=ADD=
=reftype=
14
=number=
03-05
=url=
ftp://ftp.risc.uni-linz.ac.at/pub/techreports/2003/03-05.ps.gz
=year=
2003
=month=
07
=author=
Rosenkranz; Markus
=title=
The Green's Algebra: A Polynomial Approach to Boundary Value Problems
=abstract=
Although boundary value problems (BVPs) are among the most important problem types
coming from physics, chemistry and even finance, their coverage in symbolic
computation is still rather weak. Up to now, even the simplest BVPs are usually
solved either numerically or by some hand-crafted calculations, possibly supported
by some computer algebra package in an ad-hoc way. Symbolic computation does have
several tools for solving differential equations, but their application to BVPs is
largely unsatisfactory.
In this thesis, we present a new method for solving BVPs for linear ordinary
differential equations with constant coefficients. Unlike existing methods that
reduce everything to the functional level via the Green's function, our approach
works on the level of operators throughout. Our method proceeds by representing the
operators needed as noncommutative polynomials using as indeterminates basic operators
like differentation, integration, and boundary evaluation.
The crucial step for solving the BVP is to understand the desired Green's operator as a
suitable oblique Moore-Penrose inverse. The resulting equations are then solved for
the Green's operator using a carefully compiled noncommutative Gr{\"o}bner basis
that reflects the essential interactions between the basic operators.
We have implemented our method as a Mathematica package embedded into the Theorema system
developed in B.~Buchberger's group. Part of the thesis may also be regarded as a user's
manual for this implementation.
=note=
PhD Thesis
=sponsor=
FWF project F1302
=keywords=
symbolic ODE solution, noncommutative polynomials, Gr{\"o}bner bases