An asymptotic initial value method for boundary value problems for a system of singularly perturbed second order ordinary differential equations

摘要

In this paper a numerical method is suggested to solve a class of boundary value problems for a weakly coupled system of singularly perturbed second order ordinary differential equations of reaction-diffusion type. First, in this method, an asymptotic expansion approximation of the solution of the boundary value problem is constructed using the basic ideas of the well known perturbation method. Then initial value problems and terminal value problems (TVPs) are formulated such that their solutions are the terms of this asymptotic expansion. These problems are happened to be singularly perturbed problems and therefore exponentially fitted finite difference schemes are used to solve these problems. Since the boundary value problem is converted into a set of initial and TVPs and an asymptotic expansion approximation is used, the present method is termed as an asymptotic initial value method. Necessary error estimates are derived and examples provided to illustrate the method. The present method is easy to implement and well suited for parallel computing.