Spatial Contrast Structure For Several Singularly Perturbed Delay Differential Equations

In the middle of last century,researchers had found that singularly perturbed delay differential equation plays an important role in the mathematical modeling of different types of practical problems in science and engineering.Recently,lots of scholars have devoted themselves to this research field,which greatly promote the development of singular perturbation theory and enrich its research content.By employing the classical singular perturbation theory and method as well as the phase plane analysis of the aux-iliary problems,the study of this dissertation is aimed to consider the spatial contrast structure for several singularly perturbed delay differential equations.This dissertation is divided into five chapters and organized as follows.In the 1st chapter,the history of singular perturbation theory and some fundamental theorems as well as the concept of spatial contrast structure are sketched.And then,the singularly perturbed delay differential equation and their research progresses are briefly introduced.The detailed contents of this dissertation are described at the end of this chapter.In the 2nd chapter,a kind of quasilinear problem which can be transformed into a fast–slow system is carried out.The boundary layer function method is firstly applied by Vasil'eva [Comput.Math.Math.Phys.,35(4):411–419,1995] to study the asymp-totic solution for a second order quasilinear singularly perturbed differential equation.In [Differ.Equ.,53(12):1567–1577,2017],this kind of equation with discontinuous right–hand side is brought out by Ni et al.,which has extended the work of Vasil'eva to discontinuous case.By means of the spatial contrast structure theory,the internal layer phenomenon for a kind of quasilinear problem with a time–delay will be further studied in this chapter.And asymptotic expression for the uniformly valid smooth asymptotic solution is constructed.In the meantime,the existence of smooth solution is proved and the remainder estimation is obtained.Finally,a concrete example and some numerical simulations are presented to illustrate the practicability of the construction algorithm.In the 3rd chapter,a class of weakly nonlinear problem is considered.The spatial contrast structure theory is firstly used by Vasil'eva & Davydova in the pioneering work[Comput.Math.Math.Phys.,38(6):900–908,1998] to consider weakly nonlinear sin-gularly perturbed differential equation.Subsequently,the internal layer for a weakly nonlinear singularly perturbed differential–difference equation has been considered by Wang & Ni [Acta Math.Sci.,32(2):695–709,2012].On the basis of the existing lit-eratures,the effect of the first derivative dy/dt on the original problem will be further enhanced in this chapter,the smooth asymptotic solution of the proposed problem with left and right boundary layers as well as internal transition layers will be deeply stud-ied.A uniformly valid smooth asymptotic solution for this problem is constructed,the existence of smooth solution is proved and the remainder estimation of the solution will be obtained.At the end of this chapter,an example and numerical simulations are presented to verify the feasibility of the algorithm.In the 4th chapter,a singularly perturbed differential system with time–delay as well as the same “velocity” will be discussed.The boundary layer function method is firstly applied to consider singularly perturbed ODEs by Vasil'eva in [Comput.Math.Math.Phys.,34(10):1215–1223,1994].Recently,contrast structure for first–order singularly perturbed ODEs with discontinuous right–hand side has been investigated by Pang et al.[Differ.Equ.,54(12):1583–1594,2018],in which the existing results are generalized to the nonsmooth case.The corresponding results of the singularly perturbed ODEs will be extended to the time–delay situation in the current chapter.By virtue of the standard Vasil'eva's boundary layer function method and spatial contrast structure theory,the construction algorithm of asymptotic solution to the considered system is established.Then the asymptotic solution will be sewn by multiple sewing connection method and the obtained solution is proved to be existing and uniformly valid in the total interval.Finally,the main result of this chapter is demonstrated by a specific example as well as some numerical simulations.Unlike the previous two chapters,only a continuous rather than smooth asymptotic solution can be achieved according to the algorithm in this chapter.In the 5th chapter,some briefly summaries about this dissertation are exhibited,and the future investigation is directed.