We consider the inverse problem of reconstructing the interior boundary curve of a doubly connected bounded 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 Fréchet 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-posedness 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.

中文翻译：

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热方程逆边值问题的非线性积分方程法

我们考虑从外部边界曲线上的温度和热通量的知识重建双连通有界域的内部边界曲线的逆问题。及时使用拉盖尔变换会导致一系列平稳逆问题。然后，改进的单层 ansatz 的应用将问题简化为非线性边界积分方程组的序列。迭代算法被开发用于获得的积分方程的数值解。我们找到相应积分算子的 Fréchet 导数，并展示了线性化方程的唯一可解性。完全离散化是通过三角正交方法实现的。由于线性方程派生系统的继承病态，我们应用 Tikhonov 正则化。数值结果表明，所提出的方法产生准确和稳定的重建。