 SolutionDimension - Maple Help

SolutionDimension

calculate the solution dimension of a LHPDE object.

IsFiniteType

check if a LHPDE object is of finite type

IsTrivial

check if a LHPDE object has only the trivial solution Calling Sequence SolutionDimension( obj) IsFiniteType( obj) IsTrivial( obj) Parameters

 obj - a LHPDE object that is in rif-Reduced from. Description

 • The SolutionDimension method calculates the solution dimension of a LHPDE object. It returns $\mathrm{\infty }$ if the solution dimension is not finite.
 • Let S be a LHPDE object. Then IsFiniteType(S) returns true if and only if SolutionDimension(S) $\ne \mathrm{\infty }$.
 • Let S be a LHPDE object. Then IsTrivial(S) returns true if and only if SolutionDimension(S) $=0$.
 • These methods are associated with the LHPDE object. For more detail, see Overview of the LHPDE object. Examples

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

 > $S≔\mathrm{LHPDE}\left(\left[\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y,y\right)=0,\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),x\right)=-\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y\right),\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),y\right)=0,\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),x\right)=0\right],\mathrm{indep}=\left[x,y\right],\mathrm{dep}=\left[\mathrm{\xi }\left(x,y\right),\mathrm{\eta }\left(x,y\right)\right],\mathrm{inRifReducedForm}=\mathrm{true}\right)$
 ${S}{≔}\left[\frac{{{\partial }}^{{2}}}{{\partial }{{y}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){=}{0}{,}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\eta }}{}\left({x}{,}{y}\right){=}{-}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){,}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\eta }}{}\left({x}{,}{y}\right){=}{0}{,}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{}\left({x}{,}{y}\right){,}{\mathrm{\eta }}{}\left({x}{,}{y}\right)\right]$ (1)
 > $\mathrm{SolutionDimension}\left(S\right)$
 ${3}$ (2)

The system S is of finite type but not trivial:

 > $\mathrm{IsFiniteType}\left(S\right)$
 ${\mathrm{true}}$ (3)
 > $\mathrm{IsTrivial}\left(S\right)$
 ${\mathrm{false}}$ (4) Compatibility

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