\begin{center}
\LARGE
The impact of computer algebras on the theory and
        applications of conditional symmetries of PDE
\end{center}

\begin{center}
\Large
E.M. Vorob'ev

\end{center}

\noindent
Date:  July 20th (Saturday)\\
Time: 11:30-12:00\\

\begin{center}
\large
Abstract
\end{center}

Nonclassical conditional symmetries of PDE, which are one of the possible generalizations of
Lie's classical symmetries, proved to be a fruitful tool for obtaining exact explicit solutions of
nonlinear PDE. They constitute a rapidly developing area of symmetry research at least during the
last decade. 

Similar to their classical counterpart, obtaining such symmetries is based on solving the so called
``determining equations'' (DE) for the infinitesimal symmetries. In the classicalcase, these equtions
form the overdetermined linear systems of PDEs, and the degree of their redundancy permits one
to easily solve them. In contrast to Lie's symmetries, the DE for conditional symmetries are
nonlinear and less overdetermined. This explains that as a rule only particular solutions of
nonclassical DE were obtained even in the case of PDEs in two independent variables. The
situation was essentially improved after several authors had developed computer algebras based
algorithms for reducing the overdetermined systems to passive, or involutive forms. 

But even with such programs, it takes too much time and sometimes too many computer resources
to get the general solution of nonclassical DE. Here two ways out are possible: either to restrict
oneself to particular but well determined classes of conditional symmetries or to develop more
efficient algorithms and packages.One of the most attractive ways in the spirit of symmetry theory
is to restrict the family of conditional symmetries by taking into account the symmetry properties
of the conditional symmetries w.r.t. the classical ones. This allows to get conditional symmetries
with an additional advantage that the quotient equations for the invariant solutions inherit the
classical symmetries. After conditional symmetries are obtained, the problem of solving the
quotient equations (QE) appears. In the case of PDEs in two independent variables, the QE are
ODEs but in general they are not explicitly solvable. Here we again can take advantage of
computer algebra systems supplied with the means for numerical solutionof ODEs and graphical
representation of the approximate solutions. 

All the considerations made above are illustrated with the author's results in symmetry research
with the ``Mathematica'' program SYMMAN. 

