IntegrabilityConditions

\documentstyle[12pt]{article}
\input{latlib}
\parindent 0mm
\pagestyle{empty}
\topmargin 5mm
\oddsidemargin 0pt
\evensidemargin 60pt
\headheight 0cm
\headsep 0cm
\footheight 0cm
\footskip 0mm
\textheight 28.7cm
\textwidth 18cm
\begin{document}
\begin{large}


\centerline{\fbox{\parbox{11.1cm}{\Large\bf UserFunction IntegrabilityConditions} }}

\vspace{5mm}\noindent{\bf IntegrabilityConditions}$(u:LDFMOD(C,zs,xs,O))$
The single argument $u$ has the type $LDFMOD$ which abbreviates
{\em LinearDifferentialFormModule}, 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.}
The integrability conditions for $u$ are 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 ls:=\{Df(z,y)+a\_1*Df(z,x)+a\_2*z, Df(z,x,2)+b\_1*Df(z,x)+b\_2*z\}; }

\4{\tt T==|LDFMOD(DFRATF(Q,\{a\_1,a\_2,b\_1,b\_2\},
                       \{x,y\},GRLEX),\{z\},\{y,x\},GRLEX)|; }

\4{\tt ics:=IntegrabilityConditions(ls|T|); }

\vspace{3mm}

\vspace{1mm}\hspace*{30mm}\fbox{\parbox{9.5cm}{

$\displaystyle a_{2,x,x}-2a_{1,x}b_2+a_{2,x}b_1-b_{2,x}a_1-b_{2,y}=0$

\vspace*{2mm}
$\displaystyle a_{1,x,x}-a_{1,x}b_1+2a_{2,x}-b_{1,x}a_1-b_{1,y}=0$ }}


\vspace*{4mm}
\4{\tt ls:=\{Df(z\_1,x)+a\_1*z\_2+a\_2*z\_1, Df(z\_1,y)+b\_1*z\_2+b\_2*z\_1,
 
\hspace*{15mm}
             Df(z\_2,x)+c\_1*z\_2+c\_2*z\_1, Df(z\_2,y)+d\_1*z\_2+d\_2*z\_1\};}

\4{\tt T==|LDFMOD(DFRATF(Q,\{a\_1,a\_2,b\_1,b\_2,c\_1,c\_2,d\_1,d\_2\},\{x,y\},GRLEX),
                         
\hspace*{60mm}   \{z\_1,z\_2\},\{y,x\},GRLEX)|; }

\4{\tt ics:=IntegrabilityConditions(ls|T|); }

 


\vspace*{5mm}

{\bf References}


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

 

\end{large}
\end{document}

 

User Interface Functions

Go back to parent