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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. [ABSTRACT FROM PUBLISHER]

This paper deals with an inverse problem of determining a diffusion coefficient and a spatially dependent source term simultaneously in one-dimensional (1-D) space fractional advection–diffusion equation with final observations using the optimal perturbation regularization algorithm. An implicit finite difference scheme for solving the forward problem is set forth, and a fine estimation to the spectrum radius of the coefficient matrix of the difference scheme is given with which unconditional stability and convergence are proved. The simultaneous inversion problem is transformed to a minimization problem, and existence of solution to the minimum problem is proved by continuity of the input–output mapping. The optimal perturbation algorithm is introduced to solve the inverse problem, and numerical inversions are performed with the source function taking on different forms and the diffusion coefficient taking on different values, respectively. The inversion solutions give good approximations to the exact solutions demonstrating that the optimal perturbation algorithm with the Sigmoid-type regularization parameter is efficient for the simultaneous inversion problem in the space fractional diffusion equation. [ABSTRACT FROM AUTHOR]