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Overview of the LHPDE Object

 

Description

LHPDE Object Methods

Examples

Description

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The LHPDE object is designed and created to represent a collection of linear homogeneous PDEs (LHPDEs) in both rif-reduced or non-rif-reduced form.

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There are collections of methods that are available for a LHPDE object, including (i) methods for exploring properties of LHPDEs system such as solution dimension, (ii) utility methods for manipulating DEs system (e.g. rif-reducing the DEs system, solving DEs,..), and (iii) exploring relationship between solution spaces of two LHPDE objects. Some Maple existing builtins are extended for allowing LHPDE object.

• 

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

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The LHPDE object is the main Maple object exported by the LieAlgebrasOfVectorFields package. See Overview of the LieAlgebrasOfVectorFields package for more detail.

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A LHPDE object is mathematically represented by the minimum of three data attributes: the "DEs system", the "independent variables" and the "dependent variables". These data attributes can be accessed via the GetSystem, GetIndependents and GetDependents methods.

• 

To represent a LHPDEs system that is in rif-reduced form with respect to a given ranking, a LHPDE object has two additional data attributes: a boolean variable "inRifReducedForm" and the "ranking". These two attributes can be accessed via the IsRifReduced and GetRanking methods.

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

LHPDE Object Methods

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After a LHPDE object is constructed, the following methods are available:

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

 

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The following Maple builtins functions are extended so that they work for a LHPDE object: type, expand, has, hastype, indets, normal, simplify, convert. See LHPDE Object Overloaded Builtins for more detail.

Examples

withLieAlgebrasOfVectorFields:

Typesetting:-Settingsuserep=true:

Typesetting:-Suppressξx,y,ηx,y,ux,y,vx,y:

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

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

(1)

Inserting option `static` gives a list of exports that are available for S as a LHPDE object.

exportsS,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

(2)

Basic properties of S can be got:

GetSystemS

ξy,y=0,ηx+ξy=0,ηy=0,ξx=0

(3)

GetIndependentsS

x,y

(4)

GetDependentsS

η,ξ

(5)

QRifReduceS

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

(6)

For a LHPDE object that is being reduced, we can find more information about the solution space of Q:

SolutionDimensionQ

3

(7)

ParametricDerivativesQ

ξ,η,ξy

(8)

RCopyQ,u,v

Rvy,y=0,ux=vy,vx=0,uy=0,indep=x,y,dep=u,v

(9)

Q and R should regard as same LHPDEs system in the sense that they have same partial differential operator forms.

AreSameQ,R,criteria=sameOperator

true

(10)

We can simplify the dependency of their dependent variables, for example, a new LHPDE object R1 is constructed with minimal dependencies.

R1AdjustDependenciesR,dep=least

R1ⅆ2ⅆy2vy=0,ⅆⅆxux=ⅆⅆyvy,indep=x,y,dep=ux,vy

(11)

A partial depended LHPDE object R1 has no trouble to be manipulated. For example, solving it..

LHSolveR1

ux=_C1x+_C3,vy=_C1y+_C2

(12)

AreSameSpaceS,Q,R,R1

true

(13)

See Also

LieAlgebrasOfVectorFields

LHPDE (Object Overview)

LHPDO (Object Overview)

LieAlgebrasOfVectorFields[LHPDE]

LieAlgebrasOfVectorFields[LHPDO]

GetSystem

GetIndependents

GetDependents

GetRanking

IsRifReduced