Details:
Title  Transforming problems from analysis to algebra: A case study in linear boundary problems  Author(s)  Bruno Buchberger, Markus Rosenkranz  Type  Article in Journal  Abstract  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 WenTsun 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.  Keywords  Symbolic analysis, Linear boundary problems, WenTsun Wu, Wolfgang Groebner, Theorema, Algorithmic functors, Groebner bases, Integrodifferential algebras, Baxter algebras  ISSN  07477171 
URL 
http://www.sciencedirect.com/science/article/pii/S074771711100215X 
Language  English  Journal  Journal of Symbolic Computation  Volume  47  Number  6  Pages  589  609  Year  2012  Note  Advances in Mathematics Mechanization Mathematics Mechanization  Edition  0  Translation 
No  Refereed 
No 
