# Solving the Axisymmetric Inverse Heat Conduction Problem by a Wavelet Dual Least Squares Method

- Wei Cheng
^{1}and - Chu-Li Fu
^{2}Email author

**2009**:260941

**DOI: **10.1155/2009/260941

© W. Cheng and C.-L. Fu. 2009

**Received: **17 August 2008

**Accepted: **10 March 2009

**Published: **17 March 2009

## Abstract

We consider an axisymmetric inverse heat conduction problem of determining the surface temperature from a fixed location inside a cylinder. This problem is ill-posed; the solution (if it exists) does not depend continuously on the data. A special project method—dual least squares method generated by the family of Shannon wavelet is applied to formulate regularized solution. Meanwhile, an order optimal error estimate between the approximate solution and exact solution is proved.

## 1. Introduction

Inverse heat conduction problems (IHCP) have become an interesting subject recently, and many regularization methods have been developed for the analysis of IHCP [1–13]. These methods include Tikhonov method [1, 2], mollification method [3, 4], optimal filtering method [5], lines method [6], wavelet and wavelet-Galerkin method [7–11], modified Tikhonov method [12] and "optimal approximations" [13], and so forth. However, most analytical and numerical methods were only used to dealing with IHCP in semiunbounded region. Some works of numerical methods were presented for IHCP in bounded domain [14–19].

Chen et al. [14] applied the hybrid numerical algorithm of Laplace transform technique to the IHCP in a rectangular plate. Busby and Trujillo [15] used the dynamic programming method to investigate the IHCP in a slab. Alifanov and Kerov [16] and Louahlia-Gualous et al. [17] researched IHCP in a cylinder. However to the authors' knowledge, most of them did not give any stability theory and convergence proofs.

where the functions and belong to for every fixed , is the radial coordinate, denotes the temperature history at one fixed radius of cylinder. We want to recover for . This problem is ill-posed problem; a small perturbation in the data may cause dramatically large errors in the solution (The details can be seen in Section 2).

To the authors' knowledge, up to now, there is no regularization theory with error estimate for problem (1.1) in the interval . The major objective of this paper is to do the theoretic stability and convergence estimates for problem (1.1).

Xiong and Fu [11] and Regińska [20] solved the sideways heat equation in semi-unbounded region by applying the wavelet dual least squares method, which is based on the family of Meyer wavelet. In this paper, we will apply a wavelet dual least squares method generated by the family of Shannon wavelet to problem (1.1) in bounded domain for determining surface temperature. According to the optimality results of general regularization theory, we conclude that our error estimate on surface temperature is order optimal.

## 2. Formulation of Solution of Problem (1.1)

Then we have the following lemma.

Lemma 2.1.

Proof.

Substitution of in (2.16) into (2.12), we obtain (2.8).

In order to obtain ill-posedness of problem (1.1) for , we need the following lemma.

Lemma 2.2.

Proof.

Finally, combining inequalities (2.22) with (2.23), we can see that there exist others constants and such that, for , inequalities (2.18) are valid. Similarly, we obtain inequalities (2.19).

From (2.26), we can prove the following lemma.

Lemma 2.3.

Proof.

Hence the conclusion of Lemma 2.3 is proved.

This implies that , which is Fourier transform of exact data , must decay rapidly at high frequencies since . But such a decay is not likely to occur in the Fourier transform of the measured noisy data at . So, small perturbation of in high frequency components can blow up and completely destroy the solution given by (2.17) for .

## 3. Wavelet Dual Least Squares Method

### 3.1. Dual Least Squares Method

### 3.2. Shannon Wavelets

From (3.9), can be seen as a low-pass filter. The frequencies with greater than are filtered away.

Theorem 3.1.

Proof.

Hence the conclusion of Theorem3.1 is proved.

## 4. Error Estimates via Dual Least Squares Method Approximation

Now we will devote to estimating the error .

Theorem 4.1.

Proof.

Hence the conclusion of Theorem 4.1 is proved.

The following is the main result of this paper.

Theorem 4.2.

where .

Proof.

Hence the conclusion of Theorem 4.2 is proved.

- (i)When and , estimate (4.8) is a Hölder stability estimate given by(4.12)

- (ii)
When , estimate (4.8) is a logarithmical Hölder stability estimate.

- (iii)When , estimate (4.3) becomes(4.13)

This is a logarithmical stability estimate.

Remark 4.4.

where .

## Declarations

### Acknowledgments

The work is supported by the National Natural Science Foundation of China (No. 10671085), the Hight-level Personnel fund of Henan University of Technology (2007BS028), and the Fundamental Research Fund for Natural Science of Education Department of Henan Province of China (No. 2009B110007).

## Authors’ Affiliations

## References

- Carasso A: Determining surface temperatures from interior observations.
*SIAM Journal on Applied Mathematics*1982, 42(3):558–574. 10.1137/0142040MATHMathSciNetView Article - Fu C-L: Simplified Tikhonov and Fourier regularization methods on a general sideways parabolic equation.
*Journal of Computational and Applied Mathematics*2004, 167(2):449–463. 10.1016/j.cam.2003.10.011MATHMathSciNetView Article - Murio DA:
*The Mollification Method and the Numerical Solution of Ill-Posed Problems, A Wiley-Interscience Publication*. John Wiley & Sons, New York, NY, USA; 1993:xvi+254.View Article - Hào DN, Reinhardt H-J: On a sideways parabolic equation.
*Inverse Problems*1997, 13(2):297–309. 10.1088/0266-5611/13/2/007MATHMathSciNetView Article - Seidman TI, Eldén L: An 'optimal filtering' method for the sideways heat equation.
*Inverse Problems*1990, 6(4):681–696. 10.1088/0266-5611/6/4/013MATHMathSciNetView Article - Eldén L: Solving the sideways heat equation by a method of lines.
*Journal of Heat Transfer*1997, 119: 406–412. 10.1115/1.2824112View Article - Eldén L, Berntsson F, Regińska T: Wavelet and Fourier methods for solving the sideways heat equation.
*SIAM Journal on Scientific Computing*2000, 21(6):2187–2205. 10.1137/S1064827597331394MATHMathSciNetView Article - Regińska T, Eldén L: Solving the sideways heat equation by a wavelet-Galerkin method.
*Inverse Problems*1997, 13(4):1093–1106. 10.1088/0266-5611/13/4/014MATHMathSciNetView Article - Regińska T, Eldén L: Stability and convergence of the wavelet-Galerkin method for the sideways heat equation.
*Journal of Inverse and Ill-Posed Problems*2000, 8(1):31–49.MATHMathSciNet - Fu C-L, Qiu CY: Wavelet and error estimation of surface heat flux.
*Journal of Computational and Applied Mathematics*2003, 150(1):143–155. 10.1016/S0377-0427(02)00657-XMATHMathSciNetView Article - Xiong X-T, Fu C-L: Determining surface temperature and heat flux by a wavelet dual least squares method.
*Journal of Computational and Applied Mathematics*2007, 201(1):198–207. 10.1016/j.cam.2006.02.014MATHMathSciNetView Article - Cheng W, Fu C-L, Qian Z: A modified Tikhonov regularization method for a spherically symmetric three-dimensional inverse heat conduction problem.
*Mathematics and Computers in Simulation*2007, 75(3–4):97–112. 10.1016/j.matcom.2006.09.005MATHMathSciNetView Article - Tautenhahn U: Optimal stable approximations for the sideways heat equation.
*Journal of Inverse and Ill-Posed Problems*1997, 5(3):287–307. 10.1515/jiip.1997.5.3.287MATHMathSciNetView Article - Chen H-T, Lin S-Y, Fang L-C: Estimation of surface temperature in two-dimensionnal inverse heat conduction problems.
*International Journal of Heat and Mass Transfer*2001, 44(8):1455–1463. 10.1016/S0017-9310(00)00212-XMATHView Article - Busby HR, Trujillo DM: Numerical soluition to a two-dimensionnal inverse heat conduction problem.
*International Journal for Numerical Methods in Engineering*1985, 21(2):349–359. 10.1002/nme.1620210211MATHView Article - Alifanov OM, Kerov NV: Determination of external thermal load parameters by solving the two-dimensional inverse heat-conduction problem.
*Journal of Engineering Physics*1981, 41(4):1049–1053. 10.1007/BF00824760View Article - Louahlia-Gualous H, Panday PK, Artyukhin EA: The inverse determination of the local heat transfer coefficients for nucleate boiling on horizontal cylinder.
*Journal of Heat Transfer*2003, 125(1):1087–1095.View Article - Hon YC, Wei T: A fundamental solution method for inverse heat conduction problem.
*Engineering Analysis with Boundary Elements*2004, 28(5):489–495. 10.1016/S0955-7997(03)00102-4MATHView Article - Shidfar A, Pourgholi R: Numerical approximation of solution of an inverse heat conduction problem based on Legendre polynomials.
*Applied Mathematics and Computation*2006, 175(2):1366–1374. 10.1016/j.amc.2005.08.040MATHMathSciNetView Article - Regińska T: Application of wavelet shrinkage to solving the sideways heat equation.
*BIT Numerical Mathematics*2001, 41(5):1101–1110. 10.1023/A:1021909816563MathSciNetView Article - Abramowitz M, Stegun IA (Eds):
*Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables*. Dover, New York, NY, USA; 1972.MATH - Wang JR: The multi-resolution method applied to the sideways heat equation.
*Journal of Mathematical Analysis and Applications*2005, 309(2):661–673. 10.1016/j.jmaa.2004.11.025MATHMathSciNetView Article

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.