The Sinc Collocation Methods For Nonlinear Integro-differential Equation And Fractional Differential Equations

In the recent thirty years, various numerical methods based on Sincapproximation have been developed rapidly. Sinc methods excel for problems withsingularities, for boundary-layers problems and for problems over infinite orsemin-infinite ranges. They are particularly suited for solving partial differential andintegral equations. This paper mainly studies solving the boundary value problem ofnonlinear integro-differential equation and initial value problem of fractionaldifferential equation by means of Sinc collocation method.The paper is divided into three parts. The structure is arranged as follows:In the first chapter, we mainly describe the background and the significance ofnonlinear integro-differential equation and fractional differential equation.Corresponding to the two aspects, we also introduce the results of Sinc collocationmethod in recent years.In the second chapter, we introduce in detail that the Sinc collocation methodfor second-order boundary value problems of nonlinear integro-differential equation.We consider the nonlinear equation with homogeneous boundary conditions. Someproperties of the Sinc collocation method for our subsequent development are givenand are utilized to reduce the computation of solution of the second order boundaryvalue problems of nonlinear integro-differential equation to some algebraicequations. Finally, we give numerical results to demonstrate the accuracy andefficiency of the method.In the third chapter, we study the Sinc collocation method for the initial valueproblem of fractional differential equation. The differnence between DE-Sinccollocation method and Sinc collocation method is that during the process of makingthe interval of the problem into infinite interval the former uses a double exponentialtransformation (hereinafter referred to as DE transformation). We first use thecontent of literature to reduce the computation of solution of the initial valueproblem of fractional differential equation to Volterra integral equation, then we useDE-Sinc collocation to approximate the unknown function and the weakly singularpart, which can transform the problem into solving nonlinear equations. We useNewton iteration method to complete our solving process. Finally, we make erroranalysis of the DE-Sinc method, and derive a nearly exponential convergence rate.At the same time, numerical result verifies our conclusion well.