\begin{problem}{MATH 294}{\spr{83}}{FINAL}{5}{}
\probb{Solve the wave equation with wave speed $c=1$, boundary
conditions: $u(0,t) = u(6,t) = 0$ and initial conditions $u(x,0)
= 0, u_t(x,0) = 5 \sin{\brc{\frac{\pi x}{3}}}.$} \probpart{Make a
clearly labeled graph of $u(3,t)$ vs. $t$ for your solution in
part (a) above. }
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{94}}{FINAL}{14}{}
\prob{Verify that $u(t,x) = \frac{1}{2} [f(x+t) + f(x-t)]$ solves
the initial value problem:} \probpartnnn{$\ndpd{u}{t} =
\ndpd{u}{x} \; \; t > 0 , \; \; -\infty < x < \infty,$}
\probpartnnn{$u(t=0,x) = f(x)$} \probpartnnn{$u_t(t=0,x) = 0.$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{86}}{FINAL}{9}{}
\probb{Solve $ \ndpd{u}{x} + \ndpd{u}{y} = 0, \; (0<x<1, 0 < y<1)
$ where $u = u(x,y)$ and $u(0,y)=0, u(1,y)=0, u(x,0)=0,$ and
$u(x,1)=2 \sin(2 \pi x)$.} \probpart{Use your result from part
(a) to solve \[ \ndpd{u}{x} + \ndpd{u}{y} = 0, (0 < x < 1, 0 < y
< 1)
\] where $u = u(x,y)$ and $u(0,y) = 0, u(1,y) = 2 \sin(2 \pi y), u(x,0) = 0, u(x,1) =
0.$} \probpart{Use your result from part (a) and (b) to solve \[
\ndpd{u}{x} + \ndpd{u}{y} = 0, (0 < x < 1, 0 < y < 1)\] where $u
= u(x,y)$ and $u(0,y) = 0, u(x,0) = 0, u(1,y) = 2 \sin( 2 \pi y),
u(x,1) = 2 \sin(2 \pi x).$ }
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{86}}{FINAL}{12}{}
\prob{Find the solution to the initial/boundary value problem}
\probpartnnn{$\ndpd{u}{t} = C^2 \ndpd{u}{x}, 0 < x < L, t > 0
$}\probpartnnn{$u(0,t) = u(L,t) = 0, t > 0
$}\probpartnnn{$u(x,0)=0, 0 < x <
L$}\probpartnnn{$\stpd{u}{t}(x,0) = \sin{\brc{36 \pi
\frac{x}{L}}}, 0 < x<L.$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{87}}{\pr{2}}{2}{}
\prob{Find the value of $u$ at $x=t=1$ if $u(x,t)$ satisfies:
\[ \ndpd{u}{t} = 2 \ndpd{u}{x}\]
\[ 0 = u(0,t) = u(3 \pi, t) \]
with
\[ u(x,0) = \sin(5 x) \]
\[ \stpd{u}{t} (x,0) = \sin{x} \]}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{87}}{FINAL}{7}{}
\prob{Find \emph{any} non-zero solution $u(x,t)$ to \[
\ndpd{u}{t} = 4 \ndpd{u}{x} \mbox{with} 0 = \stpd{u}{x}(0,t) =
\stpd{u}{x}(1,t)
\] and the extra restriction that $u(0,0) \neq u(1,0).$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{87}}{\pr{2}}{3}{}
\prob{Find the solution of the initial-boundary-value problem}
\probpartnnn{$\ndpd{u}{t} = 4\ndpd{u}{x} \; \; 0 < x < 1 \; \; t
> 0 $} \probpartnnn{$u(0,t) = u(1,t) = 0 \; \; t > 0$}\probpartnnn{$\begin{array}{c}
  u(x,0) = 0 \\
  \stpd{u}{t}(x,0) = x.
\end{array} \;\; 0 < x < 1$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{88}}{\pr{2}}{5}{}
\prob{Once released, the deflection $u$ of a taught string
satisfies the wave equation \[ \ndpd{u}{t} = 4 \ndpd{u}{x} \]
where $x$ is position along the string and $t$ is time. It is
held fixed (no deflection) at its ends at $x=0$ and $x=2$.  At
time $t=0$ it is released from rest with the deflected shape $u =
3 \sin\brc{\frac{\pi x}{2}}. $ Make a plot of $u(1,t)$ versus $t$
for $0 \leq t \leq 2.$ Label the axes at points of intersection
with the curve. (You may quote any results that you remember.)}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{91}}{FINAL}{3}{}
\prob{The displacement $u(x,y)$ of a vibrating string satisfies
\[ \ndpd{x}{t} - \ndpd{u}{x} = 0 \] in $0 \leq x \leq 4, \; t\geq 0$ and the boundary and initial conditions \[ u(0,t) = 0, u(4,t) = 0, u(x,0) = 0, \stpd{u}{t} (x,0) = f(x),\] where \[ f(x) = \begin{cases}
1, & \mbox{when } 0\leq x \leq 2 \\ 0, & \mbox{ when } 2 \leq x
\leq 4
\end{cases} \]} \probpart{Find a series representation for the
solution.}\probpart{Write down the equation for the displacement
of the string at $t = 4.$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{93}}{FINAL}{14}{}
\probb{Solve the wave equation $(a^2 u_{xx} = u_{tt}$ to find the
displacement $u(x,t)$ of an elastic string of length $\ell$. Both
ends of the string are always free [$u_x(0,t) = 0 ; \;
u_x(\ell,t) = 0$] and the string is set in motion from its
equilibrium position, $u(x,0) = 0,$ with an initial velocity,
$u_t(x,0) = V_0 \cos{\frac{3 \pi x}{\ell}}$. Assume for this
problem that it is legitimate to differentiate any Fourier series
term-by-term.  If you use separation of variables, consider only
the case with a negative separation constant. } \probpart{Write
the solution to the wave equation $\brc{a^2 u_{xx} = u_{tt}}$ for
the boundary conditions $u(0,t) = h_L$ and $u(\ell,t) = h_R$ with
initial conditions $u(x,0) = h_L + (h_R - h_L)\frac{x}{\ell}$ and
$u_t(x,0) = 0.$ }
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{94}}{FINAL}{8}{}
\prob{Consider \[ u_{xx} = u_{tt} \; \; -\infty < x < \infty,
\;\; t>0 \] \[ u(x,0) = 0 , u_t(x,0) = g(x), \] where $g(x)$ is a
given function.} \probpart{Show that $u(x,t) = G(x+t)-G(x-t)$
satisfies the above wave equation and initial conditions for a
suitable function $G(x)$.  How are $G(x)$ and $g(x)$ related?}
\probpart{Find $u(x,t)$ if $u_t(x,0) = g(x) = \frac{x}{1+x^2}.$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{93}}{FINAL}{15}{}
\probb{The solution to} \probpartnnn{$u_{tt} = u_{xx} \; \;
-\infty < x < \infty$} \probpartnnn{$u(x,0) =
e^{-x^2}$}\probpartnnn{$u_t(x,0) = 0$ is of the form $u(x,t) =
\varphi(x+t) + \varphi(x-t).$ Find the solution without using
Fourier series.} \probpart{Find the solution of}
\probpartnnn{$u_{xx} = u_t \;\; 0 \leq x \leq 1
$}\probpartnnn{$u(0,t) =
1$}\probpartnnn{$u(1,t)=2$}\probpartnnn{$u(x,0)=1+x$}
\probpartnn{Hint: The solution may be time-independent.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{95}}{FINAL}{15}{}
\prob{If $u(x,t) = F(x+t) + G(x-t)$ for some functions $F$ and
$G$,} \probpart{Find expressions for $u(x,0)$ and $u_t(x,0)$ in
terms of $F$ and $G$.} \probpart{If also $ \left\{
\begin{array}{c}
  u_{tt} = u_{xx} \; \; -\infty < x < \infty \\
  u(x,0) = e^{-x^2} \\
  u_t{x,0} = 0
\end{array} \right. $
find expressions for $F$ and $G$, and sketch the graph of
$u(x,t)$ when $t = 0, 1,$ and $2$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{98}}{\pr{1}}{4}{}
\prob{Consider the following partial differential equation for
$u(x,t),$ \[ \ndpd{u}{t} - \ndpd{u}{x} = 0, \;\; 0 \leq x \leq 1,
\] with boundary conditions $u(0,t) = u(1,t) = 0, \; t > 0,$}
\probnn{and initial conditions $u(x,0) = f(x)$ and $\stpd{u}{t}
(x,0) = 0, \; \; 0 \leq x \leq 1.$} \probnn{which, if any, of the
functions below is a solution to the initial/boundary-value
problem? Justify your answer. } \probpart{$u(x,t) = \sumn b_n
e^{-\pi^2 n^2 t} \sin{n \pi x}, \; b_n = 2 \int^1_0 f(x) \sin{n
\pi x} d x$} \probpart{$u(x,t) = \sumn b_n cos{n \pi t}\sin{n \pi
x}, \; b_n = 2 \int^1_0 f(x) \sin{n \pi x} d x$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SUMMER 1990}{\pr{2}}{5}{}
\prob{Consider the partial differential equation \[ (*) \;
\ndpd{u}{x} = \frac{1}{c^2} \ndpd{u}{t}, \mbox{ for } 0 \leq x
\leq L,
\] with conditions} \probparts{$u(0,t) = 0,$} \probparts{$u(\ell,t) =
0$}\probparts{$\stpd{u}{t} (x,0) = 0,$} \probpartnn{and}
\probparts{$u(x,0) = f(x).$} \probpart{Verify that $u(x,t) =
\sumn C_n \sin \brc{\frac{n \pi x}{L}} \cos\brc{\frac{n \pi c t
}{L}}$ is a solution to (*) and the conditions (i), (ii), and
(iii).} \probpart{Suppose $f(x) = \sin\brc{\frac{\pi x}{L}}$.
What values for the $C_n$'s will satisfy condition (iv)?}
\probpart{For a general piecewise smooth function $f(x);$
Determine the formula for the $C_n$ so that condition (iv) is
satisfied.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{92}}{UNKNOWN}{4}{}
\prob{Solve the initial-boundary-value problem \[ u_{tt} =
u_{xx}, 0 < x < 1, t > 0,
\] \[ u(0,t) = u(1,t) = 0, t > 0, \] \[ u(x,0) = 8\sin 13 \pi x - 2 \sin 31 \pi x, \] \[ u_t (x,0) = -\sin 8 \pi x + 12 \sin 88 \pi x. \]}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{96}}{FINAL}{5 MAKE-UP}{}
\prob{Consider $u(x,y,z,t) = w(a x + b y + c z + d t)$ where $w$
is some differentiable function of one variable, and the
expression $a x + b y + c z + d t$ has been substituted for that
variable.} \probpart{Find restrictions on the constants $a,b,c,
\and d$ so that $u$ will be a solution to the three dimensional
wave equation $u_{xx} + u_{yy} + u_{zz} = u_{tt}$} \probpart{Find
a solution to the wave equation if (a) having $u(x,y,z,0) = 5
\cos x \and u_t(x,y,z,0) = 0.$ }
\end{problem}