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\year=1999 \month=2 \day=9
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    AS554\quad Irreversible Processes in Plasmas\\
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\section*{Problem Set \#1 (due Feb.~16, 1999)}
\subsection*{Question 1}

Solve the PDE
$$yu_x+xu_y=u-1$$
subject to the initial conditions
$$u=x^2+y+1\hbox{ on } y=2x.$$
(Here, $u_x={\partial u/ \partial x}$, etc.)  [Hint: use method of
characteristics.]

\subsection*{Question 2}

Suppose $X_1$ and $X_2$ are independent random variables uniformly
distributed in $[0,1]$.  What are the distributions of

\noindent (a) $X_1 + X_2$,

\noindent (b) $\max(X_1,X_2)$,

\noindent (c) $\sqrt{X_1}$?

\subsection*{Question 3}

Use Fourier transform techniques to solve the diffusion equation
$$u_t=Du_{xx}$$
with initial condition $u(t=0)=\delta(x)$.
Make sure (a) that you know where the $k$ and $\omega$ contours go,
(b) that the solution satisfies causality, (c) that you obtain the
same answer if you do the $k$ integration first and if you do the
$\omega$ integration first.

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