Regular singular Volterra equations on complex domains

TL;DR

This paper investigates regular singular Volterra equations on complex domains, demonstrating the existence and uniqueness of solutions for a specific class, which aids in solving differential equations via Laplace transforms.

Contribution

It establishes a theoretical result on the existence and uniqueness of solutions for regular singular Volterra equations on complex domains, extending the understanding of their structure.

Findings

Unique solutions exist for the considered class of equations.

The results facilitate solving differential equations with irregular singularities.

The approach simplifies complex differential equations through Laplace transform techniques.

Abstract

The inverse Laplace transform can turn a linear differential equation on a complex domain into an equivalent Volterra integral equation on a real domain. This can make things simpler: for example, a differential equation with irregular singularities can become a Volterra equation with regular singularities. It can also reveal hidden structure, especially when the Volterra equation extends to a complex domain. Our main result is to show that for a certain kind of regular singular Volterra equation on a complex domain, there is always a unique solution of a certain form. As a motivating example, this kind of Volterra equation arises when using Laplace transform methods to solve a level 1 differential equation.