Abstract
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.
Similar content being viewed by others
References
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 55–69
Chapko R, Kress R, Yoon JR (1999) An inverse boundary value problem for the heat equation: the Neumann condition. Inverse Probl 15(4):1033
Colton D, Kress R (2013) Inverse acoustic and electromagnetic scattering theory, 3rd edn, vol 93. Applied mathematical sciences. Springer, Berlin
Ivanyshyn O, Johansson BT (2007) Nonlinear integral equation methods for the reconstruction of an acoustically sound-soft obstacle. J Integral Equ Appl 19:289–308
Ivanyshyn O, Kress R (2006) Nonlinear integral equations for solving inverse boundary value problems for inclusions and cracks. J Integral Equ Appl 18:13–38
Kress R, Rundell W (2005) Nonlinear integral equations and the iterative solution for an inverse boundary value problem. Inverse Probl 21(4):1207
Heck H, Nakamura G, Wang H (2012) Linear sampling method for identifying cavities in a heat conductor. Inverse Probl 28(7):075014
Ikehata M, Kawashita M (2009) The enclosure method for the heat equation. Inverse Probl 25(7):075005
Nakamura G, Wang H (2013) Linear sampling method for the heat equation with inclusions. Inverse Probl 29(10):104015
Nakamura G, Wang H (2015) Reconstruction of an unknown cavity with robin boundary condition inside a heat conductor. Inverse Probl 31(12):125001
Nakamura G, Wang H (2017) Numerical reconstruction of unknown Robin inclusions inside a heat conductor by a non-iterative method. Inverse Probl 33(5):055002
Wang H, Li Y (2018) Numerical solution of an inverse boundary value problem for the heat equation with unknown inclusions. J Comput Phys 369:1–15
Altundag A, Kress R (2012) On a two-dimensional inverse scattering problem for a dielectric. Appl Anal 91(4):757–771
Chapko R, Gintides D, Mindrinos L (2018) The inverse scattering problem by an elastic inclusion. Adv Comput Math 44:453–476
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 2:7–20
Gintides D, Mindrinos L (2019) The inverse electromagnetic scattering problem by a penetrable cylinder at oblique incidence. Appl Anal 98:781–798
Johansson BT, Sleeman BD (2007) Reconstruction of an acoustically sound-soft obstacle from one incident field and the far-field pattern. IMA J Appl Math 72(1):96–112
Chapko R, Kress R, Yoon JR (1998) On the numerical solution of an inverse boundary value problem for the heat equation. Inverse Probl 14(4):853
Chapko R, Johansson BT (2018) A boundary integral equation method for numerical solution of parabolic and hyperbolic Cauchy problems. Appl Numer Math 129:104–119
Chapko R, Kress R (1997) Rothe’s method for the heat equation and boundary integral equations. J Integral Equ Appl 9:47–69
Friedman A (1964) Partial differential equations of parabolic type. Prentice-Hall, Englewood Cliffs
Kress R (2014) Linear integral equations, 3rd edn. Springer, New York
Ladyzenskaja OA, Solonnikov VA, Uralceva NN (1968) Linear and quasilinear equations of parabolic type, vol 23. AMS Publications, Providence
Lions JL, Magenes E (1972) Non-homogeneous boundary value problems and applications, vol 2. Springer, Berlin
Bryan K, Caudill LF Jr (1997) Uniqueness for a boundary identification problem in thermal imaging. Electron J Differ Equ 1:23–39
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 DC
Potthast R (1994) Fréchet differentiability of boundary integral operators in inverse acoustic scattering. Inverse Probl 10(2):431
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 130:7–17
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Chapko, R., Mindrinos, L. On the non-linear integral equation approach for an inverse boundary value problem for the heat equation. J Eng Math 119, 255–268 (2019). https://doi.org/10.1007/s10665-019-10028-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10665-019-10028-4