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Mathematics in Computer Science

, Volume 7, Issue 2, pp 201–227 | Cite as

A Noncommutative Algebraic Operational Calculus for Boundary Problems

  • M. Rosenkranz
  • A. Korporal
Article

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.

Keywords

Linear boundary problems Differential algebra Mikusiński calculus Integro-differential operators Ring of fractions 

Mathematics Subject Classification (2000)

34B10 13N10 44A40 

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Authors and Affiliations

  1. 1.School of Mathematics, Statistics and Actuarial ScienceUniversity of KentCanterburyUK
  2. 2.Research Institute for Symbolic ComputationJohannes Kepler UniversityLinzAustria

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