International Journal of Nonlinear Analysis and Applications
The extended tanh method for solving conformable space-time fractional KdV equations
Document Type : Research Paper
Authors
Department of Mathematics, Pamukkale University, Denizli, 20070, Turkey
Abstract
In this study, we obtain exact traveling wave solutions of the conformable space-time fractional Sawada-Kotera-Ito, Lax and Kaup-Kupershmidt equations by using the extended tanh method. The obtained traveling wave solutions are expressed by the hyperbolic, trigonometric, exponential and rational functions. Simulation of the obtained solutions are given at the end of the paper.
Keywords
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[32] R. Khalil, MA. Horani, A. Yousef and M. Sababheh, A new defnition of fractional derivative, J. Comput. Appl.
[5] BR. Sontakke and A. Shaikh, Approximate solutions of time fractional Kawahara and modified Kawahara equations by Fractional complex transform, CNA 2 (2016) 218–229.
[6] BR. Sontakke and A. Shaikh, Numerical Solutions of Time Fractional Fornberg-whitham And Modified Fornbergwhitham Equations Using New Iterative Method, Asian J. Math. Comput. Res. 13 (2) (2016) 66–76.
[7] M. Inc, The approximate and exact solutions of the space- and time-fractional Burgers equations with initial conditions by variational iteration method, J. Math. Anal. Appl. 345 (2008) 476–484.
[8] BP. Moghaddam and JAT. Machado, A stable three-level explicit spline finite difference scheme for a class of nonlinear time variable order fractional partial differential equations, Comput. Math. Appl. 73 (2017) 1262–1269.
[9] IE. Inan, S. Duran and Y. Ugurlu, tan(F(ξ2))-expansion method for traveling wave solutions of AKNS and Burgerslike equations Modified method of simplest equation and its applications to the Bogoyavlenskii equation, Optik 138 (2017) 15–20.
[10] NH. Sweilam, SM. Al-Mekhlafi and AO. Albalawi, A novel variable-order fractional nonlinear Klein Gordon model: A numerical approach. Numer. Methods Partial Differ. Equ. 35 (2019) 1617–1629.
[11] Y. Shekari, A. Tayebi and MH. Heydari, A meshfree approach for solving 2D variable-order fractional nonlinear diffusion-wave equation, Comput. Methods Appl. Mech. Engrg. 350 (2019) 154–168.
[12] DJ. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves, Philos. Mag. 39 (1895) 422–443.
[13] A. Jeffrey and T. Kakutani, Weak nonlinear dispersive waves: a discussion centered around the Korteweg-de Vries equation, SIAM Rev. 14 (1972) 582–643.
[14] AC. Scott, FYF. Chu and DW. McLaughlin, The soliton: a new concept in applied science, Proc. IEEE 61 (1973) 1443–1483.
[15] RM. Miura, The Korteweg-de Vries equation: a survey of results, SIAM Rev. 18 (1976) 412–459.
[16] R. Grimshaw, E. Pelinovsky and T. Talipova, Solitary wave transformation in a medium with sign-variable quadratic nonlinearity and cubic nonlinearity, Phys. D 132 (1999) 40–62.
[17] R. Grimshaw, D. Pelinovsky, E. Pelinovsky and T. Talipova, Wave group dynamics in weakly nonlinear long-wave models, Phys. D 159 (2001) 35–57.
[18] Y. Pomeau, A. Ramani and B. Grammaticos, Structural stability of the Korteweg-de Vries solitons under a singular perturbation, Physica D 31 (1988) 127–134.
[19] AM. Wazwaz. Partial differential equations and solitary waves theory, Springer Science and Business Media, 2010.
[20] EME. Zayed and KAE. Alurrfi, The modified Kudryashov method for solving some seventh order nonlinear PDEs in mathematical physics, WJMS. 11 (2015) 308–319.
[21] DD. Ganji, AG. Davodi and YA. Geraily, SawadaKoteraIto, Lax and KaupKupershmidt equations using Expfunction method, Math. Meth. Appl. Sci. 33 (2010) 167–176.
[22] J. Feng, New Traveling Wave Solutions to the Seventh-Order Sawada-Kotera Equation, J. Appl. Math. Infor. 28 (2010) 1431–1437.
[23] YJ. Shen, YT. Gao, X. Yu, G. Meng and Y. Qin, Bell-polynomial approach applied to the seventh-order SawadaKotera Ito equation, Appl. Math. Comput. 227 (2014) 502–508.
[24] AM. Wazwaz, The Hirota’s direct method and the tanh-coth method for multiple-soliton solutions of the SawadaKotera-Ito seventh-order equation, Appl. Math. Comput. 199 (2008) 133–138.
[25] H. Nemati, Z. Eskandari, F. Noori and M. Ghorbanzadeh, Application of the homotopy perturbation method to seven-order Sawada-Kotara equations, J. Eng. Sci. Technol. Rev. 4 (2011) 101–104.
[26] AH. Salas, CA. Gomez and BA. Frias, Computing exact solutions to a generalized Lax-Sawada-Kotera-Ito seventhorder KdV equation, Math. Probl. Eng. 2010 (2010) 7 pages.
[27] M. Saravi, A. Nikkar, M. Hermann, J. Vahidi and R. Ahari, A new modified approach for solving seven-order Sawada-Kotara equations, J. Math. Computer Sci. 6 (2013) 230–237.
[28] AA. Al-Shawba, A. Gepreel, FA. Abdullah and A. Azmi, Abundant closed form solutions of the conformable time fractional Sawada-Kotera-Ito equation using G0/G-expansion method, Results Phys. 9 (2018) 337–343.
[29] E. Yassar, Y. Yldrm and CM. Khalique, Lie symmetry analysis, conservation laws and exact solutions of the seventh-order time fractional SawadaKoteraIto equation, Results Phys. 6 (2016) 322–328.
[30] E. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A 277 (2000) 212–218.
[31] UM. Abdelsalam, Exact travelling solutions of two coupled (2 + 1)-dimensional equations, Egypt. Math. Soc. 25 (2017) 125–128.
[32] R. Khalil, MA. Horani, A. Yousef and M. Sababheh, A new defnition of fractional derivative, J. Comput. Appl.
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Volume 12, Issue 1
May 2021Pages 1181-1194
- Receive Date: 09 July 2018
- Revise Date: 18 October 2019
- Accept Date: 28 October 2019
- First Publish Date: 05 April 2021