Abstract
The article studies the application of the extended Fan sub-equation method to \((1+1)\)-dimensional nonlinear dispersive modified Benjamin-Bona-Mahony equation with fractional evolution. This equation describes the hydromagnetic waves in cold plasma, acoustic waves in inharmonic crystals and acoustic gravity waves in compressible fluids. The structure of the extended Fan sub-equation method on the basis of time-fractional derivative is presented. The main idea of the method is to take full advantage of the general elliptic equation involving five parameters. The fractional derivatives are taken as in the sense of Jumarie’s modified Riemann–Liouville derivative. The method is reliable and gives more general exact solutions than the existing methods.
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Akram, G., Batool, F.: Solitary wave solutions of the Schäfer-Wayne short-pulse equation using two reliable methods. Opt. Quantum Electron. 49(1), 14 (2017)
Akram, G., Batool, F.: A class of traveling wave solutions for space-time fractional biological population model in mathematical physics. Indian. J. Phys. (2017). doi:10.1007/s12648-017-1007-1
Baskonus, H.M.: New acoustic wave behaviors to the Davey–Stewartson equation with power-law nonlinearity arising in fluid dynamics. Nonlinear Dyn. 86(1), 177–183 (2016)
Baskonus, Haci Mehmet, Hasan, Bulut: Analytical studies on the \((1+1)-\)dimensional nonlinear dispersive modified Benjamin-Bona-Mahony equation defined by seismic sea waves. Waves in Random and Complex Media 25(4), 576–586 (2015)
Batool, F., Akram, G.: A novel approach for solitart wave solutions of the Generalized fractional Zakharov–Kuznetsov equation. Indian. J. Phys. (2017). doi:10.1007/s12648-017-1071-6
Batool, F., Akram, G.: Solitary wave solutions of (2+ 1)-dimensional soliton equation arising in mathematical physics. Optik-Int. J. Light and Electron. Opt. 144, 156–162 (2017). doi:10.1016/j.ijleo.2017.06.079
Batool, F., Akram, G.: On the solitary wave dynamics of complex Ginzburg-Landau equation with cubic nonlinearity. Opt. Quantum Electron. 49(4), 129 (2017)
Benjamin, T.B., Bona, J.L., Mahony, J.J.: Model equations for long waves in nonlinear dispersive system. Philos. Trans. R. Soc. Lond. Ser. A: Math. Phys. Sci. 272, 47–78 (1972)
Dai, C.Q., Wang, Y., Liu, J.: Spatiotemporal Hermite–Gaussian solitons of a \((3 + 1)-\)dimensional partially nonlocal nonlinear Schrodinger equation. Nonlinear Dynamics 84, 1157–1161 (2016)
Guner, O., Atik, H., Kayyrzhanovich, A.A.: New exact solution for space-time fractional differential equations via \((G/G)-\)expansion method. Optik-Int. J. Light and Electron. Opt. 130, 696–701 (2017)
Gurefe, Y., Misirli, E.: Exp-function method for solving nonlinear evolution equations with higher order nonlinearity. Comput. Math. Appl. 61(8), 2025–2030 (2011)
Jumarie, G.: Table of some basic fractional calculus formulae derived from a modified Riemann–Liouville derivative for non differentiable functions. Appl. Math. Lett. 22(3), 378–385 (2009)
Li, Z.B., He, J.H.: Fractional complex transform for fractional differential equations. Math. Comput. Appl. 15(5), 970–973 (2010)
Mirzazadeh, M., Biswas, A.: Optical solitons with spatio-temporal dispersion by first integral approach and functional variable method. Optik-Int. J. Light and Electron. Opt. 125(19), 5467–5475 (2014)
Mirzazadeh, M., Eslami, M., Zerrad, E., Mahmood, M.F., Biswas, A., Belic, M.: Optical solitons in nonlinear directional couplers by sine-cosine function method and Bernoullis equation approach. Nonlinear Dyn. 81(4), 1933–1949 (2015)
Tariq, H., Akram, G.: New traveling wave exact and approximate solutions for the nonlinear CahnAllen equation: evolution of a nonconserved quantity. Nonlinear Dyn. 88(581), 1–14 (2016)
Tariq, H., Akram, G.: New approach for exact solutions of time fractional Cahn–Allen and time fractional Phi-4 equation. Physica A 473, 352–362 (2017)
Triki, H., Wazwaz, A.M.: Bright and dark soliton solutions for a K\((m, n)\) equation with t-dependent coefficients. Phys. Lett. A 373(25), 2162–2165 (2009)
Wazwaz, A.M.: A sine-cosine method for handlingnonlinear wave equations. Math. Comput. Model. 40(5), 499–508 (2004)
Yomba, E.: The extended Fan sub-equation method and its application to the \((2+ 1)-\)dimensional dispersive long wave and Whitham–Broer–Kaup equations. Chin. J. Phys. 43(4), 789–805 (2005)
Yomba, E.: The extended Fan’s sub-equation method and its application to KdV-MKdV. BKK and Var. Boussinesq Equ. Phys. Lett. A 336(6), 463–476 (2005)
Yusufolu, E.: New solitonary solutions for the MBBM equations using exp-function method. Phys. Lett. A 372, 442446 (2008)
Zayed, E.M.E., Al-Joudi, S.: Applications of an extended \((\frac{G}{G})-\)expansion method to find exact solutions of nonlinear PDEs in mathematical physics. Math. Prob. Eng. 2010(2010), 19 (2010)
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Batool, F., Akram, G. Application of extended Fan sub-equation method to \(\mathbf {(1+1)}\)-dimensional nonlinear dispersive modified Benjamin-Bona-Mahony equation with fractional evolution. Opt Quant Electron 49, 375 (2017). https://doi.org/10.1007/s11082-017-1212-3
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DOI: https://doi.org/10.1007/s11082-017-1212-3