Geodesic
Analytical solutions of a nonlinear convection-diffusion equation with polynomial sources
Modelirovanie i analiz informacionnyh sistem, Tome 23 (2016) no. 3, pp. 309-316.

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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
Mots-clés : nonlocal transformations, Liénard equations. 
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N. A. Kudryashov; D. I. Sinelshchikov. Analytical solutions of a nonlinear convection-diffusion equation with polynomial sources. Modelirovanie i analiz informacionnyh sistem, Tome 23 (2016) no. 3, pp. 309-316. http://geodesic.mathdoc.fr/item/MAIS_2016_23_3_a6/

[1] J. D. Murray, Mathematical Biology, v. I, An Introduction, Springer-Verlag, Berlin, 2001, 556 pp. | MR

[2] N.A. Kudryashov, A.S. Zakharchenko, “A note on solutions of the generalized Fisher equation”, Appl. Math. Lett., 32 (2014), 43–56 | DOI | MR

[3] A.D. Polyanin, V.F. Zaitsev, Handbook of Nonlinear Partial Differential Equations, Chapman and Hall/CRC, Boca Raton–London–New York, 2011, 112 pp. | MR

[4] M. Sabatini, “On the period function of $x^{''}+f(x)x^{'2}+g(x)=0$”, J. Differ. Equ., 196 (2004), 151–168 | DOI | MR | Zbl

[5] V.K. Chandrasekar, M. Senthilvelan, M. Lakshmanan, “On the complete integrability and linearization of certain second-order nonlinear ordinary differential equations”, Proc. R. Soc. A Math. Phys. Eng. Sci., 461 (2005), 2451–2476 | DOI | MR | Zbl

[6] A.K. Tiwari, S.N. Pandey, M. Senthilvelan, M. Lakshmanan, “Classification of Lie point symmetries for quadratic Lienard type equation $\ddot{x}+f(x)\dot{x}^{2}+g(x)=0$”, J. Math. Phys., 54 (2013), 053506 | DOI | MR | Zbl

[7] N.A. Kudryashov, D.I. Sinelshchikov, “On the criteria for integrability of the Lienard equation”, Appl. Math. Lett., 57 (2016), 114–120 | DOI | MR | Zbl

[8] N.A. Kudryashov, D.I. Sinelshchikov, “On the connection of the quadratic Lienard equation with an equation for the elliptic functions”, Regul. Chaotic Dyn., 20 (2015), 486–496 | DOI | MR | Zbl

[9] N.A. Kudryashov, D.I. Sinelshchikov, “Analytical solutions for problems of bubble dynamics”, Phys. Lett. A, 379 (2015), 798–802 | DOI | MR

[10] N.A. Kudryashov, D.I. Sinelshchikov, “Analytical solutions of the Rayleigh equation for empty and gas-filled bubble”, J. Phys. A Math. Theor., 47 (2014), 405202 | DOI | MR | Zbl

[11] W. Nakpim, S.V. Meleshko, “Linearization of Second-Order Ordinary Differential Equations by Generalized Sundman Transformations”, Symmetry, Integr. Geom. Methods Appl., 6 (2010), 051, 11 pp. | MR

[12] S. Moyo, S.V. Meleshko, “Application of the generalised Sundman transformation to the linearisation of two second-order ordinary differential equations”, J. Nonlinear Math. Phys., 12 (2011), 213–236 | DOI | MR

[13] E.L. Ince, Ordinary differential equations, Dover, New York, 1956, 558 pp. | MR

[14] E. Hille, Ordinary Differential Equations in the Complex Domain, Dover Publications, Mineola, 1997, 484 pp. | MR | Zbl

[15] E.T. Whittaker, G.N. Watson, A Course of Modern Analysis, Cambridge University Press, Cambridge, 1927, 620 pp. | MR | Zbl