International Journal of Analysis and Applications ISSN 2291-8639 (Online)

Research Article

Exact Solutions of Kupershmidt Equation, Approximate Solutions for Time-Fractional Kupershmidt Equation: A Comparison Study

  • Medjahed Djilali1
  • Ali Hakem1
  • Abdelkader Benali2*
  1. Laboratory ACEDP, Djillali Liabes University, 22000 SIDI-BEL-ABBES, Algeria.
  2. Faculty of The Exact sciences And computer, Mathematics Department, University of Hassiba Benbouali, B.P. 151 Hay Essalem, Chlef 02000, Algeria

* Corresponding author

Abstract

In this article, a technique namely Tanh method is applied to obtain some traveling wave solutions for Kupershmidt equation, and by using LADM we obtain an approximate solution to timefractional Kupershmidt equation.

A comparison between the traveling wave solution (exact solution) and the approximate one of equation under study, indicate that Laplace Adomian Decomposition Method (LADM) is highly accurate and can be considered a very useful and valuable method.

How to Cite

M. Djilali, A. Hakem, A. Benali, Exact Solutions of Kupershmidt Equation, Approximate Solutions for Time-Fractional Kupershmidt Equation: A Comparison Study, Int. J. Anal. Appl. 18 (2020), 493–512. https://doi.org/10.28924/2291-8639-18-2020-493

BibTeX
@article{Djilali2020,
  author  = {Djilali, Medjahed and Hakem, Ali and Benali, Abdelkader},
  title   = {Exact Solutions of Kupershmidt Equation, Approximate Solutions for Time-Fractional Kupershmidt Equation: A Comparison Study},
  journal = {Int. J. Anal. Appl.},
  volume  = {18},
  year    = {2020},
  pages   = {493--512},
  doi     = {10.28924/2291-8639-18-2020-493}
}

References

  • [1] A. Abdelrazec, D. Pelinovsky , Convergence of the Adomian Decomposition Method for Initial-Value Problems, Numer. Methods Partial Differ. Equations, 27 (2011), 749-766.
  • [2] G. Adomian , A Review of the Decomposition Method in Applied Mathematics , J. Math. Anal. Appl. 135 (1988), 501-544.
  • [3] G. Adomian, System of nonlinear partial differential equations , J. Math. Anal. Appl. 115 (1) (1986), 235-238.
  • [4] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publication, Boston, 1994.
  • [5] E. Babolian a, J. Biazar b, A.R. Vahidi, A new computational method for Laplace transforms by decomposition method, Appl. Math. Comput. 150 (2004), 841-846.
  • [6] E. Babolian, S. Javadi , New method for calculating Adomian polynomials , Appl. Math. Comput. 153 (2004), 253-259.
  • [7] K. Charalambous , A. K. Halder and Peter G. L. Leach, A note on analysis of the Kaup-Kupershmidt equation, AIP Conf. Proc. 2153 (2019), 020006.
  • [8] J. Fadaei, Application of Laplace-Adomian Decomposition Method on Linear and Nonlinear System of PDEs, Appl. Math. Sci. 5 (27) (2011), 1307 - 1315.
  • [9] E. Fan, Uniformly constructing a series of explicit exact solutions to nonlinear equations in mathematical physics, Chaos Solitons Fractals, 16 (2003), 819-839.
  • [10] E. G. Fan, Traveling Wave Solutions for Nonlinear Equations Using Symbolic Computation, Computers Math. Appl. 42 (6-7) (2002), 671-680.
  • [11] A. Fordy and J. Gibbons, Some Remarkable Nonlinear Transfomations, Phys. Lett. A, 75 (5) (1980), 325.
  • [12] B. Fuchssteiner, W. Oevel, The bi-Hamiltonlan structure of some nonlinear fifth- and seventh-order differential equatlons and recursion formulas for their symmetries and conserved covarlants, J. Math. Phys. 23 (1982), 358-363.
  • [13] B. Fuchssteiner, W. Oevel and W. Wiwianka, Computer-Algebra Methods For Investigation Of Hereditary Operators Of Higher Order Soliton Equations, Computer Phys. Commun. 44 (1987), 47-55.
  • [14] O. G-Gaxiola · J. R. Chávez, R. B-Jaquez, Solution of the Nonlinear Kompaneets Equation Through the Laplace-Adomian Decomposition Method, Int. J. Appl. Comput. Math. 3 (2017), 489-504.
  • [15] M. A. Helal, M. S. Mehanna , The tanh method and Adomian decomposition method for solving the foam drainage equation, Appl. Math. Comput. 190 (2007), 599-609.
  • [16] W. Hereman , A. Nuseir , Symbolic methods to construct exact solutions of nonlinear partial differential equations , Math. Computers Simul. 43 (1997), 13-27.
  • [17] X. B. Hu , D. L. Wang and X. M. Qian , Soliton solutions and symmetries of the 2 + 1 dimensional Kaup-Kupershmidt equation, Phys. Lett. A, 262 (1999), 409-415.
  • [18] M. Hussain and M. Khan, Modified Laplace Decomposition Method, Appl. Math. Sci. 36 (4) (2010), 1769 - 1783.
  • [19] M. Inc, On numerical soliton solution of the Kaup-Kupershmidt equation and convergence analysis of the decomposition method, Appl. Math. Comput. 172 (2006), 72-85.
  • [20] K. Jaradat , D. ALoqali, W. Alhabashene, Using Laplace Decomposition Method to Solve Nonlinear Klien-Gordan Equation, U.P.B. Sci. Bull., Ser. D, 80 (2) (2018), 213-222.
  • [21] K. Khan, M. Ali Akbar, A. H. Arnous, Exact traveling wave solutions for system of nonlinear evolution equations, Springer Plus, 5 (2016), 663.
  • [22] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006.
  • [23] S. Kumar & A. Yildirim & Y. Khanc & L. Weid, A fractional model of the diffusion equation and its analytical solution using Laplace transform Sci. Iran. 19 (4) (2012), 1117-1123.
  • [24] W. Malfliet, The tanh method: a tool for solving certain classes of nonlinear evolution and wave equations, J. Comput. Appl. Math. 164-165 (2004), 529-541.
  • [25] W. Malfliet, The tanh method: A tool for solving certain classes of non-linear PDEs, Math. Meth. Appl. Sci. 28 (2005), 2031-2035.
  • [26] Z. M. Odibat, S. Momani, Approximate solutions for boundary value problems of time-fractional wave equation, Appl. Math. Comput. 181 (2006), 767-774.
  • [27] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974.
  • [28] A. Parker, On soliton solutions of the Kaup-Kupershmidt equation. II. 'Anomalous' N-soliton solutions, Physica D 137 (2000), 34-48.
  • [29] I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1999.
  • [30] G. Qingling, Exact Solutions of the mBBM Equation, Appl. Math. Sci. 5 (25) (2011), 1209 - 1215.
  • [31] S. S. Ray, R. K. Bera, An approximate solution of nonlinear fractional differential equation by Adomians decomposition method, Appl. Math. Comput. 167 (2005), 561-571.
  • [32] E. G. Reyes, Nonlocal symmetries and the Kaup-Kupershmidt equation, J. Math. Phys. 46 (2005), 073507.
  • [33] S. S. Ray, R. K. Bera , Analytical solution of a fractional diffusion equation by Adomian decomposition method, Appl. Math. Comput. 174 (2006), 329-336.
  • [34] A. H. Salas, Solving the Generalized Kaup-Kupershmidt Equation, Adv. Studi. Theor. Phys. 6 (18) (2012), 879 - 885.
  • [35] J. L. Schiff, The Laplace Tranform, Theory and Applications, Springer-Verlag, New York, 1999.
  • [36] M. R. Spiegel, Laplace Tranforms, McGraw-Hill, New York, 1965.
  • [37] T. Sumbal Shaikh, N. Ahmed, N. Shahid, Z. Iqbal, Solution of the Zabolotskaya-Khokholov Equation by Laplace Decomposition Method, Int. J. Sci. Eng. Res. 9 (2) (2018), 1811-1816.
  • [38] P. Wang, Bilinear form and soliton solutions for the fifth-order Kaup-Kupershmidt, Mod. Phys. Lett. B, 31 (6) (2017), 1750057.
  • [39] P. Wang and S. H. Xiao, Soliton solutions for the fifth-order Kaup-Kupershmidt equation, Phys. Scr. 93 (10), 105201.
  • [40] A. M. Wazwaz, The combined Laplace transform-Adomian decomposition method for handling nonlinear Volterra integrodifferential equations, Appl. Math. Comput. 216 (4) (2010), 1304-1309.
  • [41] A. M. Wazwaz, A new algorithm for calculating adomian polynomials for nonlinear operators , Appl. Math. Comput. 111 (2000), 53-69.
  • [42] A. M. Wazwaz, Partial Differential Equations and Solitary Waves Theory, Higher Education Press, Berlin (2009).
  • [43] L. Yan, Numerical Solutions of Fractional Fokker-Planck Equations Using Iterative Laplace Transform Method, Abstr. Appl. Anal. 2013(2013), Article ID 465160.
  • [44] Q. Yu , F. Liu, V. Anh and I. Turner, Solving linear and non-linear space-time fractional reaction-diffusion equations by the Adomian decomposition method, Int. J. Numer. Methods Eng. 74 (2008), 138-158.
  • [45] S. A. Zarea, The tanh method: A tool for solving some mathematical models, Chaos Solitons Fractals, 41 (2009), 979-988.