Research Article | OPEN ACCESS
A New Homotopy Analysis Method for Approximating the Analytic Solution of KdV Equation
2Vahid Barati, 2Mojtaba Nazari, 2, 3Vincent Daniel David and 1, 2Zainal Abdul Aziz
1Centre for Industrial and Applied Mathematics
2Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia,
81310 UTM Johor Bahru, Johor, Malaysia
3Faculty of Computer and Mathematical Sciences, Universiti Teknologi Mara,
40450 Shah Alam, Selangor, Malaysia
Research Journal of Applied Sciences, Engineering and Technology 2014 4:826-831
Received: April 26, 2013 | Accepted: May 07, 2013 | Published: January 27, 2014
Abstract
In this study a new technique of the Homotopy Analysis Method (nHAM) is applied to obtain an approximate analytic solution of the well-known Korteweg-de Vries (KdV) equation. This method removes the extra terms and decreases the time taken in the original HAM by converting the KdV equation to a system of first order differential equations. The resulted nHAM solution at third order approximation is then compared with that of the exact soliton solution of the KdV equation and found to be in excellent agreement.
Keywords:
Approximate analytic solution, h-curve, KdV equation , new homotopy analysis method, system of first order differential equation,
References
-
Ablowitz, M.J. and H. Segur, 1981. Solitons and the Inverse Scattering Transform. Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA.
CrossRef
-
Ablowitz, M.J. and P.A. Clarkson, 1991. Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, New York.
CrossRef PMid:9905818
-
Adomian, G., 1976. Nonlinear stochastic differential equations. J. Math. Anal. Appl., 55: 441-452.
CrossRef
-
Adomian, G., 1994. Solving Frontier Problems of Physics: The Decomposition Method. Kluwer Academic Publishers, Boston and London.
CrossRef
-
Bush, A.W., 1992. Perturbation Methods for Engineers and Scientists: CRC Press Library of Engineering Mathematics. CRC Press, Boca Raton, Florida.
-
Cole, J.D., 1968. Perturbation Methods in Applied Mathematics. Blasdell Publishing Co., Waltham, Massachusetts.
-
Hassan, H.N. and M.A. El-Tawil, 2011. A new technique of using homotopy analysis method for solving high-order nonlinear differential equations. Math. Methods Appl. Sci., 34: 728-742.
CrossRef -
Hassan, H.N. and M.A. El-Tawil, 2012. A new technique of using homotopy analysis method for second order nonlinear differential equations. Appl. Math. Comput., 219: 708-728.
CrossRef
-
Hinch, E.J., 1991. Perturbation Methods: Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge.
CrossRef PMid:1860046
-
Kahn, P.B. and Y. Zarmi, 1998.Nonlinear Dynamics: Exploration through Normal Forms. John Wiley and Sons, Inc., New York.
-
Liao, S.J., 1992. The proposed homotopy analysis technique for the solution of nonlinear problems. Ph.D. Thesis, Shanghai Jiao Tong University.
-
Liao, S.J., 2003. Beyond Perturbation: Introduction to the Homotopy Analysis Method. Chapman and Hall, Boca Raton, FL.
CrossRef
-
Murdock, J.A., 1991. Perturbations: Theory and Methods. John Wiley and Sons, New York.
-
Nayfeh, A.H., 1981. Introduction to Perturbation Techniques. John Wiley and Sons, New York.
-
Nayfeh, A.H., 1985. Problems in Perturbation. John Wiley and Sons, New York.
-
Nayfeh, A.H., 2000. Perturbation Methods. John Wiley and Sons, New York.
CrossRef PMid:11051476
-
Nazari, M., F. Salah, Z.A. Aziz and M. Nilashi, 2012a. Approximate analytic solution for the KdV and burger equations with the homotopy analysis method. J. Appl. Math., Vol. 2012, Article ID 878349, pp: 13.
-
Nazari, M., F. Salah and Z.A. Aziz, 2012b. Analytic approximate solution for the KdV equation with the homotopy analysis method. Matematika., 28(1): 53-61.
CrossRef
-
Von-Dyke, M., 1975. Perturbation Methods in Fluid Mechanics. The Parabolic Press, Stanford, California.
-
Wazwaz, A.M., 2001. Construction of solitary wave solution and rational solutions for the KdV equation by Adomian decomposition method. Chaos Soliton. Fract., 12: 2283-2293.
CrossRef
Competing interests
The authors have no competing interests.
Open Access Policy
This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Copyright
The authors have no competing interests.
|
|
![](http://www.maxwellsci.com/images/RJASET-maxw.jpg) |
ISSN (Online): 2040-7467
ISSN (Print): 2040-7459 |
![Submit Manuscript](../images/sub.jpg) |
Information |
|
|
|
Sales & Services |
|
|
|