Fourier regularization method of a sideways heat equation for determining surface heat flux

We consider a non-standard inverse heat conduction problem in a quarter plane which appears in some applied subjects. We want to know the surface heat flux in a body from a measured temperature history at a fixed location inside the body. This is an exponentially ill-posed problem in the sense that the solution (if it exists) does not depend continuously on the data. A Fourier regularization method together with order optimal logarithmic stability estimates is given. A numerical example shows that the theoretical results are valid.

[1]  Xiang-Tuan Xiong,et al.  Central difference schemes in time and error estimate on a non-standard inverse heat conduction problem , 2004, Appl. Math. Comput..

[2]  Xiang-Tuan Xiong,et al.  Wavelet and spectral regularization methods for a sideways parabolic equation , 2005, Applied Mathematics and Computation.

[3]  P. Knabner,et al.  STABILITY ESTIMATES FOR ILL-POSED CAUCHY PROBLEMS FOR PARABOLIC EQUATIONS , 1987 .

[4]  Lars Eldén,et al.  Numerical solution of the sideways heat equation by difference approximation in time , 1995 .

[5]  Heinz W. Engl,et al.  Inverse and Ill-Posed Problems , 1987 .

[6]  Thomas I. Seidman,et al.  An 'optimal filtering' method for the sideways heat equation , 1990 .

[7]  Hans-Jürgen Reinhardt,et al.  On a sideways parabolic equation , 1997 .

[8]  P. Mathé,et al.  Geometry of linear ill-posed problems in variable Hilbert scales Inverse Problems 19 789-803 , 2003 .

[9]  B. Blackwell,et al.  Inverse Heat Conduction: Ill-Posed Problems , 1985 .

[10]  Fredrik Berntsson,et al.  Wavelet and Fourier Methods for Solving the Sideways Heat Equation , 1999, SIAM J. Sci. Comput..

[11]  Ulrich Tautenhahn,et al.  Optimality for ill-posed problems under general source conditions , 1998 .

[12]  Fu Chuli,et al.  Wavelet and error estimation of surface heat flux , 2003 .

[13]  Thorsten Hohage,et al.  Regularization of exponentially ill-posed problems , 2000 .

[14]  J. Beck.  Nonlinear estimation applied to the nonlinear inverse heat conduction problem , 1970 .

[15]  U. Tautenhahn,et al.  Optimal stable approximations for the sideways heat equation , 1997 .

[16]  Chu-Li Fu,et al.  Wavelet regularization for an inverse heat conduction problem , 2003 .

[17]  C. Fu,et al.  Wavelets and regularization of the sideways heat equation , 2003 .

[18]  H. Engl,et al.  Regularization of Inverse Problems , 1996 .

[19]  A. Carasso.  Determining Surface Temperatures from Interior Observations , 1982 .

[20]  Chu-Li Fu,et al.  Simplified Tikhonov and Fourier regularization methods on a general sideways parabolic equation , 2004 .

[21]  L. Eldén,et al.  Solving an inverse heat conduction problem by a `method of lines' , 1997 .