Lie group method and fractional differential equations

Volume 10, Issue 8, pp 4175--4180 http://dx.doi.org/10.22436/jnsa.010.08.13

Publication Date: August 11, 2017 Submission Date: March 28, 2017

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Authors

M. M. Alshamrani - Department of Mathematics, Faculty of Science, Northern Border University, Arar, Saudi Arabia. H. A. Zedan - Department of Mathematics, Faculty of Science, Kafrelsheikh University, Egypt. M. Abu-Nawas - Department of Mathematics, Faculty of Science, Northern Border University, Arar, Saudi Arabia.

Abstract

In this paper, Lie group method is applied to investigate and solve some classes of nonlinear fractional differential equations. In addition, we use the obtained symmetries to induce exact solutions for the equations under consideration.

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Keywords

• Lie group method
• nonlinear partial differential equations
• symmetry analysis.

MSC

•  22EXX
•  17BXX
•  20GXX

References

• [1] W. M. Abd-Elhameed, Y. H. Youssri, Spectral solutions for fractional differential equations via a novel Lucas operational matrix of fractional derivatives, Rom. J. Phys., 61 (2016), 795–813.
  • Google Scholar


• [2] M. A. Abdelkawy, M. A. Zaky, A. H. Bhrawy, D. Baleanu , Numerical simulation of time variable fractional order mobile-immobile advection-dispersion model , Rom. Rep. Phys., 67 (2015), 773–791.
  • Google Scholar


• [3] A. Agila, D. Baleanu, R. Eid, B. Irfanoglu , Applications of the extended fractional Euler-Lagrange equations model to freely oscillating dynamical systems, Rom. J. Phys., 61 (2016), 350–359.
  • Google Scholar


• [4] W. F. Ames, R. L. Anderson, V. A. Dorodnitsyn, E. V. Ferapontov, R. K. Gazizov, N. H. Ibragimov, S. R. Svirshchevskiĭ, CRC handbook of Lie group analysis of differential equations, Vol. 1, Symmetries, exact solutions and conservation laws. CRC Press, Boca Raton, FL (1994)
  • MathSciNet
  • Google Scholar
  • MATH


• [5] A. H. Bhrawy , A new spectral algorithm for time-space fractional partial differential equations with subdiffusion and superdiffusion, Proc. Rom. Acad. Ser. A Math. Phys. Tech. Sci. Inf. Sci., 17 (2016), 39–47.
  • MathSciNet
  • Google Scholar


• [6] E. Buckwar, Y. Luchko , Invariance of a partial differential equation of fractional order under the Lie group of scaling transformations , J. Math. Anal. Appl., 227 (1997), 81–97.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [7] E. H. El Kinani, A. Ouhadan, Lie symmetry analysis of some time fractional partial differential equations, Int. J. Mod. Phys., 38 (2015), 8 pages.
  • View Article
  • Google Scholar


• [8] M. Gaur, K. Singh, On group invariant solutions of fractional order Burgers-Poisson equation, Appl. Math. Comput., 244 (2014), 870–877.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [9] R. K. Gazizov, A. A. Kasatkin , Construction of exact solutions for fractional order differential equations by the invariant subspace method , Comput. Math. Appl., 66 (2013), 576–584.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [10] R. K. Gazizov, A. A. Kasatkin, S. Y. Lukashchuk, Continuous transformation groups of fractional differential equations , (Russian) Vestnik USATU, 9 (2007), 125–135.
  • Google Scholar


• [11] R. K. Gazizov, A. A. Kasatkin, S. Y. Lukashchuk, Symmetry properties of fractional diffusion equations, Phys. Scr., T136 (2009), 5 pages. [12] R. K. Gazizov, A. A. Kasatkin, S. Y. Lukashchuk, Group-invariant solutions of fractional differential equations, Nonlinear science and complexity, Springer, Dordrecht, (2011), 51–59.
  • View Article
  • Google Scholar


• [12] R. K. Gazizov, A. A. Kasatkin, S. Y. Lukashchuk , Symmetry properties of fractional diffusion equations, Phys. Scr., T136 (2009), 5 pages.
  • View Article
  • Google Scholar


• [13] R. K. Gazizov, A. A. Kasatkin, S. Y. Lukashchuk, Group-invariant solutions of fractional differential equations, Nonlinear science and complexity, Springer, Dordrecht, (2011), 51–59.
  • View Article
  • MathSciNet
  • Google Scholar


• [14] Q. Huang, R. Zhdanov , Symmetries and exact solutions of the time fractional Harry-Dym equation with Riemann-Liouville derivative , Phys. A, 409 (2014), 110–118.
  • View Article
  • MathSciNet
  • Google Scholar


• [15] D. Kumar, J. Singh, D. Baleanu, A fractional model of convective radial fins with temperature-dependent thermal conductivity, Rom. Rep. Phys., 69 (2017), 13 pages.
  • Google Scholar


• [16] K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York (1993)
  • MathSciNet
  • Google Scholar


• [17] K. Nouri, S. Elahi-Mehr, L. Torkzadeh, Investigation of the behavior of the fractional Bagley-Torvik and Basset equations via numerical inverse Laplace transform, Rom. Rep. Phys., 68 (2016), 503–514.
  • Google Scholar


• [18] K. B. Oldham, J. Spanier, The fractional calculus, Theory and applications of differentiation and integration to arbitrary order, With an annotated chronological bibliography by Bertram Ross, Mathematics in Science and Engineering, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London (1974)
  • Google Scholar


• [19] I. Podlubny, Fractional differential equations, An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, Academic Press, Inc., San Diego, CA (1999)
  • Google Scholar


• [20] R. Sahadevan, T. Bakkyaraj , Invariant analysis of time fractional generalized Burgers and Korteweg-de Vries equations , J. Math. Anal. Appl., 393 (2012), 341–347.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [21] R. Sahadevan, T. Bakkyaraj, Invariant subspace method and exact solutions of certain nonlinear time fractional partial differential equations, Fract. Calc. Appl. Anal., 18 (2015), 146–162.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [22] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives , Theory and applications, Edited and with a foreword by S. M. Nikol'skiĭ, Translated from the 1987 Russian original, Revised by the authors, Gordon and Breach Science Publishers, Yverdon (1993)
  • Google Scholar


• [23] G.-W. Wang, X.-Q. Liu, Y.-Y. Zhang, Lie symmetry analysis to the time fractional generalized fifth-order KdV equation, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 2321–2326.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [24] G.-W. Wang, T.-Z. Xu, Symmetry properties and explicit solutions of the nonlinear time fractional KdV equation, Bound. Value Probl., 232 (2013), 13 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [25] G.-W.Wang, T.-Z. Xu, Invariant analysis and explicit solutions of the time fractional nonlinear perturbed Burgers equation , Nonlinear Anal. Model. Control, 20 (2015), 570–584.
  • View Article
  • MathSciNet
  • Google Scholar


• [26] G.-C. Wu , A fractional Lie group method for anonymous diffusion equations , Commun. Frac. Calc., 1 (2010), 27–31.
  • Google Scholar


• [27] G.-C. Wu, Lie group classifications and non-differentiable solutions for time-fractional Burgers Equation, Commun. Theor. Phys., 55 (2011), 1073–1076.
  • View Article
  • Google Scholar
  • MATH


• [28] X.-J. Yang, F. Gao, H. M. Srivastava , New rheological models within local fractional derivative, Rom. Rep. Phys., 69 (2017), 12 pages.
  • Google Scholar


• [29] H. Zedan, M. M. Alshamrani, A novel class of solutions for the (2 + 1)-dimensional higher-order Broer-Kaup system, Comput. Math. Appl., 69 (2015), 67–80.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [30] L. M. Zelenyi, A. V. Milovanov , Fractal topology and strange kinetics: from percolation theory to problems in cosmic electrodynamics, Phys. Usp., 47 (2004), 749–788.
  • View Article
  • Google Scholar


• [31] Y. Zhang, D. Baleanu, X.-J. Yang, New solutions of the transport equations in porous media within local fractional derivative, Proc. Rom. Acad. Ser. A Math. Phys. Tech. Sci. Inf. Sci., 17 (2016), 230–236.
  • MathSciNet
  • Google Scholar


• [32] M. Zurigat, S. Momani, Z. Odibat, A. Alawneh , The homotopy analysis method for handling systems of fractional differential equations, Appl. Math. Model., 34 (2010), 24–35.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH