Reconstruction of a moving boundary from Cauchy data in one-dimensional heat equation

Abstract

In this article, we propose a new numerical method for determining a moving boundary from Cauchy data in a one-dimensional heat equation where an initial temperature is not required. The numerical scheme is based on the use of fundamental solutions of the heat equation as basis functions. In order to regularize the ill-conditioned linear system of equations resulting by collocating boundary data, we successfully apply the Tikhonov regularization with the generalized cross-validation parameter choice rule to obtain a stable numerical approximation to the moving boundary. The uniqueness of the moving boundary is also proved under some assumptions.