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 problems are the Cauchy data specified at time $t=0$ and the redefinition condition on the hyperplane $x_n=0.$ Problems are equivalently reduced to problems that are convenient for further study. Using the fundamental solution of 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.
Keywords:Gerasimov-Caputo fractional derivative, Fox $H$-function, Mittag–Leffler function, integral equation, fixed point theorem, existence, uniqueness, stability.
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