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Gellerstedt Problem with a Nonlocal Oddness Boundary Condition for the Lavrent’ev–Bitsadze Equation

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Abstract

We study the Gellerstedt problem for the Lavrent’ev–Bitsadze equation with the oddness boundary condition on the boundary of the ellipticity domain. All eigenvalues and eigenfunctions are obtained in closed form. It is proved that the system of eigenfunctions is complete in the elliptic part of the domain and incomplete in the entire domain. The unique solvability of the problem is also proved; the solution is written in the form of a series if the spectral parameter is not equal to an eigenvalue. For the spectral parameter coinciding with an eigenvalue, solvability conditions are obtained under which the family of solutions is found in the form of a series. A condition for the solvability of the problem depending on the eigenvalues is obtained. The constructed analytical solutions can be used efficiently in numerical modeling of transonic gas dynamics problems.

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REFERENCES

  1. Ponomarev, S.M., Spectral theory of the principal boundary value problem for the Lavrent’ev–Bitsadze equation, Doctoral (Phys.-Math.) Dissertation, Moscow, 1981.

  2. Tikhonov, A.N. and Samarskii, A.A., Uravneniya matematicheskoi fiziki (Equations of Mathematical Physics), Moscow: Nauka, 1966.

    Google Scholar 

  3. Watson, G.N., A Treatise on the Theory of Bessel Functions, Cambridge: Cambridge Univ. Press, 1945. Translated under the title: Teoriya besselevykh funktsii. T. 1 , Moscow: Izd. Inostr. Lit., 1949.

    Google Scholar 

  4. Bateman, H. and Erdélyi, A., Higher Transcendental Functions, New York–Toronto–London: McGraw-Hill, 1953. Translated under the title Vysshie transtsendentnye funktsii. T. 2, Moscow: Nauka, 1965.

  5. Moiseev, E.I., On the differential properties of expansions in the system of sines and cosines, Differ. Uravn., 1996, vol. 32, no. 1, pp. 117–126.

    MathSciNet  Google Scholar 

  6. Moiseev, E.I., The basis property for systems of sines and cosines, Dokl. Akad. Nauk SSSR, 1984, vol. 275, no. 4, pp. 794–798.

    MathSciNet  Google Scholar 

  7. Zygmund, A., Trigonometric Series, Cambridge: Cambridge Univ. Press, 1959. Translated under the title: Trigonometricheskie ryady. T. 1 , Moscow, 1985.

    MATH  Google Scholar 

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Funding

This work was carried out with financial support from the Ministry of Science and Higher Education of the Russian Federation within the framework of the program of the Moscow Center for Fundamental and Applied Mathematics under agreement no. 075-15-2022-284.

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Authors and Affiliations

  1. Lomonosov Moscow State University, Moscow, 119991, Russia

    T. E. Moiseev

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  1. T. E. Moiseev
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Correspondence to T. E. Moiseev.

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Translated by V. Potapchouck

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Moiseev, T.E. Gellerstedt Problem with a Nonlocal Oddness Boundary Condition for the Lavrent’ev–Bitsadze Equation. Diff Equat 59, 1372–1383 (2023). https://doi.org/10.1134/S00122661230100051

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  • DOI: https://doi.org/10.1134/S00122661230100051

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