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

Autores: Hermann Brunner
Localización: Journal of computational and applied mathematics, ISSN 0377-0427, Vol. 228, Nº 2, 2009 (Ejemplar dedicado a: Innovative methods for solving evolutionary problems), págs. 524-537
Idioma: inglés
DOI: 10.1016/j.cam.2008.03.024
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Resumen
- 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.