Advertisement

On the fractional order space-time nonlinear equations arising in plasma physics

  • M A Abdou
Original Paper
  • 23 Downloads

Abstract

In this study, the \(\exp (-\phi (\xi ))\)-expansion function method is considered for solving two classes of space-time fractional partial differential equations of very special interest. The two classes, namely the higher dimensional Kadomtsev–Petviashvili and Boussinesq equations, have a wide range applications in different areas of complex nonlinear physics such as plasma physics, fluid dynamics and nonlinear optics. As a result, the \(\exp (-\phi (\xi ))\)-expansion function method yields a different class of traveling solutions mapped to trigonometric functions, rational functions and hyperbolic functions. Also, the behavior of these solutions has been significantly affected by changing the values of fractional order where the obtained solutions go back to those obtained previously to the normal case, i.e.,\(\alpha =\beta =1\). Finally, our finding may be of wide relevance and helpful to better understand the main features and propagation of the nonlinear waves in fractal medium.

Keywords

Fractional complex transform Modified \(\exp (-\phi (\xi ))\)-expansion function method Plasma physics Exact solutions 

PACS Nos.

02.90.+p 02.30.Jr 02.30.Mv 

References

  1. [1]
    M Horonyi Astron. Astrophys. 34 383–418 (1996)ADSCrossRefGoogle Scholar
  2. [2]
    F Verheest Space Sci. Rev. 77(3–4) 267–302 (1996)ADSGoogle Scholar
  3. [3]
    P K Shukla and A A Mamun Inst. Phys. Bristol (2002)Google Scholar
  4. [4]
    D A Mendis and M Rosenberg Astron. Astrophys. 32 419–463 (1994)ADSCrossRefGoogle Scholar
  5. [5]
    N Akhtar, S Mahmood and H Saleem Phys. Lett. A 361 126 (2007)ADSCrossRefGoogle Scholar
  6. [6]
    J R Asbridge, S J Bame and I B Strong J. Geophys. Res. 73 5777 (1968)ADSCrossRefGoogle Scholar
  7. [7]
    W C Feldman, S J Anderson, S J Bame, S P Gary, J T Gosling, D J McComas, M F Thomsen, G Paschmann and M M Hoppe J. Geophys. Res. Space Phys. 88 96 (1983)ADSCrossRefGoogle Scholar
  8. [8]
    R Lundlin, A Zakharov, R Pellinen, H Borg, B Hultqvist, N Pissarenko, E M Dubinin, S W Barabash, I Liede and H Koskinen Nature 341 609 (1998)Google Scholar
  9. [9]
    Y Futaana,S Machida, Y Saito, A Matsuoka and H Hayakawa J. Geophys. Res. 108 15 (2003)Google Scholar
  10. [10]
    S A Elwakil, A Elgarayhi, E K El-Shewy, A A Mahmoud and M A El-Attafi Astrophys. Space Sci. 343 661 (2013)ADSCrossRefGoogle Scholar
  11. [11]
    Y Nakamura and A Sarma Phys. Plasmas 8 3921 (2001)ADSCrossRefGoogle Scholar
  12. [12]
    S S Duha, M G M Anowar and A A Mamun Phys. Plasmas 17 03711 (2010)CrossRefGoogle Scholar
  13. [13]
    K B Oldman and J Spanier It The Fractional Calculus (New York: Academic Press) (1974)Google Scholar
  14. [14]
    R Hilfer Applications of Fractional Calculus in Physics (Singapore: World Scientific) (2000)CrossRefGoogle Scholar
  15. [15]
    A A Kilbas, H M Srivastava and J Trujillo Theory and Applications of Fractional Differential Equations (Amsterdam: Elsevier) (2006)Google Scholar
  16. [16]
    M D Ortigueira Fractional Calculus for Scientists and Engineers Springer Mathematics (2011)Google Scholar
  17. [17]
    D Baleanu, K Diethelm, E Scalas and J J Trujillo (Boston: World Scientific) (2012)Google Scholar
  18. [18]
    I Podlubny Fractional Differential Equations, Mathematics in Science and Engineering (San Diego, CA: Academic Press) (1999)zbMATHGoogle Scholar
  19. [19]
    Z Dahmani, M M Mesmoudi and R Bebbouchi Theor. Differ. Equ. 31 1 (2008)Google Scholar
  20. [20]
    G Jumarie Comput. Math. Appl. 51 1367 (2006)MathSciNetCrossRefGoogle Scholar
  21. [21]
    G Jumarie Appl. Math. Lett. 22 378 (2009)MathSciNetCrossRefGoogle Scholar
  22. [22]
    R Cimpoiasu and R Constantinescu Nonlinear Anal. Ser. A Theory Methods Appl. 68 2261 (2008)CrossRefGoogle Scholar
  23. [23]
    R Cimpoiasu and R Constantinescu Nonlinear Anal. Ser A Theor. Methods Appl. 57 147 (2010)CrossRefGoogle Scholar
  24. [24]
    S Zhang, Q A Zong, D Liu and Q Gao Commun. Fract. Calculus 1 48 (2010)Google Scholar
  25. [25]
    B Lu J. Math. Anal.Appl. 395 684 (2012)CrossRefGoogle Scholar
  26. [26]
    L Bin Commun. Theor. Phys. 58 623 (2012)CrossRefGoogle Scholar
  27. [27]
    K A Gepreel and S Omran Chin. Phys. B 21 110204 (2012)CrossRefGoogle Scholar
  28. [28]
    S S Ray and R K Bera Appl. Math. Comput. 167 561 (2005)MathSciNetGoogle Scholar
  29. [29]
    S Zhang and H Q Zhang Phys. Lett. A 375 1069 (2011)ADSCrossRefGoogle Scholar
  30. [30]
    J Zhao, B Tang, S Kumar and Y Hou Math. Probl. Eng.  https://doi.org/10.1155/2012/924956 (2012)
  31. [31]
    B Tong, Y He, L Wei and X Zhang Phys. Lett. A 376 2588 (2012)ADSCrossRefGoogle Scholar
  32. [32]
    S Guo, L Mei, Y Li, Y Sun Phys. Lett. A 376 407 (2012)ADSCrossRefGoogle Scholar
  33. [33]
    L Bin Phys. Lett. A 376 2045 (2012)MathSciNetCrossRefGoogle Scholar
  34. [34]
    M Cui J. Comput. Phys. 228 7792 (2009)ADSMathSciNetCrossRefGoogle Scholar
  35. [35]
    Q Hung,G Huang and H Zhan Adv. Water Resour. 31 1578 (2008)ADSCrossRefGoogle Scholar
  36. [36]
    M A Abdou, A Elgarayhi and E El Shewy Nonlinear Sci. Lett. A 5 31 (2014)Google Scholar
  37. [37]
    M A Abdou and A Elhanbaly Nonlinear Sci. Lett. A 6 10 (2015)Google Scholar
  38. [38]
    A Elgarayhi, M A Abdou and A T Attia Nonlinear Sci. Lett. A 5 35 (2014)Google Scholar
  39. [39]
    M A Abdou and A Yildirim Int. J. Numer. Methods Heat Fluid Flow 22 829 (2012)CrossRefGoogle Scholar
  40. [40]
    M N Alam, M G Hafez, M A Akbar and H O Roshid J. Sient. Res. 7 1 (2015)CrossRefGoogle Scholar
  41. [41]
    M Kaplan and A Bekir Optik 12 8209 (2016)ADSCrossRefGoogle Scholar
  42. [42]
    H O Roshid and Md A Rahman Res. Phys. 4 150 (2014)Google Scholar
  43. [43]
    M A Abdou J. Oce. Eng. Sci. 2 1 (2017)Google Scholar
  44. [44]
    M A Abdou and A A Soliman Results iPhys. 9 1497 (2018)ADSCrossRefGoogle Scholar
  45. [45]
    M A Abdou, A A Soliman, A Biswas, M Ekic and S P Moshokoa Optik 171 463 (2018)ADSCrossRefGoogle Scholar
  46. [46]
    M A Abdou Wave Random Complex Media.  https://doi.org/10.1080/17455030.2018.1517951
  47. [47]
    M A Abdou Int. J. Nonlinear Sci. 26(2) 89 (2018)Google Scholar

Copyright information

© Indian Association for the Cultivation of Science 2018

Authors and Affiliations

  1. 1.Physics Department, College of ScienceUniversity of BishaBishaKingdom of Saudi Arabia
  2. 2.Theoretical Research Group, Physics Department, Faculty of ScienceMansoura UniversityMansouraEgypt

Personalised recommendations