10.1007/s40096-021-00420-6

High-order exponentially fitted and trigonometrically fitted explicit two-derivative Runge–Kutta-type methods for solving third-order oscillatory problems

  • Khai Chien Lee*1,
  • Norazak Senu2,
  • Ali Ahmadian3,
  • Siti Nur Iqmal Ibrahim2,
  1. Institute for Mathematical Research, Universiti Putra Malaysia, UPM Serdang, Selangor, 43400, MY
  2. Institute for Mathematical Research and Department of Mathematics and Statistics, Universiti Putra Malaysia, UPM Serdang, Selangor, 43400, MY
  3. Institute of Industry Revolution 4.0, The National University of Malaysia, UKM Bangi, Selangor, 43600, MY

Published in Issue 2021-07-23

How to Cite

Lee, K. C., Senu, N., Ahmadian, A., & Ibrahim, S. N. I. (2021). High-order exponentially fitted and trigonometrically fitted explicit two-derivative Runge–Kutta-type methods for solving third-order oscillatory problems. Mathematical Sciences, 16(3 (September 2022). https://doi.org/10.1007/s40096-021-00420-6
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Abstract

Abstract Three stage sixth-order exponentially fitted and trigonometrically fitted explicit two-derivative Runge–Kutta-type methods are proposed for solving u′′′(t)=f(t,u(t),u′(t)).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{'''}(t) = \, f(t,u(t),u'(t)).$$\end{document} The idea of construction is based on linear composition of the set functions eωt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e^{\omega t}$$\end{document} and e-ωt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e^{-\omega t}$$\end{document} for exponentially fitted and eiωt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e^{i\omega t}$$\end{document} and e-iωt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e^{-i\omega t}$$\end{document} for trigonometrically fitted with ω∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in \mathbb {R}$$\end{document} to integrate initial value problems. The selected coefficients of two-derivative Runge–Kutta-type method are modified to depend on the principle frequency of the numerical problems to construct exponentially fitted and trigonometrically fitted Runge–Kutta-type direct methods, denoted as EFTDRKT6 and TFTDRKT6 methods. The numerical experiments illustrate competence of the new exponentially fitted and trigonometrically fitted method compared to existing methods for solving special type third-order ordinary differential equations with initial value problems.

Keywords

  • Runge–Kutta-type methods,
  • Third-order oscillatory differential equations,
  • Initial value problems,
  • Exponentially fitted,
  • Trigonometrically fitted

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