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
\[
\begin{gathered}
  \phi(x,0)=0,\qquad \frac{\partial\phi}{\partial t}(x,0)=0,\\
  \phi(0,t)=0,\qquad \phi(l,t)=1.
\end{gathered}
\]
\begin{enumerate}
\item[(a)] Show that the Laplace transform of the solution \(\widehat{\Phi}(x,s)\) is given by
\[
  \widehat{\Phi}(x,s)=\frac{\sinh sx}{s\sinh sl}.
\]
\item[(b)] Obtain the solution \(\phi(x,t)\) by inverting the Laplace transform using a Bromich
integral and show that it can be expressed as the infinite series
\[
  \phi(x,t)=\frac{x}{l}+\sum_{n=1}^{\infty}\frac{2(-1)^n}{n\pi}
  \sin\left(\frac{n\pi x}{l}\right)\cos\left(\frac{n\pi t}{l}\right).
\]
\end{enumerate}