On Integro-Differential Algebras

TL;DR

This paper generalizes the concept of integro-differential algebras to include a weight parameter, constructing free objects and exploring their properties to advance the algebraic framework for boundary problems of differential equations.

Contribution

It introduces weighted integro-differential algebras and provides explicit constructions of free objects generated by a base differential algebra.

Findings

Constructed free commutative integro-differential algebras with weight.

Provided explicit construction on one generator.

Studied properties of these free objects.

Abstract

The concept of integro-differential algebra has been introduced recently in the study of boundary problems of differential equations. We generalize this concept to that of integro-differential algebra with a weight, in analogy to the differential Rota-Baxter algebra. We construct free commutative integro-differential algebras with weight generated by a base differential algebra. This in particular gives an explicit construction of the integro-differential algebra on one generator. Properties of these free objects are studied.