International Journal of Partial Differential Equations and Applications
ISSN (Print): 2376-9548 ISSN (Online): 2376-9556 Website: http://www.sciepub.com/journal/ijpdea Editor-in-chief: Mahammad Nurmammadov
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International Journal of Partial Differential Equations and Applications. 2013, 1(1), 6-12
DOI: 10.12691/ijpdea-1-1-2
Open AccessArticle

Enhanced (G’/G)-Expansion Method to Find the Exact Solutions of Nonlinear Evolution Equations in Mathematical Physics

Md. Ekramul Islam1, Kamruzzaman Khan1, , M. Ali Akbar2 and Rafiqul Islam1

1Department of Mathematics, Pabna University of Science and Technology, Pabna, Bangladesh

2Department of Applied Mathematics, University of Rajshahi, Rajshahi, Bangladesh

Pub. Date: November 11, 2013

Cite this paper:
Md. Ekramul Islam, Kamruzzaman Khan, M. Ali Akbar and Rafiqul Islam. Enhanced (G’/G)-Expansion Method to Find the Exact Solutions of Nonlinear Evolution Equations in Mathematical Physics. International Journal of Partial Differential Equations and Applications. 2013; 1(1):6-12. doi: 10.12691/ijpdea-1-1-2

Abstract

In the present paper, we construct the traveling wave solutions involving parameters for the (2+1)-dimensional cubic Klein-Gordon equation (cKG) via Enhanced (G’/G)-expansion method. The efficiency of this method for finding these exact solutions has been demonstrated. As a result, a set of solitary wave solutions are derived, which are expressed by the combinations of rational, hyperbolic and trigonometric functions involving several parameters. It is shown that the method is effective and can be used for many other nonlinear evolution equations (NLEEs) in mathematical physics.

Keywords:
enhanced (G’/G)-expansion method cKG equation solitary wave traveling wave

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References:

[1]  R. Hirota, Exact envelope soliton solutions of a nonlinear wave equation. J. Math. Phy. 14 (1973) 805-810.
 
[2]  R. Hirota, J. Satsuma, Soliton solutions of a coupled KDV equation. Phy. Lett. A. 85 (1981) 404-408.
 
[3]  M. Malfliet, Solitary wave solutions of nonlinear wave equations. Am. J. Phys. 60, (1992)650-654.
 
[4]  H.A.Nassar, M.A. Abdel-Razek, A.K. Seddeek, Expanding the tanh-function method for solving nonlinear equations, Appl. Math. 2(2011) 1096-1104.
 
[5]  E.G. Fan, Extended tanh-method and its applications to nonlinear equations. Phy. Lett. A. 277 (2000) 212-218.
 
[6]  M.A. Abdou, The extended tanh-method and its applications for solving nonlinear physical models. App. Math. Comput. 190(2007) 988-996.
 
[7]  J.H. He, X.H. Wu, Exp-function method for nonlinear wave equations, Chaos, Solitons and Fract. 30 (2006) 700-708.
 
[8]  M.A. Akbar, N.H.M. Ali, Exp-function method for Duffing Equation and new solutions of (2+1) dimensional dispersive long wave equations. Prog. Appl. Math. 1(2) (2011) 30-42.
 
[9]  H. Naher, A.F. Abdullah, M.A. Akbar, The Exp-function method for new exact solutions of the nonlinear partial differential equations, Int. J. Phys. Sci., 6(29): (2011)6706-6716.
 
[10]  H. Naher, A.F. Abdullah, M.A. Akbar, New traveling wave solutions of the higher dimensional nonlinear partial differential equation by the Exp-function method, J. Appl. Math., Article ID 575387, 14 pages.
 
[11]  A. Bekir, A. Boz, Exact solutions for nonlinear evolution equations using Exp-function method. Phy. Lett. A. 372(2008) 1619-1625.
 
[12]  M. A. Abdou, A.A.Soliman and S. T. Basyony, New application of exp-function method for improved Boussinesq equation. Phys. Lett.A, 369(2007), 469-475.
 
[13]  S. A. El-Wakil, M. A. Madkour and M. A. Abdou, Application of exp-function method for nonlinear evolution equations with variable co-efficient, Phys. Lett. A, 369(2007), 62-69.
 
[14]  S. T. Mohyud-Din, M. A. Noor and A. Waheed, Exp-function method for generalized travelling solutions of Calogero-Degasperis-Fokas equation, Zeitschrift für Naturforschung A- A Journal of Physical Sciences, 65a (2010), 78-84.
 
[15]  G. Adomian, Solving frontier problems of physics: The decomposition method. Boston (1994), M A: Kluwer Academic.
 
[16]  Y.B. Zhou, M.L. Wang, Y.M. Wang, Periodic wave solutions to coupled KdV equations with variable coefficients, Phys. Lett. A. 308(2003) 31-36.
 
[17]  Sirendaoreji, New exact travelling wave solutions for the Kawahara and modified Kawahara equations. Chaos Solitons Fract. 19(2004) 147-150.
 
[18]  A.T. Ali, New generalized Jacobi elliptic function rational expansion method. J. Comput. Appl. Math. 235(2011) 4117-4127.
 
[19]  Y. He, S. Li, Y. Long, Exact solutions of the Klein-Gordon equation by modified Exp-function method. Int. Math. Forum. 7(4) (2012) 175-182.
 
[20]  M.A. Akbar, N.H.M. Ali, E.M.E. Zayed, Abundant exact traveling wave solutions of the generalized Bretherton equation via (G’/G)-expansion method. Commun. Theor. Phys. 57(2012a) 173-178.
 
[21]  M.A. Akbar, N.H.M. Ali, E.M.E. Zayed, A generalized and improved (G’/G)-expansion method for nonlinear evolution equations, Math. Prob. Engr., Vol. 2012, 22 pages.
 
[22]  M.A. Akbar, N.H.M. Ali, S.T. Mohyud-Din, The alternative (G’/G)-expansion method with generalized Riccati equation: Application to fifth order (1+1)-dimensional Caudrey-Dodd-Gibbon equation. Int. J. Phys. Sci. 7(5) (2012c) 743-752.
 
[23]  M.A. Akbar, N.H.M. Ali, S.T. Mohyud-Din, Some new exact traveling wave solutions to the (3+1)-dimensional Kadomtsev-Petviashvili equation. World Appl. Sci. J. 16(11) (2012d) 1551-1558.
 
[24]  E. M. E. Zayed and A.J. Shorog, Applications of an Extended (G’/G)-Expansion Method to Find Exact Solutions of Nonlinear PDEs in Mathematical Physics, Hindawi Publishing Corporation,Mathematical Problems in Engineering, Article ID 768573, 19 pages.
 
[25]  E.M.E. Zayed, Traveling wave solutions for higher dimensional nonlinear evolution equations using the (G’/G)-expansion method. J. Appl. Math. Informatics, 28 (2010) 383-395.
 
[26]  E.M.E. Zayed, K.A. Gepreel, The (G’/G)-expansion method for finding the traveling wave solutions of nonlinear partial differential equations in mathematical physics. J. Math. Phys. 50(2009) 013502-013514.
 
[27]  M. Wang, X. Li, J. Zhang, The (G’/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Phys. Lett. A. 372(2008) 417-423.
 
[28]  M.A. Akbar, N.H.M. Ali, The alternative (G’/G)-expansion method and its applications to nonlinear partial differential equations. Int. J. Phys. Sci. 6(35) (2011) 7910-7920.
 
[29]  A.R. Shehata, The traveling wave solutions of the perturbed nonlinear Schrodinger equation and the cubic-quintic Ginzburg Landau equation using the modified (G’/G) -expansion method. Appl. Math. Comput. 217(2010), 1-10.
 
[30]  M.S. Liang, et al., A method to construct Weierstrass elliptic function solution for nonlinear equations, Int. J. Modern Phy. B. 25(4) (2011) 1931-1939.
 
[31]  S.T. Mohiud-Din, Homotopy perturbation method for solving fourth-order boundary value problems, Math. Prob. Engr. Vol. 2007, 1-15, Article ID 98602.
 
[32]  S. T. Mohyud-Din and M. A. Noor, Homotopy perturbation method for solving partial differential equations, Zeitschrift für Naturforschung A- A Journal of Physical Sciences, 64a (2009), 157-170.
 
[33]  S. T. Mohyud-Din, A. Yildirim, S. Sariaydin, Numerical soliton solutions of the improved Boussinesq equation, International Journal of Numerical Methods for Heat and Fluid Flow 21 (7) (2011):822-827.
 
[34]  S. T. Mohyud-Din, A. Yildirim, G. Demirli, Analytical solution of wave system in Rn with coupling controllers, International Journal of Numerical Methods for Heat and Fluid Flow, Emerald 21 (2) (2011), 198-205.
 
[35]  S. T. Mohyud-Din, A. Yildirim, S. Sariaydin, Numerical soliton solution of the Kaup-Kupershmidt equation, International Journal of Numerical Methods for Heat and Fluid Flow, Emerald 21 (3) (2011), 272-281.
 
[36]  M.Wang, Solitary wave solutions for variant Boussinesq equations. Phy. Lett. A. 199(1995) 169-172.
 
[37]  E.M.E. Zayed, H.A. Zedan, K.A. Gepreel, On the solitary wave solutions for nonlinear Hirota-Sasuma coupled KDV equations, Chaos, Solitons and Fractals, 22(2004) 285-303.
 
[38]  A.J. M. Jawad, M.D. Petkovic, A. Biswas, Modified simple equation method for nonlinear evolution equations. Appl. Math. Comput. 217(2010), 869-877.
 
[39]  E.M.E. Zayed, A note on the modified simple equation method applied to Sharma-Tasso-Olver equation. Appl. Math. Comput. 218 (2011) 3962-3964.
 
[40]  E.M.E. Zayed, S.A.H. Ibrahim, Exact solutions of nonlinear evolution equations in Mathematical physics using the modified simple equation method. Chinese Phys. Lett. 29(6) (2012) 060201.
 
[41]  K. Khan, M.A. Akbar and N.H.M. Ali. The Modified Simple Equation Method for Exact and Solitary Wave Solutions of Nonlinear Evolution Equation: The GZK-BBM Equation and Right-Handed Noncommutative Burgers Equations, ISRN Mathematical Physics, Hindawi Publishing Corporation, Volume 2013.
 
[42]  K. Khan and M. Ali Akbar, Exact and solitary wave solutions for the Tzitzeica-Dodd-Bullough and the modified KdV-Zakharov-Kuznetsov equations using the modified simple equation method, Ain Shams Engineering Journal.
 
[43]  M.T. Ahmed, K. Khan and M.A. Akbar, Study of Nonlinear Evolution Equations to Construct Traveling Wave Solutions via Modified Simple Equation Method, Physical Review & Research International, 3(4): 490-503, 2013.
 
[44]  S. T. Mohyud-Din, M. A. Noor and K. I. Noor, Travelling wave solutions of seventh-order generalized KdV equations using He's polynomials, International Journal of Nonlinear Sciences and Numerical Simulation, 10 (2) (2009), 223-229.
 
[45]  J. H. He, An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering, Int. J. Mod. Phys. B 22 (21) (2008), 3487-3578.
 
[46]  S. T. Mohyud-Din, M. A. Noor and K. I. Noor, Some relatively new techniques for nonlinear problems, Mathematical Problems in Engineering, Hindawi, 25 pages.
 
[47]  S. T. Mohyud-Din, M. A. Noor, K. I. Noor and M. M. Hosseini, Solution of singular equations by He's variational iteration method, International Journal of Nonlinear Sciences and Numerical Simulation, 11 (2) (2010),81-86.
 
[48]  W. X. Ma and Y. You, Rational solutions of the Toda lattice equation in Casoratian form, Chaos, Solitons & Fractals, 22 (2004), 395-406.
 
[49]  W. X. Ma, H. Y. Wu and J. S. He, Partial differential equations possessing Frobenius integrable decompositions, Phys. Lett. A, 364 (2007), 29-32.
 
[50]  K. A. Gepreel, Exact Complexiton Soliton Solutions for Nonlinear Partial Differential Equations, International Mathematical Forum, Vol. 6( 2011) no. 26, 1261-1272.
 
[51]  K. A. Gepreel, A. R. Shehata, Exact complexiton soliton solutions for nonlinear partial differential equations in mathematical physics, Scientific Research and Essays Vol. 7(2), pp. 149-157.
 
[52]  K. Khan and M. Ali Akbar. Traveling Wave Solutions of Nonlinear Evolution Equations via the Enhanced (G'/G)-expansion Method. Journal of the Egyptian Mathematical Society, 2013,JOEMS-D-13-00172. (Accepted for publication).