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%==================== END PREAMBLE ==============
\begin{document}
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\vspace*{-50pt}
{\bf  Math 241, Spring 2015\hfill Jerry L. Kazdan}
\medskip

\begin{center}
{\large\bf Problem Set 3}
\end{center}

{\sc Due:} Thurs. Feb. 5 in class. [Late papers will be accepted until 1:00 PM Friday.]
\medskip

{\bf This week}. Please re-read all of Chapter 2 and read Chapter 4.4
in the Haberman text. 

\bigskip

\begin{problems}

\item p. 51 \#2.3.3 (b, d)

\item p. 52 \#2.3.4

\item p. 65 \# 2.4.1 (a, b)

\item p. 66 \# 2.4.2

\item p. 66 \# 2.4.6

\item p. 81 \# 2.5.1 (a,b)

\item p. 81 \# 2.5.2

\item p. 82 \# 2.5.3

\item p. 82 \# 2.5.5 (b)

\item p. 82 \# 2.5.7 (a)

\item This is now Bonus \B-3.


\end{problems}

%\comments{
\vskip 20pt

\begin{center}{\large \bf Bonus Problems}
			
[Please give this directly to Professor Kazdan]
\end{center}

\begin{problems}

\item [B-1] p. 82 \# 2.5.4


\item [B-2] Let $u(x,t)$ be a solution of the heat equation $u_t=u_{xx}$ for
$0\le x \le \pi$, Say the initial and boundary conditions are
$$
u(x,0) = f(x), \qquad u(0,t)=0,\qquad u(\pi,t)=0
$$
and assume that $\abs{f(x)}\le M$ for $0\le x \le \pi$.  Then the
solution this is
\begin{equation*}
u(x,t) = \sum_{n=1}^\infty b_n\sin nx \ e^{-n^2t},
\end{equation*}
where the $b_n=\frac{2}{\pi}\int_0^\pi f(x)\sin nx\,dx$ 
are the Fourier coefficients of $f$.  

Say we approximate this solution only by using the first $(n=1)$ term.
The question is to estimate the error in doing so.
\begin{parts}
\item Show that 
$$
\int_0^\pi \abs{\sin nx}\,dx = 2
$$ 
and use this to show that
$$
\abs{b_n}\le \frac{4M}{\pi}.
$$
\item Estimate the terms beginning with $n=2$ by showing that
$$
\left|\sum_{n=2}^\infty b_n\sin nx \ e^{-n^2t}\right| \le
\frac{4M}{\pi}\sum_{n=2}^\infty e^{-n^2t}.
$$
\item  Assume the first  term is roughly $Me^{-t}$.  Show that the
ratio between the above error and the first term is roughly
$$
\frac{4}{\pi}\sum_{n=2}^\infty e^{(1-n^2)t}
$$
and give a numerical estimate of this at $t=1$.
\end{parts}

\item [B-3] p. 82 \# 2.5.9 (b) 

[See \url{http://www.math.upenn.edu/~kazdan/241S15/hw/hw3-2.5.9b.pdf} ]


 \end{problems}
%}

\medskip
[Last revised: \today]
\end{document}

