On singular perturbation boundary-value problem of coupling type system of convection-diffusion equations

Abstract

In this paper we consider the singular perturbation boundary-value problem of the following coupling type system of convection-diffusion equations

$$\left\{ {\begin{array}{*{20}c} \begin{gathered} \varepsilon u_1 ^{\prime \prime } + a_1 (x)u_1 ^\prime + b_{11} (x)u_1 + b_{12} (x)u_2 = f_1 (x) \hfill \\ \varepsilon u_2 ^{\prime \prime } + a_2 (x)u_2 ^\prime + b_{21} (x)u_1 + b_{22} (x)u_2 = f_2 (x) \hfill \\ u_1 (0) = a_1 , u_1 (1) = \beta _1 \hfill \\ u_2 (0) = a_2 , u_2 (1) = \beta _2 \hfill \\ \end{gathered} & {(0< x< 1)} \\ \end{array} } \right.$$

We advance two methods: the first one is the initial value solving method, by which the original boundary-value problem is changed into a series of unperturbed initial-value problems of the first order ordinary differential equation or system so that an asymptotic expansion is obtained; the second one is the boundary-value solving method, by which the original problem is changed into a few boundary-value problems having no phenomenon of boundary-layer so that the exact solution can be obtained and any classical numerical methods can be used to obtain the numerical solution of consismethods can be used to obtain the numerical solution of consistant high accuracy with respect to the perturbation parameter ɛ