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LieAlgebrasOfVectorFields

 LHPDE
 construct a LHPDE object

 Calling Sequence LHPDE( sys, options) LHPDE( rifTable, options) LHPDE( str, dep = vars, options) LHPDE( [], dep = vars, options)

Parameters

 sys - a list or set of linear homogeneous PDEs or ODEs or expressions rifTable - a table as returned by DEtools[rifsimp] str - a string: either "trivial" or "universal" vars - a list of dependent variables as functions options - optional equations controlling details of the first input argument

Options

 • dep = deps

This option specifies the dependent variables of the DEs system sys, as a list of functions (or a list of names if the functional dependencies are apparent from the system)

 • indep = indeps

This option specifies the independent variables of the DEs system sys, as a list of names

 • inRifReducedForm = true or inRifReducedForm = false

This option indicates whether the DEs system sys is in rif-reduced form or not

 • ranking = rk

This option specifies the ranking that is used on the rif-reduced DEs system sys; as a list (or list of lists) of dependent variable names.  The specification of this option is consistent with the ranking used by the DEtools[rifsimp] command. See ranking for more detail. For the first calling sequence, this option requires that the option inRifReducedForm = true be specified.

Description

 • The command LHPDO(...) is for constructing a LHPDO object. It returns a LHPDO object if successful. A valid LHPDO object has access to various methods which allow it to be manipulated and its contents queried. For more detail, see Overview of the LHPDO object.
 • A LHPDEs system $E$ consists of independent variables $x=\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)$, dependent variables $u=\left({u}_{1},{u}_{2},..,{u}_{m}\right)$ and a collection of linear homogeneous PDEs $f=\left({f}_{1},{f}_{2},..,{f}_{s}\right)$ . Here, each ${f}_{i}$ is an equation that depends on the ${u}_{i}$ and derivatives of ${u}_{i}$ of order up to $k$, such that ${f}_{i}$ is linear homogeneous in each ${u}_{i}$.
 • In the first calling sequence, the items in the input argument sys must be linear homogeneous with respect to the dependent variables. If sys is a list of expressions then any expression $f$ will be turned into an equation $f=0$.
 • In the second calling sequence, the input argument rifTable must be a table as returned by DEtools[rifsimp]. The rifTable R must only contain a single case and include no more than "Solved", "Pivots", and "Case" indices (see DEtools[rifsimp] Algorithm Output for more detail). The equations in R[Solved] will used as the DEs system for constructing the LHPDE object. A LHPDE object that is constructed from a rifTable is automatically marked as a rif-reduced LHPDE object (see IsRifReduced).
 • The third calling sequence is a special constructor for either a trivial LHPDE object or a universal LHPDE object. A trivial LHPDE has only the zero solution. A universal LHPDE has empty system (i.e. no restriction on solutions). The second input argument dep = vars must be provided. A LHPDE object that is constructed using this calling sequence is automatically marked as a rif-reduced LHPDE object.
 • The fourth calling sequence is a special constructor for a universal LHPDE object. The second input argument dep = vars must be provided. A LHPDE object that is constructed using this calling sequence is automatically marked as a rif-reduced LHPDE object.
 • The options (dep, indep, inRifReducedForm, and ranking) are used for fully specifying properties of the DEs system.
 • This 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 in the form LieAlgebrasOfVectorFields:-LHPDE(...).

Examples

 > $\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right):$
 > $\mathrm{Typesetting}:-\mathrm{Settings}\left(\mathrm{userep}=\mathrm{true}\right):$
 > $\mathrm{Typesetting}:-\mathrm{Suppress}\left(\left[\mathrm{\xi }\left(x,y\right),\mathrm{\eta }\left(x,y\right)\right]\right):$
 > $\mathrm{detSys}≔\left[\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y,y\right)=0,\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),x\right)+\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y\right)=0,\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),y\right)=0,\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),x\right)=0\right]$
 ${\mathrm{detSys}}{≔}\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{+}{{\mathrm{\xi }}}_{{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}\right]$ (1)
 > $S≔\mathrm{LHPDE}\left(\mathrm{detSys}\right)$
 ${S}{≔}\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{+}{{\mathrm{\xi }}}_{{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\eta }}{,}{\mathrm{\xi }}\right]$ (2)
 > $S≔\mathrm{LHPDE}\left(\mathrm{detSys},\mathrm{dep}=\left[\mathrm{\xi },\mathrm{\eta }\right],\mathrm{indep}=\left[x,y\right]\right)$
 ${S}{≔}\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{+}{{\mathrm{\xi }}}_{{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{,}{\mathrm{\eta }}\right]$ (3)

Rif-table as input argument, the LHPDE object is automatically marked as rif-reduced.

 > $R≔\mathrm{DEtools}\left[\mathrm{rifsimp}\right]\left(\mathrm{detSys},\left[\mathrm{\xi },\mathrm{\eta }\right]\right)$
 ${R}{≔}{table}{}\left(\left[{\mathrm{Solved}}{=}\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{-}{{\mathrm{\xi }}}_{{y}}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}\right]\right]\right)$ (4)
 > $\mathrm{S1}≔\mathrm{LHPDE}\left(R,\mathrm{ranking}=\left[\mathrm{\xi },\mathrm{\eta }\right]\right)$
 ${\mathrm{S1}}{≔}\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{-}{{\mathrm{\xi }}}_{{y}}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\eta }}{,}{\mathrm{\xi }}\right]$ (5)
 > $\mathrm{IsRifReduced}\left(\mathrm{S1}\right)$
 ${\mathrm{true}}$ (6)
 > $\mathrm{GetRanking}\left(\mathrm{S1}\right)$
 $\left[{\mathrm{\xi }}{,}{\mathrm{\eta }}\right]$ (7)

Constructing a trivial LHPDE and a universal LHPDE.

 > $T≔\mathrm{LHPDE}\left("trivial",\mathrm{dep}=\left[\mathrm{\alpha }\left(x\right),\mathrm{\beta }\left(y\right)\right]\right)$
 ${T}{≔}\left[{\mathrm{\alpha }}{}\left({x}\right){=}{0}{,}{\mathrm{\beta }}{}\left({y}\right){=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\alpha }}{}\left({x}\right){,}{\mathrm{\beta }}{}\left({y}\right)\right]$ (8)
 > $U≔\mathrm{LHPDE}\left("universal",\mathrm{dep}=\left[\mathrm{\alpha }\left(x\right),\mathrm{\beta }\left(y\right)\right]\right)$
 ${U}{≔}\left[\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\alpha }}{}\left({x}\right){,}{\mathrm{\beta }}{}\left({y}\right)\right]$ (9)

Compatibility

 • The LieAlgebrasOfVectorFields[LHPDE] command was introduced in Maple 2020.