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This topic is concerned with inverse source problems related to fractional operators. Three problems have been addressed in this research study.

The first one is related to the fractional Laplacian operator. The aim is to reconstruct an unknown source term from internal noisy measured data. The leading term of the mathematical model equation is governed by the fractional spectral Laplacian. The inverse problem is formulated as a regularized optimiza- tion one minimizing a least square type functional. The existence, uniqueness and stability of the unknown source term have been established. In the numer- ical part, we develop a numerical reconstruction approach for identifying the unknown source term and solving the inverse problem.

The second inverse problem is governed by a time-fractional diffusion equa- tion. It consists in identifying an unknown source term support from boundary measurements of the potential field. In this study, we have proposed a fast and accurate approach combining the robustness of the Kohn-Vogelius formulation and the rapidity of the topological gradient method. In the theoretical part of this work, We have derived a topological asymptotic expansion, with respect to a small geometric perturbation of the source term, valid for a large class of source functions. In the numerical part, we have developed a reconstruction algorithm for solving the considered geometric inverse source problem and iden- tifying the location, size and shape of the unknown sub-domains.

The third inverse source problem is related to a space-time fractional diffu- sion equation. Our aim is to identify an unknown source term from partially observed data. The employed model involves the Caputo fractional derivative in time and the non-local fractional Laplacian operator in space. The well- posedness of the forward problem is discussed. The considered ill-posed inverse source problem is formulated as a minimization one. The existence, uniqueness and stability of the solution of the minimization problem are examined. An iter- ative process is developed for identifying the unknown source term. A numerical implementation of the proposed approach is performed. The convergence of the discretized fractional derivatives is analyzed. The efficiency and accuracy of the proposed identification algorithm are confirmed by some numerical experiments.