Lie group method and fractional differential equations
-
2697
Downloads
-
4440
Views
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.
Share and Cite
ISRP Style
M. M. Alshamrani, H. A. Zedan, M. Abu-Nawas, Lie group method and fractional differential equations, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 8, 4175--4180
AMA Style
Alshamrani M. M., Zedan H. A., Abu-Nawas M., Lie group method and fractional differential equations. J. Nonlinear Sci. Appl. (2017); 10(8):4175--4180
Chicago/Turabian Style
Alshamrani, M. M., Zedan, H. A., Abu-Nawas, M.. "Lie group method and fractional differential equations." Journal of Nonlinear Sciences and Applications, 10, no. 8 (2017): 4175--4180
Keywords
- Lie group method
- nonlinear partial differential equations
- symmetry analysis.
MSC
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.
-
[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.
-
[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.
-
[4]
W. F. Ames, R. L. Anderson, V. A. Dorodnitsyn, E. V. Ferapontov, R. K. Gazizov, N. H. Ibragimov, S. R. SvirshchevskiÄ, CRC handbook of Lie group analysis of differential equations, Vol. 1, Symmetries, exact solutions and conservation laws. CRC Press, Boca Raton, FL (1994)
-
[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.
-
[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.
-
[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.
-
[8]
M. Gaur, K. Singh, On group invariant solutions of fractional order Burgers-Poisson equation, Appl. Math. Comput., 244 (2014), 870–877.
-
[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.
-
[10]
R. K. Gazizov, A. A. Kasatkin, S. Y. Lukashchuk, Continuous transformation groups of fractional differential equations , (Russian) Vestnik USATU, 9 (2007), 125–135.
-
[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.
-
[12]
R. K. Gazizov, A. A. Kasatkin, S. Y. Lukashchuk , Symmetry properties of fractional diffusion equations, Phys. Scr., T136 (2009), 5 pages.
-
[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.
-
[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.
-
[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.
-
[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)
-
[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.
-
[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)
-
[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)
-
[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.
-
[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.
-
[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'skiÄ, Translated from the 1987 Russian original, Revised by the authors, Gordon and Breach Science Publishers, Yverdon (1993)
-
[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.
-
[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.
-
[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.
-
[26]
G.-C. Wu , A fractional Lie group method for anonymous diffusion equations , Commun. Frac. Calc., 1 (2010), 27–31.
-
[27]
G.-C. Wu, Lie group classifications and non-differentiable solutions for time-fractional Burgers Equation, Commun. Theor. Phys., 55 (2011), 1073–1076.
-
[28]
X.-J. Yang, F. Gao, H. M. Srivastava , New rheological models within local fractional derivative, Rom. Rep. Phys., 69 (2017), 12 pages.
-
[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.
-
[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.
-
[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.
-
[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.