A reduction algorithm for Volterra integral equations

TL;DR

This paper introduces a systematic reduction algorithm for separable Volterra integral equations, transforming them into an operator-linear form using algebraic structures and combinatorial tools.

Contribution

It presents a novel algorithm that standardizes Volterra integral equations into a form with only iterated integrals, utilizing twisted Rota-Baxter identity and decorated trees.

Findings

Algorithm effectively reduces integral equations to operator-linear form

Standardizes the presentation of Volterra integral equations

Uses algebraic and combinatorial methods for the reduction process

Abstract

An integral equation is a way to encapsulate the relationships between a function and its integrals. We develop a systematic way of describing Volterra integral equations -- specifically an algorithm that reduces any separable Volterra integral equation into an equivalent one in operator-linear form, i.e. one that only contains iterated integrals. This serves to standardize the presentation of such integral equations so as to only consider those containing iterated integrals. We use the algebraic object of the integral operator, the twisted Rota-Baxter identity, and vertex-edge decorated rooted trees to construct our algorithm.