Central difference schemes in time and error estimate on a non-standard inverse heat conduction problem. (English) Zbl 1068.65117

Authors’ summary: Inverse heat conduction problems (IHCP) are severely ill-posed in the sense that the solution (if it exists) does not depend continuously on the standard sideways heat equation. This paper remedies this by a central difference scheme in time which itself has a regularizing effect for a non-standard IHCP which appears in some applied subjects. An error estimate is obtained and the error estimate also gives information about how to choose the step length in the time discretization. A numerical example shows that the computational effect of this method is satisfactory.


65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35K05 Heat equation
35R30 Inverse problems for PDEs
Full Text: DOI


[1] Beck, J.V.; Blackwell, B.; Clair, S.R., Inverse heat conduction: ill-posed problems, (1985), Wiley New York · Zbl 0633.73120
[2] Carrasso, A., Determining surface temperature from interior observations, SIAM J. appl. math., 42, 558-574, (1982)
[3] Eldén, L.; Berntsson, F.; Regińska, T., Wavelet and Fourier methods for solving the sideways heat equation, SIAM J. sci. comput., 21, 6, 2187-2205, (2000) · Zbl 0959.65107
[4] Seidman, T.; Eldén, L., An optimal filtering method for the sideways heat equation, Inverse problems, 6, 681-696, (1990) · Zbl 0726.35053
[5] Eldén, L., Numerical solution of the sideways heat equation by difference approximation in time, Inverse problems, 11, 913-923, (1995) · Zbl 0839.35143
[6] Eldén, L., Solving an inverse heat conduction problem by “a method of lines”, J. heat. transfer, trans. ASME, 119, 406-412, (1997)
[7] Tautenhahn, U., Optimal stable approximations for the sideways heat equation, J. inv. ill-posed problems, 5, 287-307, (1997) · Zbl 0879.35158
[8] Regińska, T., Sideways heat equation and wavelets, J. comput. appl. math., 63, 209-214, (1995) · Zbl 0858.65099
[9] Fu, C.L.; Qiu, C.Y.; Zhu, Y.B., A note on “sideways heat equation and wavelets” and constant \( e\^{}\{∗\}\), Comput. math. appl., 43, 1125-1134, (2002) · Zbl 1051.65090
[10] Fu, C.L.; Qiu, C.Y., Wavelet and error estimate of surface heat flux, J. comput. appl. math., 150, 143-155, (2003) · Zbl 1019.65074
[11] Beck, J.V., Nonlinear estimation applied to the nonlinear inverse heat conduction problem, Int. J. heat. mass. transfer, 13, 703-716, (1970)
[12] Háo, D.N.; Reinhardt, H.-J., On a sideways parabolic equation, Inverse problems, 13, 297-309, (1997) · Zbl 0871.35105
[13] Qiu, C.Y.; Fu, C.L., Wavelet regularization for an ill-posed problem of parabolic equation, Acta math. sci., 22A, 3, 361-372, (2002) · Zbl 1043.35142
[14] Háo, D.N.; Reinhardt, H.-J.; Schneider, A., Numerical solution to a sideways parabolic equation, Int. J. numer. methods eng., 50, 1253-1267, (2001) · Zbl 1082.80003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.