An Order (K+5) Block Hybrid Backward Differentiation Formula for Solution of Fourth Order Ordinary Differential Equations

Year 2024, Volume: 21 Issue: 2, 113 - 132, 01.11.2024

Raihanatu Muhammad , Hajara Hussaini , Abdulmalik Oyedeji

Abstract

This paper covers the derivation and implementation of the 4-step linear Multistep method of Block Hybrid Backward Differentiation Formula (BHBDF) for solving fourth-order initial value problems in ordinary differential equations. In the derivation of the proposed numerical method, the utilization of collocation and interpolation points was adopted with Legendre polynomials serving as the fundamental basis function. The 4-step BHBDF developed to solve fourth-order IVPs has a higher order of accuracy
(p=9). Furthermore, the proposed numerical block methods are employed directly to solve fourth-order ODEs. In comparison to some existing methods examined in the prior studies, the proposed method has a robust implementation strategy and demonstrate a higher level of accuracy.

Keywords

Backward Differentiation Formula, Block Method, Interpolation methods, Linear Multistep Methods, Collocation Method

Ethical Statement

I Abdulmalik Oyedeji the corresponding author of the manuscript titled “An order (K+5) Block Hybrid Backward Differentiation Formula for Solution of Fourth Order Ordinary Differential Equations,” hereby declare that the work reported in this article is original and has not been previously published, nor is it under consideration for publication elsewhere. All authors have made significant contributions to the research, have read and approved the final manuscript, and agree to its submission to Cankaya University Journal of Science and Engineering. We confirm that the research was conducted following ethical standards. All informed consent was obtained from all participants involved in the study. There are no conflicts of interest to declare, and no financial or personal relationships that could inappropriately influence or bias the research. We acknowledge the importance of integrity and transparency in the publication process and are committed to maintaining the highest ethical standards in research and publication.

References

• W. Clarence de Silva, Modeling of Dynamic Systems with Engineering Applications, 2nd ed. CRC Press, 2022.
• W.E. Boyce, R.C. DiPrima, and D.B. Meade, Elementary Differential Equations and Boundary Value Problems, John Wiley & Sons, 2021.
• E.W. Schiesser and G.W. Griffiths, A Compendium of Partial Differential Equations: Model Method of Lines Analysis with MATLAB, Cambridge University Press, 2009.
• G.D. Bryne and J.D. Lambert, “Pseudorunge-Kutta methods involving two points,” J. Assoc. Comput. Mathematics, vol. 40, pp. 114–123, Jan. 1996, doi: 10.1145/321312.321321.
• A. J. Salgado and S. M. Wise, "Classical Numerical Analysis: A Comprehensive Course," Cambridge University Press, 2022, ch. 20, p. 555-556.
• G. Söderlind, I. Fekete, and I. Faragó, "On the zero-stability of multistep methods on smooth nonuniform grids," SIAM Journal on Numerical Analysis, vol. 58, pp. 1125–1143, Dec. 2018, doi: 10.1007/s10543-018-0716-y.
• R. D’Ambrosio, "Linear Multistep Methods," Numerical Approximation of Ordinary Differential Problems, vol. 148, Springer, Cham, 2023.
• A.C. Cardone and R. D'Ambrosio, “Collocation methods for Volterra integral and integro-differential equations: A review,” Department of Mathematics, University of Salerno, 84084 Fisciano, 2018.
• A. B. Familua and E. O. Omole, “Five Points Mono Hybrid Point Linear Multistep Method for Solving Nth Order Ordinary Differential Equations Using Power Series Function,” Asian Res. J. Math., vol. 3, no. 1, pp. 1–17, Jan. 2017, doi: 10.9734/ARJOM/2017/31190.
• E.O. Adeyefa and J.O. Kuboye, “Derivation of new numerical model capable of solving second and third order ordinary differential equations directly,” IEANG Int. J. of Applied Mat., vol. 50, no. 2, pp. 1-9, Jun. 2020.
• J.O. Kuboye, O.R. Elusakin, and O.F. Quadri, “Numerical algorithm for direct solution of fourth order ordinary differential equations,” J. Nigerian Society of Physical Sciences, vol. 2, pp. 218–227, Nov. 2020, doi: 10.46481/jnsps.2020.100.
• K. J. Audu, J. Garba, A. T. Tiamiyu, and B. A. Thomas, "Application of Backward Differentiation Formula on Fourth-Order Differential Equations," J. Sci.Techno., vol. 14, no. 2, pp. 52–65, Dec. 2022, doi: 10.30880/jst.2022.14.02.006.
• O.E. Abolarin and B.G. Ogunware, “New hybrid method for direct numerical solution of nonlinear second, third and fourth orders ordinary differential equations,” Int. J. Mathematics in Operational Research, vol. 23, no. 3, pp. 285-315, Nov. 2022, doi.org/10.1504/IJMOR.2022.127378.
• L. Ukpebor and O. Ezekiel, “Three-step optimized backward differentiation formulae (TOBBDF) for solving stiff ordinary differential equations,” African Journal of Mathematics and Computer Science Research, Vol. 13, No.1, pp. 51-57, Apr. 2020. doi: 10.5897/AJMCSR2019.0811.