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

Abstract

We consider a linear Volterra integral equation of the second kind with a sum kernel $K\left(t^{\prime },t\right)={\sum }_i{K}_i\left(t^{\prime },t\right)$ 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.