DChange - Maple Help

DChange

perform change of variables on a LHPDE object

 Calling Sequence DChange (tr, obj, indep = ivars, dep = dvars, rifReduce = r, options) dchange (tr, obj, indep = ivars, dep = dvars, rifReduce = r, options)

Parameters

 tr - set of equations corresponding to the transformation from the old variables on the left-hand side of the equations to the new variables on the right-hand side. obj - a LHPDE object ivars - (optional) a list of new independent variable names dvars - (optional) a list of new dependent variables as functions or names r - (optional) true or false options - optional arguments that will be passed down to PDEtools[dchange] command

Description

 • The DChange method performs change of variables tr in the LHPDE object obj.
 • The method returns a new LHPDE object written with respect to new dependent and independent variables that are on the right hand side of tr.
 • The new dependent and independent variables can also be fully specified via the optional arguments indep = ivars and dep = dvars.
 • The returned LHPDE object can be rif-reduced if the option rifReduce = true is specified.
 • The method name dchange is provided as alias.
 • This DChange method is based on the existing Maple function PDEtools[dchange].
 • This method is associated with the LHPDE object. For more detail, see Overview of the LHPDE object.

Examples

 > $\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right):$

Example 1:

 > $S≔\mathrm{LHPDE}\left(\left[\frac{\partial }{\partial x}u\left(x,y\right)+\frac{\partial }{\partial y}u\left(x,y\right)=0\right]\right)$
 ${S}{≔}\left[\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}{,}{y}\right){+}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}{,}{y}\right){=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{u}{}\left({x}{,}{y}\right)\right]$ (1)
 > $\mathrm{DChange}\left(\left\{x=r+s,y=r-s,u\left(x,y\right)=v\left(r,s\right)\right\},S\right)$
 $\left[\frac{{\partial }}{{\partial }{r}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{v}{}\left({r}{,}{s}\right){=}{0}\right]{,}{\mathrm{indep}}{=}\left[{r}{,}{s}\right]{,}{\mathrm{dep}}{=}\left[{v}{}\left({r}{,}{s}\right)\right]$ (2)

Example 2:

 > $S≔\mathrm{LHPDE}\left(\left[\frac{ⅆ}{ⅆx}u\left(x\right)+a\left(x\right)u\left(x\right)=0\right],\mathrm{dep}=\left[u\left(x\right)\right]\right)$
 ${S}{≔}\left[\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}\right){+}{a}{}\left({x}\right){}{u}{}\left({x}\right){=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}\right]{,}{\mathrm{dep}}{=}\left[{u}{}\left({x}\right)\right]$ (3)
 > $\mathrm{DChange}\left(\left\{x=t,u\left(x\right)=\frac{ⅆ}{ⅆt}v\left(t\right)\right\},S,\mathrm{dep}=\left[v\left(t\right)\right]\right)$
 $\left[\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{t}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{v}{}\left({t}\right){+}{a}{}\left({t}\right){}\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{v}{}\left({t}\right)\right){=}{0}\right]{,}{\mathrm{indep}}{=}\left[{t}\right]{,}{\mathrm{dep}}{=}\left[{v}{}\left({t}\right)\right]$ (4)

By specifying rifReduce = true, a rif-reduced LHPDE object is returned.

 > $\mathrm{Sp}≔\mathrm{DChange}\left(\left\{x=t,u\left(x\right)=\frac{ⅆ}{ⅆt}v\left(t\right)\right\},S,\mathrm{dep}=\left[v\left(t\right)\right],\mathrm{rifReduce}=\mathrm{true}\right)$
 ${\mathrm{Sp}}{≔}\left[\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{t}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{v}{}\left({t}\right){=}{-}{a}{}\left({t}\right){}\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{v}{}\left({t}\right)\right)\right]{,}{\mathrm{indep}}{=}\left[{t}\right]{,}{\mathrm{dep}}{=}\left[{v}{}\left({t}\right)\right]$ (5)
 > $\mathrm{IsRifReduced}\left(\mathrm{Sp}\right)$
 ${\mathrm{true}}$ (6)

Example 3:

 > $S≔\mathrm{LHPDE}\left(\left[\right],\mathrm{dep}=\left[u\left(x,y\right),v\left(x,y\right)\right]\right)$
 ${S}{≔}\left[\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{u}{}\left({x}{,}{y}\right){,}{v}{}\left({x}{,}{y}\right)\right]$ (7)
 > $\mathrm{DChange}\left(\left\{x=s,y=t,u\left(x,y\right)=\mathrm{φ}\left(s,t\right)\right\},S,\mathrm{dep}=\left[\mathrm{φ}\left(s,t\right),v\left(s,t\right),\mathrm{δ}\left(s,t\right)\right]\right)$
 $\left[\right]{,}{\mathrm{indep}}{=}\left[{s}{,}{t}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\phi }}{}\left({s}{,}{t}\right){,}{v}{}\left({s}{,}{t}\right){,}{\mathrm{\delta }}{}\left({s}{,}{t}\right)\right]$ (8)

Compatibility

 • The DChange command was introduced in Maple 2020.