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Year 2016, Volume: 4 Issue: 4, 51 - 66, 31.12.2016

Samil Akcagil Tugba Aydemir Omer Faruk Gozukizil

Abstract

References

  • 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).

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

Year 2016, Volume: 4 Issue: 4, 51 - 66, 31.12.2016

Samil Akcagil Tugba Aydemir Omer Faruk Gozukizil

Abstract

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.

Keywords

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

References

  • 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).
There are 34 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Samil Akcagil This is me

Tugba Aydemir

Omer Faruk Gozukizil

Publication Date December 31, 2016
Published in Issue Year 2016 Volume: 4 Issue: 4

Cite

APA Akcagil, S., Aydemir, T., & Gozukizil, O. F. (2016). Exact travelling wave solutions of nonlinear pseudoparabolic equations by using the G′/G Expansion Method. New Trends in Mathematical Sciences, 4(4), 51-66.
AMA Akcagil S, Aydemir T, Gozukizil OF. Exact travelling wave solutions of nonlinear pseudoparabolic equations by using the G′/G Expansion Method. New Trends in Mathematical Sciences. December 2016;4(4):51-66.
Chicago Akcagil, Samil, Tugba Aydemir, and Omer Faruk Gozukizil. “Exact Travelling Wave Solutions of Nonlinear Pseudoparabolic Equations by Using the G′/G Expansion Method”. New Trends in Mathematical Sciences 4, no. 4 (December 2016): 51-66.
EndNote Akcagil S, Aydemir T, Gozukizil OF (December 1, 2016) Exact travelling wave solutions of nonlinear pseudoparabolic equations by using the G′/G Expansion Method. New Trends in Mathematical Sciences 4 4 51–66.
IEEE S. Akcagil, T. Aydemir, and O. F. Gozukizil, “Exact travelling wave solutions of nonlinear pseudoparabolic equations by using the G′/G Expansion Method”, New Trends in Mathematical Sciences, vol. 4, no. 4, pp. 51–66, 2016.
ISNAD Akcagil, Samil et al. “Exact Travelling Wave Solutions of Nonlinear Pseudoparabolic Equations by Using the G′/G Expansion Method”. New Trends in Mathematical Sciences 4/4 (December 2016), 51-66.
JAMA Akcagil S, Aydemir T, Gozukizil OF. Exact travelling wave solutions of nonlinear pseudoparabolic equations by using the G′/G Expansion Method. New Trends in Mathematical Sciences. 2016;4:51–66.
MLA Akcagil, Samil et al. “Exact Travelling Wave Solutions of Nonlinear Pseudoparabolic Equations by Using the G′/G Expansion Method”. New Trends in Mathematical Sciences, vol. 4, no. 4, 2016, pp. 51-66.
Vancouver Akcagil S, Aydemir T, Gozukizil OF. Exact travelling wave solutions of nonlinear pseudoparabolic equations by using the G′/G Expansion Method. New Trends in Mathematical Sciences. 2016;4(4):51-66.