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Quasi-Reversibility Regularization Method for Solving a Backward Heat Conduction Problem
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American Journal of Computational Mathematics·Volume 01 (2011)·Pages 159–162·Published 19 September 2011·DOI10.4236/ajcm.2011.13018
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Abstract
Non-standard backward heat conduction problem is ill-posed in the sense that the solution(if it exists) does not depend continuously on the data. In this paper, we propose a regularization strategy-quasi-reversibility method to analysis the stability of the problem. Meanwhile, we investigate the roles of regularization parameter in this method. Numerical result show that our algorithm is effective and stable.
KeywordsBack Heat ConductionIll-Posed ProblemQuasi-ReversibilityRegularization
- R. Latter and J. L. Lions, “Methode de Quasi-Reversibility et Applications,” Dunod, Paris,1967.
- R. E. Showalter, “The Final Value Problem for Evolution Equations,” Journal of Mathematical Analysis and Applications, Vol. 47, 1974, pp. 563-572. doi:10.1016/0022-247X(74)90008-0
- K. A. Ames, W. C. Gordon, J. F. Epperson and S. F. Oppenhermer, “A Compari-son of Regularizations for an Ill-Posed Problem,” Mathematics of Computation, Vol. 67, No. 224, 1998, pp. 1451-1471. doi:10.1090/S0025-5718-98-01014-X
- K. Miller, “Stabi-lized Quasireversibility and Other Nearly Best Possible Methods for Non-Well-Posed Problems,” Symposium on Non-Well-Posed Problems and Logarithmic Convexity, Lecture Notes in Mathematics, Springer-Verlag, Berlin, Vol. 316, 1973, pp. 161-176.
- T. Schr?ter and U. Tautenhahn, “On Optimal Regularization Methods for the Backward Heat Equation,” Zeitschrift für Analysis und ihre Anwendungen, Vol. 15, 1996, pp. 475-493.
- N. S. Mera, L. Elliott, D. B. Ingham and D. Lesnic, “An Iterative Boundary Element Method for Solving the One Dimensional Backward Heat Conduction Problem,” International Journal of Heat and Mass Transfer, Vol. 44, 2001, pp. 1946-1973.
- M. Jourhmane and N. S. Mera, “An Iterative Algorithm for the Backward Heat Conduction Problem Based on Variable Relaxtion Factors,” Inverse Problems in Engineering, Vol. 10, No. 4, 2002, pp. 293-308. doi:10.1080/10682760290004320
- N. S. Mera, “The Method of Fundamental Solutions for the Backward Heat Conduction Problem,” Inverse Problems in Engineering, Vol. 13, No. 1, 2005, pp. 79-98. doi:10.1080/10682760410001710141
- D. N. Ha?, “A Mollification Method for Ill-Posed Problems,” Numerische Mathematik, Vol. 68, No. 4, 1994, pp. 469-506. doi:10.1007/s002110050073
- S. M. Kirkup and M. Wadsworth, “Solution of Inverse Diffusion Problems by Operator-Splitting Methods,” Applied Mathematical Modelling, Vol. 26, No. 10, 2002, pp. 1003-1018. doi:10.1016/S0307-904X(02)00053-7
- C. L. Fu, X. T. Xiong and Z. Qian, “Fourier Regularization for a Backward Heat Equation,” Journal of Mathematical Analysis and Appli-cations, Vol. 33, No. 1, 2007, pp. 472-481. doi:10.1016/j.jmaa.2006.08.040
- L. Elden, “Numerical Solutiono of the Sideways Heat Equation by Diference Approximation in Time,” Inverse Problems, Vol. 201, No. 11, 1995, pp. 913-923. doi:10.1088/0266-5611/11/4/017