LHSolve - Maple Help

LHSolve

attempts to solve a LHPDEs system of finite type.

 Calling Sequence LHSolve( obj, output = out, consts = c)

Parameters

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

Description

 • The LHSolve method attempts to solve the linear homogeneous PDEs in a LHPDE object.
 • If solving is successful then the method returns a list of equations as the general solution.
 • By specifying output = "basis", the returned output will be a list of lists of equations representing each solution in a basis.
 • By specifying output = "lhpde", the returned output will be a new LHPDE object that is fully integrated.
 • For a returned LHPDE object S that involves constants of integration variables, these variables are treated as additional dependent variables of S. The default names are _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 ${\mathrm{\alpha }}_{1},{\mathrm{\alpha }}_{2},{\mathrm{\alpha }}_{3},\dots$
 – 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 existing pdsolve command (for partial DEs system) and the dsolve command (for ordinary DEs system) of finite type.
 • The method throws an exception if the LHPDEs system is not of finite type.
 • This method is associated with the LHPDE object. For more detail, see Overview of the LHPDE object.

Examples

 > $\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right):$
 > $\mathrm{Typesetting}:-\mathrm{Settings}\left(\mathrm{userep}=\mathrm{true}\right):$
 > $\mathrm{Typesetting}:-\mathrm{Suppress}\left(\left\{\mathrm{\alpha },\mathrm{\beta },\mathrm{\eta },\mathrm{\xi }\right\}\left(x,y\right)\right):$
 > $S≔\mathrm{LHPDE}\left(\left[\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y,y\right)=0,\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),x\right)=-\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y\right),\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),y\right)=0,\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),x\right)=0\right],\mathrm{indep}=\left[x,y\right],\mathrm{dep}=\left[\mathrm{\xi },\mathrm{\eta }\right]\right)$
 ${S}{≔}\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{-}{{\mathrm{\xi }}}_{{y}}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{,}{\mathrm{\eta }}\right]$ (1)
 > $\mathrm{IsFiniteType}\left(S\right)$
 ${\mathrm{true}}$ (2)
 > $\mathrm{LHSolve}\left(S\right)$
 $\left[{\mathrm{\xi }}{=}{-}{\mathrm{_C1}}{}{y}{+}{\mathrm{_C3}}{,}{\mathrm{\eta }}{=}{\mathrm{_C1}}{}{x}{+}{\mathrm{_C2}}\right]$ (3)
 > $\mathrm{LHSolve}\left(S,\mathrm{consts}=\left[\mathrm{\alpha },\mathrm{\beta },\mathrm{\delta }\right]\right)$
 $\left[{\mathrm{\xi }}{=}{-}{\mathrm{\alpha }}{}{y}{+}{\mathrm{\delta }}{,}{\mathrm{\eta }}{=}{\mathrm{\alpha }}{}{x}{+}{\mathrm{\beta }}\right]$ (4)
 > $\mathrm{LHSolve}\left(S,\mathrm{output}="basis"\right)$
 $\left[\left[{\mathrm{\xi }}{=}{-}{y}{,}{\mathrm{\eta }}{=}{x}\right]{,}\left[{\mathrm{\xi }}{=}{0}{,}{\mathrm{\eta }}{=}{1}\right]{,}\left[{\mathrm{\xi }}{=}{1}{,}{\mathrm{\eta }}{=}{0}\right]\right]$ (5)
 > $\mathrm{LHSolve}\left(S,\mathrm{output}="lhpde",\mathrm{consts}=\mathrm{\alpha }\right)$
 $\left[{\mathrm{\xi }}{=}{-}{y}{}{{\mathrm{\alpha }}}_{{1}}{+}{{\mathrm{\alpha }}}_{{3}}{,}{\mathrm{\eta }}{=}{x}{}{{\mathrm{\alpha }}}_{{1}}{+}{{\mathrm{\alpha }}}_{{2}}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{,}{\mathrm{\eta }}{,}{{\mathrm{\alpha }}}_{{1}}{,}{{\mathrm{\alpha }}}_{{2}}{,}{{\mathrm{\alpha }}}_{{3}}\right]$ (6)

Compatibility

 • The LHSolve command was introduced in Maple 2020.