Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 184253, 16 pages
http://dx.doi.org/10.1155/2012/184253
Research Article

Direct Two-Point Block One-Step Method for Solving General Second-Order Ordinary Differential Equations

Zanariah Abdul Majid,1,2 Nur Zahidah Mokhtar,1 and Mohamed Suleiman1,2

1Institute for Mathematical Research, Universiti Putra Malaysia, Selangor Darul Ehsan, 43400 Serdang, Malaysia
2Mathematics Department, Faculty of Science, Universiti Putra Malaysia, Selangor Darul Ehsan, 43400 Serdang, Malaysia

Received 25 May 2011; Revised 19 September 2011; Accepted 30 September 2011

Academic Editor: Gerhard-Wilhelm Weber

Copyright © 2012 Zanariah Abdul Majid 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

A direct two-point block one-step method for solving general second-order ordinary differential equations (ODEs) directly is presented in this paper. The one-step block method will solve the second-order ODEs without reducing to first-order equations. The direct solutions of the general second-order ODEs will be calculated at two points simultaneously using variable step size. The method is formulated using the linear multistep method, but the new method possesses the desirable feature of the one-step method. The implementation is based on the predictor and corrector formulas in the 𝑃 𝐸 ( 𝐶 𝐸 ) 𝑚 mode. The stability and precision of this method will also be analyzed and deliberated. Numerical results are given to show the efficiency of the proposed method and will be compared with the existing method.