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This talk is based on the papers [1] and [2]. In [1], an inverse source problem (ISP) for a time-fractional diffusion equation of order $\alpha \in (0,1)$ is discussed. The missing solely time-dependent source is recovered from an additional integral measurement. An additional challenge is that the coefficients of the elliptic operator considered are dependent on spatial and time variables. Two research questions are of concern in this talk: (i) the existence and uniqueness of a (weak) solution to the ISP for exact data, and (ii) the numerical reconstruction of the unknown source. Moreover, the extension of the results to multiterm time-fractional diffusion equation [2], and directions for future work will be discussed.

[1] Hendy, A. S. and Van Bockstal, K. On a reconstruction of a solely time-dependent source in a time-fractional diffusion equation with non-smooth solutions, J. Sci. Comput., vol. 90, Art. no. 1, 2022.

[2] Hendy, A. S. and Van Bockstal, K. A solely time-dependent source reconstruction in a multiterm time-fractional order diffusion equation with non-smooth solutions, Numer. Algorithms, vol. 90, Art. no. 2, 2022.