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Superconvergent interpolants for collocation methods applied to Volterra integro-differential equations with delay

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Abstract

Standard software based on the collocation method for differential equations delivers a continuous approximation (called the collocation solution) which augments the high order discrete approximate solution that is provided at mesh points. This continuous approximation is less accurate than the discrete approximation. For ‘non-standard’ Volterra integro-differential equations with constant delay, that often arise in modeling predator-prey systems in Ecology, the collocation solution is C 0 continuous. The accuracy is O(h s+1) at off-mesh points and O(h 2s) at mesh points where s is the number of Gauss points used per subinterval and h refers to the stepsize. We will show how to construct C 1 interpolants with an accuracy at off-mesh points and mesh points of the same order (2s). This implies that even for coarse mesh selections we achieve an accurate and smooth approximate solution. Specific schemes are presented for s=2, 3, and numerical results demonstrate the effectiveness of the new interpolants.

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Acknowledgements

This research was supported in part by the Natural Sciences and Engineering Research Council of Canada. The authors would like to thank two anonymous referees for their helpful comments and suggestions that led to improvements in the paper.

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Authors and Affiliations

  1. Department of Computer Science, University of Toronto, Toronto, Canada

    Mohammad Shakourifar & Wayne Enright

Authors
  1. Mohammad Shakourifar
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  2. Wayne Enright
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Correspondence to Wayne Enright.

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Communicated by Mechthild Thalhammer.

This research was supported in part by the Natural Sciences and Engineering Research Council of Canada.

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Shakourifar, M., Enright, W. Superconvergent interpolants for collocation methods applied to Volterra integro-differential equations with delay. Bit Numer Math 52, 725–740 (2012). https://doi.org/10.1007/s10543-012-0373-5

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  • DOI: https://doi.org/10.1007/s10543-012-0373-5

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