phone +7 (3412) 91 60 92

Archive of Issues


Uzbekistan Bukhara; Tashkent
Year
2022
Volume
32
Issue
3
Pages
383-402
<<
>>
Section Mathematics
Title The problem of determining the memory of an environment with weak horizontal heterogeneity
Author(-s) Durdiev D.K.ab, Safarov Zh.Sh.bc
Affiliations Bukhara State Universitya, Institute of Mathematics, National Academy of Sciences of Uzbekistanb, Tashkent University of Information Technologyc
Abstract The problem of determining the convolutional kernel $k(t,x)$, $t>0$, $x \in {\Bbb R}$, included in a hyperbolic integro-differential equation of the second order, is investigated in a domain bounded by a variable $z$ and having weakly horizontal heterogeneity. It is assumed that this kernel weakly depends on the variable $x$ and decomposes into a power series by degrees of a small parameter $\varepsilon$. A method for finding the first two coefficients $k_{0}(t)$, $k_{1}(t)$ of this expansion is constructed according to the given first two moments in the variable $x$ of the solution of the direct problem at $z=0$.
Keywords integro-differential equation, inverse problem, the Dirac delta function, the kernel of the integral, the norm
UDC 517.958
MSC 35L70, 45Q05
DOI 10.35634/vm220303
Received 13 April 2022
Language Russian
Citation Durdiev D.K., Safarov Zh.Sh. The problem of determining the memory of an environment with weak horizontal heterogeneity, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2022, vol. 32, issue 3, pp. 383-402.
References
  1. Romanov V.G. Ustoichivost' v obratnykh zadachakh (Stability in inverse problems), Moscow: Nauchnyi Mir, 2005.
  2. Romanov V.G. Obratnye zadachi matematicheskoi fiziki (Inverse problems of mathematical physics), Moscow: Nauka, 1984.
  3. Kabanikhin S.V. Obratnye i nekorrektnye zadachi (Inverse and incorrect problems), Novosibirsk: Sibirskoe Nauchnoe Izdatel'stvo, 2009.
  4. Durdiev D.K. Obratnye zadachi dlya sred s posledeistviyami (Inverse problems for environments with aftereffects), Tashkent: Turon-Iqbol, 2014.
  5. Totieva Zh.D. One-dimensional inverse coefficient problems of anisotropic viscoelasticity, Sibirskie Èlektronnye Matematicheskie Izvestiya, 2019, vol. 16, pp. 786-811 (in Russian). https://doi.org/10.33048/semi.2019.16.053
  6. Safarov Zh.Sh., Durdiev D.K. Inverse problem for an integro-differential equation of acoustics, Differential Equations, 2018, vol. 54, no. 1, pp. 134-142. https://doi.org/10.1134/S0012266118010111
  7. Safarov J.Sh. Global solvability of the one-dimensional inverse problem for the integro-differential equation of acoustics, Journal of Siberian Federal University. Mathematics and Physics, 2018, vol. 11, no. 6, pp. 753-763. https://doi.org/10.17516/1997-1397-2018-11-6-753-763
  8. Durdiev U.D. An inverse problem for the system of viscoelasticity equations in homogeneous anisotropic media, Journal of Applied and Industrial Mathematics, 2019, vol. 13, pp. 623-628. https://doi.org/10.1134/S1990478919040057
  9. Durdiev D.K., Rakhmonov 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, no. 6, pp. 923-937. https://doi.org/10.1134/S0040577918060090
  10. Durdiev D.K., Totieva Zh.D. The problem of determining the one-dimensional kernel of the electroviscoelasticity equation, Siberian Mathematical Journal, 2017, vol. 58, no. 3, pp. 427-444. https://doi.org/10.1134/S0037446617030077
  11. Safarov Zh.Sh. Evaluation of the stability of some inverse problems solutions for integro-differential equations, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2014, issue 3, pp. 75-82 (in Russian). https://doi.org/10.20537/vm140307
  12. Durdiev D.K., Nuriddinov Zh.Z. On investigation of the inverse problem for a parabolic integro-differential equation with a variable coefficient of thermal conductivity, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2020, vol. 30, issue 4, pp. 572-584. https://doi.org/10.35634/vm200403
  13. Durdiev D.K., 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. 867-877. https://doi.org//10.1134/S0001434615050223
  14. Karchevsky A.L., Turganbayev Y.M., Rakhmetullina S.G., Beldeubayeva Zh.T. Numerical solution of an inverse problem of determining the parameters of a source of groundwater pollution, Eurasian Journal of Mathematical and Computer Applications, vol. 5, issue 1, pp. 53-73. https://doi.org/10.32523/2306-3172-2017-5-1-53-73
  15. Durdiev U.D. Numerical method for determining the dependence of the dielectric permittivity on the frequency in the equation of electrodynamics with memory, Sibirskie Èlektronnye Matematicheskie Izvestiya, 2020, vol. 17, pp. 179-189 (in Russian). https://doi.org/10.33048/semi.2020.17.013
  16. Bozorov Z.R. Numerical determining a memory function of a horizontally-stratified elastic medium with aftereffect, Eurasian Journal of Mathematical and Computer Applications, 2020, vol. 8, issue 2, pp. 28-40. https://doi.org/10.32523/2306-6172-2020-8-2-28-40
  17. Durdiev D.K., Totieva Zh.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. https://doi.org/10.1134/S0037446621020038
  18. Romanov V.G. On the determination of the coefficients in the viscoelasticity equations, Siberian Mathematical Journal, 2014, vol. 55, no. 3, pp. 503-510. https://doi.org/10.1134/S0037446614030124
  19. Romanov V.G. On justification of the Gelfand-Levitan-Krein method for a two-dimensional inverse problem, Siberian Mathematical Journal, 2021, vol. 62, no. 2, pp. 908-924. https://doi.org/10.1134/S003744662105013X
  20. Romanov V.G. Phaseless inverse problems for Schrödinger, Helmholtz, and Maxwell equations, Computational Mathematics and Mathematical Physics, 2020, vol. 60, issue 6, pp. 1045-1062. https://doi.org/10.1134/s0965542520060093
  21. Klibanov M.V., Romanov V.G. Uniqueness of a 3-D coefficient inverse scattering problem without the phase information, Inverse Problems, 2017, vol. 33, no. 9, 095007. https://doi.org/10.1088/1361-6420/aa7a18
  22. Klibanov M.V., Romanov V.G. Two reconstruction procedures for a 3D phaseless inverse scattering problem for the generalized Helmholtz equation, Inverse Problems, 2016, vol. 32, no. 1, 015005. https://doi.org/10.1088/0266-5611/32/1/015005
  23. Blagoveshchenskii A.S., Fedorenko D.A. The inverse problem for an acoustic equation in a weakly horizontally inhomogeneous medium, Journal of Mathematical Sciences, 2008, vol. 155, issue 3, pp. 379-389. https://doi.org/10.1007/s10958-008-9221-1
  24. Blagoveshchenskii A.S. The quasi-two-dimensional inverse problem for the wave equation, Proceedings of the Steklov Institute of Mathematics, 1971, vol. 115, pp. 63-76.
  25. Totieva Zh.D. Determining the kernel of the viscoelasticity equation in a medium with slightly horizontal homogeneity, Siberian Mathematical Journal, 2020, vol. 61, issue 2, pp. 359-378. https://doi.org/10.1134/S0037446620020172
  26. Durdiev D.K., Safarov J.Sh. 2D kernel identification problem in viscoelasticity equation with a weakly horizontal homogeneity, Sibirskii Zhurnal Industrial'noi Matematiki, 2022, vol. 25, no. 1, pp. 14–38. https://doi.org/10.33048/SIBJIM.2022.25.102
  27. Durdiev D.K., 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). http://mi.mathnet.ru/eng/dvmg264
  28. Durdiev D.K., Rahmonov A.A. A 2D kernel determination problem in a visco-elastic porous medium with a weakly horizontally inhomogeneity, Mathematical Methods in the Applied Sciences, 2020, vol. 43, isue 15, pp. 8776-8796. https://doi.org/10.1002/mma.6544
  29. Akhmatov Z.A., Totieva Zh.D. Quasi-two-dimensional coefficient inverse problem for the wave equation in a weakly horizontally inhomogeneous medium with memory, Vladikavkazskii Matematicheskii Zhurnal, 2021, vol. 23, no. 4, pp. 15-27 (in Russian). https://doi.org/10.46698/l4464-6098-4749-m
  30. Durdiev D.K., Bozorov Z.R. Quasi-two-dimensional inverse problem of determining the kernel of an integral term in the viscoelasticity equation, Nauchnyi Vestnik Bukharskogo Gosudarstvennogo Universiteta, 2020, vol. 3, no. 79, pp. 10-21 (in Russian).
Full text
<< Previous article
Next article >>