A Compact Difference Method For Fractional Cattaneo Equation With Neumann Boundary Conditions

In this paper,compact finite difference methods are proposed for solving a class of fractional Cattaneo model with Neumann boundry conditions.Because of high precision,high resolution and low grid requirements,the compact finite difference schemes are widely used in numerial approximation of partial differential equations.This paper is divided into four chapters,The first chapter is an introduction,which primaryly introduce the physical background of the fractional Cattaneo model and states of contents,and results of research in this paper.In the second chapter,the compact finite difference scheme of Cattaneo method with Neumann boundry is proposed.Compact difference methods are constructed for inner node and boundary node respectively.The linear algebraic equations formed by this scheme are easy to be solved.The error estimations of the scheme are carried out,and the validity is verified by a numerial example.In the third chapter,the finite difference scheme and the compact finite difference scheme are obtained for the fractional Cattaneo equations under semi-unbounded domain.An artificial boundary is constructed to make the model equivalent to a problem where both ends are Neumann boundry conditions.Stability and convergence are analyzed respectively,and the validity is verified by numercial examples.Finally,the convergence order of two schemes are compared.The fourth chapter is the summary of the whole work and a prospect of the future reserch.