adjust dependencies in a LHPDEs system

 Calling Sequence AdjustDependencies( obj, dep = vars)

Parameters

 obj - a LHPDE object that is in rif-reduced form (see RifReduce) vars - (optional) either the strings "least" or "full", or a list of functions or names

Description

 • The AdjustDependencies method adjusts dependencies in a LHPDE object. It returns a new LHPDE object with specified dependencies.
 • The call AdjustDependencies(obj, dep = "least") returns a new LHPDE object whose dependent variables have smallest possible dependencies on the independent variables of obj.
 • The call AdjustDependencies(obj, dep = "full") returns a new LHPDE object whose dependent variables depend on all the independent variables of obj.
 • If the optional argument dep =  vars is specified with vars being a list of functions or names then it represents new dependencies that are to be adjusted in a LHPDEs system.
 • The call AdjustDependencies(S) is equivalent to the call AdjustDependencies(S, dep = "least").
 • This method is associated with the LHPDE object. For more detail, see Overview of the LHPDE object.

Examples

 > $\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right):$
 > $\mathrm{Typesetting}:-\mathrm{Settings}\left(\mathrm{userep}=\mathrm{true}\right):$
 > $S≔\mathrm{LHPDE}\left(\left[\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y,y\right)=0,\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),x\right)=0,\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y\right)=-\mathrm{diff}\left(\mathrm{\eta }\left(x\right),x\right)\right],\mathrm{indep}=\left[x,y\right],\mathrm{dep}=\left[\mathrm{\xi }\left(x,y\right),\mathrm{\eta }\left(x\right)\right]\right)$
 ${S}{≔}\left[\frac{{{\partial }}^{{2}}}{{\partial }{{y}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){=}{0}{,}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){=}{0}{,}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){=}{-}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\eta }}{}\left({x}\right)\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{}\left({x}{,}{y}\right){,}{\mathrm{\eta }}{}\left({x}\right)\right]$ (1)

Adjust the dependent variables to have the smallest possible dependencies on the independent variables (x,y).

 > $\mathrm{S1}≔\mathrm{AdjustDependencies}\left(S\right)$
 ${\mathrm{S1}}{≔}\left[\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{y}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({y}\right){=}{0}{,}\frac{{ⅆ}}{{ⅆ}{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({y}\right){=}{-}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\eta }}{}\left({x}\right)\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{}\left({y}\right){,}{\mathrm{\eta }}{}\left({x}\right)\right]$ (2)

When dep= is specified as list of functions or names, only those dependent variables with new dependencies are required.

 > $\mathrm{AdjustDependencies}\left(S,\mathrm{dep}=\left[\mathrm{\xi }\left(y\right)\right]\right)$
 $\left[\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{y}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({y}\right){=}{0}{,}\frac{{ⅆ}}{{ⅆ}{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({y}\right){=}{-}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\eta }}{}\left({x}\right)\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{}\left({y}\right){,}{\mathrm{\eta }}{}\left({x}\right)\right]$ (3)

Adjust the dependent variables to depend on all independent variables.

 > $\mathrm{S2}≔\mathrm{AdjustDependencies}\left(S,\mathrm{dep}="full"\right)$
 ${\mathrm{S2}}{≔}\left[\frac{{{\partial }}^{{2}}}{{\partial }{{y}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){=}{0}{,}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){=}{0}{,}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){=}{-}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\eta }}{}\left({x}{,}{y}\right){,}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\eta }}{}\left({x}{,}{y}\right){=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{}\left({x}{,}{y}\right){,}{\mathrm{\eta }}{}\left({x}{,}{y}\right)\right]$ (4)
 > $\mathrm{S3}≔\mathrm{AdjustDependencies}\left(S,\mathrm{dep}=\left[\mathrm{\xi }\left(y\right),\mathrm{\eta }\left(x,y\right)\right]\right)$
 ${\mathrm{S3}}{≔}\left[\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{y}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({y}\right){=}{0}{,}\frac{{ⅆ}}{{ⅆ}{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({y}\right){=}{-}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\eta }}{}\left({x}{,}{y}\right){,}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\eta }}{}\left({x}{,}{y}\right){=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{}\left({y}\right){,}{\mathrm{\eta }}{}\left({x}{,}{y}\right)\right]$ (5)

Any invalid adjustment to the dependencies will trigger an error. For example, xi has to depend at least on y.

 > $\mathrm{AdjustDependencies}\left(S,\mathrm{dep}=\left[\mathrm{\xi }\right]\right)$

Note: the second argument dep = [xi] specifies the dependent variable xi to have no dependencies (i.e. to be a constant). These four LHPDEs should be essentially the same (i.e. they have same LHPDOs).

 > $\mathrm{AreSame}\left(S,\mathrm{S1}\right)$
 ${\mathrm{true}}$ (6)
 > $\mathrm{AreSame}\left(\mathrm{S1},\mathrm{S2}\right)$
 ${\mathrm{true}}$ (7)
 > $\mathrm{AreSame}\left(\mathrm{S2},\mathrm{S3}\right)$
 ${\mathrm{true}}$ (8)

Compatibility

 • The AdjustDependencies command was introduced in Maple 2020.