Two regularization methods to identify a space-dependent source for the time-fractional diffusion equation

In this paper, the inverse problem of identifying a space-dependent source for the time-fractional diffusion equation is investigated. Such a problem is obtained from the classical diffusion equation in which the time derivative is replaced with a Caputo derivative of order α ∈ ( 0 , 1 ] . We show that such a problem is ill-posed and apply the Tikhonov regularization method and a simplified Tikhonov regularization method to solve it based on the solution given by the separation of variables. Convergence estimates are presented under an a priori parameter choice rule and an a posteriori parameter choice rule, respectively. Finally, numerical examples are given to show that the regularization methods are effective and stable.