DOI QR코드

DOI QR Code

SIMPLIFIED TIKHONOV REGULARIZATION FOR TWO KINDS OF PARABOLIC EQUATIONS

  • Jing, Li ;
  • Fang, Wang
  • Received : 2009.10.04
  • Published : 2011.03.01

Abstract

This paper is devoted to simplified Tikhonov regularization for two kinds of parabolic equations, i.e., a sideways parabolic equation, and a two-dimensional inverse heat conduction problem. The measured data are assumed to be known approximately. We concentrate on the convergence rates of the simplified Tikhonov approximation of u(x, t) and its derivative $u_x$(x, t) of sideways parabolic equations at 0 $\leq$ x < 1, and that of two-dimensional inverse heat conduction problem at 0 < x $\leq$ 1, respectively.

Keywords

Fourier transformation;simplified Tikhonov regularization;convergence rate;sideways parabolic equations;inverse heat conduction problems

References

  1. L. Elden, Numerical solution of the sideways heat equation by difference approximation in time, Inverse Problems 11 (1995), no. 4, 913-923. https://doi.org/10.1088/0266-5611/11/4/017
  2. L. Elden, F. Berntsson, and T. Reginska, Wavelet and Fourier methods for solving the sideways heat equation, SIAM J. Sci. Comput. 21 (2000), no. 6, 2187-2205. https://doi.org/10.1137/S1064827597331394
  3. H. W. Engl, Regularization of Inverse Problem, Kluwer Academic Publishers, Bostom, 2000.
  4. C. L. Fu, Simplied Tikhonov and Fourier regularization methods on a general sideways parabolic equation, J. Comput. Appl. Math. 167 (2004), no. 2, 449-463. https://doi.org/10.1016/j.cam.2003.10.011
  5. L. Guo and D. A. Murio, A mollied space-marching nite-difference algorithm for the two-dimensional inverse heat conduction problem with slab symmetry, Inverse Problems 7 (1991), no. 2, 247-259. https://doi.org/10.1088/0266-5611/7/2/008
  6. D. N. Hao, A mollication method for ill-posed problems, Numer. Math. 68 (1994), no. 4, 469-506. https://doi.org/10.1007/s002110050073
  7. D. N. Hao and H. J. Reinhardt, On a sideways parabolic equation, Inverse Problems 13 (1997), no. 2, 297-309. https://doi.org/10.1088/0266-5611/13/2/007
  8. D. N. Hao, H. J. Reinhardt, and A. Schneider, Numerical solution to a sideways parabolic equation, Internat. J. Numer. Methods Engrg. 50 (2001), no. 5, 1253-1267. https://doi.org/10.1002/1097-0207(20010220)50:5<1253::AID-NME81>3.0.CO;2-6
  9. Z. H. Liu, Browder-Tikhonov regularization of non-coercive evolution hemivariational inequalities, Inverse Problems 21 (2005), no. 1, 13-20. https://doi.org/10.1088/0266-5611/21/1/002
  10. Z. H. Liu, J. Li, and Z. W. Li, Regularization method with two parameters for nonlinear ill-posed problems, Sci. China Ser. A 51 (2008), no. 1, 70-78. https://doi.org/10.1007/s11425-007-0131-3
  11. L. Liu and D. A. Murio, Numerical experiments in 2-D IHCP on bounded domains. I. The "interior" cube problem, Comput. Math. Appl. 31 (1996), no. 1, 15-32. https://doi.org/10.1016/0898-1221(95)00180-7
  12. D. A. Murio, The Mollication Method and the Numerical Solution of Ill-Posed Problems, A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1993.
  13. M. T. Nair and U. Tautenhahn, Lavrentiev regularization for linear ill-posed problems under general source conditions, Z. Anal. Anwendungen 23 (2004), no. 1, 167-185.
  14. Z. Qian and C. L. Fu, Regularization strategies for a two-dimensional inverse heat conduction problem, Inverse Problems 23 (2007), no. 3, 1053-1068. https://doi.org/10.1088/0266-5611/23/3/013
  15. T. Reginska and L. Elden, Stability and convergence of the wavelet-Galerkin method for the sideways heat equation, J. Inverse Ill-Posed Probl. 8 (2000), no. 1, 31-49.
  16. M. Tadi, An iterative method for the solution of ill-posed parabolic systems, Appl. Math. Comput. 201 (2008), no. 1-2, 843-851. https://doi.org/10.1016/j.amc.2007.12.048
  17. A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems, John Wiley & Sons, New York-Toronto, Ont.-London, 1977.
  18. J. S. Tu and J. V. Beck, Solution of inverse heat conduction problems using a com- bined function specication and regularization method and a direct nite element solver,Proc. 2nd Ann. Inverse Problems in Engineering Seminar ed J. V. Beck (East Lansing, MI:College of Engineering, Michigan State University), (1989), pp. 1-25.

Cited by

  1. A Modified Kernel Method for Solving Cauchy Problem of Two-Dimensional Heat Conduction Equation vol.7, pp.01, 2015, https://doi.org/10.4208/aamm.12-m12113
  2. A Revised Tikhonov Regularization Method for a Cauchy Problem of Two-Dimensional Heat Conduction Equation vol.2018, pp.1563-5147, 2018, https://doi.org/10.1155/2018/1216357

Acknowledgement

Supported by : NNSF of China, Hunan Provincial Natural Science Foundation of China