Buchberger, Bruno and Rosenkranz, Markus (2012) Transforming boundary problems from analysis to algebra: A case study in boundary problems. Journal of Symbolic Computation, 47 (6). pp. 589-609. ISSN 0747-7171.
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In this paper, we summarize our recent work on establishing, for the first time, an algorithm for the symbolic solution of linear boundary problems. We put our work in the frame of Wen-Tsun Wu's approach to algorithmic problem solving in analysis, geometry, and logic by mapping the significant aspects of the underlying domains into algebra. We briefly compare this with the lines of thought of Wolfgang Groebner. For building up the necessary tower of domains in a generic and flexible way, we use the machinery of algorithmic functors introduced in our Theorema project. The essence of this concept is explained in the first section of the paper. The main part of the paper then describes our symbolic analysis approach to linear boundary problems, which hinges on three basic principles: (1) Differentiation as well as integration is treated axiomatically, setting up an algebraic data structure that can encode the problem statement (differential equation and boundary conditions) and suitable symbolic expressions for their solution (Green's operators qua integral operators). (2) Abstract boundary problems are introduced as pairs consisting of an epimorphism on a vector space (abstract differential operator) and a subspace of its dual (abstract boundary conditions). (3) Operator algebras are treated by noncommutative polynomials, modulo Groebner bases for certain relation ideals.
|Uncontrolled keywords:||Symbolic analysis, linear boundary problems, Wen-Tsun Wu, Wolfgang Groebner, Theorema, algorithmic functors, Groebner bases, integro-differential algebras, Baxter algebras|
|Subjects:||Q Science > QA Mathematics (inc Computing science) > QA150 Algebra
Q Science > QA Mathematics (inc Computing science) > QA372 Ordinary differential equations
|Divisions:||Faculties > Science Technology and Medical Studies > School of Mathematics Statistics and Actuarial Science > Applied Mathematics|
|Depositing User:||M.G. Rosenkranz|
|Date Deposited:||03 Apr 2012 14:53|
|Last Modified:||04 Apr 2012 08:09|
|Resource URI:||http://kar.kent.ac.uk/id/eprint/29247 (The current URI for this page, for reference purposes)|
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