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 Model. Anal. Inform. Sist., 2016, Volume 23, Number 3, Pages 309–316 (Mi mais500)

Analytical solutions of a nonlinear convection-diffusion equation with polynomial sources

N. A. Kudryashov, D. I. Sinelshchikov

National Research Nuclear University MEPhI, Kashirskoe shosse, 31, Moscow, 115409, Russia

Abstract: Nonlinear convection–diffusion equations are widely used for the description of various processes and phenomena in physics, mechanics and biology. In this work we consider a family of nonlinear ordinary differential equations which is a traveling wave reduction of a nonlinear convection–diffusion equation with a polynomial source. We study a question about integrability of this family of nonlinear ordinary differential equations. We consider both stationary and non–stationary cases of this equation with and without convection. In order to construct general analytical solutions of equations from this family we use an approach based on nonlocal transformations which generalize the Sundman transformations. We show that in the stationary case without convection the general analytical solution of the considered family of equations can be constructed without any constraints on its parameters and can be expressed via the Weierstrass elliptic function. Since in the general case this solution has a cumbersome form we find some correlations on the parameters which allow us to construct the general solution in the explicit form. We show that in the non–stationary case both with and without convection we can find a general analytical solution of the considered equation only imposing some correlation on the parameters. To this aim we use criteria for the integrability of the Lienard equation which have recently been obtained. We find explicit expressions in terms of exponential and elliptic functions for the corresponding analytical solutions.

Keywords: analytical solutions, elliptic function, nonlocal transformations, Liénard equations.

 Funding Agency Grant Number Russian Science Foundation 14-11-00258 This research was supported by Russian Science Foundation grant No. 14–11–00258.

DOI: https://doi.org/10.18255/1818-1015-2016-3-309-316

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English version:
Automatic Control and Computer Sciences, 2017, 51:7, 621–626

Bibliographic databases:

UDC: 517.9

Citation: N. A. Kudryashov, D. I. Sinelshchikov, “Analytical solutions of a nonlinear convection-diffusion equation with polynomial sources”, Model. Anal. Inform. Sist., 23:3 (2016), 309–316; Automatic Control and Computer Sciences, 51:7 (2017), 621–626

Citation in format AMSBIB
\Bibitem{KudSin16} \by N.~A.~Kudryashov, D.~I.~Sinelshchikov \paper Analytical solutions of a nonlinear convection-diffusion equation with polynomial sources \jour Model. Anal. Inform. Sist. \yr 2016 \vol 23 \issue 3 \pages 309--316 \mathnet{http://mi.mathnet.ru/mais500} \crossref{https://doi.org/10.18255/1818-1015-2016-3-309-316} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3520852} \elib{http://elibrary.ru/item.asp?id=26246296} \transl \jour Automatic Control and Computer Sciences \yr 2017 \vol 51 \issue 7 \pages 621--626 \crossref{https://doi.org/10.3103/S0146411617070148} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. N. A. Kudryashov, D. I. Sinelshchikov, “On the Integrability Conditions for a Family of Liénard-type Equations”, Regul. Chaotic Dyn., 21:5 (2016), 548–555
2. L. F. Spevak, A. L. Kazakov, P. A. Kuznetsov, “Trekhmernaya teplovaya volna, porozhdennaya kraevym rezhimom, zadannym na podvizhnom mnogoobrazii”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika, 26 (2018), 16–34
3. A. A. Kosov, È. I. Semenov, “Exact solutions of the nonlinear diffusion equation”, Siberian Math. J., 60:1 (2019), 93–107
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