The Internal Problems For Singularly Perturbed Integro-Differential Equation And Difference-Differential Equation

In singular perturbation theory, a solution with a internal layer is one of the most important object, and the connection of singular perturbation theory with other mathematical equations is a main manner to use the singularly perturbed method in practice. When study the singularly perturbed problems, the context often is difference-differential equations, integro-differential equations and PDEs and so on, so the scope of its application is very wide. The person who does't do theoretical studies only takes attention to the form of the solution, but the theorists should construct the asymptotic solution and prove the existence of it.In this paper, under some conditions, a kind of singularly perturbed second-order integro-differential equation is considered. For some specificity of the reduced solution, lead its solution has a corner layer in the domain [a,b],an asymptotic ex-pansion of the solution is developed using boundary layer method, justification of the existence of the solution and error estimates are given, and then an example is showed. And further discussion is the Tikhonov system with an integro-differential equation. For some particularity of this problem, we could use some conclusion of the former problem to this system. For example, when prove the existence of the solution by differential inequality, the method of constructing the supper and lower solution of the former problem could be used in this.At the last of this paper, in view of another internal layer, has talked about the Tikhonov system with a difference-differential equation, this time the internal layer is like the Contrast Steplike Structure. We have constructed the asymptotic solutions of the left and right problems, and proved the existence of the solution of the initial problem in "connection" method.