Skip to main content
SpringerLink
Log in

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

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)

    MATH  Google Scholar 

  2. 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)

    MathSciNet  MATH  Google Scholar 

  3. 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)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bellen, A., Zennaro, M.: Numerical Methods for Delay Differential Equations. Oxford University Press, Oxford (2003)

    Book  MATH  Google Scholar 

  5. Brunner, H.: Collocation Methods for Volterra Integral and Related Functional Equations. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  6. 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)

    Article  MathSciNet  MATH  Google Scholar 

  7. Brunner, H.: Iterated collocation methods for Volterra integral equtions with delay arguments. Math. Comput. 62, 581–599 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  8. Brunner, H., Zhang, W.: Primary discontinuities in solutions for delay integro-differential equations. Methods Appl. Anal. 6, 525–533 (1999)

    MathSciNet  MATH  Google Scholar 

  9. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods Fundamentals in Single Domains. Springer, Berlin (2006)

    MATH  Google Scholar 

  10. 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)

    Article  MATH  Google Scholar 

  11. Chen, Y., Tang, T.: Spectral methods for weakly singular Volterra integral equtions with smooth solutions. J. Comput. Appl. Math. 233, 938–950 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Enright, W.H., Hu, M.: Continuous Runge–Kutta methods for neutral Volterra integro-differential equations with delay. Appl. Numer. Math. 24, 175–190 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gan, S.Q.: Dissipativity of \(\theta \)-methods for nonlinear Volterra delay-integro-differential equations. J. Comut. Appl. Math. 206, 898–907 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Guo, B.Y., Wang, L.L.: Jacobi approzimations in non-uniformly Jacobi-weighted Sobolev spaces. J. Approx. Theory 1, 1–41 (2004)

    Article  MATH  Google Scholar 

  15. Guo, B.Y., Wang, Z.Q.: Legendre–Gauss collocation methods for ordinary differential equations. Adv. Comput. Math. 30, 249–280 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. 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)

    Article  MathSciNet  MATH  Google Scholar 

  17. 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)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jiang, Y.J.: On spectral methods for Volterra-type integro-differential equations. J. Comput. Appl. Math. 230, 333–340 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. 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)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lambert, J.D.: Numerical Methods for Ordinary Differential Systems: The Initial Value Problem. Wiley, Chichester (1991)

    MATH  Google Scholar 

  21. Li, Y.K., Kuang, Y.: Periodic solutions of periodic delay Lotka–Volterra equations and systems. J. Math. Anal. Appl. 255, 260–280 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  22. Linz, P., Wang, R.L.C.: Error bounds for the solution of Volterra and delay equations. Appl. Numer. Math. 9, 201–207 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  23. Shen, J., Tang, T.: Spectral Methods: Algorithms, Analysis and Applications. Springer, Berlin (2011)

    Book  MATH  Google Scholar 

  24. 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  Google Scholar 

  25. 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)

    Article  MathSciNet  MATH  Google Scholar 

  26. 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)

    Article  MathSciNet  MATH  Google Scholar 

  27. Wei, Y.X., Chen, Y.P.: Legendre spectral collocation methods for pantograph Volterra delay-integro-differential equations. J. Sci. Comput. 53, 672–688 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. 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)

    Article  MathSciNet  MATH  Google Scholar 

  29. 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)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the anonymous referees for their valuable comments which helped us to improve the present paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China

    Jingjun Zhao, Yang Cao & Yang Xu

Authors
  1. Jingjun Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yang Cao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yang Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yang Xu.

Additional information

This work was supported by the National Natural Science Foundation of China (11271102, 11101109), the Fundamental Research Funds for the Central Universities and Program for Innovation Research of Science in Harbin Institute of Technology (A201405).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhao, J., Cao, Y. & Xu, Y. Legendre Spectral Collocation Methods for Volterra Delay-Integro-Differential Equations. J Sci Comput 67, 1110–1133 (2016). https://doi.org/10.1007/s10915-015-0121-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-015-0121-5

Keywords

Mathematics Subject Classification

Search

Navigation