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Home » Issues » Archive of Issues » 2023 - 2 issue » pp. 312-328

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Uzbekistan Ferghana; Tashkent
Year
2023
Volume
33
Issue
2
Pages
312-328
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Section Mathematics
Title On one problem for a fourth-order mixed-type equation that degenerates inside and on the boundary of a domain
Author(-s) Urinov A.K.ab, Usmonov D.A.a
Affiliations Ferghana State Universitya, Institute of Mathematics, National Academy of Sciences of Uzbekistanb
Abstract In the article, a nonlocal boundary value problem has been investigated for a fourth-order mixed-type equation degenerating inside and on the boundary of a domain. Applying the method of separation of variables to the problem under study, the spectral problem for an ordinary differential equation is obtained. The Green function of the last problem is constructed, with the help of which it is equivalently reduced to the Fredholm integral equation of the second kind with a symmetric kernel, which implies the existence of eigenvalues and the system of eigenfunctions for the spectral problem. The theorem of expansion of a given function into a uniformly convergent series with respect to the system of eigenfunctions is proved. Using the found integral equation and Mercer's theorem, a uniform convergence of some bilinear series depending on the found eigenfunctions is proved. The order of the Fourier coefficients is established. The solution of the problem under study is written as the sum of the Fourier series with respect to the system of eigenfunctions of the spectral problem. An estimate for the problem's solution is obtained, from which its continuous dependence on the given functions follows.
Keywords degenerate mixed-type equations, spectral problem, Green's function, integral equation, Fourier series, method of separation of variables
UDC 517.956
MSC 35G15
DOI 10.35634/vm230209
Received 29 December 2022
Language Russian
Citation Urinov A.K., Usmonov D.A. On one problem for a fourth-order mixed-type equation that degenerates inside and on the boundary of a domain, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2023, vol. 33, issue 2, pp. 312-328.
References
  1. Vakhaniya N.N. A special problem for an equation of mixed type, Trudy Akademii Nauk Gruzinskoi SSR, 1963, vol. 3. pp. 69-80 (in Russian). https://zbmath.org/03261145
  2. Cannon J.R. A Dirichlet problem for an equation of mixed type with a discontinuous coefficient, Annali di Matematica Pura ed Applicata, 1963, vol. 61, issue 1, pp. 371-377. https://doi.org/10.1007/BF02410656
  3. Khachev M.M. The Dirichlet problem for a certain equation of mixed type, Differentsial'nye Uravneniya, 1976, vol. 12, no. 1, pp. 137-143 (in Russian). https://www.mathnet.ru/eng/de2658
  4. Sokhadze R.I. The first boundary value problem for an equation of mixed type with weighted glueing conditions along a line of parabolic degeneration, Differentsial'nye Uravneniya, 1981, vol. 17, no. 1, pp. 150-156 (in Russian). https://www.mathnet.ru/eng/de4178
  5. Sabitov K.B., Suleimanova A.Kh. The Dirichlet problem for a mixed-type equation of the second kind in a rectangular domain, Russian Mathematics, 2007, vol. 51, issue 4, pp. 42-50. https://doi.org/10.3103/S1066369X07040068
  6. Khairullin R.S. Dirichlet problem for a mixed type equation of the second kind in exceptional cases, Differential Equations, 2018, vol. 54, issue 4, pp. 562-565. https://doi.org/10.1134/S0012266118040134
  7. Khairullin R.S. Problem with a periodicity condition for an equation of the mixed type with strong degeneration, Differential Equations, 2019, vol. 55, issue 8, pp. 1105-1117. https://doi.org/10.1134/S0012266119080111
  8. Sabitova Yu.K. Boundary-value problem with nonlocal integral condition for mixed-type equations with degeneracy on the transition line, Mathematical Notes, 2015, vol. 98, issue 3, pp. 454-465. https://doi.org/10.1134/S0001434615090114
  9. Safina R.M. Keldysh problem for a mixed-type equation of the second kind with the Bessel operator, Differential Equations, 2015, vol. 51, issue 10, pp. 1347-1359. https://doi.org/10.1134/S0012266115100109
  10. Irgashev B.Yu. On the conditions for the solvability of boundary-value problems for higher-order equations with discontinuous coefficients, Mathematical Notes, 2022, vol. 111, issue 2, pp. 217-229. https://doi.org/10.1134/S0001434622010254
  11. Irgashev B.Yu. On a boundary value problem for a high order mixed type equation, Sibirskie Èlektronnye Matematicheskie Izvestiya, 2020, vol. 17, pp. 899-912. https://doi.org/10.33048/semi.2020.17.066
  12. 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. https://doi.org/10.35634/vm200103
  13. Fayazov K.S., Khudayberganov Ya.K. Ill-posed boundary value problem for mixed type system equations with two degenerate lines, Sibirskie Èlektronnye Matematicheskie Izvestiya, 2020, vol. 17, pp. 647-660 (in Russian). https://doi.org/10.33048/semi.2020.17.043
  14. Sabitov K.B. Initial boundary and inverse problems for the inhomogeneous equation of a mixed parabolic-hyperbolic equation, Mathematical Notes, 2017, vol. 102, issue 3, pp. 378-395. https://doi.org/10.1134/S0001434617090085
  15. Sidorov S.N. Nonlocal problems for an equation of mixed parabolic-hyperbolic type with power degeneration, Russian Mathematics, 2015, vol. 59, issue 12, pp. 46-55. https://doi.org/10.3103/S1066369X15120051
  16. Sidorov S.N. Inverse problems for a mixed parabolic-hyperbolic equation with a degenerate parabolic part, Sibirskie Èlektronnye Matematicheskie Izvestiya, 2019, vol. 16, pp. 144-157 (in Russian). https://doi.org/10.33048/semi.2019.16.007
  17. Yuldashev T.K., Kadirkulov B.J. Nonlocal problem for a mixed type fourth-order differential equation with Hilfer fractional operator, Ural Mathematical Journal, 2020, vol. 6, issue 1, pp. 153-167. https://doi.org/10.15826/umj.2020.1.013
  18. Kadirkulov B.J., Jalilov M.A. On a nonlocal problem for a fourth-order mixed-type equation with the Hilfer operator, Bulletin of the Karaganda University-Mathematics, 2021, vol. 104, issue 4, pp. 89-102. https://doi.org/10.31489/2021m4/89-102
  19. Jalilov M.A. On a problem for a nonlocal mixed-type equation of fractional order with degeneration, Lobachevskii Journal of Mathematics, 2021, vol. 42, issue 15, pp. 3652-3660. https://doi.org/10.1134/S1995080222030118
  20. Feng Pengbin, Karimov E.T. Inverse source problems for time-fractional mixed parabolic-hyperbolic-type equations, Journal of Inverse and Ill-posed Problems, 2015, vol. 23, issue 4, pp. 339-353. https://doi.org/10.1515/jiip-2014-0022
  21. Karimov E., Al-Salti N., Kerbal S. An inverse source non-local problem for a mixed type equation with a Caputo fractional differential operator, East Asian Journal of Applied Mathematics, 2017, vol. 7, issue 2, pp. 417-438. https://doi.org/10.4208/eajam.051216.280217a
  22. Yuldashev T.K., Karimov E.T. Inverse problem for a mixed type integro-differential equation with fractional order Caputo operators and spectral parameters, Axioms, 2020, vol. 9, issue 4, 121. https://doi.org/10.3390/axioms9040121
  23. Karimov E., Ruzhansky M., Toshtemirov B. Solvability of the boundary-value problem for a mixed equation involving hyper-Bessel fractional differential operator and bi-ordinal Hilfer fractional derivative, Mathematical Methods in the Applied Sciences, 2022, vol. 46, issue 1, pp. 54-70. https://doi.org/10.1002/mma.8491
  24. Yuldashev T.K., Kadirkulov B.J. Inverse boundary value problem for a fractional differential equations of mixed type with integral redefinition conditions, Lobachevskii Journal of Mathematics, 2021, vol. 42, issue 3, pp. 649-662. https://doi.org/10.1134/S1995080221030227
  25. Irgashev B.Yu. On a problem with conjugation conditions for an equation of even order involving a Caputo fractional derivative, Mathematical Notes, 2022, vol. 112, issue 2, pp. 215-222. https://doi.org/10.1134/S0001434622070252
  26. Irgashev B.Yu. A boundary value problem with conjugation conditions for a degenerate equation with the Caputo fractional derivative, Russian Mathematics, 2022, vol. 66, issue 4, pp. 24-31. https://doi.org/10.3103/S1066369X2204003X
  27. Kilbas A.A., Srivastava H.M., Trujillo J.J. Theory and applications of fractional differential equations, Amsterdam: Elsevier, 2006.
  28. Bateman H. Higher transcendental functions. Vol. 1, New York: McGraw-Hill, 1953.
  29. Naimark M.A. Lineinye differentsial'nye operatory (Linear differential operators), Moscow: Fizmatlit, 1969.
  30. Mikhlin S.G. Lektsii po lineinym integral'nym uravneniyam (Lectures on linear integral equations), Moscow: Fizmatlit, 1959.
  31. Boudabsa L., Simon T. Some properties of the Kilbas-Saigo function, Mathematics, 2021, vol. 9, issue 3, 217. https://doi.org/10.3390/math9030217
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