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Exact solutions of nonlinear diffusion reaction equation with quadratic, cubic and quartic nonlinearities

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Abstract

In this work, using a generalized ansatz, we have presented the extended tanh-method for constructing a variety of exact traveling wave solutions of the nonlinear diffusion reaction equation with quadratic, cubic and quartic nonlinearities. We have examined the density independent and dependent nonlinear diffusion reaction equation with a convective flux term and successfully obtain some new and more general solutions like kink and antikink solitons. The present work confirms the significant features of the employed method and shows the variety of the obtained solutions.

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References

  1. L Debnath Nonlinear Partial Differential Equations for Scientists and Engineers (Bosten: Birkhauser) (1997)

    MATH  Google Scholar 

  2. M J Ablowitz and P A Clarkson Solitons, Nonlinear Evolution Equations and Inverse Scattering Transform (Cambridge: Cambridge University Press) (1990)

    Google Scholar 

  3. R Hirota The Direct Method in Soliton Theory (Cambridge: Cambridge University Press) (2004)

    Book  MATH  Google Scholar 

  4. A L Sakhnovich Inverse Probl. 10 699 (1994)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. A S Fokas and V E Zakharov J. Nonlinear Sci. 2 109 (1992)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  6. M J Ablowitz, A Ramani and H Segur Lett. Nuovo Cimento 23 333 (1978)

    Article  MathSciNet  Google Scholar 

  7. A Ramani, B Grammaticos and T Buntis Phys. Rep.180 159 (1989)

    Article  MathSciNet  ADS  Google Scholar 

  8. P Chatterjee, B Das, G Mondal, S V Muniandy and C S Wong Phys. Plasmas 17 103705 (2010)

    Article  ADS  Google Scholar 

  9. P Chatterjee, G Mondal, K Roy, S V Muniandy, S L Yap et al Phys. Plasmas 16 072102 (2009)

    Article  ADS  Google Scholar 

  10. W Malfliet and W Hereman Phys. Scr. 54 563 (1996)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  11. E Fan and H Zhang Phys. Lett. A 246 403 (1998)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  12. M L Wang Phys. Lett. A 199 169 (1995)

    Article  MathSciNet  ADS  Google Scholar 

  13. M L Wang Phys. Lett. A 213 279 (1996)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  14. Y B Zhou, M L Wang and Y M Wang Phys. Lett. A 308 31 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  15. J Lin, L Yang and K Yang Chaos Solit. Fract. 20 1175 (2004)

    ADS  Google Scholar 

  16. Z Y Yan Phys. Lett. A 224 77 (1996)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  17. A Malik, F Chand and S C Mishra Appl. Math. Comput. 216 2596 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. A Malik, F Chand, H Kumar and S C Mishra Indian J. Phys. 86 129 (2012)

  19. H R Pakzad Indian J. Phys. 84 867 (2010)

    Article  ADS  Google Scholar 

  20. Z Emami and H R Pakzad Indian J. Phys. 85 1643 (2011)

    Article  ADS  Google Scholar 

  21. H R Pakzad and K Javidan Indian J. Phys.83 349 (2009)

    Article  Google Scholar 

  22. P Chatterjee, B Das and C S Wong Indian J. Phys. doi:10.1007/s12648-012-0074-6

  23. K Roy and P Chatterjee Indian J. Phys. 85 1653 (2011)

    Google Scholar 

  24. J D Murray Mathematical Biology (New York: Springer) (1993)

    Book  MATH  Google Scholar 

  25. A M Turing Philos. Trans. R. Soc. Lond. 237 37 (1952)

    Article  ADS  Google Scholar 

  26. J Smoller Shock Waves and Reaction Diffusion Equations, (New York: Springer) (1994)

    Book  MATH  Google Scholar 

  27. R A Fisher Ann. Eng. 7 355 (1937)

    Google Scholar 

  28. E Fan Phys. Lett. A 277 212 (2000)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  29. E Fan and Y C Hon Math. Comput. 141 351 (2003)

    MathSciNet  MATH  Google Scholar 

  30. A M Wazwaz Appl. Math. Comput. 184 1002 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  31. A M Wazwaz Appl. Math. Comput. 188 1467 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  32. A M Wazwaz Appl. Math. Comput. 187 1131 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  33. A M Wazwaz Appl. Math. Comput. 195 24 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  34. A Bekir and A C Cevikel Commun. Nonlinear Sci. Numer. Simulat. 14 1804 (2009)

    Article  ADS  Google Scholar 

  35. C A G Sierra and A H Salas Appl. Math. Comput. 203 873 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  36. J R King J. Phys. A 24 3213 (1991)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  37. J D Logan An Introduction to Nonlinear Partial Differential Equations, 2nd edn. (New York: Wiley Interscience) (2008)

    Book  Google Scholar 

  38. R S Kaushal J. Phys. A 38 3897 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  39. R S Banarjee Int. J. Theo. Phys. 32 879 (1993)

    Article  Google Scholar 

  40. V A Galaktionov Phys. D 238 1717 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  41. V M Kenkre and M N Kuperman Phys. Rev. E 67 051921 (2003)

    Article  ADS  Google Scholar 

  42. D R Nelson and N M Shnerb Phys. Rev. E 58 1383 (and references theirin) (1998)

  43. J T Chalker and Z J Wang Phys. Rev. Lett. 79 1797 (1997)

    Article  ADS  Google Scholar 

  44. N Moiseyev and M Gluck Phys. Rev. E 63 041103 (2001)

    Article  ADS  Google Scholar 

  45. M Remoissent Waves Called Solitons: Concept and Experiments (Berlin–Heidelberg: Springer) (1999)

    Google Scholar 

  46. C Zhai-Xiong and G Ben-Yu IMA J. Appl. Math. 48 107 (1992)

    Article  MathSciNet  ADS  Google Scholar 

  47. R S Kaushal, R Kumar and A Prasad Pramana J. Phys. 67 249 (2006)

    Article  ADS  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Physics, Kurukshetra University, Kurukshetra, 136 119, Haryana, India

    Hitender Kumar, Anand Malik, Fakir Chand & S. C. Mishra

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  1. Hitender Kumar
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  2. Anand Malik
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  3. Fakir Chand
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  4. S. C. Mishra
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Correspondence to Hitender Kumar.

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Kumar, H., Malik, A., Chand, F. et al. Exact solutions of nonlinear diffusion reaction equation with quadratic, cubic and quartic nonlinearities. Indian J Phys 86, 819–827 (2012). https://doi.org/10.1007/s12648-012-0126-y

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  • DOI: https://doi.org/10.1007/s12648-012-0126-y

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