A numerical method for KdV equation using rational radial basis functions

Document Type : Research Paper

Authors

1 Department of Mathematics, Payame Noor University (PNU), Tehran, Iran.

2 Department of Applied Mathematics, University of Kurdistan, Sanandaj, Iran.

3 Faculty of Mathematical Sciences and Computer, Kharazmi University, Tehran, Iran.

Abstract

In this paper, we use the rational radial basis functions ( RRBFs) method to solve the Korteweg-de Vries (KdV) equation, particularly when the equation has a solution with steep front or sharp gradients. We approximate the spatial derivatives by RRBFs method then we apply an explicit fourth-order Runge-Kutta method to advance the resulting semi-discrete system in time. Numerical examples show that the presented scheme preserves the conservation laws and the results obtained from this method are in good agreement with analytical solutions. 

Keywords

References

  • [1] E. N. Aksan and A. O¨ zdes, Numerical solution of Korteweg–de Vries equation by Galerkin B-spline finite element method, Appl. Math. Comput., 175 (2006), 1256-1265.
  • [2] A. Bashan, An effective application of differential quadrature method based on modified cubic B-splines to numer- ical solutions of the KdV equation, Turk. J. Math., 42 (2018), 373-394.
  • [3] A. Bashan and A. Esen, Single soliton and double soliton solutions of the quadratic-nonlinear Korteweg-de Vries equation for small and long-times, Numer. Methods Partial Differential Equations, (2020), 1-22.
  • [4] R. C. Cascaval, Variable coefficient KdV equations and waves in elastic tubes, Lect. Notes Pure Appl. Math., 234 (2003), 57-70.
  • [5] D. G. Crighton, Applications of KdV, Acta Appl. Math., 39 (1995), 39-67.
  • [6] I. Da˘g and Y. Dereli, Numerical solutions of KdV equation using radial basis functions, Appl. Math. Model., 32 (2008), 535-546.
  • [7] M. Dehghan and A. Shokri, A numerical method for KdV equation using collocation and radial basis functions, Nonlinear Dyn., 50 (2007), 111-120.
  • [8] S. De Marchi, A. Martinez, and E. Perracchione, Fast and stable rational RBF-based partition of unity interpola- tion, J. Comput. Appl. Math., 349 (2019), 331-343.
  • [9] A. Dur´an and M. A. Lopez-Marcos, Conservative numerical methods for solitary wave interactions, J. Phys. A., 36 (2003), 7761-7770.
  • [10] G. E. Fasshauer, Meshfree Approximation Methods with Matlab, World Scientific, 2007.
  • [11] B. Fornberg and G. B. Whitham, A numerical and theoretical study of certain nonlinear wave phenomena, Philos. Trans. Roy. Soc., 289 (1978), 373-404.
  • [12] C. S. Gardner and G. K. Morikawa, Similarity in the Asymptotic Behavior of Collision-free Hydromagnetic Waves and Water Waves, New York Univ., Inst. of Mathematical Sciences, Technical Report, 1960.
  • [13] K. Goda, On instability of some finite difference schemes for Korteweg-de Vries equation, J. Phys. Soc. Japan, 39 (1975), 229-236.
  • [14] S. Y. Hao, S. S. Xie, and S. C. Yi, The Galerkin method for the KdV equation using a new basis of smooth piecewise cubic polynomials, Appl. Math. Comput., 218 (2012), 8659-8671.
  • [15] M. Heidari, M. Mohammadi, and S. De Marchi, A shape preserving quasi-interpolation operator based on a new transcendental RBF, Dolomites Research Notes on Approximation, 14(1) (2021), 56-73.
  • [16] S. Jakobsson, B. Andersson, and F. Edelvik, Rational radial basis function interpolation with applications to antenna design, J. Comput. Appl. Math., 233 (2009), 889-904.
  • [17] A. Jeffrey and T. Kakutani, Weak nonlinear dispersive waves: a discussion centered around the Korteweg–de Vries equation, SIAM Rev., 14(4) (1972), 582-643.
  • [18] S. B. G. Karakoc and K. K. Ali, New exact solutions and numerical approximations of the generalized KdV equation, Comput. Methods Differ. Equ., 9(3) (2021), 670-691.
  • [19] D. Kong, Y. Xu, and Z. Zheng, A hybrid numerical method for the KdV equation by finite difference and sinc collocation method, Appl. Math. Comput., 355 (2019) 61-72.
  • [20] D. J. Korteweg and G. De Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philos. Mag. 39(240) (1895), 422-443.
  • [21] M. Lakestani, Numerical solutions of the KdV equation using B-spline functions, Iran. J. Sci. Technol. Trans. A Sci., 41 (2017), 409-417.
  • [22] M. Mohammadi, R. Mokhtari, and R. Schaback, A meshless method for solving the 2d brusselator reaction- diffusion system, Comput. Model., Eng. Sci., 101(2014), 113–138.
  • [23] R. Najafi, Approximate nonclassical symmetries for the time-fractional KdV equations with the small parameter, Comput. Methods Differ. Equ., 8(1) (2020), 111-118.
  • [24] S. L. Naqvi, J. Levesley, and S. Ali, Adaptive radial basis function for time dependent partial differential equations, J. Prime Res. Math., 13 (2017), 90-106.
  • [25] S. Niknam and H. Adibi, A numerical solution of two-dimensional hyperbolic telegraph equation based on moving least square meshless method and radial basis functions, Comput. Methods Differ. Equ., 10(4) (2022), 969-985.
  • [26] O¨ . Oru¸c, F. Bulut, and A. Esen, Numerical solution of the KdV equation by Haar wavelet method, Pramana-J. Phys., 87(94) (2016), 1-11.
  • [27] B. Saka, Cosine expansion-based differential quadrature method for numerical solution of the KdV equation, Chaos Solitons Fractals, 40 (2009), 2181-2190.
  • [28] S. A. Sarra and Y. Bai, A rational radial basis function method for accuratly resolving discontinuities and steep gradients, Appl. Numer. Math., 130 (2018), 131-142.
  • [29] R. Schaback, Kernel-Based Meshless Methods, Gottingen, 2011.
  • [30] M. Shiralizadeh, A. Alipanah, and M. Mohammadi, Numerical solution of one-dimensional Sine-Gordon equation using rational radial basis functions, J. Math. Model., 10 (2022), 1-16.
  • [31] L. Van Wijngaarden, On the equations of motion for mixtures of liquid and gas bubbles, J. Fluid Mech., 33(3) (1968), 465-474.
  • [32] A. C. Vliengenthart, On finite difference methods for the Korteweg–de Vries equation, J. Eng. Math., 5 (1971), 137-155.
  • [33] H. Washimi and T. Taniuti, Propagation of ion-acoustic solitary waves of small amplitude, Phys. Rev. Lett., 17 (1966), 996-998.
  • [34] H. Wendland, Scattered data approximation, Cambridge University Press, 2004.
  • [35] N. J. Zabusky, A Synergetic Approach to Problem of Nonlinear Dispersive Wave Propagation and Interaction, in: W. Ames (Ed.), Proc. Symp. Nonlinear Partial Diff. Equations, Academic Press, (1967), 223-258.
Computational Methods for Differential Equations
Volume 11, Issue 2
April 2023
Pages 303-318

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History

  • Receive Date: 06 June 2022
  • Revise Date: 23 October 2022
  • Accept Date: 30 October 2022

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