Asymptotic And Numerical Analysis For Two Types Of Singularly Perturbed Differential Equations Initial-boundary Value Problems

The theory and method of application for singular perturbation is a very broad range of subject. The singular perturbation is a method to find approximate analytical solutions of nonlinear, high order, or a mathematical equation with variable coefficients. The current research is very active and constantly expanding .Its main idea is to find approximate solutions with differential equations containing small parameter which is from some of the original equation by solving the simple equation, so it is called the approximate analytic solution. This method can be used in mathematical physics of the original quantitative or qualitative analysis and discussion. At present, the theory for singular perturbation has been gradually established a number of effective methods, such as matched asymptotic expansion method, multi-variable expansion method, the boundary layer function method, V-L method, PLK method, WKB method, compensation compact method, multiple scales, KBM method, the average variational method, the constant field theory and the technique of diagonalization and so on. All sorts of methods for singular perturbation have been widely applied in many fields in natural science, which play a crucial role in solving practical problems. Most of dynamic mathematical models contain small parameters play a particularly important role in obtaining the uniformly valid asymptotic (approximate) solution for the complex nonlinear equations under the premise of being unable to get the accurate solution. In practical application, the numerical calculation and asymptotic methods are methods to find valid approximate solutions and complement each other.The properties of the initial-boundary value problem for the differential equation containing small parameters are studied. The main contents of this paper are outlined as follows:1. The nonlinear initial-boundary value problems for a class of singularly perturbed parabolic equations are considered. Under suitable conditions, firstly, the outer solution of original problem is constructed. And then, using the method of multiple scales variables and the expanding theory of power series we seek solution of the problem. Finally, the existence, uniqueness and asymptotic behavior of the generalized solution for the problems are studied.2. The nonlinear singularly perturbed Robin problems for reaction diffusion equations with boundary perturbation are considered in the paper. By the method of differential inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation are discussed under some suitable conditions.3. Based on shishkin-type mesh, a singularly perturbed nonlinear neutral differential difference equation with negative shift initial boundary value problem is studied. B-spline collocation method and finite difference method is applied on the problem. Estimation of the error between exact solution and approximation solution is given.