Vestnik КRAUNC. Fiz.-Mat. nauki. 2022. vol. 38. no. 1. P. 28-39. ISSN 2079-6641

Contents of this issue

Read Russian Version US Flag

MSC 35G30, 35Q53

Research Article

On a boundary value problem for an odd-order equation with multiple characteristics

O. T. Kurbanov

Tashkent State Economic University, Uzbekistan, 100003, Tashkent, Chilanzar district, Ave. Islam Karimov, 49

E-mail: odil69@inbox.ru

A nonlinear boundary value problem for a third-order nonlinear equation with multiple characteristics is studied in the article in a curvilinear domain. The unique solvability of the problem is proved. The uniqueness of the solution to the boundary value problem is proved by the energy integral method using some elementary inequalities. An auxiliary problem is considered for the existence of a solution, for which the Green function is constructed. By solving an auxiliary problem, the original problem is reduced to a system of Hammerstein integral equations. The solvability of a nonlinear system is proved by the contracting mapping method.

Key words: nonlinearity, uniqueness, existence, system of Hammerstein equations.

DOI: 10.26117/2079-6641-2022-38-1-28-39

Original article submitted: 09.02.2022

Revision submitted: 16.03.2022

For citation. Курбанов О.Т. On a boundary value problem for an odd order equation with multiple characteristics. Vestnik KRAUNC. Fiz.-mat. nauki. 2022, 38: 1, 28-39. DOI: 10.26117/2079-6641-2022-38-1-28-39

The content is published under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0/deed.ru)

© Kurbanov O. T., 2022

Competing interests. The author declares that there are no conflicts of interest with respect to authorship and publication.

Contribution and responsibility. The author contributed to the writing of the article and is solely responsible for submitting the final version of the article to the press. The final version of the manuscript was approved by the author.

References

  1. Кorteweg D. J., De Vries G.On the change of form of long waves advancing in a rectangular channel, and on a new type of long stationary waves, Phil. Mag., 1985. vol. 39, pp. 422–443.
  2. Baranov V. B., Krasnobaev K. V. Gidrodinamicheskaya teoriya kosmicheskoy plazmy [Hydrodynamic theory of space plasma]. Moscow: Nauka, 1977 (In Russian).
  3. Karpman V. I. Nelineynyye volny v dispergiruyushchikh sredakh [Nonlinear waves in dispersive media]. Moscow: Nauka, 1973 (In Russian).
  4. Cattabriga L Un problem al contorno per una equazione parabolica di ordin dispari.,Amali della Souola Normale Superiore di Pisa a Matematica, 1959. vol. XIII. Fasc, no. II. Series III., pp. 163-203.
  5. Juraev T. J. Krayevyye zadachi dlya uravneniy smeshannogo i smeshanno-sostavnogo tipov [Boundary value problems for mixed and mixed-composite types equations]. Tashkent: Fan, 1979 (In Russian).
  6. Abdinazarov S., Khashimov A. R.On boundary value problems for equations of the third order with multiple characteristics and discontinuous coefficients, Uzb. Mat. Jour, 1993. vol. 1, pp. 3-12 (In Russian).
  7. Khashimov A. R. Nonlinear boundary value problems for the equation of the third order with multiple characteristics, Uzb. Mat. Jour, 1993. vol. 2, pp. 97-102 (In Russian).
  8. Kurbanov O. T., Kholboev B. M.On a non-linear boundary value problem for an odd-order equation with multiple characteristics, Uzb. Mat. Jour, 2003. vol. 3, pp. 35-40 (In Russian).
  9. Ladyzhenskaya O. A., Solonnikov V. A., Uraltseva N. N. Lineynyye i kvazilineynyye uravneniya parabolicheskogo tipa [Linear and quasilinear equations of parabolic type]. Moscow: Nauka, 1967 (In Russian).

Kurbanov Odilzhan Tukhtamuradovich – PhD (Phys. & Math.), Associate Professor, Associate Professor, Dept. Apl. Math., Tashkent State University of Economics, Tashkent, Uzbekistan, ORCID 0000-0003-1360-325x.

Download article: Курбанов О.Т. On a boundary value problem for an odd order equation with multiple characteristics