Abstract
The some of the well-known nonlinear time fractional parabolic partial differential equations is studied in this paper. The fractional complex transform and the first integral method are employed to construct one-soliton solutions of these equations. The power of this manageable method is confirmed. The obtained solutions include solitary wave solutions, periodic wave solutions and combined formal solutions.
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Razborova, P., Kara, A.H., Biswas, A.: Additional conservation laws for Rosenau–KdV–RLW equation with power law nonlinearity by Lie symmetry. Nonlinear Dyn. 79(1), 743–748 (2014)
Biswas, A., Kara, A.H., Bokhari, A.H., Zaman, F.D.: Solitons and conservation laws of Klein–Gordon equation with power law and log law nonlinearities. Nonlinear Dyn. 73(4), 2191–2196 (2013)
Biswas, A., Ranasinghe, A.: 1-Soliton solution of Kadomtsev–Petviashvili equation with power law nonlinearity. Appl. Math. Comput. 214, 645–647 (2009)
Ismail, M.S., Petkovic, M.D., Biswas, A.: 1-Soliton solution of the generalized KP equation with generalized evolution. Appl. Math. Comput. 216(7), 2220–2225 (2010)
Biswas, A., Zerrad, E.: 1-Soliton solution of the Zakharov–Kuznetsov equation with dual-power law nonlinearity. Commun. Nonlinear Sci. Numer. Simul. 14(9), 3574–3577 (2009)
Biswas, A.: Solitary wave solution for the generalized KdV equation with time-dependent damping and dispersion. Commun. Nonlinear Sci. Numer. Simul. 14(9), 3503–3506 (2009)
Sturdevant, B.J.M., Lott, D.A., Biswas, A.: Dynamics of topological optical solitons with time-dependent dispersion, nonlinearity and attenuation. Commun. Nonlinear Sci. Numer. Simul. 14(8), 3305–3308 (2009)
Biswas, A.: 1-Soliton solution of the generalized Camassa–Holm Kadomtsev–Petviashvili equation. Commun. Nonlinear Sci. Numer. Simul. 14(6), 2524–2527 (2009)
Feng, Z.S.: The first integral method to study the Burgers–KdV equation. J. Phys. A Math. Gen. 35, 343–349 (2002)
Lu, B.: The first integral method for some time fractional differential equations. J. Math. Anal. Appl. 395(2), 684–693 (2012)
Bekir, A., Unsal, O.: Analytic treatment of nonlinear evolution equations using first integral method. Pramana J. Phys. 79, 3–17 (2012)
Tascan, F., Bekir, A., Koparan, M.: Travelling wave solutions of nonlinear evolutions by using the first integral method. Commun. Nonlinear Sci. Numer. Simul. 14, 1810–1815 (2009)
Aslan, I.: Exact and explicit solutions to nonlinear evolution equations using the division theorem. Appl. Math. Comput. 217, 8134–8139 (2011)
Aslan, I.: The first integral method for constructing exact and explicit solutions to nonlinear evolution equations. Math. Methods Appl. Sci. 35, 716–722 (2012)
Aslan, I.: Travelling wave solutions to nonlinear physical models by means of the first integral method. Pramana J. Phys. 76, 533–542 (2011)
Jafari, H., Sooraki, A., Talebi, Y., Biswas, A.: The first integral method and traveling wave solutions to Davey–Stewartson equation. Nonlinear Anal. Model. Control 17(2), 182–193 (2012)
Taghizadeh, N., Mirzazadeh, M., Farahrooz, F.: Exact solutions of the nonlinear Schrödinger equation by the first integral method. J. Math. Anal. Appl. 374, 549–553 (2011)
Mirzazadeh, M., Eslami, M.: Exact solutions of the Kudryashov–Sinelshchikov equation and nonlinear telegraph equation via the first integral method. Nonlinear Anal. Model. Control 17(4), 481–488 (2012)
Biswas, A.: Topological 1-soliton solution of the nonlinear Schrodinger’s equation with Kerr law nonlinearity in \(1+2\) dimensions. Commun. Nonlinear Sci. Numer. Simul. 14, 2845–2847 (2009)
Ebadi, G., Biswas, A.: The \(G^{\prime }/G\) method and topological soliton solution of the K(m, n) equation. Commun. Nonlinear Sci. Numer. Simul. 16, 2377–2382 (2011)
Biswas, A.: 1-Soliton solution of the K(m, n) equation with generalized evolution. Phys. Lett. A 372(25), 4601–4602 (2008)
Ma, W.X.: Travelling wave solutions to a seventh order generalized KdV equation. Phys. Lett. A 180, 221–224 (1993)
Malfliet, W.: Solitary wave solutions of nonlinear wave equations. Am. J. Phys. 60(7), 650–654 (1992)
Ma, W.X., Huang, T.W., Zhang, Y.: A multiple exp-function method for nonlinear differential equations and its application. Phys. Scr. 82, 065003 (2010)
Ma, W.X., Lee, J.-H.: A transformed rational function method and exact solutions to the (\(3+1\))-dimensional Jimbo–Miwa equation. Chaos Solitons Fract. 42, 1356–1363 (2009)
Taghizadeh, N., Mirzazadeh, M.: The simplest equation method to study perturbed nonlinear Schrodinger’s equation with Kerr law nonlinearity. Commun. Nonlinear Sci. Numer. Simul. 17, 1493–1499 (2012)
Wazwaz, A.M.: The tanh–coth method for solitons and kink solutions for nonlinear parabolic equations. Appl. Math. Comput. 188, 1467–1475 (2007)
Ebadi, G., Biswas, A.: Application of the \(G^{\prime }/G\)-expansion method for nonlinear diffusion equations with nonlinear source. J. Frankl. Inst. 347, 1391–1398 (2010)
Taghizadeh, N., Mirzazadeh, M., Tascan, F.: The first integral method applied to the Eckhaus equation. Appl. Math. Lett. 25(5), 798–802 (2012)
Taghizadeh, N., Mirzazadeh, M., Paghaleh, A.S., Vahidi, J.: Exact solutions of nonlinear evolution equations by using the modified simple equation method. Ain Shams Eng. J. 3, 321–325 (2012)
Jawad, A.J.M., Petkovic, M.D., Biswas, A.: Modified simple equation method for nonlinear evolution equations. Appl. Math. Comput. 217, 869–877 (2010)
Zayed, E.M.E.: A note on the modified simple equation method applied to Sharma–Tasso–Olver equation. Appl. Math. Comput. 218, 3962–3964 (2011)
Jumarie, G.: Modified Riemann–Liouville derivative and fractional Taylor series of nondifferentiable functions further results. Comput. Math. Appl. 51, 1367–1376 (2006)
Jumarie, G.: Table of some basic fractional calculus formulae derived from a modified Riemann–Liouville derivative for nondifferentiable functions. Appl. Math. Lett. 22, 378–385 (2009)
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Mirzazadeh, M. Analytical study of solitons to nonlinear time fractional parabolic equations. Nonlinear Dyn 85, 2569–2576 (2016). https://doi.org/10.1007/s11071-016-2845-7
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DOI: https://doi.org/10.1007/s11071-016-2845-7