Asymptotic Approximation Of Solutions For The Singularly Perturbed Problem With Degenerate Equation Having Multiple Roots

In this paper,several types of singularly perturbed problem with degenerate equation having multiple roots are studied using the method of boundary layer functions and the theory of differential inequalities.The article consists of three parts.The first chapter briefly introduces the research background,purposes,progress and also some preliminary concept and results which will be required later.In the second chapter,singularly perturbed problem with a power-law layer is studied.We consider the Dirichlet boundary value for the second order singularly perturbed problem with degenerate question having a double root.The method of boundary layer functions is used to construct the formal solutions,and get the decaying boundary layer functions of power-law.The existence and uniformly valid approximation of solutions are obtained by upper and lower solutions method.In the third chapter we study several types of singularly perturbed problems subject to the Neumann boundary condition in the case of a double root of the degenerate equation.Due to the particularity of the equation type,the research results show that this kind of problem has multi-zone boundary layer.In the beginning,the boundary layer functions decrease by a power-law,however,finally,they decay exponentially.Thus in section 1,we consider a singularly perturbed ordinary differential problem.The method of modified boundary layer function is used to construct the formal asymptotic expansion of the solution and obtain the more precise boundary layer functions which decay exponentially.Finally,an example is presented.In section 2,the asymptotic solutions to singularly perturbed reaction-diffusion problem is studied.The formal solution is constructed using the method of modified boundary layer function.The existence and uniformly valid approximation of solutions are obtained by the theory of differential inequalities.In fourth chapter,we consider the asymptotic solutions to the problem of singularly perturbed elliptic equation with degenerate equation having triple root.The formal asymptotic expansion of solution is constructed and the existence and uniformly valid approximation of solution are gained.