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LieAlgebrasOfVectorFields

  

LHPDO

  

construct a LHPDO object

 

Calling Sequence

Parameters

Options

Description

Examples

Compatibility

Calling Sequence

LHPDO( S )

LHPDO( sys, options)

LHPDO( str, dep = vars, options)

Parameters

S

-

an LHPDE object

sys

-

a list or set of linear homogeneous PDEs or ODEs or expressions, or a table as returned by DEtools[rifsimp]

str

-

a string: either "trivial" or "identity"

vars

-

a list of dependent variables as functions

options

-

optional equations controlling details of the first input argument

Options

The LHPDO constructor accepts a number of options, which are all passed through to the constructor of the LHPDE object (see LieAlgebrasOfVectorFields[LHPDE] for more detail).

Description

• 

The command LHPDO(...) is for constructing a LHPDO object. It returns a LHPDO object if successful. A valid LHPDO object has access to various methods which allow it to be manipulated and its contents queried. For more detail, see Overview of the LHPDO object.

• 

A LHPDO Δ consists of independent variables x=x1,x2,..,xn,  and a sequence of linear homogeneous PDOs Δ1,Δ2,Δs. Here, each Δj is a function / operator that applies to a list of m  scalar expressions y1,y2,..ym, and computes a linear homogenous combination of various derivatives of yi of order up to k.

• 

The LHPDO constructor command works by converting a list of s differential expressions linear homogeneous with respect to dependent variables u1,u2,..,um each of which is an indeterminate function of its arguments -- which must be some nonempty subset of x1,x2,..,xn. These differential expressions are specified in various ways by the first argument to LHPDO.

• 

In the first calling sequence, an LHPDE object S is provided.  The differential expressions are formed from taking (lhs - rhs) for each equation in the system of PDEs in S (see GetSystem). An LHPDE object fully specifies other quantities required for conversion to LHPDO, so no further arguments are allowed.

• 

In the second calling sequence, the first input argument is a list or set of scalar expressions or equations, or alternatively a table as returned by DEtools[rifsimp].  In fact the sequence of arguments for this calling sequence is exactly what is accepted by the LHPDE constructor.  See LieAlgebrasOfVectorFields[LHPDE] for the detailed specification of arguments and options for this calling sequence.

• 

In the third calling sequence, the first input argument is the string "trivial" or "identity", in which case, the identity LHPDO (i.e. one which maps u1,u2,..,umu1,u2,..,um)   is constructed. In this calling sequence, the second argument dep= vars is required. Further options are as per the LHPDE constructor (see LieAlgebrasOfVectorFields[LHPDE]).

• 

This command is part of the LieAlgebrasOfVectorFields package. For more detail, see Overview of the LieAlgebrasOfVectorFields package.

• 

This command can be used in the form LHPDO(...) only after executing the command with(LieAlgebrasOfVectorFields), but can always be used in the form LieAlgebrasOfVectorFields:-LHPDO(...).

Examples

withLieAlgebrasOfVectorFields:

Typesetting:-Settingsuserep=true:

Typesetting:-Suppressξx,y,ηx,y:

List of differential expressions...

diffExprdiffξx,y,y,y,diffηx,y,x+diffξx,y,y,diffηx,y,y,diffξx,y,x

diffExprξy,y,ηx+ξy,ηy,ξx

(1)

ΔLHPDOdiffExpr

Δη,ξ→yyξ,xη+yξ,yη,xξ

(2)

List of differential equations...

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

detSysξy,y=0,ηx+ξy=0,ηy=0,ξx=0

(3)

ΔLHPDOdetSys

Δη,ξ→yyξ,xη+yξ,yη,xξ

(4)

Rif-table...

RDEtoolsrifsimpdetSys

RtableSolved=ξy,y=0,ηx=ξy,ξx=0,ηy=0

(5)

ΔLHPDOR

Δη,ξ→yyξ,xη+yξ,xξ,yη

(6)

LHPDE object...

SLHPDEdetSys

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

(7)

ΔLHPDOS

Δη,ξ→yyξ,xη+yξ,yη,xξ

(8)

Identity operator...

idLHPDOidentity,dep=ux,y,vx,y

idu,vu,v

(9)

A LHPDO depends on how its arguments are ordered, so the operator constructed may differ according to the specification of dependent variables....

ELHPDOdetSys,indep=x,y,dep=ξ,η

Eξ,η→yyξ,xη+yξ,yη,xξ

(10)

The main purpose of the LHPDO object is to act as an operator...

Exx02,yy02

0,0,2y2y0,2x2x0

(11)

Compatibility

• 

The LieAlgebrasOfVectorFields[LHPDO] command was introduced in Maple 2020.

• 

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

See Also

LHPDO (Object overview)

LieAlgebrasOfVectorFields (Package overview)

LieAlgebrasOfVectorFields[LHPDE]

LHPDO as operator

GetSystem

DEtools[rifsimp]