Predictor-corrector scheme for solving second order ordinary differential equations
Abstract
In this research, the direct method of Adam Moulton two-step method was proposed for solving initial value problem (IVPs) of second order ordinary differential equations (ODEs) directly. The current approach for solving second order ODEs is to reduce to first order ODEs. However, the direct method in this research will solved the second order ODEs directly. The Lagrange interpolation polynomial was applied in the derivation of the proposed method. The implementation will be in predictor-corrector scheme. Numerical results shown that the method gave comparable accuracy and faster execution time compared to the existing method. The proposed direct method of Adams Moulton type is suitable for solving second order ODEs.
References
Anake, T. A. (2011). Continuous implicit hybrid one-step methods for the solution of initial value problems of general second-order ordinary differential equations. PhD thesis, Covenant University, Ota.
Awoyemi, D., Adebile, E., Adesanya, A., and Anake, T. A. (2011). Modified block method for the direct solution of second order ordinary differential equations. International Journal of Applied Mathematics and Computation, 3(3), 181–188.
Badmus, A. M. and Yahaya, Y.A. (2009). An accurate uniform order 6 block method for direct solution of general second order ordinary differential equations. The Pacific Journal of Science and Technology, 10:273-277.
Fairuz, A.N. and Majid, Z.A. (2021) Rational methods for solving first-order initial value problems, International Journal of Computer Mathematics, 98:2, 252-270, DOI: 10.1080/00207160.2020.1737862
Fatunla, S. O. (1991). Block methods for second order odes. International Journal of Computer Mathematics, 41(No. 1-2):55–63.
Khiyal, M. and Thomas, R. (1997). Variable-order, variable-step methods for second-order initial-value problems. Journal of computational and applied mathematics, 79(2):263–276.
Lambert, J. D. (1973). Computational methods in ordinary differential equations. New York: John Wiley & Sons
Majid, Z. A., Azmi, N. A., and Suleiman, M. (2009). Solving second order ordinary differential equations using two-point four step direct implicit block method. European Journal of Scientific Research, 31(1):29–36.
Majid, Z.A., Mokhtar, N. Z., and Suleiman, M. (2012). Direct two-point block one-step method for solving general second-order ordinary differential equations. Mathematical Problems in Engineering. Volume 2012 |Article ID 184253. https://doi.org/10.1155/2012/184253
Majid, Z.A., See, P.P. & Suleiman, M. (2011). Solving directly two-point non-linear boundary value problem. Journal of Mathematics and Statistics, 7(2), 124-128.
Omar, Z. and Alkasassbeh, M. (2016). Generalized one-step third derivative implicit hybrid block method for the direct solution of second order ordinary differential equation. Applied Mathematical Sciences, 10(9):417–430.
Rasedee, A.F.N., M.B. Suleiman,M.B. and Ibrahim, Z.B. (2014). Solving nonstiff higher order ODEs using variable order step size backward difference directly, Math. Probl. Eng., Article ID 565137
Van Der Houmen, P. and Someijer, B. (1987). Predictor-corrector methods for periodic second-order initial-value problems. IMA Journal of Numerical Analysis, 7(4):407–422.
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