Time fractional Kupershmidt equation: symmetry analysis and explicit series solution with convergence analysis

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Abstract

In this work, the fractional Lie symmetry method is applied for symmetry analysis of time fractional Kupershmidt equation. Using the Lie symmetry method, the symmetry generators for time fractional Kupershmidt equation are obtained with Riemann-Liouville fractional derivative. With the help of symmetry generators, the fractional partial differential equation is reduced into the fractional ordinary differential equation using Erdélyi-Kober fractional differential operator. The conservation laws are determined for the time fractional Kupershmidt equation with the help of new conservation theorem and fractional Noether operators. The explicit analytic solutions of fractional Kupershmidt equation are obtained using the power series method. Also, the convergence of the power series solutions is discussed by using the implicit function theorem.

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  • [1] R. Arora A. Chauhan: Lie Symmetry Analysis and Some Exact Solutions of (2 + 1)-dimensional KdV-Burgers Equation. International Journal of Applied and Computational Mathematics 5 (1) (2019) 15.

  • [2] D. Baleanu M. Inc A. Yusuf A.I. Aliyu: Lie symmetry analysis exact solutions and conservation laws for the time fractional modified Zakharov-Kuznetsov equation. Nonlinear Analysis: Modelling and Control 22 (6) (2017) 861–876.

  • [3] D. Baleanu A. Yusuf A.I. Aliyu: Space-time fractional Rosenou-Haynam equation: Lie symmetry analysis explicit solutions and conservation laws. Advances in Difference Equations 2018 (1) (2018) 46.

  • [4] G.W. Bluman J.D. Cole: The general similarity solution of the heat equation. Journal of Mathematics and Mechanics 18 (11) (1969) 1025–1042.

  • [5] G.W. Bluman S. Kumei: Use of group analysis in solving overdetermined systems of ordinary differential equations. Journal of Mathematical Analysis and Applications 138 (1) (1989) 95–105.

  • [6] K. Diethelm N.J. Ford A.D. Freed: A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dynamics 29 (1-4) (2002) 3–22.

  • [7] R.A. El-Nabulsi: Fractional functional with two occurrences of integrals and asymptotic optimal change of drift in the Black-Scholes model. Acta Mathematica Vietnamica 40 (4) (2015) 689–703.

  • [8] L.L. Feng S.F. Tian X.B. Wang T.T. Zhang: Lie Symmetry Analysis Conservation Laws and Exact Power Series Solutions for Time-Fractional Fordy-Gibbons Equation. Communications in Theoretical Physics 66 (3) (2016) 321.

  • [9] R.K. Gazizov A.A. Kasatkin S.Y. Lukashchuk: Symmetry properties of fractional diffusion equations. Physica Scripta 2009 (T136) (2009) 014016.

  • [10] R. Hilfer: Applications of fractional calculus in physics. World Scientific (2000).

  • [11] M. Inc A. Yusuf A.I. Aliyu D. Baleanu: Lie symmetry analysis explicit solutions and conservation laws for the space-time fractional nonlinear evolution equations. Physica A: Statistical Mechanics and its Applications 496 (2018) 371–383.

  • [12] A.A. Kilbas H.M. Srivastava J.J. Trujillo: Fractional differential equations: A emergent field in applied and mathematical sciences. In: Factorization Singular Operators and Related Problems. Springer (2003) 151–173.

  • [13] V.S. Kiryakova: Generalized fractional calculus and applications. CRC Press (1993).

  • [14] S. Lie: Theorie der Transformationsgruppen I. Mathematische Annalen 16 (4) (1880) 441–528.

  • [15] W. Liu K. Chen: The functional variable method for finding exact solutions of some nonlinear time-fractional differential equations. Pramana 81 (3) (2013) 377–384.

  • [16] Y. Luchko R. Gorenflo: Scale-invariant solutions of a partial differential equation of fractional order. Fractional Calculus and Applied Analysis 3 (1) (1998) 63–78.

  • [17] S.Y. Lukashchuk: Conservation laws for time-fractional subdiffusion and diffusion-wave equations. Nonlinear Dynamics 80 (1–2) (2015) 791–802.

  • [18] E. Noether: Invariant variation problems. Transport Theory and Statistical Physics 1 (3) (1971) 186–207.

  • [19] P.J. Olver: Applications of Lie groups to differential equations. Springer Science & Business Media (2000).

  • [20] M.D. Ortigueira J.A.T. Machado: What is a fractional derivative? Journal of computational Physics 293 (2015) 4–13.

  • [21] T.J. Osler: Leibniz rule for fractional derivatives generalized and an application to infinite series. SIAM Journal on Applied Mathematics 18 (3) (1970) 658–674.

  • [22] Y. Pandir Y. Gurefe E. Misirli: New exact solutions of the time-fractional nonlinear dispersive KdV equation. International Journal of Modeling and Optimization 3 (4) (2013) 349–351.

  • [23] I. Podlubny: Fractional differential equations: An introduction to fractional derivatives fractional differential equations to methods of their solution and some of their applications. Elsevier (1998).

  • [24] Ch.Y. Qin Sh.F. Tian X.B. Wang T.T. Zhang: Lie symmetries conservation laws and explicit solutions for time fractional Rosenau-Haynam equation. Communications in Theoretical Physics 67 (2) (2017) 157.

  • [25] S.S. Ray S. Sahoo S. Das: Formulation and solutions of fractional continuously variable order mass-spring-damper systems controlled by viscoelastic and viscous-viscoelastic dampers. Advances in Mechanical Engineering 8 (5) (2016) 1–17.

  • [26] H. Richard: Fractional Calculus: an introduction for physicists. World Scientific (2014).

  • [27] Y.A. Rossikhin M.V. Shitikova: Analysis of dynamic behaviour of viscoelastic rods whose rheological models contain fractional derivatives of two different orders. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik: Applied Mathematics and Mechanics 81 (6) (2001) 363–376.

  • [28] Y.A. Rossikhin M.V. Shitikova: Application of fractional calculus for dynamic problems of solid mechanics: novel trends and recent results. Applied Mechanics Reviews 63 (1) (2010) 010801(1–52).

  • [29] R. Sahadevan T. Bakkyaraj: Invariant analysis of time fractional generalized Burgers and Korteweg-de Vries equations. Journal of Mathematical Analysis and Applications 393 (2) (2012) 341–347.

  • [30] S.G. Samko A.A. Kilbas O.I. Marichev: Fractional integrals and derivatives: theory and applications. Gordon and Breach Switzerland (1993)

  • [31] N. Shang B. Zheng: Exact solutions for three fractional partial differential equations by the (G/G) method. Int. J. Appl. Math 43 (3) (2013) 114–119.

  • [32] K. Singla R.K. Gupta: Space-time fractional nonlinear partial differential equations: symmetry analysis and conservation laws. Nonlinear Dynamics 89 (1) (2017) 321–331.

  • [33] B. Tang Y. He L. Wei X. Zhang: A generalized fractional sub-equation method for fractional differential equations with variable coefficients. Physics Letters A 376 (38–39) (2012) 2588–2590.

  • [34] V.E. Tarasov: On chain rule for fractional derivatives. Communications in Nonlinear Science and Numerical Simulation 30 (1–3) (2016) 1–4.

  • [35] G.W. Wang X.Q. Liu Y.Y. Zhang: Lie symmetry analysis to the time fractional generalized fifth-order KdV equation. Communications in Nonlinear Science and Numerical Simulation 18 (9) (2013) 2321–2326.

  • [36] X.B. Wang S.F. Tian Ch.Y. Qin T.T. Zhang: Lie symmetry analysis conservation laws and exact solutions of the generalized time fractional Burgers equation. EPL (Europhysics Letters) 114 (2) (2016) 20003.

  • [37] X.B. Wang S.F. Tian Ch.Y. Qin T.T. Zhang: Lie symmetry analysis conservation laws and analytical solutions of a time-fractional generalized KdV-type equation. Journal of Nonlinear Mathematical Physics 24 (4) (2017) 516–530.

  • [38] X.B. Wang S.F. Tian: Lie symmetry analysis conservation laws and analytical solutions of the time-fractional thin-film equation. Computational and Applied Mathematics (2018) 1–13.

  • [39] A. Yıldırım: An algorithm for solving the fractional nonlinear Schrödinger equation by means of the homotopy perturbation method. International Journal of Nonlinear Sciences and Numerical Simulation 10 (4) (2009) 445–450.

  • [40] A. Yusuf A.I. Aliyu D. Baleanu: Lie symmetry analysis and explicit solutions for the time fractional generalized Burgers-Huxley equation. Optical and Quantum Electronics 50 (2) (2018) 94.

  • [41] S. Zhang: A generalized Exp-function method for fractional Riccati differential equations. Communications In Fractional Calculus 1 (2010) 48–51.

  • [42] Y. Zhang J. Mei X. Zhang: Symmetry properties and explicit solutions of some nonlinear differential and fractional equations. Applied Mathematics and Computation 337 (2018) 408–418.

  • [43] R.Z. Zhdanov: Conditional Lie-Backlund symmetry and reduction of evolution equations. Journal of Physics A: Mathematical and General 28 (13) (1995) 3841.

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