Asymptotic initial-value method for a system of singularly perturbed second-order ordinary differential equations of convection–diffusion type

Abstract

In this paper, a numerical method is suggested to solve a class of boundary value problems (BVPs) for a weakly coupled system of singularly perturbed second-order ordinary differential equations of convection–diffusion type. First, in this method, an asymptotic expansion approximation of the solution of the BVP is constructed by using the basic ideas of a well known perturbation method namely Wentzal, Kramers and Brillouin (WKB). Then, some initial value problems (IVPs) are constructed such that their solutions are the terms of this asymptotic expansion. These problems happen to be singularly perturbed problems and, therefore, exponentially fitted finite difference schemes are used to solve these problems. As the BVP is converted into a set of IVPs and an asymptotic expansion approximation is used, the present method is termed as asymptotic initial-value method. The necessary error estimates are derived and examples provided to illustrate the method.

Keywords:

• Singular perturbation problem
• Weakly coupled system
• Boundary value problems
• Initial-value method
• Initial-value problem
• Terminal-value problem