OICC Press Mathematical Sciences 2251-7456 16 3 (September 2022) 2021 07 23 High-order exponentially fitted and trigonometrically fitted explicit two-derivative Runge–Kutta-type methods for solving third-order oscillatory problems 10.1007/s40096-021-00420-6 EN Khai Chien Lee Institute for Mathematical Research, Universiti Putra Malaysia, UPM Serdang, Selangor, 43400, MY Norazak Senu Institute for Mathematical Research and Department of Mathematics and Statistics, Universiti Putra Malaysia, UPM Serdang, Selangor, 43400, MY Ali Ahmadian Institute of Industry Revolution 4.0, The National University of Malaysia, UKM Bangi, Selangor, 43600, MY Siti Nur Iqmal Ibrahim Institute for Mathematical Research and Department of Mathematics and Statistics, Universiti Putra Malaysia, UPM Serdang, Selangor, 43400, MY Journal Article 2021 07 23 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.