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  • Research Article
  • Open Access

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

Boundary Value Problems20082008:963753

  • Received: 12 May 2008
  • Accepted: 9 July 2008
  • Published:


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.


  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Equation
  • Full Article

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Authors’ Affiliations

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


© Chein-Shan Liu. 2008

This article is published under license to BioMed Central Ltd. 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.