ANALYTICAL APPROXIMATE SOLUTION FOR THE FORCED KORTEWEG-DE VRIES (FKDV) ON CRITICAL FLOW OVER A HOLE USING HOMOTOPY ANALYSIS METHOD
DOI:
https://doi.org/10.11113/jt.v78.7823Keywords:
fKdV equation, homotopy analysis method, approximate analytical solution, holed bottom topographyAbstract
Free surface flows in a two-dimensional channel past over a hole is studied using shallow water forced Korteweg-de Vries (fKdV) equation. The forcing term of fKdV equation represents the hole shaped bottom topography. Froude number (Fr), which represents the ratio of flow speed to the wave speed, will also be used in solving fKdV equation. The fKdV equation is solved using Homotopy Analysis Method (HAM). HAM is an approximate analytical technique used to obtain series of solutions for the nonlinear problems where HAM has an auxiliary parameter coto adjust and control the convergence region of the series solution. Solitary wave solutions are obtained from the series of solutions of HAM and wave flows are observed at particular time. The HAM solution shows the hole shaped bottom topography plays an important role in determining the evolution of solitary waves.Â
References
Hereman, W. 2011. Shallow Water Waves And Solitary Waves. In Mathematics of Complexity and Dynamical Systems. Springer New York. 1520-1532.
Zabusky, N. J., & Kruskal, M. D. 1965. Interaction Of Solitons In A Collisionless Plasma And The Recurrence Of Initial States. Physical Review Letters. 15(6):240-243.
Gardner, C. S., Greene, J. M., Kruskal, M. D., & Miura, R. M. 1967. Method For Solving The Korteweg-Devries Equation. Physical Review Letters. 19(19): 1095.
Hirota, R. 2004. The Direct Method In Soliton Theory. Cambridge University Press.
Wahlquist, H. D., & Estabrook, F. B. 1973. Bäcklund Transformation For Solutions Of The Korteweg-De Vries Equation. Physical Review Letters. 31(23):1386.
Matveev, V. B., & Salle, M. A. 1991. Darboux Transformations And Solitons. Berlin: Springer-Verlag.
Wazwaz, A. M. 2001. Construction Of Solitary Wave Solutions And Rational Solutions For The Kdv Equation By Adomian Decomposition Method. Chaos, Solitons & Fractals. 12(12):2283-2293.
Jun-Xiao, Z., & Bo-Ling, G. 2009. Analytic Solutions To Forced Kdv Equation. Communications in Theoretical Physics. 52(2): 279.
Lee, S. J., Yates, G. T., & Wu, T. Y. 1989. Experiments And Analyses Of Upstream-Advancing Solitary Waves Generated By Moving Disturbances. Journal of Fluid Mechanics. 199: 69-593.
Camassa, R., & Wu, T. Y. T. 1991. Stability Of Forced Steady Solitary Waves. Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences. 337(1648): 429-466.
Zhang, Y., & Zhu, S. 1997. Subcritical, transcritical and supercritical flows over a step. Journal of Fluid Mechanics.333:257-271.
Zhang, D. H., & Chwang, A. T. 2001. Generation Of Solitary Waves By Forward-And Backward-Step Bottom Forcing. Journal of Fluid Mechanics. 432:341-350.
Grimshaw, R. H. J., Zhang, D. H., & Chow, K. W. 2009. Transcritical flow over a hole. Studies in Applied Mathematics. 122(3): 235-248.
Ee, B. K., Grimshaw, R. H. J., Zhang, D. H., & Chow, K. W. 2010. Steady Transcritical Flow Over A Hole: Parametric Map Of Solutions Of The Forced Korteweg–De Vries Equation. Physics of Fluids (1994-present). 22(5): 056602.
Liao, S. J. 2003. Beyond Perturbation: Introduction To The Homotopy Analysis Method. CRC press.
Liao, S. J. 1992. The Proposed Homotopy Analysis Technique For The Solution Of Nonlinear Problems (Doctoral dissertation, Ph. D. Thesis, Shanghai Jiao Tong University).
Liao, S. J. 2009. Notes On The Homotopy Analysis Method: Some Definitions And Theorems. Communications In Nonlinear Science And Numerical Simulation. 14(4): 983-997.
Abbasbandy, S. 2007. The Application Of Homotopy Analysis Method To Solve A Generalized Hirota–Satsuma Coupled Kdv Equation. Physics Letters A. 361(6):478-483.
Liao, S. J., & Cheung, K. F. 2003. Homotopy Analysis Of Nonlinear Progressive Waves In Deep Water. Journal of Engineering Mathematics. 45(2):105-116.
Nazari, M., Salah, F., Aziz, Z. A., & Nilashi, M. 2012. Approximate Analytic Solution For The Kdv And Burger Equations With The Homotopy Analysis Method. Journal of Applied Mathematics. Article ID 878349, 13 pages.
David, V. D., Nazari, M., Barati, V., Salah, F., & Aziz, Z. A. 2013. Approximate Analytical Solution for the Forced Korteweg-de Vries Equation. Journal of Applied Mathematics. Article ID 795818, 9 pages.
Wu, T. 1987. Generation Of Upstream Advancing Solitons By Moving Disturbances. Journal Of Fluid Mechanics. 184: 75-99
Downloads
Published
Issue
Section
License
Copyright of articles that appear in Jurnal Teknologi belongs exclusively to Penerbit Universiti Teknologi Malaysia (Penerbit UTM Press). This copyright covers the rights to reproduce the article, including reprints, electronic reproductions, or any other reproductions of similar nature.