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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 878349, 13 pages
Approximate Analytic Solution for the KdV and Burger Equations with the Homotopy Analysis Method
1Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, Johor, 81310 Johor Bahru, Malaysia
2Department of Mathematics, Faculty of Science, University of Kordofan, North Kordofan State, Elobeid 51111, Sudan
3Ibnu Sina Institute for Fundamental Science Studies, Universiti Teknologi Malaysia, Johor, 81310 Johor Bahru, Malaysia
4Department of Computer Science and Information System, Universiti Teknologi Malaysia, Johor, 81310 Johor Bahru, Malaysia
Received 11 June 2012; Accepted 25 July 2012
Academic Editor: Saeid Abbasbandy
Copyright © 2012 Mojtaba Nazari et al. 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.
- S. J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems [Ph.D. thesis], Shanghai Jiao University, 1992.
- S. J. Liao, Ed., Beyond Perturbation: Introduction to the Homotopy Analysis Method Boca Raton, Chapman & Hall, Boca Raton, Fla, USA, 2003.
- S. J. Liao, “On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids over a stretching sheet,” Journal of Fluid Mechanics, vol. 488, pp. 189–212, 2003.
- S. J. Liao, “A new branch of solutions of boundary-layer flows over an impermeable stretched plate,” International Journal of Heat and Mass Transfer, vol. 48, no. 12, pp. 2529–2539, 2005.
- S. J. Liao, J. Su, and A. T. Chwang, “Series solutions for a nonlinear model of combined convective and radiative cooling of a spherical body,” International Journal of Heat and Mass Transfer, vol. 49, no. 15-16, pp. 2437–2445, 2006.
- S. J. Liao and E. Magyari, “Exponentially decaying boundary layers as limiting cases of families of algebraically decaying ones,” Zeitschrift für Angewandte Mathematik und Physik, vol. 57, no. 5, pp. 777–792, 2006.
- S. J. Liao, “Series solutions of unsteady boundary-layer flows over a stretching flat plate,” Studies in Applied Mathematics, vol. 117, no. 3, pp. 239–263, 2006.
- S. Abbasbandy, “The application of homotopy analysis method to nonlinear equations arising in heat transfer,” Physics Letters A, vol. 360, no. 1, pp. 109–113, 2006.
- S. Abbasbandy, “The application of homotopy analysis method to solve a generalized Hirota-Satsuma coupled KdV equation,” Physics Letters A, vol. 361, no. 6, pp. 478–483, 2007.
- S. Abbasbandy, “Homotopy analysis method for heat radiation equations,” International Communications in Heat and Mass Transfer, vol. 34, no. 3, pp. 380–387, 2007.
- M. Ayub, A. Rasheed, and T. Hayat, “Exact flow of a third grade fluid past a porous plate using homotopy analysis method,” International Journal of Engineering Science, vol. 41, no. 18, pp. 2091–2103, 2003.
- T. Hayat and M. Khan, “Homotopy solutions for a generalized second-grade fluid past a porous plate,” Nonlinear Dynamics, vol. 42, no. 4, pp. 395–405, 2005.
- T. Hayat, M. Khan, and M. Ayub, “On non-linear flows with slip boundary condition,” Zeitschrift für Angewandte Mathematik und Physik, vol. 56, no. 6, pp. 1012–1029, 2005.
- S. Asghar, M. Mudassar Gulzar, and T. Hayat, “Rotating flow of a third grade fluid by homotopy analysis method,” Applied Mathematics and Computation, vol. 165, no. 1, pp. 213–221, 2005.
- M. Sajid, T. Hayat, and S. Asghar, “On the analytic solution of the steady flow of a fourth grade fluid,” Physics Letters A, vol. 355, no. 1, pp. 18–26, 2006.
- Y. Tan and S. Abbasbandy, “Homotopy analysis method for quadratic Riccati differential equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 13, no. 3, pp. 539–546, 2008.
- S. Abbasbandy and T. Hayat, “Solution of the MHD Falkner-Skan flow by homotopy analysis method,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 9-10, pp. 3591–3598, 2009.
- C. Wang, Y. Y. Wu, and W. Wu, “Solving the nonlinear periodic wave problems with the Homotopy Analysis Method,” Wave Motion, vol. 41, no. 4, pp. 329–337, 2005.
- S. Abbasbandy and A. Shirzadi, “A new application of the homotopy analysis method: solving the Sturm-Liouville problems,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 1, pp. 112–126, 2011.
- S. Abbasbandy and A. Shirzadi, “Homotopy analysis method for multiple solutions of the fractional Sturm-Liouville problems,” Numerical Algorithms, vol. 54, no. 4, pp. 521–532, 2010.
- S. Abbasbandy and A. Shirzadi, “Homotopy analysis method for a nonlinear chemistry problem,” Studies in Nonlinear Sciences, vol. 1, no. 4, pp. 127–132, 2010.
- S. Abbasbandy and A. Shirzadi, “The series solution of problems in the calculus of variations via the homotopy analysis method,” Zeitschrift fur Naturforschung, vol. 64, no. 1-2, pp. 30–36, 2009.
- H. Bateman, “Some recent researches on the motion of fluids,” Monthly Weather Review, vol. 43, pp. 163–170, 1915.
- J. M. Burgers, Mathematical Examples Illustrating Relations Occurring in the Theory of Turbulent Fluid Motion, vol. 17, Transitions of Royal Netherlands Academy of Arts and Sciences, Amsterdam, The Netherlands, 1939, Reprinted in F. T. M. Nieuwstadt and J. A. Steketee, Selected papers of J. M. Burgers, Kluwer Academic, Dordrecht, The Netherlands, pp. 281–334, 1995.
- C. A. J. Fletcher, “Burgers' equation: a model for all reasons,” in Numerical Solutions of Partial Differential Equations, North-Holland, Amsterdam, The Netherlands, 1982.
- M. J. Ablowitz and H. Segur, Eds., Solitons and the Inverse Scattering Transform, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 1981.
- M. J. Ablowitz and P. A. Clarkson, Eds., Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, New York, NY, USA, 1991.
- A. M. Wazwaz, “Construction of solitary wave solutions and rational solutions for the KdV equation by Adomian decomposition method,” Chaos, Solitons and Fractals, vol. 12, no. 12, pp. 2283–2293, 2001.
- P. G. Drazin and R. S. Jonson, Soliton: An Introduction, Cambridge University Press, New York, NY, USA, 1993.