\documentclass[leqno]{article}
\usepackage{hyperref}
\usepackage{graphicx}
\input{linalg.sty}
\parskip=2pt
\def\Pi{\pi}
\begin{document}
\thispagestyle{empty}
 {\bf Name~}\underline{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}\hfill
 %{Class Time~}\underline{~~~~~~~~}
 \newline


\begin{center}
\bf Math 3150 Midterm 1 \\
{\small\bf Sample Exam}
\end{center}


\Prob{Problem 1}{(Heat Conduction in a Rod, Ends at Different Temperatures)}
Consider the heat conduction problem in a laterally insulated bar of length 1 with one end at zero Celsius and the other end at 100 Celsius. The initial temperature along the bar is given by function $f(x)$.

$$\renewcommand{\arraystretch}{1}
 \left\{ \begin{array}{lclll}
 u_{t} &=& c^2 u_{xx}, & 0<x<1, & t>0, \\
 u(0,t) &=& 0, & & t>0,\\
 u(1,t)&=& 100, && t>0,\\
 u(x,0) &=& f(x), &0 < x < 1.
 \end{array}\right.
$$

(a) [25\%] Find the steady-state temperature $u_1(x)$.

(b) [50\%] Solve the bar problem with zero Celsius temperatures at both ends, but $f(x)$
     replaced by $f(x) - u_1(x)$. Call the answer $u_2(x,t)$. Besides the series answer for $u_2(x,t)$,
     which is a superposition of product solutions, please display the Fourier coefficient formula
     in integral form, unevaluated.

(c) [25\%] Explain why the bar temperature is $u(x,t) = u_1(x)+u_2(x,t)$.


\Prob{Problem 2}{(Total Thermal Energy in a Rod)}
If the temperature $u(x,t)$ is known, then give an expression for the time-dependent
(because energy escapes at the ends) total thermal
       energy $\int_0^L e(x,t)\,A(x)\,dx$ contained in a rod $x=0$ to $x=L$, with cross-sectional area $A(x)$. Symbol $e(x,t)$ is the thermal energy per unit volume at location $x$ and time $t$, known to equal the specific heat times the mass density per unit volume times the temperature.

Validate the answer using a uniform rod of length $L$ and constant cross-sectional
area $A$, held at steady-state temperature $u=u_0$.

\Prob{Problem 3}{(Steady-State Heat Conduction on a Rectangular Plate)}
Solve Laplace's equation
$\dd\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}$
on the rectangle $0 < x < L$, $0 < y < H$
subject to the boundary conditions.
$u_x(x,y)=0$ for $x=0$ and $x=L$, $u(x,y)=0$ for $y=0$, $u(x,y)=f(x)$ for $y=H$.

\Prob{Problem 4}{(Poisson's Formula, Mean Value Theorem and the Maximum Principle)}
For Laplace's equation inside a circular disk ($r < a$), the series solution formula can be re-arranged into Poisson's integral formula
$$\begin{array}{l}
\dd u(r,\theta)=\frac{1}{2\pi}\int_0^{2\pi} f(\phi)K(r,\theta,\phi)d\phi,\quad \mbox{where} \\
K(r,\theta,\phi) = \dd\frac{a^2-r^2}{a^2-2ar\cos(\theta-\phi)+r^2} = \mbox{~Poisson's Kernel.}
\end{array}
$$
Poisson's formula says that $u(r,\theta)$ is a weighted average of the boundary data $f(\theta)$
on the circle, with weight function $K$, the Poisson kernel. 

(a) [50\%] Compute $K$ at $r=0$, then show that $u(r,\theta)$ at $r=0$ is the average value of $f$ on the circle (Mean Value Theorem).

(b) [25\%] Explain why solution $u=100$ for $f(\theta)=100$ does not contradict the Maximum Principle.

(c) [25\%] Explain why $K$ must have integral $1$ over $0\le \theta \le 2\pi$.

\Prob{Problem 5}{(Steady-State Heat Conduction on a Disk)}

Consider the problem
$$\renewcommand{\arraystretch}{1}
 \left\{ \begin{array}{lllll}
 \dd u_{rr}(r,\theta) + \frac{1}{r}u_r(r,\theta)+\frac{1}{r^2} u_{\theta\theta}(r,\theta) = 0,
  & 0<r<a, &0<\theta<2\pi,\\[2ex]
 \dd u(a,\theta)=f(\theta), \quad 0<\theta < 2\pi.
 \end{array}\right.
$$
Solve for $u(r,\theta)$ when $a=1$ and  $f(\theta)=100$ on $0\le\theta<\pi$, $f(\theta)=0$ on $\pi\le\theta<2\pi$.


\end{document}