Current work and open problems in the numerical analysis of Volterra functional equations with vanishing delays

Abstract

The aims of this paper are (i) to present a survey of recent advances in the analysis of superconvergence of collocation solutions for linear Volterra-type functional integral and integro-differential equations with delay functions θ(t) vanishing at the initial point of the interval of integration (with θ(t) = qt (0 < q< 1, t ≥ 0) being an important special case), and (ii) to point, by means of a list of open problems, to areas in the numerical analysis of such Volterra functional equations where more research needs to be carried out.

Key words： Volterra functional integral and integro-differential equation; vanishing delay; pantograph equation; collocation solution; optimal order of superconvergence

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