Numerical Solution to the Space-Time Fractional Diffusion Equation and Inversion for the Space-Dependent Diffusion Coefficient

This paper deals with numerical solution for the space-time fractional diffusion equation with variable diffusion coefficient, and numerical inversion for the space-dependent diffusion coefficient by the homotopy regularization algorithm. An equivalent system to the forward problem is deduced by utilizing properties of the fractional derivatives, and an implicit finite difference scheme for solving the forward problem is set forth, and its stability and convergence are proved based on estimation to the spectrum radius of the coefficient matrix. The homotopy regularization algorithm is introduced to solve the inverse problem, and numerical inversions are performed with noisy data. The inversion solutions give good approximations to the exact solution as the noise level goes to small demonstrating a numerical stability of the inverse problem here.