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VectorSpaceSum

find a LAVF object whose solution space is the sum of the solution spaces of given LAVF objects.

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

VectorSpaceSum( L1, L2, ..., depname = vars )

Parameters

L1, L2, ...

-

a sequence of LAVF objects living on the same space (see AreSameSpace)

vars

-

(optional) a list of new dependent variable names

Description

• 

Let L1,L2, ... be a sequence of LAVF objects living on the same space. The VectorSpaceSum method finds a LAVF object whose solution space is the vector space sum of solution spaces of L1,L2,....

• 

By default, the dependent variable names of the returned object are taken from L1. The dependent variable names will be vars if the optional argument depnames = vars is specified.

• 

This method is front-end to the corresponding method of a LHPDE object. That is, let S1, S2,... be the determining systems of L1,L2,...  (i.e.  Si = GetDeterminingSystem(Li)), then the call VectorSpaceSum(L1,L2,..) is equivalent to VectorSpaceSum(S1,S2,..). All remaining input arguments will be passed down to its determining system level. See the method VectorSpaceSum of a LHPDE object for more detail.

• 

This method is associated with the LAVF object. For more detail, see Overview of the LAVF object.

Examples

withLieAlgebrasOfVectorFields:

Typesetting:-Settingsuserep=true:

Typesetting:-Suppressξx,y,ηx,y:

XVectorFieldξx,yDx+ηx,yDy,space=x,y

Xξx+ηy

(1)

The determining system for 2-dim Euclidean

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

E2ξx=0,ηy=0,ξy+ηx=0,ξy,y=0,ηx,x=0,indep=x,y,dep=ξ,η

(2)

The determining system for 2-dim translations

T2LHPDEdiffξx,y,x=0,diffξx,y,y=0,diffηx,y,x=0,diffηx,y,y=0,indep=x,y,dep=ξ,η

T2ξx=0,ξy=0,ηx=0,ηy=0,indep=x,y,dep=ξ,η

(3)

We first construct LAVFs for E(2) and T(2)

LE2LAVFX,E2

LE2ξx+ηy&whereξy,y=0,ξx=0,ηx=ξy,ηy=0

(4)

LT2LAVFX,T2

LT2ξx+ηy&whereξx=0,ηx=0,ξy=0,ηy=0

(5)

VectorSpaceSumLE2,LT2

ξx+ηy&whereξx=0,ξy=ηx,ηx,x=0,ηy=0

(6)

VectorSpaceSumLE2,LT2,depname=α,β

αx,yx+βx,yy&wherexαx,y=0,yαx,y=xβx,y,2x2βx,y=0,yβx,y=0

(7)

Compatibility

• 

The VectorSpaceSum command was introduced in Maple 2020.

• 

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

See Also

LieAlgebrasOfVectorFields (Package overview)

LAVF (Object overview)

LieAlgebrasOfVectorFields[VectorField]

LieAlgebrasOfVectorFields[LHPDE]

LieAlgebrasOfVectorFields[LAVF]

AreSameSpace

VectorSpaceSum