Overview of the liesymm Package
List of liesymm Package Commands
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.
The following is a list of available commands.
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.
to define (or redefine) a list of coordinate variables
to compute the exterior derivative with respect to the
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)).
to compute the Lie derivative of an expression involving
forms, relative to a specified vector.
to express a form as a sum of forms each multiplied by a
coefficient of wedge degree 0.
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,
convert(eqlist, forms, eqlist, w)
Generates a set of forms that
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
equations represented by the given
Extends the given list of forms to
achieve closure under application of d().
Checks if the forms list is closed under
applications of d()
Reduces a form modulo an
exterior ideal (specified by a
closed list of forms).
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
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
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.
Nonlinear Boltzmann's equation.
eq ≔ ∂2∂t⁢∂x⁢u⁡x,t+∂∂x⁢u⁡x,t+u⁡x,t2=0
forms ≔ makeforms⁡eq,u⁡x,t,w
eq ≔ mixpar⁡eq
"Harrison-Estabrook procedure." Journal of Mathematical Physics , Vol. 12. New York: American Institute of Physics. (1971): 653-665.
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