#### UserFunction JanetBasis

JanetBasis(u:LDFMOD(C,zs,xs,O)

The single argument $u$ has the type $LDFMOD$ which abbreviatesLinearDifferentialFormModule,
it represents a system of linear homogeneous partial differential equations.
The meaning of its parameters is as follows.

$C$: Coefficient type.

$zs=\{z_1,z_2,\ldots \}$: Dependent variables in decreasing order.

$xs=\{x_1,x_2,\ldots \}$: Independent variables in decreasing order.

$O=LEX|GRLEX|GREVLEX$: Term ordering.

\vspace*{1mm}\noindent{\bf Specification.}
A Janet basis for $u$ in term order $O$ is returned.

\vspace{1mm}\noindent{\bf Examples.}
The input for the examples is given in {\em Reduce} algebraic mode syntax. The output for each minisession is returned in a separate window.

\4{\tt jb:=\{Df(w,y,2)+3/(4*y)*Df(w,y),

\hspace*{10mm}  Df(z,x,2)+3*y**2*Df(z,y)-6*y**2*Df(w,x)-6*y*z,

\hspace*{10mm}  Df(z,x,y)-1/2*Df(w,x,2)-3/(4*y)*Df(z,x)-9/2*y**2*Df(w,y),

\hspace*{10mm}  Df(z,y,2)-2*Df(w,x,y)-3/(4*y)*Df(z,y)+3/(4*y**2)*z\};}

\5{\tt jb:=JanetBasis(jb|LDFMOD(RATF Q,\{w,z\},\{y,x\},GRLEX)|);}

\vspace{3mm}

\vspace{1mm}\hspace*{30mm}\fbox{\parbox{11.5cm}{

Janet Basis for Term Order $GRLEX,\hspace*{2mm}z>w,\hspace*{1mm}y>x$
\vspace{1mm}

$\displaystyle w_{x}+\fracsm{1}{2y}z=0,\2 w_{y}=0,\2 z_{x}=0,\2 z_{y}-\fracsm{1}{y}z=0$ }}

\vspace{5mm}

\5{\tt JanetBasis(jb|LDFMOD(RATF Q,\{w,z\},\{y,x\},LEX)|);}

\vspace{3mm}

\vspace{1mm}\hspace*{30mm}\fbox{\parbox{11cm}{

Janet Basis for Term Order $LEX,\hspace*{2mm}z>w,\hspace*{1mm}y>x$
\vspace{2mm}

$\displaystyle w_{x,x}=0,\2 w_{y}=0,\2 z+2yw_{x}=0$ }}

\vspace{5mm}\noindent{\bf JanetBasis}$(u:MODULE(LDO(C,zs,xs,O),ORD)$
The arguments of $LDO$ have the same meaning as in $LDFMOD$ above. The last argument $ORD$ determines the module term order according to Adams and Loustaunau.

$ORD:TOP|POT$ determines the module term ordering; it is always assumed that the components are orderd es $e_1<e_2<e_3\ldots$.

\vspace{1mm}\noindent{\bf Specification.}
A Janet basis for $u$ is returned. The term order is determined by $O$ and $ORD$.

\vspace{1mm}\noindent{\bf Examples.}
The input for the examples is given in {\em Reduce} algebraic mode syntax. The output for each minisession is returned in a separate window.

\pagebreak

{\tt
\5 sm:=\{\{D(y,2)+3/(4*y)*D(y),0\},

\hspace*{15mm} \{-6*y**2*D(x),D(x,2)+3*y**2*D(y)-6*y\},

\hspace*{15mm} \{-1/2*D(x,2)-9/2*y**2*D(y),D(y,x)-3/(4*y)*D(x)\},

\hspace*{15mm} \{-2*D(y,x),D(y,2)-3/(4*y)*D(y)+3/(4*y**2)\}\}; }

{\tt JanetBasis(sm|MODULE(LDO(RATF Q,{y,x},GRLEX),TOP)|;}

\vspace{1mm}\hspace*{30mm}
\fbox{\parbox{11cm}{

Janet Basis for Term Order $GRLEX,\hspace*{2mm}y>x,\hspace*{2mm}TOP$

\vspace{2mm}
$\left(\begin{array}{c}\partial_{x}\\[2mm]\fracsm{\Frac{1}{2}}{y}\end{array} \right)$
$\left(\begin{array}{c}0\\[2mm]\partial_{x}\end{array}\right)$$\left (\begin{array}{c}0\\[2mm] \partial_{y}-\fracsm{1}{y}\end{array}\right)$$\left(\begin{array}{c} \partial_{y}\\[2mm]0 \end{array}\right)$ }}

\vspace{5mm}
{\tt JanetBasis(sm|MODULE(LDO(RATF Q,{y,x},GRLEX),POT)|;}

\vspace{1mm}\hspace*{30mm}
\fbox{\parbox{11cm}{

Janet Basis for Term Order $GRLEX,\hspace*{2mm}y>x,\hspace*{2mm}POT$

\vspace{2mm}
$\left(\begin{array}{c}-\Frac{2}{15}\partial_{x}\partial_{x}\partial_{x}- \Frac{6}{5}y^{2}\partial_{ y}\partial_{x}+2y\partial_{x}\\[2mm]1\end{array}\right)$$\left (\begin{array}{c}\partial_{y}\\[2mm]0 \end{array}\right)$$\left(\begin{array}{c}\partial_{x}\partial_{x} \\[2mm]0\end{array}\right)$ }}

\vspace{10mm}\noindent{\bf JanetBasis}$(u:ODE)$
The single argument $u$ has the type $ODE$ which abbreviates
{\em Ordinary Differential Equation}.

\vspace*{1mm}\noindent{\bf Specification.}
The Janet basis for the determining system of its Lie symmetries is returned in a default term order.

\vspace{1mm}\noindent{\bf Examples.}
The input for the examples is given in {\em Reduce} algebraic mode syntax. The output for each minisession is returned in a separate window.

{\tt deq:=4*y*Df(y,x,2)-3*Df(y,x)**2-12*y**3;

JanetBasis deq;}

\vspace{3mm}

\vspace{1mm}\hspace*{30mm}
\fbox{\parbox{11.5cm}{

The Differential Equation:

$4yy''-3y'^2-12y^3=0$

Janet Basis for Term Order $GRLEX,\hspace*{2mm}z>w,\hspace*{1mm}y>x$

$\displaystyle \eta_{x}+\fracsm{1}{2y}\eta=0,\2 \xi_{y}=0,\2 \eta_{x}=0,\2 \eta_{y}-\fracsm{1}{y}\eta=0$

\vspace{3mm}
Term Order $GRLEX,\hspace*{2mm}z>w,\hspace*{1mm}y>x$ }}

\vspace{10mm}\noindent{\bf JanetBasis}$(u:PDE)$
The single argument $u$ has the type $PDE$ which abbreviates
{\em Partial Differential Equation}.

\vspace*{1mm}\noindent{\bf Specification.}
The Janet basis for the determining system of its Lie symmetries is returned in a default term order.

\vspace{1mm}\noindent{\bf Examples.}
The input for the examples is given in {\em Reduce} algebraic mode syntax. The output for each minisession is returned in a separate window.

{\tt deq:=Df(u,x,2)+u*Df(u,x)+Df(u,t);

jb:=JanetBasis deq;}

\vspace{3mm}

\vspace{1mm}\hspace*{30mm}
\fbox{\parbox{11cm}{
$\xi ^{t}_{x}=0$

$\xi ^{x}_{t}-u\xi ^{x}_{x}-\eta =0$

$\xi ^{t}_{t}-2\xi ^{x}_{x}=0$

$\eta _{t}+u\eta _{x}=0$

$\xi ^{x}_{u}=0$

$\xi ^{t}_{u}=0$

$\eta _{u}+\xi ^{x}_{x}=0$

$\xi ^{x}_{x,x}=0$

$\eta _{x,x}=0$

Term Order: GRLEX,
$\eta >\xi ^{t}>\xi ^{x},u>t>x$ }}

This Janet basis has been assigned to the variable jb. A new term order may be established by calling e.g.

{\tt JanetBasis(jb|LDFMOD(RATF Q,\{eta,xi x,xi t\},\{u,x,t\},LEX))|);}

\vspace{1mm}\hspace*{30mm}
\fbox{\parbox{11cm}{
$\xi ^{t}_{t,t,t}=0$

$\xi ^{t}_{x}=0$

$\xi ^{t}_{u}=0$

$\xi ^{x}_{t,t}=0$

$\xi ^{x}_{x}- \Frac{1}{2}\xi ^{t}_{t}=0$

$\xi ^{x}_{u}=0$

$\eta -\xi ^{x}_{t}+ \Frac{1}{2}u\xi ^{t}_{t}=0$

Term Order: LEX,
$\eta >\xi ^{x}>\xi ^{t},u>x>t$  }}

\vspace{10mm}\noindent{\bf JanetBasis}$(u:LFMOD,zs,xs)$
The first argument $u$ is a set of algebraic linear forms in $C_1,\ldots, C_k$.
The variables $z$ are dependent variables depending on the variables $x$.  The meaning of its parameters is as follows.

$zs=\{z_1,z_2,\ldots \}$: Dependent variables in decreasing order.

$xs=\{x_1,x_2,\ldots \}$: Independent variables in decreasing order.

\vspace*{1mm}\noindent{\bf Specification.}
Returns Janet basis for linear homogeneous system of pde's for functions on $zs$ for which $u$ is $k$-dimensional solution space.

\vspace{1mm}\noindent{\bf Examples.}
The input for the examples is given in {\em Reduce} algebraic mode syntax. The output for each minisession is returned in a separate window.

{\tt T==|LFMOD(RATF Q,$\{C_1,C_2\}$)|;}

{\tt JanetBasis($\{C_1*(x**2+2*y)+C_2\}$|T|,\{z\},\{x,y\});}

\vspace{3mm}

\vspace{1mm}\hspace*{30mm}
\fbox{\parbox{11cm}{
$z_x-xz_y=0,$

$z_{yy}=0$

Term Order: GRLEX,
$z,\1 x>y$  }}

{\tt JanetBasis($\{C_1*(x**2+2*z)+C_2*(y**2+2*z)\}$|T|,\{w\},\{x,y,z\});}

\vspace{3mm}

\vspace{1mm}\hspace*{30mm}
\fbox{\parbox{11cm}{
$w_y-\fracsm{x^2y+2yz}{x^2-y^2}w_z+\fracsm{2y}{x^2-y^2}w=0,$

$w_x+\fracsm{xy^2+2xz}{x^2-y^2}w_z-\fracsm{2x}{x^2-y^2}w=0$

$w_{zz}=0$

Term Order: GRLEX,
$w,\1 x>y>z$  }}

\vspace{6mm}\noindent{\bf Related Functions.} {\em GroebnerBasis, Syzygies}.

{\bf References}

W.~W.~Adams, P.~Loustaunau, {\em An Introduction to Gr\"{o}bner Bases}, American Mathematcical Society, 1994.

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

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