LaplaceDivisor

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\centerline{\fbox{\parbox{9cm}{\Large\bf UserFunction LaplaceDivisor } }}
\vspace*{5mm}

Description of the arguments and its types.

\vspace*{1mm}\noindent{\bf LaplaceDivisor$(u:LDO(C,vs,O),v,n:BASIC)$}

The first argument $u$ has the type $LDO$; it abbreviates
{\em LinearDifferentialOperator}. The meaning of its parameters is as follows.

$C$: Coefficient type.

$xs=\{x,y\}$: Two variables in decreasing order.

$O:LEX|GRLEX$: Term ordering.

\vspace*{1mm}\noindent{\bf Specification.}
A Laplace divisor of lowest possible order not higher than $n$ in the variable
$v$ is returned if it exists.

 

\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.


\hspace*{4mm}{\tt L:=D(x,y)+x*y*D(x)-2*y;}

\hspace*{4mm}{\tt T==|LDO(RATF Q,\{x,y\},GRLEX);}

\hspace*{4mm}{\tt LaplaceDivisor(L|T|,x,3);}

\vspace{3mm}

\vspace{1mm}\hspace*{30mm}\fbox{\parbox{6cm}{
The Laplace divisor:

\vspace*{1mm}
$\displaystyle
\langle\partial_{xy}+xy\partial_x-2y,\1\partial_{xxx}\rangle$ }}

A Laplace divisor for a third-order operator is shown in the next example.

\hspace*{4mm}{\tt L:=D(x,2,y)+x*D(x,y,2)-1/(y+1)*D(x,2)+D(x,y)+D(y,2)}

\hspace*{4mm}{\tt -1/(y+1)*D(x)-1/(y+1)*D(y);}

\hspace*{4mm}{\tt T==|LDO(RATF Q,\{x,y\},GRLEX);}

\hspace*{4mm}{\tt LaplaceDivisor(L|T|,y,2);}


\vspace{1mm}\hspace*{30mm}\fbox{\parbox{12cm}{
The Laplace divisor:

\vspace*{1mm}
$\displaystyle
\langle\partial_{xxy}+x\partial_{xyy}-\Frac{1}{y+1}\partial_{xx}
    +\partial_{xy}+\partial_{yy}-\Frac{1}{y+1}\partial_x
    -\Frac{1}{y+1}\partial_y,\1\partial_{yy}\rangle$ }}


\vspace*{1mm}\noindent{\bf Related Functions.}
FirstOrderRightFactors, LoewyDecomposition.

\vspace*{5mm}

{\bf References}
E.~Darboux, {\em Le\c{c}ons sur la
th\'{e}orie g\'{e}n\'{e}rale des surfaces}, vol II, Chelsea Publishing
Company, New York 1972.

 


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