Skip to main content
SpringerLink
Log in

Inverse Problem for Finding the Order of the Fractional Derivative in the Wave Equation

  • Research Articles
  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

The paper investigates an inverse problem for finding the order of the fractional derivative in the sense of Gerasimov–Caputo in the wave equation with an arbitrary positive self-adjoint operator \(A\) having a discrete spectrum. By means of the classical Fourier method, it is proved that the value of the projection of the solution onto some eigenfunction at a fixed time uniquely restores the order of the derivative. Several examples of the operator \(A\) are discussed, including a linear system of fractional differential equations, fractional Sturm–Liouville operators, and many others.

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

Similar content being viewed by others

References

  1. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, in North-Holland Math. Stud. (Elsevier, Amsterdam, 2006), Vol. 204.

    Book  Google Scholar 

  2. R. Gorenflo, A. A. Kilbas, F. Mainardi, and S. V. Rogozin, Mittag-Leffler Functions, Related Topics, and Applications (Springer, Heidelberg, 2014).

    Book  Google Scholar 

  3. Handbook of Fractional Calculus with Applications. Vol. 2. Fractional Differential Equations, Ed. by A. Kochubei and Yu. Luchko (De Gruyter, Berlin, 2019).

  4. Z. Li, Y. Liu and M. Yamamoto, “Inverse problems of determining parameters of the fractional partial differential equations,” in Handbook of Fractional Calculus with Applications. Vol. 2. Fractional Differential Equations (De Gruyter, Berlin, 2019), pp. 431–442.

    MathSciNet  Google Scholar 

  5. M. M. Dzhrbashyan, Integral Transformations and Representations of Functions in the Complex Domain (Nauka, Moscow, 1966) [in Russian].

    Google Scholar 

  6. A. V. Pskhu, Partial Differential Equations of Fractional Order (Nauka, Moscow, 2005) [in Russian].

    MATH  Google Scholar 

  7. A. Alsaedi, B. Ahmad, M. Kirane, and B. T. Torebek, “Blowing-up solutions of the time-fractional dispersive equations,” Adv. Nonlinear Anal. 10 (1), 952–971 (2020).

    Article  MathSciNet  Google Scholar 

  8. B. Ahmad, A. Alsaedi and M. Kirane, “Blowing-up solutions of distributed fractional differential systems,” Chaos Solitons Fractals 145 (2021), Paper No. 110747.

    Article  MathSciNet  Google Scholar 

  9. M. Kirane and B. T. Torebek, Maximum Principle for Space and Time-Space Fractional Partial Differential Equations, arXiv: 2002.09314v1 (2020).

  10. J. Cheng, J. Nakagawa, M. Yamamoto, and T. Yamazaki, “Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation,” Inverse Problems 25 (11) (2009), Paper No. 115002.

    Article  MathSciNet  Google Scholar 

  11. Z. Li and M. Yamamoto, “Uniqueness for inverse problems of determining orders of multi-term time- fractional derivatives of diffusion equation,” Appl. Anal. 94 (3), 570–579 (2015).

    Article  MathSciNet  Google Scholar 

  12. M. Yamamoto, Uniqueness in Determining the Order of Time and Spatial Fractional Derivatives, arXiv: 2006.15046v1 (2020).

  13. Z. Li, Y. Luchko and M. Yamamoto, “Analyticity of solutions to a distributed order time-fractional diffusion equation and its application to an inverse problem,” Comput. Math. Appl. 73 (6), 1041–1052 (2017).

    Article  MathSciNet  Google Scholar 

  14. J. Janno, “Determination of the order of fractional derivative and a kernel in an inverse problem for a generalized time-fractional diffusion equation,” Electron. J. Differential Equations 2016 (2016), Paper No. 199.

    MathSciNet  Google Scholar 

  15. R. Ashurov and S. Umarov, “Determination of the order of fractional derivative for subdiffusion equations,” Fract. Calc. Appl. Anal. 23 (6), 1647–1662 (2020).

    Article  MathSciNet  Google Scholar 

  16. Sh. Alimov and R. Ashurov, “Inverse problem for finding an order of the Caputo time-fractional derivative for a subdiffusion equation,” J. Inverse Ill-Posed Probl. 28 (5), 651–658 (2020).

    Article  MathSciNet  Google Scholar 

  17. R. Ashurov and Yu. Fayziev, Determination of Fractional Order and Source Term in a Fractional Subdifusion Equation, arXiv: https:// www.researchgate.net/ publication/354997348 (2021).

  18. R. R. Ashurov and Yu. E. Fayziev, “Uniqueness and existence for inverse problem for finding an order of time-fractional derivative of subdiffusion equation,” Lobachevskii J. Math 42 (3), 508–516 (2021).

    Article  MathSciNet  Google Scholar 

  19. R. Ashurov and R. Zunnunov, “Initial-boundary value and inverse problems for subdiffusion equation in \(\mathbb R^N\),” Fract. Differ. Calc. 10 (2), 291–306 (2020).

    Article  MathSciNet  Google Scholar 

  20. R. Ashurov and R. Zunnunov, Inverse Problem for Determining the Order of the Fractional Derivative in Mixed-Type Equations, arXiv: 2103.05287v1 (2021) [in Russian].

  21. R. R. Ashurov, Inverse Problems for Determining the Order of Time-Fractional Derivative in a Wave Equation (2021).

  22. C. Lizama, “Abstract linear fractional evolution equations,” in Handbook of Fractional Calculus with Applications. Vol. 2. Fractional Differential Equations (De Gruyter, Berlin, 2019), pp. 465–479.

  23. O. Novozhenova, Biography and Scientific Works of Alexey Nikiforovich Gerasimov. On Linear Operators, Elasto-Viscosity, Eleftheros, and Fractional Derivatives (Izd. “Pero”, Moscow, 2018) [in Russian].

    Google Scholar 

  24. O. P. Agrawal, “Solution for a fractional diffusion-wave equation defined in a bounded domain,” Nonlinear Dynam. 29 (1-4), 145–155 (2002).

    Article  MathSciNet  Google Scholar 

  25. A. V. Pskhu, “Initial-value problem for a linear ordinary differential equation of noninteger order,” Sb. Math. 202 (4), 571–582 (2011).

    Article  MathSciNet  Google Scholar 

  26. A. V. Pskhu, “Green function of the first boundary-value problem for the fractional diffusion wave equation in a multidimensional rectangular domain,” in Proceedings of the IV International Scientific Conference “Actual Problems of Applied Mathematics”. Kabardino-Balkar Republic, Nalchik, Elbrus Region, May 22–26, 2018. Part III, Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz. (VINITI, Moscow, 2019), Vol. 167, pp. 52–61.

    MathSciNet  Google Scholar 

  27. S. Umarov, Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols, in Dev. Math. (Springer, Cham, 2015), Vol. 41.

    Book  Google Scholar 

  28. R. Ashurov and O. Muhiddinova, “Initial-boundary-value problem for a time-fractional subdiffusion equation with an arbitrary elliptic differential operator,” Lobachevskii J. Math. 42 (3), 517–525 (2021).

    Article  MathSciNet  Google Scholar 

  29. M. Ruzhansky, N. Tokmagambetov, and B. T. Torebek, “On a nonlocal problem for a multi-term fractional diffusion-wave equation,” Fract. Calc. Appl. Anal. 23 (2), 324–355 (2020).

    Article  MathSciNet  Google Scholar 

  30. A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, Based, in part, on notes left by Harry Bateman (McGraw-Hill, New York, 1953).

    MATH  Google Scholar 

  31. V. A. Il’in, “The solvability of mixed problems for hyperbolic and parabolic equations,” Russian Math. Surveys 15 (1), 85–142 (1960).

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

The author wish to express gratitude to S. A. Alimov for discussing the results in this paper. The authors are also grateful to the anonymous referee of the journal for his/her comments, which have significantly improved the content of this paper.

Author information

Authors and Affiliations

  1. Institute of Mathematics of the Academy of Sciences of Uzbekistan, Tashkent, 100125, Uzbekistan

    R. R. Ashurov

  2. National University of Uzbekistan, Tashkent, 100174, Uzbekistan

    Yu. É. Faiziev

Authors
  1. R. R. Ashurov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yu. É. Faiziev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. R. Ashurov.

Additional information

Translated from Matematicheskie Zametki, 2021, Vol. 110, pp. 824–836 https://doi.org/10.4213/mzm13090.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ashurov, R.R., Faiziev, Y.É. Inverse Problem for Finding the Order of the Fractional Derivative in the Wave Equation. Math Notes 110, 842–852 (2021). https://doi.org/10.1134/S0001434621110213

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0001434621110213

Keywords

Search

Navigation