Advertisement

Pramana

, 93:66 | Cite as

Numerical solution of nonlinear fractional Zakharov–Kuznetsov equation arising in ion-acoustic waves

  • Amit PrakashEmail author
  • Vijay Verma
Article
  • 59 Downloads

Abstract

The main purpose of this work is to suggest an efficient hybrid computational technique, namely the q-homotopy analysis transform method (q-HATM) to find the solution of the nonlinear time-fractional Zakharov–Kuznetsov (FZK) equation in two dimensions. The uniqueness and convergence analysis of the nonlinear time-FZK equation is presented. The Laplace decomposition method (LDM) is also employed to get the approximate solution of the nonlinear FZK equation. We implemented these techniques on two numerical examples, plotted the solution and compared the absolute error with the variational iteration technique and homotopy perturbation transform technique to show the efficiency of these techniques.

Keywords

Zakharov–Kuznetsov equation q-homotopy analysis transform method Caputo fractional derivative Laplace decomposition method 

PACS Nos

02.60.–Cb 05.45.–a 

Notes

Acknowledgements

The authors are thankful to the anonymous reviewers    and editors for their valuable comments and suggestions to improve the quality of this paper.

References

  1. 1.
    J H He, Commun. Nonlinear Sci. Numer. Simul.  2, 235 (1997)ADSCrossRefGoogle Scholar
  2. 2.
    J H He, Commun. Nonlinear Sci. Numer. Simul.  3, 92 (1998)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    A Prakash, M Kumar and K K Sharma, Appl. Math. Comput.  260, 314 (2015)MathSciNetGoogle Scholar
  4. 4.
    A Prakash and M Kumar, J. Appl. Anal. Comput.  6(3), 738 (2016)MathSciNetGoogle Scholar
  5. 5.
    A Prakash and M Kumar, Open Phys.  14, 177 (2016)CrossRefGoogle Scholar
  6. 6.
    A Yildirim, Int. J. Nonlinear Sci. Numer. Simul.  10(4), 445 (2009)CrossRefGoogle Scholar
  7. 7.
    A Yildirim, J. King Saud Univ. (Sci.)  22, 257 (2010)CrossRefGoogle Scholar
  8. 8.
    S J Liao, Appl. Math. Comput.  147, 499 (2004)MathSciNetGoogle Scholar
  9. 9.
    S Kumar, H Kocak and A Yildirim, Z. Naturforsch. A  67, 389 (2012)ADSCrossRefGoogle Scholar
  10. 10.
    S Kumar, A Yildirim, Y Khan and L Wei, Sci. Iran B  19(4), 1117 (2012)CrossRefGoogle Scholar
  11. 11.
    D Kumar, J Singh and D Baleanu, J. Comput. Nonlinear Dyn.  11(6), 061004 (2016)CrossRefGoogle Scholar
  12. 12.
    A Prakash, M Kumar and D Baleanu, Appl. Math. Comput.  334, 30 (2018)MathSciNetGoogle Scholar
  13. 13.
    H M Baskonus and H Bulut, Open Math.  13(1), 547 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    D G Prakasha, P Veeresha and H M Baskonus, Fractal Fract.  3(9), 1 (2019)Google Scholar
  15. 15.
    D G Prakasha, P Veeresha and H M Baskonus, Comput. Math. Methods 1(2), 1 (2019)Google Scholar
  16. 16.
    M Arshad, Aly Seadawy, Dianchen Lu and Jem Wang, Results Phys.  6, 1136 (2016)ADSCrossRefGoogle Scholar
  17. 17.
    J Singh, D Kumar and D Baleanu, Math. Model. Nat. Phenom.  14, 303 (2019)CrossRefGoogle Scholar
  18. 18.
    P Veeresha, D G Prakasha and H M Baskonus, Math. Sci. (2019),  https://doi.org/10.1007/s40096-019-0284-6 MathSciNetCrossRefGoogle Scholar
  19. 19.
    P Veeresha, D G Prakasha and H M Baskonus, AIP Chaos Interdiscip. J. Nonlinear Sci.  29(1), 1 (2019)Google Scholar
  20. 20.
    M T Gencoglu, H M Baskonus and H Bulut, AIP Conf. Proc.  1798, 1 (2017)Google Scholar
  21. 21.
    M Goyal, A Prakash and S Gupta, Pramana – J. Phys.  92: 82 (2019)ADSCrossRefGoogle Scholar
  22. 22.
    A Prakash, M Goyal and S Gupta, Pramana – J. Phys.  92(2): 1 (2019)Google Scholar
  23. 23.
    M A El-Tawil and S N Huseen, Int. J. Appl. Math. Mech.  8, 51 (2012)Google Scholar
  24. 24.
    M A El-Tawil and S N Huseen, Int. J. Contemp. Math. Sci.  8, 481 (2013)MathSciNetCrossRefGoogle Scholar
  25. 25.
    S J Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D. thesis (Shanghai Jiao Tong Univ., 1992) Google Scholar
  26. 26.
    S J Liao, Commun. Nonlinear Sci. Numer. Simul.  2, 95 (1997)ADSCrossRefGoogle Scholar
  27. 27.
    S J Liao, Beyond perturbation: Introduction to the homotopy analysis method (Chapman and Hall\(/\)CRC Press, Boca Raton, 2003)Google Scholar
  28. 28.
    D L Xu, Z L Lin, S J Liao and M Stiassnie, J. Fluid Mech.  710, 379 (2012)ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    S J Liao, Math. Comput.  147, 499 (2004)Google Scholar
  30. 30.
    H Jafari, A Golbabai, S Seifi and K Sayevand, Comput. Math. Appl.  66, 838 (2010)CrossRefGoogle Scholar
  31. 31.
    S Kumar, A Kumar and D Baleanu, Nonlinear Dyn. 85(2), 699 (2016),  https://doi.org/10.1007/s11071-016-2716-2 CrossRefGoogle Scholar
  32. 32.
    S Nadeem, A Hussain and M Khan, Z. Naturforsch.  65, 540 (2010)ADSCrossRefGoogle Scholar
  33. 33.
    M Khan, M A Gondal, I Hussain and S Karimi Vanani, Math. Comput. Model.  55, 1143 (2012)CrossRefGoogle Scholar
  34. 34.
    D Kumar, J Singh and Sushila, Rom. Rep. Phys.  65(1), 63 (2013)Google Scholar
  35. 35.
    D Kumar, J Singh, S Kumar and Sushila, Alex. Eng. J.  53(2), 469 (2014)CrossRefGoogle Scholar
  36. 36.
    S Munro and E J Parkes, J. Plasma Phys.  62(3), 305 (1999)ADSCrossRefGoogle Scholar
  37. 37.
    S Munro and E J Parkes, J. Plasma Phys.  64(4), 411 (2000)ADSCrossRefGoogle Scholar
  38. 38.
    V E Sakharov and E A Kuznetsov, Sov. Phys. JETP  39, 285 (1974)ADSGoogle Scholar
  39. 39.
    R Y Molliq, M S M Noorani, I Hashim and R R Ahmad, J. Comput. Appl. Math.  233, 103 (2009)ADSMathSciNetCrossRefGoogle Scholar
  40. 40.
    A Yildirim and Y Gulkanat, Commun. Theor. Phys.  53, 1005 (2010)ADSCrossRefGoogle Scholar
  41. 41.
    D Kumar, J Singh and S Kumar, J. Egypt. Math. Soc.  22, 373 (2014)CrossRefGoogle Scholar
  42. 42.
    D Kumar, J Singh and D Baleanu, Nonlinear Dyn. 87(1), 511 (2019).  https://doi.org/10.1007/s11071-016-3057-x CrossRefGoogle Scholar
  43. 43.
    X J Yang, J A T Machado and J Haristov, Nonlinear Dyn.  84, 3 (2016)CrossRefGoogle Scholar
  44. 44.
    E H Doha, A H Bhrawy and S S Ezz-Eldien, J. Comput. Nonlinear Dyn.  10, 1 (2015)Google Scholar
  45. 45.
    I Podlubny, Fractional differential equations (Academic Press, New York, 1999)zbMATHGoogle Scholar
  46. 46.
    M Inc, Chaos Solitons Fractals  33(5), 1783 (2007)ADSMathSciNetCrossRefGoogle Scholar
  47. 47.
    P Veeresha, D G Prakasha and D Baleanu, Mathematics  7, 265 (2019)CrossRefGoogle Scholar
  48. 48.
    I K Argyros, Convergence and applications of Newton-type iterations (Springer Science and Business Media, Berlin, Germany, 2008) zbMATHGoogle Scholar
  49. 49.
    A A Magrenam, Appl. Math. Comput.  248, 215 (2014)MathSciNetGoogle Scholar
  50. 50.
    Z M Odibat and N T Shawagfeh, Appl. Math. Comput.  186, 286 (2007)MathSciNetGoogle Scholar
  51. 51.
    H Jafari, C M Khalique and M Nazari, Appl. Math. Lett.  24, 1799 (2011)MathSciNetCrossRefGoogle Scholar
  52. 52.
    H M Srivastava, D Kumar and J Singh, Appl. Math. Model.  45, 192 (2017)MathSciNetCrossRefGoogle Scholar
  53. 53.
    A M Wazwaz and M S Mehana, Nonlinear Sci.  10, 248 (2010)Google Scholar
  54. 54.
    M Khan and M Hussain, Numer. Algorithms  56, 211 (2011)MathSciNetCrossRefGoogle Scholar

Copyright information

© Indian Academy of Sciences 2019

Authors and Affiliations

  1. 1.Department of MathematicsNational Institute of TechnologyKurukshetraIndia
  2. 2.Department of MathematicsPt. Chiranji Lal Sharma Govt. (PG) CollegeKarnalIndia

Personalised recommendations