Abstract
This article studies the inverse problem of determining a convolution kernel, which depends, in addition to the time variable, also on the first two coordinates of the spatial variable $$x=(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}$$ in the $$3D$$ diffusion equation with a fractional Caputo derivative in the time $$t$$ variable. As a overdetermination condition, the solution of the direct problem on the $$x_{3}=0$$ plane is specified. The problem reduce to the equivalent auxiliary problems with the fractional Riemann–Liouville operator in the main equation. At the beginning, a direct problem is studied, the solution of which, using the fundamental solution of the fractional diffusion operator, is represented in the form of an integral equation. Then the method of successive approximations is applied to this equation and it is proved that the direct problem has a regular solution. Further, inverse problem is equivalently replaced by an integral equation of the Volterra type of the second kind. We apply the contraction mapping principle to prove the existence of solutions of these equations. Local existence and global uniqueness theorems are proved. We also obtain a stability estimate for the solution of the inverse problem.