Research ArticleOpen AccessGoogle Scholar indexed
Existence the Solutions of Some Fifth-Order Kdv Equation by Laplace Decomposition Method
Department of Mathematics, Mahatma Basweshwar Mahavidyalaya, Latur, India
Department of Mathematics, Maharashtra Udaygiri Mahavidyalaya, Udgir, India
- 1 Department of Mathematics, Mahatma Basweshwar Mahavidyalaya, Latur, India
- 2 Department of Mathematics, Maharashtra Udaygiri Mahavidyalaya, Udgir, India
American Journal of Computational Mathematics·Volume 03 (2013)·Pages 80–85·Published 26 March 2013·DOI10.4236/ajcm.2013.31013
Copy link · social · email
Abstract
In this paper, we develop a method to calculate numerical and approximate solution of some fifth-order Korteweg-de Vries equations with initial condition with the help of Laplace Decomposition Method (LDM). The technique is based on the application of Laplace transform to some fifth-order Kdv equations. The nonlinear term can easily be handled with the help of Adomian polynomials. We illustrate this technique with the help of four examples and results of the present technique have closed agreement with approximate solutions obtained with the help of (LDM).
KeywordsLaplace Decomposition MethodNonlinear Partial Differential EquationsFifth-Order Kdv EquationThe Kawahara Equation
- P. G. Drazin and R. S. Johnson, “Solutions: An Introduction,” Cambridge University Press, Cambridge, 1989. doi:10.1017/CBO9781139172059
- M. Khan, “Application of Laplace Decomposition Method to Solve Nonlinear Coupled Partial Differential Equations,” World Applied Sciences Journal, Vol. 9, 2010, pp. 13-19.
- G. Adomian, “A Review of the Decomposition Method in Applied Mathematics,” Journal of Mathematical Analysis and Applications, Vol. 135, No. 2, 1988, pp. 501-544.
- X. Q. Liu and C. L. Bai, “Exact Solutions of Some FifthOrder Nonlinear Equations,” Applied Mathematics—A Journal of Chinese Universities, Vol. 15, No. 1, 2000, pp. 28-32.
- E. J. Parkes and B. R. Duffy, “An Automated Tanh-Function Method for Finding Solitary Wave Solutions to NonLinear Evolution Equations,” Computer Physics Communications, Vol. 98, No. 3, 1996, pp. 288-300. doi:10.1016/0010-4655(96)00104-X
- T. R. Akylas and T.-S. Yang, “On Short-Scale Oscillatory Tails of Long-Wave Disturbances,” Studies in Applied Mathematics, Vol. 94, 1995, pp. 1-20.
- J. K. Hunter and J. Scheurle, “Existence of Perturbed Solitary Wave Solutions to a Model Equation for Water Waves,” Physica D: Nonlinear Phenomena, Vol. 32, No. 2, 1988, pp. 253-268. doi:10.1016/0167-2789(88)90054-1
- J. P. Body, “Weak Non-Local Solitons for CapillaryGravity Waves: Fifth-Order Korteweg-de Vries Equation,” Physica D, Vol. 48, 1991, pp. 129-146.
- J. T. Beale, “Exact Solitary Waves with Capillary Ripples at Infinity,” Communications Pure Applied Mathematics, Vol. 44, 1991, pp. 211-247.
- G. Adomian, “Solving Frontier Problems of Physics: The Decomposition Method,” Kluwer Academic Publishers, Boston, 1994.
- Y. Cherruault, “Convergence of Adomian’s Method,” Kybernetics, Vol. 18, 1989, pp. 31-38.
- A. Repaci, “Nonlinear Dynamical Systems: On the Accuracy of Adomian’s Decomposition Method,” Applied Mathematics Letters, Vol. 3, No. 4, 1990, pp. 35-39.
- Y. Cherruault and G. Adomian, “Decomposition Methods: A New Proof of Convergence,” Mathematical and Computer Modelling, Vol. 18, No. 12, 1993, pp. 103-106. doi:10.1016/0895-7177(93)90233-O
- A. M. Wazwaz, “A Reliable Modification of Adomian Decomposition Method,” Applied Mathematics and Computation, Vol. 102, No. 1, 1999, pp. 77-86. doi:10.1016/S0096-3003(98)10024-3