Math 6420 • PDE

\begin{enumerate}
\item[(a)] Introducing the discrete Fourier transform
\(U(k,t)=\sum_{n=-\infty}^{\infty}e^{ikn}u_n(t)\), \(0\le k<2\pi\), show that the
general solution to the discretized heat equation
\[
  \frac{du_n}{dt}=\frac{1}{h^2}\{u_{n+1}-2u_n+u_{n-1}\},\qquad -\infty<n<\infty,
\]
can be written in the form
\[
  u_n(t)=\int_0^{2\pi} U(k,0)e^{-\epsilon(k)t}e^{-ikn}\,\frac{dk}{2\pi}
\]
and calculate the function \(\epsilon(k)\).
\item[(b)] Using the fact that \(U(k,t)\) and \(u_n(t)\) form a transform pair for fixed \(t\),
prove the identity
\[
  \sum_{n=-\infty}^{\infty}e^{i(k-k')n}=2\pi\delta(k-k').
\]
Hence, find \(u_n(t)\) given the initial data \(u_n(0)=\sin(2\pi n/p)\) for some fixed integer
\(p\).
\end{enumerate}