  We present an efficient integral equation approach to solve the heat
equation, $u_t (\x) - \Delta u(\x) = F(\x,t)$, in a two-dimensional, multiply
connected domain, and with Dirichlet boundary conditions. Instead of using
integral equations based on the heat kernel, we take the approach of
discretizing in time, first. This leads to a non-homogeneous modified Helmholtz
equation that is solved at each time step. The solution to this equation is
formulated as a volume potential plus a double layer potential.The volume
potential is evaluated using a fast multipole-accelerated solver. The boundary
conditions are then satisfied by solving an integral equation for the
homogeneous modified Helmholtz equation. The integral equation solver is also
accelerated by the fast multipole method (FMM). For a total of $N$ points in
the discretization of the boundary and the domain, the total computational cost
per time step is $O(N)$ or $O(N\log N)$.
