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LieAlgebrasOfVectorFields

 LHPDO
 construct a LHPDO object

 Calling Sequence LHPDO( S ) LHPDO( sys, options) LHPDO( str, dep = vars, options)

Parameters

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

 Options The LHPDO constructor accepts a number of options, which are all passed through to the constructor of the LHPDE object (see LieAlgebrasOfVectorFields[LHPDE] for more detail).

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 LHPDO $\mathrm{\Delta }$ consists of independent variables $x=\left({x}_{1},{x}_{2},..,{x}_{n}\right)$,  and a sequence of linear homogeneous PDOs $\left({\mathrm{\Delta }}_{1},{\mathrm{\Delta }}_{2},\dots {\mathrm{\Delta }}_{s}\right)$. Here, each ${\mathrm{\Delta }}_{j}$ is a function / operator that applies to a list of $m$  scalar expressions $\left({y}_{1},{y}_{2},..{y}_{m}\right)$, and computes a linear homogenous combination of various derivatives of ${y}_{i}$ of order up to $k$.
 • The LHPDO constructor command works by converting a list of s differential expressions linear homogeneous with respect to dependent variables $\left({u}_{1},{u}_{2},..,{u}_{m}\right)$ each of which is an indeterminate function of its arguments -- which must be some nonempty subset of $\left({x}_{1},{x}_{2},..,{x}_{n}\right)$. These differential expressions are specified in various ways by the first argument to LHPDO.
 • In the first calling sequence, an LHPDE object S is provided.  The differential expressions are formed from taking (lhs - rhs) for each equation in the system of PDEs in S (see GetSystem). An LHPDE object fully specifies other quantities required for conversion to LHPDO, so no further arguments are allowed.
 • In the second calling sequence, the first input argument is a list or set of scalar expressions or equations, or alternatively a table as returned by DEtools[rifsimp].  In fact the sequence of arguments for this calling sequence is exactly what is accepted by the LHPDE constructor.  See LieAlgebrasOfVectorFields[LHPDE] for the detailed specification of arguments and options for this calling sequence.
 • In the third calling sequence, the first input argument is the string "trivial" or "identity", in which case, the identity LHPDO (i.e. one which maps $\left({u}_{1},{u}_{2},..,{u}_{m}\right)\to \left({u}_{1},{u}_{2},..,{u}_{m}\right)$)   is constructed. In this calling sequence, the second argument dep= vars is required. Further options are as per the LHPDE constructor (see LieAlgebrasOfVectorFields[LHPDE]).
 • 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 LHPDO(...) only after executing the command with(LieAlgebrasOfVectorFields), but can always be used in the form LieAlgebrasOfVectorFields:-LHPDO(...).

Examples

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

List of differential expressions...

 > $\mathrm{diffExpr}≔\left[\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y,y\right),\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),x\right)+\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y\right),\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),y\right),\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),x\right)\right]$
 ${\mathrm{diffExpr}}{≔}\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{,}{{\mathrm{\eta }}}_{{x}}{+}{{\mathrm{\xi }}}_{{y}}{,}{{\mathrm{\eta }}}_{{y}}{,}{{\mathrm{\xi }}}_{{x}}\right]$ (1)
 > $\mathrm{\Delta }≔\mathrm{LHPDO}\left(\mathrm{diffExpr}\right)$
 ${\mathrm{\Delta }}{≔}\left({\mathrm{η}}{,}{\mathrm{ξ}}\right){→}\left[\frac{{\partial }}{{\partial }{y}}{}\left(\frac{{\partial }}{{\partial }{y}}{}{\mathrm{ξ}}\right){,}\frac{{\partial }}{{\partial }{x}}{}{\mathrm{η}}{+}\frac{{\partial }}{{\partial }{y}}{}{\mathrm{ξ}}{,}\frac{{\partial }}{{\partial }{y}}{}{\mathrm{η}}{,}\frac{{\partial }}{{\partial }{x}}{}{\mathrm{ξ}}\right]$ (2)

List of differential equations...

 > $\mathrm{detSys}≔\left[\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y,y\right)=0,\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),x\right)+\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y\right)=0,\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),y\right)=0,\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),x\right)=0\right]$
 ${\mathrm{detSys}}{≔}\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{+}{{\mathrm{\xi }}}_{{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}\right]$ (3)
 > $\mathrm{\Delta }≔\mathrm{LHPDO}\left(\mathrm{detSys}\right)$
 ${\mathrm{\Delta }}{≔}\left({\mathrm{η}}{,}{\mathrm{ξ}}\right){→}\left[\frac{{\partial }}{{\partial }{y}}{}\left(\frac{{\partial }}{{\partial }{y}}{}{\mathrm{ξ}}\right){,}\frac{{\partial }}{{\partial }{x}}{}{\mathrm{η}}{+}\frac{{\partial }}{{\partial }{y}}{}{\mathrm{ξ}}{,}\frac{{\partial }}{{\partial }{y}}{}{\mathrm{η}}{,}\frac{{\partial }}{{\partial }{x}}{}{\mathrm{ξ}}\right]$ (4)

Rif-table...

 > $R≔\mathrm{DEtools}\left[\mathrm{rifsimp}\right]\left(\mathrm{detSys}\right)$
 ${R}{≔}{table}{}\left(\left[{\mathrm{Solved}}{=}\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{-}{{\mathrm{\xi }}}_{{y}}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}\right]\right]\right)$ (5)
 > $\mathrm{\Delta }≔\mathrm{LHPDO}\left(R\right)$
 ${\mathrm{\Delta }}{≔}\left({\mathrm{η}}{,}{\mathrm{ξ}}\right){→}\left[\frac{{\partial }}{{\partial }{y}}{}\left(\frac{{\partial }}{{\partial }{y}}{}{\mathrm{ξ}}\right){,}\frac{{\partial }}{{\partial }{x}}{}{\mathrm{η}}{+}\frac{{\partial }}{{\partial }{y}}{}{\mathrm{ξ}}{,}\frac{{\partial }}{{\partial }{x}}{}{\mathrm{ξ}}{,}\frac{{\partial }}{{\partial }{y}}{}{\mathrm{η}}\right]$ (6)

LHPDE object...

 > $S≔\mathrm{LHPDE}\left(\mathrm{detSys}\right)$
 ${S}{≔}\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{+}{{\mathrm{\xi }}}_{{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\eta }}{,}{\mathrm{\xi }}\right]$ (7)
 > $\mathrm{\Delta }≔\mathrm{LHPDO}\left(S\right)$
 ${\mathrm{\Delta }}{≔}\left({\mathrm{η}}{,}{\mathrm{ξ}}\right){→}\left[\frac{{\partial }}{{\partial }{y}}{}\left(\frac{{\partial }}{{\partial }{y}}{}{\mathrm{ξ}}\right){,}\frac{{\partial }}{{\partial }{x}}{}{\mathrm{η}}{+}\frac{{\partial }}{{\partial }{y}}{}{\mathrm{ξ}}{,}\frac{{\partial }}{{\partial }{y}}{}{\mathrm{η}}{,}\frac{{\partial }}{{\partial }{x}}{}{\mathrm{ξ}}\right]$ (8)

Identity operator...

 > $\mathrm{id}≔\mathrm{LHPDO}\left("identity",\mathrm{dep}=\left[u\left(x,y\right),v\left(x,y\right)\right]\right)$
 ${\mathrm{id}}{≔}\left({u}{,}{v}\right){↦}\left[{u}{,}{v}\right]$ (9)

A LHPDO depends on how its arguments are ordered, so the operator constructed may differ according to the specification of dependent variables....

 > $E≔\mathrm{LHPDO}\left(\mathrm{detSys},\mathrm{indep}=\left[x,y\right],\mathrm{dep}=\left[\mathrm{\xi },\mathrm{\eta }\right]\right)$
 ${E}{≔}\left({\mathrm{ξ}}{,}{\mathrm{η}}\right){→}\left[\frac{{\partial }}{{\partial }{y}}{}\left(\frac{{\partial }}{{\partial }{y}}{}{\mathrm{ξ}}\right){,}\frac{{\partial }}{{\partial }{x}}{}{\mathrm{η}}{+}\frac{{\partial }}{{\partial }{y}}{}{\mathrm{ξ}}{,}\frac{{\partial }}{{\partial }{y}}{}{\mathrm{η}}{,}\frac{{\partial }}{{\partial }{x}}{}{\mathrm{ξ}}\right]$ (10)

The main purpose of the LHPDO object is to act as an operator...

 > $E\left(\left[{\left(x-\mathrm{x0}\right)}^{2},{\left(y-\mathrm{y0}\right)}^{2}\right]\right)$
 $\left[{0}{,}{0}{,}{2}{}{y}{-}{2}{}{\mathrm{y0}}{,}{2}{}{x}{-}{2}{}{\mathrm{x0}}\right]$ (11)

Compatibility

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