Main Article Content
This paper focuses on developing a self-starting numerical approach that can be used for direct integration of higher-order initial value problems of Ordinary Differential Equations. The method is derived from power series approximation with the resulting equations discretized at the selected grid and off-grid points. The method is applied in a block-by-block approach as a numerical integrator of higher-order initial value problems. The basic properties of the block method are investigated to authenticate its performance and then implemented with some tested experiments to validate the accuracy and convergence of the method.
Published Online First 20/1/2022
This work is licensed under a Creative Commons Attribution 4.0 International License.
Arqub OA, Maayah B. Modulation of reproducing kernel Hilbert space method for numerical solutions of Riccati and Bernoulli equations in the Atangana-Baleanu fractional sense. Chaos Solitons Fractals. 2019 Aug 1;125:163-70.https://doi.org/10.1016/j.chaos.2019.05.025
Arqub O A, Maayah B. Fitted fractional reproducing kernel algorithm for the numerical solutions of ABC – Fractional Volterra integro-differential equations. Chaos Solitons Fractals. 2019 Sep 1;126:394-402. https://doi.org/10.1016/j.chaos.2019.07.023
Arqub O A, Maayah B. Numerical solutions of integrodifferential equations of Fredholm operator type in the sense of the Atangana–Baleanu fractional operator. Chaos Solitons Fractals. 2018 Dec 1;117:117-24.. https://doi.org/10.1016/j.chaos.2018.10.007
Arqub O A, Al-Smadi M. Atangana–Baleanu fractional approach to the solutions of Bagley–Torvik and Painlevé equations in Hilbert space. Chaos Solitons Fractals. 2018 Dec 1;117:161-7. https://doi.org/10.1016/j.chaos.2018.10.013
Omar Z, Kuboye J O. New Seven-Step Numerical Method for Direct Solution of Fourth Order Ordinary Differential Equations. J. Math. Fundam. Sci. 2016; 48(2): 94-105.
Waeleh N, Majid Z A. A 4-point Block Method for Solving Higher Order Ordinary Differential Equations Directly. Int. j. math. sci. 2016; Article ID: 9823147, 8 pages.
Hussain K, Ismail F, Senu N. Two Embedded Pairs of Runge-Kutta Type Methods for Direct Solution of Special Fourth-Order Ordinary Differential Equations. Math Probl Eng. 2015; Volume 2015, Article ID 196595, 12 pages. Available from: http://dx.doi.org/10.1155/2015/196595
Akinfenwa O A, Ogunseye H A, Okunuga S A. Block Hybrid Method for Solution of Fourth Order Ordinary Differential Equations. Nig. J. Math. & Appl. 2016; 25: 140 − 150.
Duromola M K. An Accurate Five Off-Step Points Implicit Block Method for Direct Solution of Fourth Order Differential Equations. OALibJ. 2016; 3(6). http://dx.doi.org/10.4236/oalib.110266
Kuboye J O, Omar Z, Abolarin O E, Abdelrahim R. Generalized Hybrid Block Method for Solving Second Order Ordinary Differential Equations. Res. Rep. Math. 2018; 2(2): 1 - 7.
Kuboye J O, Omar Z. New Zero-stable Block Method for Direct Solution of Fourth Order Ordinary Differential Equations. Indian J Sci Technol. 2015; 8(12): 1-8.
Omar Z, Abdelrahim R. Direct Solution of Fourth Order Ordinary Differential Equations using A One Step Hybrid Block Method of Order Five. Int. J. Pure Appl. Math. 2016; 109(4): 763 – 777.
Majid ZA, Azmi N A, Suleiman M, Ibrahaim ZB. Solving Directly General Third Order Ordinary Differential Equations using Two-Point Four Step Block Method. Sains Malays. 2012;41(5): 623 - 632.
Modebei M I, Adeniyi R B, Jator S N, Ramos HA. Block hybrid integrator for numerically solving fourth-order Initial Value Problems. Appl. Math. Comput. 2019; 346: 680 - 694.
Waeleh N, Majid Z A. Ismail F. and Suleiman, M. Numerical Solution of Higher Order Ordinary Differential Equations by Direct Block Code. J. Math. Stat. 2012; 8 (1): 77 – 81.
Anake T A, Adesanya AO, Oghonyon G J, Agarana MC. Block Algorithm for General Third Order Ordinary Differential Equation. ICASTOR, J. Math. Sci. 2013; 7(2), 127 – 136.
Adeyeye O, Omar Z. Implicit Five-Step Block Method with Generalized Equidistant Points for Solving Fourth Order Linear and nonlinear Initial Value Problems. Ain Shams Eng. J. 2017; https://doi.org/10.1016/j.asej.2017.11.011
Mehrkanoon S, Suleiman M, Majid Z A. Block Method for Numerical Solution of Fuzzy Differential Equations. Int. Math. Forum. 2009;4(46): 2269 – 2280.
Adeyefa E O, Kuboye J O. Derivation of New Numerical Model Capable of Solving Second and Third Order Ordinary Differential Equations Directly. IAENG Int. J. Appl. Math. 2020; 50:2, 9 pages.
Lambert JD. Computational Methods in Ordinary Differential Equations. New York: John Wiley, 1973.
LeVeque R J. Finite Difference Methods for Differential Equations. University of Washington, 2006.