Overview of Linear Homogeneous PDE Commands - Maple Programming Help

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Overview of Linear Homogeneous PDE Commands

 

Description

List of Linear Homogeneous PDE-Related Commands

List of Methods from LHPDE, LHPDO, IDBasis Objects

Examples

Description

• 

The Linear Homogeneous PDE commands in the LieAlgebrasOfVectorFields package provide a general purpose toolkit for dealing with linear homogeneous partial differential equations (LHPDE). To be more specific, with these commands you can:

1. 

reduce a LHPDEs system using the rifsimp command and with ranking recorded with the system.

2. 

find properties of a LHPDEs system such as solution dimension, parametric derivatives, and whether a total degree ranking was used.

3. 

explore relationships between solutions of two LHPDEs systems including whether one solution space is a subspace of the other, finding intersection of solution spaces of two LHPDEs, and finding vector space sum of solution spaces of two LHPDEs.

4. 

access to utility commands for manipulating LHPDEs.

5. 

manipulate as a partial differential operator form.

6. 

manipulate a LHPDEs system that has partial dependencies (i.e. the dependent variables may depend on only some of the independent variables).

7. 

manipulate a LHPDEs system that is in partially or fully integrated form (i.e. a system with constants of integration involved).

• 

The linear homogeneous PDE functionality have been written focusing on the key LHPDE object, and almost all methods are within the LHPDE object. There only five exports relating to Linear Homogeneous PDE in the LieAlgebrasOfVectorFields package, four of them are for constructing a Maple object, and one is a subpackage containing utility commands.

• 

The existing Maple builtins normal, expand, simplify, indets, has, hastype, type and convert are extended to work for LHPDE and LHPDO objects.

• 

Some of the internal routines for Linear Homogeneous PDE are also available for use as LieAlgebrasOfVectorFields:-LHLibrary:-Routine. For more details, see overview of the LHLibrary subpackage.

• 

To use this functionality, first a LHPDE object must be constructed (e.g. via LieAlgebrasOfVectorFields[LHPDE] command). Then once a LHPDE object is constructed successfully, a variety of methods will be available. For more detail, see Overview of the LHPDE object.

List of Linear Homogeneous PDE-Related Commands

• 

There are five related exports, namely:

LHPDE

A constructor to create a LHPDE object.

LHPDO

A constructor to create a LHPDO object.

IDBasis

A constructor to create an IDBasis object.

EliminationSystem

A command for eliminating some dependent variables from a PDEs system.

LHLibrary

A subpackage containing low-level utilities for DEs

List of Methods from LHPDE, LHPDO, IDBasis Objects

• 

The following methods are available for a LHPDE object.

AdjustDependencies

AreSame

AreSameSpace

Augment

Copy

DChange

EliminationSystem

GetDependents

GetIDBasis

GetIndependents

GetRanking

GetSystem

InitialData

Intersection

IsFiniteType

IsRifReduced

IsSubspace

IsTotalDegreeRanking

IsTrivial

LHSolve

OrderOfInvolution

ParametricDerivatives

ReducedForm

RifReduce

SetIDBasis

SolutionDimension

VectorSpaceSum

 

• 

The following methods are available for a LHPDO object.

GetIndependents

GetDependentsCount

GetDependencies

GetSystemCount

• 

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

• 

The following Maple builtins are extended for allowing both LHPDE and LHPDO objects: type, expand, has, hastype, indets, normal, simplify, convert. See LHPDE Object Overloaded Builtins and LHPDO Object Overloaded Builtins for more detail.

• 

The following methods are available for a IDBasis object.

GetChangeBasis

GetParametricDerivatives

Copy

Examples

withLieAlgebrasOfVectorFields

Differential,DisplayStructure,Distribution,EliminationLAVF,EliminationSystem,IDBasis,IsLinearizable,LAVF,LHLibrary,LHPDE,LHPDO,OneForm,SymmetryLAVF,VFPDO,VectorField

(1)

Construct an LHPDE object.  (This is the infinitesimal determining system for the vector fields of the 2-D Euclidean group (i.e. rotations and translations).)

detSysdiffξx,y,y,y=0,diffηx,y,x+diffξx,y,y=0,diffηx,y,y=0,diffξx,y,x=0

detSys2y2ξx,y=0,xηx,y+yξx,y=0,yηx,y=0,xξx,y=0

(2)

E2LHPDEdetSys,indep=x,y,dep=ξ,η

E22y2ξx,y=0,xηx,y+yξx,y=0,yηx,y=0,xξx,y=0,indep=x,y,dep=ξx,y,ηx,y

(3)

E2 is a LHPDE object containing a collection of methods.

exportsE2,static

type,convert,normal,expand,simplify,indets,has,hastype,GetIndependents,GetDependents,GetSystem,GetRanking,SetIDBasis,GetIDBasis,Copy,Augment,SolutionDimension,IsFiniteType,IsTrivial,ParametricDerivatives,OrderOfInvolution,IsRifReduced,IsTotalDegreeRanking,AreSameSpace,AreSame,RifReduce,ReducedForm,AdjustDependencies,Intersection,VectorSpaceSum,EliminationSystem,IsSubspace,DChange,dchange,LHSolve,initialdata,InitialData,ModulePrint,ModuleCopy,ModuleApply

(4)

The object has an associated type:

typeE2,LHPDE

true

(5)

Conversion of an LHPDE object to a partial differential operator LHPDO object:

ΔconvertE2,LHPDO

Δξ,η→yyξ,xη+yξ,yη,xξ

(6)

An LHPDO acts as an operator.  In this case it annihilates rotation vector field about the point (x0,y0)...

Δyy0,xx0

0,0,0,0

(7)

Apply the differential reduction and completion method of DEtools[rifsimp]...

E2pRifReduceE2

E2p2y2ξx,y=0,xξx,y=0,xηx,y=yξx,y,yηx,y=0,indep=x,y,dep=ξx,y,ηx,y

(8)

...which puts the system into a form where the dimension of its solution space can be deduced:

SolutionDimensionE2p

3

(9)

There is a library of low level commands that are useful for PDE:

LHLibrary:-IsHomogeneousGetSystemE2p,ξx,y,ηx,y

true

(10)

The LHPDE package fluently handles PDE with partial dependency.  There is a method to adjust the dependencies...

AdjustDependenciesE2p,dep=least

ⅆ2ⅆy2ξy=0,ⅆⅆxηx=ⅆⅆyξy,indep=x,y,dep=ξy,ηx

(11)

PDE (or ODE) systems of finite type can be passed to Maple's PDE (respectively ODE) solvers pdsolve and dsolve.

LHSolveE2p

ξx,y=_C1y+_C3,ηx,y=_C1x+_C2

(12)

See Also

LHPDE (Object Overview)

LHPDO (Object Overview)

IDBasis (Object Overview)

LieAlgebrasOfVectorFields[LHPDE]

LieAlgebrasOfVectorFields[LHPDO]

LieAlgebrasOfVectorFields[IDBasis]