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\begin{center}\Large\bf
Partial Differential Equations 3150 \\
\small Sample Final Exam\\
\small Exam Date: Friday, April 25, 2014
\end{center}

{\bf Instructions}: This exam is timed for 120 minutes. You may be given
extra time to complete the exam (10-15 extra minutes). No calculators, notes, tables or
books. Problems use topics from the required textbook which were covered in lectures. Details count 3/4, answers count 1/4.

\begin{exercises}
\prob{1.}{(Chapters H1-H2. Heat Conduction in a Bar)} ~\newline

Considered is the heat conduction problem 
\begin{equation}\label{bar-problem}
 \left\{ \begin{array}{lclll}
 u_{t} &=& \dd\frac14 u_{xx}, & 0<x<10, & t>0, \\
 u(0,t) &=& 50, & & t>0,\\
 u(10,t)&=& 80, && t>0,\\
 u(x,0) &=& f(x), &0 < x < 10.
 \end{array}\right.
\end{equation}
It represents a laterally insulated uniform bar of length 10 with one end at 50 Celsius and the other end at 80 Celsius, and initial temperature $f(x)$.

\BF{ 1(a)} [30\%] Show the details for finding the the steady-state temperature $u_1(x)=50+3x$.

\BF{ 1(b)} [40\%] Show the details for the solution $u_2(x,t)=\sin(\pi x/10)\,e^{-\pi^2 t/400}$ of the ice-pack ends bar problem 
$$
 \left\{ \begin{array}{lclll}
 u_{t} &=& \dd\frac14 u_{xx}, & 0<x<10, & t>0, \\
 u(0,t) &=& 0, & & t>0,\\
 u(10,t)&=& 0, && t>0,\\
 u(x,0) &=& \sin (\pi x/10), &0 < x < 10.
 \end{array}\right.
$$


\BF{ 1(c)} [30\%] Superposition implies $u(x,t)=u_1(x,t)+u_2(x,t) = 50+3x+\sin(\pi x/10)\,e^{-\pi^2 t/400}$ is a solution of (\ref{bar-problem}) with $f(x)=-50-3x+\sin(\pi x/10)$. Show the details of an answer check for this solution $u(x,t)$.
\end{exercises}

\answer{
}

\EJECT
\begin{exercises}
\prob{2.}{(Chapter H3. Fourier Series)} ~\newline

\BF{ 2(a)} [20\%] Find and display the nonzero terms in the Fourier series expansion of
 $f(x)$, formed as the even $2\pi$-periodic extension
of the function $f_0(x)=\sin^2(x) + 4\cos(2x)$ on $0 < x < \pi$.

\BF{ 2(b)} [40\%] Let $g(x)$ be the Fourier sine series for the 
the period $2$ odd extension of the function $g_0(x)=1$ on $0\le x \le 1$. Complete the following.
\begin{quote}\sf
 \begin{itemize}
\item[\BF{(1)}] Graph of $g_0$ with its odd extension on $|x|<1$. 
\item[\BF{(2)}] Graph of $g(x)$ over 4 periods.
\item[\BF{(3)}] Fourier sine series coefficient formulas.
\item[\BF{(4)}] Numerical values for the coefficients.
\item[\BF{(5)}] Gibb's overshoot graphic on $|x|<2$.
 \end{itemize}
 \end{quote}
 
\BF{ 2(c)} [20\%]  Define $h_0(x)=\left\{ \begin{array}{lcl} \sin(2x) & 0 \le x < \pi ,\\ x-\pi & \pi \le x \le 2\pi, \end{array}\right.$ and let $h(x)$ be the $4\pi$ odd periodic extension of $h_0(x)$ to the whole real line. Compute
the sum $h(-5.25\pi)+h(1.5\pi)$.

\BF{ 2(d)} [20\%] Compute the smallest period, for those functions which are periodic.
 \begin{quote}\sf
  \begin{itemize}
    \item[{\bf (1)}] $f_1(x)=2\sin(x)+5\cos(x)$
    \item[{\bf (2)}] $f_2(x)=\sin(x)+\cos(5x)$
    \item[{\bf (3)}] $f_3(x)=\sin(x)+\cos(\pi x)$
    \item[{\bf (4)}] $f_4(x)=\sin(x)+\sin(x)\cos(x)$
  \end{itemize}
 \end{quote}
\end{exercises}

\answer{
%assume(n::integer,n::positive);
%getassumptions(n);
%#f1:=x->sin(Pi*x)^2*(-100);
%f1:=x->-50;
%int(f1(x)*sin(n*Pi*x/10),x=0..10);
%int(f1(x)*sin(20*Pi*x/10),x=0..10);
%int(1*sin(n*Pi*x/10),x=0..10);
%B:=n->-2*(-1+(-1)^n)/n/Pi;
%F:=x->sum(B(n)*sin(n*Pi*x),n=1..infinity);
%plot(F(x),x=-2..2);
%B(1);
%plot(add(B(n)*sin(n*Pi*x),n=1..5),x=-2..2);
}

\EJECT

\begin{exercises}
\prob{3.}{(CH H4. Finite String: Fourier Series Solution)} ~\newline
\BF{ 3(a)} [50\%] Complete the following for the finite string problem
$$
 \left\{ \begin{array}{lclll}
 u_{tt}(x,t) &=& \frac14 u_{xx}(x,t), & 0<x<2, & t>0, \\
 u(0,t) &=& 0, & & t>0,\\
 u(2,t)&=& 0, && t>0,\\
 u(x,0) &=& f(x), & 0 < x < 2, \\
 u_t(x,0)&=& g(x), & 0<x<2.
 \end{array}\right.
$$
\begin{quote}\sf
\begin{itemize}
\item[\BF{(1)}] Separation of variables details. 
\item[\BF{(2)}] Product solutions and boundary conditions. 
\item[\BF{(3)}] The normal modes.
\item[\BF{(4)}] Superposition.
\item[\BF{(5)}] Series solution $u(x,t)$.
\end{itemize}
\end{quote}

\BF{ 3(b)} [25\%] Display explicit formulas for the generalized Fourier coefficients which
contains the symbols $f(x)$, $g(x)$.

\BF{ 3(c)} [25\%] Evaluate the coefficients when $f(x)=100$ and $g(x)=0$.
\end{exercises}
\EJECT


\begin{exercises}
\prob{4.}{(CH H4. Rectangular Membrane)} ~\newline
Complete the following for the general membrane problem
$$
 \left\{ \begin{array}{lcllll}
 u_{tt}(x,y,t) &=& c^2 \left(u_{xx}(x,y,t)+u_{yy}(x,y,t)\right), & 0<x<a, & 0 < y < b, & t>0, \\
 u(x,y,t) &=& 0 & \multicolumn{3}{l}{\mbox{on the boundary,}}\\
 u(x,y,0)&=& f(x,y), & 0 < x < a, & 0 < y< b,\\
 u_t(x,y,0) &=& g(x,y), & 0 < x < a, & 0<y<b.
 \end{array}\right.
$$
under the assumptions $a=b=c=1$, $f(x,y)=1$, $g(x,y)=0$.
\begin{quote}\sf
\begin{itemize}
\item[\BF{(1)}] Separation of variables. 
\item[\BF{(2)}] Product solution boundary value problems.
\item[\BF{(3)}] Product solutions (the normal modes).
\item[\BF{(4)}] Superposition for $u(x,y,t)$.
\item[\BF{(5)}] Generalized Fourier coefficient formulas.
\item[\BF{(6)}] Explicit numerical values for the coefficients.
\end{itemize}
\end{quote}
\end{exercises}
\EJECT

\begin{exercises}
\prob{5.}{(CH H5. Heat Equation and Gauss' Heat Kernel)} ~\newline
\BF{\bf 5(a)} [10\%] Define a Fourier transform pair so as to include as many definitions as possible, in particular, Haberman's definition.

\medskip

\BF{\bf 5(b)} [20\%] Assume $f(x)=100$ on $-1\le x \le 1$ and $f(x)=0$ otherwise. Compute the Fourier transform $F(w)$ of $f(x)$.

{\bf Graded Details}: (1) Transform formula, (2) Integration details, (3) Answer.

\medskip

\BF{\bf 5(c)} [50\%] The heat kernel $g(x)$ and the error function $\erf(x)$
are defined by the equations
$$
 g(x)=\sqrt{\frac{\pi}{kt}}\,e^{-\frac{x^2}{4kt}},\quad
  \erf(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-z^2}dz.
$$
   Solve the infinite
rod heat conduction problem
$$
 \left\{ \begin{array}{lcllll}
 u_t(x,t) &=& \dd\frac{1}{16} u_{xx}(x,t), & -\infty < x < \infty, \quad t>0,\\[2ex]
 u(x,0)&=& f(x), & -\infty < x < \infty, \\[1.5ex]
 f(x) &=& \left\{\begin{array}{ll} 50 & 0<x<1, \\ 100 & -1 < x < 0,\\ 0 & \mbox{otherwise}\end{array}\right.
 \end{array}\right.
$$
{\bf Graded Details}: (1) Fourier Transform method, (2) Convolution, (3) Heat kernel use, (4) Error function methods, (5) Final answer, expressed in terms of the error function.

\BF{\bf 5(d)} [10\%] Compute the limits at $x$ equal to infinity and minus infinity of $u(x,t)$.

\BF{\bf 5(e)} [10\%] Compute the limit $u(x,0+)$ for each $x$ in $-\infty < x < \infty$.

\end{exercises}
\EJECTEND
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