Skip to main content
Log in

New exact solutions of generalized convection-reaction-diffusion equation

  • Regular Article
  • Published:
The European Physical Journal Plus Aims and scope Submit manuscript

Abstract.

In this paper, we explain how to construct a complete classification of invariant subspaces for the generalized nonlinear convection-reaction-diffusion equation. Also, we have explicitly shown that the convection-reaction-diffusion equation admits more than one invariant subspaces in different dimensions which in turn helps to derive more than one different types of exact solution.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. B.H. Gilding, R. Kersner, Travelling Waves in Nonlinear Diffusion-Convection-Reaction (Birkhauser, 2004)

  2. L.A. Richards, Physica 1, 318 (1931)

    Google Scholar 

  3. J.D. Murray, Mathematical Biology (Springer-Verlag, Berlin, 1989)

  4. P. Glansdorff, I.R. Prigogine, Thermodynamic Theory of Structure, Stability and Fluctuations (Wiley Interscience, New York, 1971)

  5. H. Haken, Synergetics: An Introduction (Springer-Verlag, Berlin, New York, 1978)

  6. R. Cherniha, M. Serov, I. Rassokha, J. Math. Anal. Appl. 342, 1363 (2008)

    Article  MathSciNet  Google Scholar 

  7. P.A. Clarkson, E.L. Mansfield, Physica D 70, 250 (1993)

    Article  ADS  Google Scholar 

  8. V.A. Vladimirov, Cz. Maczka, Chaos, Solitons Fractals 44, 677 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  9. H. Jia, W. Xu, X.Zhao, Z. Li, J. Math. Anal. Appl. 339, 982 (2008)

    Article  MathSciNet  Google Scholar 

  10. S. Sharifi, J. Rashidinia, J. King Saud Univ.: Science (2019) https://doi.org/10.1016/j.jksus.2018.10.004

  11. G.R. Barrenechea, A.H. Poza, H. Yorston, Comput. Methods Appl. Mech. Eng. 339, 389 (2018)

    Article  ADS  Google Scholar 

  12. M. Hayek, Appl. Math. Comput. 218, 2407 (2011)

    MathSciNet  Google Scholar 

  13. V.A. Galaktionov, S.R. Svirshchevskii, Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics (Chapman and Hall/CRC, London, 2007)

  14. J.R. King, Physica D 64, 35 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  15. S.R. Svirshchevskii, Phys. Lett. A 199, 344 (1995)

    Article  ADS  MathSciNet  Google Scholar 

  16. S.R. Svirshchevskii, Commun. Nonlinear Sci. Numer. Simul. 9, 105 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  17. R. Hirota, B. Grammaticos, A. Ramani, J. Math. Phys. 27, 1499 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  18. W.X. Ma, Nonlinear Anal. 63, e2461 (2005)

    Article  Google Scholar 

  19. W.X. Ma, E.G. Fan, Comput. Math. Appl. 61, 950 (2011)

    Article  MathSciNet  Google Scholar 

  20. W.X. Ma, Sci. China Math. 55, 1769 (2012)

    Article  MathSciNet  Google Scholar 

  21. W.X. Ma, Y. Liu, Commun. Nonlinear Sci. Numer. Simul. 17, 3795 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  22. Y. Ye, W.X. Ma, S. Shen, D. Zhang, J. Nonlinear Math. Phys. 21, 132 (2014)

    Article  MathSciNet  Google Scholar 

  23. C.Z. Qu, C.R. Zhu, J. Phys. A: Math. Theor. 42, 475201 (2009)

    Article  ADS  Google Scholar 

  24. R. Sahadevan, P. Prakash, Chaos, Solitons Fractals 104, 107 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  25. H. Liu, Appl. Math. Lett. 83, 164 (2018)

    Article  MathSciNet  Google Scholar 

  26. A.D. Polyanin, A.I. Zhurov, Appl. Math. Lett. 37, 43 (2014)

    Article  MathSciNet  Google Scholar 

  27. R. Sahadevan, P. Prakash, Commun. Nonlinear Sci. Numer. Simul. 42, 158 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  28. R. Sahadevan, P. Prakash, Nonlinear Dyn. 85, 659 (2016)

    Article  Google Scholar 

  29. W. Rui, Appl. Math. Comput. 339, 158 (2018)

    MathSciNet  Google Scholar 

  30. R. Sahadevan, T. Bakkyaraj, Fract. Calc. Appl. Anal. 18, 146 (2015)

    Article  MathSciNet  Google Scholar 

  31. P. Artale Harris, R. Garra, Nonlinear Stud. 20, 471 (2013)

    MathSciNet  Google Scholar 

  32. S.F. Shen, C.Z. Qu, Y.Y. Jin, L.N. Ji, Chin. Ann. Math. Ser. B 33, 161 (2012)

    Article  MathSciNet  Google Scholar 

  33. C.R. Zhu, C.Z. Qu, J. Math. Phys. 52, 043507 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  34. J. Song, S. Shen, Y. Jin, J. Zhang, Commun. Nonlinear Sci. Numer. Simul. 18, 2984 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  35. M.S. Hashemi, Chaos, Solitions Fractals 107, 161 (2018)

    Article  ADS  Google Scholar 

  36. W.X. Ma, Y. Zhou, J. Differ. Equ. 264, 2633 (2018)

    Article  ADS  Google Scholar 

  37. W.X. Ma, J. Li, C.M. Khalique, Complexity 2018, 9059858 (2018)

    Google Scholar 

  38. S.T. Chen, W.X. Ma, Comput. Math. Appl. 76, 1680 (2018)

    Article  MathSciNet  Google Scholar 

  39. W.X. Ma, J. Appl. Anal. Comput. 9, 1 (2019)

    MathSciNet  Google Scholar 

  40. W.X. Ma, X. Yong, H.Q. Zhang, Comput. Math. Appl. 75, 289 (2018)

    Article  MathSciNet  Google Scholar 

  41. J.Y. Yang, W.X. Ma, Z. Qin, Anal. Math. Phys. 8, 427 (2018)

    Article  MathSciNet  Google Scholar 

  42. J.Y. Yang, W.X. Ma, Z. Qin, East Asian J. Appl. Math. 8, 224 (2018)

    Article  MathSciNet  Google Scholar 

  43. W.X. Ma, J. Geom. Phys. 133, 10 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  44. W.X. Ma, East Asian J. Appl. Math. 9, 185 (2019)

    Article  MathSciNet  Google Scholar 

  45. W.X. Ma, Acta Math. Sci. B 39, 498 (2019)

    Google Scholar 

  46. W.X. Ma, Phys. Lett. A 301, 35 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  47. V.A. Dorodnitsyn, Ufimsk. Math. J. 4, 186 (2012)

    Google Scholar 

  48. W.X. Ma, Y. Zhang, Y.N. Tang, J. Tu, Appl. Math. Comput. 218, 7174 (2012)

    MathSciNet  Google Scholar 

  49. W.X. Ma, Integrability, in Encyclopedia of Nonlinear Science, edited by A. Scott (Taylor & Francis, New York, 2005) pp. 250--253

  50. W.X. Ma, T.W. Huang, Y. Zhang, Phys. Scr. 82, 065003 (2010)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to P. Prakash.

Additional information

Publisher's Note

The EPJ Publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Prakash, P. New exact solutions of generalized convection-reaction-diffusion equation. Eur. Phys. J. Plus 134, 261 (2019). https://doi.org/10.1140/epjp/i2019-12657-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjp/i2019-12657-3

Navigation