Some Numerical Methods For Fractional Diffusion Equations

Despite the history of fractional calculus is almost as long as integer-order cal-culus. However, because of lack of application background, fractional calculus was developed very slowly. It is known that the classical calculus provides a powerful tool for explaining and modeling important dynamic processes in many areas of the applied sciences. But experiments and reality teach us that there are many complex systems in nature with anomalous dynamics, which can not be characterized by classical deriva-tive models. Therefore, during the last decade fractional calculus has been applied to almost every field of science, engineering, and mathematics.Anomalous diffusion is perhaps the most frequently studied complex problem. Classical partial differential equation of diffusion and wave has been extended to the equation with fractional time and/or space by means of fractional operator. Fur-thermore, it has been extended to the problems of every kind of nonlinear fractional differential equation. And to present the solutions to the problems of initial and bound-ary values subject to above equations is another rapidly developing field of fractional operator applications. In general, all of these equations have important background of practice applications, such as dispersion in fractals and porous media, semiconductor physics, turbulence and condensed matter physics.The paper focuses on some fractional diffusion equations and their numerical meth-ods. It is composed of four chapters, which are independent and correlative to one another. The first chapter contains a brief introduction to fractional calculus and some elementary knowledge. The second and third chapters deal with the space-fractional advection-diffusion equations by some different finite difference methods, such as, the fractional weight average method and the characteristic finite difference method, etc. In the last chapter, we propose a new implicit numerical solution for the time-fractional diffusion equation. The chapter 1 is introduction. Firstly, the history and the development of the fractional calculus and its applications is introduced. We also introduce some differ-ent kinds of fractional operators, such as Riemann-Liouville, Caputo and Griinwald-Letnikov fractional operators, and so on.At the same time, there are some properties of the fractional operators, too. Next, we present the Mittag-Leffler function, which usually contains Mittag-Leffler function and generalized Mittag-Leffler function in two parameters, etc. It is the elementary solution of many fractional differential equations. Similarly, there are still two different special functions, Wright function and H-fox function.Additionally, in§1.4, we include some numerical methods relative researches of fractional differential equations, which are known so far. Such as, finite difference method, finite element method, differential transform method, Adomian decomposition method, variational iteration method, homotopy perturbation method, etc. In the last section of the chapter, we introduce in details the applications of the fractional calculus in almost every field of nonlinear complex physical systems.In the following chapters, two different models of abnormal diffusion are studied. In chapter 2, we study the one-dimensional space fractional advection-diffusion equa-tion. Based on the shifted Grunwald approximation to the Riemann-Liouville fractional derivative, we propose the fractional weight average (FWA) method in this chapter. we can see that some results obtained previously are special cases of the method. In§2.3, stability of the FWA method is proved by Gerschgorin theorem and matrix analysis, and the conclusion is given in the form of Theorem 2.1.Then, we also discuss a new improved FWA scheme and its stability in§2.4. At last, some numerical examples are carried out to confirm our theory. At the same time, as a special case of the FWA method, the fractional Crank-Nicholson (FCN) method is much better, which is not only unconditionally stable, but also second-order accurate in time. The result of this chapter has been published on Physics Letters A.In chapter 3, to the best knowledge of the authors, the numerical methods devel-oped so far for two-sided space fractional advection-diffusion equations are all Eulerian methods. Consequently, these methods suffer from the same numerical limitations as their analogue for second-order advection-diffusion equations. In§3.3 of this chap-ter, we firstly develop a characteristic finite difference method (CFDM) for fractional advection-diffusion equations, by combining the shifted Griinwald-Letnikov fractional finite difference procedures with the Lagrangian treatment. The proposed method re- tains all the numerical advantages of characteristic methods for second-order advection-diffusion equations and the finite difference methods for fractional advection-diffusion equations. Then we prove that this method is unconditionally stability, consistent and convergence in§3.4. The maximum error estimate is derived, too.Numerical solutions and exact solutions of a special fractional diffusion problem are shown in§3.5 and a comparison between the fractional CFDM and the standard finite difference methods (SFDM) is given. Finally, we draw our conclusions in§3.6. Obviously, the fractional CFDM is better in both the accuracy and the stability than the known fractional SFDM, included the fractional explicit upwind finite difference method and the fractional implicit upwind finite difference method. Especially for the convection-dominated problems, this new method is very efficient and superior. The result of this chapter has been submitted to Journal of Computational Physics.In chapter 4, we study the time fractional diffusion equation. Subdiffusive mo-tion is particularly important in the context of complex systems such as glassy and disordered materials, in which pathways are constrained for geometric or energetic rea-sons. For anomalous subdiffusive random walkers, the continuum description via the ordinary diffusion equation is replaced by the Riemann-Liouville fractional diffusion equation. This fractional-order model tends to be more appropriate than the tradi-tional integer-order model.In this chapter, firstly, we propose a new kind of implicit difference method based on the shifted Grunwald approximation to the time fractional derivative, and use the central difference scheme for the time first-order derivative and space second-order derivative, which is a three-layer difference scheme. The numerical solution of the first time layer can be calculated by the fully implicit scheme or the Crank-Nicholson scheme, which are both unconditionally stable. Next, stability and accuracy are dis-cussed by means of the generalized Fourier-Von Neumann method. Finally, numerical solutions and exact analytical solutions of a typical fractional diffusion problem are compared. The result of this chapter has been submitted to Applied Mathematics and Computation.