Advances in Computational Mathematics

, Volume 42, Issue 5, pp 1015–1030 | Cite as

A spectral collocation method for a weakly singular Volterra integral equation of the second kind

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Abstract

The solution y of a weakly singular Volterra equation of the second kind posed on the interval −1 ≤ t ≤ 1 has in general a certain singular behaviour near t = −1: typically, \(|y^{\prime }(t)| \sim (1+t)^{-\mu }\) for a parameter μ ∈ (0, 1). Various methods have been proposed for the numerical solution of these problems, but up to now there has been no analysis that takes into account this singularity when a spectral collocation method is applied directly to the problem. This gap in the literature is filled by the present paper.

Keywords

Weakly singular Volterra equation of the second kind Spectral collocation method 

Mathematics Subject Classification (2010)

65R20 
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References

  1. 1.
    Atkinson, K.: The numerical solution of integral equation of the second kind, volume 4 of Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (1997)CrossRefGoogle Scholar
  2. 2.
    Brunner, H.: Collocation methods for Volterra integral and related functional differential equations, volume 15 of Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (2004)Google Scholar
  3. 3.
    Brunner, H., Pedas, A., Vainikko, G.: The piecewise polynomial collocation method for nonlinear weakly singular Volterra equations. Math. Comp. 68(227), 1079–1095 (1999)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Brutman, L.: On the Lebesgue function for polynomial interpolation. SIAM J. Numer. Anal. 15(4), 694–704 (1978)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Cao, Y., Herdman, T., Xu, Y.: A hybrid collocation method for Volterra integral equations with weakly singular kernels. SIAM J. Numer. Anal. 41(1), 364–381 (2003). (electronic)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Chen, S., Shen, J., Wang, L.: Generalized Jacobi functions and their applications to fractional differential equations. Math. Comp. (to appear)Google Scholar
  7. 7.
    Chen, Y., Tang, T.: Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel. Math. Comp. 79(269), 147–167 (2010)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Conway, J.B.: A course in functional analysis, volume 96 of Graduate Texts in Mathematics, 2nd edn. Springer, New York (1990)Google Scholar
  9. 9.
    Davis, P.J.: Interpolation and approximation. Dover Publications, Inc., New York (1975). Republication, with minor corrections, of the 1963 original, with a new preface and bibliographyGoogle Scholar
  10. 10.
    Diethelm, K.: The analysis of fractional differential equations, volume 2004 of Lecture Notes in Mathematics. Springer, Berlin (2010). An application-oriented exposition using differential operators of Caputo typeGoogle Scholar
  11. 11.
    Gui, W., Babuška, I.: The h, p and h-p versions of the finite element method in 1 dimension. I. The error analysis of the p-version. Numer. Math. 49(6), 577–612 (1986)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Huang, C., Tang, T., Zhang, Z.: Supergeometric convergence of spectral collocation methods for weakly singular Volterra and Fredholm integral equations with smooth solutions. J. Comput. Math. 29(6), 698–719 (2011)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Lubich, Ch.: Runge-Kutta theory for Volterra and Abel integral equations of the second kind. Math. Comp. 41(163), 87–102 (1983)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Prudnikov, A.P., Brychkov, Yu.A., Marichev, O.I.: Integrals and series. Vol. 2. Gordon & Breach Science Publishers, New York (1986). Special functions, Translated from the Russian by N. M. QueenGoogle Scholar
  15. 15.
    Shen, J., Tang, T., Wang, L.-L.: Spectral methods, volume 41 of Springer Series in Computational Mathematics. Springer, Heidelberg (2011). Algorithms, analysis and applicationsGoogle Scholar
  16. 16.
    Szegő, G.: Orthogonal polynomials. American Mathematical Society, Providence, R.I., fourth edition, 1975. American Mathematical Society, Colloquium Publications, Vol. XXIIIGoogle Scholar
  17. 17.
    Trefethen, L.N.: Is Gauss quadrature better than Clenshaw-Curtis?. SIAM Rev. 50(1), 67–87 (2008)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling & High Performance Scientific ComputingXiamen UniversityFujianChina
  2. 2.Beijing Computational Science Research CenterBeijingChina

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