Analytical Solution of Two Extended Model Equations for Shallow Water Waves By Adomian’S Decomposition Method

Author(s)
Mehdi. Safari

ABSTRACT

In this paper, we consider two extended model equations for shallow water waves. We use Adomian’s decomposition method (ADM) to solve them. It is proved that this method is a very good tool for shallow water wave equations and the obtained solutions are shown graphically.

In this paper, we consider two extended model equations for shallow water waves. We use Adomian’s decomposition method (ADM) to solve them. It is proved that this method is a very good tool for shallow water wave equations and the obtained solutions are shown graphically.

Cite this paper

nullM. Safari, "Analytical Solution of Two Extended Model Equations for Shallow Water Waves By Adomian’S Decomposition Method,"*Advances in Pure Mathematics*, Vol. 1 No. 4, 2011, pp. 238-242. doi: 10.4236/apm.2011.14042.

nullM. Safari, "Analytical Solution of Two Extended Model Equations for Shallow Water Waves By Adomian’S Decomposition Method,"

References

[1] P. A. Clarkson and E.L. Mansfield, “On a shallow water wave equation”, Nonlinearity, Vol. 7, 1994, pp.975-1000.

[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.

[4] G.Adomian, “An analytical solution of the stochastic Navier-Stokes system”, Foundations of Physics, Vol. 21, No. 7, 1991, pp.831-843.

[5] G.Adomian and R. Rach, “Linear and nonlinear Schr?dinger equations”, Foundations of Physics, Vol. 21, 1991, pp. 983-991.

[6] G. Adomian, “Solution of physical problems by decomposition”, Computers & Mathematics with Applications, Vol. 27, No. 9/10, 1994, pp.145-154.

[7] G. Adomian, “Solutions of nonlinear PDE”, Applied Mathematics Letters, Vol. 11, No. 3, 1998, pp.121-123.

[8] K. Abbaoui and Y. Cherruault, “The decomposition method applied to the Cauchy problem”, Kybernetes , Vol. 28, 1999, pp.103-108.

[9] D. Kaya and A. Yokus, “A numerical comparison of partial solutions in the decomposition method for linear and nonlinear partial differential equations”, Mathematics and Computers in Simulation, Vol. 60, No. 6, 2002, pp. 507-512.

[10] A.M.Wazwaz, “Partial Differential Equations: Methods and Applications”, Balkema, Rottesdam, 2002.

[11] A. M. Wazwaz, “The extended tanh method for new solitons solutions for many forms of the fifth-order KdV equations”, Applied Mathematics and Computation, Vol. 184, No. 2, 2007, pp.1002-1014.

[12] A.M. Wazwaz, “The tanh–coth method for solitons and kink solutions for nonlinear parabolic equations”, Applied Mathematics and Computation, Vol. 188, 2007, pp.1467-1475.

[13] D.D. Ganji, E. M. M. Sadeghi, M. Safari, “Application of He’s variational iteration method and adomian’s decom- position method method to Prochhammer Chree equa- tion”, International Journal of Modern Physics B, Vol. 23, No. 3, 2009, pp.435-446.

[14] M. Safari, D.D. Ganji, M. Moslemi , “Application of He’s variational iteration method and Adomian’s decomposition method to the fractional KdV-Burgers- Kuramoto equation”, Computers and Mathematics with Applications, Vol. 58, 2009, pp. 2091-2097.

[15] D.D. Ganji, M. Safari, R. Ghayor, “Application of He’s variational iteration method and Adomian’s decomposi- tion method to Sawada-Kotera-Ito seventh-order equation”, Numerical Methods for Partial Differential Equations, Vol. 27, No. 4, 2011, pp: 887-897.

[16] A.M.Wazwaz, “A reliable modification of Adomian decomposition method”, Applied Mathematics and Computation, Vol. 102, No. 1, 1999, pp.77-86.

[1] P. A. Clarkson and E.L. Mansfield, “On a shallow water wave equation”, Nonlinearity, Vol. 7, 1994, pp.975-1000.

[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.

[4] G.Adomian, “An analytical solution of the stochastic Navier-Stokes system”, Foundations of Physics, Vol. 21, No. 7, 1991, pp.831-843.

[5] G.Adomian and R. Rach, “Linear and nonlinear Schr?dinger equations”, Foundations of Physics, Vol. 21, 1991, pp. 983-991.

[6] G. Adomian, “Solution of physical problems by decomposition”, Computers & Mathematics with Applications, Vol. 27, No. 9/10, 1994, pp.145-154.

[7] G. Adomian, “Solutions of nonlinear PDE”, Applied Mathematics Letters, Vol. 11, No. 3, 1998, pp.121-123.

[8] K. Abbaoui and Y. Cherruault, “The decomposition method applied to the Cauchy problem”, Kybernetes , Vol. 28, 1999, pp.103-108.

[9] D. Kaya and A. Yokus, “A numerical comparison of partial solutions in the decomposition method for linear and nonlinear partial differential equations”, Mathematics and Computers in Simulation, Vol. 60, No. 6, 2002, pp. 507-512.

[10] A.M.Wazwaz, “Partial Differential Equations: Methods and Applications”, Balkema, Rottesdam, 2002.

[11] A. M. Wazwaz, “The extended tanh method for new solitons solutions for many forms of the fifth-order KdV equations”, Applied Mathematics and Computation, Vol. 184, No. 2, 2007, pp.1002-1014.

[12] A.M. Wazwaz, “The tanh–coth method for solitons and kink solutions for nonlinear parabolic equations”, Applied Mathematics and Computation, Vol. 188, 2007, pp.1467-1475.

[13] D.D. Ganji, E. M. M. Sadeghi, M. Safari, “Application of He’s variational iteration method and adomian’s decom- position method method to Prochhammer Chree equa- tion”, International Journal of Modern Physics B, Vol. 23, No. 3, 2009, pp.435-446.

[14] M. Safari, D.D. Ganji, M. Moslemi , “Application of He’s variational iteration method and Adomian’s decomposition method to the fractional KdV-Burgers- Kuramoto equation”, Computers and Mathematics with Applications, Vol. 58, 2009, pp. 2091-2097.

[15] D.D. Ganji, M. Safari, R. Ghayor, “Application of He’s variational iteration method and Adomian’s decomposi- tion method to Sawada-Kotera-Ito seventh-order equation”, Numerical Methods for Partial Differential Equations, Vol. 27, No. 4, 2011, pp: 887-897.

[16] A.M.Wazwaz, “A reliable modification of Adomian decomposition method”, Applied Mathematics and Computation, Vol. 102, No. 1, 1999, pp.77-86.