Math 6410 • ODE

Consider the scalar linear equation
\[
  a\frac{\partial u}{\partial x}+b\frac{\partial u}{\partial y}=\alpha u
\]
with \(a,b,\alpha\) differentiable functions of \(x,y\). Suppose that \(u\) is prescribed
on some arc \(\Gamma\) (Cauchy data).
\begin{enumerate}
\item[(a)] By introducing a test function \(\psi\) that vanishes on an arbitrary curve
\(\gamma\) and applying Green's theorem to the domain \(D\) bounded by
\(\partial D=\Gamma\cup\gamma\), derive the integral equation for a weak solution to
the Cauchy problem.

\item[(b)] Extend the analysis to the case where \(u\) is discontinuous across an open
curve \(C_0\) within the domain \(D\). Hence show that \(C_0\) is a characteristic projection.

\item[(c)] Write down the generalization of part (b) to the case of the quaslinear
equation
\[
  \frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}=c
\]
where \(P,Q,c\) are differentiable functions of \(x,y,u\), and hence derive the
Rankine--Hugonoit condition.
\end{enumerate}