Exponentially Fitted and Trigonometrically Fitted Two-Derivative Runge-Kutta-Nyström Methods for Solving y′′(x)=fx,y,y′

Abstract

Two exponentially fitted and trigonometrically fitted explicit two-derivative Runge-Kutta-Nyström (TDRKN) methods are being constructed. Exponentially fitted and trigonometrically fitted TDRKN methods have the favorable feature that they integrate exactly second-order systems whose solutions are linear combinations of functions {exp⁡(wx),exp⁡(-wx)} and {sin⁡(wx),cos⁡(wx)} respectively, when w∈R, the frequency of the problem. The results of numerical experiments showed that the new approaches are more efficient than existing methods in the literature.