  We consider the inverse problem of reconstructing the interior boundary curve
of a doubly connected domain from the knowledge of the temperature and the
thermal flux on the exterior boundary curve. The use of the Laguerre transform
in time leads to a sequence of stationary inverse problems. Then, the
application of the modified single-layer ansatz, reduces the problem to a
sequence of systems of non-linear boundary integral equations. An iterative
algorithm is developed for the numerical solution of the obtained integral
equations. We find the Fr\'echet derivative of the corresponding integral
operator and we show the unique solvability of the linearized equation. Full
discretization is realized by a trigonometric quadrature method. Due to the
inherited ill-possedness of the derived system of linear equations we apply the
Tikhonov regularization. The numerical results show that the proposed method
produces accurate and stable reconstructions.
