Boundary Value Problems
Volume 2008 (2008), Article ID 963753, 22 pages
A Two-Stage LGSM for Three-Point BVPs of Second-Order ODEs
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 , , and in a general setting, where . 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 . The best 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.