Exact travelling wave solutions of nonlinear pseudoparabolic equations by using the G′/G Expansion Method

Yıl 2016, Cilt: 4 Sayı: 4, 51 - 66, 31.12.2016

Samil Akcagil Tugba Aydemir , Omer Faruk Gozukizil

Öz

In this paper, the ( G′/G)expansion method with the aid of computer algebraic system Maple, is proposed for seeking the travelling wave solutions for the a class of nonlinear  pseudoparabolic equations. The method is straightforward and concise, and it be also applied to other nonlinear pseudoparabolic equations. We studied mostly important four nonlinear pseudoparabolic physical models : the Benjamin-Bona-Mahony-Peregrine-Burger(BBMPB) equation, the Oskolkov-Benjamin-Bona-Mahony-Burgers(OBBMB) equation, the one-dimensional Oskolkov equation and the generalized hyperbolic-elastic-rod wave equation.

Anahtar Kelimeler

The ( G′/G) expansion method , Travelling wave solution; Nonlinear pseudoparabolic equation Benjamin-Bona-Mahony-Peregrine-Burger(BBMPB) equation , Oskolkov-Benjamin-Bona-Mahony-Burgers (OBBMB) equation

Kaynakça

• M.J.Ablowitz, H.Segur, Solitons and the inverse scattering transform, SIAM, Philadelphia, Pa, USA, (1981).
• M.R.Miura, Backlund transformation, Springer, Berlin, Germany, (1978).
• C.Rogers, W.F. Shadwick, Backlund transformations, Academic Press, New York, NY, USA, (1982).
• V.B.Matveev, M.A.Salle, Darboux transform and Solitons, Springer, Berlin, Germany, (1991).
• R.Hirota, The direct method in soliton theory, Cambridge University Press, Cambridge, (2004).
• S¸ .Akcagıl, O.F.Gozukızıl, The tanh-coth method for some nonlinear pseudoparabolic equations with exact solutions, Advances in Difference Equations 143, (2013).
• W.Malfiet, The tanh method: a tool for solving certain classes of nonlinear PDEs, Mathematical Methods in the Applied Sciences , 28(17), 2013-2935, (2005).
• C.T.Yan, A simple transformation for nonlinear waves, Physics Letter A 224, 77-84 (1996) .
• J.H.He, X.H.Wu, Exp-function method for nonlinear wave equations, Chaos,Solitons and Fractals, 30(3), 700-708, (2006).
• Z.Y.Yan, H.Q.Zhang, New explicit solitary wave solutions and periodic wave solutions for Whitham-Broer-Kaup equation in shallow water, Physics Letter A 285 355-362 (2001).
• M.L.Wang, Exact solutions for a compound KdV-Burgers equation, Physics Letter A 213, 279-287 (1996).
• F.Tascan, A.Bekir, M.Koparan, Travelling wave solutions of nonlinear evolution equations by using the first integral method, Commun. Nonlinear Sci. Numer. Simul. 14, 1810-1815 (2009).
• Z.S.Feng, The first integral method to the two dimensional Burgers-Korteweg-de-Vries equation, Physics Letter A 308, 173-178 (2003).
• M.L.Wang, X.Li, J.Zhang, The (frac(G’,G))-expansion method and travelling wave solutions of nonlinear evolutions equations in mathematical physics, Physics Letter A 372, 417-423 (2008) .
• E.M.E.Zayed, Travelling wave solutions for higher dimensional nonlinear evolution equations using the (G’/G)-expansion method, Journal of Applied Mathematics and Informatics, 28, 383-395 (2010).
• A.J.M.Jawad,M.D.Petkovic, A.Biswas,Modified simple equation method for nonlinear evolution equations, Applied Mathematics and Computation, 217 (2), 869-877 (2010).
• E.M.E.Zayed, A note on the modified simple equation method applied to Sharma-Tasso-Olver equation, Applied Mathematics and Computation, 218 (7), 3962-3964 (2011).
• E.M.E.Zayed, S.A.H.Ibrahim, Exact solutions of nonlinear evolution equations in mathematical physics using the modified simple equation method, Chinese Physics Letters, 29(6), (2012).
• M.Wang, X.Li, J.Zhang, The (G′/G) expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Physics Letter A 372, 417-423 (2008).
• A.Bekir, Application of the (G′/G) expansion method for nonlinear evolution equations, Physics Letter A 372, 3400-3406 (2008).
• J.Zhang, X.Wei, A generalized (G′/G) expansion method and its applications, Physics Letter A 372, 3653-3658 (2008).
• I.Aslan, T.¨Ozis¸, Analytical study on two nonlinear evolution equations by using (G′/G) expansion method, Appl. Math. Comp. 209, 425-429 (2009).
• E.M.E.Zayed, The (G′/G) expansion method and its application to some nonlinear evolution equations, J. Appl. Math. Comp. 30, 89-103 (2009).
• E.M.E.Zayed, K.A.Gepreel, Some applications of the (G′/G) expansion method to nonlinear partial differential equations, Appl. Math. Comp. 212 , 1-13 (2009).
• A.Borhanifar, A.M.Zamiri, Application of the (G′/G) expansion method for the Zhiber-Shabat equation and other related equations, Math. Comp. Model. 549 (9-10), 2109-2116, (2011).
• Serbay Duran, Yavuz U˘gurlu and Ibrahim Enam Inan, (G′/G) Expansion Method for (3+1)-dimensional Burgers and Burgers Like Equations, World Applied Sciences Journal 20 (12),1607-1611 (2012) .
• Dumitru Baleanu, Yavuz U˘gurlu, Mustafa Inc and B¨ulent Kilic, Improved (G’/G)-expansion method for the time-fractional biological population model and Cahn-Hilliard equation, Journal of Computational and Nonlinear Dynamics, 10(5) / 051016-1 (2015).
• Mustafa Inc, Bulent Kılıc¸, Yavuz U˘gurlu, Soliton Solutions for Bogoyavlensky-Konoplechenko and Jaulent–Miodek Equations Via Extended (G’/G)-Expansion Method, Romanian Journal of Physics, 60 (9-10),1395-1408 (2015).
• Ibrahim E. Inan, Yavuz Ugurlu and Mustafa Inc, New Applications of the (G’/G,1/G)-Expansion Method Acta Physica Polonica A, 128 (3) 245-251 (2015).
• A.Quarteroni, Fourier spectral methods for pseudo-parabolic equations, SIAM J.Numer.Anal.24 (2), 323-335 (1987).
• Korpusov, MO,Sveshnikov, AG: Blow-up solutions of strongly nonlinear equations of pseudoparabolic type, J.Math.Sci. 148 (1), 1-142 (2008).
• SA.Dubey, Numerical solution for nonlocal Sobolev-type differential equations, Electron J. Differ.Eq.Conf. 19, 75-83 (2010).
• El.Kalkina, Pl. Naumkin, IA.Shishmarev, The Cauchy Problem for an equation of Sobolev-type with power non-linearity, Izv.Math. 69 (1), 59-111 (2005).
• G.Karch, Asymtotic behaviour of solutions to some pseudoparabolic equations, Math.Methods Appl.Sci.20,271-289 (1997).