Multiscale Analysis On A Kind Of Singularly Perturbed Reaction–diffusion Equations

Based on the theory and method of singular perturbation,this dissertation studies the boundary value problem of a kind of reaction-diffusion equation with time-independent steady-state and singular perturbation,and then studies the initial boundary value problem of a kind of parabolic reaction-diffusion equation with singular perturbation,and the relevant conclusions are obtained and verified by specific examples.In the first chapter,the history of the singularly perturbed theory are reviewed and the development are listed.And then,via simple introduction,some well-known theorems and lemmas in this field,which will be played important roles in the research for this article,are carried out.On the same time,some contents that are used and refered are briefly reviewed.The aim of this dissertation will be introduced in the end of this chapter.In the second chapter,based on the actual background,studying a kind of boundary value problem of steady-state singularly perturbed time-independent reaction-diffusion equation.The formal asymptotic solution of the problem is constructed by using the boundary layer function method of singularly perturbed theory.The solution is analyzed in multiple scales.The existence of the solution is proved by the differential inequality method and the remainder estimation is completed,the asymptotic state of the solution is obtained.Through numerical simulation,the numerical solution and the asymptotic solution are compared,and the conclusion is verified.In the third chapter,based on the conclusion of the second chapter,studying the initial boundary value problem of a kind of singularly perturbed parabolic reaction-diffusion equation.The existence of the solution of the problem is proved by using the differential inequality method.The asymptotic stability of the solution is obtained on the basis of the existing conclusions in the relevant literature,and the range of the attractive region of the boundary layer solution is determined.Through numerical simulation,the numerical solution and the asymptotic solution of the specific singularly perturbed parabolic reaction-diffusion initial boundary value problem are obtained,and the conclusion is verified by comparison.In the fourth chapter,giving a comprehensive summary of the paper,and puts forward some deficiencies and prospects.