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Nonlinear Dynamics

, Volume 81, Issue 3, pp 1569–1574 | Cite as

Fractional Lie group method of the time-fractional Boussinesq equation

  • Hossein Jafari
  • Nematollah Kadkhoda
  • Dumitru Baleanu
Original Paper

Abstract

Finding the symmetries of the nonlinear fractional differential equations is a topic which has many applications in various fields of science and engineering. In this manuscript, firstly, we are interested in finding the Lie point symmetries of the time-fractional Boussinesq equation. After that, by using the infinitesimal generators, we determine their corresponding invariant solutions.

Keywords

Fractional differential equation  Lie group Time-fractional Boussinesq equation Riemann–Liouville derivative Group-invariant solutions 

References

  1. 1.
    Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J.: Fractional Calculus Models and Numerical Methods (Series on Complexity, Nonlinearity and Chaos). World Scientific, Singapore (2012)Google Scholar
  2. 2.
    Baumann, G.: Symmetry Analysis of Differential Equations with Mathematica. Telos, Springer, New York (2000)CrossRefMATHGoogle Scholar
  3. 3.
    Bluman, G.W., Anco, S.C.: Symmetry and Integration Methods for Differential Equations. Springer, New York (2002)MATHGoogle Scholar
  4. 4.
    Bluman, G.W., Kumei, S.: Symmetries and Differential Equations, Applied Mathematics and Sciences, vol. 81. Springer, New York (1989)CrossRefGoogle Scholar
  5. 5.
    Buckwar, E., Luchko, Y.: Invariance of a partial differential equation of fractional order under the Lie group of scaling transformations. J. Math. Anal. Appl. 227, 8197 (1998)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chen, J., Liu, F., Anh, V.: Analytical solution for the time-fractional telegraph equation by the method of separating variables. J. Math. Anal. Appl. 338(2), 1364–1377 (2008)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Dehghan, M., Manafian, J., Saadatmandi, A.: Solving nonlinear fractional partial differential equations using the homotopy analysis method. Numer. Meth. Part Differ. Equ. 26(2), 448–479 (2010)MathSciNetMATHGoogle Scholar
  8. 8.
    Djordjevic, V.D., Atanackovic, T.M.: Similarity solutions to nonlinear heat conduction and Burgers/Korteweg–deVries fractional equations. J. Comput. Appl. Math. 212, 701714 (2008)MathSciNetGoogle Scholar
  9. 9.
    Gazizov, R.K., Kasatkin, A.A., Lukashchuk, S.Y.: Symmetry properties of fractional diffusion equations. Phys. Scr. T136, 014016 (2009)CrossRefGoogle Scholar
  10. 10.
    Gazizov, R.K., Kasatkin, A.A., Lukashchuk, SYu.: Continuous transformation groups of fractional differential equations Vestnik. USATU 9, 125135 (2007). [in Russian]Google Scholar
  11. 11.
    Gazizov, R.K., Kasatkin, A.A., Lukashchuk, SYu.: Symmetry properties of fractional diffusion equations. Phys. Scr. T136, 014016 (2009)CrossRefGoogle Scholar
  12. 12.
    He, J.H., Wu, G.C., Austin, F.: The variational iteration method which should be followed. Nonlinear Sci. Lett. A 1(1), (2010)Google Scholar
  13. 13.
    He, J.H.: Analytical methods for thermal science-An elementary introduction. Therm. Sci. 15(S1), (2011)Google Scholar
  14. 14.
    Ibragimov, N.H.: Handbook of Lie Group Analysis of Differential Equations, vols. 1, 2, 3. CRC Press, Boca Raton (1994, 1995,1996)Google Scholar
  15. 15.
    Jafari, H., Kadkhoda, N., Tajadodi, H., Hosseini Matikolai, S.A.: Homotopy perturbation Pade technique for solving fractional Riccati differential equations. Int. J. Nonlin. Sci. Numer. Simul. 11, 271–275 (2010)MathSciNetGoogle Scholar
  16. 16.
    Jefferson, G.F., Carminati, J.: FracSym: automated symbolic computation of Lie symmetries of fractional differential equations. Comp. Phys. Commun. 185, 430–441 (2014)CrossRefGoogle Scholar
  17. 17.
    Kasatkin, A.A.: Symmetry properties for systems of two ordinary fractional differential equations. Ufa Math. J. 4, 65–75 (2012)Google Scholar
  18. 18.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies. Elsevier, Amsterdam (2006)Google Scholar
  19. 19.
    Liu, C.S.: Counterexamples on Jumarie’s two basic fractional calculus formula. Commun. Nonlinear Sci. Numer. Simul. 22(1–3), 9294 (2015)Google Scholar
  20. 20.
    Lu, B.: The first integral method for some time fractional differential equations. J. Math. Anal. Appl. 395, 684–693 (2012)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Mainardi, F.: The fundamental solutions for the fractional diffusion-wave equation. Appl. Math. Lett. 9, 23–28 (1996)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Machado, J.A.T., Galhano, A.: Approximating fractional derivatives in the perspective of system control. Nonlinear Dyn. 56, 401–407 (2009)CrossRefMATHGoogle Scholar
  23. 23.
    Nadjafikhah, M., Ahangari, F.: Symmetry reduction of two-dimensional damped Kuramoto–Sivashinsky equation. Commun. Theor. Phys. 56, 211–217 (2011)CrossRefMATHGoogle Scholar
  24. 24.
    Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974)MATHGoogle Scholar
  25. 25.
    Olver, P.J.: Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics, vol. 107, 2nd edn. Springer, Berlin (1993)CrossRefGoogle Scholar
  26. 26.
    Ovsyannikov, L.V.: Group Analysis of Differential Equations. Academic Pres, New York (1982)MATHGoogle Scholar
  27. 27.
    Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, fractional differential equations, to Methods of Their Solution and some of Their Applications, vol. 198 of Mathematics in Science and Engineering. Academic Press, San Diego (1999)Google Scholar
  28. 28.
    Samko, G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Yverdon (1993)MATHGoogle Scholar
  29. 29.
    Sahadevan, R., Bakkyaraj, T.: Invariant analysis of time fractional generalized Burgers and Korteweg–deVries equations. J. Math. Anal. Appl. 393, 341347 (2012)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Tarasov, V.E.: No violation of the Leibniz rule. No fractional derivative. Commun. Nonlinear Sci. Numer. Simul. 18, 2945–2948 (2013)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Wang, G.W., Liu, X.Q., Zhang, Y.Y.: Lie symmetry analysis to the time fractional generalized fifth-order KdV equation. Commun. Nonlinear Sci. (2013). doi: 10.1016/j.cnsns.2012.11.032
  32. 32.
    Wu, G.C.: A fractional Lie group method for anomalous diffusion equations. Commun. Fract. Calc. 1, 27–31 (2010)Google Scholar
  33. 33.
    Wu, G.C., Baleanu, D.: Discrete fractional logistic map and its chaos. Nonlinear Dyn. 75(1–2), 283–287 (2014)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Xing-Jiang, Y., Xi-Qiang, L.: Lie symmetry analysis of the time fractional Boussinesq equation. Acta Phys. Sin. 62(23), 230201 (2013). in ChineseGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • Hossein Jafari
    • 1
    • 2
  • Nematollah Kadkhoda
    • 3
  • Dumitru Baleanu
    • 4
    • 5
  1. 1.Department of MathematicsUniversity of MazandaranBabolsarIran
  2. 2.Department of Mathematical SciencesUniversity of South Africa (UNISA)PretoriaSouth Africa
  3. 3.Department of Mathematics, Faculty of Basic SciencesBozorgmehr University of QaenatQaenatIran
  4. 4.Department of Mathematics, Faculty of Art and SciencesCankaya UniversityBalgatTurkey
  5. 5.Institute of Space SciencesMagurele, BucharestRomania

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