A Unified Approach to Convergence Analysis of Discretization Methods for Volterra-typeEquations

Abstract

This paper presents a general convergence analysis of numerical methods for solving ordinary differential equations and non-linear Voltcrra integral and integrodifferential equations. The concept of analytic and discrete fundamental forms is introduced. Prolongation and restriction operators reduce the problem of comparing the analytic and numerical solutions to that of considering the effect of perturbations in the fundamental forms. Integral inequalities and their discrete analogues are then employed to derive error estimates. The theory is illustrated by a convergence proof of a collocation method for solving Volterra integral equations of the second kind.