Frechet Differentiability and The Gradient of The Cost Functional Of An Inverse Source Problem

The problem of determining the source term f = f(x) in a time fractional diffusion equation from the measured data at the final time is formulated. To this end, a methodology involving minimization of the cost functional is applied and proved that the Fréchet derivative ofthe cost functional can be formulated via the solution of an adjoint prob_x0002_lem. The obtained results permit one to prove existence and uniqueness of a quasi solution of the considered inverse problems, as well as to construct a monotone iteration scheme based on a gradient-type method.