Intersection - Maple Programming Help

Home : Support : Online Help : Mathematics : Differential Equations : Lie Symmetry Method : Commands for PDEs (and ODEs) : LieAlgebrasOfVectorFields : LAVF : LieAlgebrasOfVectorFields/LAVF/Intersection

Intersection

find a LAVF object whose solution space is the intersection of solution spaces of given LAVF objects.

 Calling Sequence Intersection( L1, L2, ..., depname = vars )

Parameters

 L1, L2, ... - a sequence of LAVF objects living on the same space vars - (optional) a list of new dependent variable names

Description

 • Let L1,L2, ... be a sequence of LAVF objects living on the same space (see AreSameSpace). The Intersection method returns a new LAVF object whose solution space is the intersection of solutions of L1,L2,....
 • By default, the dependent variable names of the returned object are taken from L1. The dependent variable names will be vars if the optional argument depnames = vars is specified.
 • This method is front-end to the corresponding method of a LHPDE object. That is, let S1, S2,...  be the determining systems of L1,L2,...  (i.e. Si = GetDeterminingSystem(Li)), then the call Intersection(L1,L2,..) is equivalent to Intersection(S1,S2,..). All remaining input arguments will be passed down to its determining system level. See the method Intersection of a LHPDE object for more detail.
 • This method is associated with the LAVF object. For more detail, see Overview of the LAVF object.

Examples

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

The determining system for 2-dim Euclidean

 > E2:= LHPDE([diff(xi(x,y),x)=0, diff(eta(x,y),y)=0, diff(xi(x,y),y)+diff(eta(x,y),x)=0, diff(xi(x,y),y,y)=0, diff(eta(x,y),x,x)=0], indep = [x,y], dep = [xi,eta]);
 ${\mathrm{E2}}{≔}\left[{{\mathrm{\xi }}}_{{x}}{=}{0}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{y}}{+}{{\mathrm{\eta }}}_{{x}}{=}{0}{,}{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}{,}{x}}{=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{,}{\mathrm{\eta }}\right]$ (2)

The determining system for 2-dim translations

 > T2 := LHPDE([diff(xi(x,y),x) = 0, diff(xi(x,y),y)=0, diff(eta(x,y),x) = 0, diff(eta(x,y),y)=0], indep = [x,y], dep = [xi,eta]);
 ${\mathrm{T2}}{≔}\left[{{\mathrm{\xi }}}_{{x}}{=}{0}{,}{{\mathrm{\xi }}}_{{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{0}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{,}{\mathrm{\eta }}\right]$ (3)

We first construct LAVFs for E(2) and T(2)

 > LE2 := LAVF(X,E2);
 ${\mathrm{LE2}}{≔}\left[{\mathrm{\xi }}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\eta }}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{-}{{\mathrm{\xi }}}_{{y}}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}\right]\right\}$ (4)
 > LT2 := LAVF(X,T2);
 ${\mathrm{LT2}}{≔}\left[{\mathrm{\xi }}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\eta }}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{{\mathrm{\xi }}}_{{x}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{0}{,}{{\mathrm{\xi }}}_{{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}\right]\right\}$ (5)
 > Intersection(LE2,LT2);
 $\left[{\mathrm{\xi }}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\eta }}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{{\mathrm{\xi }}}_{{x}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{0}{,}{{\mathrm{\xi }}}_{{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}\right]\right\}$ (6)
 > Intersection(LE2, LT2, depname = [alpha, beta]);
 $\left[{\mathrm{\alpha }}{}\left({x}{,}{y}\right){}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\beta }}{}\left({x}{,}{y}\right){}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\alpha }}{}\left({x}{,}{y}\right){=}{0}{,}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\beta }}{}\left({x}{,}{y}\right){=}{0}{,}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\alpha }}{}\left({x}{,}{y}\right){=}{0}{,}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\beta }}{}\left({x}{,}{y}\right){=}{0}\right]\right\}$ (7)

Compatibility

 • The Intersection command was introduced in Maple 2020.