- About this Journal
- Abstracting and Indexing
- Aims and Scope
- Annual Issues
- Article Processing Charges
- Articles in Press
- Author Guidelines
- Bibliographic Information
- Citations to this Journal
- Contact Information
- Editorial Board
- Editorial Workflow
- Free eTOC Alerts
- Publication Ethics
- Reviewers Acknowledgment
- Submit a Manuscript
- Subscription Information
- Table of Contents
Mathematical Problems in Engineering
Volume 2011 (2011), Article ID 147327, 21 pages
Approximate Method for Studying the Waves Propagating along the Interface between Air-Water
1Department of Mathematics, Faculty of Science, Benha University, Benha 13518, Egypt
2Department of Mathematics, Faculty of Science, Umm Al-Qura University, 21955, Saudi Arabia
Received 23 November 2010; Accepted 6 January 2011
Academic Editor: Ezzat G. Bakhoum
Copyright © 2011 M. M. Khader and R. F. Al-Bar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- D. J. Korteweg and G. de Vries, “On the change form of long waves advancing in a rectangular canal, and on a new type of long stationary waves,” Philosophical Magazine, vol. 39, p. 422, 1895.
- T. B. Benjamin, “Internal waves of finite amplitude and permanent form,” The Journal of Fluid Mechanics, vol. 25, p. 241, 1966.
- D. J. Benney, “Long non-linear waves in fluid flows,” Journal of Mathematical Physics, vol. 45, pp. 52–63, 1966.
- P. G. Drazin and R. S. Johnson, Solitons: An Introduction, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, UK, 1996.
- C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, “Method for solving the Korteweg-deVries equation,” Physical Review Letters, vol. 19, no. 19, pp. 1095–1097, 1967.
- M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, vol. 4 of SIAM Studies in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 2000.
- A. Chertock and D. Levy, “A particle method for the KdV equation,” Journal of Scientific Computing, vol. 17, no. 1–4, pp. 491–499, 2002.
- G. B. Whitham, Linear and Nonlinear Waves, Wiley-Interscience, New York, NY, USA, 1974.
- W. Craig, “An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits,” Communications in Partial Differential Equations, vol. 10, no. 8, pp. 787–1003, 1985.
- G. Schneider and C. E. Wayne, “The long-wave limit for the water wave problem. I. The case of zero surface tension,” Communications on Pure and Applied Mathematics, vol. 53, no. 12, pp. 1475–1535, 2000.
- G. Schneider and C. E. Wayne, “The rigorous approximation of long-wavelength capillary-gravity waves,” Archive for Rational Mechanics and Analysis, vol. 162, no. 3, pp. 247–285, 2002.
- T. Iguchi, “A mathematical justification of the forced Korteweg-de Vries equation for capillary-gravity waves,” Kyushu Journal of Mathematics, vol. 60, no. 2, pp. 267–303, 2006.
- T. Iguchi, “A long wave approximation for capillary-gravity waves and an effect of the bottom,” Communications in Partial Differential Equations, vol. 32, no. 1–3, pp. 37–85, 2007.
- W. Craig, P. Guyenne, D. P. Nicholls, and C. Sulem, “Hamiltonian long-wave expansions for water waves over a rough bottom,” Proceedings of The Royal Society of London. Series A, vol. 461, no. 2055, pp. 839–873, 2005.
- A. M. Abourabla, M. A. Mahmoud, and G. M. Khedr, “Korteweg-de Vries type equations for waves propagating along the interface between air-water,” Canadian Journal of Physics, vol. 86, no. 12, pp. 1427–1435, 2008.
- B. Deconinck, Solitons and Nonlinear Waves, Washington University, St. Louis, Mo, USA, 2007.
- S. Abbasbandy and M. T. Darvishi, “A numerical solution of Burgers' equation by modified Adomian method,” Applied Mathematics and Computation, vol. 163, no. 3, pp. 1265–1272, 2005.
- G. Adomian, Nonlinear Stochastic Systems Theory and Applications to Physics, vol. 46 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1989.
- R. K. Bhattacharyya and R. K. Bera, “Application of Adomian method on the solution of the elastic wave propagation in elastic bars of finite length with randomly and linearly varying Young's modulus,” Applied Mathematics Letters, vol. 17, no. 6, pp. 703–709, 2004.
- S. M. El-Sayed and D. Kaya, “On the numerical solution of the system of two-dimensional Burgers' equations by the decomposition method,” Applied Mathematics and Computation, vol. 158, no. 1, pp. 101–109, 2004.
- S. Guellal, P. Grimalt, and Y. Cherruault, “Numerical study of Lorenz's equation by the Adomian method,” Computers & Mathematics with Applications, vol. 33, no. 3, pp. 25–29, 1997.
- D. Kaya and I. E. Inan, “A convergence analysis of the ADM and an application,” Applied Mathematics and Computation, vol. 161, no. 3, pp. 1015–1025, 2005.
- D. Kaya and A. Yokus, “A decomposition method for finding solitary and periodic solutions for a coupled higher-dimensional Burgers equations,” Applied Mathematics and Computation, vol. 164, no. 3, pp. 857–864, 2005.
- T. A. Abassy, M. A. El-Tawil, and H. El Zoheiry, “Solving nonlinear partial differential equations using the modified variational iteration Padé technique,” Journal of Computational and Applied Mathematics, vol. 207, no. 1, pp. 73–91, 2007.
- J. H. He, “Variation iteration method-a kind of non-linear analytical technique: some examples,” International Journal of Non-Linear Mechanics, vol. 34, p. 699, 1999.
- N. H. Sweilam, M. M. Khader, and R. F. Al-Bar, “Numerical studies for a multi-order fractional differential equation,” Physics Letters A, vol. 371, no. 1-2, pp. 26–33, 2007.
- N. H. Sweilam and M. M. Khader, “On the convergence of variational iteration method for nonlinear coupled system of partial differential equations,” International Journal of Computer Mathematics, vol. 87, no. 5, pp. 1120–1130, 2010.
- H. N. Sweilam, M. M. Khader, and F. R. Al-Bar, “On the numerical simulation of population dynamics with density-dependent migrations and the Allee effects,” Journal of Physics: Conference Series, vol. 96, no. 1, 2008.
- N. H. Sweilam, M. M. Khader, and R. F. Al-Bar, “Nonlinear focusing Manakov systems by variational iteration method and Adomian decomposition method,” Journal of Physics: Conference Series, vol. 96, no. 1, 2008.
- J.-H. He, “Application of homotopy perturbation method to nonlinear wave equations,” Chaos, Solitons and Fractals, vol. 26, no. 3, pp. 695–700, 2005.
- A.-M. Wazwaz, “Necessary conditions for the appearance of noise terms in decomposition solution series,” Applied Mathematics and Computation, vol. 81, no. 2-3, pp. 265–274, 1997.
- G. Adomian and R. Rach, “Noise terms in decomposition solution series,” Computers & Mathematics with Applications, vol. 24, no. 11, pp. 61–64, 1992.
- S. A. Khuri, “A new approach to Bratu's problem,” Applied Mathematics and Computation, vol. 147, no. 1, pp. 131–136, 2004.
- M. Hussain and M. Khan, “Modified Laplace decomposition method,” Applied Mathematical Sciences, vol. 4, no. 33-36, pp. 1769–1783, 2010.
- H. Liu and J. Yan, “A local discontinuous Galerkin method for the Korteweg-de Vries equation with boundary effect,” Journal of Computational Physics, vol. 215, no. 1, pp. 197–218, 2006.
- D. Rannacher and A. Engel, “Cylindrical Korteweg-de Vries solitons on a ferrofluid surface,” New Journal of Physics, vol. 8, pp. 108.1–108.16, 2006.