On singular perturbation boundary-value problem of coupling type systems of convection-diffusion equations (Q1822258)

The author considers the singular perturbation boundary value problem of the form \[ \epsilon u_ 1''+a_ 1(x)u_ 1'+b_{11}(x)u_ 1+b_{12}(x)u_ 2=f_ 1(x) \] \[ \epsilon u_ 2''-a_ 2(x)u_ 2'+b_{21}(x)u_ 1+b_{22}(x)u_ 2=f_ 2(x)\quad 0<x<1 \] with boundary conditions \(u_ 1(0)=\alpha_ 1\), \(u_ 1(1)=\beta_ 1\); \(u_ 2(0)=\alpha_ 2\), \(u_ 2(1)=\beta_ 2\). The author presents two methods. First, an asymptotic expansion of the power series form is obtained, in which the coefficients satisfy a few initial-value or boundary-vaue problems of the first order differential equations or system which have explicit form and can be calculated easily. 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.