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(...).

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

$\mathrm{Typesetting}:-\mathrm{Settings}\left(\mathrm{userep}=\mathrm{true}\right)\:$

$\mathrm{Typesetting}:-\mathrm{Suppress}\left(\left[\mathrm{\xi }\left(x\,y\right)\,\mathrm{\eta }\left(x\,y\right)\right]\right)\:$

$\mathrm{diffExpr}≔\left[\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,y\,y\right)\,\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,x\right)+\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,y\right)\,\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,y\right)\,\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,x\right)\right]$

$\mathrm{diffExpr}≔\left[{\mathrm{\xi }}_{y,y}\,{\mathrm{\eta }}_x+{\mathrm{\xi }}_y\,{\mathrm{\eta }}_y\,{\mathrm{\xi }}_x\right]$

$\mathrm{\Delta }≔\mathrm{LHPDO}\left(\mathrm{diffExpr}\right)$

$\mathrm{\Delta }≔\left(\mathrm{\η}\,\mathrm{\ξ}\right)\→\left[\frac{\partial }{\partial y}\left(\frac{\partial }{\partial y}\mathrm{\ξ}\right)\,\frac{\partial }{\partial x}\mathrm{\η}\+\frac{\partial }{\partial y}\mathrm{\ξ}\,\frac{\partial }{\partial y}\mathrm{\η}\,\frac{\partial }{\partial x}\mathrm{\ξ}\right]$

$\mathrm{detSys}≔\left[\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,y\,y\right)=0\,\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,x\right)+\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,y\right)=0\,\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,y\right)=0\,\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,x\right)=0\right]$

$\mathrm{detSys}≔\left[{\mathrm{\xi }}_{y,y}=0\,{\mathrm{\eta }}_x+{\mathrm{\xi }}_y=0\,{\mathrm{\eta }}_y=0\,{\mathrm{\xi }}_x=0\right]$

$\mathrm{\Delta }≔\mathrm{LHPDO}\left(\mathrm{detSys}\right)$

$\mathrm{\Delta }≔\left(\mathrm{\η}\,\mathrm{\ξ}\right)\→\left[\frac{\partial }{\partial y}\left(\frac{\partial }{\partial y}\mathrm{\ξ}\right)\,\frac{\partial }{\partial x}\mathrm{\η}\+\frac{\partial }{\partial y}\mathrm{\ξ}\,\frac{\partial }{\partial y}\mathrm{\η}\,\frac{\partial }{\partial x}\mathrm{\ξ}\right]$

$R≔\mathrm{DEtools}\left[\mathrm{rifsimp}\right]\left(\mathrm{detSys}\right)$

$R≔table\left(\left[\mathrm{Solved}=\left[{\mathrm{\xi }}_{y,y}=0\,{\mathrm{\eta }}_x=-{\mathrm{\xi }}_y\,{\mathrm{\xi }}_x=0\,{\mathrm{\eta }}_y=0\right]\right]\right)$

$\mathrm{\Delta }≔\mathrm{LHPDO}\left(R\right)$

$\mathrm{\Delta }≔\left(\mathrm{\η}\,\mathrm{\ξ}\right)\→\left[\frac{\partial }{\partial y}\left(\frac{\partial }{\partial y}\mathrm{\ξ}\right)\,\frac{\partial }{\partial x}\mathrm{\η}\+\frac{\partial }{\partial y}\mathrm{\ξ}\,\frac{\partial }{\partial x}\mathrm{\ξ}\,\frac{\partial }{\partial y}\mathrm{\η}\right]$

$S≔\mathrm{LHPDE}\left(\mathrm{detSys}\right)$

$S≔\left[{\mathrm{\xi }}_{y,y}=0\,{\mathrm{\eta }}_x+{\mathrm{\xi }}_y=0\,{\mathrm{\eta }}_y=0\,{\mathrm{\xi }}_x=0\right],\mathrm{indep}=\left[x\,y\right],\mathrm{dep}=\left[\mathrm{\eta }\,\mathrm{\xi }\right]$

$\mathrm{\Delta }≔\mathrm{LHPDO}\left(S\right)$

$\mathrm{\Delta }≔\left(\mathrm{\η}\,\mathrm{\ξ}\right)\→\left[\frac{\partial }{\partial y}\left(\frac{\partial }{\partial y}\mathrm{\ξ}\right)\,\frac{\partial }{\partial x}\mathrm{\η}\+\frac{\partial }{\partial y}\mathrm{\ξ}\,\frac{\partial }{\partial y}\mathrm{\η}\,\frac{\partial }{\partial x}\mathrm{\ξ}\right]$

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

$\mathrm{id}≔\left(u\,v\right)\mapsto \left[u\,v\right]$

$E≔\mathrm{LHPDO}\left(\mathrm{detSys}\,\mathrm{indep}=\left[x\,y\right]\,\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\right]\right)$

$E≔\left(\mathrm{\ξ}\,\mathrm{\η}\right)\→\left[\frac{\partial }{\partial y}\left(\frac{\partial }{\partial y}\mathrm{\ξ}\right)\,\frac{\partial }{\partial x}\mathrm{\η}\+\frac{\partial }{\partial y}\mathrm{\ξ}\,\frac{\partial }{\partial y}\mathrm{\η}\,\frac{\partial }{\partial x}\mathrm{\ξ}\right]$

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

$\left[0\,0\,2y-2\mathrm{y0}\,2x-2\mathrm{x0}\right]$

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

For more information on Maple 2020 changes, see Updates in Maple 2020.