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LieAlgebrasOfVectorFields

 SymmetryLAVF
 construct a LAVF object for the determining equations of a given list of DEs

 Calling Sequence SymmetryLAVF(DEs, depVars, V)

Parameters

 DEs - a list of differential equations depVars - (optional) a function or a list of functions of names V - a VectorField object

Description

 • The command SymmetryLAVF(...) constructs a LAVF object for the symmetry determining system of given list of DEs, using infinitesimal specification from the vector field V. A valid LAVF object then has access to at least 60 methods methods which allow it to be manipulated and its contents queried. For more detail, see Overview of the LAVF object.
 • The second input argument (depVars) is only used when the first input argument DEs have unclear dependent variables (e.g. trouble distinguishing dependent variables from arbitrary coefficients).
 • The command ultimately calls PDEtools[DeterminingPDE] command for finding the symmetry determining system DQ. It then calls LAVF(V,DQ) to construct a LAVF object.
 • 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 SymmetryLAVF(...) only after executing the command with(LieAlgebrasOfVectorFields), but can always be used in the form :-LieAlgebrasOfVectorFields:-SymmetryLAVF(...).

Examples

 > $\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right):$
 > $\mathrm{Typesetting}:-\mathrm{Settings}\left(\mathrm{userep}=\mathrm{true}\right):$
 > $\mathrm{Typesetting}:-\mathrm{Suppress}\left(\left\{\mathrm{\eta }\left(x,u\right),\mathrm{\xi }\left(x,u\right)\right\}\right):$
 > $V≔\mathrm{VectorField}\left(\mathrm{\xi }\left(x,u\right)\mathrm{D}\left[x\right]+\mathrm{\eta }\left(x,u\right)\mathrm{D}\left[u\right],\mathrm{space}=\left[x,u\right]\right)$
 ${V}{≔}{\mathrm{\xi }}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\eta }}{}\frac{{\partial }}{{\partial }{u}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (1)

The one-dimensional Blasius equation

 > $\mathrm{Blasius}≔\left[2\mathrm{diff}\left(u\left(x\right),x,x,x\right)+u\left(x\right)\mathrm{diff}\left(u\left(x\right),x,x\right)=0\right]$
 ${\mathrm{Blasius}}{≔}\left[{2}{}\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}\right){+}{u}{}\left({x}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}\right)\right){=}{0}\right]$ (2)

An LAVF object with the determining system for symmetries of the Blasius equation can be constructed by

 > $L≔\mathrm{SymmetryLAVF}\left(\mathrm{Blasius},V\right)$
 ${L}{≔}\left[{\mathrm{\xi }}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\eta }}{}\frac{{\partial }}{{\partial }{u}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{{\mathrm{\xi }}}_{{x}{,}{x}}{=}{0}{,}{{\mathrm{\xi }}}_{{u}}{=}{0}{,}{\mathrm{\eta }}{=}{-}\left({{\mathrm{\xi }}}_{{x}}\right){}{u}\right]\right\}$ (3)
 > $\mathrm{SolutionDimension}\left(L\right)$
 ${2}$ (4)
 > $\mathrm{LAVFSolve}\left(L,'\mathrm{output}'="basis"\right)$
 $\left[{-}{x}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{u}{}\frac{{\partial }}{{\partial }{u}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{,}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right]$ (5)

Compatibility

 • The LieAlgebrasOfVectorFields[SymmetryLAVF] command was introduced in Maple 2020.