This paper devoted to the obtaining the explicit solution of $n$-dimensional wave equation with Gerasimov–Caputo fractional derivative in the infinite domain with non-zero initial condition and vanishing condition at infinity. It is shown that this equation can be derived from the classical homogeneous hyperbolic integro-differential equation with memory in which the kernel is $t^{1-\alpha}E_{2-\alpha, 2-\alpha}(-t^{2-\alpha}), \ \alpha\in(1, 2),$ where $E_{\alpha, \beta}$ is the Mittag-Liffler function. Based on Laplace and Fourier transforms the properties of the Fox H-function and convolution theorem, explicit solution for the solution of the considered problem is obtained.

We consider the problems of simultaneously identifying two unknowns. The wave propagation velocity and the memory of the layered medium will be determined. For their determination, two observations are used for how the boundary of the domain fluctuates. Main results are stability estimates and uniqueness theorems for the problems under consideration.

In this paper, we consider two dimensional inverse problem for a fractional diffusion equation. The inverse problem is reduced to the equivalent integral equation. For solving this equation the contracted mapping principle is applied. The local existence and global uniqueness results are proven. Also the stability estimate is obtained.