The Dirichlet-to-Neumann map for the heat equation on a moving boundary

Abstract

We construct the Dirichlet-to-Neumann map for a moving initial/boundary value problem for the linear heat equation. The unknown Neumann boundary value is expressed in terms of the Dirichlet boundary value and of the initial condition through the solution of a linear Volterra integral equation of the second type. This equation involves an exponentially decaying kernel, and this leads to efficient numerical integration, as illustrated by some concrete examples.