Skip to main content
SpringerLink
Log in

Finding the Two-Dimensional Relaxation Kernel of an Integro-Differential Wave Equation

  • PARTIAL DIFFERENTIAL EQUATIONS
  • Published:
Differential Equations Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

We consider the multidimensional inverse problem of determining the kernel of the integral term in an integro-differential wave equation. In the direct problem, it is required to find the displacement function from an initial–boundary value problem, and in the inverse one, to determine the kernel of the integral term depending on both time and one of the spatial variables. The local unique solvability of the problem in the class of functions continuous in one of the variables and analytic in the other one is proved on the basis of the method of scales of Banach spaces of real analytic functions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

REFERENCES

  1. Durdiev, D.K. and Totieva, Zh.D., The problem of determining the one-dimensional kernel of the viscoelasticity equation, Sib. Zh. Ind. Mat., 2013, vol. 16, no. 2, pp. 72–82.

    MathSciNet  MATH  Google Scholar 

  2. Durdiev, D.K. and Safarov, Zh.Sh., Inverse problem of determining the one-dimensional kernel of the viscoelasticity equation in a bounded domain, Math. Notes, 2015, vol. 97, no. 6, pp. 867–877.

    Article  MathSciNet  MATH  Google Scholar 

  3. Safarov, Zh.Sh. and Durdiev, D.K., Inverse problem for an integro-differential equation of acoustics, Differ. Equations, 2018, vol. 54, no. 1, pp. 134–142.

    Article  MathSciNet  MATH  Google Scholar 

  4. Safarov, J.Sh., Global solvability of the one-dimensional inverse problem for the integro-differential equation of acoustics, J. Sib. Fed. Univ. Math. & Phys., 2018, vol. 11, no. 6, pp. 753–763.

    Article  MathSciNet  MATH  Google Scholar 

  5. Ovsyannikov, L.V., Nonlinear Cauchy problem in scales of Banach spaces, Dokl. Akad. Nauk SSSR, 1971, vol. 200, no. 4, pp. 789–792.

    MathSciNet  Google Scholar 

  6. Nirenberg, L., Topics in Nonlinear Functional Analysis, New York: Courant Inst. Math. Sci., 1974.

    MATH  Google Scholar 

  7. Romanov, V.G., On the local solvability of some multidimensional inverse problems for equations of hyperbolic type, Differ. Uravn., 1989, vol. 25, no. 2, pp. 275–283.

    MathSciNet  Google Scholar 

  8. Romanov, V.G., Problem of determining the speed of sound, Sib. Math. J., 1989, vol. 30, no. 4, pp. 598–605.

    Article  MATH  Google Scholar 

  9. Romanov, V.G., On the solvability of inverse problems for hyperbolic equations in the class of functions analytic with respect to part of the variables, Dokl. Math., 1989, vol. 39, no. 1, pp. 160–164.

    MathSciNet  MATH  Google Scholar 

  10. Durdiev, D.K., A multidimensional inverse problem for an equation with memory, Sib. Math. J., 1994, vol. 35, no. 3, pp. 514–521.

    Article  MathSciNet  MATH  Google Scholar 

  11. Durdiev, D.K. and Safarov, Zh.Sh., Local solvability of the problem of determining the spatial part of a multidimensional kernel in an integro-differential equation of hyperbolic type, Vestn. Samarsk. Gos. Tekh. Univ. Fiz.-Mat. Nauki, 2012, no. 4 (29), pp. 37–47.

  12. Durdiev, D.K. and Totieva, Zh.D., The problem of determining the multidimensional kernel of the viscoelasticity equation, Vladikavkaz. Mat. Zh., 2015, vol. 17, no. 4, pp. 18–43.

    MathSciNet  MATH  Google Scholar 

  13. Durdiev, D.K. and Totieva, Zh.D., About global solvability of a multidimensional inverse problem for an equation with memory, Sib. Math. J., 2021, vol. 62, no. 2, pp. 215–229.

    Article  MathSciNet  MATH  Google Scholar 

  14. Romanov, V.G., Ustoichivost’ v obratnykh zadachakh (Stability in Inverse Problems), Moscow: Nauchn. Mir, 2005.

    Google Scholar 

Download references

Funding

The work of the first author was supported by the Ministry of Science and Higher Education of the Russian Federation, project no. 075-02-2023-939.

Author information

Authors and Affiliations

  1. Romanovskii Institute of Mathematics, Uzbekistan Academy of Sciences, Tashkent, 100170, Uzbekistan

    D. K. Durdiev & J. Sh. Safarov

  2. Bukhara State University, Bukhara, 705018, Uzbekistan

    D. K. Durdiev

  3. Diplomat University, Tashkent, 100047, Uzbekistan

    J. Sh. Safarov

Authors
  1. D. K. Durdiev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Sh. Safarov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to D. K. Durdiev or J. Sh. Safarov.

Additional information

Translated by V. Potapchouck

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Durdiev, D.K., Safarov, J.S. Finding the Two-Dimensional Relaxation Kernel of an Integro-Differential Wave Equation. Diff Equat 59, 214–229 (2023). https://doi.org/10.1134/S0012266123020064

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0012266123020064

Search

Navigation