Automatic Control and Computer Sciences

, Volume 51, Issue 7, pp 621–626 | Cite as

Analytical Solutions for Nonlinear Convection–Diffusion Equations with Nonlinear Sources



Nonlinear equations of the convection–diffusion type with nonlinear sources are used for the description of many processes and phenomena in physics, mechanics, and biology. In this work, we consider the family of nonlinear differential equations which are the traveling wave reductions of the nonlinear convection–diffusion equation with polynomial sources. The question of the construction of the general analytical solutions to these equations is studied. The steady-state and nonstationary cases with and without account of convection are considered. The approach based on the application of nonlocal transformations generalizing the Sundman transformation is applied for construction of the analytical solutions. It is demonstrated that in the steady-state case without account of convection, the general analytical solution can be found without any constraints on the equation parameters and it is expressed in terms of the Weierstrass elliptic function. In the general case, this solution has a cumbersome form; the constraints on the parameters for which it has a simple form are found, and the corresponding analytical solutions are obtained. It is shown that in the nonstationary case, both with and without account of convection, the general solution to the studied equations can be constructed under certain constraints on the parameters. The integrability criteria for the Liénard-type equations are used for this purpose. The corresponding general analytical solutions to the studied equations are explicitly constructed in terms of exponential or elliptic functions.


analytical solutions elliptic functions nonlocal transformations Liénard equations 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Murray, J.D., Mathematical Biology. I. An Introduction, Springer-Verlag, Berlin, 2001.Google Scholar
  2. 2.
    Kudryashov, N.A. and Zakharchenko, A.S., A note on solutions of the generalized Fisher equation, Appl. Math. Lett., 2014, vol. 32, pp. 43–56.MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Polyanin, A.D. and Zaitsev, V.F., Handbook of Nonlinear Partial Differential Equations, Boca Raton–London–New York: Chapman and Hall/CRC, 2011.MATHGoogle Scholar
  4. 4.
    Sabatini, M., On the period function of x'' + f(x)(x')2 + g(x) = 0, J. Differ. Equations, 2004, vol. 196, pp. 151–168.CrossRefMATHGoogle Scholar
  5. 5.
    Chandrasekar, V.K., Senthilvelan, M., and Lakshmanan, M., On the complete integrability and linearization of certain second-order nonlinear ordinary differential equations, Proc. R. Soc. A Math. Phys. Eng. Sci., 2005, vol. 461, pp. 2451–2476.MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Tiwari, A.K., Pandey, S.N., Senthilvelan, M., and Lakshmanan, M., Classification of Lie point symmetries for quadratic Liénard type equation x'' + (x')2 + g(x) = 0, J. Math. Phys., 2013, vol. 54.Google Scholar
  7. 7.
    Kudryashov, N.A. and Sinelshchikov, D.I., On the criteria for integrability of the Liénard equation, Appl. Math. Lett., 2016, vol. 57, pp. 114–120.MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Kudryashov, N.A. and Sinelshchikov, D.I., On the connection of the quadratic Liénard equation with an equation for the elliptic functions, Regul. Chaotic Dyn., 2015, vol. 20, pp. 486–496.MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Kudryashov, N.A. and Sinelshchikov, D.I., Analytical solutions for problems of bubble dynamics, Phys. Lett. A, 2015, vol. 379, pp. 798–802.MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kudryashov, N.A. and Sinelshchikov, D.I., Analytical solutions of the Rayleigh equation for empty and gasfilled bubble, J. Phys. A Math. Theor., 2014, vol. 47.Google Scholar
  11. 11.
    Nakpim, W. and Meleshko, S.V., Linearization of second-order ordinary differential equations by generalized Sundman transformations, Symmetry Integr. Geom. Methods Appl., 2010, vol. 6, pp. 1–11.MathSciNetMATHGoogle Scholar
  12. 12.
    Moyo, S. and Meleshko, S.V., Application of the generalised Sundman transformation to the linearisation of two second-order ordinary differential equations, J. Nonlinear Math. Phys., 2011, vol. 12, pp. 213–236.MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Ince, E.L., Ordinary Differential Equations, New York: Dover, 1956.MATHGoogle Scholar
  14. 14.
    Hille, E., Ordinary Differential Equations in the Complex Domain, Mineola: Dover Publications, 1997.MATHGoogle Scholar
  15. 15.
    Whittaker, E.T. and Watson, G.N., A Course of Modern Analysis, Cambridge: Cambridge University Press, 1927.MATHGoogle Scholar

Copyright information

© Allerton Press, Inc. 2017

Authors and Affiliations

  1. 1.National Nuclear Research University MEPhI (Moscow Engineering Physics Institute)MoscowRussia

Personalised recommendations