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In this work, we derive two new Runge-Kutta-Nystrom (RKN) methods for solving periodic second order ordinary differential equations. We constructed these methods based on the RKN method of three-stage fourth-order derived by Garcia and RKN method of four-stage fifth-order derived by Hairer. The numerical results show that the efficiency of the new methods is more than the other existing methods.

NEW EXPLICIT TRIGONOMETRICALLY-FITTED FOURTH-ORDER AND FIFTH-ORDER RUNGE--KUTTA--NYSTROM METHODS FOR PERIODIC INITIAL VALUE PROBLEMS