Quadratic Spline Collocation for Volterra Integral Equation

Abstract

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.