ON THE NON-LINEAR INTEGRAL EQUATION METHOD FOR THE INVERSE BOUNDARY RECONSTRUCTION PROBLEM

Анотація

This paper is concerned with the reconstruction of an interior boundary curve in a planar double-connected domain from given Cauchy data of the harmonical function prescribed on the outer boundary. By employing single-layer potential representations, we reduce the problem to a system of non-linear boundary integral equations. Two iterative Newton's methods are proposed and implemented for solving the system. The Frechet derivatives of the corresponding integral operators are computed. Full discretization is
achieved using quadrature and collocation methods. To resolve ill-posedness of the resulting linear system, we apply Phillips-Tikhonov regularization. An initial approximation for the interior boundary is found by the line method and a size estimation algorithm. Numerical examples confirm the stability, accuracy, and computational eficiency of the proposed approaches.