FAR EASTERN BRANCH OF THE RUSSIAN ACADEMY OF SCIENCES
INSTITUTE OF APPLIED MATHEMATICS KHABAROVSK DIVISION
Far Eastern Mathematical Journal
The problem of finding the kernels in the system of integro-differential acoustics equations |
Durdiev D.K., Turdiev Kh.Kh. |
2023, issue 2, P. 190-210 DOI: https://doi.org/10.47910/FEMJ202317 |
Abstract |
| For reduced to the canonical system of integro-differential equations of acoustics, a direct problem is posed, which consists in determining the velocity of the perturbed medium and the pressure and the inverse problem of finding the diagonal memory matrix. The problems are reduced to a closed system of integral equations of the second kind of the Volterra type with respect to the solution of the direct problem and unknowns of the inverse problem. The method of contraction mappings in the space of continuous functions with an exponential weighted norm is applied to this system. Existence and uniqueness theorems for solutions to problems in the global sense are proved. |
Keywords: hyperbolic system, system of acoustics equations, integral equation, contraction mapping principle. |
Download the article (PDF-file) |
References |
| [1] V. Vol'terra, Teoriia funktsionalov, integral'nykh i integro–differentsial'nykh uravnenii, Nauka, Gl. red. fiz.–mat. liter, M., 1982. [2] Mura. Toshio, Micromechanics of defects in solids, Second, Revised Edition, IL, USA, Northwestern University, Evanston, 1987 [3] L. D. Landau, E. M. Lifshits, Elektrodinamika sploshnykh sred, Nauka, M, 1959. [4] A. Lorenzi, “An identification problem related to a nonlinear hyperbolic integro-differential equation”, Nonlinear Anal., Theory, Methods Appl., 22:1 (1994), 21–44. [5] Z. S. Safarov, D. K. Durdiev, “Inverse Problem for an Integro-Differential Equation of Acoustics”, Differential Equations, 54:1 (2018), 134–142. [6] J. Janno, L. Von Wolfersdorf, “Inverse problems for identification of memory kernels in viscoelasticity”, Math. Methods Appl. Sci., 20:4 (1997), 291–314. [7] D. K. Durdiev, A. A. Rakhmonov, “Obratnaia zadacha dlia sistemy integro-differentsial'nykh uravnenii SH-voln v viazkouprugoi poristoi srede: global'naia razreshimost'”, TMF., 195:3 (2018), 491–506. [8] V. G. Romanov, “Otsenki ustoichivosti resheniia v zadache ob opredelenii iadra uravneniia viazkouprugosti”, Sib. zhurn. industr. matem., 15:1 (2012), 86–98. [9] Z. D. Totieva, D. K. Durdiev, “The Problem of Finding the One-Dimensional Kernel of the Thermoviscoelasticity Equation”, Mathematical Methods in the Applied Sciences, 103:1–2 (2018), 118–132. [10] D. K. Durdiev, Zh.Sh. Safarov, “Obratnaia zadacha ob opredelenii odnomernogo iadra uravneniia viazkouprugosti v ogranichennoi oblasti”, Matem. zametki., 97:6 (2015), 855–867. [11] D. K. Durdiev, Z. D. Totieva, “The problem of determining the one-dimensional kernel of viscoelasticity equation with a source of explosive type”, Journal of Inverse and Ill-Posed Problems, 28:1 (2020), 43–52. [12] U. D. Durdiev, “Obratnaia zadacha dlia sistemy uravnenii viazkouprugosti v odnorodnykh anizotropnykh sredakh”, Sib. zhurn. industr. matem., 22:4 (2019), 26–32. [13] D. K. Durdiev, Zh. D. Totieva, “Zadacha ob opredelenii mnogomernogo iadra uravneniia viazkouprugosti”, Vladikavk. matem. zhurn., 17:4 (2015), 18–43. [14] D. K. Durdiev, Z. R. Bozorov, “Zadacha opredeleniia iadra integrodifferentsial'nogo volnovogo uravneniia so slabo gorizontal'noi odnorodnost'iu”, Dal'nevost. matem. zhurn., 13:2 (2013), 209–221. [15] V. G. Romanov, “Zadacha ob opredeleniia iadra v uravnenii viazkouprugosti”, Dokl. AN., 446:1 (2012), 18–20. [16] D. K. Durdiev, A. A. Rahmonov, “A 2D kernel determination problem in a visco-elastic porous medium with a weakly horizontally inhomogeneity”, Mathematical Methods in the Applied Sciences, 43:15 (2020), 8776–8796. [17] V. G. Romanov, “Inverse problems for equation with a memory”, Eurasian Jour. of Math. and Computer Applications, 2:4 (2014), 51–80. [18] D. K. Durdiev, A. A. Rakhmonov, “Zadacha ob opredelenii dvumernogo iadra v sisteme integro-differentsial'nykh uravnenii viazkouprugoi poristoi sredy”, Sib. zhurn. industr. matem., 23:2 (2020), 63–80. [19] M. K. Teshaev, I. I. Safarov, M. Mirsaidov, “Oscillations of multilayer viscoelastic composite toroidal pipes”, Journal of the Serbian Society for Computational Mechanics, 13:2 (2019), 104–115. [20] M. K. Teshaev, “Realization of servo-constraints by electromechanical servosystems”, Russian Mathematics, 54 (2010), 38–44. [21] U. D. Durdiev, “Numerical method for determining the dependence of the dielectric permittivity on the frequency in the equation of electrodynamics with memory”, Sib. Electron. Mat. Izv., 17 (2020), 179–189. [22] Z. R. Bozorov, “Numerical determining a memory function of a horizontally-stratified elastic medium with aftereffect”, Eurasian journal of mathematical and computer applications, 8:2 (2020), 4–16. [23] A. L. Karchevskii, A. G. Fat'ianov, “Chislennoe reshenie obratnoi zadachi dlia sistemy uprugosti s posledeistviem dlia vertikal'no neodnorodnoi sredy”, Sib. zhurn. vychisl. matem., 4:3 (2001), 259–268. [24] D. K. Durdiev, Kh. Kh. Turdiev, “Obratnaia zadacha dlia giperbolicheskoi sistemy pervogo poriadka s pamiat'iu”, Differentsial'nye uravneniia, 56:12 (2020), 1666–1675. [25] D. K. Durdiev, Kh. Kh. Turdiev, “The problem of finding the kernels in the system of integro-differential Maxwell’s equations”, Sib. Zh. Ind. Math., 24:2 (2021), 38–61. [26] S. K. Godunov, Uravneniia matematicheskoi fiziki (2-e izd.), Nauka, M., 1979. [27] V. G. Romanov, Obratnye zadachi matematicheskoi fiziki, Nauka, M., 1984. [28] A. N. Kolmogorov, S. V. Fomin, Elementy teorii funktsii i funktsional'nogo analiza, Nauka, Gl. red. fiz.-mat. lit., M., 1989. [29] A. A. Kilbas, Integral'nye uravneniia: kurs lektsii, Minsk, 2005. [30] F. R. Gantmakher, Teoriia matrits, Nauka, Gl. red. fiz.-mat. lit., M, 1988. |