Abstract
The inverse problem of determining the solution and the kernel of the integral term in an inhomogeneous two-dimensional integrodifferential wave equation in a rectangular domain is considered. First, the uniqueness of the solution of the direct problem is established using the completeness of the eigenfunction system of the corresponding homogeneous Dirichlet problem for the two-dimensional Laplace operator, and the existence of a solution of the direct problem is proved. Using additional information about the solution of the direct problem, we obtain a Volterra integral equation of the second kind for the kernel of the integral term. The existence and uniqueness of a solution of this equation is proved by the contraction mapping method in the space of continuous functions with a weighted norm.
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References
A. H. Hasanov and V. G. Romanov, Introduction to Inverse Problems for Differential Equations (Springer, Cham, 2017).
V. Isakov, Inverse Problems for Partial Differential Equations, in Appl. Math. Sci. (Springer, New York, 2006), Vol. 127.
A. A. Samarskii and P. N. Vabishchevich, Numerical Methods for Solving Inverse Problems of Mathematical Physics, in Inverse and Ill-posed Probl. Ser. (de Gruyter, Berlin, 2007), Vol. 52.
A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, in Appl. Math. Sci. (Springer, Cham, 2021), Vol. 120.
D. Lesnic, Inverse Problems with Aaplications in Science and Engineering (CRC Press, Boca Raton, FL, 2022).
S. I. Kabanikhin, Inverse and Ill-Posed Problems (Sibirsk. Nauchn. Izd., Novosibirsk, 2009) [in Russian].
A. Lorenzi and E. Sinestrari, “Stability results for a partial integrodifferential inverse problem,” in Volterra Integrodifferential Equations in Banach Spaces and Applications (Trento, 1987), Pitman Res. Notes Math. Ser., (Longman, Harlow, 1989), Vol. 190, pp. 271–294.
A. Lorenzi and E. Paparoni, “Direct and inverse problems in the theory of materials with memory,” Rend. Sem. Mat. Univ. Padova 87, 105–138 (1992).
A. Lorenzi, “An identification problem related to a nonlinear hyperbolic integro-differential equation,” Nonlinear Anal. 22 (1), 21–44 (1994).
Zh. Sh. Safarov and D. K. Durdiev, “Inverse problem for an integro-differential equation of acoustics,” Differ. Equ. 54 (1), 134–142 (2018).
J. Sh. Safarov, “Global solvability of the one-dimensional inverse problem for the integro-differential equation of acoustics,” J. Sib. Fed. Univ. Math. Phys. 11 (6), 753–763 (2018).
V. G. Romanov, “On the determination of the coefficients in the viscoelasticity equations,” Siberian Math. J. 55 (3), 503–510 (2014).
D. K. Durdiev and Zh. Sh. Safarov, “Inverse problem of determining the one-dimensional kernel of the viscoelasticity equation in a bounded domain,” Math. Notes 97 (6), 867–877 (2015).
Zh. D. Totieva and D. K. Durdiev, “The problem of finding the one-dimensional kernel of the thermoviscoelasticity equation,” Math. Notes 103 (1), 118–132 (2018).
D. K. Durdiev and Z. D. Totieva, “The problem of determining the one-dimensional matrix kernel of the system of viscoelasticity equations,” Math. Methods Appl. Sci. 41 (17), 8019–8032 (2018).
D. Guidetti, “Reconstruction of a convolution kernel in a parabolic problem with a memory term in the boundary conditions,” in Bruno Pini Mathematical Analysis Seminar, Bruno Pini Math. Anal. Semin. (Univ. Bologna, Bologna, 2013), Vol. 4, pp. 47–55.
C. Cavaterra and D. Guidetti, “Identification of a convolution kernel in a control problem for the heat equation with a boundary memory term,” Ann. Mat. Pura Appl. (4) 193 (3), 779–816 (2014).
J. Janno and L. von Wolfersdorf, “Inverse problems for identification of memory kernels in viscoelasticity,” Math. Methods Appl. Sci. 20 (4), 291–314 (1997).
D. K. Durdiev and A. A. Rahmonov, “The problem of determining the 2D-kernel in a system of integro-differential equations of a viscoelastic porous medium,” J. Appl. Industr. Math. 14 (2), 281–295 (2020).
D. K. Durdiev and J. Sh. Safarov, “2D kernel identification problem in viscoelasticity equation with a weakly horizontal homogeneity,” Sib. Zh. Ind. Mat. 25 (1), 14–38 (2022).
D. K. Durdiev and A. A. Rahmonov, “The problem of determining the 2D-kernel in a system of integro-differential equations of a viscoelastic porous medium,” J. Appl. Industr. Math. 14 (2), 281–295 (2020).
A. L. Karchevsky and A. G. Fatianov, “Numerical solution of the inverse problem for a system of elasticity with the aftereffect for a vertically inhomogeneous medium,” Sib. Zh. Vychisl. Mat. 4 (3), 259–268 (2001).
U. D. Durdiev, “Numerical method for determining the dependence of the dielectric permittivity on the frequency in the equation of electrodynamics with memory,” Sib. Èlektron. Mat. Izv. 17, 179–189 (2020).
Z. R. Bozorov, “Numerical determining a memory function of a horizontally-stratified elastic medium with aftereffect,” Eurasian J. Math. Comp. Appl. 8 (2), 4–16 (2020).
K. B. Sabitov and A. R. Zainullov, “Inverse problems for a two-dimensional heat equation with unknown right-hand side,” Russian Math. (Iz. VUZ) 65 (3), 75–88 (2021).
A. N. Kolmogorov and S. V. Fomin, Elements of Function Theory and Functional Analysis (Nauka, Moscow, 1976) [in Russian].
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Translated from Matematicheskie Zametki, 2023, Vol. 114, pp. 244–259 https://doi.org/10.4213/mzm13686.
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Durdiev, D.K., Safarov, J.S. Inverse Problem for an Integrodifferential Equation of the Hyperbolic Type in a Rectangular Domain. Math Notes 114, 199–211 (2023). https://doi.org/10.1134/S0001434623070210
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DOI: https://doi.org/10.1134/S0001434623070210