=ADD=
=reftype=
14
=number=
02-26
=url=
ftp://ftp.risc.uni-linz.ac.at/pub/techreports/2002/02-26.ps.gz
=year=
2002
=month=
12
=author=
Hillgarter; Erik
=title=
A Contribution to the Symmetry Classification Problem for 2nd Order PDEs in one Dependent and two Independent Variables
=abstract=
In this thesis, we provide a contribution to the symmetry classification
problem for partial differential equations of order two in one dependent
and two independent variables, i.e. to an overview of all possible
symmetry groups admitted by this class of equations.

Symmetries are the most important means for finding closed form
solutions of non-linear differential equations (DEs for short). The
study of symmetries has been initiated by Sophus Lie (1842-1899).
Roughly speaking, a symmetry is a point- or contact-transformation that
does not change the form of a DE. Obviously, the entirety of symmetry
transformations of any given DE forms a group. The term symmetry group
is applied to the largest group of transformations with this property.
The symmetry problem for DEs comes in various versions. We are concerned
with the classification problem for point symmetries. It aims at
obtaining a complete survey of all possible symmetries for a class of
given DE's, e.g. DEs of a given order in a predetermined number of
dependent and independent variables. The starting point of this approach
is always a listing of groups, whose differential invariants determine
the general form of a DE that may be invariant under the respective
group.

Sophus Lie determined all continuous transformation groups of the two
dimensional plane and gave normal forms for any ordinary DE that is
invariant under one of those groups. We start with the point symmetries
of the three dimensional dimensional space and compute the differential
invarints for a big part of them.

=note=
PhD Thesis
=sponsor=
SFB project F1304, RISC PhD scholarship program of the government of Upper Austria
=keywords=
partial differential equations