Abstract
This paper fills an important gap in the convergence analysis of collocation solutions in spaces of continuous piecewise polynomials for Volterra integral equations of the second kind. Our analysis is then extended to Volterra functional integral equations of the second kind with constant delays.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Brunner, H.: The use of splines in the numerical solution of differential and Volterra integral equations. In: Dubuc, S., Deslauriers, G. (eds.) Spline Functions and the Theory of Wavelets (Montreal 1996), CRM Proceedings and Lecture Notes, vol. 18, pp. 15–31. American Mathematical Society, Providence (1999)
Brunner, H.: Collocation Methods for Volterra Integral and Related Functional Differential Equations. Cambridge University Press, Cambridge (2004)
Brunner, H.: On the divergence of collocation solutions in smooth piecewise polynomial spaces for Volterra integral equations. BIT 44, 631–650 (2004)
El Tom, M.E.A.: On the numerical stability of spline function approximations to the solutions of Volterra integral equations of the second kind. BIT 14, 136–143 (1974)
Hairer, E., Lubich, Ch., N\(\phi \)rsett, S.P.: Order of convergence of cne-step methods for Volterra integral equations of the second kind. SIAM J. Numer. Anal. 20, 569–579 (1983)
Hung, H.S.: The numerical solution of differential and integral equations by spline functions, MRC Tech. Summary Report, No. 1053, University of Wisconsin, Madison (1970)
Kauthen, J.P., Brunner, H.: Continuous collocation approximations to solutions of first kind Volterra equations. Math. Comput. 66, 1441–1459 (1997)
N\(\phi \)rsett, S.P.: Splines and collocation for ordinary initial value problems. In: Singh, S.P. (ed.) Approximation Theory and Spline Functions (St. John’s, Nfld., 1983), NATO Adv. Sci. Inst. Ser. C Math. Phys., vol. 136, pp. 397–417. Reidel, Dordrecht (1984)
Acknowledgments
The research of Hermann Brunner was supported by the Hong Kong Research Grants Council (GRF Grant HKBU 200207, 200210), and the Natural Sciences and Engineering Research Council of Canada (NSERC Discovery Grant No. 9406). The research of Hui Liang was supported by the National Nature Science Foundation of China (No. 11101130), the Research Fund of the Heilongjiang Provincial Education Department for the Academic Backbone of the Excellent Young People (No. 1254G044), Science and Technology Innovation Team in Higher Education Institutions of Heilongjiang Province (No. 2014TD005), the Heilongjiang University Science Funds for Distinguished Young Scholars (No. JCL201303), the Natural Science Foundation of Heilongjiang Province (No. A201211). Part of the work of the first author was carried out while she was a Visiting Research Scholar at Hong Kong Baptist University (March 2010 and August 2011); she gratefully acknowledges the hospitality extended to her by HKBU’s Department of Mathematics. Especially, she is also thankful to Professor Tao Tang and Professor Hermann Brunner for their invitation to visit HKBU. The authors thank the anonymous referee for his/her careful reading of the manuscript and for the valuable comments and suggestions. They greatly improved the presentation of the results.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Anne Kværnø.
Rights and permissions
About this article
Cite this article
Liang, H., Brunner, H. On the convergence of collocation solutions in continuous piecewise polynomial spaces for Volterra integral equations. Bit Numer Math 56, 1339–1367 (2016). https://doi.org/10.1007/s10543-016-0609-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10543-016-0609-x
Keywords
- Volterra integral equations
- Collocation solutions
- Continuous piecewise polynomials
- Convergence
- Volterra functional integral equations with constant delays