Collocation Methods for Weakly Singular Second-kind Volterra Integral Equations with Non-smooth Solution

Abstract

Collocation type methods are studied for the numerical solution of the weakly singular Volterra integral equation of the second kind: $f(t)=g(t)+∫0tK(s,f(s))(t−s)−1ds,t∈[0,T]$ where the solution ƒ(t) is assumed to have the form f(t) = x(t)+r^½ψ(t), x and ψ being sufficiently smooth. The solution is approximated near zero by a linear combination of powers of t½, and away from zero by the usual polynomial representation. Convergence is proved and many numerical experiments are carried out with examples from the literature. A comparison is made with a method of Brunner & Norsett (1981), originally developed for (1) with a smooth solution. Special attention is paid to the numerical approximation of the so-called moment integrals which emerge in the collocation scheme.