Superconvergence results for linear second-kind Volterra integral equations

Original Research


In this paper, Galerkin method is applied to approximate the solution of Volterra integral equations of second kind with a smooth kernel, using piecewise polynomial bases. We prove that the approximate solutions of the Galerkin method converge to the exact solution with the order \({\mathcal {O}}(h^{r}),\) whereas the iterated Galerkin solutions converge with the order \({\mathcal {O}}(h^{2r})\) in infinity norm, where h is the norm of the partition and r is the smoothness of the kernel. We also consider the multi-Galerkin method and its iterated version, and we prove that the iterated multi-Galerkin solution converges with the order \({\mathcal {O}}(h^{3r})\) in infinity norm. Numerical examples are given to illustrate the theoretical results.


Volterra integral equations Smooth kernels Galerkin method Multi-Galerkin method Piecewise polynomials Superconvergence rates 

Mathematics Subject Classification

45B05 45G10 65R20 


  1. 1.
    Ahues, M., Largillier, A., Limaye, B.: Spectral computations for bounded operators. CRC Press, Boca Raton (2001)CrossRefMATHGoogle Scholar
  2. 2.
    Atkinson, K.: The numerical solution of integral equations of the second kind, vol. 4. Cambridge University Press, Cambridge (1997)CrossRefMATHGoogle Scholar
  3. 3.
    Brunner, H.: Iterated collocation methods and their discretizations for Volterra integral equations. SIAM J. Numer. Anal. 21(6), 1132–1145 (1984)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Brunner, H.: Collocation methods for Volterra integral and related functional differential equations, vol. 15. Cambridge University Press, Cambridge (2004)CrossRefMATHGoogle Scholar
  5. 5.
    Brunner, H., Ningning, Y.: On global superconvergence of iterated collocation solutions to linear second-kind Volterra integral equations. J. Comput. Appl. Math. 67(1), 185–189 (1996)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Brunner, H., Qun, L., Ningning, Y.: The iterative correction method for Volterra integral equations. BIT 36(2), 221–228 (1996)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Chatelin, F.: Spectral approximation of linear operators ci. SIAM, Philadelphia (2011)CrossRefMATHGoogle Scholar
  8. 8.
    Chen, Z., Long, G., Nelakanti, G.: The discrete multi-projection method for Fredholm integral equations of the second kind. J. Integral Equ. Appl. 19(2), 143–162 (2007)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Graham, I.G., Sloan, I.H.: On the compactness of certain integral operators. J. Math. Anal. Appl. 68(2), 580–594 (1979)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Kulkarni, R.P.: A superconvergence result for solutions of compact operator equations. Bull. Aust. Math. Soc. 68(3), 517–528 (2003)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Lin, Q., Shi, J.: Iterative corrections and a posteriori error estimate for integral-equations. J. Comput. Math. 11(4), 297–300 (1993)MathSciNetMATHGoogle Scholar
  12. 12.
    Long, G., Nelakanti, G.: Iteration methods for Fredholm integral equations of the second kind. Comput. Math. Appl. 53(6), 886–894 (2007)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Rudin, W.: Real and complex analysis. Tata McGraw-Hill Education, New York (1987)MATHGoogle Scholar
  14. 14.
    Sloan, I.H.: Improvement by iteration for compact operator equations. Math. Comput. 30(136), 758–764 (1976)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Tang, T., Xu, X., Cheng, J.: On spectral methods for volterra integral equations and the convergence analysis. J. Comput. Math. 26(6), 825–837 (2008)MathSciNetMATHGoogle Scholar
  16. 16.
    Wan, Z., Chen, Y., Huang, Y.: Legendre spectral Galerkin method for second-kind Volterra integral equations. Front. Math. China 4(1), 181–193 (2009)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Xie, Z., Li, X., Tang, T.: Convergence analysis of spectral Galerkin methods for Volterra type integral equations. J. Sci. Comput. 53(2), 414–434 (2012)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Zhang, S., Lin, Y., Rao, M.: Numerical solutions for second-kind Volterra integral equations by Galerkin methods. Appl. Math. 45(1), 19–39 (2000)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2017

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology KharagpurKharagpurIndia

Personalised recommendations