Paper Info
Title
Transforming problems from analysis to algebra: A case study in linear boundary problems
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 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.
Year
DOI
Venue
2012
10.1016/j.jsc.2011.12.022
J. Symb. Comput.
Keywords
DocType
Volume
Wolfgang Groebner,symbolic solution,linear boundary problem,boundary condition,suitable symbolic expression,algorithmic functors,Transforming problem,abstract boundary problem,abstract boundary condition,symbolic analysis approach,case study,abstract differential operator
Journal
47
Issue
ISSN
Citations 
6
0747-7171
1
PageRank 
References 
Authors
0.36
9
2
Name
Order
Citations
PageRank
Bruno Buchberger1847168.26
Markus Rosenkranz217516.66