Hostname: page-component-745bb68f8f-kw2vx Total loading time: 0 Render date: 2025-01-10T08:09:38.988Z Has data issue: false hasContentIssue false
  • English
  • Français

Legendre-Gauss Spectral Collocation Method for Second Order Nonlinear Delay Differential Equations

Published online by Cambridge University Press:  28 May 2015

Lijun Yi  and
Zhongqing Wang
Lijun Yi*
Affiliation:
Department of Mathematics, Shanghai Normal University, Shanghai 200234, China Division of Computational Science of E-institute of Shanghai Universities, Shanghai 200234, China
Zhongqing Wang*
Affiliation:
Department of Mathematics, Shanghai Normal University, Shanghai 200234, China Division of Computational Science of E-institute of Shanghai Universities, Shanghai 200234, China
*
Corresponding author.Email address:[email protected]
Corresponding author.Email address:[email protected]
Get access

Abstract

In this paper, we present and analyze a single interval Legendre-Gauss spectral collocation method for solving the second order nonlinear delay differential equations with variable delays. We also propose a novel algorithm for the single interval scheme and apply it to the multiple interval scheme for more efficient implementation. Numerical examples are provided to illustrate the high accuracy of the proposed methods.

Keywords

65M7065L0565L7041A10Legendre-Gauss spectral collocation methodsecond order nonlinear delay differential equationserror analysis
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 7 , Issue 2 , May 2014 , pp. 149 - 178
Copyright
Copyright © Global Science Press Limited 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Ali, I., Brunner, H. and Tang, T., A spectral method for pantograph-type delay differential equations and its convergence analysis, J. Comput. Math., 27 (2009), pp. 254–265.Google Scholar
[2] Baker, C. T. H., Paul, C. A. H. and Willé, D. R., A bibliography on the numerical solution of delay differential equations, NA Report 269, Dept. of Mathematics, University of Manchester, 1995.Google Scholar
[3] Bellen, A. and Zennaro, M., Numerical Methods for Delay Differential Equations, Oxford University Press, Oxford, 2003.Google Scholar
[4] Bernardi, C. and Maday, Y., Spectral methods, in Handbook of Numerical Analysis, edited by Ciarlet, P. G. and Lions, J.L., North-Holland, Amsterdam, 1997.Google Scholar
[5] Boyd, J. P., Chebyshev and Fourier Spectral Methods, Second edition, Dover Publications, Inc., Mineola, NY, 2001.Google Scholar
[6] Brunner, H., Collocation Methods for Volterra Integral and Related Functional Equations, Cambridge University Press, Cambridge, 2004.Google Scholar
[7] Butcher, J. C., The Numerical Analysis of Ordinary Differential Equations, Runge-Kutta and General Linear Methods, John Wiley & Sons, Chichester, 1987.Google Scholar
[8] Canuto, C., Hussaini, M. Y., Quarteroni, A. and Zang, T. A., Spectral Methods: Fundamentals in Single Domains, Springer-Verlag, Berlin, 2006.Google Scholar
[9] Erdogan, F. and Amiraliyev, G. M., Fitted finite difference method for singularly perturbed delay differential equations, Numer. Algor., 59 (2012), pp. 131–145.CrossRefGoogle Scholar
[10] Funaro, D., Polynomial Approximations of Differential Equations, Springer-Verlag, Berlin, 1992.CrossRefGoogle Scholar
[11] Gottlieb, D. and Orszag, S. A., Numerical Analysis of Spectral Methods: Theory and Applications, SIAM-CBMS, Philadelphia, 1977.Google Scholar
[12] Guo, B. Y., Spectral Methods and Their Applications, World Scientific, Singapore, 1998.Google Scholar
[13] Guo, B. Y. and Wang, Z. Q., Legendre-Gauss collocation methods for ordinary differential equations, Adv. Compt. Math., 30 (2009), pp. 249–280.Google Scholar
[14] Guo, B. Y. and Wang, Z. Q., A spectral collocation method for solving initial value problems of first order ordinary differential equations, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), pp. 1029–1054.Google Scholar
[15] Guo, B. Y. and Yan, J. P., Legendre-Gauss collocation methods for initial value problems of second ordinary differential equations, Appl. Numer. Math., 59 (2009), pp. 1386–1408.CrossRefGoogle Scholar
[16] Hairer, E., NØrsett, S.P. and Wanner, G., Solving Ordinary Differential Equation I: Nonstiff Problems, Second edition, Springer-Verlag, Berlin, 1993.Google Scholar
[17] Hairer, E. and Wanner, G., Solving Ordinary Differential Equation II: Stiff and Differential-Algebraic Problems, Second edition, Springer-Verlag, Berlin, 1996.Google Scholar
[18] Ito, K., Tran, H. T. and Manitius, A., A fully-discrete spectral method for delay-differential equations, SIAM J. Numer. Anal., 28 (1991), pp. 1121–1140.CrossRefGoogle Scholar
[19] Kanyamee, N. and Zhang, Z., Comparison of a spectral collocation method and symplectic methods for Hamiltonian systems, Int. J. Numer. Anal. Model., 8 (2011), pp. 86–104.Google Scholar
[20] Lambert, J. D., Numerical Methods for Ordinary Differential Systems: The Initial Value Problem, John Wiley & Sons, Chichester, 1991.Google Scholar
[21] Minorsky, N., Nonlinear Oscillations, D. Van Nostrand Company, Princeton, 1962.Google Scholar
[22] Oberle, H. J. and Pesch, H. J., Numerical treatment of delay differential equations by Hermite interpolation, Numer. Math., 37 (1981), pp. 235–255.Google Scholar
[23] Pinney, E., Ordinary Difference-Differential Equations, University of California Press, Berkeley, 1958.Google Scholar
[24] Shen, J. and Tang, T., Spectral and High-order Methods with Applications, Science Press, Beijing, 2006.Google Scholar
[25] Shen, J., Tang, T. and Wang, L. L., Spectral Methods: Algorithms, Analysis and Applications, Springer Series in Computational Mathematics, Vol. 41, Springer, Heidelberg, 2011.Google Scholar
[26] Wang, Z. Q. and Guo, B. Y., Legendre-Gauss-Radau collocation method for solving initial value problems of first order ordinary differential equations, J. Sci. Comput., 52 (2012), pp. 226–255.CrossRefGoogle Scholar
[27] Wang, Z. Q. and Wang, L. L., A Legendre-Gauss collocation method for nonlinear delay differential equations, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), pp. 685–708.Google Scholar
[28] Wei, Y. X. and Chen, Y. P., Convergence analysis of the Legendre spectral collocation methods for second order Volterra integro-differential equations, Numer. Math. Theory Methods Appl., 4 (2011), pp. 419–438.Google Scholar
[29] Wei, Y. X. and Chen, Y. P., Legendre spectral collocation methods for pantograph Volterra delay-integro-differential equations, J. Sci. Comput., 53 (2012), pp. 672–688.Google Scholar
[30] Yi, L. J., Liang, Z. Q. and Wang, Z. Q., Legendre-Gauss-Lobatto collocation method for nonlinear delay differential equations, Math. Methods Appl. Sci., 36 (2013), pp. 2476–2491.Google Scholar
[31] Yi, L. J. and Wang, Z. Q., Legendre-Gauss-type collocation algorithms for nonlinear ordinary/partial differential equations, Int. J. Comput. Math., DOI:10.1080/00207160.2013.841901.Google Scholar
[32] Yi, L. J. and Wang, Z. Q., Legendre spectral collocation method for second order nonlinear ordinary/partial differential equations, Discrete Contin. Dyn. Syst. Ser. B., 19 (2014), pp. 299–322.Google Scholar
[33] Zennaro, M., Delay Differential Equations: Theory and Numerics, in Theory and Numerics of Ordinary and Partial Differential Equations, edited by Ainsworth, M., Levesley, J., Light, W.A. and Marietta, M., Clarendon Press, Oxford, 1995.Google Scholar