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LHPDO Object as Operator

 Calling Sequence Delta( exprList )

Parameters

 Delta - a LHPDO object exprList - a list of differential expressions

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right):$
 > $\mathrm{\Delta }≔\mathrm{LHPDO}\left(\left[\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y,y\right),\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),x\right)+\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y\right),\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),y\right),\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),x\right)\right],\mathrm{indep}=\left[x,y\right],\mathrm{dep}=\left[\mathrm{\xi },\mathrm{\eta }\right]\right)$
 ${\mathrm{\Delta }}{≔}\left({\mathrm{ξ}}{,}{\mathrm{η}}\right){→}\left[\frac{{\partial }}{{\partial }{y}}{}\left(\frac{{\partial }}{{\partial }{y}}{}{\mathrm{ξ}}\right){,}\frac{{\partial }}{{\partial }{x}}{}{\mathrm{η}}{+}\frac{{\partial }}{{\partial }{y}}{}{\mathrm{ξ}}{,}\frac{{\partial }}{{\partial }{y}}{}{\mathrm{η}}{,}\frac{{\partial }}{{\partial }{x}}{}{\mathrm{ξ}}\right]$ (1)

The LHPDO Delta acts on an ordered pair of expressions, and returns a list of 4 elements.

 > $\mathrm{\Delta }\left(\left[\mathrm{\xi }\left(x,y\right),\mathrm{\eta }\left(x,y\right)\right]\right)$
 $\left[\frac{{{\partial }}^{{2}}}{{\partial }{{y}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){,}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\eta }}{}\left({x}{,}{y}\right){+}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){,}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\eta }}{}\left({x}{,}{y}\right){,}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right)\right]$ (2)
 > $\mathrm{\Delta }\left(\left[-\left(y-\mathrm{y0}\right),x-\mathrm{x0}\right]\right)$
 $\left[{0}{,}{0}{,}{0}{,}{0}\right]$ (3)
 > $\mathrm{GetDependentsCount}\left(\mathrm{\Delta }\right)$
 ${2}$ (4)
 > $\mathrm{GetSystemCount}\left(\mathrm{\Delta }\right)$
 ${4}$ (5)
 > $\mathrm{GetDependencies}\left(\mathrm{\Delta }\right)$
 $\left[\left[{x}{,}{y}\right]{,}\left[{x}{,}{y}\right]\right]$ (6)

Operator constructed from differential expressions with partial dependencies:

 > $U≔\mathrm{LHPDO}\left(\left[\mathrm{diff}\left(u\left(x\right),x,x\right)-u\left(x\right),\mathrm{diff}\left(v\left(y\right),y,y\right)\right],\mathrm{indep}=\left[x,y\right],\mathrm{dep}=\left[u,v\right]\right)$
 ${U}{≔}\left({u}{,}{v}\right){→}\left[\frac{{\partial }}{{\partial }{x}}{}\left(\frac{{\partial }}{{\partial }{x}}{}{u}\right){-}{u}{,}\frac{{\partial }}{{\partial }{y}}{}\left(\frac{{\partial }}{{\partial }{y}}{}{v}\right)\right]$ (7)
 > $\mathrm{GetDependencies}\left(U\right)$
 $\left[\left[{x}\right]{,}\left[{y}\right]\right]$ (8)
 > $U\left(\left[\mathrm{exp}\left(x\right),\mathrm{exp}\left(y\right)\right]\right)$
 $\left[{0}{,}{{ⅇ}}^{{y}}\right]$ (9)

The operator U is constructed to act on functions [u(x),v(y)], and it is an error to ask it to act on expressions with more general dependencies:

 > $U\left(\left[\mathrm{exp}\left(x+y\right),\mathrm{exp}\left(y\right)\right]\right)$

Compatibility

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