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 language | English |
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Pages (from-to) | 524-537 |
Number of pages | 14 |
Journal | Journal of Computational and Applied Mathematics |
Volume | 228 |
Issue number | 2 |
DOIs | |
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