 LAVFSolve - Maple Help

LAVFSolve

find a LAVF object whose solution space is the sum of the solution spaces of given LAVF objects. Calling Sequence LAVFSolve( obj, output = out, consts = c) Parameters

 obj - a LAVF object that is of finite type (see IsFiniteType) out - (optional) a string: either "solution", "basis", or "lavf" c - (optional) a name or a list of names Description

 • The LAVFSolve method attempts to solve the determining system in a LAVF object.
 • If solving is successful then by default the method returns a list of solution vector fields.
 • The returned output can be as a basis (by specifying output = "basis") or a new LAVF object (by specifying output = "lavf").
 • For a returned output that involves constants of integration variables, by default these variables are labeled as  _C1, _C2, ...
 • The constant of integration variables can be renamed by specifying the optional argument consts = c.
 – By specifying consts = alpha (i.e. a name), the constants of integration will be named as ${\mathrm{\alpha }}_{1},{\mathrm{\alpha }}_{2},{\mathrm{\alpha }}_{3},..$
 – By specifying consts = [alpha, beta, phi...] (i.e. a list of names), the constants of integration will be named as $\mathrm{\alpha },\mathrm{\beta },\mathrm{\phi },\dots$
 • This is a front-end to the LHSolve method for solving the determining system. LHSolve is associated with the LHPDE object, see Overview of the LHPDE object for more detail.
 • The method throws an exception if the LAVF is not of finite type.
 • This method is associated with the LAVF object. For more detail, see Overview of the LAVF object. Examples

 > $\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right):$
 > $\mathrm{Typesetting}:-\mathrm{Settings}\left(\mathrm{userep}=\mathrm{true}\right):$
 > $\mathrm{Typesetting}:-\mathrm{Suppress}\left(\left[\mathrm{ξ}\left(x,y\right),\mathrm{η}\left(x,y\right)\right]\right):$
 > $V≔\mathrm{VectorField}\left(\mathrm{ξ}\left(x,y\right){\mathrm{D}}_{x}+\mathrm{η}\left(x,y\right){\mathrm{D}}_{y},\mathrm{space}=\left[x,y\right]\right)$
 ${V}{≔}{\mathrm{\xi }}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right){+}{\mathrm{\eta }}{}\left(\frac{{ⅆ}}{{ⅆ}{y}}\right)$ (1)
 > $\mathrm{E2}≔\mathrm{LHPDE}\left(\left[\frac{{\partial }^{2}}{\partial {y}^{2}}\mathrm{ξ}\left(x,y\right)=0,\frac{\partial }{\partial x}\mathrm{η}\left(x,y\right)=-\left(\frac{\partial }{\partial y}\mathrm{ξ}\left(x,y\right)\right),\frac{\partial }{\partial y}\mathrm{η}\left(x,y\right)=0,\frac{\partial }{\partial x}\mathrm{ξ}\left(x,y\right)=0\right],\mathrm{indep}=\left[x,y\right],\mathrm{dep}=\left[\mathrm{ξ},\mathrm{η}\right]\right)$
 ${\mathrm{E2}}{≔}\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{-}{{\mathrm{\xi }}}_{{y}}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{,}{\mathrm{\eta }}\right]$ (2)

Construct a vector fields system for E(2).

 > $L≔\mathrm{LAVF}\left(V,\mathrm{E2}\right)$
 ${L}{≔}\left[{\mathrm{\xi }}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right){+}{\mathrm{\eta }}{}\left(\frac{{ⅆ}}{{ⅆ}{y}}\right)\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{-}{{\mathrm{\xi }}}_{{y}}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}\right]\right\}$ (3)
 > $\mathrm{LAVFSolve}\left(L\right)$
 $\left({-}\mathrm{c__1}{}{y}{+}\mathrm{c__3}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right){+}\left(\mathrm{c__1}{}{x}{+}\mathrm{c__2}\right){}\left(\frac{{ⅆ}}{{ⅆ}{y}}\right)$ (4)
 > $\mathrm{LAVFSolve}\left(L,\mathrm{consts}=\left[\mathrm{α},\mathrm{β},\mathrm{δ}\right]\right)$
 $\left({-}{\mathrm{\alpha }}{}{y}{+}{\mathrm{\delta }}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right){+}\left({\mathrm{\alpha }}{}{x}{+}{\mathrm{\beta }}\right){}\left(\frac{{ⅆ}}{{ⅆ}{y}}\right)$ (5)
 > $\mathrm{LAVFSolve}\left(L,\mathrm{output}="basis"\right)$
 $\left[{-}{y}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right){+}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{y}}\right){,}\frac{{ⅆ}}{{ⅆ}{y}}{,}\frac{{ⅆ}}{{ⅆ}{x}}\right]$ (6)
 > $\mathrm{LAVFSolve}\left(L,\mathrm{output}="lavf",\mathrm{consts}=\mathrm{α}\right)$
 $\left[{\mathrm{\xi }}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right){+}{\mathrm{\eta }}{}\left(\frac{{ⅆ}}{{ⅆ}{y}}\right)\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{\mathrm{\xi }}{=}{-}{y}{}{{\mathrm{\alpha }}}_{{1}}{+}{{\mathrm{\alpha }}}_{{3}}{,}{\mathrm{\eta }}{=}{x}{}{{\mathrm{\alpha }}}_{{1}}{+}{{\mathrm{\alpha }}}_{{2}}\right]\right\}$ (7) Compatibility

 • The LAVFSolve command was introduced in Maple 2020.