Quasi-two-dimensional coefficient inverse problem for the wave equation in a weakly horizontally inhomogeneous medium with memory

The paper studies the inverse problem of sequentially determining the two unknowns: the coefficient characterizing the properties of a medium with weakly horizontal inhomogeneity and the kernel of some integral operator describing the memory of the medium. The direct initial-boundary value problem contains zero data and the Neumann boundary condition. As additional information, the trace of the Fourier image of the direct problem solution at the boundary of the medium is given. To study inverse problems, it is assumed that the unknown coefficient decomposes into an asymptotic series. In this paper, a method is constructed for finding (taking into account the memory of the medium) the coefficient with accuracy $O(\epsilon^2)$. At the first stage, the solution of the direct problem in the zero approximation and the kernel of the integral operator are simultaneously determined. The inverse problem is reduced to solving a system of nonlinear Volterra integral equations of the second kind. At the second stage, the kernel is considered to be given, and the first approximation solution of the direct problem and the unknown coefficient are determined. In this case, the inverse problem and the problem of solving a linear system of Volterra integral equations of the second kind will be equivalent. Two theorems on unique local solvability of the inverse problems are proved. Numerical results on the kernel function and coefficient are presented.