% ch03slidesa.tex

\begin{slide}
\begin{center}\large\bf
Partial Differential Equations in Physics and Engineering
\end{center}

\begin{itemize}  
\item {Superposition}
\item {Vibration of Strings and the Wave Equation}
\item {The Method of Separation of Variables}
\item {Solution of the $1$-dimensional Wave Equation}
\item {A Stretched String with Fixed Edges}
\item {D'Alembert's Method}
\item {The $1$-Dimensional Heat Equation}
\item {Steady-State Heat Problem}
\item {Heat Conduction in a Bar: Fourier's Problem}
\item {Two-Dimensional Wave Equation: Membrane}
\item {Two-Dimensional Heat Equation: Rectangular Plate}
\item {Laplace's Equation: Dirichlet Problem}
\item {Poisson's Equation and Eigenfunction Expansions}
\end{itemize}

\end{slide}

\begin{slide}
\header{Superposition}

\bigskip
\begin{Theorem}[Section 3.1]
If $U_1$, $U_2$ are two solutions of  a linear homogeneous partial differential equation,
then any linear combination $u=c_1 u_1 + c_2 u_2$, where $c_1$, $c_2$
are constants, is also a solution. 


\bigskip

If in addition $u_1$ and $u_2$ satisfy a linear homogeneous boundary
condition, then so will $u=c_1 u_1 + c_2 u_2$.
\end{Theorem}

\header{}

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{\bf Example}. Consider Laplace's equation 
$\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0$.

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Let $u_1= x+y$, $u_2=x^2-y^2$, which are two harmonic functions.
They are solutions of Laplace's equation satisfying the boundary
condition $u(0,0)=0$. Then $u=2(x+y)+3(x^2-y^2)$ is also a solution of
Laplace's equation satisfying the same boundary condition $u(0,0)=0$.

\end{slide}
\begin{slide}
\header{Vibration of Strings and the Wave Equation}

\begin{itemize}
\item Free vibrations
\item Forced vibrations
\item Rectangular membrane
\end{itemize}

\bigskip

\end{slide}

\begin{slide}
\header{The Method of Separation of Variables}

\bigskip

\end{slide}

\begin{slide}
\header{Solution of the $1$-dimensional Wave Equation}

\bigskip

\end{slide}

\begin{slide}
\header{A Stretched String with Fixed Edges}

\bigskip

\end{slide}

\begin{slide}
\header{D'Alembert's Method}

\bigskip

\end{slide}

\begin{slide}
\header{The $1$-Dimensional Heat Equation}

\bigskip

\end{slide}

\begin{slide}
\header{Steady-State Heat Problem}

\bigskip

\end{slide}

\begin{slide}
\header{Heat Conduction in a Bar: Fourier's Problem}

\bigskip

\end{slide}

\begin{slide}
\header{Two-Dimensional Wave Equation: Membrane}

\bigskip

\end{slide}

\begin{slide}
\header{Two-Dimensional Heat Equation: Rectangular Plate}

\bigskip

\end{slide}

\begin{slide}
\header{Laplace's Equation: Dirichlet Problem}

\bigskip

\end{slide}

\begin{slide}
\header{Poisson's Equation and Eigenfunction Expansions}

\bigskip

\end{slide}
