TY  - EJOU
AU  - Liu, Chein-Shan 

TI  - A Fictitious Time Integration Method for Solving m-Point Boundary Value Problems
T2  - Computer Modeling in Engineering \& Sciences

PY  - 2009
VL  - 39
IS  - 2
SN  - 1526-1506

AB  - We propose a new numerical method for solving the boundary value problems of ordinary differential equations (ODEs) under multipoint boundary conditions specified at <i>t = T<sub>i</sub>, i = 1,...,m, where T<sub>1</sub> < ... < T<sub>m</sub></i>. 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.
KW  - Multi-point boundary value problems
KW  -  Periodic solution
KW  -  Euler-Bernoulli beam
KW  -  Fictitious Time Integration Method (FTIM)

DO  - 10.3970/cmes.2009.039.125