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Solitons and periodic solutions to a couple of fractional nonlinear evolution equations

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Abstract

This paper studies a couple of fractional nonlinear evolution equations using first integral method. These evolution equations are foam drainage equation and Klein–Gordon equation (KGE), the latter of which is considered in (2 + 1) dimensions. For the fractional evolution, the Jumarie’s modified Riemann–Liouville derivative is considered. Exact solutions to these equations are obtained.

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References

  1. K S Miller and B Ross, An introduction to the fractional calculus and fractional differential equations (Wiley, New York, 1993)

  2. A A Kilbas, H M Srivastava and J J Trujillo, Theory and applications of fractional differential equations (Elsevier, San Diego, 2006)

  3. I Podlubny, Fractional differential equations (Academic Press, San Diego, 1999)

  4. A Biswas, C Zony and E Zerrad, Appl Math Comput 203(1), 153 (2008)

  5. A Biswas, Int. J. Theor. Phys. 48, 256 (2009)

    Google Scholar 

  6. A Biswas, Nonlinear Dyn. 58, 345 (2009)

    Google Scholar 

  7. A Biswas, Phys. Lett. A 372, 4601 (2008)

    Google Scholar 

  8. A Biswas, Appl. Math. Lett. 22, 208 (2009)

    Google Scholar 

  9. W X Ma, Phys. Lett. A 180, 221 (1993)

    Google Scholar 

  10. W Malfliet, Am. J. Phys. 60(7), 650 (1992)

    Google Scholar 

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

  12. W X Ma and Z N Zhu, Appl. Math. Comput. 218, 11871 (2012)

  13. N K Vitanov and Z I Dimitrova, Commun. Nonlinear Sci. Numer. Simul. 15(10), 2836 (2010)

    Google Scholar 

  14. N K Vitanov, Z I Dimitrova and H Kantz, Appl. Math. Comput. 216(9), 2587 (2010)

  15. R Hirota, Phys. Rev. Lett. 27, 1192 (1971)

    Google Scholar 

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

  17. W X Ma, Stud Nonlinear Sci. 2, 140 (2011)

  18. W X Ma and J-H Lee, Chaos, Solitons and Fractals 42, 1356 (2009)

  19. G Jumarie, Comput. Math. Appl. 51, 1367 (2006)

    Google Scholar 

  20. Z S Feng, J. Phys. A: Math. Gen. 35, 343 (2002)

    Google Scholar 

  21. B Lu, J. Math. Anal. Appl. 395(2), 684 (2012)

    Google Scholar 

  22. A Bekir and O Unsal, Pramana – J. Phys. 79, 3 (2012)

  23. F Tascan, A Bekir and M Koparan, Commun. Non. Sci. Numer. Simulat. 14, 1810 (2009)

    Google Scholar 

  24. I Aslan, Appl. Math. Comput. 217, 8134 (2011)

    Google Scholar 

  25. I Aslan, Math. Meth. Appl. Sci. 35, 716 (2012)

    Google Scholar 

  26. I Aslan, Pramana – J. Phys. 76, 533 (2011)

    Google Scholar 

  27. I Aslan, AU.P.B. Sci. Bull., Ser. A 75, 13 (2013)

    Google Scholar 

  28. N Taghizadeh, M Mirzazadeh and F Farahrooz, J. Math. Anal. Appl. 374, 549 (2011)

    Google Scholar 

  29. M Mirzazadeh and M Eslami, Nonlin. Anal. Model Control 17(4), 481 (2012)

    Google Scholar 

  30. T R Ding and C Z Li, Ordinary differential equations, (Peking University Press, Peking, 1996)

  31. Y Zhang and Q Feng, Appl. Math. Inf. Sci. 7(4), 1575 (2013)

  32. D Weaire, S Hutzler, S Cox, N Kern, M D Alonso and W Drenckhan, J. Phys.: Condens. Matter 15, S65 (2003)

  33. W X Ma and B Fuchssteiner, Int. J. Nonlinear Mech. 31, 329 (1996)

  34. K C Basak, P C Ray and R K Bera, Commun. Nonlinear Sci. Numer. Simulat. 14(3), 718 (2009)

    Google Scholar 

  35. A Biswas, C Zony and E Zerrad, Appl. Math. Comput. 203(1), 153 (2008)

  36. G Chen, Phys. Lett. A 339(3–5), 300 (2005)

  37. A Biswas, A Yildirim, T Hayat, O M Aldossary and R Sassaman, Proceedings of the Romanian Academy, Series A 13(1), 32 (2012)

    Google Scholar 

  38. R Sassaman and A Biswas, Nonlinear Dyn. 61, 23 (2010)

  39. P Chen and Y Li. Existence of mild solutions of fractional evolution equations with mixed monotone local conditions, ZAMP, DOI: 10.1007/s00033-013-0351-z

  40. G Verbist, D Weaire and A M Kraynik. J Phys Condens Matter 8(21), 3715 (1996)

    Google Scholar 

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MIRZAZADEH, M., Eslami, M. & BISWAS, A. Solitons and periodic solutions to a couple of fractional nonlinear evolution equations. Pramana - J Phys 82, 465–476 (2014). https://doi.org/10.1007/s12043-013-0679-0

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  • DOI: https://doi.org/10.1007/s12043-013-0679-0

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