Simultaneous determination of the zeroth-order coefficient and source term in a time-fractional diffusion equation

This paper investigates the simultaneous inversion problem of recovering the zeroth-order coefficient and source term in a time-fractional diffusion equation based on nonlocal integral observations. Compared with the linear simultaneous inversion problem, this problem is more difficult to solve. In terms of theoretical analysis, we define the weak solution for the forward problem and establish its existence and uniqueness, and the ill-posedness of the simultaneous inversion problem is analyzed. To address this ill-posed problem, we employ the Tikhonov regularization method to reformulate it as a variational problem. The existence of a minimizer for the variational functional is rigorously proved, and convergence rate estimates are derived under both the a priori and a posteriori regularization parameter choice strategies, subject to an appropriate source condition. For the numerical implementation, by introducing the sensitivity problems and the adjoint problem, we obtain the Fréchet gradients of the functional with respect to the zeroth-order coefficient and the source term. The variational problem is then solved using the conjugate gradient algorithm. Finally, several numerical examples are provided to demonstrate the feasibility and effectiveness of the proposed method.