Abstract
Volterra integral equations of the second kind with weakly singular kernels possess, in general, solutions which are not smooth near the left endpoint of the interval of integration. Since ordinary polynomial spline collocation cannot lead to high-order convergence we introduce special nonpolynomial spline spaces which are modelled after the structure of these solutions near the point of nonsmooth behavior; collocation in these spaces will once more lead to high-order methods. Analogous results are derived for Volterra integro-differential equations with weakly singular kernels.
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