Trigonometrically-fitted explicit Runge-Kutta-Nystrom methods for solving special second order ordinary differential equations with periodic solutions

In this study, a trigonometrically-fitted explicit Runge-Kutta-Nystrom (RKN) methods are proposed for the integration of initial-value problems (IVPs) of special secondorder ordinary differential equations (ODEs) with periodic behavior. The derivation of fourth and fifth-order trigonometrically-fitted explicit RKN methods using constant step length and an embedded trigonometrically-fitted explicit 4(3) and 5(4) pairs of RKN methods for variable step length have been developed. The numerical results obtained show that the new trigonometrically-fitted explicit RKN methods developed for constant and variable step length are more accurate and efficient than several existing methods in the literature. Meanwhile, a symplectic trigonometrically-fitted explicit RKN methods for solving Hamiltonian system with periodic solutions were derived. However, it is well known that the local error of a non-symplectic method is smaller than that of the symplectic method, the error produce during the integration process is slower for the symplectic method. Thus, for a large interval of integration of Hamiltonian systems the symplectic method will be more efficient than the non-symplectic method. The numerical results obtained show that the symplectic methods incorporated with trigonometric fitting technique are more efficient than the non-symplectic methods when solving IVPs with periodic character. In conclusion, a trigonometrically-fitted explicit RKN methods were derived for solving special second-order ODEs with periodic solutions. The local truncation error (LTE) of each method derived was computed, the absolute stability interval of the methods derived were discussed. Numerical experiment performed show the accuracy and efficiency in terms of function evaluation per step of the new methods in comparison with other existing methods.

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