Geodesic
Inverse coefficient problems for a time-fractional wave equation with the generalized Riemann--Liouville time derivative
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 10 (2023), pp. 46-59.

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This paper considers the inverse problem of determining the time-dependent coefficient in the fractional wave equation with Hilfer derivative. In this case, the direct problem is initial-boundary value problem for this equation with Cauchy type initial and nonlocal boundary conditions. As overdetermination condition nonlocal integral condition with respect to direct problem solution is given. By the Fourier method, this problem is reduced to equivalent integral equations. Then, using the Mittag-Leffler function and the generalized singular Gronwall inequality, we get apriori estimate for solution via unknown coefficient which we will need to study of the inverse problem. The inverse problem is reduced to the equivalent integral of equation of Volterra type. The principle of contracted mapping is used to solve this equation. Local existence and global uniqueness results are proved.
Keywords: fractional derivative, Riemann–Liouville fractional integral, inverse problem, integral equation, Fourier series, Banach fixed point theorem. 
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H. H. Turdiev. Inverse coefficient problems for a time-fractional wave equation with the generalized Riemann--Liouville time derivative. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 10 (2023), pp. 46-59. http://geodesic.mathdoc.fr/item/IVM_2023_10_a3/

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