Numerical Estimation of Traveling Wave Solution of Two-Dimensional K-dV Equation Using a New Auxiliary Equation Method

DOI: 10.4236/ajcm.2013.31004   PDF   HTML   XML   4,218 Downloads   7,140 Views   Citations

Abstract

Korteweg de-Vries (K-dV) has wide applications in physics, engineering and fluid mechanics. In this the Korteweg de-Vries equation with traveling solitary waves and numerical estimation of analytic solutions have been studied. We have found some exact traveling wave solutions with relevant physical parameters using new auxiliary equation method introduced by PANG, BIAN and CHAO. We have solved the set of exact traveling wave solution analytically. Some numerical results of time dependent wave solutions have been presented graphically and discussed. This procedure has a potential to be used in more complex system of many types of K-dV equation.

Share and Cite:

R. Karim, A. Alim and L. Andallah, "Numerical Estimation of Traveling Wave Solution of Two-Dimensional K-dV Equation Using a New Auxiliary Equation Method," American Journal of Computational Mathematics, Vol. 3 No. 1, 2013, pp. 27-36. doi: 10.4236/ajcm.2013.31004.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] Yu. N. Zaiko, “Quasiperiodic Solutions of the Kortewegd Vries Equation,” Technical Physics Letters, Vol. 28. No. 3, 2002, pp. 235-236. doi:10.1134/1.1467286
[2] F. L. Qu and W. Q. Wang, “Alternating Segment Explicit—Implicit Scheme for Nonlinear Third-Order KdV Equation,” Applied Mathematics and Mechanics, Vol. 28, No. 7, 2007, pp. 973-980. doi:10.1007/s10483-007-0714-y
[3] N. E. Zhukovskii, “Hydraulic Shock in water Pipelines,” Gostekhteorizdal, Moscow, 1949.
[4] A. C. Newell, “Solitons in Mathematics and Physics,” SIAM, Philadelphia, 1985.
[5] V. A. Rukavishnikov and O. P. Tkachenko, “The Korteweg-de Vries Equation in a Cylindrical Pipe,” Computational Mathematics and Mathematical Physics, Vol. 48, No. 1, 2008, pp. 139-146. doi:10.1134/S0965542508010107
[6] N. Smaoui and R. H. Al-Jamal, “Boundary Control of the Generalized Korteweg-de Vries-Burgers Equation,” Nonlinear Dynamics, Vol. 51, 2008, pp. 439-446.
[7] J. Pang, C.-Q. Bian and L. Chao, “A New Auxiliary Equation Method for Finding Traveling Wave Solution to K-dV Equation,” Applied Mathematics and Mechanics (English Edition), Vol. 31, No. 7, 2010, pp. 929-936.
[8] L. Debnath, “Linear Partial Differential Equations for Scientists and Engineers,” 4th Edition, Tyn Myint-U, 2007, pp. 573-580.
[9] R. L. Herman, “Solitary Waves,” American Scientist, Vol. 80, 1992, pp. 350-361.
[10] P. A. Clarkson and M. D. Kruskal, “New Similarity Reductions of the Boussinesq Equation,” Journal of Mathematical Physics, Vol. 30, No. 10, 1989, pp. 2201-2213. doi:10.1063/1.528613

  
comments powered by Disqus

Copyright © 2020 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.