This article considered the development of a two-point hybrid method for the numerical solution of initial value problems of second order ordinary differential Equations (ODEs) using power series and exponentially-fitted basis function. Interpolation and collocation techniques were used to derive the method. The method was implemented in predictor-corrector mode. In order to increase the accuracy of the results of the method, the predictor was designed to have same order of accuracy as the corrector. The method is symmetric, consistent, zero-stable and has small error constant and has better accuracy over other methods in the reviewed literature when tested with some numerical examples.
Cite this paper
Kayode, S. J. , Obarhua, F. O. and Osuntope, O. C. (2023). Series and Exponentially-Fitted Two-Point Hybrid Method for General Second Order Ordinary Differential Equations. Open Access Library Journal, 10, e258. doi: http://dx.doi.org/10.4236/oalib.1110258.
Awoyemi, D.O. (2003) A P-Stable Linear Multistep Method for Solving General Third Order Ordinary Differential Equations. International Journal of Computer Mathematics, 80, 987-993. https://doi.org/10.1080/0020716031000079572
Kayode, S.J. (2009) A Zero Stable Method for Direct Solution of Fourth Order Ordinary Differential Equations. American Journal of Applied Sciences, 5, 1461-1466.
Kayode, S.J. and Adeyeye, O. (2013) Two-Step Two-Point Hybrid Methods for General Second Order Differential Equations. African Journal of Mathematics and Computer Science Research, 6, 191-196.
Adesanya, A.O., Odekunle, M.R. and Udoh, M.O. (2013) Four-Steps Continuous Method for the Solution of y''=f(x, y, y'). American Journal of Computational Mathematics, 3, 169-174. https://doi.org/10.4236/ajcm.2013.32025
Kayode, S.J. and Obarhua, F.O. (2015) 3-Step y-Function Hybrid Methods for Direct Numerical Integration of Second Order IVPs in Ordinary Differential Equations. Theoretical Mathematics & Application, 5, 39-51.
Areo, E.A. and Rufai. M.A., (2016a): A New Uniform Fourth Order One-Third Step Continuous Block Method for the Direct Solutions of y''=f(x, y, y'). British Journal of Mathematics & Computer Science, 15, 1-12.
https://doi.org/10.9734/BJMCS/2016/24310
Adesanya, A.O., Udoh, D.M. and Ajileye, A.M. (2013) A New Hybrid Block Method for the Solution of General Third Order Initial Value Problems of Ordinary Differential Equations. International Journal of Pure and Applied Mathematics, 86, 365-375. https://doi.org/10.12732/ijpam.v86i2.11
Olabode, B.T. (2009) A Six-Step Scheme for the Solution of Fourth Order Ordinary Differential Equations. Pacific Journal of Science and Technology, 10, 143-148.
Omar, Z., Abdullahi, Y.A. and Kuboye, J.O. (2016) Predictor-Corrector Block Method of Order Seven for Solving Third Order Ordinary Differential Equations. International Journal of Mathematical Analysis, 10, 223-235.
https://doi.org/10.12988/ijma.2016.510266
Kayode, S.J. and Adeyeye, O. (2011) A 3-Step Hybrid Method for Direct Solution of Second Order Initial Value Problems. Journal of Basic and Applied Science, 5, 2121-2126.
Alabi, M.O., Oladipo, A.T. and Adesanya, A.O. (2008) Initial Value Solvers for Second Order Ordinary Differential Equations Using Chebyshev Polynomial as Bases Functions. Journal of Modern Mathematics and Statistics, 2, 18-27.
https://doi.org/10.37745/irjns.13/vol10n2pp1838
Obarhua, F.O. and Kayode, S.J. (2020) Continuous Explicit Hybrid Method for Solving Second Order Ordinary Differential Equations. Pure and Applied Mathematics Journal, 9, 26-31. https://doi.org/10.11648/j.pamj.20200901.14
Omar, Z. and Alkasassbeh, M.F. (2016) Generalized One-Step Third Derivative Implicit Hybrid Block Method for Direct Solution of Second Order Ordinary Differential Equations. Applied Mathematical Sciences, 10, 417-430.
https://doi.org/10.12988/ams.2016.510667
Adeyeye, O. and Omar, Z. (2018) New Generalized Algorithm for Developing k-Step Higher Derivative Block Methods for Solving Higher Order Ordinary Differential Equations. Journal of Mathematical and Fundamental Sciences, 50, 40-58.
https://doi.org/10.5614/j.math.fund.sci.2018.50.1.4
