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To the Problem of a Point Source in an Inhomogeneous Medium

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Abstract

This paper studies the asymptotic behavior with respect to the complex parameter of the fundamental solution for a second-order elliptic operator with smooth compact coefficients obtained by the V. P. Maslov canonical operator method using the results of V. V. Kucherenko.

It is shown that the singular part of the asymptotics can be represented as a series in Hankel functions of the first kind. The asymptotics are constructed under the assumption that all trajectories of the corresponding Hamiltonian system go to infinity.

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References

  1. M. V. Keldysh, “On the completeness of the eigenfunctions of some classes of non-selfadjoint linear operators,” Russian Math. Surveys 26 (4), 15–44 (1971).

    Article  ADS  Google Scholar 

  2. V. V. Kucherenko, “Quasiclassical asymptotics of a point-source function for the stationary Schrödinger equation,” Theor. Math. Phys. 1 (3), 294–310 (1969).

    Article  Google Scholar 

  3. T. M. Gataulin, “Asymptotic behavior of the fundamental solution of an elliptic equation with respect to a complex parameter,” Math. Notes 21 (3), 210–217 (1977).

    Article  MathSciNet  Google Scholar 

  4. V. P. Maslov, Perturbation Theory and Asymptotic Methods (Izd. Mosk. Univ., Moscow, 1965) [in Russian].

    Google Scholar 

  5. V. P. Maslov, Operator Methods (Nauka, Moscow, 1973) [in Russian].

    Google Scholar 

  6. V. V. Kucherenko, “Some properties of the short wave asymptotic behavior of the fundamental solution of the equation \([\Delta+k^2 n^2(x)]u=0\),” Trudy MIEM, No. 25, 32–55 (1972).

    MathSciNet  Google Scholar 

  7. V. V. Kucherenko, “Short wave asymptotics of the Green’s function for the \(N\)-dimensional wave equation in an inhomogeneous medium,” U.S.S.R. Comput. Math. Math. Phys., 8 (4), 294–302 (1965).

    Article  MathSciNet  Google Scholar 

  8. I. S. Gradshtein and I. M. Ryzhik, Tables of Integrals, Sums, Series, and Products (GIFML, Moscow, 1963) [in Russian].

    Google Scholar 

  9. V. M. Babich, “The short wave asymptotic form of the solution for the problem of a point source in an inhomogeneous medium,” Comput. Math. Math. Phys., No. 5, 247–251 (1965).

    Article  Google Scholar 

  10. V. M. Babich, “A point source of electromagnetic waves in an inhomogeneous medium: a high frequency ansatz and the dual nonstationary singular solution,” J. Math. Sci. (N. Y.) 243, 634–639 (2019).

    Article  MathSciNet  Google Scholar 

  11. A. Yu. Anikin, S. Yu. Dobrokhotov, V. E. Nazaikinskii, and M. Rouleux, “Lagrangian manifolds and the construction of asymptotics for (pseudo)differential equations with localized right-hand sides,” Theor. Math. Phys. 214 (1), 1–23 (2023).

    Article  MathSciNet  Google Scholar 

  12. A. Yu. Anikin, S. Yu. Dobrokhotov, V. E. Nazaikinskii, and M. Rouleux, “The Maslov canonical operator on a pair of Lagrangian manifolds and asymptotics of the solutions of stationary equations with localized right-hand sides,” Dokl. Math. 96 (1), 406–410 (2017).

    Article  MathSciNet  Google Scholar 

  13. R. Melrose and G. A. Uhlmann, “Lagrangian intersection and the Cauchy problem,” Comm. Pure Appl. Math. 32 (4), 483–519 (1979).

    Article  MathSciNet  Google Scholar 

  14. S. G. Pyatkov and L. V. Neustroeva, “On some asymptotic representations of solutions to elliptic equations and their applications,” Complex Var. Elliptic Equ. 66 (6–7), 964–987 (2021).

    Article  MathSciNet  Google Scholar 

  15. E. M. Kartashev, “New operational relationships for mathematical models of locally nonequilibrium heat exchange,” Russ. Tech. J. 10 (1), 68–79 (2022).

    Google Scholar 

  16. J. Zhang, X. Kang, Y. Liu, H. Ma, T. Li, Z. Ma, and S. Gataullin, “A secure and lightweight multi-party private intersection-sum scheme over a symmetric cryptosystem,” Symmetry 15 (2) (2023).

    Article  Google Scholar 

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Acknowledgments

The authors are grateful to S. Yu. Dobrokhotov for useful discussions.

Funding

This work was supported by ongoing institutional funding. No additional grants to carry out or direct this particular research were obtained.

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Authors and Affiliations

  1. MIREA—Russian Technological University, Moscow, 119454, Russia

    S. T. Gataullin

  2. Moscow Technical University of Communications and Informatics, Moscow, 111024, Russia

    T. M. Gataullin

Authors
  1. S. T. Gataullin
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  2. T. M. Gataullin
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Corresponding author

Correspondence to S. T. Gataullin.

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The authors of this work declare that they have no conflicts of interest.

Additional information

Translated from Matematicheskie Zametki, 2023, Vol. 114, pp. 822–826 https://doi.org/10.4213/mzm13923.

The paper is dedicated to the memory of V. V. Kucherenko

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Gataullin, S.T., Gataullin, T.M. To the Problem of a Point Source in an Inhomogeneous Medium. Math Notes 114, 1212–1216 (2023). https://doi.org/10.1134/S0001434623110524

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  • DOI: https://doi.org/10.1134/S0001434623110524

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