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

Hermann BRUNNER*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

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

Original languageEnglish
Pages (from-to)524-537
Number of pages14
JournalJournal of Computational and Applied Mathematics
Volume228
Issue number2
DOIs
Publication statusPublished - 15 Jun 2009

Scopus Subject Areas

  • Computational Mathematics
  • Applied Mathematics

User-Defined Keywords

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

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