SolutionDimension - Maple Help

SolutionDimension

calculate the solution dimension for a LAVF object.

IsFiniteType

check if a LAVF object is of finite type

IsTrivial

check if a LAVF object has only the trivial solution

 Calling Sequence SolutionDimension( obj) IsFiniteType( obj) IsTrivial( obj)

Parameters

 obj - a LAVF object

Description

 • The SolutionDimension method calculates the solution dimension of the determining system for a LAVF object. It returns $\mathrm{\infty }$ if the solution dimension is not finite.
 • Let L be a LAVF object. Then IsFiniteType(L) returns true if and only if SolutionDimension(L) < $\mathrm{\infty }$.
 • Let L be a LAVF object. Then IsTrivial(L) returns true if and only if SolutionDimension(L) = 0.
 • These methods are front-end to the corresponding methods of a LHPDE object. That is, let L be a LAVF object and S be its determining system (i.e.  S = GetDeterminingSystem(L)), then SolutionDimension(L) equals SolutionDimension(S), IsFiniteType(L) equals IsFiniteType(S) and IsTrivial(L) equals IsTrivial(S). For more detail, see the corresponding methods: SolutionDimension, IsFiniteType, IsTrivial of a LHPDE object.
 • These methods are associated with the LAVF object. For more detail, see Overview of the LAVF object.

Examples

 > $\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right):$
 > $\mathrm{Typesetting}:-\mathrm{Settings}\left(\mathrm{userep}=\mathrm{true}\right):$
 > $\mathrm{Typesetting}:-\mathrm{Suppress}\left(\left[\mathrm{\xi }\left(x,y\right),\mathrm{\eta }\left(x,y\right)\right]\right):$

First, construct an indeterminate vector field and a determining system, then construct an LAVF object from them...

 > $V≔\mathrm{VectorField}\left(\mathrm{\xi }\left(x,y\right)\mathrm{D}\left[x\right]+\mathrm{\eta }\left(x,y\right)\mathrm{D}\left[y\right],\mathrm{space}=\left[x,y\right]\right)$
 ${V}{≔}{\mathrm{\xi }}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\eta }}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (1)
 > $\mathrm{E2}≔\mathrm{LHPDE}\left(\left[\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y,y\right)=0,\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),x\right)=-\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y\right),\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),y\right)=0,\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),x\right)=0\right],\mathrm{indep}=\left[x,y\right],\mathrm{dep}=\left[\mathrm{\xi },\mathrm{\eta }\right]\right)$
 ${\mathrm{E2}}{≔}\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{-}{{\mathrm{\xi }}}_{{y}}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{,}{\mathrm{\eta }}\right]$ (2)
 > $L≔\mathrm{LAVF}\left(V,\mathrm{E2}\right)$
 ${L}{≔}\left[{\mathrm{\xi }}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\eta }}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{-}{{\mathrm{\xi }}}_{{y}}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}\right]\right\}$ (3)

Now we can apply the methods to see properties of L.

 > $\mathrm{SolutionDimension}\left(L\right)$
 ${3}$ (4)

The determining system of L is of finite type but not trivial:

 > $\mathrm{IsFiniteType}\left(L\right)$
 ${\mathrm{true}}$ (5)
 > $\mathrm{IsTrivial}\left(L\right)$
 ${\mathrm{false}}$ (6)

Compatibility

 • The SolutionDimension, IsFiniteType and IsTrivial commands were introduced in Maple 2020.