Identification of a space-dependent source term in a nonlocal problem for the general time-fractional diffusion equation

Abstract

Abstract The diffusion equation with a general convolutional derivative in time is considered on a bounded domain, as one of the boundary conditions is nonlocal. We are concerned with the inverse source problem of recovery of a space-dependent source term from given final time data. To find the source term and the solution, we resort to generalized eigenfunction expansion, using a bi-orthogonal pair of bases. Estimates for the time-dependent components in the spectral expansions are established and applied to prove uniqueness and existence in the classical sense. Analytical and numerical examples are provided.