Discrete Singular Convolution Method for Numerical Solutions of Fifth Order Korteweg-De Vries Equations


A new computational method for solving the fifth order Korteweg-de Vries (fKdV) equation is proposed. The nonlinear partial differential equation is discretized in space using the discrete singular convolution (DSC) scheme and an exponential time integration scheme combined with the best rational approximations based on the Carathéodory-Fejér procedure for time discretization. We check several numerical results of our approach against available analytical solutions. In addition, we computed the conservation laws of the fKdV equation. We find that the DSC approach is a very accurate, efficient and reliable method for solving nonlinear partial differential equations.

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Pindza, E. and Maré, E. (2013) Discrete Singular Convolution Method for Numerical Solutions of Fifth Order Korteweg-De Vries Equations. Journal of Applied Mathematics and Physics, 1, 5-15. doi: 10.4236/jamp.2013.17002.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] P. D. Lax, “Integrals of Nonlinear Equations of Evolution and Solitary Waves,” Communications on Pure and Applied Mathematics, Vol. 21, No. 5, 1968, pp. 467-490.
[2] K. Sawada and T. Kotera, “A Method for Finding Soliton Solutions for the KdV Equation and KdV-Like Equations,” Progress of Theoretical Physics, Vol. 51, No. 5, 1974, pp. 1355-1367.
[3] A. P. Fordy and J. Gibons, “Some Remarkable Nonlinear Transformations,” Physics Letters, Vol. A75, No. 5, 1980, p. 325. http://dx.doi.org/10.1016/0375-9601(80)90829-4
[4] M. Ito, “An Extension of Nonlinear Evolution Equations of the KdV (mKdV) Type to Higher Orders,” Journal of the Physical Society of Japan, Vol. 49, 1980, pp. 771-778.
[5] J. Shen, “A New Dual-Petrov-Galerkin Method for Third and Higher Odd-Order Differential Equations: Application to the KdV Equation,” SIAM Journal on Numerical Analysis, Vol. 41, No. 5, 2003, pp. 1595-1619.
[6] J. Shen and L. L. Wang, “Laguerre and Composite Legendre-Laguerre Dual-Petrov-Galerkin Methods for ThirdOrder Equations,” DCDS-B, Vol. 6, No. 6, 2006, pp. 1381-1402. http://dx.doi.org/10.3934/dcdsb.2006.6.1381
[7] J. M. Yuan, J. Shen and J. Wu, “A dual-Petrov-Galerkin Method for the Kawahara-Type Equation,” Journal of Scientific Computing, Vol. 34, No. 1, 2008, pp. 48-63.
[8] G. W. Wei, “Discrete Singular Convolution for the Fokker-Planck Equation,” Journal of Chemical Physics, Vol. 110, 1999, pp. 8930-8942.
[9] W. Bao, F. Sun and G. W. Wei, “Numerical Methods for the Generalized Zakharov System,” Journal of Computational Physics, Vol. 190, No. 1, 2003, pp. 201-228.
[10] G. W. Wei, Y. B. Zhao and Y. Xiang, “Discrete Singular Convolution and Its Application to the Analysis of Plates with Internal Supports. Part 1: Theory and Algorithm,” International Journal Numerical Methods in Engineering, Vol. 55, No. 8, 2002, pp. 913-946.
[11] G. W. Wei, “Vibration Analysis by Discrete Singular Convolution,” Journal of Sound Vibration, Vol. 244, No. 3, 2001, pp. 535-553.
[12] G. W. Wei, “A New Algorithm for Solving Some Mechanical Problems,” Computational Methods in Applied Mechanical Engineering, Vol. 190, No. 15, 2001, pp. 2017-2030. http://dx.doi.org/10.1016/S0045-7825(00)00219-X
[13] Y. C. Zhou and G. W. Wei, “High-Resolution Conjugate Filters for the Simulation of Flows,” Journal of Computational Physics, Vol. 189, No. 1, 2003, pp. 150-179.
[14] G. Bao, G. W. Wei and S. Zhao, “Numerical Solution of the Helmholtz Equation with High Wave Numbers,” International Journal of Numerical Methods in Engineering, Vol. 59, No. 3, 2004, pp. 389-408.
[15] G. Bao, G. W. Wei and S. Zhao, “Local Spectral Time-Domain Method for Electromagnetic Wave Propagation,” Optic Letters, Vol. 28, No. 7, 2003, pp. 513-515.
[16] Z. J. Hou and G. W. Wei, “A New Approach for Edge Detection,” Pattern Recognition, Vol. 35, No. 7, 2002, pp. 1559-1570.
[17] E. Pindza and E. Maré, “Discrete Singular Convolution and Exponential Time Integrators for Solving the Generalized Korteweg-de Vries Equation,” Technical Report UPWT 2013/14, University of Pretoria, Pretoria.
[18] S. Y. Yang, Y. C. Zhou and G. W. Wei, “Comparison of the Discrete Singular Convolution Algorithm and the Fourier Pseudospectral Methods for Solving Partial Differential Equations,” Computer Physics Communications, Vol. 143, No. 2, 2002, pp. 113-135.
[19] G. W. Wei, Y. B. Zhao and Y. Xiang, “A Novel Approach for the Analysis of High Frequency Vibrations,” Journal of Sound and Vibration, Vol. 257, No. 2, 2002, pp. 207-246. http://dx.doi.org/10.1006/jsvi.2002.5055
[20] A. K. Kassam and L. N. Trefethen, “Fourth-Order Time Stepping for Stiff PDEs,” SIAM Journal of Scientific Computing, Vol. 26, No. 4, 2005, pp. 1214-1233.
[21] L. N. Trefethen and H. M. Gutknecht, “The Carathéodory-Fejér Method for Real Rational Approximation,” SIAM Journal on Numerical Analysis, Vol. 20, No. 2, 1983, pp. 420-436. http://dx.doi.org/10.1137/0720030
[22] G. W. Wei, “Discrete Singular Convolution for the Sine-Gordon Equation,” Physica D, Vol. 137, No. 3, 2000, pp. 247-259.
[23] L. W. Qian, “On the Regularized Whittaker-Kotel’nikov-Shannon Sampling Formula,” Proceedings of American Mathematical Society, Vol. 131, No. 4, 2003, pp. 1169-1176. http://dx.doi.org/10.1090/S0002-9939-02-06887-9
[24] R. X. Yao, C. Z. Qu and Z. B. Li, “On Properties of New Parameterized 5th-Order Nonlinear Evolution Equation,” Chaos, Solitons and Fractals, Vol. 21, No. 5, 2004, pp. 1145-1152. http://dx.doi.org/10.1016/j.chaos.2003.12.078
[25] E. Pindza, “Spectral Difference Methods for Solving Equation of the KdV Hierarchy,” MSc Thesis, University of Stellenbosch, Stellenbosch, 2008.
[26] B. Minchev and W. Wright, “A Review of Exponential Integrators for First Order Semi-Linear Problems,” Technical Report 2, The Norwegian University of Science and Technology, 2005.
[27] S. M. Cox and P. C. Matthews, “Exponential Time Differencing for Stiff Systems,” Journal of Computational Physics, Vol. 176, No. 2, 2002, pp. 430-455.
[28] Y. Saad, “Analysis of Some Krylov Subspace Approximations to the Matrix Exponential Operator,” SIAM Journal of Numerical Analysis, Vol. 29, No. 1, 1992, pp. 209-228.
[29] M. Hochbruck and C. Lubich, “On Krylov Subspace Approximations to the Matrix Exponential Operator,” SIAM Journal of Numerical Analysis, Vol. 34, No. 5, 1997, pp. 1911-1925.
[30] T. Schmelzer and L. N. Trefethen, “Evaluating Matrix Functions for Exponential Integrators via CarathéodoryFejér Approximation and Contour Integrals,” Electronic Transactions on Numerical Analysis, Vol. 29, 2007, pp. 1-18.
[31] A. Nuseir, “Symbolic Computation of Exact Solutions of Nonlinear Partial Differential Equations Using Direct Methods,” PhD Thesis, Colorado School of Mines, 1997.

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