Discrete Boundary Problems via Integro-differential Algebra

Undergraduate #90
Discipline: Mathematics and Statistics
Subcategory: Mathematics and Statistics

Sieu Tran - Virginia Polytechnic Institute and State University
Co-Author(s): Markus Rosenkranz, University of Kent, Canterbury, England Daniel Orr, Virginia Polytechnic Institute and State University, Blacksburg, VA

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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. 515544] 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, ∆, ∑), where ∆: (fk) ↦ (fk+1 − fk) is the forward difference operator and ∑: (fk) ↦, accord ingly 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 Grӧbner 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 [∆, ∑] can be reduced to a linear combination of monomials f φ ∑ g ψ ∆i, where i ≤ 0 and each of f, φ, g, ∑, and ψ may also be absent. Additionally, every boundary condition of |Φ), denoting the right ideal of Φ, has the normal form with aφ, i ∈ K and f ∈ F almost all zero. Finally, we always have the direct decomposition FΦ[∆,∑] = F[∆] ⊕ F[∑]⊕ (Φ), where (Φ) is a left F module. Using these ingredients and a given fundamental system of the difference operator, we construct a solution algorithm for linear boundary problems over a discrete ordinary integro-differential algebra (this algorithm closely resembles the corresponding algorithm for differential equations). We conclude with an example that might be called a discrete analog of an ill-posed boundary problem from which we extract its Green’s function.

Funder Acknowledgement(s): N/A

Faculty Advisor: Daniel Orr,