On the Divergence of Collocation Solutions in Smooth Piecewise Polynomial Spaces for Volterra Integral Equations

Abstract

In this paper we present a characterization of those smooth piecewise polynomial collocation spaces that lead to divergent collocation solutions for Volterra integral equations of the second kind. The key to these results is an equivalence result between such collocation solutions and collocation solutions in slightly smoother spaces for initial-value problems for ordinary differential equations. For the latter problems ) established a complete divergence (and convergence) theory. Our analysis can be extended to furnish divergence results for smooth collocation solutions to Volterra integral equations of the first kind. (2000): 65R20, 65L20, 65L60. AMS subject classification

Numerical approximations of second-order matrix differential equations using higher degree splines

“…Recent work in this field can be found in Refs. [3,4,18]. Recently, numerical schemes with cubic splines were extended to the resolution of second-order matrix problems without any additional increase in dimensionality of the problem [22].…”