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On the Divergence of Collocation Solutions in Smooth Piecewise Polynomial Spaces for Volterra Integral Equations

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Abstract

In this paper we present a characterization of those smooth piecewise polynomial collocation spaces that lead to divergent collocation solutions for Volterra integral equations of the second kind. The key to these results is an equivalence result between such collocation solutions and collocation solutions in slightly smoother spaces for initial-value problems for ordinary differential equations. For the latter problems Mülthei (1979/1980) established a complete divergence (and convergence) theory. Our analysis can be extended to furnish divergence results for smooth collocation solutions to Volterra integral equations of the first kind.

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Authors and Affiliations

  1. Dept. of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, Canada

    Hermann Brunner

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  1. Hermann Brunner
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Correspondence to Hermann Brunner.

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AMS subject classification (2000)

65R20, 65L20, 65L60.

Received May 2004. Accepted September 2004. Communicated by Tom Lyche.

Hermann Brunner: This research was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).

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Brunner, H. On the Divergence of Collocation Solutions in Smooth Piecewise Polynomial Spaces for Volterra Integral Equations. Bit Numer Math 44, 631–650 (2004). https://doi.org/10.1007/s10543-004-3828-5

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