Preconditioners For The Linear Systems Arising From The Fractional Diffusion Equations

Fractional derivative and fractional integral have a very wide range of applications in many areas such as physics, biology, chemistry, image processing, and even finance. Particularly, fractional diffusion equation can model the phenomena exhibiting anoma-lous diffusion that can not be modeled accurately by the second-order diffusion equa-tions. As a closed-form analytical solution is usually not available, numerical meth-ods have become an important means for computing the approximate solution. With a proper finite difference discretization, a system of linear equations is obtained, which is then be solved numerically.In this thesis, we consider the preconditioned iterative methods for two class of fractional diffusion equations with different boundary conditions, including the steady-state and time-dependent cases. Our main work is as follow:(1) For the fractional diffusion equation with Dirichlet boundary condition, by trans-forming the resulted system of linear equations into an equivalent block 2-by-2 linear system, we proposed the block triangular splitting iterative method. The convergence of such method is proved under suitable conditions. Meanwhile, we also studied the corresponding preconditioner, and numerical experiments are car-ried out to show the performance of the new preconditioner.(2) For the fractional diffusion equation with fractional Neumann boundary condi-tion, the coefficient matrix of the resulted system of linear equations is a sum of a diagonal matrix, a Toeplitz matrix and a rank-1 matrix. We proposed the block triangular preconditioner for the equivalent block 2-by-2 linear system. Numerical results show that the proposed preconditioner is quite efficient.