Skip to main content
Erschienen in: Journal of Scientific Computing 3/2016

15.10.2015

Legendre Spectral Collocation Methods for Volterra Delay-Integro-Differential Equations

verfasst von: Jingjun Zhao, Yang Cao, Yang Xu

Erschienen in: Journal of Scientific Computing | Ausgabe 3/2016

Einloggen

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

search-config
loading …

Abstract

This work is to analyze a Legendre spectral collocation approximation for Volterra delay-integro-differential equations. An error analysis is provided for the proposed methods. Moreover, it is proved that the errors of approximate solutions decay exponentially. The numerical results verify our theoretical analysis.

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 Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)MATH Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)MATH
2.
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
3.
Zurück zum Zitat Ali, I., Brunner, H., Tang, T.: Spectral methods for pantograph-type differential and integral equations with multiple delays. Front. Math. China 4, 49–61 (2009)MathSciNetCrossRefMATH Ali, I., Brunner, H., Tang, T.: Spectral methods for pantograph-type differential and integral equations with multiple delays. Front. Math. China 4, 49–61 (2009)MathSciNetCrossRefMATH
4.
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
5.
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
6.
Zurück zum Zitat Brunner, H.: Current work and open problems in the numerical analysis of Volterra functional equations with vanishing delays. Front. Math. China 4, 3–22 (2009)MathSciNetCrossRefMATH Brunner, H.: Current work and open problems in the numerical analysis of Volterra functional equations with vanishing delays. Front. Math. China 4, 3–22 (2009)MathSciNetCrossRefMATH
7.
Zurück zum Zitat Brunner, H.: Iterated collocation methods for Volterra integral equtions with delay arguments. Math. Comput. 62, 581–599 (1994)MathSciNetCrossRefMATH Brunner, H.: Iterated collocation methods for Volterra integral equtions with delay arguments. Math. Comput. 62, 581–599 (1994)MathSciNetCrossRefMATH
8.
Zurück zum Zitat Brunner, H., Zhang, W.: Primary discontinuities in solutions for delay integro-differential equations. Methods Appl. Anal. 6, 525–533 (1999)MathSciNetMATH Brunner, H., Zhang, W.: Primary discontinuities in solutions for delay integro-differential equations. Methods Appl. Anal. 6, 525–533 (1999)MathSciNetMATH
9.
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
10.
Zurück zum Zitat Chen, Y., Tang, T.: Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equtions with a weakly singular kernel. Math. Comput. 79, 147–167 (2010)CrossRefMATH Chen, Y., Tang, T.: Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equtions with a weakly singular kernel. Math. Comput. 79, 147–167 (2010)CrossRefMATH
11.
Zurück zum Zitat Chen, Y., Tang, T.: Spectral methods for weakly singular Volterra integral equtions with smooth solutions. J. Comput. Appl. Math. 233, 938–950 (2009)MathSciNetCrossRefMATH Chen, Y., Tang, T.: Spectral methods for weakly singular Volterra integral equtions with smooth solutions. J. Comput. Appl. Math. 233, 938–950 (2009)MathSciNetCrossRefMATH
12.
Zurück zum Zitat Enright, W.H., Hu, M.: Continuous Runge–Kutta methods for neutral Volterra integro-differential equations with delay. Appl. Numer. Math. 24, 175–190 (1997)MathSciNetCrossRefMATH Enright, W.H., Hu, M.: Continuous Runge–Kutta methods for neutral Volterra integro-differential equations with delay. Appl. Numer. Math. 24, 175–190 (1997)MathSciNetCrossRefMATH
13.
Zurück zum Zitat Gan, S.Q.: Dissipativity of \(\theta \)-methods for nonlinear Volterra delay-integro-differential equations. J. Comut. Appl. Math. 206, 898–907 (2007)MathSciNetCrossRefMATH Gan, S.Q.: Dissipativity of \(\theta \)-methods for nonlinear Volterra delay-integro-differential equations. J. Comut. Appl. Math. 206, 898–907 (2007)MathSciNetCrossRefMATH
14.
Zurück zum Zitat Guo, B.Y., Wang, L.L.: Jacobi approzimations in non-uniformly Jacobi-weighted Sobolev spaces. J. Approx. Theory 1, 1–41 (2004)CrossRefMATH Guo, B.Y., Wang, L.L.: Jacobi approzimations in non-uniformly Jacobi-weighted Sobolev spaces. J. Approx. Theory 1, 1–41 (2004)CrossRefMATH
15.
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
16.
Zurück zum Zitat Guo, B.Y., Yan, J.P.: Legendre–Gauss collocation methods for initial value problems of second ordinary differential equations. Appl. Numer. Math. 59, 1386–1408 (2009)MathSciNetCrossRefMATH Guo, B.Y., Yan, J.P.: Legendre–Gauss collocation methods for initial value problems of second ordinary differential equations. Appl. Numer. Math. 59, 1386–1408 (2009)MathSciNetCrossRefMATH
17.
Zurück zum Zitat Huang, C.M., Vandewalle, S.: Stability of Runge–Kutta–Pouzet methods for Volterra integro-differential equation with delays. Front. Math. China 4, 63–87 (2009)MathSciNetCrossRefMATH Huang, C.M., Vandewalle, S.: Stability of Runge–Kutta–Pouzet methods for Volterra integro-differential equation with delays. Front. Math. China 4, 63–87 (2009)MathSciNetCrossRefMATH
18.
Zurück zum Zitat Jiang, Y.J.: On spectral methods for Volterra-type integro-differential equations. J. Comput. Appl. Math. 230, 333–340 (2009)MathSciNetCrossRefMATH Jiang, Y.J.: On spectral methods for Volterra-type integro-differential equations. J. Comput. Appl. Math. 230, 333–340 (2009)MathSciNetCrossRefMATH
19.
Zurück zum Zitat Jiang, Y.J., Ma, J.T.: Spectral collocation methods for Volterra-integro differential equations with noncompact kernels. J. Comput. Appl. Math. 224, 115–124 (2013)MathSciNetCrossRefMATH Jiang, Y.J., Ma, J.T.: Spectral collocation methods for Volterra-integro differential equations with noncompact kernels. J. Comput. Appl. Math. 224, 115–124 (2013)MathSciNetCrossRefMATH
20.
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
21.
Zurück zum Zitat Li, Y.K., Kuang, Y.: Periodic solutions of periodic delay Lotka–Volterra equations and systems. J. Math. Anal. Appl. 255, 260–280 (2001)MathSciNetCrossRefMATH Li, Y.K., Kuang, Y.: Periodic solutions of periodic delay Lotka–Volterra equations and systems. J. Math. Anal. Appl. 255, 260–280 (2001)MathSciNetCrossRefMATH
22.
Zurück zum Zitat Linz, P., Wang, R.L.C.: Error bounds for the solution of Volterra and delay equations. Appl. Numer. Math. 9, 201–207 (1992)MathSciNetCrossRefMATH Linz, P., Wang, R.L.C.: Error bounds for the solution of Volterra and delay equations. Appl. Numer. Math. 9, 201–207 (1992)MathSciNetCrossRefMATH
23.
Zurück zum Zitat Shen, J., Tang, T.: Spectral Methods: Algorithms, Analysis and Applications. Springer, Berlin (2011)CrossRefMATH Shen, J., Tang, T.: Spectral Methods: Algorithms, Analysis and Applications. Springer, Berlin (2011)CrossRefMATH
24.
Zurück zum Zitat Tang, T., Xu, X., Chen, J.: On spectral methods for Volterra integral equtions and the convergence analysis. J. Comput. Math. 26, 825–837 (2008)MathSciNet Tang, T., Xu, X., Chen, J.: On spectral methods for Volterra integral equtions and the convergence analysis. J. Comput. Math. 26, 825–837 (2008)MathSciNet
25.
Zurück zum Zitat Wang, W.S., Li, S.F.: Convergence of Runge–Kutta methods for neural Volterra delay-integro-differential equations. Front. Math. China 4, 195–216 (2009)MathSciNetCrossRefMATH Wang, W.S., Li, S.F.: Convergence of Runge–Kutta methods for neural Volterra delay-integro-differential equations. Front. Math. China 4, 195–216 (2009)MathSciNetCrossRefMATH
26.
Zurück zum Zitat Wang, Z.Q., Wang, L.L.: A Legendre–Gauss collocation method for nonlinear delay differential equations. Discret. 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. Discret. Contin. Dyn. Syst. Ser. B 13, 685–708 (2010)MathSciNetCrossRefMATH
27.
Zurück zum Zitat Wei, Y.X., Chen, Y.P.: Legendre spectral collocation methods for pantograph Volterra delay-integro-differential equations. J. Sci. Comput. 53, 672–688 (2012)MathSciNetCrossRefMATH Wei, Y.X., Chen, Y.P.: Legendre spectral collocation methods for pantograph Volterra delay-integro-differential equations. J. Sci. Comput. 53, 672–688 (2012)MathSciNetCrossRefMATH
28.
Zurück zum Zitat Wu, S.F., Gan, S.Q.: Analytical and numerical stability of neutral delay integro-differential equations and neutral delay partial differential equations. Comput. Math. Appl. 55, 2426–2443 (2008)MathSciNetCrossRefMATH Wu, S.F., Gan, S.Q.: Analytical and numerical stability of neutral delay integro-differential equations and neutral delay partial differential equations. Comput. Math. Appl. 55, 2426–2443 (2008)MathSciNetCrossRefMATH
29.
Zurück zum Zitat Zhang, C.J., Vandewalle, S.: Stability analysis of Runge–Kutta methods for nonlinear Volterra delay-integro-differential equtions. IMA J. Numer. Anal. 24, 193–214 (2004)MathSciNetCrossRefMATH Zhang, C.J., Vandewalle, S.: Stability analysis of Runge–Kutta methods for nonlinear Volterra delay-integro-differential equtions. IMA J. Numer. Anal. 24, 193–214 (2004)MathSciNetCrossRefMATH
Metadaten
Titel
Legendre Spectral Collocation Methods for Volterra Delay-Integro-Differential Equations
verfasst von
Jingjun Zhao
Yang Cao
Yang Xu
Publikationsdatum
15.10.2015
Verlag
Springer US
Erschienen in
Journal of Scientific Computing / Ausgabe 3/2016
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI
https://doi.org/10.1007/s10915-015-0121-5

Weitere Artikel der Ausgabe 3/2016

Journal of Scientific Computing 3/2016 Zur Ausgabe

Premium Partner