LieAlgebrasOfVectorFields - Maple Programming Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Mathematics : Differential Equations : Lie Symmetry Method : Commands for PDEs (and ODEs) : LieAlgebrasOfVectorFields : IDBasis : LieAlgebrasOfVectorFields/IDBasis/constructor

LieAlgebrasOfVectorFields

  

IDBasis

  

construct an IDBasis object

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

IDBasis(S, Vs)

Parameters

S

-

a LHPDE object that is of finite type (see IsFiniteType)

Vs

-

a list of Vectors, a Matrix, a list of linear combination of parametric derivatives of S, or the string "standardBasis", representing a change-of-basis for the initial data of S

Description

• 

Let S be a LHPDE object of finite type whose solution dimension is r. The IDBasis command constructs an IDBasis object representing a basis for the initial data of S.

• 

The IDBasis object is managed with respect to a fixed standard basis, namely the parametric derivatives of the LHPDE object S.

• 

The second input argument, Vs, represents a change-of-basis matrix to give a new basis.

– 

If Vs is a list of Vectors, then there must be r Vectors of length r in the list.

– 

If Vs is a Matrix, then it must be r x r invertible matrix.

– 

If Vs is a list of linear combination of parametric derivatives of S, then there must be r items in the list.

• 

The call IDBasis(S, "standardBasis") returns a IDBasis object containing the standard initial data basis (i.e. the change-of-basis matrix is the identity matrix).

• 

Some methods become available once a valid IDBasis object is constructed. See Overview of the IDBasis object for more detail.

• 

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 IDBasis(...) only after executing the command with(LieAlgebrasOfVectorFields), but can always be used by executing LieAlgebrasOfVectorFields:-IDBasis(...).

Examples

withLieAlgebrasOfVectorFields:

Typesetting:-Suppressξx,y,ηx,y:

Typesetting:-Settingsuserep=true:

E2LHPDEdiffξx,y,y,y=0,diffηx,y,x=diffξx,y,y,diffηx,y,y=0,diffξx,y,x=0,dep=ξ,η,indep=x,y

E2ξy,y=0,ηx=ξy,ηy=0,ξx=0,indep=x,y,dep=ξ,η

(1)

ParametricDerivativesE2

ξ,ξy,η

(2)

The solution dimension of E2 is 3, so we must have three vectors of length 3.

BIDBasisE2,1,0,0,y,x,1,0,1,0

Bξyξy,xξy+η,ξy

(3)

B looks like a list but it is really an IDBasis object.

typeB,list

false

(4)

typeB,IDBasis

true

(5)

The IDBasis can be constructed from a change-of-basis 3 by 3 matrix

IDBasisE2,Matrix1,y,0,0,x,1,0,1,0

ξyξy,xξy+η,ξy

(6)

... or can be constructed from three linear combinations of the parametric derivatives of E2

IDBasisE2,ξx,yydiffξx,y,y,ηx,yxdiffξx,y,y,diffξx,y,y

ξyξy,xξy+η,ξy

(7)

Or we can construct a standard IDBasis object whose change-of-basis is the identity matrix.

IDBasisE2,standardBasis

ξ,ξy,η

(8)

Compatibility

• 

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

• 

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

See Also

LieAlgebrasOfVectorFields (Package Overview)

LieAlgebrasOfVectorFields[LHPDE]

IDBasis (Object Overview)

LHPDE (Object Overview)

IsFiniteType

ParametricDerivatives