Indian Journal of Physics

, Volume 88, Issue 2, pp 177–184 | Cite as

Application of first integral method to fractional partial differential equations

  • M. Eslami
  • B. Fathi Vajargah
  • M. Mirzazadeh
  • A. Biswas
Original paper

Abstract

In this paper, fractional derivatives in the sense of modified Riemann-Liouville derivative and first integral method are applied for constructing exact solutions of nonlinear fractional generalized reaction duffing model and nonlinear fractional diffusion reaction equation with quadratic and cubic nonlinearity. Our approach provides first integrals in polynomial form with a high accuracy for two-dimensional plane autonomous systems. Exact soliton solutions are constructed through established first integrals.

Keywords

First integral method Solitons Fractional generalized reaction duffing model 

PACS Nos.

02.30.Jr 05.45.Yv 

References

  1. [1]
    A H Bhrawy, M A Abdelkawy, S Kumar, S Johnson and A Biswas Indian J. Phys. 87 455 (2013)ADSCrossRefGoogle Scholar
  2. [2]
    H Kumar, A Malik, F Chand and S C Mishra Indian J. Phys. 86 819 (2012)ADSCrossRefGoogle Scholar
  3. [3]
    R S Kaushal, D Parashar, S Gupta and S C Mishra Ann. Phys. 259 233 (1997)ADSCrossRefMATHMathSciNetGoogle Scholar
  4. [4]
    A Biswas, S Konar and E Zerrad Int. J. Modern Math. 2 35 (2007)MATHMathSciNetGoogle Scholar
  5. [5]
    A Biswas and S Konar Appl. Math. Lett. 20 1122 (2007)CrossRefMATHMathSciNetGoogle Scholar
  6. [6]
    A Biswas and S Konar Commun. Nonlinear Sci. Numer. Simul. 13 703 (2008)ADSCrossRefMATHMathSciNetGoogle Scholar
  7. [7]
    A Biswas, E Zerrad and S Konar Commun. Nonlinear Sci. Numer. Simul. 13 1281 (2008)ADSCrossRefMATHMathSciNetGoogle Scholar
  8. [8]
    A Biswas and S Konar Introduction to Non-Kerr law Optical Solitons (Boca Raton, FL: CRC Press) (2006)CrossRefGoogle Scholar
  9. [9]
    A Fabian, R Kohl and A Biswas Commun. Nonlinear Sci. Numer. Simul. 14 1227 (2009)ADSCrossRefMATHMathSciNetGoogle Scholar
  10. [10]
    W X Ma, T W Huang and Y Zhang Phys. Scr. 82 065003 (2010)ADSCrossRefGoogle Scholar
  11. [11]
    W X Ma and Z N Zhu Appl. Math. Comput. 218 11871 (2012)CrossRefMATHMathSciNetGoogle Scholar
  12. [12]
    W X Ma Stud. Nonlinear Sci. 2 140 (2011)Google Scholar
  13. [13]
    W X Ma J. Phys. Conf. Ser. 411 012021 (2013)ADSCrossRefGoogle Scholar
  14. [14]
    W X Ma and J H Lee Chaos Solitons Fract. 42 1356 (2009)ADSCrossRefMATHMathSciNetGoogle Scholar
  15. [15]
    G Jumarie Comput. Math. Appl. 51 1367 (2006)CrossRefMATHMathSciNetGoogle Scholar
  16. [16]
    Z S Feng J. Phys. A: Math. Gen. 35 343 (2002)ADSCrossRefMATHGoogle Scholar
  17. [17]
    Z S Feng Phys. Lett. A 293 57 (2002)ADSCrossRefMATHMathSciNetGoogle Scholar
  18. [18]
    Z S Feng and X Wang Phys. Lett. A 308 173 (2003)ADSCrossRefMATHMathSciNetGoogle Scholar
  19. [19]
    I Aslan Acta Phys. Polonica A 123 16 (2013)CrossRefGoogle Scholar
  20. [20]
    B Lu J. Math. Anal. Appl. 395 684 (2012)CrossRefMATHMathSciNetGoogle Scholar
  21. [21]
    M Mirzazadeh and M Eslami Nonlinear Anal. Model. Control 17 481 (2012)MathSciNetGoogle Scholar
  22. [22]
    W X Ma and B Fuchssteiner Int. J. Nonlinear Mech. 31 329 (1996)CrossRefMATHMathSciNetGoogle Scholar
  23. [23]
    J R King J. Phys. A 24 3213 (1991)ADSCrossRefMATHMathSciNetGoogle Scholar
  24. [24]
    J D Logan An Introduction to Nonlinear Partial Differential Equations 2nd edn. (New York: Wiley Interscience) (2008)CrossRefMATHGoogle Scholar
  25. [25]
    R S Kaushal J. Phys. A 38 3897 (2005)ADSCrossRefMATHMathSciNetGoogle Scholar
  26. [26]
    R S Banarjee Int. J. Theo. Phys. 32 879 (1993)CrossRefGoogle Scholar
  27. [27]
    V A Galaktionov Phys. D 238 1717 (2009)CrossRefMATHMathSciNetGoogle Scholar
  28. [28]
    J D Murray Math. Biol. (New York: Springer) (1993)CrossRefGoogle Scholar
  29. [29]
    V M Kenkre and M N Kuperman Phys. Rev. E 67 051921 (2003)ADSCrossRefGoogle Scholar
  30. [30]
    D R Nelson and N M Shnerb Phys. Rev. E 58 1383 (and references theirin) (1998)Google Scholar
  31. [31]
    T Chalker and Z J Wang Phys. Rev. Lett. 79 1797 (1997)ADSCrossRefGoogle Scholar
  32. [32]
    N Moiseyev and M Gluck Phys. Rev. E 63 041103 (2001)ADSCrossRefGoogle Scholar
  33. [33]
    M Remoissent Waves Called Solitons: Concept and Experiments (Berlin-Heidelberg: Springer) (1999)CrossRefGoogle Scholar

Copyright information

© Indian Association for the Cultivation of Science 2013

Authors and Affiliations

  • M. Eslami
    • 1
  • B. Fathi Vajargah
    • 2
  • M. Mirzazadeh
    • 2
  • A. Biswas
    • 3
    • 4
  1. 1.Department of Mathematics, Faculty of Mathematical SciencesUniversity of MazandaranBabolsarIran
  2. 2.Department of Mathematics, Faculty of Mathematical SciencesUniversity of GuilanRashtIran
  3. 3.Department of Mathematical SciencesDelaware State UniversityDoverUSA
  4. 4.Department of Mathematics, Faculty of ScienceKing Abdulaziz UniversityJeddahSaudi Arabia

Personalised recommendations