Determination of Initial Distribution for a Space-Fractional Diffusion Equation with Time-Dependent Diffusivity

In the present paper, we devote our aspiration to some initial and final value problems for a class of space-fractional diffusion equation with time-dependent diffusivity factor. For the initial value problem (IVP), we investigate the stability of the solution concerning the data and the fractional order. For the final value problem, we prove the ill-posedness and suggest a filter method to regularize the problem. Explicit convergence rate of Hölder type is established. Finally, several numerical examples based on the finite difference approximation and the discrete Fourier transform are performed to demonstrate the effectiveness of the proposed method.