An LHPDO object is appliable, and can act as a partial differential operator

An LHPDO object Delta is a function that accepts a list of $m$ expressions as argument, and returns a list of $s$ expressions. Here, the integers $1\le m$ and $0\le s$ are properties of Delta determined at the time it is constructed, and which can be queried from the object. See example below.

Where an LHPDO Delta is constructed from differential expressions with respect to dependent variables with partial dependencies, the operator Delta may only act on a list of expressions with these same dependencies. An error message is emitted if this condition is not met. See example below.

These methods are associated with the LHPDO object. For more detail, see Overview of the LHPDO object.

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

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

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

$\mathrm{\Delta }\left(\left[\mathrm{\xi }\left(x\,y\right)\,\mathrm{\eta }\left(x\,y\right)\right]\right)$

$\left[\frac{{\partial }^2}{\partial y^2}\mathrm{\xi }\left(x\,y\right)\,\frac{\partial }{\partial x}\mathrm{\eta }\left(x\,y\right)+\frac{\partial }{\partial y}\mathrm{\xi }\left(x\,y\right)\,\frac{\partial }{\partial y}\mathrm{\eta }\left(x\,y\right)\,\frac{\partial }{\partial x}\mathrm{\xi }\left(x\,y\right)\right]$

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

$\left[0\,0\,0\,0\right]$

$\mathrm{GetDependentsCount}\left(\mathrm{\Delta }\right)$

$2$

$\mathrm{GetSystemCount}\left(\mathrm{\Delta }\right)$

$4$

$\mathrm{GetDependencies}\left(\mathrm{\Delta }\right)$

$\left[\left[x\,y\right]\,\left[x\,y\right]\right]$

$U≔\mathrm{LHPDO}\left(\left[\mathrm{diff}\left(u\left(x\right)\,x\,x\right)-u\left(x\right)\,\mathrm{diff}\left(v\left(y\right)\,y\,y\right)\right]\,\mathrm{indep}=\left[x\,y\right]\,\mathrm{dep}=\left[u\,v\right]\right)$

$U≔\left(u\,v\right)\→\left[\frac{\partial }{\partial x}\left(\frac{\partial }{\partial x}u\right)-u\,\frac{\partial }{\partial y}\left(\frac{\partial }{\partial y}v\right)\right]$

$\mathrm{GetDependencies}\left(U\right)$

$\left[\left[x\right]\,\left[y\right]\right]$

$U\left(\left[\exp \left(x\right)\,\exp \left(y\right)\right]\right)$

$\left[0\,{\ⅇ}^y\right]$

$U\left(\left[\exp \left(x+y\right)\,\exp \left(y\right)\right]\right)$

Error, (in U) invalid input: 1st argument to LHPDO operator depends on extra variables [y]

The LHPDO Object as Operator command was introduced in Maple 2020.

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