Research On Some Singularly Perturbed Boundary Value Problems With Discontinuous Right Hand Side

In recent years,the research of internal layer solution is well developed and thus it will provide theoretical basis for the research of internal layer solution of singularly perturbed boundary value problems with discontinuous right hand side.In this dissertation,by means of further investigation of state solution limiting behavior of singularly perturbed boundary value problems,we discuss the existence of internal layer solution of singularly perturbed boundary value problems with discontinuous right hand side.The internal layer solution is called contrast structure in singularly perturbed prob-lems,it is mainly classified as a step like internal layer and a spike like internal layer.In this dissertation,we consider the step like internal layer solution of the singularly perturbed boundary value problem with discontinuous right hand side.Its fundamental characteristic is that there exists t₀?or multiple t₀?within the domain of interest,t₀is called as an internal transition point.The discussion at each t₀is exactly the same,so we only consider the case that there exists one internal transition point.The position of t₀is known in advance and the variable y?t₀?needs to be determined thereafter.In the neighborhood of t₀,the solution will have an abrupt structure change and the solution will approach different reduced solutions when the small parameter??0.In the chapter one,we introduce the development of singularly perturbed boundary value problem,give some basic concepts and lemmas which are relevant to our study,and show the main research work and innovative points in this thesis.In the chapter two,we consider singularly perturbed second-order quasilinear bound-ary value problem with Neumann and Dirichlet boundary conditions.For the discontinu-ity of right hand terms,the solution will change rapidly nearby the discontinuous points.We prove the existence of the inter layer solution by the matching method and construct the uniformly valid formal asymptotic solution.In chapter three,we consider second-order singularly perturbed boundary value prob-lem with discontinuous right hand terms and Robin boundary condition.We not only prove the existence of the solution,determine the transition point,but also construct the asymptotic solution by the boundary function method.In chapter four,we consider singularly perturbed boundary value problem with re-action diffusion advection.The existence of step like internal layer solution for the above problem is proved,and we give the uniformly valid formal asymptotic solution for the internal transition layer solution.In chapter five,we investigate the singularly perturbed equations with Robin bound-ary condition.Base on the structure of the solution,we not only prove the existence of internal transition layer solution by the matching method,but also construct the uni-formly valid asymptotic solution by the boundary function method.