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{ \large \textbf{\textsc{Math 5587 -- Homework 5 (Due Thursday Oct 13)}}}
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In any question that asks for a series representation for the solution of a PDE, you do not need to give formulas for the coefficients $A_n$ and $B_n$. 
\begin{enumerate}
\item Let $\phi(x) = x^2$ for $0 \leq x \leq 1=\ell$. 
\begin{enumerate}
\item Calculate the Fourier sine series for $\phi$.
\item Calculate the Fourier cosine series for $\phi$.
\end{enumerate}
\item  Find the Fourier cosine series of the function $\sin(x)$ in the interval $(0,\pi)$. Use it to compute the sums
\[ \sum_{n=1}^\infty \frac{1}{4n^2-1} \ \ \text{ and } \ \ \sum_{n=1}^\infty \frac{(-1)^n}{4n^2 - 1}.\]
\item \begin{enumerate}
\item Use the Fourier expansion to explain why the note produced by a violin string rises by one octave when the string is clamped exactly at its midpoint. [Each increase by an octave corresponds to a doubling of the frequency.]
\item Explain why the pitch of the note rises when the string is tightened.
\end{enumerate}
\item A quantum-mechanical particle on the line with an infinite potential outside the interval $(0,\ell)$ (``particle in a box'') is described by Schr\"odinger's equation 
\[u_t = iu_{xx} \ \ \text{ on } (0,\ell),\]
where $i=\sqrt{-1}$, with Dirichlet conditions $u(0,t)=u(\ell,t) = 0$ at the ends. Use separation of variables to find a representation formula for $u(x,t)$ as a series.
\item Consider waves in a resistant medium, which satisfy the equation
\[\text{(W)} \ \left\{ \begin{aligned}
u_{tt} - c^2u_{xx} + ru_t &= 0,&&   0 <  x < \ell, \ t>0\\
u(x,0) &=\phi(x),&&  0 < x < \ell\\
u_t(x,0) &=\psi(x),&&  0 < x < \ell\\
u(0,t)=u(\ell,t) &=0&&  t>0,
\end{aligned}\right.\]
where $r$ is a constant, $0 < r < 2\pi c/\ell$. Use separation of variables to write down the series expansion of the solution.
\item Consider the wave equation $u_{tt} = c^2u_{xx}$ for $0 < x <\ell$ with boundary conditions $u_x(0,t) = u(\ell,t) = 0$. 
\begin{enumerate}
\item Show that the eigenfunctions are $X_n(x) = \cos\left((n+\tfrac{1}{2})\pi x/\ell\right)$.
\item Write down the series expansion for a solution $u(x,t)$.
\end{enumerate}
\item Consider diffusion inside an enclosed circular tube. Let its length (circumference) be $2\ell$. Let $x$ denote the arc length parameter where $-\ell \leq x \leq \ell$. Then the concentration of the diffusing substance satisfies
\[\left\{ \begin{aligned}
u_{t} - ku_{xx} &= 0,&&   -\ell <  x < \ell, \ t>0\\
u(-\ell,t)&=u(\ell,t)&&  t>0\\
u_x(-\ell,t)&=u_x(\ell,t)&&  t>0.
\end{aligned}\right.\]
These are called \emph{periodic boundary conditions}.
\begin{enumerate}
\item Show that the eigenvalues are $\lambda = (n\pi/\ell)^2$ for $n=0,1,2,3,\dots$.
\item Show that the concentration is
\[u(x,t) = \frac{1}{2}A_0 + \sum_{n=1}^\infty \left( A_n \cos\left(\frac{\pi n x}{\ell}\right) + B_n \sin\left(\frac{\pi n x}{\ell}\right) \right) \exp\left(\frac{-n^2\pi^2kt}{\ell^2} \right).\]
\end{enumerate}
\end{enumerate}


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