Math 6410 • ODE

Find and sketch the solution of the differential equation
\[
  u_t+uu_x=0
\]
on the domain \(-\infty<x<\infty\), \(t>0\) subject to the initial condition
\(u(0,x)=u_0(x)\), where
\begin{enumerate}
\item[(a)]
\[
  u_0(x)=
  \begin{cases}
    0; & \text{if }x<0,\\
    x, & \text{if }0<x<1,\\
    1, & \text{if }x>1;
  \end{cases}
\]
\item[(b)]
\[
  u_0(x)=
  \begin{cases}
    1; & \text{if }x<0,\\
    1-x, & \text{if }0<x<1,\\
    0, & \text{if }x>1.
  \end{cases}
\]
\end{enumerate}
For which (if any) of these initial profiles does the solution of the problem
\[
  u_t+uu_x=\epsilon u_{xx},
\]
\(\epsilon>0\), converges to a traveling wave profile? What is the speed of the
travelling wave solution?