In the traditional step-by-step collocation method with quadratic splines for Volterra integral equations an initial condition is replaced by a not-a-knot boundary condition at the other end of the interval. Such a nonlocal method gives the uniform boundedness of collocation projections for all parameters c in (0,1) characterizing the position of collocation points between spline knots. For c = 1 the projection norms have linear growth and, therefore, for any choice of c some general convergence theorems may be applied to establish the convergence with two-sided error estimates. The numerical tests supporting the theoretical results are also presented.
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Peeter Oja, Darja Saveljeva, Quadratic Spline Collocation for Volterra Integral Equation. Z. Anal. Anwend. 23 (2004), no. 4, pp. 833–854