Solution With Internal Layer Of Two Classes Of Three Order Discontinuous Singularly Perturbed Equations With Boundary Value

In recent years,piece-wise continuous singularly perturbed problems have received extensive attention and have important applications in mechanics and engineering.For discontinuous singularly perturbed differential systems,traditional singularly perturbed methods are difficult to deal with discontinuities at discontinuous points.Many problems need to be solved urgently.The piece-wise continuous third-order singular perturbation equation is widely used in solving bridge engineering problems,but there are few studies on it.Therefore,this paper studies two classes of piece-wise continuous singularly perturbed problems based on the singular perturbation theory and the sewing connection method.The existence and uniqueness of the solutions can be proved,and further generalizes singularly perturbed theory.The specific contents of this paper are as follows:The first chapter briefly introduces the development and achievements of the singular perturbation theory.It explains the important theorems and carries out the basic knowledge,lemmas and concepts that need to be used later in the article.The following is an overview of the issues that will be studied in this paper.The second chapter discusses the solution of a class of piece-wise continuous singularly perturbed mixed boundary value problem,and uses the boundary layer function method to study the solutions of third-order and first-order equations.Through the spatial contrast structure theory,the expression of the asymptotic solution with parameters is obtained.The solutions of the left and right problems at the discontinuous point are sewed together for obtaining a smooth,consistent and effective asymptotic solution,and the upper and lower solutions are constructed by the differential inequality method.The existence of the solution are proved.Numerical simulation is carried out on the example,and validity of the conclusion are verified by comparing the numerical solution with the asymptotic solution image.The third chapter,on the basis of chapter two,discusses the solution of a class of piece-wise continuous singularly perturbed mixed boundary value problem.The asymptotic solution is obtained by the same theory and method.The difference is that two sets of undetermined parameters need to be constructed,and getting the parameters is more complicated.The differential inequality method is used to prove the existence and uniqueness of the solution,and the expression of the asymptotic solution is obtained.Numerical simulation of the example is carried out,and the image of the numerical and asymptotic solution is compared to verify the accuracy of conclusion.The full text is summarized and explained,as well as the insufficiency and the prospect of the research direction.