Discrete Boundary Problems via Integro-Differential Algebra

The notion of integro-differential algebra was introduced in [Rosenkranz, M. and Regensburger, G.: Solving and Factoring Boundary Problems for Linear Ordinary Differential Equations in Differential Algebras, J. Symbolic Comput., 2008(43/8), pp. 515–544] to facilitate the algebraic study of boundary problems for linear ordinary differential equations. In this report, we construct a discrete analog in order to investigate boundary problems for difference equations. We restrict ourselves to the standard setting (F ,∆,Σ), where ∆∶ (fk) ↦ (fk+1 − fk) is the forward difference operator and Σ∶ (fk) ↦ (∑ i=0 fi) accordingly the left Riemann sum. We work here with sequences f ∶Z → C, which we write in the variable k. Key properties of the (discrete) integro-differential algebra are proven, including the discrete analog of the variation-of-constants formula.. Our next goal is to build up an algorithmic structure for specifying difference equations as well as the boundary conditions, and to solve them via integro-differential operators. We have written the relations between these operators in the form of rewrite rules, and we prove that the resulting reduction system is Noetherian and confluent. Thus it corresponds to a noncommutative Gröbner basis for the relation ideal of the operator ring. We derive the normal forms modulo this reduction system. Let F be a commutative K-algebra with f, g ∈ F and Φ be the set of all characters with φ,ψ ∈ Φ. We show that every discrete operator in FΦ[∆,Σ] can be reduced to a linear combination of monomials f φΣ g ψ∆i, where i ≥ 0 and each of f,φ,Σ, g, and ψ may also be absent. Additionally, every boundary condition of ∣Φ), denoting the right ideal of Φ, has the normal form