Abstract: The paper studies the inverse problem of sequentially determining the two unknowns: the coefficient characterizing the properties of a medium with weakly horizontal inhomogeneity and the kernel of some integral operator describing the memory of the medium. The direct initial-boundary value problem contains zero data and the Neumann boundary condition. As additional information, the trace of the Fourier image of the direct problem solution at the boundary of the medium is given. To study inverse problems, it is assumed that the unknown coefficient decomposes into an asymptotic series. In this paper, a method is constructed for finding (taking into account the memory of the medium) the coefficient with accuracy \(O(\epsilon^2)\). At the first stage, the solution of the direct problem in the zero approximation and the kernel of the integral operator are simultaneously determined. The inverse problem is reduced to solving a system of nonlinear Volterra integral equations of the second kind. At the second stage, the kernel is considered to be given, and the first approximation solution of the direct problem and the unknown coefficient are determined. In this case, the inverse problem and the problem of solving a linear system of Volterra integral equations of the second kind will be equivalent. Two theorems on unique local solvability of the inverse problems are proved. Numerical results on the kernel function and coefficient are presented.
For citation: Akhmatov, Z. A. and Totieva, Zh. D. Quasi-Two-Dimensional Coefficient Inverse Problem for the Wave Equation in a Weakly Horizontally Inhomogeneous Medium with Memory, Vladikavkaz Math. J., 2021, vol. 23, no. 4, pp. 15-27 (in Russian). DOI 10.46698/l4464-6098-4749-m
1. Romanov, V. G. Obratnye zadachi matematicheskoj fiziki
[Inverse Problems of Mathematical Physics], Moscow, Nauka, 1984 (in Russian).
2. Lorenzi, A. and Sinestrari, E. An Inverse Problem in the Theory
of Materials with Memory I, Nonlinear
Analysis: Theory, Methods and Applications, 1988, vol. 12, no. 12,
pp. 1317-1335. DOI: 10.1016/0362-546X(88)90080-6.
3. Lorenzi, À. An Inverse Problem in the Theory of Materials with
Memory II, Journal of Semigroup Theory and Applications, Series
on Pure and Applied Mathematics, 1989, vol. 116, pp. 261-290.
4. Durdiev, D. K. The Inverse Problem for a Three-Dimensional Wave Equation in a Memory
Environmentu, Matematicheskij analiz i diskretnaya matematika,
Novosibirsk, Izd-vo Novosibirskogo Universiteta, 1989, pp. 19-27 (in Russian).
5. Lorenzi, A. and Paparoni, E. Direct and Inverse Problems in the Theory of Materials with Memory,
The Mathematical Journal of the University of Padua, 1992, vol. 87, pp. 105-138.
6. Bukhgeym, A. L. Inverse Problems of Memory Reconstruction,
Journal of Inverse and Ill-posed Problems, 1993, vol. 1, no. 3,
pp. 193-206. DOI: 10.1515/jiip.19126.96.36.199.
7. Durdiev, D. K. A Multidimensional Inverse Problem for an Equation with Memory, Siberian
Mathematical Journal, 1994, vol. 35, pp. 514-521. DOI: 10.1007/BF02104815.
8. Bukhgeim, A. L. and Dyatlov, G. V. Inverse Problems for Equations with Memory, SIAM J. Math. Fool., 1998, vol. 1, no. 2, pp. 1-17.
9. Durdiev, D. K. Obratnye zadachi dlya sred s posledejstviem, Tashkent, Turon-Ikbol, 2014.
10. Durdiev, D. K. and Safarov, Zh. Sh. Inverse Problem of Determining the One-Dimensional Kernel of the Viscoelasticity Equation in a Bounded Domain, Mathematical Notes, 2015, vol. 97, no. 6, pp. 855-867. DOI: 10.4213/mzm10659.
11. Durdiev, D. K. and Totieva, Zh. D. The Problem of Determining the Multidimensional Kernel
of Viscoelasticity Equation, Vladikavkaz Mathematical Journal, 2015, vol. 17, no. 4, pp. 18-43 (in Russian). DOI: 10.23671/VNC.2015.4.5969.
12. Durdiev, D. K. and Totieva, Z. D.
The Problem Of Determining The One-Dimensional Kernel of
the Electroviscoelasticity Equation, Siberian Mathematical Journal, 2017, vol. 58, no. 3, pp. 427-444.
13. Durdiev, D. K. and Rahmonov, A. A. Inverse Problem for a System of Integro-Differential
Equations for SH Waves in a Visco-Elastic Porous Medium: Global Solvability, Theoretical and Mathematical
Physics, 2018, vol. 195, pp. 923-937. DOI: 10.1134/S0040577918060090.
14. Durdiev, U. D. A Problem of Identification of a Special 2D Memory Kernel in an Integro-Differential Hyperbolic Equation, Eurasian J. Math. Comp. App., 2019, vol. 7, no. 2, pp. 4-19.
15. Durdiev, U. D. and Totieva, Z. D. A Problem of Determining a Special Spatial Part of 3D Memory Kernel
in an Integro-Differential Hyperbolic Equation, Mathematical Methods in Applied Sciences, 2019, vol. 42, no. 18, pp. 7440-7451. DOI: 10.1002/mma.5863.
16. Durdiev, D. K. and Totieva, Z. D. About Global Solvability of a Multidimensional Inverse Problem for an Equation with Memory, Siberian Mathematical Journal, 2021, vol. 62, no. 2, pp. 215-229. DOI: 10.1134/S0037446621020038.
17. Kumar, P., Kinra, R. and Mohan, M. A Local in Time Existence and Uniqueness Result of an Inverse Problem for the Kelvin-Voigt Fluids, Inverse Problems, 2021, vol. 37, no. 8, 085005. DOI: 10.1088/1361-6420/ac1050.
18. Blagoveshchenskii, D. A. and Fedorenko, A. S. The Inverse Problem for the Acoustic Equation in a Weakly Horizontally Inhomogeneous Medium, Journal of Mathematical Sciences, 2008, vol. 155, no. 3, pp. 379-389, DOI: 10.1007/s10958-008-9221-1.
19. Durdiev, D. K. and Bozorov, Z. R. A Problem of Determining the Kernel of Integrodifferential Wave
Equation with Weak Horizontal Properties, Dal’nevostochnyi matematicheskii zhurnal, 2013, vol. 13,
no. 2, pp. 209-221 (in Russian).
20. Durdiev, D. K. The Inverse Problem of Determining Two Coefficients in One Integro Differential Wave Equation, Sib. zhurnal industrialnoy matematiki [Siberian Journal of Industrial Mathematics], 2009, vol. 12, no. 3, pp. 28-40 (in Russian).
21. Courant, R. and Hilbert, D. Methods of Mathematical Physics, II, New York-London, Interscience Publ., 1962.
22. Kolmogorov, A. N. and Fomin, S. V. Elementy teorii funktsiy i funktsionalnogo analiza
[Elements of the Function's Theory and Functional Analysis], Moscow, Fizmatlit, 2006, 572 p. (in Russian).
23. Yakhno, V. G. Obratnyye zadachi dlya differentsialnykh uravneniy uprugosti [Inverse Problems for Differential Equations of Elasticity], Novosibirsk, Nauka, 1988, 304 p. (in Russian).