Honam Mathematical Journal (호남수학학술지)

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NUMERICAL SOLUTION OF THE NONLINEAR KORTEWEG-DE VRIES EQUATION BY USING CHEBYSHEV WAVELET COLLOCATION METHOD

  • BAKIR, Yasemin (Department of Computer Engineering, Dogus University)
  • Received : 2020.12.07
  • Accepted : 2021.06.17
  • Published : 2021.09.25
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Abstract

In this study, a numerical method deals with the Chebyshev wavelet collocation and Adomian decomposition methods are proposed for solving Korteweg-de Vries equation. Integration of the Chebyshev wavelets operational matrices is derived. This problem is reduced to a system of non-linear algebraic equations by using their operational matrix. Thus, it becomes easier to solve KdV problem. The error estimation for the Chebyshev wavelet collocation method and ADM is investigated. The proposed method's validity and accuracy are demonstrated by numerical results. When the exact and approximate solutions are compared, for non-linear or linear partial differential equations, the Chebyshev wavelet collocation method is shown to be acceptable, efficient and accurate.

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References

  1. A. Ali, M.A. Iqbal, S.T. Mohyud-Din, Chebyshev wavelets method for boundary value problems, Scientific Research and Essays, (2013), 8.46: 2235-2241.
  2. J. Aminuddin, Numerical Solution of The Korteweg de Vries Equation, Int. J. of Basic and Appl. Sciences IJBAS-IJENS, (2011), 11, 2.
  3. I. Aziz, B. Sarler, The numerical solution of second-order boundary value problems by collocation method with the Haar wavelets, Math Comput Model, (2010), 52:1577-90. https://doi.org/10.1016/j.mcm.2010.06.023
  4. E. Babolian, F. Fattahzadeh, Numerical solution of differential equations by using Chebyshev wavelet operational matrix of integration, Applied Mathematics and Computation, (2007), 188:417-426. https://doi.org/10.1016/j.amc.2006.10.008
  5. I. Celik, Numerical solution of differential equations by using Chebyshev wavelet collocation method, Cankaya University Journal of Science and Engineering, (2013), 10(2): 169-184.
  6. I. Daubechies, Ten Lectures on Wavelets, SIAM: Philadelphia, PA., 1992.
  7. A. Farouk, Numerical Solution for Non-linear Korteweg-de Vries-Burger's Equation Using the Haar Wavelet Method, Iraqi Journal of Statistical Science, (2011), 20, pp [93-110].
  8. L. Fox, I.B. Parker, Chebyshev Polynomials in Numerical Analysis, Oxford University Press, London, 1968.
  9. M.W. Frazier, An introduction to wavelets through linear algebra, Springer Science & Business Media, 2006.
  10. M.R. Hooshmandasl, M.H. Heydari, F.M.M. Ghaini, Numerical Solution of the One-Dimention Heat Equation by Using Chebyshev Wavelets Method, Applied &Computational Mathematics, (2012), 2168-9676.
  11. A. Kilicman, Z.A.A. Al Zhour, Kronecker operational matrices for fractional calculus and some applications, Applied Mathematics and Computation, (2007), 187.1: 250-265. https://doi.org/10.1016/j.amc.2006.08.122
  12. U. Lepik, Application of the Haar wavelet transform to solving integral and differential equations, Proc Estonian Acad Sci Phys Math, (2007), 56(1):28-46. https://doi.org/10.3176/phys.math.2007.1.03
  13. W.H. Luo, T.Z. Huang, X.M. Gu, Y. Liu Barycentric rational collocation methods for a class of nonlinear parabolic partial differential equations, Applied Mathematics Letters, Volume 68, June 2017, Pages 13-19. https://doi.org/10.1016/j.aml.2016.12.011
  14. F. Saadi, M.J. Azizpour, S.A. Zahedi, Analytical Solutions of Kortweg-de Vries (KdV) Equation, World Academy of Science, Engineering and Technology, (2010),9: 171-175.
  15. J. Villegas, J. Castano, J. Duarte, E. Fierro, Wavelet-Petrov-Galerkin Method for the Numerical Solution of the KdV Equation, Appl. Math. Sci, (2012), 6,69, 3411 - 3423.
  16. L.I. Yuanlu, Solving a nonlinear fractional differential equation using Chebyshev wavelets, Commun Nonlinear Sci Numer Simulat, (2010), 2284-2292.
  17. A.M. Wazwaz, Construction of solitary wave solutions and rational solutions for the KdV equation by Adomian decomposition method, Chaos, Solitons & Fractals 12.12 (2001): 2283-2293. https://doi.org/10.1016/S0960-0779(00)00188-0
  18. JF. Wang, S. Feng-Xin, C. Rong-Jun, Element-free Galerkin method for a kind of KdV equation, Chin. Phys. B, 19.6 (2010).