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Advances in Numerical Analysis
Volume 2011 (2011), Article ID 513148, 14 pages
http://dx.doi.org/10.1155/2011/513148
Research Article

3-Point Block Methods for Direct Integration of General Second-Order Ordinary Differential Equations

1Department of Mathematics, Faculty of Science, University of Lagos, Akoka, Yaba, Lagos, Nigeria
2Department of Computer Sciences, Faculty of Science, University of Lagos, Akoka, Yaba, Lagos, Nigeria

Received 3 February 2011; Revised 18 April 2011; Accepted 26 May 2011

Academic Editor: István Faragó

Copyright © 2011 J. O. Ehigie 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 Multistep collocation techniques is used in this paper to develop a 3-point explicit and implicit block methods, which are suitable for generating solutions of the general second-order ordinary differential equations of the form 𝑦 = 𝑓 ( 𝑥 , 𝑦 , 𝑦 ) , 𝑦 ( 𝑥 0 ) = 𝑎 , 𝑦 ( 𝑥 0 ) = 𝑏 . The derivation of both explicit and implicit block schemes is given for the purpose of comparison of results. The Stability and Convergence of the individual methods of the block schemes are investigated, and the methods are found to be 0-stable with good region of absolute stability. The 3-point block schemes derived are tested on standard mechanical problems, and it is shown that the implicit block methods are superior to the explicit ones in terms of accuracy.