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 $K(t^{\prime },t)={\sum }_i{K}_i(t^{\prime },t)$ 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.

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• Heun Functions and Some of Their Applications in Physics
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Cited by in corpus (3)

• Elementary Integral Series for Heun Functions. With an Application to Black-Hole Perturbation Theory
• A Fréchet Lie group on distributions
• Exact solutions for the time-evolution of quantum spin systems under arbitrary waveforms using algebraic graph theory