Symmetries

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\centerline{\fbox{\parbox{7.7cm}{\Large\bf UserFunction Symmetries} }}

\vspace{5mm}\noindent{\bf Symmetries}$(u:ODE1(C,y,x))$
The single argument $u$ has the type $ODE1$ which abbreviates
{\em OrdinarDifferentialEquation} of order 1. The meaning of its parameters is as follows.

$C:$ Coefficient type.

$y:$ Dependent variable.

$x:$ Independent variable.

\vspace*{1mm}\noindent{\bf Specification.}
Symmetry generators with coefficients that are polynomial in both $x$ and $y$ of total degree not higher than 4 are returned.

\vspace*{1mm}\noindent{\bf Examples.}
The input for the examples is given in {\em Reduce} algebraic mode syntax. The output is returned in a separate window. The first example is a first-order ode.

\4{\tt deq:=Df(y,x)-x*y**2+1/x*y+20/x**3=0;}


\4{\tt T==|ODE1(RATF Q,y,x)|;}

\4{\tt Symmetries(deq|T|);}

\vspace{3mm}

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

Symmetry generators:
\vspace*{1mm}

$U_1=x\partial_x-2y\partial_y$ }}

\vspace{5mm}\noindent{\bf Symmetries}$(u:ODE1(C,y,x),n)$.
The first argument $u$ is the same as above. The second argument $n$ is a non-negative integer.


\vspace*{1mm}\noindent{\bf Specification.}
Symmetry generators with coefficients that are polynomial in both $x$ and $y$ of total degree not higher than $n$ are returned.

{\bf Example.} The same differential equation as above with $n=7$.


\4{\tt s:=Symmetries(deq|ODE1|,7);}

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

Symmetry generators:
\vspace*{1mm}

$U_1=x\partial_x-2y\partial_y$
\vspace*{1mm}

$U_2=(x^3y-10x)\partial_x+(x^4y^3-x^2y^2)\partial_y$ }}

 


\vspace{5mm}\noindent{\bf Symmetries}$(u:ODE2(C,y,x))$
The single argument $u$ has the type $ODE2$ which abbreviates
{\em OrdinaryDifferentialEquation} of order 2. The meaning of its parameters is the same as for $ODE1(C,y,x)$.

\vspace*{1mm}\noindent{\bf Specification.}
Symmetry generators with Liouvillian coefficients are returned.

\vspace*{5mm}
{\bf Example.} The following second-order ode is considered.

\4{\tt deq:=Df(y,x,2)+8/9*Df(y,x)**4;}

At first its symmetry class should be determined by calling

\4{\tt SymmetryClass deq;}

 


\vspace{1mm}\hspace*{10mm}
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The differential equation:

$y''+\frac{8}{9}y'^2=0$

Symmetry Class: ${\cal S}^2_{3,3}(\frac{3}{2})$ }}

Now the three-parameter symmetries may be determined as follows.

\4{\tt sym:=Symmetries deq;}

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\fbox{\parbox{13cm}{

Symmetry generators:

$U_{1}=\partial_x$

$U_{2}=\partial_y$

$U_{3}=x\partial_x+ \Frac{2}{3}{y}\partial_y$ }}


\vspace{5mm}\noindent{\bf Symmetries}$(u:ODE3(C,y,x))$
The single argument $u$ has the type $ODE3$ which abbreviates
{\em OrdinaryDifferentialEquation} of order 3. The meaning of its parameters is the same as for $ODE2(C,y,x)$.

\vspace*{1mm}\noindent{\bf Specification.}
Symmetry generators with Liouvillian coefficients are returned.


\vspace{5mm}\noindent{\bf Symmetries}$(u:PDE(C,z,x))$
The single argument $u$ has the type $PDE$ which abbreviates
{\em PartialDifferentialEquation}. The meaning of its parameters is as follows.

$C$: Coefficient type;

$z=\{z_1,\ldots,z_m\}$: List of dependent variables.

$x=\{x_1,\ldots,x_n\}$: List of independent variables.


\vspace*{1mm}\noindent{\bf Specification.}
Symmetry generators with rational coefficients or undetermined functions are returned.


\vspace*{5mm}
{\bf Examples.} The symmetry generators of Burgers' equation are determined first.

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

\4{\tt Symmetries deq;} or {\tt Symmetries(deq|PDE(RATF Q,{u},{t,x})|);}

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Symmetry generators:
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$U_{1}=\partial_x$

$U_{2}=\partial_t$

$U_{3}=t\partial_x+\partial_u$

$U_{4}=tx\partial_x+t^{2}\partial_t-(tu-x)\partial_u$

$U_{5}=x\partial_x+2t\partial_t-u\partial_u$  }}


\vspace*{1mm}\noindent{\bf Related Functions.} SymmetryClass, JanetBase, DeterminingSystem, CommutatorTable.


\vspace*{5mm}

{\bf References}

P. Olver, Application of Lie Groups to Differential Equations, Springer, 1986.

G. W. Bluman, S. Kumei, Symmetries and Differential Equations, Springer, 1989.

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

 

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