Two-Step Optimized Technique with Two Hybrid Points for Solving Fourth-Order Initial Value Problems
DOI:
https://doi.org/10.64321/jcr.v2i6.03Keywords:
Two-step, Optimization, Free-parameter, Exponential function, Fourth orderAbstract
This article presents an optimized two-step, two-off-grid hybrid point for solving fourth order initial value problems. The method uses an exponential function as the basis function for a chosen two hybrid points, appropriately optimizing one of the two off-grid points by setting the principal term of the local truncation error to zero and using the local truncation error to determine the approximate values of the unknown parameter. Basic properties were examined, and the developed method was experimented to work out some fourth order initial value problems of ordinary differential equations. From the numerical results, it is clear that our new approach provides a better approximation than the existing method when compared to our result.
References
Blessing Iziegbe Akinnukawe, John Olusola Kuboye, and Solomon Adewale Okunuga, “Numerical Solution of Fourth Order Initial Value Problems using Novel Fourth-Order Block Algorithm,” Journal of Nepal Mathematical Society, vol. 6, no. 2, pp. 7-18, 2024.
J. O. Kuboye, O. R. Elusakin, and O. F. Quadri “Numerical Algorithm for Direct Solution of Fourth Order Ordinary Differential Equations,” Journal of Nigerian Society of Physical Sciences, vol. 2, no. 4, pp. 218-227, 2020.
S. J. Kayode, and O. Adeyeye, “A 3-Step Hybrid Method for Direct Solution of Second Order Initial Value Problems," Australian Journal of Basic and Applied Sciences, vol. 5, no. 12, pp. 2121-2126, 2011.
Lambert, J.D. (1973) Computational Methods in ODEs. John Wiley & Sons, New York.
Fatunla, S.O (1991) Block Method for Second Order IVPs. International Journal of Computer Mathematics, 41, 55-63. http://dx.doi.org/10.1080/00207169108804026
Brujnano, L. and Trigiante, D. (1998) Solving Differential Problems by Multistep Initial and Boundary Value Methods.
Vigo-Aguiar, J. and Ramos, H. (2006) Variable Step Size Implementation of Multistep Method for y f xyy ′′ ′ = ( , , ) Journal of Computational and Applied Mathematics, 192, 114-131. http://dx.doi.org/10.1016/j.cam.2005.04.043
Omar, Z. (1999) Developing Parallel Block Method for Solving Higher Orders ODES Directly. PhD Thesis, University Putra, Malaysia.
Mohammed, U. (2010) A Six Step Block Method for Solution of Fourth Order Ordinary Differential Equations. The Pacific Journal of Science and Technology, 11, 258-265
Ademiluyi, R.A, Duromola, M.K. and Bolaji, B. (2014) Modified Block Method for the Direct Solution of Initial Value Problems of Fourth Order Ordinary Differential Equations. Australian Journal of Basic and Applied Sciences, 8, 389-394.
Adesanya A Olaide, Fasansi M Kolawole, and Odekunle M Remilekun, “One Step, Three Hybrid Block Predictor-Corrector Method for the Solution of y''' = f (x, y, y ', y''),” Journal of Applied & Computational Mathematics, vol. 2, no. 4, 2013.
Adebayo Oluwadare Adeniran, and Adebola Evelyn Omotoye, “One Step Hybrid Method for the Numerical Solution of General Third Order Ordinary Differential Equations,” International Journal of Mathematical Sciences, vol. 2, no. 5, pp. 1-12, 2016.
Adoghe Lawrence Osa, and Omole Ezekiel Olaoluwa, “A Fifth-Fourth Continuous Block Implicit Hybrid Method for the Solution of Third Order Initial Value Problems in Ordinary Differential Equations,” Applied and Computational Mathematics, vol. 8, no. 3, 2019.
Bothayna S. H. Kashkari, and Sadeem Alqarni, “Optimization of Two-Step Block Method with Three Hybrid Points for Solving Third Order Initial Value Problems,” Journal of Nonlinear Sciences and Applications, vol. 12, no. 7, pp. 450-469, 2019.
S.H. Bothayna Kashkari, and I. Muhammed Syam, “Optimization of One-Step Bloch Method with Three Hybrid Points for Solving First-Order Ordinary Differential Equations,” Results in Physics, vol. 12, pp. 592-596, 2019
Gurjinder Singh et al., “An Efficient Optimized Adaptive Step Size Hybrid Block Methods for Integrating Differential Systems,” Applied Mathematics and Computation, vol. 362, 2019.
Joshua Sunday, “Optimized Two-Step Second Derivative Methods for the Solutions of Stiff Systems,” Journal of Physics Communications, vol. 6, no. 5, 2022.
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