Document Type : Research Paper

**Authors**

School of Mathematics, Iran University of Science and Technology, Tehran, Iran.

**Abstract**

In this paper, we consider an inverse boundary value problem for two-dimensional heat equation in an annular domain. This problem consists of determining the temperature on the interior boundary curve from the Cauchy data (boundary temperature and heat flux) on the exterior boundary curve. To this end, the boundary integral equation method is used. Since the resulting system of linear algebraic equations is ill-posed, the Tikhonov first-order regularization procedure is employed to obtain a stable solution. Determination of regularization parameter is based on L-curve technique. Some numerical examples for the feasibility of the proposed method are presented.

**Keywords**

**Main Subjects**

O. M. Alifanov, E. A. Artyukhin and S. V. Rumyantsev, Extreme Methods for Solving Ill-Posed Problems with Applications to Inverse Heat Transfer Problems, Begell House, New York, 1995.

S. Andrieux, T. Baranger and A. Ben Abda, Solving Cauchy problems by minimizing an energy-like functional, Inverse Problems **22 **(2006), no. 1, 115--133.

G. Bastay, V. A. Kozlov and B. O. Turesson, Iterative methods for an inverse heat conduction problem, J. Inverse Ill-Posed Probl. **9 **(2001), no. 4, 375--388.

C. Babenko, R. Chapko and B. T. Johansson, On the numerical solution of the Laplace equation with complete and incomplete Cauchy data using integral equations, CMES: Comput. Model. Eng. Sci. **101 **(2014), no. 5, 299--317.

A.S. Carasso, Determining surface temperatures from interior observations, SIAM J. Appl. Math. 42 (1982), no. 3, 558--574.

H. T. Chen, S. Y. Lin and L. C. Fang, Estimation of surface temperature in two-dimensional inverse heat conduction problems, Int. J. Heat Mass Transfer **44 **(2001) 1455--1463.

R. Chapko, R. Kress and J. R. Yoon, On the numerical solution of an inverse boundary value problem for the heat equation, Inverse Problems **14 **(1998), no. 4, 853--867.

R. Chapko and R. Kress, On a quadrature method for a logarithmic integral equation of the first kind, Contributions in numerical mathematics, 127--140, World Sci. Ser. Appl. Anal., 2, World Sci. Publ., River Edge, NJ, 1993.

R. Chapko, B. T. Johansson and Y. Savka, On the use of an integral equation approach for the numerical solution of a Cauchy problem for Laplace equation in a doubly connected planar domain, Inverse Probl. Sci. Eng. **22 **(2014), no. 1, 130--149.

R. Chapko, B. T. Johansson, On the numerical solution of a Cauchy problem for the

Laplace equation via a direct integral equation approach, Inverse Probl. Imaging **6 **(2012), no. 1, 25--38.

L. B. Drenchev and J. Sobczak, Inverse Heat Conduction Problems and Application to Estimate of Heat Parameters in 2-D Experiments, in 2nd Int. Conf. High Temperature Capillarity, Cracow, Poland, 29 June-2 July 1997, Foundry Research Institute, Krakow (Poland), 1998, pp. 355-361.

H. W. Engl, Discrepancy principles for Tikhonov regularization of ill-posed problems leading to optimal convergence rates, J. Optim. Theory Appl. **52 **(1987), no. 2, 209--215.

H. Fenga and D. Jiangb, Convergence rates of Tikhonov regularization for parameter identi_cation in a Maxwell system, Appl. Anal. **94 **(2015), no. 2, 361--375.

R. B. Guenther and J. W. Lee, Partial Differential Equations of Mathematical Physics and Integral Equations, Dover Publications, Inc., Mineola, 1996.

D. N. H_ao, Methods for inverse heat conduction problems Habilitationsschrift, University of Siegen, Siegen, 1996, Methoden und Verfahren der Mathematischen Physik, 43, Peter Lang, Frankfurt am Main, 1998.

P. C. Hansen, Analysis of discrete ill-posed problems by means of the L-curve, SIAM Rev. **34 **(1992), no. 4, 561--580.

P. C. Hansen and D. P. O'Leary, The use of the L-curve in the regularization of discrete ill-posed problems, SIAM J. Sci. Comput. **14 **(1993), no. 6, 1487--1503.

Y. C. Hon and T. Wei, A fundamental solution method for inverse heat conduction problem, Eng. Anal. Bound. Elem. **28 **(2004) 489--495.

G. C. Hsiao and J. Saranen, Boundary integral solution of the two-dimensional heat equation, Math. Methods Appl. Sci. **16 **(1993), no. 2, 87--114.

X. Z. Jia and Y. B. Wang, A Boundary Integral Method for Solving Inverse Heat Conduction Problem, J. Inverse Ill-Posed Probl. **14 **(2006), no. 4, 375--384.

S. I. Kabanikhin, Definitions and examples of inverse and ill-posed problems, J. Inverse Ill-Posed Probl. **16 **(2008), no. 4, 317--357.

R. Kress and I. H. Sloan, On the numerical solution of a logarithmic integral equation of the first kind for the Helmholtz equation, Numer. Math. **66 **(1993), no. 2, 199--214.

K. Kunisch and J. Zou, Iterative choices of regularization parameters in linear inverse problems, Inverse Problems **14 **(1998), no. 5, 1247--1264.

D. Lesnic, L. Elliott and D. B. Ingham, Application of the boundary element method to inverse heat conduction problems, Int. J. Heat Mass Transfer **39 **(1996) 1503--1517.

N. N. Lebedev, Special Functions and Their Applications, Revised English edition, Translated and edited by Richard A. Silverman, Prentice Hall Inc., Englewood Cliffs, 1965.

C. S. Liu, A dynamical Tikhonov regularization for solving ill-posed linear algebraic systems, Acta Appl. Math. **123 **(2013) 285--307.

D. A. Murio, The mollification method and the numerical solution of ill-posed problems, John Wiley & Sons, Inc., New York, 1993.

H. J. Reinhardt, D. N. Hao, J. Frohne and F. T. Suttmeier, Numerical solution of inverse heat conduction problems in two spatial dimensions, J. Inverse Ill-Posed Probl. **15 **(2007), no. 2, 181--198.

J. C. Saut and B. Scheurer, Remarques sur un theoreme de prolongement unique de Mizohata, C. R. Acad. Sci. Paris Ser. I Math. **296 **(1983), no. 6, 307--310.

A. N. Tikhonov and V. Y. Arsenin, Solutions of ill-posed problems, Translated from the Russian, Preface by translation editor Fritz John, Scripta Series in Mathematics, V. H. Winston & Sons, Washington, D.C., John Wiley & Sons, New York-Toronto, Ont.-London, 1977.

October 2016

Pages 1039-1057

**Receive Date:**21 February 2015**Revise Date:**13 June 2015**Accept Date:**20 June 2015