VectorSpaceSum - Maple Help

VectorSpaceSum

find a LHPDE object whose solution space is the sum of the solution spaces of given LHPDE objects.

 Calling Sequence VectorSpaceSum( obj1, obj2, ..., depname = vars )

Parameters

 obj1, obj2, ... - a sequence of LHPDE objects living on the same space vars - (optional) a list of new dependent variable names

Description

 • Let obj1, obj2, ... be a sequence of LHPDE objects living on the same space (see AreSameSpace). The VectorSpaceSum method finds a LHPDEs system whose solution space is the vector space sum of solution spaces of obj1,obj2,....
 • The method returns a rif-reduced LHPDE object.
 • By default, the dependent variable names of the returned object are taken from obj1. The dependent variable names will be vars if the optional argument depname = vars is specified.
 • This method is associated with the LHPDE object. For more detail, see Overview of the LHPDE object.

Examples

 > $\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right):$
 > $\mathrm{Typesetting}:-\mathrm{Settings}\left(\mathrm{userep}=\mathrm{true}\right):$
 > $\mathrm{Typesetting}:-\mathrm{Suppress}\left(\left\{\mathrm{ξ},\mathrm{η},\mathrm{α},\mathrm{β},\mathrm{φ},\mathrm{ψ}\right\}\left(x,y\right)\right):$
 > $S≔\mathrm{LHPDE}\left(\left[\frac{\partial }{\partial x}\mathrm{ξ}\left(x,y\right)=0,\frac{\partial }{\partial y}\mathrm{η}\left(x,y\right)=0,\frac{\partial }{\partial y}\mathrm{ξ}\left(x,y\right)+\frac{\partial }{\partial x}\mathrm{η}\left(x,y\right)=0,\frac{{\partial }^{2}}{\partial {y}^{2}}\mathrm{ξ}\left(x,y\right)=0,\frac{{\partial }^{2}}{\partial {x}^{2}}\mathrm{η}\left(x,y\right)=0\right]\right)$
 ${S}{≔}\left[{{\mathrm{\xi }}}_{{x}}{=}{0}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{y}}{+}{{\mathrm{\eta }}}_{{x}}{=}{0}{,}{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}{,}{x}}{=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\eta }}{,}{\mathrm{\xi }}\right]$ (1)
 > $\mathrm{S1}≔\mathrm{LHPDE}\left(\left[\frac{{\partial }^{2}}{\partial {x}^{2}}\mathrm{α}\left(x,y\right)=0,\frac{\partial }{\partial y}\mathrm{α}\left(x,y\right)=0,\frac{\partial }{\partial x}\mathrm{β}\left(x,y\right)=0,\frac{{\partial }^{2}}{\partial {y}^{2}}\mathrm{β}\left(x,y\right)=0,\frac{\partial }{\partial x}\mathrm{α}\left(x,y\right)-\left(\frac{\partial }{\partial y}\mathrm{β}\left(x,y\right)\right)=0\right],\mathrm{indep}=\left[x,y\right],\mathrm{dep}=\left[\mathrm{α},\mathrm{β}\right]\right)$
 ${\mathrm{S1}}{≔}\left[{{\mathrm{\alpha }}}_{{x}{,}{x}}{=}{0}{,}{{\mathrm{\alpha }}}_{{y}}{=}{0}{,}{{\mathrm{\beta }}}_{{x}}{=}{0}{,}{{\mathrm{\beta }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\alpha }}}_{{x}}{-}{{\mathrm{\beta }}}_{{y}}{=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\alpha }}{,}{\mathrm{\beta }}\right]$ (2)
 > $\mathrm{VectorSpaceSum}\left(S,\mathrm{S1}\right)$
 $\left[{{\mathrm{\eta }}}_{{x}{,}{x}}{=}{0}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\eta }}{,}{\mathrm{\xi }}\right]$ (3)
 > $\mathrm{VectorSpaceSum}\left(S,\mathrm{S1},\mathrm{depname}=\left[\mathrm{φ},\mathrm{ψ}\right]\right)$
 $\left[{{\mathrm{\phi }}}_{{x}{,}{x}}{=}{0}{,}{{\mathrm{\phi }}}_{{y}}{=}{0}{,}{{\mathrm{\psi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\psi }}}_{{x}}{=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\phi }}{,}{\mathrm{\psi }}\right]$ (4)

Compatibility

 • The VectorSpaceSum command was introduced in Maple 2020.