Higher-order multi-step Runge-Kutta-Nystr\"om methods with frequency-dependent coefficients for second-order initial value problem $u^{\prime \prime}=f(x,u,u^{\prime})$

Document Type : Research Paper

Authors

1 1- Department of Mathematics and Computer Applications, College of Sciences, Al-Nahrain University, Jadriya, Baghdad, Iraq.\\ 2- Institute for Mathematical Research, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia.

2 1- Institute for Mathematical Research, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia.\\ 2- Department of Mathematics and Statistics, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia.

Abstract

In this study, for the numerical solution of general second-order ordinary differential equations (ODEs) that exhibit oscillatory or periodic behavior, fifth- and sixth-order explicit multi-step Runge-Kutta-Nystr¨om (MSGRKN) methods, respectively, are constructed. The parameters of the proposed methods rely on the frequency ω of each problem whose solution is a linear combination of functions {e(iωx), e(−iωx)} or {cos(ωx), sin(ωx)}. The study also includes an analysis of the linear stability of the suggested methods. The numerical results indicate the efficiency of the proposed methods in solving such problems compared to methods with similar characteristics in the literature.

Keywords

Main Subjects

References

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Computational Methods for Differential Equations
Volume 12, Issue 4
October 2024
Pages 808-826

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History

  • Receive Date: 24 July 2023
  • Revise Date: 01 February 2024
  • Accept Date: 27 March 2024

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