Collocation methods for functional integro-differential equations with proportional delays

Hermann Brunner

Department of Mathematics & Statistics, Memorial University of Newfoundland, St. John's, NF, Canada

hermann@math.mun.ca

Contributed talk

While the global convergence properties of piecewise polynomial collocation solutions to Volterra functional integro-differential equations of the form

$$y^{(k)}(t) = a(t)y(t) + b(t)y(qt) + (V_{q}y)(t), \quad t \in I := [0,T] \;\; (k \geq 1),$$ (1)

with

$$(V_{q}y)(t) := \int _{qt}^{t} K(t,s)y(s)ds, \quad 0 < q < 1,$$

are now well understood (e.g. [2]), the analysis of optimal superconvergence and asymptotic stability for uniform meshes remains largely open. In this talk I shall survey recent work on local and global superconvergence results and numerical stability for certain types of geometric meshes ([3,1]) and then describe current approaches for dealing with the analogous analysis on uniform meshes. A discussion of open problems, e.g. the convergence of collocation solutions for $(V_{q}y)(t) = g(t)$ (which may for example occur as part of a system of ``integral-algebraic'' equations), will complement this presentation.