The Analysis of Product Integration Methods for Abel's Equation using Discrete Fractional Differentiation

Abstract

This paper is concerned with the analysis of convergence of product integration methods for the generalized Abel's integral equation of the first kind. A discrete analogue of fractional differentiation is introduced and used to provide a new definition of consistency of product integration methods for this equation. A concept of zero-stability is introduced which is related to a root condition. It is shown that consistency together with zero-stability implies convergence. A specific class of high accuracy methods, the explicit backward difference product integration methods, are introduced as an example and their associated root conditions examined. It was found numerically that all members of this family of methods up to and including order s are convergent for all values of the exponent α ε (0, 1).