OrderOfInvolution - Maple Programming Help

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OrderOfInvolution

find the (possible) minimal differential order that a LHPDEs system is in involution

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

OrderOfInvolution( obj)

Parameters

obj

-

a LHPDE object that is in rif-reduced form with respect to a total degree ranking (see IsTotalDegreeRanking, IsRifReduced)

Description

• 

The OrderOfInvolution method returns the order at which a LHPDE object is involutive, or a bound for this order. Note that the LHPDE object must be rif-reduced, with respect to a total degree ranking (see IsTotalDegreeRanking).

• 

So far, the only implementation in this method is the Mansfield bound (ref: E. Mansfield. A Simple Criterion for Involutivity. Journal of the London Mathematics Society 54: 323-345,1996).  This gives an upper bound for the order of involutivity. It is often -- but not always -- exact.

• 

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

Examples

withLieAlgebrasOfVectorFields:

Typesetting:-Settingsuserep=true:

Typesetting:-Suppressξx,y,ηx,y:

Create an LHPDE object (these are the determining equations for the Euclidean group E(2))...

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

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

(1)

IsFiniteTypeE2

true

(2)

ParametricDerivativesE2

ξ,ξy,η

(3)

Create another LHPDE object that is rif-reduced with respect to a total degree ranking....

E2redRifReduceE2,ξ,η

E2redξy,y=0,ξx=0,ηx=ξy,ηy=0,indep=x,y,dep=ξ,η

(4)

Now this can be checked for the order at which it becomes involutive....

OrderOfInvolutionE2red

2

(5)

Compatibility

• 

The OrderOfInvolution command was introduced in Maple 2020.

• 

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

See Also

LHPDE (Object overview)

LieAlgebrasOfVectorFields[LHPDE]

IsFiniteType

IsRifReduced