J. Nig. Soc. Phys. Sci. 3 (2021) 308–333 Journal of the Nigerian Society of Physical Sciences A New Special 15-Step Block Method for Solving General Fourth Order Ordinary Differential Equations Victor Oboni Ataboa,∗, Solomon Ortwer Adeeb aDepartment of Mathematics, Ahmadu Ribadu College, Yola, Nigeria bDepartment of Mathematics, Modibbo Adama University, Yola, Nigeria Abstract A new higher-implicit block method for the direct numerical solution of fourth order ordinary differential equation is derived in this research paper. The formulation of the new formula which is 15-step, is achieved through interpolation and collocation techniques. The basic numerical properties of the method such as zero-stability, consistency and A-stability have been examined. Investigation showed that the new method is zero stable, consistent and A-stable, hence convergent. Test examples from recent literature have been used to confirm the accuracy of the new method. DOI:10.46481/jnsps.2021.337 Keywords: Consistency, Zero stability, A-stability, Implicit block formula Article History : Received: 11 August 2021 Received in revised form: 22 October 2021 Accepted for publication: 13 November 2021 Published: 29 November 2021 ©2021 Journal of the Nigerian Society of Physical Sciences. All rights reserved. Communicated by: T. Latunde 1. Introduction Ordinary differential equations (ODEs) have important applications and are a powerful tool in the study of many problems in the natural sciences and in technology; they are extensively employed in mechanics, astronomy, physics, and in many problems of chemistry and biology. Mathematical models in these vast range of disciplines, describe how quantities change. This leads naturally to the language of ordinary differential equations (ODEs). For instance, Newton’s laws in mechanics make it possible to reduce the description of the motion of mass points or solid bodies to solving ordinary differential equations. The computation of radiotechnical circuits or satellite tragetories, studies of the stability of a plane in flight, and explaining the course of chemical ∗Corresponding author tel. no: +2348061362898 Email address: atabovictor@gmail.com (Victor Oboni Atabo ) reactions are all carried out by studying and solving ordinary differential equations. The most interesting and most important applications of these equations are in the theory of oscillations, physical ship dynamic theory and in automatic control theory. These applied problems in turn produce new formulations of problems in the theory of ordinary differential equations from first, second, third, fourth, fifth, e.t.c, to other higher derivatives in ODEs Conti, Graffi and Sansone[1]. However, the solutions to some of these varieties of higher order ODEs problems do not exist explicitly. Hence, the need to develop numerical methods in the form of implicit linear multistep methods (LMMs) to solve these numerous problems in the form of ODEs. Therefore, in this research paper, we shall consider the general fourth order ordinary differential equation of the form: 308 Atabo and Adee / J. Nig. Soc. Phys. Sci. 3 (2021) 308–333 309 y(iv) = f (x, y, y′, y′′, y′′′), y(a) = �0, y ′(a) = �1, y ′′(a) = �2, y ′′′(a) = �3 (1) where, R × Rm × Rm → Rm and �0, �1, �2, �3 ∈ R. However, interpolation and collocation techniques have been used overtime for developing implicit linear multistep methods using power series as a basis function. Such authors as Raymond, Skwame and Adiku [2] formulated a four-step one off-grid block method using interpolation and collocation approach for the solution of fourth derivative ordinary differential equations. Two-step hybrid linear multistep block method for solving second, third and fourth order initial value problems of ordinary differential equations directly has also been derived by Abolarin, Kuboye, Adeyefa and Ogunware [3]. Hermite interpolation polynomial as a basis function has extensively been used also for the formulation of implicit block methods. Two point implicit block method of uniform order 6 has been derived for solving fourth-order initial value problems directly by Allogmany, Ismail, Majid and Ibrahim [4] using the aforementioned basis function. In addition, in order to avoid order reduction at solving higher order ODEs, such authors as Kuboye [5], Jena, Mohanty and Mishra [6], Adoghe and Omole [7], Omar and Kuboye [8], Duromola [9], Adesanya, Momoh, Adamu and Tahir [10], Awoyemi, Kayode and Adoghe [11], Awoyemi [12] and Ukpebor, Omole and Adoghe [13] have solved equation (1) directly. Similarly, Omole and Ukpebor [14] and Kuboye, Elusakin and Quadri [15] have both formulated a 4-point and 5-point hybrids block formulae for the solution of system and linear fourth order initial value problems in ordinary differential equations using power series as a basis function via interpolation and collocation techniques with uniform order 4 and 7 respectively. Therefore, we are motivated to improve on Jena et al. [6] who have formulated a 9-point block method of a uniform order 12 for the direct solution of equation (1) by proposing a new special 15-step block numerical method for the direct approximation of (1). Consequently, in section two, we carry out the formulation of 15-step block formula, section three provides the analysis of basic numerical properties, section four considers numerical examples, its implementation and discussion of results and in section five, the conclusion is drawn. 2. Materials and Methods of 15-Step Block Method In this section, we shall consider the formulation of a 15- step block method for the numerical approximation of equation (1). 2.1. Formulation of 15-Step Block Method Let us consider the power series approximations of the form: y(x) = k+4∑ j=0 a j x j (2) which approximates equation (1), so that a j′s, j = 0(1)(k+4) are parameters to be found and k is the step-number. Thus, the first, second, third and fourth derivatives of equation (2) become: y′(x) = k+4∑ j=0 ja j x j−1 (3) y′′(x) = k+4∑ j=0 j( j − 1)a j x j−2 (4) y′′′(x) = k+4∑ j=0 j( j − 1)( j − 2)a j x j−3 (5) y(iv)(x) = k+4∑ j=0 j( j − 1)( j − 2)( j − 3)a j x j−4 (6) Therefore, equating equation (1) and (6) gives: k+4∑ j=0 j( j − 1)( j − 2)( j − 3)a j x j−4 = f (x, y, y′, y′′, y′′′) (7) Thus, equation (2), (3), (4) and (5) are then interpolated at xn+i, i = (k−15) and equation (7) is collocated at xn+i, i = 0(1)k to give system of linear equations in the form: CX = D (8) Where, C =  1 xn x2n x 3 n x 4 n x 5 n ... x k+4 n 0 1 2xn 3x2n 4x 3 n 5x 4 n ... (k + 4)x k+3 n 0 0 2 6xn 12x2n 20x 3 n ... (k + 4)(k + 3)x k+2 n 0 0 0 6 24xn 60x2n ... (k + 4)(k + 3)(k + 2)x k+1 n 0 0 0 0 24 120xn ... (k + 4)(k + 3)(k + 2)(k + 1)xkn 0 0 0 0 24 120xn+1 ... (k + 4)(k + 3)(k + 2)(k + 1)x k n+1 . . . . . . . . . . . . . . . . . . . . . . . . 0 0 0 0 24 120xn+k · · · (k + 4)(k + 3)(k + 2)(k + 1)x k n+k  , X =  a0 a1 a2 a3 . . . ak+4  , D =  yn y′n y′′n y′′′n fn . . . fn+k  By solving equation (8) with k = 15 and j = 0(1)(k+4) ( where C is a 20-by-20 matrix), using Gaussian elimination method, ak+4′s are obtained and substitution into equation (2) is made to give a continuous linear multistep method(LMM) of the form: y(x) = α0yn+hα1y′n+h 2α2y′′n +h 3α3y′′′n +h 4 k∑ j=0 β j fn+ j (9) Where, ξ = x−xn α0 = 1 (10) α1 = ξ (11) α2 = 1 2 ξ2 (12) α3 = 1 6 ξ3 (13) 309 Atabo and Adee / J. Nig. Soc. Phys. Sci. 3 (2021) 308–333 310 β0 = 1 24 ξ4− 1195757 ξ5 43243200 h + 13215487 ξ6 1009008000 h2 − 35118025721 ξ7 7628100480000 h3 + 2065639 ξ8 1676505600 h4 − 277382447 ξ9 1086375628800 h5 + 2271089 ξ10 54867456000 h6 − 54576553 ξ11 10346434560000 h7 + 4783 ξ12 9053130240 h8 − 324509 ξ13 7846046208000 h9 + 109 ξ14 43589145600 h10 − 26921 ξ15 235381386240000 h11 + ξ16 261534873600 h12 − 47 ξ17 533531142144000 h13 + ξ18 800296713216000 h14 − ξ19 121645100408832000 h15 (14) β1 = 1 8 ξ5 h − 835397 ξ6 8648640 h2 + 60461593 ξ7 1412611200 h3 − 146723651 ξ8 11176704000 h4 + 8548607 ξ9 2874009600 h5 − 28162523 ξ10 54867456000 h6 + 1475953 ξ11 21555072000 h7 − 7346057 ξ12 1034643456000 h8 + 5973601 ξ13 10461394944000 h9 − 24599 ξ14 697426329600 h10 + 223 ξ15 135862272000 h11 − 2329 ξ16 41845579776000 h12 + 71 ξ17 54721142784000 h13 − ξ18 53801459712000 h14 + ξ19 8109673360588800 h15 (15) β2 = − 7 ξ5 16 h + 1015577 ξ6 2471040 h2 − 20848547 ξ7 100900800 h3 + 1546697837 ξ8 22353408000 h4 − 149466307 ξ9 8941363200 h5 + 664545493 ξ10 219469824000 h6 − 289311887 ξ11 689762304000 h7 + 5795249 ξ12 129330432000 h8 − 1379251 ξ13 373621248000 h9 + 40549 ξ14 174356582400 h10 − 5231 ξ15 475517952000 h11 + 989 ξ16 2615348736000 h12 − 61 ξ17 6840142848000 h13 + 59 ξ18 457312407552000 h14 − ξ19 1158524765798400 h15 (16) β3 = 91 ξ5 72 h − 1075637 ξ6 855360 h2 + 281523559 ξ7 419126400 h3 − 23857083361 ξ8 100590336000 h4 + 32534861339 ξ9 543187814400 h5 − 617568593 ξ10 54867456000 h6 + 207888451 ξ11 129330432000 h7 − 5522731 ξ12 31352832000 h8 + 66465877 ξ13 4483454976000 h9 − 1989607 ξ14 2092278988800 h10 + 390953 ξ15 8559323136000 h11 − 66571 ξ16 41845579776000 h12 + 6229 ξ17 164163428352000 h13 − 13 ξ18 23451918336000 h14 + ξ19 267351869030400 h15 (17) β4 = − 91 ξ5 32 h + 1105667 ξ6 380160 h2 − 18107311 ξ7 11289600 h3 + 3269363347 ξ8 5588352000 h4 − 6129294103 ξ9 40236134400 h5 + 3229334839 ξ10 109734912000 h6 − 2970738131 ξ11 689762304000 h7 + 31183253 ξ12 64665216000 h8 − 15450997 ξ13 373621248000 h9 + 235087 ξ14 87178291200 h10 − 62483 ξ15 475517952000 h11 + 12121 ξ16 2615348736000 h12 − 139 ξ17 1243662336000 h13 + 29 ξ18 17588938752000 h14 − ξ19 89117289676800 h15 (18) β5 = 1001 ξ5 200 h − 224737 ξ6 43200 h2 + 103207793 ξ7 35280000 h3 − 221496517 ξ8 203212800 h4 + 176798693 ξ9 609638400 h5 − 3138934651 ξ10 54867456000 h6 + 230071601 ξ11 26943840000 h7 − 201455717 ξ12 206928691200 h8 + 126849403 ξ13 1494484992000 h9 − 3917983 ξ14 697426329600 h10 + 1319249 ξ15 4755179520000 h11 − 1507 ξ16 152165744640 h12 + 1201 ξ17 4974649344000 h13 − 23 ξ18 6395977728000 h14 + ξ19 40507858944000 h15 (19) 310 Atabo and Adee / J. Nig. Soc. Phys. Sci. 3 (2021) 308–333 311 β6 = − 1001 ξ5 144 h + 1135697 ξ6 155520 h2 − 158534377 ξ7 38102400 h3 + 28776650641 ξ8 18289152000 h4 − 84007238759 ξ9 197522841600 h5 + 2079145531 ξ10 24385536000 h6 − 26754892001 ξ11 2069286912000 h7 + 193584983 ξ12 129330432000 h8 − 21199931 ξ13 160123392000 h9 + 4645801 ξ14 523069747200 h10 − 1900739 ξ15 4279661568000 h11 + 1271 ξ16 79252992000 h12 − 737 ξ17 1865493504000 h13 + 19 ξ18 3197988864000 h14 − ξ19 24304715366400 h15 (20) β7 = 429 ξ5 56 h − 1144277 ξ6 141120 h2 + 15356659 ξ7 3292800 h3 − 603869969 ξ8 338688000 h4 + 297351533 ξ9 609638400 h5 − 1811075923 ξ10 18289152000 h6 + 218600713 ξ11 14370048000 h7 − 4306835671 ξ12 2414168064000 h8 + 557256047 ξ13 3487131648000 h9 − 2522719 ξ14 232475443200 h10 + 174023 ξ15 317011968000 h11 − 25411 ξ16 1268047872000 h12 + 827 ξ17 1658216448000 h13 − 113 ξ18 14923948032000 h14 + ξ19 18903667507200 h15 (21) β8 = − 429 ξ5 64 h + 143839 ξ6 20160 h2 − 108870653 ξ7 26342400 h3 + 33726391 ξ8 21168000 h4 − 178809037 ξ9 406425600 h5 + 825125941 ξ10 9144576000 h6 − 460176289 ξ11 32845824000 h7 + 2237897 ξ12 1347192000 h8 − 131166143 ξ13 871782912000 h9 + 75043 ξ14 7264857600 h10 − 83719 ξ15 158505984000 h11 + 193 ξ16 9906624000 h12 − 29 ξ17 59222016000 h13 + ξ18 133249536000 h14 − ξ19 18903667507200 h15 (22) β9 = 1001 ξ5 216 h − 128413 ξ6 25920 h2 + 109950473 ξ7 38102400 h3 − 3413334089 ξ8 3048192000 h4 + 15383980811 ξ9 49380710400 h5 − 3534853769 ξ10 54867456000 h6 + 652662223 ξ11 64665216000 h7 − 1249849571 ξ12 1034643456000 h8 + 495342143 ξ13 4483454976000 h9 − 1780879 ξ14 232475443200 h10 + 3386083 ξ15 8559323136000 h11 − 56057 ξ16 3804143616000 h12 + 5581 ξ17 14923948032000 h13 − 37 ξ18 6395977728000 h14 + ξ19 24304715366400 h15 (23) β10 = − 1001 ξ5 400 h + 1159721 ξ6 432000 h2 − 13852667 ξ7 8820000 h3 + 249027661 ξ8 406425600 h4 − 1254757531 ξ9 7315660800 h5 + 1567631503 ξ10 43893964800 h6 − 19442163553 ξ11 3448811520000 h7 + 17594483 ξ12 25866086400 h8 − 23443657 ξ13 373621248000 h9 + 765371 ξ14 174356582400 h10 − 543937 ξ15 2377589760000 h11 + 409 ξ16 47551795200 h12 − 137 ξ17 621831168000 h13 + 11 ξ18 3197988864000 h14 − ξ19 40507858944000 h15 (24) 311 Atabo and Adee / J. Nig. Soc. Phys. Sci. 3 (2021) 308–333 312 β11 = 91 ξ5 88 h − 105727 ξ6 95040 h2 + 10139863 ξ7 15523200 h3 − 2860697731 ξ8 11176704000 h4 + 160975931 ξ9 2235340800 h5 − 827528203 ξ10 54867456000 h6 + 103295831 ξ11 43110144000 h7 − 301248457 ξ12 1034643456000 h8 + 40445413 ξ13 1494484992000 h9 − 1331119 ξ14 697426329600 h10 + 1049441 ξ15 10461394944000 h11 − 159209 ξ16 41845579776000 h12 + 5381 ξ17 54721142784000 h13 − 109 ξ18 70355755008000 h14 + ξ19 89117289676800 h15 (25) β12 = − 91 ξ5 288 h + 1165727 ξ6 3421440 h2 − 672835853 ξ7 3353011200 h3 + 3969230807 ξ8 50295168000 h4 − 24226741543 ξ9 1086375628800 h5 + 515061979 ξ10 109734912000 h6 − 1551629011 ξ11 2069286912000 h7 + 1976179 ξ12 21555072000 h8 − 9616213 ξ13 1120863744000 h9 + 14491 ξ14 23775897600 h10 − 138169 ξ15 4279661568000 h11 + 3229 ξ16 2615348736000 h12 − 1321 ξ17 41040857088000 h13 + ξ18 1954326528000 h14 − ξ19 267351869030400 h15 (26) β13 = 7 ξ5 104 h − 89849 ξ6 1235520 h2 + 8665381 ξ7 201801600 h3 − 189340169 ξ8 11176704000 h4 + 96662087 ξ9 20118067200 h5 − 55723553 ξ10 54867456000 h6 + 1757123 ξ11 10777536000 h7 − 20736923 ξ12 1034643456000 h8 + 402847 ξ13 213497856000 h9 − 94109 ξ14 697426329600 h10 + 6847 ξ15 951035904000 h11 − 11611 ξ16 41845579776000 h12 + 5189 ξ17 711374856192000 h13 − 107 ξ18 914624815104000 h14 + ξ19 1158524765798400 h15 (27) β14 = − ξ5 112 h + 1170017 ξ6 121080960 h2 − 8077807 ξ7 1412611200 h3 + 50565173 ξ8 22353408000 h4 − 17264887 ξ9 26824089600 h5 + 29969773 ξ10 219469824000 h6 − 15185081 ξ11 689762304000 h7 + 2461967 ξ12 905313024000 h8 − 673219 ξ13 2615348736000 h9 + 3229 ξ14 174356582400 h10 − 473 ξ15 475517952000 h11 + 101 ξ16 2615348736000 h12 − ξ17 977163264000 h13 + 53 ξ18 3201186852864000 h14 − ξ19 8109673360588800 h15 (28) β15 = ξ5 1800 h − 1171733 ξ6 1945944000 h2 + 30946717 ξ7 86682960000 h3 − 406841 ξ8 2874009600 h4 + 21939781 ξ9 543187814400 h5 − 3209 ξ10 373248000 h6 + 899683 ξ11 646652160000 h7 − 3247 ξ12 18811699200 h8 + 515261 ξ13 31384184832000 h9 − 71 ξ14 59779399680 h10 + 2747 ξ15 42796615680000 h11 − ξ16 398529331200 h12 + ξ17 14923948032000 h13 − ξ18 914624815104000 h14 + ξ19 121645100408832000 h15 (29) Substituting equation (10) – (29) into equation (9), interpolating and collocating at xn+ j, j = 0, 1; 0, 2; 0, 3; ..., 0, k¡ and xn+ j, j = 0(1)k gives the solution to equation (1) and is given explicitly below: 312 Atabo and Adee / J. Nig. Soc. Phys. Sci. 3 (2021) 308–333 313 yn+1 = yn+hy ′ n+ 1 2 h2y′′n + 1 6 h3y′′′n + 27312614539002931 1161157776629760000 h4 fn+ 215021456509297 3547982095257600 h4 fn+1 − 2405950254864953 13516122267648000 h4 fn+2+ 228535928736331 465478700544000 h4 fn+3− 43830916431996773 40548366802944000 h4 fn+4 + 1983716565322187 1055947052160000 h4 fn+5− 188870621369290423 72987060245299200 h4 fn+6+ 16765512493838707 5913303492096000 h4 fn+7 − 77901096585757121 31537618624512000 h4 fn+8+ 38820512227495919 22808456326656000 h4 fn+9− 26549037149067547 28963119144960000 h4 fn+10 + 63857023621987 168951528345600 h4 fn+11− 42069828441973351 364935301226496000 h4 fn+12+ 31069509811453 1267136462592000 h4 fn+13 − 307336703482477 94612855873536000 h4 fn+14+ 14651877357209 72572361039360000 h4 fn+15 (30) yn+2 = yn+2 hy ′ n+2 h 2y′′n + 4 3 h3y′′′n + 776157610312243 3118343638410000 h4 fn+ 12110365177663 11549420883000 h4 fn+1 − 320666666657 123743795175 h4 fn+2+ 313723318722181 44547766263000 h4 fn+3− 7257112564561 471404934000 h4 fn+4 + 659444520473071 24748759035000 h4 fn+5− 815371875214873 22273883131500 h4 fn+6+ 30824521197751 769961392200 h4 fn+7 − 804553592927429 23098841766000 h4 fn+8+ 1068040546872449 44547766263000 h4 fn+9− 26608028437433 2062396586250 h4 fn+10 + 537006152783 101015343000 h4 fn+11− 2625137049893 1619918773200 h4 fn+12+ 568425874883 1649917269000 h4 fn+13 − 790392890503 17324131324500 h4 fn+14+ 4419708470941 1559171819205000 h4 fn+15 (31) yn+3 = yn+3 hy ′ n+ 9 2 h2y′′n + 9 2 h3y′′′n + 602530652428169 648922193920000 h4 fn+ 19434314988567 4055763712000 h4 fn+1 − 189942041947311 18540634112000 h4 fn+2+ 63328896091 2228441600 h4 fn+3− 1152715866154893 18540634112000 h4 fn+4 + 311770415766357 2896974080000 h4 fn+5− 2741097040974839 18540634112000 h4 fn+6+ 1311392773004949 8111527424000 h4 fn+7 − 3650761245166821 25956887756800 h4 fn+8+ 56087920654093 579394816000 h4 fn+9− 4828796962028649 92703170560000 h4 fn+10 + 12434976965421 579394816000 h4 fn+11− 121294082904151 18540634112000 h4 fn+12+ 14650769187 10534451200 h4 fn+13 − 23901979612131 129784438784000 h4 fn+14+ 464062820041 40557637120000 h4 fn+15 (32) yn+4 = yn+4 hy ′ n+8 h 2y′′n + 32 3 h3y′′′n + 451207826917664 194896477400625 h4 fn+ 57491991503264 4331032831125 h4 fn+1 − 5241628601168 206239658625 h4 fn+2+ 58358490710752 795495826125 h4 fn+3− 1525291110688 9518753475 h4 fn+4 + 286110321363872 1031198293125 h4 fn+5− 303253476312368 795495826125 h4 fn+6+ 601876165585312 1443677610375 h4 fn+7 − 174546287711456 481225870125 h4 fn+8+ 278040058459552 1113694156575 h4 fn+9− 59370049162288 441942125625 h4 fn+10 + 11415845225504 206239658625 h4 fn+11− 93955520375296 5568470782875 h4 fn+12+ 317044076768 88388425125 h4 fn+13 − 137148361136 288735522075 h4 fn+14+ 5751617237536 194896477400625 h4 fn+15 (33) 313 Atabo and Adee / J. Nig. Soc. Phys. Sci. 3 (2021) 308–333 314 yn+5 = yn+5 hy ′ n+ 25 2 h2y′′n + 125 6 h3y′′′n + 95264979269192125 20436376868683776 h4 fn + 168412149190625 5913303492096 h4 fn+1− 16289893064271875 324386934423552 h4 fn+2+ 2509888032209375 16587968237568 h4 fn+3 − 35503567768034375 108128978141184 h4 fn+4+ 80665948618375 141777506304 h4 fn+5− 2282582909693346875 2919482409811968 h4 fn+6 + 13484181814896875 15768809312256 h4 fn+7− 563136843495371875 756902846988288 h4 fn+8+ 46722464671271875 91233825306624 h4 fn+9 − 29797815959624875 108128978141184 h4 fn+10+ 2302123472646875 20274183401472 h4 fn+11− 101052862907884375 2919482409811968 h4 fn+12 + 3825296084375 519850856448 h4 fn+13− 2212679175659375 2270708540964864 h4 fn+14+ 7029876481375 116115777662976 h4 fn+15 (34) yn+6 = yn+6 hy ′ n+18 h 2y′′n +36 h 3y′′′n + 1302374156467 158428270000 h4 fn+ 166010903703 3168565400 h4 fn+1 − 98189047503 1131630500 h4 fn+2+ 615303609259 2263261000 h4 fn+3− 2646014872329 4526522000 h4 fn+4 + 11487075212241 11316305000 h4 fn+5− 1103785049 791350 h4 fn+6+ 24170242181949 15842827000 h4 fn+7 − 42060841828767 31685654000 h4 fn+8+ 2068042917719 2263261000 h4 fn+9− 2782160441883 5658152500 h4 fn+10 + 91711309509 452652200 h4 fn+11− 279567458927 4526522000 h4 fn+12 + 29717015319 2263261000 h4 fn+13− 3443405418 1980353375 h4 fn+14+ 8557581379 79214135000 h4 fn+15 (35) yn+7 = yn+7 hy ′ n+ 49 2 h2y′′n + 343 6 h3y′′′n + 493224413054950241 37238296043520000 h4 fn + 102260630103289 1175451264000 h4 fn+1− 7568023997612911 55167845990400 h4 fn+2+ 206970176772057433 465478700544000 h4 fn+3 − 783475292841925019 h4 fn+4 827517689856000 h4 fn+4+ 8886369763068371 5387484960000 h4 fn+5 − 16869125122720872113 7447659208704000 h4 fn+6+ 29871704176307 12055910400 h4 fn+7− 198233731133162081 91946409984000 h4 fn+8 + 43176101394439517 29092418784000 h4 fn+9− 3304470142638045703 4137588449280000 h4 fn+10+ 5673457014546587 17239951872000 h4 fn+11 − 149427441213124277 1489531841740800 h4 fn+12+ 275759583050483 12929963904000 h4 fn+13− 59927250442381 21218402304000 h4 fn+14 + 290057854303 1652978340000 h4 fn+15 (36) yn+8 = yn+8 hy ′ n+32 h 2y′′n + 256 3 h3y′′′n + 3895558043809792 194896477400625 h4 fn + 193850692468736 1443677610375 h4 fn+1− 125956008558592 618718975875 h4 fn+2+ 108065421131776 159099165225 h4 fn+3 − 295522157750272 206239658625 h4 fn+4+ 7751983865397248 3093594879375 h4 fn+5− 19137382555205632 5568470782875 h4 fn+6 + 1809910071427072 481225870125 h4 fn+7− 6607665740288 2019129525 h4 fn+8+ 12544381621239808 5568470782875 h4 fn+9 − 25512731459584 21044863125 h4 fn+10+ 309076971618304 618718975875 h4 fn+11− 847973274318848 5568470782875 h4 fn+12 + 1335378968576 41247931725 h4 fn+13− 18568423555072 4331032831125 h4 fn+14+ 51915343659008 194896477400625 h4 fn+15 (37) 314 Atabo and Adee / J. Nig. Soc. Phys. Sci. 3 (2021) 308–333 315 yn+9 = yn+9 hy ′ n+ 81 2 h2y′′n + 243 2 h3y′′′n + 18625133626628643 648922193920000 h4 fn + 1591587721573497 8111527424000 h4 fn+1− 5340901247722377 18540634112000 h4 fn+2+ 1141699342205313 1158789632000 h4 fn+3 − 1091696657937021 529732403200 h4 fn+4+ 10481199823834029 2896974080000 h4 fn+5− 91850956406711577 18540634112000 h4 fn+6 + 44009977111307523 8111527424000 h4 fn+7− 612567442391345667 129784438784000 h4 fn+8+ 5266011862773 1620684800 h4 fn+9 − 162099315257981103 92703170560000 h4 fn+10+ 119277892927593 165541376000 h4 fn+11− 4072438241603397 18540634112000 h4 fn+12 + 211374969363 4526522000 h4 fn+13− 14592591850059 2359717068800 h4 fn+14+ 15583153071177 40557637120000 h4 fn+15 (38) yn+10 = yn+10 hy ′ n+50 h 2y′′n + 500 3 h3y′′′n + 197774498840375 4989349821456 h4 fn + 76168854565625 277186101192 h4 fn+1− 1296060059375 3299834538 h4 fn+2+ 44460274821875 32398375464 h4 fn+3 − 20508142934375 7199638992 h4 fn+4+ 66257006246125 13199338152 h4 fn+5− 1222568114959375 178191065052 h4 fn+6 + 694980202403125 92395367064 h4 fn+7− 402852494434375 61596911376 h4 fn+8+ 1604910232878125 356382130104 h4 fn+9 − 83883071750 34613649 h4 fn+10+ 13181028190625 13199338152 h4 fn+11− 216981310746875 712764260208 h4 fn+12 + 2562778803125 39598014456 h4 fn+13− 395952809375 46197683532 h4 fn+14+ 1328461659125 2494674910728 h4 fn+15 (39) yn+11 = yn+11 hy ′ n+ 121 2 h2y′′n + 1331 6 h3y′′′n + 61603729660715825561 1161157776629760000 h4 fn + 19997710920159577 53757304473600 h4 fn+1− 1916068546902768599 3686215163904000 h4 fn+2 + 39142139354400203 21158122752000 h4 fn+3− 4686113275972795271 1228738387968000 h4 fn+4 + 3884821540842611237 575971119360000 h4 fn+5− 8722148090842645789 947883899289600 h4 fn+6 + 1809824212823561759 179191014912000 h4 fn+7− 75477320193022950763 8601168715776000 h4 fn+8 + 6267565444696350157 1036748014848000 h4 fn+9− 2854635393056124029 877670277120000 h4 fn+10 + 1187685734092033 886109414400 h4 fn+11− 13555618279103961511 33175936475136000 h4 fn+12 + 68072706364399 783634176000 h4 fn+13− 22834016763745847 1984885088256000 h4 fn+14 + 51871823638164079 72572361039360000 h4 fn+15 (40) 315 Atabo and Adee / J. Nig. Soc. Phys. Sci. 3 (2021) 308–333 316 yn+12 = yn+12 hy ′ n+72 h 2y′′n +288 h 3y′′′n + 62288947936 900160625 h4 fn+ 970010779104 1980353375 h4 fn+1 − 37981121616 56581525 h4 fn+2+ 686872050784 282907625 h4 fn+3− 1407334462848 282907625 h4 fn+4 + 12486203584416 1414538125 h4 fn+5− 3400616961584 282907625 h4 fn+6+ 5232260492064 396070675 h4 fn+7 − 22708918277856 1980353375 h4 fn+8+ 2236670113376 282907625 h4 fn+9− 6013168148016 1414538125 h4 fn+10 + 495777441312 282907625 h4 fn+11− 2324971232 4352425 h4 fn+12+ 32128912224 282907625 h4 fn+13 − 29783960496 1980353375 h4 fn+14+ 841151264 900160625 h4 fn+15 (41) yn+13 = yn+13 hy ′ n+ 169 2 h2y′′n + 2197 6 h3y′′′n + 86779718953850369369 982518118686720000 h4 fn + 430007608906153603 682304249088000 h4 fn+1− 883011992382639353 1039701712896000 h4 fn+2 + 1093285607607322649 350899328102400 h4 fn+3− 19802369214482134397 3119105138688000 h4 fn+4 + 3671517181341961757 324906785280000 h4 fn+5− 431204822493929017859 28071946248192000 h4 fn+6 + 7688899400227183177 454869499392000 h4 fn+7− 7111838608315340341 485194132684800 h4 fn+8 + 4437011288736058799 438624160128000 h4 fn+9− 84741976626999074461 15595525693440000 h4 fn+10 + 145708237149829601 64981357056000 h4 fn+11− 1743000544456694861 2551995113472000 h4 fn+12 + 5662729165476331 38988814233600 h4 fn+13− 139985003179042333 7277911990272000 h4 fn+14 + 73385695962908641 61407382417920000 h4 fn+15 (42) yn+14 = yn+14 hy ′ n+98 h 2y′′n + 1372 3 h3y′′′n + 1006280506843003 9091380870000 h4 fn + 26776412350933 33671781000 h4 fn+1− 53335962667519 50507671500 h4 fn+2+ 3566435834783353 909138087000 h4 fn+3 − 107133265333577 13468712400 h4 fn+4+ 7171100242899187 505076715000 h4 fn+5− 2189555205249664 113642260875 h4 fn+6 + 238278143310809 11223927000 h4 fn+7− 1238370377602421 67343562000 h4 fn+8+ 2310124038552169 181827617400 h4 fn+9 − 573731688450277 84179452500 h4 fn+10+ 284604385256219 101015343000 h4 fn+11− 1556780681533049 1818276174000 h4 fn+12 + 558564964069 3061071000 h4 fn+13− 121946014477 5050767150 h4 fn+14+ 6819244114711 4545690435000 h4 fn+15 (43) 316 Atabo and Adee / J. Nig. Soc. Phys. Sci. 3 (2021) 308–333 317 yn+15 = yn+15 hy ′ n+ 225 2 h2y′′n + 1125 2 h3y′′′n + 141760259867125 1038275510272 h4 fn + 16008433396875 16223054848 h4 fn+1− 191857931184375 148325072896 h4 fn+2+ 6435135446875 1324331008 h4 fn+3 − 1454924082628125 148325072896 h4 fn+4+ 1271375605125 72424352 h4 fn+5− 3527747563259375 148325072896 h4 fn+6 + 212840309896875 8111527424 h4 fn+7− 23572142846465625 1038275510272 h4 fn+8+ 1137431715625 72424352 h4 fn+9 − 178243328212875 21189296128 h4 fn+10+ 32313630965625 9270317056 h4 fn+11− 156469002934375 148325072896 h4 fn+12 + 525121903125 2317579264 h4 fn+13− 30805101646875 1038275510272 h4 fn+14+ 60144050875 32446109696 h4 fn+15 (44) Thus, equation (30) – (44) is the new method named (NS4O15M). Therefore, we take the first derivative of equation (9) and evaluate at xn+ j, j = 0(1)k to have: y′n+1 = y ′ n+hy ′′ n + 1 2 h2y′′′n + 2646946687904237 32011868528640000 h3 fn + 9331210633373 35568742809600 h3 fn+1− 525466618744679 711374856192000 h3 fn+2+ 11892772256669 5884534656000 h3 fn+3 − 50067223151677 11291664384000 h3 fn+4+ 6838723295804269 889218570240000 h3 fn+5− 13547754274847623 1280474741145600 h3 fn+6 + 6130273402717 529296768000 h3 fn+7− 2391130153285807 237124952064000 h3 fn+8+ 11115802946972267 1600593426432000 h3 fn+9 − 13298258493701081 3556874280960000 h3 fn+10+ 195767508553 127031224320 h3 fn+11− 3008598640512121 6402373705728000 h3 fn+12 + 5923791011249 59281238016000 h3 fn+13− 855970682491 64670441472000 h3 fn+14+ 822811814369 1000370891520000 h3 fn+15 (45) y′n+2 = y ′ n+2 hy ′′ n +2 h 2y′′′n + 6414418122631 15630795180000 h3 fn+ 176725029073 86837751000 h3 fn+1 − 653662511 141775920 h3 fn+2+ 9816667434881 781539759000 h3 fn+3− 866478322811 31577364000 h3 fn+4 + 404117238667 8513505000 h3 fn+5− 101892006563273 1563079518000 h3 fn+6+ 412585117387 5789183400 h3 fn+7 − 7177626750949 115783668000 h3 fn+8+ 680466388363 15949791000 h3 fn+9− 19933118316169 868377510000 h3 fn+10 + 821247749563 86837751000 h3 fn+11− 1802310446791 625231807200 h3 fn+12+ 53212387759 86837751000 h3 fn+13 − 1565842507 19297278000 h3 fn+14+ 19699416731 3907698795000 h3 fn+15 (46) y′n+3 = y ′ n+3 hy ′′ n + 9 2 h2y′′′n + 690720887139 697016320000 h3 fn+ 32059975167 5544448000 h3 fn+1 − 981108537597 88711168000 h3 fn+2+ 16959465003 536166400 h3 fn+3− 67574744254179 975822848000 h3 fn+4 + 746266958199 6223360000 h3 fn+5− 160777027856499 975822848000 h3 fn+6+ 2585730381813 14350336000 h3 fn+7 − 30594531344121 195164569600 h3 fn+8+ 6580669587693 60988928000 h3 fn+9− 40468694141943 697016320000 h3 fn+10 + 5836036102983 243955712000 h3 fn+11− 1016547199431 139403264000 h3 fn+12+ 18909103959 12197785600 h3 fn+13 − 200320234209 975822848000 h3 fn+14+ 15557090403 1219778560000 h3 fn+15 (47) 317 Atabo and Adee / J. Nig. Soc. Phys. Sci. 3 (2021) 308–333 318 y′n+4 = y ′ n+4 hy ′′ n +8 h 2y′′′n + 891141341312 488462349375 h3 fn+ 5951251456 516891375 h3 fn+1 − 395392552 20138625 h3 fn+2+ 5846193073216 97692469875 h3 fn+3− 443144332 3408075 h3 fn+4 + 12223798950016 54273594375 h3 fn+5− 30236186795768 97692469875 h3 fn+6+ 1224829520192 3618239625 h3 fn+7 − 1065684738736 3618239625 h3 fn+8+ 6657405184 32837805 h3 fn+9− 657887719304 6030399375 h3 fn+10 + 9958015424 221524875 h3 fn+11− 1338659514532 97692469875 h3 fn+12+ 31620742784 10854718875 h3 fn+13 − 837478216 2170943775 h3 fn+14+ 900559936 37574026875 h3 fn+15 (48) y′n+5 = y ′ n+5 hy ′′ n + 25 2 h2y′′′n + 149078112485525 51218989645824 h3 fn+ 27357607800625 1422749712384 fn+1 − 57524077045625 1896999616512 h3 fn+2+ 9768839123125 100037089152 h3 fn+3− 108220654263125 517363531776 h3 fn+4 + 516319144825 1421328384 h3 fn+5− 3653176292400625 7316998520832 h3 fn+6+ 32375202664375 59281238016 h3 fn+7 − 901457665911875 1896999616512 h3 fn+8+ 4188606865466875 12804747411456 h3 fn+9− 1001792026947025 5690998849536 h3 fn+10 + 252938010625 3487131648 h3 fn+11− 14708026245625 665181683712 h3 fn+12+ 955425548125 203249958912 h3 fn+13 − 3542687313125 5690998849536 h3 fn+14+ 1934553625 50018544576 h3 fn+15 (49) y′n+6 = y ′ n+6 hy ′′ n +18 h 2y′′′n + 10124252319 2382380000 h3 fn+ 689167593 23823800 h3 fn+1 − 10275078321 238238000 h3 fn+2+ 2468567643 17017000 h3 fn+3− 145912136193 476476000 h3 fn+4 + 318410446077 595595000 h3 fn+5− 34946139 47600 h3 fn+6+ 95665826601 119119000 h3 fn+7 − 47568246489 68068000 h3 fn+8+ 57304332759 119119000 h3 fn+9− 6293482551 24310000 h3 fn+10 + 2541447099 23823800 h3 fn+11− 15494777187 476476000 h3 fn+12+ 6920469 1001000 h3 fn+13 − 218119509 238238000 h3 fn+14+ 33879999 595595000 h3 fn+15 (50) y′n+7 = y ′ n+7 hy ′′ n + 49 2 h2y′′′n + 346932898878727 59391221760000 h3 fn + 36848690199839 907365888000 h3 fn+1− 168555897445747 2903570841600 h3 fn+2+ 6601074966881017 32665171968000 h3 fn+3 − 555890398779737 1319804928000 h3 fn+4+ 3356052363748651 4536829440000 h3 fn+5− 132440457899414987 130660687872000 h3 fn+6 + 625723050379 564019200 h3 fn+7− 4671757797600943 4839284736000 h3 fn+8+ 5427096160617551 8166292992000 h3 fn+9 − 25961087688408481 72589271040000 h3 fn+10+ 534888408252611 3629463552000 h3 fn+11− 234803718886513 5226427514880 h3 fn+12 + 787863604577 82487808000 h3 fn+13− 18363232338017 14517854208000 h3 fn+14+ 755030443391 9607403520000 h3 fn+15 (51) 318 Atabo and Adee / J. Nig. Soc. Phys. Sci. 3 (2021) 308–333 319 y′n+8 = y ′ n+8 hy ′′ n +32 h 2y′′′n + 16988198848 2210236875 h3 fn+ 53560565248 986792625 h3 fn+1 − 815067367168 10854718875 h3 fn+2+ 1050018913792 3907698795 h3 fn+3− 859150752128 1550674125 h3 fn+4 + 53114085282304 54273594375 h3 fn+5− 130629987954944 97692469875 h3 fn+6+ 757517948416 516891375 h3 fn+7 − 2150503744 1686825 h3 fn+8+ 85740073246208 97692469875 h3 fn+9− 25634459993344 54273594375 h3 fn+10 + 301807773184 1550674125 h3 fn+11− 5796336118912 97692469875 h3 fn+12+ 27384514048 2170943775 h3 fn+13 − 18132713216 10854718875 h3 fn+14+ 50697789952 488462349375 h3 fn+15 (52) y′n+9 = y ′ n+9 hy ′′ n + 81 2 h2y′′′n + 47735374876317 4879114240000 h3 fn + 1312259291757 18765824000 h3 fn+1− 10036420749 106496000 h3 fn+2+ 10518533830593 30494464000 h3 fn+3 − 137554269574383 195164569600 h3 fn+4+ 1526337505582101 1219778560000 h3 fn+5− 1663567405083603 975822848000 h3 fn+6 + 57085662261447 30494464000 h3 fn+7− 1587493105268517 975822848000 h3 fn+8+ 7800788673 6963200 h3 fn+9 − 2941612400903409 4879114240000 h3 fn+10+ 473504266647 1905904000 h3 fn+11− 73906005035817 975822848000 h3 fn+12 + 3928134597579 243955712000 h3 fn+13− 416165679981 195164569600 h3 fn+14+ 57389067 433160000 h3 fn+15 (53) y′n+10 = y ′ n+10 hy ′′ n +50 h 2y′′′n + 43351425625 3572753184 h3 fn+ 60830018125 694702008 h3 fn+1 − 12345544375 106877232 h3 fn+2+ 384753416875 893188296 h3 fn+3− 47592424375 54486432 h3 fn+4 + 22085668025 14177592 h3 fn+5− 26465861018125 12504636144 h3 fn+6+ 539270756875 231567336 h3 fn+7 − 1871692800625 926269344 h3 fn+8+ 8709301139375 6252318072 h3 fn+9− 1040166025 1388016 h3 fn+10 + 214532249375 694702008 h3 fn+11− 336344910625 3572753184 h3 fn+12+ 421341875 21051576 h3 fn+13 − 283286875 106877232 h3 fn+14+ 1029672275 6252318072 h3 fn+15 (54) y′n+11 = y ′ n+11 hy ′′ n + 121 2 h2y′′′n + 42887155137009007 2910169866240000 h3 fn + 12377607151943 115482931200 h3 fn+1− 183316518915541 1319804928000 h3 fn+2+ 76566704845774583 145508493312000 h3 fn+3 − 1399043605047649 1319804928000 h3 fn+4+ 4262048618111119 2245501440000 h3 fn+5− 3522716671290113 1369491701760 h3 fn+6 + 15283394277950909 5389203456000 h3 fn+7− 52954438598926117 21556813824000 h3 fn+8+ 8814989815349839 5196731904000 h3 fn+9 − 294676003269851291 323352207360000 h3 fn+10+ 1907340759533 5076172800 h3 fn+11− 66668404334979491 582033973248000 h3 fn+12 + 98429973965983 4041902592000 h3 fn+13− 69521484528617 21556813824000 h3 fn+14+ 11214181843469 55964805120000 h3 fn+15 (55) 319 Atabo and Adee / J. Nig. Soc. Phys. Sci. 3 (2021) 308–333 320 y′n+12 = y ′ n+12 hy ′′ n +72 h 2y′′′n + 1309780608 74449375 h3 fn+ 1917572544 14889875 h3 fn+1 − 489571704 2977975 h3 fn+2+ 9399103872 14889875 h3 fn+3− 18828258012 14889875 h3 fn+4+ 24164791488 10635625 h3 fn+5 − 6534821544 2127125 h3 fn+6+ 10105364736 2977975 h3 fn+7− 2570011632 875875 h3 fn+8 + 30227141184 14889875 h3 fn+9− 81008278056 74449375 h3 fn+10+ 6696008064 14889875 h3 fn+11 − 4481292 32725 h3 fn+12+ 61932096 2127125 h3 fn+13− 5219352 1353625 h3 fn+14+ 17836032 74449375 h3 fn+15 (56) y′n+13 = y ′ n+13 hy ′′ n + 169 2 h2y′′′n + 50976734501772521 2462451425280000 h3 fn + 231607232521561 1520031744000 h3 fn+1− 10507261454113931 54721142784000 h3 fn+2+ 327984650433739 439723468800 h3 fn+3 − 7396667135771797 4974649344000 h3 fn+4+ 183307899755534437 68401428480000 h3 fn+5− 254414562335962769 70355755008000 h3 fn+6 + 570182964741793 142502976000 h3 fn+7− 1800041137343969 521153740800 h3 fn+8+ 294815320789342379 123122571264000 h3 fn+9 − 140128717971047 109486080000 h3 fn+10+ 454258655878469 855017856000 h3 fn+11− 79225841723025829 492490285056000 h3 fn+12 + 2682311572421 78173061120 h3 fn+13− 248596418804357 54721142784000 h3 fn+14+ 21720378795557 76951607040000 h3 fn+15 (57) y′n+14 = y ′ n+14 hy ′′ n +98 h 2y′′′n + 7676045327671 318995820000 h3 fn+ 28667764727 161109000 h3 fn+1 − 23816981209 107406000 h3 fn+2+ 13878519596837 15949791000 h3 fn+3− 489732836761 283551840 h3 fn+4 + 27659304286957 8860995000 h3 fn+5− 134110052796167 31899582000 h3 fn+6+ 2752397206979 590733000 h3 fn+7 − 9488406575333 2362932000 h3 fn+8+ 8897984343731 3189958200 h3 fn+9− 26342918970319 17721990000 h3 fn+10 + 122149393583 196911000 h3 fn+11− 696997533883 3752892000 h3 fn+12+ 71589567763 1772199000 h3 fn+13 − 3740727473 708879600 h3 fn+14+ 26185092311 79748955000 h3 fn+15 (58) y′n+15 = y ′ n+15 hy ′′ n + 225 2 h2y′′′n + 216067472925 7806582784 h3 fn+ 100262210625 487911424 h3 fn+1 − 179965535625 709689344 h3 fn+2+ 1959327448125 1951645696 h3 fn+3− 2213858930625 1115226112 h3 fn+4 + 1754723396925 487911424 h3 fn+5− 37748275074375 7806582784 h3 fn+6+ 1496243626875 278806528 h3 fn+7 − 36049237025625 7806582784 h3 fn+8+ 1568033338125 487911424 h3 fn+9− 13327761065625 7806582784 h3 fn+10 + 199792130625 278806528 h3 fn+11− 1651019848125 7806582784 h3 fn+12+ 23381533125 487911424 h3 fn+13 − 2568718125 459210752 h3 fn+14+ 752826975 1951645696 h3 fn+15 (59) Similarly, the second derivative of equation (9) is found and then evaluated at xn+ j, j = 0(1)k to give: 320 Atabo and Adee / J. Nig. Soc. Phys. Sci. 3 (2021) 308–333 321 y′′n+1 = y ′′ n +hy ′′′ n + 3206819325906617 16005934264320000 h2 fn + 111956703448001 133382785536000 h2 fn+1− 780902869927541 355687428096000 h2 fn+2+ 9502268460740177 1600593426432000 h2 fn+3 − 1975509756311887 152437469184000 h2 fn+4+ 9965996481056899 444609285120000 h2 fn+5− 98538747853228853 3201186852864000 h2 fn+6 + 5985224981757391 177843714048000 h2 fn+7− 3470876036880139 118562476032000 h2 fn+8+ 251938134205319 12504636144000 h2 fn+9 − 57837526128808793 5335311421440000 h2 fn+10+ 113475037891729 25406244864000 h2 fn+11− 4358164674336769 3201186852864000 h2 fn+12 + 77205249783577 266765571072000 h2 fn+13− 13631425781543 355687428096000 h2 fn+14+ 19055094997949 8002967132160000 h2 fn+15 (60) y′′n+2 = y ′′ n +2 hy ′′′ n + 7100763759181 15630795180000 h2 fn + 79686372907 28945917000 h2 fn+1− 198060940481 37216179000 h2 fn+2+ 11595391593263 781539759000 h2 fn+3 − 3762629808143 115783668000 h2 fn+4+ 73285372020391 1302566265000 h2 fn+5− 120811992265001 1563079518000 h2 fn+6 + 815470210927 9648639000 h2 fn+7− 25538181924601 347351004000 h2 fn+8+ 807085379107 15949791000 h2 fn+9 − 492563604418 18091198125 h2 fn+10+ 2922359996183 260513253000 h2 fn+11− 10689180138419 3126159036000 h2 fn+12 + 21039751529 28945917000 h2 fn+13− 4559016221 47366046000 h2 fn+14+ 23367229601 3907698795000 h2 fn+15 (61) y′′n+3 = y ′′ n +3 hy ′′′ n + 1724745170011 2439557120000 h2 fn+ 1156246932303 243955712000 h2 fn+1 − 3669218612307 487911424000 h2 fn+2+ 15741501473 670208000 h2 fn+3− 25023934662201 487911424000 h2 fn+4 + 15482577601863 174254080000 h2 fn+5− 59583820501187 487911424000 h2 fn+6+ 16294185200871 121977856000 h2 fn+7 − 56712794491323 487911424000 h2 fn+8+ 19519814663081 243955712000 h2 fn+9− 15006216511971 348508160000 h2 fn+10 + 8453922669 476476000 h2 fn+11− 376990411261 69701632000 h2 fn+12+ 25501122657 22177792000 h2 fn+13 − 74295242361 487911424000 h2 fn+14+ 5770019551 609889280000 h2 fn+15 (62) y′′n+4 = y ′′ n +4 hy ′′′ n + 468835748086 488462349375 h2 fn+ 31276065416 4652022375 h2 fn+1 − 237066268 24613875 h2 fn+2+ 3223897961912 97692469875 h2 fn+3− 25089869444 357847875 h2 fn+4 + 2201175147736 18091198125 h2 fn+5− 16340498219092 97692469875 h2 fn+6+ 1986215722936 10854718875 h2 fn+7 − 21338011906 134008875 h2 fn+8+ 1529765976712 13956067125 h2 fn+9− 9604572197884 162820783125 h2 fn+10 + 4188081368 172297125 h2 fn+11− 723896144432 97692469875 h2 fn+12+ 51299745704 32564156625 h2 fn+13 − 754844036 3618239625 h2 fn+14+ 6331431944 488462349375 h2 fn+15 (63) 321 Atabo and Adee / J. Nig. Soc. Phys. Sci. 3 (2021) 308–333 322 y′′n+5 = y ′′ n +5 hy ′′′ n + 31054788444685 25609494822912 h2 fn + 6194003019125 711374856192 h2 fn+1− 100287550861375 8536498274304 h2 fn+2+ 545635775073875 12804747411456 h2 fn+3 − 22793304846625 258681765888 h2 fn+4+ 164744717905 1065996288 h2 fn+5− 776841970197875 3658499260416 h2 fn+6 + 110166567234875 474249904128 h2 fn+7− 575215088287625 2845499424768 h2 fn+8+ 890981273520125 6402373705728 h2 fn+9 − 213109965786455 2845499424768 h2 fn+10+ 131726502560375 4268249137152 h2 fn+11− 3129099276125 332590841856 h2 fn+12 + 3176113625 1587890304 h2 fn+13− 2261231861125 8536498274304 h2 fn+14+ 210742341295 12804747411456 h2 fn+15 (64) y′′n+6 = y ′′ n +6 hy ′′′ n + 3491237903 2382380000 h2 fn+ 1273496769 119119000 h2 fn+1 − 3302866863 238238000 h2 fn+2+ 888545417 17017000 h2 fn+3− 50509176723 476476000 h2 fn+4+ 112104615069 595595000 h2 fn+5 − 30620993 119000 h2 fn+6+ 33546751263 119119000 h2 fn+7− 116779083507 476476000 h2 fn+8 + 20098920059 119119000 h2 fn+9− 15452545149 170170000 h2 fn+10+ 4457401749 119119000 h2 fn+11 − 5435403101 476476000 h2 fn+12+ 41270877 17017000 h2 fn+13− 4782384 14889875 h2 fn+14+ 11885633 595595000 h2 fn+15 (65) y′′n+7 = y ′′ n +7 hy ′′′ n + 561270290962031 326651719680000 h2 fn+ 138008915043671 10888390656000 h2 fn+1 − 115995161213911 7258927104000 h2 fn+2+ 1009684768894753 16332585984000 h2 fn+3− 2698346531507767 21776781312000 h2 fn+4 + 4028977853553919 18147317760000 h2 fn+5− 19697678682126767 65330343936000 h2 fn+6+ 2100668420753 6345216000 h2 fn+7 − 696945974650877 2419642368000 h2 fn+8+ 6477310161851851 32665171968000 h2 fn+9− 11619838717354259 108883906560000 h2 fn+10 + 79805888878681 1814731776000 h2 fn+11− 875850118609039 65330343936000 h2 fn+12+ 2387301668339 837568512000 h2 fn+13 − 2740021526389 7258927104000 h2 fn+14+ 3740727473 159497910000 h2 fn+15 (66) y′′n+8 = y ′′ n +8 hy ′′′ n + 962791153472 488462349375 h2 fn+ 159116760448 10854718875 h2 fn+1 − 589264474048 32564156625 h2 fn+2+ 6977664005248 97692469875 h2 fn+3− 219901261024 1550674125 h2 fn+4 − 33760546806464 97692469875 h2 fn+6+ 1379590216064 3618239625 h2 fn+7− 25115068768 75907125 h2 fn+8 + 22258735804544 97692469875 h2 fn+9− 6655353722816 54273594375 h2 fn+10+ 235080913792 4652022375 h2 fn+11 − 1504988356384 97692469875 h2 fn+12+ 35552129152 10854718875 h2 fn+13− 14124842944 32564156625 h2 fn+14 + 13164251008 488462349375 h2 fn+15 (67) 322 Atabo and Adee / J. Nig. Soc. Phys. Sci. 3 (2021) 308–333 323 y′′n+9 = y ′′ n +9 hy ′′′ n + 5425276143393 2439557120000 h2 fn+ 507508999443 30494464000 h2 fn+1− 1408763359683 69701632000 h2 fn+2 + 1797053294583 22177792000 h2 fn+3− 77925040957971 487911424000 h2 fn+4+ 176608903026609 609889280000 h2 fn+5 − 190121626816461 487911424000 h2 fn+6+ 105298772316291 243955712000 h2 fn+7− 181947002167377 487911424000 h2 fn+8 + 15689842209 60928000 h2 fn+9− 337995172999251 2439557120000 h2 fn+10+ 13927989639699 243955712000 h2 fn+11 − 8492014219401 487911424000 h2 fn+12+ 451361269269 121977856000 h2 fn+13− 18392351427 37531648000 h2 fn+14 + 37139996961 1219778560000 h2 fn+15 (68) y′′n+10 = y ′′ n +10 hy ′′′ n + 61940195105 25009272288 h2 fn+ 38819735375 2084106024 h2 fn+1 − 7755391625 347351004 h2 fn+2+ 11564784875 127598328 h2 fn+3− 1480662560375 8336424096 h2 fn+4 + 32091122155 99243144 h2 fn+5− 5423884076375 12504636144 h2 fn+6+ 334822805875 694702008 h2 fn+7 − 384255384875 926269344 h2 fn+8+ 1800509413375 6252318072 h2 fn+9− 321370705 2082024 h2 fn+10 + 44211042625 694702008 h2 fn+11− 69319839125 3572753184 h2 fn+12+ 8597247125 2084106024 h2 fn+13 − 759050375 1389404016 h2 fn+14+ 212230705 6252318072 h2 fn+15 (69) y′′n+11 = y ′′ n +11 hy ′′′ n + 3971658023273897 1455084933120000 h2 fn + 2266813707059 109983744000 h2 fn+1− 48389901241627 1979707392000 h2 fn+2+ 1823189496513649 18188561664000 h2 fn+3 − 100348334174341 513257472000 h2 fn+4+ 86610066741761629 242514155520000 h2 fn+5− 139054076635086833 291016986624000 h2 fn+6 + 53121794226743 99800064000 h2 fn+7− 14769476593075457 32335220736000 h2 fn+8+ 6621237757011361 20786927616000 h2 fn+9 − 9127089742965557 53892034560000 h2 fn+10+ 9366906277091 133249536000 h2 fn+11− 6230159494371769 291016986624000 h2 fn+12 + 8175398913517 1796401152000 h2 fn+13− 58463252748769 97005662208000 h2 fn+14+ 13621603946467 363771233280000 h2 fn+15 (70) y′′n+12 = y ′′ n +12 hy ′′′ n + 222031178 74449375 h2 fn+ 336427848 14889875 h2 fn+1 − 395464572 14889875 h2 fn+2+ 1635581336 14889875 h2 fn+3− 3177823608 14889875 h2 fn+4 + 4157834328 10635625 h2 fn+5− 1110232588 2127125 h2 fn+6+ 8675883864 14889875 h2 fn+7 − 7426541862 14889875 h2 fn+8+ 5198849816 14889875 h2 fn+9− 13721678676 74449375 h2 fn+10 + 1158088968 14889875 h2 fn+11− 3815012 163625 h2 fn+12+ 10574568 2127125 h2 fn+13 − 9806508 14889875 h2 fn+14+ 3046952 74449375 h2 fn+15 (71) 323 Atabo and Adee / J. Nig. Soc. Phys. Sci. 3 (2021) 308–333 324 y′′n+13 = y ′′ n +13 hy ′′′ n + 3983186767192217 1231225712640000 h2 fn + 504351825845881 20520428544000 h2 fn+1− 261504344043007 9120190464000 h2 fn+2+ 2100826446847817 17588938752000 h2 fn+3 − 18984767728152889 82081714176000 h2 fn+4+ 1210240131830197 2850059520000 h2 fn+5− 19906199411300267 35177877504000 h2 fn+6 + 8656591000332301 13680285696000 h2 fn+7− 1642958954761241 3040063488000 h2 fn+8+ 23360194872458951 61561285632000 h2 fn+9 − 11668896830043479 58629795840000 h2 fn+10+ 388626555599243 4560095232000 h2 fn+11− 5998735795932529 246245142528000 h2 fn+12 + 8030528674181 1465744896000 h2 fn+13− 594992073871 829108224000 h2 fn+14+ 27410337567119 615612856320000 h2 fn+15 (72) y′′n+14 = y ′′ n +14 hy ′′′ n + 1112609593123 318995820000 h2 fn+ 47075937433 1772199000 h2 fn+1 − 327517459171 10633194000 h2 fn+2+ 2059176926639 15949791000 h2 fn+3− 1767851783719 7088796000 h2 fn+4 + 12198755857183 26582985000 h2 fn+5− 4871439361817 7974895500 h2 fn+6+ 404310610109 590733000 h2 fn+7 − 4139166288943 7088796000 h2 fn+8+ 6562564086043 15949791000 h2 fn+9− 3810288983977 17721990000 h2 fn+10 + 496658084279 5316597000 h2 fn+11− 1636772074721 63799164000 h2 fn+12+ 1113970361 161109000 h2 fn+13 − 3740727473 5316597000 h2 fn+14+ 3740727473 79748955000 h2 fn+15 (73) y′′n+15 = y ′′ n +15 hy ′′′ n + 14605252055 3903291392 h2 fn+ 55679491125 1951645696 h2 fn+1 − 127956427875 3903291392 h2 fn+2+ 134732597875 975822848 h2 fn+3− 147877626375 557613056 h2 fn+4 + 951630836355 1951645696 h2 fn+5− 2519631003875 3903291392 h2 fn+6+ 175914493875 243955712 h2 fn+7 − 2381638669875 3903291392 h2 fn+8+ 834744780875 1951645696 h2 fn+9− 856960688385 3903291392 h2 fn+10 + 13247717625 139403264 h2 fn+11− 93624917875 3903291392 h2 fn+12+ 14479300125 1951645696 h2 fn+13 + 1501410375 3903291392 h2 fn+14+ 50188465 487911424 h2 fn+15 (74) Also, we take the third derivative of equation (9) and evaluate at xn+ j, j = 0(1)k to give: y′′′n+1 = y ′′′ n + 25221445 98402304 h fn+ 105145058757073 62768369664000 h fn+1 − 20999287611259 5706215424000 h fn+2+ 612744541065337 62768369664000 h fn+3− 189568380436867 8966909952000 h fn+4 + 2285168598349733 62768369664000 h fn+5− 3129453071993581 62768369664000 h fn+6+ 1138313909617631 20922789888000 h fn+7 − 988788576755233 20922789888000 h fn+8+ 679781959848881 20922789888000 h fn+9− 1096355235402331 62768369664000 h fn+10 + 64486158419069 8966909952000 h fn+11− 137515713789319 62768369664000 h fn+12+ 29219384284087 62768369664000 h fn+13 − 3867689367599 62768369664000 h fn+14+ 2639651053 689762304000 h fn+15 (75) 324 Atabo and Adee / J. Nig. Soc. Phys. Sci. 3 (2021) 308–333 325 y′′′n+2 = y ′′′ n + 631693279 2501928000 h fn+ 15268129873 7662154500 h fn+1− 302093934503 122594472000 h fn+2 + 31510558777 3831077250 h fn+3− 2248853768341 122594472000 h fn+4+ 245046332159 7662154500 h fn+5 − 1802000069137 40864824000 h fn+6+ 30858062633 638512875 h fn+7− 1720640706481 40864824000 h fn+8 + 222285459167 7662154500 h fn+9− 1915228492981 122594472000 h fn+10+ 24675139849 3831077250 h fn+11 − 240803930183 122594472000 h fn+12+ 3200718769 7662154500 h fn+13− 6783778481 122594472000 h fn+14 + 3012 875875 h fn+15 (76) y′′′n+3 = y ′′′ n + 1674820979 6623232000 h fn+ 56902877331 28700672000 h fn+1 − 60273315579 28700672000 h fn+2+ 794168588449 86102016000 h fn+3− 550323797343 28700672000 h fn+4 + 135645163833 4100096000 h fn+5− 1303414182247 28700672000 h fn+6+ 203566719249 4100096000 h fn+7 − 1239549466761 28700672000 h fn+8+ 2559362190283 86102016000 h fn+9− 459042547569 28700672000 h fn+10 + 189136929441 28700672000 h fn+11− 3529763983 1757184000 h fn+12+ 12256162053 28700672000 h fn+13 − 1622998893 28700672000 h fn+14+ 9166839 2609152000 h fn+15 (77) y′′′n+4 = y ′′′ n + 1328507 5255250 h fn+ 3800835068 1915538625 h fn+1− 579872212 273648375 h fn+2 + 18445487108 1915538625 h fn+3− 70153091887 3831077250 h fn+4+ 62404505692 1915538625 h fn+5 − 86024128376 1915538625 h fn+6+ 31404619756 638512875 h fn+7− 54690363709 1277025750 h fn+8 + 2690307652 91216125 h fn+9− 30412523756 1915538625 h fn+10+ 12534712108 1915538625 h fn+11 − 7643539093 3831077250 + 812610068 1915538625 h fn+13 − 107625136 1915538625 h fn+14+ 53500 15324309 h fn+15 (78) y′′′n+5 = y ′′′ n + 887775845 3511517184 h fn+ 10946710975 5518098432 h fn+1 − 81687827825 38626689024 h fn+2+ 4819615460525 502146957312 h fn+3− 8971075692025 502146957312 h fn+4 + 16743731110985 502146957312 h fn+5− 1081685073325 23911759872 h fn+6+ 8274717147275 167382319104 h fn+7 − 7198523492725 167382319104 h fn+8+ 14864967700175 502146957312 h fn+9− 7999362395575 502146957312 h fn+10 + 3296252627975 502146957312 h fn+11− 1004846299475 502146957312 h fn+12+ 30518628325 71735279616 h fn+13 − 28291221275 502146957312 h fn+14+ 803745 229605376 h fn+15 (79) 325 Atabo and Adee / J. Nig. Soc. Phys. Sci. 3 (2021) 308–333 326 y′′′n+6 = y ′′′ n + 340115 1345344 h fn+ 6950787 3503500 h fn+1− 118634421 56056000 h fn+2 + 2404277 250250 h fn+3− 1004179551 56056000 h fn+4+ 118528077 3503500 h fn+5 − 7491091147 168168000 h fn+6+ 43106373 875875 h fn+7− 2403399321 56056000 h fn+8 + 310373351 10510500 h fn+9− 127297257 8008000 h fn+10+ 11476851 1751750 h fn+11 − 10179391 5096000 h fn+12+ 135273 318500 h fn+13− 450453 8008000 h fn+14+ 1312 375375 h fn+15 (80) y′′′n+7 = y ′′′ n + 148080429 585728000 h fn+ 2541298850209 1280987136000 h fn+1 − 2709977083289 1280987136000 h fn+2+ 1118294324987 116453376000 h fn+3− 22922743552933 1280987136000 h fn+4 + 43245378870197 1280987136000 h fn+5− 56382862132093 1280987136000 h fn+6+ 21283573174031 426995712000 h fn+7 − 18372161728753 426995712000 h fn+8+ 12638994737537 426995712000 h fn+9− 20401231357003 1280987136000 h fn+10 + 8406116820059 1280987136000 h fn+11− 232954185989 116453376000 h fn+12+ 544780276327 1280987136000 h fn+13 − 72145233503 1280987136000 h fn+14+ 4482518383 1280987136000 h fn+15 (81) y′′′n+8 = y ′′′ n + 69181108 273648375 h fn+ 3800295904 1915538625 h fn+1− 368501776 174139875 h fn+2 + 18400755872 1915538625 h fn+3− 4900116016 273648375 h fn+4+ 64738663904 1915538625 h fn+5 − 28171120816 638512875 h fn+6+ 32195047904 638512875 h fn+7− 27104604508 638512875 h fn+8 + 56498981024 1915538625 h fn+9− 30436015696 1915538625 h fn+10+ 1792450592 273648375 h fn+11 − 3825896576 1915538625 h fn+12+ 813518752 1915538625 h fn+13− 107747312 1915538625 h fn+14+ 1312 375375 h fn+15 (82) y′′′n+9 = y ′′′ n + 148080429 585728000 h fn+ 56938145427 28700672000 h fn+1 − 60718067451 28700672000 h fn+2+ 275617326987 28700672000 h fn+3− 513628536159 28700672000 h fn+4 + 969126008463 28700672000 h fn+5− 1264262164839 28700672000 h fn+6+ 1442792036727 28700672000 h fn+7 − 1200258046089 28700672000 h fn+8+ 861743554713 28700672000 h fn+9− 458089367409 28700672000 h fn+10 + 188548120737 28700672000 h fn+11− 57454723701 28700672000 h fn+12+ 12212476677 28700672000 h fn+13 − 1617125229 28700672000 h fn+14+ 803745 229605376 h fn+15 (83) 326 Atabo and Adee / J. Nig. Soc. Phys. Sci. 3 (2021) 308–333 327 y′′′n+10 = y ′′′ n + 340115 1345344 h fn+ 121610725 61297236 h fn+1− 2075551475 980755776 h fn+2 + 294433325 30648618 h fn+3− 17564076025 980755776 h fn+4+ 295984445 8756748 h fn+5 − 43273790575 980755776 h fn+6+ 36779675 729729 h fn+7− 13743878725 326918592 h fn+8 + 627611225 20432412 h fn+9− 15176096665 980755776 h fn+10+ 18169175 2786238 h fn+11 − 39878275 20015424 h fn+12+ 25987525 61297236 h fn+13− 55091525 980755776 h fn+14+ 53500 15324309 h fn+15 (84) y′′′n+11 = y ′′′ n + 887775845 3511517184 h fn+ 11319967290643 5706215424000 h fn+1 − 132626474849 62705664000 h fn+2+ 54784127878667 5706215424000 h fn+3− 102073339405279 5706215424000 h fn+4 + 192568525522703 5706215424000 h fn+5− 83717504794957 1902071808000 h fn+6+ 95525302183421 1902071808000 h fn+7 − 79485015923203 1902071808000 h fn+8+ 24771909147359 815173632000 h fn+9− 83924967291121 5706215424000 h fn+10 + 39756270033953 5706215424000 h fn+11− 11548005399029 5706215424000 h fn+12+ 2443388534917 5706215424000 h fn+13 − 322980342509 5706215424000 h fn+14+ 9166839 2609152000 h fn+15 (85) y′′′n+12 = y ′′′ n + 1328507 5255250 h fn+ 248268 125125 h fn+1− 12972 6125 h fn+2 + 25263044 2627625 h fn+3− 31416549 1751750 h fn+4+ 29661108 875875 h fn+5− 5536136 125125 h fn+6 + 44305452 875875 h fn+7− 74018277 1751750 h fn+8+ 81177812 2627625 h fn+9− 13324692 875875 h fn+10 + 6858276 875875 h fn+11− 8502661 5255250 h fn+12+ 51204 125125 h fn+13− 48192 875875 h fn+14+ 3012 875875 h fn+15 (86) y′′′n+13 = y ′′′ n + 1674820979 6623232000 h fn+ 9574328794069 4828336128000 h fn+1− 10180906367021 4828336128000 h fn+2 + 6600937081531 689762304000 h fn+3− 85872829106857 4828336128000 h fn+4+ 161714860824569 4828336128000 h fn+5 − 210181855059553 4828336128000 h fn+6+ 79669188209819 1609445376000 h fn+7− 65878891642213 1609445376000 h fn+8 + 47601806972213 1609445376000 h fn+9− 9733223634433 689762304000 h fn+10+ 33795680175959 4828336128000 h fn+11 − 2990410218211 4828336128000 h fn+12+ 3733892296147 4828336128000 h fn+13− 44874067901 689762304000 h fn+14 + 2639651053 689762304000 h fn+15 (87) 327 Atabo and Adee / J. Nig. Soc. Phys. Sci. 3 (2021) 308–333 328 y′′′n+14 = y ′′′ n + 631693279 2501928000 h fn+ 311056753 156370500 h fn+1− 5395044599 2501928000 h fn+2 + 765940609 78185250 h fn+3− 46375653541 2501928000 h fn+4+ 5525678207 156370500 h fn+5 − 39205297537 833976000 h fn+6+ 712193069 13030875 h fn+7− 39205297537 833976000 h fn+8 + 5525678207 156370500 h fn+9− 46375653541 2501928000 h fn+10 + 765940609 78185250 h fn+11− 5395044599 2501928000 h fn+12+ 311056753 156370500 h fn+13+ 631693279 2501928000 h fn+14 (88) y′′′n+15 = y ′′′ n + 25221445 98402304 h fn+ 442589775 229605376 h fn+1− 388226775 229605376 h fn+2+ 1746295975 229605376 h fn+3 − 372104925 32800768 h fn+4+ 4103141115 229605376 h fn+5− 10001664025 688816128 h fn+6+ 1698012675 229605376 h fn+7 + 1698012675 229605376 h fn+8− 10001664025 688816128 h fn+9+ 4103141115 229605376 h fn+10− 372104925 32800768 h fn+11 + 1746295975 229605376 h fn+12− 388226775 229605376 h fn+13+ 442589775 229605376 h fn+14+ 25221445 98402304 h fn+15 (89) 3. Analysis of the Basic Properties Theorem 3.1 (Source: Lambert [21]). No zero-stable linear multistep method of step number k can have order exceeding k+1 when k is odd, or k+2 when k is even. Remark: For a proof of this theorem, the reader is referred to [18]. 3.1. Order of the Block Method We define the linear operator associated with the new method, (NS4O15M) as: L(y(x); h) = A0Ym−A1Ym−1−hA2Y ′ m−1−h 2 B0Y ′′ m−1−h 3 B1Y ′′′ m−1−h 4(C0 Fm+C1 Fm−1) (90) where, Ym =  yn+1 yn+2 . . . yn+k  , Ym−1 =  yn−k+1 yn−k+2 . . . yn  , Y′m−1 =  y′n−k+1 y′n−k+2 . . . y′n  , Y′′m−1 =  y′′n−k+1 y′′n−k+2 . . . y′′n  , Y′′′m−1 =  y′′′n−k+1 y′′′n−k+2 . . . y′′′n  , Fm =  fn+1 fn+2 . . . fn+k  , Fm−1 =  fn−k+1 fn−k+2 . . . fn  Taylor series expansion of equation (90) gives: L(y(x); h) = E0y(x) + hE1y ′(x) + h22 E2y ′′(x) + . . . + h( p)p E py ( p)(x) + h( p+1)p+1 E p+1y ( p+1)(x) + . . . (91) The new method, (NS4O15M) and its linear operator equation (90) are said to be of order p if and only if E0 = E1 = E2 = . . . = E p = E p+1 = E p+2 = 0, E p+3 = 0, E p+4 , 0 in equation (91). Thus, the new method, (NS4O15M) is of a uniform order 16 with the error constants: 328 Atabo and Adee / J. Nig. Soc. Phys. Sci. 3 (2021) 308–333 329 E p+4 =− 97656794693207524009663299584000 ,− 43313735102631663116607152000 ,− 109088471551231038275510272000 ,− 27085200034999469114875 ,− 4328588356506257785286426165248 ,− 25150957551253485232000 , 320104985832634119860424556544000 ,− 2890186588161181190772125 ,− 3664651642057711038275510272000 ,− 650885653187513304932857216 , − 31281013654043197 47637242118144000 ,− 242877078 282907625 ,− 575350492228736471 524009663299584000 ,− 514075874669 372979728000 ,− 14142409239375 8306204082176  T Hence, our new derived method (NS4O15M) is inline with theorem 3.1. 3.2. Consistency A numerical method, according to Lambert [20], [21], is said to be consistent if its order, that is, p ≥ 1. Therefore, the new method (NS4O15M), whose order is 16, is certainly consistent. 3.3. Zero Stability The new method (NS4O15M) is said to be zero stable if no roots of the first characteristic polynomial, |rs| > 1,∀s = 1, 2, . . . , N and that one of its roots is simple. That is, ρ(r) = ∣∣∣∣∣∣(rA0 − A1) ∣∣∣∣∣∣ (92) Where, A0 =  1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1  , A1 =  0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1  Figure 1. Absolute Stability Region of NS4O15M(16) So that equation (92) is evaluated and solved for r to give: r = 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0. Thus, the new method (NS4O15M) is zero stable. 3.4. Convergency According to Lambert [21], a numerical method is convergent if it is both zero stable and consistent. Hence, the new method (NS4O15M) is certainly convergent inline with (3.2) and (3.3) 3.5. Absolute Stability Definition 3.5.1 (Source: Lambert [20]). The LMM (NS4O15M) is said to be A-Stable if the region of absolute stability includes the entire left half of the z-plane (that is z ∈ (−∞, 0)). Therefore, the absolute stability region for the new method (NS4O15M) is plotted in the spirit Lambert [20], [21] and Okuonghae and Ikhile [22] using MATLAB. Hence, the stability polynomial is given by: R(t; h) = 141760259867125 1038275510272 t14(h)4 + 1125 2 t14(h)3 + 225 2 t14(h)2 + 15 t14(h) + (t − 1) t14 (93) Where, h = λh Since A-stability is a severe property that is desired by all numerical methods, by Definition (3.5.1) the new method (NS4O15M) is A-Stable. Therefore, it is certain that Dahlquist barrier no longer posses restrictions on the use of step sizes in the method. Also, it is clear from Figure 1 that the new method has small unstable region and a wider region of absolute stability which proved its convergence with approximate solutions having little or no deviation from the true solution as a result of small change in input data Lambert [20]. Hence, the new method is numerically stable and it is investigated in section four below. 329 Atabo and Adee / J. Nig. Soc. Phys. Sci. 3 (2021) 308–333 330 4. Numerical Examples The new method (NS4O15M) employed Taylor series approach to incorporate all the initial values for the solution of (1). The following numerical examples are used to test the accuracy of the new method and comparison are made with some selected numerical methods in recent literature. All numerical iterations are carried out on MATLAB Software environment. Problem 4.1 (Source: Jena et al. [6]). y(iv) = ez ( z4 + 14 z3 + 49 z2 + 32 z − 12 ) , h = 0.1, y (0) = 0, y′ (0) = 0, y′′ (0) = 2, y′′′ (0) = −6, Exact Solution: y (z) = z2 (1 − z)2 ez, The numerical results for Problem 4.1 is as shown in Table 1 Problem 4.2 (Source: Ukpebor et al. [13]). Consider the special fourth order below u(iv) = x, u (0) = 0, u′ (0) = 1, u′′ (0) = 0, u′′′ (0) = 0, h = 0.1 Exact Solution: u (0) = x 5 120 + x The numerical results for Problem 4.2 is as shown in Table 2 Problem 4.3 (Source: Ukpebor et al. [13]). Consider the linear differential equation of fourth order: u(iv) + u′′ = 0, u (0) = 0, u′ (0) = −1.172−50π , u ′′ (0) = 1144−100π , u ′′′ (0) = 1.2144−100 π , h = 0.01, Exact Solution: u (x) = 1−x−cos(x)−1.2 sin(x)144−100 π The numerical results for Problem 4.3 is as shown in Table 3 Problem 4.4. To confirm the application of the new method NS4O15M(16), we solve a physical problem from ship dynamics. As stated by Familua and Omole [17], when a sinusoidal wave of frequency Ω passes long a ship or offshore structure, the resultant fluid actions vary with time t. Therefore, consider the fourth-order problem as: y(iv) + 3y′′ + y(2 + ε cos(Ωt)) = 0, t > 0 Is subjected to the following initial conditions: y(0) = 1, y′(0) = 0, y′′(0) = 0, y′′′(0) = 0, h = 1320 where ε = 0 , for the existence of the theoretical solution: y(t) = 2 cos(t) − cos(t √ (2)). Problem 4.5 (Source: Allogmany et al., [4]). We shall consider a nonlinear initial value problem of the form: yiv = (y′)2 − yy′′′ − 4x2 + ex(1 − 4x + x2), 0 ≤ x ≤ 1 y(0) = 1, y′(0) = 1, y′′(0) = 3, y′′′(0) = 1 Exact Solution: y(x) = x2 + ex We also compare the results for this problem that has been solved by [19]. The results are shown in Table 5. To further illustrate the Table 1. Comparison of Absolute Maximum Error for Problem 4.1 when h=0.1 x Error in Error in Error in ODE45 Jena et al. [6] NS4O15M 0.1 1.0326 e-07 1.5370 e-14 1.7347 e-18 0.2 1.6792 e-07 8.2021 e-14 1.6653 e-16 0.3 2.5340 e-07 3.6666 e-13 4.3715 e-16 0.4 3.6477 e-07 6.3424 e-13 8.3267 e-16 0.5 5.0816 e-07 6.7024 e-13 5.2736 e-16 0.6 6.9093 e-07 5.2608 e-13 3.7748 e-15 0.7 9.2191 e-07 3.3906 e-13 4.6629 e-15 0.8 1.2116 e-06 1.9011 e-13 5.1209 e-15 0.9 1.5727 e-06 9.6152 e-14 1.3854 e-14 1.0 4.1649 e-06 4.4983 e-14 1.9791 e-14 Figure 2. Efficiency Curves for Test Problem 4.1 accuracy of (NS4O15M) with comparison to other recent existing methods such as Jena et al. [6], Ukpebor et al. [13] and Familua et al. [17], we have also presented the efficiency curves for Problem 4.1, 4.3 and 4.4 and are given below: 5. Results and Discussions This research paper has considered five special fourth order ODEs problems in recent literature. Problems 4.1 and 4.2 are solved using a step size, h=0.1, Problem 4.3 has been solved using a step-size, h=0.01 while Problem 4.4 and 4.5 have been solved using a step- 330 Atabo and Adee / J. Nig. Soc. Phys. Sci. 3 (2021) 308–333 331 Table 2. Comparison of Absolute Maximum Error for Problem 4.2 when h=0.1 x Error in Error in Error in Omar et al. [8] Ukpebor et al. [13] NS4O15M 0.1 1.002087 e-12 0.00 e-00 0.00 e+00 0.2 0.000000 e+00 0.00 e-00 0.00 e+00 0.3 0.000000 e+00 0.00 e-00 0.00 e+00 0.4 0.000000 e+00 0.00 e-00 0.00 e+00 0.5 1.002087 e-12 0.00 e-00 0.00 e+00 0.6 2.755907 e-12 0.00 e-00 0.00 e+00 0.7 3.507306 e-12 0.00 e-00 0.00 e+00 0.8 3.507306 e-12 2.00 e-18 0.00 e+00 0.9 4.175569 e-12 2.00 e-18 1.11 e-16 1.0 4.175569 e-12 1.00 e-17 0.00 e+00 Table 3. Comparison of Absolute Maximum Error for Problem 4.3 when h=0.01 x Error in Error in Error in Adesanya et al. [10] Ukpebor et al. [13] NS4O15M 0.01 8.5052 e-19 8.0000 e-20 5.4210 e-20 0.02 1.3010 e-18 9.1500 e-19 5.4210 e-20 0.03 4.7704 e-18 3.4150 e-18 2.7105 e-19 0.04 1.7347 e-17 8.2220 e-18 1.0842 e-19 0.05 4.3368 e-17 1.5965 e-17 3.2526 e-19 0.06 9.5409 e-17 2.7404 e-17 3.2526 e-19 0.07 1.8127 e-16 3.4329 e-17 3.2526 e-19 0.08 3.1571 e-16 3.2551 e-17 4.3368 e-19 0.09 5.1868 e-16 6.5927 e-17 2.1684 e-19 0.10 8.0491 e-16 1.1919 e-16 6.5052 e-19 size of h = 1320 Jena et al. [6] has solved Problem 4.1 using 9-step block method with uniform order 12 in comparison with ODE45 and Awoyemi et al. [11]. Similarly, Ukpebor et al. [13] derived a 7-step block formula for solving fourth order ODEs with a uniform order 4 and results compared with Duromola [9]. However, results for our test problems are shown in Tables 1, 2 and 3 respectively. Also, Familua et al. [17] formulated a five point mono hybrid block method for the solution of nth order ordinary differential equations and results compared in Table 4 with direct block and predictor-corrector block methods and also compared with our new method. Results from Tables 4, 5 and Figure 4, showed that the new method is considerably efficient at solving application problems as ship dynamics and nonlinear IVPs. It is certain from the tables that the new derived method (NS4O15M) gave better accuracy than Jena et al. [6], ODE45 Omar et al. [8], Adesanya et al. [10], Awoyemi et al. [11], Ukpebor et al. [13] and Familua et al. [17] respectively. Generally, the new method, (NS4O15M) showed improved approximations as h → 0 on some of the test examples considered. Further analysis using the efficiency Table 4. Comparison of Absolute Maximum Error for Problem 4.4 when h = 1 320 (Application problem arising from ship dynamics) x Error in block Error in PC- Error in method method, p=7, in [17] method, p=7, [17] NS4O15M(16) 0.003125 6.685763 e-13 5.685763 e-10 0.000000e+00 0.006250 1.458489 e-11 1.767654 e-10 1.110223e-16 0.009375 1.082968 e-10 5.909878 e-09 0.000000e+00 0.001250 3.917803 e-10 5.767654 e-09 2.220446e-16 0.015625 1.025145 e-09 1.100202 e-08 7.771561e-16 0.018750 2.217319 e-09 6.898767 e-08 2.886580e-15 0.021875 4.226068 e-09 4.636354 e-08 8.548717e-15 0.025000 7.358019 e-09 5.787654 e-07 2.187139e-14 0.028125 1.196868 e-08 2.245763 e-07 4.951595e-14 0.031250 1.846249 e-08 2.846249 e-07 1.035838e-13 Table 5. Comparison of Absolute Maximum Error for Problem 4.5 when h = 1 320 x IHB8 [19] I2PBDO6 [4] NS4O15M(16) 0.003125 4.261834 e-14 0.0000000 e+00 2.220446 e-16 0.006250 7.314820 e-13 3.9099252 e-16 0.000000 e+00 0.009375 1.625321 e-12 7.2441945 e-16 2.220446 e-16 0.001250 6.568972 e-12 9.9936929 e-16 0.000000 e+00 0.015625 2.849147 e-12 1.2947886 e-15 0.000000 e+00 0.018750 4.231457 e-12 0.0000000 e+00 2.220446 e-16 0.021875 1.246852 e-11 1.5163355 e-15 6.661338 e-16 0.025000 5.146142 e-11 2.8632323 e-15 1.332268 e-15 0.028125 3.765423 e-11 4.5622733 e-15 2.886580 e-15 Figure 3. Efficiency Curves for Test Problem 4.3 curves in Figures 2, 3 and 4 showed that the new method has small scale errors unlike other existing methods compared. 331 Atabo and Adee / J. Nig. Soc. Phys. Sci. 3 (2021) 308–333 332 Figure 4. Efficiency Curves for Test Problem 4.4 6. Conclusion In this research paper, a new block method that approximates 15-step simultaneously for the solution of (1) has been formulated. The new method is derived through interpolation and collocation techniques using power series as a basis function. The numerical properties of the method have been investigated and proven to be zero stable, consistent and A-stable. Test results showed that the new method is capable of solving general fourth order ODEs including physical problems from ship dynamics and nonlinear IVPs with better accuracy than other existing methods considered in this research paper. Therefore, the new method 15-step with uniform higher order 16, proved better accuracy over 7-step with uniform order 4 and 9-step with uniform higher order 12, among others. Hence, the new method (NS4O15M) should be considered as a viable alternative for solving general fourth order ODEs. Acknowledgements The authors wish to gracefully thank all editors and peer reviewers for their constructive comments at making this research paper a success. More grease to your elbows! References [1] R. Conti, D. Graffi & G. 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