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\title{Exam 1: Math 330, Fall 2017}
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%%%%%%%%%% TAKE-HOME EXAM %%%%%%%%%%
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\textbf{\large{Exam 1}} \hfill Name: \underline{\hspace{2.5in}}\\
Math 330\\
Due Tuesday, October 10 at \textbf{8am}
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\textbf{Instructions:}
\vspace{-4pt}
\begin{itemize}[itemsep=-2pt]\begin{small}
\item Solve any 5 of the following 6 problems.
\item You may use your textbook, your notes, the course web site, \emph{Mathematica}, \emph{Wolfram Alpha}, and homework assignments/solutions.
\item \emph{Do not consult other sources, web sites, or people other than the professor.}
\item Type your solutions in \LaTeX. If you use technology to compute something, indicate what you computed. Make sure to explain your solutions clearly, check your work, and proofread.
\item \emph{Make sure you attend to the pledge that at the end of this exam.}
\end{small}\end{itemize}
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%%%%% QUESTIONS %%%%%
\begin{enumerate}[label=\bf{\arabic*.},leftmargin=*,itemsep=10pt]
\item Consider the following simple 1-D model for traffic flow in the interval from $x=a$ to $x=b$.
Let $x$ be the position on a number line and $t$ be the time.
We define two functions: $\rho(x,t)$ is the density of cars (cars per unit length) at position $x$ and time $t$ and $v(x,t)$ is the velocity of the cars at position $x$ and time $t$.
Use the convention that a positive velocity implies cars are moving in the direction from $a$ to $b$.
Assume cars can only enter/exit this region at the endpoints $x=a$ and $x=b$.
Write a conservation equation and derive the corresponding partial differential equation.
\item Suppose $u(x,t)$ is the temperature of a rod for $00$, subject to
\begin{align*}
& \text{PDE:} & & \pd{u}{t} = \pd{^2u}{x^2} + x - \beta \\
& \text{Boundary conditions:} & & \pd{u}{x}(0,t) = \pd{u}{x}(L,t) = 0 \\
& \text{Initial condition:} & & u(x,0) = f(x)
\end{align*}
\begin{enumerate}
\item Calculate the value of the constant $\beta$ such that an equilibrium temperature exists.
Briefly explain why this value of $\beta$ makes sense based on the physical context.
\item Using the $\beta$ value found in part (a), determine the equilibrium temperature in terms of the initial condition.
\end{enumerate}
\item Consider a circular annulus defined on $a0,x\in(0,L),\\
&u(t,0)=0, u(t,L)=0,\\
&u(0,x)=u_0(x),\qquad x\in(0,L).
\end{align*}
Recall in Exercise 1.5.2 you considered convection as another process that introduces flux. The term $4u_x$ is the convective term in this problem. Here, you will find the solution of the equation with the following steps:
\begin{enumerate}
\item Use separation of variables method to show that if $u(t, x) = G (t)\phi (x)$ is a solution, then for some constant $\lambda$, $G$ and $\phi$ satisfy
\[G'=\lambda G,\qquad \phi''+4\phi'+8\phi=\lambda\phi.\]
\item Find the eigenvalues and eigenfunctions of
\[\phi''+4\phi'+8\phi=\lambda\phi,\qquad \phi(0)=\phi(L)=0.\]
(You will need to consider three cases for $\lambda$ separately.)
\item Find the solution of the equation in a series form. Make sure that you write your coefficients in terms of the initial condition.
\end{enumerate}
\item Find $u(x,t)$ that solves the following:
\begin{align*}
& \text{PDE:} & & \pd{^2u}{t^2} = \pd{^2u}{x^2} - 4u, \quad 00 & \\
& \text{Boundary conditions:} & & \pd{u}{x}(0,t) = \pd{u}{x}(1,t)=0 & \\
& \text{Initial condition:} & & u(x,0) = 0, \quad \pd{u}{t}(x,0) = 5 + 3\cos(2\pi x)
\end{align*}
\item Solve Laplace's equation inside a rectangle $0 < x < L$, $0 < y < H$, with the boundary conditions
\[ \pd{u}{x}(0,y)=0, \qquad \pd{u}{x}(L,y) = 0, \qquad u(x,0) = 0, \qquad \pd{u}{y}(x,H) = \begin{cases} 0 & x > L/2 \\ 1 & x < L/2 \end{cases}. \]
\end{enumerate}
\vfill
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\textbf{St.\ Olaf Honor Pledge}:
I pledge my honor that on this examination I have neither given
nor received assistance not explicitly approved by the professor
and that I have seen no dishonest work.
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Signed: \underline{\hspace{2.5in}}
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\tikz{\draw (0,0) rectangle (9pt,9pt);} I have intentionally not signed the pledge. (Check the box if appropriate.)
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