Layer Phenomena For Several Classes Of Singularly Perturbed Problems With Nonlinear Boundary Conditions

In this thesis, layer phenomena for several classes of singularly perturbed prob-lems with nonlinear boundary conditions are studied by the method of boundary layer functions and the theory of differential inequalities.This thesis consists of four chapters:In the first chapter, we mainly introduce the research significance and situation of the singularly perturbed problem and summarize some achievements of singularly perturbed boundary value problems which related to this thesis in recent years. Then, we give the main work and the innovation of this thesis.In the second chapter, we mainly discuss a class of singularly perturbed problems for nonlinear equations with turning point and nonlinear boundary conditions ε2y"=f(x,y,y’), -1<x<1, g1(y, z)|y=y(-1),z=y’(-1) = 0, g2(y, z)|y=y(1), z=y’(1) = 0 by comparing the equations, selecting appropriate bounding functions and using the techniques of inequalities amplifying. The existence of three classes of problems’ solu-tions which exhibit interior layer behavior are proved and the asymptotic estimation of solutions are given using the method of the theory of differential inequalities. Then, it pointed out that each class of problem have the situation of exponential decay or algebra decay in the place of interior layer under different conditions. Finally, an example is given to illustrate the applying value of the study achievements.In the third chapter, we construct the formal asymptotic solutions of a class of singularly perturbed nonlinear mixed boundary value problems for third-order semi-linear differential equations ε2y’’’= f(x, y, y’,ε), a<x<c, y(b)= A, y’(a)= y’(c), g(y(a), y(c), y’(a), y’(c))= 0 by introduce the stretched variable and the method of boundary layer functions. Ac-cording to the theory of differential inequalities, the existence of solutions are proved and the error estimate of asymptotic solutions are given. Then, the exponential decay of boundary layer functions for the problems are concluded.In the fourth chapter, we construct the formal asymptotic solutions of a class of singularly perturbed boundary value problems for forth-order nonlinear differential equations with nonlinear mixed boundary conditions εy(4)= f(t, y, y’,y’’,y’’’), a< t <c, y(b)= A, y’(b)=B, g(y"(a), y"(b), y"(c), y’’’(a))= 0, y"(c)= C by introduce the stretched variable and the method of boundary layer functions. According to the theory of differential inequalities, the existence of solutions for the problems are proved and the error estimate of asymptotic solutions are given. Then, the exponential decay of boundary layer functions for the problems are concluded.