The Shock Solution For Several Nonlinear Singularly Perturbed Boundary Value Problems

Currently, nonlinear singular perturbation problem is a very concerned subject in theinternational academic community. Its theory and methods are used in many subjects, suchas the vibration theory, fluid mechanics, solid mechanics, modern physics, automaticcontrol, ocean engineering, biology, chemistry, and economics, population and so on.There are many questions of nonlinear model that are in need of being solved in these areas.Perturbation methods are one of main mathematical tools to find out the analyticapproximate solutions of these questions. And we can achieve very high precision incertain situations. At present, the research about singular perturbation method is veryactive, many singular perturbation methods have been developed, including the method ofmatched asymptotic expansion, boundary layer function method and the method of phaseplane analysis,etc. This paper will use some of the above singular perturbation method andtheory to study several class of nonlinear singularly perturbed boundary value problemwith which shock layer may be occurred in the solutions.The main contents of this paper are as follows:1、The shock layer in the solution of a class of nonlinear singularly perturbedboundary value problem is considered. Under suitable conditions, the existence and theuniform validity of asymptotic solution of the problem with shock layer are proved by thetheory of differential inequality.2、The asymptotic expansion of the shock solution for a class of singularly perturbedboundary value problem is explored by the method of matching asymptotic expansion. Theshock solution and shock location are stronger dependent upon the boundary value by theresearch. At last the shock solution and the relationship of the shock location and boundaryvalue are obtained.3、The shock location for a class of nonlinear singularly perturbed boundary valueproblem is quantized by the method of phase plane analysis. And the problem of boundaryperturbation is analyzed in this paper，we study the sensitive dependence of the shocklocation on small changes in the boundary values. At last we draw some conclusion.