Abstract and Applied Analysis
Volume 2013 (2013), Article ID 136961, 10 pages
http://dx.doi.org/10.1155/2013/136961
Research Article

Semi Implicit Hybrid Methods with Higher Order Dispersion for Solving Oscillatory Problems

S. Z. Ahmad,1 F. Ismail,2 N. Senu,2 and M. Suleiman2

1Department of Mathematics, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia
2Department of Mathematics and Institute for Mathematical Research, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia

Received 11 January 2013; Accepted 4 March 2013

Academic Editor: Juan Carlos Cortés López

Copyright © 2013 S. Z. Ahmad et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We constructed three two-step semi-implicit hybrid methods (SIHMs) for solving oscillatory second order ordinary differential equations (ODEs). The first two methods are three-stage fourth-order and three-stage fifth-order with dispersion order six and zero dissipation. The third is a four-stage fifth-order method with dispersion order eight and dissipation order five. Numerical results show that SIHMs are more accurate as compared to the existing hybrid methods, Runge-Kutta Nyström (RKN) and Runge-Kutta (RK) methods of the same order and Diagonally Implicit Runge-Kutta Nyström (DIRKN) method of the same stage. The intervals of absolute stability or periodicity of SIHM for ODE are also presented.