Athraa Jasim - ۱- Department of Mathematics and Computer Applications, College of Sciences, Al-Nahrain University, Jadriya, Baghdad, Iraq.\\ ۲- Institute for Mathematical Research, Universiti Putra Malaysia, ۴۳۴۰۰ Serdang, Selangor, Malaysia.
Norazak Senu - ۱- Institute for Mathematical Research, Universiti Putra Malaysia, ۴۳۴۰۰ Serdang, Selangor, Malaysia.\\ ۲- Department of Mathematics and Statistics, Universiti Putra Malaysia, ۴۳۴۰۰ Serdang, Selangor, Malaysia.
Zanariah Abdul Majid - ۱- Institute for Mathematical Research, Universiti Putra Malaysia, ۴۳۴۰۰ Serdang, Selangor, Malaysia.\\ ۲- Department of Mathematics and Statistics, Universiti Putra Malaysia, ۴۳۴۰۰ Serdang, Selangor, Malaysia.
Nik Mohd Asri Nik Long - ۱- Institute for Mathematical Research, Universiti Putra Malaysia, ۴۳۴۰۰ Serdang, Selangor, Malaysia.\\ ۲- Department of Mathematics and Statistics, Universiti Putra Malaysia, ۴۳۴۰۰ Serdang, Selangor, Malaysia.

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.

Explicit methods, trigonometrical fitting, Multi-step Runge-Kutta-Nystr\"om methods, initial value problems, general second-order oscillatory differential equations