Finite Difference Methods For Fractional Diffusion Equations And Their Applications

The main contribution of this dissertation is to propose some high order finite difference schemes for three kinds of fractional diffusion equations in one space dimension,and establish an e?cient computational platform for the optimal control problems of pest anomalous diffusion. Firstly, we investigate the high order numerical discretization for αth(0 < α < 1) order Caputo derivative, and then a new high order difference scheme for Caputo type diffusion–advection equation is proposed. Secondly, we discuss the solutions of fractional subdiffusion equations, and construct high order compact finite difference schemes for both constant order and variable order fractional subdiffusion equations. Thirdly, we study the numerical solutions of time–space fractional diffusion equations. Two finite difference schemes are proposed for two different diffusion cases respectively. Finally, by using numerical methods, we establish an e?cient computational platform to solve the optimal control problem of anomalous diffusion of pest spreading in agriculuture.In the first chapter, we firstly introduce the history and development of fractional calculus, then the common used definitions and properties of fractional integrals and derivatives are given. We also present some special functions together with their properties. At last, we highlight the main works of this thesis.Chapter 2 is devoted to developing a high order approximation to Caputo derivative. In Section 1, we summarize those already existed numerical algorithms for approximating Caputo derivative. In Section 2, the derivation of a high order numerical approximation to Caputo derivative is given in detail. Finally, we develop a high order finite difference scheme to numerically solve Caputo type diffusion–advection equation by using the derived alogirithm.In chapter 3, we investigate high order compact finite difference schemes for both constant order and variable order fractional subdiffusion equations. In Section 1, we give a high order compact finite difference scheme for constant order fractional subdiffusion equation. The high order compact finite difference method for variable order fractional subdiffusion equation is discussed in Section 2. Finally, several numerical examples are given to demonstrate the theoretical analysis.Chapter 4 is devoted to studying the numerical scheme for time-space fractional diffusion equation. In Section 1, we construct a finite difference scheme for time-space fractional diffusion equation with α ∈(0, 1). Another finite difference scheme for timespace fractional diffusion equation with α ∈(1, 2) is established in Section 2. Finally,numerical results are given to show the e?ciency of our schemes and also verify the theoretical analysis.In chapter 5, we describe the anomalous diffusion process of pest spreading by using fractional diffusion equations, and then an e?cient computational platform for the optimal control of anomalous diffusion is designed and established. Firstly, three kinds of fractional diffusion models with control input are established. Based on the computational platform Diff-MAS2 D for classical diffusion control, a new platform called FO-Diff-MAS2 D is developed to model and control fractional diffusion problems. Finally, three examples are given to illustrate the e?ciency of this platform.