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On the numerical evaluation for studying the fractional KdV, KdV-Burgers and Burgers equations

  • M. M. Khader
  • Khaled M. SaadEmail author
Regular Article

Abstract.

This paper is devoted to present an accurate numerical procedure to solve fractional (Caputo sense) Korteweg-de Vries, Korteweg-de Vries-Burgers and Burgers equations by using the spectral Chebyshev collocation method and finite difference method (FDM). The proposed problem is reduced to a system of ODEs with the help of the properties of Chebyshev polynomials of the third kind. This system is solved by using the FDM. Some theorems about the convergence analysis are stated and proved. A numerical simulation and a comparison with the previous work are presented.

References

  1. 1.
    K. Diethelm, Electron. Trans. Numer. Anal. 5, 1 (1997)MathSciNetGoogle Scholar
  2. 2.
    M. Inc, J. Math. Anal. Appl. 345, 476 (2008)MathSciNetCrossRefGoogle Scholar
  3. 3.
    I. Podlubny, Fractional Differential Equations (Academic Press, New York, 1999)Google Scholar
  4. 4.
    J.P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd edition (Dover, New York, NY, USA, 2000)Google Scholar
  5. 5.
    M.M. Khader, N.H. Sweilam, Appl. Math. Model. 37, 9819 (2013)MathSciNetCrossRefGoogle Scholar
  6. 6.
    M.M. Khader, Commun. Nonlinear Sci. Numer. Simul. 16, 2535 (2011)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    O. Abdulaziz, I. Hashim, E.S. Ismail, Math. Comput. Model. 49, 136 (2009)CrossRefGoogle Scholar
  8. 8.
    W.L. Kath, N.F. Smyth, Phys. Rev. E 51, 661 (1995)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    N.J. Zabusky, M.D. Kruskal, Phys. Rev. Lett. 15, 240 (1965)ADSCrossRefGoogle Scholar
  10. 10.
    R.S. Johnson, J. Fluid Mech. 42, 49 (1970)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Z. Feng, Nonlinearity 20, 343 (2007)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    G.B. Whitham, Linear and Nonlinear Waves (John Wiley and Sons Inc., New York, 1974)Google Scholar
  13. 13.
    Y. Zhang, D. Baleanu, X.J. Yang, Proc. Romanian Acad. 17, 230 (2016)Google Scholar
  14. 14.
    C. Lubich, SIAM J. Math. Anal. 17, 704 (1986)MathSciNetCrossRefGoogle Scholar
  15. 15.
    M.A. Snyder, Chebyshev Methods in Numerical Approximation (Prentice-Hall, Inc. Englewood Cliffs, N. J., 1966)Google Scholar
  16. 16.
    J.C. Mason, D.C. Handscomb, Chebyshev Polynomials (Chapman and Hall, CRC, New York, NY, Boca Raton, 2003)Google Scholar
  17. 17.
    N.H. Sweilam, A.M. Nagy, A. El-Sayed, J. King Saud Univ. Sci. 28, 41 (2016)CrossRefGoogle Scholar
  18. 18.
    C.Y. Handan, J. Eng. Technol. Appl. Sci. 2, 1 (2017)Google Scholar
  19. 19.
    K. Parand, M. Delkhosh, Gazi Univ. J. Sci. 29, 895 (2016)Google Scholar
  20. 20.
    M. Safari, D.D. Ganji, M. Moslemi, Comput. Math. Appl. 58, 2091 (2009)MathSciNetCrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Statistics, College of ScienceAl Imam Mohammad Ibn Saud Islamic University (IMSIU)RiyadhSaudi Arabia
  2. 2.Department of Mathematics, Faculty of ScienceBenha UniversityBenhaEgypt
  3. 3.Department of Mathematics, College of Arts and SciencesNajran UniversityNajranSaudi Arabia
  4. 4.Department of Mathematics, Faculty of Applied ScienceTaiz UniversityTaizYemen

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