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\title{Exam 2: Math 330, Fall 2017}
\begin{document} \flushleft
%%%%%%%%%% TAKE-HOME EXAM %%%%%%%%%%
\thispagestyle{empty}
\textbf{\large{Exam 2}} \hfill Name: \underline{\hspace{2.5in}}\\
Math 330\\
Due Tuesday, November 21 at \textbf{8am}
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\textbf{Instructions:}
\vspace{-4pt}
\begin{itemize}[itemsep=-2pt]\begin{small}
\item Solve any 5 of the following 6 problems.
\item You may use your textbook, your notes, the course web site, \emph{Mathematica}, \emph{Wolfram Alpha}, and homework assignments/solutions.
\item \emph{Do not consult other sources, web sites, or people other than the professor.}
\item Type your solutions in \LaTeX. If you use technology to compute something, indicate what you computed. Make sure to explain your solutions clearly, check your work, and proofread.
\item \emph{Make sure you attend to the pledge that at the end of this exam.}
\end{small}\end{itemize}
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%%%%% QUESTIONS %%%%%
\begin{enumerate}[label=\bf{\arabic*.},leftmargin=*,itemsep=10pt]
%%%%% QUESTION 1 %%%%%
\item Suppose that $f(x)$ and $df/dx$ are piecewise smooth. Prove that the Fourier series of $f(x)$ can be differentiated term by term if the Fourier series of $f(x)$ is continuous.
%%%%% QUESTION 2 %%%%%
\item Consider the heat equation for a disk of radius $a$ with constant thermal properties and a circularly symmetric temperature distribution (i.e., temperature $u(r, t)$ does not depend on the angle):
\begin{align*}
&\pd{u}{t} = \frac{k}{r} \pd{}{r}\left( r\pd{u}{r} \right), \quad 0 < r < a, t > 0 \\
&u(0,t) \text{ is bounded} \\
&u(a,t) = 0 \\
&u(r,0) = f(r)
\end{align*}
\begin{enumerate}
\item Separate variables and show that the spatial equation is in Sturm-Liouville form.
What are $p$, $q$, and $\sigma$?
\item Solve for $u(r,t)$ assuming that the eigenfunctions $\phi_n(r)$ are known (and therefore the corresponding $\lambda_n$ are known).
Write down an expression for the coefficients.
\item Prove that the eigenfunctions of this Sturm-Liouville problem are orthogonal.
\item What is the dominant term in $u(r, t)$ for large $t$? What is $\dps \lim_{t \to \infty}u(r,t)$?
\end{enumerate}
%%%%% QUESTION 3 %%%%%
\item Consider the first-order wave equation
\[ \pd{u}{t} + c\pd{u}{x} = 0. \]
\begin{enumerate}
\item Determine a partial difference equation by using a forward difference in time and a centered difference in space.
\item Analyze the stability of this scheme.
\end{enumerate}
%%%%% QUESTION 4 %%%%%
\item Derive a second-order finite difference approximation for $\frac{d^4f}{dx^4}$.\\
\emph{Hint}: Apply the centered difference approximation for the second derivative two times.
%%%%% QUESTION 5 %%%%%
\item Solve the following nonhomogeneous problem:
\begin{align*}
&\pd{u}{t} = k \pd{^2u}{x^2} + e^{-2t}\sin(4\pi x) \\
&u(0,t) = 0 \\
&u(1,t) = 0 \\
&u(x,0) = x - x^2
\end{align*}
You may assume $2 \ne k4^2\pi^2$.
\emph{Hint}: Use the method of eigenfunction expansion (Section 3.4).
%%%%% QUESTION 6 %%%%%
\item Consider the eigenvalue problem
\begin{align*}
&\frac{d^2y}{dx^2} + \lambda y = 0, \qquad 0 < x < 3 \\
&y(0) = 0 \\
&y(3) + y'(3) = 0
\end{align*}
Based on what you've learned this semester, say as much as you can about the eigenvalues and eigenfunctions for this problem.
\end{enumerate}
\vfill
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\textbf{St.\ Olaf Honor Pledge}:
I pledge my honor that on this examination I have neither given
nor received assistance not explicitly approved by the professor
and that I have seen no dishonest work.
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Signed: \underline{\hspace{2.5in}}
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\tikz{\draw (0,0) rectangle (9pt,9pt);} I have intentionally not signed the pledge. (Check the box if appropriate.)
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