Advances in Computational Mathematics

, Volume 39, Issue 3–4, pp 611–644

Multilevel correction for collocation solutions of Volterra integral equations with proportional delays

Article
  • 205 Downloads

Abstract

In this paper, we propose a convergence acceleration method for collocation solutions of the linear second-kind Volterra integral equations with proportional delay qt\((0<q<1)\). This convergence acceleration method called multilevel correction method is based on a kind of hybrid mesh, which can be viewed as a combination between the geometric meshes and the uniform meshes. It will be shown that, when the collocation solutions are continuous piecewise polynomials whose degrees are less than or equal to \({m} (m \leqslant 2)\), the global accuracy of k level corrected approximation is \(O(N^{-(2m(k+1)-\varepsilon)})\), where N is the number of the nodes, and \(\varepsilon\) is an arbitrary small positive number.

Keywords

Delay integral equation Geometric mesh Collocation method Superconvergence High order interpolation operator Multilevel correction Hybrid meshes 

Mathematics Subject Classifications (2010)

65R20 34K28 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Andreoli, G.: Sulle equazioni integrali. Rend. Circ. Mat. Palermo 37, 76–112 (1914)CrossRefMATHGoogle Scholar
  2. 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)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Baker, C.T.H., Derakhshan, M.S.: Convergence and stability of quadrature methods applied to Volterra equations with delay. IMA J. Numer. Anal. 13, 67–91 (1993)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bellen, A., Brunner, H., Maset, S., Torelli, L.: Superconvergence in collocation methods on quasi-graded meshes for functional differential equations with vanishing delay. BIT Numer. Math. 46, 229–247 (2006)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Brunner, H.: On the discretization of differential and Volterra integral equations with variable delay. BIT Numer. Math. 37, 1–12 (1997)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Brunner, H.: Collocation Methods for Volterra Integral and Related Functional Equations. Cambridge University Press, Cambridge (2004)CrossRefMATHGoogle Scholar
  7. 7.
    Brunner, H., Hu, Q.Y., Lin, Q.: Geometric meshes in collocation methods for Volterra integral equations with proportional delays. IMA J. Numer. Anal. 21, 783–798 (2001)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Burton, T.A.: Volterra integral and differential equations, 2nd edn. Elsevier B. V., New York (2005)MATHGoogle Scholar
  9. 9.
    Chambers, Ll.G.: Some properties of the functional equation \(\phi (x)= f(x)+\int ^{\lambda x}_{0}g(x,y,\phi (y))dy\). Int. J. Math. Math. Sci. 14, 27–44 (1990)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Cerha, J.: On some linear Volterra delay equations. Časopis Pešt Mat 101, 111–123 (1976)MathSciNetMATHGoogle Scholar
  11. 11.
    Chatelin, F., Lebbar, R.: Superconvergence results for the iterated projection method applied to a Fredholm integral equation of the second kind and the corresponding eigenvalue problem. J. Integr. Equ. 6, 71–91 (1984)MathSciNetMATHGoogle Scholar
  12. 12.
    Hu, Q.Y.: The stepwise collocation methods based on the higher order interpolation for Volterra integral equations with multiple delays. Math. Numer. Sini. 19, 353–358 (1994)Google Scholar
  13. 13.
    Hu, Q.Y.: Extrapolation for collocation solutions of Volterra integro-differential equations. Chin. J. Numer. Math. Appl. 18, 28–37 (1996)Google Scholar
  14. 14.
    Hu, Q.Y.: Supercovergence of numerical solutions to Volterra integral equations with singularities. SIAM J. Numer. Anal. 34, 1698–1707 (1997)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Hu, Q.Y.: Geometric meshes and their application to Volterra integro- differential equations with singularities. IMA J. Numer. Anal. 18, 151–164 (1998)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Hu, Q.Y.: Interpolation correction for collocation solutions of Fredholm integro-differential equations. Math. Comput. 67, 987–999 (1998)CrossRefMATHGoogle Scholar
  17. 17.
    Hu, Q.Y.: Multilevel correction for discrete collocation solutions of Volterra integral equations with delay arguments. Appl. Numer. Math. 31, 159–171 (1999)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Hu, Q.Y., Peng, L.: Multilevel correction for collocation solutions of Volterra delay integro-differential equations. J. Syst. Sci. Math. Sci. 19, 134–141 (1999)MathSciNetMATHGoogle Scholar
  19. 19.
    Lin, Q., Sloan, I.H., Xie, R.: Extrapolation of the iterated-collocation method for integral equations of the second kind. SIAM J. Numer. Anal. 27, 1535–1541 (1990)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Lin, Q., Zhang, S.H., Yan, N.N.: An acceleration method for integral equations by using interpolation post-processing. Adv. Comput. Math. 9, 117–129 (1998)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Marchuk, G.I., Shaidurov, V.V.: Difference Methods and their Extrapolation. Springer-Verlag, New York (1983)CrossRefGoogle Scholar
  22. 22.
    Mureşan, V.: On a class of Volterra integral equations with deviating argument. Stud. Univ. Babeş-Bolyai Math. 44, 47–54 (1999)Google Scholar
  23. 23.
    Piila, J.: Characterization of the membrane theory of a clamped shell: the hyperbolic case. Math. Meth. Appl. Sci. 6, 169–194 (1996)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Sloan, I.H.: Improvement by iteration for compact operator equations. Math. Comput 30, 758–764 (1976)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Shi, J.: Higher accuracy algorithm of the second-kind integral equations and application to boundary integral equations. PhD Thesis, Institute of Systems Science, Academia Sinica, Beijing (1990)Google Scholar
  26. 26.
    Takama, N., Muroya, Y., Ishiwata, E.: On the attainable order of collocation methods for the delay differential equation with proportional delay. BIT 40, 374–394 (2000)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Volterra, V.: Leçons sur les équations intégrales et les équations intégro-différentielles. Gauthier-Villars, Paris (1913)MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematical and System Sciences, Chinese Academy of SciencesBeijingChina

Personalised recommendations