Math 6420 • PDE

\begin{enumerate}
\item[(a)] Find the spherically symmetric eigenfunctions and corresponding eigenvalues for the
Laplacian on a spherical domain of radius \(R\) with Dirichlet boundary conditions \(u(R)=0\).

\item[(b)] Use these eigenfunctions to solve the heat equation on a spherical domain of radius
\(R\), with \(u(R,t)=U_0\) and \(u(r,0)=0\).

\item[(c)] Estimate the time \(t\) at which \(u(0,t)\) is \(\frac{U_0}{2}\). How does this time depend
on the radius \(R\)?
\end{enumerate}