Direct and Inverse Problems for Nonlocal Diffusion Equation in ℝ3

ABSTRACT We considered the direct problem (DP) and inverse problem (IP) of determining a reaction coefficient of the diffusion equation involving Dzherbashian‐Nersesian integro‐differential operator (DNIDO). The solutions of direct and IPs are written in the form of Fox H‐function. The reaction coefficient is supposed to depend only on the first two components of the spatial variable and on the time variable . The over‐specified condition for the IP is the projection of diffusion concentration in the ‐plane. The IP is reduced to an equivalent integral equation (IE), and existence and uniqueness for the solution of the IP is proved by applying the Banach fixed point theorem (BFPT). The stability of the solution of the direct and IPs, when diffusion coefficient, source term, and initial data are perturbed, is proved and some examples are provided. We have generalized our main problem from to . Some numerical examples with an exponentially decaying source and periodic reaction coefficient term are presented.