Overview - Maple Help

Overview of the IDBasis Object

Description

 • 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).

List of IDBasis object Methods

 • Three methods are available after an IDBasis object is constructed:

Examples

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

Inserting option static gives a list of exports of the IDBasis object.

 > $\mathrm{exports}\left(\mathrm{IDBasis},'\mathrm{static}'\right)$
 ${\mathrm{GetChangeBasis}}{,}{\mathrm{GetParametricDerivatives}}{,}{\mathrm{Copy}}{,}{\mathrm{ModulePrint}}{,}{\mathrm{ModuleCopy}}{,}{\mathrm{ModuleApply}}$ (1)
 > $\mathrm{Typesetting}:-\mathrm{Suppress}\left(\left[\mathrm{\xi }\left(x,y\right),\mathrm{\eta }\left(x,y\right),\mathrm{\alpha }\left(x,y\right),\mathrm{\beta }\left(x,y\right)\right]\right)$
 > $\mathrm{E2}≔\mathrm{LHPDE}\left(\left[\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y,y\right)=0,\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),x\right)=-\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y\right),\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),y\right)=0,\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),x\right)=0\right],\mathrm{dep}=\left[\mathrm{\xi },\mathrm{\eta }\right],\mathrm{indep}=\left[x,y\right]\right)$
 ${\mathrm{E2}}{≔}\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{-}{{\mathrm{\xi }}}_{{y}}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{,}{\mathrm{\eta }}\right]$ (2)

Constructing an IDBasis B relevant to the LHPDEs system E2,

 > $B≔\mathrm{IDBasis}\left(\mathrm{E2},\left[⟨1,0,0⟩,⟨-y,-x,-1⟩,⟨0,1,0⟩\right]\right)$
 ${B}{≔}\left[{\mathrm{\xi }}{-}{y}{}\left({{\mathrm{\xi }}}_{{y}}\right){,}{-}{x}{}\left({{\mathrm{\xi }}}_{{y}}\right){+}{\mathrm{\eta }}{,}{-}{{\mathrm{\xi }}}_{{y}}\right]$ (3)
 > $\mathrm{GetChangeBasis}\left(B\right)$
 $\left[{\mathrm{\xi }}{-}{y}{}\left({{\mathrm{\xi }}}_{{y}}\right){,}{-}{x}{}\left({{\mathrm{\xi }}}_{{y}}\right){+}{\mathrm{\eta }}{,}{-}{{\mathrm{\xi }}}_{{y}}\right]$ (4)
 > $\mathrm{GetChangeBasis}\left(B,'\mathrm{output}'="matrix"\right)$
 $\left[\begin{array}{ccc}{1}& {-}{y}& {0}\\ {0}& {-}{x}& {1}\\ {0}& {-1}& {0}\end{array}\right]$ (5)

Returns the inverse of the change-of-basis matrix

 > $\mathrm{GetChangeBasis}\left(B,"newToOld"\right)$
 $\left[\begin{array}{ccc}{1}& {0}& {-}{y}\\ {0}& {0}& {-1}\\ {0}& {1}& {-}{x}\end{array}\right]$ (6)
 > $\mathrm{Copy}\left(B,\left[\mathrm{\alpha },\mathrm{\beta }\right]\right)$
 $\left[{\mathrm{\alpha }}{-}{y}{}\left({{\mathrm{\alpha }}}_{{y}}\right){,}{-}{x}{}\left({{\mathrm{\alpha }}}_{{y}}\right){+}{\mathrm{\beta }}{,}{-}{{\mathrm{\alpha }}}_{{y}}\right]$ (7)

The IDBasis B is constructed based on the parametric derivatives of the LHPDE object E2,

 > $\mathrm{GetParametricDerivatives}\left(B\right)$
 $\left[{\mathrm{\xi }}{,}{{\mathrm{\xi }}}_{{y}}{,}{\mathrm{\eta }}\right]$ (8)
 > $\mathrm{ParametricDerivatives}\left(\mathrm{E2}\right)$
 $\left[{\mathrm{\xi }}{,}{{\mathrm{\xi }}}_{{y}}{,}{\mathrm{\eta }}\right]$ (9)

The initial data basis of a LHPDEs system E2 can be recorded as being the IDBasis object B.

 > $\mathrm{SetIDBasis}\left(\mathrm{E2},B\right)$
 $\left[{\mathrm{\xi }}{-}{y}{}\left({{\mathrm{\xi }}}_{{y}}\right){,}{-}{x}{}\left({{\mathrm{\xi }}}_{{y}}\right){+}{\mathrm{\eta }}{,}{-}{{\mathrm{\xi }}}_{{y}}\right]$ (10)
 > $\mathrm{GetIDBasis}\left(\mathrm{E2}\right)$
 $\left[{\mathrm{\xi }}{-}{y}{}\left({{\mathrm{\xi }}}_{{y}}\right){,}{-}{x}{}\left({{\mathrm{\xi }}}_{{y}}\right){+}{\mathrm{\eta }}{,}{-}{{\mathrm{\xi }}}_{{y}}\right]$ (11)