On the solvability of direct and inverse problems for a generalized diffusion equation

Abstract

This paper investigates direct and two inverse source problems for a diffusion equation involving integral convolution in time, subject to non-classical boundary conditions. The inverse source problems are proved to be ill-posed in the sense of Hadamard. A bi-orthogonal system of functions is used to write the series solutions of the inverse source problems. The unique existence of series solutions is obtained by imposing certain conditions on the given data. Some specific cases for the diffusion equation are given depending on the type of memory kernel. Furthermore, some particular examples are also formulated to illustrate the obtained results for the inverse source problems.