Overview of the liesymm Package - Maple Programming Help

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Overview of the liesymm Package

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.

 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,

$\mathrm{flist}=\left[\mathrm{f1},\mathrm{f2},\mathrm{...},\mathrm{fn}\right]\left(\mathrm{x1},\mathrm{...},\mathrm{xj}\right)$

 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.

 > $\mathrm{with}\left(\mathrm{liesymm}\right):$
 > $\mathrm{setup}\left(\right)$
 $\left[{}\right]$ (1)
 > $\mathrm{eq}≔\frac{{{\partial }}^{2}}{{\partial }t{\partial }x}u\left(x,t\right)+\frac{{\partial }}{{\partial }x}u\left(x,t\right)+{u\left(x,t\right)}^{2}=0$
 ${\mathrm{eq}}{:=}\frac{{{\partial }}^{{2}}}{{\partial }{t}{}{\partial }{x}}{}{u}{}\left({x}{,}{t}\right){+}\frac{{\partial }}{{\partial }{x}}{}{u}{}\left({x}{,}{t}\right){+}{{u}{}\left({x}{,}{t}\right)}^{{2}}{=}{0}$ (2)
 > $\mathrm{forms}≔\mathrm{makeforms}\left(\mathrm{eq},u\left(x,t\right),w\right)$
 ${\mathrm{forms}}{:=}\left[{d}{}\left({u}\right){-}{\mathrm{w1}}{}{d}{}\left({x}\right){-}{\mathrm{w2}}{}{d}{}\left({t}\right){,}\left({d}{}\left({\mathrm{w2}}\right)\right){&^}\left({d}{}\left({t}\right)\right){+}\left({{u}}^{{2}}{+}{\mathrm{w1}}\right){}\left({d}{}\left({x}\right)\right){&^}\left({d}{}\left({t}\right)\right)\right]$ (3)
 > $\mathrm{eq}≔\mathrm{mixpar}\left(\mathrm{eq}\right)$
 ${\mathrm{eq}}{:=}\frac{{{\partial }}^{{2}}}{{\partial }{x}{}{\partial }{t}}{}{u}{}\left({x}{,}{t}\right){+}\frac{{\partial }}{{\partial }{x}}{}{u}{}\left({x}{,}{t}\right){+}{{u}{}\left({x}{,}{t}\right)}^{{2}}{=}{0}$ (4)
 > $\mathrm{determine}\left(\mathrm{eq},V,u\left(x,t\right),w\right)$
 $\left\{\frac{{\partial }}{{\partial }{t}}{}{\mathrm{V1}}{}\left({x}{,}{t}{,}{u}\right){=}{0}{,}\frac{{\partial }}{{\partial }{u}}{}{\mathrm{V1}}{}\left({x}{,}{t}{,}{u}\right){=}{0}{,}\frac{{{\partial }}^{{2}}}{{\partial }{{u}}^{{2}}}{}{\mathrm{V1}}{}\left({x}{,}{t}{,}{u}\right){=}{0}{,}\frac{{\partial }}{{\partial }{u}}{}{\mathrm{V2}}{}\left({x}{,}{t}{,}{u}\right){=}{0}{,}\frac{{\partial }}{{\partial }{x}}{}{\mathrm{V2}}{}\left({x}{,}{t}{,}{u}\right){=}{0}{,}\frac{{{\partial }}^{{2}}}{{\partial }{{u}}^{{2}}}{}{\mathrm{V2}}{}\left({x}{,}{t}{,}{u}\right){=}{0}{,}\frac{{{\partial }}^{{2}}}{{\partial }{u}{}{\partial }{t}}{}{\mathrm{V1}}{}\left({x}{,}{t}{,}{u}\right){=}{0}{,}\frac{{{\partial }}^{{2}}}{{\partial }{x}{}{\partial }{t}}{}{\mathrm{V1}}{}\left({x}{,}{t}{,}{u}\right){=}\frac{{{\partial }}^{{2}}}{{\partial }{u}{}{\partial }{t}}{}{\mathrm{V3}}{}\left({x}{,}{t}{,}{u}\right){+}\frac{{\partial }}{{\partial }{t}}{}{\mathrm{V2}}{}\left({x}{,}{t}{,}{u}\right){,}\frac{{{\partial }}^{{2}}}{{\partial }{x}{}{\partial }{u}}{}{\mathrm{V1}}{}\left({x}{,}{t}{,}{u}\right){=}\frac{{{\partial }}^{{2}}}{{\partial }{{u}}^{{2}}}{}{\mathrm{V3}}{}\left({x}{,}{t}{,}{u}\right){-}\left(\frac{{{\partial }}^{{2}}}{{\partial }{u}{}{\partial }{t}}{}{\mathrm{V2}}{}\left({x}{,}{t}{,}{u}\right)\right){,}\frac{{{\partial }}^{{2}}}{{\partial }{x}{}{\partial }{t}}{}{\mathrm{V2}}{}\left({x}{,}{t}{,}{u}\right){=}\frac{{{\partial }}^{{2}}}{{\partial }{x}{}{\partial }{u}}{}{\mathrm{V3}}{}\left({x}{,}{t}{,}{u}\right){,}\frac{{{\partial }}^{{2}}}{{\partial }{x}{}{\partial }{u}}{}{\mathrm{V2}}{}\left({x}{,}{t}{,}{u}\right){=}{0}{,}\frac{{{\partial }}^{{2}}}{{\partial }{x}{}{\partial }{t}}{}{\mathrm{V3}}{}\left({x}{,}{t}{,}{u}\right){=}{-}\left(\frac{{\partial }}{{\partial }{x}}{}{\mathrm{V1}}{}\left({x}{,}{t}{,}{u}\right)\right){}{{u}}^{{2}}{-}\left(\frac{{\partial }}{{\partial }{t}}{}{\mathrm{V2}}{}\left({x}{,}{t}{,}{u}\right)\right){}{{u}}^{{2}}{+}\left(\frac{{\partial }}{{\partial }{u}}{}{\mathrm{V3}}{}\left({x}{,}{t}{,}{u}\right)\right){}{{u}}^{{2}}{-}{2}{}{\mathrm{V3}}{}\left({x}{,}{t}{,}{u}\right){}{u}{-}\left(\frac{{\partial }}{{\partial }{x}}{}{\mathrm{V3}}{}\left({x}{,}{t}{,}{u}\right)\right)\right\}$ (5)
 > $\mathrm{value}\left(\right):$
 > $\mathrm{wedgeset}\left(0\right)$
 ${x}{,}{t}{,}{u}{,}{\mathrm{w1}}{,}{\mathrm{w2}}$ (6)
 > $\mathrm{close}\left(\mathrm{forms}\right)$
 $\left[{d}{}\left({u}\right){-}{\mathrm{w1}}{}{d}{}\left({x}\right){-}{\mathrm{w2}}{}{d}{}\left({t}\right){,}\left({d}{}\left({\mathrm{w2}}\right)\right){&^}\left({d}{}\left({t}\right)\right){+}\left({{u}}^{{2}}{+}{\mathrm{w1}}\right){}\left({d}{}\left({x}\right)\right){&^}\left({d}{}\left({t}\right)\right){,}{-}\left({d}{}\left({\mathrm{w1}}\right)\right){&^}\left({d}{}\left({x}\right)\right){-}\left({d}{}\left({\mathrm{w2}}\right)\right){&^}\left({d}{}\left({t}\right)\right)\right]$ (7)
 > $\mathrm{annul}\left(,\left[x,t\right]\right)$
 $\left[\frac{{\partial }}{{\partial }{t}}{}{u}{}\left({x}{,}{t}\right){-}{\mathrm{w2}}{}\left({x}{,}{t}\right){=}{0}{,}\frac{{\partial }}{{\partial }{x}}{}{u}{}\left({x}{,}{t}\right){-}{\mathrm{w1}}{}\left({x}{,}{t}\right){=}{0}{,}{{u}{}\left({x}{,}{t}\right)}^{{2}}{+}\frac{{\partial }}{{\partial }{x}}{}{\mathrm{w2}}{}\left({x}{,}{t}\right){+}{\mathrm{w1}}{}\left({x}{,}{t}\right){=}{0}{,}\frac{{\partial }}{{\partial }{t}}{}{\mathrm{w1}}{}\left({x}{,}{t}\right){-}\left(\frac{{\partial }}{{\partial }{x}}{}{\mathrm{w2}}{}\left({x}{,}{t}\right)\right){=}{0}\right]$ (8)

References

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