Current work and open problems in the numerical analysis of Volterra functional equations with vanishing delays. (English) Zbl 1396.65161

Brunner, Hermann

Front. Math. China 4, No. 1, 3-22 (2009).

Summary: 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 \(\theta(t)\) vanishing at the initial point of the interval of integration (with \(\theta(t) = qt\) \((0 < q < 1\), \(t \geq 0\)) 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.

MSC:

65R20

Numerical methods for integral equations

Keywords:

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

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