An asymptotic numerical method for fourth order singular perturbation problems with a discontinuous source term

Authors: Shanthi, V.; Ramanujam, N.

Source: International Journal of Computer Mathematics, Volume 85, Number 7, July 2008 , pp. 1147-1159(13)

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Abstract:

Singularly perturbed two-point boundary-value problems (BVPs) for fourth-order ordinary differential equations (ODEs) with a small positive parameter multiplying the highest derivative with a discontinuous source term is considered. The given fourth-order BVP is transformed into a system of weakly coupled systems of two second-order ODEs, one without the parameter and the other with the parameter ε multiplying the highest derivative, and suitable boundary conditions. In this paper a computational method for solving this system is presented. In this method we first find the zero-order asymptotic approximation expansion of the solution of the weakly coupled system. Then the system is decoupled by replacing the first component of the solution by its zero-order asymptotic approximation expansion of the solution in the second equation. Then the second equation is solved by the numerical method, which is constructed for this problem and which involves an appropriate piecewise-uniform mesh.

Keywords: fourth-order ordinary differential equation; singularly perturbed problem; discontinuous source term; asymptotic expansion; finite difference scheme

Document Type: Research article

DOI: http://dx.doi.org/10.1080/00207160701478862

Affiliations: 1: Department of Mathematics, Bharathidasan University, Tiruchirappalli, Tamilnadu, India

Publication date: 2008-07-01

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