The Algebra of Integro-Differential Polynomials
by
Georg Regensburger
Johann Radon Institute for Computational and Applied Mathematics (RICAM)
Coauthors: Markus Rosenkranz

Motivated by boundary value problems for linear ordinary differential equations, we introduced the notion of an integro-differential algebra: a differential algebra together with an integral operator satisfying the Baxter rule (ïntegration by parts"). For studying extensions of such algebras, we use the algebra of integro-differential polynomials. They can be understood in the sense of Lausch-Nöbauer as the polynomial algebra in the variety of integro-differential algebras. In order to allow computations in this domain, we have constructed a canonical simplifier that solves the corresponding word problem. While most operations can be defined in a straight-forward way, the multiplication is defined by the shuffle product and the integral by a careful case distinction on the differential exponents. Passing to a suitable quotient of the resulting polynomial algebra, we can compute (sum, product, derivative, integral) with formal solutions of a given nonlinear differential equation with intitial conditions.

Date received: April 26, 2008