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Neumann problem for the Lavrent’ev–Bitsadze equation with two type-change lines in a rectangular domain

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Abstract

The Neumann problem for an equation with two perpendicular internal type-change lines in a rectangular domain is investigated. Uniqueness and existence theorems are proved by applying the spectral method. The separation of variables yields an eigenvalue problem for an ordinary differential equation. This problem is not self-adjoint, and the system of its eigenfunctions is not orthogonal. A corresponding biorthogonal system of functions is constructed and proved to be complete. The completeness result is used to prove a necessary and sufficient uniqueness condition for the problem under study. Its solution is constructed in the form of the sum of a biorthogonal series.

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References

  1. F. I. Frankl, Izv. Akad. Nauk SSSR, Ser. Mat. 9, 387–422 (1945).

    MathSciNet  Google Scholar 

  2. C. S. Morawetz, Michigan Math. J. 4 1, 5–14 (1957).

    Article  MathSciNet  Google Scholar 

  3. L. Bers, Mathematical Aspects of Subsonic and Transonic Gas Dynamics (New York Univ., New York, 1952; Inostrannaya Literatura, Moscow, 1961).

    MATH  Google Scholar 

  4. K. B. Sabitov and A. A. Akimov, Russ. Math. 45 10, 69–76 (2001).

    MathSciNet  Google Scholar 

  5. E. I. Moiseev, Differ. Uravn. 35 8, 1094–1100 (1999).

    MathSciNet  Google Scholar 

  6. K. B. Sabitov, Dokl. Math. 75 2, 193–196 (2007).

    Article  MathSciNet  Google Scholar 

  7. A. A. Gimaltdinova, Dokl. Math. 91 1, 41–46 (2015).

    Article  MathSciNet  Google Scholar 

  8. V. A. Il’in, Math. Notes 17 1, 53–58 (1975).

    Article  Google Scholar 

  9. V. I. Arnol’d, Izv. Akad. Nauk SSSR, Ser. Mat. 25, 21–86 (1961).

    MathSciNet  Google Scholar 

  10. I. S. Lomov, Differ. Uravn. 29 12, 2079–2089 (1993).

    MathSciNet  Google Scholar 

  11. I. S. Lomov, Differ. Equations 47 3, 355–362 (2011).

    Article  MathSciNet  Google Scholar 

  12. M. V. Keldysh, Russ. Math. Surv. 26 4, 15–44 (1971).

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. Sterlitamak Branch, Bashkortostan State University, pr. Lenina 49, Sterlitamak, 453103, Bashkortostan, Russia

    A. A. Gimaltdinova

  2. Institute of Applied Studies, Academy of Sciences of Republic Bashkortostan, Odesskaya ul. 68, Sterlitamak, 453103, Bashkortostan, Russia

    A. A. Gimaltdinova

Authors
  1. A. A. Gimaltdinova
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Correspondence to A. A. Gimaltdinova.

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Original Russian Text © A.A. Gimaltdinova, 2016, published in Doklady Akademii Nauk, 2016, Vol. 466, No. 1, pp. 7–11.

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Gimaltdinova, A.A. Neumann problem for the Lavrent’ev–Bitsadze equation with two type-change lines in a rectangular domain. Dokl. Math. 93, 1–5 (2016). https://doi.org/10.1134/S1064562416010038

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  • DOI: https://doi.org/10.1134/S1064562416010038

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