Asymptotic Analysis Of Solutions For Some Classes Of Singularly Perturbed Boundary Value Problems

The theory and method of application for singular perturbation problem is a very broad range of subject. The singularly perturbed method is used to find approximate analytical solutions of nonlinear, high order, or variable coefficients mathematical physical equations. Classical numerical methods usually give unsatisfactory numerical results when the singular perturbation parameterεis small. The current research is very active and constantly expanding .Recently, many approximate methods have been developed, including the method of matched asymptotic expansion, multi-variable expansion method, the boundary layer function method and the method of multiple scales.The main contents of this paper are outlined as follows:1. The nonlinear singularly perturbed Robin problems for reaction diffusion equations with boundary perturbation are considered in the paper. By using the method of differential inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation are discussed under some suitable conditions.2. A class of quasi-linear singularly perturbed boundary value problems with discontinuous of differential systems is considered. Using the method of boundary functions, the formal asymptotic expansions of the solution is constructed with step-wise method. At the same time, the solution is continuous under some suitable conditions. At last, the existence of the solution is proved and the uniformly efficiency of this asymptotic solution is obtained.3. A class of singularly perturbed boundary value problems with delay is considered. Using the method of boundary functions, the formal asymptotic expansions of the solution is constructed with step-wise method. Based on differential inequality techniques and the method of multiple scales variables, the existence of the solution is proved and the uniform validity of the asymptotic expansion is proved.