Asymptotic And Numerical Analysis For Several Classes Of Singularly Perturbed Boundary Value Problems

The theory and method for singular perturbation is a very active and constantly broadened subject. All sorts of methods for singular perturbation have been widely applied in many fields of natural science, which play a crucial role in solving practical problems. Most of dynamic mathematical models contain small parameters, so obtaining the uniformly valid asymptotic solution or numerical approximate solution is particularly important for the complex nonlinear equations under the premise of being unable to get the accurate solution. In practical application, the numerical calculation and asymptotic method are not mutually exclusive but complement each other.The properties of the initial-boundary value problem for the ordinary 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 solved. And then, by using the method of stretched variable and the expanding theory of power series the higher order formal asymptotic expansions of the solutions are constructed.2. A class of singularly perturbed partly dissipative reaction diffusion systems in case of exchange of stabilities are studied under suitable conditions. Firstly, the families of equilibria of the degenerate problems are constructed. The proofs of the results are based on the method of lower and upper solutions. Secondly, by using the stretched variable, the initial layer term of solution is constructed. And then, by using the theory of differential inequalities the asymptotic behaviors of solutions for the initial boundary value problems are studied. Also, the existence and uniqueness of solutions for original problem are discussed.3. A singularly perturbed advection–diffusion two point Robin boundary value problem whose solution has a single boundary layer is studied. Based on finite element method is applied on the problem. Estimation of the error between solution and the finite element approximation are given in energy norm on shishkin-type mesh.