\begin{problem}{MATH 294}{FALL 1982}{FINAL}{6}{}
\probb{Determine the solution to Laplace's equation} \probpartnn{$
\ndpd{u}{x} + \ndpd{u}{y} = 0 \mbox{ on } 0 < x < 1, \; 0<y<1,$
subject to the boundary conditions $u(0,y) = u(x,0) = u(1,y) = 0$
, $u(x,1) = 2x$} \probpart{Use this solution and the linearity of
Laplace's equation to obtain the solution to the boundary value
problem $u(0,y) = u(x,0) = 0,$  $u(1,y) = 2y,$  $u(x,1) = 2x$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1983}{FINAL}{6}{}
\prob{Solve Laplace's equation on the rectangle $0 \leq x \leq 4,
0 \leq y \leq 3$ with the given boundary conditions.}
\probpartnn{$u_{xx} + u_{yy} = 0,$} \probpartnn{$u(x,0) = u(0,y) =
0,$} \probpartnn{$u(4,y) = 2\sin{\left( \frac{\pi y}{3}\right)},$}
\probpartnn{$u(x,3) = 5\sin{\left( \frac{\pi x}{4}\right)}$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1990}{PRELIM 3}{6}{}
\probb{Solve Laplace's equation $\delta^2 u = 0$ on the square $0
< x < \pi, 0 < y < \pi, $ subject to $u=0$ on the three sides
$x=0, y = 0$ and $x = \pi$, and $u(x,\pi) = g(x)$, where $g$ is
defined in 4(b). (Hint: $u_n(x,y) = \sin{(n x)}\sinh{(n y)},
n=1,2,\ldots)$} \probpart{Repeat (a) if the b.c. $u(\pi,y) = 0$ is
replaced by $u(\pi,y) = 2\sin{3y} - 14\sin{9y}$ }

\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1993}{FINAL}{5}{}
\prob{The solution to any partial differential equation depends
on the domain in which the solution is to be valid as well as the
boundary conditions and/or initial conditions that the solution
must satisfy. We wish to consider physically plausible (i.e. $u$
does not approach $\infty$) solutions to Laplace's equation in
circular regions, $u_{rr} + r^{-1} u_r + r^{-2} u_{\theta \theta}
= 0$, for various boundary conditions. Below you are given some
functions that satisfy the Laplace equation and certain
conditions. You are asked to choose (and to give arguments that
led to that choice) what problem is being discussed.
\underline{You do not necessarily have to solve any equation}.}
\probpart{$u(r, \theta) = frac{C_n}{2} + \sumn r^{-n} (C_n \cos{n
\theta } + K_n \sin{n \theta} )$} \probpartnn{$C_n =
\frac{a^n}{\pi} \int^{2 \pi}_0 f(\theta) \cos{n \theta} d \theta
; \; K_n =  \frac{a^n}{\pi} \int^{2 \pi}_0 f(\theta) \sin{n
\theta} d \theta$} \probparts{a disk $(r \leq a)$ with $u(a,
\theta) = f(\theta)$} \probparts{an annulus of inner radius a $(a
\leq r < infty)$ with $\stpd{u}{r} (a, \theta) = f(\theta)$}
\probparts{an annulus of inner radius a $(a \leq r < infty)$ with
$u (a, \theta) = f(\theta)$} \probparts{none of above}
\probpart{$u(r,\theta) = \sumn C-n r^n \sin{n \theta}; \; C_n =
\frac{2}{\pi a^n} \int^\pi_0 f(\theta) \sin{n \theta} d \theta$}
\probparts{An annulus $(a \leq r < \infty)$, satisfying $u(a,
\theta) = f(\theta), \; 0 \leq \theta < \pi$} \probparts{A
half-disk $(r \leq a; \; 0 \leq \theta \leq \pi),$ with $u(r,
\pi) = 0, u(a,\theta) = f(\theta)$} \probparts{A disk $(r \leq a)$
with $u(a,\theta) = f(\theta)$ for $0 \leq \theta < \pi$ }
\probparts{none of the above} \probpart{$u(r, \theta) = \frac{u_b
\ln{\brc{\frac{r}{a}}}+u_b
\ln{\brc{\frac{b}{r}}}}{\ln{\brc{\frac{b}{a}}}}$} \probparts{a
pie-shaped wedge $(0 \leq r \leq a)$ of angle $\tan{\theta_0} =
\frac{b}{a}$ with $u(a, \theta) = u_a \theta + u_b
\brc{\frac{b}{r}}$} \probparts{An annulus $\brc{ a \leq r \leq b
}$ with constant specified values on the inner and outer bounding
circles } \probparts{a disk $\brc{r \leq a}$ satisfying $u(a,
\theta) = u_a + \theta u_b$} \probparts{none of the above}
\probpart{Describe a situation where Laplace's equation arises in
cartesian coordinates. Discuss the meaning of the appropriate
boundary and initial conditions for the problem that you have
chosen.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{88}}{\pr{2}}{4}{}
\probb{Find constants $A, B, C, D, E, \and F$ (real numbers) so
that the function \[ u = A + B x + C x^2 + D x^3 + E y^2 + F x y
\] is} \probparts{a solution to Laplace's equation in the rectangle shown,
and}\probparts{satisfies the three boundary conditions shown.}
%------------------------------------------------------------------------------------------------------picture p.320
\probpart{What boundary condition is satisfied on the fourth
boundary? $$\centerline{\epsfxsize=2.4in \epsfbox{5_3_320.jpg}}$$}
\end{problem}
%----------------------------------
\begin{problem}{UNKNOWN}{UNKNOWN}{UNKNOWN}{UNKNOWN}{}
\prob{Solve: $U_{xx} + u_{yy} = 0, 0 < x < \pi, 0 < y < 1$
subject to the boundary conditions $u_x (0,y) = u_x (\pi,y) = 0,
0 \leq y \leq 1, u(x,0) = 4 \cos(6 x) + \cos(7 x), 0 \leq x \leq
\pi, u(x,1) = 0, 0 \leq x \leq \pi. $}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{96}}{FINAL}{6 MAKE-UP}{}
\prob{Consider the Laplace equation in the semi-infinite strip $0
< x < L, y > 0, $ with boundary conditions $u(0,y) = 0, u(L,y) =
0, u(x,0) = f(x), $ and $u$ must not approach $\infty$ as $y$
approaches $\infty.$} \probpart{Find a general solution for these
conditions. } \probpart{Write out the solution for this problem
in the case that $f(x) = x.$ You are given the Fourier sine and
cosine series for $x, (0 < x <L)$ \[ x = \frac{L}{2} - \frac{4
L}{\pi^2} \sum_{n = 1,3,5,\ldots} \frac{1}{n^2} \cos\brc{\frac{n
\pi x }{L}}
\] \[ x = \frac{2 L}{\pi} \sumn \frac{(-1)^{n+1}}{n}\sin\brc{\frac{n \pi x}{L}} \]}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{87}}{FINAL}{3 MAKE-UP}{}
\prob{Find the solution of the boundary-value problem \[
\ndpd{u}{x} + \ndpd{u}{y} = 0 \left\{ \begin{array}{l}
  0 < x < \pi \\
  0 < y < \pi
\end{array} \right. ,
\] \[ \stpd{u}{x} (0,y) = \stpd{u}{x} (\pi,y) = 0, \; 0 < y < \pi,\] \[ u(x,0) = 0, u(x,\pi) = 9, \; 0 < x < \pi. \]}
\end{problem}
