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

Hermann BRUNNER

doi:10.1016/j.cam.2008.03.024

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

Hermann BRUNNER^*

^*Corresponding author for this work

Department of Mathematics

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

34 Citations (Scopus)

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.

Original language	English
Pages (from-to)	524-537
Number of pages	14
Journal	Journal of Computational and Applied Mathematics
Volume	228
Issue number	2
DOIs	https://doi.org/10.1016/j.cam.2008.03.024
Publication status	Published - 15 Jun 2009

User-Defined Keywords

Collocation solutions
Optimal order of superconvergence
Pantograph equation
Proportional delays
Vanishing delays
Volterra functional integro-differential equations

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10.1016/j.cam.2008.03.024

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title = "Recent advances in the numerical analysis of Volterra functional differential equations with variable delays",

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 uh on uniform meshes for pantograph-type functional equations, and the analysis of collocation methods for pantograph-type functional equations with advanced arguments.",

keywords = "Collocation solutions, Optimal order of superconvergence, Pantograph equation, Proportional delays, Vanishing delays, Volterra functional integro-differential equations",

author = "Hermann BRUNNER",

note = "Funding Information: This paper is based on two invited lectures presented at the Workshop on Innovative Methods for Solving Evolution Problems with Memory (Capri, June 19–21, 2006). The research was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada. The author also gratefully acknowledges the financial support extended to him by the Universit{\`a} degli Studi di Napoli “Federico II” during a visit in June 2006.",

year = "2009",

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PY - 2009/6/15

Y1 - 2009/6/15

N2 - 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 uh on uniform meshes for pantograph-type functional equations, and the analysis of collocation methods for pantograph-type functional equations with advanced arguments.

AB - 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 uh on uniform meshes for pantograph-type functional equations, and the analysis of collocation methods for pantograph-type functional equations with advanced arguments.

KW - Collocation solutions

KW - Optimal order of superconvergence

KW - Pantograph equation

KW - Proportional delays

KW - Vanishing delays

KW - Volterra functional integro-differential equations

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DO - 10.1016/j.cam.2008.03.024

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SN - 0377-0427

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SP - 524

EP - 537

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

IS - 2

ER -

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

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