The Classification And Asymptotic Analysis For The Solutions Of A Nonlinear Singularly Perturbed Equation

Even in a simple looking nonlinear problem, the solutions may emerge a wide variety of phenomenons, including the boundary layer, the internal layer, the corner layer or the multiple situations. In this paper, a second-order nonlinear boundary value problem with small parameters is mainly discussed: The main problem we are interested is the study of the dependence of the solutions on the boundary values A and B which are not depended on ε. Since the problem is nonlinear, the qualitative nature of the solution will be changed when the boundary conditions alter.This paper will refer to Cole’s idea and divide the A-B plane into nine parts. After using Lienard-Transformation and converting the original equation to the equivalent differ-ential equations, we will apply the phase plane analysis method to analyze the character of the solution and calculate the moment when the the boundary layer, the internal layer or the corner layer occurs. The following three aspects will be done in five regions:firstly, the first-order formal asymptotic solution will be constructed; secondly, the images of the asymptotic solution and numerical solution will be compared; thirdly, the existence and uniqueness of the solution will be proved and the remainder term will be estimated.