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A Noncommutative Algebraic Operational Calculus for Boundary Problems

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Abstract

We set up a left ring of fractions over a certain ring of boundary problems for linear ordinary differential equations. The fraction ring acts naturally on a new module of generalized functions. The latter includes an isomorphic copy of the differential algebra underlying the given ring of boundary problems. Our methodology employs noncommutative localization in the theory of integro-differential algebras and operators. The resulting structure allows to build a symbolic calculus in the style of Heaviside and Mikusiński, but with the added benefit of incorporating boundary conditions where the traditional calculi allow only initial conditions. Admissible boundary conditions include multiple evaluation points and nonlocal conditions. The operator ring is noncommutative, containing all integrators initialized at any evaluation point.

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Authors and Affiliations

  1. School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury, CT2 7NF, UK

    M. Rosenkranz

  2. Research Institute for Symbolic Computation, Johannes Kepler University, 4040, Linz, Austria

    A. Korporal

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  1. M. Rosenkranz
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Correspondence to M. Rosenkranz.

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The first author acknowledges support from the EPSRC First Grant EP/I037474/1.

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Rosenkranz, M., Korporal, A. A Noncommutative Algebraic Operational Calculus for Boundary Problems. Math.Comput.Sci. 7, 201–227 (2013). https://doi.org/10.1007/s11786-013-0154-9

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