On the solutions of linear Volterra equations of the second kind with sum kernels

TL;DR

This paper presents a new method for solving linear Volterra equations with sum kernels, providing a series representation with better convergence and applying it to solve equations involving Heun's confluent functions.

Contribution

It introduces a novel solution approach for Volterra equations with sum kernels and offers a new series representation with improved convergence properties.

Findings

Derived solutions in terms of separate kernel equations

Provided the first known solution involving Heun's confluent functions

Demonstrated widespread applicability in physics

Abstract

We consider a linear Volterra integral equation of the second kind with a sum kernel $K(t',t)=\sum_i K_i(t',t)$ and give the solution of the equation in terms of solutions of the separate equations with kernels $K_i$, provided these exist. As a corollary, we obtain a novel series representation for the solution with improved convergence properties. We illustrate our results with examples, including the first known Volterra equation solved by Heun's confluent functions. This solves a long-standing problem pertaining to the representation of such functions. The approach presented here has widespread applicability in physics via Volterra equations with degenerate kernels.