Combining Asymptotic Solution and Numerical Solution for Differential Equation with Small Parameter

Ordinary differential equation with small parameter is considered in this paper. This kind of problem changes rapidly in both side of boundary layer. Firstly, the asymptotic solution of the problem is presented in order one. The asymptotic solution is used to solve the problem outside the boundary layer. Secondly, the analytical solution is decomposed into the smooth component and the singular component in order to improve the computational effect. The bounds for the derivatives of the smooth component and the singular component are studied. Thirdly, the fitted operator method is constructed for both side of boundary layer. The error estimation of numerical method is given also. Finally, numerical experiment is given to support the theoretical result.