Asymptotic Properties Of Solutions For Some Singularly Perturbed Problems Possessing Singularity

The matching asymptotic expansion method and differential inequality theory are used to study some singularly perturbed problems with singularity.This article consists of four parts:The first chapter briefly introduces the research background,purposes and also some preliminary concept and results which will be required later.In the second chapter,the second-order linear singularly perturbed boundary value problem with singularities before the sub-higher derivative of the equation is studied.The research results show that this type of problem has a double-layer phenomenon.The asymptotic solution of the equation is constructed by the matched asymptotic expansion method,and the uniform validity of the asymptotic solution is proved by the differential inequality theory.In the third chapter,the first-order nonlinear initial value singular perturbation problem with singularity and multiple reduced root is studied.The research results show that this kind of problem also has a double-layer phenomenon.In the fourth chapter,a class of non-linear singularly perturbed boundary value problems with delays is studied.The singularity of this type of problem is generally located at a certain fixed point within the interval.The research results show that such problems have a shock phenomenon.At the same time,the matching asymptotic expansion method is used to construct the formal asymptotic expansion.The differential inequality theory is used to prove the uniform validity of the formal solution.