Overview of the VFPDO Object - Maple Programming Help

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Overview of the VFPDO Object

 

Description

VFPDO Object Methods

Examples

Description

• 

The VFPDO object is designed and created to represent partial differential operators (PDO) that are applied to vector fields.

• 

There is a collection of methods that are available for a VFPDO object, including (i) method allowing the VFPDO object to act as an operator / function  (ii) methods for exploring properties of VFPDO (e.g. specification of domain and codomain). Some existing Maple builtins are extended for allowing VFPDO object.

• 

Methods of the VFPDO object become available only once a valid VFPDO object is constructed successfully. To construct a VFPDO object, see LieAlgebrasOfVectorFields[VFPDO].

• 

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

• 

A VFPDO Delta acts as an operator on an m-tuple (m1) of scalar expressions, mapping it to an s-tuple (s0) of scalars.  Thus the input to Delta is a list of m elements, and it returns a list of s elements acting on a vector field of the same components.

• 

The space where the vector field that the VFPDO object is operated on may be accessed via the GetSpace method.  The integers m, s respectively are accessed via the GetSystemCount methods.

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After a VFPDO object Delta is successfully constructed, each method of Delta can be accessed by either the short form method(Delta, arguments) or the long form Delta:-method(Delta, arguments).

VFPDO Object Methods

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The following is a list of available methods for a VFPDO object.

AreSameSpace

GetSystemCount

GetSpace

 

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A VFPDO object can also act as a differential operator. See VFPDO Object as Operator for more detail.

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The following Maple builtins are extended to allow VFPDO object: has, hastype, indets, type. See VFPDO Object Overloaded Builtins for more detail.

Examples

withLieAlgebrasOfVectorFields:

XVectorFieldξx,yDx+ηx,yDy,space=x,y

Xξx,yx+ηx,yy

(1)

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

Sxξx,y=0,2y2ξx,y=0,xηx,y=yξx,y,yηx,y=0,indep=x,y,dep=ξx,y,ηx,y

(2)

LLAVFX,S

Lξx,yx+ηx,yy&where2y2ξx,y=0,xξx,y=0,xηx,y=yξx,y,yηx,y=0

(3)

Can easily build an VFPDO using a LAVF L.

ΔVFPDOL

ΔX→yyXx,ⅆⅆxXx,xXy+yXx,ⅆⅆyXy

(4)

GetSystemCountΔ

4

(5)

GetSpaceΔ

x,y

(6)

AreSameSpaceΔ,L,X

true

(7)

It should kill rotations:

RVectorFieldyy0Dx+xx0Dy,space=x,y

Ry+y0x+xx0y

(8)

ΔR

0,0,0,0

(9)

See Also

LieAlgebrasOfVectorFields (Package overview)

LAVF (Object overview)

LHPDE (Object overview)

VectorField (Object overview)

LieAlgebrasOfVectorFields[VectorField]

LieAlgebrasOfVectorFields[LHPDE]

LieAlgebrasOfVectorFields[LAVF]