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\pagestyle{fancy} \lhead{{\sf Homework 11 $|$ Due July 26 (Thursday) }}
\rhead{\thepage} \cfoot{{\sf Math 4340/6340 $|$ Advanced Engineering
    Mathematics $|$ Summer II 2018 (online) $|$ M.~Macauley}}


\begin{document}

$\;$ \vspace{-6mm}

\begin{center}
  \textsc{Topics: PDEs on unbounded domains.}% Laplace \& Fourier transforms.}
\end{center}

%\noindent In the process of solving these problems, you will encounter
%several ODEs, Sturm-Liouville problems, PDEs, and Fourier series, many
%of which you have encountered before. You do not need to re-derive the
%solutions of anything you have previously solved.

\begin{enumerate}

% Logan 2.1.1(a) (modified)
\item Solve the initial value problem for the heat equation, where $-\infty<x<\infty$ and $t>0$:
  \[
  u_t=c^2u_{xx},\qquad u(x,0)=\begin{cases}1 & |x|\leq 1 \\ 0 & |x|>1.\end{cases}
  \]
  Write your solution in terms of the erf function. Sketch the heat distribution for $t=0$ and several larger values of $t$ on the same set of axes. \medskip



% Logan 2.2.2 (modified)
\item Solve the initial value problem for the wave equation, where $-\infty<x<\infty$ and $t>0$:
  \[
  u_{tt}=c^2u_{xx},\qquad u(x,0)=\frac{1}{1+x^2/4},\qquad u_t(x,0)=0.
  \]
  Then repeat for the following IVP:
  \[
  u_{tt}=c^2u_{xx},\qquad u(x,0)=0,\qquad u_t(x,0)=\frac{1}{1+x^2/4}.
  \]
  Describe what each IVP models. Compare and contrast the differences
  in the solutions. For each one, sketch the function $u(x,t)$ for
  $t=0$, and several larger values of $t$ on the same set of
  axes. \medskip



% Logan 2.4.2 (modified)
\item Solve the I/BVP for the heat equation on the semi-infinite domain $x>0$, $t>0$:
  \[
  u_t=c^2u_{xx},\qquad u(0,t)=0,\qquad u(x,0)=1.
  \]
  Write your answer in terms of the erf function. Sketch a graph of
  $u(x,t)$ for $t=0$ and several larger values of $t$. What is the
  long-term behavior? \medskip


%% Logan 2.6.1 or 2.6.6
%\item Add a problem about Laplace transforms. \medskip


%% Logan 2.7.8
%\item Use a Fourier transform to solve the following initial value problem for the free Schr\"odinger equation, where $-\infty<x<\infty$ and $t>0$:
%  \[
%  u_t=iu_{xx},\qquad u(x,0)=e^{-x^2}.
%  \]


\end{enumerate}

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