Researches On Singularly Perturbed Frontier Problems

This dissertation studies several non-smooth singularly perturbed problems and a stochastic singularly perturbed problem.In the past few years,non-smooth and stochastic singularly perturbed problems received widespread attention.It has gained great development in studying these problems by means of geometric singularly perturbed theory,but the application of asymptotical theory to these problems remains to be further developed and improved.In this dissertation,problems are studied by means of geometric qualitative analysis and asymptotic theory,some results of predecessors are improved.The dissertation is divided into six chapters.The main results are outlined as follows:In Chapter I,the history,development,and actuality of singularly perturbed system,non-smooth dynamic system,and slow-fast system which are perturbed by noise are reviewed.The work of thesis are introduced and some remaining problems are given.In Chapter II-IV,the contrast structure for the non-smooth singularly perturbed systems whose reduced equation has simple root,including the second order quasi-linear Dirichlet problem,weakly nonlinear Dirichlet problem,and the first order ordinary differential equations with homogeneous Neumann condition are discussed.The asymptotic solutions of these problems are constructed by means of boundary layer function method and phase plane analysis of the auxiliary equation.By sewing smoothly for the solutions of quasi-linear and weakly nonlinear problems,the existence of smooth solutions for the problems are shown,and by sewing continuously for the solutions of ordinary differential equations,the existence of continuous solutions for the problem is shown,and the asymptotic solutions of these problems are proved to be uniformly valid in the whole domain.In Chapter V,the contrast structure for the non-smooth singular singularly perturbed problem whose reduced equation has multiple root is considered.The study is focus on discussing the case that the reduced equation has a double root on the left-side of the discontinuous point and a simple root on the right-side.By virtue of nonstandard boundary layer function method and sewing connection method,the existence of solution is shown and the asymptotic solution is proved to be uniformly valid on the whole interval.In particular,the internal layer is multi-zonal,which means in different zones of internal layer the decaying nature of the coefficient functions of the internal layer series transfers from algebra decaying to exponential decaying.In Chapter VI,the slow-fast system of differential equations with initial value condition is considered,in which both the slow and fast variables are perturbed by noise.The method to construct asymptotic solution of stochastic slow-fast system is shown.It is necessary to be mentioned that the high order coefficient functions of boundary layer don't meet the exponential decay character.Meanwhile,when the root of reduced equations of deterministic system satisfies stable condition,the asymptotic solution with a boundary layer of the singularly perturbed problem about parameter is shown to be uniformly valid on the whole interval,and the asymptotic solution of the regularly perturbed problem about parameter is shown to be uniformly valid on the whole interval.Meanwhile,the stochastic Vasileva theory is given.