Symbolic Computation with Integro-Differential Operators

The algebraic and algorithmic study of integro-differential algebras and operators has only started in the past decade. Integro-differential operators allow us in particular to study initial value and boundary problems for linear ODEs from an algebraic point of view. Differential operators already provide a rich algebraic structure with a wealth of results and algorithmic methods. Adding integral operators and evaluations, many new phenomena appear, including zero devisors and non-finitely generated ideals. In this tutorial, we give an introduction to symbolic methods for integro-differential operators and boundary problems developed over the last years. In particular, we discuss normal forms, basic algebraic properties, and the computation of polynomial solutions for ordinary integro-differential equations with polynomial coefficients. We will also outline methods for manipulating and solving linear boundary problems and illustrate them with an implementation.