Journal of Applied Mathematics

F-Expansion Method and Its Application for Finding New Exact Solutions to the Kudryashov-Sinelshchikov Equation

Yun-Mei Zhao

Full-text: Open access

Abstract

Based on the F-expansion method, and the extended version of F-expansion method, we investigate the exact solutions of the Kudryashov-Sinelshchikov equation. With the aid of Maple, more exact solutions expressed by Jacobi elliptic function are obtained. When the modulus m of Jacobi elliptic function is driven to the limits 1 and 0, some exact solutions expressed by hyperbolic function solutions and trigonometric functions can also be obtained.

Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 895760, 7 pages.

Dates
First available in Project Euclid: 14 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1394807958

Digital Object Identifier
doi:10.1155/2013/895760

Mathematical Reviews number (MathSciNet)
MR3049439

Zentralblatt MATH identifier
1266.76056

Citation

Zhao, Yun-Mei. F -Expansion Method and Its Application for Finding New Exact Solutions to the Kudryashov-Sinelshchikov Equation. J. Appl. Math. 2013 (2013), Article ID 895760, 7 pages. doi:10.1155/2013/895760. https://projecteuclid.org/euclid.jam/1394807958


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References

  • M. L. Wang, Y. B. Zhou, and Z. B. Li, “Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics,” Physics Letters, vol. 216, no. 1–5, pp. 67–75, 1996.
  • M. L. Wang, “Exact solutions for a compound KdV-Burgers equation,” Physics Letters A, vol. 213, no. 5-6, pp. 279–287, 1996.
  • S. Zhang and T. C. Xia, “A generalized new auxiliary equation method and its applications to nonlinear partial differential equations,” Physics Letters A, vol. 363, no. 5-6, pp. 356–360, 2007.
  • Sirendaoreji and S. Jiong, “Auxiliary equation method for solving nonlinear partial differential equations,” Physics Letters A, vol. 309, no. 5-6, pp. 387–396, 2003.
  • X.-H. (Benn) Wu and J.-H. He, “EXP-function method and its application to nonlinear equations,” Chaos, Solitons & Fractals, vol. 38, no. 3, pp. 903–910, 2008.
  • X.-H. (Benn) Wu and J.-H. He, “Solitary solutions, periodic solutions and compacton-like solutions using the Exp-function method,” Computers & Mathematics with Applications, vol. 54, no. 7-8, pp. 966–986, 2007.
  • H.-C. Hu, X.-Y. Tang, S.-Y. Lou, and Q.-P. Liu, “Variable separation solutions obtained from Darboux transformations for the asymmetric Nizhnik-Novikov-Veselov system,” Chaos, Solitons & Fractals, vol. 22, no. 2, pp. 327–334, 2004.
  • S. B. Leble and N. V. Ustinov, “Darboux transforms, deep reductions and solitons,” Journal of Physics A, vol. 26, no. 19, pp. 5007–5016, 1993.
  • H. A. Abdusalam, “On an improved complex tanh-function method,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 6, no. 2, pp. 99–106, 2005.
  • J. Lee and R. Sakthivel, “New exact travelling wave solutions of bidirectional wave equations,” Pramana Journal of Physics, vol. 76, no. 6, pp. 819–829, 2011.
  • E. J. Parkes, B. R. Duffy, and P. C. Abbott, “The Jacobi elliptic-function method for finding periodic-wave solutions to nonlinear evolution equations,” Physics Letters A, vol. 295, no. 5-6, pp. 280–286, 2002.
  • S. K. Liu, Z. T. Fu, S. D. Liu, 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.
  • Y. B. Zhou, M. L. Wang, and Y. M. Wang, “Periodic wave solutions to a coupled KdV equations with variable coefficients,” Physics Letters A, vol. 308, no. 1, pp. 31–36, 2003.
  • M. L. Wang and X. Z. Li, “Applications of F-expansion to periodic wave solutions for a new Hamiltonian amplitude equation,” Chaos, Solitons and Fractals, vol. 24, no. 5, pp. 1257–1268, 2005.
  • W.-W. Li, Y. Tian, and Z. Zhang, “F-expansion method and its application for finding new exact solutions to the sine-Gordon and sinh-Gordon equations,” Applied Mathematics and Computation, vol. 219, no. 3, pp. 1135–1143, 2012.
  • S. Zhang and T. C. Xia, “A generalized F-expansion method and new exact solutions of Konopelchenko-Dubrovsky equations,” Applied Mathematics and Computation, vol. 183, no. 2, pp. 1190–1200, 2006.
  • Y.-J. Ren and H.-Q. Zhang, “A generalized F-expansion method to find abundant families of Jacobi elliptic function solutions of the (2 + 1)-dimensional Nizhnik-Novikov-Veselov equation,” Chaos, Solitons and Fractals, vol. 27, no. 4, pp. 959–979, 2006.
  • G. L. Cai, Q. C. Wang, and J. J. Huang, “A modified F-expansion method for solving breaking soliton equation,” International Journal of Nonlinear Science, vol. 2, no. 2, pp. 122–128, 2006.
  • E. Yomba, “The extended F-expansion method and its application for solving the nonlinear wave, CKGZ, GDS, DS and GZ equations,” Physics Letters A, vol. 304, no. 1–4, pp. 149–160, 2005.
  • J. L. Zhang, M. L. Wang, Y. M. Wang, and Z. D. Fang, “The improved F-expansion method and its applications,” Physics Letters A, vol. 350, no. 1-2, pp. 103–109, 2006.
  • N. A. Kudryashov and D. I. Sinelshchikov, “Nonlinear waves in bubbly liquids with consideration for viscosity and heat transfer,” Physics Letters A, vol. 374, no. 19-20, pp. 2011–2016, 2010.
  • P. N. Ryabov, “Exact solutions of the Kudryashov-Sinelshchikov equation,” Applied Mathematics and Computation, vol. 217, no. 7, pp. 3585–3590, 2010.
  • D. J. Korteweg and G. de Vries, “On the change of long waves advancing in a rectangular canal, and on a new type of long stationary waves,” Philosophical Magazine Series, vol. 39, no. 240, pp. 422–443, 1895.
  • G. B. Whitham, Linear and Nonlinear Waves, Wiley-Interscience, New York, NY, USA, 1974.
  • V. E. Nakoryakov, V. V. Sobolev, and L. R. Shreiber, “Long waves perturbations in a gas-liquid mixture,” Fluid Dynamics, vol. 7, pp. 763–768, 1972.
  • 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, “One method for finding exact solutions of nonlinear differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 6, pp. 2248–2253, 2012.
  • P. N. Ryabov, D. I. Sinelshchikov, and M. B. Kochanov, “Application of the Kudryashov method for finding exact solutions of the high order nonlinear evolution equations,” Applied Mathematics and Computation, vol. 218, no. 7, pp. 3965–3972, 2011.
  • J. Li, Y. Zhang, and G. Chen, “Exact solutions and their dynamics of traveling waves in three typical nonlinear wave equations,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 19, no. 7, pp. 2249–2266, 2009.
  • H. Bin, L. Jibin, L. Yao, and R. Weiguo, “Bifurcations of travelling wave solutions for a variant of Camassa-Holm equation,” Nonlinear Analysis. Real World Applications, vol. 9, no. 2, pp. 222–232, 2008.
  • B. He, Q. Meng, and Y. Long, “The bifurcation and exact peakons, solitary and periodic wave solutions for the Kudryashov-Sinelshchikov equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 11, pp. 4137–4148, 2012.
  • M. A. Abdou, “The extended F-expansion method and its application for a class of nonlinear evolution equations,” Chaos, Solitons & Fractals, vol. 31, no. 1, pp. 95–104, 2007.