Math 6410 • ODE

\begin{enumerate}
\item State d'Alembert's formula for the solutions \(u\) of the wave equation
\(u_{tt}=u_{xx}\).
\item Consider the initial boundary value problem
\[
\begin{aligned}
  u_{tt}&=u_{xx} &&\text{for }0<x<\pi,\ t>0,\\
  u(x,0)&=f(x),\qquad u_t(x,0)=0 &&\text{for }0<x<\pi,\\
  u(0,t)&=0,\qquad u(\pi,t)=0 &&\text{for }t\ge0.
\end{aligned}
\]
Use d'Alembert's formula and the parallelogram rule to define a weak solution solution
to this problem. HINT: divide the region
\(R=\{(x,t):0<x<\pi,\ t>0\}\) into subregions. What compatibility conditions are needed
upon \(f\) to ensure that the above solution is \(C\), is \(C^1\) and is \(C^2\)?
\item Show that the energy
\[
  E(t)=\frac{1}{2}\int_0^\pi (u_t^2+u_x^2)\,dx
\]
is conserved, and use this to prove that a \(C^2\) solution to the above initial boundary
value problem is unique.
\end{enumerate}