 reduce - Maple Help

liesymm

 reduce
 reduce a set of differential forms Calling Sequence reduce(eqns) Parameters

 eqns - set of determining equations for the symmetries of a given system of PDEs Description

 • If we begin with a system of PDEs rather than a set of differential forms then knowledge of the variable dependencies can be used to simplify the system of determining equations as produced by determine().  This command allows us to make use of that information. It is primarily of use when determine(..., 'Extended'); has been used to set up the system of determining equations, or when the investigation began with differential forms, but knowledge of the variable dependencies is available from some other source.
 • Its effect is to force the isovector components to be independent of any extended'' variables that have been introduced during the construction of the determining equations.
 • This routine is ordinarily invoked automatically by determine()
 • This routine is ordinarily loaded via with(liesymm) but can be used in the package style'' as liesymm[reduce]() Examples

 > $\mathrm{with}\left(\mathrm{liesymm}\right):$
 > $\mathrm{e1}≔\frac{{{\partial }}^{2}}{{\partial }x{\partial }t}u\left(t,x\right)+\frac{{\partial }}{{\partial }x}u\left(t,x\right)+{u\left(t,x\right)}^{2}=0$
 ${\mathrm{e1}}{≔}\frac{{{\partial }}^{{2}}}{{\partial }{t}{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({t}{,}{x}\right){+}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({t}{,}{x}\right){+}{{u}{}\left({t}{,}{x}\right)}^{{2}}{=}{0}$ (1)
 > $\mathrm{eqns1}≔\mathrm{determine}\left(\mathrm{e1},V,u\left(t,x\right),\left[v,w\right],'\mathrm{Extended}'\right):$

The result is a system of eight equations, one of which is:

 > $\mathrm{e2}≔\frac{{\partial }}{{\partial }w}\mathrm{V4}\left(t,x,u,v,w\right)+\left(w+{u}^{2}\right)\left(\frac{{\partial }}{{\partial }w}\mathrm{V2}\left(t,x,u,v,w\right)\right)$
 ${\mathrm{e2}}{≔}\frac{{\partial }}{{\partial }{w}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{V4}}{}\left({t}{,}{x}{,}{u}{,}{v}{,}{w}\right){+}\left({{u}}^{{2}}{+}{w}\right){}\left(\frac{{\partial }}{{\partial }{w}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{V2}}{}\left({t}{,}{x}{,}{u}{,}{v}{,}{w}\right)\right)$ (2)

Many of these new equations can be simplified by using the fact that the V's are only dependent on t, x, and u.  For example, the above simplifies to

 > $\mathrm{e3}≔\mathrm{reduce}\left(\mathrm{e2}\right)$
 ${\mathrm{e3}}{≔}{-}\frac{{\partial }}{{\partial }{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{V2}}{}\left({t}{,}{x}{,}{u}\right){-}{v}{}\left(\frac{{\partial }}{{\partial }{u}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{V2}}{}\left({t}{,}{x}{,}{u}\right)\right)$ (3)