On The Study Of Some Discontinuous Singularly Perturbed Problems

This thesis aims to study the application of singular perturbation theory in the piecewise-continuous problems and the fractional-order problems.In recent years,due to the promotion of industrialization and the development of singular perturbation theory and its application,the mathematical models are always discontinuous,or piecewise-continuous,thus,close attention has been paid to the application of singular perturbation theory in piecewise-continuous systems.In chapters 2 to 5,some classes of piecewise-continuous singularly perturbed problems are studied by using the theory of contrast structures.Similarly,as was having been the special use partner of mathematician,fractional-order differential problem was widely applied in the fields of fluid mechanics,Rheology,Viscoelastic mechanics and so on,in the sixth chapter,a class of fractional-order differential problem is studied.In chapter one,the history and the present situation of singular perturbation theory and its application in piecewise-continuous problem and fractional-order problem are introduced,and a summary of the problems discussed in this thesis are given.In the second chapter,a class of second-order piecewise-continuous slow-fast system with Dirichlet boundary value conditions has been discussed,the asymptotic solution has been constructed based on boundary layer function method,the existence of first-order continuous solution of this problem has been proved by using the theory of contrast structures,and the estimation of remainder has been given.It is worth to be noticed that the two-dimension piecewise-continuous problem means the problem does not satisfy the continuity on a vertical plane of time axis,appears as being discontinuous on a vertical line of time axis in the one-dimension problem.Unfortunately,the discontinuous line is not vertical to time axis in many cases,therefore,the discussion of more general cases is needed.In the third chapter,a class of semi-linear singularly perturbed problem has been studied,being different from chapter two,the discontinuous line of this problem can be any monotonic curve.In this general case,the asymptotic solution has been constructed by boundary layer function method,the existence of first-order continuous solution has been proved through the theory of contrast structures and the estimation of remainder has been given.Thus,the case has been generalized.However,the high dimension problem appears everywhere,and that will be the problem discussed in the forth and fifth chapter,being limited to the personal ability,only the case that the discontinuous“plane” is vertical to time axis has been discussed when it comes to high dimension.In chapter four,a class of first-order 3-dimension system has been discussed,asymptotic solution has been constructed,the existence of continuous solution has been proved,and the estimation of remainder has been given.In chapter five,the problem has been generalized to high-dimension case,the content of chapter four can be seen as special case of chapter five.Unfortunately,since the expression of solution of high-dimension system is hard to be found,the exact expressions of coefficients of asymptotic solution won't be given,instead,the existence of coefficients has been proved.In the sixth chapter,a class of fractional-order singularly perturbed problem has been studied,Being independent of piecewise-continuous case,close attention has been paid to the fractional-order problem owing to its effective application in Viscoelastic model and the field of digital signal processing.A class of problem with Caputo-Type fractional derivative has been discussed,the boundary layer function method has been generalized to fractional-order problems,the property of decay(not exponentially decay)has been proved by using the final value theorem from the method of Laplace Transformation,the existence of solution has been proved based on fixed-point theorem,and the estimation of remainder has been given.