Closed form solutions to the coupled space-time fractional evolution equations in mathematical physics through analytical method


M. Nurul Islam,M. Ali Akbar,



Coupled mKdV equations, coupled WBK equation,nonlinear evolution equations,ractional differential equations,


In this article, we consider the space-time fractional coupled modified Korteweg-de-Vries (mKdV) equations and the space-time fractional coupled Whitham-Broer-Kaup (WBK) equations which are important mathematical model to depict the propagation of wave in shallow water under gravity, combined formal solitary wave, internal solitary waves in a density and current stratified shear flow with a free surface, ion acoustic waves in plasma, turbulent motion, quantum mechanics and also in financial mathematics. We examine new, useful and further general exact wave solutions to the above mentioned space-time fractional equations by means of the generalized -expansion method by using of fractional complex transformation and discuss the examined results with other method. This method is more general, powerful, convenient and direct and can be used to establish new solutions for other kind nonlinear fractional differential equations arising in mathematical physics. Keywords: Coupled mKdV equations; coupled WBK equation; nonlinear evolution equations; fractional differential equations.


I.Alam, M.N. and Akbar, M.A. “The new approach of the generalized -expansion method for nonlinear evolution equations”. Ain Shams Eng. J., Vol. 5, pp 595-603 (2014).

II.Alam, M.N. and Akbar, M.A. “Application of the new approach of generalized )/(GG-expansion method to find exact solutions of nonlinearPDEs in mathematical physics”. BIBECHANA, Vol. 10, pp 58-70 (2014).

III.Ahmad, J., Mushtaq, M. and Sajjad, N. “Exact solution of Whitham-Broer-Kaup shallow water equations”. J. Sci. Arts, Vol. 1, No. 30, pp 5-12 (2015).

IV.Ali, A.H.A “The modified extended tanh-function method for solving coupled mKdV and coupled Hirota-Satsuma coupled KdV equations”. Phys. Lett. A, Vol. 363, No. (5-6), pp 420-425 (2007).

V.Atchi, L.M. and Appan, M.K., “A Review of the homotopy analysis method and its applications to differentialequations of fractional order”.Int. J. Pure Appl. Math., Vol. 113, No. (10), pp 369-384 (2017).

VI.Bekir, A. and Guner, O. “Exact solutions of nonlinear fractional differential equation by -expansion method”. Chin. Phys. B, Vol. 22, No. (11), pp 1-6 (2013).

VII.Bekir, A., Kaplan, M. “Exponential rational function method for solving nonlinear equations arising in various physical models”. Chin. J. Phys.54(3), 365–370(2016).

VIII.Bulut, H. Baskonus, H.M. and Pandir, Y. “The modified trial equation methodfor fractional wave equation and time fractional generalized Burgers equation”. Abst. Appl. Anal., 2013, Article ID 636802, (2013).

IX.Caputo, M. and Fabrizio, M.A. “A new definition of fractional derivatives without singular kernel”. Math. Comput. Model., Vol. 1, pp. 73-85 (2015).

X.Deng, W. “Finite element method for the space and time fractional Fokker-Planck equation”.Siam J. Numer. Anal., Vol. 47, No. (1), pp 204-226 (2009).

XI.Ege, S.M. and Misirli, E. “Solutions of space-time fractional foam drainage equation and the fractional Klein-Gordon equation by use of modified Kudryashov method”.Int. J. Res. Advent Tech., Vol. 2, No. (3), pp 384-388 (2014).

XII.El-Sayed, A.M.A., Behiry, S.H. and Raslan, W.E. “The Adomin’s decomposition method for solving an intermediate fractional advection-dispersion equation”. Comput. Math. Appl., Vol. 59, No. (5), pp 1759-1765 (2010).

XIII.El-Borai, M.M., El-Sayed, W.G. and Al-Masroub, R.M. “Exact solutions for time fractional coupled Whitham-Broer-Kaup equations via exp-function method”.Int. Res. J. Eng. Tech., Vol. 2, No. (6), pp 307-315 (2015).

XIV.Gomez-Aguilar, J.F., Yepez-Martnrez, H., Escober-Jimenez, R.F., Olivarer-Peregrino, V.H., Reyes, J.M. and Sosa, I.O. “Series solution for the time-fractional coupled mKdV equation using the homotopy analysis method”. Math. Prob. Eng., Vol. 2016, Article ID 7047126, 8 pages2016.

XV.He, J.H., Elagan, S.K. and Li, Z.B. “Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus”. Phys. Lett. A, Vol. 376, No. (4), pp 257-259 (2012).

XVI.He, J.H. “Asymptotic methods for solitary solutions and compacts”. Abst. Appl. Anal., Volume2012, Article ID916793, 130 pages(2012).

XVII.Helal, M.A. and Mehanna, M.S. “The tanh-function method and Adomin decomposition method for solving the foam drainage equation”. App. Math. Comput., vol. 190, No. (1), 599-609 (2007).

XVIII.Inc, M. “The approximate and exact solutions of the space and time-fractional Burgers equations with initial conditions by the variational iteration method”. J. Math. Anal. Appl., Vol. 345, No. (1), pp 476-484 (2008).

XIX.Jumarie, G. “Modified Riemann-Liouville derivative and fractional Taylor series of non-differentiable functions further results”. Comput. Math. Appl., Vol. 51, No. (9-10), pp. 1367-1376 (2006).

XX.Kadem, A. and Baleanu, D. “On fractional coupled Whitham-Broer-Kaup equations”. Rom. J. Phys., Vol. 56, No. (5-6), pp 629-635 (2011).

XXI.Kaplan, M. Bekir, A. Akbulut, A. and Aksoy, E. “The modified simple equation method for nonlinear fractional differential equations”. Rom. J. Phys., Vol. 60, No. (9-10), pp 1374-1383 (2015).

XXII.Lu, B. “Backlund transformation of fractional Riccati equation and its applications to nonlinear fractional partial differential equations”.Phys. Lett. A, Vol. 376, pp 2045-2048 ( 2012).

XXIII.Lu, B. “The first integral method for some time fractional differential equations”. J. Math. Appl., Vol. 395, pp. 684-693 (2012).

XXIV.Lu, D., Yue, C. and Arshad, M. “Traveling wave solutions of space-time fractional generalized fifth-order KdV equation”. Advances Math. Phys., Volume 2017, Article ID 6743276, 6 pages, (2017).

XXV.Moatimid, G.M., El-Shiekh, R.M., Ghani, A. and Al-Nowehy, A.A.H. “Modified Kudryashov method for finding exact solutions of the (2+1) dimensional modified Korteweg-de Varies equations and nonlinear Drinfeld-Sokolov system”. American J. Comput. Appl. Math., Vol. 1, No. 1 (2011).

XXVI.Odibat, Z. and Monani, S. “The variational iteration method: An efficient scheme for handling fractional partial differential equations in fluid mechanics”. Coumpt. Math. Appl., Vol. 58, No. (11-12), pp 2199-2208 (2009).

XXVII.Rabtah, A.A., Erturk, R.S. and Momani, S. “Solution of fractional oscillator by using differential transformation method”.Comput. Math. Appl., Vol. 59, pp 1356-1362 (2010).

XXVIII.Saad, M.,Ehgan, S.K.,Hamed, Y.S. and Sayed, M. “Using a complex transformation to get an exact solution for fractional generalized coupled mKdV and KDV equations”.Int. J. Basic Appl. Sci., Vol. 13, No. (01), pp 23-25(2014).

XXIX.Wang, G.W. and Xu, T.Z. “The modified fractional sub-equation method and its applications to nonlinear fractional partial differential equations”.Rom. J. Phys., Vol. 59, No. (7-8), pp 636-645 (2014).

XXX.Yan, Z. and Zhang, H. “New explicit solitary wave solutions and periodic wave solutions for Whitham-Broer-Kaup equations in shallow water”. Phys. Lett. A, Vol. 285, No. (5-6), pp 355-362 (2001).

XXXI.Yepez-Martinez, H., J.M. Reyes and I.O. Sosa, “Fractional sub-equation method and analytical solutions to the Hirota-Satsuma coupled KdV equation and mKdv equation”. British J. Math. Coumpt. Sci., Vol. 4, No. (4), pp 572-589 (2014).

XXXII.Younis, M. “The first integral method for time-space fractional differential equations”. J. Adv. Phys., Vol. 2, pp 220-223 (2013).

XXXIII.Younis, M. and Zafar, A. “Exact solutions to nonlinear differential equations of fractional order via -expansion method”. Appl. Math., Vol. 2014, No. (5), pp 1-6 (2014).

XXXIV.Zayed, E.M.E, Amer, Y.A. and Al-Nowehy, A.G. “The modified simple equation method and the multiple ex-function method for solving nonlinear fractional Sharma-Tasso-Olver equation”. Acta Mathematicae Applicatae Sinica, English Series, Vol. 32, No. (4), pp 793-812 (2016).

XXXV.Zayed, E.M.E., Amer, Y.A and Shohib, R.M.A. “The fractional complex transformation for nonlinear fractional partial differential equations in the mathematical physics”. J. Association Arab Uni. Basic Appl. Sci., Vol. 19, pp 59-69 (2016).

XXXVI.Zheng, B. “Exp-function method for solving fractional partial differential equations”.Sci. World J., DOI: 10.1155/2013/465723(2013).

XXXVII.Zheng, B. and Feng,Q. “The Jacobi elliptic equation method for solving fractional partial differential equations”.Abst. Appl. Anal.,2014, 9 pages, Article ID249071(2014).

Author(s): M. Nurul Islam and M. Ali Akbar View Download