Math 6410 • ODE

Paint flowing down a wall has thickness \(u(x,t)\) satisfying (for \(t>0\))
\[
  \frac{\partial u}{\partial t}+u^2\frac{\partial u}{\partial x}=0.
\]
\begin{enumerate}
\item[(a)] Show that the characteristics are straight lines and that the Rankine--Hugoniot
condition on a shock \(x=S(t)\) is
\[
  \frac{dS}{dt}=\frac{\left[u^3/3\right]_-^+}{[u]_-^+}.
\]

\item[(b)] A stripe of paint is applied at \(t=0\) so that
\[
  u(x,0)=
  \begin{cases}
    0, & x<0\text{ or }x>1\\
    1, & 0<x<1
  \end{cases}
\]
For sufficiently small \(t\), determine \(u\) in the domains \(x<0\), \(0<x<t\),
\(t<x<S(t)\) and \(S(t)<x\), where the shock is \(x=S(t)=1+t/3\)

\item[(c)] Explain why the solution changes at \(t=3/2\) and show that thereafter
\(\dot{S}=S/3t\).
\end{enumerate}