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\title{To the Final: Problems for practice. M 3150-001, Spring 2017}
\author{Andrej Cherkaev}
\date{}
\maketitle

~~

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







\item 
Determine the equilibrium temperature distribution $u(r, \theta)$ {\em outside} 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

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


\item

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

A function $g(x)$, $x\in (0, 1)$ is presented by its Fourier series 
$$ g(x)=\sum_{n=1}^\infty a_n sin(\pi \, n\, \) $$ with some unknown coefficients $a_n$. This function  safisfies the differential equation (harmonic oscillator)
$$ g''(x)+ c \, g(x) = \sum_{n=1}^\infty \frac{1}{\pi^3 n^3} sin(\pi \, n\, \)$$
Find the coefficients $a_n$. 


 

\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 $u(x,t)$ is
$u(x, 0) = 2\sin(5 \pi x)$, and \newline
the initial speed is zero, $\frac{\partial u}{\partial t}|_{t=0}
=0. $\newline
Find $u(x, t)$.
\item 
The initial distribution of the temperature in an infinite
bar is

 $$
u(x,0)= \left\{ \begin{array}{ll}
1& \quad \mbox{if } x \in [-1, 1] \\
0 &\quad \mbox{if } x \not\in [-1, 1]
\end{array} \right. $$

Find the temperature distribution $u(x, t)$ at any time $t$. The diffusivity constant is one. 
\end{enumerate}



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