Research On The Two Classes Of Singularly Perturbed Problems With Internal Layers

In recent years, the singularly perturbed problem with Inner layers has gained more and more attention for its applicability. As the Inner layers become complex, non-smooth problems and exchange of stabilities ones arise, both of which need to be further addressed.Firstly, a class of non-smooth singularly perturbed problems, which do not match the Tikhonov system, is discussed in this paper. In order to get the smoothiy asymptotic solution, this problem is divided into left and right two pure boundary issues. With the boundary layer function method, the asymptotic solutions of left and right problems are constructed, the estimate exponential of boundary layer function is proved. Then useing "Sewing connection method" and smoothly sewing the discontinuous point of left and right problems. What’s more, the existence and uniformly valid estimation of the solution are given in the whole interval. At last, with the help of computer programming and mapping, as well as the concrete example, the conclusion is verified. This paper extend the "Sewing connection method" to the problem which do not match the Tikhonov system.In addition, the paper focuses on a class of singularly perturbed problems with exchange of stabilities. The asymptotic solution of the problem is consided accord-ing to different intervals. Using the differential inequality method, the existence and uniformly valid O((?)) estimation of the solution are proved in the interval where the turning point lies. Then according to boundary layer function method, the high-order asymptotic solution are shown in the stable intervals, the existence and uniformly valid estimation of the solution are given in the whole interval. At last, with the help of computer programming and mapping, as well as the concrete example, the conclusion is verified.