Generalized Reynolds algebras from Volterra integrals and their free construction by complete shuffle product

This paper introduces algebraic structures for Volterra integral operators with separable kernels, in the style of differential algebra for derivations and Rota-Baxter algebra for operators with kernels dependent solely on a dummy variable. We demonstrate that these operators satisfy a generalization of the algebraic identity defining the classical Reynolds operator, which is rooted in Reynolds's influential work on fluid mechanics. To study Volterra integral operators and their integral equations through this algebraic lens, particularly in providing a general form of these integral equations, we construct free objects in the category of algebras equipped with generalized Reynolds operators and the associated differential operators, termed differential Reynolds algebras. Due to the cyclic nature of the Reynolds identity, the natural rewriting rule derived from it does not terminate. To address this challenge, we develop a completion for the underlying space, where a complete shuffle product is defined for the free objects. We also include examples and applications related to Volterra integral equations.