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Symmetry analysis, explicit power series solutions and conservation laws of the space-time fractional variant Boussinesq system

  • Baljinder Kour
  • Sachin KumarEmail author
Regular Article

Abstract.

In this study, the classical Lie symmetry method is successfully applied to investigate the symmetries of the space-time fractional variant Boussinesq system which was introduced as a model of water waves. With the help of the obtained symmetries, the governing system is reduced into the system of nonlinear fractional ordinary differential equations (NLFODEs) which contains Erdèlyi-Kober fractional differential operators via Riemann-Liouville fractional derivative. The system is also studied for the explicit power series solution. The obtained power series solution is further examined for the convergence. The conservation laws of the governing system are constructed by using the new conservation theorem and generalization of the Noether operators. The numerical approximation for the fractional system is also found by using the residual power series method (RSPM). Some figures are also presented to explain the physical understanding for both explicit and approximate solutions.

References

  1. 1.
    M. Dehghan, J. Manafian, A. Saadatmandi, Numer. Methods Partial Differ. Equ. 26, 448 (2010)Google Scholar
  2. 2.
    B. Zheng, Commun. Theor. Phys. (Beijing) 58, 623 (2012)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    S. Zhang, H.-Q. Zhang, Phys. Lett. A 375, 1069 (2011)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    J.-H. He, X.-H. Wu, Chaos, Solitons Fractals 30, 700 (2006)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    C. Chen, Y.-L. Jiang, Commun. Nonlinear Sci. Numer. Simul. 26, 24 (2015)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    R.A. Leo, G. Sicuro, P. Tempesta, Fract. Calc. Appl. Anal. 20, 212 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    G. Wang, A.H. Kara, K. Fakhar, Nonlinear Dyn. 82, 281 (2015)CrossRefGoogle Scholar
  8. 8.
    G.W. Bluman, S.C. Anco, Symmetry and Integration Methods for Differential Equations (Springer-Verlag, New York, 2002)Google Scholar
  9. 9.
    S. Kumar, Nonlinear Dyn. 87, 1153 (2017)CrossRefGoogle Scholar
  10. 10.
    P.J. Olver, Applications of Lie Groups to Differential Equations (Springer-Verlag, New York, 1986)Google Scholar
  11. 11.
    W. Rui, X. Zhang, Commun. Nonlinear Sci. Numer. Simul. 34, 38 (2016)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Q. Zhou, Q. Zhu, A. Bhrawy, L. Moraru, A. Biswas, Optoelectron. Adv. Mater. 8, 800 (2014)Google Scholar
  13. 13.
    N.K. Ibragimov, E.D. Avdonina, Uspekhi Mat. Nauk 68, 111 (2013)CrossRefGoogle Scholar
  14. 14.
    K. Singla, R.K. Gupta, Commun. Nonlinear Sci. Numer. Simul. 53, 10 (2017)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    G. Wang, A.H. Kara, K. Fakhar, J. Vega-Guzman, A. Biswas, Chaos, Solitons Fractals 86, 8 (2016)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    N.H. Ibragimov, J. Math. Anal. Appl. 333, 311 (2007)MathSciNetCrossRefGoogle Scholar
  17. 17.
    M. Arshad, D. Lu, J. Wang, Commun. Nonlinear Sci. Numer. Simul. 48, 509 (2017)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    M. Gaur, K. Singh, Appl. Math. Comput. 244, 870 (2014)MathSciNetGoogle Scholar
  19. 19.
    G.-W. Wang, T.-Z. Xu, Nonlinear Dyn. 76, 571 (2014)CrossRefGoogle Scholar
  20. 20.
    B. Gao, Wave Random Complex 27, 700 (2017)ADSCrossRefGoogle Scholar
  21. 21.
    F. Tchier et al., Eur. Phys. J. Plus 133, 240 (2018)CrossRefGoogle Scholar
  22. 22.
    D.-X. Meng, Y.-T. Gao, X.-L. Gai, L. Wang, X. Yu, Z.-Y. Sun, M.-Z. Wang, X. Lü, Appl. Math. Comput. 215, 1744 (2009)MathSciNetGoogle Scholar
  23. 23.
    W. Yuan, F. Meng, Y. Huang, Y. Wu, Appl. Math. Comput. 268, 865 (2009)Google Scholar
  24. 24.
    E. Yaş, Acta Phys. Pol. A 128, 252 (2015)CrossRefGoogle Scholar
  25. 25.
    V. Kiryakova, Generalized Fractional Calculus and Applications (John Wiley, New York, 1994)Google Scholar
  26. 26.
    I. Podlubny, Fractional Differential Equations (Academic Press, Inc., San Diego, 1999)Google Scholar
  27. 27.
    M. Inc, A. Yusuf, A.I. Aliyu, D. Baleanu, Phys. A 496, 371 (2015)CrossRefGoogle Scholar
  28. 28.
    A.S. Nuseir, A. Al-Hasoon, Appl. Math. Sci. 6, 5147 (2012)MathSciNetGoogle Scholar
  29. 29.
    C.-Y. Qin, S.-F. Tian, X.-B. Wang, T.-T. Zhang, Waves Random Complex Media 27, 308 (2017)ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    W. Rudin, Principles of Mathematical Analysis (McGraw-Hill, New York, 1964)Google Scholar
  31. 31.
    K. Singla, R.K. Gupta, Nonlinear Dyn. 89, 321 (2017)CrossRefGoogle Scholar
  32. 32.
    I. Podlubny, Y. Chen, in ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering (2007) 1385Google Scholar
  33. 33.
    H. Tariq, G. Akram, J. Appl. Math. Comput. 55, 683 (2017)MathSciNetCrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsCentral University of PunjabPunjabIndia

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