Non-linear integral equation approach for the boundary reconstruction in double-connected planar domains
We consider the reconstruction of an interior curve from the given Cauchy data of a harmonic function on the exterior boundary of the given planar domain. With the help of Green’s function and potential theory the non-linear boundary problem is reduced to the system of non-linear boundary integral equations. The three iterative algorithms are developed for its numerical solution. We find the Frech´et derivatives for the corresponding operators and show unique solviability of the linearized systems. Full discretization of the systems is realized by a trigonometric quadrature method. Due to the inherited ill-possedness in the obtained system of linear equations we apply the Tikhonov regularization. The numerical results show that the proposed methods give a good accuracy of reconstructions with an economical computational cost.