Exp-Function Method for Finding Some Exact Solutions of Rosenau Kawahara and Rosenau Korteweg-de Vries Equations

In this paper, we apply the Exp-function method to
Rosenau-Kawahara and Rosenau-KdV equations. Rosenau-Kawahara
equation is the combination of the Rosenau and standard Kawahara
equations and Rosenau-KdV equation is the combination of the
Rosenau and standard KdV equations. These equations are nonlinear
partial differential equations (NPDE) which play an important role
in mathematical physics. Exp-function method is easy, succinct and
powerful to implement to nonlinear partial differential equations
arising in mathematical physics. We mainly try to present an
application of Exp-function method and offer solutions for common
errors wich occur during some of the recent works.

• Ehsan Mahdavi

• Exp-function method
• Rosenau Kawahara equation
• Rosenau Korteweg-de Vries equation
• nonlinear partial differential equation.

[1] A.M. Wazwaz. The tanh method: exact solutions of the sine-Gordon
and the sinh-Gordon equations. Applied Mathematics and Computation,
167:1196–1210, 2005.
[2] A.M. Wazwaz. The variable separated ODE and the tanh methods
for solving the combined and the double combined sinh-cosh-Gordon
equations. Applied Mathematics and Computation, 177:745–754, 2006.
[3] A.M. Wazwaz. Exact solutions for the generalized sine-Gordon and
the generalized sinh-Gordon equations. Chaos, Solitons and Fractals,
28:127–135, 2006.
[4] I. Hashim, M. S. M. Noorani, and M. R. S. Hadidi. Solving the
generalized Burgers-Huxley equation using the Adomian decomposition
method. Mathematical and Computer Modelling, 43:1404–1411, 2006.
[5] M. Tatari, M. Dehghan, and M. Razzaghi. Application of the Adomian
decomposition method for the Fokker-Planck equation. Mathematical
and Computer Modelling, 45:639–650, 2007.
[6] M.M. Rashidi, D.D. Ganji, and S. Dinarvand. Explicit analytical
solutions of the generalized burger and burger-fisher equations by
homotopy perturbation method. Numerical Methods for Partial
Differential Equations, 25:409–417, 2009.
[7] J. Biazar, F. Badpeimaa, and F. Azimi. Application of the homotopy
perturbation method to Zakharov-Kuznetsov equations. Computers and
Mathematics with Applications, 58:2391–2394, 2009.
[8] M.E. Berberler and A. Yildirim. He’s homotopy perturbation method
for solving the shock wave equation. Applicable Analysis, 88:997–1004,
2009.
[9] F. Shakeri and M. Dehghan. Numerical solution of the Klein-Gordon
equation via He’s variational iteration method. Nonlinear Dynamics,
51:89–97, 2008.
[10] A.A. Soliman and M.A. Abdou. Numerical solutions of nonlinear
evolution equations using variational iteration method. Journal of
Computational and Applied Mathematics, 207:111–120, 2007.
[11] E.Yusufoglu and A. Bekir. The variational iteration method for solitary
patterns solutions of gBBM equation. Physics Letters A, 367:461–464,
2007.
[12] K. Parand and A. Taghavi. Rational scaled generalized Laguerre
function collocation method for solving the Blasius equation. Journal
of Computational and Applied Mathematics, 233:980–989, 2009.
[13] K. Parand, M. Dehghan, and A. Pirkhedri. Sinc-collocation method
for solving the Blasius equation. Physics Letters, Section A: General,
Atomic and Solid State Physics, 373:4060–4065, 2009.
[14] K. Parand, M. Dehghan, A. R. Rezaei, and S. M. Ghaderi. An
approximation algorithm for the solution of the nonlinear Lane-Emden
type equations arising in astrophysics using Hermite functions
collocation method. Comput Phys Commun, 181:1096–1108, 2010.
[15] F. Tascan and A. Bekir. Analytic solutions of the (2 + 1)-dimensional
nonlinear evolution equations using the sine-cosine method. Applied
Mathematics and Computation, 215:3134–3139, 2009.
[16] A. M. Wazwaz. New solitary wave solutions to the modified Kawahara
equation. Physics Letters, Section A: General, Atomic and Solid State
Physics, 360:588–592, 2007.
[17] J. H. He and X. H. Wu. Exp-function method for nonlinear wave
equations. Chaos, Solitons and Fractals, 30:700–708, 2006.
[18] J.H. He. An elementary introduction to recently developed asymptotic
methods and nanomechanics in textile engineering. International journal
of Modern Physics B, 22:3487–3578, 2008.
[19] S. Zhang. Application of Exp-function method to a KdV equation with
variable coefficients. Physics Letters, Section A: General, Atomic and
Solid State Physics, 365:448–453, 2007.
[20] S. Zhang. Exp-function method for solving Maccari’s system. Physics
Letters, Section A: General, Atomic and Solid State Physics, 371:65–71,
2007.
[21] M. A. Abdou, A. A. Soliman, and S. T. El-Basyony. New application
of Exp-function method for improved Boussinesq equation. Physics
Letters, Section As : General, Atomic and Solid State Physics,
369:469–475, 2007.
[22] A. Ebaid. Exact solitary wave solutions for some nonlinear evolution
equations via Exp-function method. Physics Letters, Section As :
General, Atomic and Solid State Physics, 365:213–219, 2007.
[23] J. Biazar and Z. Ayati. Extension of the Exp-function method for systems
of two-dimensional Burger’s equations. Computers and Mathematics
with Applications, 58:2103–2106, 2009.
[24] A.E.H. Ebaid. Generalization of He’s Exp-Function method and new
exact solutions for burgers equation. Zeitschrift fur Naturforschung -
Section A Journal of Physical Sciences, 64:604–608, 2009.
[25] G. Domairry, A. G. Davodi, and A. G. Davodi. Solutions for the
Double Sine-Gordon equation by Exp-function, Tanh, and extended
Tanh methods. Numerical Methods for Partial Differential Equations,
26:384–398, 2010.
[26] J. H. He and M. A. Abdou. New periodic solutions for nonlinear
evolution equations using Exp-function method. Chaos, Solitons and
Fractals, 34:1421–1429, 2007
[27] F. Khani, S. Hamedi-Nezhad, and A. Molabahrami. A reliable treatment
for nonlinear schr¨odinger equations. Physics Letters A, 371:234–240,
2007.
[28] J. H. He and L. Zhang. Generalized solitary solution and compacton-like
solution of the Jaulent-Miodek equations using the Exp-function method.
Physics Letters, Section As : General, Atomic and Solid State Physics,
372:1044–1047, 2008.
[29] B.C. Shin, M.T. Darvishi, and A. Barati. Some exact and new solutions
of the Nizhnik-Novikov-Vesselov equation using the Exp-function
method. Computers and Mathematics with Applications, 58:2147–2151,
2009.
[30] A. Bekir and A. Boz. Exact solutions for a class of nonlinear partial
differential equations using exp-function method. International Journal
of Nonlinear Sciences and Numerical Simulation, 8:505–512, 2007.
[31] K. Parand and J. A. Rad. Exp-function method for some nonlinear
PDE’s and a nonlinear ODE’s. Journal of King Saud University-Science,
24:1–10, 2012.
[32] K. Parand and J. A. Rad. Some solitary wave solutions of generalized
Pochhammer-Chree equation via Exp-function method. International
Journal of Computational and Mathematical Sciences, 4:142–147, 2010.
[33] J. A. Rad and K. Parand. Analytical solution of Gas Flow
Through a Micro-Nano Porous Media by Homotopy Perturbation
method. International Journal of Mathematical and Computer Sciences,
6:184–188, 2010.
[34] N. A. Kudryashov. Seven common errors in finding exact solutions of
nonlinear differential equations. Commun Nonlinear Sci Numer Simulat,
14:3507–3529, 2009.
[35] P. Rosenau. A quasi-continuous description of a nonlinear transmission
line. Physica Scripta, 34:827–829, 1986.
[36] P. Rosenau. Dynamics of dense discrete systems. Progress of Theoretical
Physics, 79:1028–1042, 1988.
[37] T. Bridges and G. Derks. Linear instability of solitary wave solutions
of the Kawahara equation and its generalizations. SIAM J. Math. Anal.,
33:1356–1378, 2002.
[38] J. H. He and X. H. Wu. Exp-function method and its application to
nonlinear equations. Chaos, Solitons and Fractals, 38:903–910, 2008.
[39] J. M. Zuo. Solitons and periodic solutions for the Rosenau-KdV and
Rosenau-Kawahara equations. Applied Mathematics and Computation,
215:835–840, 2009.