Regularization of the Final Value Problem for the Time-Fractional Diffusion Equation

We consider the backward problem of reconstructing the initial condition of a nonhomogeneous time-fractional diffusion equation from final measurements. The proposed method relies on the eigenfunction expansion of the forward solution and the Tikhonov regularization to control the instability of the underlying inverse problem. We establish stability results and we provide convergence rates under a priori and a posteriori parameter choice rules. The resulting algorithm is robust and computationally inexpensive. Two examples are included to illustrate the effectiveness and accuracy of the proposed method.