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

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.

Keywords:

• Singularly perturbed problem
• Discontinuous source term
• Third-order differential equation
• Asymptotic expansion approximation
• Boundary value problem

Additional information

Notes on contributors

T. Valanarasu

Email: [email protected]