Asymptotic Expansion Of Solutions For Singularly Perturbed Problems Different Initial Boundary Values Problems In Several Critical Cases

At present,singular perturbation problem is usually studied under the condition that the degenerate system has isolated roots.In contrast,studies on degenerate systems with no isolated root and a series of solutions that depend on one or more parameters are rare.Such problems are called critical cases.Such problems are widely used as mathematical models in the fields of chemical kinetics,enzyme kinetics and semiconductor simulation.Moreover,the critical case is always a difficult problem in singular perturbation.The difficulties are as follows:firstly,the zeroth regular asymptotic solution is an arbitrary unknown function,which can only be determined in the equation related to the k(k≥1)regular part.Secondly,the solution process of the zero-order asymptotic solution of the boundary layer is very complicated,and it requires specific problems to find specific corresponding methods to solve.Finally,the equations related to the boundary layer part of the k(k≥1)degree asymptotic solution need to find a specific diagonalization transformation,to reduce the coupling degree of the equations.Therefore,it is very challenging and significant to study singularly perturbed different initial boundary value problems in critical cases.This paper is divided into five chapters.In the first chapter of this paper,the background and research status of singular perturbation in critical cases are described,and the work and innovation of this paper are described.In the second chapter,the weak nonlinear singularly perturbed integral boundary problem for a class of critical cases is studied.Firstly,the integral boundary conditions are converted into the first type of initial boundary value conditions by using variable transformation and exponential decay property of boundary layer function,thus reducing the complexity of calculation.Secondly,the formal asymptotic solution of the problem is constructed using the boundary layer function method,and the existence of the solution is proved by the successive approximation principle.Then,the theoretical results are verified by numerical simulation.Finally,some relevant conclusions are given in the course of research.In Chapter 3,a class of singularly perturbed problems with initial integral values and Robin boundary values in critical cases is studied.First,variable substitution and exponential decay property of boundary layer function are also used to transform the initial integral value into the first type of initial value.Secondly,the formal asymptotic solution of the original problem is constructed by means of boundary layer function method.Then,the existence proof of the solution is completed by the proof of the compressed fixed point theorem.Finally,an example is given to verify the correctness of the above theoretical derivation.In chapter 4,a class of singularly perturbed differential equations with Robin initial boundary value conditions is studied.This kind of problem is the original critical case obtained by dimensionless analysis based on mathematical model of enzyme kinetics.In the process of constructing formal asymptotic solutions by boundary layer function method,since the coupling of the equations in the primary boundary layer is very strong,the cubic diagonalization transformation is used to reduce the coupling degree to construct formal asymptotic solutions.Then,the existence of the solution is proved by successive approximation theorem.In addition,an example is given to prove the rigor and correctness of the theoretical results.Finally,some problems are found in the research process and summarized.In Chapter 5,the study of singular perturbations with different initial boundary value problems in the above three critical cases is summarized.At the same time,it gives the direction of future research,the method of concrete implementation,and the expected conclusion.