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%TCIDATA{Created=Tuesday, November 15, 2005 13:57:08}
%TCIDATA{LastRevised=Monday, November 21, 2005 10:39:15}
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\begin{document}
\begin{center}
Second Order equations
Type
\end{center}
\bigskip
\begin{equation*}
Au_{xx}+2Bu_{xy}+Cu_{yy}+Du_{x}+Eu_{y}+Fu=G(x,y)
\end{equation*}
\bigskip
\begin{definition}
Elliptic: \ \ \ \ \ \ \ $B^{2}AC$ \ \ \ \ \ \ \ \ \ \
${\Huge u}_{xx}{\Huge -u}_{yy}$
\ \ \ \ \ \ \ \ \ \ \ \ \ Parabolic: \ \ \ \ \ $B^{2}=AC$ \ \ \ \ \ \ \ \ \
{\Huge \ }${\Huge u}_{xx}{\Huge -u}_{y}$
\end{definition}
\bigskip
\begin{center}
\textbf{Canonical Forms}
\end{center}
\medskip
We can divide the equation by $A$ or equivalently assume $A=1$. Since low
order terms don't affect the type we choose $D=E=F=G=0$. Then
\begin{equation*}
u_{xx}+2Bu_{xy}+Cu_{yy}=0
\end{equation*}
\begin{equation*}
(\frac{\partial }{\partial x}+B\frac{\partial }{\partial y})^{2}u+(C-B^{2})%
\frac{\partial ^{2}u}{\partial y^{2}}=0
\end{equation*}
CASE I: elliptic $B^{2}0$
we now change variables
\begin{align*}
x &= \xi \qquad y=B \xi +b \eta \\
\xi &= x \qquad \eta =\frac{y-Bx}{b}
\end{align*}
Then%
\begin{align*}
\frac{\partial }{\partial \xi }& =\frac{\partial x}{\partial \xi }\frac{%
\partial }{\partial x}+\frac{\partial y}{\partial \xi }\frac{\partial }{%
\partial y}=\partial _{x}+B\partial _{y}\qquad \frac{\partial ^{2}}{\partial
\xi ^{2}}=\partial _{x}^{2}+2B\partial _{x}\partial _{y}+B^{2}\partial
_{y}^{2} \\
\frac{\partial }{\partial \eta }& =\frac{\partial x}{\partial \eta }\frac{%
\partial }{\partial x}+\frac{\partial y}{\partial \eta }\frac{\partial }{%
\partial y}=0+b\partial _{y}\qquad \frac{\partial ^{2}}{\partial \eta ^{2}}%
=b^{2}\partial _{y}^{2}
\end{align*}
and so%
\begin{align*}
\frac{\partial^2 u}{\partial \xi^2}+\frac{\partial^2 u}{\partial \eta^2} &= %
\left[ \partial _{x}^{2}+2B\partial _{x}\partial _{y}+
\left(B^{2}+b^{2}\right) \partial _{y}^{2}\right] u \\
&= \left( \partial _{x}^{2}+2B\partial _{x}\partial _{y}+C\partial
_{y}^{2}\right) u \\
&= 0
\end{align*}
\bigskip
CASE II: hyperbolic $B^{2}>C$
\begin{equation*}
u_{xx}+2Bu_{xy}+Cu_{yy}=0
\end{equation*}
\begin{equation*}
(\frac{\partial }{\partial x}+B\frac{\partial }{\partial y})^{2}u-(B^{2}-C)%
\frac{\partial ^{2}u}{\partial y^{2}}=0
\end{equation*}
\noindent let $b=\sqrt{B^{2}-C}>0$ we now change variables
\begin{align*}
x &= \xi \qquad y=B\xi +b\eta \\
\xi &= x \qquad \eta =\frac{y-Bx}{b}
\end{align*}
Then%
\begin{align*}
\frac{\partial }{\partial \xi } &= \partial _{x}+B\partial _{y}\qquad \frac{%
\partial ^{2}}{\partial \xi ^{2}}=\partial _{x}^{2}+2B\partial _{x}\partial
_{y}+B^{2}\partial _{y}^{2} \\
\frac{\partial }{\partial \eta } &= b\partial _{y}\qquad \frac{\partial ^{2}%
}{\partial \eta ^{2}}=b^{2}\partial _{y}^{2} \\
\qquad \qquad \frac{\partial ^{2}}{\partial \xi \partial \eta } &=
b\partial_{x}\partial _{y}+B\partial _{y}^{2}
\end{align*}
and so%
\begin{align*}
\frac{\partial ^{2}u}{\partial \xi ^{2}}-\frac{\partial ^{2}u}{\partial \eta
^{2}}& =\left[ \partial _{x}^{2}+2B\partial _{x}\partial _{y}+\left(
B^{2}-b^{2}\right) \partial _{y}^{2}\right] u \\
& =\left( \partial _{x}^{2}+2B\partial _{x}\partial _{y}+C\partial
_{y}^{2}\right) u \\
& =0
\end{align*}
Similarly for systems. Consider%
\begin{equation*}
\dsum_{i=1}^{n}\dsum_{j=1}^{n}a_{ij}\frac{\partial ^{2}u}{\partial
x_{i}\partial x_{j}}+\dsum_{i=1}^{n}a_{i}\frac{\partial u}{\partial x_{i}}%
+a_{0}u=0\qquad a_{ij}=a_{ji}
\end{equation*}
Let $A=(a_{ij})$
\begin{definition}
elliptic: The eigenvalues of $A$ are all positive (or all negative)
hyperbolic: one is negative and the others positive (or the opposite)
ultrahyperbolic : 2 are negative and the others positive
parabolic: one eigenvalue is zero and the others have the same sign
\end{definition}
\newpage
\bigskip
\begin{equation*}
(\ast )\qquad L\left[ u\right]
=au_{xx}+2bu_{xy}+cu_{yy}+du_{x}+eu_{y}+fu=g(x,y)
\end{equation*}
Consider a general change of variables \ $\xi =\xi (x,y)\qquad \eta =\eta
(x,y)$ Then%
\begin{equation*}
Au_{\xi \xi }+2Bu_{\xi \eta }+Cu_{\eta \eta }+Du_{\xi }+Eu_{\eta }+Fu =G
\end{equation*}
\begin{align*}
A(\xi ,\eta )&=a\xi _{x}^{2}+2b\xi _{x}\xi _{y}+c\xi _{y}^{2} \\
B(\xi ,\eta )&=a\xi _{x}\eta _{x}+b\left( \xi _{x}\eta _{y}+\xi _{y}\eta
_{x}\right) +c\xi _{y}\eta _{y} \\
C(\xi ,\eta )&=a\eta _{x}^{2}+2b\eta _{x}\eta _{y}+c\eta _{y}^{2}
\end{align*}
and $AC-B^{2}=J^{2}\left( ac-b^{2}\right)$. \qquad
$J=\xi _{x}\eta _{y}-\xi_{y}\eta _{x}$.
\noindent Hence, the type of equation is invariant under nonsingular transformations.
\noindent This leads to a second canonical form for hyperbolic equations given by
$\xi_{xy}=0$
i.e. we choose $\xi $ and $\eta $ so that
\bigskip
\begin{eqnarray*}
a\xi _{x}^{2}+2b\xi _{x}\xi _{y}+c\xi _{y}^{2} &=&0 \\
a\eta _{x}^{2}+2b\eta _{x}\eta _{y}+c\eta _{y}^{2} &=&0
\end{eqnarray*}
by factoring we have two solutions.
\begin{equation*}
a\xi _{x}^{2}+2b\xi _{x}\xi _{y}+c\xi _{y}^{2}=\frac{\left[ a\xi _{x}+\left(
b+\sqrt{b^{2}-ac}\right) \xi _{y}\right] \left[ a\xi _{x}+\left( b-\sqrt{%
b^{2}-ac}\right) \xi _{y}\right] }{a}
\end{equation*}
We designate one as $\xi $ and the other as $\eta $
\begin{eqnarray*}
a\xi _{x}+\left( b+\sqrt{b^{2}-ac}\right) \xi _{y} &=&0 \\
a\eta _{x}+\left( b-\sqrt{b^{2}-ac}\right) \eta _{y} &=&0
\end{eqnarray*}
These are again called characteristic curves. So these obey%
\begin{eqnarray*}
\frac{dx}{ds} &=&a \\
\frac{dy}{ds} &=&\left( b+\sqrt{b^{2}-ac}\right) \\
\frac{d\xi }{ds} &=&0
\end{eqnarray*}
and%
\begin{eqnarray*}
\xi \text{ is constant on the characteristic }\frac{dy}{dx} &=&\frac{\left(
b+\sqrt{b^{2}-ac}\right) }{a}\qquad \\
\eta \text{ is constant on the characteristic}\frac{dy}{dx} &=&\frac{\left(
b-\sqrt{b^{2}-ac}\right) }{a}
\end{eqnarray*}
\bigskip
CASE III: parabolic $B^{2}=C$
\begin{equation*}
u_{xx}+2Bu_{xy}+Cu_{yy}=0
\end{equation*}
\begin{equation*}
(\frac{\partial }{\partial x}+B\frac{\partial }{\partial y})^{2}u-(B^{2}-C)%
\frac{\partial ^{2}u}{\partial y^{2}}=(\frac{\partial }{\partial x}+B\frac{%
\partial }{\partial y})^{2}u=0
\end{equation*}
choose%
\begin{align*}
x& =\xi \qquad y=B\xi +\eta \\
\xi & =x\qquad \eta =y-Bx
\end{align*}
Then
\begin{equation*}
\frac{\partial ^{2}u}{\partial ^{2}\xi }=0
\end{equation*}
\end{document}