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\pagestyle{fancy} \lhead{{\sf Homework 10 $|$ Due July 27 (Thursday) }}
\rhead{\thepage} \cfoot{{\sf Math 4340/6340 $|$ Advanced Engineering
    Mathematics $|$ Summer II 2017 (online) $|$ M.~Macauley}}


\begin{document}

$\;$ \vspace{-6mm}

\begin{center}
  \textsc{Topics: The transport, wave, and Schr\"odinger 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, 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}

\item Recall from ODEs that $y'=-\lambda y$ models exponential
  decay. The equation
  \[
  u_t+cu_x=-\lambda u
  \]
  is often called the \emph{convection-decay} equation, and it could
  model the concentration of a radioactive chemical dissolved in water
  that flows at speed $c$. Find the general solution to this PDE by
  making the following change of variables
  \[
  \xi=x-ct,\qquad \tau=t.
  \]


\item Consider the following B/IVP for the wave equation, defined for
  $0\leq x\leq 1$ and $t\geq 0$:
  \[
  u_{tt}=c^2u_{xx},\qquad
  u(0,t)=u(1,t)=0,\qquad u(x,0)=x(1-x),\quad u_t(x,0)=0.
  \]
  \begin{enumerate}
  \item Briefly describe, and sketch, a physical situation which this
    models. Be sure to explain the effect of both boundary conditions
    and both initial conditions.
  \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 $u(x,t)$, and one initial condition for
    $g(t)$. (The three BC/ICs equal to zero will give you these.)
  \item Plug $u=fg$ back into the PDE and divide both sides by
    $c^2fg$, i.e., ``separate variables.''  Briefly justify why this
    quantity must be a constant. Call this constant $-\lambda$. Write
    down two ODEs: One for $g(t)$ with one initial condition, and a
    Sturm-Lioville problem for $f(x)$.
  \item Determine $f(x)$, $\lambda$, and $g(t)$.
  \item Using your solution to Part~(d) and the principle of
    superposition, find the general solution to this PDE.
  \item Use the last initial condition, $u(x,0)=x(1-x)$, to find the
    particular solution that satisfies the initial and boundary
    conditions.
  \item What is the long-term behavior of this system? Give both a
    mathematical and physical justification.
  \end{enumerate}



\item We will solve for the function $u(x,t)$, defined for $0\leq
  x\leq 1$ and $t\geq 0$, which satisfies the following conditions:
  \[
  u_{tt}=c^2u_{xx},\qquad
  u(0,t)=u(1,t)=0,\qquad u(x,0)=0,\quad u_t(x,0)=x(1-x).
  \]
  The only difference between this and the previous equation are the
  initial conditions. Carry out each step (a)--(g) from the previous
  problem, unless it is idential (many of them will be), in which case
  you should say ``same as the previous problem'' instead.



\item Let $u(x,t)$ be defined for $0\leq x\leq 1$ and $t>0$, and
  consider the following PDE
  \[
  u_{tt}+2\beta u_t=u_{xx},\qquad u(0,t)=u(1,t)=0,\quad
  u(x,0)=x(1-x),\quad u_t(x,0)=1.
  \]
  where $0<\beta<1$ is a constant. This is the wave equation, where
  the transverse vibrations take place in a medium that imparts a
  resistance proportional to the instantaneous velocity.
  \begin{enumerate}
  \item Describe and sketch this situation at $t=0$. 
  \item Assume that there is a solution of the form
    $u(x,t)=f(x)g(t)$. Plug this back into the PDE and get an ODE for
    $g(t)$ and a Sturm-Liouville problem for $f(x)$.
  \item The Sturm-Liouville problem should be familiar: $f''=-\lambda
    f$, $f(0)=f(1)=0$, and we've seen that the eigenvalues are
    $\lambda_n=n^2$ for $n=1,2,\dots$ and the eigenfunctions are
    $f_n(x)=b_n\sin(n\pi x)$. The equation for $g(t)$ should be
    $g''+2\beta g'+n^2g=0$. Solve this ODE for $g(t)$. 
  \item Write down the general solution to this PDE. 
  \item Use the initial conditions to find the particular solution
    solving the boundary and initial conditions. (Feel free to use
    WolframAlpha to compute any derivatives you may need.)
  \item What is the long-term behavior of this system? Give both a
    mathematical and physical justification. \bigskip
  \end{enumerate}



  % Olver Exercise 9.5.20, 23.
\item In quantum mechanics, the function $\psi(x,t)$ represents the
  wave function of a quantum system. One can interpret this as a
  probability density on th epossible quantum states.
  \begin{enumerate}
    \item Solve the following initial boundary value problem to
      Schr\"odiner's equation:
      \[
      i\hbar\psi_t=-\psi_{xx},\qquad \psi(t,0)=\psi(t,1),\qquad \psi(0,x)=1. 
      \]
      The Dirchlet boundary conditions represent the case when the
      particle cannot escape from the interval $[0,1]$. (Note the
      difference from what this represents in the heat equation!)
    \item Solve the same initial boundary value problem, but with
      Neumann boundary conditions:
      \[
      i\hbar\psi_t=-\psi_{xx},\qquad \psi_x(t,0)=\psi_x(t,1),\qquad \psi(0,x)=1. 
      \]
  \end{enumerate}


\end{enumerate}

\end{document}
