We present the inverse problem of successive determination of the two unknowns that are a coefficient characterizing the properties of a medium with weakly horizontal inhomogeneity and the kernel of an integral operator describing the memory of the medium. The direct initial-boundary value problem involves the zero data and the Neumann boundary condition. The trace of the Fourier image of a solution to the direct problem on the boundary of the medium serves as additional information. Studying inverse problems, we assume that the unknown coefficient is expanded into an asymptotic series in powers of a small parameter. Also, we construct some method for finding the coefficient that accounts for the memory of the environment to within an error of order \( O(\varepsilon^{2}) \). At the first stage, we determine a solution to the direct problem in the zero approximation and the kernel of the integral operator, while the inverse problem reduces to an equivalent problem of solving the system of nonlinear Volterra integral equations of the second kind. At the second stage, we consider the kernel given and recover a solution to the direct problem in the first approximation and the unknown coefficient. In this case, the solution to the equivalent inverse problem agrees with a solution to the linear system of Volterra integral equations of the second kind. We prove some theorems on the unique local solvability of the inverse problems and present the results of numerical calculations of the kernel and the coefficient.

我们提出了连续确定两个未知数的逆问题，这两个未知数是表征具有弱水平不均匀性的介质的属性的系数和描述介质的记忆的积分算子的核。直接初边值问题涉及零数据和诺伊曼边界条件。介质边界上的直接问题的解的傅里叶图像的轨迹用作附加信息。研究反问题时，我们假设未知系数展开为小参数幂的渐近级数。此外，我们构建了一些方法来查找解释环境记忆的系数，使其误差在\( O(\varepsilon^{2}) \) 的范围内 。第一阶段，我们确定零逼近和积分算子核中正问题的解，而反问题则简化为求解第二类非线性Volterra积分方程组的等价问题。在第二阶段，我们考虑给定的核并恢复第一近似中的直接问题和未知系数的解。在这种情况下，等效反问题的解与第二类 Volterra 积分方程的线性系统的解一致。我们证明了反问题唯一局部可解性的一些定理，并给出了核和系数的数值计算结果。