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Collocation methods for second-kind Volterra integral equations with weakly singular kernels

Published online by Cambridge University Press:  14 November 2011

Teresa Diogo ,
Sean McKee  and
Tao Tang
Teresa Diogo
Affiliation:
Department of Mathematics, University of Strathclyde, Glasgow G1 1XH, Scotland, U.K.
Sean McKee
Affiliation:
Department of Mathematics, University of Strathclyde, Glasgow G1 1XH, Scotland, U.K.
Tao Tang
Affiliation:
Department of Mathematics and Statistics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
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Abstract

In this paper it is shown that the use of uniform meshes leads to optimal convergence rates provided that the analytic solutions of a particular class of Volterra integral equations (VIEs) are smooth. If the exact solutions are not smooth, however, suitable transformations can be made so that the new VIEs possess smooth solutions. Spline collocation methods with uniform meshes applied to these new VIEs are then shown to be able to yield optimal (global) convergence rates. The general theory is applied to a typical case, i.e. the integral kernels consisting of the singular term (t − s) −½.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 124 , Issue 2 , 1994 , pp. 199 - 210
Copyright
Copyright © Royal Society of Edinburgh 1994

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