Initial boundary value problems for a multi-term time fractional diffusion equation with generalized fractional derivatives in time

For a multi-term time-fractional diffusion equation comprising Hilfer fractional derivatives in time variables of different orders between 0 and 1, we have studied two problems (direct problem and inverse source problem). The spectral problem under consideration is self-adjoint. The solution to the given direct and inverse source problems is formulated utilizing the spectral problem. For the solution of the given direct problem, we proposed existence, uniqueness, and stability results. The existence, uniqueness, and consistency effects for the solution of the given inverse problem were addressed, as well as an inverse source for recovering space-dependent source term at certain T. For the solution of the challenges, we proposed certain relevant cases.