Study On The Asymptotic Behavior Of Solutions To Some Singular Perturbation Nonlinear Models

This thesis aims to use the singular perturbation method mainly studies several classes of nonlinear problems with boundary layer and inner spike layer phenomena:the spike layer solution to singular perturbation Robin boundary value problem for higher order elliptic equation,generalized solution to singular perturbation problem for hyper parabolic equation with two parameters,the generalized spike layer solution to singular perturbation nonlinear parabolic system with two parameters and a class of shock wave solution to singular perturbation nonlinear time-delay evolution equations.By using the multi-scale method,stretched variable method,composite expansion method such as singular perturbation method,set up local coordinate system,structure spike layer cor-rective term,the boundary layer corrective term and the initial layer corrective term,then the formal asymptotic solutions to the corresponding problems are given,by using the functional analysis fixed point theorem,the existence,uniqueness and uniform validity of the solutions to these nonlinear differential equations in the corresponding regions are proved.remainder estimation are given.Finally,the asymptotic approximation solutions to four specific nonlinear differential equations are given by using the improved singular perturbation method,and the asymptotic behavior of the solutions is analyzed,and the physical meaning of the asymptotic solution to the corresponding model is expounded..By changing the parameter values for simulation,the images of approximate and exact solutions to the corresponding models are given.The contents of the work are summarized as follows:The first and second chapters briefly review the nonlinear problems,the asymptotic theory of singular perturbation,the perturbation methods,the singular perturbation prob-lems of partial differential equations,and the research status of the theory and method of singular perturbation.The research ideas of this paper and the research arrangement of the main chapters are also described.The chapter 3,the spike layer solution to singular perturbation Robin boundary value problem for higher order elliptic equation is studied.Firstly,construct the external solution to singular perturbation Robin boundary value problem,set up local coordinate system in the corresponding neighborhood,by using the multi-scale method,inner spike layer corrective term and boundary layer corrective term are constructed,then they are synthesized,furthermore the uniform validity of its asymptotic expansion is proved by using the fixed point theorem.In chapter 4 and 5,the generalized solution to the nonlinear hyper-parabolic sys-tem with two parameters and the generalized spike layer solution to the nonlinear hyper-parabolic system are studied.Firstly,the inner product is defined.The corresponding external asymptotic expansion solution is constructed.Since the boundary conditions and initial conditions are not satisfied,the spike layer corrective term,boundary layer corrective term and initial layer corrective term are constructed by using the stretched variable method and multi-scale method.Furthermore,the existence,uniqueness and uniform validity of the generalized solution and the generalized spike layer solution are proved by using the functional analysis fixed point theorem,remainder estimation are given.The obtained asymptotic solution can be used to analyze the generalized solu?tion,and more characteristics of the generalized solution can be understood.Therefore,the asymptotic expansion solution is obtained in this paper which has broad application prospects.In chapter 6,a class of shock wave solution to singularly perturbed nonlinear time-delay evolution equations are studied.Firstly,the parameter with time-delay is intro-duced,the parameters with time-delay is expanded into power series forms,the external asymptotic solution is constructed by the degenerate problem,since the external solution is discontinuous at somewhere,and it might not satisfy the original initial conditions,therefore,the spike layer,the boundary layer and the initial layer correction function are constructed.Secondly,the spike layer correction term,the boundary layer correc-tion term and the initial layer correction term are constructed respectively by using the multi-scale method and the stretched variable method,and then they are synthesized,the remainder is estimated.Furthermore,by using the functional analysis fixed point theo-rem,the existence,uniformly valid asymptotic expansion for the solution are proved.In addition,the method is convenient to use.In chapter 7,the asymptotic approximation solutions to the four specific nonlinear evolution equations are studied,mainly involves the models:singularly perturbed non-linear time-delay ocean wind model,nonlinear epidemic contagion ecological model,quantum plasma nonlinear dynamic model,nonlinear Duffing oscillator with time-delay stochastic resonance mechanism.By using the improved perturbation method,the homotopic mapping iterative,the generalized variational iteration,the extended tanh function method and so on,the first integral function is found out,repeat using the same iteration,nth iterative solution is obtained,so the corresponding asymptotic approxi-mation solution to the model is found.Furthermore,by changing the specific values of small parameters,numerical simulation is carried out,the images of approximate and exact solutions to the corresponding models are given,the physical meanings of the asymptotic solutions are expounded.