Uniform asymptotic approximations of Green's functions in a long rod

Asymptotic approximations for Green's function for the operator −Δ in a long rod are derived. These approximations are uniformly valid over the whole domain including the end regions of the rod. Connections are established between the asymptotic approximations in a long rod and the asymptotics in thin domains. Overview of asymptotic approximations of Green's kernels in a domain with a small hole and domains with singularly perturbed smooth or conical boundaries is also given. Copyright © 2008 John Wiley & Sons, Ltd.