annul - Maple Help

liesymm

 annul
 annul a set of differential forms

 Calling Sequence annul(forms) annul(forms, vlist)

Parameters

 forms - list or set of differential forms vlist - list of those coordinates that are to be treated as independent on the solution manifold

Description

 • This routine is part of the liesymm package and is loaded via with(liesymm) .
 • Given a set of differential forms and a list of coordinates, this command sections the forms and sets them equal to 0.  The result is a set of partial differential equations corresponding to the differential forms.
 • If only one argument is provided then the coordinates indicated by the command indepvars() are used. In particular, if the list of forms has been produced by use of the makeforms() command then a system of PDEs equivalent to the original system of partial differential equations is produced.
 • If the set of forms is closed then the resulting equations include the integrability conditions.

Examples

 > $\mathrm{with}\left(\mathrm{liesymm}\right):$
 > $\mathrm{setup}\left(t,x,u,\mathrm{w1},\mathrm{w2}\right)$
 $\left[{t}{,}{x}{,}{u}{,}{\mathrm{w1}}{,}{\mathrm{w2}}\right]$ (1)
 > $\mathrm{a1}≔d\left(u\right)-\mathrm{w1}d\left(t\right)-\mathrm{w2}d\left(x\right)$
 ${\mathrm{a1}}{≔}{d}{}\left({u}\right){-}{\mathrm{w1}}{}{d}{}\left({t}\right){-}{\mathrm{w2}}{}{d}{}\left({x}\right)$ (2)
 > $\mathrm{a2}≔\left(\mathrm{w2}+{u}^{2}\right)d\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&ˆ\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}d\left(t\right)-d\left(\mathrm{w2}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&ˆ\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}d\left(x\right)$
 ${\mathrm{a2}}{≔}{-}\left({{u}}^{{2}}{+}{\mathrm{w2}}\right){}{d}{}\left({t}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&ˆ}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{d}{}\left({x}\right){+}{d}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&ˆ}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{d}{}\left({\mathrm{w2}}\right)$ (3)
 > $\mathrm{annul}\left(\left[\mathrm{a1},\mathrm{a2}\right],\left[t,x\right]\right)$
 $\left[\frac{{\partial }}{{\partial }{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({t}{,}{x}\right){-}{\mathrm{w1}}{}\left({t}{,}{x}\right){=}{0}{,}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({t}{,}{x}\right){-}{\mathrm{w2}}{}\left({t}{,}{x}\right){=}{0}{,}{-}{{u}{}\left({t}{,}{x}\right)}^{{2}}{-}\frac{{\partial }}{{\partial }{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{w2}}{}\left({t}{,}{x}\right){-}{\mathrm{w2}}{}\left({t}{,}{x}\right){=}{0}\right]$ (4)
 > $\mathrm{close}\left(\left[\mathrm{a1},\mathrm{a2}\right]\right)$
 $\left[{d}{}\left({u}\right){-}{\mathrm{w1}}{}{d}{}\left({t}\right){-}{\mathrm{w2}}{}{d}{}\left({x}\right){,}{-}\left({{u}}^{{2}}{+}{\mathrm{w2}}\right){}{d}{}\left({t}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&ˆ}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{d}{}\left({x}\right){+}{d}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&ˆ}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{d}{}\left({\mathrm{w2}}\right){,}{d}{}\left({t}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&ˆ}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{d}{}\left({\mathrm{w1}}\right){+}{d}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&ˆ}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{d}{}\left({\mathrm{w2}}\right)\right]$ (5)
 > $\mathrm{annul}\left(,\left[t,x\right]\right)$
 $\left[\frac{{\partial }}{{\partial }{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({t}{,}{x}\right){-}{\mathrm{w1}}{}\left({t}{,}{x}\right){=}{0}{,}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({t}{,}{x}\right){-}{\mathrm{w2}}{}\left({t}{,}{x}\right){=}{0}{,}{-}{{u}{}\left({t}{,}{x}\right)}^{{2}}{-}\frac{{\partial }}{{\partial }{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{w2}}{}\left({t}{,}{x}\right){-}{\mathrm{w2}}{}\left({t}{,}{x}\right){=}{0}{,}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{w1}}{}\left({t}{,}{x}\right){-}\frac{{\partial }}{{\partial }{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{w2}}{}\left({t}{,}{x}\right){=}{0}\right]$ (6)
 > $\mathrm{eq1}≔\mathrm{Diff}\left(h\left(t,x\right),x,x\right)=\mathrm{Diff}\left(h\left(t,x\right),t\right)$
 ${\mathrm{eq1}}{≔}\frac{{{\partial }}^{{2}}}{{\partial }{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{h}{}\left({t}{,}{x}\right){=}\frac{{\partial }}{{\partial }{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{h}{}\left({t}{,}{x}\right)$ (7)
 > $\mathrm{makeforms}\left(\mathrm{eq1},h\left(t,x\right),k\right)$
 $\left[{d}{}\left({h}\right){-}{\mathrm{k1}}{}{d}{}\left({t}\right){-}{\mathrm{k2}}{}{d}{}\left({x}\right){,}{-}{d}{}\left({\mathrm{k2}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&ˆ}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{d}{}\left({t}\right){-}{\mathrm{k1}}{}{d}{}\left({t}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&ˆ}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{d}{}\left({x}\right)\right]$ (8)
 > $\mathrm{annul}\left(\right)$
 $\left\{\frac{{{\partial }}^{{2}}}{{\partial }{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{h}{}\left({t}{,}{x}\right){-}\frac{{\partial }}{{\partial }{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{h}{}\left({t}{,}{x}\right){=}{0}\right\}$ (9)