M. SAFARI ET AL.

238

Figure 2. For the second extended model of shallow water

wave equation with the first initial condition (32) of Equa-

tion (16), when c = 2.

21

3

11 1

1

2

0

11 1

0

311

1

3

,,

,,

3,

,, ,

3d

,,

6, d

t

x

uxtuxt

ux uxux

ux

x

ux uux

xx

ux ux

ux xx

,

(34)

In Figure 2 we can see the 3-D result of second ex-

tended model of shallow water wave equation by VIM.

5. Acknowledgements

In this paper, He’s variational iteration method has been

successfully applied to find the solution of two extended

model equations for shallow water. The obtained results

were showed graphically it is proved that He’s varia-

tional iteration method is a powerful method for solving

these eq uatio ns . I n our work ; w e use d th e Map l e P ack ag e

to calculate the functions obtained from the He’s varia-

tional iteration method.

6. References

[1] P. A. Clarkson and E. L. Mansfield, “On a Shall ow Wate r

Wave Equation,” Nonlinearity, Vol. 7, No. 3, 1994, pp.

975-1000. doi:10.1088/0951-7715/7/3/012

[2] M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur,

“The Inverse Scattering Transform-Fourier Analysis for

Nonlinear Problems,” Studies in Applied Mathematics,

Vol. 53, 1974, pp.249-315.

[3] R. Hirota and J. Satsuma,” Soliton Solutions of Model

Equations for Shallow Water Waves,” Journal of the

Physical Society of Japan, Vol. 40, No. 2, 1976, pp. 611-

612. doi:10.1143/JPSJ.40.611

[4] J. H. He, “Some Asymptotic Methods for Strongly Non-

linear Equations,” International Journal of Modern Phy-

sics B, Vol. 20, No. 10, 2006, pp. 1141-1199.

doi:10.1142/S0217979206033796

[5] J. H. He, “Approximate Analytical Solution for Seepage

Flow with Fractional Derivatives in Porous Media,” Com-

puter Methods in Applied Mechanics and Engineering,

Vol. 167, No. 1-2, 1998, pp. 57-68.

doi:10.1016/S0045-7825(98)00108-X

[6] J. H. He, “Variational Iteration Method for Autonomous

Ordinary Differential Systems,” Applied Mathematics and

Computation, Vol. 114, No. 2-3, 2000, pp. 115-123.

doi:10.1016/S0096-3003(99)00104-6

[7] J. H. He and X. H. Wu, “Construction of Solitary Solu-

tion and Compacton-Like Solution by Variational Itera-

tion Method,” Chaos Solitons & Fractals, Vol. 29, No. 1,

2006, pp. 108-113. doi:10.1016/j.chaos.2005.10.100

[8] J. H. He, “A New Approach to Nonlinear Partial Differ-

ential Equations”, Communications in Nonlinear Science

and Numerical Simulation, Vol. 2, No. 4, 1997, pp. 203-

205. doi:10.1016/S1007-5704(97)90007-1

[9] J. H. He, “Variational Iteration Method—A Kind of Non-

linear Analytical Technique: Some Examples,” Interna-

tional Journal of Nonlinear Mechanics, Vol. 34, No. 4,

1999, pp. 699-708. doi:10.1016/S0020-7462(98)00048-1

[10] J. H. He, “A Generalized Variational Principle in Micro-

morphic Thermoelasticity,” Mechanics Research Commu-

nications, Vol. 32, No. 1, 2005, pp. 93-98.

doi:10.1016/j.mechrescom.2004.06.006

[11] D. D. Ganji, M. Jannatabadi and E. Mohseni, “Applica-

tion of He’s Variational Iteration Method to Nonlinear

Jaulent-Miodek Equations and Comparing It with ADM,”

Journal of Computional and Applied Mathematics, Vol.

207, No. 1, 2007, pp. 35-45.

[12] D. D. Ganji, E. M. M. Sadeghi and M. Safari, “Appli-

cation of He’s Variational Iteration Method and Ado-

mian’s Decomposition Method Method to Prochhammer

Chree Equation,” International Journal of Modern Phy-

sics B, Vol. 23, No. 3, 2009, pp. 435-446.

doi:10.1142/S0217979209049656

[13] M. Safari, D. D. Ganji and M. Moslemi, “Application of

He’s Variational Iteration Method and Adomian’s De-

composition Method to the Fractional KdV-Burgers-

Kuramoto Equation,” Computers and Mathematics with

Applications, Vol. 58, No. 11-12, 2009, pp. 2091-2097.

[14] M. Safari, D. D. Ganji and E. M. M. Sadeghi, “Appli-

cation of He’s Homotopy Perturbation and He’s Varia-

tional Iteration Methods for Solution of Benney-Lin

Equation,” International Journal of Computer Mathema-

tics, Vol. 87, No. 8, pp. 1872-1884.

doi:10.1080/00207160802524770

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