 EliminationSystem - Maple Help

LieAlgebrasOfVectorFields[EliminationSystem]

construct a LHPDE object that only contain some specific variables from a given DEs system.

EliminationSystem

construct a LHPDE object that only contain some specific variables from a given LHPDE object Calling Sequence EliminationSystem(system, dvars, varsToKeep, indep = ivars) EliminationSystem(obj, varsToKeep) Parameters

 system - a list or set of linear homogeneous PDEs or ODEs dvars - a list of dependent variables from the system, as functions or names varsToKeep - a list of dependent variable names that are to be kept after elimination in the system ivars - (optional) a list of independent variables from the system obj - a LHPDE object Description

 • The EliminationSystem command finds a LHPDE object that only contain some specific variables from a given LHPDEs system.
 • The command returns a LHPDE object that is in rif-reduced form (see IsRifReduced), and with varsToKeep as the dependent variables.
 • The command computes a differentially reduced form for the elimination module with respect to some specific variables, this is achieved by performing rif-reduce on a LHPDE object (see RifReduce) with a block ranking.
 • In the first sequence, the command is part of the LieAlgebrasOfVectorFields package. For more detail, see Overview of the LieAlgebrasOfVectorFields package.
 • This command can be used in the form LHPDE(...) only after executing the command with(LieAlgebrasOfVectorFields), but can always be used by executing LieAlgebrasOfVectorFields:-LHPDE(...).
 • The second calling sequence is an extension of the first, so that the command works for a LHPDE object. That is, a call EliminationSystem(obj, varsToKeep) is equivalent to EliminationSystem(GetSystem(obj), GetDependents(obj), varsToKeep, indep = GetIndependents(obj)).
 • In the second calling sequence, the method is associated with the LHPDE object. For more detail, see Overview of the LHPDE object. Examples

 > $\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right):$
 > $\mathrm{Typesetting}:-\mathrm{Settings}\left(\mathrm{userep}=\mathrm{true}\right):$
 > $\mathrm{Typesetting}:-\mathrm{Suppress}\left(\left\{\mathrm{\eta },\mathrm{\xi }\right\}\left(x,y\right)\right)$
 > $\mathrm{DEs}≔\left[\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),x\right)=0,\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),y\right)=0,\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y\right)+\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),x\right)=0,\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y,y\right)=0,\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),x,x\right)=0\right]$
 ${\mathrm{DEs}}{≔}\left[{{\mathrm{\xi }}}_{{x}}{=}{0}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{y}}{+}{{\mathrm{\eta }}}_{{x}}{=}{0}{,}{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}{,}{x}}{=}{0}\right]$ (1)

The variable eta is eliminated from the system. The system only contains xi.

 > $\mathrm{EliminationSystem}\left(\mathrm{DEs},\left[\mathrm{\xi }\left(x,y\right),\mathrm{\eta }\left(x,y\right)\right],\left[\mathrm{\xi }\right]\right)$
 $\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}\right]$ (2)
 > $S≔\mathrm{LHPDE}\left(\mathrm{DEs},\mathrm{dep}=\left[\mathrm{\xi },\mathrm{\eta }\right],\mathrm{indep}=\left[x,y\right]\right)$
 ${S}{≔}\left[{{\mathrm{\xi }}}_{{x}}{=}{0}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{y}}{+}{{\mathrm{\eta }}}_{{x}}{=}{0}{,}{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}{,}{x}}{=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{,}{\mathrm{\eta }}\right]$ (3)

Similar output, but the input argument is now a LHPDE object.

 > $\mathrm{EliminationSystem}\left(S,\left[\mathrm{\xi }\right]\right)$
 $\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}\right]$ (4) Compatibility

 • The LieAlgebrasOfVectorFields[EliminationSystem] and EliminationSystem commands were introduced in Maple 2020.