Overview of the liesymm Package - Maple Help

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Overview of the liesymm Package

 

Description

List of liesymm Package Commands

Examples

References

Description

• 

Each command in the liesymm package can be accessed by using either the long form or the short form of the command name in the command calling sequence.

• 

This is an implementation of the Harrison-Estabrook procedure (see References section). It obtains the determining equations leading to the similarity solutions of a system of partial differential equations using a number of important refinements and extensions as developed by J. Carminati.

• 

To construct the determining equations for the isovector using Cartan's geometric formulation of partial differential equations in terms of differential ideals use determine(). Other commands help to convert the set of equations to an equivalent set of differential forms or vice versa.

• 

You can compute or check for closure of a given set of forms and annul to a specified sublist of independent coordinates. Modding lists are used to eliminate those parts of a differential form belonging to the ideal.

• 

The implementation makes use of the exterior derivative (d) and wedge product (&^) but is completely independent of the Maple difforms package. It requires a specific coordinate system as defined by setup().  Unknowns default to constants, and automatic simplifications take into account a consistent ordering of the 1-forms and the extraction of coefficients.

List of liesymm Package Commands

• 

The following is a list of available commands.

&mod

annul

autosimp

close

d

depvars

determine

dvalue

Eta

extvars

getcoeff

getform

hasclosure

hook

indepvars

Lie

Lrank

makeforms

mixpar

prolong

reduce

setup

TD

translate

vfix

wcollect

wdegree

wedge

wedgeset

wsubs

 

 

  

To display the help page for a particular liesymm command, see Getting Help with a Command in a Package.

• 

A brief description of the functionality available follows.

setup

to define (or redefine) a list of coordinate variables

 

(0-forms).

 

 

d

to compute the exterior derivative with respect to the

 

specified coordinates.

 

 

&^

to compute the wedge product.  It automatically simplifies

 

relative to an "address" ordering of the basis variables

 

to sums of expressions of the form c*(d(x)&^d(y)&^d(z)).

 

 

Lie

to compute the Lie derivative of an expression involving

 

forms, relative to a specified vector.

 

 

wcollect

to express a form as a sum of forms each multiplied by a

 

coefficient of wedge degree 0.

 

 

wsubs

to substitute an expression for a k-form that is part

 

of an n-form.

• 

Various other commands such as choose, getcoeff, mixpar, wdegree, wedgeset, and value are used in manipulating the forms and results.

• 

Let eqn be a set or list of partial differential equations involving functions,

flist=f1,f2,...,fnx1,...,xj

convert(eqlist, forms, eqlist, w)

Generates a set of forms that

or

when closed characterize the equations

makeforms(eqns, flist, w)

in eqlist in the sense of Cartan.

 

 

convert(forms, system, vlist)

Generates a set of partial differential

or

equations represented by the given

annul(forms, vlist)

forms.

 

 

close(forms)

Extends the given list of forms to

 

achieve closure under application of d().

 

 

hasclosure(forms)

Checks if the forms list is closed under

 

applications of d()

 

 

&mod

Reduces a form modulo an

 

exterior ideal (specified by a

 

closed list of forms).

 

 

determine(forms, V)

Given a list of forms describing a

 

particular set of partial differential

 

equations with coordinates

 

the calling sequence produces a set of first

 

order equations for the isovector

 

vector (V1, ..., Vn). The resulting

 

equations are expressed using alias and

 

an inert Diff rather than diff but

 

evaluation can be forced by using

 

value().

 

 

determine(f, V, h(t, x), w)

As above, but with f as an

 

equation and with the extra

 

arguments used by makeforms()

 

to construct the initial

 

forms list.

• 

You need not work with the differential forms directly. When given a list of partial differential equations instead of a forms list, the command determine() sets up the coordinates and differential forms as required.

• 

Partial derivatives should be expressed in terms of Diff() rather than diff() or D().  The command mixpar() may be used to force mixed partials to a consistent ordering.

• 

Use value() to convert Diff() to diff() when interpreting or using the result of determine.

Examples

Nonlinear Boltzmann's equation.

withliesymm:

setup

(1)

eq2txux,t+xux,t+ux,t2=0

eq:=2txux,t+xux,t+ux,t2=0

(2)

formsmakeformseq,ux,t,w

forms:=duw1dxw2dt,dw2 &^ dt+u2+w1dx &^ dt

(3)

eqmixpareq

eq:=2xtux,t+xux,t+ux,t2=0

(4)

determineeq,V,ux,t,w

tV1x,t,u=0,uV1x,t,u=0,2u2V1x,t,u=0,uV2x,t,u=0,xV2x,t,u=0,2u2V2x,t,u=0,2utV1x,t,u=0,2xtV1x,t,u=2utV3x,t,u+tV2x,t,u,2xuV1x,t,u=2u2V3x,t,u2utV2x,t,u,2xtV2x,t,u=2xuV3x,t,u,2xuV2x,t,u=0,2xtV3x,t,u=xV1x,t,uu2tV2x,t,uu2+uV3x,t,uu22V3x,t,uuxV3x,t,u

(5)

value:

wedgeset0

x,t,u,w1,w2

(6)

closeforms

duw1dxw2dt,dw2 &^ dt+u2+w1dx &^ dt,dw1 &^ dxdw2 &^ dt

(7)

annul,x,t

tux,tw2x,t=0,xux,tw1x,t=0,ux,t2+xw2x,t+w1x,t=0,tw1x,txw2x,t=0

(8)

References

  

"Harrison-Estabrook procedure." Journal of Mathematical Physics , Vol. 12. New York: American Institute of Physics. (1971): 653-665.

See Also

dsolve

UsingPackages

 


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