Online Help

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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.

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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.

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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.

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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.

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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.

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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 .

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A brief description of the functionality available follows.

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

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

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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.

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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.

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

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$\mathrm{with}\left(\mathrm{liesymm}\right)\:$

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$\mathrm{setup}\left(\right)$

$\left[\right]$

(1)

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$\mathrm{eq}≔\mathrm{Diff}\left(u\left(x\,t\right)\,x\,t\right)+\mathrm{Diff}\left(u\left(x\,t\right)\,x\right)+{u\left(x\,t\right)}^2=0$

$\mathrm{eq}≔\frac{{\partial }^2}{\partial x\partial t}u\left(x\,t\right)+\frac{\partial }{\partial x}u\left(x\,t\right)+{u\left(x\,t\right)}^2=0$

(2)

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$\mathrm{forms}≔\mathrm{makeforms}\left(\mathrm{eq}\,u\left(x\,t\right)\,w\right)$

$\mathrm{forms}≔\left[d\left(u\right)-\mathrm{w1}d\left(x\right)-\mathrm{w2}d\left(t\right)\,d\left(\mathrm{w2}\right)\&ˆd\left(t\right)+\left(u^2+\mathrm{w1}\right)d\left(x\right)\&ˆd\left(t\right)\right]$

(3)

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$\mathrm{eq}≔\mathrm{mixpar}\left(\mathrm{eq}\right)$

$\mathrm{eq}≔\frac{\partial }{\partial x}\left(\frac{\partial }{\partial t}u\left(x\,t\right)\right)+\frac{\partial }{\partial x}u\left(x\,t\right)+{u\left(x\,t\right)}^2=0$

(4)

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$\mathrm{determine}\left(\mathrm{eq}\,V\,u\left(x\,t\right)\,w\right)$

$\left\{\frac{\partial }{\partial u}\left(\frac{\partial }{\partial t}\mathrm{V1}\left(x\,t\,u\right)\right)=0\,\frac{\partial }{\partial x}\left(\frac{\partial }{\partial t}\mathrm{V1}\left(x\,t\,u\right)\right)=\frac{\partial }{\partial u}\left(\frac{\partial }{\partial t}\mathrm{V3}\left(x\,t\,u\right)\right)+\frac{\partial }{\partial t}\mathrm{V2}\left(x\,t\,u\right)\,\frac{{\partial }^2}{\partial u^2}\mathrm{V1}\left(x\,t\,u\right)=0\,\frac{\partial }{\partial x}\left(\frac{\partial }{\partial u}\mathrm{V1}\left(x\,t\,u\right)\right)=\frac{{\partial }^2}{\partial u^2}\mathrm{V3}\left(x\,t\,u\right)-\frac{\partial }{\partial u}\left(\frac{\partial }{\partial t}\mathrm{V2}\left(x\,t\,u\right)\right)\,\frac{\partial }{\partial x}\left(\frac{\partial }{\partial t}\mathrm{V2}\left(x\,t\,u\right)\right)=\frac{\partial }{\partial x}\left(\frac{\partial }{\partial u}\mathrm{V3}\left(x\,t\,u\right)\right)\,\frac{{\partial }^2}{\partial u^2}\mathrm{V2}\left(x\,t\,u\right)=0\,\frac{\partial }{\partial x}\left(\frac{\partial }{\partial u}\mathrm{V2}\left(x\,t\,u\right)\right)=0\,\frac{\partial }{\partial x}\left(\frac{\partial }{\partial t}\mathrm{V3}\left(x\,t\,u\right)\right)=-\left(\frac{\partial }{\partial x}\mathrm{V1}\left(x\,t\,u\right)\right)u^2-\left(\frac{\partial }{\partial t}\mathrm{V2}\left(x\,t\,u\right)\right)u^2+\left(\frac{\partial }{\partial u}\mathrm{V3}\left(x\,t\,u\right)\right)u^2-2\mathrm{V3}\left(x\,t\,u\right)u-\frac{\partial }{\partial x}\mathrm{V3}\left(x\,t\,u\right)\,\frac{\partial }{\partial t}\mathrm{V1}\left(x\,t\,u\right)=0\,\frac{\partial }{\partial u}\mathrm{V1}\left(x\,t\,u\right)=0\,\frac{\partial }{\partial u}\mathrm{V2}\left(x\,t\,u\right)=0\,\frac{\partial }{\partial x}\mathrm{V2}\left(x\,t\,u\right)=0\right\}$

(5)

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$\mathrm{value}\left(\right)\:$

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$\mathrm{wedgeset}\left(0\right)$

$x,t,u,\mathrm{w1},\mathrm{w2}$

(6)

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$\mathrm{close}\left(\mathrm{forms}\right)$

$\left[d\left(u\right)-\mathrm{w1}d\left(x\right)-\mathrm{w2}d\left(t\right)\,d\left(\mathrm{w2}\right)\&ˆd\left(t\right)+\left(u^2+\mathrm{w1}\right)d\left(x\right)\&ˆd\left(t\right)\,-d\left(\mathrm{w1}\right)\&ˆd\left(x\right)-d\left(\mathrm{w2}\right)\&ˆd\left(t\right)\right]$

(7)

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$\mathrm{annul}\left(\,\left[x\,t\right]\right)$

$\left[\frac{\partial }{\partial t}u\left(x\,t\right)-\mathrm{w2}\left(x\,t\right)=0\,\frac{\partial }{\partial x}u\left(x\,t\right)-\mathrm{w1}\left(x\,t\right)=0\,{u\left(x\,t\right)}^2+\frac{\partial }{\partial x}\mathrm{w2}\left(x\,t\right)+\mathrm{w1}\left(x\,t\right)=0\,\frac{\partial }{\partial t}\mathrm{w1}\left(x\,t\right)-\frac{\partial }{\partial x}\mathrm{w2}\left(x\,t\right)=0\right]$

(8)

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