Open Access

ARTICLE

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

Chein-Shan Liu¹

1 Department of Civil Engineering, National Taiwan University, Taipei, Taiwan. E-mail: liuchei.nshan@msa.hinet.net

Computer Modeling in Engineering & Sciences 2009, 39(2), 125-154. https://doi.org/10.3970/cmes.2009.039.125

Abstract

We propose a new numerical method for solving the boundary value problems of ordinary differential equations (ODEs) under multipoint boundary conditions specified at t = T_i, i = 1,...,m, where T₁ < ... < T_m. 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.

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