An asymptotic numerical method for singularly perturbed third-order ordinary differential equations with a weak interior layer

Authors:
T. Valanarasu;N. Ramanujam
Affiliations:
Department of Mathematics, Bharathidasan University, Tiruchirappalli, Tamilnadu, India;Department of Mathematics, Bharathidasan University, Tiruchirappalli, Tamilnadu, India
Venue:
International Journal of Computer Mathematics
Year:
2007

Citing 6
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Abstract

A class of singularly perturbed two point boundary value problems (BVPs) of convection-diffusion type for third-order ordinary differential equations (ODEs) with a small positive parameter (ε) multi-plying the highest derivative and a discontinuous source term is considered. The BVP is reduced to a weakly coupled system consisting of one first-order ordinary differential equation with a suitable initial condition and one second-order singularly perturbed ODE subject to boundary conditions. In order to solve this system, a computational method is suggested. In the proposed method we first find a zero-order asymptotic expansion approximation 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 expansion approximation in the second equation. Then the second equation is solved by a finite difference method on a Shishkin mesh (a fitted mesh method). Examples are provided to illustrate the method.