Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 32726
Traveling Wave Solutions for the (3+1)-Dimensional Breaking Soliton Equation by (G'/G)- Expansion Method and Modified F-Expansion Method

Authors: Mohammad Taghi Darvishi, Maliheh Najafi, Mohammad Najafi


In this paper, using (G/G )-expansion method and modified F-expansion method, we give some explicit formulas of exact traveling wave solutions for the (3+1)-dimensional breaking soliton equation. A modified F-expansion method is proposed by taking full advantages of F-expansion method and Riccati equation in seeking exact solutions of the equation.

Keywords: Exact solution, The (3+1)-dimensional breaking soliton equation, ( G G )-expansion method, Riccati equation, Modified Fexpansion method.

Digital Object Identifier (DOI):

Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 2658


[1] L. Debtnath, Nonlinear partial differential equations for scientists and engineers, Birkhauser, Boston, 1997.
[2] A.M. Wazwaz, Partial differential equations methods and applications, Rotterdam, Balkema, 2002.
[3] J.H. He, New interpretation of homotopy perturbation method, Int. J. Mod. Phys. B 20(18) (2006) 2561-2568.
[4] J.H. He, Application of homotopy perturbation method to nonlinear wave equations, Chaos, Solitons and Fractals 26(3) (2005) 695-700.
[5] J.H. He, Homotopy perturbation method for bifurcation of nonlinear problems, Int. J. Nonlinear Sci. Numer. Simul. 6(2) (2005) 207-208.
[6] M.T. Darvishi, F. Khani, Application of He-s homotopy perturbation method to stiff systems of ordinary differential equations, Zeitschrift fur Naturforschung A, 63a (1-2) (2008) 19-23.
[7] M.T. Darvishi, F. Khani, S. Hamedi-Nezhad, S.-W. Ryu, New modification of the HPM for numerical solutions of the sine-Gordon and coupled sine- Gordon equations, Int. J. Comput. Math. 87(4) (2010) 908-919.
[8] J.H. He, Variational iteration method-a kind of non-linear analytical technique: some examples, Int. J. Non-linear Mech., 34(4) (1999) 699- 708.
[9] M.T. Darvishi, F. Khani, A.A. Soliman, The numerical simulation for stiff systems of ordinary differential equations, Comput. Math. Appl., 54(7-8) (2007) 1055-1063.
[10] M.T. Darvishi, F. Khani, Numerical and explicit solutions of the fifthorder Korteweg-de Vries equations, Chaos, Solitons and Fractals, 39 (2009) 2484-2490.
[11] J.H. He, Bookkeeping parameter in perturbation methods, Int. J. Nonlin. Sci. Numer. Simul., 2 (2001) 257-264.
[12] M.T. Darvishi, A. Karami, B.-C. Shin, Application of He-s parameterexpansion method for oscillators with smooth odd nonlinearities, Phys. Lett. A, 372(33) (2008) 5381-5384.
[13] B.-C. Shin, M.T. Darvishi, A. Karami, Application of He-s parameterexpansion method to a nonlinear self-excited oscillator system, Int. J. Nonlin. Sci. Num. Simul., 10(1) (2009) 137-143.
[14] M.T. Darvishi, S. Kheybari, A. Yildirim, Application of He-s parameterpxpansion method to a system of two van der Pol oscillators coupled via a bath, Nonlin. Science Lett. A., 1(4) (2010) 399-405.
[15] M.T. Darvishi, Preconditioning and domain decomposition schemes to solve PDEs, Int-l J. of Pure and Applied Math., 1(4) (2004) 419-439.
[16] M.T. Darvishi, S. Kheybari and F. Khani, A numerical solution of the Korteweg-de Vries equation by pseudospectral method using Darvishi-s preconditionings, Appl. Math. Comput., 182(1) (2006) 98-105.
[17] M.T. Darvishi, M. Javidi, A numerical solution of Burgers- equation by pseudospectral method and Darvishi-s preconditioning, Appl. Math. Comput., 173(1) (2006) 421-429.
[18] M.T. Darvishi, F. Khani and S. Kheybari, Spectral collocation solution of a generalized Hirota-Satsuma KdV equation, Int. J. Comput. Math., 84(4) (2007) 541-551.
[19] M.T. Darvishi, F. Khani, S. Kheybari, Spectral collocation method and Darvishi-s preconditionings to solve the generalized Burgers-Huxley equation, Commun., Nonlinear Sci. Numer. Simul., 13(10) (2008) 2091- 2103.
[20] S.J. Liao, An explicit, totally analytic approximate solution for Blasius viscous flow problems, Int. J. Non-Linear Mech., 34 (1999) 759-778.
[21] S.J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman & Hall/CRC Press, Boca Raton, 2003.
[22] S.J. Liao, On the homotopy analysis method for nonlinear problems, Appl. Math. Comput., 147 (2004) 499-513.
[23] S.J. Liao, A new branch of solutions of boundary-layer flows over an impermeable stretched plate, Int. J. Heat Mass Transfer, 48 (2005) 2529- 2539.
[24] S.J. Liao, A general approach to get series solution of non-similarity boundary-layer flows, Commun. Nonlinear Sci. Numer. Simul., 14(5) (2009) 2144-2159.
[25] M.T. Darvishi, F. Khani, A series solution of the foam drainage equation, Comput. Math. Appl., 58 (2009) 360-368.
[26] J.H. He, X.H. Wu, Construction of solitary solution and compacton-like solution by variational iteration method, Chaos, Solitons and Fractals, 29 (2006) 108-113.
[27] F. Khani, S. Hamedi-Nezhad, M.T. Darvishi, S.-W. Ryu, New solitary wave and periodic solutions of the foam drainage equation using the Expfunction method, Nonlin. Anal.: Real World Appl., 10 (2009) 1904-1911.
[28] B.-C. Shin, M.T. Darvishi, A. Barati, Some exact and new solutions of the Nizhnik-Novikov-Vesselov equation using the Exp-function method, Comput. Math. Appl. 58(11/12) (2009) 2147-2151.
[29] X.H. Wu, J.H. He, Exp-function method and its application to nonlinear equations, Chaos, Solitons and Fractals, 38(3) (2008) 903-910.
[30] J.H. He, X.H. Wu, Exp-function method for nonlinear wave equations, Chaos Solitons Fractals, 30(3) (2006) 700-708.
[31] J.H. He, M.A. Abdou, New periodic solutions for nonlinear evolution equations using Exp-function method, Chaos Solitons Fractals, 34 (2007) 1421-1429.
[32] Ma S-H, Peng J and Zhang C, New exact solutions of the (2+1)- dimensional breaking soliton system via an extended mapping method, Chaos Solitons Fractals, 46 (2009) 210-214.
[33] A.M. Wazwaz, Integrable (2+1)-dimensional and (3+1)-dimensional breaking soliton equations, Phys. Scr., 81 (2010) 1-5.
[34] M.-L. Wang., X.-Z. Li, J.-L. Zhang, The (G G )-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics,, Physics Letters A, 372 (2008) 417-423.
[35] M.T. Darvishi, M. Najafi, M. Najafi, New exact solutions for the (3+1)-dimensional breaking soliton equation, International Journal of Engineering and Mathematical Sciences, 6(2) (2010) 137-140.
[36] J. Wang, Construction of new exact traveling wave solutions to (2+1)- dimensional mVN equation, International Journal of Nonlinear Science, 9(3) (2010) 325-329.