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

Hermann BRUNNER

doi:10.1007/s11464-009-0001-0

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

Hermann BRUNNER^*

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal › Journal article › peer-review

29 Citations (Scopus)

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 ia(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.

Original language	English
Pages (from-to)	3-22
Number of pages	20
Journal	Frontiers of Mathematics in China
Volume	4
Issue number	1
DOIs	https://doi.org/10.1007/s11464-009-0001-0
Publication status	Published - Mar 2009

Scopus Subject Areas

Mathematics (miscellaneous)

User-Defined Keywords

Collocation solution
Optimal order of superconvergence
Pantograph equation
Vanishing delay
Volterra functional integral and integro-differential equation

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10.1007/s11464-009-0001-0

Cite this

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title = "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 ia(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.",

keywords = "Collocation solution, Optimal order of superconvergence, Pantograph equation, Vanishing delay, Volterra functional integral and integro-differential equation",

author = "Hermann BRUNNER",

note = "Funding Information: Acknowledgements The research was supported in part by the Natural Sciences and Engineering Research Council of Canada and by the Hong Kong Research Grants Council. The author also gratefully acknowledges the hospitality extended to him by the Department of Mathematics of the Hong Kong Baptist University where a substantial portion of this research was carried out.",

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doi = "10.1007/s11464-009-0001-0",

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AU - BRUNNER, Hermann

N1 - Funding Information: Acknowledgements The research was supported in part by the Natural Sciences and Engineering Research Council of Canada and by the Hong Kong Research Grants Council. The author also gratefully acknowledges the hospitality extended to him by the Department of Mathematics of the Hong Kong Baptist University where a substantial portion of this research was carried out.

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N2 - 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 ia(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.

AB - 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 ia(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.

KW - Collocation solution

KW - Optimal order of superconvergence

KW - Pantograph equation

KW - Vanishing delay

KW - Volterra functional integral and integro-differential equation

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SN - 1673-3452

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JO - Frontiers of Mathematics in China

JF - Frontiers of Mathematics in China

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ER -

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

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