On the non-linear integral equation approach for an inverse boundary value problem for the heat equation
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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.
KeywordsBoundary reconstruction Laguerre transform Non-linear boundary integral equations Tikhonov regularization Trigonometric quadrature method
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The authors declare that they have no conflict of interest.
- 1.Chapko R, Kress R (2000) On the numerical solution of initial boundary value problems by the Laguerre transformation and boundary integral equations. In: Agarwal RP, O’Regan D (eds) Integral and integrodifferential equations: theory, methods and applications, vol 2. Series in mathematical analysis and application. Gordon and Breach Science Publishers, Amsterdam, pp 55–69zbMATHGoogle Scholar
- 15.Chapko R, Ivanyshyn YO, Kanafotskyi TS (2016) On the non-linear integral equation approaches for the boundary reconstruction in double-connected planar domains. J Numer Appl Math 2:7–20Google Scholar
- 26.Abramowitz M, Stegun IA (1972) Handbook of mathematical functions with formulas, graphs, and mathematical tables, vol 55. National Bureau of Standards Applied Mathematics series, Washington DCGoogle Scholar
- 28.Chapko R, Ivanyshyn YO, Vavrychuk V (2019) On the non-linear integral equation method for the reconstruction of an inclusion in the elastic body. J Numer Appl Math 130:7–17Google Scholar