Singularly Perturted Phenomenoa In Differential Equations

In this paper, by the theory of differential inequalities (upper and lower solutions method), we study the existence(or uniqueness) results of some classes of boundary value problems for nonlinear differential equations(without small parameter). Furthermore, we apply the results to singularly perturbed BVPs(Boundary Value Problem for short) which involve small parameters. By the method of boundary layer function, we construct the higher order asymptotic solution and get the error estimate of asymptotic solution and exact solution. This paper is arranged as follows.In chapter 1, firstly, we introduce the background of singularly perturbation theory and some important results of former scholars have studied. Secondly, the concept of upper solution, lower solution and Nagumo condition are showed, two principal theorems of second-order and third-order differential inequality are introduced, and some principal lemmals are cited which will be utilized in the following chapters.In chapter 2, we study a class of two-point boundary value problems for third-order ordinary differential equation with a turning point. we consturct high-order asymptotic expansion of the solution to the probelm by using the boundary layer function method, and prove the existence of solution,then obtain the error estimate of the solution by using the theory of differential inqualities.In chapter 3, we study the theory of differential inequalities and the existence of solutions of a class of three-pointed boundary value problem for second order nonlinear differential equations. Then using the results obtained, we study the singular perturbation differential equation of three-pointed boundary value problem for second order quasilinear differential equations. Furthermore,we get the error estimate of the solution.In chapter 4, we first obtain existence of solution to some n-point boundary value problem for third-order differential equations using upper and lower solutions method. Based on the results, we explore singular perturbation of another n-point boundary value problem for third-order differential equations with a small positive parameter. Finally, a uniformly valid asymptotic solution is constructed and the error estimation is given.In chapter 5, by the theory of boundary layer correction method, we construct the higher-order formal sulution to the singularly perturbed BVP for third-order quasilinear differential equation. Then, existence of solution and the estimate of asymptotic solutiong about exact solution are achieved via the application of Banach contracion mapping principle.