Sinc-collocation Methods For The Time-fractional Advection-dispersion Equation

As a new tool of mathematical modeling,fractional order calculus has been widely used in various fields of life,such as viscoelastic material,hydrol-ogy,finance and control system,etc.Compared with the traditional integral order model,fractional order model given by fractional order partial differential equations is better to describe memory effects and genetic effects in material engineering.Fractional order derivative plays a very important role in the simulated anomalous diffusion in porous media.When the first or second or-der derivative with respect to time in classical diffusion equation was replaced by a order derivative(α is a fraction),one gets a time-fractional differential equation.There are many methods to solve time-fractional diffusion equation at home and abroad.Sun etc[17]give an absolutely discrete finite difference format.By introducing two new variables,they turn the original equation into an equation with lower order and present an error analysis.By using the finite difference method in the direction of time and finite element method in spa-tial direction,Li etc[3]solve time-fractional diffusion wave equation.At the same time they also give a simi-discrete and fully discrete format of numerical analysis.How to solve the time-fractional advection-dispersion equation by sinc-collocation method is also considered in this article.The concrete nu-merical examples in this article tell us that sinc-colloction method is reliable and effective.In this article,Euler’s method is used in time direction and sinc-collocation method is used in space direction.This article is organized as follows:Chapter 1 and Chapter 2 present the current research status at home and abroad for time-fractional order partial differential equation.In these t-wo chapters,we will also introduce some related theorems of sinc-collocation method and the definition of fractional order derivative.Chapter 3 gives some simple applications of sinc-collocation method.Chapter 4 construc-t a semi-discrete scheme for the equation and presented the sinc-collocation method fully discrete format for the time-fractional advection-dispersion equa-tion.Through some numerical examples,chapter 5 will demonstrate that the sinc-collocation method is effective and reliable in solving the time-fractional advection-dispersion equation.