The command VFPDO(...) is for constructing a VFPDO object. It returns a VFPDO object if successful. A valid VFPDO object has access to various methods which allow it to be manipulated and its contents queried. For more detail, see Overview of the VFPDO object.

The VFPDO constructor command works by taking a list of linear homogeneous differential expressions and return essentially the same thing, but as a differential operator on the vector field V.

In the first calling sequence VFPDO("trivial", V) returns the zero VFPDO operator on the vector field V.

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] or a LHPDE object, and their dependant variables must be components of the vector field V.

In the third sequence, it builds a VFPDO from LAVF L. Note here if the L is integrated, the VFPDO will be built from the implicit form of L.

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 VFPDO(...) only after executing the command with(LieAlgebrasOfVectorFields), but can always be used in the form LieAlgebrasOfVectorFields:-VFPDO(...).

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

$X≔\mathrm{VectorField}\left(\mathrm{\xi }\left(x\,y\right)\mathrm{D}\left[x\right]+\mathrm{\eta }\left(x\,y\right)\mathrm{D}\left[y\right]\,\mathrm{space}=\left[x\,y\right]\right)$

$X≔\mathrm{\xi }\left(x\,y\right)\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(x\,y\right)\left(\frac{\ⅆ}{\ⅆy}\right)$

$S≔\mathrm{LHPDE}\left(\left[\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,x\right)=0\,\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,y\,y\right)=0\,\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,x\right)=-\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,y\right)\,\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,y\right)=0\right]\,\mathrm{indep}=\left[x\,y\right]\,\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\right]\right)$

$S≔\left[\frac{\partial }{\partial x}\mathrm{\xi }\left(x\,y\right)=0\,\frac{{\partial }^2}{\partial y^2}\mathrm{\xi }\left(x\,y\right)=0\,\frac{\partial }{\partial x}\mathrm{\eta }\left(x\,y\right)=-\frac{\partial }{\partial y}\mathrm{\xi }\left(x\,y\right)\,\frac{\partial }{\partial y}\mathrm{\eta }\left(x\,y\right)=0\right],\mathrm{indep}=\left[x\,y\right],\mathrm{dep}=\left[\mathrm{\xi }\left(x\,y\right)\,\mathrm{\eta }\left(x\,y\right)\right]$

$L≔\mathrm{LAVF}\left(X\,S\right)$

$L≔\left[\mathrm{\xi }\left(x\,y\right)\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(x\,y\right)\left(\frac{\ⅆ}{\ⅆy}\right)\right]\&where\left\{\left[\frac{{\partial }^2}{\partial y^2}\mathrm{\xi }\left(x\,y\right)=0\,\frac{\partial }{\partial x}\mathrm{\xi }\left(x\,y\right)=0\,\frac{\partial }{\partial x}\mathrm{\eta }\left(x\,y\right)=-\frac{\partial }{\partial y}\mathrm{\xi }\left(x\,y\right)\,\frac{\partial }{\partial y}\mathrm{\eta }\left(x\,y\right)=0\right]\right\}$

$\mathrm{VFPDO}\left(''trivial''\,X\right)$

$X\mapsto \left[X\left(x\right)\,X\left(y\right)\right]$

$\mathrm{VFPDO}\left(L\right)$

$X\mapsto \left[\frac{{\partial }^2}{\partial y^2}X\left(x\right)\,\frac{\ⅆ}{\ⅆx}X\left(x\right)\,\frac{\partial }{\partial x}X\left(y\right)+\frac{\partial }{\partial y}X\left(x\right)\,\frac{\ⅆ}{\ⅆy}X\left(y\right)\right]$

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

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