Question

Using separation of variables, write down a complete list of L^2 eigenfunctions and of eigenvalues for the Laplacian on the cylinder D X [-1, 1], with homogeneous Dirichlet boundary conditions, where D is the (two-dimensional) disk centered at the origin of radius 2.

b) Use this to solve the heat equation

partial u / partial t = Delta u

on this cylinder with homogeneous Dirichlet boundary conditions, with initial data u(x, y, z, 0) = z, where z is the third coordinate, corresponding to the height of the cylinder.

Answer 1