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Vladikavkaz. Mat. Zh., 2015, Volume 17, Number 4, Pages 18–43 (Mi vmj560)  

This article is cited in 22 scientific papers (total in 22 papers)

The problem of determining the multidimensional kernel of viscoelasticity equation

D. Q. Durdieva, Zh. D. Totievabc

a Bukhara State University, Bukhara, Uzbekistan
b Center of Geophysical Investigations of Vladikavkaz Scientific Center of the Russian Academy of Sciences and the Government of Republic of North Ossetia-Alania, Vladikavkaz, Russia
c North Ossetian State University after Kosta Levanovich Khetagurov, Vladikavkaz, Russia

Abstract: The integro-differential system of viscoelasticity equations is considered. The direct problem of determining of the displacements vector from the initial-boundary problem for this system is formulated. It is assumed that the kernel in the integral part depends on both the time and the space variable $x_2$. For its determination an additional condition relative to the first component of the displacements vector with $x_3=0$ is posed. The inverse problem is replaced by the equivalent system of integral equations. The study is based on the method of scales of Banach spaces of analytic functions. The theorem on local unique solvability of the inverse problem is proved in the class of functions analytic on the variable $x_2$ and continuous on $t$.

Key words: inverse problem, stability, delta function, Lame's coefficients, kernel.

Full text: PDF file (296 kB)
References: PDF file   HTML file
UDC: 517.958
Received: 09.02.2015

Citation: D. Q. Durdiev, Zh. D. Totieva, “The problem of determining the multidimensional kernel of viscoelasticity equation”, Vladikavkaz. Mat. Zh., 17:4 (2015), 18–43

Citation in format AMSBIB
\Bibitem{DurTot15}
\by D.~Q.~Durdiev, Zh.~D.~Totieva
\paper The problem of determining the multidimensional kernel of viscoelasticity equation
\jour Vladikavkaz. Mat. Zh.
\yr 2015
\vol 17
\issue 4
\pages 18--43
\mathnet{http://mi.mathnet.ru/vmj560}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Durdiev D.K., Zhumaev Zh.Zh., “Memory Kernel Reconstruction Problems in the Integro-Differential Equation of Rigid Heat Conductor”, Math. Meth. Appl. Sci.  crossref  isi  scopus
    2. Zh. D. Totieva, “Mnogomernaya zadacha ob opredelenii funktsii plotnosti dlya sistemy uravnenii vyazkouprugosti”, Sib. elektron. matem. izv., 13 (2016), 635–644  mathnet  crossref
    3. D. K. Durdiev, U. D. Durdiev, “The problem of kernel determination from viscoelasticity system integro-differential equations for homogeneous anisotropic media”, Nanosyst.-Phys. Chem. Math., 7:3 (2016), 405–409  crossref  zmath  isi
    4. D. K. Durdiev, Zh. D. Totieva, “The problem of determining the one-dimensional kernel of the electroviscoelasticity equation”, Siberian Math. J., 58:3 (2017), 427–444  mathnet  crossref  crossref  isi  elib  elib
    5. Zh. D. Totieva, D. K. Durdiev, “The Problem of Finding the One-Dimensional Kernel of the Thermoviscoelasticity Equation”, Math. Notes, 103:1 (2018), 118–132  mathnet  crossref  crossref  mathscinet  isi  elib
    6. D. K. Durdiev, A. A. Rakhmonov, “Inverse problem for a system of integro-differential equations for SH waves in a visco-elastic porous medium: Global solvability”, Theoret. and Math. Phys., 195:3 (2018), 923–937  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    7. Jurabek Sh. Safarov, “Global solvability of the one-dimensional inverse problem for the integro-differential equation of acoustics”, Zhurn. SFU. Ser. Matem. i fiz., 11:6 (2018), 753–763  mathnet  crossref
    8. D. K. Durdiev, Zh. D. Totieva, “The problem of determining the one-dimensional matrix kernel of the system of viscoelasticity equations”, Math. Meth. Appl. Sci., 41:17 (2018), 8019–8032  crossref  mathscinet  zmath  isi
    9. Zh. D. Totieva, “The problem of determining the piezoelectric module of electroviscoelasticity equation”, Math. Meth. Appl. Sci., 41:16 (2018), 6409–6421  crossref  mathscinet  zmath  isi  scopus
    10. Zh. Sh. Safarov, D. K. Durdiev, “Inverse problem for an integro-differential equation of acoustics”, Differ. Equ., 54:1 (2018), 134–142  crossref  mathscinet  zmath  isi  scopus
    11. Zh. D. Totieva, “Odnomernye obratnye koeffitsientnye zadachi anizotropnoi vyazkouprugosti”, Sib. elektron. matem. izv., 16 (2019), 786–811  mathnet  crossref
    12. Zh. D. Totieva, “K voprosu issledovaniya zadachi opredeleniya matrichnogo yadra sistemy uravnenii anizotropnoi vyazkouprugosti”, Vladikavk. matem. zhurn., 21:2 (2019), 58–66  mathnet  crossref  elib
    13. U. D. Durdiev, “Obratnaya zadacha dlya sistemy uravnenii vyazkouprugosti v odnorodnykh anizotropnykh sredakh”, Sib. zhurn. industr. matem., 22:4 (2019), 26–32  mathnet  crossref
    14. U. Durdiev, Zh. Totieva, “A problem of determining a special spatial part of 3D memory kernel in an integro-differential hyperbolic equation”, Math. Meth. Appl. Sci., 42:18 (2019), 7440–7451  crossref  mathscinet  zmath  isi  scopus
    15. D. K. Durdiev, 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 (2020), 281–295  mathnet  crossref  crossref  elib
    16. Zh. D. Totieva, “Opredelenie yadra uravneniya vyazkouprugosti v slabo gorizontalno-neodnorodnoi srede”, Sib. matem. zhurn., 61:2 (2020), 453–475  mathnet  crossref
    17. D. K. Durdiev, Zh. D. Totieva, “Inverse problem for a second-order hyperbolic integro-differential equation with variable coefficients for lower derivatives”, Sib. elektron. matem. izv., 17 (2020), 1106–1127  mathnet  crossref
    18. D. K. Durdiev, Zh. Zh. Zhumaev, “Problem of determining the thermal memory of a conducting medium”, Differ. Equ., 56:6 (2020), 785–796  crossref  mathscinet  zmath  isi  scopus
    19. D. K. Durdiev, Zh. D. Totieva, “The problem of determining the one-dimensional kernel of viscoelasticity equation with a source of explosive type”, J. Inverse Ill-Posed Probl., 28:1 (2020), 43–52  crossref  mathscinet  zmath  isi  scopus
    20. A. A. Rakhmonov, U. D. Durdiev, Z. R. Bozorov, “Problem of determining the speed of sound and the memory of an anisotropic medium”, Theoret. and Math. Phys., 207:1 (2021), 494–513  mathnet  crossref  crossref  isi
    21. D. K. Durdiev, Zh. D. Totieva, “O globalnoi razreshimosti odnoi mnogomernoi obratnoi zadachi dlya uravneniya s pamyatyu”, Sib. matem. zhurn., 62:2 (2021), 269–285  mathnet  crossref  elib
    22. D. K. Durdiev, Kh. Kh. Turdiev, “Zadacha opredeleniya yader v sisteme integrodifferentsialnykh uravnenii Maksvella”, Sib. zhurn. industr. matem., 24:2 (2021), 38–61  mathnet  crossref
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