The IDBasis object is designed and created to represent an initial data basis for linear homogeneous partial differential equations (LHPDEs).

The IDBasis object provides possibility of changing the initial data basis on the LHPDE object.

A LHPDE object must be provided in order to construct an IDBasis object. Here the LHPDE object must be of finite type (i.e. there are a finite number of parametric derivatives). See LieAlgebrasOfVectorFields[IDBasis] for more detail.

The IDBasis object is exported by the LieAlgebrasOfVectorFields package. See Overview of the LieAlgebrasOfVectorFields package for more detail.

The default initial data basis is based on the parametric derivatives of a LHPDE object. Then the IDBasis object stores the change-of-basis matrix (and its inverse) with respect to this fixed basis.

The data attributes of the IDBasis object are "change-of-basis matrix" and the "parametric derivatives". These can be accessed via the GetChangeBasis and GetParametricDerivatives methods.

After an IDBasis object B is successfully constructed, each method in the IDBasis object can be accessed by either the short form method(B, arguments) or the long form B:-method(B, arguments).

Three methods are available after an IDBasis object is constructed:

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

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

$\mathrm{exports}\left(\mathrm{IDBasis}\,'\mathrm{static}'\right)$

$\mathrm{GetChangeBasis},\mathrm{GetParametricDerivatives},\mathrm{Copy},\mathrm{ModulePrint},\mathrm{ModuleCopy},\mathrm{ModuleApply}$

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

$\mathrm{E2}≔\mathrm{LHPDE}\left(\left[\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,y\,y\right)=0\,\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,x\right)=-\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,y\right)\,\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,y\right)=0\,\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,x\right)=0\right]\,\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\right]\,\mathrm{indep}=\left[x\,y\right]\right)$

$\mathrm{E2}≔\left[{\mathrm{\xi }}_{y,y}=0\,{\mathrm{\eta }}_x=-{\mathrm{\xi }}_y\,{\mathrm{\eta }}_y=0\,{\mathrm{\xi }}_x=0\right],\mathrm{indep}=\left[x\,y\right],\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\right]$

$B≔\mathrm{IDBasis}\left(\mathrm{E2}\,\left[⟨1\,0\,0⟩\,⟨-y\,-x\,-1⟩\,⟨0\,1\,0⟩\right]\right)$

$B≔\left[\mathrm{\xi }-y\left({\mathrm{\xi }}_y\right)\,-x\left({\mathrm{\xi }}_y\right)+\mathrm{\eta }\,-{\mathrm{\xi }}_y\right]$

$\mathrm{GetChangeBasis}\left(B\right)$

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

$\mathrm{GetChangeBasis}\left(B\,'\mathrm{output}'=''matrix''\right)$

$\left[\begin{array}{ccc}1& -y& 0\\ 0& -x& 1\\ 0& \mathrm{-1}& 0\end{array}\right]$

$\mathrm{GetChangeBasis}\left(B\,''newToOld''\right)$

$\left[\begin{array}{ccc}1& 0& -y\\ 0& 0& \mathrm{-1}\\ 0& 1& -x\end{array}\right]$

$\mathrm{Copy}\left(B\,\left[\mathrm{\alpha }\,\mathrm{\beta }\right]\right)$

$\left[\mathrm{\alpha }-y\left({\mathrm{\alpha }}_y\right)\,-x\left({\mathrm{\alpha }}_y\right)+\mathrm{\beta }\,-{\mathrm{\alpha }}_y\right]$

$\mathrm{GetParametricDerivatives}\left(B\right)$

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

$\mathrm{ParametricDerivatives}\left(\mathrm{E2}\right)$

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

$\mathrm{SetIDBasis}\left(\mathrm{E2}\,B\right)$

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

$\mathrm{GetIDBasis}\left(\mathrm{E2}\right)$

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