BSU Bulletin. Mathematics, Informatics
Bibliographic description:
, , A GENERALIZED BOUSSINESQ-TYPE EQUATION AND ITS EXACT MULTIDIMENSIONAL SOLUTIONS // BSU Bulletin. Mathematics, Informatics. - 2020. №1. . - С. 3-10.
Title:
A GENERALIZED BOUSSINESQ-TYPE EQUATION AND ITS EXACT MULTIDIMENSIONAL SOLUTIONS
Financing:
Работа выполнена при частичной финансовой поддержке РФФИ (проект № 19- 08-00746).
Codes:
Annotation:
The article studies a nonlinear fourth-order partial differential equation. The right part of the equation contains multidimensional analogs of Boussinesq equation, ex- pressed in terms of two-fold Laplace operators and squares of gradients of the re- quired function. To find the time-dependent components of the original system so- lution a system of nonlinear ordinary differential equations has been created. This system is reduced to a single fourth-order equation for which partial solutions are found. We give the examples of the constructed exact solutions of the initial system of Boussinesq-type equations, including those expressed in terms of Jacobi and Weierstrass elliptic functions in time and anisotropic ones in spatial variables. The exact solutions found have not only theoretical, but also applied value, since they can be used for testing and verifying numerical methods and algorithms for con- structing approximate solutions of boundary value problems for fourth-order nonlinear partial differential equations modeling hydrodynamic processes and phe- nomena.
Keywords:
system of ordinary differential equations; Laplace operator; nonlinear Boussinesq-type equations; reduction; exact solutions.
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Polyanin A. D., Zhurov A. I. Resheniya s funktsionalnym razdeleniem pere- mennykh dvukh klassov nelineinykh uravnenii matematicheskoi fiziki [Solutions with Functional Separation of Variables of Two Classes of Nonlinear Equations in Mathe- matical Physics]. Doklady AN, 2019. Vol. 486, no. 3. Pp. 287–291.
Pavlov M. V. Uravnenie Bussineska i preobrazovanie Miury [Boussinesq Equa- tion and Miura Transformation]. Journal of Mathematical Sciences. 2004. Vol. 10, no. 1. Pp. 175–182.
Polyanin A. D., Zaitsev V. F. Spravochnik po nelineinym uravneniyam matematicheskoi fiziki: Tochnye resheniya [Reference on Nonlinear Equations of Mathematical Physics: Exact Solutions]. Moscow, Fizmatlit Publ., 2002, 432 p.
Polyanin A. D., Zaitsev V. F., Zhurov A. I. Metody resheniya nelineinykh uravnenii matematicheskoi fiziki i mekhaniki [Methods for Solving Nonlinear Equations of Mathematical Physics and Mechanics]. Moscow: Fizmatlit Publ., 2005, 256 p.
Galactionov V. A., Svirshchevskii S. R. Subspaces of Nonlinear Partial Differen- tial Equations in Mechanics and Physics. Chapman & Hall/CRC, 2007. 493 p.
Kosov A. A., Semenov E. I. O tochnykh mnogomernykh resheniyakh sistemy uravnenii reaktsii-diffuzii so stepennymi nelineinostyami [On Exact Multidimensional Solutions of a Nonlinear System of Reaction–Diffusion Equations with Power-Law Nonlinear Terms]. Siberian Math. J. 2017. Vol. 58, no. 4. Pp. 619–632. https://doi.org/10.1134/S0037446617040085
Kosov A. A., Semyonov E. I. O tochnykh mnogomernykh resheniyakh odnoi nelineinoi sistemy uravnenii reaktsii-diffuzii [On Exact Multidimensional Solutions of a Nonlinear System of Reaction–Diffusion Equations.]. 2018. Vol. 54, no. 1. Pp. 106– 120. https://doi.org/10.1134/S0012266118010093
Kosov A. A., Semenov E. I., Tirskikh V. V. On Exact Multidimensional Solu- tions of a Nonlinear System of First Order Partial Differential Equation. Izvestiya Irkut- skogo gosudarstvennogo universiteta. Ser. Matematika. 2019. Vol. 28. Pp. 53–68. https://doi.org/10.26516/1997-7670.2019.28.53
Kosov A. A., Semenov E. I., Tirskikh V. V. Mnogomernye tochnye resheniya sistemy nelineinykh uravnenii tipa Bussineska [Multidimensional Exact Solutions for a System of Nonlinear Boussinesq-Type Equations]. Izvestiya Irkutskogo gosudarstven- nogo universiteta. Ser. Matematika. 2019. Vol. 30. Pp. 114–124. https://doi.org/10.26516/1997-7670.2019.30.114
Polyanin A. D., Zhurov A. I. Resheniya s funktsionalnym razdeleniem pere- mennykh dvukh klassov nelineinykh uravnenii matematicheskoi fiziki [Solutions with Functional Separation of Variables of Two Classes of Nonlinear Equations in Mathe- matical Physics]. Doklady AN, 2019. Vol. 486, no. 3. Pp. 287–291.
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