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Uzbekistan Ferghana
Year
2020
Volume
30
Issue
1
Pages
31-48
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Section Mathematics
Title Keldysh problem for a three-dimensional equation of mixed type with three singular coefficients in a semi-infinite parallelepiped
Author(-s) Karimov K.T.a
Affiliations Ferghana State Universitya
Abstract This article studies the Keldysh problem for a three-dimensional equation of mixed type with three singular coefficients in a semi-infinite parallelepiped. Based on the completeness property of eigenfunction systems of two one-dimensional spectral problems, the uniqueness theorem is proved. To prove the existence of a solution to the problem, the Fourier spectral method based on the separation of variables is used. The solution to this problem is constructed in the form of a sum of a double Fourier-Bessel series. In substantiating the uniform convergence of the constructed series, we used asymptotic estimates of the Bessel functions of the real and imaginary argument. Based on them, estimates were obtained for each member of the series, which made it possible to prove the convergence of the series and its derivatives to the second order inclusive, as well as the existence theorem in the class of regular solutions.
Keywords Keldysh problem, mixed type equation, spectral method, singular coefficient, Bessel function
UDC 517.956.6
MSC 35J15, 35J75
DOI 10.35634/vm200103
Received 25 September 2019
Language Russian
Citation Karimov K.T. Keldysh problem for a three-dimensional equation of mixed type with three singular coefficients in a semi-infinite parallelepiped, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2020, vol. 30, issue 1, pp. 31-48.
References
  1. Keldysh M.V. On some cases of degenerate elliptic equations on the boundary of a domain, Dokl. Akad. Nauk SSSR, 1951, vol. 77, no. 2, pp. 181-183 (in Russian).
  2. Pul'kin S.P. On uniqueness of solution of singular Hellerstedt problem, Izv. Vyssh. Uchebn. Zaved., Mat., 1960, no. 6, pp. 214-225 (in Russian).
  3. Vostrova L.E., Pul'kin S.P. Singular problem with a normal derivative, Volzhsk. Matem. Sborn., 1966, no. 5, pp. 49-57 (in Russian).
  4. Sabitov K.B. K teorii uravnenii smeshannogo tipa (On the theory of equations of the mixed type), Moscow: Fizmatlit, 2014.
  5. Safina R.M. Keldysh problem for Pul'kins equation in rectangular domaim, Vestnik of Samara university, 2015, no. 3, pp. 53-64. https://journals.ssau.ru/index.php/est/article/view/4491/43907
  6. Safina R.M. Keldysh problem for a mixed-type equation of the second kind with the Bessel operator, Differential Equations, 2015, vol. 51, no. 10, pp. 1347-1359. https://doi.org/10.1134/S0012266115100109
  7. Safina R.M. Keldysh problem for mixed type equation with strong characteristic degeneration and singular coefficient, Russian Mathematics, 2017, vol. 61, issue 8, pp. 46-54. https://doi.org/10.3103/S1066369X17080059
  8. Zaitseva N.V. Keldysh type problem for B-hyperbolic equation with integral boundary value condition of the first kind, Lobachevskii Journal of Mathematics, 2017, vol. 38, no. 1, pp. 162-169. https://doi.org/10.1134/S199508021701022X
  9. Urinov A.K., Karimov K.T. The unique solvability of boundary value problems for a 3D elliptic equation with three singular coefficients, Russian Mathematics, 2019, vol. 63, issue 2, pp. 62-73. https://doi.org/10.3103/S1066369X19020087
  10. Urinov A.K., Karimov K.T. The Keldysh problem for a three-dimensional equation of mixed type with three singular coefficients, Aktual'nye problemy prikladnoi matematiki: sbornik statei (Actual problems of applied mathematics), Kabardino Balkar Scientific Center of the Russian Academy of Sciences, Nalchik, 2018, p. 253 (in Russian). https://elibrary.ru/item.asp?id=36308506
  11. Tikhonov A.N., Samarsky A.A. Uravneniya matematicheskoi fiziki (Equations of mathematical physics), Moscow: Nauka, 1972.
  12. Watson G.N. A treatise on the theory of Bessel functions, Cambridge: Cambridge University Press, 1922. Translated under the title Teoriya besselevykh funktsii, Moscow: Inostr. Literat., 1949.
  13. Olver F.W.J. Introduction to asymptotics and special functions, New York: Academic Press, 1974. Translated under the title Vvedenie v asimptoticheskie metody i spetsial'nye funktsii, Moscow: Mir, 1986.
  14. Lebedev N.N. Spetsial'nye funktsii i ikh prilozheniya (Special functions and their applications), Moscow: Fizmatlit, 1963.
  15. Urinov A.K., Karimov K.T. The Dirichlet problem for a three-dimensional equation of mixed type with three singular coefficients, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki, 2017, vol. 21, no. 4, pp. 665-683. https://doi.org/10.14498/vsgtu1559
  16. Tolstov G.P. Ryady Fur'e (Fourier Series), Moscow: Nauka, 1980.
  17. Fikhtengol'ts G.M. Kurs differentsial'nogo i integral'nogo ischisleniya. Tom 2 (A course of differential and integral calculus. Vol. 2), Moscow: Fizmatlit, 1963.
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