\begin{problem}{MATH 294}{\spr{85}}{FINAL}{1}{}
\probb{Find the solution to the partial differential equation
given by \[ \ndpd{u}{x} = \stpd{u}{t}, \mbox{where} u = u(x,t) \]
with boundary conditions} \probpartnnn{$u(0,t) = 0$}
\probpartnnn{$u(L,t) = 0$} \probpartnn{and initial condition}
\probpartnnn{$u(x,0) = 1; \; 0 < x < L$} \probpart{What does the
solution $u(x,t)$ approach as $t \rightarrow \infty$. Briefly
explain this answer. (No credit will be given for guessing.)}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{86}}{FINAL}{10}{}
\prob{Solve for $u = u(x,t)$ where \[ \stpd{u}{t} = \ndpd{u}{x},
(0 < x < 2, t> 0), \] \[ \left.
\begin{array}{c}
  \stpd{u}{x}(0,t) = 0 \\
  \stpd{u}{x}(2,t) = 0
\end{array} \right\} t
> 0 \] \[ u(x,0) = \cos{(2 \pi x)}, 0 \leq x \leq 2.\]}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{86}}{FINAL}{13}{}
\prob{Find the solution to the initial/boundary value problem \[
\stpd{u}{t} = a^2 \ndpd{u}{x}, 0 < x < L, t >0\] \[ u(0,t) =
\stpd{u}{x}(L,t) = 0, t > 0
\] \[ u(x,0) \equiv 1, 0 < x < L. \]} \probnn{You may use symmetry to solve a more familiar problem on $0 < x < 2L$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{87}}{\pr{1}}{8}{}
\prob{Find \textbf{any} non-zero solution to the heat equation \[
\stpd{u}{t} = 3 \ndpd{u}{x} \] that satisfies the boundary
conditions $u(0,t) = u(5,t) = 0.$ (You need not go into great
detail explaining how you find this solution.)}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{83}}{\pr{3}}{3}{}
\prob{Let $\stpd{u}{t} = \alpha^2 \ndpd{u}{x}, 0 < x < 10$ have
boundary conditions \[ u_x(0,t) = u_x (10,t) = 0 \]}
\probpart{What is the most general solution to this boundary
value problem you can find.} \probpart{Given $u(x,0) = x^2, $
what is $u(5,t \rightarrow \infty)$ in the above problem?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{83}}{\pr{3}}{4}{}
\prob{Let $u_t = u_{xx}, 0 < x < \pi$ have boundary conditions
$u_x(0,t) = 0, u(\pi,t) = 0$.} \probpart{What is the most general
solution you can find to this equation and the given boundary
conditions. } \probpart{For the initial condition $u(x,0) = x^2 -
e^x \sin{x} $ what is $u(1,t \rightarrow \infty)$?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{83}}{FINAL}{4}{}
\prob{A bat of length 1 is assumed to satisfy the heat equation
with $\alpha^2 = 1$.  The ends of the bar are in ice water at
temperature $u=0$. At time $t=0$ the temperature of the bar is
$u(x,0) = 100 \sin{\pi x}$. What is the temperature in the middle
of the bar at $t=2$?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{84}}{FINAL}{16}{}
\prob{Heat conduction in a closed-loop wire of radius 1 can be
described by $u(t,x)$ where \[ \stpd{u}{t} = \ndpd{u}{x} \; 0
\leq x < 2 \pi, \; t > 0
\] and $u(t,x=0) = u(t, x = 2 \pi) \; \; t \geq 0.$ The initial distribution of temperature is \[ u(t=0,x) = \left\{ \begin{array}{cc}
  0 & 0 < x \leq \frac{\pi}{2} \\
  1 & \frac{\pi}{2} < x \leq \frac{3 \pi}{2} \\
  0 & \frac{3 \pi}{2} < x \leq 2 \pi
\end{array} \right. \] find $u(t,x)$ by separation of variables, what is the temperature distribution as $t \rightarrow \infty$?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{84}}{FINAL}{14}{}
\partinc{} \probpart{Solve the following initial-boundary value problem for the heat
equation}\probpartnnn{$\stpd{u}{t} - \ndpd{u}{x} = 0 \; 0 \leq x
\leq L, \; t \geq 0$} \probpartnnn{$\stpd{u}{x} (0,t) = 0, \;
\stpd(L,t) = 0$} \probpartnnn{$u(x,0) = f(x)$ where $f(x)$ is the
function given in IV A.} \probpart{Determine the value of $H(t)$
given by $H(t) = \int_0^L u(x,t) d x$. If you have been unable to
find the solution in IV B, then form $\stpd{H}{t}$ and use the
differential equation for $u$ to find an ordinary differential
equation for $H$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{87}}{\pr{3}}{3}{}
\prob{Find the solution of the initial-boundary-value problem}
\probpartnnn{$\stpd{u}{t} = \ndpd{u}{x} \; \; 0 < x <
\frac{\pi}{2} , \; t > 0,$} \probpartnnn{$u(0,t) = \stpd{u}{x}
\brc{\frac{\pi}{2},t} = 0 \; t > 0,$} \probpartnnn{$u(x,0) = 7
\sin{x} - 14 \sin{9 x}$} \probnn{(Hint: symmetry and
superposition)}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{87}}{\pr{2}}{5}{}
\prob{Find the solution of the initial-boundary-value
problem}\prob{Find the solution of the initial-boundary-value
problem} \probpartnnn{$\stpd{u}{t} = \ndpd{u}{x} \; \; 0 < x <
\pi , \; t > 0,$} \probpartnnn{$\stpd{u}{x}(0,t) = \stpd{u}{x}
\brc{\pi,t} = 0 \; t > 0,$} \probpartnnn{$u(x,0) = 10 \sin{^2
4x}$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{87}}{\pr{3}}{4}{}
\probb{Find the solution of the initial-boundary-value problem}
\probpartnnn{$\stpd{u}{t} = \ndpd{u}{x} \; \; 0 < x < \pi , \; t
> 0,$} \probpartnnn{$\stpd{u}{x}(0,t) = \stpd{u}{x} \brc{\pi,t} = 0 \; t >
0,$} \probpartnnn{$u(x,0) = 1 + x \; \; 0 < x < \pi$}
\probpart{What is $\lim_{t\to \infty} u(x,t),$ where $u(x,t)$ is
the solution of part (a).} \probpart{Show that your result from
part (b) is a time-independent (equilibrium) solution of the
first two equations in part (a).}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{88}}{\pr{3}}{5}{}
\prob{Solve $\stpd{u}{t} = \ndpd{u}{x}$ with boundary conditions
$\stpd{0,t} = 0$ and $\stpd{u}{x}(\pi,t) = 0$ and initial
condition $u(x,0) = 4 \cos(x) - 5 \cos(4x).$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{89}}{FINAL}{5}{}
\prob{Let $\alpha > 0$. Find the solution $u(x,t)$ of}
\probpartnnn{$u_{xx} = \frac{1}{\alpha} u_t , \; t > 0 \; , 0 < x
< 1,$} \probpartnnn{$u_x (0,t) = u_x(1,t) = 0, \; t> 0,$}
\probpartnnn{$u(x,0) = 1 + 3 \cos(2 \pi x) - 2 \cos (5 \pi x), \;
0 < x < 1.$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{91}}{FINAL}{2}{}
\prob{Consider the conduction of heat through a wire of unit
length that is insulated on its lateral surface \underline{and at
its ends}.} \probpart{Use the method of separation of variables
to show that the solution of the initial value problem}
\probpartnnn{$\stpd{u}{t} = \ndpd{u}{x}$ for $0 \leq 1 \leq0, 0
\leq t \leq \infty ;$} \probpartnnn{with $\stpd{u}{x}(0,t) =
\stpd{u}{x}(1,t) = 0 ; $ and $u(x,0) = f(x)$} \probpartnn{is
given in the form} \probpartnnn{$u(x,t) = \frac{a_0}{2} + \sumn
a_n e^{-n^2 \pi^2 t} \cos{n \pi x}.$} \probpartnn{Hint: The
equation $X^{\prime \prime} + \lambda X = 0 , 0 \leq x \leq 1,$
with $X^{\prime}(0) = X^{\prime}(1) = 0$ has nonzero solutions
only for an infinite number of constants $\lambda = n^2 \pi^2,$
for $n = 0, 1, 2, 3, \ldots .$ The corresponding solutions are
$X_n(x) = A_n \cos{n \pi x}.$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{94}}{FINAL}{5}{}
\prob{Solve} \probpartnnn{$\left\{ \begin{array}{c}
  u_t = u_{xx} \mbox{on} 0 < x < \ell \\
  u(0,t) = u(\ell,t) = 0 \\
  u(x,0) = \left\{ \begin{array}{cc}
    1 & \frac{\ell}{4} \leq x \leq \frac{3 \ell}{4} \\
    0 & \mbox{otherwise} \
  \end{array} \right.
\end{array} \right.$} \probnn{You may leave Fourier coefficients unsimplified after doing the integrals.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{93}}{FINAL}{5}{}
\probb{The solution to \eqnr{U_{tt} &=& u_{xx} ; \; \; -\infty < x
< \infty \\ u(x,0) &=& e^{-x^2} \\ u_t(x,0) &=& 0 } is the form
$u(x,t) = \varphi(x+t) + \varphi(x-t).$ Find the solution without
using Fourier series.}\probpart{Find the solution of
\begin{eqnarray*} u_{xx} & = & u_t \; \; \; 0 \leq x \leq 1 \\
u(0,t) & = & 1 \\ u(1,t) & = & 2 \\ u(x,0) & = & 1+x
\end{eqnarray*} Hint: The solution may be time-independent.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{94}}{FINAL}{7}{}
\prob{Consider \eqnr{u_{xx} &=& u_t, \; \; 0 < x < 1, \; \; t > 0
\\ u(0,t) &=& 0, \\ u(1,t) &=& -1, \\ u(x,0) &=& 0 }} \probpart{Find $v(x)$ if $u(x,t) \to v(x)$ as $t \to
\infty$} \probpart{Find a Fourier series solution for $u(x,t).$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{95}}{\pr{3}}{2}{}
\prob{Consider the condition of heat through a wire of unit
length that is insulated on its lateral surface and at its ends.
This implies boundary conditions $u_x(0,t) = 0 = u_x(1,t), t \geq
0.$} \probpart{Verify that solutions $u(x,t)$ to the heat
equation with the initial condition $u(x,0) = f(x)$ piecewise
continuous first derivatives may be given in the form \[ u(x,t) =
\frac{a_0}{2} + \sumn  a_n e^{-n^2 \pi^2 t} \cos(n \pi x)\]}
\probpart{Find $u(x,t)$ when $f(x) = 2 + 5 \cos(3 \pi x).$?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{96}}{\pr{3}}{3}{}
\prob{Consider the heat equation \eqnr{
u_t &=& 0.04 u_{xx} \\
u(x,0) &=& \sin{ \brc{\frac{\pi x}{2}} } - \frac{1}{2} \sin(\pi x)
\\
u(0,t) &=& 0
\\ u(2,t) &=& 0}} \probpart{Find the solution to this problem.}
\probpart{Verify by substitution that your answer to part (a)
does in fact satisfy all four of these equations. (You can get
full credit for this part by checking everything, even if your
answer to (a) is \emph{wrong}.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{96}}{\pr{3}}{3}{}
\probb{Find the full solution to \eqnr{u_{xx} &=& u{t} \; \; 0 <
x < \pi, \; \; t > 0
\\ u(0,t) &=& u(\pi,t) = 0 \\ u(x,0) &=& x}You may find problem 2 helpful in solving
this.} \probpart{Using only the first two terms in your solution,
write out $u(x,0)$ and $u(x,1)$. Sketch these terms and their sum.
Comment on your plot. \eqnr{e^0 &=&1 \\ e^{-1} &=& .368 \\
e^{-2} &=& .135 \\ e^{-3} &=& .050 \\ e^{-4} &=& 0.018 \\ e^{-5}
&=& .007}}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{98}}{\pr{1}}{3}{}
\prob{Consider the one dimensional heat transfer problem \[ u_{xx}
= u_t, \; 0 \leq x \leq 1
\] with boundary conditions $u(0,t)=0, u(1,t)=1, t>0,$} \probnn{and initial conditions $u(x,0)=0, \; 0 \leq x \leq
1.$} \probpart{Find the long time, i.e. time independent,
solution reached as $ t \to \infty$.} \probpart{Find the
time-dependent solution $u(x,t)$ that satisfies the given
boundary and initial conditions.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{98}}{FINAL}{2}{}
\prob{Find the solution of the initial value problem \[
\stpd{u}{t} = 2 \ndpd{u}{x} \mbox{on} 0<x<1 \mbox{for} t>0, \]
with \[ \stpd{u}{x} (0,t) = \stpd{u}{x} (1,t) = 0\] and \[ u(x,0)
= \left\{ \begin{array}{cc}
  1,& 0 < x < \frac{1}{2} \\
  0,& \frac{1}{2} < x < 1
\end{array} \right.
\]}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{98}}{FINAL}{2}{}
\prob{Solve, for $t>0$ and $0<x<\pi$, the partial differential
equation $2 \ndpd{u}{x} = \stpd{u}{t}$ with the boundary
conditions that $u(0,t) = 0, \; u(\pi,t) = \pi,$} \probnn{and the
initial condition that $u(x,0) = x + \sin(x).$ } \probnn{(This
problem can be solved completely without any integrations.)}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{UNKNOWN 1990}{UNKNOWN}{5}{}
\prob{Solve the initial boundary value problem \[ u_t = u_{xx},
\; 0 < x < \pi, T > 0,
\] \[ u(0,t) = u_x(\pi,t) = 0, \] \[ u(x,0) = 18 \sin\brc{\frac{9 x}{2}}. \] (Hint: $u(x,y) = X(x) T(t);$ you may use the fact that the only nontrivial solution here occurs when $X$ is a linear combination of a cosine and a sine.)}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{90}}{FINAL}{5}{}
\prob{Find the steady state temperature distribution in the plate
shown.$$\centerline{\epsfxsize=2.4in \epsfbox{5_4_328_01.jpg}}$$}
%------------------------------------------------------------------------------------------------------picture p.328
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{90}}{FINAL}{4 MAKE-UP}{}
\prob{Find the steady state temperature distribution in the plate
shown.$$\centerline{\epsfxsize=2.4in \epsfbox{5_4_328_02.jpg}}$$}
%------------------------------------------------------------------------------------------------------picture p.328
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{94}}{\pr{1}}{5}{}
\prob{Find the steady state temperature distribution in the plate
shown.$$\centerline{\epsfxsize=2.4in \epsfbox{5_4_330.jpg}}$$}
%------------------------------------------------------------------------------------------------------picture p.330
\end{problem}
