Kernel identification problem in a time-fractional wave equation

In this paper, the inverse problem of determining convolution kernel in the time-fractional wave equation with the Caputo derivative is studied. To express the solution of the Cauchy problem, the fundamental solution of the corresponding equation is systematically formulated, with a detailed investigation into the properties of this solution. The fundamental solution contains a Fox's function, which is widely used in the theory of diffusion-wave equation. Using the formulas of asymptotic expansions for the fundamental solution and its derivatives, an estimate for the solution of the direct problem is obtained. A priori estimate contains the norm of the unknown kernel function and it was used for studying the inverse problem. The inverse problem is reduced to the equivalent integral equation, By the fixed point argument in suitable functional classes the local solvability is proven. The global uniqueness results and also the stability estimate for solution to the inverse problem are established.