\chapter[APPLYSYM: Infinitesimal symmetries]{APPLYSYM: Infinitesimal symmetries of differential equations}
\label{APPLYSYM}
\typeout{[APPLYSYM: Infinitesimal symmetries]}
{\footnotesize
\begin{center}
Thomas Wolf \\
School of Mathematical Sciences, Queen Mary and Westfield College \\
University of London \\
London E1 4NS, England \\[0.05in]
e--mail: T.Wolf@maths.qmw.ac.uk
\end{center}
}
The investigation of infinitesimal symmetries of differential equations
(DEs) with computer algebra programs attracted considerable attention
over the last years. The package {\tt APPLYSYM} concentrates on the
implementation of applying symmetries for calculating similarity
variables to perform a point transformation which lowers the order of
an ODE or effectively reduces the number of explicitly occuring
independent variables of a PDE(-system) and for generalising given
special solutions of ODEs/PDEs with new constant parameters.
A prerequisite for applying symmetries is the solution of first order
quasilinear PDEs. The corresponding program
{\tt QUASILINPDE}\ttindex{QUASILINPDE} can as well be used without
{\tt APPLYSYM}\ttindex{APPLYSYM} for solving first order PDEs which are
linear in their first order derivative and otherwise at most rationally
non-linear. The following two PDEs are equations (2.40) and (3.12)
taken from E. Kamke, "Loesungsmethoden und Loesungen von Differential-
gleichungen, Partielle Differentialgleichungen erster Ordnung",
B.G. Teubner, Stuttgart (1979).
\newpage
{\small
\begin{verbatim}
------------------------ Equation 2.40 ------------------------
2 3 4
The quasilinear PDE: 0 = df(z,x)*x*y + 2*df(z,y)*y - 2*x
2 2 2
+ 4*x *y*z - 2*y *z .
The equivalent characteristic system:
3 4 2 2 2
0=2*(df(z,y)*y - x + 2*x *y*z - y *z )
2
0=y *(2*df(x,y)*y - x)
for the functions: x(y) z(y) .
The general solution of the PDE is given through
4 2 2
log(y)*x - log(y)*x *y*z - y *z sqrt(y)*x
0 = ff(----------------------------------,-----------)
4 2 y
x - x *y*z
with arbitrary function ff(..).
------------------------ Equation 3.12 ------------------------
The quasilinear PDE: 0 = df(w,x)*x + df(w,y)*a*x + df(w,y)*b*y
+ df(w,z)*c*x + df(w,z)*d*y + df(w,z)*f*z.
The equivalent characteristic system:
0=df(w,x)*x
0=df(z,x)*x - c*x - d*y - f*z
0=df(y,x)*x - a*x - b*y
for the functions: z(x) y(x) w(x) .
The general solution of the PDE is given through
a*x + b*y - y
0 = ff(---------------,( - a*d*x + b*c*x + b*f*z - b*z - c*f*x
b b
x *b - x
2 f f f 2 f
- d*f*y + d*y - f *z + f*z)/(x *b*f - x *b - x *f + x *f)
,w)
with arbitrary function ff(..).
\end{verbatim}
}
The program {\tt DETRAFO}\ttindex{DETRAFO} can be used to perform
point transformations of ODEs/PDEs (and -systems).
For detailed explanations the user is
referred to the paper {\em Programs for Applying Symmetries of PDEs}
by Thomas Wolf, supplied as part of the Reduce documentation as {\tt
applysym.tex} and published in the Proceedings of ISSAC'95 - 7/95
Montreal, Canada, ACM Press (1995).