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Facing application in real world, a simultaneous identification problem of determining the initial diffusion time (or the length of diffusion time) and source term in a time-fractional diffusion equation is investigated. Firstly, the simultaneous reconstruction problem is proposed by translating the Caputo fractional derivative. Then the uniqueness results for the simultaneous identification problem are proven by the technique of analytic continuation and the Laplace transformation method. Next, the Lipschitz continuousness of the observation operator is derived, and an alternating direction inversion algorithm is proposed to solve the simultaneous identification problem. At last, several numerical examples are computed to show the efficiency and stability of the reconstruction algorithm.

On the simultaneous reconstruction of the initial diffusion time and source term for the time-fractional diffusion equation