ON THE SOLUTIONS OF LINEAR VOLTERRA EQUATIONS OF THE SECOND KIND WITH SUM KERNELS

P.-L Giscard (1)

1 LMPA - Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville

We consider a linear Volterra integral equation of the second kind with a sum kernel Kpt 1 , tq " ř i K i pt 1 , tq 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.

Mots clés: Linear Volterra integral equations of the second kind; sum kernels; separable kernel; degenerate kernels; Neumann series; Heun functions

Domaines: Analyse classique [math.CA]

P.-L Giscard. ON THE SOLUTIONS OF LINEAR VOLTERRA EQUATIONS OF THE SECOND KIND WITH SUM KERNELS. Journal of Integral Equations and Applications, 2020, 32 (4), pp.429-445. ⟨10.1216/jie.2020.32.429⟩. ⟨hal-03597299⟩