Recent advances in the numerical analysis of Volterra functional differential equations with variable delays

Authors:
Hermann Brunner
Affiliations:
Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, NL, Canada A1C 5S7
Venue:
Journal of Computational and Applied Mathematics
Year:
2009

Citing 13
Cited 2

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Optimal Superconvergence Orders of Iterated Collocation Solutions for Volterra Integral Equations with Vanishing Delays

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Reliable Approximate Solution of Systems of Volterra Integro-Differential Equations with Time-Dependent Delays

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Legendre spectral-collocation method for Volterra integral equations with non-vanishing delay

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Abstract

The numerical analysis of Volterra functional integro-differential equations with vanishing delays has to overcome a number of challenges that are not encountered when solving 'classical' delay differential equations with non-vanishing delays. In this paper I shall describe recent results in the analysis of optimal (global and local) superconvergence orders in collocation methods for such evolutionary problems. Following a brief survey of results for equations containing Volterra integral operators with non-vanishing delays, the discussion will focus on pantograph-type Volterra integro-differential equations with (linear and nonlinear) vanishing delays. The paper concludes with a section on open problems; these include the asymptotic stability of collocation solutions u"h on uniform meshes for pantograph-type functional equations, and the analysis of collocation methods for pantograph-type functional equations with advanced arguments.