Reconstruction of the Source Term in a Time-Fractional Diffusion Equation from Partial Domain Measurement

Time-fractional diffusion equations draw attention of many mathematicians from various fields in the recent past because of their widely known applicable aspects. In this paper, a time-fractional inverse source problem is considered and analyzed through two interconnected streams for a broader understanding. Firstly, we establish the identifiability of this inverse problem by proving the existence of its unique solution with respect to the observed data inside the domain. Later on, in the second phase, our inverse problem is rewritten in its weaker form of a topology optimization problem involving a quadratic mismatch functional enhanced with a regularization term which penalizes the perimeter of the support of the source to be reconstructed. Existence of the minimizer is proved using the classical techniques of calculus of variations. To the end, a noniterative reconstruction algorithm is devised with the help of the topological derivative method. Finally, some numerical experiments are presented to support our findings. To conclude, few remarks are mentioned about the significance and benefits of the use of topological derivatives for the analysis of the problems as considered in this article.