dep = deps

indep = indeps

inRifReducedForm = true or inRifReducedForm = false

ranking = rk

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(...).

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

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

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

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

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

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

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

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

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

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

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

$\mathrm{S1}≔\mathrm{LHPDE}\left(R\,\mathrm{ranking}=\left[\mathrm{\xi }\,\mathrm{\eta }\right]\right)$

$\mathrm{S1}≔\left[{\mathrm{\xi }}_{y,y}=0\,{\mathrm{\xi }}_x=0\,{\mathrm{\eta }}_x=-{\mathrm{\xi }}_y\,{\mathrm{\eta }}_y=0\right],\mathrm{indep}=\left[x\,y\right],\mathrm{dep}=\left[\mathrm{\eta }\,\mathrm{\xi }\right]$

$\mathrm{IsRifReduced}\left(\mathrm{S1}\right)$

$\mathrm{true}$

$\mathrm{GetRanking}\left(\mathrm{S1}\right)$

$\left[\mathrm{\xi }\,\mathrm{\eta }\right]$

$T≔\mathrm{LHPDE}\left(''trivial''\,\mathrm{dep}=\left[\mathrm{\alpha }\left(x\right)\,\mathrm{\beta }\left(y\right)\right]\right)$

$T≔\left[\mathrm{\alpha }\left(x\right)=0\,\mathrm{\beta }\left(y\right)=0\right],\mathrm{indep}=\left[x\,y\right],\mathrm{dep}=\left[\mathrm{\alpha }\left(x\right)\,\mathrm{\beta }\left(y\right)\right]$

$U≔\mathrm{LHPDE}\left(''universal''\,\mathrm{dep}=\left[\mathrm{\alpha }\left(x\right)\,\mathrm{\beta }\left(y\right)\right]\right)$

$U≔\left[\right],\mathrm{indep}=\left[x\,y\right],\mathrm{dep}=\left[\mathrm{\alpha }\left(x\right)\,\mathrm{\beta }\left(y\right)\right]$

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

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