Let L1,L2, ... be a sequence of LAVF objects living on the same space (see AreSameSpace). The Intersection method returns a new LAVF object whose solution space is the intersection of solutions of L1,L2,....

By default, the dependent variable names of the returned object are taken from L1. The dependent variable names will be vars if the optional argument depnames = vars is specified.

This method is front-end to the corresponding method of a LHPDE object. That is, let S1, S2,...  be the determining systems of L1,L2,...  (i.e. Si = GetDeterminingSystem(Li)), then the call Intersection(L1,L2,..) is equivalent to Intersection(S1,S2,..). All remaining input arguments will be passed down to its determining system level. See the method Intersection of a LHPDE object for more detail.

This method is associated with the LAVF object. For more detail, see Overview of the LAVF object.

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

$\mathrm{Typesetting}:-\mathrm{Settings}\left(\mathrm{userep}=\mathrm{true}\right)\:$

$\mathrm{Typesetting}:-\mathrm{Suppress}\left(\left[\mathrm{\xi }\left(x\,y\right)\,\mathrm{\eta }\left(x\,y\right)\right]\right)\:$

$X≔\mathrm{VectorField}\left(\mathrm{\xi }\left(x\,y\right)\mathrm{D}\left[x\right]+\mathrm{\eta }\left(x\,y\right)\mathrm{D}\left[y\right]\,\mathrm{space}=\left[x\,y\right]\right)$

$X≔\mathrm{\xi }\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(\frac{\ⅆ}{\ⅆy}\right)$

$\mathrm{E2}≔\mathrm{LHPDE}\left(\left[\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,x\right)=0\,\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,y\right)=0\,\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,y\right)+\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,x\right)=0\,\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,y\,y\right)=0\,\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,x\,x\right)=0\right]\,\mathrm{indep}=\left[x\,y\right]\,\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\right]\right)$

$\mathrm{E2}≔\left[{\mathrm{\xi }}_x=0\,{\mathrm{\eta }}_y=0\,{\mathrm{\xi }}_y+{\mathrm{\eta }}_x=0\,{\mathrm{\xi }}_{y,y}=0\,{\mathrm{\eta }}_{x,x}=0\right],\mathrm{indep}=\left[x\,y\right],\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\right]$

$\mathrm{T2}≔\mathrm{LHPDE}\left(\left[\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,x\right)=0\,\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,y\right)=0\,\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,x\right)=0\,\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,y\right)=0\right]\,\mathrm{indep}=\left[x\,y\right]\,\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\right]\right)$

$\mathrm{T2}≔\left[{\mathrm{\xi }}_x=0\,{\mathrm{\xi }}_y=0\,{\mathrm{\eta }}_x=0\,{\mathrm{\eta }}_y=0\right],\mathrm{indep}=\left[x\,y\right],\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\right]$

$\mathrm{LE2}≔\mathrm{LAVF}\left(X\,\mathrm{E2}\right)$

$\mathrm{LE2}≔\left[\mathrm{\xi }\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(\frac{\ⅆ}{\ⅆy}\right)\right]\&where\left\{\left[{\mathrm{\xi }}_{y,y}=0\,{\mathrm{\xi }}_x=0\,{\mathrm{\eta }}_x=-{\mathrm{\xi }}_y\,{\mathrm{\eta }}_y=0\right]\right\}$

$\mathrm{LT2}≔\mathrm{LAVF}\left(X\,\mathrm{T2}\right)$

$\mathrm{LT2}≔\left[\mathrm{\xi }\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(\frac{\ⅆ}{\ⅆy}\right)\right]\&where\left\{\left[{\mathrm{\xi }}_x=0\,{\mathrm{\eta }}_x=0\,{\mathrm{\xi }}_y=0\,{\mathrm{\eta }}_y=0\right]\right\}$

$\mathrm{Intersection}\left(\mathrm{LE2}\,\mathrm{LT2}\right)$

$\left[\mathrm{\xi }\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(\frac{\ⅆ}{\ⅆy}\right)\right]\&where\left\{\left[{\mathrm{\xi }}_x=0\,{\mathrm{\eta }}_x=0\,{\mathrm{\xi }}_y=0\,{\mathrm{\eta }}_y=0\right]\right\}$

$\mathrm{Intersection}\left(\mathrm{LE2}\,\mathrm{LT2}\,\mathrm{depname}=\left[\mathrm{\alpha }\,\mathrm{\beta }\right]\right)$

$\left[\mathrm{\alpha }\left(x\,y\right)\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\beta }\left(x\,y\right)\left(\frac{\ⅆ}{\ⅆy}\right)\right]\&where\left\{\left[\frac{\partial }{\partial x}\mathrm{\alpha }\left(x\,y\right)=0\,\frac{\partial }{\partial x}\mathrm{\beta }\left(x\,y\right)=0\,\frac{\partial }{\partial y}\mathrm{\alpha }\left(x\,y\right)=0\,\frac{\partial }{\partial y}\mathrm{\beta }\left(x\,y\right)=0\right]\right\}$

The Intersection command was introduced in Maple 2020.

For more information on Maple 2020 changes, see Updates in Maple 2020.