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\chapter[SPDE: Symmetry groups of {PDE}'s]%
        {SPDE: A package for finding symmetry groups of {PDE}'s}
\label{SPDE}
\typeout{{SPDE: A package for finding symmetry groups of {PDE}'s}}

{\footnotesize
\begin{center}
Fritz Schwarz \\ 
GMD, Institut F1 \\
Postfach 1240 \\ 
5205 St. Augustin, Germany \\[0.05in]
e--mail: fritz.schwarz@gmd.de
\end{center}
}
\ttindex{SPDE}

The package SPDE provides a set of functions which may be applied
to determine the symmetry group of Lie- or point-symmetries of a
given system of partial differential equations.  Preferably it is
used interactively on a computer terminal. In many cases the
determining system is solved completely automatically. In some
other cases the user has to provide some additional input
information for the solution algorithm to terminate. 

\section{System Functions and Variables}

The symmetry analysis of partial differential equations logically
falls into three parts. Accordingly the most important functions
provided by the package are:

\begin{table}
\begin{center}
\begin{tabular}{| c | c | }\hline
Function name & Operation \\ \hline \hline
\ttindex{CRESYS}
CRESYS(\s{arguments}) & Constructs determining system \\ \hline
\ttindex{SIMPSYS}
SIMPSYS() & Solves determining system \\ \hline
\ttindex{RESULT}
RESULT() & Prints infinitesimal generators \\
&  and commutator table \\ \hline
\end{tabular}
\end{center}
\caption{SPDE Functions}
\end{table}

Some other useful functions for obtaining various kinds of output
are:

\begin{table}
\begin{center}
\begin{tabular}{| c | c |} \hline
Function name & Operation \\ \hline \hline
\ttindex{PRSYS}
PRSYS() & Prints determining system \\ \hline
\ttindex{PRGEN}
PRGEN() & Prints infinitesimal generators \\ \hline
\ttindex{COMM}
COMM(U,V) & Prints commutator of generators U and V \\ \hline
\end{tabular}
\end{center}
\caption{SPDE Useful Output Functions}\label{spde:useful}
\end{table}

SPDE expects a system of differential equations to be defined as the
values of the operator {\tt deq} and other operators.  A simple
example follows.

\begin{verbatim}
load_package spde;

deq 1:=u(1,1)+u(1,2,2);

deq(1) := u(1,2,2) + u(1,1)

CRESYS deq 1;

PRSYS();


GL(1):=2*df(eta(1),u(1),x(2)) - df(xi(2),x(2),2) - df(xi(2),x(1))

GL(2):=df(eta(1),u(1),2) - 2*df(xi(2),u(1),x(2))

GL(3):=df(eta(1),x(2),2) + df(eta(1),x(1))

GL(4):=df(xi(2),u(1),2)

GL(5):=df(xi(2),u(1)) - df(xi(1),u(1),x(2))

GL(6):=2*df(xi(2),x(2)) - df(xi(1),x(2),2) - df(xi(1),x(1))

GL(7):=df(xi(1),u(1),2)

GL(8):=df(xi(1),u(1))

GL(9):=df(xi(1),x(2))


The remaining dependencies

xi(2) depends on u(1),x(2),x(1)

xi(1) depends on u(1),x(2),x(1)

eta(1) depends on u(1),x(2),x(1)
\end{verbatim}

A detailed description can be found in the SPDE documentation and
examples.