In this paper, we simultaneously reconstruct the source term and initial value in a time-space fractional diffusion equation in an unbounded domain based on measurements at two different times \(T_{1}\) and \(T_{2}\) . Through the Fourier transform, the exact solution of the direct problem is obtained and the inverse problem is transformed into an operator equation. To overcome the ill-posedness of the inverse problem, the Tikhonov regularization method is proposed, and the convergence estimates of the regularized solutions under the a priori and the a posteriori regularization parameter selection strategies are derived in detail. In order to further verify the rationality of the theoretical results, we conduct numerical tests. The experimental results show that the proposed method is feasible and effective for solving the simultaneous inversion problem of the source term and initial value in the time-space fractional diffusion equation.
