Journal of Engineering Mathematics

, Volume 119, Issue 1, pp 255–268 | Cite as

On the non-linear integral equation approach for an inverse boundary value problem for the heat equation

  • Roman Chapko
  • Leonidas MindrinosEmail author


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.


Boundary reconstruction Laguerre transform Non-linear boundary integral equations Tikhonov regularization Trigonometric quadrature method 


Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Faculty of Applied Mathematics and InformaticsIvan Franko National University of LvivLvivUkraine
  2. 2.Johann Radon Institute for Computational and Applied Mathematics (RICAM)Austrian Academy of SciencesLinzAustria

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