[Home ] [Archive]    
:: Main :: About :: Current Issue :: Archive :: Search :: Submit :: Contact ::
:: Volume 2, Issue 4 (2-2019) ::
ijcoe 2019, 2(4): 37-46 Back to browse issues page
Comparison between Homotopy Analysis Method (HAM) and Variational Iteration Method (VIM) in Solving the Nonlinear Wave Propagation Equations in Shallow Water
Mohsen Soltani, Rouhollah Amirabadi
University of Qom
Abstract:   (76 Views)
This study aims to investigate the capability of two common numerical methods, Homotopy Analysis Method (HAM) and Variational Iteration Method (VIM), and to suggest more efficient approximate solution method to the governing equations of nonlinear surface wave propagation in shallow water. To do so, semi-flat, moderate, and sharp slope of shore which are connected to an open ocean with a uniform depth are exposed to a solitary wave with initial wave height H=2 and stationary elevation d=20. Then, the surface elevation and velocity curves for these profiles are determined and compared by HAM and VIM.  To verify the numerical modeling, two slopes i.e. semi-flat and moderate slope are considered and modeled in Flow-3D. Afterwards, the results of surface elevations are compared to each other by using correlation coefficient. The correlation coefficients for the slopes represent that the results coincide well. Ultimately, although the results of both methods are quite similar, using HAM is highly recommend rather than VIM since it makes solution procedure fast-converging and more abridged.
Keywords: Homotopy Analysis Method (HAM), Variational Iteration Method (VIM), Shallow water equations
Full-Text [PDF 1295 kb]   (38 Downloads)    
Type of Study: Research | Subject: Coastal Engineering
Received: 2019/02/3 | Accepted: 2019/03/17 | Published: 2019/04/20
References
1. N. Gedik, E. Irtem, S. Kabdasli, (2001), Laboratory investigation on tsunami run-up, Ocean Engineering vol. 32, p. 513-528. [DOI:10.1016/j.oceaneng.2004.10.013]
2. Dezvareh, R., Bargi, K., & Moradi, Y, (2012), Assessment of Wave Diffraction behind the Breakwater Using Mild Slope and Boussinesq Theories, International Journal of Computer Applications in Engineering Sciences, 2(2).
3. Keskin, Y., & Oturanc, G, (2010), Reduced differential transform method for solving linear and nonlinear wave equations, Iranian Journal of Science and Technology (Sciences), 34(2), 113-122.
4. O. Kiymaz, A. Cetinkaya, 2010, Variational Iteration method for class of Nonlinear Differential Equations, international journal of contemporary mathematics sciences Vol.5, p.1819-1826.
5. E. Yusufoglu, A. Bekir, 2006, Application of the variational iteration method to the regularized long wave equation, an international journal computers & mathematics with applications, p. 1154-1161. [DOI:10.1016/j.camwa.2006.12.073]
6. A. Hemeda, 2007, Variatinal iteration method for solving wave equation, an international journal computers and mathematics with applications, p. 1948-1953. [DOI:10.1016/j.camwa.2008.04.010]
7. S. Mohyud-Din, M. Noor, F. Jabeen, 2011, modified variational iteration method for traveling wave solutions of seventh-order generalized KdV equations, p. 107-113. [DOI:10.1080/15502287.2011.564266]
8. Younesian, M., Askari, H., Saadatnia, Z. and Yildirim, A., (2012), Analytical solution for nonlinear wave propagation in shallow media using the variational iteration method, Waves in Random and Complex Media, vol. 22, no. 2, p. 133-142. [DOI:10.1080/17455030.2011.633578]
9. S.-J. Liao, (1992), The proposed homotopy analysis technique for the solutions of nonlinear problems, Ph.D. thesis, Shanghai Jiao Tong University, Shanghai, China.
10. S.-J. Liao, (1995), An approximate solution technique not depending on small parameters: a special example, International Journal of Non-Linear Mechanics, vol. 30, no. 3, p. 371-380. [DOI:10.1016/0020-7462(94)00054-E]
11. S.-J. Liao, (1997), A kind of approximate solution technique which does not depend upon small parameters-II: an application in fluid mechanics, International Journal of Non-Linear Mechanics, vol. 32, no. 5, p. 815-822.
12. S.-J. Liao, (1999), An explicit, totally analytic approximate solution for Blasius' viscous flow problems, International Journal of Non-Linear Mechanics, vol. 34, no. 4, p. 759-778. [DOI:10.1016/S0020-7462(98)00056-0]
13. S.-J. Liao, K. F. CHEUNG, (2003), Homotopy analysis of nonlinear progressive waves in deep water, Journal of Engineering Mathematics, vol. 45, no. 2, p. 105-116.
14. S.-J. Liao, (2004), Beyond Perturbation: Introduction to the Homotopy Analysis Method, vol. 2 of CRC Series:Modern Mechanics and Mathematics, Chapman & Hall/CRC, Boca Raton, Fla, USA.
15. S.-J. Liao, (2004), On the homotopy analysis method for nonlinear problems, Applied Mathematics and Computation, vol. 147, no. 2, p. 499-513. [DOI:10.1016/S0096-3003(02)00790-7]
16. Rashidi M.M., Ganji D.D., Dinarvand S., (2008), Approximate traveling wave solutions of coupled Whitham-Broer-Kaup shallow water equations by homotopy analysis method, Differential Eq Nonlinear Mechanics, p. 1-8. [DOI:10.1155/2008/243459]
17. W. Wu, S. Liao, 2005, Solving solitary waves with discontinuity by means of the Homotopy Analysis Method, chaos solitons & fractals, p.177-185. [DOI:10.1016/j.chaos.2004.12.016]
18. E. Yusufoglu, C. Selam, 2009, the homotopy analysis method to solve the modified equal width wave equation, numerical methods for partial differential equations. [DOI:10.1002/num.20498]
19. M. Shaiq, Z. Iqbal, S. Mohyud-Din, 2013, Homotopy Analysis Method for time-fractional wave-like equations, computational mathematics and modeling, Vol.24, NO. 4. [DOI:10.1007/s10598-013-9201-2]
20. Araghi, F., & Naghshband, S, (2013), On convergence of Homotopy Analysis Method to solve the Schrodinger equation with a power law nonlinearity, International Journal of Industrial Mathematics, 5(4), 367-374.
21. Izadian, J., Abrishami, R., & Jalili, M, (2014), A new approach for solving nonlinear system of equations using Newton method and HAM, Iranian Journal of Numerical Analysis and Optimization, 4(2), 57-72.
22. X. Yin, S. Kumar, D. Kumar, 2015, A modified Homotopy Analysis Method for solution of fractional wave equations, Advances in mechanical engineering, Vol. 7, p. 1-8. [DOI:10.1177/1687814015620330]
23. S. Waewcharoen, S. Boonyapibanwong, and S. Koonprasert, (2008), Application of 2D-Nonlinear Shallow Water Model of Tsunami by using Adomian Decomposition Method, AIP Conf. Proc. 1048, p. 580-584.
Send email to the article author

Add your comments about this article
Your username or Email:

CAPTCHA code


XML     Print


Download citation:
BibTeX | RIS | EndNote | Medlars | ProCite | Reference Manager | RefWorks
Send citation to:

Soltani M, Amirabadi R. Comparison between Homotopy Analysis Method (HAM) and Variational Iteration Method (VIM) in Solving the Nonlinear Wave Propagation Equations in Shallow Water. ijcoe. 2019; 2 (4) :37-46
URL: http://ijcoe.org/article-1-131-en.html


Volume 2, Issue 4 (2-2019) Back to browse issues page
International Journal of Coastal and Offshore Engineering International Journal of Coastal and Offshore Engineering
Persian site map - English site map - Created in 0.06 seconds with 32 queries by YEKTAWEB 3877