The Expression of Large Amplitude Wave equations using Homotopy Analysis Method

Ejinkonye Ifeoma O

doi:https://doi.org/10.14445/22315373/IJMTT-V66I5P530

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 66 | Issue 5 | Year 2020 | Article Id. IJMTT-V66I5P530 | DOI : https://doi.org/10.14445/22315373/IJMTT-V66I5P530

The Expression of Large Amplitude Wave equations using Homotopy Analysis Method

Ejinkonye Ifeoma O

Citation :

Ejinkonye Ifeoma O, "The Expression of Large Amplitude Wave equations using Homotopy Analysis Method," International Journal of Mathematics Trends and Technology (IJMTT), vol. 66, no. 5, pp. 219-226, 2020. Crossref, https://doi.org/10.14445/22315373/IJMTT-V66I5P530

Abstract

This work focuses on some mechanisms that concern large amplitude ocean wave event. The existence of the rogue wave is thus partly due to ocean current wave interaction and partly due to the inter-crossing of a large number of quasi-monochromatic wave components with appropriate frequencies, wave numbers and randomly distributed phase angles. In this study the equations of water waves are solved by means of an analytic technique, namely the Homotopy Analysis Method (HAM). HAM is a capable and straight forward analytic tool for solving nonlinear differential equations and does not require small/large parameters in the governing equations unlike other well-known analytic approach. We use HAM to obtain an approximate solution to the governing water wave equations. The free surface displacement η (x,t) and velocity potential, ø(x,z,t) obtained are compared with similar results.

Keywords

Homotopy Analysis Method, free surface displacement, velocity potential

References

[1] Dysthe, K., Krogstad, H.E., and Muller, P.,Oceanic Rogue Waves; Annu. Rev. Fluid Mech. (2008)Vol. 40:287-310.
[2] Lawton, G.,; Monsters of deep sea. New scientist (2001)170,(2297); 28-32.
[3] Lavrenov, I.V., and Porubor, A.V., Three reasons for freak wave generation in the non uniform current. Eur J. Mech B/fluids, vol.25 pp574-585.
[4] White, B.S., Fornberg B.,(1998); On the chance of freak waves at the sea, J. Fluid Mech. (2006) Vol. 255 PP 113-138
[5] Brown, M.G.,); The Meslov integral representation of slowly varying dispersive wavetrains in inhomogeneous moving media, Waves motion, (2000 Vol.32; 247-266.
[6] Brown, M.G.,;Space-time surface gravity waves caustics; structurally stable extreme wave events, Wave motion (2001) Vol. 33; 117-143.
[7] Lavrenov, I.V.,; The wave energy concentration at the Agulhas current of South Africa. Hazard (1998) Vol. 17; 171-177.
[8] Arena, F, and Fedele, F “Nonlinear Space-Time Evolution of wave Groups with a Hight Crest J. of Offshore Mech. And Arctic Eng(2005).Vol 127 pp 46-51
[9] Fedele, F., and Arena, F., “Weakly Nonlinear Statistics of High random waves” Phys fluids (2005).Vol 17 pp 0266 07-1610
[10] Fedele, F. “Extreme Events in Nonlinear Random Seas”. J.A Offshore Mech. And Arctic Eng (2006). Vol 128 pp 11-16
[11] Ejinkonye, I. O. and Ifidon, E.O., The Higher Order effects of the Rogue wave events. ABACUS (2013)pp 141-155, Vol. 40 No. 2
[12] Ejinkonye I.O. The effect of interaction of Large Amplitude wave on sea with its Application.ScientiaAfricana (2019) Vol. 18 No.1 pp 59-69
[13] Liao, S. J. Proposed homotopy analysis techniques for the solution of nonlinear problems.PhD thesis, Shanghai Jiao Tong University. 1992
[14] Liao, S. J. An approximate solution technique which does not depend upon small parameters (2): an application in fluid mechanics. Intl J. Non-Linear Mech. 1997 Vol. 32, 815–822.
[15] Liao, S. J.. Beyond Perturbation: Introduction to the Homotopy Analysis Method. Chapman & Hall, CRC, Boca Raton. (2003)
[16] Liao, S. J. On the homotopy analysis method for nonlinear problems.Appl. MathsComput.(2004) Vol.147, 499–513.
[17] Liao, S. J. On the homotopy multiple-variable method and its applications in the interactions of nonlinear gravity waves.Commun.Nonlinear Sci. Numer.Simul.(2011) Vol.16 (3), 1274–1303.
[18] Liao, S. J. Homotopy Analysis Method in Nonlinear Differential Equations. Springer & Higher Education.(2012)