Identifying the Anomalous Diffusion Process and Source Term in a Space-Time Fractional Diffusion Equation With Sturm-Liouville Operator

For a diffusion equation characterized by n is an element of & Nopf;$$ n\in \mathbb{N} $$ parameters as the order of fractional derivatives in time and one parameter for the spatial fractional derivative, we addressed the inverse problem of simultaneously determining the concentration function (i.e., the diffusion process) and a time-dependent source term. The proposed model incorporates features of both Riemann-Liouville and Caputo fractional derivatives, owing to the involvement of n$$ n $$ parameters in the time-fractional derivatives. To ensure the unique solvability of the inverse problem, an integral-type overspecified condition is considered. Eigenfunctions derived from the fractional-order Sturm-Liouville operator, subject to zero Dirichlet boundary conditions, are employed for the eigenfunction expansion of the solution. We have established results regarding the existence, uniqueness, and stability of the solution for the inverse problem.