Analytical Solutions of Kortweg-de Vries(KdV) Equation

Authors: Foad Saadi, M. Jalali Azizpour, S.A. Zahedi

Abstract:

The objective of this paper is to present a comparative study of Homotopy Perturbation Method (HPM), Variational Iteration Method (VIM) and Homotopy Analysis Method (HAM) for the semi analytical solution of Kortweg-de Vries (KdV) type equation called KdV. The study have been highlighted the efficiency and capability of aforementioned methods in solving these nonlinear problems which has been arisen from a number of important physical phenomenon.

Keywords: Variational Iteration Method (VIM), HomotopyPerturbation Method (HPM), Homotopy Analysis Method (HAM), KdV Equation.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1083205

References:

[1] D.J. Korteweg, G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary wave, Philos. Mag. Vol.39, 1895, pp. 422-443.
[2] Luwai Wazzan, A modified tanh-coth method for solving the KdV and the KdV-Burgers equations, Journal of Communication in nonlinear science and numerical simulation, (2007)
[3] A.J. Khattak, Siraj-ul-Islam, A comparative study of numerical solutions of a class of KdV equation , Journal of Computnational Applied Mathematical, Vol. 199, 2008 , pp.425-434.
[4] T. Ozis, S. Ozer S, A simple similarity-transformation-iterative scheme applied to Korteweg-de Vries equation, Journal of Applied Mathematical Compution, Vol. 173, 2006, pp.19-32.
[5] Abdul-Majid Wazwaz, The variational iteration method for rational solutions for KdV, K(2,2) Burgers and cubic Boussinesq equations, Journal of Computional Applied Mathematical, Article, (2006)
[6] P. Rosenau, J. M. Hyman, Compactons Solitons with finite wavelengths, Physics. Review Letter. Vol.70, No.5, 1993, pp. 564 -567.
[7] M.A. Abdou, A.A. Soliman, Variational iteration method for solving Burger's and coupled Burger's equation, Journal in Computensional Applied Mathematical, Vol.181, 2005, pp.245-251
[8] E.M. Aboulvafa, M.A. Abdou, A.A. Mahmoud, The solution of nonlinear coagulation problem with mass loss, Chaos Solitons And Fractals Vol.29, 2006, pp.313-330
[9] J.H. He, A new approach to nonlinear partial differential equations, Comm. Nonlinear Science and Numereical Simulation, Vol.2, No.4, 1997, pp.203-205.
[10] S.J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, PhD thesis Shanghai Jiao Tong University, 1992
[11] N. Tolou. I. Khatami. B. Jafari. D.D. Ganji. Analytical Solution of Nonlinear Vibrating Systems. American journal of applied Sciences, Vol.5, No.9, 2008, pp.1219-1224.
[12] M.J. Ablowitz, P.A. Clarkson, Solitions, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, 1991
[13] A. Coely, (Eds.), Backlund and Darboux Transformations, American Mathematical Society, Providence, Rhode Island, 2001
[14] M. Wadati, H. Sanuki, K. Konno, Relationships among inverse method, backlund transformation and an infinite number of conservation laws, Prog. Theoret. Phys. Vol.53, 1975, pp.419-436
[15] C.S. Gardner, J.M. Green, M.D. Kruskal, R.M. Miura, Method for solving the Korteweg-deVries equation, Phys. Rev. Lett. Vol.19, 1967, pp.1095-1097
[16] J.H. He, Homotopy perturbation method for bifurcation of nonlinear problems", Int. J. Non-linear Sci. Num. Sim., 6(2): 207-208 (2005)
[17] S.J. Liao, Beyond Perturbation: Introduction to the homotopy Analysis Method, Chapman & Hall/CRC press, Boca Raton, (2003)
[18] S. Abbasbandy, The application of homotopy analysis method to nonlinear equations arising in heat transfer, Phys. Lett. A