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— Some historical releases purely for archival purposes
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\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).