makeforms - Maple Help

liesymm

 makeforms
 construct a set of differential forms from a PDE

 Calling Sequence makeforms(eqns, fncs, rootname) makeforms(eqns, fncs, [Extd])

Parameters

 eqns - PDE, or a list or set of PDEs fcns - list of functions, for example, u(t,x) rootname - name used to construct the names for the extended coordinates Extd - name or list of names for constructing the extended variable names

Description

 • This routine is part of the liesymm package and is loaded via with(liesymm).
 • Given one or more partial differential equations this routine constructs a set of differential forms which after closure is equivalent to the original system of equations in the sense of Cartan. The forms are obtained by first reducing the system of PDEs to a system of first order equations.  It will handle systems which are quasi-linear.
 • If more than one equation is given, they must be given as a list or a set.
 • The second argument specifies the dependent and independent variables as in $h\left(t,x\right)$.  If there is more than one dependent variable, these must be specified as a set or list.  The Maple expression $\left[h,u\right]\left(t,x,y\right)$ can be used to construct the required list when more than one dependent variable is involved.
 • The third argument is used to construct the names of the extended variables. If it is the name k then the names k1, k2, k3, ..., kn are constructed as needed.  If a list of names is provided then these names are used in place of k1, k2, ..., kn.
 • The dependent variables (as defined by $\left[h,u\right]\left(t,x\right)$) are processed in the order given.  For each dependent variable we introduce extended variables corresponding to the partial derivatives with respect to  t and x in that order.  These extended variables are in turn treated as dependent variables until the reduction to first order is achieved. The resulting coordinate list  is $\left[t,x,h,\mathrm{k1},\mathrm{k2},u,\mathrm{k3},\mathrm{w4}\right]$ with k1 and k2 being the partials of h with respect to t and x and w3 and w4 being the partials of u with respect to t and x.

Examples

 > $\mathrm{with}\left(\mathrm{liesymm}\right):$
 > $\mathrm{eq1}≔\mathrm{Diff}\left(h\left(t,x\right),x,x\right)=\mathrm{Diff}\left(h\left(t,x\right),t\right)$
 ${\mathrm{eq1}}{≔}\frac{{{\partial }}^{{2}}}{{\partial }{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{h}{}\left({t}{,}{x}\right){=}\frac{{\partial }}{{\partial }{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{h}{}\left({t}{,}{x}\right)$ (1)
 > $\mathrm{makeforms}\left(\mathrm{eq1},h\left(t,x\right),w\right)$
 $\left[{d}{}\left({h}\right){-}{\mathrm{w1}}{}{d}{}\left({t}\right){-}{\mathrm{w2}}{}{d}{}\left({x}\right){,}{-}{d}{}\left({\mathrm{w2}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&ˆ}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{d}{}\left({t}\right){-}{\mathrm{w1}}{}{d}{}\left({t}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&ˆ}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{d}{}\left({x}\right)\right]$ (2)
 > $\mathrm{makeforms}\left(\mathrm{eq1},h\left(t,x\right),\left[p,q\right]\right)$
 $\left[{d}{}\left({h}\right){-}{p}{}{d}{}\left({t}\right){-}{q}{}{d}{}\left({x}\right){,}{-}{d}{}\left({q}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&ˆ}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{d}{}\left({t}\right){-}{p}{}{d}{}\left({t}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&ˆ}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{d}{}\left({x}\right)\right]$ (3)
 > $\mathrm{annul}\left(,\left[t,x\right]\right)$
 $\left[\frac{{\partial }}{{\partial }{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{h}{}\left({t}{,}{x}\right){-}{p}{}\left({t}{,}{x}\right){=}{0}{,}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{h}{}\left({t}{,}{x}\right){-}{q}{}\left({t}{,}{x}\right){=}{0}{,}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{q}{}\left({t}{,}{x}\right){-}{p}{}\left({t}{,}{x}\right){=}{0}\right]$ (4)
 > $\mathrm{eq3}≔\mathrm{Diff}\left(U\left(t,r\right),r,r\right)+\frac{1}{r}\mathrm{Diff}\left(U\left(t,r\right),r\right)-\mathrm{Diff}\left(U\left(t,r\right),t,t\right)=\mathrm{exp}\left(-2U\left(t,r\right)\right)\left({\mathrm{Diff}\left(C\left(t,r\right),t\right)}^{2}-{\mathrm{Diff}\left(C\left(t,r\right),r\right)}^{2}\right)$
 ${\mathrm{eq3}}{≔}\frac{{{\partial }}^{{2}}}{{\partial }{{r}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{U}{}\left({t}{,}{r}\right){+}\frac{\frac{{\partial }}{{\partial }{r}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{U}{}\left({t}{,}{r}\right)}{{r}}{-}\frac{{{\partial }}^{{2}}}{{\partial }{{t}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{U}{}\left({t}{,}{r}\right){=}{{ⅇ}}^{{-}{2}{}{U}{}\left({t}{,}{r}\right)}{}\left({\frac{{\partial }}{{\partial }{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{C}{}\left({t}{,}{r}\right)}^{{2}}{-}{\frac{{\partial }}{{\partial }{r}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{C}{}\left({t}{,}{r}\right)}^{{2}}\right)$ (5)
 > $\mathrm{eq4}≔\mathrm{Diff}\left(C\left(t,r\right),r,r\right)+\frac{1}{r}\mathrm{Diff}\left(C\left(t,r\right),r\right)-\mathrm{Diff}\left(C\left(t,r\right),t,t\right)=2\left(\mathrm{Diff}\left(C\left(t,r\right),r\right)\mathrm{Diff}\left(U\left(t,r\right),r\right)-\mathrm{Diff}\left(C\left(t,r\right),t\right)\mathrm{Diff}\left(U\left(t,r\right),t\right)\right)$
 ${\mathrm{eq4}}{≔}\frac{{{\partial }}^{{2}}}{{\partial }{{r}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{C}{}\left({t}{,}{r}\right){+}\frac{\frac{{\partial }}{{\partial }{r}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{C}{}\left({t}{,}{r}\right)}{{r}}{-}\frac{{{\partial }}^{{2}}}{{\partial }{{t}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{C}{}\left({t}{,}{r}\right){=}{2}{}\frac{{\partial }}{{\partial }{r}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{C}{}\left({t}{,}{r}\right){}\frac{{\partial }}{{\partial }{r}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{U}{}\left({t}{,}{r}\right){-}{2}{}\frac{{\partial }}{{\partial }{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{C}{}\left({t}{,}{r}\right){}\frac{{\partial }}{{\partial }{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{U}{}\left({t}{,}{r}\right)$ (6)
 > $\mathrm{makeforms}\left(\left[\mathrm{eq3},\mathrm{eq4}\right],\left[U\left(t,r\right),C\left(t,r\right)\right],\left[A,B,F,G\right]\right)$
 $\left[{d}{}\left({C}\right){-}{A}{}{d}{}\left({t}\right){-}{B}{}{d}{}\left({r}\right){,}{d}{}\left({U}\right){-}{F}{}{d}{}\left({t}\right){-}{G}{}{d}{}\left({r}\right){,}{d}{}\left({G}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&ˆ}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{d}{}\left({t}\right){+}{d}{}\left({F}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&ˆ}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{d}{}\left({r}\right){+}\left(\frac{{G}}{{r}}{-}\frac{{{A}}^{{2}}}{{\left({{ⅇ}}^{{U}}\right)}^{{2}}}{+}\frac{{{B}}^{{2}}}{{\left({{ⅇ}}^{{U}}\right)}^{{2}}}\right){}{d}{}\left({r}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&ˆ}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{d}{}\left({t}\right){,}{d}{}\left({B}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&ˆ}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{d}{}\left({t}\right){+}{d}{}\left({A}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&ˆ}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{d}{}\left({r}\right){+}\left(\frac{{B}}{{r}}{-}{2}{}{B}{}{G}{+}{2}{}{A}{}{F}\right){}{d}{}\left({r}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&ˆ}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{d}{}\left({t}\right)\right]$ (7)
 > $\mathrm{annul}\left(,\left[t,r\right]\right)$
 $\left[\frac{{\partial }}{{\partial }{r}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{C}{}\left({t}{,}{r}\right){-}{B}{}\left({t}{,}{r}\right){=}{0}{,}\frac{{\partial }}{{\partial }{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{C}{}\left({t}{,}{r}\right){-}{A}{}\left({t}{,}{r}\right){=}{0}{,}\frac{{\partial }}{{\partial }{r}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{U}{}\left({t}{,}{r}\right){-}{G}{}\left({t}{,}{r}\right){=}{0}{,}\frac{{\partial }}{{\partial }{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{U}{}\left({t}{,}{r}\right){-}{F}{}\left({t}{,}{r}\right){=}{0}{,}\frac{{\partial }}{{\partial }{r}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{G}{}\left({t}{,}{r}\right){-}\frac{{\partial }}{{\partial }{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{F}{}\left({t}{,}{r}\right){+}\frac{{G}{}\left({t}{,}{r}\right)}{{r}}{-}\frac{{{A}{}\left({t}{,}{r}\right)}^{{2}}}{{\left({{ⅇ}}^{{U}{}\left({t}{,}{r}\right)}\right)}^{{2}}}{+}\frac{{{B}{}\left({t}{,}{r}\right)}^{{2}}}{{\left({{ⅇ}}^{{U}{}\left({t}{,}{r}\right)}\right)}^{{2}}}{=}{0}{,}\frac{{\partial }}{{\partial }{r}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{B}{}\left({t}{,}{r}\right){-}\frac{{\partial }}{{\partial }{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{A}{}\left({t}{,}{r}\right){+}\frac{{B}{}\left({t}{,}{r}\right)}{{r}}{-}{2}{}{B}{}\left({t}{,}{r}\right){}{G}{}\left({t}{,}{r}\right){+}{2}{}{A}{}\left({t}{,}{r}\right){}{F}{}\left({t}{,}{r}\right){=}{0}\right]$ (8)