Journal of Applied Mathematics

Exact Solutions for a New Nonlinear KdV-Like Wave Equation Using Simplest Equation Method and Its Variants

Yinghui He, Yun-Mei Zhao, and Yao Long

Full-text: Open access

Abstract

The simplest equation method presents wide applicability to the handling of nonlinear wave equations. In this paper, we focus on the exact solution of a new nonlinear KdV-like wave equation by means of the simplest equation method, the modified simplest equation method and, the extended simplest equation method. The KdV-like wave equation was derived for solitary waves propagating on an interface (liquid-air) with wave motion induced by a harmonic forcing which is more appropriate for the study of thin film mass transfer. Thus finding the exact solutions of this equation is of great importance and interest. By these three methods, many new exact solutions of this equation are obtained.

Article information

Source
J. Appl. Math., Volume 2014 (2014), Article ID 578362, 8 pages.

Dates
First available in Project Euclid: 2 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.jam/1425305851

Digital Object Identifier
doi:10.1155/2014/578362

Mathematical Reviews number (MathSciNet)
MR3228133

Citation

He, Yinghui; Zhao, Yun-Mei; Long, Yao. Exact Solutions for a New Nonlinear KdV-Like Wave Equation Using Simplest Equation Method and Its Variants. J. Appl. Math. 2014 (2014), Article ID 578362, 8 pages. doi:10.1155/2014/578362. https://projecteuclid.org/euclid.jam/1425305851


Export citation

References

  • M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, New York, NY, USA, 1991.
  • V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons, Springer, Berlin, Gemany, 1991.
  • R. Hirota, “Exact solution of the korteweg–-de vries equation for multiple Collisions of solitons,” Physical Review Letters, vol. 27, no. 18, pp. 1192–1194, 1971.
  • M. Wang, Y. Zhou, and Z. Li, “Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics,” Physics Letters, Section A: General, Atomic and Solid State Physics, vol. 216, no. 1–5, pp. 67–75, 1996.
  • S. Liu, Z. Fu, and Q. Zhao, “Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations,” Physics Letters A, vol. 289, no. 1-2, pp. 69–74, 2001.
  • E. J. Parkes and B. R. Duffy, “An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations,” Computer Physics Communications, vol. 98, no. 3, pp. 288–300, 1996.
  • J. He and X. Wu, “Exp-function method for nonlinear wave equations,” Chaos, Solitons and Fractals, vol. 30, no. 3, pp. 700–708, 2006.
  • M. Wang, X. Li, and J. Zhang, “The ${G }^{\prime }/G$-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics,” Physics Letters. A, vol. 372, no. 4, pp. 417–423, 2008.
  • S. Guo and Y. Zhou, “The extended ${G}^{\prime }/G$-expansion method and its applications to the Whitham-Broer-Kaup-like equations and coupled Hirota-Satsuma KDV equations,” Applied Mathematics and Computation, vol. 215, no. 9, pp. 3214–3221, 2010.
  • M. V. Demina and N. A. Kudryashov, “From Laurent series to exact meromorphic solutions: the Kawahara equation,” Physics Letters A, vol. 374, no. 39, pp. 4023–4029, 2010.
  • M. V. Demina and N. A. Kudryashov, “Explicit expressions for meromorphic solutions of autonomous nonlinear ordinary differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1127–1134, 2011.
  • N. Jamshidi and D. D. Ganji, “Application of energy balance method and variational iteration method to an oscillation of a mass attached to a stretched elastic wire,” Current Applied Physics, vol. 10, no. 2, pp. 484–486, 2010.
  • M. Rafei, H. Daniali, and D. D. Ganji, “Variational interation method for solving the epidemic model and the prey and predator problem,” Applied Mathematics and Computation, vol. 186, no. 2, pp. 1701–1709, 2007.
  • F. Fouladi, E. Hosseinzadeh, A. Barari, and G. Domairry, “Highly nonlinear temperature-dependent fin analysis by variational iteration method,” Heat Transfer Research, vol. 41, no. 2, pp. 155–165, 2010.
  • M. Esmaeilpour and D. D. Ganji, “Application of He\textquotesingle s homotopy perturbation method to boundary layer flow and convection heat transfer over a flat plate,” Physics Letters A, vol. 372, no. 1, pp. 33–38, 2007.
  • D. D. Ganji, M. J. Hosseini, and J. Shayegh, “Some nonlinear heat transfer equations solved by three approximate methods,” International Communications in Heat and Mass Transfer, vol. 34, no. 8, pp. 1003–1016, 2007.
  • Z. Z. Ganji, D. D. Ganji, and M. Esmaeilpour, “Study on nonlinear Jeffery-Hamel flow by He\textquotesingle s semi-analytical methods and comparison with numerical results,” Computers & Mathematics with Applications, vol. 58, no. 11-12, pp. 2107–2116, 2009.
  • S. H. H. Nia, A. N. Ranjbar, D. D. Ganji, H. Soltani, and J. Ghasemi, “Maintaining the stability of nonlinear differential equations by the enhancement of HPM,” Physics Letters A: General, Atomic and Solid State Physics, vol. 372, no. 16, pp. 2855–2861, 2008.
  • M. Omidvar, A. Barari, M. Momeni, and D. Ganji, “New class of solutions for water infiltration problems in unsaturated soils,” Geomechanics and Geoengineering, vol. 5, no. 2, pp. 127–135, 2010.
  • E. Fan, “Uniformly constructing a series of explicit exact solu-tions to nonlinear equations in mathematical physics,” Chaos, Solitons and Fractals, vol. 16, no. 5, pp. 819–839, 2003.
  • N. A. Kudryashov, “Exact solitary waves of the Fisher equation,” Physics Letters A, vol. 342, no. 1-2, pp. 99–106, 2005.
  • N. A. Kudryashov, “Simplest equation method to look for exact solutions of nonlinear differential equations,” Chaos, Solitons and Fractals, vol. 24, no. 5, pp. 1217–1231, 2005.
  • A. J. Mohamad Jawad, M. D. Petkovic, and A. Biswas, “Modified simple equation method for nonlinear evolution equations,” Applied Mathematics and Computation, vol. 217, no. 2, pp. 869–877, 2010.
  • N. A. Kudryashov and N. B. Loguinova, “Extended simplest equation method for nonlinear differential equations,” Applied Mathematics and Computation, vol. 205, no. 1, pp. 396–402, 2008.
  • E. M. E. Zayed and S. A. H. Ibrahim, “Exact solutions of non-linear evolution equations in mathematical physics using the modified simple equation method,” Chinese Physics Letters, vol. 29, no. 6, Article ID 060201, 2012.
  • S. Bilige and T. Chaolu, “An extended simplest equation method and its application to several forms of the fifth-order KdV equation,” Applied Mathematics and Computation, vol. 216, no. 11, pp. 3146–3153, 2010.
  • S. Bilige, T. Chaolu, and X. Wang, “Application of the extended simplest equation method to the coupled Schrödinger-Boussinesq equation,” Applied Mathematics and Computation, vol. 224, pp. 517–523, 2013.
  • J. M. Rees and W. B. Zimmerman, “An intermediate wavelength, weakly nonlinear theory for the evolution of capillary gravity waves,” Wave Motion, vol. 48, no. 8, pp. 707–716, 2011.
  • A. Sohail, J. M. Rees, and W. B. Zimmerman, “Analysis of capillary-gravity waves using the discrete periodic inverse scattering transform,” Colloids and Surfaces A: Physicochemical and Engineering Aspects, vol. 391, no. 1–3, pp. 42–50, 2011.
  • A. Sohail, K. Maqbool, and T. Hayat, “Painlevé property and approximate solutions using Adomian decomposition for a nonlinear KdV-like wave equation,” Applied Mathematics and Computation, vol. 229, pp. 359–366, 2014. \endinput