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

• Published in: International journal of computer mathematics Vol. 84; no. 3; pp. 333 - 346
• Main Authors: Valanarasu, T., Ramanujam, N.
• Format: Journal Article
• Language: English
• Published: Taylor & Francis 01.03.2007
• Subjects: Asymptotic expansion approximation; Boundary value problem; Discontinuous source term; Singularly perturbed problem; Third-order differential equation
• ISSN: 0020-7160; 1029-0265
• DOI: 10.1080/00207160601177200

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.