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\title{Make-Up Midterm. M 3150-001, Spring 2017}
%\author{Andrej Cherkaev}
\date{}
\maketitle

Your name (please print) ..................................................

~~

\noindent
Solve the following problems. Show the method (the derivation), \underline{do not simply copy} the final formulas.
\begin{enumerate}
\item
Find $u(x, t)$, that satisfies the following conditions:
\begin{eqnarray*}
& \mbox{Heat Equation  } &\frac{\partial u}{\partial t}= 2 \frac{\partial^2 u}{\partial x ^2}, \quad x  \in [0, 2], ~ t \in [0, \infty).
\\
& \mbox{Boundary cond.} & u(0, t)=0, \quad u(2, t)=0. \\
& \mbox{Initial cond.}~~~~ & u(x, 0) = 3 \sin \left( \pi  x\,\right).
\end{eqnarray*}








\newpage
\item 
Determine the equilibrium temperature distribution $u(r, \theta)$ in a disk of
radius a = 1, if the boundary temperature is $u(1, \theta) = 3 \cos(3\theta)$.
Note: Laplacian in polar coordinates has the form

$$
\nabla u(r, \theta)=
\frac{1}{r} \frac{\partial}{\partial r}\left( r  \frac{\partial u}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 u}{\partial \theta^2}.
$$

\newpage
\item

 Find the sine  Fourier series of
 $$  f(x)= \left\{ \begin{array}{ll}
1& \quad x \in [0, 1] \\
0 &\quad x \in [1,3]
\end{array} \right., \quad x \in (0, 3). $$
 Find the first three nonzero coefficients of the series.

\newpage
\item A string vibrates according to the equation 
$$\frac{\partial^2 u}{\partial t ^2}= 9
\frac{\partial^2 u}{\partial x ^2},
$$
\newline
where
$u(x, t)$ is the deflection, $x\in [0, 1]$, $t\in [0, \infty)$,
\newline
the ends are fixed: $u(0, t)=0$, $ u(1, t)=0$,
\newline
thye initial shape $u(x,t)$ is
$u(x, 0) = 2\sin(5 \pi x)$, \newline
the initial speed is zero, $\frac{\partial u}{\partial t}|_{t=0}
=0. $\newline
Find $u(x, t)$.



\end{enumerate}



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