Let L1,L2, ... be a sequence of LAVF objects living on the same space. The VectorSpaceSum method finds a LAVF object whose solution space is the vector space sum of solution spaces 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 VectorSpaceSum(L1,L2,..) is equivalent to VectorSpaceSum(S1,S2,..). All remaining input arguments will be passed down to its determining system level. See the method VectorSpaceSum of a LHPDE object for more detail.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$\left[\mathrm{\alpha }\left(x\,y\right)\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\beta }\left(x\,y\right)\left(\frac{\ⅆ}{\ⅆy}\right)\right]\&where\left\{\left[\frac{\partial }{\partial x}\mathrm{\alpha }\left(x\,y\right)=0\,\frac{\partial }{\partial y}\mathrm{\alpha }\left(x\,y\right)=-\frac{\partial }{\partial x}\mathrm{\beta }\left(x\,y\right)\,\frac{{\partial }^2}{\partial x^2}\mathrm{\beta }\left(x\,y\right)=0\,\frac{\partial }{\partial y}\mathrm{\beta }\left(x\,y\right)=0\right]\right\}$

The VectorSpaceSum command was introduced in Maple 2020.

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