Iterative Method Based on the Truncated Technique for Backward Heat Conduction Problem with Variable Coefficient

Abstract

We consider a backward heat conduction problem (BHCP) with variable coefficient. This problem is severely ill-posed in the sense of Hadamard and the regularization techniques are required to stabilize numerical computations. We use an iterative method based on the truncated technique to treat it. Under an a-priori and an a-posteriori stopping rule for the iterative step number, the convergence estimates are established. Some numerical results show that this method is stable and feasible.

Share and Cite:

Zhang, H. and Zhang, X. (2015) Iterative Method Based on the Truncated Technique for Backward Heat Conduction Problem with Variable Coefficient. Open Access Library Journal, 2, 1-11. doi: 10.4236/oalib.1101501.

Conflicts of Interest

The authors declare no conflicts of interest.

 [1] Hanke, M., Engle, H.W. and Neubauer, A. (1996) Regularization of Inverse Problems, Volume 375 of Mathematics and Its Applications. Kluwer Academic Publishers Group, Dordrecht. [2] Kirsch, A. (1996) An Introduction to the Mathematical Theory of Inverse Problems, Volume 120 of Applied Mathematical Sciences. Springer-Verlag, New York. [3] Cheng, J. and Liu, J.J. (2008) A Quasi Tikhonov Regularization for a Two-Dimensional Backward Heat Problem by a Fundamental Solution. Inverse Problems, 24, Article ID: 065012. http://dx.doi.org/10.1088/0266-5611/24/6/065012 [4] Feng, X.L., Qian, Z. and Fu, C.L. (2008) Numerical Approximation of Solution of Nonhomogeneous Backward Heat Conduction Problem in Bounded Region. Mathematics and Computers in Simulation, 79, 177-188.http://dx.doi.org/10.1016/j.matcom.2007.11.005 [5] Liu, J.J. (2002) Numerical Solution of Forward and Backward Problem for 2-D Heat Conduction Equation. Journal of Computational and Applied Mathematics, 145, 459-482. http://dx.doi.org/10.1016/S0377-0427(01)00595-7 [6] Qian, Z., Fu, C.L. and Shi, R. (2007) A Modified Method for a Backward Heat Conduction Problem. Applied Mathematics and Computation, 185, 564-573.http://dx.doi.org/10.1016/j.amc.2006.07.055 [7] Shidfar, A., Damirchi, J. and Reihani, P. (2007) An Stable Numerical Algorithm for Identifying the Solution of an Inverse Problem. Applied Mathematics and Computation, 190, 231-236. http://dx.doi.org/10.1016/j.amc.2007.01.022 [8] Feng, X.L., Eld’en, L. and Fu, C.L. (2010) Stability and Regularization of a Backward Parabolic PDE with Variable Coefficients. Journal of Inverse and Ill-Posed Problems, 18, 217-243. http://dx.doi.org/10.1016/j.jmaa.2004.08.001 [9] Ames, K.A., Clark, G.W., Epperson, J.F. and Oppenheimer, S.F. (1998) A Comparison of Regularizations for an Ill-Posed Problem. Mathematics of Computation, 67, 1451-1472. http://dx.doi.org/10.1090/S0025-5718-98-01014-X [10] Clark, G.W. and Oppenheimer, S.F. (1994) Quasireversibility Methods for Non-Well-Posed Problems. Electronic Journal of Differential Equations, 8, 1-9. [11] Denche, M. and Bessila, K. (2005) A Modified Quasi-Boundary Value Method for Ill-Posed Problems. Journal of Mathematical Analysis and Applications, 301, 419-426. http://dx.doi.org/10.1016/j.jmaa.2004.08.001 [12] Marbán, J.M. and Palencia, C. (2003) A New Numerical Method for Backward Parabolic Problems in the Maximum-Norm Setting. SIAM Journal on Numerical Analysis, 40, 1405-1420. [13] Kozlov, V.A. and Maz’ya, V.G. (1989) On Iterative Procedures for Solving Ill-Posed Boundary Value Problems That Preserve Differential Equations. Algebra I Analiz, 1, 144-170. [14] Baumeister, J. and Leiteao, A. (2001) On Iterative Methods for Solving Ill-Posed Problems Modeled by Partial Differential Equations. Journal of Inverse and Ill-Posed Problems, 9, 13-30. http://dx.doi.org/10.1515/jiip.2001.9.1.13 [15] Jourhmane, M. and Mera, N.S. (2002) An Iterative Algorithm for the Backward Heat Conduction Problem Based on Variable Relaxation Factors. Inverse Problems in Engineering, 10, 293-308. http://dx.doi.org/10.1080/10682760290004320 [16] Louis, A.K. (1989) Inverse und schlecht gestellte Probleme. B.G. Teubner, Leipzig.http://dx.doi.org/10.1007/978-3-322-84808-6 [17] Vainikko, G.M. and Veretennikov, A.Y. (1986) Iteration Procedures in Ill-Posed Problems. Nauka, Moscow. [18] Morozov, V.A., Nashed, Z. and Aries, A.B. (1984) Methods for Solving Incorrectly Posed Problems. Springer, New York. http://dx.doi.org/10.1007/978-1-4612-5280-1 [19] Tautenhahn, U. (1998) Optimality for Ill-Posed Problems under General Source Conditions. Numerical Functional Analysis and Optimization, 19, 377-398. http://dx.doi.org/10.1080/01630569808816834