An Inverse Problem for a Non-Homogeneous Time-Space Fractional Equation

We consider the inverse problem of finding the solution of a generalized time-space fractional equation and the source term knowing the spatial mean of the solution at any times t is an element of (0, T], as well as the initial and the boundary conditions. The existence and the continuity with respect to the data of the solution for the direct and the inverse problem are proven by Fourier's method and the Schauder fixed-point theorem in an adequate convex bounded subset. In the published articles on this topic, the incorrect use of the estimates in the generalized Mittag-Leffler functions is commonly performed. This leads to false proofs of the Fourier series' convergence to recover the equation satisfied by the solution, the initial data or the boundary conditions. In the present work, the correct framework to recover the decay of fractional Fourier coefficients is established; this allows one to recover correctly the initial data, the boundary conditions and the partial differential equations within the space-time domain.