AreSame - Maple Help

AreSame

check if two LAVF objects are the same

 Calling Sequence AreSame( obj1, obj2, criteria = crit)

Parameters

 obj1,obj2, ... - LAVF objects crit - (optional) a string: "sameOperator", "sameSystem", or "sameSolutions"

Description

 • The AreSame method checks if the two LAVF objects obj1 and obj2 are the same.
 • This method is front-end to the AreSame method of a LHPDE object. That is, let S1, S2 be the determining systems of LAVF objects L1, L2 respectively, then AreSame(L1, L2) equals AreSame(S1,S2).
 • All optional arguments in this method will pass down to the AreSame method of a LHPDE object. See AreSame of a LHPDE object for how to apply optional arguments.
 • In the second calling sequence, the word criterion is provided as alias for criteria.
 • This method is associated with the LAVF object. For more detail, see Overview of the LAVF object.

Examples

 > $\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right):$
 > $V≔\mathrm{VectorField}\left(\mathrm{ξ}\left(x,y\right){\mathrm{D}}_{x}+\mathrm{η}\left(x,y\right){\mathrm{D}}_{y},\mathrm{space}=\left[x,y\right]\right)$
 ${V}{≔}{\mathrm{\xi }}{}\left({x}{,}{y}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right){+}{\mathrm{\eta }}{}\left({x}{,}{y}\right){}\left(\frac{{ⅆ}}{{ⅆ}{y}}\right)$ (1)
 > $\mathrm{Vp}≔\mathrm{VectorField}\left(\mathrm{α}\left(x,y\right){\mathrm{D}}_{x}+\mathrm{β}\left(x\right){\mathrm{D}}_{y},\mathrm{space}=\left[x,y\right]\right)$
 ${\mathrm{Vp}}{≔}{\mathrm{\alpha }}{}\left({x}{,}{y}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right){+}{\mathrm{\beta }}{}\left({x}\right){}\left(\frac{{ⅆ}}{{ⅆ}{y}}\right)$ (2)
 > $\mathrm{E2}≔\mathrm{LHPDE}\left(\left[\frac{{\partial }^{2}}{\partial {y}^{2}}\mathrm{ξ}\left(x,y\right)=0,\frac{\partial }{\partial x}\mathrm{η}\left(x,y\right)=-\left(\frac{\partial }{\partial y}\mathrm{ξ}\left(x,y\right)\right),\frac{\partial }{\partial y}\mathrm{η}\left(x,y\right)=0,\frac{\partial }{\partial x}\mathrm{ξ}\left(x,y\right)=0\right],\mathrm{indep}=\left[x,y\right],\mathrm{dep}=\left[\mathrm{ξ},\mathrm{η}\right]\right)$
 ${\mathrm{E2}}{≔}\left[\frac{{{\partial }}^{{2}}}{{\partial }{{y}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){=}{0}{,}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\eta }}{}\left({x}{,}{y}\right){=}{-}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){,}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\eta }}{}\left({x}{,}{y}\right){=}{0}{,}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{}\left({x}{,}{y}\right){,}{\mathrm{\eta }}{}\left({x}{,}{y}\right)\right]$ (3)
 > $\mathrm{E2p}≔\mathrm{LHPDE}\left(\left[\frac{{\partial }^{2}}{\partial {y}^{2}}\mathrm{α}\left(x,y\right)=0,\frac{ⅆ}{ⅆx}\mathrm{β}\left(x\right)=-\left(\frac{\partial }{\partial y}\mathrm{α}\left(x,y\right)\right),\frac{\partial }{\partial x}\mathrm{α}\left(x,y\right)=0\right],\mathrm{indep}=\left[x,y\right],\mathrm{dep}=\left[\mathrm{α},\mathrm{β}\right]\right)$
 ${\mathrm{E2p}}{≔}\left[\frac{{{\partial }}^{{2}}}{{\partial }{{y}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\alpha }}{}\left({x}{,}{y}\right){=}{0}{,}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\beta }}{}\left({x}\right){=}{-}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\alpha }}{}\left({x}{,}{y}\right){,}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\alpha }}{}\left({x}{,}{y}\right){=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\alpha }}{}\left({x}{,}{y}\right){,}{\mathrm{\beta }}{}\left({x}\right)\right]$ (4)

Constructing two LAVFs that are essentially E(2) but their systems are different looking..

 > $\mathrm{LE2}≔\mathrm{LAVF}\left(V,\mathrm{E2}\right)$
 ${\mathrm{LE2}}{≔}\left[{\mathrm{\xi }}{}\left({x}{,}{y}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right){+}{\mathrm{\eta }}{}\left({x}{,}{y}\right){}\left(\frac{{ⅆ}}{{ⅆ}{y}}\right)\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[\frac{{{\partial }}^{{2}}}{{\partial }{{y}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){=}{0}{,}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){=}{0}{,}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\eta }}{}\left({x}{,}{y}\right){=}{-}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){,}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\eta }}{}\left({x}{,}{y}\right){=}{0}\right]\right\}$ (5)
 > $\mathrm{LE2p}≔\mathrm{LAVF}\left(\mathrm{Vp},\mathrm{E2p}\right)$
 ${\mathrm{LE2p}}{≔}\left[{\mathrm{\alpha }}{}\left({x}{,}{y}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right){+}{\mathrm{\beta }}{}\left({x}\right){}\left(\frac{{ⅆ}}{{ⅆ}{y}}\right)\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[\frac{{{\partial }}^{{2}}}{{\partial }{{y}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\alpha }}{}\left({x}{,}{y}\right){=}{0}{,}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\alpha }}{}\left({x}{,}{y}\right){=}{0}{,}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\beta }}{}\left({x}\right){=}{-}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\alpha }}{}\left({x}{,}{y}\right)\right]\right\}$ (6)

The two LAVFs are the same as operators:

 > $\mathrm{AreSame}\left(\mathrm{LE2},\mathrm{LE2p}\right)$
 ${\mathrm{true}}$ (7)

Clearly they have different dependent variables, so the systems are not identical.

 > $\mathrm{AreSame}\left(\mathrm{LE2},\mathrm{LE2p},\mathrm{criterion}="sameSystem"\right)$
 ${\mathrm{false}}$ (8)

Since they are same as operator, they definitely have the same solutions.

 > $\mathrm{AreSame}\left(\mathrm{LE2},\mathrm{LE2p},\mathrm{criteria}="sameSolutions"\right)$
 ${\mathrm{true}}$ (9)

Compatibility

 • The AreSame command was introduced in Maple 2020.