Fourier Regularization for a Time-Fractional Source Problem

Abstract

We study the inverse problem of identifying an unknown time-dependent source in a linear time-fractional diffusion equation, where the time derivative is taken in the Caputo sense of fractional order α . The problem is ill-posed in the sense of Hadamard: the solution does not depend continuously on the measured data, so a stable reconstruction requires regularization. We propose a Fourier truncation regularization method, where a cut-off frequency serves as the regularization parameter, and analyze both an a priori and an a posteriori (discrepancy-principle) choice of this parameter. For each choice we establish H"older-type convergence estimates, confirmed by numerical experiments.