Skip to main content
Erschienen in: Calcolo 4/2016

01.12.2016

A Legendre–Gauss–Radau spectral collocation method for first order nonlinear delay differential equations

verfasst von: Lijun Yi, Zhongqing Wang

Erschienen in: Calcolo | Ausgabe 4/2016

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

In this paper, we introduce a single-step Legendre–Gauss–Radau spectral collocation method for solving the first order nonlinear delay differential equations with variable delay, and analyze its convergence. We also propose two fast and efficient algorithms for the single-step scheme and apply them to the multiple interval case. Numerical results show that the suggested algorithms enjoy high order accuracy.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Literatur
1.
Zurück zum Zitat Ali, I., Brunner, H., Tang, T.: A spectral method for pantograph-type delay differential equations and its convergence analysis. J. Comput. Math. 27, 254–265 (2009)MathSciNetMATH Ali, I., Brunner, H., Tang, T.: A spectral method for pantograph-type delay differential equations and its convergence analysis. J. Comput. Math. 27, 254–265 (2009)MathSciNetMATH
2.
Zurück zum Zitat Allen, K., McKee, S.: Fixed step discretisation methods for delay differential equations. Comput. Math. Appl. 7, 413–423 (1981)MathSciNetCrossRefMATH Allen, K., McKee, S.: Fixed step discretisation methods for delay differential equations. Comput. Math. Appl. 7, 413–423 (1981)MathSciNetCrossRefMATH
3.
Zurück zum Zitat Baker, C.T.H., Paul, C.A.H., Willé, D.R.: Issues in the numerical solution of evolutionary delay differential equations. Adv. Comput. Math. 3, 171–196 (1995)MathSciNetCrossRefMATH Baker, C.T.H., Paul, C.A.H., Willé, D.R.: Issues in the numerical solution of evolutionary delay differential equations. Adv. Comput. Math. 3, 171–196 (1995)MathSciNetCrossRefMATH
4.
Zurück zum Zitat Baker, C.T.H., Paul, C.A.H., Willé, D.R.: A bibliography on the numerical solution of delay differential equations. NA Report 269, Department of Mathematics, University of Manchester (1995) Baker, C.T.H., Paul, C.A.H., Willé, D.R.: A bibliography on the numerical solution of delay differential equations. NA Report 269, Department of Mathematics, University of Manchester (1995)
6.
Zurück zum Zitat Bellen, A., Maset, S.: Numerical solution of constant coefficient linear delay differential equations as abstract Cauchy problems. Numer. Math. 84, 351–374 (2000)MathSciNetCrossRefMATH Bellen, A., Maset, S.: Numerical solution of constant coefficient linear delay differential equations as abstract Cauchy problems. Numer. Math. 84, 351–374 (2000)MathSciNetCrossRefMATH
7.
Zurück zum Zitat Bellen, A., Zennaro, M.: Numerical solution of delay differential equations by uniform corrections to an implicit Runge–Kutta method. Numer. Math. 47, 301–316 (1985)MathSciNetCrossRefMATH Bellen, A., Zennaro, M.: Numerical solution of delay differential equations by uniform corrections to an implicit Runge–Kutta method. Numer. Math. 47, 301–316 (1985)MathSciNetCrossRefMATH
8.
Zurück zum Zitat Bellen, A., Zennaro, M.: Numerical Methods for Delay Differential Equations. Oxford University Press, Oxford (2003)CrossRefMATH Bellen, A., Zennaro, M.: Numerical Methods for Delay Differential Equations. Oxford University Press, Oxford (2003)CrossRefMATH
9.
Zurück zum Zitat Bernardi, C., Maday, Y.: Spectral methods. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis. North-Holland, Amsterdam (1997) Bernardi, C., Maday, Y.: Spectral methods. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis. North-Holland, Amsterdam (1997)
10.
Zurück zum Zitat Boyd, J.P.: Chebyshev and Fourier Spectral Methods, 2nd edn. Dover Publications Inc, Mineola (2001)MATH Boyd, J.P.: Chebyshev and Fourier Spectral Methods, 2nd edn. Dover Publications Inc, Mineola (2001)MATH
11.
Zurück zum Zitat Brunner, H.: Collocation Methods for Volterra Integral and Related Functional Equations. Cambridge University Press, Cambridge (2004)CrossRefMATH Brunner, H.: Collocation Methods for Volterra Integral and Related Functional Equations. Cambridge University Press, Cambridge (2004)CrossRefMATH
12.
Zurück zum Zitat Brunner, H., Hu, Q., Lin, Q.: Geometric meshes in collocation methods for Volterra integral equations with proportional delays. IMA J. Numer. Anal. 21, 783–798 (2001)MathSciNetCrossRefMATH Brunner, H., Hu, Q., Lin, Q.: Geometric meshes in collocation methods for Volterra integral equations with proportional delays. IMA J. Numer. Anal. 21, 783–798 (2001)MathSciNetCrossRefMATH
13.
Zurück zum Zitat Bueler, E.: Error bounds for approximate eigenvalues of periodic-coefficient linear delay differential equations. SIAM J. Numer. Anal. 45, 2510–2536 (2007)MathSciNetCrossRefMATH Bueler, E.: Error bounds for approximate eigenvalues of periodic-coefficient linear delay differential equations. SIAM J. Numer. Anal. 45, 2510–2536 (2007)MathSciNetCrossRefMATH
15.
Zurück zum Zitat Butcher, J.C.: The Numerical Analysis of Ordinary Differential Equations, Runge-Kutta and General Linear Methods. Wiley, Chichester (1987)MATH Butcher, J.C.: The Numerical Analysis of Ordinary Differential Equations, Runge-Kutta and General Linear Methods. Wiley, Chichester (1987)MATH
16.
Zurück zum Zitat Butcher, E.A., Ma, H., Bueler, E., Averina, V., Szabo, Z.: Stability of linear time-periodic delay-differential equations via Chebyshev polynomials. Int. J. Numer. Meth. Eng. 59, 895–922 (2004)MathSciNetCrossRefMATH Butcher, E.A., Ma, H., Bueler, E., Averina, V., Szabo, Z.: Stability of linear time-periodic delay-differential equations via Chebyshev polynomials. Int. J. Numer. Meth. Eng. 59, 895–922 (2004)MathSciNetCrossRefMATH
17.
Zurück zum Zitat Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods: Fundamentals in Single Domains. Springer, Berlin (2006)MATH Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods: Fundamentals in Single Domains. Springer, Berlin (2006)MATH
18.
Zurück zum Zitat Cooke, K., van den Driessche, P., Zou, X.: Interaction of maturation delay and nonlinear birth in population and epidemic models. J. Math. Biol. 39, 332–352 (1999)MathSciNetCrossRefMATH Cooke, K., van den Driessche, P., Zou, X.: Interaction of maturation delay and nonlinear birth in population and epidemic models. J. Math. Biol. 39, 332–352 (1999)MathSciNetCrossRefMATH
19.
Zurück zum Zitat El-Safty, A., Salim, M.S., El-Khatib, M.A.: Convergence of the spline functions for delay dynamic systems. Int. J. Comput. Math. 80, 509–518 (2003)MathSciNetCrossRefMATH El-Safty, A., Salim, M.S., El-Khatib, M.A.: Convergence of the spline functions for delay dynamic systems. Int. J. Comput. Math. 80, 509–518 (2003)MathSciNetCrossRefMATH
20.
Zurück zum Zitat Guo, B.Y.: Spectral Methods and Their Applications. World Scientific, Singapore (1998)CrossRefMATH Guo, B.Y.: Spectral Methods and Their Applications. World Scientific, Singapore (1998)CrossRefMATH
21.
Zurück zum Zitat Guo, B.Y., Wang, Z.Q.: Legendre–Gauss collocation methods for ordinary differential equations. Adv. Comput. Math. 30, 249–280 (2009)MathSciNetCrossRefMATH Guo, B.Y., Wang, Z.Q.: Legendre–Gauss collocation methods for ordinary differential equations. Adv. Comput. Math. 30, 249–280 (2009)MathSciNetCrossRefMATH
22.
Zurück zum Zitat Guo, B.Y., 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, 1029–1054 (2010)MathSciNetCrossRefMATH Guo, B.Y., 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, 1029–1054 (2010)MathSciNetCrossRefMATH
23.
Zurück zum Zitat Hairer, E., Norsett, S.P., Wanner, G.: Solving Ordinary Differential Equation I: Nonstiff Problems, 2nd edn. Springer, Berlin (1993)MATH Hairer, E., Norsett, S.P., Wanner, G.: Solving Ordinary Differential Equation I: Nonstiff Problems, 2nd edn. Springer, Berlin (1993)MATH
24.
Zurück zum Zitat Hairer, E., Wanner, G.: Solving Ordinary Differential Equation II: Stiff and Differential-Algebraic Problems, 2nd edn. Springer, Berlin (1996)CrossRefMATH Hairer, E., Wanner, G.: Solving Ordinary Differential Equation II: Stiff and Differential-Algebraic Problems, 2nd edn. Springer, Berlin (1996)CrossRefMATH
25.
Zurück zum Zitat Ito, K., Tran, H.T., Manitius, A.: A fully-discrete spectral method for delay-differential equations. SIAM J. Numer. Anal. 28, 1121–1140 (1991)MathSciNetCrossRefMATH Ito, K., Tran, H.T., Manitius, A.: A fully-discrete spectral method for delay-differential equations. SIAM J. Numer. Anal. 28, 1121–1140 (1991)MathSciNetCrossRefMATH
26.
Zurück zum Zitat Jung, J.-H., Song, Y.F.: On a polynomial chaos method for differential equations with singular sources. Int. J. Uncertain. Quantif. 1, 77–98 (2011)MathSciNetCrossRefMATH Jung, J.-H., Song, Y.F.: On a polynomial chaos method for differential equations with singular sources. Int. J. Uncertain. Quantif. 1, 77–98 (2011)MathSciNetCrossRefMATH
27.
Zurück zum Zitat Lambert, J.D.: Numerical Methods for Ordinary Differential Systems: The Initial Value Problem. Wiley, Chichester (1991)MATH Lambert, J.D.: Numerical Methods for Ordinary Differential Systems: The Initial Value Problem. Wiley, Chichester (1991)MATH
28.
Zurück zum Zitat Shen, J., Tang, T., Wang, L.L.: “Spectral Methods: Algorithms, Analysis and Applications”, Springer Series in Computational Mathematics, vol. 41. Springer, Heidelberg (2011)CrossRef Shen, J., Tang, T., Wang, L.L.: “Spectral Methods: Algorithms, Analysis and Applications”, Springer Series in Computational Mathematics, vol. 41. Springer, Heidelberg (2011)CrossRef
29.
Zurück zum Zitat Tang, J., Xie, Z.Q., Zhang, Z.M.: The long time behavior of a spectral collocation method for delay differential equations of pantograph type. Discrete Contin. Dyn. Syst. Ser. B 18, 797–819 (2013)MathSciNetMATH Tang, J., Xie, Z.Q., Zhang, Z.M.: The long time behavior of a spectral collocation method for delay differential equations of pantograph type. Discrete Contin. Dyn. Syst. Ser. B 18, 797–819 (2013)MathSciNetMATH
30.
Zurück zum Zitat Wang, Z.Q., Guo, B.Y.: Legendre–Gauss–Radau collocation method for solving initial value problems of first order ordinary differential equations. J. Sci. Comput. 52, 226–255 (2012)MathSciNetCrossRefMATH Wang, Z.Q., Guo, B.Y.: Legendre–Gauss–Radau collocation method for solving initial value problems of first order ordinary differential equations. J. Sci. Comput. 52, 226–255 (2012)MathSciNetCrossRefMATH
31.
Zurück zum Zitat Wang, Z.Q., Wang, L.L.: A Legendre–Gauss collocation method for nonlinear delay differential equations. Discrete Contin. Dyn. Syst. Ser. B 13, 685–708 (2010)MathSciNetCrossRefMATH Wang, Z.Q., Wang, L.L.: A Legendre–Gauss collocation method for nonlinear delay differential equations. Discrete Contin. Dyn. Syst. Ser. B 13, 685–708 (2010)MathSciNetCrossRefMATH
32.
Zurück zum Zitat Wright, E.M.: A non-linear difference-differential equation. J. Reine Angew. Math. 194, 66–87 (1955)MathSciNetMATH Wright, E.M.: A non-linear difference-differential equation. J. Reine Angew. Math. 194, 66–87 (1955)MathSciNetMATH
33.
Zurück zum Zitat Yi, L.J., Liang, Z.Q., Wang, Z.Q.: Legendre–Gauss–Lobatto spectral collocation method for nonlinear delay differential equations. Math. Methods Appl. Sci. 36, 2476–2491 (2013)MathSciNetCrossRefMATH Yi, L.J., Liang, Z.Q., Wang, Z.Q.: Legendre–Gauss–Lobatto spectral collocation method for nonlinear delay differential equations. Math. Methods Appl. Sci. 36, 2476–2491 (2013)MathSciNetCrossRefMATH
34.
Zurück zum Zitat Yi, L.J., Wang, Z.Q.: Legendre–Gauss-type collocation algorithms for nonlinear ordinary/partial differential equations. Int. J. Comput. Math. 91, 1434–1460 (2014)MathSciNetCrossRefMATH Yi, L.J., Wang, Z.Q.: Legendre–Gauss-type collocation algorithms for nonlinear ordinary/partial differential equations. Int. J. Comput. Math. 91, 1434–1460 (2014)MathSciNetCrossRefMATH
35.
Zurück zum Zitat Zennaro, M.: Delay differential equations: theory and numerics. In: Ainsworth, M., Levesley, J., Light, W.A., Marietta, M. (eds.) Theory and Numerics of Ordinary and Partial Differential Equations. Clarendon Press, Oxford (1995) Zennaro, M.: Delay differential equations: theory and numerics. In: Ainsworth, M., Levesley, J., Light, W.A., Marietta, M. (eds.) Theory and Numerics of Ordinary and Partial Differential Equations. Clarendon Press, Oxford (1995)
Metadaten
Titel
A Legendre–Gauss–Radau spectral collocation method for first order nonlinear delay differential equations
verfasst von
Lijun Yi
Zhongqing Wang
Publikationsdatum
01.12.2016
Verlag
Springer Milan
Erschienen in
Calcolo / Ausgabe 4/2016
Print ISSN: 0008-0624
Elektronische ISSN: 1126-5434
DOI
https://doi.org/10.1007/s10092-015-0169-5

Weitere Artikel der Ausgabe 4/2016

Calcolo 4/2016 Zur Ausgabe

Premium Partner