On numerical solutions of time-fraction generalized Hirota Satsuma coupled KdV equation

Volume 10, Issue 2, pp 724--733 http://dx.doi.org/10.22436/jnsa.010.02.33

Publication Date: February 20, 2017 Submission Date: June 10, 2016

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 1998 Downloads
• 4615 Views

Authors

Ebru Cavlak Aslan - Department of Mathematics, Science Faculty, Firat University, 23119, Elazig, Turkey. Mustafa Inc - Department of Mathematics, Science Faculty, Firat University, 23119, Elazig, Turkey. Maysaa’ Mohamed Al Qurashi - Department of Mathematics, King Saud University, P. O. Box 22452, Riyadh, 11495, Saudi Arabia. Dumitru Baleanu - Department of Mathematics, Cankaya University, 06530, Balgat, Ankara, Turkey.

Abstract

In this study, we obtain the approximate soliton solution of the fractional generalized Hirota-Satsuma coupled Kortewegde Vries equation (GHS-cKdV) within the homotopy analysis method (HAM). Numerical results are successfully compared with other solutions obtained by the differential transform method (DTM) and the homotopy perturbation method (HPM). The numerical results indicate that the only few terms are sufficient to get the correct solutions. Also, the results are given by tables and figures.

Share and Cite

Keywords

• HAM
• fractional partial differential equation(FPD)
• HS-cKdV equation
• time-fractional GHS-cKdV equation.

MSC

•  34A08
•  26A33

References

• [1] R. Abazari, M. Abazari , Numerical simulation of generalized Hirota-Satsuma coupled KdV equation by RDTM and comparison with DTM, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 619–629.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [2] S. Abbasbandy, Soliton solutions for the Fitzhugh-Nagumo equation with the homotopy analysis method, Appl. Math. Model., 32 (2008), 2706–2714.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [3] S. Abbasbandy, E. Shivanian, Series solution of the system of integro-differential equations, Z. Naturforsch. A, 64 (2009), 811–818.
  • View Article
  • Google Scholar


• [4] D. Baleanu, O. P. Agrawal, Fractional Hamilton formalism within Caputo’s derivative, Czechoslovak J. Phys., 56 (2006), 1087–1092.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [5] Z. Z. Ganji, D. D. Ganji, Y. Rostamiyan, Solitary wave solutions for a time-fraction generalized Hirota-Satsuma coupled KdV equation by an analytical technique, Appl. Math. Model., 33 (2009), 3107–3113.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [6] C. S. Gardner, J. M. Greene, M. D. Kruskal, R. M. Miura, Korteweg-deVries equation and generalization. VI. Methods for exact solution, Comm. Pure Appl. Math., 27 (1974), 97–133.
  • View Article
  • MathSciNet
  • Google Scholar


• [7] A. K. Golmankhaneh, A. Golmankhaneh, D. Baleanu, On nonlinear fractional Klein–Gordon equation, Signal Process., 91 (2011), 446–451.
  • Google Scholar


• [8] A. K. Golmankhaneh, N. A. Porghoveh, D. Baleanu, Mean square solutions of second-order random differential equations by using homotopy analysis method, Rom. Rep. Phys., 65 (2013), 350–362.
  • Google Scholar


• [9] R. Hirota, Exact solution of the Korteweg—de Vries equation for multiple collisions of solitons, Phys. Rev. Lett., 27 (1971), 1192–1194.
  • View Article
  • Google Scholar
  • MATH


• [10] R. Hirota, J. Satsuma, Soliton solutions of a coupled Korteweg-de Vries equation, Phys. Lett. A, 85 (1981), 407–408.
  • View Article
  • MathSciNet
  • Google Scholar


• [11] M. Inc, On exact solution of Laplace equation with Dirichlet and Neumann boundary conditions by the homotopy analysis method, Phys. Lett. A, 365 (2007), 412–415.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [12] S.-J. Liao, Beyond perturbation, Introduction to the homotopy analysis method, CRC Series: Modern Mechanics and Mathematics, Chapman & Hall/CRC, Boca Raton, FL (2003)
  • Google Scholar


• [13] Y.-P. Liu, Z.-B. Li, The homotopy analysis method for approximating the solution of the modified Korteweg-de Vries equation, Chaos Solitons Fractals, 39 (2009), 1–8.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [14] J.-C. Liu, H. Li, Approximate analytic solutions of time-fractional Hirota-Satsuma coupled KdV equation and coupled MKdV equation, Abstr. Appl. Anal., 2013 (2013 ), 11 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [15] H. Jafari, S. Das, H. Tajadodi, Solving a multi-order fractional differential equation using homotopy analysis method, J. King Saud Univ. Sci., 23 (2011), 151–155.
  • View Article
  • Google Scholar


• [16] H. Jafari, S. Seifi, Homotopy analysis method for solving linear and nonlinear fractional diffusion-wave equation, Commun. Nonlinear Sci. Numer. Simul., 14 (2009), 2006–2012.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [17] S. Momani, Z. Odibat, A. Alawneh, Variational iteration method for solving the space- and time-fractional KdV equation, Numer. Methods Partial Differential Equations, 24 (2008), 262–271.
  • View Article
  • Google Scholar
  • MATH


• [18] I. Podlubny, Fractional differential equations, An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, Academic Press, Inc., San Diego, CA (1999)


• [19] M. Shateri, D. D. Ganji, Solitary wave solutions for a time-fraction generalized Hirota-Satsuma coupled KdV equation by a new analytical technique, Int. J. Differ. Equ., 2010 (2010 ), 10 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [20] X.-J. Yang, Advanced Local Fractional Calculus and Its Applications, World Sci., NewYork, USA (2012)
  • Google Scholar


• [21] X.-J. Yang, D. Baleanu, Y. Khan, S. T. Mohyud-Din, Local fractional variational iteration method for diffusion and wave equations on Cantor sets, Romanian J. Phys., 59 (2014), 36–48.
  • MathSciNet
  • Google Scholar


• [22] X.-J. Yang, D. Baleanu, H. M. Srivastava, Local fractional integral transforms and their applications, Elsevier/Academic Press, Amsterdam (2016)
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [23] X.-J. Yang, H. M. Srivastava, J.-H. He, D. Baleanu, Cantor-type cylindrical-coordinate method for differential equations with local fractional derivatives, Phys. Lett. A, 377 (2013), 1696–1700.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH