{"title":"Group Invariant Solutions of Nonlinear Time-Fractional Hyperbolic Partial Differential Equation","authors":"Anupma Bansal, Rajeev Budhiraja, Manoj Pandey","volume":121,"journal":"International Journal of Mathematical and Computational Sciences","pagesStart":48,"pagesEnd":52,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/10006686","abstract":"In this paper, we have investigated the nonlinear
\r\ntime-fractional hyperbolic partial differential equation (PDE) for
\r\nits symmetries and invariance properties. With the application of
\r\nthis method, we have tried to reduce it to time-fractional ordinary
\r\ndifferential equation (ODE) which has been further studied for exact
\r\nsolutions.","references":"[1] K.S. Miller, B. Ross, An Introduction to Fractional Calculus and\r\nFractional Differential Equations, Wiley, New York, 1993.\r\n[2] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of\r\nFrcational Differential Equations, Elsevier, San Diego, 2006.\r\n[3] P.J. Olver, Applications of Lie Groups to Differential Equations,\r\nGraduate Texts Math., vol. 107, 1993.\r\n[4] G.W. Bluman, J.D. Cole, Similarity Methods for Differential Equations,\r\nSpringer Verlag, 1974.\r\n[5] L.V. Ovsiannikov, Group Analysis of Differential Equations, Academic\r\nPress, 1982.\r\n[6] Q. Huang, R. Zhdanov, Symmetries and Exact Solutions of the Time\r\nFractional Harry-Dym Equation with Riemann-Liouville Derivative,\r\nPhysica A, vol. 409, 2014, pp. 110-118.\r\n[7] K. Al-Khaled, Numerical Solution of Time-fractional PDEs Using\r\nSumudu Decomposition Method, Romanian Journal of Physics, vol. 60,\r\n2015, pp. 99-110.\r\n[8] Y. Zhang, Lie Symmetry Analysis to General Time-Fractional\r\nKorteweg-De Vries Equation, Fractional Differential Calculus, vol.5,\r\n2015, pp. 125-135.\r\n[9] V.D. Djordjevic, T.M. Atanackovic, Similarity Solutions to Nonlinear\r\nHeat Conduction and Burgers\/Korteweg-De Vries Fractional Equations,\r\nJournal of Computational Applied Mathematics, vol. 222, 2008, pp.\r\n701-714.\r\n[10] M. Gaur, K. Singh, On Group Invariant Solutions of Fractional Order\r\nBurgers-Poisson Equation, Applied Mathematics and Computation, vol.\r\n244, 2014, pp. 870-877.\r\n[11] H. Liu, J. Li, Q. Zhang, Lie Symmetry Analysis and Exact Explicit\r\nSolutions for General Burgers Equations, Journal of Computational\r\nApplied Mathematics, vol. 228, 2009, pp. 1-9.\r\n[12] H. Liu, Complete Group Classifications and Symmetry Reductions of\r\nthe Fractional fifth-order KdV Types of Equations, Studies in Applied\r\nMathematics, vol. 131, 2013, pp. 317-330.\r\n[13] A. Bansal, R.K. Gupta, Lie point Symmetries and Similarity Solutions\r\nof the Time-Dependent Coefficients Calogero-Degasperis Equation,\r\nPhysica Scripta, vol. 86, 2012, pp. 035005 (11 pages).\r\n[14] A. Bansal, R.K. Gupta, Modified (G'\/G)-Expansion Method for\r\nFinding Exact Wave Solutions of Klein-Gordon-Schrodinger Equation,\r\nMathematical Methods in the Applied Sciences, vol. 35, 2012, pp.\r\n1175-1187.\r\n[15] R.K. Gupta, A. Bansal, Painleve Analysis, Lie Symmetries and Invariant\r\nSolutions of potential Kadomstev Petviashvili Equation with Time\r\nDependent Coefficients, Applied Mathematics and Computation, vol.\r\n219, 2013, pp. 5290-5302.\r\n[16] R. Kumar, R.K. Gupta, S.S. Bhatia, Lie Symmetry Analysis and Exact\r\nSolutions for Variable Coefficients Generalised Kuramoto-Sivashinky\r\nEquation, Romanian Reports in Physics, vol. 66, 2014, pp. 923-928.\r\n[17] M. Pandey, Lie Symmetries and Exact Solutions of Shallow Water\r\nEquations with Variable Bottom, Internation Journal of Nonlinear\r\nScience and Numerical Simulation, vol. 16, 2015, pp. 337-342.\r\n[18] V.S. Kiryakova, Generalized Fractional Calculus and Applications, CRC\r\npress,1993.","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 121, 2017"}