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

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Abstract

This paper is concerned with the convergence properties of the Legendre spectral collocation methods when used to approximate smooth solutions of Volterra integro-differential equations with proportional (vanishing) delays. We provide a vigorous error analysis for the proposed methods. Furthermore, we prove that both the errors of approximate solutions and the errors of approximate derivatives decay exponentially in L 2-norm and L ∞-norm. Some numerical experiments are given to confirm the theoretical results.

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References

  1. 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 

  2. 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 

  3. Bernardi, C., Maday, Y.: Spectral methods. In: Ciarlet, P.G., Lions, J.L. (eds.) Techniques of Scientific Computing. Handbook of Numerical Analysis, vol. 5, pp. 209–486 (1997)

    Google Scholar 

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

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

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, Y., Tang, T.: Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel. Math. Comput. 79, 147–167 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. 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 

  10. Gan, S.Q.: Dissipativity of θ-methods for nonlinear Volterra delay-integro-differential equations. J. Comput. Appl. Math. 206, 898–907 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Guo, B.Y., Shen, J.: Laguerre-Galerkin method for nonlinear partial differential equations on a semi-infinite interval. Numer. Math. 86, 635–654 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  12. Guo, B.Y., Wang, L.L.: Jacobi interpolation approximations and their applications to singular differential equations. Adv. Comput. Math. 14, 227–276 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Springer, Berlin (1989)

    Google Scholar 

  14. Ishiwata, E., Muroya, Y.: On collocation methods for delay differential and Volterra integral equations with proportional delay. Front. Math. China 4, 89–111 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. 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 

  16. 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 

  17. Mastroianni, G., Monegato, G.: Nystrom interpolants based on zeros of Laguerre polynomials for some Weiner-Hopf equations. IMA J. Numer. Anal. 17, 621–642 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  18. Neval, P.: Mean convergence of Lagrange interpolation. III. Trans. Am. Math. Soc. 282, 669–698 (1984)

    Article  Google Scholar 

  19. Qu, C.K., Wong, R.: Szego’s conjecture on Lebesgue constants for Legendre series. Pac. J. Math. 135, 157–188 (1988)

    MathSciNet  MATH  Google Scholar 

  20. Shen, J.: Stable and efficient spectral methods in unbounded domains using Laguerre functions. SIAM J. Numer. Anal. 38, 1113–1133 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  21. Shen, J., Tang, T.: Spectral and High-Order Methods with Applications. Science Press, Beijing (2006)

    MATH  Google Scholar 

  22. Tang, T., Xu, X., Chen, J.: On spectral methods for Volterra integral equations and the convergence analysis. J. Comput. Math. 26, 825–837 (2008)

    MathSciNet  MATH  Google Scholar 

  23. Wang, W.S., Li, S.F.: Convergence of Runge-Kutta methods for neutral Volterra delay-integro-differential equations. Front. Math. China 4, 195–216 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wiederholt, L.F.: Stability of multistep methods for delay differential equations. Math. Comput. 30, 283–290 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  25. 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 

  26. Xu, C.L., Guo, B.Y.: Laguerre pseudospectral method for nonlinear partial differential equations. J. Comput. Math. 20, 413–428 (2002)

    MathSciNet  MATH  Google Scholar 

  27. Zhang, C.J., Vandewalle, S.: Stability analysis of Runge-Kutta methods for nonlinear Volterra delay-integro-differential equations. IMA J. Numer. Anal. 24, 193–214 (2004)

    Article  MathSciNet  Google Scholar 

  28. Zhao, J.J., Xu, Y., Liu, M.Z.: Stability analysis of numerical methods for linear neutral Volterra delay-integro-differential system. Appl. Math. Comput. 167, 1062–1079 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work is supported by the Foundation for Talent Introduction of Guangdong Provincial University, Guangdong Province Universities and Colleges Pearl River Scholar Funded Scheme (2008), National Science Foundation of China (10971074).

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Authors and Affiliations

  1. School of Mathematics and Computational Science, Sun Yat-sen University, Guangzhou, 510275, China

    Yunxia Wei

  2. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China

    Yanping Chen

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  1. Yunxia Wei
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  2. Yanping Chen
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Correspondence to Yanping Chen.

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Wei, Y., Chen, Y. Legendre Spectral Collocation Methods for Pantograph Volterra Delay-Integro-Differential Equations. J Sci Comput 53, 672–688 (2012). https://doi.org/10.1007/s10915-012-9595-6

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  • DOI: https://doi.org/10.1007/s10915-012-9595-6

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