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.
Cite this paper
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, e1501. doi: http://dx.doi.org/10.4236/oalib.1101501.
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.
Kirsch, A. (1996) An Introduction to
the Mathematical Theory of Inverse Problems, Volume 120 of Applied Mathematical
Sciences. Springer-Verlag, New York.
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
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
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
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
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
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
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
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
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.
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.
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
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
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
