Abstract
For a time-fractional wave equation with an integral term of the convolution type, we study the direct Cauchy problem and the inverse problem of finding a multidimensional kernel of the integral, depending not only on the time variable, but also on the first \(n - 1\) components of the spatial variable \(x = ({{x}_{1}},{{x}_{2}}, \ldots ,{{x}_{n}}) \in {{\mathbb{R}}^{n}}\) . In this case, the known parameters of the problems are the Cauchy data specified at time \(t = 0\) and the overdetermination condition on the hyperplane \({{x}_{n}} = 0.\) The problems are equivalently reduced to problems that are convenient for further study. Using the fundamental solution to the time-fractional wave operator, which contains the generalized hypergeometric Fox function, the solution to the direct problem is written in the form of an integral equation of Volterra type and its properties are studied. Using the results of the direct problem, the solution to the inverse problem is also represented as a nonlinear integral equation. By applying the contraction mapping principle to this equation, the local solvability of the problem is established.
