Group Invariant Solutions of Nonlinear Time-Fractional Hyperbolic Partial Differential Equation

In this paper, we have investigated the nonlinear
time-fractional hyperbolic partial differential equation (PDE) for
its symmetries and invariance properties. With the application of
this method, we have tried to reduce it to time-fractional ordinary
differential equation (ODE) which has been further studied for exact
solutions.

Authors:

• Anupma Bansal
• Rajeev Budhiraja
• Manoj Pandey

Keywords:

• Nonlinear time-fractional hyperbolic PDE
• Lie Classical method
• exact solutions.

References:

[1] K.S. Miller, B. Ross, An Introduction to Fractional Calculus and
Fractional Differential Equations, Wiley, New York, 1993.
[2] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of
Frcational Differential Equations, Elsevier, San Diego, 2006.
[3] P.J. Olver, Applications of Lie Groups to Differential Equations,
Graduate Texts Math., vol. 107, 1993.
[4] G.W. Bluman, J.D. Cole, Similarity Methods for Differential Equations,
Springer Verlag, 1974.
[5] L.V. Ovsiannikov, Group Analysis of Differential Equations, Academic
Press, 1982.
[6] Q. Huang, R. Zhdanov, Symmetries and Exact Solutions of the Time
Fractional Harry-Dym Equation with Riemann-Liouville Derivative,
Physica A, vol. 409, 2014, pp. 110-118.
[7] K. Al-Khaled, Numerical Solution of Time-fractional PDEs Using
Sumudu Decomposition Method, Romanian Journal of Physics, vol. 60,
2015, pp. 99-110.
[8] Y. Zhang, Lie Symmetry Analysis to General Time-Fractional
Korteweg-De Vries Equation, Fractional Differential Calculus, vol.5,
2015, pp. 125-135.
[9] V.D. Djordjevic, T.M. Atanackovic, Similarity Solutions to Nonlinear
Heat Conduction and Burgers/Korteweg-De Vries Fractional Equations,
Journal of Computational Applied Mathematics, vol. 222, 2008, pp.
701-714.
[10] M. Gaur, K. Singh, On Group Invariant Solutions of Fractional Order
Burgers-Poisson Equation, Applied Mathematics and Computation, vol.
244, 2014, pp. 870-877.
[11] H. Liu, J. Li, Q. Zhang, Lie Symmetry Analysis and Exact Explicit
Solutions for General Burgers Equations, Journal of Computational
Applied Mathematics, vol. 228, 2009, pp. 1-9.
[12] H. Liu, Complete Group Classifications and Symmetry Reductions of
the Fractional fifth-order KdV Types of Equations, Studies in Applied
Mathematics, vol. 131, 2013, pp. 317-330.
[13] A. Bansal, R.K. Gupta, Lie point Symmetries and Similarity Solutions
of the Time-Dependent Coefficients Calogero-Degasperis Equation,
Physica Scripta, vol. 86, 2012, pp. 035005 (11 pages).
[14] A. Bansal, R.K. Gupta, Modified (G'/G)-Expansion Method for
Finding Exact Wave Solutions of Klein-Gordon-Schrodinger Equation,
Mathematical Methods in the Applied Sciences, vol. 35, 2012, pp.
1175-1187.
[15] R.K. Gupta, A. Bansal, Painleve Analysis, Lie Symmetries and Invariant
Solutions of potential Kadomstev Petviashvili Equation with Time
Dependent Coefficients, Applied Mathematics and Computation, vol.
219, 2013, pp. 5290-5302.
[16] R. Kumar, R.K. Gupta, S.S. Bhatia, Lie Symmetry Analysis and Exact
Solutions for Variable Coefficients Generalised Kuramoto-Sivashinky
Equation, Romanian Reports in Physics, vol. 66, 2014, pp. 923-928.
[17] M. Pandey, Lie Symmetries and Exact Solutions of Shallow Water
Equations with Variable Bottom, Internation Journal of Nonlinear
Science and Numerical Simulation, vol. 16, 2015, pp. 337-342.
[18] V.S. Kiryakova, Generalized Fractional Calculus and Applications, CRC
press,1993.