DeterminingSystem

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\centerline{\fbox{\parbox{10.2cm}{\Large\bf UserFunction DeterminingSystem} }}

\vspace{5mm}\noindent{\bf DeterminingSystem}$(u:ODE(C,y,x))$
The single argument $u$ has the type $ODE$ which abbreviates
{\em OrdinaryDifferentialEquation}, or any of its subtypes. The meaning of its parameters is as follows.

$C$: Coefficient type.

$y$: Dependent variable.

$x$: Independent variable.

\vspace*{1mm}\noindent{\bf Specification.}
The determining system for the Lie symmetries of $u$ is returned. In general it is not a Janet basis.

 

\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 first example is a first-order ode.

\4{\tt deq:=x*Df(y,x)+y**2+x**2=0;}

 

\5{\tt DeterminingSystem deq;};

\vspace{3mm}

\vspace{1mm}\hspace*{10mm}
\fbox{\parbox{13cm}{

$\eta _{y}+\fracsm{x^{2}+y^{2}}{x}\xi _{y}-\fracsm{x}{x^{2}+y^{2}}\eta _{x}-\xi _{x}-\fracsm{2y}{x^{2
}+y^{2}}\eta -\fracsm{x^{2}-y^{2}}{x^{3}+xy^{2}}\xi =0$

Term Order: GRLEX,
$\eta >\xi ,y>x$ }}


The next example is a nonlinear second-order ode.

\4{\tt deq:=y*Df(y,x,2)-1/2*Df(y,x)**2-4*y**3=0;}

\4{\tt ds:=DeterminingSystem deq;}


\vspace{1mm}\hspace*{10mm}
\fbox{\parbox{10cm}{

$\eta _{x,x}+4y^{2}\eta _{y}-8y^{2}\xi _{x}-8y\eta =0$

$\eta _{x,y}-\Frac{1}{2}\xi_{x,x}-6y^{2}\xi_{y}
                 -\fracsm{\Frac{1}{2}}{y}\eta _{x}=0$

$\xi _{y,y}+\fracsm{\Frac{1}{2}}{y}\xi _{y}=0$

$\eta _{y,y}-2\xi _{x,y}-\fracsm{\Frac{1}{2}}{y}\eta _{y}+\fracsm{\Frac{1}{2}}{y^{2}}\eta =0$

Term Order: GRLEX,
$\eta >\xi ,y>x$  }}

This result may be transformed into a Janet base of a predefined order, e.g.
the LEX order with $\eta>\xi,y>x$.

\4{\tt JanetBase(ds|LDFMOD(RATF Q,\{eta,xi\},\{y,x\},LEX)|);}

\vspace{1mm}\hspace*{10mm}
\fbox{\parbox{10cm}{

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

$\xi _{y}=0$

$\eta +2y\xi _{x}=0$

Term Order: LEX,
$\eta >\xi ,y>x$ }}

\vspace{10mm}\noindent{\bf DeterminingSystem}$(u:PDE(C,ds,vs))$
The single argument $u$ has the type $PDE$ which abbreviates
{\em PartialDifferentialEquation}. The meaning of the parameters is as follows.

$C$ determines the type of the coefficients.

$ds$ denotes the dependent variables.

$vs$ denotes the independent variables.


\vspace*{1mm}\noindent{\bf Specification.}
The determining system for the Lie symmetries of $u$ is returned. In general it is not a Janet basis.

 


{\bf Example.}
The input is given in {\em Reduce} algebraic mode syntax. The output for each minisession is returned in a separate window. The submitted equation is Burgers' equation.

\4{\tt deq:=Df(u,t)+u*Df(u,x)+Df(u,x,2)=0;}

\4{\tt ds:=DeterminingSystem deq;}

\vspace{1mm}\hspace*{10mm}
\fbox{\parbox{10cm}{
$\xi ^{t}_{x}=0$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$\eta _{u,u}-2\xi ^{x}_{u,x}+2u\xi ^{x}_{u}=0$

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

A Janet base is obtained as

\4{\tt JanetBase(ds|LDFMOD(RATF Q,\{eta,xi t,xi x\},\{u,t,x\},LEX)|);}

\vspace{1mm}\hspace*{10mm}
\fbox{\parbox{10cm}{

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

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

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

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

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

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

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

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

\vspace{10mm}\noindent{\bf DeterminingSystem}$(u:PDESYS(C,ds,vs))$
The single argument $u$ has the type $PDESYS$ which abbreviates
{\em PartialDifferentialEquationSystem}. The meaning of the parameters is the same as above.

\vspace*{1mm}\noindent{\bf Specification.}
The determining system for the Lie symmetries of $u$ is returned. In general it is not a Janet basis.

 

{\bf Example.} The input is given in {\em Reduce} algebraic mode syntax. The output for each minisession is returned in a separate window. The nonlinear Sch\"odinger system for the real and the imaginary part $\phi$ and $\psi$.

\4{\tt deq:=\{Df(phi,x)+Df(psi,y,2)+2*phi**2*psi+2*psi**3,}

\hspace*{20mm}{\tt Df(phi,y,2)-Df(psi,x)+2*phi**3+2*phi*psi**2\};}

\4{\tt ds:=DeterminingSystem deq;}

\4{\tt T==|LDFMOD(RATF Q,\{eta phi,eta psi,xi y,xi x\},
                   \{phi,psi,y,x\},LEX)|;}

\4{\tt JanetBase(ds|T|);}

 

\vspace{1mm}\hspace*{10mm}
\fbox{\parbox{10cm}{
$\xi ^{x}_{y}=0$

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

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

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

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

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

$\xi ^{y}_{\psi }=0$

$\xi ^{y}_{\phi }=0$

$\eta ^{\psi }_{y}- \Frac{1}{2}\phi \xi ^{y}_{x}=0$

$\eta ^{\psi }_{x}=0$

$\eta ^{\psi }_{\psi }+ \Frac{1}{2}\xi ^{x}_{x}=0$

$\eta ^{\psi }_{\phi }-\fracsm{1}{\phi }\eta ^{\psi }-\fracsm{\Frac{1}{2}\psi }{\phi }\xi ^{x}_{x}=0$

$\eta ^{\phi }+\fracsm{\psi }{\phi }\eta ^{\psi }
   +\fracsm{\Frac{1}{2}\phi^{2}+\Frac{1}{2}\psi^{2}}{\phi}\xi^{x}_{x}=0$ }}

Term Order: LEX,
$\eta ^{\phi }>\eta ^{\psi }>\xi ^{y}>\xi ^{x},\phi >\psi >x>y$

 

\vspace*{1mm}\noindent{\bf Related Functions.} JanetBasis, Symmetries.


\vspace*{1mm}

{\bf References.}

P. Olver, {\em Application of Lie Groups to Differential Equations}, Springer, 1986.

G. W. Bluman, S. Kumei, {\em Symmetries and Differential Equations}, Springer, 1989.

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


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