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The Gauss Algorithm reduces systems of linear algebraic equations
to a triangular or standard form.
A first guess for an analogous method for linear systems of differential
equations is to apply the Gauss algorithm to such systems
with their derivatives and dependent variables regarded as unknowns.
An example of such a Gauss-reduced form is the system ,
, for
where denotes
and
denotes
.
Differentiation of the first equation with respect to and the
second with respect to yields the integrability condition
. When reduced subject to the
system this condition becomes .
Thus generalizing the Gauss
Algorithm to systems of linear PDEs, requires the
consideration of their differential consequences.
Such a generalization was given by Riquier [69] early this century.
In rough terms, for linear systems his algorithm repeats the steps of
Gauss reducing the system, calculating and reducing its integrability conditions,
adjoining nontrivial integrability conditions to the system,
until there are no nontrivial integrability conditions.
A number of such algorithms have been implemented (see
[77,89,9] and the extensive bibliography
of our paper [65]).
Our standard form algorithm [57] uses similar principles to produce
a coordinate dependent standard form of linear
system of differential equations.
The reduced form produced by ours and Riquier's algorithms
does not solve such systems.
However it does provide a count of the number of parameters
in their local analytic solutions [78].
It can substantially simplify them, and can be used to test
whether a differential expression is a consequence of the system.
It also yields a form of the system to which local existence-uniqueness
theorems can be applied. These theorems
generalize the Cauchy-Kovaleskaya Theorem [87,88,41].
Such algorithms have proved their worth in simplifying complicated
overdetermined systems of linear PDEs;
especially those arising in the Lie symmetry analysis of physical PDEs.
They are becoming a standard part of computer algebra
packages for simplifying and solving linear PDEs
(see the references in [65] and especially those in the review article [34]).

** Next:** Buchberger's algorithm for polynomially
** Up:** Background and Motivation
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Greg Reid
2003-11-24