Collocation Solutions of a Weakly Singular Volterra Integral Equation

Abstract

Abstract. The discrete superconvergence properties of spline collocation solutions for a certain Volterra integral equation with weakly singular kernel are analyzed. In particular, the attainable convergence orders at the collocation points are examined for certain choices of the collocation parameters.

Numerical and asymptotic analysis of a localized heat source undergoing periodic motion

“…In general there has not been rigorous numerical analysis of these methods. See [5,8,19,20] for summaries of numerical research on such VIEs that do not exhibit blow-up in finite time. See [1,[6][7][8] for brief discussions of the very limited numerical research on such VIEs that do exhibit blow-up in finite time.…”

Numerical Solutions of a Class of Nonlinear Volterra Integral Equations

“…In our work we consider examples of (1) when = 2. We use (6) to approximate the solutions considering two special cases: 1 = 1/2 (implicit midpoint method) and 1 = 1 (implicit Euler method). We also use the repeated trapezoidal and repeated Simpson's rule.…”

“…Much work has been done in the study of numerical solutions to Volterra integral equations using collocation methods [1,[3][4][5][6][7]. Benitez and Bolos [8] pointed out that collocation methods have proven to be a very suitable technique for approximating solutions to nonlinear integral equations because of their stability and accuracy.…”

On the Analysis of Numerical Methods for Nonstandard Volterra Integral Equation

“…Moreover, [3] proved that the collocation methods yield global convergence of order . Other authors such as [4][5][6] also analysed the convergence of collocation methods for Volterra integral equations with different types of kernels. In this work we study the conditions for the existence and uniqueness of the numerical solutions of (1) and perform convergence analysis for the collocation methods and repeated trapezoidal rule.…”