The Asymptotic Analysis For Boundary Value Problems Of Some Singularly Perturbed Differential Equations

The theory and method for singular perturbation is a interesting subject which hasbeen developed for nearly one century. Various of methods in the asymptotic analysisof singular petrurbation can be used to solve some problems validly. Large numbersof dynamic mathematical models could be result into differential equation problemswith small parameters. Obtaining the uniformly valid asymptotic solution fornonlinear, high order or changeful coefficient mathematic and physics equations playsa particular important role because the accurate solutions of the problems are usuallyunable to be obtained. The asymptotic solution is a kind of approximate solution ofthe accurate solution. It can be used not only in theory analysis, but also in numeircalsimulation. By the use of singularly perturbed methods, we may have qualitative oreven quantitative analysis ofr mathematical and physical problems. Nowadays，several methods for singular perturbation has been gradually developed，such asmatched asymptotic expansion method，boundary layer function method, differentialinequality method, ifxed-point theorem, plane analysis method and so on.In this dissertation, boundary value problems of some singularly perturbeddifferential equations are discussed. The construction and the main content of thedissertation are as follows:1Preface. The historical development and some method in common use of singularperturbation are introduced. The oirgin of problem and the purpose of research areintroduced also.2Preliminary knowledge. The methods and conclusions will be used in thisdissertation arc stated.3The nonlinear singularly perturbed problem is discussed. By using the method ofmatchcd asymptotic expansion, the phenomena of the inteiror layer and the boundarylayer for the solution under the condition of Dirichlet or Robin type are studied in thischapter, the corresponding conditions of the parameters arc obtained 4The boundary value problem for^-dimensional vectors is researched. The formalsolution is constructed by the method of boundary layer function. Using Nagumotheory in vectors form the existence of solution for nonlinear singularly perturbedboundary value problem is proved, and the estimation for remainder is given.5Using the method of boundary layer correction and the stretched vairables, thethree-order singularly perturbed boundary value problems of boundary perturbationare considered. By use of the method of differential inequalities, the existence of thesolution for the problem is obtained, and the uniformly validity for the constructedformal asymptotic solution is proved. At the same time, the estimation for remainderis given.