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\begin{document}

\title{Math 142-2, Homework 8}
\author{Your name here}
\date{\today}
\maketitle

\section*{Problem 77.1}
\textbf{If $u = u_{\max}(1 - \rho/\rho_{\max})$, then what is the velocity of a
  traffic shock separating densities $\rho_0$ and $\rho_1$? (Simplify the
  expression as much as possible.) Show that the shock velocity is the average
  of the density wave velocities associated with $\rho_0$ and $\rho_1$.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\section*{Problem 77.2}
\textbf{If $u = u_{\max}(1 - \rho^2/\rho_{\max}^2)$, then what is the velocity
  of a traffic shock separating densities $\rho_0$ and $\rho_1$? (Simplify the
  expression as much as possible.) Show that the shock velocity is not the
  average of the density wave velocities associated with $\rho_0$ and $\rho_1$.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\section*{Problem 77.3}
\textbf{A weak shock is a shock in which the shock strength (the difference in
  densities) is small. For a weak shock, show that the shock velocity is
  approximately the average of the density wave velocities associated with the
  two densities. [Hint: Use Taylor series methods.]}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\section*{Problem 78.2}
\textbf{Assume that $u = u_{\max}(1 - \rho/\rho_{\max})$ and that the initial
  traffic density is
$$
\rho(x,0) = \left\{
\begin{array}{ll}
\frac{\rho_{\max}}{5} & x<0 \\
\frac{3\rho_{\max}}{5} & x>0.
\end{array}
\right.
$$}

\textbf{(a) Sketch the initial density.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(b) Determine and sketch the density at later times.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(c) Determine the path of a car (in space-time) which starts at $x =
  -x_0$ (behind $x = 0$).}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(d) Determine the path of a car (in space-time) which starts at $x =
  x_0$ (ahead of $x = 0$).}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\section*{Problem 78.3}
\textbf{Assume that $u = u_{\max}(1 - \rho/\rho_{\max})$ and at $t = 0$, the
  traffic density is
$$
\rho(x,0) = \left\{
\begin{array}{ll}
\frac{\rho_{\max}}{3} & x<0 \\
\frac{2\rho_{\max}}{3} & x>0.
\end{array}
\right.
$$
Why does the density not change in time?}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\section*{Problem 78.4}
\textbf{Referring to the problem in Sec. 78, show algebraically that the value
  of the shock velocity is between the velocities of the two density waves.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\section*{Problem 79.2}
\textbf{Suppose that
$$
\rho(x,0) = \left\{
\begin{array}{ll}
\rho_0 & x>0 \\
0 & x<0
\end{array}
\right.
$$
Determine the velocity of the shock. Briefly give a physical
explanation of the result.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\section*{Problem 79.3}
\textbf{The initial traffic density on a road is
$$
\rho(x,0) = \left\{
\begin{array}{ll}
0 & x \le 0 \\
\frac{\rho_{\max} x}{L} & 0 < x < L \\
\rho_{\max} & x \ge L
\end{array}
\right.
$$
Assume that $u = u_{\max}(1 - \rho/\rho_{\max})$.}

\textbf{(a) Sketch the initial density.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(b) Show that all characteristics from the interval $0 < x < L$ (and $t
  = 0$) intersect at the point $x = L/2$, $t = L/(2u_{\max})$.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(c) A traffic shock will form at this point. Find its subsequent
  motion.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(d) Sketch the $x-t$ plane, showing the shock and the characteristics
  necessary to determine $\rho(x, t)$.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(e) Sketch $\rho(x, t)$ before and after the shock.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(f) Describe briefly how the individual automobiles behave (do not
  determine their paths mathematically).}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\section*{Problem 80.1}
\textbf{Assume that $u = u_{\max}(1 - \rho/\rho_{\max})$.}

\textbf{(a) Show that the time of intersection of neighboring characteristics
  (corresponding to the collision of two observers initially at $x_1$ and $x_2$
  moving with constant density $\rho_1$ and $\rho_2$) is
$$ t = \frac{\rho_{\max}}{2 u_{\max} \frac{\Delta\rho}{\Delta x}}, $$
where $\dx = x_2 - x_1$ and $\Delta \rho = \rho_2 - \rho_1$.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(b) Extend this result to the limit as $x_2 \to x_1$ to determine when a
  shock will form from the characteristics that originate in the vicinity of
  some location $x_1$.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(c) If at $t = 0$,
$$ \rho(x,0) = \rho_{\max} \exp\gp{-\frac{x^2}{L^2}}. $$}

\textbf{\qquad(1) Sketch the initial density.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{\qquad(2) Determine the time of the first shock.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{\qquad(3) Where does this shock first occur?}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\section*{Problem 82.1}
\textbf{Assume that $u = u_{\max}(1 - \rho/\rho_{\max})$.  If the initial
  density is
$$
\rho(x,0) = \left\{
\begin{array}{ll}
\rho_1 & x < 0 \\
\rho_2 & a > x > 0 \\
\rho_3 & x > a
\end{array}
\right.
$$
with $0 < \rho_1 < \rho_2 < \rho_3 < \rho_{\max}$, then determine the density at
later times. [Hint: See exercise 77.1. Calculate the shock between $\rho_1$ and
  $\rho_2$. Show that this shock moves faster than the shock between $\rho_2$
  and $\rho_3$. What happens after these two shocks meet?]}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\section*{Problem 82.2}
\textbf{Assume that $u = u_{\max}(1 - \rho/\rho_{\max})$ and that the initial
  traffic density is
$$
\rho(x,0) = \left\{
\begin{array}{ll}
\rho_1 & |x| > a \\
\rho_0 & |x| < a \\
\end{array}
\right.
$$
where $\rho_1 > \rho_0$. Determine the density at later times.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\section*{Additional Problem 1}
\textbf{Let $N(a,t)$ be the number of individuals with age $a$ at time $t$,
  which is valid for $a \ge -g$ and $t \ge 0$, where $g$ is the gestation time
  (time between conception and birth) is $g$.  The conception rate $b(a)$ and
  death rate $c(a)$ are functions of age.  The initial age distribution is
  $f(a)$.  In the previous homework assignment, we worked out a continuous
  model for this, which took the form
  \begin{align*}
    \px{N}{t} + \px{N}{a} &= -N c & & a > -g, t > 0 \\
    N(-g,t) &= \int_{-g}^{\infty} N(a,t) b(a) \,da & & t > 0 \\
    N(a,0) &= f(a) & & a > -g
  \end{align*}}

\textbf{(a) What are the characteristics of this PDE?}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(b) Let $M_0$ be the number of individuals who are the same age
  $a_0$ at time $t_0$.  After some time $T = t_1 - t_0$ has passed, the
  individuals in this group are now of age $a_1 = a_0 + T$, and there are now
  $M_1$ of them still alive.  Show that $M_1 = M_0 e^{-r T}$, where $r$ is the
  average death rate for individuals of the age range $[a_0, a_1]$.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(c) Show that with this model if births equal deaths
$$ \int_{-g}^\infty N(a,t) b(a) \,da = \int_{-g}^\infty N(a,t) c(a) \,da $$ then
  the total population $T(t)$ does not change.  You may assume that $N(a,t) \to
  0$ as $a \to \infty$.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\end{document}
