Optimal error bound and Fourier regularization for identifying an unknown source in the heat equation

In this paper we investigate a problem of the identification of an unknown source from one supplementary temperature measurement at a given instant of time for the transient heat equation. Under an a priori condition we answer the question concerning the best possible accuracy for the problem. The Fourier regularization method is utilized for solving the problem, and its convergent rate is analyzed. Numerical results are presented to illustrate the accuracy and efficiency of the method.

[1]  M. Yamamoto,et al.  Stability analysis for determining a source term in a 1-D advection-dispersion equation , 2006 .

[2]  Xiang-Tuan Xiong,et al.  Fourier regularization for a backward heat equation , 2007 .

[3]  J. Cannon,et al.  Structural identification of an unknown source term in a heat equation , 1998 .

[4]  M. Ivanchov.  The inverse problem of determining the heat source power for a parabolic equation under arbitrary boundary conditions , 1998 .

[5]  Masahiro Yamamoto,et al.  Conditional stability in determining a heat source , 2004 .

[6]  M. Yamamoto,et al.  Conditional stability in determination of force terms of heat equations in a rectangle , 1993 .

[7]  U. Tautenhahn,et al.  On the Optimality of Regularization Methods for Solving Linear Ill-Posed Problems , 1994 .

[8]  Chu-Li Fu,et al.  Regularization strategies for a two-dimensional inverse heat conduction problem , 2007 .

[9]  J. Cannon,et al.  An inverse problem for the heat equation , 1986 .

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

[11]  R. Gorenflo,et al.  On Optimal Regularization Methods for Fractional Differentiation , 1999 .

[12]  Daniel Lesnic,et al.  The boundary-element method for the determination of a heat source dependent on one variable , 2006 .

[13]  Leevan Ling,et al.  Identification of source locations in two-dimensional heat equations , 2006 .

[14]  A. M. Denisov,et al.  Determination of the unknown sources in the heat-conduction equation , 1997 .

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

[16]  Gongsheng Li.  Data compatibility and conditional stability for an inverse source problem in the heat equation , 2006, Appl. Math. Comput..

[17]  C. Fu,et al.  Fourier regularization method for solving a Cauchy problem for the Laplace equation , 2008 .

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

[19]  Semyon Tsynkov,et al.  Inverse source problem and active shielding for composite domains , 2007, Appl. Math. Lett..

[20]  Dang Duc Trong,et al.  Nonhomogeneous heat equation: Identification and regularization for the inhomogeneous term , 2005 .

[21]  Gennadi Vainikko,et al.  On the Optimality of Methods for Ill-Posed Problems , 1987 .

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

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

[24]  D. A. Murio,et al.  Source term identification in 1-D IHCP , 2004 .