenic (oral) in the sediment of the asunle stream ranged from 0.006 – 0.1 mg/kg/day. literature has no stipulated mrls values for various stages of toxicity for oral exposure to mn, though neurobehavioral disorder has been established in literature from intermediateand chronic-duration oral exposure to excess inorganic-mn, hence, an interim guidance value of 0.16 mg mn/kg/day is recommended by atsdr for community health assessments [30]. the level of mn recorded in this study are below any alerting values. cobalt, is vital for beings with dietary allowance of 0.1 µg, and mean regular consumption from food is valued to be 5 – 40 µg/day [31]. the chronic daily intake for co in the bed sediment ranged from 0.0002 – 0.004 co mg/kg/day. the minimal risk level (0.01 mg co/kg/day) has been stipulated for average 8 olutona / j. nig. soc. phys. sci. 5 (2023) 983 9 table 7. contaminant factor of heavy metals in spatial variation in sediment of asunle stream fe mn co ti ba zn cu as al y sr 0 0.03 0.02 0.54 0.009 nd 0.13 0.39 3.18 0.02 2.11 0.003 1 0.16 0.14 1.09 0.01 0.01 0.90 1.13 4.73 0.02 nd 0.01 2 0.06 0.05 1.20 0.006 nd 0.33 0.19 0.72 0.01 nd 0.01 3 0.11 0.20 0.53 0.006 nd 0.32 0.13 5.39 0.008 nd 0.003 4 0.18 0.29 0.54 0.01 0.03 0.27 0.15 6.35 0.02 nd 0.007 5 0.08 0.06 0.89 0.004 0.001 0.22 0.05 4.44 0.009 nd 0.007 table 8. pollution load index of heavy metals across the sampling locations sites fe mn co ti ba zn cu as al y sr 0 0.03 0.02 0.54 0.009 nd 0.13 0.39 3.18 0.02 2.11 0.003 1 0.16 0.14 1.09 0.010 0.01 0.90 1.13 4.73 0.02 nd 0.01 2 0.06 0.05 1.20 0.006 nd 0.33 0.19 0.72 0.01 nd 0.01 3 0.11 0.20 0.53 0.006 nd 0.32 0.13 5.39 0.008 nd 0.003 4 0.18 0.29 0.54 0.010 0.03 0.27 0.15 6.35 0.02 nd 0.007 5 0.08 0.06 0.89 0.004 0.001 0.22 0.05 4.44 0.009 nd 0.007 mean 0.10 0.13 0.80 0.008 0.007 0.36 0.34 4.14 0.015 0.35 0.007 length oral contact to cobalt [29]. the cdi values obtained in this study is below the mrl value for co, hence there is no cause for alarm. barium, an alkaline earth metal, with a variability of usages such as getters in electronic tubes, rodenticide, colorant in paints, and x-ray contrast medium [30]. the minimal risk level (0.2 mg ba/kg/day) has been stipulated for average length oral exposure to barium. with the cdi of ba in bed sediment ranging from 0.0002 – 0.001 mg/kg/day, the cdi value obtained in this present study was several folds (200 – 1000) below the recommended mrl. zinc, a vital metal has a recommended daily allowance between 5 mg (infant) to 15 mg for infants and adults, respectively [14]. detrimental health effects of zn range between 100 to 250 mg/day [32]. atsdr [30] recommended minimal risk level of 0.3 mg/kg/day for both intermediate and chronic duration. the cdi for zn in bed sediment ranged from 0.00008 to 0.009 mg/kg/day. the zn level recorded in this study are far below the recommended limits, hence, not cause for immediate health concern. copper, an indispensable nutrient is vital in carbohydrate, and drug metabolism, haemoglobin formation, catecholamine biosynthesis, the cross-linking of collagen, elastin, and hair keratin, and the antioxidant defence mechanism[30]. some of the evidence of cu deficit are: anaemia, leukopenia, and osteoporosis. the recommended minimal risk level of both intermediate and chronic duration is 0.01 mg/kg/day. the value obtained in this study ranged from 0.0009 to 0.003 mg/kg/day which are far below the recommended value and pose no health threat to humans. arsenic is a metalloid, and very poison. arsenic main route of exposure is via food and drinking water. dietary exposures to arsenic in female ranges between 1.01, and 1,081 µg/day and mean (50.6 µg/day); male ranges between 0.21 and 1,276 µg/day, and mean (58.5 µg/day) [30]. the minimal risk level of 0.005 and 0.0003 mg as/kg/day has been resultant for acute and chronic duration oral exposure to inorganic arsenic. the cdi of as in bed sediment ranged from 0.004 to 0.01 mg/kg/day (table 10). the values obtained in this study were above the recommended minimal risk level; hence, the risk of cancer is possible. aluminium, the third most abundant element of the earth’s crust. there is appreciable amount of human data on the toxicity of al because of oral exposure for instance, dialysis encephalopathy syndrome (a degenerative neurological syndrome), and alzheimer’s disease [30]. the minimal risk level for both intermediate and chronic duration is 1.0 mg/kg/day. the cdi of al for carcinogenic (oral) in bed sediment ranged from 0.03 – 0.24 mg/kg/day. these values were below the minimum risk level; hence no health risk is implied. the levels of strontium in fresh waters ranges between 0.5 and 1.5 mg/l. the daily exposure is estimated to be 3.3 mg/day (0.046 mg/kg/day): from inhalation (400 ng/day), drinking water (2 mg/day), and diet (1.3 mg/day) [30]. the minimal risk level of sr in bed sediment ranged from 0.00001 to 0.0006 mg/kg/day. thus, the values obtained in this study were also below the minimal risk level. table 11 presents the descriptive statistics of the cancer risk assessment of the heavy metals in bed sediment of asunle stream. these values depicted non-hazard for both young and adult. the sequence of the total cancer risk of the studied metals are fe > al > mn >as > ti > ba > zn > co > y > sr > ba. given the available toxicological profile of the studied metals, it is obvious that all the studied metals may not inevitably have any adverse effect on human. the levels obtained in this study are below the rais oral chronic reference dose (mg/kg/day). the chronic daily intake for non-carcinogenic risk of both children and adult of metals in the bed sediment of asunle stream are presented in table 12 and 13. the level of human health risk caused by non-carcinogenic pollutants in children (x 106 mg/kg/day) (table 12) ranged as follows: fe (2910 51240), mn (59.08 – 127.19), co (0.75 – 34.19), ti (22.05 – 60.23), ba (1.65 – 7.90), zn (7.69 – 80.66), cu (5.75 – 24.59), al (8.61 -139.83), as (279.40 – 8210), y (25.02.a20), and sr (0.10 – 5.37). similarly, the level of human health risk caused 9 olutona / j. nig. soc. phys. sci. 5 (2023) 983 10 table 9. correlation analysis of metals in sediment of asunle stream fe mn co ti ba zn cu as al y sr fe 1.00 mn 0.875** 1.00 co -0.266 -0.279 1.00 ti 0.481** 0.371** -0.052 1.00 ba 0.719** 0.686** -0.079 0.677** 1.00 zn 0.466** 0.226 -0.389** 0.274 0.225 1.00 cu 0.341* 0.175 -0.109 0.590** 0.284 0.535** 1.00 as 0.040 0.083 0.365* 0.013 0.023 -0.307* -0.108 1.00 al 0.646** 0.449** -0.346* 0.444** 0.650** 0.544** 0.282 -0.224 1.00 y -0.149 -0.145 -0.173 0.001 -0.076 -0.103 -0.064 -0.061 0.064 1.00 sr 0.345* 0.226 -0.066 0.292* 0.250 0.596** 0.485** -0.335* 0.366* -0.072 1.00 ** significant at p < 0.01 level (two-tailed); * significant at p < 0.05 level n= 528 table 10. chronic daily intake for carcinogenic (oral) in sediment of asunle stream fe mn co ti ba zn cu as al y sr nov 3.87 0.007 0.0002 0.002 0.005 0.0009 0.002 0.14 0.0003 dec 0.56 0.01 0.0008 0.004 0.0003 00.008 0.002 0.0009 0.18 0.0004 jan 0.75 0.01 0.0003 0.003 0.0007 0.009 0.002 0.002 0.24 0.003 0.0006 feb 0.80 0.02 0.0006 0.007 0.001 0.003 0.003 0.004 0.12 0.005 0.0003 may 0.55 0.01 0.003 0.006 0.0008 0.001 0.003 0.01 0.16 0.0005 jun 0.55 0.01 0.003 0.003 0.0003 0.002 0.001 0.005 0.11 0.00001 july 0.36 0.008 0.003 0.003 0.0002 0.0009 0.0006 0.007 0.08 aug 0.31 0.006 0.004 0.003 0.00008 0.001 0.01 0.03 0.0004 table 11. cancer risk assessment of heavy metals in sediment of asunle stream fe mn co ti ba zn cu al as y sr min 0.62 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.06 0.00 0.00 max 7.74 0.04 0.01 0.01 0.00 0.02 0.01 0.02 0.48 0.01 0.01 mean 1.94 0.02 0.004 0.008 0.0009 0.007 0.003 0.01 0.27 0.002 0.002∑ cdiks fk 15.5 0.148 0.024 0.062 0.007 0.059 0.027 2.12 0.082 0.016 0.012 by non-carcinogenic pollutants in adult (x106 mg/kg/day) (table 13)ranged as follows: fe (2480 – 6300), mn (50.22 – 182.05), co (0.63 – 215.30), ti (18.75 – 53.09), ba (1.39 – 8.52), zn (6.54 – 68.56), cu (4.89 – 21.43), al (7.31 28.98), as (8.66 – 1880), y (21.27 – 42.38), and sr (0.00 – 4.56). the implication of these values is that the number of deaths that could be attributed to non-carcinogenic pollutant in each million is very high, hence the risk is of great concern. according to the united state environmental protection agency (usepa), the acceptable health risk level is 10−6 – 10−4 per year[13]. besides, the cdi values for non-carcinogenic (oral) in children was slightly higher than the value obtained in adult. this also suggests that the children are more vulnerable to hazard than adults. from each chronic non-carcinogenic exposure, the separate chronic hazard index (hi) was first calculated from the ratios of the chronic daily intake (cdi) to the chronic reference dose (rfd) for the individual metals and then the obtained results summed up as described in the equation: chronic hazard index = ∑n k=1 cdik /r f dk [17] where the hazard index is a unit less number that is expressed as the probability of an individual suffering an adverse effect. table 14 presents the total chronic hazard quotient for both children and adults in the bed sediment. generally, the hazardous index (hi) for ingestion of sediment by children is greater in comparison to that of adults. consequently, the total chronic hazard quotient index of oral exposure to contamination in the bed sediment for adult and children are far above 1, great hazard for both young and old is depicted. 4. conclusion along with several other metals, the determination of rareearth metals and some other metals that have not been reported in earlier studies of the dumpsite was done. values of metals present in the various matrices revealed that potentially toxic metal contamination had occurred to varying degrees that could cause health hazards. in some cases, the extent of contamination had reached the pollution levels that gave cause for concern. the geo-accumulation index, enrichment factor and contaminant factor revealed that the soil and sediment samples were heavily polluted with such contaminants as as, cu and be. the effective management of the water bodies and other aqueous 10 olutona / j. nig. soc. phys. sci. 5 (2023) 983 11 table 12. cdi for non-carcinogenic (oral) for children in sediment of asunle stream (x106 mg/kg/day) fe mn co ti ba zn cu as al y sr nov 3610 69.05 1.91 22.1 50.38 9.06 14.28 1290 2.59 dec 51240 112 0.75 32.95 2.92 70.84 18.76 8.61 1690 3.93 jan 7000 127 2.97 32.1 6.09 80.66 14.68 15.65 2210 25.02 5.37 feb 7420 127 2.97 32.1 6.09 80.66 14.68 15.65 2210 25.02 0.28 may 5080 98.1 26.1 60.23 7.90 12.65 24.59 116.71 1460 0.45 jun 5080 126 21.2 28.43 3.18 16.69 13.56 46.95 1020 0.10 july 3390 76.1 31.9 25.47 1.65 8.75 5.75 63.38 717 aug 2910 59.1 34.2 27.96 7.69 12.86 139.83 279 3.30 table 13. cdi for non-carcinogenic risk (oral) for adult in sediment of asunle stream fe mn co ti ba zn cu as al y sr nov 3060 58.69 1.62 18.75 42.83 7.71 12.14 1100 2.19 dec 4460 95.81 0.634 28.02 2.48 60.22 15.95 7.31 1430 3.34 jan 5950 108.11 215.3 27.26 5.18 68.56 12.47 13.30 1880 21.27 4.56 feb 6300 182.05 4.38 58.09 8.52 27.02 21.43 2898 930 42.38 2.35 may 4320 83.36 22.14 51.20 6.72 10.75 20.90 99.20 1240 0.38 jun 4320 107.34 18.05 24.16 2.70 14.19 11.53 39.90 8.66 0.09 july 2880 64.67 27.09 21.65 1.39 7.44 4.89 53.88 609 aug 2480 50.22 29.07 23.76 6.54 10.93 118.86 237 2.81 table 14. total chronic hazard quotient index (x 106) of the heavy metals in the sediment of asunle stream minimum value maximum value mean value child adult child adult child adult mn 1284.35 1091.75 2765.00 3957.61 2161.96 2038.72 co 37.30 3.70 1709.50 10766.00 762.16 1989.28 ba 0.00 0.00 39.50 42.60 17.39 16.87 zn 384.50 327.00 8345.00 3430.00 2990.81 1484.94 cu 143.75 122.25 614.75 535.75 175441.70 330.66 as 28700.00 24366.67 466100.00 9660000.00 175441.70 1351079.00 matrices is vital to ensuring the safety of the receiving water bodies as well as public health. regular monitoring of persistent organic pollutants is highly required. future studies should be targeted on the assessment of these contaminants in the aquatic biota, crops, farmers and rural dwellers 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[32] rais. the risk assessment information system (2007). http://rais.ornl.gov/tox/rap toxp.shtml (accessed 20 march 2021). appendix supplementary data/appendix variable value used at – averaging time for noncarcinogens at – averaging time for carcinogens bwadult – body weight adult bw child −− body weight child cs – concentration in soil or sediment ef – exposure frequency et indoor – exposure time indoor et outdooorexposure time outdoor 365 days/year/ ed child or adult 565days/year/70 years 70 kg 15 kg chemical specific (mg/kg) 350 days/year 0.683 0.073 df – dilution factor indoor in – inhalation rate pef – particulate emission factor, climate specific vfvolatilization factor, chemical specific if – intake factor ir adult – ingestion rate adult ir child – ingestion rate child ed child – exposure duration childhood ed adult – exposure duration adulthood 0.4 20 m3/day m3/kg m3/kg 0.0001 kg/day 0.0002 kg/day 6 years 24 years (for general case:30 years) 12 j. nig. soc. phys. sci. 4 (2022) 842 journal of the nigerian society of physical sciences environmental, health and economic implications of emerging contaminants in nigeria environment s. a. adesokan, a. a. giwa∗, i. a. bello department of pure and applied chemistry, ladoke akintola university of technology, ogbomoso abstract the following were the identified and defined classes of emerging contaminants of concern (eccs): pharmaceutical and personal care products (ppcps), perfluorinated compounds (pfcs), plasticizers, agrochemicals, industrial additives and agents (iaas), flame retardants (frs), nanoparticles (nps), steroids and hormones, gasoline additives. from 1983 to 1990, an estimated 15,000 metric tons of pesticides were reported to have been imported annually. in 2016, a yearly application of about 130,000 metric tons of pesticides was reported for nigeria. nigeria’s pesticides imports were worth usd128.671 in that year. of the applied pesticides, about 85% ended in the environment as contaminants/pollutants. while few individuals in the households or neighbourhoods deal with pesticides, almost all human beings deal with ppcps. ppcps are taken to prevent or cure diseases and/or to sustain wellbeing. nigeria produced 30% of its ppcps demands while 70% imported. in 2012, 2013 and 2014, nigeria imported ppcps worth usd425 million, usd481 million and usd530 million respectively. in 2018, nigeria imported ppcps worth usd606.31 million, while the total amount of pharmaceuticals procured was usd866.16 million. almost all the candidates of eccs had been detected in the nigerian environment. untoward episodes of pesticides abuse ranging from abuse to death, have been profiled. some of the factors responsible for these were weak regulatory instruments on accessing these pesticides, bad economy, stigmatization and lack of resilience. doi:10.46481/jnsps.2022.842 keywords: emerging contaminants, environment-health, pharmaceuticals, economy, nigeria article history: received: 29 march 2022 received in revised form: 04 june 2022 accepted for publication: 13 june 2022 published: 19 august 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: k. sakthipandi 1. introduction contamination can be defined as the presence of an unwanted substance in an environmental medium at low concentration with the attendance of chronic effects on the exposed environmental component(s). pollution, on the other hand, is defined as the presence of unwanted substance in an environmental medium at high concentration with the attendance of acute effects on the exposed environmental component(s) [1]. ∗corresponding author tel. no: +234(0)8035065456 email address: giwa1010@gmail.com (a. a. giwa) in the united states of america, the phrase “emerging pollutants” (eps) has been substituted by: pollutants of emerging concern (pecs), emerging micro-pollutants (emps), emerging contaminants (ecs), compounds or chemicals of emerging concern (cecs) and emerging contaminants of concerns (eccs). what is “emerging” is the awareness in both the scientific community and general public that these chemicals are being released into the environment and can be detected in water, sediment, soil, and biota [2-3]. in this study, the sources of eccs include: agricultural, industrial, medical, pharmaceutical and whole all of the new technologies, productions, materials in1 adesokan et al. / j. nig. soc. phys. sci. 4 (2022) 842 2 ventions, innovations and lots more. emerging pollutants/contaminants are substances in environmental media with evolving identification or recognition and little or no characterization or diagnosis of impacts on health and ecosystem but with hazardous or toxic tendencies. it can be broadly defined as any synthetic or naturally occurring chemical or any microorganism that is not commonly monitored in the environment but has the potential to enter the environment and cause known or suspected adverse ecological and/or human health effects [1-3]. in some cases, release of emerging chemical or microbial contaminants to the environment has likely occurred for a long time, but may not have been recognized until new methods were developed. in other cases, synthesis of new chemicals or changes in use and disposal of existing chemicals can create new sources of emerging contaminants and pollutants [3]. the usepa [4] in its article: “contaminants of emerging concern” described emerging contaminants as chemicals that are currently being detected or detected at levels higher than expected in the environment. also, emerging pollutants are defined as new chemicals without regulatory status and which impact on environment and human health are poorly understood. in its understanding, norman defined emerging contaminants (ecs) as ”substances that have been detected in the environment, but which are currently not included in routine monitoring programmes at eu level and whose fate, behaviour and eco-toxicological effects are not well understood” (eu norman network). again, ecs are “those chemicals that recently have been shown to occur widely in water resources and identified as being a potential environmental or public health risk, and yet adequate data do not exist to determine their risk” (definition of the consortium for research and education on emerging contaminants). state of massachusetts defined emerging pollutants as: “hazardous materials (chemical, microbial or radiological substances) or mixtures of interest that are characterized by: a perceived or real threat to human health, public safety or environment; no currently published health standard/guideline exists or it is evolving or being re-evaluated; there is insufficient or limited available toxicological information; or, a new source, pathway, or detection limit has been discovered. emerging contaminants may be naturally occurring or man-made.” to the state of south carolina, emerging contaminants are contaminants with a potential threat to health and environment that have no regulatory standard. in the wisdom of chemical material risk management, emerging contaminant is defined as: “chemical or material that has pathways to enter the environment and presents potential unacceptable human health or environmental risks, and either does not have regulatory peer-reviewed human health standards or the regulatory standards are evolving due to science, detection capabilities, or new pathways”. “lastly, emerging contaminants” as defined by susan [2] are substances not normally tested for in water quality sampling. along with pharmaceuticals, they include a number of industrial chemicals, particularly suspected endocrine disrupting substances (edss) that are used in plastics, cleaning agents, personal care products like shampoos, and pesticides. these chemicals are of concern because of the risk to human health and the environment associated with their presence, frequency of occurrence or source may not be known. a characteristic of some contaminants is that they do not need to be persistent in the environment to cause negative effects since their high transformation/removal rate is compensated by their continuous introduction into the environment. for most of the occurring emerging contaminants, risk assessment and ecotoxicological data are not available and therefore it is difficult to predict which health effects they may pose on humans, terrestrial and aquatic organisms and ecosystems [5]. eccs are not 3 necessarily new chemicals; they may be substances that have been present in the environment for a long time but whose presence and significance are only now being recognized [6]. this article therefore reviewed the presence, concentrations and implications of eccs in the nigerian environment with a view to make recommendations for better economic decisions, environmental stewardship, and safer usage of eccs. 2. materials and method relevant research publications were sourced using available search engines like google scholar, researchgate, tailor and francis online, scopus, mpdi, web of science, sage publishing, and sciencedirect. the key words input include emerging contaminants, environment, health, economy, nigeria, pharmaceuticals and personal care products, herbicides, and pesticides. the articles generated were reviewed based on impacts of these contaminants on environment, human health and economy of nigeria. also the people in the neighbourhood were interviewed on their usage of pesticides. the information extracted was used to identify the problems and make recommendations on the way forward. 3. emerging contaminants 3.1. the contaminant candidate list (ccl) classification process united states environmental protection agency [4] has, for a long time now, engaged in researches to match the trend and the state of the environment with a view of being the master of it. in the quest to safeguard the health of the citizens through drinking water, usepa developed and published an inventory (the contaminants candidate list, ccl) of contaminants (chemicals and microbes) in environment in 2003, especially aqua medium. then the list has since been updated [7] to grab the trend of environmental contaminant contents and respond appropriately, through information and regulation. in preparing the list, some parameters were standardized and deployed in order to qualitatively and quantitatively assess the potential impacts of the listed contaminant candidates. these parameters include: severity, potency, prevalence and magnitude. these parameters are employed to characterize the suspected “eco-health” and “hazardo-toxic” features of the identified substances. the ccl is mandated to be developed by safe drinking water act (sdwa). the approach to the development of ccl 2 adesokan et al. / j. nig. soc. phys. sci. 4 (2022) 842 3 includes public participation where people were tasked to nominate substances they familiarized with and suspected to possess potential harmful features (phf) for human and ecological health. this is followed by scientific testing based on established standards. also in 1998 epa set up the endocrine disruptor screening program to identify substances possessing endocrine disrupting activities. the wisdom behind the program was to identify and take informed regulatory decision as ordered by the food quality protection act and the safe drinking water act amendments passed by congress in 1996. the program led to the release of first list in 2009, contained 67 substances as endocrine disrupting substances (eds); and in 2010 the second list listed 134 substances was released. the state of california, usa, in 2009, did also take step to address the challenge of eccs in the environment. the approach it adopted was public discussions and setting up of two panels (recycled water and ecosystem health) to float regulatory framework for eccs. the panels gathered available knowledge (information), identified knowledge gaps and made recommendations on the actions to be taken by the stakeholders. the procedure of the approach involved data collection on identified and selected eccs, exposure threshold screening and concentration threshold setting or determination. when the threshold limit is exceeded (at maximum side of the scale), the ecc in question is listed as of high priority for regulation and monitoring [8]. there were a number of agencies being set up by countries, regional governments and international communities to check, monitor and regulate chemicals. these include u.s. food and drug administration in the united states; european medicines evaluation agency (emea) in the european union; health canada in the canada; australian pesticides and veterinary medicines in the australia; national agency for food and drug administration and control (nafdac) in nigeria; registration, evaluation, authorization and restriction of chemicals (reach) regulations [9] and the veterinary international cooperation on harmonization (vich) in the us, canada, japan, australia and europe. vich approach to listing substances as eccs involves two phases. phase 1 involves use and exposure thresholds [10]. substances at the minimum sides of the thresholds are dropped while those on the maximum side are subjected to phase 2 testing. in the phase 2 the level of health and eco-toxicity are assessed to prioritize the eccs [6, 11]. human pharmaceuticals in the environment are also assessed in the eu and the us for risks, supervised by emea. for pharmaceuticals in aquatic environment, the threshold of 10 ng/l was established to be maximum tolerable level for aquatic environment. pharmaceuticals that pass the threshold test are authorized for marketing. those whose levels are above the threshold are subjected to second phase test where fate and effect screening are performed [6]. 3.2. classes of eccs the following were the identified and defined classes of eccs: pharmaceutical and personal care products (ppcps), plasticizers [12], agrochemicals, industrial additives and agents (iaas), flame retardants (frs), nanoparticles (nps), steroids and hormones, gasoline additives [13]. pharmaceuticals are designed to cure specific ailments and ensure general wellbeing of human and livestock. medicine and medical therapy have been with human from time immemorial. the greeks and the arabs of post-stone age had legacy of frontline philosophy in virtually all areas of human endeavours. however the science was at the natural levels: discovery, extraction and isolation. the 19th century witnessed turn of events. advances in technology have brought about developments in most frontiers of knowledge. population demands, dwindling natural resources, generational consciousness. . . have pressurized the stakeholders to artificially imitate the natural processes, hence advanced material synthesis. advancement in technology has occasioned changes in cultural practices: food, transportation, clothing. . . these emerging practices led to emerging medical complications and then emerging pharmaceuticals and therapies. some of the classes of the pharmaceuticals include: analgesic (ibuprofen, paracetamol, diclofenac, naproxen), antibiotic (amoxicillin, trimethoprim, clarithromycin), antacid (ranitidine), anticonvulsant (carbamazepine), lipid lowering (e.g benzafibrate), antidiabetic (metformin), β − blocker (atenolol, metoprolol) [12]. as mentioned, emerging needs created by population boost have led to generation of candidates of all classes of eccs. 3.3. volumes of eccs dispensed by nigeria of all the classes of the eccs discussed above, agrochemicals and ppcps were the most dispensed and in common place in nigeria. in the period of 1983-1990, an estimate of 15,000 metric tons of pesticides was reported to have been imported annually [13, 14]. in 2016, a yearly application of about 130,000 metric tons of pesticides was reported for nigeria [15]. according to fao [16], nigeria pesticides imports worth usd128.671 in that year [17]. of the applied pesticides, about 85% ended in the environment as contaminants/pollutants [18]. while few individuals in the households or neighbourhoods deal with pesticides, almost all human beings deal with ppcps. ppcps are taken to prevent or cure diseases and/or to sustain wellbeing. almost everyone takes ppcps to address one of those conditions. it is therefore expected that the volume of ppcps be more in production, circulation, application and contamination. in 2017, over one hundred and thirty pharmaceutical industries were operating in nigeria but just nine of them listed in the stock exchange. important also, by revenue, the average size of a nigerian pharmaceutical firm was less than $1billion when the average size of each of the top 20 pharmaceutical companies was $21billion [19]. nigeria produced 30% of its ppcps demands while 70% imported [20]. in 2012, 2013 and 2014, nigeria imported ppcps worth of usd425 million, usd481 million and usd530 million respectively [21]. in 2018, nigeria imported ppcps worth of usd606.31 million [17]. the 30% locally produced pharmaceuticals in 2018 amounted to usd259.85 million. the total amount of pharmaceuticals procured for the country in 2018 was usd866.16 million. it was estimated that nigeria pharmaceutical market grew at 13% annually. it was estimated that nigeria used 722,892 kg of drugs in livestock production between june 2014 and dcember 2015 [22]. it is important to point out that nigeria has rich 3 adesokan et al. / j. nig. soc. phys. sci. 4 (2022) 842 4 flora and fauna resources which can supply huge pharmaceutical precursors and sustain world class pharmaceutical industries but lack of appropriate equipment for production, distribution and storage are the banes [23]. it was estimated that nigeria generated about 2.5 million tonnes of plastic wastes per annum and these mostly ended in hydrosphere and lithosphere [24]. nigeria accounted for 17 percent of plastic consumed in africa between 1996 and 2017. it was estimated that nigeria produced 2.3 million tonnes of plastic between 2009 and 2015 and imported 20 million tonnes during the same period. about 130,000 tonnes of plastic contents in nigeria ended up in aquatic environment, while only 10 percent of plastic wastes in nigeria recycled. this load invariably correlates with the amount of plasticizers in nigeria environment [25]. 3.4. loads of eccs in nigeria environment in the years 1990 and 2009, 12,750 and 110,000 metric tons of pesticides contaminated or polluted nigeria environment yearly respectively. in a period of 19 years (1990-2009), nigeria population increased by 59,065,292 (table 1) and the pesticides application increased by 115,000 metric tons. by projection the nigeria population increase between 2010 and 2019 was 41,685,857 (a period of 9 years). this geometric increase in the population was expected to correlate with the pesticides application in nigeria. some of the factors that might have been responsible for the increase in the usage of pesticides were: rural-urban drift; cheaper than manual weeding; easiness; and reduction in the frequency of weeding. rural-urban drift: in the southern western region of nigeria, many villagers had left their villages for cities. this drift has thinned down the rural population to aged few. this has led to change in the cultural farming practice of using peasant tools like hoes and cutlasses to do weeding. the current trend adopted in most farms now is the use of pesticides for weeding. in the time past, pesticides were only used to disinfect cash crop plants, however only few had cash crop farms unlike food crop farms. the usage of pesticides for virtually all control on farms has increased the pesticides application in many folds. cheaper than manual weeding: the use of pesticides is more economic friendly than engagement of human labourers. three liters of forceup pesticide of one thousand three hundred naira (n1,300 ≡ usd3.25 as at 15/july/2020) was enough to control annual broad leaves weeds on one hectare of land. if human weeding labourer was to be engaged for the same weeding, the least cost would be thirty thousand naira (n30, 000 ≡ usd80). pesticide usage saves about twenty six thousand naira (n26,100 ≡ usd65.25). this is a huge economic advantage among a poor population ahead of consideration for environment and/or health. the farmers have largely embraced the pesticides usage because of the economic advantage and the scarcity of human labourers. this led more pesticides application and more pesticides in the environment. the unodc reported that nigeria was the epicenter of misuse and abuse of psychoactive drugs in west africa in particular and africa in general. nigeria was the hub of trafficking drugs like cocaine, cannabis, opioids (tramadol, codeine and pentazocine), amphetamine, and ephedrine. cannabis was extensively cultivated in the country. it was estimated 7.5 percent of the school children abused cannabis in nigeria in 2016. the 25 to 39 years old nigerians were the age bracket that mostly abused psychoactive drugs [26]. ease of usage: weeding using pesticides is comparatively less labourious and faster to human weeding. this also has endeared pesticides usage to the farmers, thereby increased the levels of environmental pesticides. an herbicide usage survey was conducted at labi 2 community (l2c). l2c consist of one hundred and fifty houses. it was a semi-urban area located via moniya, akinyele local government, ibadan, oyo state, nigeria. the residents were mixture of the middle class and the poor. 80% of the houses were not fenced and their surroundings not paved. 95% of the unfenced houses control surrounding weeds with herbicides (figure 1). in a raining season, each of the unfenced houses applied herbicides at least three times. the active ingredient in the herbicides used was glyphosate. on the average, each house used one liter of the pesticides per a raining season (360 g of glyphosate). this amounted to 41, 040 g of glyphosate application in the area per a raining season. 85% of this amount was 34, 884 g which entered the environment as degradates, contaminants or pollutants. due to the small size of the residential apartments, enough quantities of herbicides are always available and this leads to overdose applications. during raining season, neighbourhoods are always overgrown which demands for weed control multiple times. the excess herbicides in the environment quickly percolate into the ground water. in nigeria, many communities depend on well water. many of these wells are shallowed enough for the quick reach of these herbicides. exposure to herbicides, by populations living in the vicinity of neighbourhoods that control weeds with herbicides, through well water is very high. in a study conducted on cereals, legume, tubers, fruits and vegetables collected from enugu state [29] aldrin, carbofuran, chlordane, heptachlor, hexacholorobenzene, lindane, dichlorodiphenyltricholoroethane. . . were detected but at below maximum permissible limits. herbicides were detected at high concentrations in tissues of fishes caught in kainji jebba lakes [30]. figure 1. pictorial image of neighbourhood treated with herbicides. according to the cia [31], the nigeria population below 4 adesokan et al. / j. nig. soc. phys. sci. 4 (2022) 842 5 table 1. population growth and pesticides usage. year 1980a 1985a 1990a 2007b 2008b 2009b 2010b 2011b 2015a 2016a 2017a 2018a 2019a population 73,423,633 83,562,785 95,212,450 144,897,327 149,510,846 154,277,742 159,203,664 164,294,516 181,137,448 185,960,241 190,873,244 195,874,683 200,963,599 average yearly population growth (1985-1990) 2,329,933 average yearly population growth (2000-2011) 4,849,297.2 yearly pesticides application around 1990 (metric tons) (2000-2011) 15,000 yearly pesticides application around 2000 (metric tons) 130,000 a: worldmeter [27] b: national bureau of statistics [28] poverty line was 70%. poverty predisposed to many health complications, ranging from malnutrition diseases to life style hazards. poverty hinders people from seeking proper medical attention and promotes quackery and abuse of pharmaceuticals in self-medication. self-medication largely leads to wrong diagnosis and procurement of wrong drugs. this invariably leads to loads of pharmaceuticals in the environment through discard and excretion together with pharmaceutical industrial discharges, hospital waste, and deceased population left over... ibuprofen, sulfamethoxazole, erythromycin, betasitosterol, chloramphenicol, diclofenac, naproxen, sulfadiazine, trimetoprim, amoxicillin, acetaminophen, methylparaben, artemether, triclosan, estrone, phenazone and clofifbrate had been detected in surface waters and environmental compartments [32-34]; and artemether, diclofenac and ofloxacin in portable waters [35] from lagos; and paracetamol, ciprofloxacine, chloroquine and diclofenac in surface water and groundwater [36] from sango-ota, ogun state, nigeria. a report has it that industrial, hospital and domestic wastewaters collected from oyo, ogun and lagos states south-west nigeria contained antibiotics, estrogens and antilipid medicines. surprisingly, some of these drugs were present at concentrations suspected to be hazardous to ecosystem [37]. in a study carried out in lagos and ologe lagoons in lagos state, acetaminophen, diclofenac, amoxicillin, and methylparaben were detected. the study concluded that methyparaben was the most prevalent, followed by diclofenac while amoxicillin the least detected [33, 38]. in contrast, anekwe et al. [39] who determined pharmaceuticals in surface water, ground water and drinking water from lagos reported amoxicillin as the most prevalent in the samples. the report further stated that acetaminophen, codeine, nicotine and ibuprofen were almost equally prevalent. season also affected environmental concentrations of naproxen, caffeine, glyburide, diclofenac, codeine and nicotine. in another study that tested lagos surface water and some effluents, the following pharmaceuticals were detected at high environmental concentrations: paracetamol, sulfamethoxazole, cimetidine, fexofenadine, carbamazepine, metformin, diazepam, atenolol, trimethoprim, and codeine [40]. oil spills in niger-delta regions of nigeria, which contains a range of hazardous and toxic substances, have been estimated to occur 229.9 times on the average with over 3.5 million barrels in volume [41]. in a similar report, an estimated 4,635 oil spills occurred between 1976 and 1996 in nigeria which amounted to 2,369,470 barrels in volume [42]. flame retardants (frs) are ecs that are common place in nigeria. the members of this group that have been detected in nigeria environment include octabromodiphenyl ether, decabromodiphenyl ether and tetrabromobisphenol a. these substances were found in (electronic waste) e-waste like computer and television. africa in general and nigeria in particular are the dumping sites for electronic wastes with 237, 000 tonnes were reported for nigeria in 2014 [43]. also, polybrominated diphenyl ethers (pbdes) have been detected in sediments collected in from lagos lagoons. the work further attributed high concentration of pbdes in the sediments to indiscriminate dumping of refuse, high industrial activities and land filling. the study concluded that deca-bdes and penta-bdes were the major congeners in the sediment samples [44]. evaluation of in house dust samples from lagos revealed the presence of halogenated frs like pbdes, polychlorinated biphenyls (pcbs) and hexabromocyclododecane (hbcdd) [45]. pbdes were also detected in a stream at obafemi awolowo university [46] but at low concentrations. sources of frs in the environment include point source: production plants; and non-point sources: e-wastes, sewage sludge, land fill leaching and soil erosion, biota, dust and particulate matters and incineration [47]. phthalate esters, classified as plasticizers under eccs, were reported to present in sachet water from delta state, though below threshold level. the presence of these molecules in the sampled water was attributed to nylon used for packaging [48]. the level of phthalates detected in orogodo river, delta state, by edjere et al. [49] was 3.29 µg/l. this amount was higher than 3µg/l maximum permissible level for some aquatic organisms set by united states environmental protection agency (usepa). this high concentration of plasticizers in the river was blamed on indiscriminate dumping of solid wastes. a test conducted on surface sediment samples collected from cross river system revealed presence of phthalates at levels that called for environmental action while discharged of untreated effluents and burning of plastic materials were cited as factors responsible for this condition [50]. 3.5. environmental and health implications of eccs the properties of glyphosate, the active ingredient in most pesticides common among users in nigeria, were presented in table 2. from table 2, it could be deduced that glyphosate was relatively soluble in water. this raises the possibility of excess of it in the environment dissolves in water, percolates into the 5 adesokan et al. / j. nig. soc. phys. sci. 4 (2022) 842 6 groundwater and contaminates/pollutes it. decomposition temperature of 215 oc would be difficult to attain in the natural environment and this explains why it is stable. fairly stability status would also make glyphosate persist in the environment. the carboxylic group (−cooh) and the amino group (−n h) impacted glyphosate with zwitterions character. −coo− will form in basic medium and −hn+ in acidic medium. the zwitterions condition, the lone pairs of electrons on oxygen and nitrogen and the presence of high percentage of oxygen atoms suggested high environmental activities (oxidative tendency) of glyphosate. figure 2. the chemical structure of glyphosate [51]. table 2. properties of glyphosate [52]. common name glyphosate iupac name 2-(phosphonomethylamino) acetic acid molecular formula hoocch2 n hch2 po(oh)2 un number 3077 decomposition 215 oc solubility 10.5 g/l at 20 oc stability stable at ph 3, 4, 5, 6, 9 at 50 oc pka 2.0; 2.6; 5.6; 10.6 metabolite formation aminomethyl phosphonic acid (ampa) (0.3%) hazard statement toxic however it was reported that glyphosate strongly adsorbed to the soil with half life of two months [53] and broke down by microorganisms [54]. glyphosate obstructed homeostasis [55, 56]. untoward episodes of pesticides deaths, ranging from abuse to accident, have been profiled. some of the factors responsible for these were abundant availability of and ease of accessing these pesticides, bad economy, stigmatization, lack of resilience. . . many nigerians were reported to have committed suicide, abusing pesticides [57]. bbc news [58] reported suspected accidental herbicide poisoning which resulted to mass deaths. the vulnerable groups, especially children, were at most receiving end of accidental poising. many cases of pesticide child poisoning were reported in nigeria [59-62]. malathion was reported to be preferred preservative agent for stored grains in nigeria, which might not be allowed to expire before the sale to consumers. also exposure to gamalin-20 and malathion in poisoned consumables had led to death of many in nigeria [28]. the effects of ppcps on living ecosystem in nigeria are still under investigation. it is also important to note that widely reported paraquat, for its unfriendly impacts on health and environment, is still common place in chemical shops in ibadan, oyo state capital. one of the factors accounted for the outbreak of diseases is resistance of disease-causing pathogens, like bacteria, to drugs. it has been pointed out that persistent of diarrhea among patients in nigeria was due to resistance of escherichia coli to tetracyclines, penicillins, cotrimoxazole, aminoglycosides, chloramphenicol, ampicillin, gentamicin, penicillins, cephalosporins, streptomycin, and nalixidic acid. studies conducted in southwest, south-south and south-east nigeria, revealed that nontyphoidal salmonella has resisted antimicrobial activities of ofloxacin, nalixidic acid, ampicillin, amoxicillin, cotrimoxazole, tetracycline, gentamicin, ciprofloxacin, and chloramphenicol. also, resistance to quinolones, ampicillin, penicillins, gentamicin, tetracyclines, streptomycin, chloramphenicol and cotrimoxazole has been observed in shigella species in nigeria. in the same vein, resistance to chloramphenicol, ampicillin, streptomycin, nalidixic acid, spectinomycin, cloxacillin, penicillin g, trimethoprim, sulphamethoxazole, and sulfonamide by vibrio cholerae in nigeria [22]. the study further stated that drug resistant bacteria were introduced into the environment through industrial, domestic, solid waste dumpsites, and veterinary wastewaters. exposure to bonny light crude oil has been reported to cause endocrine disruption in wistar albino rats by influencing levels of testosterone and estrogen in male and female populations [63]. this would point to level of hazards that might have been impacted on aquatic animals in oil spillage regions of nigerian niger-delta. oil spillage in nigeria has led to loss of faunal, floral and other living components of ecosystem in the coastal areas. this invariably led to loss of means of livelihood of residents of the oil spillage area, famine as the farmers are no longer have access to the farms, militancy, loss of species, environmental degradation, and monumental economic loss to the nation [64]. in a study conducted to determine the determinants by nigerian women to choose skincare products, it was reported that skincare products were chosen with beauty and health consciousness but without environmental consciousness [65]. this attitude, if environment is neglected in using the products, will indirectly and eventually impact the general ecosystem including human health. abuse of psychoactive drugs left in its trail unholy episodes of health complications. the common place sexually enhanced drug abuse has led to immoral sexual behaviour like raping and promiscuity. these acts invariably predispose to increase sexually transmitted diseases (stds) including hiv/aids. abuse of psychoactive substances has led to dependence. dependence on the psychoactive substances’ usage by the users led to chronic pain, high blood pressure, heart diseases, diabetes and mentallyderailed population [26]. industries have been cited as the sources of hazards like fire outbreak, water and air pollution, smoke, and dusts which predisposed to varieties of health complications in rivers state environment [66]. chronic exposure to phthalate esters in sachet water can 6 adesokan et al. / j. nig. soc. phys. sci. 4 (2022) 842 7 predispose to hormonal imbalance and eventually birth defects in new born [48]. plasticizers, being solid organic in environment, provided adsorptive surfaces for diverse organic pollutants and contaminants which eventually entered food chain and ecosystem and wrecked health and environmental havocs [24]. imo and akwa ibom states are among the top 20 ocean pollution points in the world, while it was predicted that in the very near future lagos beaches may lose their aesthetic, leisure and commercial values to micro-plastic wastes dumped or stormed in [25]. 4. recommendations the nigerian government should promote indigenous pharmaceutical companies to produce larger percentage of pharmaceutical needs of the country as against 30% currently being produced. nigerian pharmaceutical firms need to seek size expansion (in terms of r&d and revenue) as a prerequisite for capacity enhancement by collaborative arrangement [19]. this, if achieved, will boost national gross domestic product (gdp). also, nigeria should focus on drug development rather than importation of active ingredients to gain capability for cradle to grave management of drugs and pharmaceutical waste/wastewater. that is nigeria should evolve green chemistry policy for pharmaceutical administration. other methods of pharmaceutical management stewardships include introduction of personalized drugs (pharmacogenetics), reverse distribution and standardization of natural medicine. the adoption of indigenous methods of pest and weed control should be encouraged to drastically reduce propensity of emerging contaminants into the environment. certain pests can be effectively controlled by using mixture of urine and cow dung; mixture of cow urine, ginger and garlic; traps; wood ash spreading; artificial crows and fermented extract of locust bean seeds [67]. these practices are cheap, environmental benign and human health friendly. the current indiscriminate usage of herbicides for weed control in residential areas should be urgently checked by the government with appropriate policy due to proximity of the applications to public water. extensive tests should be conducted on the neighbourhood well water for the presence of herbicides. global best practices on plastic management should be adopted which include replacement of plastic packaging with compostable materials, ban of single-use plastics, convertible plastics for multipurpose usage, economic incentives on zero plastic waste generation, and aggressive plastic waste recycling [25]. another method is conversion of plastic wastes into fuel through pyrolysis [68-89]. nigerian government should also embark on sustainable poverty alleviation programmes and policies and public education on environmental health as related to the human wellbeing. 5. conclusion this review has revealed presence of candidates of some of the classes of eccs in the ngerian environment. easy access, economy, rural-urban drift and lack of effective regulatory framework are factors responsible for increase in usage and abuse of eccs, especially agrochemicals and ppcps, in nigeria. some of the pesticides contained active ingredients, e.g., glyphosate, that had been banned in developed economies. discriminate usage of agrochemicals and ppcps had led to presence of these compounds in the environmental media. there is need for government to strengthen the frameworks and 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[69] b. n. hikon, g. g. yebpella, l. jafiya & s. ayuba “preliminary investigation of microplastic as a vector for heavy metals in bye-ma salt mine, wukari, nigeria”, journal of nigerian society of physical sciences 3 (2021) 250, doi:10.46481/jnsps.2021.259. 9 j. nig. soc. phys. sci. 5 (2023) 994 journal of the nigerian society of physical sciences modeling extreme stochastic variations using the maximum order statistics of convoluted distributions adewunmi o. adeyemia,∗, ismail. a. adelekeb, eno. e. e. akarawaka adepartment of statistics, faculty of science, university of lagos, lagos state, nigeria bdepartment of actuarial science and insurance, university of lagos, lagos state, nigeria abstract modeling extreme stochastic phenomena associated with catastrophic temperatures, heat waves, earthquakes and destructive floods is an aspect of proactive mitigation of risk. hydrologists, reliability engineers, meteorologist and researchers among other stakeholders are faced with the challenges of providing adequate model for fitting real life datasets from the extreme natural hazardous occurrences in our environment. convoluted distributions (cd) and generalized extreme value (gev) distribution for rlargest order statistics (r-los) have been some of the prominent existing techniques for modeling the extreme events. this study explored the properties of order statistics from the convoluted distribution as alternative procedure for analyzing the extreme maximum with the aim of obtaining superior modeling fit compared to some other existing techniques. the new procedure called maxos-g employed the potential properties of the maximum order statistics (maxos) and the flexibilities of convoluted distributions where g is taken to be weibull-exponential pareto (wep) and the new kumaraswamy-weibull (nkwei) distributions. the maximum order statistics of the wep distribution (maxos-wep) and nkwei distribution (maxos-nkwei) was derived and applied to three datasets consisting of annual maximum flood discharges, annual maximum precipitation and annual maximum one-day rainfall. some properties of the maxos-wep was investigated including the moment, mean, variance, skewness, and kurtosis. characterization of wep distribution by the l-moment of maximum order statistics was presented and coefficient of l-variation, l-skewness and l-kurtosis were derived. the results from the application to three datasets using r-software justified the importance of this new procedure for modeling the maximum events. the maxos-nkwei and maxos-wep models provide superior goodness-of-fit to the datasets than the wep and nkwei distributions and better than some previously proposed convoluted distributions for modeling the datasets. doi:10.46481/jnsps.2023.994 keywords: extreme convoluted distributions, maximum order statistics, maxos-g, maxos-nkwei, maxos-wep, annual maximum precipitation article history : received: 17 august 2022 received in revised form: 09 january 2023 accepted for publication: 09 january 2023 published: 24 february 2023 ©2023 journal of the nigerian society of physical sciences. all rights reserved. communicated by: o. adeyeye 1. introduction modelling of extreme value datasets from stochastic random phenomena in our environments becomes an imperative ∗corresponding author tel. no: +234 7011778377 email address: adewunyemi@yahoo.com (adewunmi o. adeyemi) and important area of studies due to the adverse effects from natural hazards associated with global warming as a result of changes in climatic conditions. humongous losses associated with the extreme catastrophic phenomena is unquantifiable as revealed by several works, some of which include [1] [3]. [1] has showed that the traditional method used by many researchers for analyzing extreme value data is based on the gen1 adeyemi et al. / j. nig. soc. phys. sci. 5 (2023) 994 2 eralized extreme value (gev) distribution. modeling of exceedances over a high threshold using generalized pareto distribution (gpd) by [4] most especially in hydrology is considered to be an alternative approach by the author. some other techniques for investigating extreme phenomena and the modeling of real life datasets from the extreme scenario includes • the use of generalized extreme value distribution for rlargest order statistics • the application of lifetime distributions derived either by convolution of two baseline distributions or by some extension of existing classical distributions. 1.1. convoluted distributions [5] revealed some areas of applications of heavy-tailed distributions which includes finance, hydrology, earthquake and engineering. [4] established a domain of attraction relating to the extreme distribution called the generalized pareto distribution (gpd). [6] harnessed the long tail feature of the gpd distribution for modeling extreme value data [5] developed the beta-pareto (bp) distribution which the authors used for modeling two flood datasets. the properties of the beta generalized pareto (bgpd) distribution was explored by [7] for modeling extreme value data. the weibull distribution has wide application for modeling survival and reliability data. generalizations from the distribution include the exponentiated-weibull distribution applied to extreme value data by [8]. the weibull distribution was used as a generator to propose the weibullg family of distribution by [9] from which the authors developed weibul−weibull, the weibull-normal and [10] developed a new weibull-pareto (wp) distributions. in another dimension, [11] introduced the weibull-x family as a special case of the t-x family of distributions. many authors defined several distributions from the weibull-x family including, the weibullpareto by [11] and weibull-rayleigh by [12] and [13]. [14] proposed a new generalized distribution denoted the kumaraswamy-g (kw-g). convoluted distributions from the family include kumaraswamy-weibull (kw-w) distribution by [15], the kumaraswamy-exponentiated frechet (kefr) by [16], [17] introduced the kumaraswamy-transmuted pareto (ktp) distribution. the improved version of (kw-g) denoted (nkwg) was discovered by [10] , the author developed the new kumaraswamy weibull distribution (nkw-w) which was applied for modeling two datasets relating to extreme phenomena representing annual maximum flood discharge and annual maximum precipitation. the convolution of exponential and pareto distributions by [18] produced the exponential pareto (ep) distribution which [19, 20, 21, 22, 23, 24] further generalized and studied with diverse applications using different approaches. [25] proposed transmuted topp-leone extended frechet distribution for modeling extreme value of dataset with some application. 1.2. order statistics the study by [26], faced with the problem of estimating the mean value of the difference between two successive values x(k+1) and x(k) of order statistics in a sample of size n from a population whose probability density has a continuous function confirmed that order statistics is not a new area of study. [27] extended on the work of karl pearson by estimating the mean of the sample range in a sample of size n. [28] used order statistics for estimation and test of hypothesis and applied the study to flood related problems; [29] employed the property of linear functions of order statistics to characterize distributions by their l-moments. [30] among other authors described order statistics as an area of statistical study that deals with properties and applications of ordered random variables and their associated functions. order statistics has its wide applications in such areas that include actuarial modeling, finance, reliability engineering, hydrology, meteorology and climatology, signal processing, auction and sports. definition 1: let x1, x2, ..., xn be a random sample of size n from distribution with cdf, f(x), the order statistics corresponding to the random sample is the rearrangement of the sample in order of magnitude denoted by x(1:n), x(2:n), ..., x(n:n) [31] discussed the importance of order statistics and some techniques for deriving statistical and asymptotic distributions. [32] established the recurrence relations for the moment of order statistics for beta distribution and the generalized beta distribution. [33] obtained recurrence relations for the moment of order statistics from generalized beta distribution and [34] derived expressions for moments and recurrence relation of order statistics from the power lomax distribution and tabulated some numerical results with application. definition 2: the largest and smallest order statistics of a random sample x1, x2, ..., xn is the maximum and minimum observation from x(1:n), x(2:n), ..., x(n:n) denoted by x(n:n) and x(1:n) respectively. the minimum and maximum are sometimes the centre of attraction in many areas of application because of some beneficial statistical measures which are of great interest to the researchers. the study of r −out −o f −n system of size n identical components characterized with independent life-lengths requires the statistical tools of order statistics; when r = 1 and r = n; it is called the parallel and series system respectively which is the maximum x(n:n) and minimum x(1:n) order statistics called the extreme order statistics. the extreme order statistics of two parameter lomax distribution was studied by [35]. 1.3. generalized extreme value distribution for r-largest order statistics [36] obtained the limiting form of the frequency of extreme values and subsequently, the generalized extreme value (gev) distributions from [37] and then [38] became popular in finance, hydrology, actuarial and insurance among other field for modeling extreme events. the impacts of extreme random phenomena associated with unfavorable climatic conditions described in [39] have been investigated using the gev distribution including the changes in australian temperatures due to extremes modelled by [40]. depending on the shape parameter, the limiting forms of the gev distribution are gumbel, frechet and weibull distributions. there was a breakthorugh by some researchers who later discovered the importance of statistical tools of order statistics 2 adeyemi et al. / j. nig. soc. phys. sci. 5 (2023) 994 3 for analyzing some extreme events against the popular gev block maxima; for instance, [1] opined that the use of extreme value analysis is a wasteful approach for modeling the block maxima and from [41], the r−largest order statistics (r-los) has the tendency to capture more useful information from extremes dataset and is a useful tool for analyzing extreme events. the use of limiting distribution of the (r-los) from gev distribution for estimating return values of significant wave height was investigated by [42]. the challenge with this approach according to [43] is that as the value of r increases, there is decrease in the rate of convergence to the limiting joint distribution and [41] observed that the variance of the estimator will be high for small r and there is bias when the value of r is too large. [44] estimated extreme wind speed using the method of (r-los) and concluded that gumbel distribution is a suitable model for the dataset. when there are few numbers of observations, [45] suggested the use of (r-los) instead of the extreme value distributions for analyzing maximum values of a given dataset consisting of few observations. in modeling the average maximum daily temperature, the (r-los) when r = 4 was fitted to data by [46]. recently [46] used the (r-los) from extreme value distributions to model daily maximum temperatures from some meteorological stations in thailand. this study is motivated by some useful applications of potential benefits from the properties of order statistics as [47] has also revealed that distributions of order statistics can generate some families of distribution. the research is aimed at exploring the distributional properties of maximum order statistics (maxos) of the weibull exponential pareto (wep) and the new kumaraswamy weibull (nkwei) distributions with application to extreme value dataset. the remaining parts of the paper are structured as follows; section 2 contains the design and formulation of the proposed distributions using the new procedure. section 3 is used for investigating some order statistical properties of the wep distribution. section 4 provides characterization of wep distribution by the l-moment of maximum order statistics and estimation of parameters. results from applications to real life datasets are presented in section 5. section 6 and 7 is for discussions and conclusions respectively. 2. materials and methods 2.1. t-x families of distribution [48] defined the cumulative distribution function (cdf) and the probability density function (pdf) for a random variable x from the t -x family of distributions respectively as follows; g (x) = ∫ −log(1−f(x)) 0 r(t)dt = r (x) {−log(1 − f(x))} (1) g(x) = f (x) 1 − f(x) r{−log(1 − f(x))} (2) where r(t) is the pdf of non negative continuous random variable t defined on [0,∞) the cdf for a random variable t from weibull distribution with parameter α and γ is given as, r (x) (t) = 1 − ex p ( − ( t γ ))α (3) the corresponding probability density function is the derivative of the cdf presented as, r(t) = α γ ( t γ )α−1 ex p ( − ( t γ ))α ; α,γ > 0; x > 0 (4) the t − x family of distributions has weibull-x family by [48] as special case when random variable t follows a weibull distribution and is given as, g (x) = 1 − ex p ( − ( − log(1 − f(x)) γ )α) (5) g(x) = α γ f (x) 1 − f(x) ( − log(1 − f(x)) γ )α−1 ex p ( − ( − log(1 − f(x)) γ )α) (6) 2.2. exponential pareto distribution let x be a pareto random variable, the cdf and pdf of exponential pareto distribution defined respectively by [18] is given by f (x) = 1 − ex p ( −λ ( x k ))θ ; λ,θ, k > 0; x > 0 (7) f (x) = θλ k ( x k )θ−1 ex p ( −λ ( x k ))θ ; λ,θ, k > 0; x > 0 (8) 2.3. weibull exponential pareto (wep) distribution let t be a weibull random variable having the cdf with scale parameter γ and shape α. substitute (7) into the cdf of weibull-x family of distribution in (5) to get; g (x) = 1 − ex p ( − {λ( xk )θ γ }α) (9) the derivative of (9) is the associated density function of wep distribution given by; g(x) = αλθ kγ ( x k )θ−1{λ( xk )θ γ }α−1 ex p ( − {λ( xk )θ γ }α) (10) 2.4. the maximum order statistics generalized distribution the behavior of the maximum order statistics x(n:n) is a special study of the extreme value theory (evt). the limiting distribution of x(n:n) for a non-degenerate distribution function g(x) belong to the family of extreme value distributions with the mathematical expressions give by pr { x(n:n) − bn an ≤ x } =⇒ g(x) as n =⇒ ∞ (11) where an > 0 and bn are sequences of constants the cdf of maxos-g distribution for a continuous baseline distribution g(x) is derived using beta generalized framework as follows; pr { x(n:n) − bn an ≤ x } = 1 b(α, 1) ∫ g(x)wep 0 tα−1dt = ( g(x) )n (12) where b(n, m) = γ(n+m) γ(n)γ(m) and b(α, 1) = b(n, 1) 3 adeyemi et al. / j. nig. soc. phys. sci. 5 (2023) 994 4 2.4.1. the maximum order statistics of the weibull exponential pareto (maxos-wep) distribution the cdf of maxos-wep distribution when the continuous baseline distribution g(x) is wep distribution is derived by substituting (9) into (12) to obtain the cdf of maxos-wep distribution for β = λ/γ defined as; f(n:n)(x) = pr { x(n:n) − bn an ≤ x } = [ 1 − ex p ( − { β ( x k )θ}α)]n (13) the corresponding pdf of maxos-wep distribution is obtained as derivative of f(n:n)(x) given by f(n:n)(x) = [ 1−ex p ( − { β ( x k )θ}α)]n−1 nαβθ k ( x k )θ−1{ β ( x k )θ}α−1 ex p ( − { β ( x k )θ}α) = ∑n−1 i=0 (−1) i(n−1i ) [ ex p ( − { β ( x k )θ}α)]i+1 nαβθ k ( x k )θ−1{ β ( x k )θ}α−1 (14) 2.4.2. the maximum order statistics of the new kumaraswamy weibull (maxos-nkww) distribution the cdf of new kumaraswamy (nkww) distribution introduced by [10] is given by f (x) = 1 − ( 1 − ( 1 − ex p(−λxθ)[1−ex p(−λx θ)] )a)b (15) the pdf of new kumaraswamy weibull distribution is obtained as f (x) =ab ( 1− ( 1−ex p(−λxθ)[1−ex p(−λx θ)] )a)b−1( [1−ex p(−λxθ)] [ex p(−λxθ)] +(λxθ) ) (16) where a = abλθxθ−1ex p(−λxθ)ex p(−λxθ)[1−ex p(−λx θ)] b = (1 − ex p(−λxθ)[1−ex p(−λx θ)] )a−1 the cdf of maxos-nkww distribution is derived by substituting cdf of new kumaraswamy weibull distribution in (15) into (12) and is given by f(n:n)(x) = [ 1− ( 1− ( 1−ex p(−λxθ)[1−ex p(−λx θ)] )a)b]n (17) theorem 2.1: let the cdf and pdf of the new kumaraswamy distribution be f(x) and f (x) defined in (15) and (16) respectively. then the density function of the maxos-nkww distribution is given by; f(n:n)(x) = a ∗ n−1∑ i=0 (−1)i(n−1i ) ∑2b−1 j=0 (−1) j(2b−1j ) ∑2a−1 k=0 (−1) k (2a−1k ) (18) where a∗ =nabλθxθ−1 ex p(−λxθ) ( ex p(−λxθ)[1−ex p(−λx θ)] )k+1( [1−ex p(−λxθ)] [ex p(−λxθ)] +(λxθ) ) proof: the derivative corresponding to the cdf of maxosnkww distribution in (17) is given by; f(n:n)(x) = abc ( 1 − ( 1 − ex p(−λxθ)[1−ex p(−λx θ)] )a)b−1 (19) where a =nabλθxθ−1 ex p(−λxθ)ex p(−λxθ)[1−ex p(−λxθ)] ( [1−ex p(−λxθ)] [ex p(−λxθ)] +(λxθ) ) b = (1 − ex p(−λxθ)[1−ex p(−λx θ)] )a−1 c = [ 1 − ( 1 − ( 1 − ex p(−λxθ)[1−ex p(−λx θ)] )a)b]n−1 using binomial expansion, f(n:n)(x) = ab n−1∑ i=0 (−1)i ( n − 1 i )( 1− ( 1−ex p(−λxθ)[1−ex p(−λx θ)] )a)2b−1 =a ∑n−1 i=0 (−1) i+ j(n−1i ) ∑2b−1 j=0 (2b−1j ) ( 1−ex p(−λxθ)[1−ex p(−λx θ)] )2a−1 =a∗ ∑n−1 i=0 (−1) i(n−1i ) ∑2b−1 j=0 (−1) j(2b−1j ) ∑2a−1 k=0 (−1) k (2a−1k ) (20) where a∗ =nabλθxθ−1 ex p(−λxθ) ( ex p(−λxθ)[1−ex p(−λx θ)] )k+1( [1−ex p(−λxθ)] [ex p(−λxθ)] +(λxθ) ) 3. some properties of the maxos-weibull exponential pareto distribution the central moments, mean order statistics and the variance is derived here 3.1. moments of maxos-wep distribution theorem 3.1 let x1, x2, ..., xn be a random sample of size n from the wep distribution with cdf and pdf denoted by f (x) and f (x) respectively and let x(1) ≤ x(2) ≤ ... ≤ x(n) be corresponding order statistics. then the tth moments of the x(n:n) order statistics for t = 1, 2, .... denoted by µ(t)n:n is given by; µ (t) n:n = n−1∑ i=0 (−1)i ( n − 1 i ) nkt ( 1 β ) t θ ( 1 i + 1 )( t αθ +1) γ ( t αθ + 1 ) (21) proof µ (t) n:n = ∫ ∞ 0 xt fn:n(x)d x (22) then substituting the pdf of maxos-wep in (14) into (22) to get µ (t) n:n ∫ ∞ 0 xt ∑n−1 i=0 (−1) i(n−1i ) [ ex p ( − { β ( x k )θ}α)]i+1 nαβθ k ( x k )θ−1{ β ( x k )θ}α−1 d x (23) µ (t) n:n = ∑n−1 i=0 (−1) i(n−1i ) ∫ ∞ 0 xt [ ex p ( − { β ( x k )θ}α)]i+1 nαβθ k ( x k )θ−1{ β ( x k )θ}α−1 d x (24) 4 adeyemi et al. / j. nig. soc. phys. sci. 5 (2023) 994 5 let y = (i + 1) ( β ( x k )θ)α , by transformation of variable we have the following quantities; x = ky 1 αθ β 1 θ (i+1) 1 αθ ; dyd x = (i+1)αβθ k ( x k )θ−1( β ( x k )θ)α−1 µ (t) n:n = n−1∑ i=0 (−1)i ( n − 1 i ) n (i + 1) ∫ ∞ 0 ( ky 1αθ β 1 θ (i + 1) 1 αθ )t e−ydy(25) µ (t) n:n = n−1∑ i=0 (−1)i ( n − 1 i ) n (i + 1) ∫ ∞ 0 ( k β 1 θ (i + 1) 1 αθ )t y a αθ e−ydy(26) using the gamma function ∫ ∞ 0 yr e−ydy = γ ( r + 1 ) µ (t) n:n = n−1∑ i=0 (−1)i ( n − 1 i ) nkt ( 1 β ) t θ ( 1 i + 1 )( t αθ +1) γ ( t αθ + 1 ) (27) the mean order statistics and the variance of order statistics are derived and given respectively as; µn:n = nk n−1∑ i=0 (−1)i ( n − 1 i )( 1 β ) 1 θ ( 1 i + 1 )( 1 αθ +1) γ ( 1 αθ + 1 ) (28) and σ (2) n:n = µ (2) n:n − ( µn:n )2 σ (2) n:n = nk 2 n−1∑ i=0 (−1)i ( n − 1 i )( 1 β ) 2 θ ( 1 i + 1 )( 2 αθ +1) γ ( 2 αθ +1 ) − ( µr:n )2 (29) the results can be applied for the prediction of expected maximum of future occurrences such as expected maximum flood level in hydrology. corollary 3.1 the result in theorem 3.1 reduces to the explicit expression of the moments and the mean of exponential pareto (ep) distribution studied by [18], if n = α = 1 as deduced and respectively given by; kt (β) t θ γ ( t θ + 1 ) (30) k (β) 1 θ γ ( 1 θ + 1 ) (31) equation (9) in ([18], p.137) 3.2. some properties of maxos-wep distribution by graphical visualization the following plots in figure 1 and figure 2 provide the visualization of some properties of the maxos-wep distributions • the pdf and cdf of maxos-wep x( j: j) for various sample sizes j = 2, 3, ..., n converges to the maximum x(n:n) which is the supremum of all the sample sizes. figure 1 reveals that the pdf of x(30:30), x(32:32) and x(34:34) converges almost to the pdf of x(35:35). figure 1. cdf and pdf of maxos-wep for arbitrary parameters (α = 0.5,β = 0.05,θ = 2.5, k = 2) and various sample sizes figure 2. pdf, cdf, h(x) and r(x) of maxos-wep for arbitrary parameters • if the sample size n is large enough, the parallel systems x(n:n) and x(n+i:n+i) for i = 1, 2, ... from the wep distribution is equivalent in distribution represented by x(n:n) d = x(n+i:n+i). in figure 2, the maximum order statistics between x(195:195) and x(200:200) which are x(196:196), x(197:197), x(198:198), x(199:199) tends to have identical distribution as it is evident that the plots of their densities will coincide and overlap the pdf plots of x(195:195) and x(200:200) with the blue and yellow colours respectively and can be approximated by a common distributional properties. 5 adeyemi et al. / j. nig. soc. phys. sci. 5 (2023) 994 6 • the shapes of maxos-wep distributions is identical for large sample sizes as revealed in figure 2. as the sample sizes increases the pdf of x(120:120), x(130:130) and x(140:140)tends to overlap with no clear difference. • the dispersive ordering between x( j: j) for small sample sizes j = 2, 3, 4, 5 is higher while for large sample sizes x( j: j), say f rom j = 30, 31, 32, ..... as shown in figure 1, the dispersive ordering becomes smaller and gradually diminishes at a faster rate with increase in n. • the pdf and cdf of maxos-wep x( j: j) is less skewed than the pdf and cdf of x( j+1: j+1) for j = 2, 3, .... figure 1 reveals from the pdf that x(2:2) ≤ x(3:3) ≤ x(4:4) in skewness. • the maxos-wep has the same kurtosis for x( j: j), j = 2, 3, ... which could be leptokurtick or mesokurtic depending on either the parameter values of the distribution or the sample size. • the shapes of the hazard rate function from bottom left of figure 2 reveals that the hazard function is identical in shape as the density function posted in the top-left of figure 2. h( j: j)(x) =⇒ f( j: j)(x) in shape ∀ j > 2 for large sample sizes 3.3. limiting properties of maxos-wep distribution the asymptotic properties of the distributions is investigated by taking limits of the density function and cumulative distribution function as x →∞ and as x → 0 in this subsection. using (13) and (14) lim x→o f(n:n)(x) = lim x→o [ 1 − ex p ( − { β ( x k )θ}α)]n = 0 (32) lim x→∞ f(n:n)(x) = lim x→∞ [ 1 − ex p ( − { β ( x k )θ}α)]n = 1 (33) =⇒ 0 ≤ f(n:n)(x) ≤ 1 lim x→o f(n:n)(x) = n−1∑ i=0 (−1)i ( n − 1 i )[ ex p ( − { β ( x k )θ}α)]i+1 x nαβθ k ( x k )θ−1{ β ( x k )θ}α−1 = 0 (34) lim x→∞ f(n:n)(x) = n−1∑ i=0 (−1)i ( n − 1 i )[ ex p ( − { β ( x k )θ}α)]i+1 x nαβθ k ( x k )θ−1{ β ( x k )θ}α−1 = 0 (35) =⇒ limx→∞ = 0 = limx→0 4. characterization of the wep distribution by l-moments of the maxima order statistics the l-moments can be derived as the expectations of extreme order statistics which has been defined in [49] and given by; λr = r∑ i=1 (−1)r−ii−1 ( r − 1 i − 1 )( r + i − 2 i − 1 ) e(x(i:i)) (36) expansion of (36) leads to the following system of equations for the first four l-moments λr, r = 1, 2, 3, 4. in terms of the maximum order statistics λ1 = e ( x1:1 ) (37) λ2 = e ( x2:2 ) − e ( x1:1 ) (38) λ3 = 2e ( x3:3 ) − 3e ( x2:2 ) + e ( x1:1 ) (39) λ4 = 5e ( x4:4 ) − 10e ( x3:3 ) + 6e ( x2:2 ) − e ( x1:1 ) (40) by application of moment of maxos-wep result in (21) when t = 1 µn:n = nk n−1∑ i=0 (−1)i ( n − 1 i )[ 1 β ] 1 θ [ 1 (i + 1) ] 1 αθ +1 γ ( 1 αθ +1 ) (41) λ1 = e(x1:1) = µ = k [ 1 β ] 1 θ γ ( 1 αθ + 1 ) (42) e(x2:2) = 2k 1∑ i=0 (−1)i ( 1 i )[ 1 β ] 1 θ [ 1 (i + 1) ] 1 αθ +1 γ ( 1 αθ + 1 ) = 2k [ 1 β ] 1 θ γ ( 1 αθ + 1 ) − 2k [ 1 β ] 1 θ [ 1 2 ] 1 αθ +1 γ ( 1 αθ + 1 ) (43)  λ2 = e(x2:2) − e(x1:1) =2k [ 1 β ] 1 θ γ ( 1 αθ +1 ) −2k [ 1 β ] 1 θ [ 1 2 ] 1 αθ +1 γ ( 1 αθ +1 ) −k [ 1 β ] 1 θ γ ( 1 αθ +1 ) = k [ 1 β ] 1 θ γ ( 1 αθ + 1 ) − 2k [ 1 β ] 1 θ [ 1 2 ] 1 αθ +1 γ ( 1 αθ + 1 ) = k [ 1 β ] 1 θ γ ( 1 αθ + 1 )[ 1 − 2 [ 1 2 ] 1 αθ +1] = k [ 1 β ] 1 θ γ ( 1 αθ + 1 )[ 1 − [ 1 2 ] 1 αθ ]  (44)  e(x3:3) = 3k ∑2 i=0(−1) i ( 2 i )[ 1 β ] 1 θ [ 1 (i+1) ] 1 αθ +1 γ ( 1 αθ + 1 ) =3k [ 1 β ] 1 θ γ ( 1 αθ +1 ) −6k [ 1 β ] 1 θ [ 1 2 ] 1 αθ +1 γ ( 1 αθ +1 ) +3k [ 1 β ] 1 θ [ 1 3 ] 1 αθ +1 γ ( 1 αθ +1 )  (45) 6 adeyemi et al. / j. nig. soc. phys. sci. 5 (2023) 994 7 λ3 = 2e(x3:3) − 3e(x2:2) + e(x1:1) =2  3k [ 1 λ ] 1 θ γ ( 1 αθ +1 ) −6k [ 1 β ] 1 θ [ 1 2 ] 1 αθ +1 γ ( 1 αθ +1 ) +3k [ 1 β ] 1 θ [ 1 3 ] 1 αθ +1 γ ( 1 αθ +1 ) −3 2k [ 1 β ] 1 θ γ ( 1 αθ +1 ) −2k [ 1 β ] 1 θ [ 1 2 ] 1 αθ +1 γ ( 1 αθ +1 )+k [ 1 β ] 1 θ γ ( 1 αθ +1 ) =6k [ 1 β ] 1 θ [ 1 3 ] 1 αθ +1 γ ( 1 αθ +1 ) −6k [ 1 β ] 1 θ [ 1 2 ] 1 αθ +1 γ ( 1 αθ +1 ) +k [ 1 β ] 1 θ γ ( 1 αθ +1 ) = k [ 1 β ] 1 θ γ ( 1 αθ + 1 ) 6[ 13 ] 1 αθ +1 − 6 [ 1 2 ] 1 αθ +1 + 1  = k [ 1 β ] 1 θ γ ( 1 αθ + 1 ) 1 − 3[ 12 ] 1 αθ + 2 [ 1 3 ] 1 αθ  (46) e(x4:4) = 4k 3∑ i=0 (−1)i ( 3 i )[ 1 β ] 1 θ [ 1 (i + 1) ] 1 αθ +1 γ ( 1 αθ + 1 ) (47) =  4k(30) [ 1 β ] 1 θ γ ( 1 αθ +1 ) −4k(31) [ 1 β ] 1 θ [ 1 2 ] 1 αθ +1 γ ( 1 αθ +1 ) +4k(32) [ 1 β ] 1 θ [ 1 3 ] 1 αθ +1 γ ( 1 αθ +1 ) −4k(33) [ 1 β ] 1 θ [ 1 4 ] 1 αθ +1 γ ( 1 αθ +1 ) (48) =  4k [ 1 β ] 1 θ γ ( 1 αθ + 1 ) − 12 [ 1 β ] 1 θ [ 1 2 ] 1 αθ +1 γ ( 1 αθ + 1 ) +12 [ 1 β ] 1 θ [ 1 3 ] 1 αθ +1 γ ( 1 αθ +1 ) −4k [ 1 β ] 1 θ [ 1 4 ] 1 αθ +1 γ ( 1 αθ +1 )  (49) λ4 = 5e(x4:4) − 10e(x3:3) + 6e(x2:2) − e(x1:1) (50) λ4 =  5 4k[ 1β] 1θ γ( 1αθ + 1) − 12[ 1β] 1θ [ 12 ] 1αθ +1γ( 1αθ + 1)  +5 12 [ 1 β ] 1 θ [ 1 3 ] 1 αθ +1 γ ( 1 αθ +1 ) −4k [ 1 β ] 1 θ [ 1 4 ] 1 αθ +1 γ ( 1 αθ +1 ) −10  3k [ 1 β ] 1 θ γ ( 1 αθ +1 ) −6k [ 1 β ] 1 θ [ 1 2 ] 1 αθ +1 γ ( 1 αθ +1 )−10  +3k [ 1 β ] 1 θ [ 1 3 ] 1 αθ +1 γ ( 1 αθ +1 ) +6 2k [ 1 β ] 1 θ γ ( 1 αθ +1 ) −2k [ 1 β ] 1 θ [ 1 2 ] 1 αθ +1 γ ( 1 αθ +1 )−k [ 1 β ] 1 θ γ ( 1 αθ +1 )  (51) =  20k[ 1β] 1θ γ( 1αθ + 1) − 30[ 1β] 1θ [ 12 ] 1αθ γ( 1αθ + 1)  + 20 [ 1 β ] 1 θ [ 1 3 ] 1 αθ γ ( 1 αθ +1 ) −5k [ 1 β ] 1 θ [ 1 4 ] 1 αθ γ ( 1 αθ +1 ) − 30k[ 1β] 1θ γ( 1αθ + 1) − 30k[ 1β] 1θ [ 12 ] 1αθ γ( 1αθ + 1)  + 12k[ 1β] 1θ γ( 1αθ + 1) − 6k[ 1β] 1θ [ 12 ] 1αθ γ( 1αθ + 1)  − 10k[ 1β] 1θ [ 13 ] 1αθ γ( 1αθ + 1)  − k[ 1β] 1θ γ( 1αθ + 1)  (52) =  k [ 1 β ] 1 θ γ ( 1 αθ + 1 ) − 6k [ 1 β ] 1 θ [ 1 2 ] 1 αθ γ ( 1 αθ + 1 ) +10 [ 1 β ] 1 θ [ 1 3 ] 1 αθ γ ( 1 αθ + 1 ) − 5k [ 1 β ] 1 θ [ 1 4 ] 1 αθ γ ( 1 αθ + 1 )  (53) = k [ 1 β ] 1 θ γ ( 1 αθ +1 ) 1 − 6[ 12 ] 1 αθ + 10 [ 1 3 ] 1 αθ − 5 [ 1 4 ] 1 αθ (54) the l-moment ratios are derived as τ2 = λ2 λ1 = k [ 1 β ] 1 θ γ ( 1 αθ + 1 )[ 1 − [ 1 2 ] 1 αθ ] k [ 1 β ] 1 θ γ ( 1 αθ + 1 ) = [1−[ 12 ] 1 αθ ] (55) τ3 = λ3 λ2 = k [ 1 β ] 1 θ γ ( 1 αθ +1 )1−3 [ 1 2 ] 1 αθ +2 [ 1 3 ] 1 αθ  k [ 1 β ] 1 θ γ ( 1 αθ +1 )[ 1− [ 1 2 ] 1 αθ ] =  1−3 [ 1 2 ] 1 αθ +2 [ 1 3 ] 1 αθ [ 1− [ 1 2 ] 1 αθ ] (56) τ4 = λ4 λ2 = k [ 1 β ] 1 θ γ ( 1 αθ +1 )1−6 [ 1 2 ] 1 αθ +10 [ 1 3 ] 1 αθ −5 [ 1 4 ] 1 αθ  k [ 1 β ] 1 θ γ ( 1 αθ +1 )[ 1− [ 1 2 ] 1 αθ ] = 1 − 6[ 12 ] 1αθ + 10[ 13 ] 1αθ − 5[ 14 ] 1αθ [ 1 − [ 1 2 ] 1 αθ ] (57) the l-moment estimates of parameters α and θ denoted respectively by α̂ and θ̂ can be obtained as the solution to nonlinear equations in (56) and (57). thereafter the computation of lmoment estimates of parameters β and k denoted respectively by β̂ and k̂ can be derived from (49) and (54). 5. results from real-life applications the usefulness of the maximum order statistics generalized (maxos-g) family of convoluted distributions where g represent wep and nkwei distributions from the study is investigated by analyzing three popular extreme value datasets. analysis of data is carried out using the r-software to generate numerical values for the goodness-of-fit-statistics. the standard and widely used goodness-of-fitstatistics for model selection criteria by researchers in making decision about competing distributions are -loglikelihood (ll), akaike information criterion (aic), bayesian information criterion (bic), consistent akaike information criterion (caic), hannan-quinn information criteria (hqic), the kolmogorov-smirnov (k-s) statistics and the p-value. they can be found in several related works including [24], [50], [51] and most recently in [56]. the goodnessof-fit-statistics are defined as: 7 adeyemi et al. / j. nig. soc. phys. sci. 5 (2023) 994 8 table 1. mles of parameters for annual maximum flood data models α β θ k λ maxos-nkwei 1.0369 1.2478 0.5195 0.5034 maxos-wep 2.8914 1.2165 0.1895 8.6417 nkwei 5.8540 0.7002 0.8877 0.0969 wep 1.0592 0.0077 1.6407 2.9449 aic = −2l + 2m; caic = −2l + 2mn (n + 1) ; aicc = aic + 2m(m + 1) (n − m − 1) , bic = −2l + m log(n); hqic = −2l + 2m(log(n)) l is the estimated log-likelihood (ll), m is the number of parameters in the model. the p-value and k-s are statistics associated with the goodness-of-fit criteria. decision: the best fitted model is identified with the smallest values of the estimated goodness-of-fit criteria or model with the highest p-value returned for the k-s statistics. 5.1. application to annual maximum flood discharges hydrological dataset the hydrological data which was taken from [10] in application to nkwei distribution was previously studied by[52, 53], the data represent the annual maximum flood discharges (in units of 1000 cubic feet per second) of the north saskachevan river at edmonton, over a period of 48 years. the data are: (19.885, 20.940, 21.820, 23.700, 24.888, 25.460, 25.760, 26.720, 27.500, 28.100, 28.600, 30.200, 30.380, 31.500, 32.600, 32.680, 34.400, 35.347, 35.700, 38.100, 39.020, 39.200, 40.000, 40.400, 40.400, 42.250, 44.020, 44.730, 44.900, 46.300, 50.330, 51.442, 57.220, 58.700, 58.800, 61.200, 61.740, 65.440, 65.597, 66.000, 74.100, 75.800, 84.100, 106.600, 109.700, 121.970, 121.970, 185.560) the dataset with line plots in fig 3 received a new analysis in this present study. applications to maxos-wep and maxos-nkwei distributions has the parameter estimates and the goodness-of-fit statistics presented in table 1 and table 2 respectively. figure 4 is the plots for the histogram with density functions and estimated cdf to assess the performance of the distributions. figure 3. annual maximum flood discharge (in units of 1000 cubic feet per second) table 2. estimated goodness-of-fit statistics for annual maximum flood data model -ll aic bic k-s p-value maxos-nkwei 216.01 440.08 447.51 0.0694 0.9749 maxos-wep 216.65 441.30 448.78 0.0712 0.9681 nkwei 216.91 441.82 449.31 0.0767 0.9405 wep 225.74 459.47 466.95 0.1465 0.2540 figure 4. histogram and fitted distribution for annual maximum flood discharges of the north saskachevan river at edmonton 5.2. application to annual maximum precipitation dataset the data with line plots in figure 4 is available in extreme package from the r-software, it has been studied by [52] and recently applied to the nkw-w distribution by [10]. the data 8 adeyemi et al. / j. nig. soc. phys. sci. 5 (2023) 994 9 figure 5. annual maximum precipitation amount at fort collins, usa, 1900 − 1999. table 3. mles of parameters for precipitation data. models α β θ k λ maxos-nkwei 0.7610 0.5318 0.5454 0.5816 maxos-wep 0.3029 0.5263 1.7755 5.4395 nkwei 11.228 0.7698 0.8310 0.0481 wep 3.0488 0.1055 0.7732 10.8781 represent the annual maximum precipitation (inches) for one rain gauge in fort collins, colorado from 1900 through 1999. the data are: (239, 232, 434, 85, 302, 174, 170, 121, 193, 168, 148, 116, 132, 132, 144, 183, 223, 96, 298, 97, 116, 146, 84, 230, 138, 170, 117, 115, 132, 125, 156, 124, 189, 193, 71, 176, 105, 93, 354, 60, 151, 160, 219, 142, 117, 87, 223, 215, 108, 354, 213, 306, 169, 184, 71, 98, 96, 218, 176, 121, 161, 321, 102, 269, 98, 271, 95, 212, 151, 136, 240, 162, 71, 110, 285, 215, 103, 443, 185, 199, 115, 134, 297, 187, 203, 146, 94, 129, 162, 112, 348, 95, 249, 103, 181, 152, 135, 463, 183, 241). the dataset has the time series plots displayed in figure 5, the results presented in table 3 and table 4 established the flexibility of the proposed maxos-g distributions from reallife application of maxos-wep and maxos-nkwei distributions to the annual maximum precipitation with comparisons to their baseline components when g is a wep and nkwei distributions. the histogram and fitted distributions posted in figure 6 supported the results in table 4 about the superior performance of the proposed technique. table 4. estimated goodness-of-fit statistics for precipitation data. model -ll aic bic k-s p-value maxos-nkwei 565.11 1138.23 1148.65 0.037 0.999 maxos-wep 565.12 1138.25 1148.67 0.042 0.995 nkwei 565.22 1138.44 1148.86 0.045 0.988 wep 576.33 1160.67 1171.09 0.096 0.314 figure 6. histogram and fitted distribution for annual maximum precipitation for one rain gauge in fort collins, colorado from 1900 through 1999. table 5. mles of parameters for annual maximum one-day rainfall data. model α β θ k λ maxos-nkwei 0.3817 0.2642 0.6303 0.7125 maxos-wep 0.6541 0.5647 1.0089 5.4882 nkwei 8.9151 0.7163 0.0555 0.9217 wep 1.6118 0.0072 1.4442 3.2738 tc 61.538 − − 19.959 5.3. application to annual maximum one-day rainfall data the data from [54] was analyzed using the transmuted-cauchy (tc) distribution developed by [55] . this present study analysed the data and performances of nkwei, wep and tc convoluted distributions were compared with the performances of models from the maxos-g and presented in table 6, estimated values of parameters are displayed in table 5. the results presented in table 6 revealed improved estimated goodness-of-fit statistics from the new procedure which indicates the superiority of the maxos-g approach over the convoluted distributions. application of maxos-wep and maxos-nkwei distributions to the annual maximum one-day rainfall justified the need to explore for new technique. 9 adeyemi et al. / j. nig. soc. phys. sci. 5 (2023) 994 10 table 6. goodness-of-fit statistics of annual maximum one-day rainfall data. model -ll aic bic k-s p-value maxos-nkwei 194.02 396.04 402.69 0.067 0.9948 maxos-wep 194.04 396.08 402.73 0.066 0.9948 nkwei 194.10 396.19 402.37 0.073 0.9846 tc 197.83 401.66 − 0.085 0.9423 wep 197.86 402.92 409.57 0.113 0.7020 figure 7. histogram and fitted distribution for annual maximum one-day rainfall at florida atlantic university 6. discussion statistical modeling of extreme stochastic inevitable phenomena in our environments is at the heart of many disciplines including meteorologist, hydrologist, climatologist, reliability engineering, medical sciences, actuarial and insurance among others. this research explored for some novel approach applicable for modeling extreme value datasets by developing two new models called the maxos-nkwei and maxos-wep distributions using the maxos-g framework obtained as a special case of the beta-g family of distribution. some statistical properties of the maxos-wep investigated includes the moment, mean, variance, and asymptotic behaviours. the skewness, kurtosis and some ordering properties were investigated using the graphical visualization from plots of the density, hazard and cumulative distribution functions. the importance of the study and the usefulness of the models was tested by way of application to three extreme value datasets representing the annual maximum flood discharges of the north saskachevan river at edmonton; the annual maximum precipitation in fort collins, colorado from 1900 through 1999 and the annual maximum one-day rainfall data from prism climate group, oregon state university. the results revealed tremendous improvement over the use of convoluted distributions proposed from the existing literatures. the blue and yellow lines from (figure 4 and 6) and the pink line from (figure 7) represent the fitted convoluted distributions to the data, the red and green lines represent the fitted maxos-g models. the two proposed models from the maxos-g provides superior modeling performance established by the smallest goodness-of-fit statistics and highest p-values compared to the results for nkwei by [10] and tc distribution proposed by [55]. the new technique explored in this research revealed superior modeling capacity of the approach over the continuing reliance on convoluted distributions for extreme value modelling which hitherto has dominated the literatures. 7. conclusion the weibull exponential pareto (wep) distribution has enormous potential for analyzing random phenomena that is right skewed and approximately symmetric associated with hydrology, reliability engineering, public health and some other area of applications that are characterized with heavy tail or large kurtosis. the new kumaraswamy weibull (nkwei) distribution provides better goodness-of-fit estimates than many established generalized weibull model existing in the literatures as revealed in [10] . however, exploring some potential properties of order statistics from the two convoluted distributions using the maxos-g framework revealed an improved performance in the modeling capacity of the convoluted distributions. the maxos-wep and the maxos-nkwei has superior performance for modeling the hydrological extreme value datasets. the results proved that the two distributions from maxos-g are the best model for the data when compared to the baseline (g) distributions. the graphical plots in (figure 4, 6 and 7) further corroborated the superiority of the proposed method for modeling the three annual maximum datasets over the existing methods. the maxos-g framework will be an important modeling tool in extreme value analysis and a good choice in several fields of applications such as hydrology, actuarial, survival analysis, economics, quality control, medical sciences and meteorology. acknowledgments the authors are grateful for the comments and suggestions of the anonymous reviewers and the editor towards production of the final manuscript. there is no funding from any source towards the success of this project. no funds, grants, or other support was received references 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[55] c. ball, b. rimal & s. chhetri, “a new generalized cauchy distribution with an application to annual oneday maximum rainfall data”, stat. optim. inf. comput. 9 (2021) 123. https://doi.org/10.19139/soic-23105070-1000 [56] b. makubate, m. matsuokwane, b. o. oluyede, l. gabaitiri & s. chamunorwa, “the type ii topp-leone-g power series distribution with applications on bladder cancer”, j. nig. soc. phys. sci. 4 (2022) 848. https://doi.org/10.46481/jnsps.2022.848 11 j. nig. soc. phys. sci. 4 (2022) 867 journal of the nigerian society of physical sciences an integral transform-weighted residual method for solving second order linear boundary value differential equations with semi-infinite domain e. i. akinolaa,∗, r. a. oderinub, s. alaob, o. e. opaleyeb a mathematics programme, bowen university, iwo osun state, nigeria. b department of pure and applied mathematics, ladoke akintola university, ogbomoso. oyo state. nigeria abstract solving boundary value problem with semi-infinite domain in a conventional way or using some of the available approximate methods poses a lot of challenges or better still seems almost impossible over the years, and some of the alternative ways for crossing this hurdle is by fixing value for infinity that is in the domain. here in this work, we present an integral transform (aboodh transform) weighted residual based method (at-wrm) to address the afore-mentioned challenge. aboodh transform was used to transform and at the same time used to find the inverse of the given differential equations while weighted residual via collocation method was used in order to avoid fixing value for infinity as usual by introducing e−ix in trial function to decay the infinity that was part of the boundary condition. the accuracy of the presented method was authenticated by solving three different problems. the excellent results obtained from the three solved problems validate the accuracy and effectiveness of the method doi:10.46481/jnsps.2022.867 keywords: aboodh transform, weighted residual, partition method, boundary value problem, semi-infinite domain. article history : received: 14 june 2022 received in revised form: 20 september 2022 accepted for publication: 21 september 2022 published: 11 november 2022 c©2022 journal of the nigerian society of physical sciences. all rights reserved. communicated by: w. a. yahya 1. introduction boundary conditions at infinity are commonly used to solve linear and nonlinear phenomena that emerge in a wide range of scientific and engineering sectors. for the solution of these types of problems, a variety of mathematical techniques and methodologies have been proposed. oderinu and aregbesola [1], for example, employed the weighted residual approach to address the challenge encountered in a semi-infinite domain ∗corresponding author tel. no: +2348062723102. email address: emmanuel.idowu@bowen.edu.ng (e. i. akinola ) by reducing the domain within (0,∞) utilizing the gausslaguerre integration formula with galerkin and moment methods. similarly, odejide and aregbesola [2] utilized the same strategy to tackle the problems in a semi-infinite domain, but the method of partition was employed. to handle problems in an unbounded domain, several researchers have employed a variety of strategies. noor and syed [3] used variational iteration method that was modified. adewumi et al. [4] used the combination of laplace transformation method and weighted residual method to provide a series solution to problems in a semi-infinite domain, peker et al. [5] employed differential transform method and pade approximant 1 e. i. akinola et al. / j. nig. soc. phys. sci. 4 (2022) 867 2 to solve blasius problem, which appears in many areas of science and engineering professions. likewise, sajid et al. [6] presented the hybrid variational iteration algorithm method based on the combination of variational iteration and shooting methods to obtain the result of the same blasius problem, akinola and ogunlaran [7] presented solution to these types of boundary-value problems in a semi-finite domain by sumudu transform decomposition method, several other methods have also been employed to solve these type of problem and they are: adomian decomposition method [8], solution of the mhd falner-skan flow by hankel-pade method [9], variational iteration approach [10] homotopy analysis decomposition method for the solution of viscous boundary layer flow due to a moving sheet by alao et al. [11], application of laplace decomposition method to boundary value equation in a semi-infinite domain by ogunsola et al. [12], a comparison study of numerical techniques for solving ordinary differential equations defined on a semi-infinite domain using rational chebyshev functions by ramadan [13], an order four continuous numerical method for solving general second order ordinary differential equations by obarhua [14], numerical algorithms for direct solution of fourth order ordinary differential equations by kuboye [15] to mention a few and many other analytical and numerical methods have also been employed to tackle problems of the nature considered. here, we would employ the combination of an integral transform proposed by aboodh [16, 17] and weighted residual method by crandall [18], finlayson and scriven [19], vichnevetsky[20] for finding the solution of problems with semi infinite domain, inspired and motivated by the existing research in this direction, the researchers decided to couple weighted-residual method with aboodh transform (at) in order to compliment the drawback of aboodh transform that cannot handle boundary value problems especially with semiinfinite domain. the suggested technique uses the aboodh transform method to generate a new form of trial function, which is then substituted into the original equation to obtain the residual function, then ultimately reduced by the proposed method. 2. basic concept of the at-wrm 2.1. basic idea of aboodh transform aboodh transform is one out of many integral transforms defined for function of exponential order [16, 17] by: a = { f (t) : ∃m, k1, k2 > 0, | f (t)| < me −vt } , (1) where k1, k2 may be finite or infinite in a given set m. the aboodh transform of a function f (t) is defined as: a[ f (t)] = k(v) = 1 v ∫ ∞ 0 f (t)e−vtdt, t ≥ 0, k1 ≤ v ≤ k2. (2) aboodh transform of some of the functions are: a(c) = c v2 , where c is a constant a[tn] = n! vn+2 , a[e±at] = 1 v2 ± av , a[sin(at)] = a v(v2 + a2) , a[cos(at)] = 1 (v2 + a2) . aboodh transform of first, second and nth derivatives are: a[ f ′(t)] = vk(v) − f (0) v a[ f ′′(0)] = v2 k(v) − f ′(0) v − f (0) a[ f n(0)] = vn k(v) − n−1∑ k=0 f k(0) v(2−n+k) 2.2. basic idea of weighted residual method given a boundary value problem m[u] = r(x), x ∈ d, bµ[u] = γk, x ∈ ∂d. (3) m[u] denotes a general differential operator, bµ is the appropriate number of boundary conditions involved and d is the domain with boundary ∂d. the solution of (3) is found by assuming an approximate solution called trial function to the dependent function u(x) in the form: ω(x, a) = w0(x) + n∑ i=1 aiwi(x), (4) where ωi(x) are prescribed and satisfy the boundary conditions. upon substitution of (4) into (3) the residual r(x, a) in the differential equation is obtained. the objective is to minimize the residual such that it becomes smaller and smaller or tend to zero as the number n of the function ωi(x) increased in the successive approximations. having obtained the residual, one of the methods, namely: collocation, partition, moment, least square and galerkin of minimizing the residual is employed. 3. methodologyaboodh transform weighted residual based method (at-wrm) suppose we have a differential equation: l[u(x)] = f (5) in the domain ψ [1] bγ[u] = ψ on ∂ψ, (6) where l[u(x)] denotes a general differential operator (linear or non-linear) involving spatial derivatives of dependent variable u, f is a known function of position, bγ[u] represents the appropriate number of boundary conditions, and ψ is the domain (semi-infinite in this case) with the boundary ∂ψ. the itemized steps below were followed in solving this type of problem: 2 e. i. akinola et al. / j. nig. soc. phys. sci. 4 (2022) 867 3 (i) we assumed a trial function that satisfies (6) in the form [10] n∑ i=0 die −ikx , the assumption above is due to the condition at infinity, where di i = 0, 1, 2, ·, n are constants to be determined and k is any constant value (ii) taking the aboodh transform of (5) to have a [ l[u(x)] ] = a[ f ] (7) (iii) substituting the trial function obtained in step 1 into (7) to give a [ l [ n∑ i=0 die −ikx − f ]] = 0. (8) (iv) taking the aboodh inverse transform of (8) to obtain a new trial function given as a−1 [ a [ l [ n∑ i=0 die −ikx − f ]] = 0 ] (9) (v) imposing the boundary condition in finite domain on (9) to give a set of equations (vi) likewise, substituting (9) into (5) to obtain the residual function which is then minimized using collocation method by making use of xk which is the zeros of the laguerre polynomial ln(x) given [2, 21] ln(x) = e x d n d xn (e−x xn) ak = (n!)2 xk[l′n(xk)]2 (10) table 1: laguerre polynomial and the corresponding weight functions n xk ak 2 0.58578644 0.85355339 3.41421356 0.14644661 4 0.32254769 0.60315410 1.74576110 0.35741869 4.53662030 0.03888791 9.39507091 0.00053929 6 0.22284660 0.45896467 1.18893210 0.41700083 2.99273633 0.11337338 5.77514357 0.01039920 9.83746742 0.00026102 15.98287398 0.00000090 (vii) the number of equations obtained equalled the number of constants to be determined and, so we solved the resulting algebraic equations through maple 18 to obtain the constants di which are substituted back into the trial function in step 1 to find the solution. 3.1. numerical illustration illustration 1: 8 d2u d x2 + 2 du d x − u = 8e− x 4 (11) subject to the boundary conditions u(0) = 1, u(∞) = 0, with exact solution given as: uexact = 9e − 1 2 x − 8e− 1 4 x. assuming the trial function u = n∑ i=0 die −ikx with n = 6 and k = 14 gives: u = d0+d1e − 1 4 x+d2e − 1 2 x+d3e − 3 4 x+d4e −x +d5e − 5 4 x+d6e − 3 2 x.(12) finding the aboodh transform of (11) and substituting (12) into the resulting equation gives: a[u] = k[v] = 1 v2 [ u′(0) v + u(0) + a [ e− x 4 + 1 8 u− 1 4 u′ ]] .(13) upon application of inverse aboodh transform on (13), we have: u = a−1 [ 1 v2 [ u′(0) v + u(0) + a [ e− x 4 + 1 8 u − 1 4 u′ ]]] . (14) substituting (14) into (11) gives: r = 3 + 5 16 d3e − 3 4 x + 3 8 d4e −x + 7 16 d5e − 5 4 x + 1 2 d6e − 3 2 x + 1 8 d0 + 1 8 d1 − e − 1 4 x − 1 2 x + 1 32 d0 − 1 128 d0(x 2 + 16) − 1 72 d3 ( 4 − 3x + 5e− 3 4 x ) − 1 64 d4 ( 5 − 5x + 3e−x ) − 1 400 d5 ( 36 − 45x + 14e− 5 4 ) − 1 144 d6 ( 14 − 21x + 4e− 3 2 x ) − 1 8 e− 1 4 x(16 + 3d1) − 1 16 d1(x − 4) + 1 36 d3 ( − 3 − 15 4 e− 3 4 x ) + 1 32 d4(−5 − 3e −x) + 1 200 d5( − 45 − 35 2 e− 5 4 x ) + 1 72 d6 ( − 21 − 6e− 3 2 x ) . (15) subjecting (14) to the initial condition u(0) = 1, and collocate (11) using the roots of laguerre polynomial of six points at x = 0.22284660, 1.18893210, 2.99273633, 5.77514357, 9.83746742, 15.98287398 to take care of the boundary condition at infinity-u(∞) = 0. this gives rise to seven algebraic equations with seven unknown altogether. solving the seven equations for the unknown constants through maple 18, we have: d0 = 1.88748 × 10 −7, d1 = −8.000002108199, d2 = 9.000009965150, d3 = −0.000023454290, d4 = 0.000028981727, d5 = −0.000018037567, d6 = 0.000004464431. (16) 3 e. i. akinola et al. / j. nig. soc. phys. sci. 4 (2022) 867 4 substituting (16) into the assumed trial function (12) gives the desired solution to (11) as: uat−wrm = 1.88748 × 10 −7 − 8.000002108199e −1 4 x + 9.000009965150e −1 2 x − 0.000023454290e −3 4 x + 0.000028981727e−x − 0.000018037567e −5 4 x + 0.000004464431e −3 2 x (17) figure 1: graphical validation of the initial boundary condition of illustration 1 illustration 2: 8 d2u d x2 + 2 du d x − u = 8e− 3x 4 , 0 ≤ x ≤∞, (18) with the exact solution uexact = −3e − 1 2 x + 4e− 3 4 x. to obtain the solution to (18), salient points or procedures itemized in section 3 were carried out one after the other with the same trial function as (12), the same number of points (n = 6) and k = 14 . hence, the constant terms of (12) and the approximate result of (18) are obtained as: d0 = −2.2517 × 10 −9, d1 = 2.5194 × 10 −8, d2 = −3.000000119291, d3 = 4.000000281182, d4 = −3.47939 × 10 −7, d5 = 2.16854 × 10 −7, d6 = −5.3749 × 10 −8 (19) and uat−wrm = −2.2517 × 10 −9 + 2.5194 × 10−8e− 1 4 x − 3.000000119291e− 1 2 x + 4.000000281182e− 3 4 x − 3.47939 × 10−7e−x + 2.16854 × 10−7e− 5 4 x − 5.3749 × 10−8e− 3 2 x (20) figure 2: graphical validation of the initial boundary condition of illustration 2 illustration 3: consider the problem solved by odejide [2] d2u d x2 + 2 du d x − 2u = −e−2x, 0 ≤ x ≤∞ (21) subject to the boundary conditions u(0) = 1, u(∞) = 0. the exact solution to (21) is: uexact = 1 2 ( e(−1+ √ i)x + e−2x ) this time around the new trial function u = n∑ i=0 die (−1+ √ i)x 4 e. i. akinola et al. / j. nig. soc. phys. sci. 4 (2022) 867 5 table 2: comparison between at-wrm and the exact for illustration 1 x uexact uat−wrm |uexact − uat−wrm| 0.0 1.000000000000 0.9999999999995 5.2791 × 10−13 0.1 0.758585524280 0.7585855246842 4.04668539 × 10−10 0.2 0.533701366318 0.5337013671228 8.04471306 × 10−10 0.3 0.324423897197 0.3244238983415 1.144766413 × 10−9 0.4 0.129877433414 0.1298774348212 1.406555289 × 10−9 0.5 -0.050768173034 -0.05076817144136 1.593182583 × 10−9 0.6 -0.218299825266 -0.2182998235440 1.721015535 × 10−9 0.7 -0.373463358686 -0.3734633568723 1.812797763 × 10−9 0.8 -0.516965610304 -0.5169656084104 1.893055570 × 10−9 0.9 -0.649476385479 -0.6494763834940 1.985061000 × 10−9 1.0 -0.771630327158 -0.7716303250490 2.1089594485 × 10−9 table 3: comparison between at-wrm and the exact for illustration 2 x uexact uat−wrm umwr[2] |uexact − uat−wrm| |uexact − umwr| 0.0 1.000000000000 0.999999999999 0.9999999995 7.0 × 10−13 5.0 × 10−10 0.1 0.857285671812 0.8572856718066 0.8572856728 5.51848747 × 10−12 1.8 × 10−9 0.2 0.728319651592 0.7283196515825 0.7283196512 1.032911322 × 10−11 8.0 × 10−10 0.3 0.611940945763 0.6119409457479 0.6119409463 1.445554232 × 10−11 3.0 × 10−10 0.4 0.507080623493 0.5070806234760 0.5070806239 1.765087242 × 10−11 1.0 × 10−10 0.5 0.412754765950 0.4127547659297 0.4127547648 1.994249358 × 10−11 1.2 × 10−9 0.6 0.328057944442 0.3280579444209 0.3280579430 2.151874020 × 10−11 1.0 × 10−9 0.7 0.252157188311 0.2521571882890 0.2521571879 2.264794352 × 10−11 1.1 × 10−9 0.8 0.184286406269 0.1842864062451 0.1842864050 2.362236467 × 10−11 1.0 × 10−9 0.9 0.123741227565 0.1237412275399 0.1237412271 2.472101384 × 10−11 1.0 × 10−10 1.0 0.069874231826 0.06987423180009 0.06987423173 2.618660748 × 10−11 2.7 × 10−10 is assumed for (21) with n = 6. following the same process as done for the illustrations 1 and 2 we have: d0 = −3.464061362 × 10 −11, d1 = 0.5000000099703, d2 = −9.79486026690 × 10 −8, d3 = 0.5000003319506, d4 = −5.104366705162 × 10 −7, d5 = 3.679792467715 −7, d6 = −1.014800109354 × 10 −8 (22) and uat−wrm = −3.464061362 × 10 −11e−x + 0.5000000099703e−2x − 9.79486026690 × 10−8e−2414213562x + 0.5000003319506e−2732050808x − 5.104366705162 × 10−7e−3.000000x + 3.679792467715−7e−3.236067977x − 1.014800109354 × 10−8e−3.449489743x (23) 3.2. discussion of results an integral transform -weighted residual based technique has been successfully used in solving second order linear boundary value problems with semiinfinite domain. in order to examine the reliability of the proposed method ,three different illustrations were considered. tables 2, 3 and 4 show the figure 3: graphical validation of the initial boundary condition of illustration 3 results of the three illustrations respectively and in all the three tables it was observed that absolute difference between exact 5 e. i. akinola et al. / j. nig. soc. phys. sci. 4 (2022) 867 6 table 4: comparison between at-wrm and the exact for illustration 3 x uexact uat−wrm umwr[2] |uexact − uat−wrm| |uexact − umwr| 0.0 1.000000000000 1.000000000001 1.00000000 1.346801 × 10−13 0.0000000 0.1 0.7898337356130 0.7898337356130 0.7898312080 8.472091 × 10−14 2.5275 × 10−6 0.2 0.6246723675307 0.6246723675306 0.6246709988 9.183796 × 10−14 1.3692 × 10−6 0.3 0.4947063913440 0.4947063913439 0.4947069736 6.180149 × 10−14 5.826 × 10−6 0.4 0.3922992773092 0.3922992773093 0.3923013693 1.0272004 × 10−13 2.0923 × 10−6 0.5 0.3114991915313 0.3114991915314 0.3115024402 9.230426 × 10−14 3.2486 × 10−6 0.6 0.2476617911460 0.2476617911461 0.2476660586 1.4779032 × 10−13 4.2673 × 10−6 0.7 0.1971585649673 0.1971585649672 0.1971637757 9.0332704 × 10−14 5.2108 × 10−6 0.8 0.1571511081549 0.1571511081545 0.1571571826 3.30057946 × 10−13 6.0744 × 10−6 0.9 0.1254162556993 0.1254162556989 0.1254231315 4.38164242 × 10−13 6.8759 × 10−6 1.0 0.1002104788741 0.1002104788736 0.1002181358 5.38651959 × 10−13 7.6569 × 10−6 solution and aboodh transform-weighted residual method is very negligible. figures 1, 2 and 3 also show graphical comparison between the exact solution and aboodh transformweighted residual method for the three illustrations and it was observed that there are no significant difference between the two sets of solutions. 4. conclusion this research work has presented aboodh transformweighted residual method (at-wrm) as a means of providing solution to three different second order linear boundary value problems in a semi-infinite domain. the results obtained so far as presented in the tables and figures showed that the method is superb, provides brilliant result and at the same time proves a vital tool to surmount the challenges of seeking how to handle the condition at infinity. moreover, it has been shown that the combination of an integral transformadoodh transform and weighted residual method demostrate an excellent mathematical tool in obtaining near exact result if not the real exact solution of any boundary value problem and most especially the one with semi-infinite domain. motivation the reason for compling the two methods is to be able to solve boundary value problems within semi-infinite domain by the process of adoodh transform method as adoodh transform cannot independently deal with boundary condition at infinity naturally. in addition,to improve the efficiency of the method of weighted residual method. acknowledgements the authors thank dn. stephen taiwo akinola who did the 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[21] r. a.oderinu & a. s.aregbesola, “using laguerres quadrature in weighted residual method for problems with semi infinite domain”, international journal of pure and applied mathematics 75 (2012) 371. 7 j. nig. soc. phys. sci. 5 (2023) 854 journal of the nigerian society of physical sciences methods for the detection and remediation of ammonia from aquaculture effluent: a review k. o. sodeinde∗, s. a. animashaun, h. o. adubiaro department of chemistry, federal university oye-ekiti, p.m.b. 373, oye-ekiti, ekiti state, nigeria abstract aquaculture practice is growing at an alarming rate in the world due to rising human population and improved agricultural activities. it is a very important sector that is contributing to the food security of various nations, generating employment and foreign exchange earnings for economic development. however, this practice produces large amount of ammonia based effluent thus threatening environmental sustainability. this review focused on the critical assessment of various physicochemical and biological treatments applied in the remediation of ammonia from aquaculture effluent. the physicochemical methods include mainly adsorption, photocatalytic and electrochemical degradation by different materials while the biological methods involve the use of plant biomass, animals and microorganisms. in addition, different detection methods of ammonia and environmental impact of climate change on aquaculture management system were discussed. doi:10.46481/jnsps.2023.854 keywords: ammonia, remediation, biological and chemical methods, aquaculture effluent article history : received: 07 june 2022 received in revised form: 08 september 2022 accepted for publication: 14 october 2022 published: 08 december 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: n. a. a. babarinde 1. introduction aquaculture is a systemic rearing of fish in a confined body of water such as tanks and ponds, where its development can be monitored and controlled [1]. it is the fastest growing food processing sector across the globe [2].with the projected world population estimates of 9.3 billion by the year 2050, it constitutes a critical agricultural sub-sector that can contribute to the food security of the world [2]. it also serves as a means of employment generation and foreign exchange earnings for economic development. however, aquaculture practices have ∗corresponding author tel. no: +234 8147773137 email address: kehinde.sodeinde@fuoye.edu.ng (k. o. sodeinde) led to pollution of the environment, as some of the fish producing factories and industries in nigeria and other developing nations do release varying degrees of untreated wastewater from ponds into different water bodies which have therefore led to water pollution problem. treatment of aquaculture wastewater is necessary for environmental protection, water. conservation (via recirculation), human health, etc[3]. aquaculture effluent can also serve as an important source of natural fertilizers, irrigation, energy generation for domestic and industrial purposes [4-7]. ammonia is one of the major constituents of aquaculture wastes, being the main product of excretion in fish. according to lazzari and baldisserotto [8], ammonia originates from the organic matter decomposition, excessive use of organic and inorganic fertilizers and death of phytoplankton. ammonia presence in the aquatic environment is always in two forms; ionized 1 sodeinde et al. / j. nig. soc. phys. sci. 5 (2023) 854 2 and unionized forms. the increase in the level of un-ionized form of ammonia in water bodies leads to decrease in the rate of release of the compound in most aquatic organisms leading to prolonged accumulation in the blood tissues [9]. furthermore, the increment in the presence of ammonia leads to defects in the aquatic organisms’ physiology and thus affect the osmoregulation, growth, oxygen transportation, excretion, and disease resistance in fish [9-10]. in order to increase the quality of water in aquaculture system, different detection and treatment methods of ammonia in effluents needs to be fully understood.[1112]. hence, this study aims at reviewing different methods for the detection and remediation of ammonia from aquaculture effluents. it also assesses the environmental impact of climate change on aquaculture management system. 2. environmental impacts of ammonia ammonia is a colourless gaseous compound with elemental composition of nitrogen (n) and hydrogen (h) in the ratio 1:3. it usually serves as a precursor to some food additives and fertilizers. ammonia dissolution in water formed ionized species called ammonium ion [13] i.e. n h3+ h2o→n h+4 + oh − 2.1. effect on aquatic organisms toxicity of ammonia results in severe losses in fish hatcheries. this can be due to the different tolerance levels among fish species [13]. ammonia in its unionized form is harmful to some species of fish even at concentration as low as 0.05 mg/l which can cause poor feed conversion and growth rates, infertility, susceptibility to diseases and bacterial infections [14]. in addition, exposure of fish to high concentration of ammonia may lead to hyperstability, equilibrium loss, uptake of oxygen, increased heart rate and other respiratory activity [13]. it also causes damage to tissue and gill, inactivity, convulsion, coma and finally death at concentration exceeding 2.0mg/l [1519]. ionic imbalance is also an effect associated with high ammonium concentration in the fish blood [20]. 2.2. eutrophication eutrophication usually arises when a water body becomes too enriched with minerals and nutrients (commonly ammonia and phosphorous) and thus inducing excessive algae growth and depletion of oxygen in the waterbody [21]. the processes involved can be broken down as follows: 1. excess nutrients accompanied the discharge of waste into the soil 2. nutrients leached to the soil, which is later drained to the waterbodies 3. the nutrients lead to excessive algal formation 4. the excessive algae formed inhibits the solar light from reaching the underground of the waterbody 5. the plants under the algae died due to their inability to photosynthesise 6. the algae also died and sinks into the underground part of the water 7. bacteria decompose the remains by consuming the available oxygen through respiration 8. the decomposition leads to the depletion of oxygen in the water 9. the waterbodies can no longer support life again and thus fish and other larger organisms suffocate and die. globally, countries have started proposing policies and programme to prevent and mitigate the effects of eutrophication in their aquatic environment. for instance, in europe and asia, efforts undertaken between 2010–2020 have resulted in the formulation of baseline guidelines on the urban wastewater treatment [22]. 2.3. formation of toxins continuous release of aquaculture effluents containing ammonia and some other pollutants into water bodies has been found to enhance the formation of certain harmful microorganisms like algae [23].the produced toxins can stay inside algal cells for long or released into the aquatic environment. thus, animals present in such environment may be affected by ingesting the algal cells via drinking or feeding. the toxins could also be bioaccumulated and biomagnified through the food chains till it reaches toxic levels in some organisms and later consumed by man [24]. 2.4. reduction in aesthetic value of the environment discharge of aquaculture effluents into soils and surrounding water lowers the aesthetic value of the environment. for instance, akinrotimi et al. [25] reported the release ofn unpleasant odour caused by the presence of ammonia in the wastewater from selected catfish farms within port harcourt metropolis, nigeria. it was opined that the continuous release of such untreated effluents could result in major outbreak of epidemic diseases in the future if not abated. 3. detection of ammonia in water and wastewater over the years, several methods have been developed by researchers for the detection and quantitation of ammonia and ammonium ion in water and wastewater. these include nesslerization, phenate, electrochemical, fluorometric methods, etc. 3.1. nesslerization method this involves the reaction of alkaline solution of mercuric potassium iodide–k2hgi4 (nessler’s reagent) with ammonia to give a coloured complex with concentration determined by uv/vis spectrophotometry. researchers have continued to improve on 2 sodeinde et al. / j. nig. soc. phys. sci. 5 (2023) 854 3 this method with some modifications. typical example was reported by phansi et al. [26] wherein nessler’s reagent was combined with a paper-based analytical device to determine levels of ammonia in fertilizers and wastewater by capturing the colour intensities and colour image through the camera and using the image-j program. also, a computer camera could be used to detect the variation in colour of the wastewater upon the reaction of nessler’s reagent and ammonia concentration could be calculated from the available data [27]. the advantages of this method are low cost and relative simplicity since it involves only one reagent. however, researchers have raised concern on the toxicity of the reagents and possible interference of the process by the presence of cations [28]. 3.2. phenate method the phenate method is based on the reaction of phenol and hypochlorite with ammonium salt in a sample to form a blue coloured compound known as indophenol via addition of a catalyst (such as nitroprusside) and further analysis by uv-visible spectrophotometer [29]. the phenate and modified phenate methods are the commonest spectrophotometric methods deployed in the determination of ammonium level [30-32]. the major challenge of this method is that phenol is odorous and toxic in nature. in the modified phenate method, phenol is replaced with more environment friendly compounds such as salicylate, o-phenylphenol (opp), etc [33-38]. however, the use of salicylate has not been satisfactory due to low sensitivity [3941]. 3.3. ammonia probe method this method involves the transfer of ammonia across a gas permeable membrane until the partial pressure in the thin film of the solution between the probe and the glass electrode membrane equals that of the sample solution. according to evans and partidge [42], a precision of 4% was achieved from the recoveries of repeated calibrations and added ammonia in the probe. the detection limit of 0·03mg/l was reported for ammonia levels above than 0·4mg/l. this method has been employed to measure ammonia concentrations in different water samples. its disadvantages include low detection limit, relatively high cost, etc. 3.4. gas diffusion method in gas diffusion method, ammonium salt is converted to ammonia in a gas diffusion unit under alkaline conditions for removal of interferences in the samples. this is followed by the diffusion of ammonia gas into an acid-base indicator solution (such as bromothymolblue) across a membrane whose absorbance can be determined spectrophotometrically [43-46]. other pollutants such as nitrate, nitrite and phosphate can also be measured with this method [43]. some of the merits of the method include moderate sensitivity, short analysis time, etc. 3.5. fluorometric method fluorometry is a rapid, simple and sensitive emission spectroscopic method. it is based on the absorption of radiation at one wavelength and its emission at longer wavelength by fluorophores. fluorophores largely contain aromatic rings, conjugated double bonds, etc. fluorometric method was first reported by roth [47] for amino acid determination. this process involved the reaction of amino acids in the sample with ophthaldialdehyde (opa) usually with the aid of a catalyst such as 2-mercaptoethanol in a basic medium to produce a strongly fluorescent compound. the modified procedure involved the use of sulphite instead of 2-mercaptoethanol which formed opasulfite-ammonia which is more sensitive for the determination of ammonia than amino acids [48]. furthermore, a modified opa-based method involving integrated system of sequential injection analysis to ascertain ammonium levels has been utilized [49]. the method was enhanced by combining with an automated micro-extraction template for pre-treatment [50]. greater sensitivity was achieved when the system was incorporated with two independent microsyringe pumps in a gas-liquid extraction procedure which generated gaseous ammonia in the headspace of the first microsyringe while there is movement into the headspace of the second microsyringe [49]. the limit of detection (lod) was 2.8 nm. also, a pre-treatment method was used to trap ammonia present in sea water samples [51]. in this method, alkaline sample was introduced into purified argon to eject the ammonia from the solution into the gaseous phase and analysed with the opa method. recently, cao and co-workers introduced a new approach by reacting benzylchloride with ammonium and sodiumbicarbonate, where a new fluorescent derivative was produced upon excitation and emission at 258 nm and 284 nm respectively [52]. the advantages of this method include low cost, high sensitivity, selectivity, relatively short analysis time, etc [53]. 3.6. electrochemical methods electrochemical methods depend on the measurement of the variations in the electrical properties (such as current, voltage, conductance, resistance, etc) as a result of chemical interaction (redox) occurring at the electrode surface in the presence of the analytes in a given matrix. advantages of electrochemical methods are short analysis time, high efficiency, relatively low cost, etc [54-57]. electrochemical methods that have been applied for ammonium determination include potentiometric, amperometric, voltammetric and conductivity methods [57].the most widely utilized electrodes are the ammonium ion-selective (ais) and nanomaterial-modified electrodes. ais electrode involves a sensitive membrane comprising polyvinylchloride (pvc) which can selectively respond to ammonium ions. a potential is developed by the membrane as the electrode is placed in an aqueous medium and thus selectively detect ammonium [58]. the last two decades have witnessed a paradigm shift towards the use of nanomaterials-modified electrodes due to their unique, tailorable electrical properties, large surface area to volume ratio, improved sensitivity, specificity and selectivity for the determination of analytes [59]. in nanomaterialmodified electrodes, modifications can be imparted through the 3 sodeinde et al. / j. nig. soc. phys. sci. 5 (2023) 854 4 figure 1. schematic representation of methods for analyzing ammonia in water and wastewater designation of nano-functional materials with specific chemical properties on the electrode surface. carbon nanotubes (cnts) are one of such nanomaterials with the capacity to completely enhance the electrode response to ammonia. cnts possess a high surface area and their composites could serve as suitable templates for advanced sensor design. baciu et al [59] utilized silver-modified cnt (ag-cnt) for electrochemical sensing of ammonium and nitrite ions in aqueous solution. zhang et al [60] carried out electrode position of platinum nanoparticles on the surface of ag/ppy-polypyrrole-ni foam for the sensitive and selective detection of ammonia (lod value of 37 nm).the result showed a significant left shift by the oxidation potential. the biosensor was found to be relatively stable and displayed reliable percentage recovery compared to nessler’s method. the list of detection methods for ammonia in water and wastewater is shown in figure 1. 4. remediation of ammonia from aquaculture effluents the recirculation of wastewater depends on an effective and efficient means of treatment due to the various impacts of ammonia in the ecosystem. to maintain the quality of water, methods such as biological, chemical, physical or combination of any two are applied for sustainable production of fish and other aquatic organisms [61].important parameters usually considered during the various treatment methods include temperature, ph, dosage, etc [62]. 4.1. physical and chemical treatment in recent years, adsorption, photocatalytic degradation and electrochemical treatments have been the main physico-chemical methods employed for ammonia remediation in aquaculture effluents. adsorption is a surface phenomenon which involves the attachment of pollutants to the surface of the adsorbent. it is very important that the material to be used for adsorption should possess certain desirable properties such as inertness, cheap cost, eco-friendly, superficial area elevation and basic centres dispersed on the surface for adsorption of pollutants [9, 12]. different materials that have been applied recently include clay, biochar, chitin, chitosan, composites, nanoparticles, etc (figure 2). 4.1.1. adsorption with clay several clay materials have been studied for the adsorption of toxic substances from aquaculture wastewater. examples infigure 2. schematic representation of materials used for chemical treatment of ammonia from aquaculture effluent clude clinoptilolite, bentonite, zeolite, smectite, etc. for instance, zadinelo et al. [9] studied the application of smectite clay for the adsorption of nh4+ from aquaculture effluent. the contact time of the smectite clay in the effluent did not lead to an appreciable increment on the adsorption of nh4+ within 1min to 3 h range. 94% ammonia removal with little concentration of dry clay was achieved [9]. dryden and weatherley [63] applied clinoptilolite and natural dry clay for the adsorption of nh4+ from aquaculture effluent. the removal efficiency of 98% and 92% respectively were obtained. furthermore, the presence of other cations is also a major factor influencing the removal of nh4+. the selectivity of clinoptilolite for different cations is in the sequence: k+>nh4+>ca2+>mg2+ [64,65]. thus, from literature survey, the presence of k+ affected the exchange of nh4+ in clinoptilolite with remarkable reduction in the exchange capacity of zeolites in a synthetic effluent experiments of nh4+: k+in1:1 [66]. also, sarioglu [67] observed the selectivity of zeolites for different cations according to this sequence: k+>nh4+>na+>ca2+>fe3+>al3+>mg2+. similar pattern was reported by dontsova et al.[68] using bentonite clays. the parameters influencing the removal of nh4+ ion with clay in the wastewater included dosage, other cations in the solution, etc. 4.1.2. adsorption with biochar biochar synthesized from rice straw was examined for its potential use for ammonium adsorption from aquaculture effluent [69]. removal efficiency was strongly affected by ph, adsorbent concentration, modification methods of rice straw. maximum ammonium removal efficiency was achieved at neutral ph. 4.1.3. adsorption with chitin and chitosan chitin is one of the most abundant polymers in nature which is the building material for exoskeletons of insects, crustaceans and can be converted to chitosan through chemical deacetylation process [70]. bernardi et al. [70] studied the adsorption efficiency of chitin and chitosan from various sources for the treatment of ammonia from natural aquaculture and synthetic effluents. chitosan sources include freshwater and marine shrimps, three different commercial chitosan and labora4 sodeinde et al. / j. nig. soc. phys. sci. 5 (2023) 854 5 tory synthesized chitosan. commercial chitosan1and 2 gave 100% efficiency for ammonia removal from synthetic effluent whereas none of chitin sources were efficient in ammonia treatment from synthetic effluent. 4.1.4. electrochemical treatment electrochemical wastewater treatment involves the application of electric field between electrodes to decontaminate toxicants found in effluents via redox processes. monica et al. [71] pioneered the application of electrochemical treatment for the removal of ammonium and organic pollutants in effluents mixed with seawater. its advantages include high efficiency, versatility, little sludge generation, etc [72-73]. electrochemical oxidation of ammonium and organic substrates can be carried out via direct or indirect anodic oxidation methods. in the former, adsorbed hydroxyl radicals are involved in the oxidation of organic compounds [74-75]. marinerc et al. [76] conducted a direct electro-oxidation of ammonium on a platinum plated anode and a titanium-plated anode. the direct electrooxidation of nammonium was reported to be more favourable. in the case of the in direct method, anodically-generated oxidizing agents were added into the wastewater to degrade organic and inorganic pollutant [77-78]. the in-situ electro-generation enhanced the degradation efficiency. mao et al. [79] developed a chlorine mediated reactive barrier comprising inert electrodes for ammonia contaminated groundwater remediation experiment in a batch scale. findings revealed that ammonia in the groundwater could be readily converted into a more desirable nitrogen. higher ammonia removal efficiency was achieved at higher current densities and bicarbonate concentrations. in general, the overall efficiency of the electrochemical treatment is influenced by parameters such as ph, current density, electric voltage applied and the nature of electrode material used [80-83]. the nature of the anodic material and electric voltage applied are the most critical parameters determining the overall cost and optimum removal efficiency of an electrochemical treatment process. some of the drawbacks of this method include high cost, high energy consumption, instability and poor electrocatalytic activities in the long term, etc [84]. 4.1.5. photocatalytic degradation through nanocomposites photocatalysis; one of the forms of advanced oxidation processes (aops), involves the interaction between radiation and a solid semiconductor in an aqueous medium. nanomaterials have been reported as excellent photocatalysts for degradation of toxicants [85]. nanocomposites exhibit excellent thermal, electrical, mechanical properties. also, they possess large surface area, efficient charge transportation and separation, etc [86]. nanocomposites involve the fabrication or synthesis of materials from two or more different constituent materials on a nanoscale to enhance their properties and functionality. due to low efficiency of chitin towards the remediation of ammonia from aquaculture effluents [70], chitin/zno nanocomposite photocatalyst powder was fabricated by lin et al [56] through sol-gel method for treatment of ammonia from aquaculture effluent under ultraviolet irradiation. factors affecting the degradation process include dosage, temperature of calcination, mass ratio rate, initial concentration of ammonia and conditions of illumination. 88.64% removal efficiency was achieved using 0.5g/l chitin/zno(2:3) photocatalyst at irradiation time of 2 h and 500 ◦c calcinations temperature. yu et al. [87] synthesized and treated ammonia with tio2/carbonfibre (cf) nanocomposite, tio2 and carbon fibre photocatalysts from aquaculture wastewater. parameters such as dosage, calcination temperature of the adsorbent, etc, were studied. the best conditions for ammonia treatment were 2.0g/l dosage for tio2/cf, calcinations temperature of 600◦c, initial ammonia concentration was 30 mg/l at illumination time of 1 h and h2o2 concentration of 0.8g/l. the results showed that the composite (tio2/cf) was most effective compared with cf or tio2 alone for ammonia treatment. 4.2. removal of ammonia through biological treatment treatment of wastewater through biological processes usually involves the conversion of ammonia and nitrate by microbes to nitrogen gas. this is a modern, cost effective method for ammonia treatment which readily converts it into nitrogen gas [62] i.e nh4+ + no3− → n2+ 2h2o however, biological treatment is usually time consuming as some of the process involves longer period of time spanning weeks and months before the treatment can be achieved. the schematic representation for the various biological methods is shown in figure 3. 4.2.1. natural biodegradation in anaerobic continuous flow system ching and redzwan [88] studied the effect of concentration of salt (nacl) in natural biodegradation of ammonia in aquaculture effluent in an aerobic continuous flow system. the ammonia removal efficiency reduced as the dilution fold of the aquaculture effluent increased. optimum % ammonia removal efficiency was obtained after 10 days. this procedure is suitable for small scale aquaculture wastewater treatment. the treatment method produced an odourless effluent which can be reused as an eco-friendly fertilizer because the main constituents are known organic substances which are non-toxic or carcinogenic. further experiments should involve the impacts on the plants, soil, the different yields of the crops through irrigation with different salt content and application in a large scale fish processing industries [88]. 4.2.2. remediation with microorganisms studies have shown that microalgae are excellent bioremediators for effluents treatment with high nutrient concentrations [89]. lananan and co-workers [89] reported the remediation of ammonia and phosphorous from aquaculture effluent through symbiotic process by using effective microorganism (em) and 5 sodeinde et al. / j. nig. soc. phys. sci. 5 (2023) 854 6 figure 3. schematic representation of various biological methods for the treatment of ammonia microalgae of chlorella specie. the treatment of aquaculture effluent was performed in batch scale comprising working volume of 2 l and treatment period of 14 days.total ammonia present was almost removed from the aquaculture after the initial 7 days. batch scale experimental data yielded varying inoculation concentrations of bioremediators and retention time which can be used in the advancement of aquaculture effluent treatment in a continuous process for real effluent application. omitoyin et al. [90] studied the application of duckweed and microorganism (bacillus species) in the removal of certain pollutants in aquaculture effluent. the measured total suspended solid, biochemical oxygen demand, total ammonia nitrogen (tan) and phosphate were above the permissible limits of wastewater discharge into surface water according to who standard [91]. the result of the remediation process showed that the bacillus sp. has the highest removal efficiency for ammonia. the duckweed is effective for toxic organic waste removal for aquaculture effluent. the duckweed technology is simpler and more cost effective than bacillus specie which required expertise for its isolation, identification, mass production, application and only effective in ammonia, nitrite and phosphate removal from wastewater. 4.2.3. filters trickling filter is the main type of filter that has been employed for ammonia remediation. it consists of a fixed bed from which filtered effluents flow down over anaerobic biofilm. important factors considered during filter selection include waterflow, surface area, etc [92]. lekang and kleppe [93] investigated different filter media such as kaldnes rings, norton rings and a rolled mat of finturf artificial grass in the treatment of ammonia. the leca filter gave the highest denitrification rate of 100% because it has longer retention times and larger surface area. the specific surface area indicates the surface required in homogenous water flow and biofilm growth. disadvantages of this method include clogging and biofilm shedding, high production cost, etc [94-96]. 4.2.4. fluidized bed reactor fluidized bed reactor is known to be a solution to clogging problems peculiar to trickling filters. it is an efficient method to remove dissolved solids and wastes from aquaculture recirculating systems when compared to bed and trickling filters [97]. the size of the particle is an important factor influencing the treatment process [98].the performance of this system is being affected by the type of medium. davidson et al. [99] studied the removal of total ammonia nitrogen (tan) and other pollutants from effluent using two sand sizes. 88% efficiency for tan removal was obtained from 0.11mm sand size. schnel et al. [100] investigated the use of different filters comprising polyvinylchloride strips and fixed particle sand for tan removal. the efficiency of the whole process for total ammonia nitrogen (tan) was 65.21%. recently, much attention has been focused on the integration of fungal bioreactors into the wastewater treatment plants [101-102]. in particular, a microbial membrane bioreactor was designed to study the removal efficiency of ammonia and some other pollutants in marine aquaculture wastewater. 85% ammoniacal nitrogen removal efficiency was achieved after 40days [103]. dalecka et al. [104] conducted a batch scale experiments and compared it with bioreactor with the use of t. versicolor and a. luchuensis for non-sterile municipal wastewater and the effect of ph on nh4-n. the results in the fluidized bed bioreactor gave contrasting performance regarding ammonia removal relative to a batch experiment where no major change on nh4-n reduction was observed. the fluidized bed bioreactor and batch scale experiments revealed a good starting point towards the optimization of fungal treatment application in wastewater. however, further development and optimization of fluidized bioreactor using fungi and other microorganisms should be studied [104]. 4.2.5. wetlands wetlands can be categorized into natural and constructed wetlands. natural wetlands have been applied to remove microorganisms, phosphorous, nitrogen, trace elements and suspended. solids contained in effluents [105]. constructed wetlands which are also referred to as artificial wetlands have replaced the loss of natural wetlands in the treatment of agricultural, municipal and industrial effluents. generally, major types of constructed waste water wetlands viz: surface flow (sf), and sub-surface flow (ssf) systems [106-108] have been applied for effluent treatment to minimise pollution. lin et al.[109] reported a combination of subsurface and surface wetlands for the treatment of phosphorus and nitrate from aquaculture effluents. the removal efficiency of 82-99% was achieved for nitrate. currently, hybrid reed bed constructed wetlands (hrbcw) is gaining more recognition due to their higher removal efficiencies as secondary and tertiary treatment of domestic waste waters [110]. jehawi and co-workers [111] constructed a scirpus grossus-planted hrbcwsystem to treat some pollutants in a domestic waste water. the result showed that significant higher performance was observed with 84.7% ammoniacal nitrogen removal efficiency while the unplanted system recorded 74.8% efficiency. the advantages of constructed wetlands are 6 sodeinde et al. / j. nig. soc. phys. sci. 5 (2023) 854 7 in its cost and removal effectiveness, lesser skilled labour but required large surface area land for construction [96]. 4.3. importance of waste water remediation towards environmental sustainability waste water treatment, reuse, and safe disposal have gotten application for industrial, agricultural, recreational purposes, drinking water supplies, energy generation and thus becoming crucial in mitigating the effects of climate change and environmental sustainability. climate change mitigation which is geared towards environmental sustainability involves actions to be taken to minimize the effects of global warming by reducing human emissions of greenhouse gases (ghg) [112]. the combustion of fossil fuel is responsible for most carbondioxide and ghg emissions [112]. it is therefore necessary to reduce or stop the use of constituents from petroleum and coal by replacing them with eco-friendly energy sources. one of the best ways to achieving this is proper utilization of water resources for power generation. furthermore, urbanization and intensive agricultural practices have led to an increase in abandoned farmland [113-114]. desertification can be described as the major environmental challenge of our time. it has led to temporary or permanent reduction in quality of soil, vegetation, water resources, livestock and wildlife and therefore threatening food security and livelihood of man. this usually arises from inadequate potable water for grazing purpose and thus, cattle are made to move outside their ranches in search of food which usually result in the destruction of vegetation. all efforts are therefore needed to ensure generation of good water and recycling of wastewater towards environmental sustainability. 5. future outlook and conclusion several chemical pollutants are found in a typical aquaculture effluent, ammonia being the major pollutant with high environmental impacts. various detection techniques as well as remediation methods are reported for ammonia in effluents. biological methods are highly recommended for ammonia treatment in terms of cost effectiveness and environmental concerns. however, the biological methods are relatively slower compared to the chemical treatments. generally, technologies have been recently developed towards the use of chemical methods that are cheap and environment friendly to overcome the challenges inherent in the biological treatment methods. among the various chemical methods, application of micro and nanomaterials for wastewater treatment is increasing due to very high global demand for freshwater. nanotechnology has proven to be a remarkable success in the detection and remediation of ammonia due to relatively larger surface area and improved physicochemical properties. however, nanomaterials might have the potential risk of leaching into the treated water mainly during application, thereby leading to 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[114] j. g. j. olivier & j. a. h. w. peters, “trends in global co2 and total greenhouse gas emissions: 2018 report the hague, netherlands”, pbl netherlands environmental assessment agency (2018) 53. 10 j. nig. soc. phys. sci. 5 (2023) 871 journal of the nigerian society of physical sciences on bivariate nadarajah-haghighi distribution derived from farlie-gumbel-morgenstern copula in the presence of covariates yakubu aliyua,∗, umar usmanb adepartment of statistics, ahmadu bello university zaria-nigeria bdepartment of mathematics, usmanu danfodiyo university sokoto-nigeria abstract an important alternative distribution to the weibull, generalized exponential and gamma distributions that is used in survival analysis is the nadarajah-haghighi exponential distribution. similar to the weibull, generalized exponential and gamma distributions, the nadarajah-haghighi exponential distribution is an extension of the well known exponential distribution. in this paper, a copula function commonly used to model very weak linear dependence was used to introduced a bivariate nadarajah-haghighi distribution. the joint survival function, joint probability density function and joint cumulative distribution were given in closed form. bayesian method of estimation was used to estimate the model parameters considering the presence of right censoring and covariates. posterior summaries of interest were obtained via standard markov monte carlo (mcmc) technique. two real data sets were used to illustrate the importance and flexibility of the bivariate model in comparison with some competing models. it was observed that, the bivariate nadarajah-haghighi distribution provides a better flt than bivariate exponential, bivariate weibull, bivariate generalized exponential and bivariate modified weibull distributions. doi:10.46481/jnsps.2023.871 keywords: exponential distribution, nadarajah-haghighi distribution, bivariate models, copula function article history : received: 17 june 2022 received in revised form: 10 august 2022 accepted for publication: 23 august 2023 published: 28 april 2023 c© 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: o. adeyeye 1. introduction exponential distribution is a well known distribution due to the constant hazard rate and memory less property it exhibits. however, in survival/ reliability studies, choosing the exponential distribution may be inappropriate since its hazard rate does not show monotone and/ or non-monotone failure rate behaviours [1]. to solve this problem, many ∗corresponding author tel. no: +2347062898441 email address: aliyuyakubu40@mail.com (yakubu aliyu ) generalizations of the exponential distribution have been developed by researchers so as to add some flexibility to the exponential distribution. these include the generalized exponential distribution developed by [2], beta-exponential by [3], nadarajah-haghighi by [4]. these generalizations are among the generalizations that have received the most attention in the literature as compared with other extensions of the exponential distribution. other generalizations of the exponential distribution include the weibull-burr iii by [5], the generalized modified weibull by [6], log-beta weibull by [7], two parameter burr x by [8] and weibull kumaraswamy 1 aliyu & usman / j. nig. soc. phys. sci. 5 (2023) 871 2 distribution by [9] to mention but a few. an extension of the exponential distribution which serves as an alternative to the weibull, generalized exponential and gamma distributions called the nadarajah-haghighi exponential distribution was introduced by [4]. the probability density function (pdf ) and cumulative distribution function (cdf ) of the univariate nadarajah-haghighi exponential (nh) distribution with parameters θ and φ are respectively given by: f (t/θ,φ) = θφ (1 + θt)φ−1 ex p ( 1 − (1 + θt)φ ) (1) and f (t/θ,φ) = 1 − ex p ( 1 − (1 + θt)φ ) (2) where θ,φ > 0 are scale and shape parameters respectively. the pdf in (1) reduces to the exponential distribution when φ = 1. the shape of the nh density can be decreasing and unimodal, while that of the hazard rate function can be monotonically increasing, monotonically decreasing or constant. the survival and hazard rate functions of the nh distribution are respectively given by: s (t/θ,φ) = ex p ( 1 − (1 + θt)φ ) (3) and h (t/θ,φ) = θφ (1 + θt)φ−1 (4) the distribution has been extended by different researchers in different directions. for instance, [10] extended the distribution to the unit nh distribution which could be used to model effectively data related to rates and proportions with excess of ones. [11] extended it to the poisson nh distribution that can be used to model reliability systems and [11] explored different classical methods of estimation to estimate the parameters of the poisson nh distribution. [12] extended it to a three parameter discrete nh distribution that could serves as an alternative to the poisson, negative binomial and zero inflated poisson distribution. it was also extended to the nadarajah-haghighi mixture cure rate model by [13], a model that could be used to model effectively information from a population of a mixture of two type of individuals: susceptible and cured individuals. however, in a situation where we are studying two lifetimes t1 and t2 associated to each unit (individual), these extensions could not be used. to solve this problem, some existing bivariate distributions in the literature such as bivariate weibull by [14, 15, 16, 17, 18], bivariate modified weibull by [19], bivariate exponentiated discrete weibull distribution by [20], bivariate generalized exponential by [21, 22, 23, 24, 25, 26], bivariate generalized rayleigh by [27], bivariate inverse weibull distribution by [28, 29, 30, 31, 32, 33], bivariate frechet by [34], bivariate inverted topp-leone distribution by [35], bivariate discrete nh by [36] and bivariate nh by [37] distributions could be used. the bivariate discrete nh distribution by [36] could only be used in modelling bivariate discrete lifetimes while the bivariate nh distribution by [37] could be used to modeled bivariate lifetimes with positive and negative dependence structure. in this paper, we generalized the exponential distribution to the bivariate exponential distribution considering the nadarajah-haghighi exponential distribution using the farlie gumbel morgestern copula function. hence, the paper aimed to introduced a bivariate nadarajah-haghighi exponential distribution that could be used for describing bivariate data that have weak correlation between variables in lifetime data. the distribution will extends the univariate exponential and nadarajah-haghighi exponential distributions, and serves as a good alternative to several bivariate distributions such as: bivariate exponential, bivariate generalized exponential and bivariate weibull distributions for modelling real-valued data. an important motivation of the article is to develop a guideline for estimating the parameters of the introduced distribution in the presence of censoring and covariates, which may be of deep interest to statisticians and practitioners. some mathematical and statistical properties of the distribution was discussed. bayesian method of estimation was employed in estimating the parameters of the distribution and finally, we evaluate the performance of the estimators by analyzing some real life data sets. the rest of the paper is organized as follows: in section 2, we derive the survival function, the probability density function and cumulative distribution function of the bivariate nadarajah-haghighi distribution, also in this section, parameters of the distribution were estimated using the bayesian method of estimation procedure. application of the introduced methodology is given in section 3 and we finally conclude in section 4. 2. bivariate nadarajah-haghighi distribution in this section, the bivariate nadarajah-haghighi distribution was developed and study some of its statistical properties. 2.1. the model copula functions are used in connecting the joint distribution function of two or more univariate distributions. the copula function is said to be bivariate when it connects the joint distribution function of only two univariate distributions. let s (tk) be the univariate survival function for the random variable tk, k = 1, 2, the joint survival function s (t1, t2) is defined as: s (t1, t2) = cλ (s (t1) s (t2)) (5) for t1, t2 > 0 where λ is a measure of dependence between the random variables t1 and t2, and c is a copula function. the farlie-gumbel-morgenstern copula (fgm) was first proposed by [38] and later by [39] and [40]. the joint survival function considering the fgm copula for random variables t1 and t2 is given by: s (t1, t2) = s (t1) s (t2) [1 + λ (1 − s (t1)) (1 − s (t2))] (6) where the dependence parameter lies between ±1 inclusive. the joint survival function in (6) reduces to the survival 2 aliyu & usman / j. nig. soc. phys. sci. 5 (2023) 871 3 function of the product copula function that is s (t1, t2) = s (t1) s (t2) when the dependence parameter takes the value of zero. in this case, t1 and t2 are said to be independent. the dependence parameter λ is related to the kendall and spearman rank correlation coefficients by: τ (λ) = 2λ 9 and ρ (λ) = λ 3 respectively. hence, the fgm copula is only appropriate in modelling weak dependences. the fgm copula is useful especially when dependence between the two marginal is modest in magnitude [41]. assume t1 and t2 be two lifetimes associated to the same individual/ device with a dependence structure given by fgm copula function. the density functions of the marginal distributions for the lifetimes t1 and t2 are given by: f1 (t1) = θ1φ1 (1 + θ1t1) φ1−1 ex p ( 1 − (1 + θ1t1) φ1 ) (7) and f2 (t2) = θ2φ2 (1 + θ2t2) φ2−1 ex p ( 1 − (1 + θ2t2) φ2 ) (8) respectively, while the survival functions are given by: s 1 (t1) = ex p ( 1 − (1 + θ1t1) φ1 ) (9) and s 2 (t2) = ex p ( 1 − (1 + θ2t2) φ2 ) (10) respectively. the joint survival function based on the fgm copula given by (6) using the marginal survival functions in (9) and (10) is given by: s (t1, t2) = ex p ( 2 − (1 + θ1t1) φ1 − (1 + θ2t2) φ2 ) [ 1 + λ ( 1 − ex p ( 1 − (1 + θ1t1) φ1 )) ( 1 − ex p ( 1 − (1 + θ2t2) φ2 ))] (11) to obtain the joint density function for the random variables t1 and t2, we obtain the second derivative of s (t1, t2) with respect to t1 and t2. that is, f (t1, t2) = ∂2 s (t1,t2 ) ∂t1∂t2 which yields: f (t1, t2) = θ1θ2φ1φ2 pq1q2 [ 1 + λ (1 + 4 p − 2p1 − 2p2) ] (12) where p = ex p ( 2 − (1 + θ1t1) φ1 − (1 + θ2t2) φ2 ) , p1 = ex p ( 1 − (1 + θ1t1) φ1 ) , p2 = ex p ( 1 − (1 + θ2t2) φ2 ) , q1 = (1 + θ1t1) φ1−1 and q2 = (1 + θ2t2) φ2−1 the joint cumulative distribution function is easily obtain as: f (t1, t2) = p (1 + λ(1 − p1)(1 − p2)) − p1 − p2 + 1 (13) the marginal distribution functions are given by f (t1) = 1 − s (t1) and f (t2) = 1 − s (t2). the first partial derivatives with respect to t1i and t2i are given by: ∂s (t1i, t2i) ∂t1i = −θ1φ1 p1 p [ 1 + λ (1 − p1) (1 − p2) ] + λθ1φ1 pq1 p1 (1 − p2) (14) and ∂s (t1i, t2i) ∂t2i = −θ2φ2q2 p [ 1 + λ (1 − p1) (1 − p2) ] + p (1 − p1) p2 (15) respectively. the survival function in (11) reduce to the survival function of the product bivariate nadarajah-haghighi distribution when the dependent parameter assumed the value zero. it also reduce to the fgm bivariate exponential distribution when φ1 = φ2 = 1 and to the product bivariate exponential distribution when φ1 = φ2 = 1 and λ = 0. 2.2. estimation in this section, the problem of estimating the parameters of the farlie-gumbel-mogenstern bivariate nadarajah-haghighi (fgmbnh) distribution based on random samples of size n was addressed using the bayesian estimation procedure. let (t11, t21), (t12, t22) , · · · , (t1n, t2n) be bivariate random sample of size n from the fgmbnh distribution, let w = (θ1, θ2, φ1, φ2, λ)′ be the vector of parameters. then, the likelihood function l (w) when the lifetimes (t11, t21), (t12, t22) , · · · , (t1n, t2n) is assumed to be non-censored can be expressed as: l (w) = n∏ i=1 f (t1i, t2i) (16) substituting equation (12) and taking natural logarithm gives: `(θ) = nlog(θ1) + nlog(θ2) + nlog(φ1) + nlog(φ2)+ 2n − n∑ i=1 [ (1 + θ1t1i) φ1 + (1 + θ2t2i) φ2 ] + (φ1 − 1) n∑ i=1 log (1 + θ1t1i) + (φ2 − 1) n∑ i=1 log (1 + θ2t2i) + n∑ i=1 [ 1 + λ [ 1 + 4ex p ( 2 − (1 + θ1t1i) φ1 − (1 + θ2t2i) φ2 ) − 2ex p ( 1 − (1 + θ1t1i) φ1 ) − 2ex p ( 1 − (1 + θ2t2i) φ2 )]] (17) on the other hand, assume the lifetimes t1 or t2 or both t1 and t2 may be right censored. assume also that, the censoring is independent of the time to the event of interest in the study. let (t11, t21) , (t12, t22) , · · · , (t1n, t2n) be a random sample from the fgmbnh distribution with parameter w where w = (θ1,θ2,φ1,φ2,λ)′ is a parameter space. then, each ith observation i = 1, 2, · · · , n fall in one of the following groups: • u1 : both t1i and t2i are uncensored observations. • u2 : t1i is uncensored and t2i is censored observation. • u3 : t1i is censored and t2i is uncensored observation. • u4 : both t1i and t2i are censored observations. 3 aliyu & usman / j. nig. soc. phys. sci. 5 (2023) 871 4 then, the likelihood based on these conditions can be expressed as: l = ∏ i∈u1 [ ∂2s (t1i, t2i) ∂t1i∂t2i ] ∏ i∈u2 [ −∂s (t1i, t2i) ∂t1i ] × ∏ i∈u3 [ ∂s (t1i, t2i) ∂t2i ] ∏ i∈u4 s (t1i, t2i) (18) assume τ1i and τ2i be indicator variables, such that{ τki = 1 when tki is an uncensored observation τki = 0 when tki is censored observation then, the likelihood function in equation (18) is written as: l = n∏ i=1 [ ∂2s (t1i, t2i) ∂t1i∂t2i ]τ1iτ2i [ −∂s (t1i, t2i) ∂t1i ]τ1i (1−τ2i ) × [ −∂s (t1i, t2i) ∂t2i ](1−τ1i )τ2i [s (t1i, t2i)] (1−τ1i )(1−τ2i ) (19) observe that, the likelihood function in expression (19) reduces to the likelihood function of the non-censored observation in expression (16) when τ1i = τ2i = 1. furthermore, in the presence of m covariates x1, x2, · · · , xm affecting the parameters of the fgmbnh distribution, a link function is assumed for the parameters φ1,φ2,θ1 and θ2. that is, φ1 = ex p (φ10 + φ11 x1 + φ12 x2 + · · · + φ1m xm) (20) φ2 = ex p (φ20 + φ21 x1 + φk2 x2 + · · · + φ2m xm) (21) θ1 = ex p (θ10 + θ11 x1 + θ12 x2 + · · · + θ1m xm) (22) and θ2 = ex p (θ20 + θ21 x1 + θ22 x2 + · · · + θ2m xm) (23) therefore, to obtained inferences from the fgmbnh distribution in the presence of covariate(s), values of φ1, φ2, θ1 and θ2 in equations (20), (21), (22) and (23) respectively are appropriately substituted in the model. in bayesian method of estimation, the joint posterior distribution of the model parameters is proportional to the product of the joint prior distribution of the parameters and the likelihood function for w given by equation (16) when the lifetimes t1i and t2i are uncensored and by equation (19) when the lifetimes t1i and t2i are censored. thus, assume the prior distributions π11(θ1) ∝ θ b1−1 1 e a1θ1 θ1 > 0 π12(θ2) ∝ θ b2−1 2 e a2θ2 θ2 > 0 π21(φ1) ∝ φ d1−1 1 e c1φ1 φ1 > 0 and π22(φ2) ∝ φ d2−1 2 e c2φ2 φ2 > 0 for the parameters θ1, θ2,φ1 and φ2 respectively, while the prior distribution of the dependence parameter (λ) was assumed to be uniform distribution. that is π3(λ) ∼ uni f ( p, q), where a1, a2, b1, b2, c1, c2, d1, d2, p and q are known hyperparameters. the hyper-parameters a1, a2, b1 b2, c1, c2, d1 and d2 are assumed to be non-negative while the parameters p and q are assumed to be between ±1. lets further assumed prior independence and let the joint prior distribution for the parameters θ1, θ2,φ1, φ2 and λ be π(w) = π11(θ1)π12(θ2)π21(φ1)π22(φ2)π3(λ) (24) moreover, assumed normal prior distribution for the covariate parameters. that is φ jk ∼ n(µ,σ2) and θ jk ∼ n(µ,σ2) for j = 1, 2 and k = 1, 2, · · ·m, where µ and σ2 are known hyperparameters. hence, the joint posterior density function for w is given by: `(w/x) ∝ l × π(w) (25) where x = (t1, t2,τ1,τ2)′ is the vector of observed lifetimes, l is the likelihood function given by equations (16) and (19), while π(w) is the product of the product of the prior probability density functions. the full conditional posterior distributions are obtained by deriving the posterior distribution of each parameter given the data and all other parameters of the model. posterior summaries of interest were obtained by using markov chain monte carlo (mcmc) technique. in our applications, 220,000 gibbs samples for each model parameter was generated. however, to minimized the effect of initial values, the first 20,000 simulated samples were discarded as burn-in. furthermore, each 20th simulated sample was stored so as to avoid auto-correlation between successive samples. the bayesian estimates for the parameters were obtained using the medians of the respective posterior distributions since some simulated distributions were quite skewed. credible intervals were also determine for each model parameter from the 2.5th and 97.5th centiles of the posterior distribution of each model parameter. auto-correlation and trace plots were used in assessing the convergence of simulated samples. different formulations were assessed using the deviance information criteria (dic). the dic a generalization of the akaike information criteria for the bayesian analysis, is obtained from the samples generated from the mcmc simulation [42]. according to [43], the dic is computed as: dic = d(ŵ) + 2np = 2d̄ − d(ŵ), where d(ŵ) is the deviance evaluated using the mean of the model parameters which is obtained from the mcmc samples, d̄ is the posterior mean of the deviance and np is the effective number of the model parameters which is computed as np = d̄ − d(ŵ). better model fits are indicated by lower dic values. all bayesian parameter estimates, their 95% credible interval (cri) and the dic for each formulation were obtained using the openbugs software version 3.2.3. 4 aliyu & usman / j. nig. soc. phys. sci. 5 (2023) 871 5 3. applications in this section, infections in kidney patients data from [44] and tobacco-stained-fingers data set from [45] were used in demonstrating the applicability of the introduced methodology. the fgmbnh model is compared with its special case (product bivariate nadarajah-haghighi (pbnh) model) and that of some competing bivariate distributions. 3.1. kidney data the kidney data show the recurrence times to infection at point of insertion of catheter using portable dialysis equipment. two recurrence times were recorded for each patient together with censoring indicator (infection occurs =1 and censored=0) and the risk variable values: age, sex (male=1, female=2) and disease type. assume t1 and t2 refers to first and second recurrence time respectively. the data was first fitted to the fgmbnh distribution in the presence of right censoring not including covariate and compared its performance with the fits of fgm bivariate exponential (fgmbe), fgm bivariate weibull (fgmbw), fgm bivariate generalized exponential (fgmbge) and fgm bivariate modified weibull (fgmbmw) distributions. it is also fitted to the special case of these bivariate distributions. that is, the product bivariate nadarajah-haghighi (pbnh), product bivariate exponential (pbe), product bivariate weibull (pbw), product bivariate generalized exponential (pbge) and product bivariate modified weibull (pbmw) distributions. in addition, the data is then fitted to the fgmbnh model in the presence of right censoring and covariates. in analyzing this data set, the procedure discussed in section 2.2 was followed. to be specific with the prior distributions, we assumed gamma(1, 1) for θ1,θ2,φ1 and φ2 while we assume u(−1, 1) for the dependence parameter(λ). furthermore, n(0, 10) was assumed for the covariate parameters. table 1 give the posterior summary statistics for the bivariate nadarajah-haghighi distribution considering the fgm copula function compared to the aforementioned bivariate distributions. the table also gives the mc error for each posterior estimate. the mc error of the posterior estimates of the fitted models were all less than 120 of standard deviation of the estimates, this showed that the posterior estimates have reasonably good precision. the results also showed that, the fgmbnh distribution is more efficient than the fgmbe, fgmbw, fgmbge and fgmbmw distributions, since it has the least information criteria value. furthermore, the estimates of the dependent parameter (λ) are similar for all the distributions considered. table 2 give the posterior summary statistics for the bivariate nadarajah-haghighi distribution considering the product copula function compared to the aforementioned bivariate distributions. similar to the results for the distributions based on the fgm copula, the fgmbnh distribution is more efficient than the aforementioned distributions since it has the least information criteria value. comparing between the estimates of the bivariate distributions considering the fgm and product table 1. posterior summaries considering fgm copula kidney data model paramedsd mc 95% dic meter ian error cri fgm λ 0.5556 0.3546 0.0059 (-0.3138, 0.9781) 682.5 bnh φ1 0.4526 0.1496 0.0039 (0.2727, 0.8525) φ2 0.6546 0.2932 0.0079 (0.3430, 1.5020) θ1 0.0351 0.0330 0.0007 (0.0094, 0.1226) θ2 0.0152 0.0139 0.0003 (0.0039, 0.0535) fgm λ 0.5225 0.3345 0.0067 (-0.2720,0.9676) 686.5 be θ1 0.0077 0.0013 0.0000 (0.0053,0.0106) θ2 0.0074 0.0015 0.0000 (0.0050,0.0106) fgm β1 0.7454 0.1022 0.0024 (0.5560,0.9589) 686.4 bw β2 0.8874 0.1225 0.0028 (0.6599,1.1390) λ 0.5608 0.3536 0.0073 (-0.3120,0.9780) θ1 0.0292 0.0190 0.0005 (0.0090,0.0797) θ2 0.0134 0.0108 0.0002 (0.0035,0.0433) fgm β1 0.7479 0.1595 0.0047 (0.4950,1.1240) 687.8 bge β2 1.0470 0.2368 0.0072 (0.6696,1.5840) λ 0.5318 0.346 0.0105 (-0.2887,0.9743) θ1 0.0062 0.0016 0.0000 (0.0036,0.0098) θ2 0.0079 0.0020 0.0000 (0.0045,0.0122) fgm α1 0.0427 0.0211 0.0016 (0.0134,0.0942) 689.3 bmw α2 0.0217 0.0189 0.0013 (0.0047,0.0770) β1 0.6246 0.1068 0.0084 (0.4233,0.8568) β2 0.7354 0.1594 0.0128 (0.4085,1.0780) λ 0.0009 0.0007 0.0000 (0.0001,0.0026) φ1 0.0011 0.0009 0.0001 (0.0001,0.0035) φ2 0.6511 0.3084 0.0263 (-0.1261,0.9863) table 2. posterior summaries considering product copula kidney data model paramedsd mc 95% dic meter ian error cri pb φ1 0.4545 0.1598 0.0043 (0.2727, 0.8799) 683.2 nh φ2 0.6627 0.2860 0.0072 (0.3551, 1.4520) θ1 0.0346 0.0308 0.0005 (0.0092, 0.1237) θ2 0.0149 0.0124 0.0002 (0.0041, 0.0513) pbe θ1 0.0077 0.0013 0.0000 (0.0054,0.0106) 687.4 θ2 0.0076 0.0015 0.0000 (0.0051,0.0110) pbw β1 0.7487 0.0972 0.0034 (0.5675,0.9525) 686.9 β2 0.8998 0.1298 0.0047 (0.6717,1.1770) θ1 0.0286 0.0172 0.0005 (0.0094,0.0754) θ2 0.0126 0.0104 0.0003 (0.0029,0.0416) pb β1 0.7498 0.1574 0.0039 (0.4968,1.1180) 688.9 ge β2 1.0510 0.2453 0.0058 (0.6632,1.6240) θ1 0.0063 0.0016 0.0000 (0.0036,0.0098) θ2 0.0080 0.0020 0.0000 (0.0045,0.0124) pb α1 0.0378 0.0247 0.0018 (0.0113,0.1058) 688.7 mw α2 0.0185 0.0170 0.0011 (0.0056,0.0695) β1 0.6436 0.1259 0.0096 (0.4083,0.9010) β2 0.7752 0.1457 0.0108 (0.4663,1.0110) φ1 0.0009 0.0007 0.0000 (0.0001,0.0026) φ2 0.0010 0.0008 0.0000 (0.0000,0.0031) copula functions reveal that, the estimates for each fgm bivariate distribution are similar to its corresponding product bivariate distribution. furthermore, comparing between the fits of the bivariate distributions considering the fgm copula and the bivariate distributions considering the product copula reveal that, except for the fgmbmw distribution, the fits of the bivariate distributions considering the fgm copula are more efficient than the bivariate distributions considering the product copula function since they have the least information criteria values. furthermore, the data is fitted to the fgmbnh distribu5 aliyu & usman / j. nig. soc. phys. sci. 5 (2023) 871 6 table 3. posterior summaries considering fgm copula in the presence of covariate kidney data model paramedsd mc 95% dic meter ian error cri model λ 0.5223 0.3526 0.0145 (-0.3288,0.9705) 682.7 i φ10 -0.3115 0.2658 0.0111 (-0.8238,0.2320) φ11 0.5135 0.2207 0.0081 (0.0716,0.9167) φ12 -0.0015 0.0070 0.0002 (-0.0149,0.0126) φ20 -0.1204 0.2739 0.0107 (-0.6340,0.4389) φ21 0.0129 0.2110 0.0074 (-0.4173,0.4083) φ22 -0.0035 0.0077 0.0002 (-0.0181,0.0124) θ1 0.0137 0.0111 0.0004 (0.0035,0.0450) θ2 0.0111 0.0076 0.0002 (0.0033,0.0323) model λ 0.7428 0.2967 0.0052 (-0.1071,0.9901) 708.9 ii φ1 0.2761 0.0417 0.0007 (0.2032,0.3684) φ2 0.3175 0.0541 0.0009 (0.2226,0.4331) θ10 -0.2323 0.3129 0.0046 (-0.8392,0.3709) θ11 0.2237 0.2903 0.0047 (-0.3455,0.7947) θ12 -0.0452 0.0111 0.0002 (-0.0662,-0.0226) θ20 -0.2378 0.3175 0.0053 (-0.8479,0.4009) θ21 0.0416 0.2903 0.0045 (-0.5374,0.6097) θ22 -0.0582 0.0105 0.0002 (-0.0780,-0.0371) model λ 0.7477 0.2645 0.0024 (0.0022,0.9893) 709.1 iii φ10 -1.1140 0.1865 0.0039 (-1.4900,-0.7674) φ11 0.3020 0.2131 0.0022 (-0.1292,0.7034) φ12 0.0010 0.0080 0.0002 (-0.0138,0.0178) φ20 -0.9954 0.1906 0.0034 (-1.3930,-0.6387) φ21 0.0777 0.2234 0.0027 (-0.3676,0.5077) φ22 0.0040 0.0087 0.0002 (-0.0125,0.0218) θ10 -0.4131 0.3111 0.0038 (-1.0280,0.2000) θ11 0.1566 0.2959 0.0032 (-0.4234,0.7434) θ12 -0.0537 0.0135 0.0003 (-0.0803,-0.0273) θ20 -0.3647 0.3090 0.0030 (-0.9615,0.2536) θ21 -0.0040 0.2932 0.0030 (-0.5818,0.5714) θ22 -0.0687 0.0127 0.0002 (-0.0935,-0.0434) table 4. posterior summaries considering fgm copula tobacco-stainedfingers data model paramedsd mc 95% dic meter ian error cri fgm λ 0.9036 0.1220 0.0019 (0.5478,0.9967) 627.4 bnh φ1 0.2200 0.0723 0.0016 (0.1334,0.4149) φ2 0.2115 0.0446 0.0007 (0.1497,0.3258) θ1 1.4660 0.8177 0.0151 (0.4833,3.6510) θ2 1.4890 0.7025 0.0123 (0.6029,3.2950) fgm λ 0.9089 0.1155 0.0014 (0.5742,0.9966) 659.4 be θ1 0.1417 0.0227 0.0003 (0.1017,0.1900) θ2 0.1037 0.0122 0.0001 (0.0817,0.1292) fgm β1 0.6686 0.0882 0.0009 (0.5051,0.8524) 631.1 bw β2 0.6192 0.0665 0.0009 (0.4964,0.7597) λ 0.9018 0.1244 0.0016 (0.5383,0.9963) θ1 0.2040 0.0369 0.0004 (0.1419,0.2863) θ2 0.2029 0.0325 0.0004 (0.1452,0.2723) fgm β1 0.6528 0.0994 0.0013 (0.486,0.8726) 634.2 bge β2 0.5807 0.0742 0.0009 (0.4479,0.7384) λ 0.9032 0.1205 0.0015 (0.5487,0.9963) θ1 0.0733 0.0258 0.0003 (0.0333,0.1336) θ2 0.0504 0.0133 0.0002 (0.0282,0.0801) fgm α1 0.1898 0.0362 0.0006 (0.1273,0.2684) 634.4 bmw α2 0.1911 0.0325 0.0005 (0.1348,0.2607) β1 0.6302 0.0926 0.0014 (0.4597,0.8225) β2 0.5728 0.0745 0.0013 (0.4284,0.7207) φ1 0.0213 0.0255 0.0004 (0.0008,0.0962) φ2 0.0173 0.0177 0.0003 (0.0008,0.0662) λ 0.9010 0.1299 0.0022 (0.5132,0.9961) tion by taking sex and age as covariates. the covariates were table 5. posterior summaries considering product copula tobacco-stainedfingers data model paramedsd mc 95% dic meter ian error cri pb φ1 0.1917 0.0626 0.0010 (0.1173,0.3612) 643.4 nh φ2 0.2027 0.0443 0.0008 (0.1434,0.3138) θ1 1.6020 0.9106 0.0132 (0.5268,4.0580) θ2 1.5930 0.7913 0.0136 (0.6229,3.5410) pbe θ1 0.1277 0.0214 0.0002 (0.0907,0.1738) 678.7 θ2 0.1027 0.0123 0.0001 (0.0806,0.1288) pbw β1 0.6412 0.0873 0.0009 (0.4799,0.8255) 647.6 β2 0.6080 0.0675 0.0007 (0.486,0.7496) θ1 0.1896 0.0353 0.0004 (0.1287,0.2658) θ2 0.2030 0.0333 0.0004 (0.1452,0.2758) pb β1 0.6278 0.0967 0.0011 (0.4634,0.8398) 650 ge β2 0.5673 0.0747 0.0008 (0.438,0.7287) θ1 0.0587 0.0227 0.0002 (0.0244,0.1125) θ2 0.0476 0.0131 0.0001 (0.0263,0.0771) pb α1 0.1771 0.0351 0.0005 (0.1186,0.2570) 650.9 mw α2 0.1916 0.0329 0.0004 (0.1343,0.2625) β1 0.6025 0.0929 0.0013 (0.4298,0.7945) β2 0.5646 0.0752 0.0009 (0.4180,0.7148) φ1 0.0211 0.0249 0.0003 (0.0008,0.0941) φ2 0.0176 0.0180 0.0002 (0.0008,0.0673) assumed to have effect on φk, θk and on both φk and θk for k = 1, 2. the posterior summary statistics of the fits of the data are given in table 3. it is observed that the covariates have effect on φk. 3.2. tobacco-stained-fingers data set in this section, we consider the tobacco stained finger data set obtained from [45]. the data consist of a sample of 143 smokers screened between march 2006 and january 2010 in a 180-bed community hospital in la chauxde-fonds, switzerland. information on death and hospital admission of the patients were collected until june 2014. for more details on this data set see [45]. to demonstrate the applicability of the introduced methodology using this data set, it was considered as t1, the time before the first hospital readmission in smokers with stains on their fingers. on the other hand, the survival time of the patient with tobacco-tar stain on their fingers was considered as t2. this data is censored in case of death before the closure date. similar procedure was followed in analyzing this data set as in section 3.1. the posterior summary statistics of the fits of the bivariate distributions to the tobacco-stained-fingers data are given in tables 4 and 5. the mc error of the posterior estimates of the fitted models considering the tobacco-stained-fingers data were all less than 120 of standard deviation of the estimates, this showed that the posterior estimates have reasonably good precision. comparing the information criteria values in tables 4 and 5 showed that, the introduced bnh distribution fits the tobacco-stained-fingers data more efficiently than the compared bivariate distributions. also, comparing between fits of the bivariate distributions considering the fgm copula and the bivariate distributions considering the product copula showed that, the bivariate distributions considering the fgm copula 6 aliyu & usman / j. nig. soc. phys. sci. 5 (2023) 871 7 fits the tobacco-stained-fingers data better than the bivariate distributions considering the product copula function. for the fgm bivariate models, the estimates of the dependent parameter are similar for all the considered bivariate distributions. moreover, the estimates of the shape and scale parameters of the bivariate distributions considering the fgm copula are similar to their bivariate distribution counterparts considering the product copula. 4. conclusion in real life applications, lifetime data of two organs affected by a given type of treatment or therapy applied to the same individual that present some type of weak dependence are usually obtained. in such a situation, the use of copula functions would be a very good alternative in modelling this type of bivariate lifetime data in presence of censored data and covariates. in this article, the analytical structures of the statistical methodology associated with the modelling of lifetimes assuming weak linear dependence was introduced. fgm copula function was used, however, there are many other copula functions that could be used to build new bivariate lifetime models depending on the assumptions established about the form of the relationship between the bivariate lifetimes (t1 and t2). also, marginal nh distributions were used since the distribution usually provide goodness of fit for lifetime data, as constant, decreasing or increasing hazard functions, features that are characteristic of the studied variables (t1 and t2). hence, other univariate lifetime distributions such as inverted nh, unit nh, discrete nh distributions could also be used. two real data sets: kidney data and tobacco-stained-fingers data sets were used in demonstrating the applicability of the introduced methodology. the obtained results were compared assuming weak dependence that can be model using copula function, with those obtained considering independence between lifetimes. moreover, the introduced methodology is further compared with some competing bivariate distributions. the proposed approach could be useful in many areas of interest, including medical sciences, physical sciences and engineering studies. acknowledgments we thank the referees for the positive enlightening comments and 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[45] g. john, c. louis, a. berner & d. genné, “tobacco stained fingers and its association with death and hospital admission: a retrospective cohort study”, plos one 10(9) (2015) e0138211. appendices the following openbugs computational code can be used to obtained the posterior estimates from bivariate nadarajahhaghighi distribution with parameters θ1, θ2, φ1, φ2 and λ. # the model assuming product copula function model { for (i in 1:n) { s1[i]< -exp(1-pow((1+theta1*t1[i]), phi1)) s2[i]< -exp(1-pow((1+theta2*t2[i]), phi2)) b1[i]<pow((1+theta1*t1[i]), phi1-1) b2[i]<pow((1+theta2*t2[i]), phi2-1) f1[i]< -theta1*phi1*b1[i]*s1[i] f2[i]< -theta2*phi2*b2[i]*s2[i] ft1t2[i]< f1[i]*f2[i] del1[i]<-f1[i]*s2[i] del2[i]<-f2[i]*s1[i] st1t2[i]<s1[i]*s2[i] l1[i]< pow(ft1t2[i], d1[i]*d2[i])*pow(st1t2[i], (1-d1[i])*(1-d2[i]))*pow(del1[i], d1[i]*(1-d2[i]))* pow( del2[i], d2[i]*(1-d1[i])) l[i]<abs(l1[i]) logl[i] < log(l[i]) zeros[i] < 0 zeros[i] ~ dloglik(logl[i]) } # prior distributions theta1 ~ dgamma(1,1) theta2 ~ dgamma(1,1) phi1 ~ dgamma(1,1) phi2 ~ dgamma(1,1) } # the model assuming fgm copula function model { for (i in 1:n) { s1[i]< -exp(1-pow((1+theta1*t1[i]), phi1)) s2[i]< -exp(1-pow((1+theta2*t2[i]), phi2)) b1[i]<pow((1+theta1*t1[i]), phi1-1) b2[i]<pow((1+theta2*t2[i]), phi2-1) f1[i]< -theta1*phi1*b1[i]*s1[i] f2[i]< -theta2*phi2*b2[i]*s2[i] f12[i]<exp(2-pow((1+theta1*t1[i]), phi1)pow((1+theta2*t2[i]), phi2)) ft1t2[i]< theta1*theta2*phi1*phi2*f12[i]* b1[i]*b2[i]*(1+lambda*(1-2*s1[i])*(1-2*s2[i])) del1[i]<f1[i]*s2[i]* (1+lambda*(1-2*s1[i]) *(1-s2[i])) del2[i]<f2[i]*s1[i]* (1+lambda*(1-2*s2[i]) *(1-s1[i])) st1t2[i]<f12[i]* (1+lambda*(1-s2[i])*(1-s1[i])) l1[i]< pow(ft1t2[i], d1[i]*d2[i])*pow(st1t2[i], (1-d1[i])*(1-d2[i]))*pow(del1[i], d1[i]*(1-d2[i])) *pow( del2[i], d2[i]*(1-d1[i])) l[i]<abs(l1[i]) logl[i] < log(l1[i]) zeros[i] < 0 zeros[i] ~ dloglik(logl[i]) } # prior distributions theta1 ~ dgamma(1,1) theta2 ~ dgamma(1,1) phi1 ~ dgamma(1,1) phi2 ~ dgamma(1,1) lambda~ dunif(-1,1) 8 aliyu & usman / j. nig. soc. phys. sci. 5 (2023) 871 9 } # the model assuming fgm copula function assuming covariate effect on the shape parameters model { for (i in 1:n) { phi1[i]<-exp(phi10+phi11*x1[i]+phi12*x2[i]) phi2[i]<-exp(phi20+phi21*x1[i]+phi22*x2[i]) s1[i]< -exp(1-pow((1+theta1*t1[i]), phi1[i])) s2[i]< -exp(1-pow((1+theta2*t2[i]), phi2[i])) b1[i]<pow((1+theta1*t1[i]), phi1[i]-1) b2[i]<pow((1+theta2*t2[i]), phi2[i]-1) f1[i]< -theta1*phi1[i]*b1[i]*s1[i] f2[i]< -theta2*phi2[i]*b2[i]*s2[i] f12[i]<exp(2-pow((1+theta1*t1[i]), phi1[i])pow((1+theta2*t2[i]), phi2[i])) ft1t2[i]< theta1*theta2*phi1[i]*phi2[i]*f12[i]* b1[i]*b2[i]*(1+lambda*(1-2*s1[i])*(1-2*s2[i])) del1[i]<f1[i]*s2[i]* (1+lambda*(1-2*s1[i]) *(1-s2[i])) del2[i]<f2[i]*s1[i]* (1+lambda*(1-2*s2[i]) *(1-s1[i])) st1t2[i]<f12[i]* (1+lambda*(1-s2[i])*(1-s1[i])) l1[i]< pow(ft1t2[i], d1[i]*d2[i])*pow(st1t2[i], (1-d1[i])*(1-d2[i]))*pow(del1[i], d1[i]*(1-d2[i])) *pow( del2[i], d2[i]*(1-d1[i])) l[i]<abs(l1[i]) logl[i] < log(l1[i]) zeros[i] < 0 zeros[i] ~ dloglik(logl[i]) } # prior distributions theta1 ~ dgamma(1,1) theta2 ~ dgamma(1,1) phi10~dnorm(0, 10) phi11~dnorm(0, 10) phi12~dnorm(0, 10) phi20~dnorm(0, 10) phi21~dnorm(0, 10) phi22~dnorm(0, 10) lambda ~ dunif(-1, 1) } in this codes, n is the sample size, s1[i] is the survival function given in equation (9), s2[i] is the survival function given in equation (10), f1[i] is the probability density function given in equation (7), f2[i] is the probability density function given in equation (8), ft1t2 [i] is the joint probability density function (pdf) given in equation (12), del1[i] is an expression given in equation (14), del2[i] is an expression given in equation (15), st1t2[i] is the joint survival function given in equation (11) and l[i] is the likelihood function given in equation (19). for the product copula function, the dependence parameter (lambda) assumes the value zero, and hence, the joint density function reduces to the product of the pdfs in equations (7) and (8). the joint survival function also reduces to the product of the survival functions in equations (9) and (10). for the models that assume covariate effect on the parameter(s) of the model, appropriate substitutions are made in the codes for the fgm copula function as discussed in the work. 9 aliyu & usman / j. nig. soc. phys. sci. 5 (2023) 871 10 table 5. kidney data patient t1 t2 δ1 δ1 sex age disease types 1 8 16 1 1 1 28 3 2 23 13 1 0 0 48 0 3 22 28 1 1 1 32 3 4 447 318 1 1 0 31.5 3 5 30 12 1 1 1 10 3 6 24 245 1 1 0 16.5 3 7 7 9 1 1 1 51 0 8 511 30 1 1 0 55.5 0 9 53 196 1 1 0 69 1 10 15 154 1 1 1 51.5 0 11 7 333 1 1 0 44 1 12 141 8 1 0 0 34 3 13 96 38 1 1 0 35 1 14 149 70 0 0 0 42 1 15 536 25 1 0 0 17 3 16 17 4 1 0 1 60 1 17 185 177 1 1 0 60 3 18 292 114 1 1 0 43.5 3 19 22 159 0 0 0 53 0 20 15 108 1 0 0 44 3 21 152 562 1 1 1 46.5 2 22 402 24 1 0 0 30 3 23 13 66 1 1 0 62.5 1 24 39 46 1 0 0 42.5 1 25 12 40 1 1 1 43 1 26 113 201 0 1 0 57.5 1 27 132 156 1 1 0 10 0 28 34 30 1 1 0 52 1 29 2 25 1 1 1 53 0 30 130 26 1 1 0 54 0 31 27 58 1 1 0 56 1 32 5 43 0 1 0 50.5 1 33 152 30 1 1 0 57 2 34 190 5 1 0 0 44.5 0 35 119 8 1 1 0 22 3 36 54 16 0 0 0 42 3 37 6 78 0 1 0 52 2 38 63 8 1 0 1 60 2 table 5. tobacco data n sex age t1 δ1 t2 δ2 control31 m 83 0.063014 1 0.013699 0 control34 m 36 6.69863 0 6.079452 0 control69 m 49 5.021918 0 4.40274 0 control55 m 39 5.59726 0 4.978082 0 control4 m 49 7.879452 0 3.876712 0 control39 f 85 5.309589 1 4.060274 0 control54 f 50 5.6 0 4.980822 0 control14 m 58 6.093151 1 2.490411 1 control24 f 80 0.065753 1 0.249315 0 control11 m 33 7.739726 0 0.646575 0 control5 m 64 1.405479 1 1.389041 1 control53 m 47 5.791781 0 3.4 0 control62 f 57 5.227397 0 4.545206 0 control27 m 46 7.435616 0 4.945206 0 control44 f 83 6.019178 0 1.654795 1 control17 m 64 0.668493 1 0.454795 1 control25 f 76 3.038356 1 0.227397 0 control16 m 64 0.347945 1 0.347945 0 control32 f 83 6.460274 0 2.753425 0 control57 m 71 5.536986 0 0.10137 1 control66 m 65 0.383562 1 0.375342 1 control8 m 50 7.605479 0 6.986301 0 control65 m 60 5.213699 0 4.594521 0 control59 f 41 5.536986 0 0.052055 0 control9 m 55 7.80274 0 7.183562 0 control33 f 35 6.506849 0 1.19726 0 control3 m 57 4.693151 1 2.139726 0 control63 m 55 0.575342 1 0.065753 0 control56 m 51 4.652055 1 4.109589 0 control7 f 51 7.80548 0 7.186301 0 control47 m 59 6.079452 0 2.767123 1 control71 m 50 4.978082 0 4.358904 0 control26 m 56 0.252055 1 0.252055 1 control15 m 70 0.005479 1 0.005479 0 control60 m 41 5.161644 0 4.542466 0 control36 f 82 2.736986 1 1.865753 0 control1 f 46 7.928767 0 0.654795 0 control68 m 74 5.410959 0 0.334247 1 control51 f 71 3.263014 1 3.126027 1 control46 f 46 0.257534 1 0.109589 0 control21 f 76 6.627397 1 2.161644 0 control41 f 59 6.243835 0 1.043836 1 control37 m 59 6.112329 0 3.271233 0 control18 f 58 7.835617 0 7.216438 0 control19 f 75 1.734247 1 0.69589 1 control67 f 68 5.367123 0 4.747945 0 control64 f 60 5.19726 0 4.578082 0 control45 f 71 6.172603 0 5.553425 0 control10 f 50 7.739726 0 7.120548 0 control61 m 64 5.254795 0 1.189041 0 control35 m 62 4.257534 1 2.553425 0 10 aliyu & usman / j. nig. soc. phys. sci. 5 (2023) 871 11 table 5. tobacco data cont. n sex age t1 δ1 t2 δ2 control13 m 37 3.512329 1 3.512329 0 control30 m 66 7.041096 0 1.252055 0 control70 f 83 0.167123 1 0.167123 0 control28 m 53 6.923288 0 0.052055 0 control2 m 50 7.969863 0 6.70411 0 control12 f 77 1.561644 1 1.49863 1 control58 f 81 5.534246 0 4.915069 0 control40 f 78 0.29589 1 0.167123 1 control29 m 65 6.926027 0 0.432877 1 control48 m 37 5.646575 0 0.638356 0 control43 m 64 0.243836 1 0.032877 1 control52 f 50 5.657534 0 5.038356 0 control49 m 76 0.682192 1 0.569863 0 control23 f 66 7.210959 0 0.216438 1 control20 m 66 7.827397 0 1.131507 0 control6 m 51 7.816438 0 0.276712 0 control50 m 40 5.619178 0 5 0 control22 f 65 7.128767 0 0.169863 0 control38 f 60 5.904109 0 3.413699 0 tache40 m 68 1.183562 1 0.687671 1 tache7 m 45 8.819178 0 1.747945 0 tache64 m 54 7.419178 0 0.550685 0 tache60 m 55 7.419178 0 1.59726 0 tache61 f 71 7.512329 0 0.339726 0 tache16 f 57 1.394521 1 0.071233 0 tache57 m 64 0.315068 1 0.150685 1 tache26 f 49 8.372602 0 0.463014 0 tache73 m 65 2.180822 1 0.027397 0 tache50 f 75 1.90411 1 1.69589 1 tache42 m 73 7.260274 1 1.863014 1 tache9 m 63 5.99726 1 0.482192 0 tache25 m 62 0.128767 1 0.120548 1 tache37 f 69 5.624658 1 0.320548 0 tache6 m 50 8.819178 0 5.969863 0 tache31 m 79 2.021918 1 0.076712 0 tache67 f 88 1.780822 1 0.90411 0 tache72 f 63 2.161644 1 0.071233 0 tache8 m 71 8.873973 0 0.567123 1 tache33 m 27 8.284931 0 7.665753 0 tache30 m 56 8.265754 0 0.030137 0 tache56 f 70 6.838356 1 2.616438 1 tache13 m 69 7.882192 0 1.139726 1 tache70 m 78 7.339726 0 1.936986 1 tache19 m 50 0.679452 1 0.679452 0 tache14 f 72 3.230137 1 0.8 0 tache59 m 74 0.30137 1 0.30137 0 tache22 m 52 2.123288 1 0.956164 1 tache36 m 85 0.550685 1 0.139726 1 tache53 m 50 3.60274 1 1.038356 0 tache2 m 47 5.323287 1 4.482192 1 table 5. tobacco data cont. n sex age t1 δ1 t2 δ2 tache18 m 69 8.452055 0 0.136986 0 tache3 m 61 2.89589 1 1.649315 0 tache1 m 67 0.071233 1 0.071233 0 tache11 m 53 3.923288 1 1.994521 0 tache35 f 44 8.30137 0 4.073973 0 tache46 m 63 7.808219 0 0.09589 0 tache47 f 66 7.769863 1 3.106849 1 tache58 f 51 7.904109 0 0.424658 0 tache21 f 69 2.232877 1 0.060274 0 tache55 m 50 0.520548 1 0.109589 1 tache29 m 45 0.052055 1 0.052055 0 tache27 m 54 8.378082 0 1.032877 0 tache5 m 38 8.821918 0 1.830137 0 tache24 m 37 8.394521 0 7.775342 0 tache51 m 62 7.547945 0 0.906849 0 tache39 m 67 0.238356 1 0.013699 1 tache54 m 53 7.616438 0 0.035616 0 tache45 m 36 8.128767 0 7.509589 0 tache20 m 49 8.380822 0 5.019178 0 tache48 m 80 0.252055 1 0.016438 0 tache4 f 64 0.136986 1 0.136986 0 tache38 m 79 3.468493 1 0.39726 0 tache49 m 67 7.654795 0 7.035616 0 tache34 m 75 0.665753 1 0.386301 0 tache65 f 63 7.128767 0 0.323288 0 tache69 m 77 0.947945 1 0.071233 1 tache17 m 83 6.306849 1 1.243836 0 tache71 m 52 0.915069 1 0.868493 0 tache41 m 59 7.846575 0 1.084931 0 tache63 m 72 7.383562 0 0.213699 0 tache15 m 50 1.20274 1 1.20274 0 tache62 m 66 7.383562 0 5.243835 1 tache12 f 75 0.021918 1 0.021918 0 tache23 f 73 2.336986 1 2.336986 0 tache66 m 44 7.189041 0 1.221918 0 tache52 m 87 0.405479 1 0.405479 0 tache10 f 69 2.386301 1 2.386301 0 tache68 m 56 2.616438 1 2.610959 0 tache28 m 54 8.40274 0 1.271233 1 tache44 m 46 4.950685 1 1.991781 0 tache43 f 69 8 0 1.30411 0 tache32 f 71 5.134246 1 0.136986 1 11 j. nig. soc. phys. sci. 5 (2023) 896 journal of the nigerian society of physical sciences annual effective dose and excess lifetime cancer risk due to ingestion and inhalation of radon in groundwater of bosso community minna, north-central nigeria matthew tikpangi koloa,∗, oyeleke olarinoyea, simon olonkwoh salihub, hyginus anayo ugwuanyia, paul onuchea, opeyemi faladea, nwachukwu chibuezea adepartment of physics, federal university of technology, minna, niger state, nigeria bdepartment of chemistry, federal university of technology, minna, niger state, nigeria abstract radon in potable water has become an issue of public health concern, especially when consumed or used directly from source for domestic purposes without any pre-treatment. in this study, 222rn concentration in 22 water samples collected from 2 groundwater sources (open wells, 12 samples and boreholes, 10 samples) in bosso town, north central nigeria were measured using durridge rad-7 radon detector with rad-h2o accessories. 222rn concentrations in open wells varied from 2.1±0.7 to 27.9±2.5 bq l−1 with a mean of 10.2±1.5 bq l−1, while that in boreholes ranged from 2.8±1.1 to 39.2±1.5 bq l−1 with a mean value of 14.3±1.7 bq l−1. these values are lower than the 100 bq l−1 upper limit proposed by the european union commission, above which any practical intervention may be necessary. mean annual committed effective dose to adults, children and infants from ingestion of water were 74.64, 71.58 and 53.17 µsv y−1 respectively for the open wells and 104.24, 99.96 and 74.26 µsv y−1 respectively for borehole water samples. mean whole body dose due to ingestion and inhalation of waterborne radon from open wells and boreholes are 27.56 and 38.48 µsv y−1 respectively, which are below the reference level of 0.1 msv y−1 for potable water recommended by the world health organization for public safety. the excess lifetime cancer risk were 0.10 × 10−3 for the open wells and 0.13 × 10−3 for the boreholes, which are lower than the world safety limit 0.29 × 10−3. water from the two groundwater sources investigated is therefore fit for consumption and other domestic usage from the point of view of radiation protection. doi:10.46481/jnsps.2023.896 keywords: 222rn concentration, groundwater, bosso community, rad-7 radon detector, committed effective dose, elcr, nigeria article history : received: 28 june 2022 received in revised form: 07 january 2023 accepted for publication: 07 january 2023 published: 19 april 2023 c© 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: m. orosun 1. introduction the daily influx of human population coupled with rapid urbanization of bosso community of minna, north-central nige∗corresponding author tel. no: +234 8105168992 email address: matthewkolo@futminna.edu.ng (matthew tikpangi kolo) ria has put increased pressure on the existing epileptic potable water supply within the community. to effectively combat this growing challenge therefore, residents of the community have turned their attention to open wells and drilled boreholes. groundwater has thus, become the primary source of potable water for majority of the population in the community. groundwater, which is the world’s largest reservoir of fresh water [1], 1 kolo et al. / j. nig. soc. phys. sci. 5 (2023) 896 2 derives its radioactivity from the underlying rocks, soil formations, dissolved radioactive and mineral substances. dissolved radon (222rn) in groundwater arises from the decay of naturally occurring uranium and radium (226ra) deposits present in soil and reservoir rocks over which the water bodies move, coupled with the emanation coefficient of the reservoir rocks [2-4]. other factors that control the distribution of 222rn in groundwater includes the type of rocks, host rock mineralogy, shear zones, degree of weathering, frequency of fractures, permeability of host rock, secondary porosity, physiochemical, and nature of the geological aquifers, depths of wells and duration of water storage times [3, 5, 6]. 222rn often finds its way out of the earth’s crust and underlying bedrocks through cracks, crevices and other openings in the foundation floor and walls, into human homes, thereby enhancing the level of human exposure. additionally, human exposure to waterborne radon occur through daily use of groundwater for bathing, laundries, cooking and other domestic purposes [7].most often, human exposure occur through direct ingestion of water containing radon and inhalation of radon gas in indoor air. waterborne radon monitoring and mitigation has been a subject of great concern and research in many countries of the world, owing to its hazardous effects [2]. world health organization (who) [8] reported that about 1-7% of deaths from lung cancer is attributed to increased levels of waterborne radon. various studies have also shown that direct inhalation and ingestion of waterborne radon over long period of time can lead to pathological effects such as the respiratory functional changes, and dna damage in sensitive lung tissues leading to lung cancer (, stomach and gastrointestinal cancer [1, 2, 5, 9]. epidemiological studies have also shown that ingestion of 222rn over a considerable length of time can be responsible for stomach and gastrointestinal cancer [10]. levels of 222rn in water should therefore be measured, checked and controlled consistently in order to protect the human population from internal exposure that may arise from inhalation and ingestion of waterborne radon isinkaye et al [11],underscores the importance of water to human existence and an expressive indices for radiological, geological and environmental studies. hence, its purity, accessibility and quality cannot be jeopardized especially from the point of view of human health and protection. the first campus of federal university of technology, minna is located in bosso community. the challenge of insufficient hostel accommodation within the campus has pushed majority of the students to rented apartments within bosso. bosso community is also the readily available home for local farmers who have been displaced from their respective communities due to the nefarious activities of bandits and herdsmen. as a result of the growing population, the community has resulted into drilling open wells and boreholes to meet the increasing demand for potable water supply. locally built apartments available for rent within the community are semi en suite and poorly ventilated, resulting in high probability of population exposure to accumulated radon from groundwater used for laundry, bathing, dishwashing and other domestic activities. moreover, radon concentration in groundwater sources in figure 1: map of niger state showing the study area bosso community have not so far been examined or reported in literature. this study is therefore a pioneering work aimed at measuring 222rn concentrations in groundwater sourced from open wells and boreholes in bosso community and to evaluate the level of exposure to radiation dose from the daily usage of groundwater within the community. results from this study will help in raising awareness on the likelihood of radiation incidence that may occur from daily exposure, through inhalation and ingestion, to waterborne radon within the community. 2. materials and method 2.1. location and geology of the study area bosso, a community in niger state, north central nigeria (fig. 1), with average area of 1592 km2, is geographically located within the guinea savanna ecological zone of nigeria. it lies between latitudes 090 40 ′ n to 090 42 ′ n and longitudes 060 30 ′ e to 060 36 ′ e. the community falls within the regional basement complex of nigeria, which according to okanlawon et al [12], is underlain by two geologic episodes viz igneous and metamorphic orogeneses. the community is underlain by undifferentiated granitic rocks which are believed to be part of the older granitic suites of nigeria and various metamorphic rocks [13, 14, 15]. the metamorphic terrain is made up of gneisses, schists, quartzites and some migmatites [16]. in hand specimens, the granitic rocks constitute feldspars (mostly plagioclase), quartz, micas and some few ferromagnesian minerals like pyroxenes, amphiboles and hornblendes. the metamorphic rocks are mostly made of biotites, amphiboles, staurolites and asbestos. structurally, most of the rocks are non-deformed to slightly deformed [17]. secondary structures such as faults, folds, foliations and lineations are obvious on the insitu rocks. the community is marked by two distinct seasons, the raining season which lasts from the month of april to october, and the dry season which lasts from november to march/april. annual rainfall values ranges between 900 mm to 1,000 mm, with mean temperature of around 32◦c. 2 kolo et al. / j. nig. soc. phys. sci. 5 (2023) 896 3 2.2. sample collection 12 groundwater samples from open wells and 10 groundwater samples from drilled boreholes within the community were collected and analysed for 222rn concentration. empty 75cl water bottles which were used for collection of water samples were purchased from commercial shops. the sample bottles were thoroughly washed with distilled water and with dilute 0.1m hcl after purchase. they were then allowed to dry before sample collection. fresh water sample from open wells were collected using bailers. clean sample bottles were gently filled with water samples by submerging them into the bailers and immediately sealed tightly with a cap underwater to avoid any formation of air bubbles. fresh water from boreholes were collected into the samples bottles directly from the borehole tap heads after they were turned on and allowed to run freely for about five minutes. this was necessary to purge out all trapped air and to ensure that the temperature of the water is stable before collection. all samples bottles were well labelled and carefully filled to prevent agitations that could lead to the escape of radon gas. 2.3. sample analysis all the samples were analysed at radioanalytical research laboratory, ekiti state university, ado ekiti, nigeria, using rad-7 radon analyzer (durridge co., usa) coupled to radh2o accessories. in the rad-7 detector setup shown in fig. 2, required portion of water sample was transferred into the 250 ml vial, from which radon gas was expelled using bubbling kit. the expelled gas was passed via air circulation into the hemisphere chamber where it decayed into its daughter nuclei, polonium (po). a silicon solid-state detector which was maintained at high electric field, collected the po nuclei which were finally counted by an in-built software to give an estimation of radon concentration in the water sample. each water sample was automatically analyzed by the detector in four cycles of 5 min each, with an initial aeration time of 5 min. this was necessary for quality control purposes and to guarantee reliability of analytical methods [11, 18]. since the counting was not immediately done at the sample collection point, radon concentration in the samples must have reduced due to radioactive decay. the results of the analysis were therefore decay-corrected back to the original concentration values at the time of sample collection, following the correction procedure outlined in durridge [19]. rad-7 is a sophisticated, active radon detector that uses a computer-driven electronic detector with pre-programmed setups. the detector is able to give the result of analysis within 30 minutes, which makes it a comparatively faster analysis procedure. 2.4. annual effective dose estimation exposure to waterborne radon by human and animal population occur on daily basis as a result of water consumption and other various domestic use, which results in the annual committed effective dose (aced) incurred. aced to different icrp figure 2: schematic diagram of rad-h20 set for measurement of waterborne radon age groups due to ingestion of waterborne radon was computed from the equation aced = crn × wcr × dcf (1) where crn is the radon concentration in groundwater measured in bq l−1, wcr is the water consumption rate per year (l yr−1) and dcf is the dose conversion factor expressed in sv bq−1. wcr and dcf values adopted in this study are given in table 1. the contribution of radon in the different sources of water to indoor radon was determined using equation [21]: χrn = crn × w × ω v ×λc (2) where χrn is the fractional contribution of waterborne radon to radon in indoor air, crn is the radon concentration in water, w is the water consumption rate per person (0.01 m3 h−1), ω (0.5) is the coefficient to indoor air, v is the bulk volume of indoor air per person taken to be 20 m3, and λc (0.7 h−1) is the air exchange rate [22, 23]. 3. results and discussion 3.1. radon concentration the measured 222rn concentration in groundwater, fractional contribution of waterborne radon to radon in indoor and the associated health hazard within the study area are presented in table 2 as shown in table 2, 222rn concentrations in lw varied between 2.1±0.7 to 27.9±2.5 bq l−1 with a mean value of 10.2±1.5 bq l−1. similarly, 222rn concentrations in bh ranged from 2.8±1.1 to 39.2±1.5 bq l−1 with a mean value of 14.3±1.7 bq l−1. according to kalip, haque and gaiya [24], and suresh et al [25], observed differences in radon concentrations in lw and bh may be due to the depth of water sources, hydrogeological state of the source location and the prevailing climatic conditions. although 33% of water samples from both lw and bh have 222rn concentration values above the usepa safety threshold of 11.1 bq l−1 as seen in figure 3, 222rn concentration in two groundwater sources were within the safety range 3 kolo et al. / j. nig. soc. phys. sci. 5 (2023) 896 4 table 1: wcr and dcf for respective icrp age groups [20] water consumption age group age range(yrs) (l/day) wcr (l/yr) dcf (sv/bq) 3 months 0 1 0.55 200 1 yr (infants) 1 2 0.71 260 2 × 10−8 3 yrs 2 7 0.82 300 10 yrs (children) 7 12 0.96 350 2 × 10−8 15 yrs 12 17 1.64 600 adults > 17 2.00 730 1 × 10−8 table 2: 222rn concentration in groundwater, fractional contribution of waterborne radon to radon in indoor and the associated health hazard within the study area aced (µsv y−1) to icrp age groups sample id 222rn(bq l−1) χrn (mbq l−1) adults children infants open wells lw 01 2.1±0.7 0.75 15.33 14.70 10.92 lw 02 27.9±2.5 9.96 203.67 195.30 145.08 lw 03 24.9±2.5 8.89 181.77 174.30 129.48 lw 04 10.2±1.6 3.64 74.46 71.40 53.04 lw 05 2.9±0.8 1.04 21.17 20.30 15.08 lw 06 4.5±1.1 1.61 32.85 31.50 23.40 lw 07 8.8±1.4 3.14 64.24 61.60 45.76 lw 08 2.4±0.8 0.86 17.52 16.80 12.48 lw 09 4.4±1.1 1.57 32.12 30.80 22.88 lw 10 3.1±2.0 1.11 22.63 21.70 16.12 lw 11 16.0±2.8 5.71 116.80 112.00 83.20 lw 12 15.5±0.8 5.53 113.15 108.50 80.60 min 2.1±0.7 0.75 15.33 14.70 10.92 max 27.9±2.5 9.96 203.67 195.30 145.08 mean 10.2±1.5 3.65 74.64 71.58 53.17 boreholes bh 01 28.7±3.1 10.25 209.51 200.90 149.24 bh 02 10.1±1.5 3.61 73.73 70.70 52.52 bh 03 22.2±1.0 7.93 162.06 155.40 115.44 bh 04 2.8±1.1 1.00 20.44 19.60 14.56 bh 05 39.2±1.5 13.99 286.16 274.40 203.84 bh 06 9.9±2.0 3.53 72.27 69.30 51.48 bh 07 12.3±4.5 4.39 89.79 86.10 63.96 bh 08 10.5±0.2 3.75 76.65 73.50 54.60 bh 09 3.7±0.8 1.32 27.01 25.90 19.24 bh 10 3.4±1.3 1.21 24.82 23.80 17.68 min 2.8±1.1 1.00 20.44 19.60 14.56 max 39.2±1.5 13.99 286.16 274.40 203.84 mean 14.3±1.7 5.10 104.24 99.96 74.26 of 4 – 40 bq l−1 recommended by the united nations scientific committee on effects of atomic radiation (unscear). according to the recommendations of european union commission, no remedial action should be required if the concentration of radon in drinking water is less than 100 bq l−1 [26, 27]. results of this study does not therefore suggest any immediate implimentation of whichever practical radon programme in bosso community. the contribution of waterborne radon to radon in indoor air varies from 0.75 to 9.96 mbq l−1 for lw water sources and from 1.00 to 13.99 mbq l−1 for the bh water. these values are lower than the estimated mean indoor radon level of 1.3 pci/l (48.1 mbq l−1), thereby indicating the paltry contribution of waterborne radon within the community to total radon in indoor air. table 2 also shows the computed annual effective dose (aced) to the adults, children and infants due to 222rn concentration in the two groundwater sources (lw and bh) of the studied community. aced due to ingestion of water from lw source by 4 kolo et al. / j. nig. soc. phys. sci. 5 (2023) 896 5 figure 3: 222rn concentration in lw and bh sources together with the usepa threshold figure 4: aced received by different icrp age groups due to waterborne radon from lw groundwater sources adult population ranged from 15.33 to 203.67 µsv y−1 with an average value of 74.64 µsv y−1, from 14.70 to 195.30 µsv y−1 with a mean value of 71.58 µsv y−1 for the children, while infants recorded aced values between 10.92 to 145.08 µsv y−1, with a mean value of 53.17 µsv y−1. similarly, average aced due to ingestion of water from bh source by adult, children and infants population of the studied community are 104.24, 99.96 and 74.26 µsv y−1 respectively. the results shows that aced increases with mean radon concentration, rate of water consumption and age; thus, infants suffer least radon exposure compared to the children and adults. this agrees with the results obtained for similar studies by kalip et al [24] and ravikumar and somashekar [10]. furthermore, the highest aced due to groundwater consumption from lw and bh sources among the icrp age groups is 0.204 msv y−1 and 0.274 msv y−1 respectively. these values are below the 1msv y−1 safety threshold recommended by unscear and who. the infants, children and adult population of bosso community are therefore not under any immediate health threat from waterborne radon. variations of aced received by different icrp age groups due to ingestion of waterborne radon from lw and bh groundwater sources in the studied community are shown in figures 4 and 5. table 3 shows the comparison of the results of this investigation with those of similar studies from different parts of figure 5: aced received by different icrp age groups due to waterborne radon from bh groundwater sources nigeria. mean radon concentration for open wells within the community (10.2 bq l−1) is higher than that reported by kalip et al [24] for kaduna (1.8 bq l−1). additionally, average value obtained for the boreholes within the community (14.3 bq l−1) is about 54% lower than the one reported by isinkaye and ajiboye [28] for ekiti state. this may be due to variations in local geology of the respective sampling areas and depth of sampling. judging from the table generally, the range of radon concentration values obtained for the two groundwater sources in bosso community is comparable to those reported in literature from other parts of nigeria. 3.2. evaluation of excess lifetime cancer risk and annual effective dose due to inhalation and ingestion radiation dose to the internal organs of human population from waterborne radon has been identified as a highly potent human health threat. human exposure to waterborne radon is principally through direct ingestion of radon-contaminated water and inhalation of radon-saturated indoor air. direct ingestion results in radiation dose delivery to the human stomach while inhalation pathway delivers dose to the lungs and respiratory track. annual committed effective dose (aced) to human stomach from direct ingestion of radon-contaminated water in the studied community was calculated from the equation [11, 35]: aceding ( ms v y−1 ) = crn × wcr × dcfing (3) where crn is the concentration of radon in water, wcr the annual water consumption for adults (730 l y−1) and dcfing is the effective dose conversion factor for ingestion (3.5 nsv bq−1). since radon is comparatively insoluble in water, continuous domestic utilization of water can result in consistent liberation of radon gas into the indoor air, thereby enhancing the average committed dose incurred via inhalation pathway [9, 35]. aced due to inhalation of waterborne radon was computed from the equation [8, 36]: acedinh ( ms v y−1 ) = crn×raw ×ef×of×dcfinh(4) where crn is the concentration of radon in water, raw is the ratio of radon-in-air to radon-in-water (10−4), ef is the equilibrium factor between radon and its progeny for indoor environment (0.4), of is the average global indoor occupancy factor 5 kolo et al. / j. nig. soc. phys. sci. 5 (2023) 896 6 table 3: comparison of radon concentrations in groundwater obtained in this study with other locations within nigeria location source of water 222rn conc (bq l−1) references ekiti state, nigeria borehole 30.9 isinkaye and ajoboye [28] well 19.5 ogbomosho, sw nigeria borehole 0.60 2.64 (1.86) oni et al. [29] ekiti state university, nigeria hand pumped boreholes 7.0 41.5 (23.3) isinkaye et al. [11] hand dug wells 0.6 36.2 (13.33) motorized boreholes 0.6 27.4 (7.4) gadau, bauchi state, nigeria well 4.92 82.89 (38.3) shu’abu et al. [30] ibadan, nigeria well 2.18 76.75 ademola and oyeleke [31] kaduna, nigeria well 0.85 2.57 (1.8) kalip et al. [24] borehole 0.35 085 (0.57) ogun state, nigeria borehole 1.23 12.68 fatoki and ademola [32] zaria, nigeria borehole 7.41 garba et al. [33] well 7.18 sabon gari, kaduna, nigeria borehole 14.9 jibril et al. [34] bosso, north-central nigeria borehole 14.3 present study open well 10.2 mean values are in parenthesis. (7000 h y−1) and dcfinh is the dose conversion factor for exposure to radon in air (9 nsv bq−1 h−1 m3). computed values for committed annual effective dose (aced) received by internal organs (stomach and lungs) due to ingestion and inhalation of waterborne radon in bosso community and presented in table 4. also presented in table 4 is the calculated excess lifetime cancer risk (elcr) due to waterborne radon in the studied community. aced delivered to the stomach as a result of ingestion of waterborne radon from lw sources varied from 0.37 to 4.88 µsv y−1, with a mean value of 1.79 µsv y−1, while the aced range to the lungs via inhalation is 5.29 to 70.31 µsv y−1 with an average value of 25.77 µsv y−1. similarly, the range of aced from bh sources delivered to the stomach via the ingestion pathway is 0.49 to 6.86 µsv y−1 with a mean of 2.50 µsv y−1 and to the lungs via inhalation route is 7.06 to 98.78 µsv y−1 with an average value of 35.99 µsv y−1. with reference to the us national academy of sciences report of 1999, unscear [37] computed mean dose from radon in drinking water to be 0.002 msv y−1 and 0.025 msv y−1 for ingestion and inhalation pathways respectively. the results further show that the highest percentage of whole body dose comes through the inhalation pathway, hence, the lung cells and respiratory track in general are more prone to cancer incidences compared to the walls of the stomach. the total annual effective doses due to ingestion and inhalation (whole body dose) incurred from waterborne radon in bosso community ranged from 5.66×10−3 to 0.075 msv y−1 for lw groundwater sources and from 7.55×10−3 to 0.106 msv y−1 for bh groundwater sources, with mean values of 0.0276 and 0.0385 msv y−1 in sequence. these values are below the threshold value of 0.1 msv y−1 recommended by unscear and who for public safety. the probability of cancer incidence was evaluated in order to assess any possible potential carcinogenic effects to human population for a specific lifetime of exposure to waterborne radon. excess lifetime cancer risk (elcr), which shows the extra risk of occurrence of cancer due to exposure to waterborne radon in the studied community was computed using the equation [38, 39]: elcr = w bd × rf × dl (5) where wbd is the whole body dose in µsv y−1, rf is the risk factor taken to be 0.05 sv−1 for stochastic effects [40] and dl is the lifetime duration of 70 years. computed elcr values which gives an indication of extra risk of cancer incidence due to exposure to waterborne radon among the studied community are presented in table 4. average elcr values for lw and bh groundwater sources in bosso community are 0.10 × 10−3 and 0.13 × 10−3 respectively, which are below the world mean value of 0.29 × 10−3. the likelihood of any cancer incidence among the population of the studied community is therefore insignificant. hence water from the two groundwater sources investigated is fit for consumption and for other domestic usage from the point of view of radiation protection. 4. conclusion water samples from two major groundwater sources (open wells and boreholes) in bosso community were collected and analysed for their radon (222rn) concentration using durridge rad-7 radon detector. the results showed that, although 222rn concentration values in about 33% of water samples from the two groundwater sources were above the usepa safety threshold, all values were within the safety range of 4 – 40 bq l−1 recommended by the united nations scientific committee on effects of atomic radiation. the results also indicated that about 94% of whole body dose comes through the inhalation pathway, which shows that inhalation of radon escaping from water is a considerable 6 kolo et al. / j. nig. soc. phys. sci. 5 (2023) 896 7 table 4: radon concentration, aced to internal organs and elcr in lw and bh water samples aced (µsv/y) to internal organs elcr (10−3) sample id 222rn (bq/l) ingestion (stomach) inhalation (lungs) whole body open wells lw 01 2.1±0.7 0.37 5.29 5.66 0.02 lw 02 27.9±2.5 4.88 70.31 75.19 0.26 lw 03 24.9±2.5 4.36 62.75 67.11 0.23 lw 04 10.2±1.6 1.79 25.70 27.49 0.10 lw 05 2.9±0.8 0.51 7.31 7.82 0.03 lw 06 4.5±1.1 0.79 11.34 12.13 0.04 lw 07 8.8±1.4 1.54 22.18 23.72 0.08 lw 08 2.4±0.8 0.42 6.05 6.47 0.02 lw 09 4.4±1.1 0.77 11.09 11.86 0.04 lw 10 3.1±2.0 0.54 7.81 8.35 0.03 lw 11 16.0±2.8 2.80 40.32 43.12 0.15 lw 12 15.5±0.8 2.71 39.06 41.77 0.15 min 2.1±0.7 0.37 5.29 5.66 0.02 max 27.9±2.5 4.88 70.31 75.19 0.26 mean 10.2±1.5 1.79 25.77 27.56 0.10 boreholes bh 01 28.7±3.1 5.02 72.32 77.35 0.27 bh 02 10.1±1.5 1.77 25.45 27.22 0.10 bh 03 22.2±1.0 3.89 55.94 59.83 0.21 bh 04 2.8±1.1 0.49 7.06 7.55 0.03 bh 05 39.2±1.5 6.86 98.78 105.64 0.37 bh 06 9.9±2.0 1.73 24.95 26.68 0.09 bh 07 12.3±4.5 2.15 31.00 33.15 0.12 bh 08 10.5±0.2 1.84 26.46 28.30 0.10 bh 09 3.7±0.8 0.65 9.32 9.97 0.03 bh 10 3.4±1.3 0.60 8.57 9.16 0.03 min 2.8±1.1 0.49 7.06 7.55 0.03 max 39.2±1.5 6.86 98.78 105.64 0.37 mean 14.3±1.7 2.50 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[40] icrp, icrp publication 60, “1990 recommendations of the international commission on radiological protection”, elsevier health sciences (1991). 8 j. nig. soc. phys. sci. 4 (2022) 883 journal of the nigerian society of physical sciences suitability analysis for yam production in nigeria using satellite and observation data tertsea igbawuaa,∗, martha hembafan gbangera, fanan ujohb adepartment of physics, federal university of agriculture, makurdi, nigeria bcenter for sustainability and resilient infrastructure and communities (saric), school of the built environment and architecture, london south bank university, london, uk abstract identification of suitable areas for yam production is critical for ensuring yield in yam production in nigeria. the study is aimed at determining suitable lands for yam production in nigeria. climate, soil, and environmental parameters that have a high contribution to yam production were used in developing a yam production suitability map using the analytical hierarchy process (ahp). the ahp was used in deriving weights through a pairwise comparison technique. according to the findings, highly suitable (hs), suitable (s), marginally suitable (ms), and not suitable (ns) regions accounted for 11.79, 82.68, 4.05, and 1.47% of the study area, respectively. the normalized difference vegetation index (ndvi), a measure of vegetation vigor, was higher in hs, followed by s regions, and then ms regions. similarly, climate variables in hs regions were more favorable for plant growth, followed by s regions and ms regions. the correlation between precipitation and temperature is high and significant only in the hs class, despite the fact that ndvi and climate variables are significantly connected in all the suitability classes. the output map, thus determined, provides information on highly suitable, suitable or marginally suitable lands that are of practical importance to agriculturists. doi:10.46481/jnsps.2022.883 keywords: analytical hierarchy process, yam production, nigeria, ndvi, yield gap, multi-criterion decision making article history : received: 22 june 2022 received in revised form: 10 october 2022 accepted for publication: 15 october 2022 published: 06 november 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: o. j. oluwadare 1. introduction the united nations world food programme’s (unwfp) live hunger map identified about 860 million individuals with insufficient food consumption and a total of 16 countries with very high levels of hunger. nigeria accounts for 52.9 million (or 6.2%) of the global population with insufficient food consumption [1]. laganda [2] asserts that of the causes of world hunger, climate change is the one that can be most accurately predicted ∗corresponding author tel. no: +234 8036952266 email address: tertsea.igbawua@uam.edu.ng (tertsea igbawua) using scientific tools. it is therefore imperative that scientific knowledge is employed in resolving the global food crisis, beginning with national and/or regional studies, especially for the food crops that thrive in specific regions of the world, such as yam. the emergence of agricultural methods came out as a result of man’s quest for food in order to survive, and the agricultural methods have occasionally continued to advance [3]. yam is a tropical tuber crop that is grown primarily for domestic consumption and commercial use in africa. generally, there are several species of yam in the genus dioscorea, but only six are proven to be economically important staple species. 1 igbawua et al. / j. nig. soc. phys. sci. 4 (2022) 883 2 according to blench [4], these include dioscorea rotundata, dioscorea bulbifera, dioscorea dumetorum, dioscorea alata, dioscorea esculenta, dioscorea cayennensis, also known as white guinea, aerial, trifoliate, purple, chinese, and yellow yam, respectively. awoniyi & omonona [5], ike & inoni [6], confirmed that the most prevalent and economically significant species in nigeria are dioscorea rotundata and dioscorea alata. in 2012 and 2019, nigeria produced approximately 65.00 [7] and 67.34% [8] of the world’s yams, which were cultivated on 2.90 [7] and 6.24 million hectares [8], respectively, and were worth billions of dollars. in addition to its value as a staple crop, in nigeria, yams are a major source of income for most individuals and have tremendous cultural importance because they are employed in numerous religious and cultural rituals. every year, festivals are held in various cultures to mark the harvest season of yams. given the current financial crisis nigeria is facing, upscaling yam production for large scale export can become a major export earner, which will in turn ease pressure on the heavily import-dependent economy. despite being the world’s top producer of yams, there are many factors that contribute to the food crisis in nigeria. they include: changing climatic variables; shrinkage in size of farm lands due to population increase; poor access to fertilizers and improved varieties; conservative approach to agricultural extension programs and services; obstructive market forces; poor terrain and soil quality; farmers’ limited financial resources; disease and pests; and improper timing of planting crops [9–14]. crop production is influenced by meteorological variables, hence it is important to understand these variables and how they interact in order to determine how they together affect agricultural productivity [15]. currently, yam is one of the most expensive food items in nigeria, yet it is also intensely sought after. in addition to improving foreign earnings from exports, the improved production and availability of yams in nigeria would significantly improve access to food for many nigerians, as the nation strives towards attaining the sustainable development goal (sdg-2) which aims to eradicate hunger, establish food security and enhanced nutrition, and advance sustainable agriculture. these sdgs can be used as tools for implementing measures that can be used for minimizing the effects of climate change that pose a serious risk to the entire ecosystem [16]. several studies on agricultural land suitability for production of different crops across nigeria have been reported by different researchers. for instance, ikusemoran and hajjatu [17] studied site suitability for yam, rice and cotton production in adamawa state, nigeria. in another study, kunda and jajere [18], worked on suitability analysis for agricultural lands in nasarawa state, nigeria. ahmed and jeb [19], researched on suitability analysis for cultivation of sorghum in kano state, nigeria. zemba et al. [20] reported on suitability analysis for decision-making in cassava cultivation in adamawa state, nigeria. ujoh et al. [14], worked on suitability analysis for the production of rice in benue state, nigeria. ajala et al. [21] studied soil suitability analysis for cassava production in kwara state, nigeria. in rivers state, ondo, katsina and oya states, peter and aumwerri [22], tenkpa and balogun [23], abdullahi et al. [24] and ayaode [25] evaluated agricultural lands for the production of citrus, cocoa, millet and rice respectively using gis, ahp or a combination of both. generally, factors affecting land suitability do not have an identical level of significance. hence, numerous factors must be taken into account in order to determine if a particular area of land is suitable for agricultural production or not. the weights (influence) of these criteria and the scores of the sub-criteria were calculated using a variety of ways [26]. the objective of the current study was to ascertain the agricultural areas suitable for yam production using ahp and the gis technique. the ahp technique is one of the multi-criteria decision-making methods that are frequently employed in agricultural land use suitability analyses [26]. yam production is irregular across nigeria. an understanding of the areas suitable for yam production would therefore become a vital starting point for addressing the insufficiency of yam production in nigeria. previous similar studies conducted (on yam specifically) in nigeria [4-5,27] either focused on some sections of the country rather than a national spatial scope, and/or did not adopt the use of geo-spatial tools, thereby suggesting a critical knowledge gap that is most likely a major contributory factor in the relatively low yam production levels across nigeria. this study therefore closes the identified knowledge gap by adopting geospatial tools in analyzing key weather and soil parameters across all sections of nigeria. 2. materials and methods 2.1. study area nigeria is located between 4◦ and 14◦ latitude and 3◦ and 15◦ longitude (figure 1). the swamp forest, tropical rainforest, guinea savannah, sudan savannah, and sahel savannah are among the vegetation types found on the 923,769 square kilometers of land [28]. the dry and wet seasons are the two basic seasons in nigeria. a dusty air flux from the sahara desert known as harmattan characterizes the dry season, while an air flux from the south atlantic ocean characterizes the rainy season [29]. projected increases in surface temperature, variable precipitation, land and vegetation degradation, rises in sea level and flooding, drought and desertification, increased frequency of severe weather events, changes in fresh water resources, and the loss of biodiversity are all signs that nigeria’s climate system is changing [30-31]. nigeria has a tropical climate in general, but there are many regions within the country that have different climate conditions. mangrove forests, swamps, and hot, humid temperatures can be found along the southern ocean coast, particularly along the niger delta, resulting in extremely wet conditions. the niger and benue river valleys meet in the inland area of nigeria, which is the country’s largest and most expansive area, with a diverse flora. 2.2. data and methods 2.2.1. data the data records used in this research work include: climate, soil, and environmental datasets. the climate data comprised of precipitation and temperature variables was obtained 2 igbawua et al. / j. nig. soc. phys. sci. 4 (2022) 883 3 figure 1. study area from the climate research unit (cru), and relative humidity from the european center for medium range weather forecast (ecmwf) version 5, reanalysis data (era-5). the soil data set is comprised of ph, cation exchange capacity (cec), and soil organic carbon (soc). land use and land cover data were obtained from the european space agency (esa). the slope data was retrieved from the shuttle radar topography mission (srtm) digital elevation model (dem). all the various data sources used in this work are shown in table 1. 2.2.2. analytical hierarchy process (ahp) the ahp method was used to develop a yam suitability map for large scale yam production. the ahp method, documented by saaty [32], is a multi-criteria decision-making (mcd) technique for resolving complex issues [33-34]. figure 2 illustrates the procedure used in modeling the suitability of yam production in nigeria. the steps are (1) identification and assessment of criteria and sub-criteria classes; (2) determination of a geography information system (gis) database with the criteria and sub-criteria, with each criterion’s relevance being weighted and obtained using the mcd-ahp technique; (3) determination of the final suitability map for yam production from the weighted overlay process in arcgis; and (4) evaluation of vegetation and climate trends in the various suitability classes. following the preparation of the thematic layers (steps 1 to 2), the ahp process was used to assign weights to all of the criteria considered in this study. the criteria used in the work are important for yam growth and development. they include temperature, precipitation, relative humidity (rh), ph, slope, soil organic carbon (soc), land cover, and cation exchange capacity (cec) presented in raster format. prior to creating the pairwise matrix, reclassification was figure 2. research flowchart. cec refers to cations exchange capacity and ph refers to potency of hydrogen used to standardize the layers of each sub-criterion. a codification of yam suitability options was then built using thresholds found in related literature, as illustrated in table 1. the sub criteria were reclassified as ”hs”, ”s”, ”ms”, and ”ns”, representing highly suitable, suitable, moderately suitable, and not suitable regions respectively. the weighting of these influencing factors is based on factors that favor yam growth and expert opinion. a parameter with a high weight indicates a layer that will have a significant impact on yam production and vice versa. the relative importance of the values used for each parameter was assigned based on saaty’s scale of preference (1–9). additionally, the weights were assigned by reviewing earlier research and field experience. table 2 provides the saaty’s relative relevance rating [32]. a pairwise wise comparison matrix will be formed which will show the thematic layers’ allocated weight and rank. to estimate the weights, the eigenvector method is used to determine the largest eigenvalue [35]. the pairwise comparison matrix (pcm) b of m criteria constructed based on saaty’s scale of importance (table 2), which is of the order (m × m), is the basis 3 igbawua et al. / j. nig. soc. phys. sci. 4 (2022) 883 4 table 1. parameters, data sources and their suitability classification parameter data source data range s1 s2 s3 ns temp. (oc) http://data.ceda.ac.uk/badc/cru/data/cru ts/cru ts 4.00/data/ 20-30 30-34 34-35 <20 >35 prec. (mm/yr) http://data.ceda.ac.uk/badc/cru/data/cru ts/cru ts 4.00/data/ 1000-2000 2000-3500 >3500 ph http://dx.doi.org/10.3334/ornldaac/1247 5-7.4 7.4-7.8 5.1-5.6 <5.1 >7.8 rel. humidity (%) http://cds.climate.copernicus.eu high mid low cec (cmol/kg) http://dx.doi.org/10.3334/ornldaac/1247 17-23 10-16 5-10 <5 land cover (esa sentinel) http://www.esa-landcover-cci.org/ croplands, tree cover mosaic natural vegetation grasslands, sparse vegetation, shrubs, herbs urban, built, water slope (%) http://srtm.csi.cgiar.org/ 0-4 4-8 8-20 >20 soc (g/kg) http://dx.doi.org/10.3334/ornldaac/1247 >50 30-50 0-30 for the factors influencing the weights [36]. b =  b11 b12 . . . . . . . . . . b1m b21 b22 . . . . . . . . . . b2m bm1 bm2 . . . . . . . . . . bmm  = (bi j)m×m (1) similarly, b = [ bi j ] , i, j = 1, 2, 3, . . . . . . .m; in general, the matrix b has the properties of reciprocality and consistency, expressed as bi j = 1/bi j(i , j) and bi j = bik/b jk for any i, j and k (2) the weight of all the parameters was obtained using the pairwise comparison matrix and the maximum characteristic root was determined as b∗v∗ = λmaxv ∗ (3) where, matrix b∗ represents the estimate of b [35], v∗ the priority vector and λmax the eigen value for b. the consistency ratio (cr) is a criterion for judging a decision made at random [37]; it is given by cr = ci/ri (4) where, ci is the consistency index while ri is the random consistency index which depends on the order n of the matrix [32]. ci is given by ci = λmax − 1/m − 1 (5) table 2. saaty’s scale of importance variable preferences 1 equal importance 3 moderate importance 5 strong importance 7 very strong importance 9 extreme importance 2, 4, 6, 8 intermediate values between adjacent values 2.2.3. validation of results the gridded advanced very high resolution radiometer (avhrr) third generation normalized difference vegetation index (ndvi3g) of the global inventory modeling and mapping studies (gimms) with a temporal resolution of 15 days was utilized for validation of the derived suitability classes [38, 39]. the ndvi spans from +1.0 to -1.0 and in locations with barren rock, sand, or snow (< 0.1), values are often relatively low. senescent crops and sparse vegetation like shrubs and grasslands (approx. 0.2 to 0.5) might contribute to moderate vegetation. high ndvi values (approx. 0.6 to 0.9) are correlated with dense vegetation, such as that seen in temperate and tropical forests, or with crops in their prime developmental stage [40]. ndvi is given [41] by n dv i = bn ir − bred bn ir + bred (6) where bn ir and bred represents the near-infrared and red bands respectively. 2.2.4. pearson correlation coefficient (r) pearson correlation analysis was employed to ascertain the link between vegetation and climate parameters in the various 4 igbawua et al. / j. nig. soc. phys. sci. 4 (2022) 883 5 suitability classes. r = ∑m k=1 ( y j − y ) (k j − k)√∑m k=1 (y j − y) 2 . ∑m k=1 (k j − k) 2 (7) where y and k are the variables and m refers to the length of the variables. 3. results and discussion 3.1. spatial distribution of variables figure 3 depicts the studied maps of (a) mean temperature (2m) from 1985 to 2015, with values ranging from 20.3 to 31.1◦ c, (b) annual mean precipitation, ranging from 286.7 to 2055.4 mm/yr, (c) relative humidity, ranging from 40.7 to 91.2 %, (d) cec from 0.0 to 85.0 cmol/kg, (e) soc from 0.0 to 4.2, (f) ph from 4.3 to 8.5, (g) various land cover classes, and (h) slope (% rise) from 0.0 to 45.0. high temperature values are most prevalent in the northern part of the study area. relative humidity and precipitation, on the other hand, are more widely distributed in the southern part of the research area. the spatial maps of cec, soc and ph revealed variable distributions across the study area. the slope map displayed in figure 3 indicates the highest values around the mambilla and jos plateaus, while low values are scattered around the niger-benue trough and niger delta area. in figure 4, the examined parameters (figure 3) are used independently for the evaluation of suitable areas for yam production. each parameter yields a varying degree of suitability for yam cultivation in nigeria based on the suitability groups or classes: highly suitable (hs), suitable (s), moderately suitable (ms) and not suitable (ns). the yam is a tropical root tuber and vine that is intolerant of cold. temperatures between 25 and 30◦c are required for yam tuber growth, while temperatures below 20◦c stifle growth [42, 43]. high temperatures exceeding 35 ◦c, according to adifon et al. [44], are detrimental to yam growth and development and may cause fluctuations in moisture content, which may have an influence on agricultural activities if the plants have adaptation problems [15]. the majority of yam varieties require an average of 7 to 9 months of growing time before harvesting. to fully exploit the yam’s production potential, it requires over 1,500 mm [42, 43] of annual precipitation distributed evenly over the vegetation period. in contrast, high precipitation amounts in excess of 3500 mm are not too good for yam cultivation and may affect its production. as a result, a long rainy season during the growing season has a positive impact on yam yield, especially if precipitation is well distributed during the wet season with adequate humidity. as a key weather factor that affects the amount of moisture in the atmosphere, humidity has a considerable impact on agricultural output [15]. seasons and regions in the northern section of the country with abrupt heavy precipitation amounts and distributed within 2 to 3 months may not be favorable for yam production. after the few months of heavy precipitation, the remaining months may be too dry for the yam, except in areas where irrigation measures are put in place. arguably, the plant can withstand longer periods of drought, but this reduces the yield significantly. in terms of storage, low humidity and dry conditions are necessary for long-term storage to avoid damage to the yields. yams also require red loam soils with a ph range of 5.0 to 7.0 [45]. the sandy clay loams are the best soils for growing yams. in nigeria, yams can be grown in almost any soil type if the ridges are raised properly and stuffed with organic matter. the organic material in the soil enhances the soil structure, works as a steady source of manure, and promotes high yam tuber production. other critical requirements for yam growth include sufficient drainage, adequate aeration, crumbly soil, adequate moisture supply, and sticks or tiny trees to serve as stacks for the leaves to creep and develop properly without interference. the gradient (slope) of the research area is very important in determining the pattern of land use, which has a direct impact on groundwater flow in any area [45]. a research work by poesen [46] reported that as the slope angle increased, so did the rate of infiltration [47]. the slope of an area influences surface water infiltration. in a given region, groundwater infiltration at the surface may not occur at the same gradient. areas with gentle slopes experience the highest infiltration and minimal amount of surface water runoff, whereas regions with steep slopes may experience high runoff and minimal infiltration of surface water [34]. 3.2. determination of pairwise matrix and integration of thematic layers using equations 1 to 5 and table 2, a pairwise comparison of all the thematic layers (rasters) was performed in order to create a pcm. the sub-classes (sub-criteria) of each layer rank were assigned scores from 1 to 9 in relation to their influence on yam production. the pairwise matrix is given in table 3. the results show that temperature had the highest influence, followed by precipitation and land cover, with weights of 30.9, 24.7, and 15.9, respectively. each thematic layer and their associated attribute classes were assigned weights and ratings before being integrated (overlayed) in arcgis software using the spatial analyst tool box. the final overlayed map is given in figure 5. during the pairwise comparison matrix via the mcd analysis, the consistency index was obtained as 7.2%, which indicates a fair judgment. the highly suitable (hs) areas are mainly scattered in the south and north-central areas, while the marginally suitable (ms) areas are mostly located in the far north. arguably, the whole country is suitable (s) for yam production, with most of the pixels scattered across the study area. a few areas that are not suitable (ns) for yam production comprise bare land (open fields), water, and urban areas. figure 6 shows the percentage composition of landmass for each suitability class. the result indicates that hs, s, ms, and ns areas account for 11.79, 82.68, 4.05, and 1.47% of the study area, respectively. hs areas are primarily found in the north-central and southern regions, while s areas are scattered all over the study area from north to south. meanwhile, ms areas are majorly observed in the country’s extreme north. ns areas are comprised of wetlands, bare areas, open spaces, and urban centers. 5 igbawua et al. / j. nig. soc. phys. sci. 4 (2022) 883 6 table 3. pcm of all the variables relevant to yam crop suitability in nigeria temp pre lcd cec soc ph r.hum slp weight temp 1.000 2.000 3.000 5.000 4.000 6.000 5.000 7.000 30.9 pre 0.500 1.000 3.000 4.000 4.000 5.000 5.000 7.000 24.7 lcd 0.333 0.333 1.000 3.000 3.000 5.000 4.000 6.000 15.9 cec 0.200 0.250 0.333 1.000 3.000 4.000 3.000 3.000 10.0 soc 0.250 0.250 0.333 0.333 1.000 4.000 3.000 5.000 8.1 ph 0.167 0.200 0.200 0.250 0.250 1.000 3.000 3.000 4.5 r.hum 0.200 0.200 0.250 0.333 0.333 0.333 1.000 1.000 3.2 slp 0.143 0.143 0.167 0.333 0.200 0.333 1.000 1.000 2.7 figure 3. maps of (a) mean temperature (oc), (b) annual mean precipitation (mm), (c) relative humidity (%) from 1985-2015. (d) cations exchange capacity (cmol/ kg), (e) soil organic carbon (%) (f) land cover, (g) slope (%). cl is for cropland, mnv mosaic natural vegetation, shrb shrubs, spv sparse vegetation, bl bare land, hbc herbaceous, tc tree cover, gl grassland, ubn urban and wt water 3.3. relationship between ndvi and climate parameters in various suitability classes according to nwankwo and ukhurebor [3], monitoring climate variability is crucial because of the functions it plays in the figure 4. raster map showing suitability criteria for yam cultivation nigeria, constraint (a) mean temperature, (b) mean precipitation, (c) relative humidity (d) cations exchange capacity, (e) soil organic carbon, (f) land cover, (g) slope agricultural system. in this work, the interaction between ndvi and climate variables was determined in the various suitability classes in nigeria. even though there are many vegetation indices used in studying vegetation activity, ndvi is one of the most extensively used. the ndvi index measures the green6 igbawua et al. / j. nig. soc. phys. sci. 4 (2022) 883 7 figure 5. map of land suitability analysis for yam field cultivation based on multi-criterion decision method figure 6. pie chart showing percentage composition of the suitability classes ness of vegetation over a given area. it can be used as a proxy for vegetation productivity [41, 48] and phenology [40]. the ndvi across the seasons in hs, s, and ms suitability classes was extracted and presented in figure 7. the results show that ndvi is above average (>0.5) from apr to dec for the hs class, and from jun to nov for the s class. in contrast, for the ms class, ndvi is below average (<0.5) for all the months. similarly, precipitation is > 100 mm/month between april and october for the hs class, between may and september for the s class, and between june and july for the ms class. temperatures are higher during the start of the season (apr-may) and lower in jul-aug for all classes. ndvi and precipitation are bimodal in hs suitability (figure 7). in contrast, ndvi and precipitation are unimodal in s and ms. also, temperatures are bimodal in all the suitability classes, with peaks in the mam and son seasons. meanwhile, ndvi for the hs class has peaks in mam and son while a single peak exists in jja for the s and ms classes. overall, the ndvi and precipitation range in the hs class is higher, followed by the s class, and then the ms class. in contrast, the temperature range in hs is lower, followed by s class, and then ms class. this shows that there is abundant precipitation in the hs class and temperature remains the limiting climatic variable. in contrast, temperature (precipitation) is normally high (low) in the ms class. meanwhile, the s class has moderate precipitation and temperatures, which are just sufficient for yam production. our method shows that there was consistency in the mcd analysis during the assignment of weights. given that temperature and rainfall are directly related to how biologically plants grow, there is a strong association between agricultural productivity and these weather factors [15]. figure 8, 9, and 10 show the scatterplot matrix of ndvi, precipitation, and temperature for the various suitability classes. the results show that in the hs class (figure 8), ndvi is positively and significantly correlated with precipitation, r = 0.552 (p < 0.01), and ndvi is negatively and significantly correlated with temperature, r = -0.254 (p < 0.01). meanwhile, precipitation correlates negatively and insignificantly with temperature in the hs class, r = -0.429 (p < 0.01). in s class (figure 9), ndvi was positively and significantly correlated with precipitation, r = 0.702 (p < 0.01); ndvi was negatively and significantly correlated with temperature, r = -0.161 (p < 0.01). meanwhile, precipitation correlates negatively and insignificantly with temperature in the s class, r = -0.048 (p = 0.325). also, in the ms class (figure 10), ndvi is positively and significantly correlated with precipitation, r = 0.685 (p < 0.01), and ndvi is negatively and significantly correlated with temperature, r = -0.129 (p < 0.01). meanwhile, precipitation correlates positively and insignificantly with temperature in the hs class, r = 0.068 (p = 0.165). the ndvi trends in the hs, s, and ms classes are 0.000228, 0.000164, and 0.000037 ndvi/month, respectively. for precipitation, the trends in the hs, s, and ms classes are -0.002459, 0.015347, and 0.0355287 mm/month, respectively. in the hs, s, and ms classes, the temperature trends are 0.000944, 0.001344, and 0.001761 ◦c/month, respectively. specifically, ndvi increases with an increase in precipitation at different lags (1 to 2 months) for all the suitability classes. in contrast, ndvi decreases with an increase in temperature for all the suitability classes. the best agreement between ndvi and temperature is at the start (mam) and end (son) of the season. the work shows that nigeria’s climate system is primarily influenced by precipitation, followed by temperature, which is in agreement with a report by igbawua et al. [49]. 3.4. yam production and yield gap in nigeria as earlier mentioned, awoniyi & omonona [5], ike & inoni [6] noted that dioscorea rotundata (white yam) and dioscorea alata (water yam) are the major yam species cultivated in nigeria. over the years, benue, taraba, nassarawa, enugu, kaduna, cross river, ondo, ekiti, kogi, and niger [50]. from figure 5 and 6, the total area of land in hs, s, and ms available for yam production is 107,059.4474, 746,718.8272, and 33,074.9333 km2 respectively, which represents 10,705,944.74, 74,671,882.72 and 3,307,493.33 hectares respectively. according to verter and bečvářová [7], the yield rate was between 20–50 tons/hectare in years with bumper harvests, while about 10 tons/hectare was in years with poor harvests. to get the yield gap, we adopted the average yield in years of poor harvest (to account for worst case scenarios) to compute the expected yield in hs, s, and ms. the expected yield for ms, s, and hs using the yield rate of 10 tons/hectare indicates 7 igbawua et al. / j. nig. soc. phys. sci. 4 (2022) 883 8 figure 7. seasonal ndvi for hs, s and ms suitability classes figure 8. scatter plot matrix for ndvi and climate parameters for hs region figure 9. scatter plot matrix for ndvi and climate parameters for s region 107.06, 746.72, and 33.07 million tons per hectare in the study area. the total area comprising hs, s, and ms is about 886.85 figure 10. scatter plot matrix for ndvi and climate parameters for ms region million tons. the expected yield is far above the actual yield of 38.00 [7] and 50.05 [8] million tons for 2012 and 2019, respectively. the yield gap between the actual and expected yield is more than 800 million tons. in general, the expected yield is not always achieved due to a variety of factors such as declining soil fertility, poor planting methods, high labor costs, increase in pest [51], crop rotation, climate change, floods, droughts, and persistent displacement of farmers from their settlements. this is in addition to other factors such as increasing population, which is driving land fragmentation, and declining soil productivity as a result of shorter fallow periods and prolonged chemical application (synthetic fertilizer and pesticides/herbicides). also, the expected yield can never be achieved because most of the fertile lands are yet to be ploughed and they are covered by other land cover types. 4. conclusion yam, arguably one of the main food items in homes across nigeria, is not cultivated all year round. it is heavily dependent and reliant on rain during the wet season and is, therefore, a mono-seasonal crop. a suitability analysis for yam production in nigeria was done to determine suitable regions for yam cultivation, labeled as highly suitable (hs), suitable (s), marginally suitable (ms) and not suitable (ns). the approach uses both subjective and objective techniques to assign weights to variables through the gis and ahp methods. the variables were later overlayed into a single suitability map in arcgis 10.5. given the high potential of yam production in nigeria, it is essential to determine the suitable regions for yam production over the study area to improve food security and maximize foreign exchange. our findings show that hs, s, ms, and ms were 11.79, 82.68, 4.05, and 1.47 % of the study area. further analysis showed that vegetation vigor (activity) in the hs regions was higher, followed by s regions, and then ms regions. similarly, climate variables in hs regions were more favorable for plant growth, followed by s regions and ms regions. this has 8 igbawua et al. / j. nig. soc. phys. sci. 4 (2022) 883 9 been confirmed by pearson correlation analysis between vegetation (ndvi) and climate parameters in the various suitability classes. our findings also show that there is a high yield gap in yam production over the study area. the yam crop is not the only crop that is produced on the land that is suitable for its production. thus, yam production is not maximized over the study area. other factors include crude agricultural practices, multiple cropping, floods, climate change, drought, urbanization, and population increase. conclusively, this approach is very useful in land suitability assessment. the methods and materials are very cheap and readily available. the results from this work can serve as a guide for future yield projections in yam production by policymakers and agriculturists. understanding the lands best suited for yam planting would reduce wastage of resources increase productivity, which will help improve food security. the optimization of yam yield in nigeria will go a long way in improving local consumption and foreign exchange. acknowledgment the authors wish to appreciate the editor and the reviewers for their valuable comments that improved the quality of this study. references [1] united nations world food programme (unwfp) live hunger map. available online at: https://hungermap.wfp.org/ accessed: 13 june 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[51] b. koigi, international institute of tropical agriculture (iita) researchers achieve 23 tonnes per hectare yield for yam in nigeria, africa agribusiness, available online on 2020 at 2022 from https://africabusinesscommunities.com/agribusiness/news/iitaresearchersaccessed on 16th june. 10 j. nig. soc. phys. sci. 4 (2022) 911 journal of the nigerian society of physical sciences performance evaluation and statistical analysis of solar energy modeling: a review and case study samy a. khalil∗ solar and space department, national research institute of astronomy and geophysics (nriag), 11421, helwan, cairo, egypt abstract the main target of this research is a quantitative review of literature on global solar radiation (gsr) models available for different stations around the world. the statistical analysis of 400 existing sunshine-based gsr models on a horizontal surface is compared using 40-year meteorological data in the selected locations in egypt. the measured data is divided into two sets. the first sub-data set from 1980 to 2019 was used to develop empirical correlation models between the monthly average daily global solar radiation fraction (h/h0) and the monthly average of desired meteorological parameters. the second sub-data set from 2015–2019 was used to validate and evaluate the derived models and correlations. the developed models were compared with each other and with the experimental data of the second subset based on the statistical error indicators such as rmse, mbe, mabe, mpe, and correlation coefficient (r). the statistical test of the correlation, coefficient (r), for all models gives very good results (above 0.92). the smallest values of t-test occur around the models (m 272, m 261, m 251, and m 238). the accuracy of each model is tested using ten different statistical indicator tests. the global performance indicator (gpi) is used to rank the selected gsr models. according to the results, the rietveld model (model 272) has shown the best capability to predict the gsr on horizontal surfaces, followed by the katiyar et al. model (model 251) and the aras et al. model (model 261). doi:10.46481/jnsps.2022.911 keywords: solar energy, models, statistical indicators, performance and sunshine duration. article history : received: 02 july 2022 received in revised form: 26 august 2022 accepted for publication: 30 august 2022 published: 30 september 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: b. j. falaye 1. introduction solar energy has remained the most dependable source of energy capable of supporting and maintaining all of the activities and cycles that support life on the planet for plants, creatures, and other material substances. solar energy is generally acknowledged as a key energy hotspot for the future all over the ∗corresponding author tel. no: +20225560645, fax: +20225548020 email address: samykhalil2014@gmail.com; samynki@yahoo.com (samy a. khalil ) planet due to the natural issues related to fossil fuels as well as their restricted stores. different wellsprings of non-sustainable power cause perceptible ecological risks. sustainable power sources, for example, geothermal, wind, sun-based and flowing, are harmless to the ecosystem since they have a lot lower natural effect than regular sources like petroleum derivatives. as a result, solar energy is widely regarded as the most promising and reasonable type of energy capable of resolving the natural issues that humanity will face in the future [1-3]. sustainable power sources, for example, solar energy, can possibly moderate a few negative ecological issues, in1 khalil / j. nig. soc. phys. sci. 4 (2022) 911 2 cluding environmental change brought about by concentrated petroleum derivative abuse. with quick mechanical advancement and decreasing costs, solar energy will play an important role in future energy frameworks. in the forecast, study, and planning of solar energy frameworks, information on solar radiation and its parts in a specific area is extremely fundamental. global solar radiation on a level surface is the most fundamental piece of information used in the planning and forecasting of the display of a solar energy device at a specific location. sunbased light can be assessed by handling pictures from satellites or by on-ground estimations with pyrometers in meteorological stations. satellite radiation estimations are not exact in assessing ground solar power radiation since they require environmental models to gauge the solar radiation at ground level [4-8]. solar radiation showing up on earth is the most basic and sustainable power source in nature. solar energy, brilliant light, and people since antiquated times, utilizing a scope of truly advancing advancements, have bridled hotness from the sun. the energy from the sun could play a key part in de-carbonizing the worldwide economy through enhancements in energy efficiency and forcing costs on ozone-harming substance producers. in solar energy research, information on sun-powered radiation and its components in a given area is a significant contribution for sun-based energy applications such as photovoltaic, sun-based warm frameworks, sun-based heaters, and aloof sun-based plans. the information ought to be dependable and promptly accessible for plan, advancement, and execution assessment of nearby planet groups for a specific area. the most effective way to decide how much worldwide sunlight-based radiation at any site is to introduce estimating instruments, for example, pyranometer and pyrheliometer, at that specific spot and to screen and record its everyday recording, which is actually an extremely drawn-out and expensive activity [5]. regardless of the significance of sunlight-based radiation estimations, in many developing countries this data is not immediately accessible due to a lack of the ability to bear the cost of the estimating equipment and procedures involved. hence, various relationships and techniques have been created to assess every day or monthly worldwide solar power radiation in view of the more promptly accessible meteorological information at a larger part of weather conditions stations. experimental models that have been utilize to work out sunoriented radiation are normally find on the accompanying elements; astronomical elements (sun based steady, earth-sun distance, sun based declination and hour point), geographical variables (scope, longitude and height of the site), geometrical elements (azimuth point of the surface, slant point of the surface, sun rise point, sun azimuth point), physical elements (dissipating of air particles, water fume content, dispersing of residue and other climatic constituents like o2, n2, co2, o and so forth) and meteorological elements (extraterrestrial sunlight based radiation, sun-sparkle term, temperature, precipitation, relative stickiness, impacts of shadiness, soil temperature, vanishing, reflection of the environs, and so on) [5]. the number of connections that have been distributed and attempted to assess the monthly average global solar radiation is generally high, making it difficult to select the most appropriate technique for a specific reason and site. choosing a suitable technique from different existing models depends on their information prerequisites and model exactness. forecasts of monthly daily worldwide sunlight-based radiation for a large number of areas are being introduced in various research works. it is essential that every one of the proposed models contain exact constants, which depend upon the season and the topographical area of a specific spot. the models examined in this review article are sequentially introduced and thus valuable for choosing the suitable model to assess worldwide sun-based radiation for a specific spot of interest [5-7]. the first model to predict the worldwide solar powered radiation was developed by angstrom, which relates the normal monthly worldwide solar powered radiation to crisp morning radiation and to the normal part of conceivable daylight hours [9]. prescott [10] modified the angstrom model. from that point on, numerous scientists have modified the angstrom model by utilizing different boundaries, for example, daylight hours, minimum temperature, mean temperature, maximum temperature, relative moistness, elevation, scope, longitude, overcast cover, and so on, or blends of the boundaries. nevertheless, the most broadly utilized boundary to date is the daylight hour. iziomon and mayer [11] have presumed that the presentation of the daylight-based model is far superior to the meteorological boundary-based models. the most commonly used boundary up to this point is daylight length because it is accessible in many areas and can be effectively and consistently estimated [12, 13]. when the first request angstrom, type relationship is examined, the second and third requests do not improve the precision to a significant level [14, 15]. in addition, almorox and hontoria [16] proposed the direct, quadratic, third degree, and the logarithmic relapse models for spain and inferred that the third request gave the general best outcome, however, re-complimented to utilize the straight relapse model since the blunder between the straight and third request relapse models is extremely small. angstrom’s worldwide sunlightbased radiation model is one of the pioneering works in the field of solar radiation. in the long term, it tends to be seen that numerous scientists have generally utilized this relationship by adjusting coefficients for diversified areas (like [17-42] and some more). numerous scientists have modified the angstrom model by utilizing different boundaries like daylight hours, scope, longitude, and height. glover and mccullum [17] have proposed a relapse model that considers the effect of scope and reasoned that the worth of “a” will be an element of scope, though the worth of “b” is effectively consistent. correspondingly, [42] was proposed for the relapse condition as far as scope and daylight hours, utilizing the metrological information of 40 areas (37 areas from pakistan and 3 areas from india). hussain et al. [19] have proposed a relapse model considering the scope effect, utilizing the meteorological information of 40 stations all over the planet, and presumed that alongside the elevation effect, the half-yearly apportionment of the year ought to be utilized. [20] used meteorological data from 40 different loca2 khalil / j. nig. soc. phys. sci. 4 (2022) 911 3 tions around the world as a component of scope, height, and the proportion of daylight hours, and concluded that the proposed model can be used in any region of the world with a 10% error. for the kingdom of saudi arabia [43], the relapse model is a component of daylight length, scope, longitude, and elevation. jin it et al. [21] proposed higher request relapse models based on meteorological data from 69 locations in china. a comparative review was finished by [22] for 86 different areas across china and proposed the higher request relapse model regarding scope, longitude and elevation. as of late, gadiwala et al. [23], onyango, and ongoma [24] have proposed the relapse model for pakistan and kenya separately. bakirci [25] proposed the direct, quadratic, and third request relapse models for the eastern anatolia region (ear) of turkey and inferred that the third request polynomial condition gives the best outcome. the temperature is based on quansah et al. [26] for the ashanti region of ghana, where the deliberate and determined values show excellent concurrence with one another. likewise, the constants “a” and “b” are sitespecific. yaniktepe and genc [27] have proposed a straight, quadratic, and cubic relapse model for osmaniye. the exhibition of the proposed model was contrasted with nine different models left in writing utilizing measurable tests (mape, mabe, and rmse). it was inferred that the cubic relapse model performs better when contrasted with direct and quadratic models. namrata et al. [28] proposed a straight relapse model for three different areas in jharkhand, india. the exhibition of the proposed model has contrasted with the deliberate information and the relapse model proposed by rietveld [29], ogleman [30], akinoglu [31], glover [17] and gopinathan [20]. it was inferred that the relapse consistent “a” and “b” does not shift with height and scope. it was seen that the amount of relapse consistent is practically the same for the three different urban areas (for example, jamshedpur 0.717, ranchi 0.700, and bokaro 0.714). das et al. [32] chose eleven nonlinear and six straight models accessible in writing to assess worldwide sunlight-based radiation over south korea. it was presumed that the nonlinear models perform far superior to the straight models, likewise utilizing the normal kriging strategy solar maps have plotted for south korea. using nine years’ worth of meteorological data hejase and assi [33] fostered a direct, quadratic, cubic, dramatic, log-straight, and logarithmic model for the united arab emirates and presumed that the cubic model gives minimal blunders among the whole proposed model. the main objective of this paper is to evaluate and perform extensive global solar radiation (gsr) models available in the present research by using the statistical indicators and the most accurate models chosen in this study, and apply these models for case study using the available meteorological parameters 2. methodology the global solar radiation models are classified according to the basis of the input parameters they employ in correlating with the clearness index. the clearness index (kt) shows the proportional depletion via the sky of the incoming photovoltaic radiation and consequently offers the stage of availability of photovoltaic radiation and adjustments in the atmospheric situation in a given environment [34]. mathematically, the clearness index is the ratio of horizontal global solar radiation to the extraterrestrial solar radiation (ho) on a monthly basis, principally calculated theoretically as given by [35]. h0 = (24/π)i sc[1 + 0.033 cos(360n/365)] ×[cosϕ cos δ sin ωs + (2πωs/360) sin ϕ sin δ (1) is c is the solar constant, (ϕ) is the latitude of the location, (δ) is the solar declination, (ωs) is the mean sunrise hour angle for the given month and n is the number of days of the year starting from firstly january. the solar declination (deltaup) and mean sunrise hour angle (omegaups) for a given month can be calculated using equations 2 and 3, respectively. δ = 23.45 sin[360(n + 284)/365] (2) ωs = cos−1(− tan ϕ tan δ) (3) it has been establish that world photovoltaic radiation is exceptionally affect by means of meteorological parameters, astronomical factors, geographical factors, and geometrical factors. this may want to be attribute to the strong point of nearby local weather in figuring out the meteorological and atmospheric parameters that exceptional match a given locality. this additionally relies upon on the availability of enter meteorological/ atmospheric parameter(s) that a given radiometric station or an person is successful of measuring robotically which in the end grew to become out to be the great enter parameter at the disposal of the researcher for predicting world photo voltaic radiation in that place [36-39]. 3. statistical indicators the accuracy and performance of the derived correlations in predicting global solar radiation were evaluated because of the following statistical error tests; mean bias error (mbe), root mean square error (rmse), mean percentage error (mpe), maximum absolute relative error (mare), mean absolute error (mae), root mean square relative error (rmsre), coefficient of determination (r2) and t-test statistic. these error indices are defined as [36]: mbe = 1 n n∑ i =1(hi,m−hi,c) (4) rmsr = √√√ 1 n n∑ i=1 (hi,m−hi,c) 2 (5) mpe = 1 n n∑ i=1 ( hi,m−hi,c hi,m )x100 (6) mare = max (∣∣∣∣∣∣ (hi,m−hi,chi,m ∣∣∣∣∣∣ ) (7) mae = 1 n n∑ i=1 |hi,m−hi,c| (8) 3 khalil / j. nig. soc. phys. sci. 4 (2022) 911 4 table 1: models of gsr used in the present research for calculation of monthly average daily of gsr on horizontal surface. no. of model modeling of gsr author/references 1 h/h0 = 0.365 + 0.351 (s/s0) el-sebaii et al. [14] 2 h/h0 = 0.332 + 0.407 (s/s0) raja et al. [40] 3 h/h0 = 0.280 + 0.493 (s/s0) hay [41] 4 h/h0 = 0.412 + 0.211 (s/s0) elagih et al. [42] 5 h/h0 = 0.272 + 0.384 (s/s0) yao et al. [43] 6 h/h0 = 0.230 + 0.620 (s/s0) glover et al. [17] 7 h/h0 = 0.361 + 0.366 (s/s0) khogali [44] 8 h/h0 = 0.315+ 0.402 (s/s0) khogali [44] 9 h/h0 = 0.278 + 0.467 (s/s0) khogali [44] 10 h/h0 = 0.357 + 0.374 (s/s0) khogali [44] 11 h/h0 = 0.433 + 0.271 (s/s0) khogali [44] 12 h/h0 = 0.350 + 0.353 (s/s0) khogali [44] 13 h/h0 = 0.339 + 0.359 (s/s0) khogali [44] 14 h/h0 = 0.402 + 0.234 (s/s0) khogali [45] 15 h/h0 = 0.211 + 0.572 (s/s0) khogali [45] 16 h/h0 = 0.208 + 0.544 (s/s0) khogali [45] 17 h/h0 = 0.460 + 0.208 (s/s0) khogali [45] 18 h/h0 = 0.325 + 0.423 (s/s0) khogali [45] 19 h/h0 = 0.325 + 0.356 (s/s0) khogali [45] 20 h/h0 = 0.287 + 0.463 (s/s0) khogali [45] 21 h/h0 = 0.341 + 0.446 (s/s0) garg [46] 22 h/h0 = 0.309 + 0.482 (s/s0) garg [46] 23 h/h0 = 0.302 + 0.464 (s/s0) garg [46] 24 h/h0 = 0.327 + 0.399 (s/s0) garg [46] 25 h/h0 = 0.293 + 0.459 (s/s0) garg [46] 26 h/h0 = 0.291 + 0.464 (s/s0) garg [46] 27 h/h0 = 0.330+ 0.453 (s/s0) garg [46] 28 h/h0 = 0.286+ 0.467 (s/s0) garg [46] 29 h/h0 = 0.279+ 0.514 (s/s0) garg [46] 30 h/h0 = 0.340+ 0.399 (s/s0) garg [46] 31 h/h0 = 0.393+ 0.357 (s/s0) garg [46] 32 h/h0 = 0.334+ 0.444 (s/s0) hussain et al. [19] 33 h/h0 = 0.217+ 0.593 (s/s0) hussain et al. [19] 34 h/h0 = 0.308+ 0.407 (s/s0) hussain et al. [19] 35 h/h0 = 0.292+ 0.473 (s/s0) hussain et al. [19] 36 h/h0 = 0.300+ 0.359 (s/s0) hussain et al. [19] 37 h/h0 = 0.291+ 0.454 (s/s0) hussain et al. [19] 38 h/h0 = 0.283+ 0.469 (s/s0) hussain et al. [19] 39 h/h0 = 0.314+ 0.421 (s/s0) hussain et al. [19] 40 h/h0 = 0.292+ 0.433 (s/s0) hussain et al. [19] 41 h/h0 = 0.274+ 0.470 (s/s0) hussain et al. [19] 42 h/h0 = 0.310+ 0.400 (s/s0) hussain et al. [19] 43 h/h0 = 0.312+ 0.418 (s/s0) hussain et al. [19] 44 h/h0 = 0.296+ 0.554 (s/s0) hussain et al. [19] 45 h/h0 = 0.375+ 0.350 (s/s0) hussain et al. [19] 46 h/h0 = 0.170+ 0.590 (s/s0) ibrahim [47] 47 h/h0 = 0.290+ 0.460 (s/s0) ibrahim [47] 48 h/h0 = 0.220+ 0.550 (s/s0) ibrahim [47] 49 h/h0 = 0.140+ 0.610 (s/s0) ibrahim [47] 50 h/h0 = 0.230+ 0.540 (s/s0) ibrahim [47] 51 h/h0 = 0.520+ 0.230 (s/s0) ibrahim [47] 52 h/h0 = 0.700+ 0.030 (s/s0) ibrahim [47] 53 h/h0 = 0.130+ 0.604 (s/s0) srivastava et al. [48] 54 h/h0 = 0.126+ 0.600 (s/s0) srivastava et al. [48] 55 h/h0 = 0.336+ 0.339 (s/s0) srivastava et al. [48] continue on the next page 4 khalil / j. nig. soc. phys. sci. 4 (2022) 911 5 table 1: models of gsr used in the present research for calculation of monthly average daily of gsr on horizontal surface. no. of model modeling of gsr author/references 56 h/h0 = 0.121+ 0.582 (s/s0) srivastava et al. [48] 57 h/h0 = 0.105+ 0.569 (s/s0) srivastava et al. [48] 58 h/h0 = 0.052+ 0.644 (s/s0) srivastava et al. [49] 59 h/h0 = 0.093+ 0.632 (s/s0) srivastava et al. [49] 60 h/h0 = 0.138+ 0.556 (s/s0) srivastava et al. [49] 61 h/h0 = 0.491+ 0.263 (s/s0) jemaa et al. [50] 62 h/h0 = 0.305+ 0.429 (s/s0) khorasantzadeh et al. [51] 63 h/h0 = 0.133+ 0.647 (s/s0) jin et al. [21] 64 h/h0 = 0.570+ 0.011 (s/s0) suthar et al. [52] 65 h/h0 = 0.269+ 0.489 (s/s0) marwal [53] 66 h/h0 = 0.228+ 0.311 (s/s0) katiyar et al. [15] 67 h/h0 = 0.262+ 0.395 (s/s0) katiyar et al. [54] 68 h/h0 = 0.223+ 0.512 (s/s0) katiyar et al. [54] 69 h/h0 = 0.229+ 0.531 (s/s0) katiyar et al. [15] 70 h/h0 = 0.228+ 0.509 (s/s0) katiyar et al. [15] 71 h/h0 = 0.190+ 0.620 (s/s0) soler [55] 72 h/h0 = 0.072+ 0.694 (s/s0) hejase et al. [33] 73 h/h0 = 0.200+ 0.510 (s/s0) flocas [56] 74 h/h0 = 0.657+ 0.023 (s/s0) ahmad et al. [7] 75 h/h0 = 0.458+ 0.175 (s/s0) ahmad et al. [7] 76 h/h0 = 0.177+ 0.692 (s/s0) jain [57] 77 h/h0 = 0.175+ 0.552 (s/s0) bahel et al. [58] 78 h/h0 = 0.360+ 0.340 (s/s0) maduekwe et al. [59] 79 h/h0 = 0.347+ 0.352 (s/s0) rehman [18] 80 h/h0 = 0.324+ 0.405 (s/s0) ahmad et al. [7] 81 h/h0 = 0.222+ 0.652 (s/s0) li et al. [60] 82 h/h0 = 0.170+ 0.680 (s/s0) yohanna et al. [61] 83 h/h0 = 0.250+ 0.500 (s/s0) gadiwala et al. [23] 84 h/h0 = 0.309+ 0.368 (s/s0) chegaar et al. [62] 85 h/h0 = 0.367+ 0.366 (s/s0) chegaar et al. [62] 86 h/h0 = 0.233+ 0.591 (s/s0) chegaar et al. [62] 87 h/h0 = 0.175+ 0.712 (s/s0) onyango et al. [24] 88 h/h0 = 0.180+ 0.620 (s/s0) onyango et al. [24] 89 h/h0 = 0.278+ 0.648 (s/s0) ezekwe and ezeilo [63] 90 h/h0 = 0.260+ 0.431 (s/s0) sambo [64] 91 h/h0 = 0.250+ 0.450(s/s0) jackson et a [65] 92 h/h0 = 0.291+ 0.306 (s/s0) kuye et al. [66] 93 h/h0 = 0.242+ 0.641 (s/s0) safari et al. [67] 94 h/h0 = 0.320+ 0.309 (s/s0) tijjani [68] 95 h/h0 = 0.239+ 0.585 (s/s0) ituen et al. [69] 96 h/h0 = 0.138+ 0.488 (s/s0) isikwue et al. [70] 97 h/h0 = 0.220+ 0.430 (s/s0) quansah et al. [26] 98 h/h0 = 0.110+ 0.790 (s/s0) nwokoye et al. [71] 99 h/h0 = 0.249+ 0.566 (s/s0) adaramola [72] 100 h/h0 = 0.422+ 0.128 (s/s0) ampratwum et al. [73] 101 h/h0 = 0.267+ 0.475 (s/s0) ulgen et al. [74] 102 h/h0 = 0.318+ 0.449 (s/s0) togrul et al. [75] 103 h/h0 = 0.352+ 0.361 (s/s0) aziz et al. [76] 104 h/h0 = 0.343+ 0.322 (s/s0) bakirci [25] 105 h/h0 = 0.288+ 0.547 (s/s0) sherif [77] 106 h/h0 = 0.244+ 0.415 (s/s0) okonkwo et al. [78] 107 h/h0 = 0.183+ 0.530 (s/s0) assi et al. [79] 108 h/h0 = 0.183+ 0.647 (s/s0) assi et al. [79] 109 h/h0 = 0.495+ 0.593 (s/s0) katiyar [15] 110 h/h0 = 0.215+ 0.527 (s/s0) said et al. [80] continue on the next page 5 khalil / j. nig. soc. phys. sci. 4 (2022) 911 6 table 1: models of gsr used in the present research for calculation of monthly average daily of gsr on horizontal surface. no. of model modeling of gsr author/references 111 h/h0 = 0.226+ 0.418 (s/s0) tiris et al. [81] 112 h/h0 = 0.340+ 0.320 (s/s0) veeran et al. [82] 113 h/h0 = 0.270+ 0.650 (s/s0) veeran et al. [82] 114 h/h0 = 0.140+ 0.570 (s/s0) lewis [83] 115 h/h0 = 0.313+ 0.474 (s/s0) jain [57] 116 h/h0 = 0.307+ 0.488 (s/s0) jain [57] 117 h/h0 = 0.309+ 0.599 (s/s0) jain [84] 118 h/h0 = 0.241+ 0.488 (s/s0) luhanga et al. [85] 119 h/h0 = 0.240+ 0.513 (s/s0) jain and jain [84] 120 h/h0 = 0.191+ 0.571 (s/s0) khogali et al. [45] 121 h/h0 = 0.262+ 0.454 (s/s0) khogali et al. [45] 122 h/h0 = 0.297+ 0.432 (s/s0) khogali et al. [45] 123 h/h0 = 0.174+ 0.615 (s/s0) alsaad [86] 124 h/h0 = 0.230+ 0.380 (s/s0) hinrichsen [87] 125 h/h0 = 0.220+ 0.420 (s/s0) hinrichsen [87] 126 h/h0 = 0.230+ 0.480 (s/s0) page [88] 127 h/h0 = 0.307+ 0.312 (s/s0) abdalla et al. [89] 128 h/h0 = 0.441+ 0.292 (s/s0) ogolo [90] 129 h/h0 = 0.248+ 0.509 (s/s0) ogolo [90] 130 h/h0 = 0.278+ 0.483 (s/s0) ogolo [90] 131 h/h0 = 0.222+ 0.580 (s/s0) ogolo [90] 132 h/h0 = 0.176+ 0.563 (s/s0) rensheng et al. [22] 133 h/h0 = 0.206+ 0.546 (s/s0) louche et al. [91] 134 h/h0 = 0.180+ 0.615 (s/s0) newland [92] 135 h/h0 = 0.154+ 0.787 (s/s0) gopinathan et al. [93] 136 h/h0 = 0.196+ 0.721 (s/s0) gopinathan et al. [93] 137 h/h0 = 0.308+ 0.417 (s/s0) aras et al. [94] 138 h/h0 = 0.275+ 0.039 (s/s0) sivamadhavi et al. [95] 139 h/h0 = 0.300+ 0.415 (s/s0) amoussa [96] 140 h/h0 = 0.238+ 0.522 (s/s0) amoussa [96] 141 h/h0 = 0.279+ 0.416 (s/s0) bakirci [25] 142 h/h0 = 0.228+ 0.527 (s/s0) el-metwally [97] 143 h/h0 = 0.174+ 0.615 (s/s0) alsaad [86] 144 h/h0 = 0.309+ 0.368 (s/s0) chegaar and chibani [62] 145 h/h0 = 0.367+ 0.367 (s/s0) chegaar and chibani [62] 146 h/h0 = 0.233+ 0.591 (s/s0) chegaar and chibani [62] 147 h/h0 = 0.340+ 0.320 (s/s0) veeran and kumar [82] 148 h/h0 = 0.270+ 0.650 (s/s0) veeran and kumar [82] 149 h/h0 = 0.267+ 0.475 (s/s0) ulgen and hepbasli [74] 150 h/h0 = 0.140+ 0.570 (s/s0) lewis [83] 151 h/h0 = 0.180+ 0.620 (s/s0) tiris et al. [81] 152 h/h0 = 0.217+ 0.545 (s/s0) almorox and hontoria [16] 153 h/h0 = 0.335+ 0.367 (s/s0) raja and twidell [40] 154 h/h0 = 0.215+ 0.527 (s/s0) said et al. [80] 155 h/h0 = 0.242+ 0.501 (s/s0) ulgen and hepbasli [74] 156 h/h0 = -2.81 + 3.78 (s/s0) el-sebaii et al. [14] 157 h/h0 = 0.324+ 0.405 (s/s0) ahmed and ulfat [98] 158 h/h0 = 0.133+ 0.647 (s/s0) jin et al. [21] 159 h/h0 = 0.230+ 0.380 (s/s0) akpabio and etuk [99] 160 h/h0 = 0.332+ 0.311 (s/s0) ampratwum and dorvlo [73] 161 h/h0 = 0.242+ 0.356 (s/s0) ampratwum and dorvlo [73] 162 h/h0 = 0.180+ 0.600 (s/s0) benson et al. [100] 163 h/h0 = 0.240+ 0.530 (s/s0) benson et al. [100] 164 h/h0 = 0.180+ 0.660 (s/s0) rietveld [29] 165 h/h0 = 0.200+ 0.600 (s/s0) rietveld [29] continue on the next page 6 khalil / j. nig. soc. phys. sci. 4 (2022) 911 7 table 1: models of gsr used in the present research for calculation of monthly average daily of gsr on horizontal surface. no. of model modeling of gsr author/references 166 h/h0 = 0.220+ 0.580 (s/s0) rietveld [29] 167 h/h0 = 0.200+ 0.620 (s/s0) rietveld [29] 168 h/h0 = 0.240+ 0.520 (s/s0) rietveld [29] 169 h/h0 = 0.240+ 0.530 (s/s0) rietveld [29] 170 h/h0 = 0.230+ 0.530 (s/s0) rietveld [29] 171 h/h0 = 0.220+ 0.550 (s/s0) rietveld [29] 172 h/h0 = 0.200+ 0.590 (s/s0) rietveld [29] 173 h/h0 = 0.290+ 0.600 (s/s0) rietveld [29] 174 h/h0 = 0.170+ 0.660 (s/s0) rietveld [29] 175 h/h0 = 0.180+ 0.650 (s/s0) rietveld [29] 176 h/h0 = 0.276+ 0.648 (s/s0) ezekwe and ezeilo [63] 177 h/h0 = 0.260+ 0.620 (s/s0) ezekwe and ezeilo [63] 178 h/h0 = 0.210+ 0.490 (s/s0) ezekwe and ezeilo [63] 179 h/h0 = 0.200+ 0.740 (s/s0) arinze and obi [101] 180 h/h0 = 0.413+ 0.241 (s/s0) sambo [64] 181 h/h0 = 0.160+ 0.530 (s/s0) folayan and ogunbiyi [102] 182 h/h0 = 0.263+ 0.374 (s/s0) banna and gnininri [103] 183 h/h0 = 0.253+ 0.357 (s/s0) banna and gnininri [103] 184 h/h0 = 0.271+ 0.300 (s/s0) banna and gnininri [103] 185 h/h0 = 0.208+ 0.748 (s/s0) falayi et al. [104] 186 h/h0 = 0.018+ 1.139 (s/s0) augustine and nnabuchi [105] 187 h/h0 = 0.295+ 0.306 (s/s0) augustine and nnabuchi [105] 188 h/h0 = 0.191+ 0.433 (s/s0) augustine and nnabuchi [105] 189 h/h0 = 0.278+ 0.331 (s/s0) augustine and nnabuchi [105] 190 h/h0 = 0.290+ 0.420 (s/s0) augustine and nnabuchi [105] 191 h/h0 = 0.290+ 0.253 (s/s0) augustine and nnabuchi [105] 192 h/h0 = 0.336+ 0.247 (s/s0) augustine and nnabuchi [105] 193 h/h0 = 0.219+ 0.638 (s/s0) olayinka [106] 194 h/h0 = 0.211+ 0.629 (s/s0) olayinka [106] 195 h/h0 = 0.163+ 0.806 (s/s0) olayinka [106] 196 h/h0 = 0.450+ 0.274 (s/s0) olayinka [106] 197 h/h0 = 0.320+ 0.308 (s/s0) tijjani [68] 198 h/h0 = 0.170+ 0.680 (s/s0) yohanna and itodo [107] 199 h/h0 = 0.239+ 0.585 (s/s0) ituen [69] 200 h/h0 = 0.249+ 0.566 (s/s0) adaramola [72] 201 h/h0 = 0.300+ 0.530 (s/s0) yakubu and medugu [108] 202 h/h0 = 0.287+ 0.547 (s/s0) musa et al. [109] 203 h/h0 = 0.138+ 0.488 (s/s0) isikwue et al. [70] 204 h/h0 = 0.239+ 0.717 (s/s0) kolebaje and mustapha [110] 205 h/h0 = 0.258+ 0.612 (s/s0) kolebaje and mustapha [110] 206 h/h0 = 0.250+ 0.643 (s/s0) kolebaje and mustapha [110] 207 h/h0 = 0.286+ 0.537 (s/s0) kolebaje and mustapha [110] 208 h/h0 = 0.318+ 0.513 (s/s0) kolebaje and mustapha [110] 209 h/h0 = 0.310+ 0.540 (s/s0) kolebaje and mustapha [110] 210 h/h0 = 0.194+ 0.398 (s/s0) ohunakin et al. [111] 211 h/h0 = 0.115+ 0.567 (s/s0) solomon [112] 212 h/h0 = 0.389+ 0.358 (s/s0) gana and akpootu [113] 213 h/h0 = 0.417+ 0.316 (s/s0) gana and akpootu [113] 214 h/h0 = 0.334+ 0.449 (s/s0) gana and akpootu [113] 215 h/h0 = 0.416+ 0.317 (s/s0) gana and akpootu [113] 216 h/h0 = 0.453+ 0.268 (s/s0) gana and akpootu [113] 217 h/h0 = 0.386+ 0.360 (s/s0) gana and akpootu [113] 218 h/h0 = 0.083+ 0.684 (s/s0) afungchui and neba [114] 219 h/h0 = 0.286+ 0.579 (s/s0) afungchui and neba [114] 220 h/h0 = 0.253+ 0.427 (s/s0) afungchui and neba [114] continue on the next page 7 khalil / j. nig. soc. phys. sci. 4 (2022) 911 8 table 1: models of gsr used in the present research for calculation of monthly average daily of gsr on horizontal surface. no. of model modeling of gsr author/references 221 h/h0 = 0.303+ 0.484 (s/s0) afungchui and neba [114] 222 h/h0 = 0.314+ 0.479 (s/s0) afungchui and neba [114] 223 h/h0 = 0.005+ 1.313 (s/s0) sarsah and uba [115] 224 h/h0 = 0.244+ 0.415 (s/s0) okonkwo and nwokoye [78] 225 h/h0 = 0.220+ 0.430 (s/s0) quansah et al. [26] 226 h/h0 = 0.288+ 0.547 (s/s0) sheriff [77] 227 h/h0 = 0.192+ 0.442 (s/s0) ike [116] 228 h/h0 = 0.045+ 0.051 (s/s0) sani et al. [117] 229 h/h0 = 0.010+ 0.750 (s/s0) adesina et al. [118] 230 h/h0 = 0.240+ 0.310 (s/s0) olatona and adeleke [119] 231 h/h0 = 0.110+ 0.790 (s/s0) okonkwo et al. [78] 232 h/h0 = 0.295+ 0.532 (s/s0) innocent et al. [120] 233 h/h0 = 0.250+ 0.522 (s/s0) boluwaji and onyedi [121] 234 h/h0 = 0.314+ 0.491 (s/s0) boluwaji and onyedi [121] 235 h/h0 = 0.315+ 0.433 (s/s0) boluwaji and onyedi [121] 236 h/h0 = 0.130+ 0.620 (s/s0) coulibaly and ouedoraogo [122] 237 h/h0 = 0.210+ 0.450 (s/s0) coulibaly and ouedoraogo [122] 238 h/h0 = 0.170+ 0.470 (s/s0) coulibaly and ouedoraogo [122] 239 h/h0 = 0.210+ 0.460 (s/s0) coulibaly and ouedoraogo [122] 240 h/h0 = 0.150+ 0.460 (s/s0) coulibaly and ouedoraogo [122] 241 h/h0 = 0.180+ 0.530 (s/s0) coulibaly and ouedoraogo [122] 242 h/h0 = 0.210+ 0.430 (s/s0) coulibaly and ouedoraogo [122] 243 h/h0 = 0.230+ 0.400 (s/s0) coulibaly and ouedoraogo [122] 244 h/h0 = 0.180+ 0.490 (s/s0) coulibaly and ouedoraogo [122] 245 h/h0 = 0.219+ 0.614 (s/s0) okundamiya et al. [123] 246 h/h0 = 0.236+ 0.621 (s/s0) okundamiya et al. [123] 247 h/h0 = 0.479+ 0.246 (s/s0) okundamiya et al. [123] 248 h/h0 = 0.270+ 0.240 (s/s0) ayodele and ogunjuyigbe [124] 249 h/h0 = 0.292+ 0.284 (s/s0) +0.211 (s/s0)2 katiyar et al. [15] 250 h/h0 = -0.197+ 0.177 (s/s0) -0.912 (s/s0)2 katiyar et al. [15] 251 h/h0 = 0.129+ 0.933 (s/s0) -0.503 (s/s0)2 katiyar et al. [15] 252 h/h0 = 0.234+ 0.465 (s/s0) -0.043 (s/s0)2 katiyar et al. [15] 253 h/h0 = 0.184+ 0.845 (s/s0) -0.280 (s/s0)2 almorox et al. [16] 254 h/h0 = 0.181+ 0.895 (s/s0) -0.361 (s/s0)2 akinoglu et al. [125] 255 h/h0 = 0.222+ 0.705 (s/s0) -0.217 (s/s0)2 ogelman [30] 256 h/h0 = 1.1111.828 (s/s0) +1.660 (s/s0)2 ogelman [30] 257 h/h0 = 0.168+ 0.835 (s/s0) +0.320 (s/s0)2 ogolo [90] 258 h/h0 = 0.154+ 0.933 (s/s0) +0.370 (s/s0)2 ogolo [90] 259 h/h0 = 0.7191.067 (s/s0) +1.145 (s/s0)2 ogolo [90] 260 h/h0 = 0.339+ 0.287 (s/s0) +0.119 (s/s0)2 singh et al. [5] 261 h/h0 = 0.154+ 1.172 (s/s0) -0.705 (s/s0)2 aras et al. [94] 262 h/h0 = 0.100+ 0.874 (s/s0) -0.255 (s/s0)2 togrul et al. [126] 263 h/h0 = 0.180+ 1.160 (s/s0) -0.910 (s/s0)2 said et al. [80] 264 h/h0 = 0.348+ 0.320 (s/s0) +0.070 (s/s0)2 maduekwe et al. [59] 265 h/h0 = 0.222+ 0.538 (s/s0) +0.152 (s/s0)2 ahmad et al. [7] 266 h/h0 = 0.069+1.326 (s/s0) 0.667 (s/s0)2 onyango et al. [24] 267 h/h0 = -2.767+ 9.207 (s/s0) -6.350 (s/s0)2 safari et al. [67] 268 h/h0 = -2.767+9.207 (s/s0) -0.674 (s/s0)2 tijjani [68] 269 h/h0 = 0.025+ 1.125 (s/s0) -0.308 (s/s0)2 nwokoye et al. [71] 270 h/h0 = 0.282+ 0.572 (s/s0) -0.224 (s/s0)2 bakirci [25] 271 h/h0 = 0.100+ 1.020 (s/s0) -0.440 (s/s0)2 rietveld [29] 272 h/h0 = 0.326+ 0.344 (s/s0) +0.102 (s/s0)2 rietveld [29] 273 h/h0 = 0.6920.068 (s/s0) +0.118 (s/s0)2 sekhar et al. [40] 274 h/h0 = 0.264+ 0.516 (s/s0) -0.005 (s/s0)2 sekhar et al. [40] 275 h/h0 = 0.494+ 0.064 (s/s0) +0.173 (s/s0)2 sekhar et al. [40] continue on the next page 8 khalil / j. nig. soc. phys. sci. 4 (2022) 911 9 table 1: models of gsr used in the present research for calculation of monthly average daily of gsr on horizontal surface. no. of model modeling of gsr author/references 276 h/h0 = 0.273+ 0.468 (s/s0) +0.018 (s/s0)2 sekhar et al. [40] 277 h/h0 = 0.263+ 0.514 (s/s0) -0.008 (s/s0)2 sekhar et al. [40] 278 h/h0 = 0.403+ 0.788 (s/s0) -0.224 (s/s0)2 sekhar et al. [40] 279 h/h0 = 0.279+ 0.037 (s/s0) +0.009 (s/s0)2 sekhar et al. [40] 280 h/h0 = 0.285+ 0.036 (s/s0) +0.002 (s/s0)2 sivamadhavi et al. [95] 281 h/h0 = -0.067+ 1.772 (s/s0) -1.249 (s/s0)2 sivamadhavi et al. [95] 282 h/h0 = 0.635+ 0.097 (s/s0) -0.026 (s/s0)2 elagib et al. [42] 283 h/h0 = 0.140+ 0.613 (s/s0) +0.035 (s/s0)2 assi et al. [79] 284 h/h0 = 0.487+ 0.612 (s/s0) -0.018 (s/s0)2 jin et al. [21] 285 h/h0 = 0.434+ 0.233 (s/s0) +0.166 (s/s0)2 katiyar et al. [15] 286 h/h0 = -0.359+ 1.763 (s/s0) -0.659 (s/s0)2 jemaa et al. [50] 287 h/h0 = 0.215+ 0.625 (s/s0) -0.221 (s/s0)2 hejase et al. [33] 288 h/h0 = 0.7480.949 (s/s0) +1.042 (s/s0)2 togrul et al. [126] 289 h/h0 = 0.231+ 0.463 (s/s0) – 0.044 (s/s0)2 ravichandran et al. [127] 290 h/h0 = 0.189+ 0.845 (s/s0) – 0.390 (s/s0)2 okonkwo et al. [78] 291 h/h0 = 0.388+ 0.580 (s/s0) -0.407 (s/s0)2 assi et al. [79] 292 h/h0 = 0.086+ 1.158 (s/s0) -0.556 (s/s0)2 suthar et al. [52] 293 h/h0 =-0.140+2.52 (s/s0) -3.71 (s/s0)2+2.24(s/s0)3 samuel [128] 294 h/h0 =-2.43+11.95 (s/s0) -16.75 (s/s0)2+7.96(s/s0)3 ertekin and yaldiz [129] 295 h/h0 =0.230+0.380 (s/s0)+0.469 (s/s0)2-0.366(s/s0)3 almorox and hontoria [16] 296 h/h0 =0.81-3.34(s/s0)+7.38 (s/s0)2-4.51(s/s0)3 lewis [83] 297 h/h0 =0.241+0.363(s/s0)+0.459 (s/s0)2-0.371(s/s0)3 ulgen and hepbasli [130] 298 h/h0 =0.152+1.133(s/s0)-1.113 (s/s0)2+0.452(s/s0)3 tahran and sari [131] 299 h/h0 =0.128+0.725(s/s0)-0.229 (s/s0)2+0.184(s/s0)3 jin et al. [21] 300 h/h0 =0.483-0.616(s/s0)+1.893 (s/s0)2-1.098(s/s0)3 aras et al. [94] 301 h/h0 =0.150+1.145(s/s0)-1.474 (s/s0)2+0.963(s/s0)3 rensheng et al. [22] 302 h/h0 =0.631-0.725(s/s0)+1.208 (s/s0)2-0.463(s/s0)3 bakirci [25] 303 h/h0 =0.171+0.026(s/s0)+2.100 (s/s0)2-1.640(s/s0)3 burari et al. [132] 304 h/h0 =0.197+0.981(s/s0)+0.053 (s/s0)2-1.317(s/s0)3 kolebaje and mustapha [110] 305 h/h0 =0.231+0.548(s/s0)+1.884 (s/s0)2-4.215(s/s0)3 kolebaje and mustapha [110] 306 h/h0 =0.400+0.137(s/s0)+0.001 (s/s0)2+1.139(s/s0)3 kolebaje and mustapha [110] 307 h/h0 =0.306+0.464(s/s0)+0.001 (s/s0)2-0.189(s/s0)3 kolebaje and mustapha [110] 308 h/h0 =0.091+1.401(s/s0)-0.928 (s/s0)2-0.001(s/s0)3 kolebaje and mustapha [110] 309 h/h0 =0.851+0.869(s/s0)+0.001 (s/s0)2-1.364(s/s0)3 kolebaje and mustapha [110] 310 h/h0 =0.050+0.971(s/s0)-0.200 (s/s0)2+0.001(s/s0)3 nwokoye and okonkwo [71] 311 h/h0 =1.561-0.365(s/s0)+0.037 (s/s0)2-0.001(s/s0)3 sani et al. [117] 312 h/h0 =0.25+0.38(s/s0)-0.210 (s/s0)2+0.074(s/s0)3 ayodele and ogunjuyigbe [124] 313 h/h0 =2.72-11.01(s/s0)+ 17.43 (s/s0)2-8.654(s/s0)3 katiyar et al. [15] 314 h/h0 =1.38-6.33(s/s0)+ 12.75 (s/s0)2-7.66(s/s0)3 katiyar et al. [15] 315 h/h0 =0.294+0.086(s/s0)+ 0.77 (s/s0)2-0.436(s/s0)3 katiyar et al. [15] 316 h/h0 =0.512-0.993(s/s0)+ 2.53 (s/s0)2-1.33(s/s0)3 katiyar et al. [15] 317 h/h0 =0.16+0.87(s/s0)0.61(s/s0)2+0.34(s/s0)3 bahel et al. [58] 318 h/h0 =0.230+ 0.381(s/s0)+ 0.469(s/s0)20.366(s/s0)3 almorox et al. [15] 319 h/h0 =0.285+0.259(s/s0)+0.617 (s/s0)2-0.483(s/s0)3 ulgen et al. [130] 320 h/h0 =0.179+0.981(s/s0)-0.296 (s/s0)2-0.266(s/s0)3 togrul et al. [126] 321 h/h0 =0.475-0.898(s/s0)+2.773 (s/s0)2-1.54(s/s0)3 onyango et al. [24] 322 h/h0 =0.400+0.214(s/s0)+0.153 (s/s0)2+0.119(s/s0)3 jemaa et al. [50] 323 h/h0 =1.324-4.942(s/s0)+8.714 (s/s0)2-4.549(s/s0)3 khorasanizadeh et al. [51] 324 h/h0 =-0.898+5.936(s/s0)-7.493 (s/s0)2+3.326(s/s0)3 khorasanizadeh et al. [51] 325 h/h0 =0.147+0.099(s/s0)-0.009 (s/s0)2+0.0005(s/s0)3 sivamadhavi et al [95] 326 h/h0 =0.160+0.093(s/s0)-0.008 (s/s0)2+0.0004(s/s0)3 sivamadhavi et al [95] 327 h/h0 =-1.88+12.65(s/s0)-21.87 (s/s0)2+12.37(s/s0)3 yaniktepe et al. [27] 328 h/h0 =0.420+0.529(s/s0)-0.777 (s/s0)2+0.521(s/s0)3 assi et al. [79] 329 h/h0 =0.441+0.829(s/s0)-0.977 (s/s0)2+0.421(s/s0)3 assi et al. [79] 330 h/h0 =0.164-683(s/s0)+1.073 (s/s0)2-0.0009(s/s0)3 soler [55] continue on the next page 9 khalil / j. nig. soc. phys. sci. 4 (2022) 911 10 table 1: models of gsr used in the present research for calculation of monthly average daily of gsr on horizontal surface. no. of model modeling of gsr author/references 331 h/h0 =10.09-38.46(s/s0)+50.75 (s/s0)2-21.83(s/s0)3 hejase et al. [33] 332 h/h0 =0.346+0.422(s/s0)+2.33 (s/s0)2-1.634(s/s0)3 marwal [53] 333 h/h0 =0.11-0.01(s/s0)+5.3 (s/s0)2-10.4(s/s0)3+5.6(s/s0)4 marwal [53] 334 h/h0 =0.40-0.85(s/s0)+4.1 (s/s0)2-4.9(s/s0)3+2.1(s/s0)4 zabara [132] 335 h/h0 =0.550 + 0.656(s/s0) + 0.721 log(s/s0) sarsah and uba [115] 336 h/h0 = -0.337 + 0.656(s/s0) + 0.406 exp(s/s0) sarsah and uba [115] 337 h/h0 = 0.809 + 0.386 log(s/s0) onyango et al. [24] 338 h/h0 = 0.8336 + 1.0573 log(s/s0) ampratwum et al. [73] 339 h/h0 = 0.6050 + 0.1917 ln(s/s0) yao et al. [43] 340 h/h0 = 0.34 + 0.40 (s/s0)+ 0.17 ln(s/s0) newland [92] 341 h/h0 = 0.753 + 0.561 ln(s/s0) hejase et al. [33] 342 h/h0 = 0.7166 + 0.2705 ln(s/s0) marwal [53] 343 h/h0 = 0.6290 + 0.1053 log(s/s0) assi et al. [79] 344 h/h0 = 0.6858 + 0.01531 log(s/s0) assi et al. [79] 345 h/h0 = 0.880 − 0.1295 (s/s0) + 0.666 ln(s/s0) hejase et al. [33] 346 h/h0 = 0.7621− 0.1627(s/s0) + 0.08576 log(s/s0) assi et al. [79] 347 h/h0 = 0.7081− 0.0127(s/s0) + 0.08576 log(s/s0) assi et al. [79] 348 h/h0 = 0.3396 e0.8985(s/so) togrul et al. [126] 349 h/h0 = 0.3593 e 0.6093(s/so) khorasanizadeh et al. [51] 350 h/h0 = 0.3593 e0.6093(s/so) yao et al. [43] 351 h/h0 = 0.2674 e1.0391(s/so) elagib et al. [42] 352 h/h0 = 0.4248 e0.3947(s/so) hejase et al. [33] 353 h/h0 = 0.263 e1.088(s/so) assi et al. [79] 354 h/h0 = 0.5399 e0.1630(s/so) marwal [53] 355 h/h0 = 0.3158 e0.9251(s/so) assi et al. [79] 356 h/h0 = 0.6416 e0.099(s/so) almorox et al. [15] 357 h/h0 = −0.0271 + 0.3096 × e(s/so) khorasanizadeh et al. [51] 358 h/h0 = 0.3661 + 0.1133 e(s/so) togrul et al. [126] 359 h/h0 = 0.7316 (s/so)0.4146 nwokoye et al. [71] 360 h/h0 = 0.6673 (s/so)0.5343 yao et al. [43 ] 361 h/h0 = 0.6128 (s/so)0.2499 elagib et al. [42] 362 h/h0 = 0.7399 (s/so)0.5201 marwal [53] 363 h/h0 = e−0.4698 (s/so)0.034 assi et al. [79] 364 h/h0 = e−0.4058 (s/so)0.00014 assi et al. [79] 365 h/h0 = e−0.1361 (s/so)1.7156 ampratwum et al. [73] 366 h/h0 = -0.864 +1.862 (s/so)2.344 el-sebaii et al. [14] 367 h/h0 = -3.386 +0.220(tmax)0.003 (tmax)2 okonkwo and nwokoye [78] 368 h/h0 = -3.388.434 +0.638(tmean)-0.012 (tmean)2 ohunakin [111] 369 h/h0 = -0.987 +5.256(tr)4.536 (tr)2 okonkwo and nwokoye [78] 370 h/h0 = 0.24 + e0.064( t/s o) ayodele and ogunjuyigbe [124] 371 h/h0 = 0.45 +0.39 log(∆t/so) ayodele and ogunjuyigbe [124] 372 h/h0 = 0.219 + 0.526 (s/so) + 0.004 w el-metwally [97] 373 h/h0 = -0.107 + 0.70 (s/s0) – 0.0025t + 0.004 rh maghrabi [12] 374 h/h0 = -0.139+0.229 (s/so)+0.01t+0.004v+0.002rh+0.002ps trabea and shaltout [134] 375 h/h0 = -1.92 + 2.60 (s/so) + 0.006t el-sebaii et al. [14] 376 h/h0 = -1.62 + 2.24 (s/so) + 0.332rh el-sebaii et al. [14] 377 h/h0 = 0.388 cosφ + 0.367 (s/so) raja et al. [40] 378 h/h0 = 0.388 cosφ + 0.407 (s/so) raja et al. [40] 379 h/h0 = 0.3092 cosφ + 0.4931 (s/so) ulgen et al. [74] 380 h/h0 = 0.12 + 0.58 (s/so) + 7.56 x 10−5h lewis [83] 381 h/h0 = 0.28 – 0.141 cosφ + 0.026h + 0.542(s/so) rensheng et al. [22] 382 h/h0 = 0.122 + 0.001 cosφ + 2.57 x 10−2h+ 0.543(s/so) rensheng et al. [22] 383 h/h0 =0.275+4.27x10−5ψ+0.141cosφ + 2.63x10−2h+0.542(s/so) rensheng et al. [22] 384 h/h0 =0.117+4.11x10−5ψ+0.001cosφ + 2.59x10−2h+0.543(s/so) rensheng et al. [22] 385 h/h0 =-0.117+4.11x10−5ψ+0.001cosφ + 2.59x10−2h+0.543(s/so) rensheng et al. [22] continue on the next page 10 khalil / j. nig. soc. phys. sci. 4 (2022) 911 11 table 1: models of gsr used in the present research for calculation of monthly average daily of gsr on horizontal surface. no. of model modeling of gsr author/references 386 h/h0 =0.001φ+2.41x10−2h+0.109+1.03(s/so)-1.22(s/so)2+0.79(s/so)3 rensheng et al. [22] 387 h/h0 = 3.614 – 0.364 (rh)2 kolebaje et al. [110] 388 h/h0 = 0.507 – 0.863 (c/co)2 augustine and nnabuchi [105] 389 h/h0 = 1.309 + 0.601 (s/so) – 0.999(tr) – 0.0129 (tmean) falayi et al. [104] 390 h/h0 = -0.839 + 0.247 (s/so) + 0.0439 (tmean) olayinka [106] 391 h/h0 = 1.395 + 1.591 (s/so) 0.046 (tmean) ituen et al. [69] 392 h/h0 = -6.874 + 7.31 (s/so)0.0237 + 0.117 (ln∆t) okundamiya et al. [123] 393 h/h0 = 5.981 – 1.991[(∆t+rh)/n]0.5 kolebaje et al. [110] 394 h/h0 = 2.931 – 0.570[(∆t+rh)/n]0.5 – 1.214 (tr) kolebaje et al. [110] 395 h/h0 = 0.00689 + 0.0221 (s/so) + 0.968 (c/co)) olayinka [106] 396 h/h0 = -0.423 + 0.301 (s/so) + 0.0256 (tmax) + 0.0725 (rh/100) olayinka [106] 397 h/h0 = -0.183 + 0.279 (s/so) + 0.170 (tr)) + 0.0179 (tmax) – 0.0128 (c) okundamiya et al. [123] 398 h/h0 = -0.012+0.001(s/so)+0.999(c/co)+0.00053(tmax) -0.001 (rh/100) olayinka [106] 399 h/h0 = -0.518 cosφ +19.219 cos n +5.513(tmax)+125.78(s/so) +21.683 (tmax/rh)+5.634(tmax/rh)2-2.693(cos φ cos n) – 33.15 ajayi et al. [135] 400 h/h0 = 4.251-4.188 cosφ +0.0437h +[(-10.577+11.451 cosφ-0.0832h) (s/so)] +[(12.725-13.099 cosφ+0.1h) (s/so)2] jin et al. [21] table 2: the variable statistical indicator results and gpi of models in the present research. model mbe rmse mpe mare mae rmsre r2 t-test u95 gpi 1 -2.314 2.985 -8.365 0.225 2.356 0.214 0.4689 1.355 3.256 0.2546 2 -2.561 3.214 -7.325 0.365 1.325 0.145 0.5221 1.214 4.235 0.3165 3 -3.214 3.654 -11.325 0.335 1.965 0.125 0.4462 1.569 3.245 0.1245 4 -1.685 4.321 -9.356 0.229 2.365 0.0985 0.4391 1.689 5.325 -0.3269 5 -2.345 3.698 -8.365 0.256 3.245 0.168 0.4506 1.474 6.325 0.2148 6 -2.865 2.654 -15.365 0.654 2.654 0.265 0.4406 1.677 7.235 -0.3215 7 -1.324 1.356 -3.985 0.452 3.126 0.354 0.4425 1.654 3.256 -0.2352 8 -2.365 3.245 -11.325 0.362 2.547 0.247 0.5397 0.792 4.325 0.3256 9 -1.954 4.325 -9.365 0.289 2.658 0.298 0.5674 0.608 4.329 0.4243 10 -2.451 5.326 -14.325 0.654 3.658 0.345 0.6025 0.476 3.456 0.5263 11 1.365 3.265 -11.365 0.345 1.654 0.145 0.5698 0.589 6.354 0.4512 12 -3.214 2.965 -8.325 0.542 1.857 0.189 0.5385 0.795 4.358 0.3252 13 -2.324 3.321 15.362 0.654 1.365 0.247 0.3256 2.397 3.658 -1.2358 14 1.895 3.478 11.325 0.356 5.326 0.268 -0.5485 2.952 2.359 -2.3145 15 -1.987 2.689 -6.325 0.412 1.325 0.214 0.3251 2.399 5.326 -1.2452 16 -2.654 1.356 -11.524 0.325 4.325 0.236 -0.4529 2.482 4.365 -1.5641 17 -2.345 1.325 -14.356 0.421 3.256 0.314 0.5374 0.796 5.326 0.3248 18 -2.852 3.654 -3.568 0.229 1.325 0.198 0.5621 0.658 6.314 0.3849 19 -1.358 4.245 -13.654 0.325 4.325 0.145 0.4658 1.361 5.325 0.2508 20 -3.256 4.321 -9.365 0.361 2.698 0.168 -0.4789 2.689 4.325 -2.1465 21 -2.378 3.265 -8.325 0.358 2.754 0.325 0.4215 1.785 3.658 -0.359 22 -1.896 2.658 -7.256 0.568 2.568 0.254 0.3551 2.363 6.325 -0.6548 23 1.325 2.965 11.356 0.689 2.314 0.315 -0.5491 3.214 4.356 -2.3149 24 1.658 3.354 -13.654 0.452 1.658 0.125 0.2398 2.416 5.326 -1.3267 25 -2.356 3.658 9.324 0.365 2.359 0.168 -0.4582 2.511 4.325 -1.6549 26 -2.089 1.965 5.326 0.654 1.325 0.215 0.4428 1.595 7.321 0.0214 27 1.325 2.314 -3.865 0.235 2.365 0.314 0.4431 1.592 5.326 0.0658 28 -1.958 3.254 -9.541 0.214 1.658 0.326 0.4457 1.572 6.325 0.1241 29 -1.689 1.658 -12.654 0.425 2.365 0.289 0.4388 1.691 5.659 -0.3275 30 -1.365 2.365 -15.325 0.325 3.256 0.135 0.3631 2.325 6.458 -0.5425 31 -2.123 3.654 -9.325 0.658 2.358 0.147 0.5369 0.798 7.321 0.3246 32 2.912 2.365 -14.235 0.552 1.358 0.168 0.4504 1.478 6.325 0.2144 continue on the next page 11 khalil / j. nig. soc. phys. sci. 4 (2022) 911 12 table 2: the variable statistical indicator results and gpi of models in the present research. model mbe rmse mpe mare mae rmsre r2 t-test u95 gpi 33 1.325 1.698 -8.325 0.356 4.325 0.128 0.6098 0.362 5.326 0.6523 34 -2.365 1.965 6.325 0.654 5.326 0.068 0.4469 1.556 4.325 0.1259 35 -2.345 2.356 7.325 0.658 3.245 0.087 0.4501 1.479 7.231 0.2141 36 5.632 3.654 -21.324 0.456 2.359 0.256 0.5405 0.785 6.325 0.3264 37 -1.356 4.325 -23.654 0.425 2.145 0.326 0.6061 0.382 5.326 0.6247 38 -2.356 4.658 -9.325 0.268 1.365 0.411 0.6023 0.479 4.365 0.5245 39 -2.314 3.265 -11.456 0.689 1.987 0.358 0.2356 2.418 5.326 -1.3269 40 -1.325 2.324 -13.698 0.754 2.365 0.265 -0.4585 2.524 6.541 -1.6587 41 -1.325 1.658 -23.654 0.452 2.895 0.124 0.5481 0.703 7.325 0.3548 42 -1.365 3.256 -22.35 0.556 3.245 0.325 0.5233 0.988 5.325 0.3179 43 -2.895 4.325 -4.325 0.542 1.658 0.258 0.4355 1.693 4.369 -0.3277 44 -3.256 3.658 5.365 0.562 5.325 0.168 0.4404 1.679 3.256 -0.3246 45 -2.365 1.689 9.356 0.345 4.235 0.178 0.4451 1.575 4.325 0.1237 46 -1.365 1.885 6.325 0.658 2.356 0.165 0.4478 1.551 4.658 0.1582 47 -1.985 2.658 -9.325 0.425 1.547 0.245 0.4597 1.367 3.256 0.2479 48 -1.568 2.365 -3.567 0.635 1.325 0.187 0.3634 2.322 3.256 -0.5247 49 -3.245 3.258 -5.632 0.411 2.356 0.314 0.4216 1.784 3.652 -0.3456 50 -3.325 3.326 -11.125 0.635 1.854 0.265 0.4509 1.472 4.214 0.2245 51 -1.235 4.457 -6.325 0.458 1.658 0.124 0.3691 2.242 5.635 -0.4154 52 1.325 3.334 -8.658 0.362 2.314 0.197 0.5036 1.239 6.785 0.2874 53 2.356 2.654 -15.452 0.578 2.224 0.311 0.4218 1.762 7.224 -0.3364 54 -1.635 1.865 -3.235 0.632 1.589 0.368 0.3685 2.254 3.568 -0.4159 55 -2.658 2.356 -3.589 0.478 2.125 0.254 0.5148 1.235 4.457 0.2958 56 -1.563 3.258 -9.785 0.365 2.256 0.234 0.5363 0.802 4.892 0.3242 57 -2.654 5.689 -11.325 0.574 3.324 0.214 0.6055 0.405 3.635 0.6217 58 1.758 3.754 -11.235 0.392 1.254 0.214 0.6235 0.263 5.325 0.7457 59 -3.635 2.635 12.356 0.457 1.541 0.136 0.6124 0.359 3.258 0.6548 60 -2.568 3.547 9.325 0.458 1.254 0.225 0.6065 0.379 4.325 0.6256 61 1.235 3.245 -3.256 0.398 5.124 0.475 -0.6165 4.214 2.865 -2.4647 62 2.356 2.365 -3.658 0.689 1.254 0.194 -0.4588 2.529 4.256 -1.6589 63 -2.235 1.568 -11.425 0.478 4.145 0.241 0.5984 0.493 4.321 0.4586 64 -2.698 1.856 -11.325 0.547 4.325 0.378 0.5511 0.689 5.785 0.3574 65 -2.445 3.224 -5.326 0.368 2.314 0.236 0.5408 0.782 5.325 0.3275 66 -1.658 4.785 -11.326 0.457 4.325 0.168 0.4892 1.245 4.925 0.2691 67 -3.568 4.658 -8.325 0.658 2.214 0.192 -0.6152 3.342 4.658 -2.4593 68 1.325 4.325 -6.324 0.457 2.325 0.378 0.3456 2.375 5.647 -0.6587 69 -1.632 2.478 -7.658 0.365 2.412 0.178 0.3325 2.386 6.478 -0.7456 70 1.785 1.547 11.785 0.745 1.389 0.265 -0.6189 4.465 5.324 -2.6582 71 1.542 1.352 -12.658 0.653 1.658 0.197 -0.4251 2.475 3.958 -1.4576 72 -2.457 3.325 4.259 0.447 2.658 0.127 -0.4325 2.476 5.236 -1.4579 73 -2.245 1.568 -4.658 0.556 1.785 0.247 0.4432 1.589 6.526 0.0854 74 1.587 2.658 -3.325 0.785 2.325 0.368 0.4467 1.558 5.458 0.1252 75 -1.458 3.564 -9.659 0.347 1.478 0.295 0.4408 1.675 5.325 -0.2569 76 -1.254 1.455 4.325 0.658 2.365 0.223 0.3614 2.337 5.952 -0.6245 77 -1.147 2.365 11.325 0.457 3.447 0.189 0.5416 0.772 6.258 0.3459 78 -2.658 3.235 6.354 0.857 2.365 0.245 0.5869 0.558 5.635 0.4563 79 2.365 2.125 -10.325 0.478 1.857 0.195 0.5098 1.238 6.854 0.2952 80 1.568 1.258 -11.325 0.457 4.447 0.168 0.6028 0.474 5.445 0.5268 81 -2.689 1.635 6.985 0.458 3.256 0.092 0.4495 1.524 4.658 0.2136 82 -2.785 2.689 7.658 0.687 3.365 0.124 0.5361 0.805 3.352 0.3238 83 5.258 3.411 -15.324 0.578 2.411 0.369 0.5686 0.596 5.935 0.4257 84 -1.658 4.857 -25.324 0.258 2.245 0.458 0.6034 0.465 5.589 0.5281 85 -2.689 4.325 -9.689 0.345 3.214 0.658 0.5924 0.495 4.578 0.4584 86 -2.457 3.635 -14.895 0.785 1.325 0.411 0.3275 2.394 5.658 -1.1954 87 -1.658 2.854 -16.321 0.658 2.658 0.365 0.3059 2.405 4.698 -1.2547 continue on the next page 12 khalil / j. nig. soc. phys. sci. 4 (2022) 911 13 table 2: the variable statistical indicator results and gpi of models in the present research. model mbe rmse mpe mare mae rmsre r2 t-test u95 gpi 88 -1.547 1.441 -25.647 0.365 2.365 0.245 0.6325 0.235 6.325 0.7548 89 -1.896 3.458 -29.356 0.348 3.478 0.265 0.3547 2.365 4.358 -0.6552 90 -2.425 4.875 -4.684 0.447 1.326 0.194 0.4671 1.356 3.256 0.2542 91 1.325 3.658 5.895 0.457 4.325 0.192 0.4325 1.695 2.958 -0.3279 92 -2.125 1.356 9.457 0.689 3.214 0.132 0.4488 1.542 4.478 0.1895 93 -1.658 1.457 6.698 0.532 2.658 0.256 0.4588 1.374 4.856 0.2458 94 -1.632 2.254 -9.784 0.547 1.892 0.124 0.5413 0.775 3.478 0.3453 95 -2.568 1.457 6.589 0.632 3.256 0.135 0.4483 1.547 5.326 0.1863 96 -1.458 1.365 9.325 0.547 2.114 0.245 0.4492 1.528 4.784 0.2132 97 -1.567 2.235 -7.365 0.365 1.214 0.158 0.5419 0.769 3.247 0.3468 98 -3.256 2.456 -7.325 0.423 2.425 0.145 0.5348 0.807 2.356 0.3235 99 -2.125 3.658 -6.245 0.449 1.658 0.245 0.5863 0.561 3.254 0.4558 100 -3.658 2.356 -12.356 0.658 1.547 0.189 0.5326 0.809 3.457 0.3231 101 -1.247 3.254 -9.556 0.523 2.457 0.089 0.3452 2.377 4.325 -0.6589 102 -2.568 4.325 -8.658 0.452 3.356 0.168 0.5852 0.563 5.326 0.4552 103 -2.145 2.325 -12.365 0.457 2.547 0.158 0.4211 1.985 6.325 -0.3566 104 -1.658 2.365 -3.658 0.589 3.589 0.257 0.3617 2.336 3.254 -0.5892 105 -2.457 3.245 -10.658 0.658 2.235 0.289 0.6162 0.281 4.568 0.7434 106 -1.356 4.568 -9.785 0.457 2.125 0.195 0.5896 0.524 4.785 0.4581 107 -2.925 4.235 -11.325 0.526 3.456 0.248 0.5631 0.641 3.658 0.4117 108 1.558 3.457 -9.325 0.457 1.358 0.235 0.5593 0.669 6.785 0.3694 109 -3.657 2.235 -8.658 0.589 1.457 0.245 0.5894 0.541 4.698 0.4576 110 -2.457 3.658 15.125 0.358 1.658 0.312 -0.3657 2.466 3.122 -1.3654 111 1.263 3.256 11.658 0.457 5.425 0.247 -0.6124 3.335 2.852 -2.456 112 -1.145 2.325 -6.365 0.658 1.578 0.158 -0.4483 2.478 5.125 -1.4582 113 -2.789 1.547 -8.325 0.658 4.452 0.125 0.2985 2.408 4.857 -1.2563 114 -2.457 1.657 -11.367 0.567 3.358 0.245 0.5592 0.672 5.245 0.3686 115 -2.536 3.856 -5.364 0.365 1.658 0.325 0.5842 0.566 5.236 0.4547 116 -1.438 4.547 -11.258 0.457 4.458 0.245 0.6021 0.482 4.325 0.5241 117 -3.625 4.658 -9.635 0.547 2.425 0.245 -0.5236 2.788 3.256 -2.1691 118 -2.547 3.635 -8.857 0.534 2.624 0.236 0.3311 2.391 3.258 -0.8571 119 -1.587 2.245 -7.568 0.349 2.425 0.189 0.4036 2.128 5.236 -0.3655 120 1.559 2.658 10.325 0.524 2.415 0.235 -0.6163 3.458 4.258 -2.4632 121 2.354 3.547 -13.124 0.658 1.524 0.158 -0.4591 2.534 5.124 -1.6592 122 1.256 3.254 9.568 0.471 2.658 0.198 0.2547 2.411 4.325 -1.2582 123 -3.256 1.568 6.345 0.511 1.578 0.195 0.5476 0.705 6.325 0.3543 124 1.635 2.547 -3.258 0.325 2.458 0.247 0.4447 1.577 4.325 0.1235 125 -1.456 3.625 -9.785 0.452 1.258 0.254 0.5422 0.767 5.236 0.3508 126 -1.325 1.245 -11.356 0.652 2.125 0.314 0.3669 2.258 4.325 -0.4163 127 -1.758 2.145 -12.689 0.725 3.415 0.245 0.3365 2.379 5.362 -0.6592 128 -2.658 3.365 -7.689 0.457 2.458 0.325 0.5671 0.609 6.325 0.4155 129 2.456 2.452 -12.458 0.657 1.625 0.245 0.5324 0.845 4.352 0.3227 130 1.558 1.457 -8.785 0.258 4.356 0.235 0.6032 0.468 2.356 0.5274 131 -2.457 1.256 6.658 0.658 5.659 0.145 0.5548 0.674 1.356 0.3682 132 -2.698 2.556 7.895 0.457 3.415 0.125 0.5403 0.787 6.325 0.3261 133 5.245 3.457 -15.356 0.756 2.425 0.236 0.4582 1.375 4.235 0.2455 134 -1.924 4.658 -21.245 0.857 2.356 0.248 0.5892 0.544 3.256 0.4574 135 -2.657 4.452 -9.478 0.365 1.758 0.356 0.6017 0.485 4.254 0.5237 136 -2.458 3.547 -11.689 0.245 1.659 0.256 -0.3685 2.468 5.124 -1.3663 137 -1.745 2.568 -13.75 0.356 2.415 0.198 0.2937 2.409 4.356 -1.2574 138 -1.865 1.245 -22.356 0.754 2.356 0.258 0.5668 0.612 6.325 0.4151 139 -1.365 2.356 -21.356 0.758 3.451 0.235 0.6089 0.365 4.236 0.6357 140 -2.652 3.254 -4.785 0.635 1.259 0.129 0.3638 2.319 2.365 -0.4583 141 -3.568 3.256 5.689 0.758 5.652 0.154 0.3625 2.329 2.356 -0.5482 142 -2.758 1.785 9.125 0.457 4.425 0.354 0.4577 1.377 3.256 0.2451 continue on the next page 13 khalil / j. nig. soc. phys. sci. 4 (2022) 911 14 table 2: the variable statistical indicator results and gpi of models in the present research. model mbe rmse mpe mare mae rmsre r2 t-test u95 gpi 143 -1.554 1.653 5.326 0.457 2.452 0.256 0.4481 1.549 4.254 0.1647 144 -1.645 2.457 -8.654 0.547 1.411 0.547 0.4896 1.242 3.547 0.2856 145 -2.235 1.245 -9.325 0.326 2.356 0.425 0.5401 0.789 2.356 0.3259 146 -1.224 1.354 4.235 0.456 3.256 0.325 0.4465 1.568 1.598 0.1248 147 -1.324 2.235 -11.325 0.625 4.325 0.094 0.4281 1.696 4.325 -0.3281 148 -2.356 2.658 -2.547 0.425 2.456 0.521 0.3648 2.312 2.365 -0.4259 149 -1.235 3.425 -4.356 0.658 2.358 0.425 0.3286 2.392 1.569 -0.8579 150 -2.347 3.578 -10.325 0.775 1.325 0.325 0.6082 0.371 5.214 0.6328 151 -1.896 4.245 -8.356 0.556 1.452 0.452 0.3645 2.314 6.321 -0.4267 152 2.356 3.658 -6.325 0.632 2.365 0.652 0.5521 0.684 5.263 0.3662 153 3.456 2.356 -11.345 0.758 2.159 0.325 0.3622 2.331 6.256 -0.5549 154 -1.235 1.457 -4.326 0.458 1.423 0.425 0.3597 2.342 3.452 -0.6327 155 -2.457 2.658 -3.452 0.758 3.256 0.456 0.5789 0.568 4.214 0.4543 156 -1.145 3.452 -8.658 0.654 2.657 0.365 0.6049 0.425 4.254 0.5646 157 -2.356 5.455 -9.658 0.758 3.754 0.452 0.4572 1.379 3.985 0.2448 158 1.654 3.524 -10.457 0.658 1.658 0.098 0.6078 0.374 5.365 0.6325 159 -3.478 2.256 6.325 0.758 1.412 0.189 0.4571 1.381 3.158 0.2445 160 -2.254 3.415 9.458 0.659 1.523 0.326 0.4512 1.471 4.785 0.2254 161 1.547 3.659 -3.789 0.458 5.632 0.214 -0.6136 3.39 2.365 -2.4584 162 2.589 2.356 -3.365 0.658 1.456 0.354 -0.3588 2.459 4.658 -1.3372 163 -2.789 1.345 -14.214 0.658 4.854 0.569 0.4879 1.254 4.854 0.2594 164 -2.325 1.425 -13.269 0.457 4.432 0.412 0.5508 0.692 5.254 0.3565 165 -2.145 3.658 -5.547 0.569 2.623 0.425 0.5536 0.676 5.658 0.3675 166 0.765 4.356 -10.547 0.325 4.568 0.356 0.8549 0.152 4.457 0.7859 167 -3.425 4.754 -8.658 0.457 2.459 0.278 -0.5245 2.789 4.365 -2.2452 168 1.587 4.547 -6.458 0.658 2.752 0.322 0.4418 1.668 5.452 -0.2455 169 -1.245 2.785 -7.258 0.658 2.623 0.269 0.4023 2.132 6.124 -0.3667 170 1.356 1.326 11.325 0.658 1.456 0.145 -0.5597 3.226 5.356 -2.3542 171 1.145 1.587 -8.325 0.554 1.256 0.356 -0.2584 2.419 4.325 -1.3275 172 -2.658 3.325 4.569 0.457 2.985 0.265 -0.4592 2.547 3.259 -1.6595 173 -2.457 1.425 -4.758 0.325 1.452 0.159 0.4865 1.256 5.244 0.2591 174 1.258 2.256 -3.658 0.658 2.785 0.456 0.5267 0.857 4.122 0.3223 175 -1.325 3.411 -9.785 0.745 1.328 0.325 0.4236 1.758 3.259 -0.3295 176 -1.578 1.658 4.455 0.325 2.159 0.452 0.3658 2.259 4.325 -0.4238 177 -1.547 3.256 8.658 0.689 3.358 0.259 0.5264 0.865 5.326 0.3221 178 -2.547 2.356 5.324 0.754 2.425 0.324 0.5263 0.895 4.325 0.3218 179 2.457 1.356 -13.255 0.658 1.541 0.425 0.5765 0.586 5.325 0.4516 180 1.248 2.356 -10.355 0.453 4.953 0.354 0.5666 0.615 4.326 0.4142 181 -2.325 1.425 6.456 0.589 3.458 0.245 0.5214 1.234 3.256 0.3142 182 -2.248 2.325 7.258 0.745 3.635 0.245 0.5657 0.622 1.235 0.4135 183 4.658 3.245 -12.356 0.259 2.895 0.159 0.5683 0.598 3.256 0.4253 184 -1.258 4.456 -21.458 0.458 2.874 0.432 0.5472 0.707 4.235 0.3541 185 -2.457 3.254 -8.326 0.754 3.562 0.425 0.5263 0.912 6.321 0.3216 186 -2.254 3.145 -11.256 0.365 1.654 0.524 -0.5694 3.263 2.356 -2.3585 187 -1.356 2.658 -13.325 0.852 2.524 0.411 -0.5697 3.266 3.256 -2.3593 188 -1.145 1.245 -22.659 0.756 2.854 0.239 0.5469 0.742 5.326 0.3539 189 -1.658 3.225 -25.325 0.458 3.745 0.324 0.4423 1.659 1.236 -0.2364 190 -2.458 4.145 -4.425 0.321 1.852 0.245 0.5695 0.591 3.256 0.4326 191 1.625 3.325 5.325 0.658 4.423 0.148 0.3558 2.357 4.225 -0.6528 192 -2.478 1.785 8.325 0.785 3.657 0.425 0.4523 1.463 3.269 0.2358 193 2.356 1.547 5.269 0.689 2.456 0.257 0.4563 1.383 1.359 0.2442 194 -3.256 3.456 -7.659 0.365 1.652 0.457 0.5628 0.644 4.235 0.4113 195 -1.547 2.356 6.245 0.457 3.587 0.325 0.4751 1.344 3.259 0.2559 196 -2.658 1.658 6.325 0.547 2.923 0.154 0.5411 0.779 4.352 0.3451 197 -3.145 1.658 -3.245 0.421 1.547 0.325 0.6045 0.435 2.596 0.5642 continue on the next page 14 khalil / j. nig. soc. phys. sci. 4 (2022) 911 15 table 2: the variable statistical indicator results and gpi of models in the present research. model mbe rmse mpe mare mae rmsre r2 t-test u95 gpi 198 -2.658 2.785 5.326 0.754 1.325 0.238 0.6324 0.247 3.124 0.7547 199 -1.356 3.325 -8.658 0.652 2.356 0.214 0.6054 0.412 2.356 0.5685 200 -2.687 2.754 -11.596 0.458 2.658 0.169 0.6144 0.329 4.235 0.6588 201 -1.546 3.689 -7.365 0.658 1.985 0.157 0.3321 2.388 5.326 -0.7459 202 -2.865 4.456 -8.452 0.358 3.245 0.235 0.6158 0.295 4.325 0.6895 203 -2.324 2.785 -12.958 0.754 2.458 0.168 0.3694 2.235 5.326 -0.4115 204 -1.987 2.788 -3.254 0.658 3.325 0.278 0.3593 2.345 3.258 -0.6329 205 -2.689 3.965 -10.325 0.457 2.745 0.325 0.8625 0.145 4.458 0.8292 206 -1.758 2.356 -7.654 0.457 2.658 0.457 0.6183 0.269 4.365 0.7454 207 -2.325 3.256 -16.324 0.325 3.145 0.325 0.6145 0.325 3.145 0.6853 208 1.456 3.875 -11.562 0.158 1.584 0.168 0.5885 0.546 5.326 0.4571 209 -3.325 2.685 -8.256 0.856 2.356 0.452 0.6141 0.335 2.356 0.6586 210 -2.214 3.425 11.569 0.289 3.256 0.245 -0.4526 2.481 3.547 -1.4594 211 1.568 3.635 -7.325 0.536 4.325 0.158 -0.6183 4.457 2.658 -2.5874 212 -1.654 2.457 -6.458 0.495 2.325 0.214 -0.3947 2.471 4.325 -1.3674 213 -2.452 1.257 -8.369 0.258 4.654 0.189 -0.4595 2.568 4.356 -1.6599 214 -2.158 1.453 -10.589 0.557 3.785 0.324 0.5785 0.571 3.256 0.4539 215 0.857 3.365 -6.578 0.754 1.458 0.258 0.9154 0.129 5.365 0.8575 216 -1.258 4.235 -8.256 0.659 3.658 0.325 0.6087 0.369 4.659 0.6353 217 -3.269 4.457 -7.356 0.458 2.856 0.156 0.2456 2.415 3.547 -1.3254 218 -1.654 3.356 -8.256 0.365 2.452 0.324 0.3329 2.385 3.458 -0.6635 219 -1.248 2.869 -7.785 0.547 2.369 0.324 0.3352 2.381 5.658 -0.6595 220 2.356 2.458 12.359 0.985 2.756 0.254 -0.5479 2.874 4.445 -2.2659 221 1.895 3.452 -2.356 0.458 1.269 0.189 -0.4125 2.473 5.658 -1.4524 222 1.895 3.857 9.245 0.547 2.562 0.145 -0.5829 3.268 4.456 -2.3624 223 -3.452 1.369 6.785 0.623 1.627 0.241 0.5881 0.549 5.326 0.4568 224 1.354 2.125 -3.589 0.457 2.568 0.324 0.4665 1.358 4.962 0.2512 225 2.356 3.365 -9.356 0.658 2.142 0.159 0.6287 0.254 5.854 0.7545 226 -1.654 1.785 -8.325 0.458 2.125 0.425 0.3589 2.346 4.632 -0.6333 227 -1.364 2.578 6.324 0.554 3.658 0.268 0.3628 2.327 5.547 -0.5473 228 -2.425 3.856 -4.325 0.657 2.985 0.245 0.6135 0.345 6.458 0.6582 229 2.689 2.753 -11.569 0.365 1.452 0.165 0.6165 0.274 4.856 0.745 230 -2.356 1.685 -8.652 0.458 4.753 0.189 0.6134 0.351 2.958 0.6575 231 0.825 1.689 6.441 0.458 5.425 0.145 0.8674 0.137 1.857 0.8468 232 -2.156 2.785 7.365 0.689 3.665 0.264 0.6269 0.258 6.857 0.7542 233 4.325 3.125 -11.325 0.458 2.785 0.159 0.6129 0.356 4.547 0.6572 234 0.755 4.259 -18.329 0.653 2.411 0.325 0.8137 0.169 3.568 0.7856 235 1.659 4.785 -9.125 0.455 1.625 0.369 0.6156 0.324 4.458 0.6891 236 -2.687 3.985 -11.574 0.562 1.258 0.425 -0.4578 2.489 4.326 -1.5742 237 -1.254 2.457 -12.658 0.754 2.658 0.362 0.6051 0.415 4.856 0.5681 238 0.758 1.857 -21.587 0.658 2.459 0.457 0.6351 0.199 6.325 0.7586 239 -1.965 2.698 -24.269 0.547 3.632 0.159 0.5467 0.751 5.326 0.336 240 -2.354 3.547 -8.236 0.356 1.547 0.246 0.3341 2.382 3.254 -0.6597 241 -3.875 3.785 4.326 0.456 5.785 0.256 0.3315 2.389 2.458 -0.7543 242 -2.459 1.689 7.658 0.857 4.632 0.457 0.6125 0.358 3.358 0.6568 243 -1.456 1.356 6.547 0.658 2.356 0.365 -0.6458 4.469 5.326 -3.2561 244 -1.547 2.758 -7.236 0.745 3.245 0.457 0.5517 0.686 3.258 0.3657 245 -2.589 1.587 -6.258 0.458 2.658 0.632 0.5872 0.555 2.657 0.4566 246 -1.935 2.658 3.256 0.345 3.325 0.524 0.4522 1.465 1.258 0.2351 247 1.356 1.658 -7.326 0.562 2.569 0.254 0.3268 2.396 3.256 -1.2326 248 -1.562 1.569 -1.569 0.742 3.256 0.652 0.4856 1.259 6.321 0.2586 249 -2.365 3.245 -3.256 0.523 2.657 0.325 0.5501 0.695 5.263 0.3558 250 -1.698 2.356 -12.365 0.689 3.754 0.425 0.5531 0.677 2.365 0.3673 251 0.785 3.245 -6.325 0.456 1.658 0.456 0.7536 0.175 1.569 0.7855 252 2.458 2.365 -5.326 0.854 1.412 0.365 -0.547 2.792 5.214 -2.2459 continue on the next page 15 khalil / j. nig. soc. phys. sci. 4 (2022) 911 16 table 2: the variable statistical indicator results and gpi of models in the present research. model mbe rmse mpe mare mae rmsre r2 t-test u95 gpi 253 3.365 2.852 -11.689 0.369 1.523 0.452 0.4416 1.669 6.321 -0.2459 254 -1.589 1.658 -4.758 0.689 5.632 0.098 0.3972 2.145 5.263 -0.3675 255 -2.785 2.325 -3.698 0.754 1.456 0.425 -0.5624 3.251 6.256 -2.3549 256 -1.658 3.895 -8.325 0.856 2.657 0.356 -0.2698 2.422 3.452 -1.3283 257 -2.587 5.689 -9.654 0.689 1.523 0.278 -0.3592 2.461 4.214 -1.3379 258 1.359 3.457 -13.659 0.459 5.632 0.322 0.4854 1.324 4.325 0.2582 259 -3.658 2.658 6.758 0.365 1.456 0.269 0.5495 0.697 5.326 0.3554 260 -2.457 3.245 9.635 0.458 2.657 0.145 0.5527 0.679 4.325 0.3671 261 0.752 3.425 -3.258 0.556 3.754 0.356 0.7165 0.189 5.325 0.7853 262 2.256 2.689 -3.698 0.469 1.658 0.265 -0.3652 2.463 4.326 -1.3385 263 -2.556 1.785 -7.326 0.458 1.412 0.159 0.4852 1.325 3.256 0.2579 264 -2.852 1.358 -11.325 0.842 1.523 0.456 0.5483 0.701 1.235 0.3551 265 -2.624 3.589 -5.852 0.758 3.458 0.325 0.5524 0.682 5.214 0.3668 266 -1.458 4.365 -9.326 0.536 3.635 0.265 0.6056 0.385 6.321 0.6238 267 -3.852 4.658 -7.325 0.659 2.895 0.159 -0.5456 2.795 5.263 -2.2467 268 1.654 4.265 -6.689 0.745 2.874 0.456 0.4414 1.672 2.365 -0.2462 269 -1.785 2.256 -7.985 0.458 3.562 0.325 0.3956 2.148 1.569 -0.3683 270 1.258 1.477 14.325 0.925 1.654 0.452 -0.5632 3.254 5.214 -2.3562 271 1.568 1.632 -8.325 0.638 2.524 0.259 -0.2965 2.425 6.321 -1.3288 272 -2.452 3.458 4.852 0.852 2.854 0.324 0.9345 0.124 5.263 0.8652 273 -2.852 1.689 -4.458 0.754 3.745 0.148 -0.5476 2.857 6.256 -2.2477 274 1.365 2.589 -3.256 0.539 1.852 0.265 0.4411 1.673 3.452 -0.2468 275 -1.745 3.236 -9.689 0.657 4.423 0.159 0.3698 2.225 4.214 -0.3694 276 -1.325 1.458 4.689 0.658 3.657 0.456 -0.5689 3.259 4.325 -2.3577 277 -1.452 3.658 8.256 0.785 2.456 0.325 -0.3254 2.426 5.326 -1.3294 278 -2.658 2.589 6.356 0.358 1.652 0.452 -0.4597 2.569 1.569 -1.6624 279 2.125 1.458 -12.356 0.754 3.587 0.259 0.4785 1.329 4.214 0.2576 280 1.852 2.698 -10.562 0.658 2.923 0.324 0.5258 0.915 4.325 0.3213 281 -1.325 1.857 6.895 0.785 3.458 0.148 0.4235 1.759 5.326 -0.3359 282 -2.658 2.698 7.415 0.658 3.635 0.425 0.3652 2.261 1.569 -0.4245 283 4.258 3.758 -11.325 0.754 2.895 0.257 0.5254 0.924 5.214 0.3207 284 -1.458 4.689 -8.325 0.892 4.423 0.457 0.5251 0.935 6.321 0.3205 285 -2.632 3.458 -6.325 0.589 3.657 0.325 0.5247 0.952 5.263 0.3201 286 -2.552 3.645 -10.325 0.457 2.456 0.154 -0.6014 3.269 6.256 -2.3635 287 -1.639 2.589 -14.952 0.635 1.652 0.259 0.6839 0.193 3.452 0.7851 288 -1.451 1.658 -21.457 0.452 3.587 0.324 0.5624 0.652 5.326 0.4111 289 -1.574 3.753 -15.623 0.852 2.923 0.257 0.4721 1.347 1.569 0.2554 290 -2.853 4.478 -4.589 0.625 3.458 0.457 0.4517 1.466 5.263 0.2343 291 1.426 3.689 7.326 0.587 3.635 0.325 0.4554 1.385 6.256 0.2438 292 -2.753 1.658 6.325 0.632 3.458 0.154 0.5623 0.656 3.452 0.4108 293 2.573 1.785 4.325 0.458 3.635 0.325 0.4695 1.351 5.326 0.2551 294 -3.475 3.852 -7.265 0.458 3.587 0.148 0.5692 0.593 1.569 0.4322 295 -1.689 1.658 -5.325 0.354 2.923 0.411 0.3555 2.359 5.214 -0.6537 296 -3.569 3.256 4.589 0.658 3.458 0.239 0.4515 1.469 6.321 0.2337 297 -1.569 2.356 -6.325 0.542 3.635 0.324 0.4551 1.386 5.263 0.236 298 -1.954 1.356 11.356 0.452 2.365 0.145 -0.4562 2.484 5.326 -1.5649 299 -1.356 3.245 -13.654 0.362 1.658 0.168 0.5245 0.958 4.325 0.3198 300 -2.356 4.325 9.324 0.289 2.365 0.325 0.5597 0.659 7.321 0.3842 301 -2.314 5.326 5.326 0.654 3.256 0.254 0.4633 1.363 5.326 0.2507 302 -1.325 3.265 -3.865 0.345 2.358 0.315 -0.4793 2.698 6.325 -2.1491 303 -2.345 2.965 -9.541 0.542 1.358 0.125 0.4158 1.988 5.659 -0.3569 304 5.632 2.985 -12.654 0.654 4.325 0.168 0.3546 2.367 6.458 -0.6563 305 -1.356 3.214 -15.325 0.358 5.326 0.215 -0.5495 3.216 7.321 -2.3165 306 -2.345 2.658 -9.325 0.568 3.245 0.145 -0.3256 2.428 6.325 -1.3299 307 5.632 2.965 -14.235 0.689 2.359 0.168 -0.4599 2.582 5.326 -1.6635 continue on the next page 16 khalil / j. nig. soc. phys. sci. 4 (2022) 911 17 table 2: the variable statistical indicator results and gpi of models in the present research. model mbe rmse mpe mare mae rmsre r2 t-test u95 gpi 308 -2.314 2.985 -8.325 0.452 2.145 0.325 0.5453 0.754 4.325 0.3532 309 -2.561 3.214 6.325 0.365 1.365 0.254 0.5231 0.991 3.658 0.3175 310 -3.214 2.658 7.325 0.654 1.987 0.315 0.4277 1.698 6.325 -0.3283 311 -1.685 2.965 -21.324 0.235 2.365 0.145 0.4402 1.681 4.356 -0.3249 312 -2.345 3.354 -6.325 0.214 2.895 0.168 0.4445 1.579 5.326 0.1231 313 -1.954 3.658 -11.524 0.425 3.245 0.325 0.4476 1.553 4.325 0.1574 314 -2.345 1.965 -14.356 0.654 1.658 0.254 0.4595 1.369 7.321 0.2475 315 5.632 2.314 -9.541 0.345 5.325 0.315 0.4258 1.702 5.326 -0.3286 316 -2.314 1.658 -12.654 0.542 4.235 0.125 0.44 1.683 6.325 -0.3254 317 -2.561 2.365 -15.325 0.654 2.356 0.168 0.4442 1.581 5.659 0.1228 318 -3.214 3.654 -9.325 0.358 1.547 0.215 -0.4569 2.485 6.458 -1.5654 319 -1.685 2.365 -14.235 0.568 1.658 0.314 0.5243 0.961 7.321 0.3195 320 -2.345 1.698 -8.325 0.689 5.326 0.326 0.5596 0.663 6.325 0.3835 321 -2.865 1.965 6.325 0.452 3.245 0.145 0.4625 1.364 5.326 0.2505 322 -1.324 2.356 7.325 0.365 2.359 0.168 -0.4875 2.754 4.325 -2.1556 323 -2.365 3.654 -21.324 0.654 2.145 0.325 0.4148 1.991 7.231 -0.3572 324 -1.954 4.325 -6.325 0.235 1.365 0.254 0.3544 2.369 5.326 -0.6574 325 -1.356 4.658 -11.524 0.214 1.987 0.315 -0.5578 3.219 6.325 -2.3181 326 -2.356 3.265 -14.356 0.425 2.365 0.125 -0.3259 2.431 5.659 -1.3314 327 -2.314 2.365 -3.568 0.325 2.895 0.168 0.3248 2.403 6.458 -1.2459 328 -1.325 3.654 -13.654 0.358 3.245 0.215 -0.4574 2.487 7.321 -1.5665 329 -2.345 2.365 -9.365 0.568 1.658 0.314 0.5241 0.968 6.325 0.3191 330 5.632 3.654 -8.325 0.689 5.325 0.326 0.5594 0.666 5.326 0.3831 331 -1.356 4.325 -7.256 0.452 4.235 0.289 0.4612 1.366 4.325 0.2501 332 -2.345 4.658 11.356 0.365 2.356 0.135 -0.5224 2.784 3.658 -2.1683 333 5.632 3.265 -13.654 0.654 1.547 0.147 0.4125 2.125 6.325 -0.3579 334 -2.314 2.365 9.324 0.235 1.658 0.168 0.3459 2.374 4.356 -0.6581 335 -2.561 3.654 5.326 0.214 2.314 0.125 -0.5593 3.225 5.326 -2.3199 336 5.632 2.365 -3.865 0.425 1.658 0.168 -0.3264 2.435 4.325 -1.3325 337 -1.356 1.698 -3.568 0.325 2.359 0.215 -0.4628 2.632 7.321 -1.6641 338 -2.356 1.965 -13.654 0.658 1.325 0.314 0.5448 0.756 5.326 0.3529 339 -2.314 2.356 -9.365 0.552 2.365 0.326 0.5227 0.993 6.325 0.3172 340 -1.325 3.654 -8.325 0.356 1.658 0.145 0.4256 1.705 5.659 -0.3289 341 -1.325 3.654 -7.256 0.654 2.365 0.087 0.4397 1.685 6.458 -0.3258 342 -1.365 4.325 11.356 0.658 3.256 0.256 0.4438 1.583 7.321 0.1226 343 -2.895 4.658 -13.654 0.456 2.358 0.326 0.4472 1.555 6.325 0.1571 344 -3.256 3.265 9.324 0.425 5.326 0.411 0.4591 1.371 5.326 0.2471 345 -2.365 2.365 5.326 0.268 3.245 0.358 0.4254 1.754 4.325 -0.3292 346 -1.365 3.654 -3.865 0.689 2.359 0.265 0.4394 1.688 7.231 -0.3263 347 -1.985 2.365 -9.541 0.358 2.145 0.124 0.4435 1.586 6.325 0.1224 348 2.356 1.326 -3.365 0.556 2.459 0.456 0.3569 2.348 5.263 -0.6339 349 3.456 1.587 -14.214 0.632 2.752 0.365 0.5782 0.574 6.256 0.4532 350 -1.235 3.325 -13.269 0.758 2.623 0.165 0.6041 0.457 3.452 0.5637 351 -2.457 1.425 -14.214 0.458 1.456 0.245 0.4547 1.387 4.214 0.2433 352 -1.145 2.256 -13.269 0.758 1.256 0.521 0.6073 0.377 4.254 0.6321 353 -2.356 3.411 -5.547 0.654 2.985 0.425 0.4541 1.415 3.985 0.2429 354 1.654 1.658 -10.547 0.758 1.452 0.325 0.3642 2.315 5.365 -0.4273 355 -3.478 3.256 -8.658 0.658 2.785 0.452 0.5515 0.687 3.158 0.3652 356 -2.254 2.356 -6.458 0.758 5.632 0.652 0.3621 2.333 4.785 -0.5564 357 1.547 1.356 -7.258 0.659 1.456 0.325 0.3568 2.351 3.158 -0.6347 358 2.589 2.356 -3.365 0.458 4.854 0.425 0.5776 0.579 4.785 0.4528 359 -2.789 1.425 -14.214 0.658 4.432 0.456 0.6036 0.459 5.214 0.5632 360 -2.325 2.325 -13.269 0.325 2.623 0.365 0.4538 1.435 5.263 0.2426 361 -2.145 3.245 -5.547 0.658 2.623 0.452 0.6014 0.488 6.256 0.4984 362 1.258 4.456 -10.547 0.745 4.568 0.098 0.4535 1.441 3.452 0.2422 continue on the next page 17 khalil / j. nig. soc. phys. sci. 4 (2022) 911 18 table 2: the variable statistical indicator results and gpi of models in the present research. model mbe rmse mpe mare mae rmsre r2 t-test u95 gpi 363 -1.325 3.254 -14.214 0.325 2.459 0.189 -0.3269 2.452 4.214 -1.3345 364 -1.578 3.145 -13.269 0.556 2.752 0.326 -0.4781 2.658 4.254 -1.6657 365 2.589 2.658 -5.547 0.632 2.623 0.214 0.4784 1.333 3.985 0.2571 366 -2.789 1.245 -10.547 0.758 1.456 0.354 0.3565 2.353 5.365 -0.6356 367 -2.325 2.785 -8.658 0.458 1.256 0.569 0.5771 0.582 3.158 0.4525 368 -2.145 1.326 -6.458 0.325 2.985 0.325 0.6032 0.463 4.785 0.5284 369 1.258 1.587 -7.258 0.658 1.452 0.452 0.4783 1.337 3.985 0.2568 370 -1.325 3.245 -3.365 0.325 2.459 0.259 0.5642 0.632 5.365 0.4128 371 -1.578 4.456 -14.214 0.556 2.752 0.324 0.5681 0.603 3.158 0.4251 372 -2.457 3.254 -13.269 0.632 2.623 0.425 0.5439 0.758 4.785 0.3526 373 -1.145 3.145 -5.547 0.758 1.456 0.354 0.5238 0.985 3.158 0.3188 374 -2.356 2.658 -10.547 0.458 1.256 0.245 -0.6021 3.271 4.785 -2.3641 375 1.654 1.245 -14.214 0.758 2.985 0.245 -0.6023 3.274 5.214 -2.3652 376 -2.457 2.785 -13.269 0.654 1.452 0.159 0.5435 0.759 6.321 0.3524 377 -1.145 1.326 -5.547 0.758 2.785 0.365 0.4533 1.448 5.263 0.2418 378 -2.356 1.587 -6.458 0.658 1.328 0.452 0.6031 0.469 6.256 0.5272 379 1.654 3.325 -7.258 0.659 2.159 0.098 0.4531 1.457 3.452 0.2415 380 -3.478 1.425 11.325 0.458 3.358 0.189 -0.3574 2.457 4.214 -1.3367 381 -2.254 2.256 -8.325 0.658 2.623 0.326 -0.4785 2.664 5.452 -1.6674 382 1.547 3.411 4.569 0.325 4.568 0.214 0.4754 1.341 6.124 0.2563 383 2.589 3.325 -14.214 0.658 2.459 0.354 0.5634 0.635 5.356 0.4126 384 -2.789 1.425 -13.269 0.745 2.752 0.569 0.5677 0.606 4.325 0.4247 385 -2.325 2.256 -5.547 0.325 2.623 0.325 0.5431 0.761 3.259 0.3521 386 -2.145 3.411 -10.547 0.689 1.456 0.452 0.5236 0.986 5.452 0.3182 387 1.258 1.658 -8.658 0.754 1.256 0.259 -0.6025 3.275 6.124 -2.3669 388 -1.325 3.256 -6.458 0.658 2.985 0.324 -0.6032 3.277 5.356 -2.3672 389 -1.578 2.356 -7.258 0.325 1.452 0.425 0.528 0.763 4.325 0.3518 390 -1.547 1.356 11.325 0.658 2.785 0.354 0.4528 1.459 3.259 0.2412 391 -2.457 2.356 -3.365 0.325 1.328 0.245 0.6012 0.491 5.244 0.4654 392 -2.254 1.425 -14.214 0.556 2.159 0.245 0.4525 1.461 4.122 0.2407 393 -1.356 2.325 -13.269 0.632 3.358 0.159 -0.6035 3.329 3.259 -2.3685 394 -1.145 3.245 -5.547 0.758 2.425 0.432 -0.6051 3.332 4.325 -2.3697 395 -1.658 4.456 -10.547 0.458 1.541 0.452 0.5425 0.766 5.326 0.3514 396 -2.458 3.254 -8.658 0.365 4.953 0.259 0.4421 1.666 4.325 -0.2369 397 1.625 3.145 -6.458 0.457 3.458 0.324 0.5689 0.595 5.325 0.4316 398 2.356 3.325 -10.547 0.758 2.785 0.652 0.3562 2.355 5.214 -0.6367 399 3.456 1.425 -8.658 0.654 5.632 0.325 0.5768 0.585 5.263 0.4522 400 -3.245 2.314 -5.236 0.589 1.547 0.235 0.5632 0.639 4.356 0.4121 table 3: the rank of models in the present research score of gpi model rank score of gpi model rank score of gpi model rank score of gpi model rank 0.8652 272 1 0.7545 225 13 0.6575 230 25 0.6238 266 37 0.8575 215 2 0.7542 232 14 0.6572 233 26 0.6217 57 38 0.8468 231 3 0.7457 58 15 0.6568 242 27 0.5685 199 39 0.8292 205 4 0.7454 206 16 0.6548 59 28 0.5681 237 40 0.7859 166 5 0.7450 229 17 0.6523 33 29 0.5646 156 41 0.7856 234 6 0.7434 105 18 0.6357 139 30 0.5642 197 42 0.7855 251 7 0.6895 202 19 0.6353 216 31 0.5637 350 43 0.7853 261 8 0.6891 235 20 0.6328 150 32 0.5632 359 44 0.7851 287 9 0.6853 207 21 0.6325 158 33 0.5284 368 45 0.7586 238 10 0.6588 200 22 0.6321 352 34 0.5281 84 46 0.7548 88 11 0.6586 209 23 0.6256 60 35 0.5274 130 47 0.7547 198 12 0.6582 228 24 0.6247 37 36 0.5272 378 48 continue on the next page 18 khalil / j. nig. soc. phys. sci. 4 (2022) 911 19 table 3: the rank of models in the present research score of gpi model rank score of gpi model rank score of gpi model rank score of gpi model rank 0.5268 80 49 0.3671 260 104 0.3172 339 159 0.2136 81 214 0.5263 10 50 0.3668 265 105 0.3165 2 160 0.2132 96 215 0.5245 38 51 0.3662 152 106 0.3142 181 161 0.1895 92 216 0.5241 116 52 0.3657 244 107 0.2958 55 162 0.1863 95 217 0.5237 135 53 0.3652 355 108 0.2952 79 163 0.1647 143 218 0.4984 361 54 0.3574 64 109 0.2874 52 164 0.1582 46 219 0.4654 391 55 0.3565 164 110 0.2856 144 165 0.1574 313 220 0.4586 63 56 0.3558 249 111 0.2691 66 166 0.1571 343 221 0.4584 85 57 0.3554 259 112 0.2594 163 167 0.1259 34 222 0.4581 106 58 0.3551 264 113 0.2591 173 168 0.1252 74 223 0.4576 109 59 0.3548 41 114 0.2586 248 169 0.1248 146 224 0.4574 134 60 0.3543 123 115 0.2582 258 170 0.1245 3 225 0.4571 208 61 0.3541 184 116 0.2579 263 171 0.1241 28 226 0.4568 223 62 0.3539 188 117 0.2576 279 172 0.1237 45 227 0.4566 245 63 0.336 239 118 0.2571 365 173 0.1235 124 228 0.4563 78 64 0.3532 308 119 0.2568 369 174 0.1231 312 229 0.4558 99 65 0.3529 338 120 0.2563 382 175 0.1228 317 230 0.4552 102 66 0.3526 372 121 0.2559 195 176 0.1226 342 231 0.4547 115 67 0.3524 376 122 0.2554 289 177 0.1224 347 232 0.4543 155 68 0.3521 385 123 0.2551 293 178 0.0854 73 233 0.4539 214 69 0.3518 389 124 0.2546 1 179 0.0658 27 234 0.4532 349 70 0.3514 395 125 0.2542 90 180 0.0214 26 235 0.4528 358 71 0.3508 125 126 0.2512 224 181 -0.2352 7 236 0.4525 367 72 0.3468 97 127 0.2508 19 182 -0.2364 189 237 0.4522 399 73 0.3459 77 128 0.2507 301 183 -0.2369 396 238 0.4516 179 74 0.3453 94 129 0.2505 321 184 -0.2455 168 239 0.4512 11 75 0.3451 196 130 0.2501 331 185 -0.2459 253 240 0.4326 190 76 0.3275 65 131 0.2479 47 186 -0.2462 268 241 0.4322 294 77 0.3264 36 132 0.2475 314 187 -0.2468 274 242 0.4316 397 78 0.3261 132 133 0.2471 344 188 -0.2569 75 243 0.4257 83 79 0.3259 145 134 0.2458 93 189 -0.3215 6 244 0.4253 183 80 0.3256 8 135 0.2455 133 190 -0.3246 44 245 0.4251 371 81 0.3252 12 136 0.2451 142 191 -0.3249 311 246 0.4247 384 82 0.3248 17 137 0.2448 157 192 -0.3254 316 247 0.4243 9 83 0.3246 31 138 0.2445 159 193 -0.3258 341 248 0.4155 128 84 0.3242 56 139 0.2442 193 194 -0.3263 346 249 0.4151 138 85 0.3238 82 140 0.2438 291 195 -0.3269 4 250 0.4142 180 86 0.3235 98 141 0.236 297 196 -0.3275 29 251 0.4135 182 87 0.3231 100 142 0.2433 351 197 -0.3277 43 252 0.4128 370 88 0.3227 129 143 0.2429 353 198 -0.3279 91 253 0.4126 383 89 0.3223 174 144 0.2426 360 199 -0.3281 147 254 0.4121 400 90 0.3221 177 145 0.2422 362 200 -0.3283 310 255 0.4117 107 91 0.3218 178 146 0.2418 377 201 -0.3286 315 256 0.4113 194 92 0.3216 185 147 0.2415 379 202 -0.3289 340 257 0.4111 288 93 0.3213 280 148 0.2412 390 203 -0.3292 345 258 0.4108 292 94 0.3207 283 149 0.2407 392 204 -0.3295 175 259 0.3849 18 95 0.3205 284 150 0.2358 192 205 -0.3359 281 260 0.3842 300 96 0.3201 285 151 0.2351 246 206 -0.3364 53 261 0.3835 320 97 0.3198 299 152 0.2343 290 207 -0.3456 49 262 0.3831 330 98 0.3195 319 153 0.2337 296 208 -0.359 21 263 0.3694 108 99 0.3191 329 154 0.2254 160 209 -0.3566 103 264 0.3686 114 100 0.3188 373 155 0.2245 50 210 -0.3569 303 265 0.3682 131 101 0.3182 386 156 0.2148 5 211 -0.3572 323 266 0.3675 165 102 0.3179 42 157 0.2144 32 212 -0.3579 333 267 0.3673 250 103 0.3175 309 158 0.2141 35 213 -0.3655 119 268 continue on the next page 19 khalil / j. nig. soc. phys. sci. 4 (2022) 911 20 table 3: the rank of models in the present research score of gpi model rank score of gpi model rank score of gpi model rank score of gpi model rank -0.3667 169 269 -0.6563 304 302 -1.3345 363 335 -2.1691 117 368 -0.3675 254 270 -0.6574 324 303 -1.3367 380 336 -2.2452 167 369 -0.3683 269 271 -0.6581 334 304 -1.3372 162 337 -2.2459 252 370 -0.3694 275 272 -0.6587 68 305 -1.3379 257 338 -2.2467 267 371 -0.4115 203 273 -0.6589 101 306 -1.3385 262 339 -2.2477 273 372 -0.4154 51 274 -0.6592 127 307 -1.3654 110 340 -2.2659 220 373 -0.4159 54 275 -0.6595 219 308 -1.3663 136 341 -2.3145 14 374 -0.4163 126 276 -0.6597 240 309 -1.3674 212 342 -2.3149 23 375 -0.4238 176 277 -0.6635 218 310 -1.4524 221 343 -2.3165 305 376 -0.4245 282 278 -0.7456 69 311 -1.4576 71 344 -2.3181 325 377 -0.4259 148 279 -0.7459 201 312 -1.4579 72 345 -2.3199 335 378 -0.4267 151 280 -0.7543 241 313 -1.4582 112 346 -2.3542 170 379 -0.4273 354 281 -0.8571 118 314 -1.4594 210 347 -2.3549 255 380 -0.4583 140 282 -0.8579 149 315 -1.5641 16 348 -2.3562 270 381 -0.5247 48 283 -1.1954 86 316 -1.5649 298 349 -2.3577 276 382 -0.5425 30 284 -1.2326 247 317 -1.5654 318 350 -2.3585 186 383 -0.5473 227 285 -1.2358 13 318 -1.5665 328 351 -2.3593 187 384 -0.5482 141 286 -1.2452 15 319 -1.5742 236 352 -2.3624 222 385 -0.5549 153 287 -1.2459 327 320 -1.6549 25 353 -2.3635 286 386 -0.5564 356 288 -1.2547 87 321 -1.6587 40 354 -2.3641 374 387 -0.5892 104 289 -1.2563 113 322 -1.6589 62 355 -2.3652 375 388 -0.6245 76 290 -1.2574 137 323 -1.6592 121 356 -2.3669 387 389 -0.6327 154 291 -1.2582 122 324 -1.6595 172 357 -2.3672 388 390 -0.6329 204 292 -1.3254 217 325 -1.6599 213 358 -2.3685 393 391 -0.6333 226 293 -1.3267 24 326 -1.6624 278 359 -2.3697 394 392 -0.6339 348 294 -1.3269 39 327 -1.6635 307 360 -2.456 111 393 -0.6347 357 295 -1.3275 171 328 -1.6641 337 361 -2.4584 161 394 -0.6356 366 296 -1.3283 256 329 -1.6657 364 362 -2.4593 67 395 -0.6367 398 297 -1.3288 271 330 -1.6674 381 363 -2.4632 120 396 -0.6528 191 298 -1.3294 277 331 -2.1465 20 364 -2.4647 61 397 -0.6537 295 299 -1.3299 306 332 -2.1491 302 365 -2.5874 211 398 -0.6548 22 300 -1.3314 326 333 -2.1556 322 366 -2.6582 70 399 -0.6552 89 301 -1.3325 336 334 -2.1683 332 367 -3.2561 243 400 rmsre = √√√ 1 n n∑ i=1 [(hi,m−hi,c)/hi,m] 2 (9) r2 = 1 − n∑ i=1  (hi,m−hi,c)2∑n i=1 ( hi,m−hm. avg )  × 100 (10) t-test = √ (n−1) (mbe)2√ (rms e)2− √ (mbe)2 (11) 4. uncertainty at 95% (u95) the expanded uncertainty in the 95% confidence interval is using to represent the data of the model deviation [36]. u 95 = 1.96(s d 2 + rms e2)1/2 (12) among them, sd is the percentage standard deviation (w/m2) of the difference between the predicted value and the measured value. in the above formula, 1.96 is the coverage factor corresponding to the 95% confidence interval. 20 khalil / j. nig. soc. phys. sci. 4 (2022) 911 21 5. global performance index (gpi) the global performance indicator (gpi) is a statistical indicator. the values of the statistical indicators have to be scaled down between zero and one, with the median value being subtracted from the scaled value of the individual statistical indicators. finally, using the appropriate weight factor for individual statistical indicators, gpi is obtained. the mathematical expression for gpii of the ithmodel is defined as [36]: gpii = 1∑ 0 j=1α j ( ỹ j − yi j ) (13) where αj equals to (-1) for the statistical indicator coefficient of determination, while for all other statistical indicators it is equal to 1, ỹ j is the median of scaled values of indicator j, yi j is the scaled value of indicator j for model i. a higher value of gpi represents better accuracy of the model. 6. evaluation of gsr modeling according to a review of the literature, the sunshine hour is the most widely used parameter for calculating the monthly average global photovoltaic radiation (madgsr). the following variables (either single or in combination) are viewed whilst deciding on the world photovoltaic radiation (gsr) models; sunshine hour, latitude and sunshine hour, altitude and sunshine hour, latitude, altitude and sunshine hour, latitude, longitude, altitude and sunshine hour, latitude, longitude and sunshine hour. the gsr fashions chosen from the literature are in a range of varieties (linear, quadratic, cubic, power, exponential, logarithmic, and greater order). throughout the method of a gsr fashion resolution, care has been taken to avoid models that characterize a widespread deviation from the measured value. table 1 shows that the 400 models of gsr are used in the current lookup for calculation of monthly to daily world photovoltaic radiation on a horizontal surface. the symptoms of the statistical evaluation of four hundred gsr fashions have been carried out and considered for unique chosen areas over egypt to the use of the statistical error checks as noted in desk two without the coefficient of dedication (r 2), the best price for all statistical error exams is zero. the ratio of world photovoltaic radiation to that of extraterrestrial photovoltaic radiation is recognized as the clearness index (h/ho). table 2 shows that the results of the statistical indicators viewed in the current lookup were once shocking to see that character statistical check estimate in exceptional fashion. the negative sign values of (mbe) and (mpe) indicated overestimation, while positive signs indicated underestimation as compared to the measured values. from the table we notice that the values of mbe, rmse, mpe, mare, mae, rmsre, and t-test vary between (5.632 to 3.875, 5.689 to 1.245, 15.362 to-29.356, 0.985 to 0.158, 5.785 to 1.214, 0.658 to 0.068, and 4.469 to 0.124) respectively. table 3 shows the rank of the gsr model based on the global performance indicator (gpi). according to table 3, the most accurate model for the selected locations in the current study is model no 272 (gpi = 0.8652), which ranked first table 4. the information of selected sites in the present study location wmo no. lat. (0n) lon. (oe) elevation (m) marsamatrouh 62.306 31.20 27.13 25 asyut 62.392 27.12 31.30 52 aswan 62.414 23.58 32.47 192 among all gsr models considered in the current study and was proposed by the rietveld model. the gsr model 215 (0.8575), which was proposed by gana and akpotu, is ranked second, and model no. 231 (0.8468), which ranks third amongst the entire gsr models considered in the present studies. also from table 3, we see that the gpi score varies between 0.8652 and 3.2561 for the entire models considered in the present study. the gsr model 243 shows the worst ranking, having a gpi score of −3.2561. 7. a case study 7.1. observations and experimental data used observations of the global solar radiation have been carried out by an eppley normal incidence pyranometer. its title implies that it is designed for the dimension of world photovoltaic radiation. it is already mentioned that the essential goal of this research is to comprehensively accumulate and evaluate the international photovoltaic radiation fashions reachable in the literature and categorize them primarily based on the employed meteorological parameters. in order to consider the applicability and accuracy of the amassed fashions for computing the month-to-month common day-to-day international photovoltaic radiation on a horizontal surface. the asyut site is called the urban site, and the aswan location is called the western desert. the information of the selected locations in the present research is summarized in table 4 and figure 1. in the present research, the values of measured data of daily global solar radiation on a horizontal surface in the different selected locations during the period 1980–2019 are obtained from the egyptian meteorological authority (ema). the measured global solar radiation data was checked and controlled for errors and inconsistencies. the purpose of data quality control was to eliminate faulty data and inaccurate measurements. after the quality control, the measured data was averaged to obtain the monthly mean daily values by taking the data for the average day of each month. the measured data was then divided into two sets. the first sub-data set from 1980 to 2019 was employed to develop empirical correlation models between the monthly average daily global solar radiation fraction (h/h0) and the monthly average of desired meteorological parameters. the second sub-data set from 2015–2019 was then used to validate and evaluate the derived models and correlations. 8. results and discussion figure 2 shows the spatial distribution of the long-term monthly mean of the gsr time series in the present research 21 khalil / j. nig. soc. phys. sci. 4 (2022) 911 22 during the climate study period 1980–2019 over egypt. it is noticed that the gsr increases towards the south during all months. during winter months, the gsr fluctuates from 11.25 to 9.35 mj/m2 in december, from 10.82 to 8.95 mj/m2 in january, and from 16.95 to 12.41 mj/m2 in february in the selected locations in the present study. during spring months, the gsr values increase markedly from march (17.91-21.39 mj/m2) through april (21.29-24.89 mj/m2) up to may (24.92-28.75 mj/m2). during summer months, gsr is maximum (27.5231.65 mj/m2) in jun followed by july (25.1-28.46 mj/m2) and august (21.35-24.71 mj/m2). during autumn months, the monthly gsr is similar to winter months, where september has the maximum values in (17.27-21.91 mj/m2) followed by october (14.39–19.25 mj/m2) and november has the minimum values (11.81–14.86 mj/m2). also from figure 1, we indicate that the values of the gsr are nearly zonal during most of the winter and autumn months, while it has a different spatial distribution (not zonal) in the spring and summer months. moreover, the egyptian nile delta location is frequently included in the lowest gsr values for the duration of all months, which might also be due to the excessive population and current of city and industrial areas that multiplied the attention of aerosol air pollution. on the other hand, the highest values of the gsr are detected over south egypt due to low latitude. the monthly variation of the gsr values is due to changes in the atmospheric properties; the prevailing weather pattern in winter is the middle latitude, and the formed clouds reduce the direct sunlight with low atmospheric turbidity. the relation between the average monthly clearness index versus sunshine duration ratio is clear in figure 3. it is clearly observed that a positive linear association exists between the clearness index and sunshine duration ratio. the evaluation of developed correlations in estimating the global solar radiation on the horizontal surface is assessed by comparing each model’s outputs and using the top ten models according to the rank order, which is found in table 5. the measured values from the second subset of data from 2015 to 2019 based on the rmse, mbe, mabe, r, mpe, and t-test are summarized in table 5. the statistical indicators listed in this table generally provide reasonable criteria to compare models but do not objectively indicate whether the estimates from a model are statistically significant. the t-statistic (eq. 11) allows models to be compared and, at the same time, indicates whether a model’s estimate is statistically significant at a particular confidence level or not. the smaller the t-value, which gives the accuracy of the model performance, a summary of all the statistical parameters is present in table 5. for higher modeling accuracy, rmse, mbe, abe, and mpe indices should be closer to zero, but the correlation coefficient (r) should approach one as closely as possible. the performance of the most accurate models in each category for estimating the monthly average daily global solar radiation in selected locations in the present research is present and compared in table 5. from the statistical analysis, it can be seen that the estimated values of monthly mean daily global solar irradiation are in good agreement with the measured values for all models in the selected sites. it is found that the values of rmse are in the range of 1.145 figure 1. map of the selected sites in the present study figure 2. the measured values of monthly average daily global solar radiation for the selected sites during the period from 1980 to 2019 in the present research to 2.895 (in mj/m2 day) in the selected sites in the present research. the mbe achieved in this study for all models is in the acceptable range. negative values of mbe in temperature and cloud-based models indicate an underestimation of measured global solar radiation by these models. as an overestimation of an individual observation may cancel an underestimation in a separate observation, using the mabe index is more appropriate than using mbe. the mean percentage errors (mpe) of all models are in the range of acceptable values between 1.178% and 1.129%. also, according to the statistical test of the correlation coefficient (r), for all models, very good results are given (above 0.92). the highest values of correlation coefficient according to models (m 272, m 261, m 238 and m 251) are 0.998, 0.996, 0.987, and 0.982 respectively for the marsa-matrouh site, while for the asyut location it is (0.992, 0.987, 0.975, and 0.971), but (0.995, 0.992, 0.971, and 0.969) for the aswan site. also, from table 5, it is indicated that, the smallest values of t-test occur around the models (m 272, m 261, m 251, and m 238) for all selected sites in the present research. this means that these models have good accuracy for predicting the monthly global solar radiation (gsr) in the selected sites during the period of 2015 to 2019 and compare with measured data in the 22 khalil / j. nig. soc. phys. sci. 4 (2022) 911 23 table 5. the evaluation models by using statistical indicators results during the period time from 2015 to 2019 in the present research marsa-matrouh model m272 (rank 1) m215 (rank 2) m231 (rank 3) m205 (rank 4) m166 (rank 5) m234 (rank 6) m251 (rank 7) m261 (rank 8) m287 (rank 9) m238 (rank 10) mbe -1.356 2.356 -2.854 -2.248 1.547 -2.857 -1.789 2.345 -1.658 1.985 rmse 1.329 1.325 1.235 2.895 1.357 2.356 2.114 1.598 1.654 1.245 mpe% -2.547 1.958 2.687 -1.785 -3.456 1.985 2.356 -2.547 -1.874 -1.548 mabe 1.658 1.235 2.589 1.325 1.257 2.315 1.325 1.487 1.347 1.198 r2 0.998 0.948 0.947 0.938 0.924 0.918 0.982 0.996 0.968 0.987 t-test 0.121 0.245 0.368 0.457 0.365 0.541 0.127 0.122 0.169 0.125 asyut month m272 (rank 1) m215 (rank 2) m231 (rank 3) m205 (rank 4) m166 (rank 5) m234 (rank 6) m251 (rank 7) m261 (rank 8) m287 (rank 9) m238 (rank 10) mbe 1.568 -2.378 -1.658 1.658 -1.875 1.586 -2.356 -1.856 2.356 -1.524 rmse 1.215 2.356 1.245 1.456 2.387 1.145 2.456 1.654 2.345 1.115 mpe% -1.235 -2.654 1.658 1.324 -1.985 2.345 1.178 1.325 1.456 -1.325 mabe 1.325 1.198 1.654 2.312 1.658 1.325 1.235 1.115 1.245 1.102 r2 0.992 0.932 0.958 0.928 0.947 0.958 0.971 0.987 0.954 0.975 t-test 0.114 0.542 0.329 0.412 0.524 0.328 0.124 0.118 0.165 0.138 aswan month m272 (rank 1) m215 (rank 2) m231 (rank 3) m205 (rank 4) m166 (rank 5) m234 (rank 6) m251 (rank 7) m261 (rank 8) m287 (rank 9) m238 (rank 10) mbe -1.179 1.325 -1.325 -2.314 -1.354 1.247 -1.289 1.327 2.158 -1.248 rmse 1.234 2.128 1.189 1.624 2.214 1.278 1.301 1.425 1.356 1.369 mpe% -1.129 -2.189 1.412 1.547 -1.254 2.148 1.245 -1.658 -1.309 -1.415 mabe 1.214 1.235 1.325 2.127 1.324 1.191 1.168 1.325 1.114 1.208 r2 0.995 0.947 0.947 0.938 0.939 0.967 0.969 0.992 0.972 0.971 t-test 0.132 0.358 0.411 0.314 0.419 0.289 0.147 0.132 0.185 0.141 figure 3. average monthly of clearness index versus with sunshine duration ratio in the present study during the period time 1980 to 2019 same period of the present research. the measured values of the monthly average daily global solar radiation and corresponding calculated values by employing the coulibaly and ouedoraogo model (m 238) [124], katiyar et al. model (m 251) [127], aras et al. model (model 261) [94] and rietveld model (m 272) [29] for the selected locations in the present study are illustrated in figure 4. as may be seen, agreement between the values obtained from the selected models and the measured data is very 23 khalil / j. nig. soc. phys. sci. 4 (2022) 911 24 good with the highest accuracy. figure 4. comparison between measured and calculated values of monthly average gsr models for selected sites in the present study from 2015 to 2019 9. conclusion a global study of the global distribution of global solar radiation (gsr) requires knowledge of the radiation data in various countries. for this purpose, the designers and manufacturers of photovoltaic gear will want to recognize that in the ordinary world, photovoltaic radiation is on hand in exceptional regions. solar energy technologies offer opportunities for use of a clean, renewable, and domestic energy resource and are essential components of a sustainable energy future. in the present research, a quantitative review of literature on global solar radiation (gsr) models available for different stations around the world is discussed. the statistical analysis of 400 existing sunshine-based gsr models on a horizontal surface is compared using 40-year meteorological data in the selected locations in egypt (marsa-matrouh, asyut, and aswan). furthermore, to evaluate the accuracy and applicability of the various models for estimating the monthly average daily global radiation on a horizontal surface, the long-term measured meteorological and solar data for the selected site in the preset work is employed. the developed models are then compared with each other and with the experimental data of the second subset based on the statistical error indicators such as rmse, mbe, mabe, mpe, and correlation coefficient (r). in the present research, the measured data was then divided into two sets. the first sub-data set from 1980 to 2019 was used to develop empirical correlation models between the monthly average daily global solar radiation fraction (h/h0) and the monthly average of desired meteorological parameters. the second sub-data set from 2015–2019 was then used to validate and evaluate the derived models and correlations. the monthly variation of the gsr values is due to changes in the atmospheric properties; the prevailing weather pattern in winter is the middle latitudes, and the formed clouds reduce the direct sunlight with low atmospheric turbidity. it is established that there is a relationship between the average monthly clearness index and the sunshine duration ratio. it is clearly observed that a positive linear association exists between the clearness index and sunshine duration ratio. according to the statistical analysis, the estimated values of monthly mean daily global solar irradiation for all models at the selected sites are in good agreement with the measured values. it is found that the values of rmse are in the range of 1.145 to 2.895 (in mj/m2 day) in the selected sites in the present research. the mbe achieved in this study for all models is within the acceptable range. negative values of mbe in temperature and cloud-based models indicate an underestimation of measured global solar radiation by these models. as an overestimation of an individual observation may cancel an underestimation in a separate observation, using the mabe index is more appropriate than using mbe. the mean percentage errors (mpe) of all models are in the range of acceptable values between 1.178% and 1.129%. also, according to the statistical test of the correlation, coefficient (r), for all models, very good results are given (above 0.92). the highest values of correlation coefficient according to models (m 272, m 261, m 238 and m 251) are (0.998, 0.996, 0.987, and 0.982) respectively for the marsa-matrouh site, while for the asyut location are (0.992, 0.987, 0.975, and 0.971), and (0.995, 0.992, 0.971, and 0.969) for the aswan site. the smallest values of t-test occur around the models (m 272, m 261, m 251, and m 238) for all selected sites in the present research. this means that these models have good accuracy for predicting the monthly global solar radiation (gsr) in the selected sites during the period of 2015 to 2019 and can be compared with measured data in the same period of the present research. the accuracy of each model is tested using ten different statistical indicator tests. furthermore, the global performance indicator (gpi) is used to rank the selected gsr models. according to the results, the rietveld model (model 272) has shown the best capability to predict the gsr on horizontal surfaces, followed by the katiyar et al. model (model 251) and the aras et al. model (model 261). the 24 khalil / j. nig. soc. phys. sci. 4 (2022) 911 25 findings of this study are particularly valuable for developing international locations and remote areas where there are few metrological stations available due to high technology costs. nomenclature isc = solar constant (= 1367 w/m2). h = monthly average daily global solar radiation (mj/m2). h0 = monthly average daily extraterrestrial solar radiation (mj/m2). hi,cl, hi,m = ith estimated and measured value of monthly average daily global solar radiation (mj/m2). s = monthly average daily bright sunshine hour. s0 = monthly average daily maximum sunshine hour. h = altitude. n = day of year. abbreviations gsr = global solar radiation. madgsr = monthly average daily global solar radiation. mbe = mean bias error. rmse = root mean squared error. mpe = mean percentage error. mare = maximum absolute relative error. mae = mean absolute error. rmsre = root mean square error. r2, correlation coefficient. t-test = t-statistic. gpi = global performance indicator. u95 = uncertainty at 95%. greek letter ψ = longitude (degree). φ = latitude (degree). ωs = sunshine hour angle (degree). δ = declination angle (degree). a & b = coefficient constant. references [1] 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[135] o. o. ajayi, o. d. ohijeagbon, c. e. nwadialo, o. olasope “new model to estimate daily global solar radiation over nigeria”, sustainable energy technol. assess. 5 (2014) 28. 28 j. nig. soc. phys. sci. 4 (2022) 887 journal of the nigerian society of physical sciences assessment of heavy metal pollution status in surface soil of a nigerian university m. a. lalaa,∗, s. kawua, o. a. adesinaa, j. a. sonibareb adepartment of chemical and petroleum engineering, afe babalola university, ado-ekiti, nigeria bdepartment of chemical engineering obafemi awolowo university, nigeria abstract the problem of urban soil contamination with heavy metals due to rapid urbanization and industrialization has been a major concern in recent years. a university can be considered as a product of industrialization and urbanization which is associated with different activities that may induce heavy metals pollution into the environment. therefore, this research work assessed the contamination level of chromium (cr), copper (cu), lead (pb), zinc (zn) and manganese (mn) in surface soils of afe-babalola university (abuad) using various indices. soil samples were taken from ten (10) different functional sites in the university. these samples were taken to the laboratory and analyzed for chromium (cr), copper (cu), lead (pb), zinc (zn) and manganese (mn) using standard method. the mean concentrations of copper (cu), chromium (cr) and lead (pb) were up to 0.75, 0.66 and 0.36 mg/g respectively, while manganese (mn) and zinc (zn) were 1.37 and 0.49 mg/g respectively. the average concentration of manganese (mn) was comparable to its corresponding natural background value, but the average concentration of chromium (cr), copper (cu), zinc (zn) and lead (pb) were higher. they were approximately of the ratio 1:7, 1:2, 1:3 and 1:2 respectively compared to their corresponding natural background value. the multivariate statistical analyses indicated that vehicles, power generating sets, petrol station, machine workshops, production plants and emissions from outdoor roasted food spots were the major sources of heavy metals contamination on the universitys’ soil. the results from contamination indices and assessment showed that the contamination level of soils within the university can generally be classified as moderately contaminated. therefore, periodic assessment of the sources and associated ecological risks of the heavy metals is highly recommended. this is to enable decision-makers to effectively manage the environment in the manner that will preserve public and ecosystem health. doi:10.46481/jnsps.2022.887 keywords: contamination indices, ecological risk assessment, multivariate techniques, afe-babalola university article history : received: 23 june 2022 received in revised form: 21 july 2022 accepted for publication: 21 july 2022 published: 15 august 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: m. orosun 1. introduction heavy metals are relatively high density (usually 3.50 to 7.00 g cm−3) metallic or metalloid elements that are capable of ∗corresponding author tel. no: +2348163280469 email address: lalam@abuad.edu.ng (m. a. lala) inducing toxicity to humans and the environment at some levels of exposure [1-2]. these heavy metals include cadmium (cd), mercury (hg), arsenic (as), chromium (cr), zinc (zn), nickel (ni), lead (pb), copper (cu) etc. they are widely present in the crust of the earth and are not degradable in nature [3]. heavy metals can be considered as trace elements because of their trace quantity in the environmental. the bioavailability 1 lala et al. / j. nig. soc. phys. sci. 4 (2022) 887 2 of heavy metals is usually influenced by physical factors (i.e temperature, adsorption and phase association), biological factors (i.e biochemical adaptation, physiological adaptation and species characteristics) and chemical factors. the chemical factor affects the formation of more ions in the course of their reactions [1-6]. sources of heavy metals pollution can generally be classified as natural and anthropogenic or manmade. the crust of the earth naturally contains heavy metals, but human or anthropogenic activities have been recognized as the major source of the pollution [1]. considering the natural courses of heavy metal pollution, weathering and volcanic eruptions have been identified as parts of the main contributors [1, 7, 8]. environmental pollution through anthropogenic activities includes agricultural and domestic use of compounds containing metals, mining operations and smelting operations. industrial activities such as burning of coal in power plants and processing of metals in refineries are also major sources of manmade pollution. moreover, petroleum combustions, textiles, plastics, microelectronics, paper processing and wood preservation also contribute immensely to heavy metals level in the environment [1]. in some years back, series of studies have been carried out on the assessment urban soils for heavy metals contamination [9-11]. this is because soil is the main geochemical reservoir for all heavy metal pollutants discharged into the environment. heavy metals are being deposited on soil which is the most important component of the biosphere through vehicular and other combustion engines emissions. careless waste disposals and oil spillage in auto-mechanic workshops, industrial production, energy generation, construction activities, coal and fuel combustion, as well as indiscriminate use of pesticides and fertilizers on farmlands also induce heavy metals into the environment [12-15]. in addition, study indicated that atmospheric deposition which occurs either by dry or wet deposition represents another major contributor of heavy metals into the topsoil in urbanized areas [16]. the most commonly found metals on soils of contaminated sites include zinc (zn), arsenic (as), chromium (cr), lead (pb), cadmium (cd), mercury (hg), copper (cu) and nickel (ni). they do not undergo chemical or microbial degradation as applicable to organic pollutants which oxidized easily by microbial actions [1, 17]. prolonged exposure to cadmium (cd) and lead (pb) even at low concentrations could cause lung damage, cancer, kidney disease, and nervous disorder. this is most common with lead (pb). chromium (cr) which exists in different valence states can cause gastrointestinal, lung and nasal irritation, small intestines and stomach ulcer, decreased sperm counts and dermatitis. nickel (ni) also causes cancer, dermatitis and lung inflammation. copper (cu) is associated with cardiomyopathy and lung irritation, and the major reported side effect of copper (cu) and zinc (zn) on health are vomiting, abdominal pain, respiratory system irritation and nausea [18]. other heavy metals also pose great adverse effects to humans, animals and plants. soils contamination in different urban sector may vary depending on the activities being carried out on them. hence, assessment of urban soils for heavy metals contamination is highly important [19-20]. in recent years, studies have been conducted on the contamination assessment of heavy metals in soil samples from auto-mechanic workshops [21-22], dust from paved and unpaved roadside [12, 18], soil samples from developing communities [11, 13] and many more. in the present study, assessment of heavy metals in surface soils of a nigerian university was considered. this assessment is important because a university can be characterized by many other activities that can induce heavy metals contaminations into the environment apart from the normal academic activities. machines workshop which comprises of welding section, automobile section, lathe machine section and other machine tools sections are prime sources of trace metals contamination in university environment. emissions from power generating sets and the maintenance of these sets are also major contributors to heavy metal pollution in universities where electric power generating sets are installed as alternative power supply. other contributors to heavy metals pollution in a university environment include the activities from; automobile transport pools [23], building construction sites, university petrol station, and university central works department. despite the significance of the discussed activities to heavy metals contamination level within a university, little or no work has been done on the assessment of heavy metals in university environment. therefore, this research work aimed at assessing the contamination level of mn, pb, cr, cu and zn in surface soils of abuad, and the specific objectives are to; (1) establish the concentrations of the selected metals in surface soil of abuad, (2) identify the source and contributions of the selected metals to the contamination level of abuad soils, and (3) evaluate the contamination and risk level of the metals using enrichment factor (ef), geo-accumulation index (igeo), potential ecological risk index (ri), pollution index (pi) and integrated pollution load index (ipi). this is to detect areas that need special attention and to suggest appropriate control measures. 2. materials and methods 2.1. study area afe babalola university (abuad) was considered for the current research work. abaud is located in ado-ekiti, ekiti state, southwestern nigeria (fig. 1) the citadel of learning is located between latitude 07◦35’59”and 36’50”n, and longitude 05◦18’0”and 18’45”e. the geographical area is characterized by relatively low relief with dome-shaped isolated hills and inserbergs. the institution has over 8,500 students across all the colleges in the university and workforce of about 1000 staff. the university community consist of many strategic places such as; residential apartments for staff and students, academic and administrative buildings, research centre, planetarium, petrol station, supermarkets, bakery, water production plant, restaurants, banks, atm points, the university works department, sports pavilion, college of engineering central workshop and guest house. 2 lala et al. / j. nig. soc. phys. sci. 4 (2022) 887 3 figure 1. study area (a) map of nigeria (b) map of ekiti state showing abaud (c) sampling locations within abuad 2.2. soil sampling twenty four (24) soil samples taken at a depth of about 0 to15 cm of the earth surface were collected using a standard soil auger [24] from ten (10) different main functional sites in abuad. two (2) control samples were also taken from an undeveloped site within abuad. this is between the university premises and the university farm. samples were collected in duplicates from each site except site 9 where 4 samples were collected. four (4) samples were collected from site 9 because the site comprises of more activities. as shown in table 1, the sampling sites were considered based on different activities that can induce heavy metal contamination into the environment. to avoid further contamination other than the ones from the sampling site, collected soil samples were kept in clean nylon and taken directly to laboratory for analysis. 2.3. sample preparation and analysis debris and all other foreign materials were manually removed from the soil samples, afterwards the samples were dried and reduced to about 1.18 mm soil particle size. to determine the selected metals concentration, 1 g from each dried samples were digested in standard erlenmeyer flask using 12 ml aquaregia solution. the solution was prepared at a ratio of 1:3 of hno3 to hcl [11, 21]. the beaker was covered and the contents were heated for 2 hr using medium hot plate heat [11, 21]. after heating, the mixture was left to cool down naturally and 3 lala et al. / j. nig. soc. phys. sci. 4 (2022) 887 4 subsequently filtered using a whatman no. 42 filter paper. the filtrate was diluted to 50 ml with de-ionized distilled water [21, 25]. these solutions were further analyzed for the selected metals concentration using aa320n atomic absorption spectrophotometer. to minimize error, all soil samples were analysed in triplicate. during the analysis, standards and blanks were run to ensure 95% reliability of the machine [11]. 2.4. descriptive statistics analysis the concentration and distribution of the selected metals in abuad soil samples were investigated using statistical analysis. the quantified data where analysed using microsoft excel software to obtain mean, standard deviation, standard error, sample variation, minimum and maximum values. also, principal component analysis (pca) was carried out with the aid of xlstat software. this was used for source identification and contributions of the selected metals to the contamination level of abuad soils. 2.5. contamination and risk assessment contamination and risk assessment levels for this study were analysed using the following indices; 1. enrichment factor (ef) the enrichment factor (ef) is a relatively straightforward and simple tool used for evaluating the extent of soils contamination and to compare the contamination of different environmental media. it involves a universal method for calculating degree of soil enrichment by dividing the ratio of a particular metal and normalizing element by the ratio of the same metals found in a baseline and normalizing element [13, 26, 27]. this can be computed using equation 1. ef = (trace metal/fe)s oil sample (trace metal/fe)natural background (1) where (trace metal/fe)soil sample= ratio of heavy metal to fe in the soil sample taken from the selected site, and (trace metal/fe)natural background= ratio of heavy metal to fe in the soil sample taken from the control site. similar to ololade [22], the normalizer used for this work was fe. this is because nigeria soils are rich in fe [2829], and also due to its conservative nature during diagenesis [30]. for the background values, the average concentrations of metals detected in control site were used according to the approach given by benhaddya and hadjel [13]. ef values of unity or very close to unity shows that the heavy metal originated from natural sources. values not up to 1.0 signify depletion or mobilization of trace metals, and values more than 1.0 implies that the trace metal is from an anthropogenic source [22, 31]. values greater than 1.0 are further interpreted and classified into different categories according to birth [32], these include; no enrichment if ef is < 1, minor enrichment if ef is < 3, moderate enrichment if ef is from 3 to 5, moderate severe enrichment if ef is from 5 to 10, severe enrichment if ef if from 10 to 25, very severe enrichment if ef if from 25 to 50, and extremely severe enrichment if ef is > 50. 2. geo-accumulation index (igeo) igeo index estimates contamination level by comparing present and the preindustrial concentrations level [13]. in this study, igeo was also used to evaluate the contamination level of abaud soils by employing equation 2 proposed by muller [33]. igeo = log2 ( cm 1.5 x cb ) (2) cm and cb = trace metals concentration in soil samples and natural background respectively. the constant value 1.5 is a feasible alteration reference value to reduce the effect of possible variation in the background value. levels of contamination are classified as; unpolluted for igeo ≤ 0, unpolluted to moderately polluted for igeo between (0–1), moderately polluted for igeo between (1–2), moderately to strongly polluted for igeo between (2–3), strongly polluted for igeo between (3–4), strongly to very strongly polluted for igeo between (4–5) and very strongly polluted for igeo = 5. 3. potential ecological risk index (ri) this risk index (ri) was used to evaluate the ecological risk of trace metal contamination in soils by considering metals toxicity and the response to the environment. this is given by equations 3, 4 and 5, and was originally given by hakanson in 1980 [13, 34]. ri = ∑ ei (3) ei = ti ∗ fi (4) fi = ci/bi (5) ei is the potential ecological risk factor, fi is the pollution factor for metals, ci is the concentration of metals in soil, bi is the natural background value of metals and ti is metal toxic factor. as given by duodu et al.,[35]; the value of ti for mn and zn = 1, cr = 2, & cu and pb =5 [34, 35]. the four (4) categories of ri used in this study are; ri ≤ 50 for low contamination, 50 < ri ≤ 100 for moderated contamination, 100 < ri ≤ 200 for considerable contamination, and ri > 200 for high contamination. 4. pollution index (pi) and integrated pollution load index (ipi) these important indexes assess the contamination level of heavy metals in soils for each metal and total metals under consideration respectively. according to chen et al, 2005 [15], pi is evaluated as the ratio of the concentration of each metal to the background value of the corresponding metal. this is given in equation 6, where cm is the measured concentration of each heavy metal, and cb is the corresponding background value which is taken in this study as the mean concentrations of the corresponding heavy metal in unaffected soils (control site). 4 lala et al. / j. nig. soc. phys. sci. 4 (2022) 887 5 table 1. sampling site and the associated activities site activity ab1 mainly residential staff quarters ab2 installed and functioning university central power generating set ab3 installed and functioning staff quarter’s power generating set ab4 college of social & management sciences/college of engineering transport pool ab5 senate building transport pool and new building construction site ab6 college of engineering central workshop which comprises of welding section, automobile section, lathe machine section and other machine tools sections ab7 university central works department which comprises of welding section, auto-mechanic’s section, vulcanizing section, plumbing section, etc. ab8 university filling station ab9 various business activities such as restaurant and cafeteria, supermarket, printing press, pure water factory, barbecue and other outdoor roasted food spots ab10 barbecue fish, meat, fried yam and other outdoor roasted food spots ab11 control site not shown on the map the contamination level for each metal can be categorized as; low, medium and high contamination when pi ≤ 1, 1 < pi ≤ 3, and pi > 3 respectively [13, 15]. pi = cm cb (6) ipi as a criterion for evaluating soil pollution can be estimated as the mean pi of all estimated heavy metals. it can be categorized as; extremely high pollution level, high pollution level, moderate pollution level and low pollution level when ipi > 5, 2 < ipi ≤ 5, 1 < ipi ≤ 2 and ipi ≤ 1 respectively. 3. results and discussion descriptive statistics of cr, cu, zn, pb and mn detected in the soil of the studied university are given in table 3. the table also presents world health organization (who) and standard guidelines in europe limits for trace metals in soils [36]. table 2 presents the mean values for all the sampling sites. minimum and maximum concentration of cr, cu, zn, pb and mn were 0.2 & 1.75, 0.39 & 1.11, 0.02 & 0.70, 0.03 & 9.40, and 0.09 & 1.03 mg/g respectively, and their mean concentrations were 0.75, 0.66, 0.36, 1.37, and 0.49 respectively. the mean concentration of the considered heavy metals in the university soil follow the trend of pb > cr > cu > mn > zn. the average concentration of zn (0.36 mg/g) was found to be slightly higher than the permissible limit while mn (0.49 mg/g) was below the limit. average concentration of cr (0.75 mg/g), cu (0.66 mg/g), and pb (1.37 mg/g) were found to be more than the limit. this may be attributed to the effects from different human activities or the high nature of the corresponding natural background value [11, 36]. considering the average concentration of heavy metals and their corresponding natural background value, mn concentrations was comparable, where cr, cu, zn and pb concentration were more than the natural background value. they were approximately of the ratio 1:7, 1:2, 1:3 and 1:2 respectively. the standard deviation and standard error for cu, zn, cr and mn were high while value of pb was very high. this showed that the concentration of the heavy metals varied greatly across the university and this may be attributed to non-uniform dispersion in the soil [11, 37]. cr and pb showed high sample variation and this indicated high human influence on the concentrations [11, 38]. furthermore, the mean concentration of metals reported in this study was generally higher than those detected in soils from some urban and industrialized areas such as hassi messaoud in algeria; bangkok in thailand; madrid in spain; hong kong in china etc. concentration of cu, pb and zn in hassi messaoud (0.01317, 0.13097 and 0.061 mg/g) [11], bangkok (0.0417, 0.0478 and 0.118 mg/g) [39], madrid (0.072, 0.161 and 0.21 mg/g) [40] and hong kong (0.0233, 0.0946 and 0.125) [41] were low compared to 0.66, 1.37 and 0.36 mg/g for this study. this may be as a result of the high natural background value of metals in the soil of the present study area [11]. also, this may be associated with the nature of the activities taken place in abuad. the university has automobile workshop and machines workshop which are constantly in use for training purposed. thus, generating related heavy metals contamination. moreover, comparing results from the present study with the assessment of metal contamination in sediment samples collected from lagos dockyard harbour by basheeru et al [42], the contamination level of lagoon sediment with lead and copper is higher compared to the contamination level of abuad soils. in another work carried out by some researchers on estuary sediments from olonkoro river, oyo state, nigeria [43], it was found that the concentration level of zn, cu, zn, and mn were also higher. this is because different industrial and domestic effluent are been discharged into the harbor and the river 5 lala et al. / j. nig. soc. phys. sci. 4 (2022) 887 6 table 2. average values of selected trace metals at each sampling site sampling point heavy metal concentrations (mg/g) cr cu zn pb mn ab1 0.73+0.04 0.70+0.01 0.13+0.03 2.65+0.15 0.84+0.05 ab2 0.42+0.22 0.92+0.19 0.24+0.23 2.02+0.38 0.63+0.21 ab3 1.33+0.42 0.58+0.06 0.28+0.10 1.02+0.58 0.38+0.16 ab4 0.31+0.07 0.42+0.03 0.45+0.26 5.26+4.13 0.76+0.14 ab5 0.72+0.02 0.68+0.29 0.28+0.09 0.38+0.09 0.29+0.04 ab6 0.65+0.05 0.48+0.04 0.41+0.07 0.06+0.04 0.19+0.01 ab7 0.77+0.57 0.74+0.18 0.19+0.05 0.23+0.05 0.41+0.32 ab8 0.74+0.38 0.65+0.13 0.52+0.02 1.13+0.38 0.49+0.01 ab9 0.77+0.23 0.59+0.07 0.42+0.20 0.49+0.15 0.65+0.29 ab10 1.05+0.07 0.94+0.05 0.63+0.05 1.28+0.05 0.10+0.01 control site 0.32+0.22 0.46+0.12 0.12+0.00 1.12+0.36 0.41+0.01 limits 0.100 0.100 0.300 0.100 2.000 table 3. descriptive statistical analysis of selected trace metals in abuad soil s/n data analysis concentrations (mg/g) cr cu zn pb mn 1 mean 0.75 0.66 0.36 1.37 0.49 2 minimum 0.20 0.39 0.02 0.03 0.09 3 maximum 1.75 1.11 0.70 9.40 1.03 4 se 0.08 0.04 0.04 0.42 0.06 5 sd 0.38 0.21 0.21 1.97 0.30 6 sv 0.15 0.04 0.04 3.88 0.09 control value 0.32 0.46 0.12 1.12 0.41 permissible limit 0.10 0.10 0.30 0.10 2.00 sv sample variation, sd standard deviation, se standard error thereby contributing to their higher heavy metal level. 3.1. heavy metal contents in different sampling sites as given in table 2, the average concentrations of cu and zn in different functional areas were more than their corresponding natural background value except cu detected in site ab4. this is similar to the result obtained for soils in hassi messaoud, algeria [11]. the average concentrations of cr from different sites were also higher compared to their natural background value except for site ab4 which was lower, and site ab2 which was comparable to the background value. also, the average concentrations of mn were comparable to the natural background value in sites ab3 and ab7, lower in sites ab5, ab6 and ab10, and higher in all other sampling sites. pb mean concentrations were below the natural value except for the residential area (ab1), the university central power generating set area (ab2), transport pool (ab4) and site ab10. the study area is characterized by high automobiles and power generating sets, and the trace metals such as mn, pb, cu, cr and zn selected for consideration in the present study are reported to be present as additives in lubricants and gasoline [22, 44]. also, pb, cu and zn are reported to be present in vehicular emissions, tires, wires, alloys, brake parts, etc. [18, 45-52]. copper (cu) present in soils from all sampling sites were higher than the natural background value (0.46 mg/g) except site ab4 which is comparable to the background value. the average concentrations ranged from 0.42 mg/g in site ab4 to 0.94 mg/g in site ab10. these were all higher than the permissible limit (0.1 mg/g) and may be attributed to the higher natural background value. also, it is notable that electrical and electronic parts of automobile wastes and alloys from corroding metal parts that have possibly leached into the soil of these various sites can be attributed to the high cu present [53]. average values of pb in soils from sampling site ab1 (2.65 mg/g), ab2 (2.02 mg/g), ab4 (5.26 mg/g) and ab10 (1.28 mg/g) were notably higher compared to the corresponding background value (1.12 mg/g). this suggested that combustion engine emissions, spill waste oils and expired motor batteries from the high number of automobiles that cluster site ab1, ab4 and ab2 can be responsible for the high pb concentration in these areas. this is similar to the result obtained for a study carried out by a group of researchers in ilorin, nigeria [54]. in the study, pb concentration was higher compared to cu and cd. higher pb level in the study was suggested to have been influenced by traffic volume and exhaust from motor vehicles. the mean zn content in all soils from different sites ranged from 0.13 mg/g in site ab1 to 0.63 mg/g in site ab10. these were all greater than the natural background value (0.12 mg/g). 6 lala et al. / j. nig. soc. phys. sci. 4 (2022) 887 7 also, average values from sites ab4, ab6, ab8, ab9 and ab10 were more than the given limit. this indicated that the activities from the different functional areas such as automobiles, machines workshops, petrol station and others are responsible for the high zn content. cr concentrations at different sites ranged from 0.31 mg/g in site ab4 to 1.33 mg/g in site ab3. the values were all higher than the permissible limit of cr and were also more than the natural value (0.32 mg/g) except site ab4 which is comparable to the background value. the low level of cr in site ab4 suggested that cr emissions were not much from the transport pool since the vehicles were always packed without any emission during the working hours and also the pool is a paved area, while high cr concentration in other areas indicated the effects of anthropogenic activities in such areas. mn detected in all sampling sites was lower in concentration compared to its permissible limit. however some areas like ab1, ab2, ab4, ab8 and ab9 shows higher mn content compared to the natural background value. this indicated that activities from automobiles, machines workshops, power generators and petrol station in these areas were responsible for the high mn content. 3.2. source identification principal component analysis (pca), a multivariate receptor model analysis is applicable for contamination sources identification [55-56]. xlstat, a statistical software for microsoft excel was used in the present study for the source identification and contributions of the selected metals to the contamination level of abuad soils. typical pca score plot will have variables with the similar source located close to each other, while those with divergent sources will be far apart [57]. figure 2 shows the pca variable plot with a cumulative variance of 75.66% where factor 1 (f1) accounts for 45.60% and factor 2 (f2) 30.06% of the total variance. variables that are close to each other and far from the centre as shown in figure 2 are considered to be significantly positively correlated. also, variables located on opposite sides are significantly negatively correlated and those located orthogonally are considered not correlated. also, figure 2 shows that there is proximity between samples taken from the same site, such as samples taken within sites ab4, ab5, ab6, ab8 and ab10. this inferred that the contaminations within these sites were from the same source. there is low proximity for samples taken within sites ab3 and ab7 which implies that the contaminations on these sites may be from different sources. this is evident from the composition of the areas. site ab7 comprises of welding workshop, auto-mechanic workshop, vulcanizing workshop and plumber workshop, and site ab3 which has functional power generating set installed on it is not too far from site ab7. four samples were taken from site ab9 where some have proximity and some do not. this is because the site has various business activities and production plants such as restaurants and cafeterias, supermarkets, printing press, pure water factory, barbecue and other outdoor roasted food spots. therefore the result on the pca variable plot implies that site with similar human activity has a single source of heavy figure 2. variable plot for pca figure 3. observation plot of pca indicating the positions of cr, cu, pb, zn and mn metals while sites with various human activities have different sources of heavy metal contaminations. the pca observation plot given in figure 3 shows no cluster between the five selected heavy metals. this indicates that the heavy metals con7 lala et al. / j. nig. soc. phys. sci. 4 (2022) 887 8 tamination originated from more than a single source, and these include automobiles, generators, petrol station, machine workshops, central works departments, production plants and other business activities. 3.3. pollution assessment 1. enrichment factors (ef) enrichment factors (efs) of cu, cr, pb, zn and mn were considered to assess abuad soils contamination level (fig. 4). the average ef of cu (0.89), pb (0.76) and mn (0.73) were found to be less than one (1), while cr (1.47) and zn (1.82) were found slightly greater than one (1). this inferred no enrichment for cu, pb and mn, and minor enrichment for cr and zn. cr enrichment factor ranges from 0.56 to 3.12, cu from 0.59 to 1.63, zn from 0.67 to 4.34, pb from 0.04 to 3.07 and mn from 0.11 to 2.00. this result shows no enrichment or moderate enrichment for cr, no enrichment or minor enrichment for cu, no enrichment or moderate enrichment for zn, no enrichment or moderate enrichment for pb and no enrichment or minor enrichment for mn. site ab9 which comprises various business centres and production plants has zn (4.34) as the metal with the highest enrichment. 2. geo-accumulation index (igeo) igeo index was considered to estimate the abuad soils contamination level by comparing the preindustrial and present concentrations of heavy metals [13]. figure 5 shows that the igeo of all the selected metals in abuad soils does not exceed moderately polluted on an average level. average igeo of cu (-0.07), pb (-0.29) and mn (0.35) indicated unpolluted while cr (0.65) and zn (0.97) indicated moderately polluted. the highest igeo value obtained in the present study is for zn (1.77). this was detected on site ab10 which majorly comprises of barbecue fish, meat, fried yam and other open roasted food spots. the findings from igeo are in agreement with the ef values showing that the sampled environment is not highly contaminated. 3. potential ecological risk index (ri) ecological risk of the selected trace metal in abuad soils was assessed. table 4 gives the risk index for the contamination caused by cr, cu, pb, zn and mn in all sampled abuad soils as low contamination. this indicated that abuad soils may not be too sensitive to heavy metals contamination and the contamination level posed a low potential ecological risk. this is also in agreement with ei and igeo assessments, showing that the sampled environment has a low level of contamination. 4. pollution index (pi) and integrated pollution load index (ipi) contamination level of abuad soils was also assessed using pollution index (pi) and integrated pollution load index (ipi). the summarized results are presented in table 5. pi of cr ranged from 0.99 to 4.18 and showed low contamination to high contamination. the given mean pi of 2.36 indicated medium cr contamination of abuad figure 4. enrichment factors for metals figure 5. geo-accumulation index (igeo) soil. the highest pi of cr was detected on site ab4 and this inferred possible contamination from gasoline and lubricant leakages from vehicles that dominated the area. a similar trend as noticed for cr was also seen in zn and pb. the highest pi of 5.12 for zn and 4.72 for pb were detected on site ab10 and ab4 respectively and indicated high contamination. the high content of zn and pb in sites ab10 and ab4 respectively may be linked to the effect of activities in these areas. pi of cu and mn showed a similar trend. their pi ranged from low contamination to medium contamination, and information on their average pi indicated medium contamination of abuad soils. considering the ipi assessment, the mean value of 1.83 indicated a moderate pollution level of abuad soils. signals were also given that sites ab4 and ab10 have high pollution levels. this is similar to the pi assessment of some heavy metals in these same locations. 8 lala et al. / j. nig. soc. phys. sci. 4 (2022) 887 9 table 4. potential ecological risk indexes for heavy metals in abuad soils sampling point ei ri pollution degree cr cu zn pb mn ab1 4.59 7.59 1.07 11.89 2.02 27.15 low ab2 2.63 9.97 1.99 9.03 1.52 25.14 low ab3 8.36 6.24 2.26 4.58 0.91 22.36 low ab4 1.98 4.53 3.64 23.58 1.84 35.56 low ab5 4.54 7.33 2.26 1.68 0.70 16.51 low ab6 4.09 5.24 3.33 0.29 0.46 13.40 low ab7 4.81 8.02 1.55 1.05 0.98 16.41 low ab8 4.68 7.02 4.24 5.07 1.18 22.18 low ab9 4.81 6.36 3.39 2.21 1.56 18.33 low ab10 6.62 10.18 5.12 5.72 0.23 27.86 low table 5. pi and ipi of heavy metals in abuad soil heavy metal pi ipi min max mean min max mean cr 0.99 4.18 2.36 cu 0.91 2.04 1.45 zn 1.07 5.12 2.88 1.35 2.42 1.83 pb 0.06 4.72 1.30 mn 0.23 2.02 1.14 4. conclusion various assessment methods and tools were used to evaluate trace metals contamination levels of abuad soil. twenty four (24) soil samples taken at a depth of about 0 to15 cm of the earth surface were collected using a standard soil auger from ten (10) different main functional sites, and were analyzed for cu, cr, pb, mn and zn. control samples were taken from an undeveloped area within abuad. these were also analyzed and served as natural background to the corresponding heavy metals collected from the different functional areas within the university. the minimum and maximum concentration of cr, cu, zn, pb and mn were 0.2 and 1.75, 0.39 and 1.11, 0.02 and 0.70, 0.03 and 9.40, and 0.09 and 1.03 mg/g respectively, and the mean concentrations were 0.75, 0.66, 0.36, 1.37, and 0.49 mg/g respectively. mn average concentration in abuad soil was comparable to natural background value; however cr, cu, zn and pb were higher than their corresponding background value and approximately of the ratio 1:7, 1:2, 1:3 and 1:2 respectively. the multivariate statistical analyses indicated that staff and administrative vehicles, power generating sets, petrol station, machine workshops, central works department, production plants and various outdoor roasted food spots activities were the main sources of metal contamination. the results of ef, igeo, ri, pi and ipi showed that the contamination level of abuad soil can be generally classified as low or moderately contaminated. also, according to some of the indices, contamination attributed to zn and pb were found to be slightly high in sites ab4, ab9 and ab10. this can be linked to possible contamination from gasoline and lubricant leakages from vehicles that dominated site ab4, and various outdoor roasted food spots together with the production plants located in site ab10 and ab9 respectively. the present study considered a university which is a community that is usually associated with a high number of people. contamination of university soils by trace metals is of great risks and hazards to the immediate ecosystem if not properly handled. exposure to heavy metals is usually by; dermal contact with suspended dust and soil particles and inhalation/ingestion 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[57] w. s. lee, g. p. chang-chien, l. c. wang, w. j. lee, p. j. tsai, k. y. wu & c. lin, “source identification of pcdd/fs for various atmospheric environments in a highly industrialized city”, environ sci technol. 38 (2004) 4937. 11 j. nig. soc. phys. sci. 1 (2019) 35–50 journal of the nigerian society of physical sciences review strong convergence theorems for split common fixed point problem of bregman generalized asymptotically nonexpansive mappings in banach spaces yusuf ibrahim∗ department of mathematics, saádatu rimi college of education, kumbotso kano, nigeria abstract in this paper, a new iterative scheme is introduced and also strong convergence theorems for solving split common fixed point problem for uniformly continuous bregman generalized asymptotically nonexpansive mappings in uniformly convex and uniformly smooth banach spaces are presented. the results are proved without the assumption of semicompactness property and or opial condition. keywords: split common fixed point problem, bregman distance, generalized asymptotically nonexpansive mapping, strong convergence, uniformly convex and uniformly smooth banach space, algorithm article history : received: 04 april 2019 received in revised form: 30 april 2019 accepted for publication: 11 may 2019 published: 16 may 2019 c©2019 journal of the nigerian society of physical sciences. all rights reserved. communicated by: t. latunde 1. introduction throughout this paper, e1 and e2 are uniformly convex and uniformly smooth banach spaces. let us make some conventions: we always use p, q ∈ (1,∞) as conjugate exponents so that 1p + 1 q = 1, where by q = p p−1 , pq = p + q, (p−1)(q−1) = 1. furthermore, for real value a, b, we write a∨b = max{a, b} and a ∧ b = min{a, b} which is to be understood componentwise in case of sequences and pointwise in case of functions. throughout this paper, let p, r ≥ 1, given sequences of nonempty closed convex subsets {ci} p i=1 and {qi} r i=1 of e1 and e2, respectively, definition 1.1. [1] multiple set split feasibility problem (mssfp) ∗corresponding author tel. no: +2348062814778 email address: danustazz@gmail.com (yusuf ibrahim) is formulated as finding a point x∗ ∈ e1 with the property x∗ ∈ p⋂ i=1 ci and ax ∗ ∈ r⋂ j=1 q j. (1) definition 1.2. [2] if in a mssep (1) p = r = 1, we get what is called the split feasibility problem (sfp), which is to find a point x∗ ∈ e, with the property x∗ ∈ c and ax∗ ∈ q. (2) definition 1.3. [3] split common fixed point problem (scfpp) is formulated as finding a point x∗ ∈ e1, with the property x∗ ∈ p⋂ i=1 fix(ui) and ax ∗ ∈ r⋂ j=1 fix(t j), where each ui : e1 −→ e1 (i = 1, 2, · · · p) and t j : e2 −→ e2 35 yusuf ibrahim / j. nig. soc. phys. sci. 1 (2019) 35–50 36 ( j = 1, 2, · · ·r) are some (nonlinear) mappings. definition 1.4. [3] two-set scfpp is formulated as finding a point x∗ ∈ e1, with the property x∗ ∈ fix(u) and ax∗ ∈ fix(t ) (3) where u : e1 −→ e1 and t : e2 −→ e2 are (nonlinear) mappings. as stated in [4], let e be a banach space with norm ‖·‖. let c be a nonempty closed convex subset of e and e∗ denote the dual space of e. let b : c −→ e∗ be a nonlinear mapping and f : c × c −→ < be a bifunction. the generalized equilibrium problem is to find x ∈ c such that f(x, y) + 〈bx, y − x〉 ≥ 0∀y ∈ c. (4) now, let f(x, y) = f (x, y) + g(x, y) (5) where f, g : c × c −→ < are two bifunctions satisfying the following special properties (a1) − (a4), (b1) − (b3)and(c); (a1) − f (x, y) = 0,∀x ∈ c; (a2) − f is maximal monotone; (a3) −∀x, y, z ∈ cwe have lim supn→0+ ( f (tz + (1 − t)x, y) ≤ f (x, y)); (a4) −∀x ∈ c, the functiony 7→ f (x, y)is convex and weakly lower semi-continuous; (b1) − g(x, x) = 0∀x ∈ c; (b2) − gis maximal monotone and monotone, and weakly upper semi-continuous in the first variable; (b3) − gis convex in the second variable; (c) − for fixedλ > 0and x ∈ c, there exists a bounded set k ⊂ canda ∈ ksuch that f (a, z) + g(z, a) + 1 λ (a − z, z − x) < 0∀x ∈ c\k. (6) the well known generalized mixed equilibrium problem is to find an x ∈ c such that f (x, y) + g(x, y) + 〈bx, y − x〉 ≥ 0∀y ∈ c. (7) if b ≡ 0, the problem (7) reduces into mixed equilibrium problem for f and g, denoted by mep( f, g), which is to find x ∈ c such that f (x, y) + g(x, y) ≥ 0,∀y ∈ c. (8) if g ≡ 0 and b ≡ 0, (7) reduces into equilibrium problem for f, denoted by ep(f), which is to find x ∈ c such that f (x, y) ≥ 0 ∀ y ∈ c. (9) in 1994, the sfp (2) was introduced by censor and elfving [1]. using the cq-algorithm for solving the sfp (2), in 2002 byrne [2] proposed that which generates the new iterate as follows choosing arbitrarily x1 ∈ h1, xn+1 = pc [xn −γa t (i − pq)axn] where γ ∈ (0, 2l ), l denotes the largest eigenvalue of the matrix at a. in 2004, yang [5] presented a relaxed cq-algorithm for solving the sfp, where at n-th iteration, the projections onto c and q were replaced with the halfspaces cn and qn, respectively. in 2005, qu and xiu [6] proposed a modified relaxed cqalgorithm choosing arbitrarily x1 ∈ h1, xn+1 = pcn [xn −αn a t (i − pqn )axn]. in 2007, frank schopfer[7] developed iterative methods for the solution of the sfp (2) in banach spaces and also analyse stability and regularizing properties. these iterative methods are as follows. choosing arbitrarily x1 ∈ e1, xn+1 = j ∗ e1 (je1 (xn) −µn a ∗je2 (axn − pq(axn)). the concept of scfpp in finite dimensional hilbert spaces, say h1 and h2, was first introduced by censor and segal in 2009[1], who invented an algorithm of the two-set scfpp which generate a sequence {xn} according to the following iterative procedure: choosing arbitrarily x1 ∈ h1, xn+1 = u(xn + γa ∗(i − t )axn), n ≥ 1 where the initial guess x0 ∈ h1 is chosen arbitrarily and 0 < γ < 1 ‖a‖2 and a : h1 −→ h2 as a bounded linear operator having a∗ as the adjoint operator of a. and u : h1 −→ h1 and t : h2 −→ h2 are (nonlinear) mappings. in 2010, moudafi [8] proposed the following iteration method to approximate a scfpp of demicontractive mappings in hilbert spaces;  x1 ∈ h1 is arbitrarily chosen, un = xn + γa∗(i − t )axn, xn+1 = αnun + (1 −αn)uun, (10) where he proved that {xn} converges weakly to a split common fixed point x∗ ∈ γ, where u : h1 −→ h1 and t : h2 −→ h2 are two demicontractive mappings, a : h1 −→ h2 is a bounded linear operator and h1 and h2 are two hilbert spaces. using the iterative scheme (10), in 2011, moudafi[8] also obtained a weak convergence theorem for the split common 36 yusuf ibrahim / j. nig. soc. phys. sci. 1 (2019) 35–50 37 fixed poin problem of quasi-nonexpansive mapping in hilbert spaces. in 2012, chang et al. [9] again, using (10) proved the weakly convergence of the sequence {xn} to the split common fixed point x∗ ∈ γ of asymptotically quasi-nonexpansive mapping in hilbert spaces. but these authors could only obtain strong convergence theorem if those mappings and spaces are semi-compacts and hilbert spaces, respectively. in 2015, zhang et al [10] introduced the iterative scheme which guarantees the strong converges for scfp of the asymptotically nonexpansive mapping in hilbert spaces, without assumption of semi-compactness. the sequence is defined as follows; choosing arbitrarily x1 ∈ h1, c1 = h1, zn = xn + λa∗(t n2 − i)axn, yn = αnzn + (1 −αn)t n1 (zn), cn+1 = {v ∈ cn : ‖yn − v‖ ≤ kn‖zn − v‖,‖zn − v‖ ≤ kn‖xn − v‖}, xn+1 = pcn+1 (x1), n ≥ 1 where a∗ denote the adjoint of a, λ ∈ (0, 1 ‖a∗‖2 ) and {αn} ⊂ (0,τ) ⊂ (0, 1) satisfies limn→∞ αn(1−αn) > 0, kn = max{k (1) n , k (2) n }, n ≥ 1. takahashi [11], in 2015, also obtained a similar result for split common null point problem by using the following hybrid and shrinking projection methods, respectively. choosing arbitrarily x1 ∈ h1, zn = pc (xn − ra∗jf (axn − pq axn)), yn = αn xn + (1 −αn)zn, cn = {z ∈ h1 : ‖yn − z‖ ≤ ‖xn − z‖}, qn = {z ∈ h2 : 〈xn − z, x1 − xn〉 ≥ 0} xn+1 = pcn∩qn x1,∀n ∈ n, and  choosing arbitrarily x1 ∈ h1, zn = pc (xn − ra∗jf (axn − pq axn)), cn = {z ∈ h1 : ‖zn − z‖ ≤ ‖xn − z‖}∩ cn, xn+1 = pcn+1 un+1,∀n ∈ n, where 0 ≤ αn ≤ a < 1 for some a ∈ r and 0 < r‖a‖2 < 2. in 2015, tang et al.[12] introduced the split common fixed point problem (scfp) for an asymptotical nonexpansive mapping s and a τ−quasi-pseudocontractive mapping t in the setting of two banach spaces, e1 and e2, by using the sequence {xn} defined as follows; choosing arbitrarily x1 ∈ e1, zn = xn + γj−11 a ∗j2(t − i)axn, xn+1 = (1 −αn)zn + αns n(zn), n ≥ 1 where {αn} ⊂ (0, 1) with lim infn→∞ αn(1 −αn) > 0, γ ⊂ (0, min{1−2k 2 ‖a‖2 , 1−τ ‖a‖2 }), l = supn≥1{ln} and ∑ ∞ n=1(ln − 1) < ∞. in this paper, the modified algorithm of zhang et al. for the solution of scfp in hilbert spaces is studied. hence, the bregman generalized asymptotically nonexpansive mapping is used to obtain the strong convergence for scfpp (6) in uniformly convex and uniformly smooth banach spaces, without the assumption of semi-compactness property and or without the assumption of opial condition. 2. preliminaries throughout this paper, the classes of banach spaces we will deal with are [13]; definition 2.1. a banach space e is said to be uniformly convex, if δe (�) = inf{1 − ‖ 1 2 (x + y)‖;‖x‖ = ‖y‖ = 1,‖x − y‖ ≥ �, where 0 ≤ � ≤ 2 and 0 < δe (�) ≤ 1}. definition 2.2. a banach space e is said to be uniformly smooth, if limr→0( ρe (r) r ) = 0 where ρe (r) = 1 2 sup{‖x + y‖ + ‖x − y‖− 2 : ‖x‖ = 1,‖y‖ ≤ r; 0 ≤ r < ∞ and 0 ≤ ρe (r) < ∞}. moreover, 1. ρe is continuous, convex and nondecreasing with ρe (0) = 0 and ρe (r) ≤ r 2. the function r 7→ ρe (r)r is nondecreasing and fulfills ρe (r) r > 0 for all r > 0 3. limr→0 ρe (r) r = 0. throughout this paper, some important mappings we will be using are; definition 2.3. [14] for each p > 1, let g(t) = t p−1 be a gauge function g : <+ −→ <+ such that g(0) = 0 and limn→∞ g(t) = ∞. we defined the ganeralized daulity map by jp : e −→ 2e ∗ by jg(t) = jp(x) = {x ∗ ∈ e∗;〈x, x∗〉 = ‖x‖‖x∗‖,‖x∗‖ = g(‖x‖) = ‖x‖p−1}. the ganeralized daulity mapping has the following basic properties [7]. lemma 2.1. for every x ∈ e the set j pe (x) is not empty and convex. lemma 2.2. j pe (x) is homogeneous of degree p − 1, i.e. j pe (λx) = |λ| p−1 sgn(λ)j pe (x) ∀x ∈ e,λ ∈<. lemma 2.3. if j pe∗(x) is the duality mapping of e ∗ with gauge function t 7→ tq−1 then x∗ ∈ j pe (x) iff x ∈ j q e∗(x ∗). lemma 2.4. [7] in smooth banach space, the bregman distance of x to y with respect to the function f (x) = 1p‖x‖ p is defined by 4p(x, y) = 1 q ‖x‖p −〈j p(x), y〉 + 1 p ‖y‖p. definition 2.4. let e be a smooth banach space. let 4p be a bregman distance. a mapping t : e −→ e is said to be a bregman generalized asymptotically nonexpansive with {kn} and {µn} if there exists nonnegative real sequences {kn} and {µn} with limn→∞ kn = 0 and limn→∞ µn = 0 such that 4p(t n(x), t n(y)) ≤ kn 4p (x, y) + 4p(x, y) + µn ∀ (x, y) ∈ e × e. 37 yusuf ibrahim / j. nig. soc. phys. sci. 1 (2019) 35–50 38 definition 2.5. [15] let e be a reflexive, strictly convex and smooth banach space. then for every closed convex subset c ⊂ e and x ∈ e there exists a unique element pc (x) ∈ c such that ‖x−pc (x)‖ = miny∈c ‖x−y‖ is called the metric projection of x onto c. moreover, if j p is a duality mapping of e, then x0 ∈ c is the metric projection of x onto c iff 〈j p(x0 − x), y − x0〉 ≥ 0 ∀y ∈ c. definition 2.6. [7] let e be a reflexive, strictly convex and smooth banach space and j p be a duality mapping of e. then for every closed convex subset c ⊂ e and x ∈ e there exists a unique element πpc (x) ∈ c such that 4p(x, π p c (x)) = miny∈c 4p(x, y). π p c (x) is called the bregman projection of x onto c, with respect to the function f (x) = 1p‖x‖ p. moreover x0 ∈ c is the bregman projection of x onto c iff 〈j p(x0) − j p(x), y − x0〉 ≥ 0 or equivalently 4p(x0, y) ≤4p(x, y) −4p(x, x0) f or every y ∈ c. lemma 2.5. [7] the bregman projection and the metric projection are related via pc (x)− x = π p c−x(0), ∀x ∈ e. especially we have pc (0) = π p c (0) and thus ‖π p c (0)‖ = miny∈c ‖y‖. the bregman distance has the following properties [7]. lemma 2.6. ∀x, y ∈ e and {xn} ∈ e, 4p(x, y) ≥ 0 and 4p(x, y) = 0 ⇔ x = y. lemma 2.7. ∀x, y ∈ e and λ ≥ 0 4p(−x,−y) = 4p(x, y) and 4p is positively homogeneous of degree p, i.e. 4p(λx,λy) = λp 4p (x, y). lemma 2.8. 4p is continuous in both arguments and it is strictly convex, weakly lower semicontinuous and gateaux differentiable with respect to the second variable with derivative ∂ ∂y 4p (x, y) = j p(y) − j p(x). lemma 2.9. 4p satisfies the three-point property that generalizes the ”law of cosines”: 4p(x, y) = 4p(x, z) + 4p(z, y) −〈x − z, j py − j pz〉 lemma 2.10. let x be reflexive, smooth and strictly convex. then for all x, y ∈ x the following holds: 4p(x, y) + 4p(y, x) = 〈j p(x) − j p(y), x − y〉 lemma 2.11. consider the following assertions; 1. limn→∞ ‖xn − x‖ = 0 2. limn→∞ ‖xn‖ = ‖x‖ and limn→∞〈j p(xn), x〉 = 〈j p(x), x〉 3. limn→∞4p(xn, x) = 0. the implication (1) =⇒ (2) =⇒ (3) are valid. if e is uniformly convex then the assertions are equivalent. lemma 2.12. let us write 4∗q(x ∗, y∗) = 1p‖x ∗‖ q e∗ − 〈j q e∗, y ∗〉 + 1 q‖y ∗‖ q e∗ for the bregman distance on the dual space e ∗ with respect to the function f ∗(x∗) = 1q‖x ∗‖ q e∗. then we have 4p(x, y) = 4 ∗ q(x ∗, y∗). f or x∗ = j pe (x) ⇔ jqe∗(x ∗) = x and y∗ = j pe (y) = y . ⇔ jqe∗(y ∗) = y. lemma 2.13. ∏p c (x) maps bounded sets onto bounded sets; more precisely we have ‖π p c (x)‖ ≤ (2 q−1 ‖x‖) ∨ (3‖πpc (0)‖)∀x ∈ e. throughout this paper, some important characteristic inequalities we will be using are; lemma 2.14. [16] in the case of uniformly convex space e,with the duality map j p of e, ∀x, y ∈ e we have ‖x − y‖p = 〈j p(x − y), x − y〉 ≥ ‖x‖p − p〈j p(x), y〉 + σp(x, y). with σp(x, y) = pkp ∫ 1 0 (‖x − ty‖∨‖x‖)p t λe ( t‖y‖ 2(‖x − ty‖∨‖x‖) ) dt where by kp = 4(2 + √ 4) min{ 1 2 p(p − 1) ∧ 1, ( 1 2 p ∧ 1)(p − 1), (p − 1)(1 − ( √ 3 − 1)q), 1 − (1 + (2 − √ 3)q)1−p}. lemma 2.15. [16] in the case of uniformly smooth space, e,with the duality map j p of e, ∀x, y ∈ e we have ‖x − y‖p = 〈j p(x − y), x − y〉 ≤ ‖x‖p − p〈j p(x), y〉 + σ̄p(x, y). with σ̄p(x, y) = pg p ∫ 1 0 (‖x − ty‖∨‖x‖)p t ρe ( t‖y‖ 2(‖x − ty‖∨‖x‖) ) dt where by g p = 8 ∨ 64ck −1 p with kp = 4(2 + √ 4) min{ 1 2 p(p − 1) ∧ 1, ( 1 2 p ∧ 1)(p − 1), (p − 1)(1 − ( √ 3 − 1)q), 1 − (1 + (2 − √ 3)q)1−p} and c = 4 λ0√ 1 + λ20 − 1 ∞∏ j=1 ( 1 + 15 2 j+2 λ0 ) 38 yusuf ibrahim / j. nig. soc. phys. sci. 1 (2019) 35–50 39 with λ0 = √ 339 − 18 30 . lemma 2.16. [17] let e be a real uniformly convex banach space. for arbitrary r > 1, let br (0) = {x ∈ e : ‖x‖ ≤ r}. then, there exists a continuous strictly increasing convex function g : [0,∞) −→ [0,∞), g(0) = 0 such that for every x, y ∈ br (0), fx ∈ jp(x), fy ∈ jp(y), the following inequality hold: ‖λx+(1−λ)y‖p ≤ λ‖x‖p+(1−λ)‖y‖p−(λp(1−λ)+(1−λ)pλ)g(‖x−y‖) and, 〈x − y, fx − fy〉 ≥ g(‖x − y‖). lemma 2.17. [17] let e be a real uniformly smooth banach space. for arbitrary r > 1, let br (0) = {x ∈ e : ‖x‖ ≤ r}. then, there exists a continuous strictly increasing convex function g : [0,∞) −→ [0,∞), g(0) = 0 such that for every x, y ∈ br (0), fx ∈ jq(x), fy ∈ jq(y), the following inequality hold: ‖λx+(1−λ)y‖q ≥ λ‖x‖q+(1−λ)‖y‖q−(λq(1−λ)+(1−λ)qλ)g(‖x−y‖) and, 〈x − y, fx − fy〉 ≤ g(‖x − y‖). 3. main results theorem 3.1. let e1 and e2 be two uniformly convex and uniformly smooth banach spaces, a : e1 −→ e2 is bounded and linear operator such that a(c), for c ⊂ e1, is closed and convex, u : e1 −→ e1 be a uniformly continuous bregman generalized asymptotically nonexpansive operator with the sequences {k(1)n } ⊂ [0,∞) and {µ (1) n } ⊂ [0,∞) satisfying lim n→∞ k(1)n = 0 and lim n→∞ µ (1) n = 0, respectively, and t : e2 −→ e2 be a uniformly continuous bregman generalized asymptotically nonexpansive operator with the sequences {k(2)n } ⊂ [0,∞) and {µ (2) n } ⊂ [0,∞) satisfying lim n→∞ k(2)n = 0 and lim n→∞ µ (2) n = 0, respectively, and, for p, q ∈ (1,∞), πpac : e2 −→ ac be a bregman projection onto a subset ac, and fix(u) , φ and fix(πpac t ) , φ respectively, and (i −u) and (i −πpac t ) be demiclosed at zero. let x1 ∈ e1 be chosen arbitrarily and let c1 = e1 and the sequence {xn} be defined as follows; zn = j q e∗1 (j pe1 xn −λn a ∗j pe2 (i − π p acn t n)axn), yn = j q e∗1 (αn j p e1 (zn) + (1 −αn)j p e1 (u n(zn))), cn+1 = {v ∈ cn : 4p(yn, v) ≤ [kn + 1] 4p (zn, v) + µn; 4p(zn, v) ≤4p(xn, v)}, xn+1 = π p cn+1 (x1), n ≥ 1 (11) where λn =  1 ‖a‖ 1 ‖j pe2 (i−πpacn t n )axn‖ , xn , 0 1 ‖a‖p 〈j pe2 (i−πpacn t n )axn,axn−π p acn t n axn〉p−1 ‖j pe2 (i−πpacn t n )axn‖p , xn = 0 (12) and γ ∈ (0, 1) and τn = 1 ‖xn‖p−1 are chosen such that ρe∗1 (τn) = γ 2qgq‖a‖ × 〈j pe2 (i − π p acn t n)axn, axn − π p acn t n axn〉 ‖xn‖p‖j p e2 (i − πpacn t n)axn‖ , (13) and a∗ denote the adjoint of a, and {αn} ⊂ (0, 1) satisfies lim inf n→∞ αn(1 − αn) > 0, kn = (k (1) n ∨ k (2) n ), n ≥ 1. if γ = {v ∈ fix(u) : av ∈ fix(πpac t )}, φ, then {xn} converges strongly to x∗ ∈ γ. proof: assume that ‖λn a∗(j p e2 (i − πpacn t n)axn)‖ = 0 ⇔ xn ∈ γ = {xn ∈ fix(u) : axn ∈ fix(π p ac t )} , φ then from 11 we have that zn = xn and 4p(xn, zn) = 0 and the conclusion follows immediately. now assume that ‖λn a∗(j p e2 (i −πpacn t n)axn)‖, 0 then we will divide the proof into five steps. step one: we first show that cn is closed and convex, for any n ≥ 1. by induction hypothesis; since c1 = e1, so c1 is closed and convex. assuming that ck is closed and convex. we show that ck+1 is closed and convex, for some k ≥ 0. we first show that ck+1 is closed. let {xm} be a sequence in ck+1 such that xm → v as n →∞. we need to show that v ∈ ck+1. from 11 and continuity of bregman distance and the boundedness of {xm}, we have that 4p(zn, v) = lim m→∞ 4p (zn, xm) ≤ lim m→∞ 4p (xn, xm) = 4p(xn, v). (14) similarly, we have that 4p(yn, v) = lim m→∞ 4p (yn, xm) ≤ lim m→∞ ([kn + 1] 4p (zn, xm) + µn) = [kn + 1](4p(xn, v) + µn. (15) from (14) and (15), we have that v ∈ ck+1. next we show that ck+1 is convex. let v1, v2 ∈ ck+1 and t ∈ (0, 1); putting v = tv1 + (1 − t)v2, it suffices to show that v ∈ ck+1. let xk , 0, then 4p(zk, v) = 4p(zk, tv1 + (1 − t)v2) = 1 q ‖j pe1 xk −λk a ∗(j pe2 (i − π p ack t k)axk)‖ q + 1 p ‖tv1 + (1 − t)v2‖ p −〈j pe1 xk −λk a ∗(j pe2 (i − π p ack t k)axk), tv1 + (1 − t)v2〉 (16) 39 yusuf ibrahim / j. nig. soc. phys. sci. 1 (2019) 35–50 40 = 1 q ‖j pe1 xk −λk a ∗(j pe2 (i − π p ack t k)axk‖ q + 1 p ‖tv1 + (1 − t)v2‖ p −〈j pe1 xk, tv1 + (1 − t)v2〉 + 〈λk(j p e2 (i − πpack t k)axk, tv1 + (1 − t)v2〉. where λk〈a ∗(j pe2 (i − π p ack t k )axk, tv1 + (1 − t)v2〉 = λk〈(j p e2 (i − πpack t k )axk, tav1 + (1 − t)av2 + π p ack t k axk − π p ack t k axk〉 = λk〈(j p e2 (i − πpack t k )axk, tav1 + (1 − t)av2 − π p ack t k axk〉 + λk〈(j p e2 (i − πpack t k )axk, π p ack t k axk〉 = λk〈(j p e2 (i − πpack t k )axk, tav1 + (1 − t)av2 − π p ack t k axk〉 + λk〈(j p e2 (i − πpack t k )axk, π p ack t k axk − axk + axk〉 = λk〈(j p e2 (i − πpack t k )axk, tav1 + (1 − t)av2 − π p ack t k axk〉 −λk〈(j p e2 (i − πpack t k )axk, π p ack t k axk − axk〉 + λk〈(j p e2 (i − πpack t k )axk, axk〉 = λk〈(j p e2 (i − πpack t k )axk, tav1 + (1 − t)av2 − axk + axk − π p ack t k axk〉 −λk〈(j p e2 (i − πpack t k )axk, axk − π p ack t k axk〉 + λk〈(j p e2 (i − πpack t k )axk, axk〉 = λk〈(j p e2 (i − πpack t k )axk, (tav1 + (1 − t)av2 − axk ) − (πpack t k axk − axk )〉−λk〈(j p e2 (i − πpack t k )axk, axk − π p ack t k axk〉 + λk〈(j p e2 (i − πpack t k )axk, axk〉 = −λk〈(j p e2 (πpack t k − i)axk, (tav1 + (1 − t)av2 − axk ) − (πpack t k axk − axk )〉−λk〈(j p e2 (i − πpack t k )axk, axk − π p ack t k axk〉 + λk〈(j p e2 (i − πpack t k )axk, axk〉. by the assumption that a(ck) is closed and convex and lemma 2.5 and by the variational inequality for the bregman projection of zero onto a(ck) − axk, as in definition 2.6, we arrive at 〈(j pe2 (π p ack t k − i)axk, (tav1 + (1 − t)av2 − axk) −(πpack t k axk − axk)〉 ≥ 0 and therefore, λk〈j p e2 (i − πpack t k)axk, tav1 + (1 − t)av2〉 ≤ λk〈j p e2 (i − πpack t k)axk, axk〉 −λk〈j p e2 (i − πpack t k)axk, axk − π p ack t k axk〉. (17) in addition, from lemma 2.15, we have that 1 q ‖j pe1 xk −λk a ∗(j pe2 (i − π p ack t k)axk‖ q ≤ 1 q ‖j pe1 xk‖ q −λk〈xk, a ∗j pe2 (i − π p ack t k)axk〉 + 1 q σ̄q(j p e1 xk,λk a ∗j pe2 (i − π p ack t k)axk) = 1 q ‖j pe1 xk‖ q −λk〈axk, j p e2 (i − πpack t k)axk〉 + 1 q σ̄q(j p e1 xk,λk a ∗j pe2 (i − π p ack t k)axk) = 1 q ‖xk‖ p −λk〈axk, j p e2 (i − πpack t k)axk〉 + 1 q σ̄q(j p e1 xk,λk a ∗j pe2 (i − π p ack t k)axk) (18) and 1 q σ̄q(j p e1 xk,λk a ∗j pe2 (i − π p ack t k)axk) = gq ∫ 1 0 (‖j pe1 xk − tλk a ∗j pe2 (i − π p ack t k)axk‖∨‖j p e1 xk‖)q t × ρe∗  t‖λk a∗j pe2 (i − πpack t k)axk‖(‖j pe1 xk − tλk a∗j pe2 (i − πpack t k)axk‖∨‖j pe1 xk‖)  dt, (19) f or every t ∈ [0, 1]. but ‖j pe1 xk−tλk a ∗j pe2 (i − π p ack t k)axk‖ ≤ ‖xk‖ p−1 + ‖λk a ∗j pe2 (i − π p ack t k)axk‖. from (12), suppose that λk = τk ‖a‖ ‖xk‖p−1 ‖j pe2 (i − π p acn t n)axn‖ then we have that ‖j pe1 xk − tλk a ∗j pe2 (i − π p ack t k)axk‖ ≤ 2‖xk‖ p−1 and that ‖xk‖ p−1 ≤ ‖j pe1 xk − tλk a ∗j pe2 (i − π p ack t k)axk‖∨‖j p e1 xk‖ ≤ 2‖xk‖ p−1. (20) from definition 2.2 (2), (12) and (20), we have that ρe∗1  t‖λk a∗j pe2 (i − πpack t k )axk‖(‖j pe1 xk − tλk a∗j pe2 (i − πpack t k )axk‖∨‖j pe1 xk‖)  ≤ ρe∗1  t‖λk a∗j pe2 (i − πpack t k )axk‖ ‖xk‖p−1  = ρe∗1 (tτk ). (21) substituting eqs. (20) and (21) into (19), and since t ≤ τk and ρe∗1 is nondecreasing function then, we have that 1 q σ̄q(j p e2 xk,λk a ∗j pe2 (i − π p ack t k)axk) ≤ 2qgq‖xk‖ ( p−1)q ∫ τk 0 ρe∗1 (t) t dt ≤ 2qgq‖xk‖ p ∫ τk 0 ρe∗1 (τk) τk dt = 2qgq‖xk‖ pρe∗1 (τk). (22) 40 yusuf ibrahim / j. nig. soc. phys. sci. 1 (2019) 35–50 41 substituting eqs. (22) into (18), we have that 1 q ‖j pe1 xk −λk a ∗j pe2 (i − π p ack t k)axk‖ q ≤ 1 q ‖xk‖ p −λk〈axk, j p e2 (i − πpack t k)axk〉 + 2 qgq‖xk‖ pρe∗1 (τk) = 1 q ‖j pe1 xk‖ q −λk〈(j p e2 (i − πpack t k)axk, axk〉 + 2qgq‖xk‖ pρe∗1 (τk). (23) substituting (23) and (17) into (16), we have that 4p(zk, tv1 + (1 − t)v2) ≤ 1 q ‖xk‖ p + 1 p ‖tv1 + (1 − t)v2‖ p −〈j pe1 xk, tv1 + (1 − t)v2〉 + 2 qgq‖xk‖ pρe∗1 (τk) −λn〈(j p e2 (i − πpack t k)axk), axk − π p ack t k axk〉 = 4p(xk, tv1 + (1 − t)v2) + 2 qgq‖xk‖ pρe∗1 (τk) −λn〈(j p e2 (i − πpack t k)axk), axk − π p ack t k)axk〉. (24) substituting (13) and (12) into (24), we have that 4p(zk, tv1 + (1 − t)v2) ≤4p(xk, tv1 + (1 − t)v2) + γ〈j pe2 (i − π p ack t k)axk, axk − π p ack t k axk〉 ‖a‖‖j pe2 (i − π p ack t k)axk)‖ (25) − 〈j pe2 (i − π p ack t k)axk, axk − π p ack t k)axk〉 ‖a∗‖‖j pe2 (i − π p ack t k)axk‖ = 4p(xk, tv1 + (1 − t)v2) − [1 −γ] × 〈j pe2 (i − π p ack t k)axk, axk − π p ack t k)axk〉 ‖a‖‖j pe2 (i − π p ack t k)axk)‖ . (26) therefore, 4p(zk, tv1 + (1 − t)v2) ≤4p(xk, tv1 + (1 − t)v2). (27) let xk = 0, we have 4p(xk, tv1 + (1 − t)v2) = 1 p ‖tv1 + (1 − t)v2‖ p (28) and from (28), we have 4p(zk, tv1 + (1 − t)v2) = 1 q ‖λk a ∗j pe2 (i − π p ack t k)axk‖ q + 4p(xk, tv1 + (1 − t)v2) + λk〈j p e2 (i − πpack t k)axk), tav1 + (1 − t)av2〉. (29) substituting (17) into (29), we have that 4p(zk, tv1 + (1 − t)v2) ≤ 1 q ‖λk a ∗j pe2 (i − π p ack t k)axk‖ q + 4p(xk, tv1 + (1 − t)v2) + λn〈j p e2 (i − πpack t k)axk, axk〉 −λn〈j p e2 (i − πpack t k)axk, axk − π p ack t k)axk〉. (30) but, from (12), we have that 1 q ‖λk a ∗j pe2 (i − π p ack t k)axk‖ q = 1 q 1 ‖a‖p 〈j pe2 (i − π p ack t k)axk, axk − π p ack t k)axk〉p ‖j pe2 (i − π p ack t k)axk‖p . (31) substituting (31) into (30), we have that 4p(zk, tv1 + (1 − t)v2) ≤ 1 q 1 ‖a‖p 〈j pe2 (i − π p ack t k)axk, axk − π p ack t k)axk〉p ‖j pe2 (i − π p ack t k)axk‖p + 4p(xk, tv1 + (1 − t)v2) + λk〈j p e2 (i − πpack t k)axk, axk〉 −λn〈j p e2 (i − πpack t k)axk, axk − π p ack t k)axk〉 ≤ (1 − 1 p ) 1 ‖a‖p 〈j pe2 (i − π p ack t k)axk, axk − π p ack t k)axk〉p ‖j pe2 (i − π p ack t k)axk‖p + 4p(xk, tv1 + (1 − t)v2) + λk〈a ∗j pe2 (i − π p ack t k)axk, xk〉 − 1 ‖a‖p 〈j pe2 (i − π p ack t k)axk, axk − π p ack t k)axk〉p ‖j pe2 (i − π p ack t k)axk‖p ×〈j pe2 (i − π p ack t k)axk, axk − π p ack t k)axk〉 ≤ (1 − 1 p ) 1 ‖a‖p 〈j pe2 (i − π p ack t k)axk, axk − π p ack t k)axk〉p ‖j pe2 (i − π p ack t k)axk‖p + 4p(xk, tv1 + (1 − t)v2) + λk‖j p e2 (i − πpack t k)axk‖‖axk‖ − 1 ‖a‖p 〈j pe2 (i − π p ack t k)axk, axk − π p ack t k)axk〉p ‖j pe2 (i − π p ack t k)axk‖p ×〈j pe2 (i − π p ack t k)axk, axk − π p ack t k)axk〉 = 4p(xk, tv1 + (1 − t)v2) − 1 p 1 ‖a‖p 〈j pe2 (i − π p ack t k)axk, axk − π p ack t k)axk〉p ‖j pe2 (i − π p ack t k)axk‖p . this implies that 4p(zk, tv1 + (1 − t)v2) ≤4p(xk, tv1 + (1 − t)v2). (32) in addition, it follows from eq. (11), definition 2.4 and lemma 2.16 that 4p(yk, tv1 + (1 − t)v2) = 4p(j q e∗1 (αk j p e1 (zk) + (1 −αk)j p e1 (u k(zk)), tv1 + (1 − t)v2) = 1 q ‖αk j p e1 (zk) + (1 −αk)j p e1 (u k(zk)‖ q −〈αk j p e1 (zk) + (1 −αk)j p e1 (u k(zk), tv1 + (1 − t)v2〉 + 1 p ‖tv1 + (1 − t)v2‖ p = 1 q ‖αk j p e1 (zk) + (1 −αk)j p e1 (u k(zk)‖ p −αk〈j p e1 zk, tv1 + (1 − t)v2〉 41 yusuf ibrahim / j. nig. soc. phys. sci. 1 (2019) 35–50 42 − (1 −αk)〈j p e1 (u k(zk)), tv1 + (1 − t)v2〉 + 1 p ‖tv1 + (1 − t)v2‖ p ≤ 1 q αk‖zk‖ p + 1 q (1 −αk)‖u kzk‖ p − (αqk (1 −αk) + (1 −αk) qαk) × g(‖zk − u kzk‖) −αk〈j p e1 (zk), tv1 + (1 − t)v2〉 − (1 −αk)〈j p e1 (u k(zk)), tv1 + (1 − t)v2〉 + 1 p ‖tv1 + (1 − t)v2‖ p = αk ( 1 p ‖tv1 + (1 − t)v2‖ p −〈j pe1 (zk), tv1 + (1 − t)v2〉 + 1 q ‖zk‖ p ) + (1 −αk) ( 1 p ‖tv1 + (1 − t)v2‖ p −〈j pe1 (u k(zk)), tv1 + (1 − t)v2〉 + 1 q ‖u kzk‖ p ) − (αqk (1 −αk) + (1 −αk) qαk)g(‖zk − u kzk‖) = αk 4p (zk, tv1 + (1 − t)v2) + (1 −αk) 4p (u k(zk), tv1 + (1 − t)v2) − (α p k (1 −αk) + (1 −αk) pαk) × g(‖zk − u kzk‖) ≤ αk 4p (zk, tv1 + (1 − t)v2) + (1 −αk)(kk 4p (zk, tv1 + (1 − t)v2) + 4p(zk, tv1 + (1 − t)v2) + µk) − (α q k (1 −αk) + (1 −αk) qαk)g(‖zk − u kzk‖) ≤ kk 4p (zk, tv1 + (1 − t)v2) + 4p(zk, tv1 + (1 − t)v2) + µk − (αqk (1 −αk) + (1 −αk) qαk)g(‖zk − u kzk‖) ≤ [kk + 1] 4p (zk, tv1 + (1 − t)v2) + µk. (33) from (27), (32) and (33), we have that v ∈ ck+1. hence, ck+1 is convex. therefore cn is closed and convex for each n ∈ n step two: we prove γ ⊂ cn for any n ≥ 1. clearly γ ⊂ c1. now assuming that γ ⊂ cn for some n ≥ 1. let v ∈ γ then from (11) and lemma 2.15, having considered xn , 0, we get 4p(zn, v) = 1 q ‖j pe1 xn −λn a ∗(j pe2 (i − π p acn t k)axn)‖ q + 1 p ‖v‖p −〈j pe1 xn −λn a ∗(j pe2 (i − π p acn t n)axn), v〉 (34) = 1 q ‖j pe1 xn −λn a ∗(j pe2 (i − π p acn t n)axn‖ q + 1 p ‖v‖p −〈j pe1 xn, v〉 + 〈λn(j p e2 (i − πpacn t n)axn, v〉. where λn〈a ∗(j pe2 (i − π p acn t n)axn, v〉 = λn〈(j p e2 (i − πpacn t n)axn, av + π p acn t n axn − π p acn t n axn〉 = λn〈(j p e2 (i − πpacn t n)axn, av − π p acn t n axn〉 + λn〈(j p e2 (i − πpacn t n)axn, π p acn t n axn〉 = λn〈(j p e2 (i − πpacn t n)axn, av − π p acn t n axn〉 + λn〈(j p e2 (i − πpacn t n)axn, π p acn t n axn − axn + axn〉 = λn〈(j p e2 (i − πpacn t n)axn, av − π p acn t n axn〉 −λn〈(j p e2 (i − πpacn t n)axn, π p acn t n axn − axn〉 + λn〈(j p e2 (i − πpacn t n)axn, axn〉 = λn〈(j p e2 (i − πpacn t n)axn, av − axn + axn − π p acn t n axn〉 −λn〈(j p e2 (i − πpacn t n)axn, axn − π p acn t n axn〉 + λn〈(j p e2 (i − πpacn t n)axn, axn〉 = λn〈(j p e2 (i − πpacn t n)axn, (av − axn) − (π p acn t n axn − axn)〉 −λn〈(j p e2 (i − πpacn t n)axn, axn − π p acn t n axn〉 + λn〈(j p e2 (i − πpacn t n)axn, axn〉 = −λn〈(j p e2 (πpacn t n − i)axn, (av − axn) − (π p acn t n axn − axn)〉 −λn〈(j p e2 (i − πpacn t n)axn, axn − π p acn t n axn〉 + λn〈(j p e2 (i − πpacn t n)axn, axn〉. by the assumption that a(cn) is closed and convex and lemma 2.5 and by the variational inequality for the bregman projection of zero onto a(cn) − axn, as in definition 2.6, we arive at 〈(j pe2 (π p acn t n − i)axn, (av − axn) − (π p acn t n axn − axn)〉 ≥ 0 and therefore, λn〈j p e2 (i − πpacn t n)axn, av〉 ≤ λn〈j p e2 (i − πpacn t n)axn, axn〉 −λn〈j p e2 (i − πpacn t n)axn, axn − π p acn t n axn〉. (35) in addition, from lemma 2.15, we have that 1 q ‖j pe1 xn −λn a ∗(j pe2 (i − π p acn t n)axn‖ q ≤ 1 q ‖j pe1 xn‖ q −λn〈xn, a ∗j pe2 (i − π p acn t n)axn〉 + 1 q σ̄q(j p e1 xn,λn a ∗j pe2 (i − π p acn t n)axn) = 1 q ‖j pe1 xn‖ q −λn〈axn, j p e2 (i − πpacn t n)axn〉 + 1 q σ̄q(j p e1 xn,λn a ∗j pe2 (i − π p acn t n)axn) = 1 q ‖xn‖ p −λn〈axn, j p e2 (i − πpacn t n)axn〉 + 1 q σ̄q(j p e1 xn,λn a ∗j pe2 (i − π p acn t n)axn) (36) and 1 q σ̄q(j p e1 xn,λn a ∗j pe2 (i − π p acn t n)axn) = gq ∫ 1 0 (‖j pe1 xn − tλn a ∗j pe2 (i − π p acn t n)axn‖∨‖j p e1 xn‖)q t × ρe∗  t‖λn a∗j pe2 (i − πpacn t n)axn‖(‖j pe1 xn − tλn a∗j pe2 (i − πpacn t n)axn‖∨‖j pe1 xn‖)  dt, (37) f or every t ∈ [0, 1]. 42 yusuf ibrahim / j. nig. soc. phys. sci. 1 (2019) 35–50 43 but ‖j pe1 xn−tλn a ∗j pe2 (i − π p acn t n)axn‖ ≤ ‖xn‖ p−1 + ‖λn a ∗j pe2 (i − π p acn t n)axn‖. from (12), suppose that λn = τn ‖a‖ ‖xn‖p−1 ‖j pe2 (i − π p acn t n)axn‖ then we have ‖j pe1 xn − tλn a ∗j pe2 (i − π p acn t n)axn‖ ≤ 2‖xn‖ p−1 and that ‖xn‖ p−1 ≤ ‖j pe1 xn − tλn a ∗j pe2 (i − π p acn t n)axn‖∨‖j p e1 xn‖ ≤ 2‖xn‖ p−1. (38) from definition 2.2(2), we have that from eqs. (38) and (13) ρe∗1  t‖λn a∗j pe2 (i − πpacn t n)axn‖(‖j pe1 xn − tλn a∗j pe2 (i − πpacn t n)axn‖∨‖j pe1 xn‖)  ≤ ρe∗1  t‖λn a∗j pe2 (i − πpacn t n)axn‖ ‖xn‖p−1  = ρe∗1 (tτn). (39) substituting eqs. (38) and (39) into (37), and since t ≤ τn and ρe∗1 is nondecreasing then, we have that 1 q σ̄q(j p e2 xn,λn a ∗j pe2 (i − π p acn t n)axn) ≤ 2qgq‖xn‖ (p−1)q ∫ τ 0 ρe∗1 (t) t dt ≤ 2qgq‖xn‖ p ∫ τ 0 ρe∗1 (τn) τn dt ≤ 2qgq‖xn‖ pρe∗1 (τn). (40) substituting eq. (40) into (36), we have that 1 q ‖j pe1 xn −λn a ∗j pe2 (i − π p acn t n)axn‖ q ≤ 1 q ‖xn‖ p −λn〈axn, j p e2 (i − πpacn t n)axn〉 + 2 qgq‖xn‖ pρe∗1 (τn) = 1 q ‖j pe1 xn‖ q −λn〈(j p e2 (i − πpacn t n)axn, axn〉 + 2qgq‖xn‖ pρe∗1 (τn). (41) substituting eqs. (35) and (41) into (34), we have that 4p(zn, v) ≤ 1 q ‖xn‖ p + 1 p ‖v‖p −〈j pe1 xn, v〉 + 2 qgq‖xn‖ pρe∗1 (τn) −λn〈(j p e2 (i − πpacn t n)axn), axn − π p acn t n)axn〉 = 4p(xn, v) + 2 qgq‖xn‖ pρe∗1 (τn) −λn〈(j p e2 (i − πpacn t n)axn), axn − π p acn t n)axn〉. (42) substituting (13) and (12) into (42), we have that 4p(zn, v) ≤4p(xn, v) + γ〈j pe2 (i − π p acn t n)axn, axn − π p acn t n)axn〉 ‖a‖‖j pe2 (i − π p acn t n)axn)‖ − 〈j pe2 (i − π p acn t n)axn, axn − π p acn t n)axn〉 ‖a∗‖‖j pe2 (i − π p acn t n)axn‖ = 4p(xn, v) − [1 −γ] × 〈j pe2 (i − π p acn t n)axn, axn − π p acn t n)axn〉 ‖a∗‖‖j pe2 (i − π p acn t n)axn‖ . (43) therefore, 4p(zn, v) ≤4p(xn, v). (44) for xn = 0 we have 4p(xn, v) = 1 p ‖v‖p (45) and from (45), we have 4p(zn, v) = 1 q ‖λn a ∗j pe2 (i − π p acn t n)axn‖ q + 4p(xn, v) + λn〈j p e2 (i − πpacn t n)axn), av〉. (46) substituting eq. (35) into eq. (46), we have that 4p(zn, v) ≤ 1 q ‖λn a ∗j pe2 (i − π p acn t n)axn‖ q + 4p(xn, v) + λn〈j p e2 (i − πpacn t n)axn, axn〉 −λn〈j p e2 (i − πpacn t n)axn, axn − π p acn t n)axn〉 (47) but, from (13), we have that 1 q ‖λn a ∗j pe2 (i − π p acn t n)axn‖ q = 1 q 1 ‖a‖p 〈j pe2 (i − π p acn t n)axn, axn − π p acn t n)axn〉p ‖j pe2 (i − π p acn t n axn‖p . (48) substituting (48) into (47), we have that 4p(zn, v) (49) ≤ 1 q 1 ‖a‖p 〈j pe2 (i − π p acn t n)axn, axn − π p acn t n axn〉p ‖j pe2 (i − π p acn t n)axn‖p + 4p(xn, v) + λn〈j p e2 (i − πpacn t n)axn, axn〉 −λn〈j p e2 (i − πpacn t n)axn, axn − π p acn t n axn〉 ≤ (1 − 1 p ) 1 ‖a‖p 〈j pe2 (i − π p acn t n)axn, axn − π p acn t n axn〉p ‖j pe2 (i − π p acn t n)axn‖p 43 yusuf ibrahim / j. nig. soc. phys. sci. 1 (2019) 35–50 44 + 4p(xn, v) + λn〈a ∗j pe2 (i − π p acn t n)axn, xn〉 − 1 ‖a‖p 〈j pe2 (i − π p ack t n)axn, axn − π p acn t n axn〉p ‖j pe2 (i − π p acn t n)axn‖p ×〈j pe2 (i − π p acn t n)axn, axn − π p acn t n axn〉 ≤ (1 − 1 p ) 1 ‖a‖p 〈j pe2 (i − π p acn t n)axn, axn − π p acn t n axn〉p ‖j pe2 (i − π p acn t n)axn‖p + 4p(xn, v) + λn‖a ∗j pe2 (i − π p acn t n)axn‖‖xn‖ − 1 ‖a‖p 〈j pe2 (i − π p acn t n)axn, axn − π p acn t n axn〉p ‖j pe2 (i − π p acn t n)axn‖p ×〈j pe2 (i − π p acn t n)axn, axn − π p acn t n axn〉 = 4p(xn, v) − 1 p ) 1 ‖a‖p 〈j pe2 (i − π p acn t n)axn, axn − π p acn t n axn〉p ‖j pe2 (i − π p acn t n)axn‖p = 4p(xn, v) − 1 p ) 1 ‖a‖p × 〈j pe2 (i − π p acn t n)axn, axn − π p acn t n)axn〉p ‖j pe2 (i − π p acn t n)axn‖p (50) this implies that 4p(zn, v) ≤4p(xn, v). (51) in addition, it follows from (11), definition 2.4, lemma 2.16 and p ∈ (1,∞) that 4p(yn, v) = 4p(j q e∗1 (αn j p e1 (zn) + (1 −αn)j p e1 (u n(zn))), v) = 1 q ‖αn j p e1 zn + (1 −αn)j p e1 u zn‖ q −〈αn j p e1 zn + (1 −αn)j p e1 u nzn, v〉 + 1 p ‖v‖p = 1 q ‖αn j p e1 zn + (1 −αn)j p e1 u nzn‖ p −αn〈j p e1 zn, v〉 − (1 −αn)〈j p e1 (u n(zn)), v〉 + 1 p ‖v‖p ≤ 1 q αn‖zn‖ p + 1 q (1 −αn)‖u nzn‖ p − (αqn(1 −αn) + (1 −αn) qαn) × g(‖zn − u nzn‖) −αn〈j p e1 (zn), v〉− (1 −αn)〈j p e1 (u n(zn)), v〉 + 1 p ‖v‖p = αn( 1 p ‖v‖p −〈j pe1 (zn), v〉 + 1 q ‖zn‖ p) + (1 −αn)( 1 p ‖v‖p −〈j pe1 (u n(zn)), v〉 + 1 q ‖u nzn‖ p) − (αqn(1 −αn) + (1 −αn) qαn)g(‖zn − u nzn‖) = αn 4p (zn, v) + (1 −αn) 4p (u n(zn), v) − (αpn (1 −αn) + (1 −αn) pαn) × g(‖zn − u nzn‖) ≤ αn 4p (zn, v) + (1 −αn)(kn 4p (zn, v) + 4p(zn, v) + µn) − (αqn(1 −αn) + (1 −αn) qαn)g(‖zn − u nzn‖) ≤ kn 4p (zn, v) + 4p(zn, v) + µn − (αqn(1 −αn) + (1 −αn) qαn)g(‖zn − u nzn‖) (52) ≤ [kn + 1] 4p (zn, v) + µn. (53) from eqs. (44), (51) and (53), we have that v ∈ cn and γ ⊂ cn for any n ≥ 1. step three: we show that {xn} is cauchy sequence. for v ∈ cn, and xn+1 = π p cn+1 (x1) ∈ cn+1 ⊂ cn, then from definition 2.6, we have that 0 ≤ 〈j pe1 π p cn+1 (x1) − j p e1 (x1), v − π p cn+1 (x1)〉 = 〈j pe1 π p cn+1 (x1), v〉− 〈j p e1 (x1), v〉 − 〈j pe1 π p cn+1 (x1), π p cn+1 (x1)〉 + 〈j p e1 (x1), π p cn+1 (x1)〉. this implies that |〈j pe1 π p cn+1 (x1), π p cn+1 (x1)〉| ≤ |〈j p e1 π p cn+1 (x1), v〉 − 〈j pe1 (x1), v〉 + 〈j p e1 (x1), π p cn+1 (x1)〉|. that is ‖π p cn+1 (x1)‖ p ≤ ‖π p cn+1 (x1)‖ p−1 ‖v‖ + ‖x1‖ p−1 ‖v‖ + ‖x1‖ p−1 ‖π p cn+1 (x1)‖. (54) observe that if ‖π p cn+1 (x1)‖ p−1 < 2‖x1‖ p−1 then we have ‖πpcn+1 (x1)‖ < 2 q−1 ‖x1‖ (55) and so we are done. otherwise, let tx1 = ‖πpcn+1 (x1)‖ ‖x1‖  p−1 ≥ 2. for x1 , 0 and using (54) we have that ‖π p cn+1 (x1)‖(‖π p cn+1 (x1)‖ p−1 −‖x1‖ p−1) ≤ ‖v‖(‖πpcn+1 (x1)‖ p−1 + ‖x1‖ p−1) and so ‖π p cn+1 (x1)‖ ≤ ‖v‖ ‖π p cn+1 (x1)‖p−1 + ‖x1‖p−1 ‖π p cn+1 (x1)‖p−1 −‖x1‖p−1 . (56) from (56), since the function h(tx1 ) = tx1 +1 tx1−1 is decreasing for tx1 > 1 we arive at ‖π p cn+1 (x1)‖ ≤ min v∈cn+1 ‖v‖h(tx1 ) ≤ min v∈cn+1 ‖v‖h(2) = 3 min v∈cn+1 ‖v‖. (57) if x1 = 0 then from (54), we have that ‖π p cn+1 (x1)‖ ≤ min v∈cn+1 ‖v‖. (58) from (55), (57) and (58), we have that xn+1 = π p cn+1 (x1) is 44 yusuf ibrahim / j. nig. soc. phys. sci. 1 (2019) 35–50 45 bounded. for any n ≥ 1, by definition 2.6, we have that 0 ≤4p(j p e1 π p cn (x1), π p cn+1 (x1)) ≤4p(x1, π p cn+1 (x1)) −4p(x1, π p cn (x1)) (59) from (59), we have that 4p(x1, π p cn (x1)) ≤4p(x1, π p cn+1 (x1)). thus {4p(x1, π p cn (x1))} is nondecreasing. therefore by boundedness of πpcn (x1), we have that limn→∞ 4p (x1, π p cn (x1)) exists. let m, n ∈ n m < n. from xn = π p cn (x1) ⊂ cn and (59), we have that 4p(π p cn (x1), π p cm (x1)) ≤4p(x1, π p cm (x1)) −4p(x1, π p cn (x1)). (60) since lim n→∞ 4p (x1, π p cn (x1)) exists, it follows from (60) that 4p(π p cm (x1), π p cn (x1)) → 0 as m, n →∞. therefore {πpcn (x1)} is a cauchy sequence. step four: we will show that lim n→∞ 4p(zn, uzn) = 0 and lim n→∞ 4p (axn, π p acn t n axn) = 0. let xn+1 = π p cn+1 (x1) ⊂ cn+1 ⊂ cn. from 11, we have that 4p(zn, v) ≤4p(xn, v) ⇔ 1 p ‖zn − v‖ p ≤ 1 p ‖xn − v‖ p. therefore {zn} is bounded as {xn} is bounded. hence, from lemma 2.9 and 2.10, we have 4p(zn, xn) = 4p (zn, xn+1) + 4p(xn+1, xn) −〈zn − xn+1, j p xn − j p xn+1〉 ≤4p(xn, xn+1) + 4p(xn+1, xn) −〈zn − xn+1, j p xn − j p xn+1〉 = 〈j p xn+1 − j p xn, xn+1 − xn,〉 − 〈zn − xn+1, j p xn − j p xn+1〉 → 0 as n →∞. (61) now, for xn , 0 for all n ∈ n, from (43), we have 4p(zn, v) ≤4p(xn, v) − [1 −γ]‖axn − π p acn t n axn‖ p−1. this implies that ‖axn − π p acn t n axn‖ ≤ ( 4p(xn, v) −4p(zn, v) [1 −γ] ) 1 p−1 =  1 q (‖zn‖ p −‖xn‖p) + 〈j p e1 zn − j p e1 xn, v〉 [1 −γ]  1 p−1 and therefore we have that lim n→∞ ‖axn − π p acn t n axn‖ = 0. (62) for xn = 0 for all n ∈ n, from (49), we have 4p(zn, v) ≤4p(xn, v) − 1 p 〈j pe2 (i − π p acn t n)axn, axn − π p acn t n axn〉p ‖a‖p‖j pe2 (i − π p acn t n)axn)‖p . t his implies that ‖axn − π p acn t n axn‖ ≤ p 1 p2−p ‖a‖p ( 4p(xn, v) −4p(zn, v) ) 1 p2−p = p 1 p2−p ‖a‖p ( 1 q (‖zn‖ p −‖xn‖ p) + 〈j pe1 zn − j p e1 xn, v〉 ) 1 p2−p . therefore, we have that lim n→∞ ‖axn − π p acn t n axn‖ = 0. (63) by lemma 2.11, we have that lim n→∞ 4p (axn, π p acn t n axn) = 0. (64) on the other hand, since 4p (yn, xn) = 4p(yn, xn+1) + 4p(xn+1, xn) −〈yn − xn+1, j p xn − j p xn+1〉 ≤ [1 + kn] 4p (zn, xn+1) + 〈j p xn+1 − j p xn, xn+1 − xn,〉 − 〈zn − xn+1, j p xn − j p xn+1〉 + µn ≤ [1 + kn] 4p (xn, xn+1) + 〈j p xn+1 − j p xn, xn+1 − xn,〉 − 〈zn − xn+1, j p xn − j p xn+1〉 + µn → 0 as n →∞ (65) and 4p(yn, zn) = 4p(yn, xn) + 4p(xn, zn) −〈yn − xn, j pzn − j p xn〉→ 0 as n →∞. (66) now, from (52), we have that 4p(yn, v) ≤ [kn + 1] 4p (zn, v) + µn (αpn (1 −αn) + (1 −αn) pαn)g(‖zn − u nzn‖). this implies, from (44), we have that g(‖zn − u nzn‖) ≤ [kn + 1] 4p (zn, v) + µn −4p(yn, v) (αpn (1 −αn) + (1 −αn)pαn) = 1 q ([kn + 1]‖zn‖ p −‖yn‖p) + 〈j pyn − j pzn, v〉− kn〈j pzn, v〉 (αpn (1 −αn) + (1 −αn)pαn) . + kn p ‖v‖ p + µn (αpn (1 −αn) + (1 −αn)pαn) . since αn(1 − αn) > 0, and g is continuous, strictly increasing and convex function, then we have that lim n→∞ g‖zn − u n(zn)‖ = 0 45 yusuf ibrahim / j. nig. soc. phys. sci. 1 (2019) 35–50 46 and lim n→∞ ‖zn − u n(zn)‖ = 0. (67) by lemma 2.11, we have that lim n→∞ 4p (zn, u nzn) = 0. (68) since t and u are continuous, then from (64) and (68), we have that lim n→∞ 4p (axn, π p ac t axn) = 0 (69) and lim n→∞ 4p (zn, uzn) = 0. (70) step five: we will show that {xn} converges strongly to an element of γ. since {xn} is cauchy sequence, we may assume that xn −→ x∗, from (61) we have zn −→ x∗, which implies that zn ⇀ x∗. so it follows from (70) and the demicloseness of (i−u) at zero that x∗ ∈ fix(u). in addition, since a is bounded linear operator, we have that lim n→∞ ‖axn−ax∗‖ = 0. hence, it follows from (69), and demicloseness of (i − πpac t ) at zero that ax∗ ∈ fix(πpac t ). this means that x ∗ ∈ γ and {xn} converges strongly to x∗ ∈ γ. the proof is completed. in theorem 3.1, as u = t , e1 = e2 = e and a = i, we have the following result. corollary 3.1.1. let e be a uniformly convex and uniformly smooth banach spaces, t : e −→ e be a uniformly continuous bregman generalized asymptotically nonexpansive operator with the sequences {kn} ⊂ [0,∞) and {µn} ⊂ [0,∞) satisfying lim n→∞ kn = 0 and lim n→∞ µn = 0, respectively, and, for p, q ∈ (1,∞), π p c : e −→ c be a bregman projection onto a subset c ∈ e, and fix(t ) , φ and (i − t ) be demiclosed at zero. let x1 ∈ e be chosen arbitrarily and let c1 = e. define a sequence {xn} as follows; zn = j q e∗1 (j pe1 xn −λn j p e2 (i − πpcn t n)xn), yn = j q e∗1 (αn j p e1 (zn) + (1 −αn)j p e1 (t n(zn))), cn+1 = {v ∈ cn : 4p(yn, v) ≤ [kn + 1] 4p (zn, v) + µn; 4p(zn, v) ≤4p(xn, v)}, xn+1 = π p cn+1 (x1), n ≥ 1 (71) where λn =  1 ‖j pe2 (i−πpcn t n )xn‖ , xn , 0 〈j pe2 (i−πpcn t n )xn,xn−π p cn t n xn〉p−1 ‖j pe2 (i−πpcn t n )xn‖p , xn = 0 and γ ∈ (0, 1) and τn = 1 ‖xn‖p−1 are chosen such that ρe∗1 (τn) = γ 2qgq × 〈j pe2 (i − π p cn t n)xn, xn − π p cn t n xn〉 ‖xn‖p‖j p e2 (i − πpcn t n)xn‖ , and {αn} ⊂ (0, 1) satisfies lim inf n→∞ αn(1 − αn) > 0. if fix(t ) = {v ∈ e : v = t v} , φ, then {xn} converges strongly to x∗ ∈ fix(t ). in theorem 3.1, when u and t are two bregman symptotically nonexpansive mappings, the following result holds. corollary 3.1.2. let e1 and e2 be two uniformly convex and uniformly smooth banach spaces, a : e1 −→ e2 is bounded and linear operator, such that a(c), for c ⊂ e1, is closed and convex, u : e1 −→ e1 be a uniformly continuous bregman asymptotically nonexpansive operator with the sequence {k(1)n } ⊂ [1,∞) satisfying limn→∞ k (1) n = 1, and t : e2 −→ e2 be a uniformly continuous bregman asymptotically nonexpansive operator with the sequence {k(2)n } ⊂ [1,∞) satisfying limn→∞ k (2) n = 1, respectively and, for p, q ∈ (1,∞), π p ac : e2 −→ ac be a bregman projection onto a subset ac, and fix(u) , φ and fix(πpac t ) , φ respectively, and (i − u) and (i − πpac t ) be demiclosed at zero. let x1 ∈ e1 be chosen arbitrarily and let c1 = e1. define a sequence {xn} as follows; zn = j q e∗1 (j pe1 xn −λn a ∗j pe2 (i − π p acn t n)axn), yn = j q e∗1 (αn j p e1 (zn) + (1 −αn)j p e1 (u n(zn))), cn+1 = {v ∈ cn : 4p(yn, v) ≤ kn 4p (zn, v) ≤ kn 4p (xn, v)}, xn+1 = π p cn+1 (x1), n ≥ 1 (72) where λn =  1 ‖a‖ 1 ‖j pe2 (i−πpacn t n )axn‖ , xn , 0 1 ‖a‖p 〈j pe2 (i−πpacn t n )axn,axn−π p acn t n axn〉p−1 ‖j pe2 (i−πpacn t n )axn‖p , xn = 0 and γ ∈ (0, 1) and τn = 1 ‖xn‖p−1 are chosen such that ρe∗1 (τn) = γ 2qgq‖a‖ × 〈j pe2 (i − π p acn t n)axn, axn − π p acn t n axn〉 ‖xn‖p‖j p e2 (i − πpacn t n)axn‖ , and a∗ denote the adjoint of a, and {αn} ⊂ (0, 1) satisfies lim inf n→∞ αn(1 − αn) > 0, kn = (k (1) n ∨ k (2) n ), n ≥ 1. if γ = {v ∈ fix(u) : av ∈ fix(πpac t )}, φ, then {xn} converges strongly to x∗ ∈ γ. in theorem 3.1, when u and t are two φ-asymptotically nonexpansive mappings, the following result holds. corollary 3.1.3. let e1 and e2 be two uniformly convex and uniformly smooth banach spaces, a : e1 −→ e2 is bounded and linear operator, such that a(c), for c ⊂ e1, is closed and convex, u : e1 −→ e1 be a uniformly continuous φasymptotically nonexpansive operator with the sequence {k(1)n } ⊂ [1,∞) satisfying limn→∞ k (1) n = 1, and t : e2 −→ e2 be a uniformly continuous φ-asymptotically nonexpansive operator with the sequence {k(2)n } ⊂ [1,∞) satisfying limn→∞ k (2) n = 1, respectively and, for p, q ∈ (1,∞), πpac : e2 −→ ac be a generalized projection onto a subset ac, and fix(u) , φ and fix(πpac t ) , φ respectively, and (i − u) and (i − π p ac t ) be demiclosed at zero. let x1 ∈ e1 be chosen arbitrarily and let 46 yusuf ibrahim / j. nig. soc. phys. sci. 1 (2019) 35–50 47 c1 = e1. define a sequence {xn} as follows; zn = j−1e∗1 (je1 xn −λn a ∗je2 (i − π p acn t n)axn), yn = j−1e∗1 (αn je1 (zn) + (1 −αn)je1 (u n(zn))), cn+1 = {v ∈ cn : φ(yn, v) ≤ knφ(zn, v) ≤ knφ(xn, v)}, xn+1 = πcn+1 (x1), n ≥ 1 (73) where λn =  1 ‖a‖ 1 ‖je2 (i−π p acn t n )axn‖ , xn , 0 1 ‖a‖p 〈je2 (i−πacn t n )axn,axn−πacn t n axn〉 ‖je2 (i−πacn t n )axn‖ , xn = 0 and γ ∈ (0, 1) and τn = 1 ‖xn‖ are chosen such that ρe∗1 (τn) = γ 2g‖a‖ × 〈je2 (i − πacn t n)axn, axn − πacn t n axn〉 ‖xn‖‖je2 (i − πacn t n)axn‖ , and a∗ denote the adjoint of a, and {αn} ⊂ (0, 1) satisfies lim inf n→∞ αn(1 − αn) > 0, kn = (k (1) n ∨ k (2) n ), n ≥ 1. if γ = {v ∈ fix(u) : av ∈ fix(πac t )}, φ, then {xn} converges strongly to x∗ ∈ γ. in theorem 3.1, when e1 and e2 are real hilbert spaces h1 and h2, respectively, we have the following result. corollary 3.1.4. let h1 and h2 be two real hilbert spaces, a : h1 −→ h2 is bounded and linear operator, u : h1 −→ h1 be a uniformly continuous generalized asymptotically nonexpansive operator with the sequences {k(1)n } ⊂ [0,∞) and {µ (1) n } ⊂ [0,∞) satisfying lim n→∞ k(1)n = 0 and lim n→∞ µ (1) n = 0, respectively, and t : h2 −→ h2 be a uniformly continuous generalized asymptotically nonexpansive operator with the sequences {k(2)n } ⊂ [0,∞) and {µ(2)n } ⊂ [0,∞) satisfying lim n→∞ k(2)n = 0 and lim n→∞ µ (2) n = 0, respectively, and (i − u) and (i − t ) be demiclosed at zero. let x1 ∈ h1 be chosen arbitrarily and let c1 = h1 and the sequence {xn} be defined as follows; zn = xn −λn a∗(i − t n)axn), yn = αnzn + (1 −αn)u n(zn), cn+1 = {v ∈ cn : ‖yn − v‖ ≤ [kn + 1]‖zn − v‖ + µn; ‖zn − v‖ ≤ [kn + 1]‖xn − v‖ + µn}, xn+1 = pcn+1 (x1), n ≥ 1 (74) where λn ∈ (0, 1 ‖a∗‖2 ) and a ∗ denote the adjoint of a, and {αn} ⊂ (0, 1) satisfies lim inf n→∞ αn(1−αn) > 0, and kn = (k (1) n ∨k (2) n ), n ≥ 1. if γ = {v ∈ fix(u) : av ∈ fix(t )} , φ, then {xn} converges strongly to x∗ ∈ γ. in theorem 3.1, when u and t are two nonexpansive mappings, the following result holds. corollary 3.1.5. let e1 and e2 be two uniformly convex and uniformly smooth banach spaces, a : e1 −→ e2 is bounded and linear operator, such that a(c), for c ⊂ e1, is closed and convex, u : e1 −→ e1 be a uniformly continuous bregman nonexpansive operator, and t : e2 −→ e2 be a uniformly continuous bregman nonexpansive operator, and, for p, q ∈ (1,∞), π p ac : e2 −→ ac be a bregman projection onto a subset ac, and fix(u) , φ and fix(πpac t ) , φ respectively, and (i − u) and (i − πpac t ) be demiclosed at zero. let x1 ∈ e1 be chosen arbitrarily and let c1 = e1 and the sequence {xn} be defined as follows; zn = j q e∗1 (j pe1 xn −λn a ∗j pe2 (i − π p acn t )axn), yn = j q e∗1 (αn j p e1 (zn) + (1 −αn)j p e1 (u(zn))), cn+1 = {v ∈ cn : 4p(yn, v) ≤4p(zn, v) ≤4p(xn, v)}, xn+1 = π p cn+1 (x1), n ≥ 1 (75) where λn =  1 ‖a‖ 1 ‖j pe2 (i−πpacn t )axn‖ , xn , 0 1 ‖a‖p 〈j pe2 (i−πpacn t )axn,axn−π p acn t axn〉p−1 ‖j pe2 (i−πpacn t )axn‖ p , xn = 0 and γ ∈ (0, 1) and τn = 1 ‖xn‖p−1 are chosen such that ρe∗1 (τn) = γ 2qgq‖a‖ × 〈j pe2 (i − π p acn t )axn, axn − π p acn t axn〉 ‖xn‖p‖j p e2 (i − πpacn t )axn‖ , and a∗ denote the adjoint of a, and {αn} ⊂ (0, 1) satisfies lim inf n→∞ αn(1−αn) > 0. if γ = {v ∈ fix(u) : av ∈ fix(π p ac t )}, φ, then {xn} converges strongly to x∗ ∈ γ. 4. application to the mixed equilibrium problem lemma 4.1. [4] let e be a reflexive, strictly convex and smooth banach space, and let c be a nonempty closed convex subset of e. let f, g : c × c −→ < be two bifunctions which satisfy the conditions (a1) − (a4), (b1) − (b3)and(c), in (6), then for every x ∈ e and r > 0, there exists a unique point z ∈ c such that f (z, y) + g(z, y) + 1 r 〈y − z, jz − jx〉 ≥ 0∀y ∈ c} in reich, s, sabach, s (2010) [18], when f (x) = 1p‖x‖ p then we have the following lemma. lemma 4.2. let e be a reflexive, strictly convex and smooth banach space, and let c be a nonempty closed convex subset of e. let f, g : c × c −→ < be two bifunctions which satisfy the conditions (a1) − (a4), (b1) − (b3)and(c), in (6), then for every x ∈ e and r > 0, define a mapping s r : e −→ c as follows; s r (x) = {x ∈ c : f (z, y)+g(z, y)+ 1 r 〈y−z, j pe z−j p e x〉 ≥ 0∀y ∈ c} then the following hold; 1. s r is a single-valued; 2. s r is a bregman firmly nonexpansive-type mapping, i.e. ∀x, y ∈ e〈s r x−s r y, j p e s r x−j p e s r y〉 ≤ 〈s r x−s r y, j p e x−j p e y〉; or equivalently 4p(s r x, s r y) + 4p(s r y, s r x) + 4p(s r x, x) + 4p(s r y, y) ≤ 4p(s r x, y) + 4p(s r y, x) 47 yusuf ibrahim / j. nig. soc. phys. sci. 1 (2019) 35–50 48 3. f(s r ) = mep( f, g); 4. mep( f, g) is closed and convex; 5. for all x ∈ e and for all v ∈ f(s r ), 4p(v, s r x)+4p(s r x, x) ≤ 4p(v, x). theorem 4.1. let e1 and e2 be two uniformly convex and uniformly smooth banach spaces, a : e1 −→ e2 is bounded and linear operator, such that a(c), for c ⊂ e1, is closed and convex, t : e2 −→ e2 be a uniformly continuous bregman nonexpansive operator, and, for p, q ∈ (1,∞), πpac : e2 −→ ac be a bregman projection onto a subset ac, and f, g : c×c −→< be two bifunctions which satisfy the conditions (a1) − (a4), (b1) − (b3) and (c), in (6), for c × c ⊂ e1 × e1, assume that c = mep( f, g) , φ and fix(t ) , φ, taking c1 = e1, the sequence {xn} is defined as follows; choosing arbitrarily x1 ∈ e1, zn = j q e∗1 (j pe1 xn −λn a ∗j pe2 (i − π p acn t n)axn), f (s r (xn), b) + g(s r (xn), b) + 1 rn 〈b − s r (xn), j p e1 (s r (xn)) −j pe1 (xn)〉 ≥ 0 ∀b ∈ e1 yn = j q e∗1 (αn j p e1 (zn) + (1 −αn)j p e1 (s r (zn))), cn+1 = {v ∈ cn : 4p(yn, v) ≤4p(zn, v) ≤4p(xn, v)} xn+1 = π p cn+1 (x1), n ≥ 1, (76) where λn =  1 ‖a‖ 1 ‖j pe2 (i−πpacn t n )axn‖ , xn , 0 1 ‖a‖p 〈j pe2 (i−πpacn t n )axn,axn−π p acn t n axn〉p−1 ‖j pe2 (i−πpacn t n )axn‖p , xn = 0 and γ ∈ (0, 1) and τn = 1 ‖xn‖p−1 are chosen such that ρe∗1 (τn) = γ 2qgq‖a‖ × 〈j pe2 (i − π p acn t n)axn, axn − π p acn t n axn〉 ‖xn‖p‖j p e2 (i − πpacn t n)axn‖ , and a∗ denote the adjoint of a, and {αn} ⊂ (0, 1) satisfies lim inf n→∞ αn(1−αn) > 0. if γ = {v ∈ mep( f, g) : av ∈ fix(π p ac t )}, φ, then {xn} converges strongly to x∗ ∈ γ. proof: it follows from the lemma 4.2 that fix(s r ) = mep( f, g) is nonempty, closed and convex and s r is a firmly nonexpansive mapping. meanwhile s r is assume to be continuous. hence all conditions in corollary 3.1.5 are satisfied. the conclusion of theorem 4.1 can be directly obtained from corollary 3.1.5. let e1 and e2 be two uniformly convex and uniformly smooth banach spaces. let c and q be nonempty closed convex subsets of e1 and e2, respectively. a : e1 −→ e2 is bounded and linear operator. assume that f, g : c ×c −→< be two bifunctions and f ′, g′ : q × q −→< be another two bifunctions. the split mixed equilibrium problem (smep) is to find an element v ∈ c such that f (v, y) + g(v, y) ≥ 0∀y ∈ c (77) and such that av ∈ q solve f (av, ay) + g(av, ay) ≥ 0∀ay ∈ q (78) let ω = {v ∈ mep( f, g) : av ∈ mep( f ′, g′)} denote the solution set of the smep. corollary 4.1.1. let e1 and e2 be two uniformly convex and uniformly smooth banach spaces. let c and q be nonempty closed convex subsets of e1 and e2, respectively. a : e1 −→ e2 is bounded and linear operator, such that a(c) is closed and convex, and, such that a(c), for c ⊂ e1, is closed and convex, for p, q ∈ (1,∞), πpac : e2 −→ ac be a bregman projection onto a subset ac. assume that f, g : c × c −→ < be two bifunctions and f ′, g′ : q × q −→ < be another two bifunctions which satisfy the conditions (a1) − (a4), (b1) − (b3) and (c), in (6), for c×c ⊂ e1×e1, assuming mep( f, g) , φ and mep(πpac, f ′, g′) , φ, respectively, taking c1 = e1, the sequence {xn} is defined as follows; choosing arbitrarily x1 ∈ e1, f (s f +gr (xn), b) + g(s f +g r (xn), b) + 1 rn 〈b − s f +gr (xn), j p e1 (s f +gr (xn)) −j pe1 (xn)〉 ≥ 0 ∀b ∈ e1 f (s f ′+g′ r (axn), b) + g(s f ′+g′ r (axn), b) + 1 rn 〈b − s f ′+g′ r (axn), j p e2 (s f ′+g′ r (axn)) −j pe2 (axn)〉 ≥ 0 ∀ b ∈ e2 zn = j q e∗1 (j pe1 (xn) −λa ∗(j pe2 (i − π p ac s f ′+g′ rn )axn)), yn = j q e∗1 (αn j p e1 (zn) + (1 −αn)j p e1 (s f +gr (zn)), cn+1 = {v ∈ cn : 4p(yn, v) ≤4p(zn, v) ≤4p(xn, v)} xn+1 = ∏p cn+1 (x1), n ≥ 1, (79) where λn =  1 ‖a‖ 1 ‖j pe2 (i−πpac s f′+g′ rn )axn‖ , xn , 0 1 ‖a‖p 〈j pe2 (i−πpac s f′+g′ rn )axn ),axn−π p ac s f′+g′ rn axn〉 p−1 ‖j pe2 (i−πpac s f′+g′ rn )axn‖ p , xn = 0 and γ ∈ (0, 1) and τn = 1 ‖xn‖p−1 are chosen such that ρe∗1 (τn) = γ 2qgq‖a‖ 〈je2 (i − π p ac s f ′+g′ rn )axn, axn − π p ac s f ′+g′ rn axn〉 ‖xn‖p‖j p e2 (i − πpac s f ′+g′ rn )axn‖ and a∗ denote the adjoint of a, and {rn} ⊂ (0, 1), and {αn} ⊂ (0, 1) satisfies limn→∞ αn(1 − αn) > 0, n ≥ 1. if ω = {v ∈ mep( f, g) : av ∈ mep(πpac, f ′, g′)} , φ, then {xn} converges strongly to x∗ ∈ ω. 5. an example in this section we discuss how to apply theorem 3.1 on the following example. let e1 = e2 = c1 = (−∞,∞), a = a∗ = i, p = q = 2, and αn = 1 2n . define u : e1 −→ e1 by u(x) = x, x ∈ (−∞, 0] = b11 2 x, x ∈ (0,∞) = b2, a : e1 −→ e2 by a(x) = x, x ∈ (−∞,∞), 48 yusuf ibrahim / j. nig. soc. phys. sci. 1 (2019) 35–50 49 t : e2 −→ e2 by t (x) = x, x ∈ (−∞, 0] = d11 4 x, x ∈ (0,∞) = d2, π 2 c : e2 −→ c by π 2 c (x) = 0, x ∈ (−∞, 0)x, x ∈ [0,∞), π 2 c t : e2 −→ c by π 2 c t (x) =  x, x ∈ {0} 0, x ∈ (−∞, 0) 1 4 x, x ∈ (0,∞). now we check whether or not the mappings u and t are uniformly continuous bregman generalized asymptotically nonexpansive. to do this it suffices to check for each map the following four cases. case i: if x, y ∈ b1 then 42(u n(x), u n(y)) = 42(x, y) ≤ [ 1 2n + 1] 42 (x, y) + 1 2n ; (80) case ii: if x, y ∈ b2 then 42(u n(x), u n(y)) = 42 ( 1 2n x, 1 2n y) = 1 22n 42 (x, y) ≤ [ 1 2n + 1] 42 (x, y) + 1 2n ; (81) case iii: if x ∈ b2 and y ∈ b1, then 42(u n(x), u n(y)) = 42( 1 2n x, y) = 1 22n 1 2 |x|2 − 1 2n 〈x, y〉 + 1 2 |y|2 ≤ 1 2 |x|2 −〈x, y〉 + 1 2 |y|2 = 42(x, y) ≤ [ 1 2n + 1] 42 (x, y) + 1 2n ; (82) case iv: if x ∈ b1 and y ∈ b2, then 4p(u n(x), u n(y)) = 42(x, 1 2n y) = 1 2 |x|2 − 1 2n 〈x, y〉 + 1 2 1 22n |y|2 ≤ 1 2 |x|2 −〈x, y〉 + 1 2 |y|2 = 42(x, y) ≤ [ 1 2n + 1] 42 (x, y) + 1 2n . (83) since it is clear that u is uniformly continuous, then from (80), (81), (82) and (83) u is a uniformly continuous bregman generalized asymptotically nonexpansive mapping. similarly, it is also clear that t is a uniformly continuous bregman generalized asymptotically nonexpansive mapping. next, we simplify the scheme as follows. clearly fix(u) = (−∞, 0], fix(t ) = (−∞, 0] and fix(π2c t ) = {0} such that γ = {x ∈ fix(u); ax ∈ fix(π2c t )} = {0}. now we have that λn =  1 |xn| , xn ∈ (−∞, 0) 4 3xn , xn ∈ (0,∞) 1, xn ∈ {0}. zn =  xn + 1, xn ∈ (−∞, 0) xn − 1, xn ∈ (0,∞) 0, xn ∈ {0}, yn =  xn + 1, xn ∈ (−∞, 0) ( 22n − 1 22n )(xn − 1), xn ∈ (0,∞) 0, xn ∈ {0}, cn+1 = {v ∈ cn;  1 2 |xn + 1 − v| 2 ≤ [ 12n + 1] 1 2 |xn + 1 − v| 2 + 12n ; 1 2 |xn + 1 − v| 2 ≤ 1 2 |xn − v| 2, xn ∈ (−∞, 0) 1 2 |( 2 2n − 1 22n )(xn − 1) − v| 2 ≤ [ 12n + 1] 1 2 |xn − 1 − v| 2 + 12n ; 1 2 |xn − 1 − v| 2 ≤ 1 2 |xn − v| 2, xn ∈ (0,∞) 1 2 |0 − v| 2 ≤ [ 12n + 1] 1 2 |0 − v| 2 + 12n , xn ∈ {0}}, xn+1 = π 2 cn+1 (x1), n ≥ 1. now, take the initial point x1 = 0.5 from positive real numbers, the numerical experiment result is given below; z1 = −0.5; y1 = −0.375; c2 = (−∞, 0]; x2 = 0; z2 = 0; y2 = 0; x3 = x2 = 0. this implies that zero is in the solution set γ. similarly, from the negative real numbers, let x1 = −0.5 then we arrive at the same fixed point zero as follows; z1 = 0.5; y1 = 0.5; c2 = [0,∞); x2 = 0; z2 = 0; y2 = 0; x3 = x2 = 0. 6. conclusion in summary, the validity of the proposed scheme is provided by step one and two in the manuscript. strong convergence of the proposed sequence is achieved in step three. approximate fixed point sequence of the proposed mappings is given by step four. application of the demicloseness principle for the proposed mappings is discussed in step five. furthermore, in this paper, the strong convergence for scfpp of bregman generalized asymptotically nonexpansive mapping is obtained in uniformly convex and uniformly smooth banach spaces, without the assumption of semi-compactness property and or without the assumption of opial condition. this infer that, the main result presented here generalizes and extends that of zhang et al. [10] and the references therein. this is true by the fact that the map presented here is more general than asymptotically nonexpansive map which was used by zhang et. al. so also the space used here generalizes the one applied in the work of zhang etal. moreso, the notion of bregman distance used here is the general case of the classical norm distance used in the result of zhang et al. acknowledgments we thank the referees for the positive enlightening comments and suggestions, which have greatly helped us in making improvements to this paper. 49 yusuf ibrahim / j. nig. soc. phys. sci. 1 (2019) 35–50 50 references [1] y. censor, t. elfving,n. kopf & t. bortfeld, “the multiple-sets split feasibility problem and its applications”, inverse probl. 21 (2005) 2071. [2] c. byrne, “iterative oblique projection onto convex subsets and the split feasibility problems”, inverse probl. 18 (2002) 441. [3] y. censor & a. segal, “the split common fixed point problem for directed operators”, j. convex anal. 16 (2009) 587. [4] b. c. deng, t. chen & y. l. yin, “strong convergence theorems for mixed equilibrium problem and asymptotically i-nonexpansive mapping in banach spaces”, abstract and applied anal. 2014 (2014) article id 965737. [5] l. yang, s. s. chang, y. j. cho & j. k. kim, multiple-set split feasibility problems for total asymptotically strict pseudocontractions mappings, fixed point theory appl. 1 (2011) 77. [6] b. qu & n. xiu, “note on the cq algorithm for the split feasibility problem”, inverse probl. 21 (2005) 1655. [7] f. schopfer, “iterative methods for the solution of the split feasibility problem in banach spaces”, der naturwissenschaftlich-technischen fakultaten, universitat des saarlandes, 2007. [8] a. moudafi, “the split common fixed point problem for demi-contractive mappings”, inverse probl. 26 (2014). [9] s. s. chang, y. j. cho, j. k. kim, w. zhang & l. yang, “multiple-set split feasibility problems for asymptotically strict pseudocontractions”, abstr. appl. anal. 2012 (2012) 491760. [10] x. f. zhang, l. wang, z. l. ma & l. j. qin “the strong convergence theorems for split common fixed point problem of asymptotically nonexpansive mappings in hlibert spaces”, journal of inequalities and applications 1 (2015) 1. [11] w. takahashi, “the split common null point problem in banach spaces”, arch. math. (basel) 104 (2015) 357. [12] j. tang, s. s. chang, l. wang & x. wang, “on the split common fixed point problem for strict pseudocontractive and asymptotically nonexpansive mappings in banach spaces”, journal of inequalities and applications 2015 (2015) 305. [13] j. a. clarkson, “uniformly convex spaces”, trans. am. math. soc. 40 (1936) 396. [14] c. e. chidume, “geometric properties in banach spaces and nonlinear iterations”, springer-verlag london limited, 2009. [15] f. shopfer, t. schuster and a. k. louis, “metri and bregman projection onto affine subspaces and theircomputation via sequential subspace optimization methods”, j. inv. iii-posed problem 15 (2007) 1. [16] z. b. xu & g. f. roach, “characteristic inequalities of uniformly convex and uniformly smooth banach spaces”, journal of mathematical analysis and applications , 1991. [17] h. k. xu, “inequalities in banach spaces with applications”, nomlinear analysis 12 (1991) 1127. [18] s. reich and s. sabach, “two strong convergence theorems for bregman strongly nonexpansive operators in reflexive banach spaces”, nonlinear anal. 73 (2010) 122. 50 j. nig. soc. phys. sci. 2 (2020) 120–127 journal of the nigerian society of physical sciences original research an examination of a second order numerical method for solving initial value problems s. e. fadugbaa,∗, s. n. ogunyebia, b. o. falodunb adepartment of mathematics, ekiti state university, ado ekiti, ekiti state bdepartment of mathematics, university of ilorin, kwara state abstract this paper presents an examination of a second order convergence numerical method (socnm) for solving initial value problems (ivps) in ordinary differential equations (odes). the socnm has been derived via the interpolating function comprises of polynomial and exponential forms. the analysis and the properties of socnm were discussed. three numerical examples have been solved successfully to examine the performance of socnm in terms of accuracy and stability. the comparative study of socnm, improved modified euler method (imem), fadugba and olaosebikan scheme (fos) and the exact solution (es) is presented. by varying the step length, the absolute relative errors at the final nodal point of the associated integration interval are computed. furthermore, the analysis of the properties of socnm shows that the method is consistent, stable, convergent and has second order accuracy. moreover, the numerical results show that socnm is more accurate than imem and fos and also compared favourably with the es. by varying the step length, there are two-order decrease in the values of the final absolute relative errors generated via socnm. hence, socnm is found to be accurate, stable and a good tool for the numerical solutions of ivps of different characteristics in odes. keywords: absolute relative error, accuracy, comparative study, stability. article history : received: 20 april 2020 received in revised form: 31 may 2020 accepted for publication: 04 june 2020 published: 01 august 2020 c©2020 journal of the nigerian society of physical sciences. all rights reserved. communicated by: b. j. falaye 1. introduction most of the real world problems in science and engineering are modelled and formulated by means of differential equations. some of these differential equations cannot be solved analytically or difficult to obtain their exact solutions. one way is to use numerical methods for approximating the solution of the differential equations using prescribed initial or boundary conditions. in the recent years, many numerical methods of various ∗corresponding author tel. no: +2348067032044 email address: sunday.fadugba@eksu.edu.ng (s. e. fadugba ) nature have been developed for the solution of ivps in odes of the form dy d x = f (x, y), y(a) = y0, −∞ < y < ∞, a ≤ x ≤ b (1) such as the euler’s method, improved euler’s method, rungekutta methods and so on. in [1], the authors developed a new one-step scheme for the solution of ivps in odes. ahmad et al. [2] studied the numerical accuracy of the runge-kutta method of second, third and fourth order for the numerical solution of differential equations. fadugba and idowu [3] derived a new numerical method of third order accuracy via the transcendental function of exponential type for the solution of ivps in odes. 120 fadugba et al. / j. nig. soc. phys. sci. 2 (2020) 120–127 121 they also analysed the properties of the derived method. in [4], accurate solutions of ivps of odes with fourth order runge kutta method were presented. islam [5] compared the numerical solutions of ivps for odes with euler’s and runge-kutta methods. refs. [6, 7, 8, 9, 10] also studied the numerical solutions of ivps in odes via several developed methods. in this paper, the performance of socnm in terms of accuracy and stability, that provides numerical solutions to ivps in odes is examined. the socnm is implemented on some selected ivps of different characteristics in odes. the rest of the paper is structured as follows; section two presents the analysis and the properties of socnm. in section three, three numerical examples were solved via socnm in the context of the imem [13], fos [14] and es. section four concludes the paper with the discussion of results. 2. analysis and the properties of a second order convergence numerical method this section presents the analysis and properties of socnm as follows. 2.1. analysis of socnm consider the ivp given by (1) whose exact solution is obtained as y(x). in [1], a new numerical method of one step of the form yn+1 = yn + h ( fn + f (1) n ) + ( e−h − 1 ) f (1)n (2) was proposed and developed by means of the interpolating function of exponential and polynomial types f(x) = 1∑ i=0 ai x i + a2e −x (3) for the numerical solution of (1). simplifying further, (2) becomes yn+1 = yn + h fn + ( h + e−h − 1 ) f (1)n (4) the mesh point is defined as xn+1 = a + (n + 1)h, n = 0, 1, 2, ... and the step length is defined as h = b−an , where n is the number of integration steps. 2.2. properties of socnm the properties of socnm such as accuracy, consistency, stability and convergence are discussed as follows: 2.2.1. accuracy of socnm consider the taylor’s series expansion for the exact solution y(x) given by y(xn + h) = y(xn) + h f (xn, y(xn)) + h2 2! f (1) (xn, y(xn)) + h3 3! f (2) (xn, y(xn)) + o(h 4) (5) the local truncation error is given by tn+1 = y(xn + h) − yn+1 (6) substituting (4) and (5) into (6) yields tn+1 = y(xn) + h f (xn, y(xn)) + h2 2! f (1)(xn, y(xn)) + h3 3! f (2)(xn, y(xn)) + o(h 4) − [ yn + h fn + (h + e −h − 1) f (1)n ] (7) simplifying further and by means of the localizing assumption (that is, assume that no errors have been committed), (7) becomes tn+1 = h3 3! [ f (2)(xn, y(xn)) + f (1)(xn, y(xn)) ] + o(h4) (8) which is the local truncation error for socnm. hence, (8) confirms the second order accuracy of socnm. 2.2.2. consistency of socnm according to [15], a one step numerical method is said to be consistent if it has at least order p = 1. it is clearly seen from (8) that socnm is consistent since it has second order accuracy and that lim h→0 tn+1 h = lim h→0  h3 3! [ f (2)(xn, y(xn)) + f (1)(xn, y(xn)) ] + o(h4) h  = 0 2.2.3. stability of socnm according to [15], a one step numerical method is said to be stable if it is capable of damping out small fluctuation carried out in input data. to discuss the stability of socnm, consider the ivp of the form dy d x = −φy, y(0) = 1 (9) the exact solution of (9) is obtained as y(x) = e−φx, φ > 0 (10) where φ is a complex constant. expanding (10) at the points x = xn and x = xn+1 and using the fact that h = xn+1 − xn, one obtains y(xn) = e −φxn (11) 121 fadugba et al. / j. nig. soc. phys. sci. 2 (2020) 120–127 122 and y(xn+1) = e −φxn+1 = e−φxn e−φh = y(xn)e −φh (12) using socnm and the series expansion of e−h, one obtains yn+1 = yn [ 1 −φh + φ2h2 2! ] (13) setting a = [ 1 − z + z2 2! ] , z = φh (14) equation (13) becomes yn+1 = ayn (15) comparing (12) and (15), it is clearly seen that (14) consists of the first-three terms of the series expansion of e−φh. hence, the stability function of socnm requires that ‖a‖ < 1 (16) equation (16) shows that socnm is stable. 2.2.4. convergence of socnm according to [16], the necessary and sufficient conditions for convergence are consistency and stability. in other words, a one step numerical method is said to be convergent if it is consistent and stable. it is clearly seen that socnm is convergent since it is consistent and stable. the convergence of socnm is of second order. 3. numerical examples here, the performance of socnm in terms of the accuracy and stability is examined. also the comparative study of socnm, imem given by yn+1 = yn + hk3, where k1 = f (xn, yn), k2 = f (xn, yn + k1), k3 = f ( xn + h 2 , yn + h 2 k2 ) , fos given by yn+1 = yn + ( h + h2 + 23 h 3 ) fn and es is presented. all the calculations were carried out with the aid of computational software ”matlab r2014a, 8.3.0.552, 32 bit (win 32)” in double precision. problem 1 consider the ivp of the form dy d x − x sin(x) = 0, y(0) = 1, x ∈ [0, 1] whose exact solution is given by y(x) = 1 + x sin(x) − x cos(x) problem 2 consider the ivp of the form dy d x − 1 − (y − x)2 = 0, y(0) = 1 2 , x ∈ [0, 1] whose exact solution is given by y(x) = x + 1 2 − x problem 3 consider the riccati equation coupled with the initial value given by dy d x + 4 − 4y + y2 = 0, y(0) = 1, x ∈ [ 0, 2 5 ] whose exact solution is given by y(x) = 2x − 1 x − 1 and has a pole in x = 1. the results generated via the socnm, imem, fos in the context of es for problems 1-3 are displayed in the figures 118. the comparative results analyses of socnm, imem, fos and es for problems 1-3 with h = 0.1 are shown in figures 19,21,23, respectively. the absolute relative errors (ares) generated via socnm, imem and fos with h = 0.1 for problems 1-3 are displayed in figures 20,22,24, respectively. the relative absolute errors (fares) generated via socnm, imem and fos for problems 1-3 at the final nodal point of the associated integration interval with different values of step length, h = 0.1, 0.01, 0.001, 0.0001, 0.00001 are displayed in the figures 25,26,27, respectively. figure 1. the results generated via socnm and es for problem 1 122 fadugba et al. / j. nig. soc. phys. sci. 2 (2020) 120–127 123 figure 2. the are generated via socnm for problem 1 figure 3. the results generated via imem and es for problem 1 figure 4. the are generated via imem for problem 1 figure 5. the results generated via fos and es for problem 1 figure 6. the are generated via fos for problem 1 figure 7. the results generated via socnm and es for problem 2 123 fadugba et al. / j. nig. soc. phys. sci. 2 (2020) 120–127 124 figure 8. the are generated via socnm for problem 2 figure 9. the results generated via imem and es for problem 2 figure 10. the are generated via imem for problem 2 figure 11. the results generated via fos and es for problem 2 figure 12. the are generated via fos for problem 2 figure 13. the results generated via socnm and es for problem 3 124 fadugba et al. / j. nig. soc. phys. sci. 2 (2020) 120–127 125 figure 14. the are generated via socnm for problem 3 figure 15. the results generated via imem and es for problem 3 figure 16. the are generated via imem for problem 3 figure 17. the results generated via fos and es for problem 3 figure 18. the are generated via fos for problem 3 figure 19. the comparative results analyses of socnm, imem, fos and es for problem 1 125 fadugba et al. / j. nig. soc. phys. sci. 2 (2020) 120–127 126 figure 20. the ares generated via socnm, imem and fos for problem 1 figure 21. the comparative results analyses of socnm, imem, fos and es for problem 2 figure 22. the ares generated via socnm, imem and fos for problem 2 figure 23. the comparative results analyses of socnm, imem, fos and es for problem 3 figure 24. the ares generated via socnm, imem and fos for problem 3 figure 25. the fares generated via socnm, imem and fos with varying step length values for problem 1 126 fadugba et al. / j. nig. soc. phys. sci. 2 (2020) 120–127 127 figure 26. the fares generated via socnm, imem and fos with varying step length values for problem 2 figure 27. the fares generated via socnm, imem and fos with varying step length values for problem 3 4. conclusion in this paper, an examination of socnm for solving ivps in odes is presented. the socnm has been derived via the interpolating function comprises of polynomial and exponential forms. the analysis and the properties of socnm were discussed extensively. three numerical examples were solved successfully to examine the performance of socnm in terms of accuracy and stability. the comparative study of socnm, imem, fos and es is presented. by varying the step length, the absolute relative errors at the final nodal point of the associated integration interval are computed. furthermore, the analysis of the properties of socnm shows that the method is consistent, stable, convergent and has second order accuracy. the comparative study of the results generated via socnm, imem, fos and es is displayed in figures 1-18. moreover, the numerical results show that socnm is more accurate than its counterparts, compared favourably and agreed with es as seen in the figures 19,21,23. it is also observed from figures 20,22,24 that the error curves for socnm show that socnm follows the es curves elegantly. also, by varying the step length (h = 0.1, 0.01, 0.001, 0.0001, 0.00001), it is observed from figures 25-27 that there are two-order decrease in the values of the final absolute relative errors generated via socnm. hence, socnm is found to be accurate, stable and a good explicit method for the numerical solutions of ivps of different characteristics in odes. references [1] s. fadugba & b. falodun, “development of a new one-step scheme for the solution of initial value problem (ivp) in ordinary differential equations”, international journal of theoretical and applied mathematics, 3 (2017) 58. [2] n. ahmad, s. charan & v. p. singh, “study of numerical accuracy of runge-kutta second, third and fourth order method”, international journal of computer and mathematical sciences, 4 (2015) 111. [3] s. e. fadugba & j. o. idowu, “analysis of the properties of a third order convergence numerical method derived via transcendental function of exponential form”, international journal of applied mathematics and theoretical physics, 5 (2019) 97. [4] md. a. islam, “accurate solutions of initial value problems for ordinary differential equations with fourth order runge kutta method”, journal of mathematics research, 7 (2015) 41. [5] md. a. islam, “a comparative study on numerical solutions of initial value problems (ivp) for ordinary differential equations (ode) with euler and runge kutta methods”, american journal of computational mathematics, 5 (2015) 393. [6] y. ansari, a. shaikh & s. qureshi, “error bounds for a numerical scheme with reduced slope evaluations”, j. appl. environ. biol. sci., 8 (2018) 67. [7] j. c. butcher, numerical methods for ordinary differential equations, 3rd edition, john wiley & sons, chichester, west sussex, 2016. [8] m. e. davis, numerical methods and modeling for chemical engineers, courier corporation, new york, 2013. [9] s. e. fadugba & j. t. okunlola, “performance measure of a new one-step numerical technique via interpolating function for the solution of initial value problem of first order differential equation”, world scientific news, 90 (2017) 77. [10] s. e. fadugba & a. o. ajayi, “comparative study of a new scheme and some existing methods for the solution of initial value problems in ordinary differential equations”, international journal of engineering and future technology, 14 (2017) 47. [11] j. d. lambert, numerical methods for ordinary differential systems: the initial value problem, john wiley & sons, inc., new york, 1991. [12] s. fadugba & s. qureshi, “convergent numerical method using transcendental function of exponential type to solve continuous dynamical systems”, punjab university journal of mathematics, 51 (2019) 45. [13] o. abraham, “improving the modified euler method”, leonardo j. sci., 10 (2007) 1. [14] s. e. fadugba & t. e. olaosebikan, “comparative study of a class of onestep methods for the numerical solution of some initial value problems in ordinary differential equations”, research journal of mathematics and computer science, 2 (2018) 1. [15] j. d. lambert, computational methods in ordinary differential equations, john wiley & sons inc., 1973. [16] j. d. lambert, numerical methods for ordinary differential systems: the initial value problem, john wiley & sons, inc., new york, 1991. 127 j. nig. soc. phys. sci. 4 (2022) 949 journal of the nigerian society of physical sciences potential of anacardic acid for nanosized cellulose preparation under different treatment conditions olugbenga o. oluwasinaa,∗, abiodun d. aderibigbea,b, damilola c. petinrina, adeyemi s. adebisia, olayinka o. oluwasinac, oluwasegun j. wahabb adepartment of chemistry, federal university of technology akure, p.m.b. 704 akure, ondo state, nigeria bdepartment of chemistry, university of warwick, coventry cv4 7al, united kingdom cschool of chemistry and physics, university of kwazulu-natal, westville campus, private bag x54001, 4000, 7 south africa abstract herein, anacardic acid was applied for the preparation of nanosized cellulose using three different treatment conditions including ultrasonication, microwave irradiation, and reflux. physico-chemical characterization was undertaken using ftir, tem, sem, and xrd. ftir, tem, and sem analyses confirm the preparation of nanosized cellulose with similar chemical but different physical properties as the cellulose starting material. in addition, calculated degrees of crystallinities from xrd data revealed crystallinities of 53.9, 54.4, and 54.7% for the nanosized cellulose prepared by ultrasonication (unc), microwave irradiation (mnc), and reflux (rnc) respectively, which all are higher than the 53.3% of the precursor cellulose. overall, the study shows that anacardic acid holds potential for the preparation of nanosized cellulose. doi:10.46481/jnsps.2022.949 keywords: anacardic acid, cellulose, nanosized cellulose, ultrasonication, reflux, microwave irradiation article history: received: 20 july 2022 received in revised form: 15 september 2022 accepted for publication: 28 september 2022 published: 05 november 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: k. sakthipandi 1. introduction nanosized cellulose, which is a material with at least one dimension < 100 nm, is traditionally prepared using mineral acids like sulfuric and hydrochloric acids [1-4]. however, the use of mineral acids comes with significant problems particularly operational hazards, generation of toxic wastes, and corrosion of reactors [5]. consequently, growing attention is being paid to the development of environmentally friendly methods for preparing nanosized cellulose. some of these methods ∗corresponding author tel. no: +234(0)8107246660 email address: oooluwasina@futa.edu.ng (olugbenga o. oluwasina) are enzymatic and organic acid-based hydrolysis. although enzymatic hydrolysis is environmentally friendly, significant mechanical energy is often required to break down the enzymatically hydrolyzed cellulose fibres into nanocrystals [6, 7]. in addition, dispersion and chemical modification of the product cellulose nanocrystal is challenging [5]. consequently, significant consideration is being paid to organic acid-based hydrolysis [8, 9]. attractive features of the organic acid-based method include ease of recovery, biodegradability, limited corrosivity, and convenient handling of the organic acid [10]. fu and coinvestigators reported access to highly crystalline nanosized cellulose material in two steps using an ionic liquid and subsequently ox1 olugbenga et al. / j. nig. soc. phys. sci. 4 (2022) 949 2 figure 1. the general structure of anacardic acids (r = saturated or unsaturated alkyl chain) alic acid under relatively mild conditions [2]. robles’s group described the ultrasonic-assisted hydrolysis of cellulose using citric, oxalic, and maleic acids. though the yields of the prepared nanosized cellulose were poor to moderate (20 to 40%), the thermal stabilities were better than those obtained using sulfuric acid [11]. holilah and coworkers observed that the yields, particles sizes, and crystallinities of the nanosized cellulose they prepared using organic acids (acetic, citric, and oxalic) were higher than that prepared using inorganic acids (hydrochloric, sulphuric, and phosphoric) [12]. investigation into the discovery and development of biobased alternatives to petroleum-derived heavy chemicals is becoming increasingly popular due to the need to achieve sustainable development. while many bio-based reagents can be employed to prepare nanosized cellulose, our attention was drawn to anacardic acid (figure 1) because of the presence of the carboxylic acid group. in addition, the extraction of anacardic acid from the cashew fruit is well established [13-15] and applications of the acid [15-18] including those exploiting the carboxylic acid group [14] are growing. as part of our work on the investigation of environmentally friendly methods for the preparation of bioderived materials, we wondered if anacardic acid could function as an effective hydrolyzing agent for the preparation of nanosized cellulose. anacardic acid, a yellow liquid obtained from the nut of the cashew fruit, is composed of both saturated and unsaturated molecules. herein, we report the preparation of nanosized cellulose by the hydrolysis of cellulose (derived from plantain inflorescence stalk) using anacardic acid under different conditions including ultrasonication, microwave irradiation, and reflux. following this, the nanosized cellulose products were characterized using ftir, tem, sem, and xrd. this study shows that anacardic acid is a potential candidate for the preparation of nanosized cellulose. 2. materials and methods 2.1. materials the cashew (anacardium occidentale) seeds were purchased from the local fruit market in akure, ondo state, nigeria. plantain (musa paradisiaca) inflorescence stalk (pis) was obtained from the teaching/demonstration plantation of the federal university of technology akure (futa). the materials were authenticated at the department of crop, soil, and pest management, futa. all reagents were purchased from sigma aldrich and used as received. 2.2. instrumentation fourier transform infrared (ftir) spectra were recorded on bruker c© alpha platinum-attenuated total reflectance ir spectrometer. x-ray diffraction (xrd) data were collected on a panalytical empyrean x-ray diffractometer employing a co kα radiation at 40 kv and 40 ma. transmission electron micrographs (tem) were captured using the jeol 2100+ machine operating an acceleration voltage of 200 kv from samples prepared on a copper em grid. scanning electron microscopy (sem) characterization was undertaken using leo 1450 sem, the samples were attached to a brass stub and coated with gold before analysis. ultrasonication treatment was undertaken using a bransonic ultrasonicator (ranson 1210emt,usa) sonicator. microwave irradiation was achieved using russell hobbs-asda (leeds) microwave operating at a frequency of 2.45 ghz. 2.3. preparation of plantain inflorescence stalk cellulose plantain inflorescence stalk (pis) cellulose was prepared following the method reported by oluwasina and coworkers [19]. briefly, pis powder and 5% naoh (1:30 ratio by mass) were charged into a 15 l autoclave and heated at 140 oc under atmospheric pressure for 1 hour. the pulp obtained was washed with water until a neutral ph and finally dried. the dried pis pulp (7.5 g), hot water (375 ml), naclo2 (4.45 g), and acetic acid (1.03 ml) in a 1 l beaker were heated at 80 oc for 1 hour. next, naclo2 (8.9 g) and acetic acid (2.06 ml) were fed into the stirring mixture, and the whole suspension was heated for a further 2 hours. the bleached sample obtained was washed with water to a neutral ph and dried to give a white fibrous material labeled as plantain inflorescence stalk (pis) cellulose. 2.4. anacardic acid isolation anacardic acid was isolated from cashew nut powder following the combination of the methods reported by shobha’s [20] and bezerra’s [14] groups. briefly, cashew seed nut powder (average particle size of 1 mm) was extracted with n-hexane for 12 hours to give a brown viscous liquid which was concentrated by distillation. a solution of the brown oil (70 g) in aqueous meoh (300 ml, 5%) was warmed to 50 oc, after which ca(oh)2 was added in portions with continuous stirring for 3 hours. the crude product obtained was filtered and the resulting residue was washed successively with meoh (150 ml), and water (200 ml) and stirred in concentrated hcl (400 ml, 11 m) for 1 hour. the organic component was extracted using petroleum ether (100 ml x 3), dried over anhydrous na2so4, and concentrated to furnish a dark brown liquid identified as crude anacardic acid. 2.5. nanosized cellulose preparation nanosized cellulose was prepared by ultrasonication, microwave irradiation, and reflux. typically, a suspension of pis cellulose (5 g) and anacardic acid in acetone (50 ml, 80% v/v) was treated by ultrasonication, microwave irradiation, or reflux for 45 min. the resulting crude product was washed consecutively with acetone (100 ml), etoh (100 ml), distilled water (100 ml), and oven dried at 60 oc for 8 hour to deliver a white 2 olugbenga et al. / j. nig. soc. phys. sci. 4 (2022) 949 3 figure 2. ftir spectra show that plantain inflorescence stalk (pis) cellulose and nanosized cellulose prepared by ultrasonication (unc) microwave irradiation (mnc) and reflux (rnc) contain identical functional groups powder labeled as ultrasonication-prepared nanosized cellulose (unc), microwave irradiation-prepared nanocellulose (mnc) or reflux-prepared nanocellulose (rnc) to represent the preparation methods. 2.6. determination of degree of crystallinity the degree of crystallinity (dc) was determined using xrd data by the method described by neto’s group [21]. degree o f crystallinity(%) = crystalline band areas (crystalline band areas + amor phous band area) × 100 (1) 3. results and discussion 3.1. chemical composition identification by ftir the chemical compositions of the pis cellulose, rnc, unc, and mnc prepared products were identified from their ftir spectra (figure 2). all the spectra show the presence of identical functional groups in all samples, an observation expected if the pis cellulose is cellulosic and the rnc, unc, and mnc are the nanosized products. indeed, all spectra contain broad bands at 3312 cm−1, weak/medium peaks at 2280 cm−1, weak peaks at 1636 cm−1, and sharp peaks at 1014 cm−1 representative of o-h stretching, c-h stretching, o-h bending, and c-o stretching vibrations respectively [22-25]. 3.2. morphology and fibre width determination by tem and sem the fibre morphology and width were determined by tem analysis (figure 3). all fibres except for that prepared by reflux (rnc) appear as hollow tubes with thick edges. as expected, the pis cellulose has the highest particle width of 102.4 figure 3. tem micrographs of (a) plantain inflorescence stalk (pis) cellulose and nanosized cellulose prepared by (b) ultrasonication (c) microwave irradiation and (d) reflux nm. unc and mnc have fibre widths of 49.6 and 54.9 nm respectively.the fibre width of rnc could not be determined as a sizeable micrograph was not obtained. the lower fibre width observed for unc compared to mnc could be because ultrasonication involves the use of microbubbles of the anacardic acid solution which may be more efficient than the bulk solution involved with microwave heating for the splitting of pis cellulose [26]. the observed nano sizes of the unc and mnc fibers widths indicate that different materials have been successfully prepared from the pis cellulose [27]. further information on the morphology of the samples was obtained after sem analysis. the pis cellulose appears as fibre bundles cemented together to give a lightly rough surface (figure 4). the nature of the pis cellulose surface was attributed to the presence of lignin and hemicellulose. on the other hand, the rnc, unc, and mnc samples appear as individual strands loosely bound into a bunch with deep longitudinal furrows and a higher degree of surface roughness than observed for the pis cellulose. the loosely bound strands in the hydrolyzed cellulose samples can be attributed to the removal of the fibrous lignin and hemicellulose from the pis cellulose by anacardic acid. overall, the sem micrographs support the tem results showing that the cellulose derivative (rnc, unc, and mnc) samples were prepared from the pis cellulose using all methods employed. 3.3. degree of crystallinity determination from xrd patterns the x-ray diffractograms displayed in figure 5 were collected to determine the degrees of crystallinity.the diffractogram for the pis cellulose is similar (save the absence of a peak at 2θ values below 20 o) to the diffractograms of cellulose sam3 olugbenga et al. / j. nig. soc. phys. sci. 4 (2022) 949 4 figure 4. sem micrographs of (a) plantain inflorescence stalk cellulose and nanosized cellulose prepared by (b) ultrasonication (c) microwave irradiation and (d) reflux figure 5. x-ray diffractograms of plantain inflorescence stalk (pis) cellulose and nanosized cellulose prepared by ultrasonication (unc), microwave irradiation (mnc), and reflux (rnc) ples reported by teixeira and coauthors [27] using sugarcane bagasseas a cellulose source. the pis cellulose diffractogram looks identical to those of unc, mnc, and rnc samples. the calculated dcs were as follows 53.3, 53.9, 54.4, and 54.7 % for the pis cellulose, unc, mnc, and rnc respectively. the dcs shows that the reflux method produced the most crystalline nanosized cellulose material. this could be because refluxing was most effective in the removal of non-cellulosic components from pis cellulose. 4. conclusion the preparation of 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[27] e. de m. teixeira, t. j. bondancia, k. b. r. teodoro, a. c. corrêa, j. m. marconcini, & l. h. c. mattoso, “sugarcane bagasse whiskers: extraction and characterizations”, industrial crops and products, 33 (2011) 63. https://doi.org/10.1016/j.indcrop.2010.08.009. 5 j. nig. soc. phys. sci. 4 (2022) 909 journal of the nigerian society of physical sciences synergistic study of reduced graphene oxide as interfacial buffer layer in htl-free perovskite solar cells with carbon electrode sherifdeen o. bolarinwaa, eli danladib,∗, andrew ichojab, muhammad y. onimisia, christopher u. achemc adepartment of physics, nigerian defence academy, kaduna, nigeria bdepartment of physics, federal university of health sciences, otukpo, benue state, nigeria ccentre for satellite technology development-nasrda, abuja, nigeria abstract perovskite solar cells (pscs) being the most advanced photovoltaic technology today have attracted global attention due to its solution processability and cost effectiveness. in this paper, we reported the performance of perovskite solar cells by integrating reduced graphene oxide (rgo) as a buffer layer. the combined effect of ultraviolet visible (uv-vis) spectroscopy, x-ray diffractometer (xrd), raman spectroscopy, scanning electron microscopy (sem), four point probe and solar simulator were used to explore the absorbance, crystallinity, morphological, resistivity and current voltage behavior of the photoanodes and devices. the rgo was deposited on top the mesoporous titanium dioxide (mtio2) (device b), on methylammonium lead triiodide (ch3nh3pbi3) perovskite absorber (device c) and on m-tio2 & ch3nh3pbi3 perovskite absorber (device d). the reference device (device a) was fabricated without rgo as a bench mark for comparison. from the results obtained, the devices with rgo show significant improvement over the device lacking rgo. the reference device gave a short circuit current density (jsc) of 8.151 macm−2, open circuit voltage (voc) of 0.575 v, fill factor (ff) of 74.4 % and power conversion efficiency (pce) of 3.49 %. the best performing device was the one with rgo incorporated on both m-tio2 and ch3nh3pbi3. the device gave a jsc of 8.737 macm−2, voc of 0.836 v, ff of 68.1% and pce of 4.98 %. the device experienced an enhancement of ∼ 7.20 % and 42.69 % in jsc and pce over the pristine device. this investigation aids the fundamental understanding of the effect of graphene materials (grms) on the optoelectronic properties of pscs, and it further confirms the promising prospects of achieving low-cost, efficient and stable pscs for commercialization with graphene materials. doi:10.46481/jnsps.2022.909 keywords: perovskite solar cells, reduced graphene oxide, buffer layer, htm article history : received: 01 july 2022 received in revised form: 02 august 2022 accepted for publication: 04 august 2022 published: 17 august 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: edward anand emile ∗corresponding author tel. no: +2348063307256 email address: danladielibako@gmail.com (eli danladi) 1. introduction perovskite solar cells (pscs) in the history of photovoltaic technology (pvt) has witnessed a meteoric rise to prominence in terms of efficiency. from 3.8 % as demonstrated by kojima and colleagues to a certified efficiency of 25.5 % [1]. this 1 bolarinwa et al. / j. nig. soc. phys. sci. 4 (2022) 909 2 powerful success can be attributed to the high carrier mobility, robust broadband absorption, long diffusion length, solution processability, design flexibility and low-cost of the metal halide perovskite [2-7]. the halide absorber is a crucial layer in psc, that is responsible for absorbing light, creating photostimulated carriers, and transfers the carriers into the psc network for electrical generation. the perovskite absorber is featured by amx3 crystal pattern, where the first letter a represents organic cation, the second letter m represents metal and the third letter x represents halogen atom. the architectural flexibility that allows for creativity has contributed to the wide acceptability and attention that pscs are witnessing [1]. one of such architectures that has gained research attention is the hole transport layer (htl) free pscs. in 2013, han’s group reported for the first time in literature, a hole transport material (htm)-free mesoscopic pscs with carbon electrode as replacement to spiro-ometad with pce of 6.64 % [8, 9]. this architecture has been widely studied with several modifications owing to their relatively low production cost and low hysteresis [2, 10]. in addition, mesoscopic htl-free pscs are characterized by their long-term stability and therefore makes them a more suitable and promising architecture for commercialization [2, 6, 11-13]. however, optimization of solar device components remains an efficient way of enhancing overall device performance [1, 2, 8]. despite these attained heights, pscs are still hindered by several challenges such as poor stability under ambient conditions like humidity, heat, oxygen and uv light [1, 14-16], and couple with disparity in forward and reverse scan of currentvoltage (i-v) characteristics [10]. these challenges sum up the basic reason why pscs are yet to get to rooftop despite attaining the rivalry efficiency of current market pvt. hence, the fabrication of efficient pscs requires the crystallinity and morphology optimization of the perovskite films. investigations has shown that grain sizes, film homogeneity plays critical roles in the optoelectronic quality of the film [17, 18]. in other words, controlling nucleation process is an efficient way of improving the properties of the perovskite absorbing material and in turn the device performance [17]. apparently, thick layers with large and uniform grains of perovskite nanocrystals as absorbers enhances solar conversion efficiency. for a htl-free psc, the absorbing material gets sandwiched between the commonly employed titania (tio2) as electron transport layer (etl) and the back contact [10]. the integration of carbon-based materials has been proposed and successfully integrated into pscs to address the aforementioned drawbacks at the perovskite interface [2, 19-22]. amongst these materials, the graphene, a 2-dimesnional material with zero bandgap appears to stand out owing to its low resistivity, efficient ambipolar charge transport, optical transparency, high thermal conductivity, mechanical strength, and flexibility [2, 20]. its hydrophobic nature is an advantage that is capable of arresting instability of the perovskite film due to ambient conditions [2]. because of its high electrical properties and presence of oxygenated moieties which makes it dispersible in polar solvents, rgo has been favored amidst other graphene materials. these factors result in high functionality of both the ch3nh3pbi3 absorbing layer and the etl. a handful of literatures have reported the integration of rgo as a buffer layer between the charge carrier transport layer and perovskite layer [2, 23-25], however, the role of rgo is yet to be clearly elucidated. herein, we reported a synergistic approach towards determining the actual role of rgo in perovskite solar cells. rather than focus on obtaining high efficiency, this investigation sets to determine the most significant interface for the incorporation of rgo in htl-free mesoporous pscs. the commonly two step sequential deposition in air protocol was utilized while the graphene oxide was thermally reduced. our approach has birthed a fundamental understanding on the utilization of rgo as buffer layer in perovskite interface. 2. materials and methods 2.1. preparation of reduced graphene oxide in principle, obtaining reduced graphene oxide from graphite powder involves three fundamental stages namely; oxidation, exfoliation and reduction. the preparation of rgo in this investigation follows from the famous tours method [26-28] with some fundamental precooling and reordering of the protocols. first, we obtained graphite ore from the raw materials research and development council (rmrdc), abuja, nigeria. the graphite ore was then mashed into powdery form. a refrigerated mixture of 180 ml of concentrated sulfuric acid (h2so4) and 20 ml of phosphoric acid (h3po4) at 15 oc was gradually emptied into a beaker containing 1.5 g of untreated graphite powder. the resulting mixture was then placed in a 35 oc water bath. while stirring on a magnetic stirrer, 9 g of potassium permanganate (kmno4) was slowly added. a rise in temperature was observed with the addition of kmno4. the mixture was left to stir continually till its temperature drops to room temperature. this process lasted for 12 hours after which a graphite oxide was obtained. next, is the dispersion of the asobtained graphite oxide into 200 ml of distilled water to end the reaction. this was then gradually heated to 50 oc for 12 hours using a temperature-controlled water bath on a magnetic stirrer. as this process extended, the mixture thickened into a black paste. 1.5 ml of hydrogen peroxide (h2o2) was added to the obtained black paste and allowed to stir for 1 hour. bubble was observed with a bright brown coloration, thus, indicating high level of oxidation. next was filtration using a filter paper and collection of the go then followed by rinsing of the go with 200 ml aqueous hydrochloric acid (hcl), distilled water and methanol each. the rinsing protocol was repeated thrice. the rinsed mixture was aliquoted, and some were thermally reduced. the gradual thermal reduction was first achieved at 120 oc for 5 minutes and then at 200 oc for 10 minutes and finally, 300 oc for 10 minutes to allow adequate reduction. unlike reported journals where the starting graphite powder is mostly pretreated, this investigation reports the successful synthesis of graphene materials from untreated graphite ore. 2 bolarinwa et al. / j. nig. soc. phys. sci. 4 (2022) 909 3 2.2. preparation of precursors 2.2.1. electron transporting material compact and mesoporous tio2 were employed as the etl. for the compact layer, the precursor was prepared using titanium isopropoxide, acetyl acetone and propanol as detailed in the well-established method of wojciechowski et al. [29], while for the mesoporous layer, the famous sol-gel method was used by dispersing titanium nano-oxide in ethanol (v/v 1:3). 2.2.2. preparation of methyl ammonium lead tri-iodide perovskite (ch3nh3pbi3) precursors precursor a (pbi2): 4.6 g lead iodide (pbi2) was mixed with dimethylformamide. this mixture was placed in a molten wax bath and heated at 250 oc for 45 minutes. precursor b (mai): 20 ml of dried isopropanol was mixed with 0.32 g of methylammonium iodide and shaked for 3 minutes until there was no traces of any whitish residue. 2.3. preparation of the photoanodes the fluorine doped tin oxide (fto) glass substrate with sheet resistance of 15 ω square−1 was cleaned using sodium laureth sulphate. the compact layer (c-tio2) film was dynamically spin-coated on the fto with the condition of 4000rpm for 20 seconds. it was dried at 120 oc for 10 minutes and thereafter annealed at 450 oc for 30 minutes. the m-tio2 was deposited using spin coating procedure at 4000rpm for 20 seconds and annealed at 500 oc for 30 minutes. for the controlled device, few drops of the prepared precursor a was dynamically spin-coated and followed immediately by the dynamic spincoating of precursor b both in the glove box. the formation of dark perovskite layer was observed after which it was immersed in a beaker containing propanol for 5 minutes to expel excess mai. the resulting photoanode containing fto/ctio2/m-tio2/mapbi3 was then annealed for 10 minutes at 120 oc. for the second device, the synthesized go colloid was deposited by drop casting procedure on the substrate containing c-tio2/m-tio2. the resulting substrate of fto/c-tio2/m-tio2/go was then treated at 120 oc for 5 minutes and then raised to 230 oc for 10 minutes to allow adequate thermal reduction of go to reduced go (rgo). thereafter, few drops of the prepared pbi2 was dynamically spin-coated and followed immediately by the dynamic spin-coating of the mai precursor both in the glove box as done in the first device and annealed for 10 minutes at 120 oc. this gave a photoanode with the architecture fto/c-tio2/m-tio2/rgo/mapbi3. for the third device, the procedure followed suit with the first device, thereafter followed by go deposition with heat treatment from 120 oc to 230 oc to reduce go to rgo. this gave a design of fto/c-tio2/m-tio2/mapbi3/rgo. the last device has the architecture fto/c-tio2/m-tio2/rgo/mapbi3/rgo which was prepared following the initial methods in device one, two and three. 2.4. preparation of counter electrode a composite material consisting of carbon black and graphite (cbg) in a ratio 3:1 respectively was utilized. this follows from the well-known protocol of zhang et. al. [30]. the cbg was cut into a circular shape of area 0.786 cm2. 2.5. assembly of the cells the four pscs were assembled by sealing the four photoanodes and the counter electrodes using ethylene-vinyl acetate (eva). 2.6. characterization and measurement 1. the morphology of rgo nanofilm was studied using the scanning electron microscope (sem) (phenom prox pw100-012 model). 2. raman spectroscopy (proraman-l-785-b1s with serial number 196166) was utilized to identify the functional groups. 3. structural analyses of the fabricated films were performed using x-ray diffractometer (xrd), (rigaku d, max 2500, japan). 4. optical spectra were characterized using ultraviolet–visible light (uv–vis) spectrometer (axiom medicals uv752 uvvis-nir). 5. the hall effect measurement was done using a four-point probe (ecopia hms-3000 version 3.52) at room temperature (300 k). 6. the current density-voltage performance of the pscs were estimated using a solar simulator (newport corporation) and a keithley 2400 (keithley, inc.) source-meter. the performance characteristics of the solar cells were evaluated under 1 sun illumination (am 1.5g standard test conditions at 100 mwcm−2). 3. results and discussion 3.1. scanning electron microscope figure 1 shows the sem images of (a) rgo with a magnification of 2000x, (b) rgo with a magnification of 500x, and (c) mesoporous tio2 with a magnification of 1000x. the obtained images of the rgo at different magnification revealed a crumpled surface with distinct edges and rippled structures with haphazardly aggregated islands at the surface of the rgo nanofilms which is attributed to the after effect of the exfoliation process of go sheets from graphite. more so, it is evident from the sem images that the rgo film is corrugated which shows the intrinsic strength nature of graphene materials. the irregularity as observed could be as a result of incomplete oxidation. however, this can serve as a good potential site for the absorbing material [31]. figure 1(c) shows the structural morphology of the mesoporous tio2. the resulting image depicts a homogeneous film with the presence of nanopores which are suitable sites for the adsorption of the absorbing layer. furthermore, it is evident from the visualized nanopores that the mesoporous layer would allow passage or tunneling of the rgo and/or the harvesting material as the case may be thereby, enhancing good ohmic contact with the compact layer and other functionalized layers. in 3 bolarinwa et al. / j. nig. soc. phys. sci. 4 (2022) 909 4 figure 1. sem image of (a) rgo with a magnification of 2000x, (b) rgo with a magnification of 500x, and (c) mesoporous tio2 with a magnification of 1000x addition, the nanopores will serve as growth nucleation sites for good crystallinity for the rgo and harvesting material during thermal annealing. this crystallinity which is a function of the nucleation growth has been found to have its immense role in the electronic performance of the devices. hence, the morphological structure of both the rgo and the mesoporous tio2 shows good morphology as expected of any material that will be classified either as a buffer layer. the sem image also shows some macroscopic defects which is due to transfer of samples during characterization. 3.2. raman spectroscopy result analysis figure 2 shows the active spectrum of raman peaks at varying raman bands (raman shift or wavelengths). by using the lorentzian deconvolution fit on originlab8, sharp raman epicenters were observed at 2076 cm−1 (d-band), 2080 cm−1 (gband) and 2886cm−1 (2d-band) for the rgo. the prominent raman signal at 2080 cm−1 designated by the g-band is attributed to the strong manifestation of sp hybridization of carbon atoms resulting from the unsaturated triple bond of carbon atoms (c≡c) which correlates with some raman bands [5]. the d-band at 2076 cm−1 is ascribed to the presence of defects in the sp lattice and related to defects (edge, vacancy and ripples) in the graphene structure which was as observed from our reported sem image in this study [32]. the 2d band peak observed at 2886 cm−1 is credited to the oxidation of the graphene structure with the presence of oxygen and hydrogen functional group. this is as evident from the active raman band standard table. it can be inferred from the sp hybridization that the rgo is conjugated which implies a high level of delocalized electrons (resulting from the π-electrons from porbitals) which in general lowers energy required for charge exciton and also increases stability thereby establishing the latter analysis on the absorbance study of the role of rgo in enhancing absorptivity of pscs. furthermore, the hydrophobic and electrophilic nature of this class of material can be attributed to the cyclo-addition from the oxygenation process resulting from the triple bond [5]. it is worthy of note that the raman data for the rgo suffices since it’s the final material used in the fabrication of the cell. the d/g band intensity ratio (id/ig ) and 2d/g band intensity ratio (i2d/ig ) provides valuable insights about the structure of graphene and graphene materials which help us appreciate the numbers particularly when structural analysis cannot be figure 2. deconvoluted intensity-wavenumber for active raman peaks of rgo achieved. the id/ig represents the defect density [33, 34] while i2d/ig denotes the numbers of layers [35] present in graphene materials. it is evident from table 1, that the d, g and 2d bands significantly have different intensity that helps in making meaningful inference in the applicability of rgo. the id/ig column implies that the rgo contains defects thereby affirming our morphological analysis. in addition, the i2d/ig band suggests that, the reduced graphene oxide as synthesized consist of multilayers of graphene with an increased number of functional groups and delocalized electrons. this further attest to our analysis on the presence of high number of delocalized electrons in our rgo film which shows prospects for photovoltaic and photonic applications. however, the increase in number of electrons and minimal defect in the rgo films makes it more applicable thereby informing our choice of which graphene material is to be incorporated in the fabrication of our solar devices. 4 bolarinwa et al. / j. nig. soc. phys. sci. 4 (2022) 909 5 table 1. peaks, intensity and intensity ratio band of synthesized films of rgo from raman spectrum sample d g 2d id ig i2d id/ig i2d/ig band band band band band band band band rgo 2076.0000 2080.0000 2886.0000 593.6047 1365.0359 875.6639 0.4348 0.6415 figure 3. x-ray diffraction pattern of ch3nh3pbi3 3.3. x-ray diffraction to ascertain the complete formation of the perovskite absorber (ch3nh3pbi3) layer in this research, the xrd measurement was carried out on the synthesized methylammonium absorber thin film. as depicted from figure 3, the main peaks at 2θ angles of 14.0◦, 28.6◦, 31.9◦ corresponding to the 110, 220 and 310 planes confirms the formation of ch3nh3pbi3 [10, 36]. the formed ch3nh3pbi3 film is of high crystalline quality. 3.4. uvvis spectrophotometry analysis the absorptivity of varying configurations on fto glasses consisting of m-tio2, rgo and the perovskite film were investigated using uv-750 series spectrophotometer within the range 230-1200 nm of wavelengths. the graphical representation of each configuration as deposited on fto glass slides for necessary investigation are presented in figure 4. however, the absorption as reported was normalized by dividing through by the observed absorbance peak thereby resulting in an absorbance value between 0 and 1. as shown in the combined absorbance of the samples, m-tio2 shows a shouldering peak at 333 nm in the ultraviolet (uv) region. the undulating absorbance between 230-290 nm can be attributed to the non-bonding to the pi bonding excitations of the material [37]. at such a wavelength, no visible light will be absorbed. with m-tio2 being an electrode, the plot justifies that the material is not capable of harvesting light within the spectrum since no significant absorbance was observed. the rgo absorbs light partly from the uv region through the near infrared (nir) region with observable peaks from 500-860 nm and a noticeable absorbance peaks of 516 nm and 640 nm within the visible region. the wide range of absorbance can be ascribed to the zero-band gap of graphene which is also in tandem with its broad range theoretical absorbance capacity up to the microwave region. this also informs the presence of several delocalized electrons (ππ∗ bonding). as earlier stated, the undulating absorbance between 230-290 nm can be attributed to the non-bonding to the pi bonding transition while the nonlinear absorption behavior (saturation absorption) of rgo can be ascribed to the exposure of the rgo films to air and possibly the presence of nitrogen [38]. also, a sharp increment was observed around 995 nm which is an attestation to the ability of this class of materials to absorb adequately at this region. furthermore, the plot shows a more stable absorption rate over a wide-range of wavelength within the visible region from 385-538 nm for the absorbing material with a maximum absorbance peak at 386 nm. this can be attributed to the high absorption coefficient of this class of material as stated earlier. a sharp shouldering pattern was observed from 530 nm up to the nir which could be as a function of the degradation pattern of this class of materials. although, an eventual increment was noticed at about 1000 nm. above all, the harvesting material demonstrated a good absorbance across the entire visible region. similarly, it is evident that in the presence of rgo, a hyperchromic effect by an average of 27.78% was observed at the same wavelength of 1000 nm after incorporating rgo on mtio2. this depicts the possibility of rgo to absorb photons due to delocalized electrons and above all proving to be of positive effect to the photoanode. however, the absorbance peak for the m-tio2/perovskite configuration was at 386 nm. it is evident that in the presence of m-tio2, the absorptivity of the harvesting material witnessed no significant boost. however, a slight flattening was detected across the peaks in the plot in comparison with the perovskite absorbance graph. the implication of this according to the energy and wavelength relationship is that more energy will be required for the excitation of electrons as the gap between the homo and lumo must have likely increased. hence, this suggests a serious need for an interlayer that will bridge the energy gap between both layers which rgo proves to be suitable. nonetheless, it was depicted that the underlayer introduction of the perovskite material with rgo resulted in a bathochromic effect (red shift). the implication of this is that in the presence of the rgo, there is an increase in the number of delocalized electrons of the absorbing material thereby leading to a quasi-fermi level as there’s likely to have been a fall in the gap between the homo and lumo. similarly, lesser energy will be required in exciting electrons as the work function must have been lessened. a close look at the uv region indicates a 5 bolarinwa et al. / j. nig. soc. phys. sci. 4 (2022) 909 6 figure 4. a stacked graph showing the absorbance spectrum of all sample configurations at a glance hyperchromic effect across the uv region with a significant rise in the minimum absorbance. in addition, adequate broadening of peaks was witnessed across the visible region. summing up all, this configuration shows that the rgo is a very good material with promising outputs for photovoltaic applications in perovskite solar cells. also, the figure also shows the absorptivity for the configuration with rgo as an interlayer between the m-tio2 and the absorbing material. it is evident that the effect of the rgo has helped in addressing the “no effect” as observed in the mtio2/perovskite architecture by enhancing an absorbance peak of 433 nm within the absorbing range of 350-535 nm with a protruding peak at 958 nm as well as a stable shouldering. the observed red shift would likely be as a result of increased delocalization of electron taking place by using rgo as an interlayer material. more so, the rise in homo and lumo inferred in the m-tio2/perovskite can as well be inferred to have fallen with an upper bound hypochromic effect and lower bound hyperchromic effect in absorptivity. what this implies is the formation of a quasi-fermi level which lessens the work function of this architecture. hence, devices with rgo as an interlayer between the mesoporous transport layer and the harvesting material in this architectural configuration will be energetically favored since lesser energy will be required for the excitation of excitons. since more work will be done with less energy, this translates to devices with this configuration having a better efficiency. figure 5. box plot describing the absorbance behavior of each sample configuration 3.5. statistical analysis of uv-vis spectrophotometry data analysis may be rendered incomplete without some statistical analysis that helps us verify our scientific inferences by using applicable statistical models. herein, statistical analysis of variance (anova) was carried out to know the significant difference in the mean rate of absorptivity of the demonstrated samples. although the set of data under consideration are skewed, affected by outliers and not normally distributed thereby requiring a non-parametric test. however, with a data frequency of 314, the outliers are believed to likely not have any significant effect on the result thus the justification for the use of anova. also, a post hoc comparison using duncan multiple range test (dmrt) was used to show specifically the variation in each sample. the duncan multiple range test as proposed by duncan is a statistical analysis tool that gives results by comparing the specific difference in the means of various data set and in our case gives result on the difference between the mean absorbance of various sample configurations. in addition, a box plot was used to show the maximum, minimum and median absorbance. with a box plot, a more crystal role of the effect of the rgo on absorbance pattern of the configuration can be deduced. the descriptive statistics in table 2 gives an overview of the entire data set. it is worthy of note to state that the statistical analysis which includes the box plot and the descriptive analysis was carried out using a statistical programing software called r. the descriptive analysis shows that the number of data set for each configuration is 314. this implies that the samples were bombarded with 314 different monochromatic light between the wavelength of 200-1200 nm. it also gives the overall mean absorbance to fall within (0.06542-0.90976) with rgo/perovskite having the highest mean rate of absorption. also, the standard deviation and error associated to each configuration, the upper and lower class bound as well as the minimum and maximum value for each sample were included. 1. descriptive statistics 6 bolarinwa et al. / j. nig. soc. phys. sci. 4 (2022) 909 7 table 2. statistical description of the absorbance behavior of all configurations rate of absorption 95% ci min max type of materials n mean std. deviation std. error lower bound upper bound m-tio2 314.0 0.06542 0.15342 0.00866 0.04838 0.08245 0.00002 0.92500 rgo 314.0 0.32990 0.18625 0.01051 0.30922 0.35058 0.12000 0.85700 perovskite 314.0 0.79691 0.61023 0.03444 0.72916 0.86467 0.06200 1.80000 m-tio2/rgo 314.0 0.19063 0.21810 0.01231 0.16642 0.21485 0.03700 1.18200 m-tio2/perovskite 314.0 0.86232 0.67983 0.03837 0.78684 0.93781 0.06300 2.06100 rgo/perovskite 314.0 0.90976 0.51627 0.02913 0.85244 0.96709 0.20900 1.76300 m-tio2/rgo/perovskite 314.0 0.89379 0.55844 0.03151 0.83178 0.95580 0.20900 1.83100 the box plot in figure 5 depicts the distribution of the rate of absorption disaggregated by the type of materials. it’s evident that doping of m-tio2 with rgo (mtio2/rgo) resulted in a significant increment in both the minimum and maximum values of m-tio2 which is in accordance with the observed hyperchromic effect. similarly, the box plot revealed the role of rgo as an underlayer material for perovskite material with a significant enhancement in the minimum absorption and slight increment in maximum absorption. this agrees with the red shift as earlier analyzed. while the m-tio2/perovskite material shows the highest absorption as a result of the blue shift, the m-tio2/rgo/perovskite also indicate a significant increment in the minimum absorbance and a slight drop in the maximum absorbance. therefore, showing the role of rgo as an efficient interlayer between the mtio2 and absorbing material. 2. analysis of variance (anova) result hypothesis: the analysis of variance carried out is based on the following two hypotheses: ho: there is no significant difference in the rate of absorption across the materials h1: there is a significant difference in the rate of absorption across the materials table 3 depicts the result of one-way between groups analysis of variance conducted at a 99 % confidence interval to explore the impact of materials on the rate of absorption. the mean rate of absorption by combining different materials were compared and the result which conforms with previous comparative analysis shows that there was a statistically significant difference between the rate of absorption across the various types of materials, with f (6, 2191) = 196.078, and a probability value (pvalue= <0.001) which is < 0.01. hence, by convention, we discard our null hypothesis (ho) and uphold the research hypothesis that says that there is a significant difference between the sample configurations. although, the post hoc comparison using duncan multiple range test will help us to know which is significantly different from the other. 3. duncan multiple range test duncan multiple range test (table 4) reveals no significant difference between rgo/perovskite and m-tio2/rgo/perovskite as well as m-tio2/perovskite and perovskite. however, the same result shows that a significant difference exists between: rgo/perovskite and perovskite; m-tio2/rgo and rgo. furthermore, it depicts that rgo/perovskite is significantly different from m-tio2/perovskite. similarly, the combinations of each of m-tio2/perovskite, rgo/perovskite and m-tio2/rgo/perovskite is significantly different from the individual usage of each of rgo, m-tio2 and perovskite. hence, the significance of rgo as inferred earlier is further justified both statistically. 3.6. hall effect result and analysis table 5 depicts that the nature of charge carriers in this material are electrons which is evident from the negative sign (-) in the bulk and sheet concentration value. the concentration of the delocalized electrons affirms our observations from the raman results and also conforms with the inference from our absorbance analysis thereby making the material suitable for solar device application. similarly, the material shows a very good electron mobility, conductivity and resistance 3.7. photovoltaic performance of the pscs in obtaining the electrical properties of the fabricated cells, device a-d were simulated using the a. m. 1.5g solar simulator. from the obtained results (js c and voc ) as measured, the fill factor (ff), power conversion efficiency (pce), were obtained from equations 1 and 2 as reported in table 6. f f = jmax × vmax js c × voc (1) pce = f f × js c × voc pirradi ance .100% (2) where ff is fill factor, pce is solar cell efficiency, v max is maximum voltage, jmax is maximum current density, j sc is short circuit current density, v oc is open circuit voltage and pirradi ance is light intensity. from table 6, it was observed that the insertion of rgo has no significant effect on the short circuit current density jsc as the devices exhibit relative jsc value of 8.007 ≤ jsc ≤ 8.737. this relativity is indicative of the optical trapping capability of all the devices for photovoltaic conversion. although, there is a slight drop in the jsc value for device b which could have been due to the thickness of the rgo film thereby limiting optical 7 bolarinwa et al. / j. nig. soc. phys. sci. 4 (2022) 909 8 table 3. one-way anova result for the absorptivity of tested samples source of variation sum of squares df mean square f p-value between groups 255.249 6 42.541 196.078 <0.0001 within groups 475.363 2191 0.217 total 730.612 2197 note that the null hypothesis is to be discarded if p-value is less than 0.01. table 4. duncan multiple range test result for the absorptivity of tested samples duncan subset for alpha = 0.05 type of material n 1 2 3 4 5 m-tio2 314 .0654172 m-tio2/rgo 314 .1906338 rgo 314 .3299045 perovskite 314 .7969140 m-tio2/perovskite 314 .8623217 .8623217 m-tio2/rgo/perovskite 314 .8937898 rgo/perovskite 314 .9097611 sig. 1.000 1.000 1.000 .079 .231 figure 6. comparative graph of current against voltage for devices a, b, c and d table 5. parametric results from the hall effect measurement system parameters result bulk concentration -9.878×1013/cm3 mobility 2.099×104 cm3/vs sheet resistance 1.434×105 ?/cm2 resistivity 3.011 ?cm sheet concentration -2.074×109/cm2 conductivity 3.321×10−1 transmittance of the device. above all, the jsc result suggests good ohmic contact amongst the layers of the fabricated devices. however, it is evident that a significant boost was witnessed in the open circuit voltage of devices with rgo interlayer. a minimum of 23.21 % increment in pce was observed between the control device without rgo and other devices with rgo. while the voc value ranges within 0.575≤ voc≤0.886, the enhancement can be attributed to the ability of the rgo to minimize interfacial recombination from both charge transport interface with the absorbing layer. the recombination suppression is due to the rough adhesive surface of the rgo film as depicted by the sem image. furthermore, the voc heightening can be credited to the ferroelectric distortion of the perovskite when in contact with graphene materials. the distortion is capable of improving charge carrier extraction along the interfaces, regeneration of the active layer through enhanced charge transfer from the perovskite absorbing layer to the electron/hole transport layers. this submission is supported by the theoretical demonstration of volonakis and guistino [39]. in their work, they ascertained that the ferroelectric distortion leads to tilt in the crystallinity of the perovskite material thereby causing a 8 bolarinwa et al. / j. nig. soc. phys. sci. 4 (2022) 909 9 table 6. photovoltaic performance parameters of devices a, b, c and d device architecture jsc(ma/cm2) voc (v) ff (%) pce (%) a glass/fto/c-tio2/m-tio2/mapbi3/cbg 8.151 0.575 74.4 3.49 b glass/fto/c-tio2/m-tio2/rgo/mapbi3/cbg 8.007 0.886 60.6 4.30 c glass/fto/c-tio2/m-tio2/mapbi3/rgo/cbg 8.659 0.811 64.0 4.49 d glass/fto/c-tio2/m-tio2/rgo/mapbi3/rgo/cbg 8.737 0.836 68.1 4.98 figure 7. variation of (a) jsc, (b) voc, (c) ff and (d) pce with device having various photoanode change in the electronic and structural properties of the perovskite. in furtherance, the device d which comprises of the synergistic architecture harnessed the properties of rgo at both the electron and hole interface in: trapping and converting photon energy, minimizing recombination, enhancing hole and electron extraction -cum tunneling and the regeneration of the harvesting material all of which resulted in its heightened pv value and enhanced conversion efficiency as observed. the observed jsc, voc, and ff values for the reference device were 8.151 ma cm−2, 0.575 v, and 74.4 %, respectively, yielding a pce of 3.49 %. the best pce in our research, was achieved using rgo incorporated on both tio2 and mapbi3. particularly, the device showed a jsc of 8.737 ma cm−2, voc of 0.836 v, and ff of 68.1 %, respectively, resulting to a pce of 4.98 % (see figure 6a). the highest device was ∼ 42.69 % higher than that of the reference device, suggesting that the addition of rgo can improve the pce of pscs. this improvement in the power conversion efficiency may be attributed to the higher electrical conductivity and suitable energy band alignment of our rgo in the device [2]. figure 6b shows the relationship that exist between the power density and open circuit voltage. figure 7 (a-d), show the metric parameters dependence with various photoanode. the optimum performance was with device having rgo on both etl and htm. 4. conclusion the effect of reduced graphene oxide on the performance metrics (pce, jsc, voc, and ff) of perovskite solar cells was investigated systematically. as a result, the device performance such as jsc and pce were enhanced with rgo introduction as compared to the pure device without rgo. the champion device was achieved with the architecture of fto/c-tio2/m-tio2/rgo/ch3nh3pbi3/rgo/cbg resulting to improvement from 8.151 to 8.737 macm−2 in jsc and from 3.49 9 bolarinwa et al. / j. nig. soc. phys. sci. 4 (2022) 909 10 to 4.98 % in pce respectively. the obtained results show an enhancement of ∼42.69 % in pce and ∼ 7.20 % in jsc over the prestine device. these results were encouraging and demonstrated good means of enhancing perovskite solar cell performance using reduced graphene oxide. acknowledgments the authors are grateful to physics advanced laboratory, sheda science technology complex (shestco) and namiroch research laboratory, abuja for the use of their equipment. references [1] z. qu, f. ma, y. zhao, x. chu, s. yu & j. you, “updated progresses in perovskite solar cells”, chinese physics letters 38 (2021) 107801. 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[39] g. volonakis & f. giustino, “ferroelectric graphene-perovskite interfaces”, journal of physical chemistry letters 6 (2015) 2496. 11 j. nig. soc. phys. sci. 2 (2020) 257–261 journal of the nigerian society of physical sciences prompt response function (prf) of lifetime measurement in the 2+ state of 192os nuclei energy levels from triple-gamma coincidence techniques t. daniela,b,∗, s. kisyovc, p. h. regana,d, n. margineanc, zs. podolyaka, r. margineanc, k. nomurae,f, m. rudigiera, r. mihaic, v. wernerg, r. j. carrolla, l. a. gurgia, a. opreac, t. berrya, a. serbanc,h, c. r. nitac, c. sottyc, r. suvailac, a. turturicac, c. costachec, l. stanc, a. olacelc, m. boromizac,h, s. tomac, s. j. gemanamb, f. gbaorunb, i. ochalai, e. c. hembaj adepartment of physics, university of surrey, guildford gu2 7xh, united kingdom bdepartment of physics, benue state university, pmb 102119, makurdi, nigeria choria hulubei national institute of physics and nuclear engineering (ifin-hh), ro-077125 bucharest, romania dair division, national physical laboratory, teddington tw11 0lw, united kingdom edepartment of physics, faculty of science, university of zagreb, bijenicka cesta 32, hr-10000 zagreb, crotia fcenter for computational sciences, university of tsukuba, tsukuba 305-8577, japan ginstitut fur kernphysik, t.u. darmstadt, 64289 darmstadt, germany huniversity of bucharest, faculty of physics, magurele-bucharest, romania idepartment of physics, kogi state university, anyigba, nigeria jdepartment of physics, federal college of education, pankshin, plateau state abstract the effective prompt response function full width at half maximum, prf fwhm of 637 ps (obtained from the prompt gamma pairs of 477 kev and 700 kev associated with the yrast 2+ state in 206po), and 1007 ps (obtained from the compton gamma pairs of 189 kev and 237 kev associated with the 192os(18o,16o)194os 2 neutron transfer reaction) were used in fitting the time difference spectra obtained from the gamma coincident pairs of 206 kev and 374 kev in a symmetrised labr3(ce) associated with the gamma transitions in 192os, using the half-life program. the values of half-life measured by fitting these prf fwhm of 637 ps and 1007 ps separately show an excellent agreement of 282(16) ps and 272(21) ps, respectively, which correspond to the global half-life value of 282(4) ps for the 192os. the mean value of 277(12) ps from these two measurements was used in calculating the b(e2; il õil-2) of 4233(114) e2fm4, which is equivalent to be 81(19) w.u. doi:10.46481/jnsps.2020.98 keywords: prompt response, full width at half maximum, lifetime, scintillators article history : received: 21 april 2020 received in revised form: 07 august 2020 accepted for publication: 10 august 2020 published: 15 november 2020 c©2020 journal of the nigerian society of physical sciences. all rights reserved. communicated by: b. j. falaye ∗corresponding author tel. no: +2348167598988 email address: tdaniel@bsum.edu.ng, terrver.daniel@yahoo.co.uk (t. daniel ) 1. introduction the atomic nucleus which forms the central part of the atom is made up of the protons together with the neutrons. in gen257 daniel et al. / j. nig. soc. phys. sci. 2 (2020) 257–261 258 eral, the atom consists of all these components together with the constant orbiting electrons. the nuclear size is estimated to be of the order of few fermis, ranging from ∼ 1.6 fm (10−15 fm) for light nuclei (for example hydrogen, with only one proton) to ∼ 15 fm in heaviest elements, such as uranium [1, 2]. the nucleus of an atom is held together by the strong nuclear force, which is strong enough to overcome the proton repulsion (at short ranges of ∼ 1 fm). figure 1 shows the entire nuclei chart with 283 stable or very long-lived nuclei [3] represented by the black squares. their neutron-to-proton ratio is loosely grouped within the ranges 1.00 õ 1.40 for 2 ≤ z ≤ 50 and 1.20 õ 1.60 for 50 < z ≤94. the addition or removal of nucleons from stable nuclei obviously will alter the n/z ratio, resulting in the formation of radioactive, unstable, neutron-rich systems [2, 3, 4]. 2. measurement of the nuclear excited states lifetimes figure 1. the chart of nuclides showing the stable nuclei in black squares with the unstable ones on either side of the band of stability. the vertical and the horizontal lines represent the magic numbers or closed shells [3]. the measurement of lifetimes of excited nuclear states can be regarded as a fundamental tool for nuclear spectroscopy [2, 5]. these play a significant role in the determination of the reduced electromagnetic transition rates which are sensitive to the intrinsic properties of the nuclear levels between which the transition proceeds [5, 6, 7]. the excited states of nuclei can be produced in many ways including light particle evaporation following a heavy-ion fusion; multi-nucleon transfer, coulomb excitation etc. [2]. the half-lives of the excited nuclear levels can be measured using the time profile of γ rays detected with (high-resolution) gamma-ray detectors. in this article, the romanian array for γ ray spectroscopy in heavy ion reactions, rosphere [5] was used. the lifetimes of the low-lying excited nuclear states typically range from femtoseconds to nanoseconds [7]. in this article, the convolution method was deployed for the measurement of the yrast halflife for the 2+ state in 192os were different prf have been used in the determination of the halflife value for the state. however, there are several techniques by which the lifetime of the nuclear excited states can be obtained apart from the approach stated above. these techniques are broadly classified based on the range of the lifetime of the excited state [6, 7, 8], and can be subdivided into two main categories; (a) the indirect methods which measure the energy width of the state, γ, and (b) the direct methods which measure the mean decay lifetime, τ. 3. the concept of convolution apart from the fact that these techniques are classified based on the range of lifetime information associated with the energy levels, convolution and the centroid shift methods can widely be employed in the measurement of these lifetimes of the nuclear levels [10, 11, 12, 13, 14, 15]. in these approaches, two or more γ ray transitions are used; the populating transition(s) to an energy level and the depopulating transition(s) out of the energy levels [2]; bringing about the gamma coincidence techniques. lifetime measurements within the picosecond to nanosecond region where the decaying γ rays are measured directly are obtained using the direct method [16]. among these direct methods is the electronic timing technique, which in this article is based on the use of fast (time) response γ-ray detectors, made from labr3(ce) scintillator material [15]. for almost three decades, the electronic timing technique of β − γ − γ coincidence method using baf2 crystals has been used for picosecond lifetime measurements in neutron-rich nuclei [17]. in this method, the desired decay path/cascade can be selected with the use of a high-resolution ge detector. more recently [13, 14, 15, 18, 19], the use of triple coincidences for lifetime measurements in the picosecond region has been developed, where the time difference, ∆t, is obtained between the coincident cerium-doped labr3 scitillators gated with a hpge energy coincidence [20, 21]. this has led to an improvement compared to the previous baf2 based analysis due to the superior energy resolution and fast decay time for labr3(ce) detectors [22, 23]. in this article, a triple-coincidence, γ1 − γ2 − γ3 technique is adopted, based on the operation of both hpge detectors and labr3(ce) scintillators. the time distribution spectrum from the measured time difference between the two coincident labr3(ce) scintillators can be obtained either using a convolution of the prompt (gaussian) response function and the exponential decay (see details in figure 2) or alternatively by using the centroid shift method [24, 25, 26]. both methods of analysis can be used when the half-life of the nuclear state is long enough to be measured by fitting the exponential nature of its decay [27] while the centroid shift method is particularly useful in cases where the half-life is of the same order or smaller than the full width at half maximum, fwhm prompt time response [27, 28, 29, 30, 31]. 258 daniel et al. / j. nig. soc. phys. sci. 2 (2020) 257–261 259 figure 2. the convolution function with the exponential and the gaussian prompt component as a function of t using arbitrary σ and τ, where τ is 5 times longer than σ [2]. figure 3. background subtracted time difference spectra for the yrast iπ =2+ state in 192os, obtained using the deconvolution method, showing the time difference between the 206 kev and 374 kev transitions. time difference spectra plotted with black lines are gated on (eγ1 , eγ2). the continuous line is the gaussian exponential convolution fit to the spectra. 4. prompt response function (prf) fwhm of 637 ps and 1000 ps in the iπ = 2+ yrast state of 192os here, we present the lifetime measurements of the yrast state iπ = 2+ from the 192os isotope in the ground state band obtained from the ’unsafe coulombs excitation’ from the bombardment of the enriched (∼ 99%) 20 mgcm−2 192os target with a 80 mev 18o beam which populates excited states associated with 194os nucleus from ground state band of 374 kev and 206 kev [2]. that is, for the time difference between the labr3(ce) at 374 kev (the feeding transition) of 4+ → 2+ and the labr3(ce) at 206 kev (depopulating transition) of 2+ → 0+ ground state with fixed prf fwhm of 1007 ps (see figure 4.1 for details) and 637 ps (see figure 4.2) [32] using the halflife program [33]. the choice of these prf fwhm is necessitated from the fact that the ∆t obtained from the gamma pairs of 477 kev and 700 kev associated to 206po for the 637 ps [31] and 1007 ps as obtained from the ∆t between the compton gamma pairs of the 189 kev and 237 kev transitions are actually very ’prompt’ or figure 4. time difference spectra for the yrast iπ = 2 + state in 192os, obtained using (upper panel) the centroid shift method and (lower left and right panels) deconvolution method, showing the time difference between the gamma pairs of 206 kev and 374 kev transitions. time difference spectra plotted with black lines are gated on (eγ1 , eγ2 ), while the red lines show the reverse gating. the continuous lines in lower panels are the gaussian exponential convolution fits to the spectra (the effective prf fwhm value of 637 ps is taken from the 206po [31]. this figure is taken from ref. [33] as figure 6 (see details in cross referencing, too). gaussian in nature because the time difference between them is very small as compared with a normal time difference obtained in a fit to the exponential slope. the relevance of this work here is to see whether there is a significant difference in the half-life measurement with these prf values. in each approach stated above, the lifetime measurements for the yrast state in 192os with fixed prf fwhm at either 637 ps [32] or 1007 ps are in agreement with each other. this points to the fact that the time difference spectra presented in either case is prompt in which the time information from them does not contribute ’significantly’ to the measured yrast state in 192os. the weighted means of the two results obtained from fixing either prf fwhm at 637 ps [32] or 1007 ps for the ∆t of 206 kev and 374 kev gamma pairs are presented in figure reff5 with each result having the calculated error associated with the fwhm used and from the transitions employed here. the error bars in the stated results are mostly attributed to the fact that there is limited counts in all the measurements. 5. discussion of results and conclusion figures 3 and 4 present the time difference spectra obtained between the gamma pairs of the 206 kev and 283 kev transitions, whose extracted half-life values are 272(21) ps and 282(22) ps, from fixing the fwhm value of 1007 ps and 637 ps, respectively, constant in the half-life program [33]. the weighted mean of the extracted half-lives from the labr3(ce)labr3(ce) coincidence pairs of (206, 283) kev and (206, 374) kev transitions, fixing fwhm values at either 1007 ps or 637 ps, is plotted in figure 5. both results agree excellently with each other approach even as the prf fwhm was varied and within 259 daniel et al. / j. nig. soc. phys. sci. 2 (2020) 257–261 260 figure 5. plot of the measured half-life values for the iπ = 2+ yrast state in 192os obtained from different approaches in the current work. the upper panel shows the weighted mean obtained from the half-life values using an effective prf fwhm value of 1007 ps, while the lower panel presents the weighted mean for an effective prf fwhm value of 637 ps. the horizontal solid lines indicate the weighted average of the half-life values from different decay paths, while the red lines represent the associated uncertainty in the weighted mean. the calculated error. this result agrees with the global half-life value of 288(4) ps [32, 34] where other measured value for the 192os 2+ state has been reported for the yrast state in 192os. in conclusion, the results obtained from the prf values of 637 ps and 1007 ps have shown a great similarity, which in turn is in agreement with the global half-life value for the 2+ state in 192os. this is an indication that once the time difference between two gamma energies is “prompt”, there is an insignificant presence of the time information (that is, in the half-life value) for such a state. acknowledgement t. d. would like to appreciates the ph.d fellowship he enjoyed from tetfund nigeria for his research work in experimental nuclear physics at ifi-hh bucharest, romania, national argonne laboratory, chicago, usa and at the university of surrey, guildford, uk. t. d also wishes to thank prof. p.h. regan for introducing him to this novel and ’clean’ electronic approach for the unveiling the intrinsic time information of the nuclear structure using the coincidence technique. references [1] w. gelletly, “radioactive ion beams: a new window on atomic nuclei”, proceedings of the american philosophical society, 145 (2001) 519. 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[34] c. m. baglin, “nuclear data sheets for a = 192”, nuclear data sheets 113 (2012) 1871. 261 j. nig. soc. phys. sci. 4 (2022) 924 journal of the nigerian society of physical sciences eyring-powell mhd nanoliquid and entropy generation in a porous device with thermal radiation and convective cooling s. o. salawua,∗, r. a. kareemb, j. o. ajilorec adepartment of mathematics, bowen university, iwo, nigeria. bdepartment of mathematics, lagos state university of science and technology, ikorodu, nigeria. cdepartment of mathematics, lagos state university of science and technology, ikorodu, nigeria. abstract this study investigates the flow of magnetohydromagnetic (mhd) eyring-powell chemical reaction nanoliquid in a permeable boundless device with wall cooling and thermal radiation. the fully developed cauchy non-newtonian fluid model is stimulated by species reaction and the stretching sheet under gravity influence. using the rosseland radiation approximation model with an appropriate similarity variable, the dimensionless coupled derivatives are obtained. a shooting numerical technique is utilized to determine the thermophysical effects on the flow characteristics. the solution results are computed and given in graphs and tables for clear demonstration and clarification. the results show that entropy is minimized by augmenting the magnetic field, porosity, and thermodynamic equilibrium. also, parameters that enhance internal heat must be monitored to prevent chemical reaction nanoliquid blowup. doi:10.46481/jnsps.2022.924 keywords: entropy generation; nanoliquid; non-newtonian; hydromagnetic; porosity article history : received: 07 july 2022 received in revised form: 20 august 2022 accepted for publication: 22 august 2022 published: 08 october 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: s. fadugba 1. introduction the dynamical flow of conducting fluid (hydromagnetic) has presently dominated the scientific research and discussion because of its many applications in power generation, agricultural technology, medicine and others, hassan et al. [1]. the flow mechanics also assist in managing flow fluid velocity and improving lubricants [2,3]. however, magnetohydrodynamic liquid flow in a stretching plate will continue to be relevant in technology and industry due to its various usages in polymer extrusions, liquid condensation, photographic coasting, etc ∗corresponding author tel. no: +2348032056439 email address: kunlesalawu2@gmail.com (s. o. salawu ) [4,5]. therefore, to improve the significance and worth of electrically conducting liquid flow in a stretching surface, it becomes necessary to consider its effect on non-newtonian fluids. among these liquids is powell-eyring material that was formulated from the theory of liquids kinetic, and at small or huge shearing rate, it behaves like a newtonian fluid. the fluid defines shear thinning properties; of such is toothpaste, ketchup, human blood and so on. thus, accompany with heat transport, salawu et al. [6,7] discussed the heat stability of eyringpowell with the impact lorentz force, variable conductivity and electromagnetic heat. parameters dependent flow characteristics and irreversibility were reported in the study. nadeem and saleem [8] examined in a rotating cone, the flow of en1 salawu et al. / j. nig. soc. phys. sci. 4 (2022) 924 2 ergy and mass transport of eyring-powell mixed convection. it was offered that the buoyancy ratio encouraged skin friction, but species and heat gradient are opposite to it. akbar et al. [9] studied in a stretching plate, the powell-eyring mhd incompressible fluid with a report on the effects of some terms on the shear stress and flow velocity. it was noticed that magnetic field and material terms resists flow velocity. zaidi et al. [10] investigated the stream of thermal dissipation of poewll-eyring in a wall jet. the study analysis was done using shooting procedures and found that the heat dissipation term boosted the overall flow dimensions. some other analysis on eyring-powell can be obtained in [11-13]. recently, due to technological advancement and its applications, nanoliquid has become the central of attraction to the engineers and scientists. the fluid with base liquid made-up of infinitesimal nanoparticles dimension, an example of such is metal, nanotubes carbon, and oxide, idowu and falodun [14]. when the base liquid is used, nanofluid is useful in improving the coefficient of heat conduction and heat convection transfer. coupled with magnetic fields, magneto-nanomaterial is remarkably applicable in design of filters wavelength, filters optical fiber, optical switches and modulators, etc, ishaq et al. [11]. as such, hayat et al. [15] presented computed outcomes for a flow of squeezing magneto-nanofluid past a far stream domain. the lorentz force is seen to have obviously impacted the flow velocity and thermal dispersion in the flow device. alao et al. [16] carried out simulation on the radiative heat and species transport of chemically reacting flow with soret and dufour. the thermophoretic and brownian terms are seen to respectively increase the temperature and species distribution in the chemical reaction region. combining nanoliquid and eyring-powell fluid is of great importance in enhancing the effectiveness of technological devices for the best possible performance. as a result, agbaje et al. [17] demonstrated the impact of radiative and thermal generation on the nanomaterial eyring-powell non-newtonian fluid in a dwindling plate. the radiation and heat generation was observed to have significantly impacted the heat conduction and convective heat transport in a moving sheet. malik et al. [18] used numerical scheme to compute the eyring-powell mixed convection nanomaterial mhd flow past a widening surface. it was observed that the chemical reaction and the lewis number term reduces the fluid chemical distribution. in a vertical stretching plate, the hydromagnetic powelleyring nanoliquid with radiation was investigated by [12,19]. homotopy analytic technique was used and the results show that the radiation, lewis and heat dissipation terms remarkably influenced the flow characteristics. however, despite the numerous utilization of nanofluid and eyring-powell liquid, entropy generation in the system can significantly affect and diminish the efficiency of the thermal technological tools if no properly managed, salawu et al. [20]. in thermodynamic structure, energy is lost due to the force of friction, the chemical reaction, the viscosity of the fluid and diffusion which resulted in entropy generation. the entropy generation measures the degree of irreversibility in a structure; this is determined using thermodynamic second law, salawu and oke [21]. therefore, to optimize technology tools, entropy generation due to irreversible process should be minimized. as such, rashidi et al. [22] discussed entropy generation of hydromagnetic swarm particle in a moving gyratory disk. the authors used optimization procedure and the neural artificial system to solve the problem and found that increasing magnetic field minimized irreversible process. hayat et al. [23] presented silver and copper nanomaterials with minimization of entropy generation. the solution to the model was analytically obtained. khan et al. [24] offered a declination of entropy generation for radiative nanoparticles in a stretching thin needle. it was seen that nusselt number and wall drag force stimulated higher volume of fraction nanoliquid. bhatti et al. [19] studied powell-eyring nanoparticles entropy generation in a porous moving plate. in the considered thermophysical terms, an enhancing function of irreversible process is observed. some other studies on thermodynamic second law can be found in [25-28]. the present analysis examines the thermodynamic second law of chemical reactive eyring-powell nanoliquid with joule heat and radiation in stretching convective cooling plate. this study is being motivated by the positive achieved results and the applications of non-newtonian nanoliquid. despite many related studies, no mathematical post or study on eyring-powell nanomaterial has been examined with ohmic heating, heat generation, natural convection, heat radiation and chemical species. in the study, the entropy generation, irreversibility ratio, and heat and species distributions is examined in a cooling porous device to prevent the materials from distortion. the dimensionless quasilinear coupled derivatives are solved by shooting techniques to investigate the thermodynamic flow characteristics of the parameters dependent. the study is significant in reducing irreversibility process in order to enhance the utilization and efficiency of nano eyring-powell liquid. also, the reaction flow behaviour to variation in parameters for proper monitored of technological devices. 2. mathematical formulation examine a hydromagnetic chemical reactive eyring-powell nanoliquid flow in permeable media. the flow is gravity-driven past a vertical stretching convective cooling plate at y = 0 and at y > 0, the mhd dominance takes place. the flow is inspired by induced lorentz force and non-newtonian material terms. as demonstrated in figure 1, the flow is offered in x-axis and y-axis is normal to it. the homogenous chemical species occurs in a boundless domain and the stretching wall is exposed to thermal convective cooling satisfying the newton’s of law of cooling. due to small effect of reynolds number, the magnetic induction impact is ignored. the liquid viscoelastic behaviour is encouraged by the eyring-powell non-newtonian cauchy formulation as presented below. the non-newtonian eyring-powell viscoelastic cauchy formulation is described as [10,29] ci j = 1 δ sinh−1 ( 1 αr ∂wi ∂x j ) + µ ∂wi ∂x j . (1) the term ci j denotes cauchy tensor, αr and δ represents the 2 salawu et al. / j. nig. soc. phys. sci. 4 (2022) 924 3 working fluid material term of powell-eyring while µ depicts the fluid coefficient viscosity. thus, sinh−1 ( 1 αr ∂wi ∂x j ) � 1 αr ∂wi ∂x j − 1 6 ( 1 αr ∂wi ∂x j )3 , ∣∣∣∣∣ 1αr ∂wi∂xi ∣∣∣∣∣ < 1.(2) the velocity of the flow is not created by species reaction but influenced by gravity and moving plate. taking from all the assumptions, the approximated equations for the boundary layer mass conservation, momentum, temperature and chemical reaction balance of eyring-powell nanoliquid flow in permeable media are given as [6,9,11,13]: ∂u ∂x + ∂v ∂y = 0, (3) u ∂u ∂x + v ∂u ∂y = ( νn f + 1 ρn f αrδ ) ∂2u ∂y2 − 1 2ρn f αrδ3 ( ∂u ∂y )2 ∂2u ∂y2 − σb20 ρn f u − νn f k u + gβt (t − t∞) + gβc(c − c∞), (4) u ∂t ∂x + v ∂t ∂y = qa (ρcp)n f (t − t∞) + α ∂2t ∂y2 + 16σst 3∞ 3k∗ ∂2t ∂y2 + τ db ( ∂c ∂y ∂t ∂y ) + ( dt t∞ ) ( ∂t ∂y )2 + σb 2 0 ρn f u2 + ν k u2, (5) u ∂c ∂x + v ∂c ∂y = db ( ∂2c ∂y2 ) + dt t∞ ( ∂2t ∂y2 ) − r0(c − c∞) n. (6) the sustaining employed boundary conditions are according to [10,29]: u = uw(x) = ax, v = 0, − k ∂t ∂y = h f (t f − t ), c = cw at y = 0, u −→ 0, t −→ t∞, c −→ c∞ at y −→∞. (7) here, the terms u, u denote the velocity modules in x and y, c, c∞ and t , t∞ are the fluid concentration, far stream concentration and fluid temperature, far stream temperature respectively. the terms νn f , µn f and ρn f are the nanofluid kinematic viscosity, fluid viscosity and density separately. also, dt , σ, db, βt,c, k, g, qa, cp, b0, α, r0 and n are correspondingly the thermophoretic diffusion, electric conductivity, brownian diffusion, thermal expansion, porosity, gravity, heat generation, heat capacity, magnetic strength, thermal conduction, chemical reaction and reaction index. meanwhile, α and δ are the powelleyring terms. the second term on the right hand side of equation (5) represent the approximated thermal rosseland radiation, where σs and k∗ are the stefan-boltzman and mean absorption terms. using the succeeding associated variables for the transformation of equations (3)-(7) θ(η) = t − t∞ t f − t∞ , φ(η) = c − c∞ cw − c∞ , η = ( a ν ) 1 2 y, u = ax f ′(η), v = − √ aν f (η),ψ(x, y) = (xν)1/2 f (η). (8) here, ψ is the stream function and its satisfied (u, v) = ( ∂ψ ∂y ,− ∂ψ ∂x ) . by applying the similarity variables of equation (8) on equations (3) to (7), equation (3) is spontaneously gratified while equations (4) to (7) gives (1 + �) f ′′′ + f f ′′ − ( f ′ )2 − �λ( f ′′ )2 f ′′′ −(m + p) f ′ + grθ + gcφ = 0, (9)( 1 + 4 3 ra ) θ ′′ + pr f θ ′ + prnbθ ′ φ ′ + brpr(m + p)( f ′ )2 + prnr(θ ′ )2 + prqθ = 0, (10) φ ′′ + le f φ ′ + nb nr θ ′′ − leλφn = 0. (11) with boundary conditions gives f (0) = 0, f ′ (0) = 1, θ ′ (0) = −bi(1 − θ(0)), φ(0) = 1, f ′ (∞) = 0, θ(∞) = 0, φ(∞) = 0. (12) where the terms � = 1 µαrδ and λ = a 3 x2 2α2r ν are the working fluid materials, m = σb20 ρn f a is magnetic field, p = νak is the porosity, gr = gβt (t f −t∞) a2 x and gc = gβc (c f −c∞) a2 x are the heat and species buoyancy number, nr = τdt (t f −t∞) t∞ν denotes thermophoretic, pr = ν α is prandtl number, ra = 4σs t 3 ∞ 3νkn f is the radiation, nb = τdb (c f −c∞) ν defines brownian motion, le = νdb connotes lewis number, q = q0a(ρc)n f the heat source, λ = rn (c f −c∞)n−1 a is chemical reaction and br = u 2 w c p(t f −t∞) denotes the heat term. the wall drag (c f ), heat gradient (nu) and mass gradient (s h) are the engineering thermophysical quantities of inquisitiveness which are given as: c f = gw ρu2w ; where gw = ( µ + 1 αrδ ) ∂u ∂y − 1 2α3r δ ( ∂u ∂y )3 ,(13) nu = aqw (t − t∞)kn f ; where qw = −kn f ( ∂t ∂y ) , (14) s h = as w (c − c∞)db ; where s w = db ( ∂c ∂y ) . (15) using the variables in equation (8) on equations (13)-(15), the equations reduces to dimensionless for as: c f = (1 + �) f ′′ (0) + �λ 3 ( f ′′ (0) )3 , nu = − ( 1 + 4 3 ra ) θ ′ (0), s h = −φ ′ (0). (16) 3. entropy generation a continuous irreversibility process is created in the flow of eyring-powell nanoliquid simulated by heat conduction, fluid 3 salawu et al. / j. nig. soc. phys. sci. 4 (2022) 924 4 friction and diffusive irreversibility, this process defined entropy generation. entropy is generated when there is an exchange of heat between device wall and the eyring-powell hydromagnetic nanoliquid. the entropy volumetric rate for the considered non-newtonian fluid with the impact of electric field, porosity, and magnetic field is defined according to [25-27]: eg = kn f t 2∞  ( ∂t ∂y )2 + 16σst 3∞ 3kn f ( ∂t ∂y )2 + µn f t∞  ( 1 + 1 ρn f αrδ ) ( ∂u ∂y )2 − 1 6ρn f αrδ ( ∂u ∂y )4 + σb20 t∞ u2 + ν kt∞ u2 + rd c0 ( ∂c ∂y )2 + rd t∞ ( ∂c ∂x ∂t ∂x + ∂c ∂y ∂t ∂y ) , (17) the characteristics of entropy production is denoted as eh = (kn f∇t )2 t 2∞a2 . (18) applying equation (8) on equation (17), the non-dimensional entropy generation takes the form nc = eg eh = ( 1 + 4 3 ra ) (θ ′ )2 + reϕγ ω ( ϕ ω (φ ′ )2 + φ ′ θ ′ ) + brre ω [ (1 + λ)( f ′′ )2 − λ� 3 ( f ′′ )4 + (m + p)( f ′ )2 ] , (19) where ω = ∆tt∞ , γ = c∞rd kn f , ϕ = ∆cc∞ , re = uw a2 ν . taken np = ( 1 + 43 ra ) ( θ ′ )2 and nq = brre ω [ (1 + λ)( f ′′ )2 − λ�3 ( f ′′ )4 + (m + p)( f ′ )2 ] + reϕγ ω ( ϕ ω (φ ′ )2 + φ ′ θ ′ ) , the bejan number (be) is expressed as be = ( 1 + 43 ra ) ( θ ′ )2 brre ω [ (1 + λ)( f ′′)2 − λ�3 ( f ′′)4 + (m + p)( f ′)2 ] +( 1 + 4 3 ra ) (θ ′ )2 + reϕγ ω ( ϕ ω (φ ′ )2 + φ ′ θ ′ ) . (20) here, np + nq = nc, and np represents the energy irreversibility, nq connotes irreversibility due to diffusion, conduction and joule heating. be defines the ratio of irreversibility. the irreversibility of nanoliquid powell-eyring is dominated with chemical diffusion, joule heat and heat transfer when be = 0 but when be = 1, the irreversibility of non-newtonian nanoliquid is controlled by thermal conducting of the fluid with changes in the temperature. figure 1. flow schematic diagram figure 2. flow rate field for various � figure 3. velocity field for different m 4 salawu et al. / j. nig. soc. phys. sci. 4 (2022) 924 5 figure 4. f (η) against η for various p figure 5. heat profile for variation of br figure 6. plot of θ(η) versus η for nr figure 7. temperature field for various q figure 8. rising ra effect on temperature figure 9. reaction field for rising le 5 salawu et al. / j. nig. soc. phys. sci. 4 (2022) 924 6 figure 10. effect of λ on concentration figure 11. φ(η) versus η for various nb figure 12. entropy profile for different figure 13. m impacts on entropy field figure 14. irreversibility for increasing br figure 15. impact of p on bejan number 6 salawu et al. / j. nig. soc. phys. sci. 4 (2022) 924 7 4. method of solution based on previous analysis, the following values � = 0.5, λ = 0.1, n = 1.0, λ = 1.0, p = 0.2, m = 0.5, pr = 0.72, ra = 0.5, nb = 0.5, nr = 0.5, le = 0.5, q = 0.5, gr = 3.0, gc = 3.0, br = 0.05, re = 1.0, ω = 0.5, ϕ = 0.2, γ = 0.3 are set as default. a numerical shooting techniques is carried out on the mhd eyring-powell nanoliquid boundary value equations. the numerical method allows the transformation of boundary value equation to an initial value equation. at far field (η −→ ∞), a finite appropriate boundary value is assumed. then the derivative of higher order is presented in first order derivative by assuming and substituting the following quantities in equations (9) to (12): a1 = f, a2 = f ′, a3 = f ′′, a4 = θ, a5 = θ ′, a6 = φ, a7 = φ ′, (21) a′3 = −a1a3 + (a2)2 + (m + p)a2 − gra4 − gca6 (1 + �) −λ�(a3)2 , (22) a′5 = −pra1a5 − prnba5a7 − brpr(m + p)(a2)2 1 + 43 ra − prnr(a5)2 + prqa4 1 + 43 ra , (23) a′7 = −lea1a7 − nb nr a′5 + leλ(a6) n, (24) the boundary conditions becomes a1(0) = 0, a2 = 1, a5(0) = −bi(1 − a4(0)), a6 = 1, a2(∞) = 0, a4(∞) = 0, a6(∞) = 0. (25) the reduced initial value equations (21) to (24) are to be solved. the unknown points a2(0) = r1, a5(0) = r2 and a7(0) = r3 are needed for computation. therefore, the unknown values are initially guessed to allow computation which is carried out using integrating runge-kutta procedures with step size ∇η = 0.0001. the same solution procedures is used for the physical quantities and the entropy generation. 5. results and discussion the comparison of computed outcomes for the heat gradient and plate drag force is presented in tables 1 and 2. the comparison was done with relevant previous related studies in order to confirm the consistency and exactness of the present computed results. table 1 denotes the compared results for heat gradient with the work of tawade et al. [4], ishaq et al. [11] and salawu and ogunseye [29]. also, table 2 represents the skin friction numerical results as compared with the previous study of hayat et al. [10] and ogunseye et al. [30]. as seen from the tables, the computational values agreed well with about 10−6 relative error. the computational values for the thermophyscial quantities of engineering interest are presented in table 3. keeping other figure 16. m effects on bejan number figure 17. irreversibility ratio field for br parameters fixed, the results demonstrated the effect of some parameters on the stretching wall. as obtained from the table, some terms have an increasing effect on the wall while some have a reducing impact on the stretching wall. the computed values depends on the momentum, thermal and reaction species boundary layers. th reaction of eyring-powell nanoliquid velocity in a vertical plate to the rising values of material term �, magnetic term m, and porosity term p are respectively demonstrated in figures 2, 3 and 4. the dimensionless velocity f ′(η) is plotted against the dimensionless distance η as seen in the figures. with fixed values of other terms, the parameters �, m and p show a decreasing effect on the viscoelastic nanoliquid flow rate near the stretching boundary wall. the reducing effect occurs close to the stretching vertical plate due to a rise in the molecular bonding force and lower internal heating. however, the parameters influence on the fluid velocity decreases steadily away from the plate. as the non-newtonian fluid flows far from the plate, the heat produced in the chemical reaction system upsurges, due to an augmentation in the heat generation and ohmic heating influencing the reaction rate that in turn reduced the viscoelastic fluid bonding force. therefore, the flow rate is progressively expanded far the velocity field and converged far away the stream. 7 salawu et al. / j. nig. soc. phys. sci. 4 (2022) 924 8 table 1. results comparison for the heat gradient. λ pr nr [4] [11] [30] new results 0.2 1.0 0.1 0.3238 0.32102 0.32186 0.32186 0.3 0.3020 0.30042 0.30071 0.30071 0.4 0.2925 0.29142 0.29092 0.29092 0.5 0.3717 0.37142 0.37129 0.37129 1.5 0.1819 0.18229 0.18210 0.18210 0.4 0.6117 0.61243 0.61238 0.61238 0.6 0.6023 0.69143 0.70016 0.70016 0.8 0.5013 0.50099 0.50083 0.50083 table 2. results comparison for the skin friction. nr nb [10] [31] present results 0.5 1.21380 1.21380229 1.21380217 0.2 1.24143 1.24142703 1.24142698 0.5 1.37544 1.37544311 1.37544302 0.0 1.79329 1.79328569 1.79328551 0.2 1.0 1.25874 1.25873960 1.25873893 1.5 1.26916 1.26016234 1.26016222 table 3. numerical values for the skin friction (c f ), nussetl number (nu) and sherwood number (s h) m λ � ra le nr nb c f nu s h 0.5 0.1 0.5 0.5 0.5 0.5 0.5 0.987613128 0.0567754810 0.937155927 0.7 0.937510305 0.0531214474 0.943378662 1.5 0.763440580 0.0377939599 0.970428985 0.3 1.008526464 0.0567917496 0.937253896 0.7 1.061377860 0.0568290633 0.937472853 1.0 0.759445493 0.0564705972 0.934027849 1.5 0.610097045 0.0562309677 0.931818099 0.03 0.922979767 0.0603941793 0.941008972 0.07 0.927870854 0.0601463768 0.940178666 0.7 0.917559030 0.0532639565 1.108694229 1.0 0.843402030 0.0497823360 1.320808092 1.5 1.069353500 0.0495253726 0.919047457 2.5 1.184217940 0.0407985312 0.920494831 1.0 1.037237502 0.0542505586 0.998303261 2.0 1.124838686 0.0498904892 1.170942782 hence, the flow rate profiles for the considered fluid rises in the neighboring of the moving plate, but in opposite direction faroff the vertical surface. the effect of some thermophysical terms on the energy distribution is confirmed in figures 5, 6, 7 and 8. the impact of heat dissipation br, thermophoretic nb, heat source q and radiation ra terms on the temperature distribution are separately presented in the plots. at different significance, the parameters vigorously influenced the thermal dispersion rate in the porous device. the strong heat encouragement is due to rises in the chemical reaction rate that leads to random motion and rapid collision of the fluid particles. also, as heat source parameters is boosted, small or no thermal propagation of chemically non-newtonian viscoplastic mixtures out of the system to the ambient. this is as a result of thermal boundary film is inspired, which in turn damped the heat transport in permeable media. in the temperature equation, the thermal source are propelled to enhance thermal transport in the eyring-powell chemical nanoliquid mixture, which leads to a complete increase in the temperature profiles. thus, the parameters br, nb, q and ra inspires the temperature propagation in the moving vertical plate as depicted in figures 5, 6, 7 and 8. concentration profiles for various effects of some parameters are established in figures 9, 10 and 11. the figures depict the plot of φ(η) versus distance η −→∞ in a boundless domain. in figures 9 and 10, the response of the species concentration to variation in the lewis number le and chemical reaction λ are illustrated. as obtained in the figures, both terms of equal variational values decrease the concentration field at different significant. the reducing impact is momentously noticeable in the lewis number le than chemical reaction λ. the diminishing in the profiles is due to lower chemical reaction rate in the 8 salawu et al. / j. nig. soc. phys. sci. 4 (2022) 924 9 system that causes the species particles to react slowly and discouraged. hence, increasing the values of both terms enhances nanoliquid chemical diffusion as the concentration boundary layer shrinks. figure 11 depicts the impact of brownian movement nr on the reacting species profile. rising values of the brownian movement term increases the reaction rate and rapid random motion of the chemical particles, this then causes a rise in the reaction mass field as presented in the figure. the respective influence of variation in the thermophysical terms porosity p, magnetic m and heat dissipation br on the entropy generation is described in figures 12, 13 and 14. in a process where irreversibility is in existence, entropy generation is defined. this measure the amount of energy loss that causes degradation of technology system performance. as seen, the parameters encourages system irreversibility due to high energy dissipation near the stretching plate as the values of the parameters is raised. however, the irreversibility due chemical diffusive, friction and conducting heat decrease regularly few distance from the moving surface because the fluid thermodynamic equilibrium is rising gradually and continuously toward the free field. the decrease in the entropy generation continues until a balance thermodynamic equilibrium is attained and energy dissipation tends to zero. hence, irreversibility reduces far the field as portrayed in figures 12, 13 and 14. the effect of porosity p, magnetic m and heat dissipation br on the irreversibility ratio is respectively presented in figures 15, 16 and 17. bejan number is the irreversibility heat transfer ratio to the sum of the irreversibility of fluid friction and heat transfer. the irreversibility ratio rises all over the considered system domain as demonstrated in figure 15 and 16. the overall rising in the magnitude of the bejan number is because of the increasing rate in the heat transfer irreversibility in the non-newtonian viscoelastic fluid regime. meanwhile, heat dissipation term br completely reduced the irreversibility ratio of the chemical diffusive system. the significant impact is very obvious within the chemical reaction due to strong fluid frictional effect that dominate the reactive system. therefore, as depicted in figure 17, the bejan number decreases momentously over the domain. 6. conclusion in the study, the entropy generation of eyring-powell chemical reaction nanoliquid was considered in a porous vertical boundless device with heat distribution. a shooting numerical technique is adopted for the solutions coupled with the rungekutta scheme due to its convergence, consistency, and unconditional stability. both quantitative and qualitative computed values were obtained in agreement 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[31] h. a. ogunseye, s.o. salawu, y.o. tijani, m. riliwan & p. sibanda, “dynamical analysis of hydromagnetic brownian and thermophoresis effects of squeezing eyring-powell nanofluid flow with variable thermal conductivity and chemical reaction”, multidiscipline modeling in materials and structures 15 (2019) 1100. 10 j. nig. soc. phys. sci. 4 (2023) 951 journal of the nigerian society of physical sciences safranin o dye removal using senna fistula activated biomass: kinetic, equilibrium and thermodynamic studies c. j. ajaelu∗, o. oyedele, a. a. ikotun, e. o. faboro industrial chemistry programme, college of agriculture, engineering and science, bowen university, iwo, nigeria abstract the availability of potable water has decreased in recent times due to the extensive discharge of effluents from some industries. this contaminated water poses a great danger to both human and aquatic life. senna fistula was activated using phosphoric acid, h3po4 and its ability to remove safranin o from aqueous solution was investigated. the characterization of senna fistula activated carbon was done by scanning electron microscopy and fourier transform infrared spectroscopy. the impacts of ph, initial dye concentration, contact time, and effect of temperature were investigated. results showed that the optimum ph for the removal of safranin o was 4.4. the adsorption capacity increased as the initial dye concentration increased from 30 130 mg/l. the dye adsorption equilibrium data were properly fitted to both freundlich and langmuir isotherms. the maximum uptake capacity for safranin o was 22.1 mg/g. the kinetic studies indicated rapid sorption dynamics via a second-order kinetic model. the thermodynamic parameter shows that the sorption of safranin o on senna fistula activated carbon was feasible, spontaneous and endothermic. senna fistula-activated carbon was found to be cheap and efficient adsorbents for the removal of safranin o dye from aqueous solutions. doi:10.46481/jnsps.2022.951 keywords: senna fistula, safranin o, adsorbent, kinetics, thermodynamics article history : received: 20 july 2022 received in revised form: 21 august 2022 accepted for publication: 07 november 2022 published: 22 december 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: n. a. a. babarinde 1. introduction the surge in deleterious dye wastewater emanating from industries is of alarming public health and environmental safety concerns. around 700,000 tons of dye are produced worldwide each year, which results in 100,000 dyes being sold in the market [1, 2]. concerns have been raised by the united nations over the fact that more than 80 % of the world’s wastewater, including more than 95 % in some of the world’s ∗corresponding author tel. no: +2348034994404. email address: ajaelujohn46@gmail.com (c. j. ajaelu ) poorest countries, is released into the environment without first being treated [3]. highly polluted effluents are produced because of the considerable amount of water employed in the dyeing processes [1]. it has been observed that between 10 and 15 % of dyes meant for fibers could not adhere to them and are ultimately found as industrial effluents. these dyes infiltrate water bodies, causing colouration, a reduction in photosynthesis and dissolved oxygen, and suppression of aquatic plant growth. carcinogenesis, mutagenesis, teratogenicity, chromosomal fractures, kidney, liver and reproductive system dysfunctions, 1 ajaelu et al. / j. nig. soc. phys. sci. 4 (2023) 951 2 watery eyes, itching, and asthma symptoms are all effects of water pollution in humans [4, 5]. safranin o is a basic dye commonly referred to as basic red 2. safranins are symmetrical 3,7-diamino2,8-dimethyl-5phenylphenazin-5-ium chloride’s azonium compounds. basic dyes are the most vibrant type of soluble dyes used in the textile industry to color acrylic, nylon, silk, and wood. they have a high tinctorial value; as low as 1 ppm of dye generates noticeable colour. due to its disposal in water bodies, it poses a threat [6]. safranin o dye causes skin irritation, eye irritation, is carcinogenic and can impact blood factors such as coagulation, induced somnolence, respiratory issues and significant alimentary tract irritations. in order to address this global challenge of dye pollution, different dye removal techniques have been adopted by researchers. these include photocatalysis, ion exchange, electrochemical, and oxidation-reduction. these methods have high operational costs and are time-consuming. most of these methods have deficiencies such as the high energy and reagent requirements, the low selectivity and the difficulty in removing the secondary waste generated [3]. a cheaper and more efficient technique for dye removal is adsorption. despite the fact that adsorption is a cheaper method than other methods earlier mentioned, the cost-effectiveness of the adsorption method also depends on the ease of availability of the adsorbent. some adsorbents are waste materials while others are purchased. the use of agricultural waste material (biosorbent) is the cheapest form of adsorbent, for it has zero cost to acquire. some of the biosorbents used to remove dyes include coconut husk [7] african border tree [8], oil palm fruit [9, 10], watermelon rinds [11], okra [12], maize stuck, teak leaf [13], sugarcane bargass [14]. senna fistula, also known as the golden shower tree, produces pods as its ripe fruits. senna fistula is a medium-sized tree native to the indian subcontinent and also found in the southern region of nigeria. it is frequently grown as an ornamental plant. ripe pods are long, virtually straight, cylindrical, dark chocolate-brown fruits with a length of 45 to 60 cm and a thickness of 20-25 mm. the surface seems smooth and shiny to the human eye but is characterized by minute transverse striations under a microscope. this study is targeted at converting senna fistula pod into activated carbon and investigating the removal of safranin o dye from aqueous solutions. contact time, temperature, adsorbent dosage, and initial safranin o concentration were all evaluated as operational parameters. different isotherm and kinetic models were used to evaluate the experimental data. thermodynamic parameters that govern the adsorption process were also investigated. 2. materials and methods 2.1. raw materials/chemicals safranin o was obtained from kem light laboratories pvt. ltd, india. the structure and the physical characteristics of safranin o are presented in figure 1 and table 1, respectively. naoh (99.0 %) was obtained from merck, germany, nacl (99.5 100 %) from merck, germany, and hcl (35.4 %) from bdh. the chemicals were of analytical quality. therefore, further purifications were not carried out on them. 2.2. preparation of adsorbate the adsorbate was prepared by dissolving 1 g of the safranin o dye (c20h19n4cl, molecular weight 350.8 g/mol) in little quantity of distilled water in a 1 l volumetric flask. it was properly dissolved and made up to the mark with distilled water. serial dilutions were carried out to prepare various concentrations needed for the adsorption studies. figure 1: the molecular structure of safranin o table 1: physical characteristics of safranin o iupac name/ molecular formula 3,7-diamino-2,8-dimethyl-5phenylphenazin-5-ium chloride/ c20h19cln4 common name safranin o, basic red 2 cas number 477 73 6 molar mass 350.85 g/mol appearance dark red storage temperature room temperature εmax 1250-1650 at 530-534 nm in 50 % ethanol solubility in water water, 1 mg/ml, clear, dark red, red purple 2 ajaelu et al. / j. nig. soc. phys. sci. 4 (2023) 951 3 2.3. adsorbent preparation the pods were washed properly and then ovendried after which the pods were split and the inside were cleaned thoroughly to get rid of unwanted materials like the seeds. as a result of drying, the pods became very hard and had to be crushed to achieve smaller particle size conducive for grinding after which the crushed pods were then ground to powder. the resultant powder was passed through a sieve of mesh size 1.0 mm after which it was stored in a plastic bag for the next stage of the process. 50 g of the powdered adsorbent was measured and gently stirred into a solution of 1.0 m phosphoric acid (h3po4) and kept for a day in a cool dry place. this was to give the adsorbent ample time to soak up the acid. the mixture was rinsed with distilled water and dried in an oven at 105 ◦c. it was then transferred into crucibles and carbonized in a muffle furnace at 350 ◦c in the absence of air for 45 minutes. the carbonized sample was cooled in a desiccator and then rinsed with distilled water to remove ash. it was further washed to ensure a nearneutral ph of 6.89. it was dried in an oven at 105 ◦c until a constant weight was achieved after which the modified senna fistula pods activated carbon msf was stored in a glass bottle. 2.4. adsorption studies in this experiment, the adsorbate was safranin o dye. for the stock solution, 1 g of safranin o was dissolved with distilled water in a 1 l volumetric flask and made up to mark. lower concentrations were made by repeated dilution from the stock solution. the effectiveness of msf in removing safranin o was measured using four different operating parameters, including ph, initial so concentration, contact time, and solution temperature. in this study, 0.1 g of msf was agitated with 20 ml of varying concentrations (30, 50, 70 and 90 mg/l) of so solutions and agitated at 200 rpm in an electric shaker. a uv spectrophotometer (shimadzu 1800 double beam) at 520 nm was used to measure the absorbance of aliquot samples at regular intervals. equation (1) was used to calculate % removal and adsorption capacity. percentage uptake capacity = co − ce co × 100 (1) qe = (co − ce) v w (2) where qe (mg/g) is the amount so adsorbed, co (mgl−1) and ce (mgl−1) are the initial and final concentrations of so respectively. v(l) and w(g) represent the volume of so and the weight of senna fistula, respectively. 2.5. characterization of senna fistula a scanning electron microscope (zeiss auriga hrsem) was used to characterize the surface structure of msf. as a thin beam of electrons scans the surface of the adsorbent in a rectangular raster, sem measures the intensity of numerous signals created by interactions between the beam electrons and the adsorbent. in order to get a magnified image of the adsorbent, the surface property, components, and scenery of the adsorbent are determined from the signals provided by sem [15]. senna fistula spectra were recorded using an agilent technologies cary 630 fourier transform infrared (ftir) spectrometer. between 4000 and 600 cm−1, spectroscopic measurements were taken. spectroscopic analysis was used to analyze the surface chemistry of senna fistula powder before and after copper adsorption. before and after copper adsorption, the ftir spectra indicated the functional group(s) on the senna fistula surfaces. 3. results and discussions 3.1. characterization senna fistula morphology was determined using a scanning electron microscope as shown in figure 2. the phosphatemodified senna fistula shows a rough, irregular, heterogeneous honeycomb-like surface structure with several pores and cavities which is due to the phosphate modification. the adsorption of so resulted in less number of pores on the absorbent surface since quite a number of the pores have been occupied by the dye. the ftir spectra of phosphate-modified senna fistula with safranin o (msf-so) and the phosphate-modified senna fistula (msf) without safranin o were compared in figure 3 and table 2 to identify functional groups through characteristic vibrational frequencies. phosphate-modified senna fistula with so showed a broad and medium band around 3327 cm−1, which is owing to an (oh) vibration band of a hydrogen bond due to the phosphoric acid, which is absent in msf. the (ch) vibrational bands are responsible for the crisp and strong bands that occur at 2982 cm−1 to 2881 cm−1. after so adsorption, these bands remained nearly constant at 2922 cm−1 and 2853 cm−1. (c=o) stretching vibration band coming from the carboxylic acid is responsible for the crisp and medium band at 1702 cm−1. after so dye adsorption, this band has shifted to a higher frequency of 1744 cm−1 and also appeared as a strong band. this denotes the chemical interaction of the oxygen atom of the c=o bond thereby forming new bonds during adsorption [16]. there is a c=n vibrational band appearing as a sharp medium band at 1593 cm−1 before so adsorption. this band has almost remained unchanged at 1595 cm−1 after so adsorption, thus signifying no chemical interaction of the nitrogen from the cn group [17, 18]. the (c=c) stretching vibrations are responsible for the emergence of a new sharp and medium band at 1685 cm−1 in msf with so dye but absent in msf without so. the p-o-cstretching vibration is responsible for the bands at 1112.6 cm−1 and 980.6 cm−1. after so adsorption, these bands changed to higher frequencies of 1157 cm−1 and 1057 cm−1, respectively. 3 ajaelu et al. / j. nig. soc. phys. sci. 4 (2023) 951 4 figure 2: typical sem of modified senna fistula (a) without and (b) with safranin o. table 2: infrared spectral data of msf preand postsorption of so dye adsorbent v(oh) (cm−1) v(ch) (cm−1) v(c=o) (cm−1) v(c=n) (cm−1) v(c=c) (cm−1) v(c−o) (cm−1) v(p-o-c) (cm−1) modified senna fistula (msf) 2982 b,m 2881 b,m 1702 m 1593 m 1165s 1034s 1113s 980s modified senna fistula (msf) after so adsorption 3326.6s 1744 s 2853 s 1744 s 1595 m 1685 m 1157s 1027m 1157s 1057m 3.2. effect of ph the role played by ph in the adsorption of dye molecules is quite revealing. the ph regulates the competition between the dye molecules and hydrogen ions on the active sites of the sorbent surface [19]. the ph of the solution affects the adsorbent’s surface charge and the ionization of the adsorbate [20]. additionally, the effectiveness of adsorption in aqueous solutions is influenced by hydrogen bonds, π − π interaction, and n − π interaction [21, 22, 23]. figure 4 demonstrates that when ph rises from 1 to 4, safranin o adsorption capacity increases significantly. this is due to the availability of a large active site. at lower ph 1, there is competition for the available site between the hydrogen ion and the positive charge dye molecule. this leads to lower adsorption. as the ph rises, the competition between the two cations reduces, and more sites are made available for the dye molecule to adsorb on the surface of msf. at ph greater than 4 the adsorption reaches equilibrium. 3.3. initial dye concentration the initial dye concentration has a significant influence on the adsorption capacity of the adsorbent. figure 5 shows that the adsorption capacity increases as the initial safranin o concentration increases from 30 -130 mg/l. this increase is due to a large number of available adsorption sites [24, 25]. in addition, the mass transfer rate of safranin o from the bulk solution to the adsorbent surface is high. the interaction between so and the adsorbent, msf, is also improved by increasing the initial dye concentration. thus, increasing the initial concentration of so improves its adsorption. this demonstrates that the relationship between the adsorption capacity and the starting so concentration is linear. a similar increase was reported by rassaq et al. 2021 [26]. 3.4. effect of contact time the time required for equilibrium to be attained is very vital as it determines the effectiveness of the adsorption process as well as the cost of the wastewater treatment. the effect of contact time on the adsorption of safranin o dye is reflected in figure 6. the figure demonstrates that for a dye concentration of 30 mg/l, a sharp increase in contact time for the first 10 minutes significantly improved so’s adsorption ability, connoting that so’s initial rate of adsorption by the msf was very fast. this is due to the functional adsorbent’s enormous number of accessible sites, which is responsible for the increased adsorption. in addition, there was an increase in the collision rate of dye molecules. there was a gradual increase from 10 25 min after which equilibrium was reached. this was a result of a drastic reduction in the number of available active sites as well as a low rate of collision of dye molecules. 4 ajaelu et al. / j. nig. soc. phys. sci. 4 (2023) 951 5 figure 3: the ftir of modified senna fistula (a) without and (b) with safranin o. figure 4: ph effect on the sorption of safranin o on modified senna fistula. 3.5. adsorption kinetics studying the kinetics of the adsorption process can help in comprehending the mechanism and phases involved in the trapping of adsorbate molecules onto the adsorbent. to learn more figure 5: initial dye effect on the sorption of safranin o on modified senna fistula about the sorption mechanism, two kinetic models, pseudo-first and pseudo-second order, were fitted to the experimental data. 5 ajaelu et al. / j. nig. soc. phys. sci. 4 (2023) 951 6 figure 6: contact time effect on the sorption of safranin o on modified senna fistula. the lagergren pseudofirst-order kinetic equation is given by log (qe − qt) = log qe − k1t 2.303 , (3) where k1 is a pseudo-first-order reaction’s rate constant, and qt is the amount of dye adsorbed per unit of adsorbent (mg g−1) at time t (h−1). the adsorption rate constant (k1) was calculated using the log(qe−qt) plot against time as presented in figure 7a. the pseudo-second-order kinetic was introduced by ho and mckay [27] as: t qt = 1 k2q2e + 1 qe t, (4) where k2 (g mg−1 h−1) is the pseudo-second-order rate constant. a linear plot of t/qt vs t yields the qe and k2 as reflected in figure 7b. the pseudo-second-order kinetic model is the one that best fits the experimental data since its correlation factor is close to 1 and its calculated adsorption capacity is of close proximity to the experimental adsorption capacity as presented in table 3. 3.6. effect of temperature temperature is another significant factor that directly impacts the adsorption of dyes and has an impact on the solidsolute interface and the mobility of the pollutants during adsorption. in this study, adsorption capacity increases with an increase in temperature from 308 to 318 k (figure 8). the increase was relatively sharp from 303 318 k and then reached equilibrium by leveling off from 318 338 k. the increase in temperature increases the dye mobility in the solution and consequently increases the probability of interaction between the adsorbate and adsorbent. this investigation shows that so adsorption is an endothermic process and favours high temperatures. 3.7. adsorption isotherm isotherm study gives details about the concentration of so adsorbed on msf surface by correlation of experimental data to three equilibrium models namely langmuir (figure 9), freundlich (figure 10) and temkin (figure 11). 3.7.1. langmuir isotherm according to the langmuir isotherm model, monolayer sorption occurs on a homogenous surface where there is no interaction between the molecules that are adsorbed. the model also assumes homogeneous energies for adsorption onto the surface [28, 29, 30]. the general and linear langmuir equations are represented by equation (5) and (6), respectively. qe = x0bce 1 + bce (5) ce qe = 1 x0b + 1 x0 ce. (6) the dimensionless separation factor is given by: rl = 1 1 + bc0 (7) x0 s the maximum uptake capacity in mg/g while b is langmuir constant. table 4 shows that the maximum uptake capacity is 22.1 mg/g while the dimensionless separation factor is much less than 1 indicating that the adsorption of so unto msf is feasible and favorable. 3.7.2. freundlich isotherm the freundlich isotherm is a nonlinear sorption model that describes multilayer adsorption with a heterogeneous energy distribution of active sites and interactions between adsorbed molecules [31]. the non-linear and the linear equations are presented in equations (8) and (9), respectively. qe = kf c 1 n e (8) log qe = log kf + 1 n log ce, (9) where kf and n are freundlich constants related to adsorption capacity and adsorption intensity, respectively. figure 10 presents the graph of log qe against log ce. the n value is greater than 1 which indicates that the physical adsorption process is feasible and favourable. 3.7.3. temkin model temkin’s isotherm model presupposes a linear decrease in all molecules’ heat of adsorption with increasing adsorbent surface coverage. temkin’s model general and linear forms are presented in equations (10) and (11), respectively. qe = rt σ ln ωce (10) qe = β ln ω + β ln ce, (11) where β = rt σ . (12) σ ( jmol−1 ) is the temkin binding constant which is correlated with the heat of adsorptions, ω is the equilibrium binding constant (lmg−1), t is thermodynamic temperature and r 6 ajaelu et al. / j. nig. soc. phys. sci. 4 (2023) 951 7 figure 7: kinetic plots showing (a) pseudo-first and (b) pseudo-second order for copper adsorption onto modified senna fistula. table 3: kinetic adsorption parameters for the adsorption of so on msf first order kinetics second order kinetics conc (mg/l) qe (calc) qe (exp) k1 r2 qe (calc) k2 r2 30 1.03 10.8 0.0016 0.9404 11.4 0.182 0.9984 50 1.2 18.2 0.0007 0.9143 19.4 0.186 0.9995 70 1.39 25.5 0.0004 0.8907 27.4 0.187 0.9997 90 1.36 32.9 0.0121 0.8011 35.3 0.188 0.9998 figure 8: effect of temperature on the adsorption of safranin o on modified senna fistula. figure 9: langmuir isotherms for the adsorption of safranin o onto senna fistula table 4: langmuir, freundlich and temkin isotherms for the adsorption of safranin o on senna fistula langmuir freundlich temkin x0 = 22.1 mg/g kf = 1.69 σ = 13657.1 c = 0.677 n = 1.33 ω = 9141.1 rl = 0.047 r2 = 0.99 r2 = 0.92 r2 = 0.987 figure 10: freundlich isotherm for the adsorption of so on modified senna fistula is molar gas constant (0.008314 kj mol−1 k−1). the temkin correlation factor (figure 11, table 4) is low compared to other adsorption isotherms, hence it is not the most appropriate for explaining the adsorption process. adsorption equilibrium isotherm demonstrates that both the freundlich and langmuir isotherms are the most suitable for describing the adsorption process. 7 ajaelu et al. / j. nig. soc. phys. sci. 4 (2023) 951 8 figure 11: temkin isotherm for the adsorption of so on modified senna fistula figure 12: thermodynamic effect of the adsorption of dye on msf 3.8. thermodynamics thermodynamic parameters such as the entropy change ∆s , enthalpy change ∆h and the free energy ∆g change were determined at different temperatures (303, 308, 313 and 318 k) using the following equations: ∆g = −rt ln k (13) ln k = − ∆g rt = ∆s r − ∆h rt (14) ln k2 = ln a − ea rt (15) k2 is the rate constant for a second-order adsorption kinetic (g/mgh), a is the arrhenius factor, ea is the activation energy (kj/mol) that must be overcome for adsorption to take place, r is the gas constant (j/molk), t is the thermodynamic temperature (k). the calculated values of ∆s , ∆h and ∆g are vital thermodynamic parameters in the adsorption process. the intercept and slope of the van’t hoff plot of ln k against 1/t , as shown in figure 12, yielded the values of ∆s , ∆h, which were consistent table 5: thermodynamic parameters for the adsorption of so on modified senna fistula adsorbent ∆h(kj/mol) ∆s (j/mol) ∆g (kj/mol) ea (j/mol) 303 k 308 k 313 k 323 k msf 108.5 369.6 -5.37 -9.06 -12.7 -16.5 28.3 table 6: comparisons of the adsorption capacities of diverse adsorbents used for so removal adsorbent dye maximum adsorption capacity (mg/g) reference nano iron oxide so 1.91 [36] mango seed integument so 34.5 [37] kaolinite clay so 16.2 [38] soybean hull so 23.8 [39] senna fistula-activated carbon so 22.1 this study with our previous research on safranin o dye adsorption employing the african border tree [8]. table 5 presents the thermodynamic parameters for the adsorption of so on modified senna fistula. the entropy change (∆s = 369.6 j/mol) is positive which implies the affinity of msf for the so dye as well as enhanced randomness at the solid-solution interface during the adsorption of so dye onto the surface of the active sites of msf. this results in higher uptake of so on msf. the enthalpy change is positive (∆h = 108.5 kj/mol), which is endothermic, which indicates that heat is absorbed during the adsorption process. the free energy change values ∆g = −5.37, -9.06, 12.7, and -16.5 kj/mol at 303, 308, 313, and 318, respectively are negative which connotes the spontaneity and the feasibility of the adsorption of so on msf. moreover, ∆g became more negative as temperature increased, indicating that higher temperatures actively support the feasibility and spontaneity of the adsorption process. these observations are incongruent with other findings described in several studies [7, 15, 35]. the energy of activation ea, was determined using equation (15). the value of ea (0.0283kj/mol) reveals that adsorption is a physical process [32, 33, 34, 35]. the adsorption capacities of various adsorbent for safranin o is presented in table 6. 4. conclusion in this study we determined the effectiveness of phosphate modified senna fistula in the adsorption of safranin o dye from aqueous solution. the adsorbent was prepared and applied for the adsorption of safranin o dye from aqueous solution as a function of ph, initial dye concentration, adsorbent dose, contact time and temperature. the maximum uptake capacity was 22.1 mg/g at an optimum ph of 4.4. the contact time demonstrates a sharp increase in adsorption capacity at less than 20 min of contact. the equilibrium adsorption process demonstrates that both freundlich and langmuir models fit appropriately the sorption model data (r2 = 0.99 and r2 = 0.988, respectively). the kinetic process followed a pseudo-second order. the negative value of ∆g is an indication that the activity is feasible and spontaneous. the positive value of ∆h revealed that the adsorption process involves the absorption of heat energy. the positive value of ∆s means that there is increased entropy between the so dye and the msf adsorbent. 8 ajaelu et al. / j. nig. soc. phys. sci. 4 (2023) 951 9 this study demonstrates that senna fistula (a waste), which is readily available, can be converted to a cost-effective and efficient adsorbent for removing so from aqueous solutions. references [1] m benjelloun, y miyah, g. a zerrouq, & s. lairini, “recent advances in adsorption kinetic models: their application to dye types”, arabian journal of chemistry 14 (2021) 103031. 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[39] v. chandane, v.k. singh, “adsorption of safranin dye from aqueous solutions using a low-cost agro-waste material soybean hull”, desalination and water treatment 57 (2016) 4122. 9 j. nig. soc. phys. sci. 4 (2022) 957 journal of the nigerian society of physical sciences the use of differential forms to linearize a class of geodesic equations j. m. orverema,b,∗, y. harunaa, b. m. abdulhamida, m. y. adamua adepartment of mathematical sciences, abubakar tafawa balewa university bauchi, bauchi state, nigeria bdepartment of mathematical sciences, federal university dutsin-ma, katsina state, nigeria abstract lie was the first to consider linearization of differential equations many years ago. since then, a great deal of research has been done on linearization of differential equations using various methodologies. surprisingly, there has not been much progress in linearizing geodesic differential equations. in particular, the use of differential forms to linearize a class of geodesic equations is not documented in the literature. differential forms are used to linearize a class of geodesic differential equations in this research. geodesics on a plane, geodesics on a cone, and geodesics on a sphere are examples. the solutions to these equations were discovered during the linearization process, as the findings of this study are distinctive, innovative, and original. doi:10.46481/jnsps.2022.957 keywords: differential forms, linearization, geodesics equations, ordinary differential equations, second order article history : received: 27 july 2022 received in revised form: 31 august 2022 accepted for publication: 01 september 2022 published: 10 october 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: j. ndam 1. introduction a geodesic is a curve that minimizes length locally. it is, in other words, a path that a particle that is not accelerating would take. geodesics are straight lines in the plane. geodesics are great circles on the sphere (like the equator). the term geodesic refers to the curve that would be formed if you continued on a straight path. you could travel a great circle on the surface of a sphere (think of the earth) if you kept traveling straight without turning left or right. geodesics are the shortest distance curves on a surface. they are useful in the transportation of products and persons at low cost of time and energy, in addition ∗corresponding author tel. no: email address: orveremjoel@yahoo.com (j. m. orverem) to their intrinsic interest. they are also critical as emergency exit routes during flights. the methods of differential geometry can be used to locate geodesics. the equator and the other great circles on a sphere are common examples. in a curved space, a geodesic is the straightest path conceivable. a straight line is what we call space that is flat. when you try to move straight in a curved space, you get geodesics in general. for example, while you are driving on the highway (and the road appears to be quite straight to you), you are not driving on a straight line; instead, you are driving on the straightest line conceivable on the earth’s curved surface. ask what curve within the surface connects two neighboring places on the surface and has the smallest length. a geodesic curve is a curve that meets the condition. in a curved space, though, it is just the opposite of straight. it’s a mathematical extremum, 1 orverem et al. / j. nig. soc. phys. sci. 4 (2022) 957 2 which means that any slight divergence from it will make it longer. the geodesic between two places on the earth’s surface is thus a great-circle path; nevertheless, the earth’s surface is curved locally as well as globally, that is, it has mountains and valleys. as a result, more than one geodesic can exist between two places. although they are not always the same length (though they can be), they are both extrema. analysis of the geodesics of the einstein-maxwell-dilaton theory’s exact solution, the four-dimensional linear dilaton black hole (ldbh) spacetime was examined in [1]. the conventional lagrangian approach is used to investigate the test particles’ geodesic movements. they demonstrated that exact analytical solutions to the radial and angular geodesic equations may be achieved after obtaining the euler-lagrange equations. in particular, it is demonstrated that one of the radial trajectory solutions may be expressed in terms of the weierstrass p-function, an elliptic-type special function. in another development as can be seen in [2], some partial differential equations which were derived from the well-known two logistic distribution parameters were solved using two alternative techniques. the first approach was the conventional one, which required the solution of three partial differential equations. the well-known darboux theory was the second strategy. it was discovered that, the geodesic equations are two minimum or isotropic curves. both approaches produced the same outcome, as was to be expected. in the article [3], the differential pursuit game problem was examined, in which a finite number of pursuers chase a finite number of evaders. the problem is expressed in a hilbert space l2 with the motions of the pursuer and evader being defined by nth and mth order differential equations, respectively. integral and geometric restrictions are placed on the control mechanisms used by the evader and pursuers, respectively. the lie subalgebras of noether symmetries associated with systems of geodesic equations are found to have one-dimensional optimum systems [4]. they also discovered invariants for every component of the deduced optimal system. it was demonstrated that the resulting invariants can transform systems of geodesic equations—nonlinear systems of quadratically semi-linear secondorder ordinary differential equations into nonlinear systems of first-order odes. the article [5] demonstrate a theorem that connects the metric collineations and the lie symmetries of the geodesic equations in a riemannian space. to einstein spaces and spaces with constant curvature, the findings were applied. the utilization of the results is then demonstrated using examples. in [6], group hydrodynamical systems with right invariant l2 or h1 metrics that may be expressed as geodesic equations on diffeomorphism groups or on extensions of diffeomorphism groups was considered. the numerical solution of geodesic equations was investigated in [7] where the initial value problem on an oblate spheroid, the direct geodesic problem was solved numerically using both geodesic and cartesian coordinates. in that study, the differential geometry theory is used to formulate the geodesic equations. the initial value problem in question is reduced to a system of first-order ordinary differential equations that is solved numerically. not so much is done in the area of linearizing geodesics equations. the relationship between isometries and symmetries of the system of geodesic equations was used in [8] to construct criteria for second order quadratically and cubically semi-linear equations and systems of equations. the geodesic deviation jacobi equation that addresses finite size effects caused by gravitational tidal forces was considered in [9]. it was shown how the jacobi problem in any spacetime that admits entirely geodesics that can be integrated can be solved. invariant wronskians for the jacobi system that are linear in the ’deviation momenta’ were derived by linearizing the geodesic equation and its conserved charges, resulting in a set of integrated first-order differential equations. the continuous hybrid numerical approach is taken into consideration in the work of [10] to solve second order initial value problems of ordinary differential equations in general. utilizing the power series as the basis function, the method of collocation of the differential system resulting from the rough solution to the problem was used. a differential form, which includes differentials, is a quantity that may be integrated. f (x)d x is the differential form of the integral ∫ b a f (x) d x. this is because it is integrated over a onedimensional region or path, this differential form has degree one. one-form refers to the differential form of degree one. the differential forms was previously used to linearize some important differential equations [11-13]. in this research, our attention is focused on the linearization of a class of geodesics equations. the class of geodesics equations is obtained from a unified equation y′′ − 2 f ′ (y) f (y) y′2 − f ′ (y) f (y) = 0, (1) as presented in [14]. given that f (y) = y, equation (1) becomes y′′ − 2 y y′2 − y = 0, (2) which describes the geodesics on a cone. for f (y) = d + y, equation (1) is now y′′ − 2 d + y y′2 − d − y = 0, (3) which describes the geodesics on a plane, where d is a constant. again, for f (y) = siny , equation (1) becomes y′′ − 2coty y′2 − siny cosy = 0. (4) equation (4) describes the geodesics on sphere. as previously mentioned, [14] addressed the class of geodesic equations (2) through (4). a characterization of sundman linearizable equations in terms of one auxiliary function and ode coefficients was the method adopted. a direct alternative method for creating the first integrals and sundman transformations was provided by using this criterion to explicitly acquire the general solutions for the first integral. equation (4) was also resolved in [15] using symmetry and integration techniques for differential equations. our results utilizing the differential forms method are consistent with those obtained by other methods in every circumstance. the approach we took in this case is simpler and gets us to the findings more quickly. 2 orverem et al. / j. nig. soc. phys. sci. 4 (2022) 957 3 2. approach of differential forms linearization through differential forms entails that, the invertible change of independent and dependent variables x = f (x, y) and y = g (x, y) , that will map the general second order nonlinear ordinary differential equation y′′ = f ( x, y, y′ ) , (5) into a linear equation, should necessarily be in the form y′′ + f0 + f1y ′ + f2y ′2 + f3y ′3 = 0, (6) and the coefficients f0, f1, f2, and f3 must satisfy the conditions f0yy + f0 ( f2y − 2 f3x ) + f2 f0y − f3 f0x + 1 3 ( f2xx − 2 f1xy + f1 f2x − 2 f1 f1y ) = 0, (7) and f3xx + f3 ( 2 f 0y − f1x ) + f0 f3y − f1 f3x + 1 3 ( f1yy − 2 f2xy + 2 f2 f2x − f2 f1y ) = 0. (8) once the conditions in equations (7) and (8) are satisfied, we proceed to construct a 3 × 3 matrix m = pd x + qdy, (9) where p = ( 1 3 )  −2 f1 −3 f0 3 f0y + 3 f0 f2 0 f1 2 f2x − f1y − 3 f0 f3 −3 0 f1  , q = ( 1 3 )  − f2 0 2 f1y − f2x + 3 f0 f3 3 f3 2 f2 3 f3x − 3 f1 f3 0 3 − f2  , and solve the equation dr = mr, (10) where r =  u v w , a special solution is usually sufficient for the three components of r. we can also construct k = u/w, l = v/w. (11) next, we construct the 2 × 2 matrix z = [ (2k − f1) d x − ldy f0d x + kdy −ld x − f3dy kd x + ( f2 − 2l) dy ] , and solve for r from the equation dr = zr, (12) where r = [ fx fy ] = [ b c ] . finally, we solve df = [ d x dy ] r; (13) the two independent solutions will be taken as f and g. what is given here is the summary of the method. for more detail please see [12], [11] or [13]. 3. linearization of a class of geodesics equations the unified class of geodesic equations is represented by equation (1). at this point, we want to linearize the three equations that make up the geodesic equations class. 3.1. geodesics on a cone equation (2) which describes the geodesics on a cone has the coefficients f0 = −y, f1 = 0, f2 = − 2 y , f3 = 0, that satisfy the linearizability conditions (7) and (8). with pd x =  0 yd x d x 0 0 0 −d x 0 0  , and qdy =  2 3y dy 0 0 0 −43y dy 0 0 dy 23y dy  , m = pd x + qdy becomes m =  2 3y dy yd x d x 0 −43y dy 0 −d x dy 23y dy  . with this situation, dr =  2u 3y dy + v yd x + wd x − 4v 3y dy −ud x + v dy + 2w3y dy  , where dr = mr and r =  u v w  . if v = 0, then du = 2u3y dy + wd x, dv = 0 and dw = −ud x + 2w3y dy. again, ux = w, uy = 2u 3y , wx = −u and wy = 2w 3y . v = 0, u = y 2/3sinx and w = y2/3cosx are given as a special solution. therefore, equation (11) is now k = y2/3sinx y2/3cosx = tanx , l = 0, and the matrix z becomes z = [ 2tanx d x −yd x + tanx dy 0 tanx d x − 2y dy ] , and dr = [ 2btanx d x − cyd x + ctanx dy ctanx d x − 2cy dy ] . from dr, we see that db = (2btanx − cy) d x + ctanx dy, and dc = c ( tanx d x − 2 y dy ) , where bx = 2btanx − cy, by = ctanx , cx = ctanx and cy = − 2c y . integrating dc = c ( tanx d x − 2y dy ) , we have lnc + lny2 − lnsecx = k. that is, cy 2 secx = e k = k, and then c = k secx y2 , where k is a constant. 3 orverem et al. / j. nig. soc. phys. sci. 4 (2022) 957 4 we notice that by = cx so that by = k secx tanx y2 . integrating, we have b = −k secx tanx y + g (x) . (14) when we differentiate equation (14) with respect to x, we get bx = −k ( secx + 2secx tan2 x ) y + g′ (x) . (15) but bx is also expressed as bx = 2btanx − cy, that is bx = −2k secx tan2 x y + 2g (x) tanx − k secx y . now comparing the above with equation (15) and simplifying, one sees that g′ (x) − 2g (x) tanx = 0. using the method of integrating factor with p (x) = −2g (x) tanx and q (x) = 0, one obtains the integrating factor as 1sec2 x . therefore, g(x)sec2 x = m, that is g (x) = msec 2 x or g (x) = mcos2 x , where m is another constant. substituting g (x) = msec2 x into equation (14), one obtains that b = −k secx tanx y + msec2 x . for df = [ d x dy ] [ b c ] , you have that df = ( −ksecx tanx y + msec2 x ) d x + k secx y2 dy. on integration, f = −ky−1secx + mtanx − ky−1secx , and finally, f = −2ksecxy + mtanx . taking the coefficients proportional to the constants k and m to be the linearizing point transformation, we have x = tanx , y = 2 secx y . with the transformation y = c1 x + c2, one sees that 2 secx y = c1tanx + c2, that is 2 secx = c1ytanx + c2y is the solution of equation (2). 3.2. geodesics on plane we now proceed to linearize the equation that describes the geodesics on plane. the equation is designated as equation (3). equation (3) is in the form of (6) and its coefficients: f0 = −d − y, f1 = 0, f2 = −2 d+y, f3 = 0 satisfy the linearizability conditions (7) and (8), and therefore, it is linearizable using differential forms. now, pd x =  0 (d + y) d x d x 0 0 0 −d x 0 0  , qdy =  2dy 3(d+y) 0 0 0 −4dy3(d+y) 0 0 dy 2dy3(d+y)  , so that equation (9) becomes m =  2dy 3(d+y) (d + y) d x d x 0 −4dy3(d+y) 0 −d x dy 2dy3(d+y)  , and dr =  2udy 3(d+y) + v (d + y) d x + wd x −4v dy 3(d+y) −ud x + v dy + 2wdy3(d+y)  . letting v = 0 gives du = 2udy3(d+y) + wd x, dv = 0, dw = −ud x + 2wdy3(d+y), and ux = w, uy = 2u 3(d+y), wx = −u, wy = 2w 3(d+y). the situation above is satisfied by a special solution u = (d + y) 2 3 sinx and w = (d + y) 2 3 cosx . therefore k = tanx , l = 0. constructing the matrix z, one sees that z = [ 2tanx d x − (d + y) d x + tanx dy 0 tanx d x − 2dyd+y ] , and equation (12) is now dr = [ 2btanx d x − (d + y) cd x + ctanx dy ctanx d x − 2cdyd+y ] . this situation becomes db = (2btanx − (d + y) c) d x + ctanx dy, (16) and dc = c ( tanx d x − 2dy d + y ) . (17) integrating (17) above, we have lnc = lnsecx − 2 ∫ dy d + y + k , so that c(d + y)2 secx = ek = k. making c the subject, we have c = ksecx (d + y)2 , (18) where k is a constant. again, from equation (16), bx = 2btanx − (d + y) c, by = ctanx , cx = ctanx and cy = − 2c d+y, where by = cx. since by = cx, we have that by = k tanx secx (d+y)2 , and on integration, we have so that b = kytanx secx d(d + y) + g (x) . (19) 4 orverem et al. / j. nig. soc. phys. sci. 4 (2022) 957 5 differentiating b with respect to x, we have bx = kysecx d (d + y) ( 2tan2 x + 1 ) + g′ (x) . one notes that bx is also expressed as bx = 2btanx − (d + y) c, that is bx = 2kytan2 x secx d (d + y) + 2tanx g (x) − ksecx d + y . comparing the two expressions of bx and simplifying, one has g′ (x) − 2tanx g (x) = −kdsecx − kysecx d (d + y) . this implies that g′ (x) − 2tanx g (x) = −ksecx d . (20) equation (20) has now been transformed into a first-order linear differential equation that can be solved using the integrating factor 1sec2 x . multiplication of the integrating factor and equation (20) gives cos2 x g′ (x) − 2sinx cosx g (x) = − kcosx d . this becomes cos2 x g (x) = −k d sinx + m, where m is another constant. simplifying, the equation above becomes g (x) = −ktanx secx d + msec2 x , and the value of b from equation (19) is now b = kytanx secx d(d + y) − ktanx secx d + msec2 x . referring to equation (13), that is, df = [ d x dy ] [ b c ] , one sees that df = ( kytanx secx d(d + y) − ktanx secx d + msec2 x ) d x + ksecx (d + y)2 dy. integrating the equation above, you have f = k ( 2ysecx d(d + y) − secx d ) + mtanx . therefore x = tanx , y = (y − d)secx d(d + y) is the linearizing point transformation of equation (3). the general solution can be readily expressed using the transformation y = c1 x + c2, as y = d + c1d (d + y) sinx + c2d (d + y) cosx . 3.3. geodesics on sphere next to be considered is equation (4) that describes the geodesics on a sphere. the equation has the coefficients f0 = −siny cosy , f1 = 0, f2 = −2coty , f3 = 0 that satisfied the linearizability conditions (7) and (8). the 3 × 3 matrix m = pd x + qdy is now m =  2 3 coty dy siny cosy d x d x 0 −43 coty dy 0 −d x dy 23 coty dy  , where pd x =  0 siny cosy d x d x 0 0 0 −d x 0 0  and qdy =  2 3 coty dy 0 0 0 −43 coty dy 0 0 dy 23 coty dy  . now, dr =  2 3 ucoty dy + v siny cosy d x + wd x − 4 3 v coty dy −ud x + v dy + 23 wcoty dy  , so that, letting v = 0, we have that du = 23 ucoty dy + wd x, dv = 0 and dw = −ud x + 2 3 wcoty dy. also, ux = w, uy = 2 3 ucoty , wx = −u and wy = 2 3 wcoty . a special solution u = sinx (siny )2/3, v = 0, w = cosx (siny )2/3 satisfies the situation above. construction of k and l shows that k = sinx (siny ) 2/3 cosx (siny )2/3 = tanx and l = 0. with k and l, the 2 × 2 matrix z becomes z = [ 2tanx d x −siny cosy d x + tanx dy 0 tanx d x − 2coty dy ] , and dr = [ 2btanx d x − csiny cosy d x + ctanx dy ctanx d x − 2ccoty dy ] , so that db = 2btanx d x − csiny cosy d x + ctanx dy, (21) and dc = ctanx d x − 2ccoty dy. (22) from equations (21) and (22), one sees that bx = 2btanx − csiny cosy , by = ctanx , cx = ctanx , cy = −2ccoty . integrating equation (22), we have that c = ksecx sin2y , (23) where k = ek is a constant. one notes that by = cx, therefore, by = ktanx secx sin2y . 5 orverem et al. / j. nig. soc. phys. sci. 4 (2022) 957 6 integrating the equation above, you see that b = −ktanx secx coty + g (x) , (24) for some function g (x) . we can see that when we differentiate equation (24) with respect to x, we get bx = −kcoty tan 2 x secx − kcoty sec3 x + g′ (x) . (25) but bx is also expressed as bx = 2btanx − csiny cosy that is bx = −2ktan 2 x secx coty + 2tanx g (x) − ksecx coty .(26) now, comparing equations (25) and (26) and simplifying, we have g′ (x) − 2tanx g (x) = −ksecx coty ( tan2 x − sec2 x + 1 ) . using the identity tan2 x + 1 = sec2 x , one finally has that g′ (x) − 2tanx g (x) = 0. (27) solving equation (27) using the integrating factor 1sec2 x , one sees that g (x) = msec2 x , where m is another constant. substituting g (x) into equation (24), we now have that b = −ktanx secx coty + msec2 x . now, df = ( −ktanx secx coty + msec2 x ) d x + ksecx sin2y , and on integration, f = −2kcoty secx + mtanx . the linearizing point transformation is now x = tanx , y = 2coty secx , and the solution of equation (4) is now 2coty = c1sinx + c2cosx . 4. conclusion the linearization of geodesics on spheres, planes, and cones is accomplished using the differential forms technique. these equations have convincing solutions as obtained from the transformation y = c1 x + c2. these three geodesics equations are considered under the umbrella geodesics equation. the solution or result of equation (4) is similar to the one obtained in [15] where symmetry and integration method was used. it is shown that the equation to be linearized must have the form (6), and the coefficients must satisfy the linearizability conditions (7) and (8). the findings of this study are distinctive, innovative, and original. acknowledgment we thank the anonymous referees for the positive enlightening comments, which have greatly helped us in making improvements to this paper. references [1] h. a. hamo & i. sakalli, “exact solutions to the geodesic equations of linear dilaton black holes”, turkish journal of physics 40 (2016) 139, https://doi.org/10.3906/fiz-1504-4. 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[7] g. panou & r. korakitis, “geodesic equations and their numerical solutions in geodetic and cartesian coordinates on an oblate spheroid”, symmetry, integrability and geometry: methods and applications 4 (2017) 31, https://doi.org/0.1515/jogs-2017-0004. [8] a. qadir, “geometric linearization of ordinary differential equations”, symmetry, integrability and geometry: methods and applications (sigma) 3 (2007) 103, https://doi.org/10.3842/sigma.2007.103. [9] m. cariglia, t. houri, p. krtouš & d. kubizňák, “on integrability of the geodesic deviation equation”, european physical journal c 78 (2018) 661, https://doi.org/10.1140/epjc/s10052-018-6133-1. [10] f. o. obarhua & o. j. adegboro, “an order four continuous numerical method for solving general second order ordinary differential equations”, journal of the nigerian society of physical sciences 3 (2021) 150, https://doi.org/110.46481/jnsps.2021.150. [11] j. m. orverem, r. k. tyokyaa & h. m. balami, “differential forms: a tool for linearizing second order ordinary differential equations”, journal of scientific and engineering research 4 (2017) 10. [12] b. k. harrison, “an old problem newly treated with differential forms : when and how can the equation y = f ( x , y , y ) be linearized ?”, proceedings of institute of mathematics of nas of ukraine 43 (2002) 2. [13] j. m. orverem, y. haruna, b. m. abdulhamid & m. y. adamu, “applying differential forms and the generalized sundman transformations in linearizing the equation of motion of a free particle in a space of constant curvature”, journal of mathematics research 13 (2021) 5, https://doi.org/110.5539/jmr.v13n5p5. [14] m. t. mustafa, a. y. al-dweik & r. a. mara’beh, “on the linearization of second-order ordinary differential equations to the laguerre form via generalized sundman transformations”, symmetry, integrability and geometry: methods and applications (sigma) 9 (2013) 41, https://doi.org/10.3842/sigma.2013.041. [15] g. w. bluman & s. c. anco, symmetry and integration methods for differential equations, springer 154 (2004) 430. 6 j. nig. soc. phys. sci. 5 (2023) 967 journal of the nigerian society of physical sciences the derivation of the riemann analytic continuation formula from the euler’s quadratic equation opeyemi o. enocha,∗, adejimi a. adenijib, lukman o. salaudeena adepartment of mathematics, federal university oye-ekiti, ekiti state, nigeria bdepartment of mathematics and statistics, tshwane university of technology, south africa abstract the analysis of the derivation of the riemann analytic continuation formula from euler’s quadratic equation is presented in this paper. the connections between the roots of euler’s quadratic equation and the analytic continuation formula of the riemann zeta equation are also considered. the method of partial summation is applied twice on the resulting series, thus leading to the riemann analytic continuation formula. a polynomial approach is anticipated to prove the riemann hypothesis; thus, a general equation for the zeros of the analytic continuation formula of the riemann zeta equation based on a polynomial function is also obtained. an expression in terms of prime numbers and their products is considered and obtained. a quadratic function, g(tn), that is required for euler’s quadratic equation (eqe) to give the analytic continuation formula of the riemann zeta equation (acf) is presented. this function thus allows a new way of defining the analytic continuation formula of the riemann zeta equation (acf) via this equivalent equation. by and large, the riemann zeta function is shown to be a type of l function whose solutions are connected to some algebraic functions. these algebraic functions are shown and presented to be connected to some polynomials. these polynomials are also shown to be some of the algebraic functions’ solutions. conclusively, ς(z) is redefined as the product of a new function which is called h(tn, z) and this new function is shown to be dependent on the polynomial function, g(tn). doi:10.46481/jnsps.2023.967 keywords: riemann zeta function, euler’s equation, meromorphic function, non-trivial zeros article history : received: 31 july 2022 received in revised form: 20 november 2022 accepted for publication: 25 november 2022 published: 20 december 2022 c© 2022 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: j. ndam 1. introduction many authors have recently presented some polynomial approaches to prove the riemann hypothesis [1-9]. some of the proofs were based on jensen polynomials, laguerre polynomials as jensen polynomials of laguerre–pólyaentire functions, and some worked on a general theorem which models such polynomials by hermite polynomials. the authors presented ∗corresponding author tel. no: +234 7066466859 email address: opeyemi.enoch@fuoye.edu.ng (opeyemi o. enoch) an approximation to zeros of the riemann zeta function using fractional calculus [10]. this allowed the authors to construct a fractional iterative method to nd the zeros of functions in which it is possible to avoid expressions that involve hypergeometric functions, mittag-leffler functions or infinite series [10], to mention a few. as good as their works are, seeking a clearer insight into these possible polynomials is expedient, knowing fully that solutions to most difficult problems may not necessarily be complex. significantly, it is known that the riemann zeta function 1 enoch et al. / j. nig. soc. phys. sci. 5 (2023) 967 2 is a type of l function whose solutions are connected to some algebraic functions, and polynomials are also types of solutions of algebraic functions. a polynomial approach can be anticipated to prove the riemann hypothesis. this study aims to present the link that connects euler’s quadratic equation (eqe) and the analytic continuation formula of the riemann zeta equation (acf) by using the partial summation on a generalized polynomial function of euler’s quadratic equation. (eqe). the remaining part of this paper is organized as follows: section two is the materials and methods. the derivation of the analytic continuation formula of the riemann zeta equation (acf) from euler’s quadratic equation (eqe) is presented in section three. section four considers the obtained expression in terms of prime numbers and their products. while section five is for obtaining a quadratic function; g (tn) , in terms of the analytic continuation formula of the riemann zeta equation (acf). section six presents a new way of defining the analytic continuation formula of the riemann zeta equation (acf) via an equivalent equation. the concluding remark is presented in section seven. 2. materials and methods the following equations and method shall be used in the derivation of the analytic continuation formula of the riemann zeta equation (acf) from the euler’s quadratic equation (eqe) [11]: ∏ ( z 2 ) = − z 2 γ ( z 2 ) = ∞∑ n≥1 ( 2n + z 2n ) (1) nz = m∏ i=1 p∝i zi (2) from the expression [14-15], ( 1 pz − 1 ) ∞∑ n=0 1 pnz = −1 (3) the method of partial summation as found in literature [14-15] is given as: ∑ p≤x log2 p + ∑ pq≤x log p log q = 2x log x + ϑ (x) (4) by partial summation, we get from equation (5) ∑ p≤x log p + ∑ pq≤x log p log q log pq = 2x + ϑ ( x log x ) , (5) the above equation (5) gives equation (6) ∑ pq≤x log p log q = ∑ p≤x log p ∑ q≤x/p log q (6) or =2x ∑ p≤x log p p − ∑ p≤x log p ∑ qr≤x/p log q log r log qr + ϑ x ∑ p≤x log p p ( 1 + log xp )  (7) = 2x log x − ∑ qr≤x/p log q log r log qr ϑ ( x pq ) + o ( x log log x ) (8) these equations above shall be used to derive the analytic continuation formula of the riemann zeta function from euler’s quadratic equation. by this derivation, it will be readily obvious to see through into the generation of the zeros of the riemann zeta function. 3. derivation of the acf from eqe the polynomial function in (1) was always discovered to be prime for 1 ≤ x ≤ 40 [12]. it was from this quadratic equation that euler obtained the first few prime numbers [13]. 1 ≤ x ≤ 40 : x2 + x + 41 (9) another known fact is that equation 10 is also a prime for x = 1 . . . , 39. x2 − x + 41 (10) interestingly, the roots of (9, 10) are x = −0.5±6.3836i and x = 0.5 ± 6.3836i respectively, which are the similitude of the non-trivial zeros of the riemann zeta function. with these similarities [16], the question that readily comes to mind is; what is the possibility of using the structure of this euler’s equation to obtain acf if the coefficients of x2 and x are taken as k and for the constant integral, 41, is replaced with g (tn)? to seek an answer to this question, a transformation of the euler’s equation, when it is multiplied by another linear equation whose roots will are −2n : n = 1, 2, 3, . . ., will be ζe (z) = ( kz2 − kz + g(tn) ) (z + 2n) (11) the roots of this polynomial will be the same as the trivial and the non-trivial zeros of the riemann zeta function under certain conditions that g(tn) is known. it has been shown that there are meromorphic functions that are equivalent to the riemann zeta function [6-7], and they are given as: 2 enoch et al. / j. nig. soc. phys. sci. 5 (2023) 967 3 ζe (z) = (z + 2n) (z − 1) ( kz2 − kz + g(n) ) ; k = 4 (12) or ζe (z) = (z + 2n) (ez−1 − 1) ( kz2 − kz + g(n) ) ; k = 4 (13) provided that g(n) is also known. the equation (12) and equation (13) are transformed into matrices whose eigenvalues are the trivial and non-trivial spectral points of the riemann zeta function [5-7, 13] provided that; g (n) = 800.162 + 968.548j (n) (14) or g (n) = 800.162 + 968.548nv(n) (15) or g (tn) = 1 + kt 2 n where k = 4 (16) such that g (tn) = g(n). from 11, let ζe (z) = ( kz2 − kz + g (tn) ) (z + 2n) (17) riemann gave the following expression in his work [11]: − ∏ ( z 2 ) = ∞∑ n≥1 ( 1 + z 2n ) (18) which is the same as z2 γ ( z 2 ) , by taking ∏ ( z 2 ) = − z 2 γ ( z 2 ) = ∞∑ n≥1 ( 2n + z 2n ) , (19) equation (17) can written as ζe (z) = 2n ( 1 + z 2n ) ( kz2 − kz + g (tn) ) (20) using the method of discretization, equation (20) becomes; γ (z) = ∞∑ n≥1 (ζe (z)) (21) = ∞∑ n≥1 [ 2n ( 1 + z 2n ) ( kz2 − kz + g (tn) )] (22) thus applying the method of partial summation in [15],as in equation (6) the resulting equation from equation (22) shall be ∑ dq≤n [ 2n ( 1 + z2n ) ( kz2 − kz + g(tn) )] = ∑ d≤n 2n [ kz2 − kz + g(tn) ] ∑ q≤n/d (1 + z 2n ), (23) where d = 2n [ kz2 − kz + g(tn) ] and q = 1 + z2n eventually, equation (23) can be written as: γ (z) = ∞∑ n≥1 (ζe (z)) (24) = ∑ d≤n [ 2n ( kz2 − kz + g(tn) )] ∑ q≤n/d ( 1 + z 2n ) (25) interestingly, since − z 2 γ( z 2 ) = ∞∑ n≥1 ( 2n + z 2n ) (26) 25 becomes: γ (z) = ∞∑ n≥1 (ζe (z)) = − z 2 γ ( z 2 ) (z − 1) ∑ d≤n 2n ( kz + g (tn) (z − 1) ) (27) with a little rearrangement and the introduction of π −z 2 π z 2 = 1, equation (27) is the same as: γ (z) = ∅(z) ∑ d≤n 2n [( kz + g(tn) (z − 1) ) π z 2 ] (28) where ∅(z) = − z2 (z − 1)π −z 2 γ( z2 ) if the principle of partial summation is again applied on the series in equation (28), equation (29) we will obtain: ∑ rb≤n 2n [( kz + g(tn) (z − 1) ) π z 2 ] = π z2 ∑ r≤n ( kz + g (tn) (z − 1) ) ∑ b≤ nr 2n  ; rb = d (29) by this, (28) becomes: γ (z) = ∅(z) π z2 ∑ r≤n ( kz + g (tn) (z − 1) ) ∑ b≤ nr 2n  (30) where r = (kz + g(tn(z−1) )π z 2 and b = 2n. (30) shall be used subsequently in this paper. 3 enoch et al. / j. nig. soc. phys. sci. 5 (2023) 967 4 4. an expression in terms of prime numbers and their products a non-conventional expression for −1 [12-13] is;( 1 pz − 1 ) ∞∑ n=0 1 pnz = −1, (31) (31) allows us to write (28) as: γ (z) = − ∅ (z) ( 1 pz − 1 ) ∞∑ n=0 1 pnz ∑ d≤n 2nf (t, z) (32) f (t, z) = [( kz + g (tn) (z − 1) ) π z 2 ] pleasantly (32) gives: γ (z)  ( 1 pz − 1 ) π z 2 ∑ d≤n 2n [f(t, z)]  −1 = z 2 (z − 1) π −z 2 γ ( z 2 ) ∞∑ n=0 1 pnz . (33) intuitively, if (33) is multiplied over prime numbers one comes to; γ (z) ∏ p  ( 1 pz − 1 ) ∑ d≤n [2nf(t, z)]  −1 = z 2 (z − 1)π −z 2 γ ( z 2 ) ∏ p ∞∑ n≥1 1 pnz . (34) conclusively, the rhs of (34) is the same as the analytic continuation formula of the riemann zeta function, while the lhs is an equivalence of the rhs. 5. obtaining g (tn) in terms of the acf riemann defines ε (z) [11] as ; ε (z) = z 2 (z − 1) π −z 2 γ ( z 2 ) ζ (z) (35) by this, ζ (z) can be represented as: ζ (z) = 2ε (z) π z 2 z (z − 1) γ ( z 2 ) (36) for the lhs of (34) to be equal to (34), we represent εe in equation (37) as εe = γ (z) ∏ p ( 1 pz − 1 ) ∞∑ d≤n 2nf(t, z)  −1 (37) using (29) on (34), we obtain: εe = γ (z) j(n) ∏ p ( 1 pz − 1 ) ∑ r≤n ∏ p f (t, z)  −1 (38) such that j (n) = ∑ b≤ nr ∏ p 2n and εe = γ (z) π − z 2 b (t, z) ∏ p ( 1 pz − 1 )−1 (39) where b (t, z) = ∏ p ∑ r≤n ( kz + g (tn) (z − 1) ) ∑ b≤nr 2n  −1 gives: εe = γ (z) π − z 2 ζ (z) b (t, z) . (40) by making the series containing g (tn) the subject of the expression in (38), we obtain: ∑ r≤n ∏ p ( kz + g (tn) (z − 1) ) = γ (z) εe π z2 ∑ b≤ nr ∏ p 2n  −1 ∏ p ( 1 pz − 1 ) −1 (41) followed by further simplification shown that (41) becomes: ∑ r≤n ∏ p ( kz + g (tn) (z − 1) ) = γ (z) εe ζ (z) π z2 ∑ b≤ nr ∏ p 2n  −1 (42) since ζ (z) = ∏ p ( 1 pz − 1 ) −1 . (43) it can be shown in existing works [1-2] that; nz = m∏ i=1 p∝i zi (44) 4 enoch et al. / j. nig. soc. phys. sci. 5 (2023) 967 5 implies that ∑ d≤n 2n = 2 ∑ d≤n m∏ i=1 p∝ii (45) with this, (42) can be expressed as (45), in terms of prime numbers such that: ∑ r≤n ∏ p ( kz + g (tn) (z − 1) ) = γ (z) εe ζ (z) π− z 2 ∑ b≤ nr ∏ p  12 m∏ i=1 p−∝ii  (46) 6. defining the acf via (34) if the lhs of (34) is written as: εe = γ (z) ∏ p [( 1 pz − 1 ) h (t, z) ]−1 (47) where h (t, z) = π z 2 ∑ d≤n 2n ( kz + g (tn) (z − 1) ) then we can write (35) as the rhs of (34) such that; εe = z 2 (z − 1) π −z 2 γ ( z 2 ) ∏ p ∞∑ n≥1 1 pnz (48) where � (z) = εe and from the nachlass of riemann [15], ε (z) is also defined as; ε (z) = 1 2 + z 2 (z − 1) d (x, z) (49) where d (x, z) = ∫ ∞ 1 ψ (x) ( x z 2 −1 + x− (z+1) 2 ) d x (50) and ψ (x) = ∞∑ n=1 e−n 2πx (51) such that if ε (z) = εe then; 1 2 + z 2 (z − 1) d (x, z) = z 2 (z − 1) π −z 2 γ ( z 2 ) ∏ p ∞∑ n≥1 1 pnz (52) or as; 1 2 + z 2 (z − 1) d (x, z) = γ (z) ∏ p [( 1 pz − 1 ) h (t, z) ]−1 (53) since h (t, z) = π z 2 ∑ d≤n [ 2n ( kz + g (tn) (z − 1) )] the evaluation of the intergrades in (52 and 53) will give; d (x, z) = 2n(z) (54) where n (z) = ∞∑ n=1 e−n2π  1( z2 −1)( 2n2π + z − 2 ) + 1( z+12 )( 2n2π− z − 1 )  (55) with (52), we can write (52) and (53) as follows; 1 2 + z (z − 1) n(z) = z 2 (z − 1) π −z 2 γ ( z 2 ) ∏ p ∞∑ n≥1 1 pnz (56) and 1 2 + z (z − 1) n(z) = γ (z) ∏ p [( 1 pz − 1 ) h(t, z) ]−1 (57) (53) and (54) now give new definitions of the ζ (z) as: [ z 2 (z − 1) π −z 2 γ ( z 2 )]−1 [ 1 2 + z (z − 1) n (z) ] = ∏ p ∞∑ n≥1 1 pnz (58) and γ (z)−1 ∏ p h(t, z) [ 1 2 + z (z − 1) n(z) ] = ∏ p [( 1 pz − 1 )]−1 (59) 5 enoch et al. / j. nig. soc. phys. sci. 5 (2023) 967 6 equating (56) and (57), (60) is obtained: [ z 2 (z − 1) π −z 2 γ ( z 2 )]−1 [ 1 2 + z (z − 1) n (z) ] = γ (z)−1 ∏ p h (t, z)  [ 1 2 + z (z − 1) n (z) ] (60) from (60), γ (z) is obtained (61) : γ (z) = [ z 2 (z − 1) π −z 2 γ ( z 2 )] ∏ p h (t, z) (61) such that h (t, z) is represented in (62 h (t, z) = π z 2 ∑ d≤n [ 2n ( kz + g (tn) (z − 1) )] (62) it can be written as equation (29). 7. conclusion with the contents of g (tn),in (13 and 14): g (tn) = 1 + kt2n where k = 4, (61) and (62) can be implemented to give the same results and zeros of the analytic continuation formula and that of the riemann zeta function. it has been shown that the analytic continuation formula of the riemann zeta function can be obtained from euler’s quadratic equation. riemann zeta function can be written as (63) provided g (tn) holds as defined in (58) and (59). ζ (z) = ∏ p h (t, z) = ∏ p ∑ d≤n [ 2nπ z 2 ( kz + g (tn) (z − 1) )] (63) enoch obtained the following for the generation of the zeros of the analytic continuation formula of the riemann zeta function [5-7]: g (tn) = kz (z − 1) σ τ−ϑ (64) where σ =  z2π−z2 γ ( z 2 ) ( 1 pz − 1 ) ∞∑ n=0 1 pnz − 1  (65) such that; ϑ = z 2 π −z 2 γ ( z 2 ) ( 1 pz − 1 ) ∞∑ n=0 1 pnz (66) and τ = [ 1 2 (z − 1) + zn (z) ] ∏ p ( 1 pz − 1 ) (67) he pointed out that (58) and (59) definition of g (tn) is the same as obtained in equations (14), (15) and (16). the functions; jn and vn are written as polynomials of order two or three for this to be possible by the authors [8-9]. he was able to obtain a general equation for the zeros of the analytic continuation formula from equation (16) as; g (tn) = 1 + kt 2 n ; k = 4 (68) by which equation (69) holds as : tn = ( g (tn) − 1 k )1/2 (69) again from: 1 + kt2n = kz (z − 1) σ τ−ϑ ; k = 4 (70) such that; tn = ± ( z (z − 1) σ (τ−ϑ) − 1 k ) 1 2 ; k = 4 (71) the k value can hold for any integer, depending on the pattern of choice. acknowledgments the authors wish to thank tshwane university of technology for their financial support and the department of higher education and training, south africa. references [1] m. griffin, k. ono, l. rolen, & d. zagier, “jensen polynomials for the riemann zeta function and other sequences”, proceedings of the national academy of sciences 116 (2019) 11103. 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[8] m. chudnovsky & p. seymour, “the roots of the independence polynomial of a claw free graph”, journal of combinatorial theory, series b 97 (2007) 350. [9] h. larson & i. wagner, “hyperbolicity of the partition jensen polynomials”, research in number theory 5 (2019) 1. [10] a. torres-hernandez & f. brambila-paz, “an approximation to zeros of the riemann zeta function using fractional calculus”, arxiv preprint arxiv:2006 (2020) 14963. [11] a. selberg, “harmonic analysis and discontinuous groups in weakly symmetric riemannian spaces with applications to dirichlet series”, the journal of the indian mathematical society 20 (1956) 47. [12] s. andreas, riemann’s second proof of the analytic continuation of the riemann zeta function (1987). [13] r. r. ben, the zeta function and its relation to the prime number theorem (2000). [14] s. j. patterson, “an introduction to the theory of the riemann zetafunctio”, cambridge university press 14 (1995). [15] a. selberg, “an elementary proof of the prime number theorem”, annals of mathematics (1949) 305. [16] o. o. enoch, t. o. ewumi, & y. skwame, “on the turning point, critical line and the zeros of riemann zeta function”, australian journal of basic and applied sciences 6 (2012) 279. 7 j. nig. soc. phys. sci. 5 (2023) 992 journal of the nigerian society of physical sciences secure health information system with blockchain technology a. e. ibora,∗, e. b. edima, a. a. ojugob a department of computer science, university of calabar, calabar, nigeria bdepartment of computer science, federal university of petroleum resources, effurun, nigeria abstract this paper focuses on highlighting the problems that are associated with the absence of privacy and security of medical records in a healthcare system. it seeks to bridge the gap between the currently used security protocols in the management of health information, and encryption algorithms that should be used. extant health information systems have always been developed with conventional databases. with all the privileges to read, write and execute assigned to the administrator, who has centralised control over all medical records, there is the likelihood of the misuse, distortion and loss of such records in the event that the administrator becomes compromised or inadvertent system failure. to solve this problem, the use of decentralised and distributed databases becomes paramount. blockchain technology has recently received much attention due to its ability to permit a peer-to-peer network with distributed databases that can be stored locally on each node in the network. subsequently, all updates on records in a database are communicated to all participating parties, hence addressing the problem of centralised control. in this paper, we propose a health information system on a blockchain to create a trust-free system for both health personnel and patients. from the results obtained, we achieved the decentralisation of the medical records’ database to enhance the security and privacy of data on the modeled peer-to-peer network. doi:10.46481/jnsps.2023.992 keywords: blockchain, health information system, distributed databases, encryption algorithms, medical records article history : received: 16 august 2022 received in revised form: 06 november 2022 accepted for publication: 06 november 2022 published: 29 april 2023 © 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: tolulope latunde (ph.d.) 1. introduction computers have been employed in the delivery of healthcare services and their usage has been helpful in enhancing operational efficiency and clinical services. however, the current paradigm shift in the healthcare industry brings to light the challenges of maintaining the integrity of medical records. with the advancement in information and communication technology, healthcare delivery has come with new formats and ∗corresponding author tel. no: +2348154213807 email address: ayei.ibor@unical.edu.ng (a. e. ibor ) structures – from evidence-based medicine to e-medicine and remote e-health services [1]. similarly, the application and use of computer-based solutions in healthcare is witnessing new dimensions, thus, giving room to novel security flaws. these emerging security flaws require effective defence mechanisms to contain likely unauthorised access, modification and distortion of medical records. the world health organisation defined a health information system (his) as a system that is able to capture, store, manage, and transmit health-related information. in this sense, achieving a functional his is paramount in enhancing 1 a. e. ibor et al. / j. nig. soc. phys. sci. 5 (2023) 992 2 the storage and retrieval of medical records. according to [2–4], a typical his must allow for the effective collection, processing, reporting, and use of health-related data across different health-based units. current approaches to health information systems rely mostly on centralised databases with administrative control and policies, as well as authoritative access. in this case, changes and updates are only effected by an administrator or super user, and any other person to which such rights and privileges are assigned. this nontransparent transaction policy creates room for the possibility to distort records and compromise the integrity of classified medical data. in recent times, the emergence of disruptive technologies such as the blockchain provides the capacity of managing distributed databases, which can be updated and locally stored across all nodes in the network. according to [5], the blockchain is a distributed database of records (or public ledger) of all transactions, which are executed and shared among participating parties. similarly, [6] argued that in a blockchain, transparency is inherent, thus, permitting that updates on records are only possible when a majority of the participants reach a consensus. new insights into the expansion of his show that it has the potentials to increase the accessibility to medical records [7–9]. however, the use of conventional and centralised databases limit such possibilities as authentication credentials are granted from a central point. in the event of a failure of the database, loss of records becomes inevitable, in which case, the operations of the target system will be truncated in real time. these limitations necessitate the need for a decentralised database that is difficult to compromise in terms of data integrity, confidentiality and availability. current implementations of the blockchain enable the use of a public ledger, which consists of the information of all the participants and transactions on the blockchain [6–10]. the information captured as well as the digital transactions that have ever been executed on the blockchain can be recorded and shared across the peer-to-peer blockchain network. similarly, [11] asserted that users can validate transactions, and also have identical copies of these transactions on their local machines. in this way, health information can be securely stored, updated and transmitted without unauthorised modification. furthermore, it is easier to track all updates as the information stored in the blockchain database is always complete and includes all records of transactions from the point of origin of the transactions. health information requires a high level of accuracy at all times as minor changes in such data can lead to very unpleasant results with a plethora of consequences. with the blockchain network being able to perform periodic self-updates, it helps to deliver a self-reviewing system that is computationally infeasible and expensive to compromise. transactions stored cannot be erased without the consensus of all participating parties. in this sense, it will be feasible to store and track medical records irrespective of the location of the health maintenance organisation (hmo), patients, nurses, doctors, and other health personnel. in addition, patients’ past illnesses, which are recorded on the blockchain can be useful for the proper treatment of patients in contrast to systems that run on a centralised database with limited access and the possibility of central point of failure. to address the problem of authoritative access and centralised control of the database of medical records, this paper proposes a secure system for managing medical records using the blockchain technology. the use of the blockchain provides distributed access to the database of medical records to forestall the problem of single pint of failure. in the same sense, it becomes impossible to modify the stored medical records without the consensus of the participating parties, thus making the storage and transmission of medical records an entirely transparent process. to implement the blockchain, we used python, go, docker engine, interplanetary file system (ipfs), and other technologies with significant results obtained from rigorous experimentation. the rest of the paper is organised as follows; the review of the related literature is discussed in section 2. in section 3, the materials and method are discussed while the results and discussion are presented in section 4. finally, the conclusion with future research direction is given in section 5. 2. review of related literature a detailed analysis of the existing healthcare information systems is provided in [12]. the authors argued that computerisation is significant in realising an efficient his. similarly, they investigated the security of medical records and were able to give the design of a computerised system used by the national heath management information system (nhmis). the implemented system was able to allow the safekeeping of patients’ records with basic components of running an effective and productive hospital. this notwithstanding, the system ran on a local server database that served as the repository for all medical data including queries and reports. in [4], the evolution of his is highlighted. areas of interest include the migration from paper-based to computer-based processing and storage, the availability of global and regional his, the use of health-related data for effective healthcare planning, and the deployment of technology for health monitoring. these aspects of hiss have witnessed tremendous improvements alongside an increase in health-related data. it is therefore pertinent to have a robust mechanism through which the escalating amount of medical records can be effectively stored, queried, reported and transmitted. furthermore, [3] highlighted the design and implementation of a his, and discussed the possibilities of improving its structure through the use of a comprehensive, integrated and decentralised system. similarly, [13] assert that the tendency to manually reconcile medical data among clinics, hospitals, laboratories, pharmacies and insurance companies has not been a successful process. this is due to the fact that no single list of the disparate locations housing patients’ data exists, and at the 2 a. e. ibor et al. / j. nig. soc. phys. sci. 5 (2023) 992 3 same time, it is difficult to ascertain the order of data entry to determine the records, which predate the others. in other words, determining the medications a patient is actually taking using antecedents of prescriptions may be unclear. in the same sense, every electronic health record uses different workflows to store data so that it becomes unclear who recorded what and when. in [14], medrec is proposed. medrec is a decentralised record management systems using blockchain technology. the proposed system is able to give patients a log of their comprehensive, accessible and credible medical history. medrec allows participants to be informed of their medical history including any form of modifications on these records. it is argued in [15] that the use of blockchain can be helpful to tackle the lagging in his, which runs on local server databases. this is a fact as patients whose medical records are managed by his running on a local server database have limited privileges to such data. consequently, the administrator may alter these records or deny patients access to same. there are several attempts by malicious users to compromise the security of medical records. cyberattacks such as sql injection [16], which targets the data aggregated on the database over time are very common. to this effect, the use of various disruptive technologies such as the blockchain, artificial intelligence, and internet of things for protecting medical records from human errors and cyberattacks is discussed in [17]. the different application areas of these technologies were also highlighted in this work including their security-related concerns. specifically, the authors identified the use of blockchain-based trust models in the implementation of data management in the healthcare sector to optimise the process flow and reduce the operational costs. the authors in [18] argued that the blockchain facilitates the use of a decentralised and distributed environment devoid of a central authority. since the transactions on a blockchain are both secure and trustworthy, its use in healthcare for realising patient-centric approach to healthcare systems and maintaining accurate electronic healthcare records is crucial. from their findings, the authors agreed that the use of blockchain technology in healthcare allows for the sharing of data, the accurate management of health records, and access control. in the same sense, the use of lightweight blockchain architecture for the management of healthcare data is proposed in [19]. the approach is able to reduce the computational and communication overhead of bitcoin network using clusters, where each copy of the ledger is maintained per cluster. each cluster consists of network participants that are able to use canal for secure and confidential transactions. their approach was also targeted at eliminating the problems of traditional client-server and cloud-based systems deployed in managing healthcare data. some of these problems include single point of failure, centralised data control, inherent system vulnerabilities, and data privacy. consequently, the benefits of using blockchain for healthcare are highlighted in [20]. some of these include distributed ledger, decentralised storage, authentication, security and immutability. the authors posited that the application of blockchain in healthcare improves data sharing capabilities, integrity, availability and authentication. in other words, the blockchain will allow patients to have control and ownership of the sharing of their medical data in a secure environment. besides, [21] identified electronic health records and personal health records as the most targeted areas that rely on blockchain technology. they claimed that access control, interoperability, and data integrity are issues that the use of the blockchain in healthcare has helped to improve. one significant feature of the blockchain remains that it cannot be easily compromised [22]. when a block is created, a one-time hash is generated. this hash is unique to the blockchain and is generated using the values of the features of each block. changes to values in the block invalidate the hash as well as the block making the blockchain unbreakable as illustrated in figure 1. figure 1: an illustration of the blockchain as shown in figure 1, putting information in the blockchain creates a block that contains the present and previous hashes. the first block is known as the genesis block, and has the present hash and previous hash. the second block is then created and the hash of this block is linked with the genesis block, and so on. in this sense, [15] asserted that the use of blockchain innovation allows a verified his to resolve certain issues, for example, moderate access to medical information, framework interoperability, improved quality of medical data, and trust-free transactions. in addition, the significant advantages of the usage of blockchain in the security of medical records are highlighted in [13, 22–25]. these include: i. distributed database: each patient on a blockchain can access the entire database from its origin and be able to confirm the records of his/her information or data straightforwardly, without a middle person. ii. peer-to-peer transmission: communication happens legitimately between nodes rather than through a focal hub. every node stores and advances data to every other node. iii. transparency with pseudonymity: every transaction and its related components are obvious to anybody with access 3 a. e. ibor et al. / j. nig. soc. phys. sci. 5 (2023) 992 4 to the blockchain. every node, or client, on the blockchain has a unique identifier of 30 or more alphanumeric code that distinguishes it. clients can stay unknown or give confirmation of their identities to other people. all transactions happen between blockchain addresses. iv. irreversibility of records: once a transaction is entered in the database and the records are refreshed, the records cannot be adjusted, in light of the fact that they are connected to each record that preceded them (subsequently the expression ”chain”). different computational algorithms and methodologies are used to guarantee that the chronicle on the database is lasting, sequentially ordered, and accessible to all others on the system. other advantages of using blockchain technology include the use of smart contracts for computational logic, scalability, and the infeasibility of a single user compromising the entire blockchain [26–30]. from the literature, the inherent problems of centralised and authoritative databases can be addressed by the use of the blockchain. some of these problems include: i. standalone system: this requires authoritative access and mostly not suitable for large scale networks. ii. cloud based systems: it may be cheap to implement but may not have the appropriate mechanism for allowing secured transparent transactions. it can also lead to data loss and theft in the event of espionage or system failure. therefore, it is envisaged that the use of blockchain will help to resolve the highlighted issues since it is a peer-to-peer network where each participant has the database stored locally in each of the nodes in the blockchain network. changes to stored records are transparent to all participants and no participant can authoritatively make a change without the consent of others. 3. materials and method in this section, the design of the proposed system is discussed. the design highlights details of the functionality of the system in terms of data storage, transmission, access and retrieval for all relevant parties. 3.1. method the blockchain is developed by creating a secure and transparent environment for medical records. using a 3-tier architecture, the blockchain serves as the underlying database for the storage and authentication of medical records. the client side of the 3-tier ensures the transmission of data to the blockchain’s network. these data is processed at the logic layer, which interfaces with the blockchain to determine the integrity or otherwise of the medical record using the hash code of each block. these hash codes are used to keep the medical records safe in the blockchain. the hash codes are generated by mapping a variable length input such as a patient record to a fixed length output. this fixed length output is the hash value of the record, and will change when the block of information holding the medical record is changed. the blockchain network is a decentralised structure with peer-to-peer nodes. these nodes inspect and authenticate the validity of any new transaction such as a storage or retrieval request. this request is then fulfilled though distributed consensus by different validating nodes. moreover, no single validating node can have centralised control of the blockchain, making it difficult for medical records to be corrupted, distorted, stolen or compromised. the architecture of the proposed system is shown in figure 2. figure 2: architecture of a secured his with blockchain technology in the 3-tier architecture as shown in figure 2, there is a client side, application server and the blockchain database, in which the server-side procedures, data memory access, data storage and user interface (ui) are created and kept up as autonomous modules on isolated stages. the 3-tier design permits any of the three layers to be redesigned or supplanted freely. the ui is actualised on a workstation and utilises a standard graphical user interface with various modules running on the application server. the three layers in the 3-tier architecture are as follows: i. client side: this is the first tier and shows data identified with services accessible on the desktop application. this layer speaks with different layers by sending results to the peer-to-peer network and different layers in the system. it is a platform that enables the client to interact with the system. the client side was built on hypertext markup language (html), cascading style sheet (css) and javascript. ii. application server: this is the second, which is additionally called the business logic or logic layer. this layer controls application usefulness by performing comprehensive processing. this layer was built on node.js, composer, and docker engine. iii. blockchain database: this is the last tier of the architecture. this layer houses the database servers where data 4 a. e. ibor et al. / j. nig. soc. phys. sci. 5 (2023) 992 5 is stored and recovered. information in this tier is kept free of application servers or business logic. this layer was built on ganache, yarn and truffle (ethereum hyperledger smart contracts). 3.2. activity diagram an activity diagram represents a series of actions or flow of control in a system similar to flowchart or a data flow diagram [31]. the activities modeled can be sequential and concurrent. figure 3 shows the activities performed by each entity/class of the system and these activities are discussed thus: i. the health personnel and patient attempt to login by entering their respective usernames and passwords, and await authorisation from the blockchain database. if the username and password is invalid it aborts the operation but if valid the users (health personnel and patient) gains access into the system and are assigned individual privileges. ii. the health personnel views patients’ medical history, diagnose, run tests on the patient and then upload the medical results into the system. the blockchain encrypts the medical result and shares to multiple participants in the network for consensus. iii. the patient views the medical result uploaded by the heath personnel and can request for modification in biodata. the request is sent to the blockchain database and propagated across the network for subsequent approval or decline of the request. if the request is approved the changes are effected otherwise the operation is aborted. one participant cannot make changes without the cosensus of other participants in the network, otherwise the data is said to be compromised. 4. experimental results and discussion an implementation of the architecture of the proposed systems is given in this section. several experiments were conducted to test the developed application for realising distributed access to medical data based on the design of figures 2 and 3. the application used is built using existing technologies such as node.js, truffle smart contract, inter-planetary. 4.1. testbed of the experiments the proposed system was implemented on a windows machine running windows 10 64-bit operating system, x64-based processor with 4gb ram and intel ® core i3 6100u cpu @2.30 ghz 2.30ghz. 4.2. software components an implementation of the architecture of the proposed systems is given in this section. the software used is built using existing technologies such as node.js, truffle smart contract, inter-planetary file system, docker engine. with the implementation of the blockchain, medical records were cryptographically stored on a peer to peer network. the components used in building the software are as follows: i. html: is the standard markup language for creating web pages. it describes the structure and elements of a web page. ii. css: it is a style sheet language used for describing the presentation of a document written in a markup language like html. iii. javascript: javascript, often abbreviated as js, is a highlevel, interpreted programming language that conforms to the ecmascript specification. iv. python: python is an interpreted, high-level, generalpurpose programming language. python has a design philosophy that emphasises code readability, notably using significant whitespace. it provides constructs that enable clear programming on both small and large scales. v. go: go is a statically typed, compiled programming language that is syntactically similar to c, but with memory safety, garbage collection, structural typing, and cspstyle concurrency. vi. yarn (extension yarn.lock): in order to get consistent installs across machines, yarn needs more information than the dependencies you configure in your package.json . yarn needs to store exactly which versions of each dependency were installed. to do this yarn uses a yarn.lock file in the root of your project. vii. docker engine: it is the underlying client-server technology that builds and runs containers using docker’s components and services. docker engine supports the tasks and workflows involved to build, ship and run containerbased applications. viii. electron.js: formerly known as atom shell is an opensource framework developed and maintained by github. electron allows for the development of desktop gui applications using front and back end components originally developed for web applications: node.js runtime for the backend and chromium for the frontend. electron is the main gui framework behind several notable opensource projects including atom, visual studio code, and light table. ix. interplanetary file system (ipfs): interplanetary file system (ipfs) is a protocol and network designed to create a content-addressable, peer-to-peer method of storing and sharing hypermedia in a distributed file system. ipfs is a peer-to-peer distributed file system that seeks to connect all computing devices with the same system of files. ipfs could be seen as a single bittorrent swarm, exchanging objects within one git repository. in other words, ipfs provides a high-throughput, contentaddressed block storage model, with content-addressed hyperlinks. 4.3. results and discussion this section discusses the results of the experiments conducted to validate the efficacy of the proposed system. at the initial stage, a folder is created and populated with any category of health records either for the health personnel 5 a. e. ibor et al. / j. nig. soc. phys. sci. 5 (2023) 992 6 figure 3: the activity diagram of the proposed system figure 4: encrypted folder of health personnel figure 5: encrypted patient’s folder (doctor, nurse etc.) or the patient. this folder is then encrypted, for which a hash value is generated. the generated hash value as shown in figure 4 can be used to perform such operations as the copying and sharing of the folder across the peer-to-peer network. both the health personnel such as the doctors and the nurses, and the patients can participate in the viewing of the figure 6: adding file to the encrypted folders figure 7: importing files with hash value shared folder. the folder at the same time can be opened in a browser to view the records stored in it including the ability to download it to a local machine. 6 a. e. ibor et al. / j. nig. soc. phys. sci. 5 (2023) 992 7 figure 8: activities on the blockchain similarly, a patient’s folder undergoes the same process as shown in figure 5. in the next stage, files are added to the created and encrypted folders for each entity participating in the blockchain. the added files are encrypted and hash values assigned to them to protect the integrity of their contents. this process is depicted in figure 6. files can then be imported across the network and stored locally using the hash value earlier created for each file as illustrated in figure 7. in this way, health personnel and patients have access to the health records transparently, and modifications to files are flagged when the hash values change. all transactions (activities) in the blockchain by the different participating parties as well as the processes on data are also transparent and available to all nodes in the blockchain. sample transactions are depicted in figure 8. furthermore, we can view the total amount of data stored by multiple nodes participating in the network as well as the number of nodes connected to the blockchain per unit time. this number increases as more nodes participate in the blockchain (see figure 9). in this sense, it is possible to maintain a transparent and distributed database of health records, which are cryptographically secure and immutable over time. figure 9: used space and peers connected to the blockchain network 5. conclusion the proposed system runs on a peer-to-peer network to eradicate the actions of a middle man or administrator. interaction between the patient and health personnel, including having access to medical records shapes a crucial piece of the focal activities underlying health information systems. the proposed system was able to achieve the decentralisation of the medical records database to enhance the security and privacy of data on the modeled peer-to-peer network. the encryption of patients’ data across the ipfs and the relevance of a public/private key pair to share, upload and download data on the blockchain network had an added advantage of the security and privacy of medical records. the proposed system was designed to aid health management organisations in the storage and retrieval of medical records. the system is modeled to tackle problems arising from data integrity, data security and the manipulations of medical records, which have been the major challenges in extant health information systems. since the 7 a. e. ibor et al. / j. nig. soc. phys. sci. 5 (2023) 992 8 blockchain is a public ledger that provides the information of all the participants and all digital transactions that have ever been executed, it helps to negate the relevance of authoritative access to a database of medical data. in this sense, the proposed system will bring about an accurate and efficient way of transferring medical records from health personnel to the patients without instances of record manipulation. for future work, we intend to provide a large scale implementation of this work on a district-wide basis to ascertain its resilience in real time. acknowledgment the authors appreciate the handling editor and the reviewers for their valuable comments that improved the quality of this paper. references [1] k. a. wager, f. w. lee, & j. p. glaser, health care information systems: a practical approach for health care management, john wiley & sons, (2017). 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[31] j. rumbaugh, i. jacobson, & g. booch, unified modeling language reference manual, pearson higher education, (2004). 8 j. nig. soc. phys. sci. 5 (2023) 999 journal of the nigerian society of physical sciences tectonic setting of the syenite around igarra, southwestern nigeria: constraints from geochemistry f. c. ugbea, o. e. ominigbo id b,∗, a. o. akanoa adepartment of geology, delta state university, abraka, delta state, nigeria bdepartment of earth sciences, federal university of petroleum resources, effurun, delta state, nigeria abstract syenites are relatively rare within the nigerian basement complex. as a result of their rarity, these rocks have been given less research attention over time and are consequently poorly understood. the syenitic rocks at igarra were studied to ascertain their tectonic evolution using geochemistry. sampling was carried out using the survey-type geological field mapping approach. a total of 10 samples of syenitic rocks were collected for laboratory analyses. compositionally, the rocks are intermediate with regards to sio2 content (58.02% – 60.58%), having al2o3 and alkali (na2o + k2o) compositional ranges of 15.34% – 15.52% and 8.99% – 9.7% respectively. the sampled rocks are similar and consistent in their trace and rare earth elements concentrations (the only exception being zr with values ranging from 4 ppm to 79 ppm). the rocks are relatively enriched in ba, k, ti, and sr but depleted in tc, nb, u, hf, yb, te and ta. the syenites also show fairly high ratios of rb/nb and rb/sr with mean values of 488.627 and 0.171 respectively. as seen from the geochemical analyses, the syenites around igarra are high-k calcalkaline, alkalic to alkalic-calcic. the rocks are peraluminous in character as shown by the bivariate plot of a/nk vs. a/cnk. sedimentary protolith with continental crustal parent magma is inferred for these rocks. the similarity and consistency of the trends of major, trace and rare earth elements is indicative of cogenetic origin for the rocks. the geochemistry and discrimination plots for the rocks indicate geodynamic setting ranging from orogenic to post-orogenic. a volcanic arc geotectonic setting is interpreted for the igarra syenites, with magma emplacement and evolution thought to have been initiated during the late stages of the pan-african reactivation and continued into post-orogenic times. doi:10.46481/jnsps.2023.999 keywords: syenite, tectonic setting, geodynamic evolution, pan-african orogeny, nigerian basement complex article history : received: 21 august 2022 received in revised form: 04 october 2022 accepted for publication: 05 november 2022 published: 24 february 2023 © 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: o. j. abimbola ∗corresponding author tel. no: +234 8075961910 email address: ominigboedafe@gmail.com (o. e. ominigbo id ) 1. introduction syenites are relatively scarce in nigeria, making them to attract less research attention. as such, these rocks are often given less research attention and are consequently poorly understood. however, the syenites within the nigerian basement complex are widely distributed amongst the crystalline coun1 https://orcid.org/0000-0002-3132-8856 https://orcid.org/0000-0002-3132-8856 ugbe et al. / j. nig. soc. phys. sci. 5 (2023) 999 2 try rocks – granites, gneisses, and schists [1 – 3]. the rarity of syenites relative to other crystalline rocks is not peculiar to the nigerian basement complex as similar occurrence and distribution pattern has been recognized elsewhere [4]. the occurrence of syenites in nigeria is restricted to two contrasting petrographic settings with a group associated with the older granites whereas the other member restricted to the younger granite series. the older granites syenites are characterized by significantly high concentrations of cao, mgo and k2o, with k2o typically exceeding na2o. on the other hand, the syenitic rocks of the younger granites show broader compositional ranges, often with na2o/k2o ratios close to unity [2]. as a result of episodicity and prolonged duration of geological events, different geological settings with diverse geochemical and mineralogical attributes have evolved in the basement complex of western nigeria. to be able to accurately decipher such diverse episodes of geological events and tectonic settings, detailed geological and geochemical studies are required [5]. more so, alkaline rocks typically occur in diverse geodynamic settings – post-orogenic, rift or intraplate tectonic environments. the understanding of these rocks and their tectonic settings provide significant insights into understanding magmatic processes within and around the continental lithosphere [6]. furthermore, it has been observed that by determining the petrogenesis of a rock suite from a given geological regime, it is possible to infer limits on the geodynamic conditions which were responsible within the mantle for the tectonic or crustal activity during the time of formation of the suite of rocks [7]. although some appreciable works have been done with regards to geological field mapping and geochemistry of the nigerian basement complex, there remains a glaring need for more studies aimed at enhancing the current understanding of the geodynamic evolution of the crystalline rocks within the region [8]. notwithstanding the series of researches carried on the igarra schist belt, there is scarcity of detailed studies on tectonic setting of the syenites in the area. it has been argued that such paucity of reliable data or studies has led to “individual prejudices” with regards to the interpretation of the evolution of the basement complex in nigeria [8 – 9]. in view of the apparent knowledge gap regarding the syenitic rocks in nigeria, the present study seeks to investigate the tectonic evolution of the syenites around igarra in south western nigeria using evidences from geochemistry. geochemical analyses were carried out using data derived from geological field mapping of the area of the study. the use of geochemical data for deciphering and reconstructing the earth’s history, processes and products is a widely acceptable practice in the earth sciences research community [5, 10]. this study is expected to give further insights into the geochemistry of the syenites in nigeria. it is also expected that findings from the present study will contribute towards the overall understanding of the geology and evolution of the nigerian basement complex. figure 1: simplified geological map of west africa showing the position of nigeria and its pan-african basement complex, the congo-gabon craton, the west african craton and the tuareg shield [modified after 11 and 14] 2. regional geological setting and previous studies on the igarra syenite the present study was carried out around igarra in southwestern nigeria. the igarra crystalline rocks are part of the nigerian basement complex generally believed to have been impacted by the pan-african orogenic event (650 ma ± 150). the nigerian basement complex is an essential component of the pan-african mobile belt occurring east of the west african craton and south of the tuareg shield, northwest of the congo craton (figure 1). the nigerian basement complex is widely thought to have resulted from the collision of the passive continental margin of the west african craton and the active tuareg shield (the pharusian belt). this orogenesis often referred to as the trans-saharan pan-african orogen led to the development of high-grade metamorphism, thrust-nappe, massive granite plutons and late orogen-parallel tectonics [7, 11 – 12]. the pan-african orogeny was accompanied by the emplacement of syn-tectonic to post-tectonic granites, adamellites, tonalities and granodiorites which are commonly referred to as the older granites [13 – 14]. the older granites are typically weakly foliated and porphyroblastic in places whilst being nonfoliated and equigranular in others. the older granites vary from leucocratic to mesocratic in colour with diverse mineralogical compositions, ranging from biotite granites, hornblendebiotite-granites, hypersthene granites, adamellites, syenites to garnetiferous granites and muscovite-biotite-granites [12]. although the older granites generally have calc-alkaline attributes, the syenites within the nigerian basement complex are typically peralkaline [3]. [15 – 16] have reported on the hydrogeological properties of parts of the southwestern nigeria. the igarra schist belt is thought to occur as a broadly n – s trending zone of low – medium grades metasediments and metavolcanics ranging from basic to ultrabasic [17]. as a result of the ubiquity of the pan-african thermotectonic reactivation, the protolith age of the basement complex in nigeria remains a subject of debate [18]. although there is a general consensus that the entire basement complex in nigeria was affected by 2 ugbe et al. / j. nig. soc. phys. sci. 5 (2023) 999 3 the pervasive pan african reactivation, there remains a debate on the status of older orogenies within the basement as some authors have opined that traces of all pre-pan-african events must have been obliterated. the policyclicity of the nigerian basement complex has made the determination of the true age of the rocks rather more difficult [8]. two distinct compressive stresses (ne – sw and e – w) have been reported to have affected the igarra schist belt during the pan-african event, with the relatively younger e –w event being more pervasive as structures related to it are more dominant within the basement [19]. according to [19], the syenite and granite in the igarra area are characterized by e – w structures whereas compared to other rock types within the belt, the ne – sw structure are fairly absent. the area of the present study is located within the igarra schist belt in southwestern nigeria. generally, the igarra area is made up of several intrusive and metasedimentary rocks. notable petrological units in the area include schist, marble, granite, calc-gneiss, quartzite and pegmatitic rocks [17, 20]. as reported by [17], the igarra syenites intrude the marble/calcic gneiss and the quartz-biotite/mica schist, and are considered to be relatively younger than the pegmatites of the igarra schist belt. greenschist to amphibolite grades of metamorphism have been reported for the igarra schist belt. the greenschist and amphibolite facies are characterized by the presence of chlorite + quartz + epidote ± actinolite and hornblende + biotite + plagioclase ± quartz respectively [17]. the syenites around igarra and other associated crystalline rocks within the igarra schist belt have been the subjects of some previous studies [e.g.12, 17, 20 – 21]. however, apart from [21], most of the existing works on the area are largely regional in scope without paying close attention to the syenites outcropping around the area. more so, the study by [21] only dwelt on the petrochemical and geotechnical attributes of the syenites. the general petrology and structural characteristics of the granitoids around the igarra have been reported by [17]. again, their study was restricted to geological field occurrence and petrographic attributes of the rocks without paying attention to the geochemistry of the rocks. thus, the present study seeks to use geochemical evidence to ascertain the tectonic setting and evolution of the syenites of the area with a view to situating the rocks within the larger context of the geodynamic evolution of the nigerian basement complex. 3. research methods geological field mapping was carried out during which ten (10) fresh rock samples were collected for laboratory analysis. field mapping and sampling were carried out following the survey-type geological field mapping technique described by [22]. rocks were observed in their in situ conditions, their field relationships and mappable mineralogical attributes noted and then, fresh rock samples were collected from unweathered surfaces for laboratory analyses. only unaltered rock samples were collected for analyses in order to ensure that such samples give the true geochemical signatures of the rocks. prior to laboratory analyses, the samples were crushed in hardened steel figure 2: simplified geological map of nigeria showing the area of the study [modified after 17] jaw crusher before being pulverized into fine powder using a disc mill. sample preparation and analyses were carried out at the activation laboratories in vancouver, canada. major elements (na2o, mgo, al2o3, k2o, cao and feo) compositions were determined using thermal desorption-mass spectrometry. sio2 concentration was analyzed using fusion-inductively coupled plasma; whereas ti was analyzed using total digestioninductively coupled plasma (td-icp). loss on ignition (loi) was determined for each of the samples using the gravimetric method. the trace elements and rare earth elements (rees) were analyzed using total digestion-inductively coupled plasma. sc was analyzed using the td-icp. for most of the major oxides, detection limit was 0.01% except for tio2, and p2o5 which had detection limits of 0.0005% and 0.001% respectively. the trace elements and rres were analyzed using a detection limit of 0.01 ppm. trace elements and rees analyzed with different detection limits include: li, ni, pb (0.5ppm), ba, zr, v, sn, cr, sc (1 ppm), ag, cs, mo, ti, eu, re (0.05 ppm), bi (0.02 ppm), zn, rb, sr, cu (0.2 ppm). although high sensitivity of analytical methods to elemental concentrations in rocks (like the ones adopted for the present study) is highly desirable in geochemical analyses, an excellent detection limit in itself does not necessarily translate into accuracy of analytical outcomes. to check for, and to minimize possible sources of error(s) therefore, internationally certified reference materials were used for instrument calibration, method validation, quality control and comparisons as suggested by [10]. rare earth element (ree) and trace element normalizations were done using the standard chondrite values and procedures given by [23] and [24] respectively. those values were then used in plotting discrimination diagrams to ascertain the ree and trace elements patterns of the syenites. 4. results the results of the geochemical analyses (major, trace and ree) are shown in tables 1, 2 and 3 below. alongside the 3 ugbe et al. / j. nig. soc. phys. sci. 5 (2023) 999 4 major elements composition, some important major oxides’ parameters and variables (na2o + k2o, n+k-c, a/nk, cnk, a/cnk, feo + mgo, feo/(feo + mgo) and k2o/na2o) have been included in table 1. similarly, some important trace elements ratios and parameters (rb/sr, rb/nb, rb/zr and y + nb) have been included in table 2. these major and trace elements parameters and ratios alongside other elemental compositions were used in plotting discriminant plots for the present study as well as form the basis for many of the inferences made. the rocks are generally intermediate with sio2 ranging from 58.02% 60.58%. there are significant al2o3 and alkali (na2o + k2o) enrichments with compositional ranges of 15.34% 15.52% and 8.99% 9.70% respectively. the trace elements distribution patterns are fairly consistently in all the samples except in zr (4 ppm – 79 ppm). generally, there is relative depletion of tc, nb, u, hf, yb, te and ta whereas there is a significant enrichment of ba, rb, k, ti and sr. (table 2). 5. discussion 5.1. classification of the igarra syenites the rocks were classified using different major elements. the application of major elements for the classification and nomenclature of igneous rocks is a widely acceptable practice in geochemistry [10]. as shown in figure 3, the sampled rocks are syenitic. the total alkali versus silica content (tas) classification scheme for igneous rocks was adopted for the present study. recommended by the international union of geological sciences (iugs), the tas scheme is suitable for classifying unaltered volcanic rocks, with the total alkalis (na2o + k2o) plotted against the silica contents (figure 3). this classification is premised on the fact that both alkalis and silica are largely mobile in a variety of deuteric processes which commonly lead to the formation of secondary and post-magmatic feldspar [25]. the igarra syenites are generally intermediate in chemical composition with regards to silica content (sio2 ranging between 58.02% 60.58%). similarly, the rocks are fairly enriched in feo, al2o3 and mgo with values in the ranges of 5.01% 5.66%; 15.34% 15.52% and 3.62% 4.78% respectively, giving the sampled rocks a magnesian characteristic (figure 4). the compositionally high concentration of al2o3 and the comparatively lower values of k2o and na2o (6.50% 6.90% and 2.49% 2.94% respectively) broadly give the syenites overall alkalic to high-k calcalkaline and peraluminous signatures recorded in the bivariate plots (figure 5a – 5d). compositionally, the major oxides data obtained from the present study compare closely with earlier ones reported for the igarra syenites [21]. meanwhile, compared to the nepheline syenites around the oyioba-uganga area in southern benue trough as reported by [26], the igarra syenitic rocks are enriched in cao and k2o but depleted in na2o and al2o3. however, the syenites in both areas are compositionally comparable in terms of tio2, mno, feo and p2o5. the high-k calcalkaline characteristics of the igarra syenites are compositionally different from those of the kanoma and sabon gida plutons in northwestern nigeria which [3] reported figure 3: tas classification of the igarra syenites [boundaries modified after 11] figure 4: feo/(feo + mgo) vs. sio2 classification plot for the igarra syenites [after 11] as peralkaline. the high-k calalcaline, relatively high cao and mgo as well as k2o > na2o shown by the igarra syenitic rocks however, closely match the geochemical signatures typical of the syenites associated with the older granites in nigeria as earlier reported by [2]. 5.2. petrogenesis the correlation of trends of major oxides against sio2 on the hacker diagram has been used to infer homogeneity or heterogeneity of rock protoliths [14, 27]. as shown in figure 6, there is a strong negative correlation for all the major oxides plotted. the consistency in trend clearly suggests a common protolith or petrogenetic history for the igarra syenite. according to [27], a strong negative correlation of feo3, al2o3 and na2o on the hacker diagram is indicative of hornblende and pyroxene fractionation. as such, the significantly strong negative correlation of the major oxides against sio2 could be inferred to be indicative that fractional crystallization played a vital role in the evolution of the syenite. the peraluminous 4 ugbe et al. / j. nig. soc. phys. sci. 5 (2023) 999 5 table 1: major elements compositions of syenites around igarra oxides (wt. %) sample 1 sample 2 sample 3 sample 4 sample 5 sample 6 sample 7 sample 8 sample 9 sample 10 s1o2 59.08 60.09 59.68 58.95 58.02 58.90 60.25 59.45 60.58 60.51 t1o2 0.82 0.70 0.68 0.76 0.98 0.80 0.76 0.84 0.74 0.88 al2o3 15.42 15.36 15.38 15.50 15.45 15.49 15.40 15.37 15.52 15.34 feo 5.45 5.27 5.23 5.40 5.66 5.20 5.01 5.31 5.09 5.15 mgo 4.78 4.05 3.98 4.21 4.42 3.70 3.68 3.74 3.62 3.64 na2o 2.49 2.65 2.59 2.75 2.90 2.94 2.77 2.67 2.82 2.78 k2o 6.50 6.54 6.61 6.80 6.80 6.90 6.69 6.72 6.66 6.70 cao 3.79 4.12 3.96 3.97 3.95 3.80 3.74 3.70 3.69 3.64 p2o5 0.30 0.35 0.33 0.34 0.51 0.44 0.36 0.40 0.38 0.42 mno 0.08 0.06 0.05 0.15 0.28 0.09 0.07 0.12 0.09 0.10 loi 0.84 1.16 1.01 0.97 0.90 1.18 1.03 0.93 0.98 1.04 total 99.55 100.35 99.45 99.80 99.87 99.44 99.76 99.25 100.17 100.20 na2o+ k2o 8.99 9.19 9.2 9.55 9.70 9.84 9.46 9.39 9.48 9.48 n + k – c 5.20 5.07 5.24 5.58 5.75 6.04 5.72 5.69 5.79 5.84 a/nk 1.72 1.67 1.67 1.62 1.59 1.57 1.63 1.64 1.64 1.62 cnk 12.78 13.31 13.16 13.52 13.65 13.64 13.20 13.09 13.17 13.12 a/cnk 1.21 1.15 1.17 1.15 1.13 1.14 1.17 1.17 1.18 1.17 feo + mgo 10.23 9.32 9.21 9.61 10.08 8.90 8.69 9.05 8.71 8.79 feo/(feo + mgo) 0.53 0.57 0.57 0.56 0.56 0.58 0.58 0.59 0.58 0.59 k2o/na2o 2.61 2.47 2.55 2.47 2.34 2.35 2.42 2.52 2.36 2.41 table 2: trace elements concentrations in (ppm) of syenites around igarra element/ sample1 sample2 sample3 sample4 sample5 sample6 sample7 sample8 sample9 sample10 ratio ppm ppm ppm ppm ppm ppm ppm ppm ppm ppm sr 1200 904 950 1200 1100 1300 942 1200 1200 935 u 1.4 1.5 1.5 1.6 1.4 1.5 1.7 1.4 1.4 1.3 k 44500 23400 24600 25700 36300 30000 30000 30000 25500 26500 rb 194 152 162 151 194 182 241 210v 167 185 cs 5.1 5.26 5.44 5.26 5.31 4.94 6.69 4.88 4.64 4.45 ba 4130 3250 3460 3360 3900 3420 3440 3530 3990 2990 th 5.3 6.5 6.7 8.4 6 8.4 8.8 6.8 6.4 6.5 ce 66.2 64.2 68.8 65.9 67 67.9 65.1 66.9 73.8 66.2 p 3490 2800 2320 2600 4360 4200 4120 4430 3240 3000 ta 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 nb 0.9 0.3 0.1 0.1 1.3 2 1.5 2.9 1.7 0.4 te ¡0.1 ¡0.1 ¡0.1 ¡0.1 ¡0.1 ¡0.1 ¡0.1 ¡0.1 ¡0.1 ¡0.1 tc ¡0.1 ¡0.1 ¡0.1 ¡0.1 ¡0.1 ¡0.1 ¡0.1 ¡0.1 ¡0.1 ¡0.1 zr 43 60 4 6 61 56 58 79 28 10 hf 0.6 0.8 0.1 0.1 1.1 1.2 1.4 1.8 0.6 0.1 ti 2570 1730 1070 1780 3760 4080 4080 4130 1840 1270 y 19 18.4 19.2 17.8 19.7 19.8 18.3 19.2 20.7 17.3 yb 1.8 1.8 1.9 1.8 1.9 1.9 1.9 1.9 2 1.8 y + nb 19.9 18.7 19.3 17.9 21 21.8 19.8 22.1 22.4 17.7 rb/sr 0.162 0.168 0.171 0.126 0.176 0.14 0.256 0.175 0.139 0.198 rb/nb 215.556 506.667 1620 1510 149.231 91 160.667 72.414 98.235 462.5 rb/zr 4.512 2.533 40.5 25.167 3.180 3.250 4.155 2.658 5.964 18.5 5 ugbe et al. / j. nig. soc. phys. sci. 5 (2023) 999 6 table 3: rare-earth element concentrations in (ppm) of syenites around igarra element sample1 sample2 sample3 sample4 sample5 sample6 sample7 sample8 sample9 sample10 ppm ppm ppm ppm ppm ppm ppm ppm ppm ppm la 35.4 34.1 36.9 36 36.4 37.7 36.3 36.4 39.8 30 ce 66.2 64.2 68.8 65.9 67 67.9 65.1 66.9 73.8 66.2 pr 8.1 7.8 8.3 7.8 8.2 8 8.1 8.1 8.4 8.2 nd 31.1 30.1 31.3 30.1 31 30.8 29.8 31.9 32.8 31.1 sm 6.1 5.9 6.1 5.4 6.4 5.7 6 6.5 7 6.4 eu 1.66 1.53 1.62 1.48 1.62 1.58 1.51 1.6 1.72 1.65 gd 5.2 5 5.1 4.9 5.5 5.2 5 5.1 5.7 5.3 tb 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.7 0.7 0.6 dy 3.5 3.5 3.6 3.5 3.8 3.6 3.6 3.6 3.6 3.7 ho 0.7 0.7 0.7 0.6 0.7 0.7 0.7 0.7 0.8 0.7 er 1.9 1.8 2 1.9 1.8 1.9 1.8 1.9 1.9 1.9 tm 0.3 0.2 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 yb 1.8 1.8 1.9 1.8 1.9 1.9 1.9 1.9 2 1.8 lu 0.3 0.3 0.3 0.2 0.3 0.3 0.3 0.3 0.3 0.3 figure 5a: modified alkali-lime index, mali vs. sio2 classification of the igarra syenites [boundaries modified after 11] signature of the rock suggests the melting of sedimentary protoliths for the igarra syenites. basement rocks derived from sedimentary sources are characterized by peraluminous chemical attributes [11, 14, 28]. trace elements and rees have also been widely used as “fingerprints” in understanding the origin and evolution of igneous rocks [e.g. 2, 29 – 31]. ideally, the compositional depletion of tc, nb and ta alongside enrichment of large-ion lithophiles (rb, k, sr and th) suggests a partial melting of crustal origin for the syenites [32]. similarly, the moderate to high fractionations and pronounced negative eu anomalies displayed in the ree patterns are typical of crust-derived granites [28]. the negative eu anomaly exhibited by the samples is suggestive of either plagioclase fractionation or its retention during the melting of the parent magma. more so, the relatively high values of rb/sr and rb/nb ratios further support the lithospheric crustal origin for the syenite. mantle-derived magmas figure 5b: k2o vs. sio2 classification of the igarra syenites [boundaries modified after 11] are characterized by low rb/sr ratios (usually ≤ 0.023) just as high rb/nb ratios are indicative of crustal origin [33]. however, the intermediate compositional concentration of sio2 in the rock poses some doubts to the crustal origin assertion as crust-derived magmas typically record silica content in excess of 60% [28] unlike the sampled rock that gave ≤60%. considering the small sample size used for the present study, it is advisable that the crustal origin suggested for the syenitic rocks at igarra should be taken with caution, albeit it is safe to conclude that the rocks had common source magma and evolutionary history based on the consistency and similarities in composition and distribution pattern, both for the major and trace elements. 5.3. tectonic setting the syenitic rocks around igarra are thought to have been emplaced in an arc environment following a major orogenic event based on the discrimination plots for the tectonic setting (figures 8a – 8c). whereas figures 8a – 8c clearly indicate that 6 ugbe et al. / j. nig. soc. phys. sci. 5 (2023) 999 7 figure 5c: na2o + k2o classification plot for the igarra syenites [boundaries modified after 11] figure 5d: a/nk vs. a/acnk plot for the igarra syenites [boundaries after 11 and 14] the emplacement and evolution of the syenites occurred in an active orogenic belt, figures 8d and 8e suggest that the development occurred in more or less transitional setting between orogenic and anorogenic geodynamic settings. given that the area is part of the basement complex in nigeria generally believed to have been affected by the pan-african reactivation event, it is our opinion that the emplacement and evolution of the igarra syenite is related to the pan-african orogeny which led to the collision of the west african craton and the tuareg shield. [11, 34] opined that the collisional event was preceded by the subduction of the lithosphere beneath an ancient oceanic crust around the eastern flank of the west african craton underneath the tuareg shield. it is believed that it is this ancient oceanic crust that gives rise to the series of basic and ultrabasic rocks of ophilitic complex and high positive gravity anomaly within the basement complex of the eastern axis of the west african craton [11]. given the orogenic to post-orogenic signatures shown from the discrimination plots for the tectonic setting, we profigure 6: hacker diagram (major oxides vs. sio2) the igarra syenites figure 7a: primitive-mantle normalized trace elements pattern for igarra syenites [chondrite normalization values after 24] pose that although the emplacement of the igarra syenite commenced during the active phase of the pan-african reactivation (probably towards the end of the orogeny), the formation and evolution of the rock continued into the early post-orogenic period. similarly, the emplacement and formation of the syenites in the area of the study have also been interpreted to have occurred in a tectonic arc as clearly revealed in figures 8c – 8f. given the significant negative eu anomalies displayed by the rees and the relatively high ratios of rb/nb, rb/sr and rb/zr, an active continental arc tectonic setting is inferred for the igarra syenite. the active continental tectonic setting is further supported by the plot of k2o/nao vs. sio2 (figure 8b). generally, 3 7 ugbe et al. / j. nig. soc. phys. sci. 5 (2023) 999 8 figure 7b: chondrite-normalized ree diagram for the igarra syenites [normalization values after 23] figure 8a: feo + mgo vs. sio2 plot for the igarra syenites [boundaries after 14 and 35] geotectonic models have been proposed for the formation and evolution of syenites gobally: those associated with attributes of mantle evolution; those linked to crustal origin; and those associated with mantle-crustal origins for parent magmas [4]. the crust-derived syenites model proposes melting of crustal substrates with the rocks showing geochemical imprints similar to those recorded in the present study [4]. the arc geodynamic setting inferred for the igarra syenite is in agreement with existing knowledge on the basement complex rocks of south western nigeria which are thought to have developed in a back-arc basin [37 – 39]. the initiation of magmatism during active orogeny and its continuation well into post-orogenic periods have also been reported for basement rocks elsewhere within nigeria [e.g. 11, 14, 40]. it is hoped that findings from the present study will spur further research interests in the syenites in nigeria just as integrated approach of combining geological and mechanical attributes is recommended for such studies. figure 8b: k2o/na2o vs. sio2 plot for the igarra syenites [after 14 and 35] figure 8c: al2o3 vs. sio2 tectonic setting bivariate plot for the igarra syenites [after 14 1nd 35] figure 8d: rb vs. y + nb bivariate for the igarra syenites [after 11 and 14] 8 ugbe et al. / j. nig. soc. phys. sci. 5 (2023) 999 9 figure 8e: rb vs. sio2 discriminant plot for the tectonic setting of the igarra syenites [boundaries modified after 14] figure 8f: tio2 vs. al2o3 bivariate plot for the igarra syenites [after 14 and 36] 6. conclusions the syenites around igarra in southwestern nigeria were examined to ascertain their tectonic setting and petrogenetic history using evidences from geochemistry. the data and resultant bivariate plots for the igarra syenites show that the rocks are high-k calcalkaline, alkali to alkali-calcic and peraluminous. based on the peraluminous character of the rocks, the melting of sedimentary protolith is inferred, with primary magma probably derived from the lithospheric crust. the syenites were emplaced and formed in an arc tectonic environment during the pan-african reactivation event and continued into the early post-orogenic periods. the findings from the study compare closely with those of earlier studies on syenites within the nigerian basement complex. the present study contributes to the existing literature on the syenites in nigeria by giving further insights into the geochemical attributes of the igarra syenites. this is important considering that the syenites in nigeria have over the years been poorly understood relative to the other crystalline rocks within the basement complex. findings from the study are also expected to enhance the overall understanding of the nigerian basement complex with regards to its geodynamic evolution and tectonic setting. references [1] m. o. oyawoye, “the petrology of a potassic syenite and its associated biotite pyroxenite at shaki, western nigeria”, contrib. to mineral. & petrol. 16 (1967) 115. 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[36] a. enterazi harsini, s. a. mazhari, s. saadat, & j. f. santos, “u-pb geochronology, srnd geochemistry, petrogenesis and tectonic setting of gandab volcanic rocks, northeastern iran”, geochronomateria 44 (2017) 269. https://doi.org/10.1515/geochr-2015-0061 [37] k. c. burke & j. f. dewey, “orogeny in africa. in: dessauragie, a. j. & whiteman, t. f. j. (eds.)”, african geology, university of ibadan press, nigeria (1972) 583. [38] a. o. oloyinde & s. b. odeyemi, “the geochemistry, tectonic setting and origin of the massive melanocratic amphibolites in ilesha schist belt, southwestern nigeria”, global journal of pure and applied sciences 7 (2001) 55. [39] f. c. ugbe, u. p. adiela & u. c. ebegbare, “major and trace element geochemistry of granites in koji, kogi state, nigeria”, research journal of environmental and earth sciences 8 (2016) 8. [40] v. u. ukaegbu & b. n. ekwueme, “petrogenesis and geotectonic setting of the pan-african basement rocks in bamenda massif, obudu plateau, southeastern nigeria: evidence from trace element geochemistry”, chinese journal of geochemistry 25 (2006) 122. 10 introduction regional geological setting and previous studies on the igarra syenite research methods results discussion classification of the igarra syenites petrogenesis tectonic setting conclusions j. nig. soc. phys. sci. 5 (2023) 915 journal of the nigerian society of physical sciences corrosion inhibition potential of thiosemicarbazide derivatives on aluminium: insight from molecular modelling and qsars approaches b. t. ogunyemia,∗, f. k. ojob aphysical and computational chemistry unit, department of chemistry, federal university otuoke, bayelsa state, nigeria bdepartment of chemistry, bingham university, karu nasarawa state, nigeria abstract the potentials of six thiosemicarbazide derivatives towards corrosion inhibition were investigated theoretically using density functional theory (dft) and quantitative structural-activity relationships (qsars) methods. their performance as corrosion inhibitors were evaluated using their calculated quantum chemical parameters such as molecular weight, softness, electronegativity, dipole moments, hardness, bandgap energy (∆e), highest occupied molecular orbital energy (ehomo), and the lowest unoccupied molecular orbital energy (elu mo). regression analysis was carried out using the ordinary least square method to develop a model that establishes the relationship between chemical parameters and inhibition efficiencies that have been measured experimentally. according to the results, quantum chemical parameters confirm the inhibition potential of tsc5 to be greater than tsc2, while the predicted inhibition efficiencies of the studied thiosemicarbazide derivatives correspond to experimentally reported values with a root mean square error (%) of 1.116 and correlation coefficient of 0.998. the high correlation demonstrates and validates the quantum chemical approach’s reliability in studying corrosion inhibition on a metal surface. the validation of the developed model internally and externally demonstrates that it is robust and stable, with high predictability. doi:10.46481/jnsps.2023.915 keywords: dft, corrosion inhibitors, qsar, thiosemicarbazide article history : received: 04 july 2022 received in revised form: 04 november 2022 accepted for publication: 07 december 2022 published: 14 january 2023 c© 2023 the author(s). published by the nigerian society of physical sciences under the terms of the creative commons attribution 4.0 international license (https://creativecommons.org/licenses/by/4.0). further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and doi. communicated by: t. owolabi 1. introduction metals are utilized in most constructional operations because of their high strength contrasted with other material classes. however, the pace at which these metals revert to their natural oxide form, a process known as corrosion, has far-reaching implications for national economies and human life in general. ∗corresponding author tel. no: +234 8035811574 email address: ogunyemibt@fuotuoke.edu.ng, btogunyemi@yahoo.com (b. t. ogunyemi) although, corrosion is unavoidable yet substantial barriers in form of corrosion control technology can be utilized to slow it down. over the years, painting, cathodic and anodic protection, galvanizing, and the use of corrosion inhibitors (cis), have been explored to alleviate the corrosion problem [1-2]. corrosion inhibitors are commonly used in industry because of their low cost and ease of application [3-4]. through molecule adsorption, the corrosion inhibitor forms a passive coating that protects the metal from aggressive corrosion thus limiting the rate of corrosion. rather than utilizing high-grade carbon steel, 1 ogunyemi & ojo / j. nig. soc. phys. sci. 5 (2023) 915 2 corrosion inhibitors have permitted lower-grade carbon steel to be utilized, thus lowering capital costs in most industries [3]. aside from this, organic inhibitors are often employed due to their low toxicity, good solubility, compatibility with metals, and effectiveness at a wide temperature range [5-6]. the efficiency of organic corrosion inhibitors to inhibit corrosion largely depends on the adsorption bond strength which in turn depends on many factors such as type of metal, type of inhibitor, the concentration of inhibitor, and the environment. besides these variables, most of the notable organic corrosion inhibitors are plane conjugated systems that include various aromatic cycles containing electronegative atoms [7]. adsorption occurs as a result of the interaction of the inhibitor’s lone pair and/or -orbitals with the metal surface atoms’ d-orbitals, resulting in higher adsorption of the inhibitor molecules onto the surface and the creation of a corrosion protection coating [7]. therefore, the electronic structure of organic inhibitors has a significant impact on how effective they are in preventing metal corrosion. the selection of effective inhibitors has mostly been based on an empirical understanding of their mechanisms of action, macroscopically physicochemical features, and capacity to donate electrons. thus, molecular structure, hydrophobicity, electron density at donor atoms, dispensability, and solubility are the most important factors to consider when choosing an inhibitor. while the experimental determination of inhibition efficiency is a critical step in identifying effective molecules in corrosion chemistry, computational techniques such as density functional theory (dft), qualitative structural activities relationship modeling, and others are becoming increasingly more popular for identifying likely effective molecules in a timely and efficient manner. for instance, quantum chemical computations are a useful tool for investigating and understanding the electronic structures and reaction mechanisms of organic corrosion inhibitors interpreting the experimental results, resolving chemical ambiguities, and predicting molecular parameters that are closely associated with the corrosion inhibition property of the chemical compound. qualitative structural activities relationship is commonly used to relate the quantum descriptors with experimental inhibition efficiencies as well as, developing models for similar molecules with unknown inhibition efficiency. in essence, it is used to investigate the structure-activity relationship of molecules. the most promising technique used to ascertain the electronic structure of matter at the moment is density functional theory, commonly known as chemical ]reactivity theory [8]. in continuation of our work on the investigation and theoretical prediction of organic molecules as inhibitors of corrosion [9], this research work investigates and predicts the corrosion of thiosemicarbazide derivatives (figure 1) in the corrosion inhibition of aluminum from the perspective of molecular modeling and quantitative-structural-activity relationships. thiosemicarbazides are schiff basesorganic compounds formed as a result of condensation of a carbonyl and an amine and according to literature, they have shown to be potential inhibitors. several authors [10-12] have observed that the presence cyano group and other heteroatoms such as oxygen and sulfur atoms make schiff bases effective inhibitors in metallic corrosion in alkaline and acidic conditions [13]. the nitrogen (n) atom’s lone pair of electrons and planarity (π) of the schiff bases are also essential structural properties that influence their adsorption on metal surfaces [14]. these molecules also are useful intermediates in the production of pharmaceutical and bioactive materials, and they are widely utilized in medicinal chemistry. the corrosion inhibition of thiosemicarbazide derivatives studied has been experimentally investigated by fouda et al. [15]. this work presents a theoretical investigation on molecular and electronic properties of thiosemicarbazide derivatives with the aim of determining the relationship between the structural properties of thiosemicarbazide derivatives and their inhibition efficiency. the model derived from the qsar relationships might be useful in predicting other derivatives of thiosemicarbazide theoretically. the quantum parameters like hardness (η), dipole moment (µ), molecular orbital energies (ehomo and elu mo), the charge distribution, total energy (etotal), a fraction of electrons (∆n) transfer and electronegativity (χ) values were estimated and correlated with inhibition efficiencies (% ie). 2. methodology 2.1. molecular modeling molecular modeling of six (6) thiosemicarbazides (figure 1) were performed using the spartan 14 program package [16]. density functional theory (dft) with the b3lyp exchange functional [17-18] at 6-311g* basis set was utilized to optimize geometrical structures. during optimization, dihedral and bond angles, as well as all bond lengths, were free of constraints while real vibrational frequencies were ensured for the entire geometries. the fukui functions of thiosemicarbazides derivatives were evaluated using electron populations for their neutral and ionic neutral species. when an inhibitor is in contact with aluminum (al), electrons flow from the inhibitor which has lower electronegativity to the metal which has higher electronegativity until the chemical potentials are equal [19]. equation 1, as a first approximation, gives the proportion of electrons transferred (∆n). ∆n = χal −χinh 2(ηal + ηinh) (1) where χinh and χal symbolize absolute electronegativities of organic inhibitor and aluminum (al) respectively while ηinh and ηal symbolize absolute hardness of the organic inhibitor and aluminum (al) respectively. these parameters are related to electron affinity (a) and ionization potential (i), both of which can be used to predict chemical behavior as shown in equations 2 to 3 [20]. η = i p − ea 2 (2) χ = −µ = i p + ea 2 (3) theoretically, aluminum (al) has absolute electronegativity(χal) and hardness (ηal) values of 3.23 ev/mole and 0 ev/mole) respectively[19, 21]. 2 ogunyemi & ojo / j. nig. soc. phys. sci. 5 (2023) 915 3 figure 1. structures of the studied thiosemicarbazide derivatives meanwhile, the inverse of hardness, softness, is likewise a quantum parameter that may be determined using equation 4 [22]. s = 1 η (4) koopman’s theorem [18] asserts that electron affinity (a) and ionization potential (i) are also related to the molecular orbital energies (ehomo and elu mo) of an inhibitor, as shown in eqs. (5) and (6) i p = −ehomo (5) ea = −elu mo (6) equation (7) was used to determine the electrophilicity index which evaluates the stabilization energy when charges are transferred to a system from an environment [23] ω = µ2 4η (7) whenever the values of chemical potential (µ) and electrophilicity index (ω) are low, it indicates that inhibitory compounds are more reactive nucleophiles, whereas high values indicate that they are more reactive electrophiles furthermore, the local reactivity index describes the reactivity of a specific chemical atom in relation to an organic inhibitor’s adherence to a certain surface of a metal. also, using the computation of the determinate variance, the difference in electron density for a nucleophile f +(r) and electrophile f − (r) as the funki functions may be computed [24]. f +(r) = qk n+1 (r) − qk n (r) (for nucleophilic attack) (8) 3 ogunyemi & ojo / j. nig. soc. phys. sci. 5 (2023) 915 4 f −(r) f k == qk n (r) − qk n−1 (r) (for electrophilic attack) (9) where qkn+1(r) , qk n (r) qk n−1 (r) are the electronic densities of anionic, neutral, and cationic species respectively. 2.2. quantitative structural activity relationship (qsar) qsar was developed to explain the structure-activity relationship of molecular descriptors based on quantum chemical calculations of six thiosemicarbazide derivatives as corrosion inhibitors. the quality of a model in this type of analysis is determined by the model’s ability to fit and predict accurately. this method looks for a relationship in the form of an equation that connects molecular descriptors to the inhibition efficiency determined experimentally. lukovits’ linear equation is commonly employed in corrosion inhibitor research to allow the correlation of quantum molecular descriptors with experimental inhibitory efficacy [25] as shown in equation 10 below %ie = ά + β1 x1 + β2 x2..........βn xn (10) x1, x2.... xn indicate are quantum chemical descriptors of the modelled inhibitors while ά and β are the regression coefficients calculated from regression analysis. 2.3. test of model the model developed was statistically validated by utilizing the squared fitting factor (r2), adjusted fitting factor (r2a ), cross-validation (cv.r2) and variation ratio (f). the adjusted fitting factor(r2a), is defined as follows: r2a = n − 1) x r2 − p n − p − 1 (11) where n represents the number of observations (study molecules) and p is the number of descriptors, cross-validation (cv.r2) is a mathematical approach for ensuring the reliability of the qsar model as given in equation 12. cv.r2 = ∑ (yobs − ycal) 2∑ ( yobs − ŷobs )2 (12) the variance ratio (f) was also used to determine the overall significance of the regression coefficients [26]. as shown in equation 13 below, it is defined as the ratio of the regression mean square to deviations mean square: f = ∑ (ycal−ŷobs)2 p∑ (yobs−ycal ) 2 n−p−1 (13) at p < 0.05, a model’s estimated f value should be significant; consequently, the f value for the overall significance of the regression coefficients should be high [27]. 3. results and discussion 3.1. quantum descriptors of inhibitors the optimized structures of the six thiosemicarbazide derivatives using dft/b3lyp/6-311 g* are displayed in figure 2. suggested quantum descriptors that may be responsible for the effective inhibition of the investigated molecules are; energy band gap (∆e), ehomo, elu mo, dipole moment (dm), polarizability, ovality, log p, affinity (ea), global electrophilicity (ω), ionization potential (ip), softness (s), electronegativity (χ), chemical hardness (µ), total energy, electron transfer (∆n) and solvation energy (esolv), (table 1). the ehomo is associated with the electron donor ability of an inhibitor. the high or elevated ehomo of inhibitors indicates a higher tendency to donate electrons to the corresponding lower molecular orbital of the metal. this improves the adsorption capacity and inhibition efficiency of inhibitors to the surface of a metal. therefore, the effectiveness of an inhibitor can be improved by enhancing the transferring process. in table 1, it is clear that the ehomo value for the six thiosemicarbazide derivatives molecules (tsc1−6) corresponds to tsc5 > tsc2 > tsc1 > tsc6 > tsc4 > tsc3. the highest ehomo (-5.54 ev) for tsc5, is indicated as the best electron-donating inhibitor to the empty d-orbital of the metal. organic inhibitors, on the other hand, not only donate electrons to the metal’s empty dorbitals but also accept electrons from the metal’s d-orbital. thus elu mo shows the capability of an inhibitor to accept electrons from the metal which would definitely improve the inhibition effect on the surface of the metal. the elu mo for tsc1−6 follows the sequence: tsc6 > tsc2 > tsc1 > tsc4 >tsc5 >tsc3, indicating that the tsc3 has an improved propensity to accept electrons from the surface of a metal. the evaluation of the distribution of electron density in molecular orbitals (fig 3) indicates that electrons are localized on the atoms of both aromatic rings of the six studied molecules. the energy difference which is also known as bandgap energy (∆e) which is the difference in elu mo and ehomo of a molecule, relates the inhibitor’s reactivity to adsorption on the metal surface. decreasing the ∆e value of inhibitors increases the reactivity between the metal and inhibiting molecule which consequently increases the binding capacity at the surface of the metal. this increase in binding capacity can result in an increase in the inhibitory efficiency (%ie) of the inhibitor since the energy required to remove the electron from the highest occupied molecular orbital will be less. the addition of -c6h5, -c6h3(no2)2 -coch2cn, and –coc6h5 substituents to semicarbazide (tsc6) to give their respective derivatives (tsc1−5) greatly have an effect band gap. a very low band gap of 4.42 ev was recorded for tsc2 when the phenyl group was attached to tsc1 and a further reduction to 2.94 ev in tsc3 when the no2 substituent was attached to the attached phenyl group could be a result of the destabilization of the highest unoccupied molecular orbital (lumo) to lower energy level. the presence of a positive charge on the nitrogen atom in no2 deactivates the aromatic system and “sucks” electron density from the aromatic system by positive inductive effect and influences the resonance stabilization of the system [28]. 4 ogunyemi & ojo / j. nig. soc. phys. sci. 5 (2023) 915 5 figure 2. optimized molecular structures of thiosemicarbazide derivatives using dft/b3lyp/6-311g* table 1. quantum chemical parameters of some of the thiosemicarbazide derivatives using dft/b3lyp/6-31 1g∗ method quantum parameters tsc 1 tsc2 tsc 3 tsc 4 tsc 5 tsc 6 ehomo (ev) -5.68 -5.56 -6.22 -6.07 -5.54 -5.76 elu mo (ev) -0.63 -1.14 -3.28 -1.13 -1.36 -0.20 ip (ev) 5.68 5.56 6.22 6.17 5.94 5.76 ea (ev) 0.63 0.53 3.28 1.13 1.36 0.20 hardness (η) 2.53 2.221 1.47 2.52 2.09 2.78 softness (s) 0.404 0.397 0.680 0.396 0.436 0.35 electronegativity (χ) 3.155 3.35 4.75 3.65 3.45 2.98 electron transfer (∆n) e− 0.0145 -0.027 -0.517 -0.083 -0.053 0.045 eback−donation -0.788 -0.838 -1.18 -0.91 -0.86 -0.75 electrophilicity index (ω)d2/ev 1.967 2.526 7.674 2.643 2.85 1.597 energy difference (∆e) ev 5.05 4.42 2.94 5.04 4.18 5.56 polarizability (α) 53.50 60.54 64.44 57.93 62.25 46.43 log p 0.33 1.11 0.37 0.34 1.14 -0.67 area (a) cm3 189.62 273.01 322.49 248.90 294.11 108.60 volume (v) cm3 164.20 250.67 292.94 218.81 270.74 78.11 ovality 1.31 1.42 1.51 1.42 1.45 1.22 solvation energy (esol) kj/mol -58.13 -41.67 -40.08 -52.90 -46.05 -67.65 dipole moment (debye) 5.65 5.02 2.13 6.07 2.06 4.07 molecular weight (amu) 167236 243.334 333.328 235.271 271.244 91.138 psa 49.083 32.635 104.823 69.777 42.974 66.43 exp. ie 27.4 94.3 92.8 60.8 95.5 27 polar surface area(psa), ionization potential (ia), electron affinity (ea) the ∆e value shown in table 1 indicates tsc3 < tsc5 < tsc2 < tsc4 < tsc1 < tsc6, suggesting that tsc3 tsc5 and tsc2 have a low energy gap, higher reactivity, and therefore, better performance than other molecules. however, the higher experimental inhibition efficiency reported for tsc3 with the lowest band gap could be a result of an external factor acting on the inhibitor. in an acidic solution, the hydrogen evolved can cause a reduction of the nitro group in tsc3 which can aid the desorption of tsc3 thus making it a less effective corrosion inhibitor than tsc2 and tsc5 absolute hardness, electronegativity, and softness are indicators of reactivity that are derived from electronic energy (e) with respect to the number of electrons (n) at a constant external potential t(r) [29]. the absolute hardness and softness reactivity descriptors are associated with the description of soft and hard solutions through the acid and base theory [30-31] 5 ogunyemi & ojo / j. nig. soc. phys. sci. 5 (2023) 915 6 established that the hardness of all atoms in a molecule is actually equalized, and molecular hardness equals the geometric mean of component atoms’ chemical hardness [32]. the transfer of electrons between chemical species occurs in every reaction and in contrast to hard molecules, soft molecules are affected by charge transfer. the principle of maximum hardness (pmh) [33-34] states that chemical stability is directly linked to chemical hardness and that molecules that are hard are more stable and less reactive. as a result, the chemical hardness of a molecule denotes the resistance of the electron cloud of ions, atoms or molecules to polarization or deformation resulting from chemical reaction perturbations. the chemical hardness of a reaction also gives vital information for predicting the reaction mechanism and estimating the products generated throughout the reaction [35]. the values of absolute hardness for tsc1−6 follows the sequence: tsc3 < tsc5 < tsc2 ≈ tsc4 < tsc1< tsc6. this result shows that tsc2, tsc3 and tsc5 have a low hardness value of 1.47, 2.09, and 2.51 ev respectively, compared to the other three studied derivatives. hence, these three molecules are less stable and more reactive. derivatives with low hardness values show low bandgap and hence the high inhibition efficiency. this corresponds to the widely held idea that soft molecules should have a small energy gap whereas hard molecules should have a high energy gap. the softness values of the studied molecules using dft is as follows the sequence: tsc3 < tsc5 < tsc2 < tsc4 < tsc1 < tsc6. this trend shows that the first three soft thiosemicarbazide derivatives (tsc2, tsc3 and tsc5) have experimental inhibition efficiencies that are more than 90%. hence low global hardness value (that is, the high global softness value) is likely to show high inhibition efficiency. the experimental inhibitory efficiency of the investigated compounds agrees with this trend. the tendency of an atom in a molecule to attract shared pair of electrons to itself is known as electronegativity and it is a key concept to understanding the nature of chemical interactions [34]. the electronegativities of atoms in a molecule are equilibrated during molecule formation and molecular electronegativity is the geometric mean of the atoms’ electronegativities [34]. the electronegativity values of studied molecules tsc1− 6 (table 1) are: tsc6 < tsc1< tsc2 < tsc5 < tsc3 < tsc4. the dipole moment (dm) is the first derivative of the energy with respect to an applied field [36] electronic parameter resulting from the unequal distribution of charges on various atoms in a molecule. it measures the polarity of a polar covalent bond, predicts the direction of the corrosion inhibition process, and gives information about the distribution of electrons in the molecules [36-38]. the total dipole moment, on the other hand, simply represents a molecule’s global polarity. the overall molecular dipole moment can be estimated as the vector sum of individual bond dipole moments for a whole molecule. a high dipole moment may increase the adsorption between the metal surface and the molecules of inhibitors [39]. this statement is consistent with tsc3. however, tsc1 and tsc4 with high dipole moment (5.65 and 6.07 debye respectively) have poor inhibition efficiency, while tsc2 and tsc5 with low dipole moment, have a high inhibition efficiency. these results show that there is inconsistency with the use of dipole moment predicting the direction of a corrosion inhibition reaction. in the examined compounds, our theoretical results confirmed that there is no substantial link between inhibitory effectiveness and dipole moment. the electrostatic potential may be used to deduce the dipole’s orientation (right panels of fig. 3). the colors represent the electrostatic potential value. colors lean toward red represent places with negative potential (where a positive charge is most likely to be attracted), while colors that lean toward blue represent areas of positive potential (where a positive charge is least likely to be attracted). the region of highest electron density is found around the sulphur atom and around the nitrogen atom. log p is responsible for the hydrophobicity (property of a molecule to repel water) of a molecule [40-41]. in corrosion studies, hydrophobicity can be related to the process at which oxide/hydroxide layers which retards the corrosion process are formed on the surface of the metal. hydrophobicity will increase when the solubility of the molecule in water decreases [42]. the results obtained for log p showed that tsc5 > tsc2 > tsc3 > tsc4 > tsc1 > tsc6. it was also found that the values of log p are closely related to the corrosion inhibition efficiencies of the investigated derivatives. the effective surface coverage and molecular size of the inhibitor on the surface of metal are determined by using molecular weight and volume quantum parameters. these parameters determine how well a molecule can be adsorbed atop and cover a metal surface, thereby isolating it from the corroding environment. the molecular volume and weight for the six studied molecules increases in the following order: tsc3 > tsc5 > tsc2 > tsc4 > tsc1 > tsc6. the presence of the phenyl group increases the molecular size of tsc1, would lead to a larger surface coverage that tsc6 on the aluminum. also, tsc2 has two phenyl groups and consequently more effective than tsc1. hence as the value of this parameter increases, so do the molecule’s corrosion inhibition potential increases the fraction of electrons transferred (∆n) is the number of electrons transferred from the inhibiting molecule to the surface of the metal. it also presents the inhibitor’s ability to transfer electrons. when the electron-donating capacity of an inhibitor is improved, its inhibitory efficiency on the surface of metal also improves. similarly, the efficiency of an inhibitor increases on the face of metal as the number of electron transfers increases. the number of electrons transferred (∆n) in an organic inhibitor should be less than 3.6 (electrons) for it to be regarded effective in increasing the corrosion inhibition efficiency, but if the ∆n is greater than 3.6 (electrons) the inhibition efficiency decreases [25]. molecules with higher electron transfer value, has a greater tendency to donate electrons to molecules that accept electrons. this implies that inhibiting molecules with higher ∆n have greater tendency to adsorb on the surface of the metal which consequently increases their inhibition efficiencies [25]. the ∆n values for the six studied molecules range from -0.5170.45 e−. for the molecules tsc1−6, the largest proportion of electron transferred (∆n) is 6 ogunyemi & ojo / j. nig. soc. phys. sci. 5 (2023) 915 7 figure 3. molecule orbital density distribution of studied thiosemicarbazides: homo (left), lumo (middle) and electrostatic potential (right) associated with molecule tsc3, while the lowest proportion is associated with tsc6 which has the lowest inhibition efficiency. ∆n for molecules tsc1−6 increases in the following order: tsc3 > tsc4 > tsc5 > tsc2 > tsc1 > tsc6. the results indicate that there is no correlation between the trend in the ∆n values of these studied compounds and the trend in the experimentally determined inhibition efficiency. the ∆n values are strongly influenced by the molecular structure and 7 ogunyemi & ojo / j. nig. soc. phys. sci. 5 (2023) 915 8 substituent groups attached to the thiosemicarbazide skeletal ring. this result demonstrates that the studied thiosemicarbazide derivatives are electron-donating molecules while the aluminum surface could be an electron-accepting molecule. it’s crucial to evaluate the circumstance involving a molecule that will acquire a particular amount of charge at one center and then back-donate the same amount of charge through the same or a different center. in a simple model of charge transfer for donation and back donation of charges [42], an electronic back donation process can be a result of the interaction between the inhibiting molecule and the surface of a metal. though, it is necessary to note that these values do not predict the occurrence of a back donation process. back-donation charges for the studied molecules are less than zero (-0.75 to -1.18 e) indicating that the charges transferred to the molecules, accompanied by a back-donation from the molecule, are energetically favoured since η > 0 and ∆eback−donation < 0. therefore, tsc3 and tsc4 could be more energetically favoured than tsc1 and tsc2. the result is consistent with the concept that states that if both charge transfer (i.e. to the molecule and back-donation processes from the molecule) occurs, the change in energy is directly proportional to the hardness of the molecule in equation 3. the global electrophilicity index (ω) gives information on the nucleophilicity and electrophilicity nature of organic inhibitors. an inhibitor with a high electrophilicity index (ω) value indicates a high tendency to act as an electrophile and conversely a low electrophilicity index (ω) value indicates a high tendency to act as a nucleophile. the electrophilicity values (table 1) show that tsc3 has the highest value and as a result, the largest ability to receive electrons from the metal thereby increasing the adsorption capacity of the tsc3 on the surface of the metal. the electrophilicity values show the following sequence: tsc3 > tsc5 > tsc4 > tsc2 > tsc1 > tsc6. the inhibitors function as lewis bases in a corroding system, whereas the metal acts as a lewis acid. the distribution of charges in the chemical structure of organic inhibitors influences adsorption. the charge distribution on a molecule might reasonably be taken as an indication for locating the positions of interaction between the inhibitor and the metal surface. charge distribution in the studied molecules is estimated using mulliken population density [9, 43]. it is a way of estimating inhibitors’ adsorption centers and consequently determining the site that corresponds to a molecular center that accepts the charge as well as a molecular center that will donate back charges via the same center or another center. moreover, in the physisorption model, electrostatic attraction is expected between the metal surface and inhibiting molecules. the charges on heteroatoms of inhibitors can apparently be utilized as an index to account for physical adsorption. according to this hypothesis, the molecule with the highest atomic charge on a heteroatom will have a greater potential to physically adsorb on the metal surface [44]. the average mulliken charges for heteroatoms present in each of the studied thiosemicarbazides (denoted as heteroatom) are -3.16, -2.57, -2.18, -1.768, -1.75, and -0.57, for tsc4, tsc3, tsc1, tsc6, tsc2, and tsc5, inhibitors respectively. inhibitors with more negatively charged heteroatom will adsorb more on the surface of the metal through the donor-acceptor reaction [45]. therefore, a higher electron density on tsc4 and tsc3 and tsc1 heteroatoms would promote its physical adsorption on the surface of the metal. condensed fukui functions are often used to investigate local reactivity in inhibitors. the fukui function reveals the location in a chemical compound where electrophilic( f −k ), nucleophilic ( f +k ) and radical reactions are most likely to occur. f + k measures the change in density when the inhibiting molecule gains electrons which is related to the reactivity with respect to nucleophilic attack. conversely, f −k is related to the reactivity when the molecule losses electrons which is reactivity related to electrophilic attack. atoms with larger fukui function values are better reactive atomic centers than lower values in a given molecule. the fukui indices ( f +k and f − k ) of the studied molecules calculated from mulliken charge population analysis are given in table 2 for tsc1, tsc2, tsc3, tsc4, tsc5, and tsc6, respectively. the highest value of f −k for the nonhydrogen atoms was found on c7 (0.06) of tsc1, c7 (0.96) and n (0.96), for tsc2, c7 (0.082), c12 (0.065) and o4 (0.0854) for tsc3, c8 (0.05) and c7 (0.047) for tsc4, c12 (0.037), c7 (0.065) and o1 (0.063) for tsc5 and c (0.042) for tsc6 which represents the most probable centers for electrophilic attack. however, the highest values for f +k for non-hydrogen atoms are found on s1 (0.323) and n1 (0.051) for tsc1, s1 (0.262) and n1 (0.048) for tsc2, s1 (0.321) and n1 (0.057) for tsc3, s1 (0.386) and n1 (0.065) for tsc4, s1 (0.195), n1 (0.064), n2 (0.044) for tsc5, s1 (0.233) and n (0.311) for tsc6. these represent the most probable centers for nucleophilic attack in the molecules. 3.2. quantitative structure-activity relationships (qsars) modeling experimentally, the potentiodynamic polarization method has been used to evaluate the inhibitory efficiency (ie) of thiosemicarbazide derivatives (tsc 1−−6) (15). however, theoretically calculated inhibition efficiencies (ie%) of studied thiosemicarbazide derivatives (tsc1−6) were predicted using the qsars model (equation 14) developed via linear regression of molecular descriptors (table 1). the experimentally measured inhibitory efficiency (ie percent) against steel served as dependent variables, whereas the appropriate molecular descriptors as determined by pearson’s matrix (figure 4) served as independent variables. these selected descriptors were used to build a linear qsar model to understand how linear regression equations might explain structural key points corresponding to differential behavior in chemical descriptors against corrosion. as indicated in equation 14, the molecular descriptors solvation energy (esolv), softness (s) electronegativity, and dipole moment define the corrosion inhibition of the thiosemicarbazide derivatives. %i e = 262 − 117 ∗ so f tness + 7.81 ∗ electronegativity + 2.90 ∗ esolv − 4.87 ∗ dipole (14) the acceptability of the quality of the model developed from the qsar investigation is determined by its predictabilities and 8 ogunyemi & ojo / j. nig. soc. phys. sci. 5 (2023) 915 9 table 2. fukui indices for the atoms of the studied thiosemicarbazide derivatives tsc 1 tsc 2 tsc 3 tsc 4 tsc 5 tsc 6 atoms f k− f k + atoms f k− f k + atom f k− f k + atom f k− f k + atoms f k− f k + atoms f k− f k + ci 0.026 0.041 ci 0.067 0.027 ci 0.001 0.029 ci 0.016 0.023 ci 0.007 0.014 n1 0.01 0.051 h2 0.032 0.026 h2 0.024 0.026 h2 0.007 0.024 h2 0.045 0.042 h2 0.018 0.031 h1 0.036 0.042 c2 0.008 0.014 c2 0.006 0.009 c2 0.003 0.011 c2 0.007 0.01 c2 0.002 0.009 c7 0.067 0.024 h4 0.071 0.069 h4 0.055 0.059 h4 0.022 0.063 h4 0.065 0.06 h4 0.038 0.052 s1 0.026 0.323 c3 0.048 0.04 c3 0.036 0.038 c3 0.006 0.044 c3 0.045 0.039 c3 0.027 0.032 n2 0.031 0.019 h3 0.089 0.08 h3 0.067 0.07 h3 0.024 0.076 h3 0.08 0.069 h3 0.051 0.062 h11 0.049 0.041 c4 0.006 0.014 c4 0.004 0.009 c4 0.002 0.011 c4 0.006 0.009 c4 0.003 0.009 n3 0.006 0.003 h6 0.076 0.073 h6 0.056 0.06 h6 0.016 0.066 h6 0.07 0.062 h6 0.046 0.056 h7 0.052 0.04 c5 0.017 0.033 c5 0.014 0.025 c5 0.001 0.028 c5 0.023 0.027 c5 0.013 0.023 h5 0.068 0.067 h5 0.042 0.051 h5 0.014 0.062 h5 0.061 0.052 h5 0.037 0.048 c6 0.039 0.021 c6 0.023 0.015 c6 0 0.02 c6 0.037 0.001 c6 0.014 0 n1 0.01 0.051 n1 0.005 0.048 n1 0.005 0.057 n1 0.007 0.069 n1 0.001 0.064 h1 0.036 0.042 h1 0.024 0.031 h1 0.013 0.04 h1 0.043 0.041 h1 0.03 0.042 c7 0.067 0.024 c7 0.096 0.029 c7 0.082 0.029 c7 0.047 0.042 c7 0.065 0.031 s1 0.026 0.323 s1 0.092 0.262 s1 0.042 0.321 s1 0.028 0.286 s1 0.042 0.295 n2 0.031 0.019 n2 0.022 0.02 n2 0.022 0.026 n2 0.019 0.03 n2 0.003 0.044 h11 0.049 0.041 h11 0.037 0.027 h11 0.117 0.125 h11 0.018 0.018 h11 0.265 0.023 n3 0.006 0.003 n3 0.094 0.836 n3 0.016 0.02 n3 0.003 0.005 n3 0.007 0.049 h7 0.052 0.04 h7 0.03 0.031 h7 0.026 0.011 h7 0.019 0.011 h7 0.011 0.026 c8 0.004 0.004 c8 0.067 0.005 c8 0.05 0.016 c8 0.065 0.014 c9 0.388 0.01 c9 0.029 0.451 c9 0.025 0.005 c9 0.009 0.001 c10 0.031 0.006 c10 0.045 0.01 n4 0.001 0.007 c10 0.023 0.011 c11 0.079 0.15 c11 0.003 0.009 n5 0.042 0.026 c11 0.005 0.008 c12 0.009 0.006 c12 0.065 0.009 o1 0.058 0.024 c12 0.037 0.017 c13 0.032 0.012 c13 0.012 0.002 c13 0.003 0.006 n4 0.052 0.005 o1 0.043 0.093 n5 0.024 0.002 c14 0.025 0.008 o1 0.133 0.03 o2 0.123 0 o3 0.071 0.019 o4 0.854 0.761 fitting capabilities (9). the developed qsar model in equation 14, reproduced the experimental %ie (r2 = 0.9982) with deviation ranging between 0.03 and 0.06 while the standard error of residuals is 0.116212. table 3 compares the predicted corrosion inhibition efficiency (% ie) of molecules 1 through 6 to their experimental %ie. figure 5 depicts a graph of experimen9 ogunyemi & ojo / j. nig. soc. phys. sci. 5 (2023) 915 10 figure 4. generated pearson’s matrix table 3. experimental and predicted inhibition efficiencies of thiosemicarbazide derivatives molecules experimental inhibition efficiency predicted inhibition efficiency residual tsc 1 27.4000 27.4352 -0.0352155 tsc 2 94.3000 94.2488 0.0512029 tsc 3 92.8000 92.7691 0.0308704 tsc 4 60.8000 60.8311 -0.0311049 tsc 5 95.5000 95.5642 -0.0642269 tsc 6 27.0000 26.9515 0.0484739 tal corrosion inhibition efficiency (% ie) vs predicted corrosion inhibition efficiencies (% ie) to illustrate the relationship between the two. the developed model was quite resilient in predicting good experimental values, therefore, the theoretical percentage inhibition efficiency for the studied compounds follows: tsc2 > tsc6 > tsc3 > tsc5> tsc4 > tsc1. the model was also statistically validated using the squared fitting factor (r2), adjusted fitting factor (r2a), cross-validation (cv. r2), and variation ratio (f). the estimated r2 (0.9982) revealed a reasonable fitness as well as the model’s efficiency, as shown in equation 14. the calculated cv.r2 (0.9342) is greater than 0.5 (standard) while the r2a (0.9143) is greater than 0.6. these results show that the model is statistically reliable and acceptable and have strong external predictability. 4. conclusion the quantum chemical descriptors of thiosemicarbazide derivatives were investigated in order to elucidate their electronic structure, reactivity, and predict their potential for corrosion inhibition using a computational approach. through dft/b3lyp/6311g* computational method, a relationship between quantum 10 ogunyemi & ojo / j. nig. soc. phys. sci. 5 (2023) 915 11 figure 5. graphical representation of the experimental and predicted inhibition efficiency (%) of the studied thiosemicarbazise derivatives descriptors of six thiosemicarbazide derivatives and their effectiveness in preventing corrosion was established. the correlations were found to be effective in developing thiosemicarbazide inhibitors with appropriate substituents capable of donating electrons to the metal’s surface. because of their high ehomo, ∆n, and low ∆e values, tsc2, tsc3 and tsc5 are projected to have the best inhibition efficiency, allowing for 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