Prolongation

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\centerline{\fbox{\parbox{8cm}{\Large\bf UserFunction Prolongation} }}

\vspace*{5mm}\noindent{\bf Prolongation}$(u:LDO(C,v,O),{iv},{dv},k)$
The meaning of the parameters is as follows.


$C$: Coefficient type.

$v$: Vector field variables.

$iv$: Independent variables.

$dv$: Dependent variables.

\vspace*{1mm}\noindent{\bf Specification.}
The first argument $u$ defines a vector field in variables $v$. The second argument $k$ is a positive integer. The prolongation of $u$ of order $k$ is returned, assuming that the variables on $dv$ depend on those on $iv$.

 

\vspace*{1mm}\noindent{\bf Examples.} The input is given in {\em Reduce} algebraic mode syntax. The output for each minisession is returned in a separate window. Partial derivative operators as e.g. $\partial_x$ or $\partial_y$ are denoted by {\tt D(x)}, {\tt D(y)} etc. in the input.


\5{\tt l:=x**2*D(x)+x*y*D(y);}

\5{\tt Prolongation(l|LDO(RATF Q,\{x,y\},GRLEX)|,\{x\},\{y\},3);}


\vspace{2mm}\hspace*{20mm}\fbox{\parbox{11.5cm}{

$\displaystyle  x^2\partial_x+xy\partial_y-(y'x-y)\partial_{y'}
        -3y''x\partial_{y''}-(5y'''x+3y'')\partial_{y'''}$ }}

\5{\tt l:=v**2*D(v)+w**2*D(w);}

\5{\tt Prolongation(l|LDO(RATF Q,\{x,y,v,w\},GRLEX)|,\{x,y\},\{v,w\},2);}

\vspace{2mm}\hspace*{20mm}\fbox{\parbox{13cm}{

$\displaystyle v\partial_v+w\partial_w
     +v_x\partial_{v_x}+w_x\partial_{w_x}
     +v_y\partial_{v_y}+w_y\partial_{w_y}
     +v_{x,x}\partial_{v_{x,x}}+w_{x,x}\partial_{w_{x,x}}$

\vspace*{3mm}\hspace*{5mm}
     $+v_{x,y}\partial_{v_{x,y}}+w_{x,y}\partial_{w_{x,y}}
     +v_{y,y}\partial_{v_{y,y}}+w_{y,y}\partial_{w_{y,y}}$ }}

\vspace*{5mm}
\noindent{\bf Prolongation}$(u:LIEVEC(C,v),iv,dv:BASIC,k:Z)$

The first argument $u$ defines the infinitesimal generators of a Lie algebra
of vector fields. The second argument $k$ is a positive integer.  The meaning of the parameters is as follows.

$C$: Coefficient type.

$v$: Vector field variables.

$iv$: Independent variables.

$dv$: Dependent variables.


\vspace*{1mm}\noindent{\bf Specification.}
The prolongation of the elements of $u$ up to order $k$ are returned, assuming that the variables on $dv$ depend on those on $iv$.

\vspace*{1mm}\noindent{\bf Examples.} The input is given in {\em Reduce} algebraic mode syntax. The output for each minisession is returned in a separate window. The partial derivative operators $\partial_x$, $\partial_y$ etc. are denoted by {\tt D(x)(}, {\tt D(y)} etc. in the input.

\5{\tt l:=\{D(x)+D(y),x*D(x)+y*D(y),x**2*D(x)+y**2*D(y)\};}

\5{\tt Prolongation(l|LIEVEC(RATF Q,\{x,y\}),\{x\},\{y\}|,2);}


\vspace{1mm}\hspace*{20mm}\fbox{\parbox{13cm}{

$\displaystyle  \frac{\partial}{\partial x}+\frac{\partial}{\partial y}$

\vspace*{1mm}
$\displaystyle   x\frac{\partial}{\partial x}+y
  \frac{\partial}{\partial y}-y''\frac{\partial}{\partial y''}$

\vspace*{1mm}
$\displaystyle
   x^{2}\frac{\partial}{\partial x}+y^{2}\frac{\partial}{\partial y}
     -(2y'x-2y'y)\frac{\partial}{\partial y'}-(4y''x-2y''y-2y'^{2}+2y')
      \frac{\partial}{\partial y''}$  }}

\vspace*{5mm}

\5{\tt l:=\{D(x)+x*(x*D(x)+y*D(y)+z*D(z)),\\
\hspace*{13mm} D(y)+y*(x*D(x)+y*D(y)+z*D(z)),\\
\hspace*{13mm} D(z)+z*(x*D(x)+y*D(y)+z*D(z)),\\
\hspace*{13mm} z*D(y)-y*D(z),\\
\hspace*{13mm} x*D(z)-z*D(x),\\
\hspace*{13mm} y*D(x)-x*D(y)\};}


\5{\tt Prolongation(l|LIEVEC(RATF Q,\{x,y,z\})|,\{x,y\},\{z\},2);}

\hspace*{20mm}\fbox{\parbox{15cm}{

$\displaystyle z\partial_{x}-x\partial_{z}-(z_{x}^{2}+1)\partial_{z_{x}}
     -z_{x}z_{y}\partial_{z_{y}}$

\vspace*{2mm}
$\displaystyle
y\partial_{x}-x\partial_{y}+z_{y}\partial_{z_{x}}-z_{x}\partial_{z_{y}}$

\vspace*{2mm}
$\displaystyle
xy\partial_{x}+(y^{2}+1)\partial_{y}+yz\partial_{z}
  -(z_{x}x+z_{y}y-z)\partial_{z_{y}}$

\vspace*{2mm}
$\displaystyle
(x^{2}+1)\partial_{x}+xy
\partial_{y}+xz\partial_{z}-(z_{x}x+z_{y}y-z)\partial_{z_{x}}$

\vspace*{2mm}
$\displaystyle
xz\partial_{x}+yz\partial_{y}+(z^{2}+1)\partial_{z}-(z_{x}^{2}x
+z_{x}z_{y}y-z_{x}z)\partial_{z_{x}}-(z_{x}z_{y}x+z_{y}^{2}y-z_{y}z)
\partial_{z_{y}}$

\vspace*{2mm}
$\displaystyle
z\partial_{y}-y\partial_{z}-z_{x}z_{y}\partial_{z_{x}}-(z_{y}^{2}+1)
\partial_{z_{y}}$ }}


\vspace*{3mm}
\vspace*{1mm}\noindent{\bf Error Messages.}

\5{\tt ***** Type of Lie vector fields cannot be determined}

This error occurs when the type of the first argument has not been completely specified, e.~g. if not all independent and dependentvariables have be submitted.

{\bf References}

F.~Schwarz, {\em Algorithmic Lie Theory for Solving Ordinary Differential Equations}, CRC Press, 2007.


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