World Applied Sciences Journal 20 (12): 1685-1695, 2012 ISSN 1818-4952 © IDOSI Publications, 2012 DOI: 10.5829/idosi.wasj.2012.20.12.2881 Corresponding Author: Faranak Rabiei, Department of Mathematics and Institute for Mathematical Research, Universiti Putra, Malaysia. 1685 Construction of Improved Runge-Kutta Nystrom Method for Solving Second-Order Ordinary Differential Equations Faranak Rabiei, Fudziah Ismail, S. Norazak and N. Abasi Department of Mathematics and Institute for Mathematical Research, Universiti Putra, Malaysia, UPM Serdang, 43400 Selangor, Malaysia Abstract: Improved Runge-Kutta Nystrom (IRKN) method for the numerical solution of second-order ordinary differential equations is constructed. The scheme arises from the classical Runge-Kutta Nystrom method also can be considered as two step method. IRKN methods require less number of stages which lead to less number of function evaluations per step, compared with the existing Runge-Kutta Nystrom (RKN) methods. Therefore, the methods are computationally more efficient at achieving the higher order of local accuracy. The algebraic order conditions of the method using the Taylor’s series expansion are obtained and the methods of order 3, 4 and 5 are derived. The stability properties of method are discussed and numerical examples are given to show the efficiency of the proposed methods compared to the existing RKN methods. Key words: Improved Runge-Kutta Nystrom method • Runge-Kutta Nystrom method • second-order ordinary differential equations • algebraic order conditions INTRODUCTION The special second-order ordinary differential equations are given by: 0 0 0 0 y f(x,y), y(x ) y , y (x ) y ′ ′ ′ = = = (1) The second-order ordinary differential equations (1), can be reduced i nto system of first order ordinary differential equations (ODEs) or can be solved by using Runge-Kutta Nystrom (RKN) methods or multistep methods, directly. Udwadia and Farahani [1] proposed the Accelerated Runge-Kutta (ARK) methods for solving autonomous first order ODEs. Rabiei et al . [2] developed the Accelerated Runge-Kutta Nystrom (ARKN) methods for solving special second-order autonomous ordinary differential equations in form of y′′ = ƒ(y). Rabiei and Ismail [3, 4] by improving the ARK methods for solving general form of ordinary differential equations, constructed the Improved Runge-Kutta method for solving first order ODEs. In this paper, we developed the Improved Runge-Kutta Nystrom (IRKN) method for solving second-order ODEs in form of y ″ = ƒ(x,y). The third-order Improved Runge-Kutta Nystrom method used only 2 stages while there is not any existing Runge-Kutta Nystrom method with two stages. Also the fourth and fifth order IRKN methods used 3 and 4 stages, respectively. In section 2, the general form of IRKN methods is constructed and the order conditions of method using Taylor's series expansion are obtained in section 3. In section 4, the derivation of the method is given followed by the stability of the method in section 5. The number of tested problems to show the efficiency of the methods compared with the existing RKN methods, are given in the last section. CONSTRUCTION OF METHOD Consider the IRK method with s-stages from [3, 4] as follows