Finite Difference Methods For Several Classes Of Fractional Partial Differential Equations

In recent years, a large number of numerical experiments show that some models based on integer-order derivatives can not describe certain natural phenomena very well, such as anomalous diffusion and complex viscoelastic materials. One of the main reason is that classical integer-order derivative is defined by the local limit of function which only reflects characteristic at a local range. It promotes the fractional calculus with nonlocal properties receiving the widespread attention. It is generally difficult to give the explicit form of ana-lytical solution, thus, effective numerical methods become necessary and important. In the current thesis, we will introduce or develop some numerical methods for several classes of space and time-space fractional partial differential equations.In Chapter 1, we briefly review the historical background of fractional calculus, then introduce the classical numerical methods to solve fractional partial differential equations, in particular, the finite difference method. Finally, preparatory work is completed.In Chapter 2, a class of space fractional Schr(o|")dinger wave equations is considered. We first derive two conserved quantities (i.e., the mass and the energy) of the problems consid-ered, and then propose a three-level linear difference scheme and analyze its conservation and convergence. Finally, several numerical experiments confirm the validity of the scheme.In Chapter 3, we extend the ideas in above chapter to a class of strongly coupled space fractional Schr(o|")dinger equations. Similarly, we first give the laws of conservation of mass and energy, and then propose a nonlinear conservative difference scheme. We show that the scheme has unique solution and is unconditionally stable with respect to the initial values. Moreover, we discuss its convergence in the maximum norm. In order to improve the com-putational efficiency, a linear difference scheme with two identities is presented. Finally, several numerical experiments are provided to confirm the theoretical results.In Chapter 4, we consider the single and coupled time-space fractional Schr(o|")dinger equations. The corresponding Crank-Nicolson difference scheme and linearized difference scheme are developed. Then we analyze the local truncation errors and stabilities of both methods in detail. Numerical experiments for the single and coupled cases are carried out to demonstrate the efficiency of both schemes. The main significance of the work is that two new and effective numerical methods are provided to solve such problems, especially for the coupled problem.In Chapter 5, we focus our attention on the numerical solutions of a class of semi-linear space fractional damped wave equations in two space dimensions. The model has a wide range of application, such as the space fractional telegraph equation, sine-Gordon equation and Klein-Gordon equation can be regarded as particular cases of such equations. A compact difference scheme with accuracy of fourth-order in space and second-order in time is proposed. The solvability, stability and convergence of the scheme are shown under Lipschitz assumption. In order to reduce the computational burden, a compact alternating direction implicit difference scheme is established. Numerical results for the aforesaid three kinds of equations are provided to verify the theoretical analysis.In Chapter 6, we propose a class of high-order numerical methods for solving space-fractional diffusion equations by combining the fractional compact difference approxima-tion in space and boundary value methods in time. The suggested methods can arrive at fourth-order in space and fourth-order, fifth-order, sixth-order or even higher-order in time. Meanwhile, Strang-type, Chan-type and P-type preconditioners are introduced. When the used boundary value method is Ak1,k2-stable, it is proven that GMRES method correspond-ing to Strang-type preconditioner is rapidly convergent. The presented numerical experi-ments further verify the high-accuracy and efficiency of the suggested methods.In the last chapter, a brief conclusion is given and some future work is listed.