Trigonometrically fitted fifth-order explicit two-derivative Runge-Kutta method with FSAL property

A new trigonometrically fitted two-derivative explicit Runge-Kutta (TFTDRK) method of order five with FSAL property for solving system of first-order ordinary differential equations (ODEs) with oscillatory solutions are derived. The new method is derived using the property of First Same As Last (FSAL). This method has the advantageous to merge totally first-order ordinary differential systems which their solutions are linear composition of the set of functions {e (u );e (−u)}, or equivalently {s (u );c (u)} when u > 0 is the dominant frequency of the problem. We analyzed the stabilityof our method. The numerical results are presented to illustrate the competence of TFTDRK method compared with some well-known TFRK methods.