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Invariant analysis of nonlinear time fractional Qiao equation

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Abstract

In this study, Lie’s invariant analysis method is applied to nonlinear time fractional Qiao equation with the Riemann–Liouville derivative. Fractional derivative value \(\alpha \) is considered for three cases and each cases symmetry generators are given. Using these generators group-invariant solutions in explicit form are obtained.

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References

  1. Jafari, H., Tajadodi, H., Baleanu, D., Al-Zahrani, A.A., Alhamed, Y.A., Zahid, A.H.: Exact solutions of Boussinesq and KdV-mKdV equations by fractional sub-equation method. Rom. Rep. Phys. 65(4), 1119–1124 (2013)

    Google Scholar 

  2. Bekir, A., Güner, Ö.: Exact solutions of nonlinear fractional differential equations by (G’/G)-expansion method. Chin. Phys. B 22(11), 110202 (2013)

    Article  Google Scholar 

  3. Eslami, M., Vajargah, B.F., Mirzazadeh, M., Biswas, A.: Application of first integral method to fractional partial differential equations. Indian J. Phys. 88(2), 177–184 (2014)

    Article  Google Scholar 

  4. Jafari, H., Kadkhoda, N., Baleanu, D.: Fractional Lie group method of the time-fractional Boussinesq equation. Nonlinear Dyn. 81, 15691574 (2015)

    Article  MathSciNet  Google Scholar 

  5. Ouhadan, A., El Kinani E.H., Invariant Subspace Method and Fractional Modified Kuramoto-Sivashinsky Equation. arXiv preprint arXiv:1503.08789 (2015)

  6. Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer Science & Business Media, Berlin (2000)

  7. Ibragimov, N.H.: CRC Handbook of Lie Group Analysis of Differential Equations, vol. 3. CRC Press, Boca Raton (1995)

    MATH  Google Scholar 

  8. Özemir, C., Güngör, F.: Group-invariant solutions of the (2 + 1)-dimensional cubic Schrödinger equation. J. Phys. A Math. Gen. 39, 29732993 (2006)

    Article  MATH  Google Scholar 

  9. Güngör, F., Özemir, C.: Lie symmetries of a generalized Kuznetsov–Zabolotskaya–Khokhlov equation. J. Math. Anal. Appl. 423, 623638 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bokhari, A.H., Al-Dweik, A.Y., Zaman, F.D., Kara, A.H., Mahomed, F.M.: Generalization of the double reduction theory. Nonlinear Anal. Real World Appl. 11, 3763–3769 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Yaşar, E., Özer, T.: Invariant solutions and conservation laws to nonconservative FP equation. Comput. Math. Appl. 59, 3203–3210 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hydon, P.E., Hydon, P.E.: Symmetry Methods for Differential Equations: A Beginner’s Guide, vol. 22. Cambridge University Press, Cambridge (2000)

    Book  MATH  Google Scholar 

  13. Kara, A.H., Mahomed, F.M.: Noether-type symmetries and conservation laws via partial Lagragians. Nonlinear Dyn. 45, 367–383 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Wang, G.W., Liu, X.Q., Zhang, Y.Y.: Lie symmetry analysis to the time fractional generalized fifth-order KdV equation. Commun. Nonlinear Sci. Numer. Simul. 18(9), 2321–2326 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hashemi, M.S.: Group analysis and exact solutions of the time fractional Fokker–Planck equation. Phys. A Stat. Mech. Appl. 417, 141–149 (2015)

    Article  MathSciNet  Google Scholar 

  16. Gazizov, R.K., Ibragimov, N.H., Lukashchuk, SYu.: Nonlinear self-adjointness, conservation laws and exact solutions of time-fractional Kompaneets equations. Commun. Nonlinear Sci. Numer. Simul. 23, 153–163 (2015)

    Article  MathSciNet  Google Scholar 

  17. Rui, W., Zhang, X.: Invariant analysis and conservation laws for the time fractional foam drainage equation. Eur. Phys. J. Plus 130(10), 1–7 (2015)

    Article  Google Scholar 

  18. Rui, W., Zhang, X.: Lie symmetries and conservation laws for the time fractional Derrida–Lebowitz–Speer–Spohn equation. Commun. Nonlinear Sci. Numer. Simul. 34, 38–44 (2016)

    Article  MathSciNet  Google Scholar 

  19. El Kinani, E.H., Ouhadan, A.: Lie symmetry analysis of some time fractional partial differential equations. Int. J. Mod. Phys. Conf. Ser. 38, 1560075 (2015)

    Article  Google Scholar 

  20. Wang, G., Kara, A.H., Fakhar, K.: Symmetry analysis and conservation laws for the class of time-fractional nonlinear dispersive equation. Nonlinear Dyn. 82, 281287 (2015)

    MathSciNet  Google Scholar 

  21. Gazizov, R.K., Kasatkin, A.A., Lukashchuk, S.Y.: Symmetry properties of fractional diffusion equations. Phys. Scr. T136, 014016 (2009)

    Article  Google Scholar 

  22. wei Wang, G., zhou Xu, T., Feng, T., Lie symmetry analysis and explicit solutions of the time fractional fifth-order KdV equation. PloS one 9(2), e88336 (2014)

  23. Gaur, M., Singh, K.: On group invariant solutions of fractional order Burgers–Poisson equation. Appl. Math. Comput. 244, 870–877 (2014)

    MathSciNet  MATH  Google Scholar 

  24. Wang, G.W., Xu, T.Z.: Invariant analysis and exact solutions of nonlinear time fractional Sharma–Tasso–Olver equation by Lie group analysis. Nonlinear Dyn. 76(1), 571–580 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  25. Qiao, Z.J.: New integrable hierarchy, its parametric solutions, cuspons, one-peak solitons, and M/W-shape peak solitons. J. Math. Phys. 48, 082701 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  26. Qiao, Z.J., Liu, L.P.: A new integrable equation with no smooth solitons. Chaos, Solitons Fractals 41, 587593 (2009)

    Article  MathSciNet  Google Scholar 

  27. Yang, Y.Q., Chen, Y.: Prolongation structure of the equation studied by Qiao. Commun. Theor. Phys. 56, 463–466 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. Li, J.B., Qiao, Z.J.: Bifurcations of traveling wave solutions for an integrable equation. J. Math. Phys. 51, 042703 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. Qiao, Z.J.: A new integrable equation with cuspons and W/M-shape-peaks solitons. J. Math. Phys. 47, 112701 (2006)

  30. Qiao, Z.J.: The Camassa–Holm hierarchy, N-dimensional integrable systems, and algebro-geometric solution on a symplectic submanifold. Commun. Math. Phys. 239, 309–341 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  31. Sahadevan, R., Bakkyaraj, T.: Invariant analysis of time fractional generalized Burgers and Korteweg-de Vries equations. J. Math. Anal. Appl. 393, 341–347 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  32. Huang, Q., Zhdanov, R.: Symmetries and exact solutions of the time fractional Harry-Dym equation with Riemann–Liouville derivative. Phys. A 409, 110–118 (2014)

    Article  MathSciNet  Google Scholar 

  33. Gazizov, R.K., Kasatkin, A.A., Lukashchuk, S.Y.: Fractional differential equations: change of variables and nonlocal symmetries. Ufa Math. J. 4(4), 54–67 (2012)

    MathSciNet  MATH  Google Scholar 

  34. Podlubny, I.: Fractional Differential Equations. Academic, San Diego (1999)

    MATH  Google Scholar 

  35. Jefferson, G.F., Carminati, J.: FracSym: automated symbolic computation of Lie symmetries of fractional differential equations. Comput. Phys. Commun. 185(1), 430–441 (2014)

    Article  Google Scholar 

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San, S. Invariant analysis of nonlinear time fractional Qiao equation. Nonlinear Dyn 85, 2127–2132 (2016). https://doi.org/10.1007/s11071-016-2818-x

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  • DOI: https://doi.org/10.1007/s11071-016-2818-x

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