An inverse problem of determining a time dependent source term along with diffusion/temperature concentration from a non-local over-specified condition for a space-time fractional diffusion equation is considered. The space-time fractional diffusion equation involve Caputo fractional derivative in space and Hilfer fractional derivatives in time of different orders between 0 and 1. Under certain conditions on the given data we proved that the inverse problem is locally well-posed in the sense of Hadamard. Our method of proof based on eigenfunction expansion for which the eigenfunctions (which are Mittag-Leffler functions) of fractional order spectral problem and its adjoint problem are considered. Several properties of multinomial Mittag-Leffler functions are proved.

We considered two inverse source problems for a space–time fractional differential equation. Firstly recovery of a space dependent source term is studied, secondly determination of a time dependent source term is considered. A bi-orthogonal system of functions consisting of Mittag–Leffler type functions, obtained from fractional order spectral and its adjoint problem, is used to construct the solutions of the inverse source problems. Existence, uniqueness and stability results are presented for inverse problem of recovery of a space dependent source term while for time dependent source term existence and uniqueness results are proved. Some special cases for these inverse source problems are discussed.

We consider a diffusion equation involving fractional derivative in time of order ( ) with a nonlocal boundary condition involving a parameter . A bi-orthogonal system of functions constructed from two Riesz basis of is used to prove the existence and uniqueness of classical solution of the direct problem. The inverse problem of determination of the temperature distribution and the unknown source term is considered. The inverse problem is proved to be well-posed in the sense of Hadamard whenever an overdetermination condition of the final temperature is given.

This paper focuses on considering two inverse source problems (ISPs) for a multi-term time-fractional evolution equation with an involution term, interpolating the heat and wave equations. The fractional derivatives are defined in Caputo's sense. The ISPs are proved to be ill-posed in the sense of Hadamard. Recovering a space dependent source term from over-specified data given at some time constitute the first ISP, while in the second ISP determination of a time dependent component of the source term is considered when over-specified condition of integral type is given. The solution of ISPs are constructed by using Fourier's method. The time-dependent components of the solutions are presented in terms of the multinomial Mittag-Leffler function. Under certain conditions, the solutions of ISPs for the multi-term time-fractional evolution equation are shown to be classical solutions. In addition, some particular examples are formulated to illustrate the obtained results for the ISPs.