We consider the inverse problem of reconstructing the interior boundary curve of an arbitrary-shaped annulus from overdetermined Cauchy data on the exterior boundary curve. For the approximate solution of this ill-posed and nonlinear problem we propose a regularized Newton method based on a boundary integral equation approach for the initial boundary value problem for the heat equation. A theoretical foundation for this Newton method is given by establishing the differentiability of the initial boundary value problem with respect to the interior boundary curve in the sense of a domain derivative. Numerical examples indicate the feasibility of our method.

We consider the inverse problem to determine the shape of an insulated inclusion within a heat conducting medium from overdetermined Cauchy data of solutions for the heat equation on the accessible exterior boundary of the medium. For the approximate solution of this ill-posed and nonlinear problem we propose a regularized Newton iteration scheme based on a boundary integral equation approach for the initial Neumann boundary value problem for the heat equation. For a foundation of the Newton method we establish the differentiability of the solution to the initial Neumann boundary value problem with respect to the interior boundary curve in the sense of a domain derivative and investigate the injectivity of the linearized mapping. Some numerical examples for the feasibility of the method are presented.

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In this paper we describe a fully discrete quadrature method for the numerical solution of a hypersingular integral equation of the rst kind for the scattering of time-harmonic elastic waves by a cavity crack. We establish convergence of the method and prove error estimates in a H older space setting. Numerical examples illustrate the convergence results.

We present a unified boundary integral approach for the stable numerical solution of the ill-posed Cauchy problem for the heat and wave equation. The method is based on a transformation in time (semi-discretisation) using either the method of Rothe or the Laguerre transform, to generate a Cauchy problem for a sequence of inhomogenous elliptic equations; the total entity of sequences is termed an elliptic system. For this stationary system, following a recent integral approach for the Cauchy problem for the Laplace equation, the solution is represented as a sequence of single-layer potentials invoking what is known as a fundamental sequence of the elliptic system thereby avoiding the use of volume potentials and domain discretisation. Matching the given data, a system of boundary integral equations is obtained for finding a sequence of layer densities. Full discretisation is obtained via a Nyström method together with the use of Tikhonov regularization for the obtained linear systems. Numerical results are included both for the heat and wave equation confirming the practical usefulness, in terms of accuracy and resourceful use of computational effort, of the proposed approach.