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%TCIDATA{<META NAME="Title" CONTENT="Partial Differential Equations">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Thu Feb 13 16:45:56 1997}
%TCIDATA{Language=American English}
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\section{\label{PDE}Partial Differential Equations}

\textsl{Scientific Notebook} can solve a variety of \emph{linear partial
differential equations of the first or second order};\label%
{Partial differential equation} 
\index{Partial differential equation}that is, equations of the form 
\[
A%
\frac{\partial u}{\partial x}+B\frac{\partial u}{\partial y}+Cu=E 
\]
or 
\[
A\frac{\partial ^{2}u}{\partial x^{2}}+B\frac{\partial ^{2}u}{\partial
x\partial y}+C\frac{\partial ^{2}u}{\partial y^{2}}+E\frac{\partial u}{%
\partial x}+F\frac{\partial u}{\partial y}+Gu=H 
\]
where $A$, $B$, $C$, $E$, $F$, $G$, and $H$ are constants or functions of $x$
and $y$. These equations can also be written using the operator notation 
\[
AD_{x}u+BD_{y}u+Cu=E 
\]
or 
\[
AD_{xx}u+BD_{xy}u+CD_{yy}u+ED_{x}u+FD_{y}u+Gu=H 
\]
Other traditional notation, such as $u_{xy}$, may not be recognized.

\strut

\subsection{\ Solving Partial Differential Equations\label{Solve PDE}}

\begin{quote}
$\blacktriangleright $ To solve a partial differential equation 
\index{Solve@Solve!differential equation@differential equation}
\end{quote}

\begin{enumerate}
\item[1.]  Type the partial differential equation using standard
mathematical notation.

\item[2.]  With the insertion point in the equation, choose \textsf{Solve
PDE.} 
\index{Solve PDE}
\end{enumerate}

\begin{quote}
$\blacktriangleright $ \textsf{Solve PDE }
\end{quote}

\begin{quotation}
$%
\frac{\partial ^{2}u}{\partial x^{2}}-\frac{\partial ^{2}u}{\partial y^{2}}%
=0 $, Exact solution is : $u(x,y)=F_{1}(y+x)+F_{2}(y-x)$
\end{quotation}

\strut

To verify that $u(x,y)=F_{1}(y+x)+F_{2}(y-x)$ is actually a solution, define 
$F_{1}(x)$ and $F_{2}(x)$ to be generic functions and define 
\[
u(x,y)=F_{1}(y+x)+F_{2}(y-x) 
\]
Then \textsf{Evaluate} yields 
\[
\frac{\partial ^{2}u(x,y)}{\partial x^{2}}=\allowbreak F_{1}^{\prime \prime
}\left( y+x\right) +F_{2}^{\prime \prime }\left( y-x\right) 
\]
and 
\[
\frac{\partial ^{2}u(x,y)}{\partial y^{2}}=\allowbreak F_{1}^{\prime \prime
}\left( y+x\right) +F_{2}^{\prime \prime }\left( y-x\right) 
\]
and hence 
\[
\frac{\partial ^{2}u(x,y)}{\partial x^{2}}-\frac{\partial ^{2}u(x,y)}{%
\partial y^{2}}=0 
\]

\strut

\begin{example}
The choice 
\[
u(x,y)=\sin (y+x)+\cos (y-x) 
\]
yields 
\[
\frac{\partial ^{2}u(x,y)}{\partial x^{2}}=\allowbreak -\sin \left(
y+x\right) -\cos \left( -y+x\right) 
\]
and 
\[
\frac{\partial ^{2}u(x,y)}{\partial y^{2}}=\allowbreak -\sin \left(
y+x\right) -\cos \left( -y+x\right) 
\]
and clearly 
\[
\frac{\partial ^{2}u(x,y)}{\partial x^{2}}-\frac{\partial ^{2}u(x,y)}{%
\partial y^{2}}=0 
\]
\end{example}

\strut

\subsection{\ Wave Equation\label{Wave equation}}

Use \textsf{Solve PDE} to solve the wave equation.

\begin{quote}
$\blacktriangleright $ Solve PDE
\end{quote}

\begin{quotation}
$\dfrac{\partial ^{2}y}{\partial t^{2}}=a^{2}\dfrac{\partial ^{2}y}{\partial
x^{2}}$, Exact solution is : $y\left( t,x\right) =F_{1}\left( -x-at\right)
+F_{2}\left( x-at\right) $
\end{quotation}

\strut

\begin{example}
Let $y(x,t)$ denote a vertical displacement of a vibrating string at
position $x$ and time $t$. Then $y(x,t)$ satisfies the wave equation 
\[
\dfrac{\partial ^{2}y}{\partial t^{2}}=a^{2}\dfrac{\partial ^{2}y}{\partial
x^{2}} 
\]
and has the solution 
\[
y(x,t)=F_{1}(-t-ax)+F_{2}(t-ax) 
\]
Take 
\[
F_{1}(x)=\sin \left( \pi x\right) +\sin \left( 3\pi x\right) 
\]
and 
\[
F_{2}(x)=\sin (5\pi x) 
\]
and define 
\begin{eqnarray*}
y(x,t) &=&F_{1}(-t-3x)+F_{2}(t-3x) \\
&=&\allowbreak \sin \left( \pi \left( -t-3x\right) \right) +\sin \left( 3\pi
\left( -t-3x\right) \right) +\sin \left( 5\pi \left( t-3x\right) \right)
\end{eqnarray*}
The following figure includes graphs of $y(0,x)$ (in red), $y(0.01,x)$ (in
blue), and $y(0.02,x)$ (in green).
\end{example}

\begin{quote}
$\blacktriangleright $ \textsf{Plot 2D + Rectangular} $y(0,x)$
\end{quote}

\begin{itemize}
\item  Select and drag to the frame $y(.01,x)$

\item  Select and drag to the frame $y(.02,x)$\FRAME{dtbpF}{3in}{2.0003in}{%
0pt}{}{}{}{\special{language "Scientific Word";type "MAPLEPLOT";width
3in;height 2.0003in;depth 0pt;display "PICT";function \TEXUX{$-\sin \pi
x-\sin 3\pi x+\sin 5\pi x$};linecolor "red";linestyle 1;linethickness
1;pointstyle "point";function \TEXUX{$\sin \left( \pi \left( -x-.\,03\right)
\right) +\sin \left( 3\pi \left( -x-.\,03\right) \right) +\sin \left( 5\pi
\left( x-.\,03\right) \right) $};linecolor "blue";linestyle 1;linethickness
1;pointstyle "point";function \TEXUX{$\sin \left( \pi \left( -x-.\,06\right)
\right) +\sin \left( 3\pi \left( -x-.0\,6\right) \right) +\sin \left( 5\pi
\left( x-.0\,6\right) \right) $};linecolor "green";linestyle 1;linethickness
1;pointstyle "point";xmin "0";xmax "1";xviewmin "0";xviewmax "1";yviewmin
"-2.612";yviewmax "1.615";rangeset"X";phi 45;theta 45;plottype 4;numpoints
49;axesstyle "normal";xis \TEXUX{x};var1name \TEXUX{$x$};}}
\end{itemize}

The following surface shows the changing shape of the string as time varies
from $0$ to $1$.

\begin{quote}
$\blacktriangleright $ \textsf{Plot 3D + Rectangular }$y(t,x)$
\end{quote}

\begin{itemize}
\item  \textsf{Plot Style = Hidden Line}, $0<x<1$, $0<t<1$\FRAME{dtbpF}{3in}{%
2.0003in}{0pt}{}{}{}{\special{language "Scientific Word";type
"MAPLEPLOT";width 3in;height 2.0003in;depth 0pt;display "PICT";function
\TEXUX{$\sin \left( \pi \left( -t-3x\right) \right) +\sin \left( 3\pi \left(
-t-3x\right) \right) +\sin \left( 5\pi \left( t-3x\right) \right)
$};linecolor "black";linestyle 1;linethickness 1;pointstyle "point";xmin
"0";xmax "1";ymin "0";ymax "1";xviewmin "-0.02";xviewmax "1.02";yviewmin
"-0.02";yviewmax "1.02";zviewmin "-2.641";zviewmax "2.643";rangeset"XY";phi
32;theta -67;plottype 5;num-x-gridlines 25;num-y-gridlines 50;plotstyle
"hidden";axesstyle "frame";plotshading "Z";xis \TEXUX{x};yis
\TEXUX{t};var1name \TEXUX{$x$};var2name \TEXUX{$t$};}}
\end{itemize}

\subsection{\ Related topics}

\begin{itemize}
\item  \hyperref{Ordinary Differential Equations}{}{}{%
DM10-1.tex#Differential equation}

\item  \hyperref{Systems of Ordinary Differential Equations}{}{}{%
DM10-2.tex#Exact solutions}

\item  \hyperref{Numerical Methods for ODEs}{}{}{DM10-3.tex#Solve ODE,
numeric}

\item  \hyperref{Two Dimensional Plots of Functions and Expressions}{}{}{%
DM6-4.tex#Plot expression}

\item  \hyperref{Three Dimensional Plots of Functions and Expressions}{}{}{%
DM6-6.tex#3D Plots}

\item  \hyperref{Exercises and Solutions}{}{}{DM10-5.tex#Exercises}
\end{itemize}

\rule{0.67in}{0.01in}

\hyperref{\FRAME{itbpF}{0.5275in}{0.3399in}{0.0519in}{}{}{index.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";display "PICT";valid_file
"F";width 0.5275in;height 0.3399in;depth 0.0519in;original-width
36.875pt;original-height 22.5625pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'index.wmf';file-properties "XNPEU";}}}{}{}{%
DMIndex.tex#Math Index}\quad {\small Index entries: \hyperref{Differential
equations}{}{}{DMINdex.tex#Differential equations}, \hyperref{Partial
differential equations}{}{}{DMIndex.tex#Partial differential equations}, 
\hyperref{Plotting 2D}{}{}{DMIndex.tex#Plotting 2D}, \hyperref{Plotting 3D}{%
}{}{DMIndex.tex#Plotting 3D}, \hyperref{Solve PDE}{}{}{DMIndex.tex#Solve PDE}%
}

\end{document}
