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\pagestyle{fancy} \lhead{{\sf Homework 9 $|$ Due July 24 (Monday) }}
\rhead{\thepage} \cfoot{{\sf Math 4340/6340 $|$ Advanced Engineering
    Mathematics $|$ Summer II 2017 (online) $|$ M.~Macauley}}


\begin{document}

$\;$ \vspace{-6mm}

\begin{center}
  \textsc{Topics: Diffusion and the heat equation}
\end{center}


%\noindent Read the following:
%\begin{itemize}
%\item Chapters 1 \& 2 of \emph{Visual Group Theory} (VGT). \vspace{-2mm}
%\item VGT Exercises 1.1--1.4, 1.8--1.12, 2.13--2.17. \vspace{-2mm}
%\item The article \emph{Group Think} by Steven Strogatz, which
%  appeared in the NY Times in 2010. \vspace{-2mm}
%\item Francis Su's essay on good mathematical writing.
%\end{itemize} 

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

\begin{enumerate}

\item We will solve the following B/IVP for $u(x,t)$, defined for $0\leq
  x\leq 1$ and $t\geq 0$:
  \[
  u_t=c^2u_{xx},\qquad u(0,t)=u(1,t)=0,\qquad u(x,0)=5\sin
  \pi x+3\sin2\pi x.
  \]
  This PDE is called the \emph{heat equation}. It is homogeneous.
  \begin{enumerate}
  \item Briefly describe, and sketch, a physical situation which this
    models. Be sure to explain the effect of both boundary conditions
    and the initial condition.
  \item Assume that $u(x,t)=f(x)g(t)$. Find $u_t$ and $u_{xx}$. Also,
    determine the boundary conditions for $f(x)$ from the boundary
    conditions for $u(x,t)$.
  \item Plug $u=fg$ back into the PDE and divide both sides by $c^2
    fg$, i.e., ``separate variables.''  Briefly justify why this
    quantity must be a constant. Call this constant $-\lambda$. Write
    down two ODEs: One for $g(t)$, and a Sturm-Lioville problem for
    $f(x)$.
  \item Determine $f(x)$, $\lambda$, and $g(t)$.
  \item Using your solution to Part~(d) and linearity (in physical
    terms, the principle of superposition), find the general solution
    to this PDE.
  \item Solve the resulting initial value problem. That is, find the
    particular solution $u(x,t)$ that satisfies the boundary and
    initial conditions.
  \item What is the steady-state solution,
    $\displaystyle\lim_{t\to\infty} u(x,t)$? Explain the physical
    interpretation. \medskip
  \end{enumerate}


%\item Consider the following partial differential equation:
%  \[
%  u_t=c^2u_{xx}\,,\qquad u(0,t)=u(1,t)=0,\qquad u(x,0)=x(1-x)\,.
%  \]
%  The only difference between this and the previous equation is the
%  \emph{initial condition}. Carry out each step (a)--(g) from the previous
%  problem, unless it is identical, in which case you should say ``same
%  as the previous problem'' instead.


%\item Consider the following partial differential equation:
%  \[
%  u_t=c^2u_{xx}\,,\qquad u_x(0,t)=u_x(1,t)=0,\qquad u(x,0)=x(1-x)\,.
%  \]
%  The only difference between this and the previous equation are the
%  \emph{boundary conditions}. Carry out each step (a)--(g) from the
%  previous problem, unless it is identical, in which case you should
%  say ``same as the previous problem'' instead.
%
%

\item Solve the following PDE:
  \[
  u_t=c^2u_{xx}\,,\qquad u_x(0,t)=u(\pi,t)=0,\qquad
  u(x,0)=40+6\cos\tfrac{5x}{2}\,.
  \]
  Make sure you describe and sketch the physical situation which this
 models, and determine the steady-state solution. Also, make it clear
  what Sturm-Liouville problem arises. \bigskip


\item Consider the following PDE with inhomogeneous boundary conditions:
  \[
  u_t=c^2u_{xx}\,,\qquad u_x(0,t)=0\,,\quad u(1,t)=32\,,\qquad
  u(x,0)=72+6\cos\tfrac{5x}{2}\,.
  \]
  \begin{enumerate}
    \item Make the substitution $v(x,t)=u(x,t)-32$, and re-write
      the PDE above, including the boundary and initial conditions, in
      terms of $v$ instead of $u$.
    \item Briefly describe, and sketch, a physical situation which
      this models. Be sure to explain the effect of both boundary
      conditions and the initial condition.
    \item Solve the PDE for $u(x,t)$. You should use the
      results of the previous problem.
    \item What is the steady-state solution? Give a physical
      interpretation of this. \bigskip
  \end{enumerate}


\item Consider the following PDE, subject to \emph{periodic boundary
  conditions}:
  \[
  u_t=c^2u_{\theta\theta},\qquad u(\theta+2\pi,t)=u(\theta,t),\qquad
  u(\theta,0)=2+4\sin 3\theta-\cos 5\theta.
  \]
%  \[
%  u_t=c^2u_{\theta\theta},\quad u(\theta+2\pi,t)=u(\theta,t),\quad
%  u_\theta(\theta+2\pi,t)=u_\theta(\theta,t),\quad u(\theta,0)=2+4\sin
%  3\theta-\cos 5\theta.
%  \]
  Briefly describe, and sketch, a physical situation which this
  models. 
  %[\emph{Hint}: $u(\theta,t)$ is a function of an \emph{angle} $\theta$, and time $t$.] 
 Solve this PDE via steps (a)--(g) from the first problem. You do not need to repeat any steps that are
  identical so something you've done before, as long as you point that
  out. However, you have \emph{not} seen this type of Sturm-Liouville
  problem before. \bigskip

  %The boundary conditions are not specified, but rather are forced to be periodic. 


\item Let $u(x,t)$ be the temperature of a bar of length 10, at
  position $x$ and time $t$ (in hours). Suppose that the left endpoint
  of the bar is not insulated, but the right endpoint is fully
  insulated, and the bar is sitting in a $70^\circ$ room. Moreover,
  suppose that initially, the temperature increases linearly from
  $70^\circ$ at the left endpoint to $80^\circ$ at the other
  end. Finally, suppose the interior of the bar is poorly insulated,
  so heat can escape.
  \begin{enumerate}
  \item Suppose that heat escapes at a constant rate of $1^\circ$ per
    hour. Write (but do not solve) an initial/boundary value problem
    for $u(x,t)$ that could model this situation.
  \item A more realistic situation would be for heat to escape not at
    a constant rate, but at a rate proportional to the
    \emph{difference} between the temperature of the bar and the
    ambient temperature of the room. Write an initial/boundary value
    problem for $u(x,t)$ that could model this situation. What is the
    steady-state solution and why? \bigskip
  \end{enumerate}



\item Consider the following B/IVP for the heat equation:
  \[
  u_t=c^2u_{xx},\qquad u(0,t)=0,\quad
    u_x(1,t)=\gamma\,u(1,t),\qquad
    u(x,0)=h(x)\,,
  \]
  where $\gamma$ is constant and $h(x)$ is an arbitrary function.
  %satisfies $h(0)=0$ and $h'(1)=\gamma\,h(1)$.
  \begin{enumerate}
  \item Describe a physical situation that this models. Be sure to
    describe the impact of the initial condition, \emph{both} boundary
    conditions, and the constant $\gamma$. [\emph{Hint}: First consider
    the special case when $\gamma=0$. How does this compare to when
    $\gamma\approx 0$?]
  \item What is the steady-state solution, and why? (Use your physical
    intuition.)
  \item Assume that $u(x,t)=f(x)g(t)$, and plug this back in to get a familiar
    ODE for $g(t)$, and a Sturm-Liouville problem for $f(x)$.
  \item Solve the Sturm-Liouville problem. You won't get a closed form
    for the eigenvalues, but you can write them as
    $\lambda_n=\omega_n^2$, where $\{\omega_n\mid n=1,2,\dots\}$ are
    the non-negative solutions to a simple algebraic equation. Write
    down this equation and graph it, clearly marking the first few
    solutions on the graph.
  \item Find the general solution to the original boundary value problem.
%  \item Find the general solution of this PDE, which will have the form
%    \[
%    u(x,t)=\sum_{n=1}^\infty b_nf_n(x)g_n(t),
%    \]
%    where $f_n(x)$ is the eigenfunction associated with eigenvalue
%    $\lambda_n$.
  \item The particular solution satisfying the initial and boundary
    conditions can be found by plugging in $t=0$ and using the initial
    condition to determine the $b_n$. You do not need to do this, but
    explain the process in a few steps in enough detail that an
    educated reader (e.g., one of your classmates) could figure out
    how to do this from your explanation. Your answer should involve
    the orthogonality of the eigenfunctions and the associated inner
    product.
  \end{enumerate}



\end{enumerate}


\end{document}
