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For a space-time fractional diffusion equation, an inverse problem of determination of a space dependent source term along with the solution is considered. The fractional derivatives in time and space are defined in the sense of Caputo. Due to an over-specified data at final time say T, we proved that there exists a unique solution of the inverse source problem. We use the eigenfunction expansion method to prove our main results. Several special cases of space-time fractional diffusion equations are discussed and results are interpolated from generalized results. Some examples are provided.

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An inverse problem of determining a time‐dependent source term from the total energy measurement of the system (the over‐specified condition) for a space‐time fractional diffusion equation is considered. The space‐time fractional diffusion equation is obtained from classical diffusion equation by replacing time derivative with fractional‐order time derivative and Sturm‐Liouville operator by fractional‐order Sturm‐Liouville operator. The existence and uniqueness results are proved by using eigenfunction expansion method. Several special cases are discussed, and particular examples are provided.

Inverse problem for a family of multi-term time fractional differential equation with non-local boundary conditions is studied. The spectral operator of the considered problem is non-self-adjoint and a bi-orthogonal set of functions is used to construct the solution. The operational calculus approach has been used to obtain the solution of the multi-term time fractional differential equations. Integral type over-determination condition is considered for unique solvability of the inverse problem. Different estimates of multinomial Mittag-Leffler functions alongside Banach fixed point theorem are used to prove the unique local existence of the solution of the inverse problem. Stability of the solution of the inverse problem on the given datum is established.