Abstract:
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.
Citation:
H. H. Turdiev, “Inverse coefficient problems for a time-fractional wave equation with the generalized Riemann–Liouville time derivative”, Izv. Vyssh. Uchebn. Zaved. Mat., 2023, no. 10, 46–59
\Bibitem{Tur23}
\by H.~H.~Turdiev
\paper Inverse coefficient problems for a time-fractional wave equation with the generalized Riemann--Liouville time derivative
\jour Izv. Vyssh. Uchebn. Zaved. Mat.
\yr 2023
\issue 10
\pages 46--59
\mathnet{http://mi.mathnet.ru/ivm9940}
\crossref{https://doi.org/10.26907/0021-3446-2023-10-46-59}
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https://www.mathnet.ru/eng/ivm9940
https://www.mathnet.ru/eng/ivm/y2023/i10/p46
This publication is cited in the following 10 articles:
Jonibek J. Jumaev, “NUMERICAL ANALYSIS OF INVERSE PROBLEMS FOR THE DIFFUSION EQUATION WITH INITIAL-BOUNDARY AND OVERDETERMINATION CONDITIONS”, J Math Sci, 2025
D. K. Durdiev, T. R. Suyarov, Kh. Kh. Turdiev, “Obratnaya koeffitsientnaya zadacha dlya drobnogo telegrafnogo uravneniya s sootvetstvuyuschei drobnoi proizvodnoi po vremeni”, Izv. vuzov. Matem., 2025, no. 2, 39–52
D. K. Durdiev, D. A. Toshev, H. H. Turdiev, “Determining a Source Function in the Mixed Parabolic–Hyperbolic Equation with Characteristic Type Change Line”, Lobachevskii J Math, 45:3 (2024), 1032
D. K. Durdiev, H. H. Turdiev, “Determining of a Space Dependent Coefficient of Fractional Diffusion Equation with the Generalized Riemann–Liouville Time Derivative”, Lobachevskii J Math, 45:2 (2024), 648
D. K. Durdiev, J. J. Jumaev, “Recovering Source Function and Kernel for a Time-fractional Diffusion Equation in the Bounded Domain”, Lobachevskii J Math, 45:4 (2024), 1691
U. D. Durdiev, A. A. Rakhmonov, “Obratnaya zadacha dlya differentsialnogo uravneniya chetvertogo poryadka s drobnym operatorom Kaputo”, Izv. vuzov. Matem., 2024, no. 9, 22–33
D. K. Durdiev, I. I. Hasanov, “Inverse coefficient problem for a partial differential equation with multi-term orders fractional Riemann–Liouville derivatives”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 34:3 (2024), 321–338
D. K. Durdiev, “Initial Boundary Value and Inverse Coefficient Problems for One-Dimensional Fractional Diffusion Equation in a Half-Line”, Lobachevskii J Math, 45:7 (2024), 3265
H. H. Turdiev, “Solvability Cauchy Problem for Time-Space Fractional Diffusion-Wave Equation with Variable Coefficient”, Lobachevskii J Math, 45:10 (2024), 5281
U. D. Durdiev, “Global Solvability of the Kernel Identification Problem for the Integro-Differential Equation of Beam Vibrations”, Lobachevskii J Math, 45:11 (2024), 5802
Citation in format AMSBIB
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