Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ahues, M., Largillier, A., Limaye, B.: Spectral computations for bounded operators. CRC Press, Boca Raton (2001)
Atkinson, K.: The numerical solution of integral equations of the second kind, vol. 4. Cambridge University Press, Cambridge (1997)
Brunner, H.: Iterated collocation methods and their discretizations for Volterra integral equations. SIAM J. Numer. Anal. 21(6), 1132–1145 (1984)
Brunner, H.: Collocation methods for Volterra integral and related functional differential equations, vol. 15. Cambridge University Press, Cambridge (2004)
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)
Brunner, H., Qun, L., Ningning, Y.: The iterative correction method for Volterra integral equations. BIT 36(2), 221–228 (1996)
Chatelin, F.: Spectral approximation of linear operators ci. SIAM, Philadelphia (2011)
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)
Graham, I.G., Sloan, I.H.: On the compactness of certain integral operators. J. Math. Anal. Appl. 68(2), 580–594 (1979)
Kulkarni, R.P.: A superconvergence result for solutions of compact operator equations. Bull. Aust. Math. Soc. 68(3), 517–528 (2003)
Lin, Q., Shi, J.: Iterative corrections and a posteriori error estimate for integral-equations. J. Comput. Math. 11(4), 297–300 (1993)
Long, G., Nelakanti, G.: Iteration methods for Fredholm integral equations of the second kind. Comput. Math. Appl. 53(6), 886–894 (2007)
Rudin, W.: Real and complex analysis. Tata McGraw-Hill Education, New York (1987)
Sloan, I.H.: Improvement by iteration for compact operator equations. Math. Comput. 30(136), 758–764 (1976)
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)
Wan, Z., Chen, Y., Huang, Y.: Legendre spectral Galerkin method for second-kind Volterra integral equations. Front. Math. China 4(1), 181–193 (2009)
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)
Zhang, S., Lin, Y., Rao, M.: Numerical solutions for second-kind Volterra integral equations by Galerkin methods. Appl. Math. 45(1), 19–39 (2000)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Mandal, M., Nelakanti, G. Superconvergence results for linear second-kind Volterra integral equations. J. Appl. Math. Comput. 57, 247–260 (2018). https://doi.org/10.1007/s12190-017-1104-5
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12190-017-1104-5
Keywords
- Volterra integral equations
- Smooth kernels
- Galerkin method
- Multi-Galerkin method
- Piecewise polynomials
- Superconvergence rates