Applicability of spline collocation to cordial volterra equations

Abstract

We study the applicability of the standard spline collocation method, on a uniform grid, to linear Volterra integral equations of the second kind with the so-called cordial operators; these operators are noncompact and the applicability of the collocation method becomes crucial in the convergence analysis. In particular, piecewise constant, piecewise linear and piecewise quadratic collocation methods are applicable under wide, quite acceptable conditions. For higher order spline collocation, it is more complicated to carry out an analytical study of the applicability of the method; however, a numerical check is rather simple and this is illustrated by some numerical examples.