Two Numerical Methods For The Time - Fractional Diffusion Equation

In recent years, with the rapid development of computer science and technology, the calculation and realization of fractional calculus is feasible, and is gradually applied to various engineering fields and shows the advantages which cannot be replaced by classical calculus. At the same time, with the application of fractional calculus theory in practical control systems, the approximate discretization method of fractional calculus is getting more and more attention of the scholars.This article will discuss the numerical methods of the time fractional diffusion equation, and two new high precision numerical formats are given. The main works are as follows:(1)Given the fractional diffusion equation is equivalent to time partial differential integral equation and a new numerical method will be constructed by using block-by-block method and classic second-order central difference approximation to discretize its temporal direction and spatial direction, respectively. The overall improvement demonstrates this new scheme can effectively solve a class of time fractional diffusion equations with initial boundary value conditions.(2) Based on partial differential- integral equation is given in the previous chapter,another new numerical method will be constructed by using block-by-block method and compact difference approximation to discretize its temporal direction and spatial direction,respectively. Finally, by comparing with the exact solution and numerical solution shows the effectiveness of the algorithm.