Boundary Value Problems
Volume 2008 (2008), Article ID 963753, 22 pages
Research Article

A Two-Stage LGSM for Three-Point BVPs of Second-Order ODEs

Chein-Shan Liu1,2

1Department of Mechanical and Mechatronic Engineering, Taiwan Ocean University, Keelung 20224, Taiwan
2Department of Harbor and River Engineering, Taiwan Ocean University, Keelung 20224, Taiwan

Received 12 May 2008; Accepted 9 July 2008

Academic Editor: Wenming Zou

Copyright © 2008 Chein-Shan Liu. 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.


The study in this paper is a numerical integration of second-order three-point boundary value problems under two imposed nonlocal boundary conditions at t=t0, t=ξ, and t=t1 in a general setting, where t0<ξ<t1. We construct a two-stage Lie-group shooting method for finding unknown initial conditions, which are obtained through an iterative solution of derived algebraic equations in terms of a weighting factor r(0,1). The best r is selected by matching the target with a minimal discrepancy. Numerical examples are examined to confirm that the new approach has high efficiency and accuracy with a fast speed of convergence. Even for multiple solutions, the present method is also effective to find them.