Numerical Solution Of Singularly Perturbed Differential Equation And Fractional Order Differential Equation

The contribution of this paper is two parts. The first part of this paper address numerical solution of singularly perturbed problems. In the second part, we consider the FPDE and it's numerical approximation.Singularly perturbed problem with boundary layer was mainly studied in this paper. In the near region of boundary layer, the solution of singularly perturbed problem changes repidly when ε→0. The differential equation has singularity related to boundary layer.Classical difference scheme usually gives unsatisfactory numerical results for singularly perturbed problem. In particular, the pointwies errors of numerical methods based on centered or upwinded difference scheme on uniform meshes depend inversely on the power of the small parameter ε. Therefore, the interest in developing and analyzing efficient numerical methods, especially unifrom convergence numerical method, which is not depend on the small parameter ε, has increased enormously.Shishkin method has become popular in recent 10 years. In order to improve the rate of convergence, Shishkin method with Bakhvalov's technique, which is called Bakhvalov-Shishkin method, is presented.Based on Shishkin method, multi-transition points method is constructed in this paper. The new method is nonequidistant mesh partition. It has the numerical accuracy as Bakhvalov-Shishkin method, while it has simple computation procedure as Shishkin method. This novel numerical method is an practical and efficient computational method.The main ideas of multi-transition point method are as follow. Firstly, the choice of multi-transition points is presented according to the property of boundary layer. Secondly, multi-segment discrete mesh functions as barrier functions for numerical solution of the singular component are constructed. Thirdly, the estimate of truncation error, especially truncation error in transition points, is given. The proof is different from proof in traditional Shishkin method. There are some new techniques in our work.Numerical techniques based on integer-order differential equation have been applied to solve fractional-order differential equation. However, theoretical results for numerical solution of fractional-order differential equation have not been developed until recent 3 years. It is just beginning in the proof of numerical solution, especially in the stability and convergence. There are many difficulty in numerical solution. This is an field under further developed and improved.