\begin{problem}{MATH 294}{SPRING 1996}{PRELIM 2}{8}{}
\prob{Consider the PDE $u_t=-6 u_x$} \probpart{What is the most
general solution to this equation you can find?}
\probpart{Consider the initial condition $u(x,0) = \sin{(x)}$.
What does $u(x,t)$ look like for a very small but not zero t?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1983}{PRELIM 3}{5}{}
\prob{Consider $u_t = u_x$. Which of the functions below are
solutions to this equation? (Show your reasoning.)} \probpart{$3
e^{-\lambda t} \sin{\sqrt{\lambda t}} $} \probpart{3 $e^{-3t}
e^{-3 x} + 5 e^{-5 t} e^{-5 x}$} \probpart{$a e^{-3t} e^{-5 x}$}
\probpart{$\sin{(x)}\cos{(t)} + \cos{(x)} \sin{(t)}$}
\probpart{$sinh^{-1} \left[ (x+t)^3 \right]$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1984}{FINAL}{12}{}
\prob{Determine if the following equation is of the form of a
linear partial differential equation. If not, explain why. \[
\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial
y^2} + \frac{\partial u}{\partial x}\frac{\partial u}{\partial y}
= 0.
\]}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1984}{FINAL}{13}{}
\prob{Verify that the given function is a solution of the given
partial differential equation \[ x \frac{\partial u}{\partial x}
+ y \frac{\partial u}{\partial y} = 0, u(x,y) = f \left(
\frac{y}{x} \right), x=0
\] $f(\cdot)$ is a differentiable function of \textbf{one} variable.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1991}{PRELIM 2}{1}{}
\probb{Find the general solution of} \probpartnn{$(y + 2 x \sin{y}
d x + (x + x^2 \cos{y} ) d y = 0.$} \probpart{Determine the
solution of the initial-value problem} \probpartnn{$(x^2+4)
\frac{d y}{d x} + 2 x y = x,$ with $y(0) = 0$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPring 1992}{FINAL}{9}{}
\prob{Consider the initial boundary value problem for the heat
equation \[ \frac{\partial u}{\partial t} = \frac{\partial^2
u}{\partial x^2}, 0 < x < L, t > 0
\] with the boundary conditions \[ u(0,t) = u(L,t) = 0, t \geq 0 \] and the initial condition \[ u(x,0) = f(x), 0 \leq x \leq L.
\]} \probpart{Use the method of separation of variables to derive the solution of this problem. You may use the fact that the equation $ \dot{X} + \lambda X = 0, 0 < x < L$ with the boundary condition $X(0) = X(L) = 0$ has nontrivial solutions only for an interface number of constants $lambda_n = \frac{n^2 \pi^2}{L^2}$ for $n= 1,2, \ldots$. This corresponding solutions are of the form $X_n = A_n \sin{\frac{n \pi
x}{L}}$.} \probpart{Find the solution when $L=1$ and $f(x) = -6
\sin{4 \pi x} + \sin{7 \pi x}$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1992}{FINAL}{4}{}
\prob{For the PDE $u_x + 4 u_y = 0$:} \probpart{Solve it by
separation of variables.} \probpart{Show that any function of the
form $u(x,y) = f(ax+y)$ is a solution if $f$ is differentiable
and the constant $a$ is chosen correctly. } \probpart{Solve the
PDE with boundary condition $u(x,0) = \cos{x}$. (You may use (b)
rather than (a).)}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1994}{FINAL}{3}{}
\prob{Let $D$ be a region in the $(x,y)$ plane and $C$ be its
boundary curve with counterclockwise orientation. If the function
$u(x,y)$ satisfies $u_{xx} + u_{yy} = 0$ in $D$, show that \[
\oint_C u u_x d y - u u_y d x = \int \int_D \left( u^2_x + u^2_y
\right) d x d y.
\]}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1984}{FINAL}{15}{}
\prob{Show that the partial differential equation \[
\frac{\partial u}{\partial t} = k \left( \frac{\partial^2
u}{\partial x^2} + A \frac{\partial u}{\partial x} + B u \right)
\] can be reduced to \[ \frac{\partial v}{\partial t} = k \frac{\partial^2 v}{\partial x^2}\] by setting $u(t,x) = e^{\alpha x + \beta t} v(t,x)$ and choosing the constants $\alpha$ and $\beta$ appropriately.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1988}{PRELIM 2}{3}{}
\prob{Find \textbf{one} non-zero solution to the equation below.
Do not leave any free constants in your solution (that is, assign
some specific numerical values to any constants in your
solution). Note that you do not have to satisfy any specific
initial conditions or boundary conditions. \[ \frac{\partial
u}{\partial t} = \frac{\partial^2 u}{\partial x^2} - u
\] )}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1993}{PRELIM 3}{3}{}
\probb{Find the two \textbf{ordinary} differential equations that
arise from the \textbf{partial} differential equation} \probnn{ $
\alpha^2 u_{x x} = u{t t}$ for $ 0 < x < \ell, t \geq 0$ when the
equation is solved by separation of variables using a separation
constant $\sigma = - \lambda^2 < 0 $. } \probpart{Solve the
ordinary differential equation which gives the $x$ dependence. Use
the boundary conditions $u(0,t) = u(\ell,t) = 0$ for $t \geq 0$}
\probpart{Solve the ordinary differential equation which gives
the time dependence.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1994}{PRELIM 3}{2}{}
\prob{Given the partial differential equation (P.D.E) \[ u_x +
u_{y y} + u = 0,
\]} \probpart{Use separation of variables to replace the equation with two ordinary differential
equations.} \probpart{Find a non-zero solution to the P.D.E.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1995}{FINAL}{4}{}
\prob{Consider the \textit{first order} partial differential
equation \[ u_t + c u_x = 0, -\infty < x < \infty, o < t < \infty,
(*)\] where $c$ is a constant. We wish to solve this in two different
ways.} \probpart{Find a general solution to (*) by first writing
the equation with the change of variables, $\xi = x - c t, \eta =
t $.} \probpart{Now solve (*) using a separation of variables
technique. What are the units of $c$ if $x$ is in meters and $t$
is in second?} \probpart{Find $u(x,t)$ if $u(x,0) = k e^{-x^2}$.}
\probpart{Discuss the nature of your solution.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1995}{PRELIM 3}{3}{}
\prob{While separating variables for a PDE, Professor X was faced
with the problem of finding positive numbers $\lambda$ and
functions $X$ which are not identically zero and \[ X^{\prime
\prime}(x) = - \lambda X(x)
\] \[ X(0) = 0, X^\prime (1) = 0 \] Find the values of $\lambda$ and corresponding functions $X$ which will solve the professor's problem.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1995}{PRELIM 3}{4}{}
\prob{For the PDE $y u_{xx} + u_y = 0,$ (which is not the heat
equation)} \probpart{Assuming the product form $u(x,y) = X(x)
Y(y)$, find ODE's satisfied by $X$ and $Y$.} \probpart{Find
solutions to the ODE's.} \probpart{Write down at least one
non-constant solution to the PDE.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1996}{PRELIM 3}{2}{}
\probb{Consider the function $f$ defined by \[ f(x) = \left\{
\begin{array}{cc}
0 & $if $ -\pi \leq x < 0 \\
3 & $if $ 0 \leq x < \pi \\
f(x+ 2 \pi) & $for all $ x.
\end{array} \right.
\] Calculate the Fourier series of $f$. Write out the first few terms of the series explicitly. Make a sketch showing the graph of the function to which the series converges on the interval $-4 \pi < x < 4 \pi$. To what values does the series converges at $x =
0$?} \probpart{Consider the partial differential equation $u_t +
u = 3 u_x $ which is not the heat equation). Assuming the product
form $u(x,t) = X(x)T(t),$ find ordinary differential equations
satisfied by $X$ and $T$, (You are not asked to solve them.) }
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1996}{FINAL}{6}{}
\prob{Consider the equation for a vibrating string moving in an
elastic medium \[ a^2 u_{xx} - b^2 u = u_{tt} \] where $a$ and
$b$ are constants. ($a$ would be the wave speed if not for the
elastic constant $b$.) Assume the ends are fixed at $x=0, L$ and
initially the string is displaced by $u(x,0) = f(x),$ but not
moving $u_t(x,o) = 0.$} \probpart{Find a general solution for
these conditions. (If you need help, you may wish to work part
(b) first.)} \probpart{If the first term in the general solution
to part (a) is \[ u_1(x,t) = c_1 \cos{(\lambda_1 t)} \sin{\left(
\frac{\pi x}{L} \right)}
\] where $(\lambda_1)^2 = \left(\frac{\pi a}{L}\right)^2 + b^2,$ find the solution when the string starts from $u(x,0) = 2 \sin{\left( \frac{\pi x}{L} \right)}$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1997}{FINAL}{9}{}
\prob{Consider the PDE \[ \frac{\partial u(x,t)}{\partial t} =
\frac{\partial^2 u(x,t)}{\partial x^2} - u(x,t) ; \; 0 \leq x <
\pi ; \; t \geq 0 \makebox[1 in]{\hfill}(7) \] with boundary
conditions \[ u(0,t) = u(\pi,t) = 0 \makebox[1 in]{\hfill}(8) \]
\[ u(x,0) = \sin{(x)} \makebox[1 in]{\hfill}(9) \]} \probpart{Define \[ v(x,t) = e^t u)x,t) \makebox[1 in]{\hfill}(10) \] If $u(x,t)$ satisfies equations (7, 8 ,9), show that $v(x,t)$ satisfies the standard heat equation \[ \frac{\partial v(x,t)}{\partial t} = \frac{\partial^2 v(x,t)}{\partial x^2} \frac{\partial v(x,t)}{\partial t} \makebox[1
in]{\hfill}(11) \] with boundary conditions and initial
conditions \[ v(0,t) = v(\pi,t) = 0 \makebox[1 in]{\hfill}(12) \]
\[ v(x,0) = \sin{x} \makebox[1 in]{\hfill}(13) \]} \probpart{The general solution of equations (11,12) is \[ v(x,t) = \sum^\infty_{n=1} C_n e^{-n^2 t} \sin{n x} \makebox[1 in]{\hfill}(14)\] Find the unique solution $v)x,t)$ of equations
(11,12,13).}\probpart{Now find the unique solution of equations
(7, 8, 9).}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1992}{FINAL}{7}{}
\prob{For each of the following Fourier series representations,}
\probnn{$1 = \frac{4}{\pi} \sum^infty_{n=1} \frac{1}{2n-1}
\sin{[(2n-1) x]}, \; 0 < x < \pi,$} \probnn{$x = 2
\sum^infty_{n=1} \frac{(-1)^{n+1}}{n} \sin{n x}, \; -\pi < x <
\pi,$} \probnn{$x = \frac{\pi}{2} - \frac{4}{\pi} \sum^infty_{n=1}
\frac{1}{(2n-1)^2} \cos{[(2n-1) x]}, \; 0 \leq x < \pi.$}
\probpart{Find the numerical value of the series at $x =
-\frac{\pi}{3}, \pi$ and $12\pi+0.2$ (9 answers required).}
\probpart{Find the Fourier series for $|x|, \; -\pi < x < \pi.$
(Think - this is easy!).}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1992}{FINAL}{8}{}
\prob{Solve the initial-boundary-value problem \[ T_t = T_{xx} ,
\; 0 < x < \pi, \; t>0,
\] \[ T_x (0,t) = T_x (\pi,t) = 0, \] \[ T(x,0) = 3x. \] (You may use information from problem 7 if this helps.)}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1998}{PRELIM 1}{5}{}
\prob{Consider the partial differential equation for $u(x,t)$ \[
\frac{\partial u}{\partial t} + \frac{\partial u}{\partial x} = 0
\] with the initial conditions \[u(x,0) = 1-x, \; 0 \leq x \leq 1, \; u(x,0) = 0 \mbox{ elsewhere.}\] (No boundary conditions are
necessary).} \probnn{Use a centered difference approximation for
the space derivative: \[ \frac{\partial u}{\partial x}(x,t)
\approx \frac{1}{2 h} [u(x+h,t) - u(x-h,t)],\] and a forward
difference approximation for the time derivative: \[ u(x,t+k)
\approx u(x,t) + k \frac{\partial u}{\partial t} (x,t).
\]
Then introduce a grid with $N+1$ spatial points $x_i = ih,$ with
$i = 0,1,2, \ldots N $ (where $h$ is the grid spacing) and times
$t_j = j k$ (where $k$ is the time step). Let $u(x_i, t_j) \equiv
u[i,j].$} \probpart{With $h = 0.25$, and $k = 0.25$, write down
the values of the initial condition at each grid point for $0
\leq x \leq 2,$ i.e. $u[i,0], \; i = 0, \ldots, 8.$}
\probpart{Obtain the expression relating $u[i,j+1]$ to $u[i-1,j]$
and $u[i+1,j].$} \probpart{Use the initial data (with $h = 0.25$
and $k = 0.25$) to determine the approximate the value of $u$ at
$x_i = 1.0, \; t_k = 0.5 $.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1998}{FINAL}{3}{}
\prob{Consider the partial differential equation $\frac{\partial u
}{\partial x} - \beta \frac{\partial u }{\partial t} = 0$ (Note
that this is not the heat equation.) with the initial condition
$u(x,0) = x^2$. In an approximate solution $u$ is to be evaluated
on a grid of points spaces by $h$ on the $x$ axis and $\delta t$
on the $t$ axis: $x_i = (i-1) h$ and $t_j = (j-1) \delta t$. The
values of $u(x_i, t_j)$ are contained in the array $\hat{u}_{ij}
\equiv \hat{u}(i,j) \equiv u(x_i,t_j).$ Here are the forward
difference approximations:} \probnn{$\frac{\partial
u(x,t)}{\partial x} = \frac{1}{h} [u(x+h,t) - u(x,t)]$ and
$\frac{\partial u(x,t)}{\partial t} = \frac{1}{\delta t}
[u(x,t+\delta t) - u(x,t)]$} \probpart{Derive a finite difference
algorithm for this equation. That is, find an expression for
$\hat{u}(i,j+1)$ in terms of $\hat{u}(i,j)$ and
$\hat{u}(i+1,j)$.} \probpart{Let $h =1 , \delta t = \frac{1}{2},$
and $\beta = 1$. Use your approximate scheme above and the given
initial condition to find approximate values for $u\left( 2,
\frac{1}{2} \right)$ , $u\left( 3, \frac{1}{2} \right)$, and
$u(2,1)$.} \probpart{Find the exact solution to the partial
differential equation and given boundary condition. (Do not waste
time with Fourier series formulae).}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1999}{PRELIM 3}{2}{}
\prob{Parts a), b), and c) are related, but each part can be done
independently of the other parts. Consider the following problem
consisting of a PDE for $u = u(x,t)$, two B.C.'s and an I.C.:\[
\frac{\partial^2 u}{\partial x^2} = u + \frac{\partial
u}{\partial t}\] \[ \mbox{B.C.'s: } u(0,t) = 0, \frac{\partial
u(2,t)}{\partial x} = 0, t>0\] \[ I.C.: u(x,0) = \sin{\frac{\pi
x}{4}}, \; 0 \leq x \leq 2.
\]} \probpart{Use separation of variables on the PDE to obtain two ODE's for $X(x)$ and $T(t)$, and the B.C. for
$X(x)$.}\probpart{In some separation of variables problem, a
student obtained the following ODE plus B.C.'s on $X(x)$: \[
\frac{d^2 X}{d x^2} + \lambda X = 0 \; X(0) = 0, \frac{d X(2)}{d
x} = 0.
\] Find all nontrivial solutions to the ODE with these B.C.'s.}
\probpart{Which, if any, of the equations given below is a
solution to the PDE's, B.C.'s and I.C. at the top of the page?
(Justification of your answer is required to get credit. Note
that you may have to check several boundary conditions as well as
the PDE.) } \probparts{$u(x,t) = e^{\left( -1-\frac{\pi^2}{16}
\right) t} \sin{\frac{\pi x}{4}}$} \probparts{$u(x,t) = e^{
-\frac{\pi^2 t}{16} } \sin{\frac{\pi x}{4}}$} \probparts{$u(x,t)
= e^{ -\frac{\pi^2 t}{16} } \cos{\pi x}$} \probparts{$u(x,t) =
\sum^\infty_{n=1} b_n e^{ -\frac{ n^2 \pi^2 t}{16} } \sin{\frac{n
\pi x}{4}} , \; b_n = \int^2_0 sin{\left( \frac{\pi x}{4}
\right)}sin{\left( \frac{n \pi x}{2} \right)}$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{83}}{\pr{3}}{2}{}
\prob{Consider the PDE $u_t = -6 u_x.$} \probpart{What is the
most general solution to this equation you can find?}
\probpart{Consider the initial condition $U(x,0) = \sin(x).$ What
does $u(x,t)$ look like for a very small but not zero $t$?
$$\centerline{\epsfxsize=2.4in \epsfbox{5_2_308.jpg}}$$}
%------------------------------------------------------------------------------------------------------picture p.308
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{87}}{\pr{2}}{5 MAKE-UP}{}
\prob{Find the solution of the boundary-value problem \[
\ndpd{u}{x} + \ndpd{u}{y} = 0 \; \begin{array}{l}
0 < x < 1 \\
0 < y < 1
\end{array},
\] \[ u(0,y) = u(x,0) \equiv 0, \] \[ u(1,y) = u(x,1) \equiv 1. \]}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{96}}{FINAL}{7 MAKE-UP}{}
\prob{Solve the problem \[ u_{rr} + \frac{1}{r} u_r +
\frac{1}{r^2} u_{\theta \theta} = 0
\] \[ u(1,\theta) = 4\sin(\theta) - 3 \cos(2 \theta) \] \[ \lim_{r \to 0} u(r,\theta) = 0 \] You may use the fact that $r^{\pm n} \cos(n \theta) \and r^{\pm n} \sin(n \theta) $ are some of the solutions to this partial differential equation.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{96}}{\pr{3}}{3 MAKE-UP}{}
\probb{Find all solutions of the form $u(x,t) = X(x) T(t)$ for \[
x u_x = 2 u_t \; ;\] subscripts indicate differentiation with
respect to that variable.}
\end{problem} \probpart{If $u(x,0) = 2 x - 3 x^2, $ find $u(x,t).$}
%----------------------------------
\begin{problem}{MATH 294}{\spr{96}}{FINAL}{7}{}
\prob{In the problem below, choose the solution that corresponds
to the given physical problem. Justify your choice. (Note that a
sketch is required with (a), and that to make the right choices
you will probably have to check several of the boundary/initial
conditions as well as the appropriate partial differential
equation.)} \probpart{A taut string stretching to infinity in
both directions has a wave speed $a$ and an initial displacement
$y(x,0) = \frac{1}{\brc{1+ 8 x^2}}$ but no initial velocity.}
\probparts{$y(x,t) = \frac{\frac{1}{2}}{1+8\brc{x-a t}^2} +
\frac{\frac{1}{2}}{1+8\brc{x+a t}^2}$} \probparts{$y(x,t) =
\frac{1}{1 + 8 x^2} + \frac{\frac{1}{2}}{1+8\brc{x-a t}^2} +
\frac{\frac{1}{2}}{1+8\brc{x+a t}^2}$} \probparts{$y(x,t) = \sumn
c_n \sin\brc{\frac{n \pi x}{L}} \sin\brc{\frac{n \pi a t}{L}}
\mbox{ where } c_n = \frac{1}{L} \int^L_0 f(x) \cos\brc{\frac{n
\pi x}{L}} d x.$} \probparts{(no choice, do this) Plot the
solution initially and when $a t = 1$} \probpart{The steady state
temperature exterior to a semicircular hole $(r > a, 0 < \theta <
\pi) $ with boundary conditions $u(r,0) = 0\ and u(r,\pi) = 0$
for $a < r < \infty \and u(a,\theta) = f(\theta) \and \lim_{r \to
\infty} u(r,\theta) = 0 $ for $0 \leq \theta \leq \pi.$
$$\centerline{\epsfxsize=2.4in \epsfbox{5_2_313_01.jpg}}$$}
%------------------------------------------------------------------------------------------------------picture p.313
\probpartnn{In all of the choices, $c_n = \frac{2 a^n}{\pi}
\int^\pi_0 f(\theta) \sin(n \theta) d \theta.$}
\probparts{$u(r,\theta) = \sumn c_n r^{-n} \sin(n \theta)$}
\probparts{$u(r,\theta) = \sumn c_n r^{-n} \cos(n \theta)$}
\probparts{$u(r,\theta) = \sumn c_n r^{n} \sin(n \theta)$}
\probparts{$u(r,\theta) = \sumn c_n r^{n} \cos(n \theta)$}
\probpart{A square copper plate with sides $L$ has all four edges
maintained at $0^\circ$ $$\centerline{\epsfxsize=2.4in
\epsfbox{5_2_313_02.jpg}}$$}
%------------------------------------------------------------------------------------------------------picture p.313
\probpartnn{A line across the plate at $x = x_1, 0 < y < L$ is
heated to $T_1$ by an external heat source until a steady state
results. The temperature in the plate is:} \probparts{$T =
\left\{ \begin{array}{lll}
T_1 \frac{x}{x_1} & $if$ & x \leq x_1 \\
T_1 \frac{L-x}{L-x_1} & $if$ & x > x_1
\end{array} \right.$} \probparts{$T = \sumn b_n e^{-\frac{n^2 \pi^2 k t}{L}}$ where $b_n = \frac{1}{L} \int_0^L \frac{T_1 x}{x_1} \sin\brc{\frac{n \pi x}{L}} d
x$}\probparts{$T = \frac{4 T_1}{\pi} \sum\limits_{n =
1,3,5,\ldots} \frac{ \sin \brc{\frac{n \pi x}{L}} \sinh
\brc{\frac{n \pi (L-y)}{L}}}{n\sinh(n \pi)}$} \probparts{None of
the above.}
\end{problem}