Numerical solution of forward and backward problem for 2-D heat conduction equation. (English) Zbl 1005.65107

Liu, Jijun

J. Comput. Appl. Math. 145, No. 2, 459-482 (2002).

The author discusses the two-dimensional inverse heat conduction problem, which determines the initial temperature distribution from transient temperature measurements. The conditional stability for this inverse problem and the error analysis for the Tikhonov regularization are presented. An implicit inversion method, which is based on the regularization technique and the successive overrelaxation iteration process, is established. This paper also develops an explicit difference scheme for a direct efficient, while the application of the successive over-relaxation technique makes this inversion convergent rapidly. Lastly some numerical examples are given.

Reviewer: Zhang Whanguo (Shanghai)

MSC:

65M32	Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
65M12	Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15	Error bounds 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

Keywords:

heat equation; difference scheme; inverse problem; successive overrelaxation; Tikhonov regularization; numerical examples; stability; error analysis

References:

[1]	Chapko, R., On the numerical solution of direct and inverse problems for the heat equations in a semi-infinite region, J. comput. appl. math., 108, 41-55, (1999) · Zbl 0943.65110
[2]	Cheng, J.; Yamamoto, M., On new strategy for a priori choice of regularising parameters in Tikhonov’s regularization, Inverse problems, 16, 31-38, (2000)
[3]	Dautray, R.; Lions, J.L., Mathematical analysis and numerical methods for science and technology, vol. 1, physical origins and classical methods, (1990), Springer Berlin
[4]	Kirsch, A., An introduction to the mathematical theory of inverse problems, (1996), Springer NewYork · Zbl 0865.35004
[5]	Kleinman, R.E.; Roach, G.F.; Schuetz, L.S.; Shirron, J.; van den Berg, P.M., An over-relaxation method for the iterative solution of integral equations in scattering problems, Wave motion, 12, 2, 161-170, (1990) · Zbl 0706.73078
[6]	R.E. Kleinman, P.M. van den Berg, Iterative methods for solving integral equations, Progr. Electromagn. Res. 5 (Elsevier, New York) (1991) (Chapter 3). · Zbl 0762.65094
[7]	Kleinman, R.E.; van den Berg, P.M., A modified gradient method for two-dimensional problems in tomography, J. comput. appl. math., 42, 17-35, (1992) · Zbl 0757.65133
[8]	Kress, R., Linear integral equations, (1989), Springer Berlin
[9]	M.M. Lavrentiev, V.G. Romanov, S.P. Shishatskij, Ill-posed problems of mathematical physics and analysis, Transl. of Math. Monographs 64 AMS, Providence, R.I., 1986.
[10]	Lesselier, D., Optimization techniques and inverse problems: reconstruction of conductivity profiles in the time domain, IEEE trans antennas propogation, Ap-30, 59-65, (1982)
[11]	Muniz, W.B., A comparison of some inverse methods for estimating the initial condition of the heat equation, J. comput. appl. math., 103, 145-163, (1999) · Zbl 0952.65068
[12]	L.E. Payne, Improperly posed problems in partial differential equations, SIAM Regional Conference Series in Applied Mathematics, Philadelphia, 1975. · Zbl 0302.35003
[13]	Smith, G.D., Numerical solutions of partial differential equations: finite difference methods, (1985), Oxford University Press New York · Zbl 0576.65089
[14]	Tikhonov, A.N., Solutions of ill-posed problems, (1977), Winston and Sons New York