Fundamental Solutions And Numerical Methods Of The Fractional Advection-Dispersion Equations

The characteristic of fractional order differential equation is containing the noninteger order derivative.It can effectively describe the memory and transmissibility of many kinds of material,and plays an increasingly important role in physics,mathematics, mechanical engineering,biology,electrical engineering,control theory,finance and other fields.All kinds of fractional models have close relation with chaotic dynamics. Anomalous diffusion in physics were originally developed from stochastic random walk models.Fractional advection-dispersion equations is powerful tool to simulate all kinds of anomalous diffusion phenomena.They are a subset of fractional kinetic equations that allow fractional derivatives in both the space and time operators.We discuss the time,space,space-time Fractional advection-dispersion equations respectively in this paper.The spatial derivatives discussed in the paper are all Riesz space fractional derivative,which include the left and right Riemann-Liouville fractional derivatives. The notable merit of Riesz space fractional derivative lies in its applicability to higher dimensional space.This thesis consists of the four chapters.Introduction presents the developmental history of fractional calculus and some important previous works at first.Then,gives some concerning fractional calculus to prepare the knowledge and present basic definitions and properties of fractional calculus.In Chapter 2,starting from the time fractional diffusion equation,we present an explicit conservative difference approximation,and give the stability and convergency analysis.Then,we extend the obtained results to the time fractional advectiondispersion equation.For the explicit conservative difference approximation of the time fractional advection-dispersion equation,we analyge the stability and convergency by using mathematical induction,and interpret it as a particle random walk.Random walks have proven to be a useful model in understanding processes across a wide spec- trum of scientific disciplines.In Chapter 3,we consider the Riesz space fractional advection-dispersion equation. It has three components.At first,we consider the case of initial value problem. Using the method of the Laplace and Fourier transforms,we obtain the fundamental solution of the equation with initial condition.The fundamental solution is represented by Green function,and can be intergreted the probability interpretation.We construct an explicit finite difference approximation for the equation by using the equivalence relation between Riemann-Liouville fractional derivative and Grünwald-Letnikovmake fractional derivative.The discrete scheme can be interpreted as a discrete random walk model,and the random walk model converges to a stable probability distribution.Secondly, we consider the case of initial-boundary problem.For the Riesz space fractional derivative can be expressed by a fractional power of the Laplacian operator,the numerical solution of our equation can be obtained by recur to matrix transfer technique and fractional method of lines.We also derive the new analytic solution by utilizing the property of eigenfunction and Laplace transform.Furthermore we compare the analytic solution and the numerical solution.Finally,we discuss the finite difference approximations in the case of initial-boundary problem.The explicit and implicit difference approximations are presented and the error analysis is also given.In Chapter 4,we consider the Riesz space-time fractional advection-dispersion equation.At first,we consider the case of initial value problem.We obtain the fundamental solution by using the method of the Laplace and Fourier transforms.The fundamental solution also be represented by Green function,and also can be proposed the probability interpretation.Using the equivalence relation between Riemann-Liouville fractional derivative and Grünwald-Letnikovmake fractional derivative,an explicit finite difference approximation for the equation is presented.The discrete scheme can be interpreted as a discrete random walk model.Then,the case of initial-boundary problem are discussed.The explicit and implicit finite difference approximations are proposed and the error analysis are also given.The non-local structure of fractional derivatives is one reason,why numerical methods for fractional differential equations are much more costly in computational time and storage requirements that their in- teger order counterparts.Thus,we propose the Richardson extrapolation which can promote the accuracy and "short-memory" principle which reduce the computational cost finally,these two methods are used to improve our numerical methods.Some numerical examples are presented in each chapter,which show the efficiency of our numerical methods.