Inverse Problem for an Integrodifferential Equation of the Hyperbolic Type protect in a Rectangular Domain

D. K. Durdiev ; J. Sh. Safarov ; J. Sh. Safarov

Matematičeskie zametki, Tome 114 (2023) no. 2, pp. 244-259.

Résumé

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.

Détail

Keywords: integrodifferential equation, integral kernel, Fourier method, eigenfunctions, eigenvalues, Banach theorem.

Bibliographie

[1] A. H. Hasanov, V. G. Romanov, Introduction to Inverse Problems for Differential Equations, Springer, Cham, 2017 | MR

[2] V. Isakov, Inverse Problems for Partial Differential Equations, Appl. Math. Sci., 127, Springer, New York, 2006 | MR

[3] A. A. Samarskii, P. N. Vabishchevich, Numerical Methods for Solving Inverse Problems of Mathematical Physics, Inverse and Ill-posed Probl. Ser., 52, de Gruyter, Berlin, 2007 | MR

[4] A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Appl. Math. Sci., 120, Springer, Cham, 2021 | MR

[5] D. Lesnic, Inverse Problems with Aaplications in Science and Engineering, CRC Press, Boca Raton, FL, 2022 | MR

[6] S. I. Kabanikhin, Obratnye i nekorrektnye zadachi, Sibirskoe nauchnoe izd-vo, Novosibirsk, 2009 | MR

[7] A. Lorenzi, E. Sinestrari, “Stability results for a partial integrodifferential inverse problem”, Volterra Integrodifferential Equations in Banach Spaces and Applications (Trento, 1987), Pitman Res. Notes Math. Ser., 190, Longman, Harlow, 1989, 271–294 | MR

[8] A. Lorenzi, E. Paparoni, “Direct and inverse problems in the theory of materials with memory”, Rend. Sem. Mat. Univ. Padova, 87 (1992), 105–138 | MR

[9] A. Lorenzi, “An identification problem related to a nonlinear hyperbolic integro-differential equation”, Nonlinear Anal., 22:1 (1994), 21–44 | DOI | MR

[10] Z. Sh. Safarov, D. K. Durdiev, “Inverse problem for an integro-differential equation of acoustics”, Differ. Equ., 54:1 (2018), 134–142 | DOI | MR

[11] 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 (2018), 753–763 | DOI | MR

[12] V. G. Romanov, “Ob opredelenii koeffitsientov v uravneniyakh vyazkouprugosti”, Sib. matem. zhurn., 55:3 (2014), 617–626 | MR

[13] D. K. Durdiev, Zh. Sh. Safarov, “Obratnaya zadacha ob opredelenii odnomernogo yadra uravneniya vyazkouprugosti v ogranichennoi oblasti”, Matem. zametki, 97:6 (2015), 855–867 | DOI | MR

[14] Zh. D. Totieva, D. K. Durdiev, “Zadacha ob opredelenii odnomernogo yadra uravneniya termovyazkouprugosti”, Matem. zametki, 103:1 (2018), 129–146 | DOI | MR

[15] D. K. Durdiev, Z. D. Totieva, “The problem of determining the one-dimensional matrix kernel of the system of viscoelasticity equations”, Math. Methods Appl. Sci., 41:17 (2018), 8019–8032 | DOI | MR

[16] D. Guidetti, “Reconstruction of a convolution kernel in a parabolic problem with a memory term in the boundary conditions”, Bruno Pini Mathematical Analysis Seminar, Bruno Pini Math. Anal. Semin., 4, Univ. Bologna, Bologna, 2013, 47–55 | MR

[17] C. Cavaterra, 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 (2014), 779–816 | DOI | MR

[18] J. Janno, L. von Wolfersdorf, “Inverse problems for identification of memory kernels in viscoelasticity”, Math. Methods Appl. Sci., 20:4 (1997), 291–314 | MR

[19] D. K. Durdiev, A. A. Rakhmonov, “Zadacha ob opredelenii dvumernogo yadra v sisteme integrodifferentsialnykh uravnenii vyazkouprugoi poristoi sredy”, Sib. zhurn. industr. matem., 23:2 (2020), 63–80 | DOI

[20] D. K. Durdiev, Zh. Sh. Safarov, “Zadacha ob opredelenii dvumernogo yadra uravneniya vyazkouprugosti so slabo gorizontalnoi neodnorodnostyu”, Sib. zhurn. industr. matem., 25:1 (2022), 14–38 | DOI

[21] D. K. Durdiev, A. A. Rakhmonov, “Zadacha ob opredelenii dvumernogo yadra v sisteme integrodifferentsialnykh uravnenii vyazkouprugoi poristoi sredy”, Sib. zhurn. industr. matem., 23:2 (2020), 63–80 | DOI

[22] A. L. Karchevskii, A. G. Fatyanov, “Chislennoe reshenie obratnoi zadachi dlya sistemy uprugosti s posledeistviem dlya vertikalno neodnorodnoi sredy”, Sib. zhurn. vychisl. matem., 4:3 (2001), 259–268 | Zbl

[23] U. D. Durdiev, “Chislennoe opredelenie zavisimosti dielektricheskoi pronitsaemosti sloistoi sredy ot vremennoi chastoty”, Sib. elektron. matem. izv., 17 (2020), 179–189 | DOI

[24] Z. R. Bozorov, “Numerical determining a memory function of a horizontally-stratified elastic medium with aftereffect”, Eurasian J. Math. Comp. Appl., 8:2 (2020), 4–16 | DOI

[25] K. B. Sabitov, A. R. Zainullov, “Obratnye zadachi dlya dvumernogo uravneniya teploprovodnosti po otyskaniyu pravoi chasti”, Izv. vuzov. Matem., 2021, no. 3, 83–97 | DOI

[26] A. R. Kolmogorov, S. V. Fomin, Elementy teorii funktsii i funktsionalnogo analiza, Nauka, M., 1976 | MR