Application of the modified simple equation method for solving two nonlinear time-fractional long water wave equations




Fractional partial differential equations, Modified simple equation method, Conformable fractional derivative, Approximate long water wave equation, Coupled Boussinesq-Burger equation


Recently, non-linear fractional partial differential equations are used to model many phenomena in applied sciences and engineering. In this study, the modified simple equation scheme is implemented to obtain some new traveling wave solutions of the non-linear conformable time-fractional approximate long water wave equation and the non-linear conformable coupled time-fractional Boussinesq-Burger equation, which are used in the expression of shallow-water waves. The time- fractional derivatives are described in terms of conformable fractional derivative sense. Consequently, new exact traveling wave solutions of both equations are achieved. The graphics and correctness of the wave solutions are obtained with the Mathematica package program.


R. Hilfer, Applications of fractional calculus in physics, (World Scientific, Singapore, 2000), pp.87–130.

A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and applications of fractional differential equations, (Elsevier Science Limited, 2006), p.204.

J. Sabatier, O.P. Agrawal and J.T. Machado, Advances in fractional calculus, (Springer, Dordrecht,2007).

M. Kaplan and A. Bekir, The modified simple equation method for solving some fractional-order nonlinear equations, Pramana, 87 (2016) 15.

Y. Gurefe, The generalized Kudryashov method for the nonlinear fractional partial differential equations with the beta-derivative, Rev. Mex. Fis., 66 (2020) 771. RevMexFis.66.771

O. Guner and A. Bekir, Soliton solutions for the time fractional Hamiltonian system by various approaches, Iran. J. Sci. Technol. Trans. A Sci., 42 (2018) 1587. s40995-018-0504-1

N. Islam, K. Khan and M.H. Islam, Travelling wave solution of Dodd-Bullough-Mikhailov equation: a comparative study between generalized Kudryashov and improved F-expansion methods, J. Phys. Commun., 3 (2019) 055004.

B. Ghanbari and J.F. G´omez-Aguilar, The generalized exponential rational function method for Radhakrishnan-Kundu-Lakshmanan equation with b-conformable time derivative, Rev. Mex. Fis., 65 (2019) 503.

H. Y´epez-Mart´ınez, J.F. G´omez-Aguilar and A. Atangana, First integral method for non-linear differential equations with conformable derivative, Math. Model. Nat. Phenom., 13 (2018) 14.

Z. Wen, The generalized bifurcation method for deriving exact solutions of nonlinear space-time fractional partial differential equations, Appl. Math. Comput., 366 (2020) 124735. https://doi. org/10.1016/j.amc.2019.124735

M. Odabasi, Traveling wave solutions of conformable time-fractional Zakharov–Kuznetsov and Zoomeron equations, Chinese J. Phys., 64 (2020) 194. 2019.11.003

A. Tozar, A. Kurt and O. Tasbozan, New wave solutions of time fractional integrable dispersive wave equation arising in ocean engineering models, Kuwait J. Sci., 47 (2020) 22.

S. Alfaqeih, G. Bakıcıerler and E. Mısırlı, Conformable double Sumudu transform with applications, J. Appl. Comput. Mech., (2020).

H. Y´epez-Mart´ınez and J.F. G´omez-Aguilar, Fractional sub-equation method for Hirota–Satsumacoupled KdV equation and coupled mKdV equation using the Atangana’s conformable derivative, Waves Random Complex Media, 29 (2019) 678.

M. Senol, New analytical solutions of fractional symmetric regularized-long-wave equation, Rev.Mex. Fis., 66 (2020) 297.

K.S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, (John-Wily and Sons, New York, 1993).

I. Podlubny, Fractional differential equations, (Academic Press, London, 1999).

G. Jumarie, Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions, Appl. Math. Lett., 22 (2009) 378. https://

M. Dalir and M. Bashour, Applications of fractional calculus, Appl. Math. Sci., 4 (2010) 1021.

R. Khalil, M. Al Horani, A. Yousef and M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014 ) 65.

T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015) 57.

M. Kaplan, A. Bekir, A. Akbulut and E. Aksoy, The modified simple equation method for nonlinear fractional differential equations, Rom. J. Phys., 60 (2015) 1374.

E.M. Zayed, Y.A. Amer and A.G. Al-Nowehy, The modified simple equation method and the multiple exp-function method for solving nonlinear fractional Sharma-Tasso-Olver equation, Acta

Math. Appl. Sin. Engl. Ser., 32 (2016) 793.

A.M. Ali, N.M.H. Ali and A.M. Wazwaz, Closed form traveling wave solutions of non-linear fractional evolution equations through the modified simple equation method, Therm. Sci., 22 (2018) 341.

M. Wang, J. Zhang and X. Li, Application of the (G0=G)-expansion to travelling wave solutions of the Broer–Kaup and the approximate long water wave equations, Appl. Math. Comput., 206 (2008) 321.

D. Shi, Y. Zhang, W. Liu and J. Liu, Some exact solutions and conservation laws of the coupled time-fractional Boussinesq-Burgers system, Symmetry, 11 (2019) 77.


S. Guo, L. Mei, Y. Li and Y. Sun, The improved fractional sub-equation method and its applications to the space–time fractional differential equations in fluid mechanics, Phys. Lett. A, 376 (2012)

L. Wang and X. Chen, Approximate analytical solutions of time fractional Whitham-Broer-Kaup equations by a residual power series method, Entropy, 17 (2015) 6519.


S.S. Ray, A novel method for travelling wave solutions of fractional Whitham-Broer-Kaup, fractional modified Boussinesq and fractional approximate long wave equations in shallow water, Math.

Methods Appl. Sci., 38 (2015) 1352.

W. Zheng-Yan and C. Ai-Hua, Explicit solutions of Boussinesq-Burgers equation, Chinese J. Phys., 16 (2007) 1233.

A.M. Wazwaz, A variety of soliton solutions for the Boussinesq-Burgers equation and the higher-order Boussinesq-Burgers equation, Filomat, 31 (2017) 831.







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