J. Nig. Soc. Phys. Sci. 2 (2020) 218–227 Journal of the Nigerian Society of Physical Sciences Original Research Numerical Algorithms for Direct Solution of Fourth Order Ordinary Differential Equations J. O. Kuboye, O. R. Elusakin∗, O. F. Quadri Department of Mathematics, Federal University Oye-Ekiti, Oye-Ekiti, Nigeria Abstract This paper examines the derivation of hybrid numerical algorithms with step length(k) of five for solving fourth order initial value problems of ordinary differential equations directly. In developing the methods, interpolation and collocation techniques are considered. Approximated power series is used as interpolating polynomial and its fourth derivative as the collocating equation. These equations are solved using Gaussian- elimination approach in finding the unknown variables a j, j=0,...,10 which are substituted into basis function to give continuous implicit scheme. The discrete schemes and its derivatives that form the block are obtained by evaluating continuous implicit scheme at non-interpolating points. The developed methods are of order seven and the results generated when the methods were applied to fourth order initial value problems compared favourably with existing methods. DOI:10.46481/jnsps.2020.100 Keywords: Interpolation, Collocation, Block methods, Fourth order, Ordinary differential equations Article History : Received: 07 May 2020 Received in revised form: 10 August 2020 Accepted for publication: 15 August 2020 Published: 01 November 2020 c©2020 Journal of the Nigerian Society of Physical Sciences. All rights reserved. Communicated by: F. Y. Eguda 1. Introduction The general fourth order initial value problem of ordinary differential equations of the form yiv = f (x, y(x), y′(x), y′′(x), y′′′(x)), y(x0) = y1, y ′(x0) = y2, y ′′(x0) = y3 (1) is considered in this article. In the past, solving fourth or- der ordinary differential equations (ODEs) requires reducing ∗Corresponding author tel. no: +23480xxxx572 Email address: opeyemielusakin21@gmail.com (O. R. Elusakin ) the differentials to systems of first order ODEs and approxi- mate numerical method for the first order would be used to solve the system. This approach is been attached with lots of setbacks which include: computational burden, lots of human effort, complexity in developing computer code which affects the accuracy of the method in terms of error. This was exten- sively discussed by researchers like Awoyemi [1], Fatunla [2] and Lambert [3]. Due to several disadvantages found in reduc- tion method, the direct method of solving ODEs of higher order was developed by lots of scholars which include Akeremale et al. [4], Abolarin et al. [5], Kuboye et. al [6], Omar & Kuboye [7], Adeyefa [8], Abdullahi et al. [9], Adeniyi & Mohammed [10], Olabode [11], Adesanya et al.[12], Omar & Suleiman [13] 218 Kuboye et al. / J. Nig. Soc. Phys. Sci. 2 (2020) 218–227 219 and Familua & Omole [14]. Specifically, numerical methods for solving equation (1) were proposed by Omar and Kuboye[15], Areo and Omole[16] and Mohammed[17]. These current methods solved directly equation (1) but its accuracy in terms of error can still be improved. Therefore, this paper examines the derivation and implementation of the efficient numerical algorithm for solving fourth order ordinary differential equations directly and it focuses on improving the accuracy of the existing methods. 2. Methodology This section considers derivation of block methods for direct solution of fourth order ODEs. 2.1. Derivation of First Block Method(FBM) Power series approximate solution of the form y(x) = k+5∑ j=0 a j x j (2) is used as interpolating polynomial where k=5.The fourth derivative of equation(2) gives: yiv(x) = k+5∑ j=4 j( j − 1)( j − 2)( j − 3)a j x j−4 (3) Equation (2) is interpolated at x = xn+i, i = 0(1)2 and 5 2 and equation (3) is collocated at x = xn+i, i = 0(1)5 and 5 2 . Interpolation and collocation equations are combined together to give a non-linear system of equations of the form: k+5∑ j=0 a j x j n+i = yn+i k+5∑ j=4 j( j − 1)( j − 2)( j − 3)a j x j−4 n+i = fn+i (4) The unknown variables a′j s in (4) are gotten with the use of Gaussian elimination approach which are substituted into equation (2) and this yields a continuous implicit scheme of the form k−3∑ j=0 α j(t)yn+ j + α 5 2 yn+ 52 = h 4 k∑ j=0 β j(t) fn+ j + h 4λ 5 2 fn+ 52 (5) where t = x−xn+k−1h α0(t) α1(t) α2(t) α 5 2 (t) = −9 5 −27 10 −13 10 −1 5 8 34 3 5 2 3 −18 45 2 17 2 1 64 5 208 15 24 5 8 15 t0 t1 t2 t3 (6) 219 Kuboye et al. / J. Nig. Soc. Phys. Sci. 2 (2020) 218–227 220 β0(t) β1(t) β2(t) β 5 2 (t) β3(t) β4(t) β5(t) = T t0 t1 t2 t3 t5 t6 t7 t8 t9 t10 (7) where T = 297000 290304000 404694 290304000 75279 290304000 −102185 290304000 72576 290304000 36288 290304000 −2880 290304000 −7200 290304000 −2080 290304000 −192 290304000 3441528 11612160 5154498 11612160 2433933 11612160 323717 11612160 32256 11612160 15232 11612160 −1728 11612160 −3072 11612160 −800 11612160 −64 11612160 6797304 5806080 10732194 5806080 5745021 5806080 1207573 5806080 −145152 5806080 −6048 5806080 10944 5806080 12384 5806080 2720 5806080 192 5806080 560520 2268000 690174 2268000 419859 2268000 267195 2268000 −129024 2268000 −46592 2268000 11520 2268000 9600 2268000 1920 2268000 128 2268000 1716984 5806080 3760866 5806080 3494877 5806080 153384 5806080 −290304 5806080 −72576 5806080 27072 5806080 16704 5806080 3040 5806080 192 5806080 187272 11612160 129150 11612160 −353709 11612160 −650629 11612160 −169344 11612160 −3136 11612160 20160 11612160 7392 11612160 1120 11612160 64 11612160 428760 290304000 508266 290304000 −76719 290304000 −322779 290304000 290304 290304000 266112 290304000 118080 290304000 2880 290304000 3680 290304000 192 290304000 The coefficient of first and higher derivatives of (5) give α′0(t) α′1(t) α′2(t) α′5 2 (t) = 27 10 26 10 −6 10 34 3 30 3 6 3 45 2 34 2 −6 2 208 15 144 15 124 15 1 h (8) 220 Kuboye et al. / J. Nig. Soc. Phys. Sci. 2 (2020) 218–227 221 β′0 β′1 β′2 β′5 2 β′3(t) β′4(t) β′5(t) = S t0 t1 t2 t4 t5 t6 t7 t8 t9 (9) where S = −134898 96768000 −50186 96768000 102185 96768000 −120960 96768000 −72576 96768000 6720 96768000 19200 96768000 6240 96768000 640 96768000 5154498 11612160 4867866 11612160 971151 11612160 161280 11612160 91392 11612160 −12096 11612160 −24576 11612160 −7200 11612160 −640 11612160 3577398 1935360 3830014 1935360 1207573 1935360 −241920 1935360 −120960 1935360 25536 1935360 33024 1935360 8160 1935360 640 1935360 690174 2268000 839718 2268000 801585 2268000 −645120 2268000 −2795520 2268000 80640 2268000 76800 2268000 17280 2268000 1280 2268000 1253622 1935360 2329918 1935360 1533845 1935360 −483840 1935360 −145152 1935360 63168 1935360 44544 1935360 9120 1935360 640 1935360 −129150 11612160 707418 11612160 1951887 11612160 846720 11612160 18816 11612160 −141120 11612160 −59136 11612160 −10080 11612160 −640 11612160 169422 96768000 −51146 96768000 −322775 96768000 4838405 96768000 532224 96768000 275520 96768000 76800 96768000 11040 96768000 640 96768000 α′′0 (t) α′′1 (t) α′′2 (t) α′′5 2 (t) = 13 5h2 −6 5h2 10 h2 4 h2 17 h2 −6 h2 48 5h2 16 5h2 t0 t1 (10) 221 Kuboye et al. / J. Nig. Soc. Phys. Sci. 2 (2020) 218–227 222 β′′0 (t) β′′1 (t) β′′2 (t) β′′5 2 (t) β′′3 (t) β′′4 (t) β′′5 (t) = U t0 t1 t3 t4 t5 t6 t7 t8 (11) where U = −25093 48384000 102185 48384000 −241920 48384000 −181440 48384000 20160 48384000 67200 48384000 24960 48384000 2880 48384000 811311 1935360 323717 1935360 107520 1935360 76160 1935360 −12096 1935360 −28672 1935360 −9600 1935360 −960 1935360 1915007 967680 1207573 967680 −483840 967680 −302400 967680 76608 967680 11558 967680 32640 967680 2880 967680 139953 378000 267195 378000 −430080 378000 −232960 378000 80640 378000 89600 378000 23040 378000 1920 378000 1164959 967680 1533845 967680 −967680 967680 −362880 967680 18950 967680 155904 967680 36480 967680 2880 967680 117903 1935360 650629 1935360 564480 1935360 15680 1935360 −141120 1935360 −68992 1935360 −13440 1935360 −960 1935360 −25573 48384000 −322775 48384000 967680 48384000 1330560 48384000 826560 48384000 268800 48384000 44160 48384000 2880 48384000 α′′′0 (t) α′′′1 (t) α′′′2 (t) α′′′5 2 (t) = −6 5 4 −6 16 5 t0 (12) β′′′0 (t) β′′′1 (t) β′′′2 (t) β′′′5 2 (t) β′′′3 (t) β′′′4 (t) β′′′5 (t) = V t0 t2 t3 t4 t5 t6 t7 (13) 222 Kuboye et al. / J. Nig. Soc. Phys. Sci. 2 (2020) 218–227 223 where V = 20437 9676800 −14515 9676800 −145152 9676800 20160 9676800 80640 9676800 3494 9676800 4608 9676800 323717 967680 322560 967680 304640 967680 −60480 967680 −172032 967680 −67200 967680 −7680 967680 1207573 967680 −1451520 967680 −1209600 967680 383040 967680 69350 967680 228480 967680 23040 967680 −53439 75600 258048 75600 186368 75600 −80640 75600 −107520 75600 −32256 75600 −3072 75600 1533845 967680 −2903040 967680 −1451520 967680 947520 967680 935424 967680 255360 967680 23040 967680 650629 1935360 1935360 1693440 62720 1935360 −705600 1935360 −413952 1935360 −94080 1935360 −7680 1935360 −64555 9676800 580608 9676800 1064448 9676800 826560 9676800 322560 9676800 61824 9676800 4608 9676800 Discrete schemes and its derivatives are derived by evaluating (5) as well as its derivatives at grid points and non-grid points which are used to form the block yn+1 yn+2 yn+ 52 yn+3 yn+4 yn+5 = 1 1 1 1 1 1 [ yn ] + 1 2 5 2 3 4 5 [ hy′n ] + 1 2 2 25 8 9 2 8 25 2 [ h2y′′n ] + 1 6 4 3 125 48 9 2 32 3 125 6 [ h3y′′′n ] + h4 0 0 0 0 0 579232268000 0 0 0 0 0 1967670875 0 0 0 0 0 1075412518579456 0 0 0 0 0 5850956000 0 0 0 0 0 18534470875 0 0 0 0 0 478759072 fn−4 fn−3 fn− 52 fn−2 fn−1 fn + h4 413 10368 −8977 90720 9292 70875 −2279 36288 29 3780 −6593 9072000 428 567 −122 81 140288 70875 −2672 2835 4 35 764 70875 33640625 18579456 −30109375 9289728 89375 20736 −19109375 9289728 171875 688128 −437875 18579456 16137 4480 3267 560 6948 875 −243 64 207 448 −4887 112000 28928 2835 −7936 567 1441792 70875 −27136 2835 32 27 −7936 70875 1609375 72576 −484375 18144 23500 567 −671875 36288 15625 6048 −2375 10368 fn+1 fn+2 fn+ 52 fn+3 fn+4 fn+5 (14) 2.2. Derivation of Second Block Method(SBM) Equation(2) is interpolated at x = xn+i, i = 0(1)2 and 9 4 and equation(3) is collocated at x = xn+i, i = 0(1)5 and 9 4 . The same steps used in deriving the first block method are also employed and this produces the block method 223 Kuboye et al. / J. Nig. Soc. Phys. Sci. 2 (2020) 218–227 224 yn+1 yn+2 yn+ 94 yn+3 yn+4 yn+5 = [ yn ] + 1 2 9 4 3 4 5 [ hy′n ] + 1 2 2 81 32 9 2 8 25 2 [ h2y′′n ] + 1 6 4 3 243 128 9 2 32 3 125 6 [ h3y′′′n ] + h4 0 0 0 0 0 690432721600 0 0 0 0 0 1169642525 0 0 0 0 0 11919084392936012800 0 0 0 0 0 57935600 0 0 0 0 0 220168505 0 0 0 0 0 568625108864 fn−4 fn−3 fn−94 fn−2 fn−1 fn + h4 76921 1814400 −1139 6480 594688 3274425 −6749 181440 7421 1270080 −11717 19958400 11248 14175 −7558 2835 8978432 3274425 −1576 2835 344 3969 −1352 155925 3705501897 2936012800 −169057287 41943040 229379121 55193600 −247763043 293601280 540947889 4110417920 −425211849 32296140800 84159 22400 −2349 224 148224 13475 −5031 2240 1377 3920 −8667 246400 150272 14175 −10496 405 8388608 297675 −15872 2835 17888 19845 −256 2835 1668125 72576 −115625 2268 7520000 130977 −378125 36288 509375 254016 −147625 798336 fn+1 fn+2 fn+ 94 fn+3 fn+4 fn+5 (15) 3. Analysis of the method Properties of the methods are examined in this section. 3.1. Order of block methods In finding the order of the block methods,the method proposed by Lambert[3] is employed whereby Taylor series expansion are used in expanding the y and f-functions and by further comparing the coefficients of h,this gives the block methods to be of order [7, 7, 7, 7, 7, 7]T . 3.2. Zero-stability A linear multi-step method is said to be zero-stable if the roots rs, s=1,2,..., N(grid and non grid points) of the first characteristics polynomial defined by p(r) = det(rA ′ − B ′ ) satisfy |rs| < 1 and the root |r| = 1 having multiplicity not exceeding one.(Lambert [3]) Where A ′ = 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 224 Kuboye et al. / J. Nig. Soc. Phys. Sci. 2 (2020) 218–227 225 , B ′ = 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 Therefore r=0,0,0,0,0,1. Hence the zero-stability of first block method is confirmed which is also applied to the second block method. 3.3. Convergence According to Awoyemi[1], Equation(5) converges if it is zero-stable and consistent.This implies that the developed methods converged. 4. Test Problems The following fourth order initial value problems[I.V.P] are solved in order to examine the accuracy of the methods Problem 1: yiv = x, y(0) = 0, y ′ (0) = 1, y ′′ (0) = y ′′′ = 0, h = 0.1 E xactsolution : y(x) = x 5 120 + x Source: Mohammed[17] Problem 2: yiv − y = 0, y(0) = 1, y ′ (0) = 0, y ′′ (0) = −2, y ′′′ (0) = 0, h = 1320 E xactsolution : y(x) = −14 e x − 1 4 e −x + 32 cos(x) Source: Areo and Omole[16] Problem 3: yiv = (y ′ )2 − yy ′′′ − 4x2 + ex(1 − 4x + x2), y(0) = 1, y ′ (0) = 1, y ′′ (0) = 3, y ′′′ (0) = 1, h = 0.132 E xactsolution : y(x) = x2 + ex Source: Familua and Omole [14] The following acronyms are used in the Tables below ES - Exact Solution CS - Computed Solution FBM – First Block Method 225 Kuboye et al. / J. Nig. Soc. Phys. Sci. 2 (2020) 218–227 226 SBM – Second Block Method EIM[17] - Error in Mohammed[17] EIAO[16]- Error in Areo and Omole[16] EIOK [15]- Error in Omar and Kuboye[15] EIFBM - Error in First Block Method EISBM - Error in Second Block Method EIFO [14] - Error in Familua and Omole [14] Table 1. ES and CS of FBM for Problem 1 x ES CS 0.1 0.100000083333333340 0.100000083333333340 0.2 0.200002666666666690 0.200002666666666690 0.3 0.300020250000000040 0.300020250000000040 0.4 0.400085333333333350 0.400085333333333400 0.5 0.500260416666666650 0.500260416666666760 0.6 0.600647999999999960 0.600648000000000070 0.7 0.701400583333333330 0.701400583333333550 0.8 0.802730666666666700 0.802730666666666700 0.9 0.904920750000000050 0.904920750000000160 1.0 1.008333333333333300 1.008333333333333500 Table 2. ES and CS of SBM for Problem 1 x ES CS 0.1 0.100000083333333340 0.100000083333333340 0.2 0.200002666666666690 0.200002666666666690 0.3 0.300020250000000040 0.300020250000000100 0.4 0.400085333333333350 0.400085333333333400 0.5 0.500260416666666650 0.500260416666666650 0.6 0.600647999999999960 0.600648000000000070 0.7 0.701400583333333330 0.701400583333333220 0.8 0.802730666666666700 0.802730666666666810 0.9 0.904920750000000050 0.904920750000000160 1.0 1.008333333333333300 1.008333333333333300 5. Discussion of Results In Tables 1 and 2, exact and computed solutions of FBM and SBM for solving problem 1 are shown. Table 3 reveals the efficiency of these block methods (EISBM and EISBM) as com- pared favourably with EIM[17] and EIOK[15]. Furthermore, exact and computed solutions of the newly developed block Table 3. Comparison of EIFBM and EISBM with EIM[17] and EIOK[15] for solving problem 1 x EIFBM EISBM EIM(2010) EIOK(2016) 0.1 0.0000000e+00 0.0000000e+00 7.000024E-10 1.002087E-12 0.2 0.0000000e+00 0.0000000e+00 8.9999912-10 0.000000E+00 0.3 0.0000000e+00 5.5511151e-17 2.599993E-09 0.000000E+00 0.4 5.5511151e-17 5.5511151e-17 5.100033E-09 0.000000E+00 0.5 1.1102230e-16 0.0000000e+00 7.799979E-09 1.002087E-12 0.6 1.1102230e-16 1.1102230e-16 1.180009E-08 2.755907E-12 0.7 2.2204460e-16 1.1102230e-16 1.180009E-08 3.507306E-12 0.8 0.0000000e+00 1.1102230e-16 1.410006E-08 3.507306E-12 0.9 1.1102230e-16 1.1102230e-16 1.880000E-08 4.175549E-12 1.0 2.2204460e-16 0.0000000e+00 1.008335E-08 4.759970E-12 Table 4. ES and CS of FBM for Problem 2 x ES CS 0.0031250 1.000009765628973500 1.000009765628973700 0.0062500 1.000039062563578400 1.000039062563578400 0.0093750 1.000087890946866900 1.000087890946867100 0.0125000 1.000156251017263000 1.000156251017263500 0.0156250 1.000244143108567400 1.000244143108567400 0.0187500 1.000351567649961900 1.000351567649962400 0.0218750 1.000478525166021100 1.000478525166021300 0.0250000 1.000625016276719800 1.000625016276719600 0.0281250 1.000791041697446400 1.000791041697446800 0.0312500 1.000976602239017000 1.000976602239017000 Table 5. ES and CS of SBM for Problem 2 x ES CS 0.0031250 1.000009765628973500 1.000009765628973900 0.0062500 1.000039062563578400 1.000039062563578400 0.0093750 1.000087890946866900 1.000087890946866900 0.0125000 1.000156251017263000 1.000156251017263500 0.0156250 1.000244143108567400 1.000244143108567100 0.0187500 1.000351567649961900 1.000351567649962100 0.0218750 1.000478525166021100 1.000478525166021100 0.0250000 1.000625016276719800 1.000625016276719600 0.0281250 1.000791041697446400 1.000791041697445900 0.0312500 1.000976602239017000 1.000976602239017200 Table 6. Comparison of EIFBM and EISBM with EIAO[16] for solving prob- lem 2 x EIFBM EISBM EIAO (2015) 0.0031250 2.2204460e-016 4.4408921e-016 4.440892e-16 0.0062500 0.0000000e+000 0.0000000e+000 2.176037e-14 0.0093750 2.2204460e-016 0.0000000e+000 .771916e-13 0.0125000 4.4408921e-016 4.4408921e-016 7.666090e-13 0.0156250 0.0000000e+000 2.2204460e-016 2.367773e-12 0.0187500 4.4408921e-016 2.2204460e-016 5.932477e-12 0.0218750 2.2204460e-016 0.0000000e+000 1.287681e-11 0.0250000 2.2204460e-016 2.2204460e-016 2.517841e-11 0.0281250 4.4408921e-016 4.4408921e-016 4.546752e-11 0.0312500 0.0000000e+000 2.2204460e-016 7.712331e-11 ’ Table 7. ES and CS of FBM for Problem 3 x ES CS 0.103125 1.119264744787591900 1.119264744969084200 0.206250 1.271599493198048500 1.271599504741302500 0.306250 1.452110907065013100 1.452111029006491400 0.406250 1.666216862500122800 1.666217515460942200 0.506250 1.915347109920916500 1.915349507140536000 0.603125 2.191581593606204900 2.191588302867649500 0.703125 2.514440293333696500 2.514456732090109900 0.803125 2.877516387746607200 2.877551937602963200 0.903125 3.282936158805099100 3.283006004031709900 1.003125 3.733049511495175400 3.733176679391747100 226 Kuboye et al. / J. Nig. Soc. Phys. Sci. 2 (2020) 218–227 227 Table 8. ES and CS of SBM for Problem 3 x ES CS 0.103125 1.119264744787591900 1.119264744966372600 0.206250 1.271599493198048500 1.271599504536039800 0.306250 1.452110907065013100 1.452111026692671300 0.406250 1.666216862500122800 1.666217502634746300 0.506250 1.915347109920916500 1.915349459022614800 0.603125 2.191581593606204900 2.191588166231517800 0.703125 2.514440293333696500 2.514456393591375100 0.803125 2.877516387746607200 2.880551715825508700 0.903125 3.282936158805099100 3.283004543615038400 1.003125 3.733049511495175400 3.733174005515217600 Table 9. Comparison of EIFBM and EISBM with EIFO [14] for solving prob- lem 3 x EIFBM EISBM EIFO[14] 0.103125 1.8149238e-010 1.7878077e-010 9.02145880e-10 0.206250 1.1543254e-008 1.1337991e-008 1.216821428e-09 0.306250 1.2194148e-007 1.1962766e-007 1.21681228e-09 0.406250 6.5296082e-007 6.4013462e-007 1.713796095e-09 0.506250 2.3972196e-006 2.3491017e-006 1.481970916e-08 0.603125 6.7092614e-006 6.5726253e-006 3.058338503e-08 0.703125 1.6438756e-005 1.6100258e-005 4.941858156e-08 0.803125 3.5549856e-005 3.5007632e-005 7.128679089e-08 0.903125 6.9845227e-005 6.8384810e-005 1.058773080e-07 1.003125 1.2716790e-004 1.2449402e-004 1.445520074e-07 methods for the solution of problems 2 and 3 are demonstrated in Tables 4, 5, 7 and 8. These methods outperform method pro- posed by Areo and Omole [16] in terms of accuracy. In addi- tion, the performance of these methods in solving problem 3 is not encouraging as the accuracy is lower when the comparison is made with EIFO [14]. However, the capability of these meth- ods in solving the nonlinear equation is established in Table 9. Finally, it is evident in Tables 3, 6 and 9 that SBM is better than FBM in solving fourth order ODEs. 6. Conclusion In this paper, new numerical algorithms for solving fourth order initial value problems of ODEs via multistep collocation approach were developed. The use of approximated power se- ries as a basis function and its fourth derivatives as collocating equation were considered. The derived methods are efficient in the solution of fourth order ODEs as depicted in Tables 3, 6 and 9. The accuracy of these numerical models is found better compared with some of the existing methods in terms of error. Hence, FBM and SBM are viable numerical methods for solv- ing fourth order initial value problems. Acknowledgments We thank the referees and editor for the creative comments in making improvements to this paper. References [1] D. A. 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