World Applied Sciences Journal 33 (10): 1614-1622, 2015 ISSN 1818-4952 © IDOSI Publications, 2015 DOI: 10.5829/idosi.wasj.2015.33.10.300 Corresponding Author: Kasim Hussain, Department of Mathematics, Faculty of Science, UniversitiPutra Malaysia 43400 UPM Serdang, Selangor, Malaysia. 1614 A New Optimized Runge-Kutta-Nyström Method to Solve Oscillation Problems Kasim Hussain, Fudziah Ismail and Norazak Senu 1, 2 1, 3 1, 3 Department of Mathematics, Faculty of Science, 1 Universiti Putra Malaysia 43400 UPM Serdang, Selangor, Malaysia Department of Mathematics, Collage of Science, Al-Mustansiriyah University Baghdad, Iraq 2 Institute for Mathematical Research, UniversitiPutra Malaysia 43400 UPM Serdang, Selangor, Malaysia 3 Abstract: In this article, a new Runge-Kutta-Nyström method is derived. The new RKN method has zero phase-lag, zero amplification error and zero first derivative of phase-lag. This method is basically based on the sixth algebraic order Runge-Kutta-Nyström method, which has proposed by Dormand, El-Mikkawy and Prince. Numerical illustrations show that the new proposed method is much efficient as compared with other Runge-Kutta-Nyström methods in the scientific literature, for the numerical integration of oscillatory problems. Key words: Runge-Kutta-Nyström method Phase-lag Amplification error Derivative Oscillatory problems INTRODUCTION The main goal of this article is to derive an explicit In this article, we deal with the numerical method for phase-lag of order infinity, first derivative of phase-lag of solving special second order ordinary differential order infinity and amplification error of order infinity. equations (ODEs) of the form The new RKN method is based on the parameters of the (1) and Prince Runge-Kutta-Nyström method [24] with FSAL for which their solutions they are known are oscillating. method reduces the number of function evaluations. This type of problem occurs in several of applied fields such as quantum mechanics, electronics physical Phase-Lag Properties of Rkn Method: The general form chemistry, molecular dynamics, astronomy, chemical of Runge-Kutta-Nyström method with S-stage for solving physics and control engineering. In the last decade, many second order ordinary differential equations ODEs (1) can researchers developed methods with minimal phase-lag or be expressed as follows: phase-lag of order infinity [1-9]. Moreover several methods derived with nullification of phase-lag, amplification factor and phase-lag’s derivative for the numerical integration of ordinary differential equations (ODEs) with oscillatory solutions [10-18]. Brusa and Nigro [19] suggested the definition of phase-lag of a method. (2) Several authors have been studied the phase-lag of numerical methods for solving (1) (Houwen and Sommeijer where, [20, 21] and Thomas [22]). Papadopoulos and Simos [23] suggested a Runge-Kutta-Nyström method with invalidation of phase-lag, amplification factor together with the invalidation of their integrals. (3) Runge-Kutta-Nyström method with sixth algebraic order, well-known sixth algebraic order of Dormand, El-Mikkawy (first stage as last) property. This means that a RKN