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\title{Final Exam \\ M 3150-001, Spring 2017}
\author{Name ..............................................................................}
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~~

\noindent
Solve the following problems. Show the method (the derivation), 
 

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\begin{enumerate}


\item

 Find the  Fourier series of
 $$  f(x)= \left\{ \begin{array}{ll}
0& \quad \mbox{if } x \in (-1, 0) \\
1 &\quad \mbox{if } x \in (0, 1)
\end{array} \right., \quad x \in (-1, 1). $$
 Evaluate the first four coefficients of the series.
 
 \newpage
 
 \item

A 1-periodic function $g(t)$, $t\in (0, 1)$ is presented by its Fourier series 
$$ g(t)=\sum_{n=1}^\infty a_n \sin(\pi \, n\, \,t) $$ 
with some unknown coefficients $a_n$. This function  satisfies the differential equation 
$$ g''(t)+ c^2 \, g(t) = \sum_{n=1}^\infty \frac{1}{\pi^3 n^3} \sin(\pi \, n\, t)$$
Find the coefficients $a_n$. 


 \newpage



\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, 1], ~ t \in [0, \infty).
\\
& \mbox{Boundary cond.} & u(0, t)=0, \quad u(1, t)=0. \\
& \mbox{Initial cond.}~~~~ & u(x, 0) = 2 \sin \left( 7  \pi  x\,\right).
\end{eqnarray*}

\newpage





\item 
Determine the equilibrium temperature distribution $u(r, \theta)$ {\em outside} of a disk of
radius $a = 1$, if the boundary temperature is $u(1, \theta) = 2+ \cos(5\theta)$, and the temperature at infinity is bounded. Show the separation of variables technique. 

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 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
the initial shape $u(x,0)=f(x)$ is
$f(x) = \sin (\pi x) - 2 \sin(5 \pi x) $, and \newline
the initial speed is zero, $\frac{\partial u}{\partial t}|_{t=0}
=0. $\newline
Find $u(x, t)$.


\newpage
\item 
The initial distribution of the temperature in an infinite
bar is
 $$
u(x,0)= e^{- 4 x^2} $$
The diffusivity constant $k$ is equal two, $k=2$. 
Find the temperature distribution $u(x, t)$ at any time $t$ using
convolution formula. 

~
\newline
{\em Hint:} The inverse Fourier transform ${\cal F}^{-1}$ of $ G(\omega)= e^{-k \omega^2 t}$ is
$$ 
~{\cal F}^{-1} (G(\omega))=
\sqrt{\frac{\pi}{k t} }e^{-\frac{x^2}{4 k t} }
$$
\end{enumerate}



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