We study electrodynamic equations in which the dielectric permittivity and the conductivity of the medium possess “memory.”Owing to this property, the solution to the equations depends on the entire history of the wave propagation process.It is assumed that the kernels of the integral operators modeling the memory effect depend on both spatial and temporal variables, and these kernels can be represented as products of two functions, one depending on the spatial variables and the other on time.The functions depending on the temporal variable are assumed to be known, while those depending on the spatial variables are unknown and have to be determined.We assume that the functions $ p({\mathbf{x}}) $ and $ q({\mathbf{x}}) $ , which correspond to the kernels describing the memory properties of the dielectric permittivity and the conductivity, respectively, are compactly supported, and their supports lie inside a ball $ B $ of finite radius.To solve the inverse problem, we consider the direct problem with completely known kernels and its special solution for a homogeneous medium corresponding to a running delta-shaped wave propagating in the direction $ \nu $ .This wave impinges on an inhomogeneity localized in $ B $ , and on the boundary of this ball we measure the amplitude of the singular part of the solution and the amplitude of the first time derivative of its regular part on the wave front.The corresponding data recorded for various directions $ \nu $ constitute the input information for solving the inverse problem.It is shown that the problems of determining the functions $ p({\mathbf{x}}) $ and $ q({\mathbf{x}}) $ reduce to successive solutions of the well-known problem of X-ray tomography.Hence, the solution to the inverse problem under consideration is unique and can be efficiently found both analytically and numerically.
