IsSubspace - Maple Programming Help

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IsSubspace

check if solution space of a LHPDE object is subspace of solution space of another LHPDE object.

 Calling Sequence IsSubspace( obj1, obj2)

Parameters

 obj1 - a LHPDE object that is assumed to be in rif-reduced form (see IsRifReduced) obj2 - a LHPDE object

Description

 • The IsSubspace method returns true if solution space of obj1 is a subspace of solution space of obj2. False otherwise.
 • More precisely, the method returns true if at each point ${x}_{0}$, the local solution space of obj1  at ${x}_{0}$ is a subspace of the local solution space of obj2 at ${x}_{0}$.
 • The input arguments obj1 and obj2 need not have the same dependent variable names or dependencies.
 • This method is associated with the LHPDE object. For more detail, see Overview of the LHPDE object.

Examples

 > $\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right):$
 > $\mathrm{C2}≔\mathrm{LHPDE}\left(\left[\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),x,x\right)=0,\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),x,y\right)=0,\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y,y\right)=0,\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),x\right)=-\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y\right),\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),y\right)=\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),x\right)\right],\mathrm{indep}=\left[x,y\right],\mathrm{dep}=\left[\mathrm{\xi },\mathrm{\eta }\right],\mathrm{inRifReducedForm}=\mathrm{true}\right)$
 ${\mathrm{C2}}{≔}\left[\frac{{{\partial }}^{{2}}}{{\partial }{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){=}{0}{,}\frac{{{\partial }}^{{2}}}{{\partial }{x}{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){=}{0}{,}\frac{{{\partial }}^{{2}}}{{\partial }{{y}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){=}{0}{,}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\eta }}{}\left({x}{,}{y}\right){=}{-}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){,}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\eta }}{}\left({x}{,}{y}\right){=}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right)\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{}\left({x}{,}{y}\right){,}{\mathrm{\eta }}{}\left({x}{,}{y}\right)\right]$ (1)
 > $\mathrm{E2}≔\mathrm{LHPDE}\left(\left[\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y,y\right)=0,\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),x\right)=-\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y\right),\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),y\right)=0,\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),x\right)=0\right],\mathrm{indep}=\left[x,y\right],\mathrm{dep}=\left[\mathrm{\xi },\mathrm{\eta }\right],\mathrm{inRifReducedForm}=\mathrm{true}\right)$
 ${\mathrm{E2}}{≔}\left[\frac{{{\partial }}^{{2}}}{{\partial }{{y}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){=}{0}{,}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\eta }}{}\left({x}{,}{y}\right){=}{-}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){,}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\eta }}{}\left({x}{,}{y}\right){=}{0}{,}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{}\left({x}{,}{y}\right){,}{\mathrm{\eta }}{}\left({x}{,}{y}\right)\right]$ (2)
 > $\mathrm{IsSubspace}\left(\mathrm{E2},\mathrm{C2}\right)$
 ${\mathrm{true}}$ (3)

This LHPDE object's dependent variables have different names and dependencies.

 > $\mathrm{E2p}≔\mathrm{LHPDE}\left(\left[\mathrm{diff}\left(\mathrm{\alpha }\left(y\right),y,y\right)=0,\mathrm{diff}\left(\mathrm{\beta }\left(x\right),x\right)=-\mathrm{diff}\left(\mathrm{\alpha }\left(y\right),y\right)\right],\mathrm{indep}=\left[x,y\right],\mathrm{dep}=\left[\mathrm{\alpha },\mathrm{\beta }\right],\mathrm{inRifReducedForm}=\mathrm{true}\right)$
 ${\mathrm{E2p}}{≔}\left[\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{y}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\alpha }}{}\left({y}\right){=}{0}{,}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\beta }}{}\left({x}\right){=}{-}\frac{{ⅆ}}{{ⅆ}{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\alpha }}{}\left({y}\right)\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\alpha }}{}\left({y}\right){,}{\mathrm{\beta }}{}\left({x}\right)\right]$ (4)
 > $\mathrm{IsSubspace}\left(\mathrm{E2p},\mathrm{C2}\right)$
 ${\mathrm{true}}$ (5)

Compatibility

 • The IsSubspace command was introduced in Maple 2020.