Symmetry analysis, conservation laws and invariant solutions of the time-fractional equal width wave equation
Subject Areas : Statistics
Ramin Najafi
1
(Department of Mathematics, Maku Branch, Islamic Azad University, Maku, Iran)
Keywords: معادله زمان-کسری موج همسان, جواب ناوردا, آنالیز تقارن لی, قوانین بقا, معادله الحاقی,
Abstract :
Lie symmetry analysis provides an efficient method to get the analytical and exact solutions of the fractional differential equations. In this paper, we discuss Lie symmetry analysis for the time-fractional equal width wave equation with Riemann–Liouville derivative. This equation is used to describe the simulation of one-dimensional wave propagation in nonlinear media with dispersion processes. By employing classical and nonclassical Lie symmetry analysis and some technical calculations, new infinitesimal generators are obtained. Then we reduce the fractional equal width wave equation to the ordinary fractional differential equation by changing the coordinates and find invariant solutions to this equation. By means of Ibragimov’s new conservation theorem and the generalization of the Noether operators, we construct the conservation laws for the equation. Also, we derive the adjoint equation and infinitesimal generator associated with Lie symmetries of the underlying equation and we reduce this equation to the ordinary fractional differential equation. In the reduced equations the derivative is in Erdelyi–Kober sense.
[1] Kilbas A.A., Srivastava H.M., Trujillo J.J., Theory and Application of Fractional Differential Equations, Elsevier, The Netherlands, (2006)
[2] Diethelm K., The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Springer, Heidelberg, (2010)
[3] Hilfer R., Applications of Fractional Calculus in Physics, World Scientific, Singapore, (2000)
[4] Barkari E., Metzler R., Klafter J., From continuous time random walks to the fractional Fokker-Planck equation, Phys. Rev. E, 61: 132-138 (2000)
[5] Yuste S.B., Acedo L., Lindenberg K., Reaction front in an A+BC reaction-subdiffusion process, Phys. Rev. E, 69: 036126 (2004)
[6] Nunno G.D., Oksendal B., Advenced Mathematical Methods for Finance, Springer, Berlin (2011)
[7] Hall M.G., Barrick T.R., From diffusion-weighted MRI to anomalous diffusion imaging. Magn. Reson. Med., 59: 447-455 (2008)
[8] Najafi R., Küçük G.D., Çelik E., Modified iteration method for solving fractional gas dynamics equation, Math. Meth. Appl. Sci., 40: 939-946 (2017)
[9] Wazwaz A.M., Rach R., Duan J.S., A study on the systems of the Volterra integral forms of the Lane–Emden equations by the Adomian decomposition method, Math. Meth. Appl. Sci., 37(1):10–19 (2014)
[10] Sakar M.G., Ergören H., Alternative variational iteration method for solving the time-fractional Fornberg–Whitham equation, Appl. Math. Model., 39(14): 3972–3979 (2015)
[11] Khan Y., Wu Q. Homotopy perturbation transform method for nonlinear equations using He’s polynomials, Comput. Math. Appl., 61:1963–1967 (2011)
[12] El-Ajoua, A., Abu Arquba, O., Momani, S., Approximate analytical solution of the nonlinear fractional KdV–Burgers equation: a new iterative algorithm, J. Comput. Phys. 293: 81–95 (2015)
[13] Kurulay M., Bayram M., Approximate analytical solution for the fractional modified KdV by differential transform method, Commun. Nonlinear Sci. Numer. Simulat., 15:1777–178 (2010)
[14] Morrison P.J., Meiss J.D., Carey J.R., Scattering of RLW solitary waves, Physica 11D 324–336 (1984)
[15] Shi D., Zhang Y., Diversity of exact solutions to the conformable space-time fractional MEW equation, Appl. Math. Lett., 99: 105994 (2020)
[16] Tariq K.U., Seadawy A.R., Younis M., Rizvi S.T.R, Dispersive traveling wave solutions to the space-time fractional equal-width dynamical equation and its applications, Opt. Quant. Electron., 50:147 (2018)
[17] Raslan K.R., Ali K.K., Shallal M.A., The modified extended tanh method with the Riccati equation for solving the space-time fractional EW and MEW equations, Chaos, Soliton. Fract., 103: 404–409 (2017)
[18] Olver P.J., Applications of Lie groups to differential equations, Second ed., Springer, New York, (1993)
[19] Bluman, G.W., Cheviakov, A.F., Anco, S.C., Applications of Symmetry Methods to Partial Differential Equations, Springer, Berlin (2010)
[20] Grigoriev, Y.N., Ibragimov, N.H., Kovalev, V.F., Meleshko, S.V., Symmetries of Integro-Differential Equations: With Applications in Mechanics and Plasma Physics, Springer, Berlin (2010)
[21] Hashemi M.S., Bahrami F., Najafi R., Lie symmetry analysis of steady-state fractional reaction-convection-diffusion equation, Optik, 138: 240-249 (2017)
[22] Qin C.Y., Tian S.F., Wang X.B., Zhang T.T., Lie symmetry analysis, conservation laws and analytical solutions for a generalized time-fractional modified KdV equation, Wave. Random Complex., 29: 456-476 (2019)
[23] Sahoo S., Ray S.S., The conservation laws with Lie symmetry analysis for time fractional integrable coupled KdV-mKdV system, Int. J. NonLin. Mech., 98: 114-121 (2018)
[24] Najafi R., Bahrami F., Hashemi M.S., Classical and nonclassical Lie symmetry analysis to a class of nonlinear time--fractional differential equations, Nonlinear Dynam., 87: 1785-1796 (2017)
[25] Bahrami F., Najafi R., Hashemi M.S., On the invariant solutions of space/time-fractional diffusion equations, Indian J. Phys., 91: 1571-1579 (2017)
[26] Najafi R., Group-invariant solutions for time-fractional Fornberg-Whitham equation by Lie symmetry analysis, 8(2): 251-258 (2020)
[27] Noether E., Invariant variational problems, Transp. Theory Stat. Phys., 1:186-207 (1971)
[28] Ibragimov N.H., A new conservation theorem, J. Math. Anal. Appl., 333(1): 311-328 (2007)
[29] Lukashchuk SY., Conservation laws for time-fractional subdiffusion and diffusion-wave equations, Nonlinear Dynam., 80: 791-802 (2015)
[30] Gazizov R.K., Ibragimov N.H., Lukashchuk SY., Nonlinear self-adjointness, conservation laws and exact solutions of time–fractional Kompaneets equations. Commun. Nonlinear Sci. Numer. Simulat., 23: 153–163 (2015)
