On a fractional inverse source problem with mixed boundary conditions through a novel method

In this study, an inverse problem including the identification of a source function in a time-fractional diffusion equation with initial and mixed boundary conditions is investigated by employing the Laplace transform and the Daftardar-Gejji and Jafari method (DJM). Two kinds of inverse problems are considered, where the source function depends either on the temporal variable or the spatial variable. Initially, the Laplace and inverse Laplace transforms are applied to transform the fractional differential equation into an algebraic form. Subsequently, DJM is utilized to establish the solution of the converted problem by using the initial condition. Then, under the mixed boundary conditions, the unknown source function is explicitly determined. The suggested method's ability to obtain the source function under no need for extra conditions and overmeasured data is one of its noteworthy features. The Banach fixed-point theorem and the energy estimate for fractional partial differential equations are also used to theoretically support the existence, uniqueness, and stability of the solution.