Math 6420 • PDE

Consider the wave equation
\[
  \frac{\partial^2\phi}{\partial t^2}=\frac{\partial^2\phi}{\partial x^2},
  \qquad 0\le x\le l
\]
with the following boundary conditions
\[
\begin{gathered}
  \phi(x,0)=0,\qquad \frac{\partial\phi}{\partial t}(x,0)=0,\\
  \phi(0,t)=0,\qquad \phi(l,t)=f(t).
\end{gathered}
\]
\begin{enumerate}
\item[(a)] Show that the Laplace transform of the solution \(\widehat{\Phi}(x,s)\) is given by
\[
  \widehat{\Phi}(x,s)=\frac{\widehat{F}(s)\sinh sx}{\sinh sl}
\]
where \(\widehat{F}(s)\) is the Laplace transform of \(f(t)\).
\item[(b)] Suppose that \(\phi_0(x,t)\) is the solution when \(f(t)=1\). Show that the solution
for general \(f\) is given by
\[
  \phi(x,t)=\int_0^t \frac{\partial\phi_0}{\partial t'}(x,t')f(t-t')\,dt'.
\]
\end{enumerate}