Numerical inversions for space-dependent diffusion coefficient in the time fractional diffusion equation

Abstract.

This paper deals with numerical inversion for space-dependent diffusion coefficient in a one-dimensional time fractional diffusion equation with additional boundary measurement. An implicit difference scheme for the forward problem is presented based on discretization of Caputo fractional derivative, and numerical stability and convergence of the linear system are proved with the help of matrix analysis. An optimal perturbation regularization algorithm is introduced to determine the space-dependent diffusion coefficient numerically in different approximate space, and numerical inversions are carried out by several numerical examples. Furthermore, impacts of the fractional order, approximate space, regularization parameter, numerical differential step, and initial iterations on the inversion algorithm are analyzed showing that the inversion is of numerical uniqueness, and the inversion algorithm is efficient at least to the inverse problem here. The order of fractional derivative not only represents a global property of the forward problem, but also indicates ill-posedness of the corresponding inverse problem.