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A Fictitious Time Integration Method for Solving m-Point Boundary Value Problems

Chein-Shan Liu1

Department of Civil Engineering, National Taiwan University, Taipei, Taiwan. E-mail:

Computer Modeling in Engineering & Sciences 2009, 39(2), 125-154.


We propose a new numerical method for solving the boundary value problems of ordinary differential equations (ODEs) under multipoint boundary conditions specified at t = Ti, i = 1,...,m, where T1 < ... < Tm. The finite difference scheme is used to approximate the ODEs, which together with the m-point boundary conditions constitute a system of nonlinear algebraic equations (NAEs). Then a Fictitious Time Integration Method (FTIM) is used to solve these NAEs. Numerical examples confirm that the new approach is highly accurate and efficient with a fast convergence. The FTIM can also be used to find the periods of nonlinear ODEs system and its corresponding periodic solutions, as the van der Pol and Duffing equations are investigated here. The numerical examples also include a vibration problem of the Euler-Bernoulli beam under three-point boundary conditions. The present method has a number advantages of easy implementation, easily to treat nonlinear multipoint boundary value problems, and easily to extend to a higher-dimensional first-order ODEs.


Cite This Article

Liu, C. (2009). A Fictitious Time Integration Method for Solving m-Point Boundary Value Problems. CMES-Computer Modeling in Engineering & Sciences, 39(2), 125–154.

This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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