autosimp - Maple Help

liesymm

 autosimp
 autosimp a set of differential forms

 Calling Sequence autosimp(eqns)

Parameters

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

Description

 • The system of PDEs characterizing the symmetries of a system of PDEs will often include a number of equations which are particularly easy to integrate (for example, first order partials equated with 0). This command will automatically carry out many of these simple integrations.  If no such integrations can be found then the system is left alone.
 • When simplifications are completed the result is returned in a form of  eqns &where equations where the first operand is the set of remaining equations and the second operand is a set of algebraic equations indicating the relationship between the new functions appearing in the set of simplified equations and the original functions.
 • This routine is ordinarily loaded via with(liesymm) but can be used in the package style'' as liesymm[autosimp]()

Examples

 > $\mathrm{with}\left(\mathrm{liesymm}\right):$
 > $\mathrm{e1}≔\mathrm{Diff}\left(u\left(t,x\right),t,x\right)+\mathrm{Diff}\left(u\left(t,x\right),x\right)+{u\left(t,x\right)}^{2}=0$
 ${\mathrm{e1}}{≔}\frac{{{\partial }}^{{2}}}{{\partial }{t}{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({t}{,}{x}\right){+}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({t}{,}{x}\right){+}{{u}{}\left({t}{,}{x}\right)}^{{2}}{=}{0}$ (1)
 > $\mathrm{eqns1}≔\mathrm{determine}\left(\mathrm{e1},V,u\left(t,x\right),k\right):$

The next step is the automatic integration of these equations.  We begin by setting up names for some regularly occurring expressions.

 > $\mathrm{eqns2}≔\mathrm{autosimp}\left(\mathrm{eqns1}\right)$
 ${\mathrm{eqns2}}{≔}\left\{\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{V1_2}}{}\left({t}\right){=}{-}{\mathrm{V3_1}}{}\left({t}\right){-}{\mathrm{C1}}{,}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{V3_1}}{}\left({t}\right){=}{\mathrm{V3_1}}{}\left({t}\right){+}{\mathrm{C1}}\right\}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{{\mathrm{V1}}{}\left({t}{,}{x}{,}{u}\right){=}{\mathrm{V1_2}}{}\left({t}\right){,}{\mathrm{V2}}{}\left({t}{,}{x}{,}{u}\right){=}{x}{}{\mathrm{C1}}{+}{\mathrm{C2}}{,}{\mathrm{V2_2}}{}\left({x}\right){=}{x}{}{\mathrm{C1}}{+}{\mathrm{C2}}{,}{\mathrm{V3}}{}\left({t}{,}{x}{,}{u}\right){=}{u}{}{\mathrm{V3_1}}{}\left({t}\right){,}{\mathrm{V3_2}}{}\left({t}{,}{x}\right){=}{0}\right\}$ (2)